Patent Publication Number: US-10326526-B2

Title: Method for muxing orthogonal modes using modal correlation matrices

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority from U.S. Provisional Application Ser. No. 62/385,038, filed on Sep. 8, 2016, and entitled MODE CROSSTALK MATRIX, which is incorporated herein in its entirety. This application is also a continuation-in-part of U.S. patent application Ser. No. 15/608,609, filed on May 30, 2017, entitled SYSTEM AND METHOD FOR TRANSMISSIONS USING ELLIPTICAL CORE FIBERS which is incorporated herein by reference in its entirety. 
    
    
     TECHNICAL FIELD 
     The present invention relates to optical signal transmissions, and more particularly, to manner for tracking mode crosstalk using a mode crosstalk matrix. 
     BACKGROUND 
     Optical communications may be carried out over an optical fiber using optical signals processed with orthogonal functions such as Hermite-Gaussian functions, Laguerre-Gaussian functions and Ince-Gaussian functions as described herein in order to improve system bandwidth. Laguerre-Gaussian (LG), Hermite-Gaussian (HG), and Ince-Gaussian (IG) signals have three important properties and advantages. The advantages include the ability to form two complete families of exact and orthogonal solutions of the paraxial wave equations. Another advantage is that the HG, LG and IG signals are transverse eigenmodes of stable resonators. Finally, the HG, LG and IG signals do not change shape on propagation and provide stable modes of propagation for signals. However, when using LG, HG and IG signals for transmissions of various types on a particular mode there is often crosstalk between the mode being used and other adjacent modes on the transmission link. The crosstalk can have effects on the communications over the transmission link so some manner for tracking and illustrating the mode crosstalk effects over the transmission link would prove useful to system designers. 
     SUMMARY 
     The present invention, as disclosed and described herein, in one aspect thereof, comprises a method for transmitting an orthogonal function processed signal over a communications link on a fiber involves generating at least one mode crosstalk matrix illustrating mode crosstalk between transmitted modes and adjacent modes within the fiber. Adjacent modes to be multiplexed together are selected based on entries within the generated mode crosstalk matrix being less than or equal to a predetermined value. The transmitted modes and the selected adjacent modes are multiplexed together into the orthogonal function processed signal for transmission on the communications link on the fiber. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       For a more complete understanding, reference is now made to the following description taken in conjunction with the accompanying Drawings in which: 
         FIG. 1  illustrates various techniques for increasing spectral efficiency within a transmitted signal; 
         FIG. 2  illustrates a particular technique for increasing spectral efficiency within a transmitted signal; 
         FIG. 3  illustrates a general overview of the manner for providing communication bandwidth between various communication protocol interfaces; 
         FIG. 4  illustrates the manner for utilizing multiple level overlay modulation with twisted pair/cable interfaces; 
         FIG. 5  illustrates a general block diagram for processing a plurality of data streams within an optical communication system; 
         FIG. 6  is a functional block diagram of a system for generating orbital angular momentum within a communication system; 
         FIG. 7  is a functional block diagram of the orbital angular momentum signal processing block of  FIG. 6 ; 
         FIG. 8  is a functional block diagram illustrating the manner for removing orbital angular momentum from a received signal including a plurality of data streams; 
         FIG. 9  illustrates a single wavelength having two quanti-spin polarizations providing an infinite number of signals having various orbital angular momentums associated therewith; 
         FIG. 10A  illustrates an object with only a spin angular momentum; 
         FIG. 10B  illustrates an object with an orbital angular momentum; 
         FIG. 10C  illustrates a circularly polarized beam carrying spin angular momentum; 
         FIG. 10D  illustrates the phase structure of a light beam carrying an orbital angular momentum; 
         FIG. 11A  illustrates a plane wave having only variations in the spin angular momentum; 
         FIG. 11B  illustrates a signal having both spin and orbital angular momentum applied thereto; 
         FIGS. 12A-12C  illustrate various signals having different orbital angular momentum applied thereto; 
         FIG. 12D  illustrates a propagation of Poynting vectors for various Eigen modes; 
         FIG. 12E  illustrates a spiral phase plate; 
         FIG. 13  illustrates a system for using to the orthogonality of an HG modal group for free space spatial multiplexing; 
         FIG. 14  illustrates a multiple level overlay modulation system; 
         FIG. 15  illustrates a multiple level overlay demodulator; 
         FIG. 16  illustrates a multiple level overlay transmitter system; 
         FIG. 17  illustrates a multiple level overlay receiver system; 
         FIGS. 18A-18K  illustrate representative multiple level overlay signals and their respective spectral power densities; 
         FIG. 19  illustrates comparisons of multiple level overlay signals within the time and frequency domain; 
         FIG. 20A  illustrates a spectral alignment of multiple level overlay signals for differing bandwidths of signals; 
         FIG. 20B-20C  illustrate frequency domain envelopes located in separate layers within a same physical bandwidth; 
         FIG. 21  illustrates an alternative spectral alignment of multiple level overlay signals; 
         FIG. 22  illustrates three different superQAM signals; 
         FIG. 23  illustrates the creation of inter-symbol interference in overlapped multilayer signals; 
         FIG. 24  illustrates overlapped multilayer signals; 
         FIG. 25  illustrates a fixed channel matrix; 
         FIG. 26  illustrates truncated orthogonal functions; 
         FIG. 27  illustrates a typical OAM multiplexing scheme; 
         FIG. 28  illustrates various manners for converting a Gaussian beam into an OAM beam; 
         FIG. 29A  illustrates a fabricated metasurface phase plate; 
         FIG. 29B  illustrates a magnified structure of the metasurface phase plate; 
         FIG. 29C  illustrates an OAM beam generated using the phase plate with l=+1; 
         FIG. 30  illustrates the manner in which a q-plate can convert a left circularly polarized beam into a right circular polarization or vice-versa; 
         FIG. 31  illustrates the use of a laser resonator cavity for producing an OAM beam; 
         FIG. 32  illustrates spatial multiplexing using cascaded beam splitters; 
         FIG. 33  illustrated de-multiplexing using cascaded beam splitters and conjugated spiral phase holograms; 
         FIG. 34  illustrates a log polar geometrical transformation based on OAM multiplexing and de-multiplexing; 
         FIG. 35  illustrates an intensity profile of generated OAM beams and their multiplexing; 
         FIG. 36A  illustrates the optical spectrum of each channel after each multiplexing for the OAM beams of  FIG. 10A ; 
         FIG. 36B  illustrates the recovered constellations of 16-QAM signals carried on each OAM beam; 
         FIG. 37A  illustrates the steps to produce 24 multiplex OAM beams; 
         FIG. 37B  illustrates the optical spectrum of a WDM signal carrier on an OAM beam; 
         FIG. 38A  illustrates a turbulence emulator; 
         FIG. 38B  illustrates the measured power distribution of an OAM beam after passing through turbulence with a different strength; 
         FIG. 39A  illustrates how turbulence effects mitigation using adaptive optics; 
         FIG. 39B  illustrates experimental results of distortion mitigation using adaptive optics; 
         FIG. 40  illustrates a free-space optical data link using OAM; 
         FIG. 41A  illustrates simulated spot sized of different orders of OAM beams as a function of transmission distance for a 3 cm transmitted beam; 
         FIG. 41B  illustrates simulated power loss as a function of aperture size; 
         FIG. 42A  illustrates a perfectly aligned system between a transmitter and receiver; 
         FIG. 42B  illustrates a system with lateral displacement of alignment between a transmitter and receiver; 
         FIG. 42C  illustrates a system with receiver angular error for alignment between a transmitter and receiver; 
         FIG. 43A  illustrates simulated power distribution among different OAM modes with a function of lateral displacement; 
         FIG. 43B  illustrates simulated power distribution among different OAM modes as a function of receiver angular error; 
         FIG. 44  illustrates a bandwidth efficiency comparison for square root raised cosine versus multiple layer overlay for a symbol rate of 1/6; 
         FIG. 45  illustrates a bandwidth efficiency comparison between square root raised cosine and multiple layer overlay for a symbol rate of 1/4; 
         FIG. 46  illustrates a performance comparison between square root raised cosine and multiple level overlay using ACLR; 
         FIG. 47  illustrates a performance comparison between square root raised cosine and multiple lever overlay using out of band power; 
         FIG. 48  illustrates a performance comparison between square root raised cosine and multiple lever overlay using band edge PSD; 
         FIG. 49  is a block diagram of a transmitter subsystem for use with multiple level overlay; 
         FIG. 50  is a block diagram of a receiver subsystem using multiple level overlay; 
         FIG. 51  illustrates an equivalent discreet time orthogonal channel of modified multiple level overlay; 
         FIG. 52  illustrates the PSDs of multiple layer overlay, modified multiple layer overlay and square root raised cosine; 
         FIG. 53  illustrates a bandwidth comparison based on −40 dBc out of band power bandwidth between multiple layer overlay and square root raised cosine; 
         FIG. 54  illustrates equivalent discrete time parallel orthogonal channels of modified multiple layer overlay; 
         FIG. 55  illustrates four MLO symbols that are included in a single block; 
         FIG. 56  illustrates the channel power gain of the parallel orthogonal channels of modified multiple layer overlay with three layers and T sym =3; 
         FIG. 57  illustrates a spectral efficiency comparison based on ACLR 1  between modified multiple layer overlay and square root raised cosine; 
         FIG. 58  illustrates a spectral efficiency comparison between modified multiple layer overlay and square root raised cosine based on OBP; 
         FIG. 59  illustrates a spectral efficiency comparison based on ACLR 1  between modified multiple layer overlay and square root raised cosine; 
         FIG. 60  illustrates a spectral efficiency comparison based on OBP between modified multiple layer overlay and square root raised cosine; 
         FIG. 61  illustrates a block diagram of a baseband transmitter for a low pass equivalent modified multiple layer overlay system; 
         FIG. 62  illustrates a block diagram of a baseband receiver for a low pass equivalent modified multiple layer overlay system; 
         FIG. 63  illustrates a channel simulator; 
         FIG. 64  illustrates the generation of bit streams for a QAM modulator; 
         FIG. 65  illustrates a block diagram of a receiver; 
         FIG. 66  is a flow diagram illustrating an adaptive QLO process; 
         FIG. 67  is a flow diagram illustrating an adaptive MDM process; 
         FIG. 68  is a flow diagram illustrating an adaptive QLO and MDM process 
         FIG. 69  is a flow diagram illustrating an adaptive QLO and QAM process; 
         FIG. 70  is a flow diagram illustrating an adaptive QLO, MDM and QAM process; 
         FIG. 71  illustrates the use of a pilot signal to improve channel impairments; 
         FIG. 72  is a flowchart illustrating the use of a pilot signal to improve channel impairment; 
         FIG. 73  illustrates a channel response and the effects of amplifier nonlinearities; 
         FIG. 74  illustrates the use of QLO in forward and backward channel estimation processes; 
         FIG. 75  illustrates the manner in which Hermite Gaussian beams and Laguerre Gaussian beams diverge when transmitted from phased array antennas; 
         FIG. 76A  illustrates beam divergence between a transmitting aperture and a receiving aperture; 
         FIG. 76B  illustrates the use of a pair of lenses for reducing beam divergence; 
         FIG. 77  illustrates the configuration of an optical fiber communication system; 
         FIG. 78A  illustrates a single mode fiber; 
         FIG. 78B  illustrates multi-core fibers; 
         FIG. 78C  illustrates multi-mode fibers; 
         FIG. 78D  illustrates a hollow core fiber; 
         FIG. 79  illustrates the first six modes within a step index fiber; 
         FIG. 80  illustrates the classes of random perturbations within a fiber; 
         FIG. 81  illustrates the intensity patterns of first order groups within a vortex fiber; 
         FIGS. 82A and 82B  illustrate index separation in first order modes of a multi-mode fiber; 
         FIG. 83  illustrates a few mode fiber providing a linearly polarized OAM beam; 
         FIG. 84  illustrates the transmission of four OAM beams over a fiber; 
         FIG. 85A  illustrates the recovered constellations of 20 Gbit/sec QPSK signals carried on each OAM beam of the device of  FIG. 84 ; 
         FIG. 85B  illustrates the measured BER curves of the device of  FIG. 84 ; 
         FIG. 86  illustrates a vortex fiber; 
         FIG. 87  illustrates intensity profiles and interferograms of OAM beams; 
         FIG. 88  illustrates a free-space communication system; 
         FIG. 89  illustrates a block diagram of a free-space optics system using orbital angular momentum and multi-level overlay modulation; 
         FIGS. 90A-90C  illustrate the manner for multiplexing multiple data channels into optical links to achieve higher data capacity; 
         FIG. 90D  illustrates groups of concentric rings for a wavelength having multiple OAM valves; 
         FIG. 91  illustrates a WDM channel containing many orthogonal OAM beams; 
         FIG. 92  illustrates a node of a free-space optical system; 
         FIG. 93  illustrates a network of nodes within a free-space optical system; 
         FIG. 94  illustrates a system for multiplexing between a free space signal and an RF signal; 
         FIG. 95  illustrates a seven dimensional QKD link based on OAM encoding; 
         FIG. 96  illustrates the OAM and ANG modes providing complementary 7 dimensional bases for information encoding; 
         FIG. 97  illustrates a block diagram of an OAM processing system utilizing quantum key distribution; 
         FIG. 98  illustrates a basic quantum key distribution system; 
         FIG. 99  illustrates the manner in which two separate states are combined into a single conjugate pair within quantum key distribution; 
         FIG. 100  illustrates one manner in which 0 and 1 bits may be transmitted using different basis within a quantum key distribution system; 
         FIG. 101  is a flow diagram illustrating the process for a transmitter transmitting a quantum key; 
         FIG. 102  illustrates the manner in which the receiver may receive and determine a shared quantum key; 
         FIG. 103  more particularly illustrates the manner in which a transmitter and receiver may determine a shared quantum key; 
         FIG. 104  is a flow diagram illustrating the process for determining whether to keep or abort a determined key; 
         FIG. 105  illustrates a functional block diagram of a transmitter and receiver utilizing a free-space quantum key distribution system; 
         FIG. 106  illustrates a network cloud-based quantum key distribution system; 
         FIG. 107  illustrates a high-speed single photon detector in communication with a plurality of users; and 
         FIG. 108  illustrates a nodal quantum key distribution network. 
         FIG. 109  illustrates the use of a reflective phase hologram for data exchange; 
         FIG. 110  is a flow diagram illustrating the process for using ROADM for exchanging data signals; 
         FIG. 111  illustrates the concept of a ROADM for data channels carried on multiplexed OAM beams; 
         FIG. 112  illustrates observed intensity profiles at each step of an add/drop operation such as that of  FIG. 111 ; 
         FIG. 113  illustrates circuitry for the generation of an OAM twisted beam using a hologram within a micro-electromechanical device; 
         FIG. 114  illustrates multiple holograms generated by a micro-electromechanical device; 
         FIG. 115  illustrates a square array of holograms on a dark background; 
         FIG. 116  illustrates a hexagonal array of holograms on a dark background; 
         FIG. 117  illustrates a process for multiplexing various OAM modes together; 
         FIG. 118  illustrates fractional binary fork holograms; 
         FIG. 119  illustrates an array of square holograms with no separation on a light background and associated generated OAM mode image; 
         FIG. 120  illustrates an array of circular holograms separated on a light background and associated generated OAM mode image; 
         FIG. 121  illustrates an array of square holograms with no separation on a dark background and associated generated OAM mode image; 
         FIG. 122  illustrates an array of circular holograms on a dark background and associated generated OAM mode image; 
         FIG. 123  illustrates circular holograms with separation on a bright background and associated generated OAM mode image; 
         FIG. 124  illustrates circular holograms with separation on a dark background and associated generated OAM mode image; 
         FIG. 125  illustrates a hexagonal array of circular holograms on a bright background and associated OAM mode image; 
         FIG. 126  illustrates an hexagonal array of small holograms on a bright background and associated OAM mode image; 
         FIG. 127  illustrates a hexagonal array of circular holograms on a dark background and associated OAM mode image; 
         FIG. 128  illustrates a hexagonal array of small holograms on a dark background and associated OAM mode image; 
         FIG. 129  illustrates a hexagonal array of small holograms separated on a dark background and associated OAM mode image; 
         FIG. 130  illustrates a hexagonal array of small holograms closely located on a dark background and associated OAM mode image; 
         FIG. 131  illustrates a hexagonal array of small holograms that are separated on a bright background and associated OAM mode image; 
         FIG. 132  illustrates a hexagonal array of small holograms that are closely located on a bright background and associated OAM mode image; 
         FIG. 133  illustrates reduced binary holograms having a radius equal to 100 micro-mirrors and a period of 50 for various OAM modes; 
         FIG. 134  illustrates OAM modes for holograms having a radius of 50 micro-mirrors and a period of 50; 
         FIG. 135  illustrates OAM modes for holograms having a radius of 100 micro-mirrors and a period of 100; 
         FIG. 136  illustrates OAM modes for holograms having a radius of 50 micro-mirrors and a period of 50; 
         FIG. 137  illustrates additional methods of multimode OAM generation by implementing multiple holograms within a MEMs device; 
         FIG. 138  illustrates binary spiral holograms; 
         FIG. 139  is a block diagram of a circuit for generating a muxed and multiplexed data stream containing multiple new Eigen channels; 
         FIG. 140  is a flow diagram describing the operation of the circuit of  FIG. 139 ; 
         FIG. 141  is a block diagram of a circuit for de-muxing and de-multiplexing a data stream containing multiple new Eigen channels; 
         FIG. 142  is a flow diagram describing the operation of the circuit of  FIG. 141 ; 
         FIG. 143  illustrates various types of orthogonal functions that may be used in optical fiber transmissions; 
         FIG. 144  illustrates a transmitter and receiver transmitting signals over the elliptical fiber; 
         FIG. 145  illustrates the process for generating an Ince-Gaussian signal for transmission on elliptical fiber; 
         FIG. 146  illustrates a elliptical-cylindrical coordinate system; 
         FIG. 147  illustrates the curves of constant value of trace confocal ellipses; 
         FIG. 148  illustrates a confocal hyperbola with a constant value of η; 
         FIGS. 149A and 149B  illustrate the frequency of even Ince-Polynomials; 
         FIG. 150  illustrates the modes and phases for even Ince-Polynomials; 
         FIGS. 151A and 151B  illustrate the frequency of odd Ince-Polynomials; 
         FIG. 152  illustrates the modes and phases of odd Ince-Polynomials; 
         FIG. 153  illustrates a parabolic index profile of an elliptical core fiber; 
         FIG. 154  illustrates a few mode fiber with an elliptical core; 
         FIG. 155  illustrates intensity diagrams for different types of beam topologies within an elliptical core fiber; 
         FIG. 156  illustrates a measurement technique for generating a mode crosstalk matrix; 
         FIG. 157  illustrates a flow diagram of the process of  FIG. 156 ; 
         FIG. 158  illustrates a generated single row of a mode crosstalk matrix; 
         FIG. 159  illustrates the results for a comparison of selectively excited modes calculated from a transmitting SLM; and 
         FIG. 160  illustrates a mode crosstalk matrix populated using Hermite-Gaussian modes; 
         FIGS. 161-163  illustrate various mode cross talk matrices; and 
         FIG. 164  illustrates a flow diagram of the process of selecting modes for multiplexing together using a mode crosstalk matrix. 
     
    
    
     DETAILED DESCRIPTION 
     Referring now to the drawings, wherein like reference numbers are used herein to designate like elements throughout, the various views and embodiments of a system and method for transmissions using elliptical core fibers are illustrated and described, and other possible embodiments are described. The Figures are not necessarily drawn to scale, and in some instances the drawings have been exaggerated and/or simplified in places for illustrative purposes only. One of ordinary skill in the art will appreciate the many possible applications and variations based on the following examples of possible embodiments. 
     Achieving higher data capacity is perhaps one of the primary interest of the communications community. This is led to the investigation of using different physical properties of a light wave for communications, including amplitude, phase, wavelength and polarization. Orthogonal modes in spatial positions are also under investigation and seemed to be useful as well. Generally these investigative efforts can be summarized in 2 categories: 1) encoding and decoding more bets on a single optical pulse; a typical example is the use of advanced modulation formats, which encode information on amplitude, phase and polarization states, and 2) multiplexing and demultiplexing technologies that allow parallel propagation of multiple independent data channels, each of which is addressed by different light property (e.g., wavelength, polarization and space, corresponding to wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM) and space division multiplexing (SDM), respectively) using Hermite Gaussian, Laguerre Gaussian and Ince Gaussian spactial orthogonal modes among others. 
     The recognition that orbital angular momentum (OAM) has applications in communication has made it an interesting research topic. It is well-known that a photon can carry both spin angular momentum and orbital angular momentum. Contrary to spin angular momentum (e.g., circularly polarized light), which is identified by the electrical field erection, OAM is usually carried by a light beam with a helical phase front. Due to the helical phase structure, an OAM carrying beam usually has an annular intensity profile with a phase singularity at the beam center. Importantly, depending on discrete twisting speed of the helical phase, OAM beams can be quantified is different states, which are completely distinguishable while propagating coaxially. This property allows OAM beams to be potentially useful in either of the 2 aforementioned categories to help improve the performance of a free space or fiber communication system. Specifically, OAM states could be used as a different dimension to encode bits on a single pulse (or a single photon), or be used to create additional data carriers in an SDM system. 
     There are some potential benefits of using OAM for communications, some specially designed novel fibers allow less mode coupling and cross talk while propagating in fibers. In addition, OAM beams with different states share a ring-shaped beam profile, which indicate rotational insensitivity for receiving the beams. Since the distinction of OAM beams does not rely on the wavelength or polarization, OAM multiplexing could be used in addition to WDM and PDM techniques so that potentially improve the system performance may be provided. 
     Referring now to the drawings, and more particularly to  FIG. 1 , wherein there is illustrated two manners for increasing spectral efficiency of a communications system. In general, there are basically two ways to increase spectral efficiency  102  of a communications system. The increase may be brought about by signal processing techniques  104  in the modulation scheme or using multiple access technique. Additionally, the spectral efficiency can be increase by creating new Eigen channels  106  within the electromagnetic propagation. These two techniques are completely independent of one another and innovations from one class can be added to innovations from the second class. Therefore, the combination of this technique introduced a further innovation. 
     Spectral efficiency  102  is the key driver of the business model of a communications system. The spectral efficiency is defined in units of bit/sec/hz and the higher the spectral efficiency, the better the business model. This is because spectral efficiency can translate to a greater number of users, higher throughput, higher quality or some of each within a communications system. 
     Regarding techniques using signal processing techniques or multiple access techniques. These techniques include innovations such as TDMA, FDMA, CDMA, EVDO, GSM, WCDMA, HSPA and the most recent OFDM techniques used in 4G WIMAX and LTE. Almost all of these techniques use decades-old modulation techniques based on sinusoidal Eigen functions called QAM modulation. Within the second class of techniques involving the creation of new Eigen channels  106 , the innovations include diversity techniques including space and polarization diversity as well as multiple input/multiple output (MIMO) where uncorrelated radio paths create independent Eigen channels and propagation of electromagnetic waves. 
     Referring now to  FIG. 2 , the present communication system configuration introduces two techniques, one from the signal processing techniques  104  category and one from the creation of new eigen channels  106  category that are entirely independent from each other. Their combination provides a unique manner to disrupt the access part of an end to end communications system from twisted pair and cable to fiber optics, to free space optics, to RF used in cellular, backhaul and satellite, to RF satellite, to RF broadcast, to RF point-to point, to RF point-to-multipoint, to RF point-to-point (backhaul), to RF point-to-point (fronthaul to provide higher throughput CPRI interface for cloudification and virtualization of RAN and cloudified HetNet), to Internet of Things (IOT), to Wi-Fi, to Bluetooth, to a personal device cable replacement, to an RF and FSO hybrid system, to Radar, to electromagnetic tags and to all types of wireless access. The first technique involves the use of a new signal processing technique using new orthogonal signals to upgrade QAM modulation using non sinusoidal functions. This is referred to as quantum level overlay (QLO)  202 . The second technique involves the application of new electromagnetic wavefronts using a property of electromagnetic waves or photon, called orbital angular momentum (QAM)  104 . Application of each of the quantum level overlay techniques  202  and orbital angular momentum application  204  uniquely offers orders of magnitude higher spectral efficiency  206  within communication systems in their combination. 
     With respect to the quantum level overlay technique  202 , new eigen functions are introduced that when overlapped (on top of one another within a symbol) significantly increases the spectral efficiency of the system. The quantum level overlay technique  302  borrows from quantum mechanics, special orthogonal signals that reduce the time bandwidth product and thereby increase the spectral efficiency of the channel. Each orthogonal signal is overlaid within the symbol acts as an independent channel. These independent channels differentiate the technique from existing modulation techniques. 
     With respect to the application of orbital angular momentum  204 , this technique introduces twisted electromagnetic waves, or light beams, having helical wave fronts that carry orbital angular momentum (OAM). Different OAM carrying waves/beams can be mutually orthogonal to each other within the spatial domain, allowing the waves/beams to be efficiently multiplexed and demultiplexed within a communications link. OAM beams are interesting in communications due to their potential ability in special multiplexing multiple independent data carrying channels. 
     With respect to the combination of quantum level overlay techniques  202  and orbital angular momentum application  204 , the combination is unique as the OAM multiplexing technique is compatible with other electromagnetic techniques such as wave length and polarization division multiplexing. This suggests the possibility of further increasing system performance. The application of these techniques together in high capacity data transmission disrupts the access part of an end to end communications system from twisted pair and cable to fiber optics, to free space optics, to RF used in cellular, backhaul and satellite, to RF satellite, to RF broadcast, to RF point-to point, to RF point-to-multipoint, to RF point-to-point (backhaul), to RF point-to-point (fronthaul to provide higher throughput CPRI interface for cloudification and virtualization of RAN and cloudified HetNet), to Internet of Things (TOT), to Wi-Fi, to Bluetooth, to a personal device cable replacement, to an RF and FSO hybrid system, to Radar, to electromagnetic tags and to all types of wireless access. 
     Each of these techniques can be applied independent of one another, but the combination provides a unique opportunity to not only increase spectral efficiency, but to increase spectral efficiency without sacrificing distance or signal to noise ratios. 
     Using the Shannon Capacity Equation, a determination may be made if spectral efficiency is increased. This can be mathematically translated to more bandwidth. Since bandwidth has a value, one can easily convert spectral efficiency gains to financial gains for the business impact of using higher spectral efficiency. Also, when sophisticated forward error correction (FEC) techniques are used, the net impact is higher quality but with the sacrifice of some bandwidth. However, if one can achieve higher spectral efficiency (or more virtual bandwidth), one can sacrifice some of the gained bandwidth for FEC and therefore higher spectral efficiency can also translate to higher quality. 
     Telecom operators and vendors are interested in increasing spectral efficiency. However, the issue with respect to this increase is the cost. Each technique at different layers of the protocol has a different price tag associated therewith. Techniques that are implemented at a physical layer have the most impact as other techniques can be superimposed on top of the lower layer techniques and thus increase the spectral efficiency further. The price tag for some of the techniques can be drastic when one considers other associated costs. For example, the multiple input multiple output (MIMO) technique uses additional antennas to create additional paths where each RF path can be treated as an independent channel and thus increase the aggregate spectral efficiency. In the MIMO scenario, the operator has other associated soft costs dealing with structural issues such as antenna installations, etc. These techniques not only have tremendous cost, but they have huge timing issues as the structural activities take time and the achieving of higher spectral efficiency comes with significant delays which can also be translated to financial losses. 
     The quantum level overlay technique  202  has an advantage that the independent channels are created within the symbols without needing new antennas. This will have a tremendous cost and time benefit compared to other techniques. Also, the quantum layer overlay technique  202  is a physical layer technique, which means there are other techniques at higher layers of the protocol that can all ride on top of the QLO techniques  202  and thus increase the spectral efficiency even further. QLO technique  202  uses standard QAM modulation used in OFDM based multiple access technologies such as WIMAX or LTE. QLO technique  202  basically enhances the QAM modulation at the transceiver by injecting new signals to the I &amp; Q components of the baseband and overlaying them before QAM modulation as will be more fully described herein below. At the receiver, the reverse procedure is used to separate the overlaid signal and the net effect is a pulse shaping that allows better localization of the spectrum compared to standard QAM or even the root raised cosine. The impact of this technique is a significantly higher spectral efficiency. 
     Referring now more particularly to  FIG. 3 , there is illustrated a general overview of the manner for providing improved communication bandwidth within various communication protocol interfaces  302 , using a combination of multiple level overlay modulation  304  and the application of orbital angular momentum  306  to increase the number of communications channels. 
     The various communication protocol interfaces  302  may comprise a variety of communication links, such as RF communication, wireline communication such as cable or twisted pair connections, or optical communications making use of light wavelengths such as fiber-optic communications or free-space optics. Various types of RF communications may include a combination of RF microwave or RF satellite communication, as well as multiplexing between RF and free-space optics in real time. 
     By combining a multiple layer overlay modulation technique  304  with orbital angular momentum (OAM) technique  306 , a higher throughput over various types of communication links  302  may be achieved. The use of multiple level overlay modulation alone without OAM increases the spectral efficiency of communication links  302 , whether wired, optical, or wireless. However, with OAM, the increase in spectral efficiency is even more significant. 
     Multiple overlay modulation techniques  304  provide a new degree of freedom beyond the conventional 2 degrees of freedom, with time T and frequency F being independent variables in a two-dimensional notational space defining orthogonal axes in an information diagram. This comprises a more general approach rather than modeling signals as fixed in either the frequency or time domain. Previous modeling methods using fixed time or fixed frequency are considered to be more limiting cases of the general approach of using multiple level overlay modulation  304 . Within the multiple level overlay modulation technique  304 , signals may be differentiated in two-dimensional space rather than along a single axis. Thus, the information-carrying capacity of a communications channel may be determined by a number of signals which occupy different time and frequency coordinates and may be differentiated in a notational two-dimensional space. 
     Within the notational two-dimensional space, minimization of the time bandwidth product, i.e., the area occupied by a signal in that space, enables denser packing, and thus, the use of more signals, with higher resulting information-carrying capacity, within an allocated channel. Given the frequency channel delta (Δf), a given signal transmitted through it in minimum time Δt will have an envelope described by certain time-bandwidth minimizing signals. The time-bandwidth products for these signals take the form:
 
Δ tΔf= ½(2 n+ 1)
 
where n is an integer ranging from 0 to infinity, denoting the order of the signal.
 
     These signals form an orthogonal set of infinite elements, where each has a finite amount of energy. They are finite in both the time domain and the frequency domain, and can be detected from a mix of other signals and noise through correlation, for example, by match filtering. Unlike other wavelets, these orthogonal signals have similar time and frequency forms. 
     The orbital angular momentum process  306  provides a twist to wave fronts of the electromagnetic fields carrying the data stream that may enable the transmission of multiple data streams on the same frequency, wavelength, or other signal-supporting mechanism. Similarly, other orthogonal signals may be applied to the different data streams to enable transmission of multiple data streams on the same frequency, wavelength or other signal-supporting mechanism. This will increase the bandwidth over a communications link by allowing a single frequency or wavelength to support multiple eigen channels, each of the individual channels having a different orthogonal and independent orbital angular momentum associated therewith. 
     Referring now to  FIG. 4 , there is illustrated a further communication implementation technique using the above described techniques as twisted pairs or cables carry electrons (not photons). Rather than using each of the multiple level overlay modulation  304  and orbital angular momentum techniques  306 , only the multiple level overlay modulation  304  can be used in conjunction with a single wireline interface and, more particularly, a twisted pair communication link or a cable communication link  402 . The operation of the multiple level overlay modulation  404 , is similar to that discussed previously with respect to  FIG. 3 , but is used by itself without the use of orbital angular momentum techniques  306 , and is used with either a twisted pair communication link or cable interface communication link  402  or with fiber optics, free space optics, RF used in cellular, backhaul and satellite, RF satellite, RF broadcast, RF point-to point, RF point-to-multipoint, RF point-to-point (backhaul), RF point-to-point (fronthaul to provide higher throughput CPRI interface for cloudification and virtualization of RAN and cloudified HetNet), Internet of Things (IOT), Wi-Fi, Bluetooth, a personal device cable replacement, an RF and FSO hybrid system, Radar, electromagnetic tags and all types of wireless access. 
     Referring now to  FIG. 5 , there is illustrated a general block diagram for processing a plurality of data streams  502  for transmission in an optical communication system. The multiple data streams  502  are provided to the multi-layer overlay modulation circuitry  504  wherein the signals are modulated using the multi-layer overlay modulation technique. The modulated signals are provided to orbital angular momentum processing circuitry  506  which applies a twist to each of the wave fronts being transmitted on the wavelengths of the optical communication channel. The twisted waves are transmitted through the optical interface  508  over an optical or other communications link such as an optical fiber or free space optics communication system.  FIG. 5  may also illustrate an RF mechanism wherein the interface  508  would comprise and RF interface rather than an optical interface. 
     Referring now more particularly to  FIG. 6 , there is illustrated a functional block diagram of a system for generating the orbital angular momentum “twist” within a communication system, such as that illustrated with respect to  FIG. 3 , to provide a data stream that may be combined with multiple other data streams for transmission upon a same wavelength or frequency. Multiple data streams  602  are provided to the transmission processing circuitry  600 . Each of the data streams  602  comprises, for example, an end to end link connection carrying a voice call or a packet connection transmitting non-circuit switch packed data over a data connection. The multiple data streams  602  are processed by modulator/demodulator circuitry  604 . The modulator/demodulator circuitry  604  modulates the received data stream  602  onto a wavelength or frequency channel using a multiple level overlay modulation technique, as will be more fully described herein below. The communications link may comprise an optical fiber link, free-space optics link, RF microwave link, RF satellite link, wired link (without the twist), etc. 
     The modulated data stream is provided to the orbital angular momentum (OAM) signal processing block  606 . The orbital angular momentum signal processing block  606  applies in one embodiment an orbital angular momentum to a signal. In other embodiments the processing block  606  can apply any orthogonal function to a signal being transmitted. These orthogonal functions can be spatial Bessel functions, Laguerre-Gaussian functions, Hermite-Gaussian functions or any other orthogonal function. Each of the modulated data streams from the modulator/demodulator  604  are provided a different orbital angular momentum by the orbital angular momentum electromagnetic block  606  such that each of the modulated data streams have a unique and different orbital angular momentum associated therewith. Each of the modulated signals having an associated orbital angular momentum are provided to an optical transmitter  608  that transmits each of the modulated data streams having a unique orbital angular momentum on a same wavelength. Each wavelength has a selected number of bandwidth slots B and may have its data transmission capability increase by a factor of the number of degrees of orbital angular momentum l that are provided from the OAM electromagnetic block  606 . The optical transmitter  608  transmitting signals at a single wavelength could transmit B groups of information. The optical transmitter  608  and OAM electromagnetic block  606  may transmit l×B groups of information according to the configuration described herein. 
     In a receiving mode, the optical transmitter  608  will have a wavelength including multiple signals transmitted therein having different orbital angular momentum signals embedded therein. The optical transmitter  608  forwards these signals to the OAM signal processing block  606 , which separates each of the signals having different orbital angular momentum and provides the separated signals to the demodulator circuitry  604 . The demodulation process extracts the data streams  602  from the modulated signals and provides it at the receiving end using the multiple layer overlay demodulation technique. 
     Referring now to  FIG. 7 , there is provided a more detailed functional description of the OAM signal processing block  606 . Each of the input data streams are provided to OAM circuitry  702 . Each of the OAM circuitry  702  provides a different orbital angular momentum to the received data stream. The different orbital angular momentums are achieved by applying different currents for the generation of the signals that are being transmitted to create a particular orbital angular momentum associated therewith. The orbital angular momentum provided by each of the OAM circuitries  702  are unique to the data stream that is provided thereto. An infinite number of orbital angular momentums may be applied to different input data streams using many different currents. Each of the separately generated data streams are provided to a signal combiner  704 , which combines/multiplexes the signals onto a wavelength for transmission from the transmitter  706 . The combiner  704  performs a spatial mode division multiplexing to place all of the signals upon a same carrier signal in the space domain. 
     Referring now to  FIG. 8 , there is illustrated the manner in which the OAM processing circuitry  606  may separate a received signal into multiple data streams. The receiver  802  receives the combined OAM signals on a single wavelength and provides this information to a signal separator  804 . The signal separator  804  separates each of the signals having different orbital angular momentums from the received wavelength and provides the separated signals to OAM de-twisting circuitry  806 . The OAM de-twisting circuitry  806  removes the associated OAM twist from each of the associated signals and provides the received modulated data stream for further processing. The signal separator  804  separates each of the received signals that have had the orbital angular momentum removed therefrom into individual received signals. The individually received signals are provided to the receiver  802  for demodulation using, for example, multiple level overlay demodulation as will be more fully described herein below. 
       FIG. 9  illustrates in a manner in which a single wavelength or frequency, having two quanti-spin polarizations may provide an infinite number of twists having various orbital angular momentums associated therewith. The l axis represents the various quantized orbital angular momentum states which may be applied to a particular signal at a selected frequency or wavelength. The symbol omega (ω) represents the various frequencies to which the signals of differing orbital angular momentum may be applied. The top grid  902  represents the potentially available signals for a left handed signal polarization, while the bottom grid  904  is for potentially available signals having right handed polarization. 
     By applying different orbital angular momentum states to a signal at a particular frequency or wavelength, a potentially infinite number of states may be provided at the frequency or wavelength. Thus, the state at the frequency Δω or wavelength  906  in both the left handed polarization plane  902  and the right handed polarization plane  904  can provide an infinite number of signals at different orbital angular momentum states Δl. Blocks  908  and  910  represent a particular signal having an orbital angular momentum Δl at a frequency Δω or wavelength in both the right handed polarization plane  904  and left handed polarization plane  910 , respectively. By changing to a different orbital angular momentum within the same frequency Δω or wavelength  906 , different signals may also be transmitted. Each angular momentum state corresponds to a different determined current level for transmission from the optical transmitter. By estimating the equivalent current for generating a particular orbital angular momentum within the optical domain and applying this current for transmission of the signals, the transmission of the signal may be achieved at a desired orbital angular momentum state. 
     Thus, the illustration of  FIG. 9 , illustrates two possible angular momentums, the spin angular momentum, and the orbital angular momentum. The spin version is manifested within the polarizations of macroscopic electromagnetism, and has only left and right hand polarizations due to up and down spin directions. However, the orbital angular momentum indicates an infinite number of states that are quantized. The paths are more than two and can theoretically be infinite through the quantized orbital angular momentum levels. 
     It is well-known that the concept of linear momentum is usually associated with objects moving in a straight line. The object could also carry angular momentum if it has a rotational motion, such as spinning (i.e., spin angular momentum (SAM)  1002 ), or orbiting around an axis  1006  (i.e., OAM  1004 ), as shown in  FIGS. 10A and 10B , respectively. A light beam may also have rotational motion as it propagates. In paraxial approximation, a light beam carries SAM  1002  if the electrical field rotates along the beam axis  1006  (i.e., circularly polarized light  1005 ), and carries OAM  1004  if the wave vector spirals around the beam axis  1006 , leading to a helical phase front  1008 , as shown in  FIGS. 10C and 10D . In its analytical expression, this helical phase front  1008  is usually related to a phase term of exp(ilθ) in the transverse plane, where θ refers to the angular coordinate, and l is an integer indicating the number of intertwined helices (i.e., the number of 2π phase shifts along the circle around the beam axis). l could be a positive, negative integer or zero, corresponding to clockwise, counterclockwise phase helices or a Gaussian beam with no helix, respectively. 
     Two important concepts relating to OAM include: 1) OAM and polarization: As mentioned above, an OAM beam is manifested as a beam with a helical phase front and therefore a twisting wavevector, while polarization states can only be connected to SAM  1002 . A light beam carries SAM  1002  of ±h/2π (h is Plank&#39;s constant) per photon if it is left or right circularly polarized, and carries no SAM  1002  if it is linearly polarized. Although the SAM  1002  and OAM  1004  of light can be coupled to each other under certain scenarios, they can be clearly distinguished for a paraxial light beam. Therefore, with the paraxial assumption, OAM  1004  and polarization can be considered as two independent properties of light. 
     2) OAM beam and Laguerre-Gaussian (LG) beam: In general, an OAM-carrying beam could refer to any helically phased light beam, irrespective of its radial distribution (although sometimes OAM could also be carried by a non-helically phased beam). LG beam is a special subset among all OAM-carrying beams, due to that the analytical expression of LG beams are eigen-solutions of paraxial form of the wave equation in a cylindrical coordinates. For an LG beam, both azimuthal and radial wavefront distributions are well defined, and are indicated by two index numbers, l and p, of which l has the same meaning as that of a general OAM beam, and p refers to the radial nodes in the intensity distribution. Mathematical expressions of LG beams form an orthogonal and complete basis in the spatial domain. In contrast, a general OAM beam actually comprises a group of LG beams (each with the same l index but a different p index) due to the absence of radial definition. The term of “OAM beam” refers to all helically phased beams, and is used to distinguish from LG beams. 
     Using the orbital angular momentum state of the transmitted energy signals, physical information can be embedded within the radiation transmitted by the signals. The Maxwell-Heaviside equations can be represented as: 
     
       
         
           
             
               ∇ 
               
                 · 
                 E 
               
             
             = 
             
               ρ 
               
                 ɛ 
                 0 
               
             
           
         
       
       
         
           
             
               ∇ 
               
                 × 
                 E 
               
             
             = 
             
               - 
               
                 
                   ∂ 
                   B 
                 
                 
                   ∂ 
                   t 
                 
               
             
           
         
       
       
         
           
             
               ∇ 
               
                 · 
                 B 
               
             
             = 
             0 
           
         
       
       
         
           
             
               ∇ 
               
                 × 
                 B 
               
             
             = 
             
               
                 
                   ɛ 
                   0 
                 
                 ⁢ 
                 
                   μ 
                   0 
                 
                 ⁢ 
                 
                   
                     ∂ 
                     E 
                   
                   
                     ∂ 
                     t 
                   
                 
               
               + 
               
                 
                   μ 
                   0 
                 
                 ⁢ 
                 
                   j 
                   ⁡ 
                   
                     ( 
                     
                       t 
                       , 
                       x 
                     
                     ) 
                   
                 
               
             
           
         
       
     
     where ∇ is the del operator, E is the electric field intensity and B is the magnetic flux density. Using these equations, one can derive 23 symmetries/conserved quantities from Maxwell&#39;s original equations. However, there are only ten well-known conserved quantities and only a few of these are commercially used. Historically if Maxwell&#39;s equations where kept in their original quaternion forms, it would have been easier to see the symmetries/conserved quantities, but when they were modified to their present vectorial form by Heaviside, it became more difficult to see such inherent symmetries in Maxwell&#39;s equations. 
     Maxwell&#39;s linear theory is of U( 1 ) symmetry with Abelian commutation relations. They can be extended to higher symmetry group SU( 2 ) form with non-Abelian commutation relations that address global (non-local in space) properties. The Wu-Yang and Harmuth interpretation of Maxwell&#39;s theory implicates the existence of magnetic monopoles and magnetic charges. As far as the classical fields are concerned, these theoretical constructs are pseudo-particle, or instanton. The interpretation of Maxwell&#39;s work actually departs in a significant ways from Maxwell&#39;s original intention. In Maxwell&#39;s original formulation, Faraday&#39;s electrotonic states (the Aμ field) was central making them compatible with Yang-Mills theory (prior to Heaviside). The mathematical dynamic entities called solitons can be either classical or quantum, linear or non-linear and describe EM waves. However, solitons are of SU( 2 ) symmetry forms. In order for conventional interpreted classical Maxwell&#39;s theory of U( 1 ) symmetry to describe such entities, the theory must be extended to SU( 2 ) forms. 
     Besides the half dozen physical phenomena (that cannot be explained with conventional Maxwell&#39;s theory), the recently formulated Harmuth Ansatz also address the incompleteness of Maxwell&#39;s theory. Harmuth amended Maxwell&#39;s equations can be used to calculate EM signal velocities provided that a magnetic current density and magnetic charge are added which is consistent to Yang-Mills filed equations. Therefore, with the correct geometry and topology, the Aμ potentials always have physical meaning 
     The conserved quantities and the electromagnetic field can be represented according to the conservation of system energy and the conservation of system linear momentum. Time symmetry, i.e. the conservation of system energy can be represented using Poynting&#39;s theorem according to the equations: 
     
       
         
           
             Hamiltonian 
             ⁢ 
             
                 
             
             ⁢ 
             
               ( 
               
                 total 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 energy 
               
               ) 
             
           
         
       
       
         
           
             H 
             = 
             
               
                 
                   ∑ 
                   i 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     m 
                     i 
                   
                   ⁢ 
                   
                     γ 
                     i 
                   
                   ⁢ 
                   
                     c 
                     2 
                   
                 
               
               + 
               
                 
                   
                     ɛ 
                     0 
                   
                   2 
                 
                 ⁢ 
                 
                   ∫ 
                   
                     
                       d 
                       3 
                     
                     ⁢ 
                     
                       x 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                                
                               E 
                                
                             
                             2 
                           
                           + 
                           
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               
                                  
                                 B 
                                  
                               
                               2 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             conservation 
             ⁢ 
             
                 
             
             ⁢ 
             of 
             ⁢ 
             
                 
             
             ⁢ 
             energy 
           
         
       
       
         
           
             
               
                 
                   dU 
                   mech 
                 
                 dt 
               
               + 
               
                 
                   dU 
                   em 
                 
                 dt 
               
               + 
               
                 
                   ∮ 
                   
                     s 
                     ′ 
                   
                 
                 ⁢ 
                 
                   
                     d 
                     2 
                   
                   ⁢ 
                   
                     x 
                     ′ 
                   
                   ⁢ 
                   
                     
                       
                         n 
                         ^ 
                       
                       ′ 
                     
                     · 
                     S 
                   
                 
               
             
             = 
             0 
           
         
       
     
     The space symmetry, i.e., the conservation of system linear momentum representing the electromagnetic Doppler shift can be represented by the equations: 
     
       
         
           
             linear 
             ⁢ 
             
                 
             
             ⁢ 
             momentum 
           
         
       
       
         
           
             p 
             = 
             
               
                 
                   ∑ 
                   i 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     m 
                     i 
                   
                   ⁢ 
                   
                     γ 
                     i 
                   
                   ⁢ 
                   
                     v 
                     i 
                   
                 
               
               + 
               
                 
                   ɛ 
                   0 
                 
                 ⁢ 
                 
                   ∫ 
                   
                     
                       d 
                       3 
                     
                     ⁢ 
                     
                       x 
                       ⁡ 
                       
                         ( 
                         
                           E 
                           × 
                           B 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             conservation 
             ⁢ 
             
                 
             
             ⁢ 
             of 
             ⁢ 
             
                 
             
             ⁢ 
             linear 
             ⁢ 
             
                 
             
             ⁢ 
             momentum 
           
         
       
       
         
           
             
               
                 
                   dp 
                   mech 
                 
                 dt 
               
               + 
               
                 
                   dp 
                   em 
                 
                 dt 
               
               + 
               
                 
                   ∮ 
                   
                     s 
                     ′ 
                   
                 
                 ⁢ 
                 
                   
                     d 
                     2 
                   
                   ⁢ 
                   
                     x 
                     ′ 
                   
                   ⁢ 
                   
                     
                       
                         n 
                         ^ 
                       
                       ′ 
                     
                     · 
                     T 
                   
                 
               
             
             = 
             0 
           
         
       
     
     The conservation of system center of energy is represented by the equation: 
     
       
         
           
             R 
             = 
             
               
                 
                   1 
                   H 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     i 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           x 
                           i 
                         
                         - 
                         
                           x 
                           0 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       m 
                       i 
                     
                     ⁢ 
                     
                       γ 
                       i 
                     
                     ⁢ 
                     
                       c 
                       2 
                     
                   
                 
               
               + 
               
                 
                   
                     ɛ 
                     0 
                   
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     H 
                   
                 
                 ⁢ 
                 
                   ∫ 
                   
                     
                       d 
                       3 
                     
                     ⁢ 
                     
                       x 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           - 
                           
                             x 
                             0 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                            
                           
                             E 
                             2 
                           
                            
                         
                         + 
                         
                           
                             c 
                             2 
                           
                           ⁢ 
                           
                              
                             
                               B 
                               2 
                             
                              
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     Similarly, the conservation of system angular momentum, which gives rise to the azimuthal Doppler shift is represented by the equation: 
                   dJ             ⁢   mech       dt     +               ⁢     dJ             ⁢   em         dt     +       ∮     s   ′       ⁢       d             ⁢   2       ⁢     x   ′     ⁢         n   ′     ^     ·   M           =   0         
conservation of angular momentum
 
     For radiation beams in free space, the EM field angular momentum J em  can be separated into two parts:
 
 J   em =ε 0 ∫ V′   d   3   x ′( E×A )+ε 0 ∫ V′   d   3   x′E   i [( x′−x   0 )×∇] A   i  
 
     For each singular Fourier mode in real valued representation: 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 J 
                 em 
               
               = 
               
                 
                   
                     - 
                     i 
                   
                   ⁢ 
                   
                     
                       ɛ 
                       0 
                     
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                     
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         V 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         d 
                         3 
                       
                       ⁢ 
                       
                         
                           x 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               E 
                               * 
                             
                             × 
                             E 
                           
                           ) 
                         
                       
                     
                   
                 
                 - 
                 
                   i 
                   ⁢ 
                   
                     
                       ɛ 
                       0 
                     
                     
                       2 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                     
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         V 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         d 
                         3 
                       
                       ⁢ 
                       
                         x 
                         ′ 
                       
                       ⁢ 
                       
                         
                           E 
                           i 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   x 
                                   ′ 
                                 
                                 - 
                                 
                                   x 
                                   0 
                                 
                               
                               ) 
                             
                             × 
                             ∇ 
                           
                           ] 
                         
                       
                       ⁢ 
                       
                         E 
                         i 
                       
                     
                   
                 
               
             
           
         
       
     
     The first part is the EM spin angular momentum S em , its classical manifestation is wave polarization. And the second part is the EM orbital angular momentum L em  its classical manifestation is wave helicity. In general, both EM linear momentum P em , and EM angular momentum J em =L em +S em  are radiated all the way to the far field. 
     By using Poynting theorem, the optical vorticity of the signals may be determined according to the optical velocity equation: 
                     ∂   U       ∂   t       +     ∇     ·   S         =   0     ,         
continuity equation
 
where S is the Poynting vector
 
 S= ¼( E×H*+E*×H ),
 
and U is the energy density
 
 U= ¼(ε| E|   2 +μ 0   |H|   2 ),
 
with E and H comprising the electric field and the magnetic field, respectively, and ε and μ 0  being the permittivity and the permeability of the medium, respectively. The optical vorticity V may then be determined by the curl of the optical velocity according to the equation:
 
     
       
         
           
             V 
             = 
             
               
                 ∇ 
                 
                   × 
                   
                     v 
                     opt 
                   
                 
               
               = 
               
                 ∇ 
                 
                   × 
                   
                     ( 
                     
                       
                         
                           E 
                           × 
                           
                             H 
                             * 
                           
                         
                         + 
                         
                           
                             E 
                             * 
                           
                           × 
                           H 
                         
                       
                       
                         
                           ɛ 
                           ⁢ 
                           
                             
                                
                               E 
                                
                             
                             2 
                           
                         
                         + 
                         
                           
                             μ 
                             0 
                           
                           ⁢ 
                           
                             
                                
                               H 
                                
                             
                             2 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Referring now to  FIGS. 11A and 11B , there is illustrated the manner in which a signal and its associated Poynting vector in a plane wave situation. In the plane wave situation illustrated generally at  1102 , the transmitted signal may take one of three configurations. When the electric field vectors are in the same direction, a linear signal is provided, as illustrated generally at  1104 . Within a circular polarization  1106 , the electric field vectors rotate with the same magnitude. Within the elliptical polarization  1108 , the electric field vectors rotate but have differing magnitudes. The Poynting vector remains in a constant direction for the signal configuration to  FIG. 11A  and always perpendicular to the electric and magnetic fields. Referring now to  FIG. 11B , when a unique orbital angular momentum is applied to a signal as described here and above, the Poynting vector S  1110  will spiral about the direction of propagation of the signal. This spiral may be varied in order to enable signals to be transmitted on the same frequency as described herein. 
       FIGS. 12A through 12C  illustrate the differences in signals having different helicity (i.e., orbital angular momentums). Each of the spiraling Poynting vectors associated with the signals  1202 ,  1204 , and  1206  provide a different shaped signal. Signal  1202  has an orbital angular momentum of +1, signal  1204  has an orbital angular momentum of +3, and signal  1206  has an orbital angular momentum of −4. Each signal has a distinct angular momentum and associated Poynting vector enabling the signal to be distinguished from other signals within a same frequency. This allows differing type of information to be transmitted on the same frequency, since these signals are separately detectable and do not interfere with each other (Eigen channels). 
       FIG. 12D  illustrates the propagation of Poynting vectors for various Eigen modes. Each of the rings  1220  represents a different Eigen mode or twist representing a different orbital angular momentum within the same frequency. Each of these rings  1220  represents a different orthogonal channel. Each of the Eigen modes has a Poynting vector  1222  associated therewith. 
     Topological charge may be multiplexed to the frequency for either linear or circular polarization. In case of linear polarizations, topological charge would be multiplexed on vertical and horizontal polarization. In case of circular polarization, topological charge would multiplex on left hand and right hand circular polarizations. The topological charge is another name for the helicity index “I” or the amount of twist or OAM applied to the signal. Also, use of the orthogonal functions discussed herein above may also be multiplexed together onto a same signal in order to transmit multiple streams of information. The helicity index may be positive or negative. In wireless communications, different topological charges/orthogonal functions can be created and muxed together and de-muxed to separate the topological charges charges/orthogonal functions. The signals having different orthogonal function are spatially combined together on a same signal but do not interfere with each other since they are orthogonal to each other. 
     The topological charges l s can be created using Spiral Phase Plates (SPPs) as shown in  FIG. 12E  using a proper material with specific index of refraction and ability to machine shop or phase mask, holograms created of new materials or a new technique to create an RF version of Spatial Light Modulator (SLM) that does the twist of the RF waves (as opposed to optical beams) by adjusting voltages on the device resulting in twisting of the RF waves with a specific topological charge. Spiral Phase plates can transform a RF plane wave (l=0) to a twisted RF wave of a specific helicity (i.e. l=+1). 
     Cross talk and multipath interference can be corrected using RF Multiple-Input-Multiple-Output (MIMO). Most of the channel impairments can be detected using a control or pilot channel and be corrected using algorithmic techniques (closed loop control system). 
     While the application of orbital angular momentum to various signals allow the signals to be orthogonal to each other and used on a same signal carrying medium, other orthogonal function/signals can be applied to data streams to create the orthogonal signals on the same signal media carrier. 
     Within the notational two-dimensional space, minimization of the time bandwidth product, i.e., the area occupied by a signal in that space, enables denser packing, and thus, the use of more signals, with higher resulting information-carrying capacity, within an allocated channel. Given the frequency channel delta (Δf), a given signal transmitted through it in minimum time Δt will have an envelope described by certain time-bandwidth minimizing signals. The time-bandwidth products for these signals take the form;
 
Δ tΔf= ½(2 n+ 1)
 
where n is an integer ranging from 0 to infinity, denoting the order of the signal.
 
     These signals form an orthogonal set of infinite elements, where each has a finite amount of energy. They are finite in both the time domain and the frequency domain, and can be detected from a mix of other signals and noise through correlation, for example, by match filtering. Unlike other wavelets, these orthogonal signals have similar time and frequency forms. These types of orthogonal signals that reduce the time bandwidth product and thereby increase the spectral efficiency of the channel. 
     Hermite-Gaussian polynomials are one example of a classical orthogonal polynomial sequence, which are the Eigenstates of a quantum harmonic oscillator. Signals based on Hermite-Gaussian polynomials possess the minimal time-bandwidth product property described above, and may be used for embodiments of MLO systems. However, it should be understood that other signals may also be used, for example orthogonal polynomials such as Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials, and Laguerre-Gaussian polynomials. Q-functions are another class of functions that can be employed as a basis for MLO signals. 
     In addition to the time bandwidth minimization described above, the plurality of data streams can be processed to provide minimization of the Space-Momentum products in spatial modulation. In this case:
 
Δ xΔp= ½
 
     Processing of the data streams in this manner create wavefronts that are spatial. The processing creates wavefronts that are also orthogonal to each other like the OAM twisted functions but these comprise different types of orthogonal functions that are in the spatial domain rather than the temporal domain. 
     The above described scheme is applicable to twisted pair, coaxial cable, fiber optic, RF satellite, RF broadcast, RF point-to point, RF point-to-multipoint, RF point-to-point (backhaul), RF point-to-point (fronthaul to provide higher throughput CPRI interface for cloudification and virtualization of RAN and cloudified HetNet), free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as a personal device cable replacement, RF and FSO hybrid system, Radar, electromagnetic tags and all types of wireless access. The method and system are compatible with many current and future multiple access systems, including EV-DO, UMB, WIMAX, WCDMA (with or without), multimedia broadcast multicast service (MBMS)/multiple input multiple output (MIMO), HSPA evolution, and LTE. 
     Hermite Gaussian Beams 
     Hermite Gaussian beams may also be used for transmitting orthogonal data streams. In the scalar field approximation (e.g. neglecting the vector character of the electromagnetic field), any electric field amplitude distribution can be represented as a superposition of plane waves, i.e. by: 
     
       
         
           
             E 
             ∝ 
             
               ∫ 
               
                 ∫ 
                 
                   
                     
                       
                         dk 
                         x 
                       
                       ⁢ 
                       
                         dk 
                         y 
                       
                     
                     
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                         
                         ) 
                       
                       2 
                     
                   
                   ⁢ 
                   
                     A 
                     ⁡ 
                     
                       ( 
                       
                         
                           k 
                           x 
                         
                         , 
                         
                           k 
                           y 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     e 
                     
                       
                         
                           ik 
                           x 
                         
                         ⁢ 
                         x 
                       
                       + 
                       
                         
                           ik 
                           y 
                         
                         ⁢ 
                         y 
                       
                       + 
                       
                         
                           ik 
                           z 
                         
                         ⁢ 
                         z 
                       
                       + 
                       
                         iz 
                         ⁢ 
                         
                           
                             
                               k 
                               2 
                             
                             - 
                             
                               k 
                               x 
                               2 
                             
                             - 
                             
                               k 
                               y 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     This representation is also called angular spectrum of plane waves or plane-wave expansion of the electromagnetic field. Here A(k x , k y ) is the amplitude of the plane wave. This representation is chosen in such a way that the net energy flux connected with the electromagnetic field is towards the propagation axis z. Every plane wave is connected with an energy flow that has direction k. Actual lasers generate a spatially coherent electromagnetic field which has a finite transversal extension and propagates with moderate spreading. That means that the wave amplitude changes only slowly along the propagation axis (z-axis) compared to the wavelength and finite width of the beam. Thus, the paraxial approximation can be applied, assuming that the amplitude function A(k x , k y ) falls off sufficiently fast with increasing values of (k x , k y ). 
     Two principal characteristics of the total energy flux can be considered: the divergence (spread of the plane wave amplitudes in wave vector space), defined as: 
     
       
         
           
             Divergence 
             ∝ 
             
               ∫ 
               
                 ∫ 
                 
                   
                     
                       
                         dk 
                         x 
                       
                       ⁢ 
                       
                         dk 
                         y 
                       
                     
                     
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                         
                         ) 
                       
                       2 
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         K 
                         x 
                         2 
                       
                       + 
                       
                         K 
                         y 
                         2 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                        
                       
                         A 
                         ⁡ 
                         
                           ( 
                           
                             
                               k 
                               x 
                             
                             , 
                             
                               k 
                               y 
                             
                           
                           ) 
                         
                       
                        
                     
                     2 
                   
                 
               
             
           
         
       
     
     and the transversal spatial extension (spread of the field intensity perpendicular to the z-direction) defined as: 
     
       
         
           
             
               
                 Transversal 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 Extention 
               
               ∝ 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   ∞ 
                 
                 ⁢ 
                 
                   dx 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         dy 
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               2 
                             
                             + 
                             
                               y 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                            
                           E 
                            
                         
                         2 
                       
                     
                   
                 
               
             
             = 
             
               ∫ 
               
                 ∫ 
                 
                   
                     
                       
                         dk 
                         x 
                       
                       ⁢ 
                       
                         dk 
                         y 
                       
                     
                     
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                         
                         ) 
                       
                       2 
                     
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         
                            
                           
                             
                               ∂ 
                               A 
                             
                             
                               ∂ 
                               x 
                             
                           
                            
                         
                         2 
                       
                       + 
                       
                         
                            
                           
                             
                               ∂ 
                               A 
                             
                             
                               ∂ 
                               y 
                             
                           
                            
                         
                         2 
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     Let&#39;s now look for the fundamental mode of the beam as the electromagnetic field having simultaneously minimal divergence and minimal transversal extension, i.e. as the field that minimizes the product of divergence and extension. By symmetry reasons, this leads to looking for an amplitude function minimizing the product: 
     
       
         
           
             
               
                 [ 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     ∞ 
                   
                   ⁢ 
                   
                     
                       
                         dk 
                         x 
                       
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       k 
                       x 
                       2 
                     
                     ⁢ 
                     
                       
                          
                         A 
                          
                       
                       2 
                     
                   
                 
                 ] 
               
               ⁡ 
               
                 [ 
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     ∞ 
                   
                   ⁢ 
                   
                     
                       
                         dk 
                         x 
                       
                       
                         ( 
                         
                           2 
                           ⁢ 
                           π 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       
                          
                         
                           
                             ∂ 
                             A 
                           
                           
                             ∂ 
                             
                               k 
                               x 
                             
                           
                         
                          
                       
                       2 
                     
                   
                 
                 ] 
               
             
             = 
             
               
                 
                    
                   A 
                    
                 
                 4 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     
                       π 
                       2 
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
     Thus, seeking the field with minimal divergence and minimal transversal extension can lead directly to the fundamental Gaussian beam. This means that the Gaussian beam is the mode with minimum uncertainty, i.e. the product of its sizes in real space and wave-vector space is the theoretical minimum as given by the Heisenberg&#39;s uncertainty principle of Quantum Mechanics. Consequently, the Gaussian mode has less dispersion than any other optical field of the same size, and its diffraction sets a lower threshold for the diffraction of real optical beams. 
     Hermite-Gaussian beams are a family of structurally stable laser modes which have rectangular symmetry along the propagation axis. In order to derive such modes, the simplest approach is to include an additional modulation of the form: 
     
       
         
           
             
               E 
               
                 m 
                 , 
                 n 
               
               H 
             
             = 
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 ∞ 
               
               ⁢ 
               
                 
                   
                     
                       dk 
                       x 
                     
                     ⁢ 
                     
                       dk 
                       y 
                     
                   
                   
                     
                       ( 
                       
                         2 
                         ⁢ 
                         π 
                       
                       ) 
                     
                     2 
                   
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       ik 
                       x 
                     
                     ) 
                   
                   m 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       ik 
                       y 
                     
                     ) 
                   
                   n 
                 
                 ⁢ 
                 
                   e 
                   S 
                 
               
             
           
         
       
       
         
           
             
               S 
               ⁡ 
               
                 ( 
                 
                   
                     k 
                     x 
                   
                   , 
                   
                     k 
                     y 
                   
                   , 
                   x 
                   , 
                   y 
                   , 
                   z 
                 
                 ) 
               
             
             = 
             
               
                 
                   ik 
                   x 
                 
                 ⁢ 
                 x 
               
               + 
               
                 
                   ik 
                   y 
                 
                 ⁢ 
                 y 
               
               + 
               
                 
                   ik 
                   z 
                 
                 ⁢ 
                 z 
               
               - 
               
                 
                   
                     W 
                     0 
                   
                   4 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       1 
                       + 
                       
                         i 
                         ⁢ 
                         
                           Z 
                           
                             Z 
                             R 
                           
                         
                       
                     
                     ) 
                   
                   ⁡ 
                   
                     [ 
                     
                       
                         k 
                         x 
                         2 
                       
                       + 
                       
                         k 
                         y 
                         2 
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
     The new field modes occur to be differential derivatives of the fundamental Gaussian mode E 0 . 
     
       
         
           
             
               E 
               
                 m 
                 , 
                 n 
               
               H 
             
             = 
             
               
                 
                   ∂ 
                   
                     m 
                     + 
                     n 
                   
                 
                 
                   
                     ∂ 
                     
                       x 
                       m 
                     
                   
                   ⁢ 
                   
                     ∂ 
                     
                       y 
                       n 
                     
                   
                 
               
               ⁢ 
               
                 E 
                 0 
               
             
           
         
       
     
     Looking at the explicit form E0 shows that the differentiations in the last equation lead to expressions of the form: 
                 ∂   P       ∂     x   p         ⁢     e     (       -   α     ⁢           ⁢     x             ⁢   2         )             
with some constant p and α. Using now the definition of Hermits&#39; polynomials,
 
     
       
         
           
             
               
                 H 
                 p 
               
               ⁡ 
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 
                   ( 
                   
                     - 
                     1 
                   
                   ) 
                 
                 p 
               
               ⁢ 
               
                 e 
                 
                   ( 
                   
                     x 
                     2 
                   
                   ) 
                 
               
               ⁢ 
               
                 
                   d 
                   P 
                 
                 
                   dx 
                   p 
                 
               
               ⁢ 
               
                 e 
                 
                   ( 
                   
                     
                       - 
                       α 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       x 
                       
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Then the field amplitude becomes 
                 E     m   ,   n     H     ⁡     (     x   ,   y   ,   z     )       =       ∑   m     ⁢           ⁢       ∑   n     ⁢           ⁢       C   mn     ⁢     E   0     ⁢       w   0       w   ⁡     (   z   )         ⁢       H   m     ⁡     (       2     ⁢     x     w   ⁡     (   z   )           )       ⁢       H   n     ⁡     (       2     ⁢     y     w   ⁡     (   z   )           )       ⁢     e       -     (       x   2     +     y   2       )           w   ⁡     (   z   )       2         ⁢     e       -     j   ⁡     (     m   +   n   +   1     )         ⁢     tan     -   1       ⁢     z   /     z   R           ⁢     e       -     (       x   2     +     y   2       )         2   ⁢     R   ⁡     (   z   )                         
Where
 
               ρ   2     =       x   2     +     y   2                   ξ   =     z     z   R             
and Rayleigh length z R 
 
               z   R     =       π   ⁢           ⁢     w   0   2       λ           
And beam diameter
 
 w (ξ)= w   0 √{square root over ((1+ξ 2 ))}
 
     In cylindrical coordinates, the filed takes the form: 
                 E     l   ,   p     L     ⁡     (     ρ   ,   φ   ,   z     )       =                           ⁢           ∑       l             ⁢       ∑   np             ⁢       C   lp     ⁢     E   0     ⁢       w   0       w   ⁡     (   z   )         ⁢       (       2     ⁢     ρ     w   ⁡     (   z   )           )     l     ⁢       L   p   l     ⁡     (       2     ⁢     ρ     w   ⁡     (   z   )           )       ⁢     e       -     ρ   2           w   ⁡     (   z   )       2         ⁢     e       -     j   ⁡     (       2   ⁢   p     +   l   +   1     )         ⁢     tan     -   1       ⁢     z   /     z   R           ⁢     e     jl   ⁢           ⁢   φ       ⁢     e         -   jk     ⁢           ⁢     ρ   2         2   ⁢     R   ⁡     (   z   )                         
Where L p   l  is Laguerre functions.
 
     Mode division multiplexing (MDM) of multiple orthogonal beams increases the system capacity and spectral efficiency in optical communication systems. For free space systems, multiple beams each on a different orthogonal mode can be transmitted through a single transmitter and receiver aperture pair. Moreover, the modal orthogonality of different beans enables the efficient multiplexing at the transmitter and demultiplexing at the receiver. 
     Different optical modal basis sets exist that exhibit orthogonality. For example, orbital angular momentum (OAM) beams that are either Laguerre Gaussian (LG or Laguerre Gaussian light modes may be used for multiplexing of multiple orthogonal beams in free space optical and RF transmission systems. However, there exist other modal groups that also may be used for multiplexing that do not contain OAM. Hermite Gaussian (HG) modes are one such modal group. The intensity of an HG m,n  beam is shown according to the equation: 
                 I   ⁡     (     x   ,   y   ,   z     )       =       C     m   ,   n       ⁢       H   m   2     ⁡     (         2     ⁢   x       w   ⁡     (   z   )         )       ⁢       H   n   2     ⁡     (         2     ⁢   y       w   ⁡     (   z   )         )       ×     exp   ⁡     (       -       2   ⁢     x   2           w   ⁡     (   z   )       2         -       2   ⁢     y   2           w   ⁡     (   z   )       2         )           ,     
     ⁢       w   ⁡     (   z   )       =       w   0     ⁢       1   +     [     λ   ⁢           ⁢     z   /   π     ⁢           ⁢     w   0   2       ]                   
in which H m (*) and H n (*) are the Hermite polynomials of the mth and nth order. The value w 0  is the beam waist at distance Z=0. The spatial orthogonality of HG modes with the same beam waist w 0  relies on the orthogonality of Hermite polynomial in x or y directions.
 
     Referring now to  FIG. 13 , there is illustrated a system for using the orthogonality of an HG modal group for free space spatial multiplexing in free space. A laser  1302  is provided to a beam splitter  1304 . The beam splitter  1304  splits the beam into multiple beams that are each provided to a modulator  1306  for modulation with a data stream  1308 . The modulated beam is provided to collimators  1310  that provides a collimated light beam to spatial light modulators  1312 . Spatial light modulators (SLM&#39;s)  1312  may be used for transforming input plane waves into HG modes of different orders, each mode carrying an independent data channel. These HG modes are spatially multiplexed using a multiplexer  1314  and coaxially transmitted over a free space link  1316 . At the receiver  1318  there are several factors that may affect the demultiplexing of these HG modes, such as receiver aperture size, receiver lateral displacement and receiver angular error. These factors affect the performance of the data channel such as signal-to-noise ratio and crosstalk. 
     With respect to the characteristics of a diverged HG m,0  beam (m=0-6), the wavelength is assumed to be 1550 nm and the transmitted power for each mode is 0 dBm. Higher order HG modes have been shown to have larger beam sizes. For smaller aperture sizes less power is received for higher order HG modes due to divergence. 
     Since the orthogonality of HG modes relies on the optical field distribution in the x and y directions, a finite receiver aperture may truncate the beam. The truncation will destroy the orthogonality and cost crosstalk of the HG channels. When an aperture is smaller, there is higher crosstalk to the other modes. When a finite receiver is used, if an HG mode with an even (odd) order is transmitted, it only causes cross talk to other HG modes with even (odd) numbers. This is explained by the fact that the orthogonality of the odd and even HG modal groups remains when the beam is systematically truncated. 
     Moreover, misalignment of the receiver may cause crosstalk. In one example, lateral displacement can be caused when the receiver is not aligned with the beam axis. In another example, angular error may be caused when the receiver is on axis but there is an angle between the receiver orientation and the beam propagation axis. As the lateral displacement increases, less power is received from the transmitted power mode and more power is leaked to the other modes. There is less crosstalk for the modes with larger mode index spacing from the transmitted mode. 
     Referring now to  FIG. 14 , the reference number  1400  generally indicates an embodiment of a multiple level overlay (MLO) modulation system, although it should be understood that the term MLO and the illustrated system  1400  are examples of embodiments. The MLO system may comprise one such as that disclosed in U.S. Pat. No. 8,503,546 entitled Multiple Layer Overlay Modulation which is incorporated herein by reference. In one example, the modulation system  1400  would be implemented within the multiple level overlay modulation box  504  of  FIG. 5 . System  1400  takes as input an input data stream  1401  from a digital source  1402 , which is separated into three parallel, separate data streams,  1403 A- 1403 C, of logical is and Os by input stage demultiplexer (DEMUX)  1404 . Data stream  1401  may represent a data file to be transferred, or an audio or video data stream. It should be understood that a greater or lesser number of separated data streams may be used. In some of the embodiments, each of the separated data streams  1403 A- 1403 C has a data rate of 1/N of the original rate, where N is the number of parallel data streams. In the embodiment illustrated in  FIG. 14 , N is 3. 
     Each of the separated data streams  1403 A- 1403 C is mapped to a quadrature amplitude modulation (QAM) symbol in an M-QAM constellation, for example, 16 QAM or 64 QAM, by one of the QAM symbol mappers  1405 A-C. The QAM symbol mappers  1405 A-C are coupled to respective outputs of DEMUX  1404 , and produced parallel in phase (I)  1406 A,  1408 A, and  1410 A and quadrature phase (Q)  1406 B,  1408 B, and  1410 B data streams at discrete levels. For example, in 64 QAM, each I and Q channel uses 8 discrete levels to transmit 3 bits per symbol. Each of the three I and Q pairs,  1406 A- 1406 B,  1408 A- 1408 B, and  1410 A- 1410 B, is used to weight the output of the corresponding pair of function generators  1407 A- 1407 B,  1409 A- 1409 B, and  1411 A- 1411 B, which in some embodiments generate signals such as the modified Hermite polynomials described above and weights them based on the amplitude value of the input symbols. This provides 2N weighted or modulated signals, each carrying a portion of the data originally from income data stream  1401 , and is in place of modulating each symbol in the I and Q pairs,  1406 A- 1406 B,  1408 A- 1408 B, and  1410 A- 1410 B with a raised cosine filter, as would be done for a prior art QAM system. In the illustrated embodiment, three signals are used, SH 0 , SH 1 , and SH 2 , which correspond to modifications of H 0 , H 1 , and H 2 , respectively, although it should be understood that different signals may be used in other embodiments. 
     While the description relates to the application of QLO modulation to improve operation of a quadrature amplitude modulation (QAM) system, the application of QLO modulation will also improve the spectral efficiency of other legacy modulation schemes. 
     The weighted signals are not subcarriers, but rather are sublayers of a modulated carrier, and are combined, superimposed in both frequency and time, using summers  1412  and  1416 , without mutual interference in each of the I and Q dimensions, due to the signal orthogonality. Summers  1412  and  1416  act as signal combiners to produce composite signals  1413  and  1417 . The weighted orthogonal signals are used for both I and Q channels, which have been processed equivalently by system  1400 , and are summed before the QAM signal is transmitted. Therefore, although new orthogonal functions are used, some embodiments additionally use QAM for transmission. Because of the tapering of the signals in the time domain, as will be shown in  FIGS. 18A through 18K , the time domain waveform of the weighted signals will be confined to the duration of the symbols. Further, because of the tapering of the special signals and frequency domain, the signal will also be confined to frequency domain, minimizing interface with signals and adjacent channels. 
     The composite signals  1413  and  1417  are converted to analogue signals  1415  and  1419  using digital to analogue converters  1414  and  1418 , and are then used to modulate a carrier signal at the frequency of local oscillator (LO)  1420 , using modulator  1421 . Modulator  1421  comprises mixers  1422  and  1424  coupled to DACs  1414  and  1418 , respectively. Ninety degree phase shifter  1423  converts the signals from LO  1420  into a Q component of the carrier signal. The output of mixers  1422  and  1424  are summed in summer  1425  to produce output signals  1426 . 
     MLO can be used with a variety of transport mediums, such as wire, optical, and wireless, and may be used in conjunction with QAM. This is because MLO uses spectral overlay of various signals, rather than spectral overlap. Bandwidth utilization efficiency may be increased by an order of magnitude, through extensions of available spectral resources into multiple layers. The number of orthogonal signals is increased from 2, cosine and sine, in the prior art, to a number limited by the accuracy and jitter limits of generators used to produce the orthogonal polynomials. In this manner, MLO extends each of the I and Q dimensions of QAM to any multiple access techniques such as GSM, code division multiple access (CDMA), wide band CDMA (WCDMA), high speed downlink packet access (HSPDA), evolution-data optimized (EV-DO), orthogonal frequency division multiplexing (OFDM), world-wide interoperability for microwave access (WIMAX), and long term evolution (LTE) systems. MLO may be further used in conjunction with other multiple access (MA) schemes such as frequency division duplexing (FDD), time division duplexing (TDD), frequency division multiple access (FDMA), and time division multiple access (TDMA). Overlaying individual orthogonal signals over the same frequency band allows creation of a virtual bandwidth wider than the physical bandwidth, thus adding a new dimension to signal processing. This modulation is applicable to twisted pair, coaxial cable, fiber optic, RF satellite, RF broadcast, RF point-to point, RF point-to-multipoint, RF point-to-point (backhaul), RF point-to-point (fronthaul to provide higher throughput CPRI interface for cloudification and virtualization of RAN and cloudified HetNet), free-space optics (FSO), Internet of Things (TOT), Wifi, Bluetooth, as a personal device cable replacement, RF and FSO hybrid system, Radar, electromagnetic tags and all types of wireless access. The method and system are compatible with many current and future multiple access systems, including EV-DO, UMB, WIMAX, WCDMA (with or without), multimedia broadcast multicast service (MBMS)/multiple input multiple output (MIMO), HSPA evolution, and LTE. 
     Referring now back to  FIG. 15 , an MLO demodulator  1500  is illustrated, although it should be understood that the term MLO and the illustrated system  1500  are examples of embodiments. The modulator  1500  takes as input an MLO signal  1526  which may be similar to output signal  1526  from system  1400 . Synchronizer  1527  extracts phase information, which is input to local oscillator  1520  to maintain coherence so that the modulator  1521  can produce base band to analogue I signal  1515  and Q signal  1519 . The modulator  1521  comprises mixers  1522  and  1524 , which, coupled to OL 1520  through 90 degree phase shifter  1523 . I signal  1515  is input to each of signal filters  1507 A,  1509 A, and  1511 A, and Q signal  1519  is input to each of signal filters  1507 B,  1509 B, and  1511 B. Since the orthogonal functions are known, they can be separated using correlation or other techniques to recover the modulated data. Information in each of the I and Q signals  1515  and  1519  can be extracted from the overlapped functions which have been summed within each of the symbols because the functions are orthogonal in a correlative sense. 
     In some embodiments, signal filters  1507 A- 1507 B,  1509 A- 1509 B, and  1511 A- 1511 B use locally generated replicas of the polynomials as known signals in match filters. The outputs of the match filters are the recovered data bits, for example, equivalence of the QAM symbols  1506 A- 1506 B,  1508 A- 1508 B, and  1510 A- 1510 B of system  1500 . Signal filters  1507 A- 1507 B,  1509 A- 1509 B, and  1511 A- 1511 B produce 2n streams of n, I, and Q signal pairs, which are input into demodulators  1528 - 1533 . Demodulators  1528 - 1533  integrate the energy in their respective input signals to determine the value of the QAM symbol, and hence the logical 1s and 0s data bit stream segment represented by the determined symbol. The outputs of the modulators  1528 - 1533  are then input into multiplexers (MUXs)  1505 A- 1505 C to generate data streams  1503 A- 1503 C. If system  1500  is demodulating a signal from system  1400 , data streams  1503 A- 1503 C correspond to data streams  1403 A- 1403 C. Data streams  1503 A- 1503 C are multiplexed by MUX  1504  to generate data output stream  1501 . In summary, MLO signals are overlayed (stacked) on top of one another on transmitter and separated on receiver. 
     MLO may be differentiated from CDMA or OFDM by the manner in which orthogonality among signals is achieved. MLO signals are mutually orthogonal in both time and frequency domains, and can be overlaid in the same symbol time bandwidth product. Orthogonality is attained by the correlation properties, for example, by least sum of squares, of the overlaid signals. In comparison, CDMA uses orthogonal interleaving or displacement of signals in the time domain, whereas OFDM uses orthogonal displacement of signals in the frequency domain. 
     Bandwidth efficiency may be increased for a channel by assigning the same channel to multiple users. This is feasible if individual user information is mapped to special orthogonal functions. CDMA systems overlap multiple user information and views time intersymbol orthogonal code sequences to distinguish individual users, and OFDM assigns unique signals to each user, but which are not overlaid, are only orthogonal in the frequency domain. Neither CDMA nor OFDM increases bandwidth efficiency. CDMA uses more bandwidth than is necessary to transmit data when the signal has a low signal to noise ratio (SNR). OFDM spreads data over many subcarriers to achieve superior performance in multipath radiofrequency environments. OFDM uses a cyclic prefix OFDM to mitigate multipath effects and a guard time to minimize intersymbol interference (ISI), and each channel is mechanistically made to behave as if the transmitted waveform is orthogonal. (Sync function for each subcarrier in frequency domain.) 
     In contrast, MLO uses a set of functions which effectively form an alphabet that provides more usable channels in the same bandwidth, thereby enabling high bandwidth efficiency. Some embodiments of MLO do not require the use of cyclic prefixes or guard times, and therefore, outperforms OFDM in spectral efficiency, peak to average power ratio, power consumption, and requires fewer operations per bit. In addition, embodiments of MLO are more tolerant of amplifier nonlinearities than are CDMA and OFDM systems. 
       FIG. 16  illustrates an embodiment of an MLO transmitter system  1600 , which receives input data stream  1401 . System  1600  represents a modulator/controller  1601 , which incorporates equivalent functionality of DEMUX  1604 , QAM symbol mappers  1405 A-C, function generators  1407 A- 1407 B,  1409 A- 1409 B, and  1411 A- 1411 B, and summers  1412  and  1416  of system  1400 , shown in  FIG. 14 . However, it should be understood that modulator/controller  1601  may use a greater or lesser quantity of signals than the three illustrated in system  1400 . Modulator/controller  1601  may comprise an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), and/or other components, whether discrete circuit elements or integrated into a single integrated circuit (IC) chip. 
     Modulator/controller  1601  is coupled to DACs  1604  and  1607 , communicating a 10 bit I signal  1602  and a 10 bit Q signal  1605 , respectively. In some embodiments, I signal  1602  and Q signal  1605  correspond to composite signals  1413  and  1417  of system  1400 . It should be understood, however, that the 10 bit capacity of I signal  1602  and Q signal  1605  is merely representative of an embodiment. As illustrated, modulator/controller  1601  also controls DACs  1604  and  1607  using control signals  1603  and  1606 , respectively. In some embodiments, DACs  1604  and  1607  each comprise an AD 5433 , complementary metal oxide semiconductor (CMOS) 10 bit current output DAC. In some embodiments, multiple control signals are sent to each of DACs  1604  and  1607 . 
     DACs  1604  and  1607  output analogue signals  1415  and  1419  to quadrature modulator  1421 , which is coupled to LO  1420 . The output of modulator  1420  is illustrated as coupled to a transmitter  1608  to transmit data wirelessly, although in some embodiments, modulator  1421  may be coupled to a fiber-optic modem, a twisted pair, a coaxial cable, or other suitable transmission media. 
       FIG. 17  illustrates an embodiment of an MLO receiver system  1700  capable of receiving and demodulating signals from system  1600 . System  1700  receives an input signal from a receiver  1708  that may comprise input medium, such as RF, wired or optical. The modulator  1521  driven by LO  1520  converts the input to baseband I signal  1515  and Q signal  1519 . I signal  1515  and Q signal  1519  are input to analogue to digital converter (ADC)  1709 . 
     ADC  1709  outputs 10 bit signal  1710  to demodulator/controller  1701  and receives a control signal  1712  from demodulator/controller  1701 . Demodulator/controller  1701  may comprise an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), and/or other components, whether discrete circuit elements or integrated into a single integrated circuit (IC) chip. Demodulator/controller  1701  correlates received signals with locally generated replicas of the signal set used, in order to perform demodulation and identify the symbols sent. Demodulator/controller  1701  also estimates frequency errors and recovers the data clock, which is used to read data from the ADC  1709 . The clock timing is sent back to ADC  1709  using control signal  1712 , enabling ADC  1709  to segment the digital I and Q signals  1517  and  1519 . In some embodiments, multiple control signals are sent by demodulator/controller  1701  to ADC  1709 . Demodulator/controller  1701  also outputs data signal  1301 . 
     Hermite-Gaussian polynomials are a classical orthogonal polynomial sequence, which are the Eigenstates of a quantum harmonic oscillator. Signals based on Hermite-Gaussian polynomials possess the minimal time-bandwidth product property described above, and may be used for embodiments of MLO systems. However, it should be understood that other signals may also be used, for example orthogonal polynomials such as Jacobi polynomials, Gegenbauer polynomials, Legendre polynomials, Chebyshev polynomials, and Laguerre-Gaussian polynomials. Q-functions are another class of functions that can be employed as a basis for MLO signals. 
     In quantum mechanics, a coherent state is a state of a quantum harmonic oscillator whose dynamics most closely resemble the oscillating behavior of a classical harmonic oscillator system. A squeezed coherent state is any state of the quantum mechanical Hilbert space, such that the uncertainty principle is saturated. That is, the product of the corresponding two operators takes on its minimum value. In embodiments of an MLO system, operators correspond to time and frequency domains wherein the time-bandwidth product of the signals is minimized. The squeezing property of the signals allows scaling in time and frequency domain simultaneously, without losing mutual orthogonality among the signals in each layer. This property enables flexible implementations of MLO systems in various communications systems. 
     Because signals with different orders are mutually orthogonal, they can be overlaid to increase the spectral efficiency of a communication channel. For example, when n=0, the optimal baseband signal will have a time-bandwidth product of ½, which is the Nyquist Inter-Symbol Interference (ISI) criteria for avoiding ISI. However, signals with time-bandwidth products of 3/2, 5/2, 7/2, and higher, can be overlaid to increase spectral efficiency. 
     An embodiment of an MLO system uses functions based on modified Hermite polynomials, 4n, and are defined by: 
                 ψ   n     ⁡     (     t   ,   ξ     )       =           (     tanh   ⁢           ⁢   ξ     )       n   /   2           2     n   /   2       ⁢       (       n   !     ⁢   cosh   ⁢           ⁢   ξ     )       1   /   2           ⁢     e       1   2     ⁢       t   2     ⁡     [     1   -     tanh   ⁢           ⁢   ξ       ]           ⁢       H   n     ⁡     (     1       2   ⁢   cosh   ⁢           ⁢   ξsinhξ         )               
where t is time, and ξ is a bandwidth utilization parameter. Plots of Ψ n  for n ranging from 0 to 9, along with their Fourier transforms (amplitude squared), are shown in  FIGS. 5A-5K . The orthogonality of different orders of the functions may be verified by integrating:
 
∫∫ψ n (t,ξ)ψ m (t,ξ)dtdξ
 
     The Hermite polynomial is defined by the contour integral: 
                 H   n     ⁡     (   z   )       =         n   !       2   ⁢     π   !         ⁢     ∮       e       -     t   2       +     2   ⁢   t   ⁢           ⁢   2         ⁢     t       -   n     -   1       ⁢   dt               
where the contour encloses the origin and is traversed in a counterclockwise direction. Hermite polynomials are described in Mathematical Methods for Physicists, by George Arfken, for example on page 416, the disclosure of which is incorporated by reference.
 
       FIGS. 18A-18K  illustrate representative MLO signals and their respective spectral power densities based on the modified Hermite polynomials Ψ n  for n ranging from 0 to 9.  FIG. 18A  shows plots  1801  and  1804 . Plot  1801  comprises a curve  1827  representing Ψ 0  plotted against a time axis  1802  and an amplitude axis  1803 . As can be seen in plot  1801 , curve  1827  approximates a Gaussian curve. Plot  1804  comprises a curve  1837  representing the power spectrum of Ψ 0  plotted against a frequency axis  1805  and a power axis  1806 . As can be seen in plot  1804 , curve  1837  also approximates a Gaussian curve. Frequency domain curve  1807  is generated using a Fourier transform of time domain curve  1827 . The units of time and frequency on axis  1802  and  1805  are normalized for baseband analysis, although it should be understood that since the time and frequency units are related by the Fourier transform, a desired time or frequency span in one domain dictates the units of the corresponding curve in the other domain. For example, various embodiments of MLO systems may communicate using symbol rates in the megahertz (MHz) or gigahertz (GHz) ranges and the non-0 duration of a symbol represented by curve  1827 , i.e., the time period at which curve  1827  is above 0 would be compressed to the appropriate length calculated using the inverse of the desired symbol rate. For an available bandwidth in the megahertz range, the non-0 duration of a time domain signal will be in the microsecond range. 
       FIGS. 18B-18J  show plots  1807 - 1824 , with time domain curves  1828 - 1836  representing Ψ 1  through Ψ 9 , respectively, and their corresponding frequency domain curves  1838 - 1846 . As can be seen in  FIGS. 18A-18J , the number of peaks in the time domain plots, whether positive or negative, corresponds to the number of peaks in the corresponding frequency domain plot. For example, in plot  1823  of  FIG. 18J , time domain curve  1836  has five positive and five negative peaks. In corresponding plot  1824  therefore, frequency domain curve  1846  has ten peaks. 
       FIG. 18K  shows overlay plots  1825  and  1826 , which overlay curves  1827 - 1836  and  1837 - 1846 , respectively. As indicated in plot  1825 , the various time domain curves have different durations. However, in some embodiments, the non-zero durations of the time domain curves are of similar lengths. For an MLO system, the number of signals used represents the number of overlays and the improvement in spectral efficiency. It should be understood that, while ten signals are disclosed in  FIGS. 18A-18K , a greater or lesser quantity of signals may be used, and that further, a different set of signals, rather than the Ψ n  signals plotted, may be used. 
     MLO signals used in a modulation layer have minimum time-bandwidth products, which enable improvements in spectral efficiency, and are quadratically integrable. This is accomplished by overlaying multiple demultiplexed parallel data streams, transmitting them simultaneously within the same bandwidth. The key to successful separation of the overlaid data streams at the receiver is that the signals used within each symbols period are mutually orthogonal. MLO overlays orthogonal signals within a single symbol period. This orthogonality prevents ISI and inter-carrier interference (ICI). 
     Because MLO works in the baseband layer of signal processing, and some embodiments use QAM architecture, conventional wireless techniques for optimizing air interface, or wireless segments, to other layers of the protocol stack will also work with MLO. Techniques such as channel diversity, equalization, error correction coding, spread spectrum, interleaving and space-time encoding are applicable to MLO. For example, time diversity using a multipath-mitigating rake receiver can also be used with MLO. MLO provides an alternative for higher order QAM, when channel conditions are only suitable for low order QAM, such as in fading channels. MLO can also be used with CDMA to extend the number of orthogonal channels by overcoming the Walsh code limitation of CDMA. MLO can also be applied to each tone in an OFDM signal to increase the spectral efficiency of the OFDM systems. 
     Embodiments of MLO systems amplitude modulate a symbol envelope to create sub-envelopes, rather than sub-carriers. For data encoding, each sub-envelope is independently modulated according to N-QAM, resulting in each sub-envelope independently carrying information, unlike OFDM. Rather than spreading information over many sub-carriers, as is done in OFDM, for MLO, each sub-envelope of the carrier carries separate information. This information can be recovered due to the orthogonality of the sub-envelopes defined with respect to the sum of squares over their duration and/or spectrum. Pulse train synchronization or temporal code synchronization, as needed for CDMA, is not an issue, because MLO is transparent beyond the symbol level. MLO addresses modification of the symbol, but since CDMA and TDMA are spreading techniques of multiple symbol sequences over time. MLO can be used along with CDMA and TDMA. 
       FIG. 19  illustrates a comparison of MLO signal widths in the time and frequency domains. Time domain envelope representations  1901 - 1903  of signals SH 0 -SH 3  are illustrated as all having a duration T S . SH 0 -SH 3  may represent PSI 0 -PSI 2 , or may be other signals. The corresponding frequency domain envelope representations are 1905-1907, respectively. SH 0  has a bandwidth BW, SH 1  has a bandwidth three times BW, and SH 2  has a bandwidth of 5BW, which is five times as great as that of SH 0 . The bandwidth used by an MLO system will be determined, at least in part, by the widest bandwidth of any of the signals used. The highest order signal must set within the available bandwidth. This will set the parameters for each of the lower order signals in each of the layers and enable the signals to fit together without interference. If each layer uses only a single signal type within identical time windows, the spectrum will not be fully utilized, because the lower order signals will use less of the available bandwidth than is used by the higher order signals. 
       FIG. 20A  illustrates a spectral alignment of MLO signals that accounts for the differing bandwidths of the signals, and makes spectral usage more uniform, using SH 0 -SH 3 . Blocks  2001 - 2004  are frequency domain blocks of an OFDM signal with multiple subcarriers. Block  2003  is expanded to show further detail. Block  2003  comprises a first layer  2003   x  comprised of multiple SH 0  envelopes  2003   a - 2003   o . A second layer  2003   y  of SH 1  envelopes  2003   p - 2003   t  has one third the number of envelopes as the first layer. In the illustrated example, first layer  2003   x  has 15 SH 0  envelopes, and second layer  2003   y  has five SH 1  envelopes. This is because, since the SH 1  bandwidth envelope is three times as wide as that of SH 0 , 15 SH 0  envelopes occupy the same spectral width as five SH 1  envelopes. The third layer  2003   z  of block  2003  comprises three SH 2  envelopes  2003   u - 2003   w , because the SH 2  envelope is five times the width of the SH 0  envelope. 
     The total required bandwidth for such an implementation is a multiple of the least common multiple of the bandwidths of the MLO signals. In the illustrated example, the least common multiple of the bandwidth required for SH 0 , SH 1 , and SH 2  is 15BW, which is a block in the frequency domain. The OFDM-MLO signal can have multiple blocks, and the spectral efficiency of this illustrated implementation is proportional to (15+5+3)/15. 
       FIGS. 20B-20C  illustrate a situation wherein the frequency domain envelopes  2020 - 2024  are each located in a separate layer within a same physical band width  2025 . However, each envelope rather than being centered on a same center frequency as shown in  FIG. 19  has its own center frequency  2026 - 2030  shifted in order to allow a slided overlay. The purposed of the slided center frequency is to allow better use of the available bandwidth and insert more envelopes in a same physical bandwidth. 
     Since each of the layers within the MLO signal comprises a different channel, different service providers may share a same bandwidth by being assigned to different MLO layers within a same bandwidth. Thus, within a same bandwidth, service provider one may be assigned to a first MLO layer, service provider two may be assigned to a second MLO layer and so forth. 
       FIG. 21  illustrates another spectral alignment of MLO signals, which may be used alternatively to alignment scheme shown in  FIG. 20 . In the embodiment illustrated in  FIG. 21 , the OFDM-MLO implementation stacks the spectrum of SH 0 , SH 1 , and SH 2  in such a way that the spectrum in each layer is utilized uniformly. Layer  2100 A comprises envelopes  2101 A- 2101 D, which includes both SH 0  and SH 2  envelopes. Similarly, layer  2100 C, comprising envelopes  2103 A- 2103 D, includes both SH 0  and SH 2  envelopes. Layer  2100 B, however, comprising envelopes  2102 A- 2102 D, includes only SH 1  envelopes. Using the ratio of envelope sizes described above, it can be easily seen that BW+5BW=3BW+3BW. Thus, for each SH 0  envelope in layer  2100 A, there is one SH 2  envelope also in layer  2100 C and two SH 1  envelopes in layer  2100 B. 
     Three Scenarios Compared:
     1) MLO with 3 Layers defined by:   

                   f   0     ⁡     (   t   )       =       W   0     ⁢     e     -       t   2     4             ,       W   0     =   0.6316                       f   1     ⁡     (   t   )       =       W   1     ⁢     e     -       t   2     4             ,       W   1     ≈   0.6316                       f   2     ⁡     (   t   )       =         W   2     ⁡     (       t   2     -   1     )       ⁢     e     -       t   2     4             ,     W   ≈   0.4466           
(The current FPGA implementation uses the truncation interval of [−6, 6].)
     2) Conventional scheme using rectangular pulse   3) Conventional scheme using a square-root raised cosine (SRRC) pulse with a roll-off factor of 0.5   

     For MLO pulses and SRRC pulse, the truncation interval is denoted by [−t 1 , t 1 ] in the following FIGS. For simplicity, we used the MLO pulses defined above, which can be easily scaled in time to get the desired time interval (say micro-seconds or nano-seconds). For the SRRC pulse, we fix the truncation interval of [−3T, 3T] where T is the symbol duration for all results presented in this document. 
     Bandwidth Efficiency 
     The X-dB bounded power spectral density bandwidth is defined as the smallest frequency interval outside which the power spectral density (PSD) is X dB below the maximum value of the PSD. The X-dB can be considered as the out-of-band attenuation. 
     The bandwidth efficiency is expressed in Symbols per second per Hertz. The bit per second per Hertz can be obtained by multiplying the symbols per second per Hertz with the number of bits per symbol (i.e., multiplying with log 2 M for M-ary QAM). 
     Truncation of MLO pulses introduces inter-layer interferences (ILI). However, the truncation interval of [−6, 6] yields negligible ILI while [−4, 4] causes slight tolerable ILI. Referring now to  FIG. 22 , there is illustrated the manner in which a signal, for example a superQAM signal, may be layered to create ILI.  FIG. 22  illustrates  3  different superQAM signals  2202 ,  2204  and  2206 . The superQAM signals  2202 - 2206  may be truncated and overlapped into multiple layers using QLO in the manner described herein above. However, as illustrated in  FIG. 66 , the truncation of the superQAM signals  2202 - 2206  that enables the signals to be layered together within a bandwidth T d    2302  creates a single signal  2304  having the interlayer interference between each of the layers containing a different signal produced by the QLO process. The ILI is caused between a specific bit within a specific layer having an effect on other bits within another layer of the same symbol. 
     The bandwidth efficiency of MLO may be enhanced by allowing inter-symbol interference (ISI). To realize this enhancement, designing transmitter side parameters as well as developing receiver side detection algorithms and error performance evaluation can be performed. One manner in which ISI may be created is when multilayer signals such as that illustrated in  FIG. 23  are overlapped with each other in the manner illustrated in  FIG. 24 . Multiple signal symbols  2402  are overlapped with each other in order to enable to enable more symbols to be located within a single bandwidth. The portions of the signal symbols  2402  that are overlapping cause the creation of ISI. Thus, a specific bit at a specific layer will have an effect on the bits of nearby symbols. 
     The QLO transmission and reception system can be designed to have a particular known overlap between symbols. The system can also be designed to calculate the overlaps causing ISI (symbol overlap) and ILI (layer overlay). The ISI and ILI can be expressed in the format of a NM*NM matrix derived from a N*NM matrix. N comprises the number of layers and M is the number of symbols when considering ISI. Referring now to  FIG. 25 , there is illustrated a fixed channel matrix H xy  which is a N*NM matrix. From this we can calculate another matrix which is H yx  which is a NM*NM matrix. The ISI and ILI can be canceled by (a) applying a filter of H yx   −1  to the received vector or (b) pre-distorting the transmitted signal by the SVD (singular value decomposition) of H yx   −1 . Therefore, by determining the matrix H xy  of the fixed channel, the signal may be mathematically processed to remove ISL and ILI. 
     When using orthogonal functions such as Hermite Guassian (HG) functions, the functions are all orthogonal for any permutations of the index if infinitely extended. However, when the orthogonal functions are truncated as discussed herein above, the functions become pseudo-orthogonal. This is more particularly illustrated in  FIG. 26 . In this case, orthogonal functions are represented along each of the axes. At the intersection of the same orthogonal functions, functions are completely correlated and a value of “1” is indicated. Thus, a diagonal of “1” exists with each of the off diagonal elements comprising a “0” since these functions are completely orthogonal with each other. When truncated HG choose functions are used the 0 values will not be exactly 0 since the functions are no longer orthogonal but are pseudo-orthogonal. 
     However, the HG functions can be selected in a manner that the functions are practically orthogonal. This is achieved by selecting the HG signals in a sequence to achieve better orthogonality. Thus, rather than selecting the initial three signals in a three signal HG signal sequence (P 0  P 1  P 2 ), various other sequences that do not necessarily comprise the first three signals of the HG sequence may be selected as shown below. 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                   
                 P0 P1 P4 
                 P0 P3 P6 
               
               
                   
                 P0 P1 P6 
                 P0 P4 P5 
               
               
                   
                 P0 P2 P3 
                 P0 P5 P6 
               
               
                   
                 P0 P2 P5 
                 P1 P3 P6 
               
               
                   
                 P0 P3 P4 
                 P2 P5 P6 
               
               
                   
               
            
           
         
       
     
     Similar selection of sequences may be done to achieve better orthogonality with two signals, four signals, etc. 
     The techniques described herein are applicable to a wide variety of communication band environments. They may be applied across the visible and invisible bands and include RF, Fiber, Freespace optical and any other communications bands that can benefit from the increased bandwidth provided by the disclosed techniques. 
     Application of OAM to Optical Communication 
     Utilization of OAM for optical communications is based on the fact that coaxially propagating light beams with different OAM states can be efficiently separated. This is certainly true for orthogonal modes such as the LG beam. Interestingly, it is also true for general OAM beams with cylindrical symmetry by relying only on the azimuthal phase. Considering any two OAM beams with an azimuthal index of l  1  and l  2 , respectively:
 
 U   1 ( r,θ,z )= A   1 ( r,z )exp( il   1 θ)  (12)
 
where r and z refers to the radial position and propagation distance respectively, one can quickly conclude that these two beams are orthogonal in the sense that:
 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       
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                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
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                       θ 
                     
                   
                   = 
                   
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                               A 
                               2 
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                               if 
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                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     There are two different ways to take advantage of the distinction between OAM beams with different l states in communications. In the first approach, N different OAM states can be encoded as N different data symbols representing “0”, “1”, . . . , “N−1”, respectively. A sequence of OAM states sent by the transmitter therefore represents data information. At the receiver, the data can be decoded by checking the received OAM state. This approach seems to be more favorable to the quantum communications community, since OAM could provide for the encoding of multiple bits (log 2(N)) per photon due to the infinitely countable possibilities of the OAM states, and so could potentially achieve a higher photon efficiency. The encoding/decoding of OAM states could also have some potential applications for on-chip interconnection to increase computing speed or data capacity. 
     The second approach is to use each OAM beam as a different data carrier in an SDM (Spatial Division Multiplexing) system. For an SDM system, one could use either a multi-core fiber/free space laser beam array so that the data channels in each core/laser beam are spatially separated, or use a group of orthogonal mode sets to carry different data channels in a multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s data transmission in a multi-core fiber and up to 6 linearly polarized (LP) modes each with two polarizations in a single core multi-mode fiber has been reported. Similar to the SDM using orthogonal modes, OAM beams with different states can be spatially multiplexed and demultiplexed, thereby providing independent data carriers in addition to wavelength and polarization. Ideally, the orthogonality of OAM beams can be maintained in transmission, which allows all the data channels to be separated and recovered at the receiver. A typical embodiments of OAM multiplexing is conceptually depicted in  FIG. 27 . An obvious benefit of OAM multiplexing is the improvement in system spectral efficiency, since the same bandwidth can be reused for additional data channels. 
     OAM Beam Generation and Detection 
     Many approaches for creating OAM beams have been proposed and demonstrated. 
     One could obtain a single or multiple OAM beams directly from the output of a laser cavity, or by converting a fundamental Gaussian beam into an OAM beam outside a cavity. The converter could be a spiral phase plate, diffractive phase holograms, metalmaterials, cylindrical lens pairs, q-plates or fiber structures. There are also different ways to detect an OAM beam, such as using a converter that creates a conjugate helical phase, or using a plasmonic detector. 
     Mode Conversion Approaches 
     Referring now to  FIG. 28 , among all external-cavity methods, perhaps the most straightforward one is to pass a Gaussian beam through a coaxially placed spiral phase plate (SPP)  2802 . An SPP  2802  is an optical element with a helical surface, as shown in  FIG. 12E . To produce an OAM beam with a state of l, the thickness profile of the plate should be machined as lλθ/2π(n−1), where n is the refractive index of the medium. A limitation of using an SPP  2802  is that each OAM state requires a different specific plate. As an alternative, reconfigurable diffractive optical elements, e.g., a pixelated spatial light modulator (SLM)  2804 , or a digital micro-mirror device can be programmed to function as any refractive element of choice at a given wavelength. As mentioned above, a helical phase profile exp(ilθ) converts a linearly polarized Gaussian laser beam into an OAM mode, whose wave front resembles an l-fold corkscrew  2806 , as shown at  2804 . Importantly, the generated OAM beam can be easily changed by simply updating the hologram loaded on the SLM  2804 . To spatially separate the phase-modulated beam from the zeroth-order non-phase-modulated reflection from the SLM, a linear phase ramp is added to helical phase code (i.e., a “fork”-like phase pattern  2808  to produce a spatially distinct first-order diffracted OAM beam, carrying the desired charge. It should also be noted that the aforementioned methods produce OAM beams with only an azimuthal index control. To generate a pure LG_(l,p) mode, one must jointly control both the phase and the intensity of the wavefront. This could be achieved using a phase-only SLM with a more complex phase hologram. 
     Some novel material structures, such as metal-surface, can also be used for OAM generation. A compact metal-surface could be made into a phase plate by manipulation of the structure caused spatial phase response. As shown in  FIGS. 29A and 29B , a V-shaped antenna array  2902  is fabricated on the metal surface  2904 , each of which is composed of two arms  2906 ,  2908  connected at one end  2910 . A light reflected by this plate would experience a phase change ranging from 0 to 2π, determined by the length of the arms and angle between two arms. To generate an OAM beam, the surface is divided into 8 sectors  2912 , each of which introduces a phase shift from 0 to 7π/4 with a step of π/4. The OAM beam with l=+1 is obtained after the reflection, as shown in  FIG. 29C . 
     Referring now to  FIG. 30 , another interesting liquid crystal-based device named “q-plate”  3002  is also used as a mode converter which converts a circularly polarized beam  3004  into an OAM beam  3006 . A q-plate is essentially a liquid crystal slab with a uniform birefringent phase retardation of π and a spatially varying transverse optical axis  3008  pattern. Along the path circling once around the center of the plate, the optical axis of the distributed crystal elements may have a number of rotations defined by the value of q. A circularly polarized beam  3004  passing through this plate  3002  would experience a helical phase change of exp (ilθ) with l=2q, as shown in  FIG. 30 . 
     Note that almost all the mode conversion approaches can also be used to detect an OAM beam. For example, an OAM beam can be converted back to a Gaussian-like non-OAM beam if the helical phase front is removed, e.g., by passing the OAM beam through a conjugate SPP or phase hologram. 
     Intra-Cavity Approaches 
     Referring now to  FIG. 31 , OAM beams are essentially higher order modes and can be directly generated from a laser resonator cavity. The resonator  3100  supporting higher order modes usually produce the mixture of multiple modes  3104 , including the fundamental mode. In order to avoid the resonance of fundamental Gaussian mode, a typical approach is to place an intra-cavity element  3106  (spiral phase plate, tiled mirror) to force the oscillator to resonate on a specific OAM mode. Other reported demonstrations include the use of an annular shaped beam as laser pump, the use of thermal lensing, or by using a defect spot on one of the resonator mirrors. 
     OAM Beams Multiplexing and Demultiplexing 
     One of the benefits of OAM is that multiple coaxially propagating OAM beams with different l states provide additional data carriers as they can be separated based only on the twisting wavefront. Hence, one of the critical techniques is the efficient multiplexing/demultiplexing of OAM beams of different l states, where each carries an independent data channel and all beams can be transmitted and received using a single aperture pair. Several multiplexing and demultiplexing techniques have been demonstrated, including the use of an inverse helical phase hologram to down-convert the OAM into a Gaussian beam, a mode sorter, free-space interferometers, a photonic integrated circuit, and q-plates. Some of these techniques are briefly described below. 
     Beam Splitter and Inverse Phase Hologram 
     A straightforward way of multiplexing is simply to use cascaded 3-dB beam splitters (BS)  3202 . Each BS  3202  can coaxially multiplex two beams  3203  that are properly aligned, and cascaded N BSs can multiplex N+1 independent OAM beams at most, as shown in  FIG. 32 . Similarly, at the receiver end, the multiplexed beam  3205  is divided into four copies  3204  by BS  3202 . To demultiplex the data channel on one of the beams (e.g., with l=1_i), a phase hologram  3206  with a spiral charge of  −1 _i is applied to all the multiplexed beams  3204 . As a result, the helical phase on the target beam is removed, and this beam evolves into a fundamental Gaussian beam, as shown in  FIG. 33 . The down-converted beam can be isolated from the other beams, which still have helical phase fronts by using a spatial mode filter  3308  (e.g., a single mode fiber only couples the power of the fundamental Gaussian mode due to the mode matching theory). Accordingly, each of the multiplexed beams  3304  can be demultiplexed by changing the spiral phase hologram  3306 . Although this method is very power-inefficient since the BSs  3302  and the spatial mode filter  3306  cause a lot of power loss, it was used in the initial lab demonstrations of OAM multiplexing/demultiplexing, due to the simplicity of understanding and the reconfigurability provided by programmable SLMs. 
     Optical Geometrical Transformation-Based Mode Sorter 
     Referring now to  FIG. 34 , another method of multiplexing and demultiplexing, which could be more power-efficient than the previous one (using beam splitters), is the use of an OAM mode sorter. This mode sorter usually comprises three optical elements, including a transformer  3402 , a corrector  3404 , and a lens  3406 , as shown in  FIG. 34 . The transformer  3402  performs a geometrical transformation of the input beam from log-polar coordinates to Cartesian coordinates, such that the position (x,y) in the input plane is mapped to a new position (u,v) in the output plane, where 
               u   =       -   a     ⁢           ⁢     ln   ⁡     (           x   2     +     y   2         b     )           ,         
and v=a arctan(y/x). Here, a and b are scaling constants. The corrector  3404  compensates for phase errors and ensures that the transformed beam is collimated. Considering an input OAM beam with a ring-shaped beam profile, it can be unfolded and mapped into a rectangular-shaped plane wave with a tilted phase front. Similarly, multiple OAM beams having different 1 states will be transformed into a series of plane waves each with a different phase tilt. A lens  3406  focuses these tilted plane waves into spatially separated spots in the focal plane such that all the OAM beams are simultaneously demultiplexed. As the transformation is reciprocal, if the mode sorter is used in reverse it can become a multiplexer for OAM. A Gaussian beam array placed in the focal plane of the lens  3406  is converted into superimposed plane waves with different tilts. These beams then pass through the corrector and the transformer sequentially to produce properly multiplexed OAM beams.
 
Free Space Communications
 
     The first proof-of-concept experiment using OAM for free space communications transmitted eight different OAM states each representing a data symbol one at a time. The azimuthal index of the transmitted OAM beam is measured at the receiver using a phase hologram modulated with a binary grating. To effectively use this approach, fast switching is required between different OAM states to achieve a high data rate. Alternatively, classic communications using OAM states as data carriers can be multiplexed at the transmitter, co-propagated through a free space link, and demultiplexed at a receiver. The total data rate of a free space communication link has reached 100 Tbit/s or even beyond by using OAM multiplexing. The propagation of OAM beams through a real environment (e.g., across a city) is also under investigation. 
     Basic Link Demonstrations 
     Referring now to  FIGS. 35-36B , initial demonstrates of using OAM multiplexing for optical communications include free space links using a Gaussian beam and an OAM beam encoded with OOK data. Four monochromatic Gaussian beams each carrying an independent 50.8 Gbit/s (4×12.7 Gbit/s) 16-QAM signal were prepared from an IQ modulator and free-space collimators. The beams were converted to OAM beams with l=−8, +10, +12 and −14, respectively, using 4 SLMs each loaded with a helical phase hologram, as shown in  FIG. 30A . After being coaxially multiplexed using cascaded 3 dB-beam splitters, the beams were propagated through ˜1 m distance in free-space under lab conditions. The OAM beams were detected one at a time, using an inverse helical phase hologram and a fiber collimator together with a SMF. The 16-QAM data on each channel was successfully recovered, and a spectral efficiency of 12.8 bit/s/Hz in this data link was achieved, as shown in  FIGS. 36A and 36B . 
     A following experiment doubled the spectral efficiency by adding the polarization multiplexing into the OAM-multiplexed free-space data link. Four different OAM beams (l=+4, +8, −8, +16) on each of two orthogonal polarizations (eight channels in total) were used to achieve a Terabit/s transmission link. The eight OAM beams were multiplexed and demultiplexed using the same approach as mentioned above. The measured crosstalk among channels carried by the eight OAM beams is shown in Table 1, with the largest crosstalk being ˜−18.5 dB. Each of the beams was encoded with a 42.8 Gbaud 16-QAM signal, allowing a total capacity of ˜1.4 (42.8×4×4×2) Tbit/s. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                   
                   
                 OAM +4  (dB) 
                 OAM +8  (dB) 
                 OAM −8  (dB) 
                 OAM +16  (dB) 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 Measured Crosstalk 
                 X-Pol. 
                 Y-Pol. 
                 X-Pol. 
                 Y-Pol. 
                 X-Pol. 
                 Y-Pol. 
                 X-Pol. 
                 Y-Pol. 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
            
               
                 OAM +4  (dB) 
                 X-Pol. 
                 — 
                 −23.2 
                 −26.7 
                 −30.8 
                 −30.5 
                 −27.7 
                 −24.6 
                 −30.1 
               
               
                   
                 Y-Pol. 
                 −25.7 
                 — 
                   
                   
                   
                   
                   
                   
               
               
                 OAM +8  (dB) 
                 X-Pol. 
                 −26.6 
                 −23.5 
                 — 
                 −21.6 
                 −18.9 
                 −25.4 
                 −23.9 
                 −28.8 
               
               
                   
                 Y-Pol. 
                   
                   
                 −25 
                 — 
                   
                   
                   
                   
               
               
                 OAM −8  (dB) 
                 X-Pol. 
                 −27.5 
                 −33.9 
                 −27.6 
                 −30.8 
                 — 
                 −20.5 
                 −26.5 
                 −21.6 
               
               
                   
                 Y-Pol. 
                   
                   
                   
                   
                 −26.8 
                 — 
                   
                   
               
               
                 OAM +16  (dB) 
                 X-Pol. 
                 −24.5 
                 −31.2 
                 −23.7 
                 −23.3 
                 −25.8 
                 −26.1 
                 — 
                 −30.2 
               
               
                   
                 Y-Pol. 
                   
                   
                   
                   
                   
                   
                 −24 
                 — 
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 Total from other OAMs* (dB) 
                 −21.8 
                 −21 
                 −21.2 
                 −21.4 
                 −18.5 
                 −21.2 
                 −22.2 
                 −20.7 
               
               
                   
               
            
           
         
       
     
     The capacity of the free-space data link was further increased to 100 Tbit/s by combining OAM multiplexing with PDM (phase division multiplexing) and WDM (wave division multiplexing). In this experiment, 24 OAM beams (l=±4, ±7, ±10, ±13, ±16, and ±19, each with two polarizations) were prepared using 2 SLMs, the procedures for which are shown in  FIG. 37A  at  3702 - 3706 . Specifically, one SLM generated a superposition of OAM beams with l=+4, +10, and +16, while the other SLM generated another set of three OAM beams with l=+7, +13, and +19 ( FIG. 37A ). These two outputs were multiplexed together using a beam splitter, thereby multiplexing six OAM beams: l=+4, +7, +10, +13, +16, and +19 ( FIG. 37A ). Secondly, the six multiplexed OAM beams were split into two copies. One copy was reflected five times by three mirrors and two beam splitters, to create another six OAM beams with inverse charges ( FIG. 37B ). There was a differential delay between the two light paths to de-correlate the data. These two copies were then combined again to achieve 12 multiplexed OAM beams with l=±4, ±7, ±10, ±13, ±16, and ±19 ( FIG. 37B ). These 12 OAM beams were split again via a beam splitter. One of these was polarization-rotated by 90 degrees, delayed by ˜33 symbols, and then recombined with the other copy using a polarization beam splitter (PBS), finally multiplexing 24 OAM beams (with l=±4, ±7, ±10, ±13, ±16, and ±19 on two polarizations). Each of the beam carried a WDM signal comprising 100 GHz-spaced 42 wavelengths (1,536.34-1,568.5 nm), each of which was modulated with 100 Gbit/s QPSK data. The observed optical spectrum of the WDM signal carried on one of the demultiplexed OAM beams (l=+10). 
     Atmospheric Turbulence Effects on OAM Beams 
     One of the critical challenges for a practical free-space optical communication system using OAM multiplexing is atmospheric turbulence. It is known that inhomogeneities in the temperature and pressure of the atmosphere lead to random variations in the refractive index along the transmission path, and can easily distort the phase front of a light beam. This could be particularly important for OAM communications, since the separation of multiplexed OAM beams relies on the helical phase-front. As predicted by simulations in the literature, these refractive index inhomogeneities may cause inter-modal crosstalk among data channels with different OAM states. 
     The effect of atmospheric turbulence is also experimentally evaluated. For the convenience of estimating the turbulence strength, one approach is to emulate the turbulence in the lab using an SLM or a rotating phase plate.  FIG. 38A  illustrates an emulator built using a thin phase screen plate  3802  that is mounted on a rotation stage  3804  and placed in the middle of the optical path. The pseudo-random phase distribution machined on the plate  3802  obeys Kolmogorov spectrum statistics, which are usually characterized by a specific effective Fried coherence length r 0 . The strength of the simulated turbulence  3806  can be varied either by changing to a plate  3802  with a different r 0 , or by adjusting the size of the beam that is incident on the plate. The resultant turbulence effect is mainly evaluated by measuring the power of the distorted beam distributed to each OAM mode using an OAM mode sorter. It was found that, as the turbulence strength increases, the power of the transmitted OAM mode would leak to neighboring modes and tend to be equally distributed among modes for stronger turbulence. As an example,  FIG. 38B  shows the measured average power (normalized) l=3 beam under different emulated turbulence conditions. It can be seen that the majority of the power is still in the transmitted OAM mode  3808  under weak turbulence, but it spreads to neighboring modes as the turbulence strength increases. 
     Turbulence Effects Mitigation Techniques 
     One approach to mitigate the effects of atmospheric turbulence on OAM beams is to use an adaptive optical (AO) system. The general idea of an AO system is to measure the phase front of the distorted beam first, based on which an error correction pattern can be produced and can be applied onto the beam transmitter to undo the distortion. As for OAM beams with helical phase fronts, it is challenging to directly measure the phase front using typical wavefront sensors due to the phase singularity. A modified AO system can overcome this problem by sending a Gaussian beam as a probe beam to sense the distortion, as shown in  FIG. 39A . Due to the fact that turbulence is almost independent of the light polarization, the probe beam is orthogonally polarized as compared to all other beams for the sake of convenient separation at beam separator  3902 . The correction phase pattern can be derived based on the probe beam distortion that is directly measured by a wavefront sensor  3804 . It is noted that this phase correction pattern can be used to simultaneously compensate multiple coaxially propagating OAM beams.  FIG. 39  at  3910 - 3980  illustrate the intensity profiles of OAM beams with l=1, 5 and 9, respectively, for a random turbulence realization with and without mitigation. From the far-field images, one can see that the distorted OAM beams (upper), up to l=9, were partially corrected, and the measured power distribution also indicates that the channel crosstalk can be reduced. 
     Another approach for combating turbulence effects is to partially move the complexity of optical setup into the electrical domain, and use digital signal processing (DSP) to mitigate the channel crosstalk. A typical DSP method is the multiple-input-multiple-output (MIMO) equalization, which is able to blindly estimate the channel crosstalk and cancel the interference. The implementation of a 4×4 adaptive MIMO equalizer in a four-channel OAM multiplexed free space optical link using heterodyne detection may be used. Four OAM beams (l=+2, +4, +6 and +8), each carrying 20 Gbit/s QPSK data, were collinearly multiplexed and propagated through a weak turbulence emulated by the rotating phase plate under laboratory condition to introduce distortions. After demultiplexing, four channels were coherently detected and recorded simultaneously. The standard constant modulus algorithm is employed in addition to the standard procedures of coherent detection to equalize the channel interference. Results indicate that MIMO equalization could be helpful to mitigate the crosstalk caused by either turbulence or imperfect mode generation/detection, and improve both error vector magnitude (EVM) and the bit-error-rate (BER) of the signal in an OAM-multiplexed communication link. MIMO DSP may not be universally useful as outage could happen in some scenarios involving free space data links. For example, the majority power of the transmitted OAM beams may be transferred to other OAM states under a strong turbulence without being detected, in which case MIMO would not help to improve the system performance. 
     OAM Free Space Link Design Considerations 
     To date, most of the experimental demonstrations of optical communication links using OAM beams took place in the lab conditions. There is a possibility that OAM beams may also be used in a free space optical communication link with longer distances. To design such a data link using OAM multiplexing, several important issues such as beam divergence, aperture size and misalignment of two transmitter and receiver, need to be resolved. To study how those parameters affect the performance of an OAM multiplexed system, a simulation model was described by Xie et al, the schematic setup of which is shown in  FIG. 40 . Each of the different collimated Gaussian beams  4002  at the same wavelength is followed by a spiral phase plate  4004  with a unique order to convert the Gaussian beam into a data-carrying OAM beam. Different orders of OAM beams are then multiplexed at multiplexor  4006  to form a concentric-ring-shape and coaxially propagate from transmitter  4008  through free space to the receiver aperture located at a certain propagation distance. Propagation of multiplexed OAM beams is numerically propagated using the Kirchhoff-Fresnel diffraction integral. To investigate the signal power and crosstalk effect on neighboring OAM channels, power distribution among different OAM modes is analyzed through a modal decomposition approach, which corresponds to the case where the received OAM beams are demultiplexed without power loss and the power of a desired OAM channel is completely collected by its receiver  4010 . 
     Beam Divergence 
     For a communication link, it is generally preferable to collect as much signal power as possible at the receiver to ensure a reasonable signal-to-noise ratio (SNR). Based on the diffraction theory, it is known that a collimated OAM beam diverges while propagating in free space. Given the same spot size of three cm at the transmitter, an OAM beam with a higher azimuthal index diverges even faster, as shown in  FIG. 41A . On the other hand, the receiving optical element usually has a limited aperture size and may not be able to collect all of the beam power. The calculated link power loss as a function of receiver aperture size is shown in  FIG. 41B , with different transmission distances and various transmitted beam sizes. Unsurprisingly, the power loss of a 1-km link is higher than that of a 100-m link under the same transmitted beam size due to larger beam divergence. It is interesting to note that a system with a transmitted beam size of 3 cm suffers less power loss than that of 1 cm and 10 cm over a 100-m link. The 1-cm transmitted beam diverges faster than the 3 cm beam due to its larger diffraction. However, when the transmitted beam size is 10 cm, the geometrical characteristics of the beam dominate over the diffraction, thus leading larger spot size at the receiver than the 3 cm transmitted beam. A trade-off between the diffraction, geometrical characteristics and the number of OAMs of the beam therefore needs to be carefully considered in order to achieve a proper-size received beam when designing a link. 
     Misalignment Tolerance 
     Referring now to  FIGS. 42A-42C , besides the power loss due to limited-size aperture and beam divergence, another issue that needs further discussion is the potential misalignment between the transmitter and the receiver. In an ideal OAM multiplexed communication link, transmitter and receiver would be perfectly aligned, (i.e., the center of the receiver would overlap with the center of the transmitted beam  4202 , and the receiver plane  4204  would be perpendicular to the line connecting their centers, as shown in  FIG. 42A ). However, due to difficulties in aligning because of substrate distances, and jitter and vibration of the transmitter/receiver platform, the transmitter and receiver may have relative lateral shift (i.e., lateral displacement) ( FIG. 42B ) or angular shift (i.e., receiver angular error) ( FIG. 42C ). Both types of misalignment may lead to degradation of system performance. 
     Focusing on a link distance of 100 m,  FIGS. 43A and 43B  show the power distribution among different OAM modes due to lateral displacement and receiver angular error when only l=+3 is transmitted with a transmitted beam size of 3 cm. In order to investigate the effect of misalignment, the receiver aperture size is chosen to be 10 cm, which could cover the whole OAM beam at the receiver. As the lateral displacement or receiver angular error increases, power leakage to other modes (i.e., channel crosstalk) increases while the power on l=+3 state decreases. This is because larger lateral displacement or receiver angular causes larger phase profile mismatch between the received OAM beams and receiver. The power leakage to l=+1 and l=+5 is greater than that of l=+2 and l=+3 due to their larger mode spacing with respect to l=+3. Therefore, a system with larger mode spacing (which also uses higher order OAM states suffers less crosstalk. However, such a system may also have a larger power loss due to the fast divergence of higher order OAM beams, as discussed above. Clearly, this trade-off between channel crosstalk and power loss shall be considered when choosing the mode spacing in a specific OAM multiplexed link. 
     Referring now to  FIG. 44 , there is a bandwidth efficiency comparison versus out of band attenuation (X-dB) where quantum level overlay pulse truncation interval is [−6,6] and the symbol rate is 1/6. Referring also to  FIG. 45 , there is illustrated the bandwidth efficiency comparison versus out of band attenuation (X-dB) where quantum level overlay pulse truncation interval is [−6,6] and the symbol rate is 1/4. 
     The QLO signals are generated from the Physicist&#39;s special Hermite functions: 
     
       
         
           
             
               
                 
                   f 
                   n 
                 
                 ⁡ 
                 
                   ( 
                   
                     t 
                     , 
                     α 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     α 
                     
                       
                         π 
                       
                       ⁢ 
                       
                         n 
                         ! 
                       
                       ⁢ 
                       
                         2 
                         n 
                       
                     
                   
                 
                 ⁢ 
                 
                   
                     H 
                     n 
                   
                   ⁡ 
                   
                     ( 
                     
                       α 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   e 
                   
                     - 
                     
                       
                         
                           α 
                           2 
                         
                         ⁢ 
                         
                           t 
                           2 
                         
                       
                       2 
                     
                   
                 
               
             
             , 
             
               α 
               &gt; 
               0 
             
           
         
       
     
     Note that the initial hardware implementation is using 
             α   =     1     2             
and for consistency with his part,
 
             α   =     1     2             
is used in all FIGS. related to the spectral efficiency.
 
     Let the low-pass-equivalent power spectral density (PSD) of the combined QLO signals be X(f) and its bandwidth be B. Here the bandwidth is defined by one of the following criteria. 
     ACLR 1  (First Adjacent Channel Leakage Ratio) in dBc equals: 
               ACLR   ⁢           ⁢   1     =         ∫     B   /   2       3   ⁢     B   /   2         ⁢       X   ⁡     (   f   )       ⁢   df           ∫     -   ∞     ∞     ⁢       X   ⁡     (   f   )       ⁢   df               
ACLR 2  (Second Adjacent Channel Leakage Ratio) in dBc equals:
 
               ACLR   ⁢           ⁢   2     =         ∫     3   ⁢     B   /   2         5   ⁢     B   /   2         ⁢       X   ⁡     (   f   )       ⁢   df           ∫     -   ∞     ∞     ⁢       X   ⁡     (   f   )       ⁢   df               
Out-of-Band Power to Total Power Ratio is:
 
               2   ⁢       ∫     B   /   2     ∞     ⁢       X   ⁡     (   f   )       ⁢   df             ∫     -   ∞     ∞     ⁢       X   ⁡     (   f   )       ⁢   df             
The Band-Edge PSD in dBc/100 kHz equals:
 
     
       
         
           
             
               
                 ∫ 
                 
                   B 
                   / 
                   2 
                 
                 
                   
                     B 
                     2 
                   
                   + 
                   
                     10 
                     5 
                   
                 
               
               ⁢ 
               
                 
                   X 
                   ⁡ 
                   
                     ( 
                     f 
                     ) 
                   
                 
                 ⁢ 
                 df 
               
             
             
               
                 ∫ 
                 
                   - 
                   ∞ 
                 
                 ∞ 
               
               ⁢ 
               
                 
                   X 
                   ⁡ 
                   
                     ( 
                     f 
                     ) 
                   
                 
                 ⁢ 
                 df 
               
             
           
         
       
     
     Referring now to  FIG. 46  there is illustrated a performance comparison using ACLR 1  and ACLR 2  for both a square root raised cosine scheme and a multiple layer overlay scheme. Line  4602  illustrates the performance of a square root raised cosine  4602  using ACLR 1  versus an MLO  4604  using ACLR 1 . Additionally, a comparison between a square root raised cosine  4606  using ACLR 2  versus MLO  4608  using ACLR 2  is illustrated. Table 2 illustrates the performance comparison using ACLR. 
     
       
         
           
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Criteria: 
                   
                   
               
               
                 ACLR1 ≤−30 dBc 
                   
                   
               
               
                 per bandwidth 
                   
                   
               
               
                 ACLR2 ≤−43 dBc 
                 Spectral Efficiency 
                   
               
               
                 per bandwidth 
                 (Symbol/sec/Hz) 
                 Gain 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 SRRC [−8T, 8T] β = 0.22 
                 0.8765 
                 1.0 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                 Symbol Duration 
                   
                   
               
               
                   
                 N Layers 
                 (Tmol) 
               
               
                   
               
               
                 QLO 
                 N = 3 
                 Tmol = 4 
                 1.133 
                 1.2926 
               
               
                 [−8, 8] 
                 N = 4 
                 Tmol = 5 
                 1.094 
                 1.2481 
               
               
                   
                   
                 Tmol = 4 
                 1.367 
                 1.5596 
               
               
                   
                 N = 10 
                 Tmol = 8 
                 1.185 
                 1.3520 
               
               
                   
                   
                 Tmol = 7 
                 1.355 
                 1.5459 
               
               
                   
                   
                 Tmol = 6 
                 1.580 
                 1.8026 
               
               
                   
                   
                 Tmol = 5 
                 1.896 
                 2.1631 
               
               
                   
                   
                 Tmol = 4 
                 2.371 
                 2.7051 
               
               
                   
               
            
           
         
       
     
     Referring now to  FIG. 47 , there is illustrated a performance comparison between a square root raised cosine  4702  and a MLO  4704  using out-of-band power. Referring now also to Table 3, there is illustrated a more detailed comparison of the performance using out-of-band power. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Table 3: Performance Comparison Using Out-of-Band Power 
               
            
           
           
               
               
               
            
               
                 Criterion: 
                   
                   
               
               
                 Out-of-band Power/Total 
                 Spectral Efficiency 
                   
               
               
                 Power ≤−30 dB 
                 (Symbol/sec/Hz) 
                 Gain 
               
               
                   
               
            
           
           
               
               
               
            
               
                 SRRC [−8T, 8T] β = 0.22 
                 0.861 
                 1.0 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                 Symbol Duration 
                   
                   
               
               
                   
                 N Layers 
                 (Tmol) 
               
               
                   
               
               
                 QLO 
                 N = 3 
                 Tmol = 4 
                 1.080 
                 1.2544 
               
               
                 [−8, 8] 
                 N = 4 
                 Tmol = 5 
                 1.049 
                 1.2184 
               
               
                   
                   
                 Tmol = 4 
                 1.311 
                 1.5226 
               
               
                   
                 N = 10 
                 Tmol = 8 
                 1.152 
                 1.3380 
               
               
                   
                   
                 Tmol = 7 
                 1.317 
                 1.5296 
               
               
                   
                   
                 Tmol = 6 
                 1.536 
                 1.7840 
               
               
                   
                   
                 Tmol = 5 
                 1.844 
                 2.1417 
               
               
                   
                   
                 Tmol = 4 
                 2.305 
                 2.6771 
               
               
                   
               
            
           
         
       
     
     Referring now to  FIG. 48 , there is further provided a performance comparison between a square root raised cosine  4802  and a MLO  4804  using band-edge PSD. A more detailed illustration of the performance comparison is provided in Table 4. 
     
       
         
           
               
             
               
                 TABLE 4 
               
             
            
               
                   
               
               
                 Table 4: Performance Comparison Using Band-Edge PSD 
               
            
           
           
               
               
               
            
               
                 Criterion: 
                   
                   
               
               
                 Band-Edge PSD = 
                 Spectral Efficiency 
                   
               
               
                 −50 dBc/100 kHz 
                 (Symbol/sec/Hz) 
                 Gain 
               
               
                   
               
            
           
           
               
               
               
            
               
                 SRRC [−8T, 8T] β = 0.22 
                 0.810 
                 1.0 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                 Symbol Duration 
                   
                   
               
               
                   
                 N Layers 
                 (Tmol) 
               
               
                   
               
               
                 QLO 
                 N = 3 
                 Tmol = 4 
                 0.925 
                 1.1420 
               
               
                 [−8, 8] 
                 N = 4 
                 Tmol = 5 
                 0.912 
                 1.1259 
               
               
                   
                   
                 Tmol = 4 
                 1.14 
                 1.4074 
               
               
                   
                 N = 10 
                 Tmol = 8 
                 1.049 
                 1.2951 
               
               
                   
                   
                 Tmol = 7 
                 1.198 
                 1.4790 
               
               
                   
                   
                 Tmol = 6 
                 1.398 
                 1.7259 
               
               
                   
                   
                 Tmol = 5 
                 1.678 
                 2.0716 
               
               
                   
                   
                 Tmol = 4 
                 2.097 
                 2.5889 
               
               
                   
               
            
           
         
       
     
     Referring now to  FIGS. 49 and 50 , there are more particularly illustrated the transmit subsystem ( FIG. 49 ) and the receiver subsystem ( FIG. 50 ). The transceiver is realized using basic building blocks available as Commercially Off The Shelf products. Modulation, demodulation and Special Hermite correlation and de-correlation are implemented on a FPGA board. The FPGA board  5002  at the receiver  5000  estimated the frequency error and recovers the data clock (as well as data), which is used to read data from the analog-to-digital (ADC) board  5006 . The FGBA board  5000  also segments the digital I and Q channels. 
     On the transmitter side  4900 , the FPGA board  4902  realizes the special hermite correlated QAM signal as well as the necessary control signals to control the digital-to-analog (DAC) boards  4904  to produce analog I&amp;Q baseband channels for the subsequent up conversion within the direct conversion quad modulator  4906 . The direct conversion quad modulator  4906  receives an oscillator signal from oscillator  4908 . 
     The ADC  5006  receives the I&amp;Q signals from the quad demodulator  5008  that receives an oscillator signal from  5010 . 
     Neither power amplifier in the transmitter nor an LNA in the receiver is used since the communication will take place over a short distance. The frequency band of 2.4-2.5 GHz (ISM band) is selected, but any frequency band of interest may be utilized. 
     MIMO uses diversity to achieve some incremental spectral efficiency. Each of the signals from the antennas acts as an independent orthogonal channel. With QLO, the gain in spectral efficiency comes from within the symbol and each QLO signal acts as independent channels as they are all orthogonal to one another in any permutation. However, since QLO is implemented at the bottom of the protocol stack (physical layer), any technologies at higher levels of the protocol (i.e. Transport) will work with QLO. Therefore one can use all the conventional techniques with QLO. This includes RAKE receivers and equalizers to combat fading, cyclical prefix insertion to combat time dispersion and all other techniques using beam forming and MIMO to increase spectral efficiency even further. 
     When considering spectral efficiency of a practical wireless communication system, due to possibly different practical bandwidth definitions (and also not strictly bandlimited nature of actual transmit signal), the following approach would be more appropriate. 
     Referring now to  FIG. 51 , consider the equivalent discrete time system, and obtain the Shannon capacity for that system (will be denoted by Cd). Regarding the discrete time system, for example, for conventional QAM systems in AWGN, the system will be:
 
 y[n]=a x[n]+w[n] 
 
where a is a scalar representing channel gain and amplitude scaling, x[n] is the input signal (QAM symbol) with unit average energy (scaling is embedded in a), y[n] is the demodulator (matched filter) output symbol, and index n is the discrete time index.
 
     The corresponding Shannon capacity is:
 
 C   d =log 2 (1+| a|   2 /σ 2 )
 
where σ 2  is the noise variance (in complex dimension) and |a|2/σ2 is the SNR of the discrete time system.
 
     Second, compute the bandwidth W based on the adopted bandwidth definition (e.g., bandwidth defined by −40 dBc out of band power). If the symbol duration corresponding to a sample in discrete time (or the time required to transmit C d  bits) is T, then the spectral efficiency can be obtained as:
 
 C/W=C   d /( TW )bps/Hz
 
     In discrete time system in AWGN channels, using Turbo or similar codes will give performance quite close to Shannon limit C d . This performance in discrete time domain will be the same regardless of the pulse shape used. For example, using either SRRC (square root raised cosine) pulse or a rectangle pulse gives the same C d  (or C d /T). However, when we consider continuous time practical systems, the bandwidths of SRRC and the rectangle pulse will be different. For a typical practical bandwidth definition, the bandwidth for a SRRC pulse will be smaller than that for the rectangle pulse and hence SRRC will give better spectral efficiency. In other words, in discrete time system in AWGN channels, there is little room for improvement. However, in continuous time practical systems, there can be significant room for improvement in spectral efficiency. 
     Referring now to  FIG. 52 , there is illustrated a PSD plot (BLANK) of MLO, modified MLO (MMLO) and square root raised cosine (SRRC). From the illustration in  FIG. 30 , demonstrates the better localization property of MLO. An advantage of MLO is the bandwidth.  FIG. 30  also illustrates the interferences to adjacent channels will be much smaller for MLO. This will provide additional advantages in managing, allocating or packaging spectral resources of several channels and systems, and further improvement in overall spectral efficiency. If the bandwidth is defined by the −40 dBc out of band power, the within-bandwidth PSDs of MLO and SRRC are illustrated in  FIG. 53 . The ratio of the bandwidths is about 1.536. Thus, there is significant room for improvement in spectral efficiency. 
     Modified MLO systems are based on block-processing wherein each block contains N MLO symbols and each MLO symbol has L layers. MMLO can be converted into parallel (virtual) orthogonal channels with different channel SNRs as illustrated in  FIG. 54 . The outputs provide equivalent discrete time parallel orthogonal channels of MMLO. 
     Referring now to  FIG. 55 , there are illustrated four MLO symbols that are included in a single block  5500 . The four symbols  5502 - 5508  are combined together into the single block  5500 . The adjacent symbols  5502 - 5508  each have an overlapping region  5510 . This overlapping region  5510  causes intersymbol interference between the symbols which must be accounted for when processing data streams. 
     Note that the intersymbol interference caused by pulse overlapping of MLO has been addressed by the parallel orthogonal channel conversion. As an example, the power gain of a parallel orthogonal virtual channel of MMLO with three layers and 40 symbols per block is illustrated in  FIG. 56 .  FIG. 56  illustrates the channel power gain of the parallel orthogonal channels of MMLO with three layers and T sim =3. By applying a water filling solution, an optimal power distribution across the orthogonal channels for a fixed transmit power may be obtained. The transmit power on the k th  orthogonal channel is denoted by P k . Then the discrete time capacity of the MMLO can be given by: 
     
       
         
           
             
               C 
               d 
             
             = 
             
               
                 ∑ 
                 
                   k 
                   = 
                   1 
                 
                 k 
               
               ⁢ 
               
                 
                   
                     log 
                     2 
                   
                   ⁡ 
                   
                     ( 
                     
                       1 
                       + 
                       
                         
                           
                             P 
                             k 
                           
                           ⁢ 
                           
                             
                                
                               
                                 a 
                                 k 
                               
                                
                             
                             2 
                           
                         
                         
                           σ 
                           k 
                           2 
                         
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 bits 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 per 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 block 
               
             
           
         
       
     
     Note that K depends on the number of MLO layers, the number of MLO symbols per block, and MLO symbol duration. 
     For MLO pulse duration defined by [−t 1 , t 1 ], and symbol duration T mlo , the MMLO block length is:
 
 T   block =( N− 1) T   mlo +2 t   1  
 
     Suppose the bandwidth of MMLO signal based on the adopted bandwidth definition (ACLR, OBP, or other) is W mmlo , then the practical spectral efficiency of MMLO is given by: 
     
       
         
           
             
               
                 C 
                 d 
               
               
                 
                   W 
                   mmlo 
                 
                 ⁢ 
                 
                   T 
                   block 
                 
               
             
             = 
             
               
                 1 
                 
                   
                     W 
                     mmlo 
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           ( 
                           
                             N 
                             - 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         
                           T 
                           mlo 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           t 
                           1 
                         
                       
                     
                     } 
                   
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   K 
                 
                 ⁢ 
                 
                   
                     
                       log 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             
                               P 
                               k 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   a 
                                   k 
                                 
                                  
                               
                               2 
                             
                           
                           
                             σ 
                             k 
                             2 
                           
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     bps 
                     Hz 
                   
                 
               
             
           
         
       
     
       FIGS. 57-58  show the spectral efficiency comparison of MMLO with N=40 symbols per block, L=3 layers, T mlo =3, t 1 =8, and SRRC with duration [−8T, 8T], T=1, and the roll-off factor β=0.22, at SNR of 5 dB. Two bandwidth definitions based on ACLR 1  (first adjacent channel leakage power ratio) and OBP (out of band power) are used. 
       FIGS. 59-60  show the spectral efficiency comparison of MMLO with L=4 layers. The spectral efficiencies and the gains of MMLO for specific bandwidth definitions are shown in the following tables. 
     
       
         
           
               
               
               
             
               
                 TABLE 5 
               
               
                   
               
               
                   
                 Spectral Efficiency 
                 Gain with 
               
               
                   
                 (bps/Hz) based on ACLR1  
                 reference 
               
               
                   
                 ≤30 dBc per bandwidth 
                 to SRRC 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 SRRC 
                 1.7859 
                 1 
               
               
                 MMLO (3 layers, Tmlo = 3) 
                 2.7928 
                 1.5638 
               
               
                 MMLO (4 layers, Tmlo = 3) 
                 3.0849 
                 1.7274 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                 TABLE 6 
               
               
                   
               
               
                   
                 Spectral Efficiency 
                   
               
               
                   
                 (bps/Hz) based on 
                 Gain with reference 
               
               
                   
                 OBP ≤ −40 dBc 
                 to SRRC 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 SRRC 
                 1.7046 
                 1 
               
               
                 MMLO (3 layers, Tmlo = 3) 
                 2.3030 
                 1.3510 
               
               
                 MMLO (4 layers, Tmlo = 3) 
                 2.6697 
                 1.5662 
               
               
                   
               
            
           
         
       
     
     Referring now to  FIGS. 61 and 62 , there are provided basic block diagrams of low-pass-equivalent MMLO transmitters ( FIG. 61 ) and receivers ( FIG. 62 ). The low-pass-equivalent MMLO transmitter  6100  receives a number of input signals  6102  at a block-based transmitter processing  6104 . The transmitter processing outputs signals to the SH(L−1) blocks  6106  which produce the I&amp;Q outputs. These signals are then all combined together at a combining circuit  6108  for transmission. 
     Within the baseband receiver ( FIG. 62 )  6200 , the received signal is separated and applied to a series of match filters  6202 . The outputs of the match filters are then provided to the block-based receiver processing block  6204  to generate the various output streams. 
     Consider a block of N MLO-symbols with each MLO symbol carrying L symbols from L layers. Then there are NL symbols in a block. Define c(m, n)=symbol transmitted by the m-th MLO layer at the n-th MLO symbol. Write all NL symbols of a block as a column vector as follows: c=[c(0,0), c(1,0), . . . , c(L−1, 0), c(0,1), c(1,1), . . . , c(L−1, 1), . . . , c(L−1, N−1)]T. Then the outputs of the receiver matched filters for that transmitted block in an AWGN channel, defined by the column vector y of length NL, can be given as y=H c+n, where H is an NL X NL matrix representing the equivalent MLO channel, and n is a correlated Gaussian noise vector. 
     By applying SVD to H, we have H=U D VH where D is a diagonal matrix containing singular values. Transmitter side processing using V and the receiver side processing UH, provides an equivalent system with NL parallel orthogonal channels, (i.e., y=H Vc+n and UH y=Dc+UH n). These parallel channel gains are given by diagonal elements of D. The channel SNR of these parallel channels can be computed. Note that by the transmit and receive block-based processing, we obtain parallel orthogonal channels and hence the ISI issue has be resolved. 
     Since the channel SNRs of these parallel channels are not the same, we can apply the optimal Water filling solution to compute the transmit power on each channel given a fixed total transmit power. Using this transmit power and corresponding channel SNR, we can compute capacity of the equivalent system as given in the previous report. 
     Issues of Fading, Multipath, and Multi-Cell Interference 
     Techniques used to counteract channel fading (e.g., diversity techniques) in conventional systems can also be applied in MMLO. For slowly-varying multi-path dispersive channels, if the channel impulse response can be fed back, it can be incorporated into the equivalent system mentioned above, by which the channel induced ISI and the intentionally introduced MMLO ISI can be addressed jointly. For fast time-varying channels or when channel feedback is impossible, channel equalization needs to be performed at the receiver. A block-based frequency-domain equalization can be applied and an oversampling would be required. 
     If we consider the same adjacent channel power leakage for MMLO and the conventional system, then the adjacent cells&#39; interference power would be approximately the same for both systems. If interference cancellation techniques are necessary, they can also be developed for MMLO. 
     Channel fading can be another source of intersymbol interference (ISI) and interlayer interference (ILI). One manner for representing small-scale signal fading is the use of statistical models. White Gaussian noise may be used to model system noise. The effects of multipath fading may be modeled using Rayleigh or Rician probability density functions. Additive white Gaussian noise (AWGN) may be represented in the following manner. A received signal is:
 
 r ( t )= s ( t )+ n ( t )
 
where: r(t)=a received signal; s(t)=a transmitted signal; and n(t)=random noise signal
 
     Rayleigh fading functions are useful for predicting bit error rate (BER) any multipath environment. When there is no line of sight (LOS) or dominate received signal, the power the transmitted signal may be represented by: 
                 P   r     ⁡     (   r   )       =     {             r     σ   2       ⁢     e       -     r   2         2   ⁢     σ   2                   r   ≥   0               0   ,           r   &lt;   0                   
where: σ=rms value of received signal before envelope detection,
         σ=time average power of the received signal before envelope detection.       

     In a similar manner, Rician functions may be used in situations where there is a line of sight or dominant signal within a transmitted signal. In this case, the power of the transmitted signal can be represented by: 
                 P   r     ⁡     (   r   )       =     {               r     σ   2       ⁢     e       -     (       r   2     +     A   2       )         2   ⁢     σ   2           ⁢       II   0     ⁡     (       A   r       σ   2       )         ,           A   ≥   r   ≥   0               0   ,           r   &lt;   0                   
where A=peak amplitude of LOS component
         II 0 =modified Bessel Function of the first kind and zero-order       

     These functions may be implemented in a channel simulation to calculate fading within a particular channel using a channel simulator such as that illustrated in  FIG. 63 . The channel simulator  6302  includes a Bernoulli binary generator  6304  for generating an input signal that is provided to a rectangular M-QAM modulator  6306  that generates a QAM signal at baseband. Multipath fading channel block  6308  uses the Rician equations to simulate multipath channel fading. The simulated multipath fading channel is provided to a noise channel simulator  6310 . The noise channel simulator  6310  simulates AWGN noise. The multipath fading channel simulator  6308  further provides channel state information to arithmetic processing block  6312  which utilizes the simulated multipath fading information and the AWGN information into a signal that is demodulated at QAM demodulator block  6314 . The demodulated simulated signal is provided to the doubler block  6316  which is input to a receive input of an error rate calculator  6318 . The error rate calculator  6318  further receives at a transmitter input, the simulated transmission signal from the Bernoulli binary generator  6304 . The error rate calculator  6318  uses the transmitter input and the received input to provide in error rate calculation to a bit error rate block  6320  that determines the channel bit error rate. This type of channel simulation for determining bit error rate will enable a determination of the amount of QLO that may be applied to a signal in order to increase throughput without overly increasing the bit error rate within the channel. 
     Scope and System Description 
     This report presents the symbol error probability (or symbol error rate) performance of MLO signals in additive white Gaussian noise channel with various inter-symbol interference levels. As a reference, the performance of the conventional QAM without ISI is also included. The same QAM size is considered for all layers of MLO and the conventional QAM. 
     The MLO signals are generated from the Physicist&#39;s special Hermite functions: 
                 f   n     ⁡     (     t   ,   α     )       =         α       π     ⁢     n   !     ⁢     2   n           ⁢       H   n     ⁡     (     α   ⁢           ⁢   t     )       ⁢     e     -         α   2     ⁢     t   2       2                 
where Hn(αt) is the n th  order Hermite polynomial. Note that the functions used in the lab setup correspond to
 
             α   =     1     2             
and, for consistency,
 
             α   =     1     2             
is used in this report.
 
     MLO signals with 3, 4 or 10 layers corresponding to n=0˜2, 0˜3, or 0˜9 are used and the pulse duration (the range of t) is [−8, 8] in the above function. 
     AWGN channel with perfect synchronization is considered. 
     The receiver consists of matched filters and conventional detectors without any interference cancellation, i.e., QAM slicing at the matched filter outputs. 
               %   ⁢           ⁢   pulse   ⁢     -     ⁢   overlapping     =           T   p     -     T   sym         T   p       ×   100   ⁢   %           
where Tp is the pulse duration (16 in the considered setup) and Tsym is the reciprocal of the symbol rate in each MLO layer. The considered cases are listed in the following table.
 
                             TABLE 7               % of Pulse Overlapping   T sym     T p                                                0%   16   16       12.5%   14   16       18.75%    13   16         25%   12   16       37.5%   10   16       43.75%    9   16         50%   8   16       56.25%    7   16       62.5%   6   16         75%   4   16                    
Derivation of the Signals Used in Modulation
 
     To do that, it would be convenient to express signal amplitude s(t) in a complex form close to quantum mechanical formalism. Therefore the complex signal can be represented as: 
               ψ   ⁡     (   t   )       =       s   ⁡     (   t   )       +     j   ⁢           ⁢     σ   ⁡     (   t   )                         where   ⁢           ⁢     s   ⁡     (   t   )         ≡     real   ⁢           ⁢   signal                   σ   ⁡     (   t   )       =     imaginary   ⁢             ⁢             ⁢   signal   ⁢           ⁢     (   quadrature   )                     σ   ⁡     (   t   )       =         1   π     ⁢       ∫     -   ∞     ∞     ⁢       s   ⁡     (   τ   )       ⁢       d   ⁢           ⁢   τ       τ   -   t       ⁢     
     ⁢     s   ⁡     (   t   )             =       -     1   π       ⁢       ∫     -   ∞     ∞     ⁢       σ   ⁡     (   τ   )       ⁢       d   ⁢           ⁢   τ       τ   -   t                     
Where s(t) and σ(t) are Hilbert transforms of one another and since σ(t) is qudratures of s(t), they have similar spectral components. That is if they were the amplitudes of sound waves, the ear could not distinguish one form from the other.
 
     Let us also define the Fourier transform pairs as follows: 
     
       
         
           
             
               ψ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 1 
                 π 
               
               ⁢ 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   ∞ 
                 
                 ⁢ 
                 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                     e 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                   
                   ⁢ 
                   df 
                 
               
             
           
         
       
       
         
           
             
               ψ 
               ⁡ 
               
                 ( 
                 f 
                 ) 
               
             
             = 
             
               
                 1 
                 π 
               
               ⁢ 
               
                 
                   ∫ 
                   
                     - 
                     ∞ 
                   
                   ∞ 
                 
                 ⁢ 
                 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                     e 
                     
                       j 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                   
                   ⁢ 
                   dt 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   ψ 
                   * 
                 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
               ⁢ 
               
                 ψ 
                 ⁡ 
                 
                   ( 
                   t 
                   ) 
                 
               
             
             = 
             
               
                 
                   
                     [ 
                     
                       s 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ] 
                   
                   2 
                 
                 + 
                 
                   
                     [ 
                     
                       σ 
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ] 
                   
                   2 
                 
                 + 
                 … 
               
               ≡ 
               
                 signal 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 power 
               
             
           
         
       
     
     Let&#39;s also normalize all moments to M 0 : 
     
       
         
           
             
               M 
               0 
             
             = 
             
               
                 ∫ 
                 0 
                 τ 
               
               ⁢ 
               
                 
                   s 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 ⁢ 
                 dt 
               
             
           
         
       
       
         
           
             
               M 
               0 
             
             = 
             
               
                 ∫ 
                 0 
                 τ 
               
               ⁢ 
               
                 
                   φ 
                   * 
                 
                 ⁢ 
                 φ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 df 
               
             
           
         
       
     
     Then the moments are as follows: 
     
       
         
           
             
               M 
               0 
             
             = 
             
               
                 ∫ 
                 0 
                 τ 
               
               ⁢ 
               
                 
                   s 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 ⁢ 
                 dt 
               
             
           
         
       
       
         
           
             
               M 
               1 
             
             = 
             
               
                 ∫ 
                 0 
                 τ 
               
               ⁢ 
               
                 
                   ts 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 ⁢ 
                 dt 
               
             
           
         
       
       
         
           
             
               M 
               2 
             
             = 
             
               
                 ∫ 
                 0 
                 τ 
               
               ⁢ 
               
                 
                   t 
                   2 
                 
                 ⁢ 
                 
                   s 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 ⁢ 
                 dt 
               
             
           
         
       
       
         
           
             
               M 
               
                 N 
                 - 
                 1 
               
             
             = 
             
               
                 ∫ 
                 0 
                 τ 
               
               ⁢ 
               
                 
                   t 
                   
                     N 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                   s 
                   ⁡ 
                   
                     ( 
                     t 
                     ) 
                   
                 
                 ⁢ 
                 dt 
               
             
           
         
       
     
     In general, one can consider the signal s(t) be represented by a polynomial of order N, to fit closely to s(t) and use the coefficient of the polynomial as representation of data. This is equivalent to specifying the polynomial in such a way that its first N “moments” M j  shall represent the data. That is, instead of the coefficient of the polynomial, we can use the moments. Another method is to expand the signal s(t) in terms of a set of N orthogonal functions φ k (t), instead of powers of time. Here, we can consider the data to be the coefficients of the orthogonal expansion. One class of such orthogonal functions are sine and cosine functions (like in Fourier series). 
     Therefore we can now represent the above moments using the orthogonal function  104   with the following moments: 
     
       
         
           
             
               t 
               _ 
             
             = 
             
               
                 ∫ 
                 
                   
                     
                       ψ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   t 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   dt 
                 
               
               
                 ∫ 
                 
                   
                     
                       ψ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   dt 
                 
               
             
           
         
       
       
         
           
             
               
                 t 
                 2 
               
               _ 
             
             = 
             
               
                 ∫ 
                 
                   
                     
                       ψ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   
                     t 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   dt 
                 
               
               
                 ∫ 
                 
                   
                     
                       ψ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   dt 
                 
               
             
           
         
       
       
         
           
             
               
                 t 
                 n 
               
               _ 
             
             = 
             
               
                 ∫ 
                 
                   
                     
                       ψ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   
                     t 
                     n 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   dt 
                 
               
               
                 ∫ 
                 
                   
                     
                       ψ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     ψ 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ⁢ 
                   dt 
                 
               
             
           
         
       
     
     Similarly, 
     
       
         
           
             
               f 
               _ 
             
             = 
             
               
                 ∫ 
                 
                   
                     
                       φ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   f 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   df 
                 
               
               
                 ∫ 
                 
                   
                     
                       φ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   df 
                 
               
             
           
         
       
       
         
           
             
               
                 f 
                 _ 
               
               2 
             
             = 
             
               
                 ∫ 
                 
                   
                     
                       φ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                     f 
                     2 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   df 
                 
               
               
                 ∫ 
                 
                   
                     
                       φ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   df 
                 
               
             
           
         
       
       
         
           
             
               
                 f 
                 _ 
               
               n 
             
             = 
             
               
                 ∫ 
                 
                   
                     
                       φ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                     f 
                     n 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   df 
                 
               
               
                 ∫ 
                 
                   
                     
                       φ 
                       * 
                     
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     φ 
                     ⁡ 
                     
                       ( 
                       f 
                       ) 
                     
                   
                   ⁢ 
                   df 
                 
               
             
           
         
       
     
     If we did not use complex signal, then:
 
   f = 0
 
     To represent the mean values from time to frequency domains, replace: 
     
       
         
           
             
               φ 
               ⁡ 
               
                 ( 
                 f 
                 ) 
               
             
             → 
             
               ψ 
               ⁡ 
               
                 ( 
                 t 
                 ) 
               
             
           
         
       
       
         
           
             f 
             → 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   j 
                 
               
               ⁢ 
               
                 d 
                 dt 
               
             
           
         
       
     
     These are equivalent to somewhat mysterious rule in quantum mechanics where classical momentum becomes an operator: 
     
       
         
           
             
               P 
               x 
             
             → 
             
               
                 h 
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   j 
                 
               
               ⁢ 
               
                 ∂ 
                 
                   ∂ 
                   x 
                 
               
             
           
         
       
     
     Therefore using the above substitutions, we have: 
     
       
         
           
             
               f 
               _ 
             
             = 
             
               
                 
                   ∫ 
                   
                     
                       
                         φ 
                         * 
                       
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     f 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     df 
                   
                 
                 
                   ∫ 
                   
                     
                       
                         φ 
                         * 
                       
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     df 
                   
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       
                         
                           ψ 
                           * 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             j 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ψ 
                             ⁡ 
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         dt 
                       
                       ⁢ 
                       dt 
                     
                   
                   
                     ∫ 
                     
                       
                         
                           ψ 
                           * 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       
                         ψ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ⁢ 
                       dt 
                     
                   
                 
                 = 
                 
                   
                     ( 
                     
                       1 
                       
                         2 
                         ⁢ 
                         π 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         
                           ψ 
                           * 
                         
                         ⁢ 
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ψ 
                           
                           dt 
                         
                         ⁢ 
                         dt 
                       
                     
                     
                       ∫ 
                       
                         
                           ψ 
                           * 
                         
                         ⁢ 
                         ψ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         dt 
                       
                     
                   
                 
               
             
           
         
       
     
     And: 
     
       
         
           
             
               
                 f 
                 _ 
               
               2 
             
             = 
             
               
                 
                   ∫ 
                   
                     
                       
                         φ 
                         * 
                       
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     
                       
                         f 
                         ⁢ 
                         
                             
                         
                       
                       2 
                     
                     ⁢ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     df 
                   
                 
                 
                   ∫ 
                   
                     
                       
                         φ 
                         * 
                       
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     
                       φ 
                       ⁡ 
                       
                         ( 
                         f 
                         ) 
                       
                     
                     ⁢ 
                     df 
                   
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       
                         
                           
                             ψ 
                             * 
                           
                           ⁡ 
                           
                             ( 
                             
                               1 
                               
                                 2 
                                 ⁢ 
                                 π 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 j 
                               
                             
                             ) 
                           
                         
                         2 
                       
                       ⁢ 
                       
                         
                           d 
                           2 
                         
                         
                           dt 
                           2 
                         
                       
                       ⁢ 
                       ψ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       dt 
                     
                   
                   
                     ∫ 
                     
                       
                         ψ 
                         * 
                       
                       ⁢ 
                       ψ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       dt 
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         1 
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                           ⁢ 
                           
                               
                           
                         
                       
                       ) 
                     
                     2 
                   
                   ⁢ 
                   
                     
                       ∫ 
                       
                         
                           ψ 
                           * 
                         
                         ⁢ 
                         
                           
                             d 
                             2 
                           
                           
                             dt 
                             2 
                           
                         
                         ⁢ 
                         ψ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         dt 
                       
                     
                     
                       ∫ 
                       
                         
                           ψ 
                           * 
                         
                         ⁢ 
                         ψ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         dt 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   t 
                   2 
                 
                 _ 
               
               = 
               
                 
                   ∫ 
                   
                     
                       ψ 
                       * 
                     
                     ⁢ 
                     
                       t 
                       2 
                     
                     ⁢ 
                     ψ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     dt 
                   
                 
                 
                   ∫ 
                   
                     
                       ψ 
                       * 
                     
                     ⁢ 
                     ψ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     dt 
                   
                 
               
             
           
         
       
     
     We can now define an effective duration and effective bandwidth as: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               t 
             
             = 
             
               
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           t 
                           - 
                           
                             t 
                             _ 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       -- 
                       
                         -- 
                         -- 
                       
                     
                   
                 
               
               = 
               
                 2 
                 ⁢ 
                 
                   π 
                   · 
                   rms 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 in 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 time 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 
                   2 
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ( 
                         
                           f 
                           - 
                           
                             f 
                             _ 
                           
                         
                         ) 
                       
                       2 
                     
                     
                       -- 
                       
                         -- 
                         
                           -- 
                           - 
                         
                       
                     
                   
                 
               
               = 
               
                 2 
                 ⁢ 
                 
                   π 
                   · 
                   rms 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 in 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 frequency 
               
             
           
         
       
     
     But we know that: 
     
       
         
           
             
               
                 
                   ( 
                   
                     t 
                     - 
                     
                       t 
                       _ 
                     
                   
                   ) 
                 
                 2 
               
               
                 -- 
                 
                   -- 
                   -- 
                 
               
             
             = 
             
               
                 
                   t 
                   2 
                 
                 -- 
               
               - 
               
                 
                   ( 
                   
                     t 
                     _ 
                   
                   ) 
                 
                 2 
               
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   
                     f 
                     - 
                     
                       f 
                       _ 
                     
                   
                   ) 
                 
                 2 
               
               
                 -- 
                 
                   -- 
                   
                     -- 
                     -- 
                   
                 
               
             
             = 
             
               
                 
                   f 
                   2 
                 
                 -- 
               
               - 
               
                 
                   ( 
                   
                     f 
                     _ 
                   
                   ) 
                 
                 2 
               
             
           
         
       
     
     We can simplify if we make the following substitutions:
 
τ= t− t   
 
Ψ(τ)=ψ( t ) e   −j ω t  
 
ω 0 = ω =2 π f = 2π f   0  
 
     We also know that:
 
(Δ t ) 2 (Δ f ) 2 =(Δ tΔf ) 2  
 
     And therefore: 
     
       
         
           
             
               
                 ( 
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   f 
                 
                 ) 
               
               2 
             
             = 
             
               
                 
                   1 
                   4 
                 
                 ⁡ 
                 
                   [ 
                   
                     4 
                     ⁢ 
                     
                       
                         ∫ 
                         
                           
                             
                               Ψ 
                               * 
                             
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           
                             τ 
                             2 
                           
                           ⁢ 
                           
                             Ψ 
                             ⁡ 
                             
                               ( 
                               τ 
                               ) 
                             
                           
                           ⁢ 
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           τ 
                           ⁢ 
                           
                             ∫ 
                             
                               
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     Ψ 
                                     * 
                                   
                                 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   τ 
                                 
                               
                               ⁢ 
                               
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Ψ 
                                 
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   τ 
                                 
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               τ 
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             ∫ 
                             
                               
                                 
                                   Ψ 
                                   * 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ψ 
                                 ⁡ 
                                 
                                   ( 
                                   τ 
                                   ) 
                                 
                               
                               ⁢ 
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               τ 
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                   ] 
                 
               
               ≥ 
               
                 ( 
                 
                   1 
                   4 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               ( 
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 t 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 f 
               
               ) 
             
             ≥ 
             
               ( 
               
                 1 
                 2 
               
               ) 
             
           
         
       
     
     Now instead of (Δt Δf)≥(½) we are interested to force the equality (Δt Δf)=(½) and see what signals satisfy the equality. Given the fixed bandwidth Δf, the most efficient transmission is one that minimizes the time-bandwidth product (Δt Δf)=(½) For a given bandwidth Δf, the signal that minimizes the transmission in minimum time will be a Gaussian envelope. However, we are often given not the effective bandwidth, but always the total bandwidth f 2 −f 1 . Now, what is the signal shape which can be transmitted through this channel in the shortest effective time and what is the effective duration? 
                 Δ   ⁢           ⁢   t     ==         1       (     2   ⁢   π     )     2       ⁢       ∫     f   1       f   2       ⁢         d   ⁢           ⁢     φ   *       df     ⁢       d   ⁢           ⁢   φ     df               ∫     f   1       f   2       ⁢       φ   *     ⁢   φ   ⁢           ⁢   df           →   min         
Where φ(f) is zero outside the range f 2 −f 1 .
 
     To do the minimization, we would use the calculus of variations (Lagrange&#39;s Multiplier technique). Note that the denominator is constant and therefore we only need to minimize the numerator as: 
     
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   t 
                 
                 → 
                 
                   min 
                   → 
                   
                     δ 
                     ⁢ 
                     
                       
                         ∫ 
                         
                           f 
                           1 
                         
                         
                           f 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     φ 
                                     * 
                                   
                                 
                                 df 
                               
                               ⁢ 
                               
                                 
                                   d 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   φ 
                                 
                                 df 
                               
                             
                             + 
                             
                               
                                 Λφ 
                                 * 
                               
                               ⁢ 
                               φ 
                             
                           
                           ) 
                         
                         ⁢ 
                         df 
                       
                     
                   
                 
               
               = 
               0 
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               First 
               ⁢ 
               
                   
               
               ⁢ 
               Trem 
             
           
         
       
       
         
           
             
               δ 
               ⁢ 
               
                 
                   ∫ 
                   
                     f 
                     1 
                   
                   
                     f 
                     2 
                   
                 
                 ⁢ 
                 
                   
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         φ 
                         * 
                       
                     
                     df 
                   
                   ⁢ 
                   
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       φ 
                     
                     df 
                   
                   ⁢ 
                   df 
                 
               
             
             = 
             
               
                 ∫ 
                 
                   
                     ( 
                     
                       
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               φ 
                               * 
                             
                           
                           df 
                         
                         ⁢ 
                         δ 
                         ⁢ 
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             φ 
                           
                           df 
                         
                       
                       + 
                       
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             φ 
                           
                           df 
                         
                         ⁢ 
                         δ 
                         ⁢ 
                         
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               φ 
                               * 
                             
                           
                           df 
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   df 
                 
               
               = 
               
                 
                   ∫ 
                   
                     
                       ( 
                       
                         
                           
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 φ 
                                 * 
                               
                             
                             df 
                           
                           ⁢ 
                           
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               δφ 
                             
                             df 
                           
                         
                         + 
                         
                           
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               φ 
                             
                             df 
                           
                           ⁢ 
                           δ 
                           ⁢ 
                           
                             
                               d 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 φ 
                                 * 
                               
                             
                             df 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     df 
                   
                 
                 = 
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 d 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   φ 
                                   * 
                                 
                               
                               df 
                             
                             ⁢ 
                             δφ 
                           
                           + 
                           
                             
                               
                                 d 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 φ 
                               
                               df 
                             
                             ⁢ 
                             
                               δφ 
                               * 
                             
                           
                         
                         ] 
                       
                       
                         f 
                         1 
                       
                       
                         f 
                         2 
                       
                     
                     - 
                     
                       ∫ 
                       
                         
                           ( 
                           
                             
                               
                                 
                                   
                                     
                                       d 
                                       ⁢ 
                                       
                                           
                                       
                                     
                                     2 
                                   
                                   ⁢ 
                                   
                                     φ 
                                     * 
                                   
                                 
                                 
                                   df 
                                   2 
                                 
                               
                               ⁢ 
                               δφ 
                             
                             + 
                             
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   φ 
                                 
                                 
                                   df 
                                   2 
                                 
                               
                               ⁢ 
                               
                                 δφ 
                                 * 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         df 
                       
                     
                   
                   = 
                   
                     ∫ 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   
                                     d 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                   2 
                                 
                                 ⁢ 
                                 
                                   φ 
                                   * 
                                 
                               
                               
                                 df 
                                 2 
                               
                             
                             ⁢ 
                             δφ 
                           
                           + 
                           
                             
                               
                                 
                                   d 
                                   2 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 φ 
                               
                               
                                 df 
                                 2 
                               
                             
                             ⁢ 
                             
                               δφ 
                               * 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       df 
                     
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               Second 
               ⁢ 
               
                   
               
               ⁢ 
               Trem 
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 δ 
                 ⁢ 
                 
                   
                     ∫ 
                     
                       f 
                       1 
                     
                     
                       f 
                       2 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Λφ 
                           * 
                         
                         ⁢ 
                         φ 
                       
                       ) 
                     
                     ⁢ 
                     df 
                   
                 
               
               = 
               
                 Λ 
                 ⁢ 
                 
                   
                     ∫ 
                     
                       f 
                       1 
                     
                     
                       f 
                       2 
                     
                   
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           
                             φ 
                             * 
                           
                           ⁢ 
                           δφ 
                         
                         + 
                         
                           φδφ 
                           * 
                         
                       
                       ) 
                     
                     ⁢ 
                     df 
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               BothTrems 
               ⁢ 
               
                 
 
               
               ⁢ 
               
                   
               
               = 
               
                 
                   ∫ 
                   
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁢ 
                                   
                                     φ 
                                     * 
                                   
                                 
                                 
                                   df 
                                   2 
                                 
                               
                               + 
                               
                                 Λφ 
                                 * 
                               
                             
                             ) 
                           
                           ⁢ 
                           δφ 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     d 
                                     2 
                                   
                                   ⁢ 
                                   φ 
                                 
                                 
                                   df 
                                   2 
                                 
                               
                               + 
                               Λφ 
                             
                             ) 
                           
                           ⁢ 
                           
                             δφ 
                             * 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     df 
                   
                 
                 = 
                 0 
               
             
           
         
       
     
     This is only possible if and only if: 
     
       
         
           
             
               ( 
               
                 
                   
                     
                       d 
                       2 
                     
                     ⁢ 
                     φ 
                   
                   
                     df 
                     2 
                   
                 
                 + 
                 Λφ 
               
               ) 
             
             = 
             0 
           
         
       
     
     The solution to this is of the form 
     
       
         
           
             
               φ 
               ⁡ 
               
                 ( 
                 f 
                 ) 
               
             
             = 
             
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               k 
               ⁢ 
               
                   
               
               ⁢ 
               
                 π 
                 ⁡ 
                 
                   ( 
                   
                     
                       f 
                       - 
                       
                         f 
                         1 
                       
                     
                     
                       
                         f 
                         2 
                       
                       - 
                       
                         f 
                         1 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Now if we require that the wave vanishes at infinity, but still satisfy the minimum time-bandwidth product:
 
(Δ tΔf )=(½)
 
     Then we have the wave equation of a Harmonic Oscillator: 
     
       
         
           
             
               
                 
                   
                     d 
                     2 
                   
                   ⁢ 
                   
                     Ψ 
                     ⁡ 
                     
                       ( 
                       τ 
                       ) 
                     
                   
                 
                 
                   d 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     τ 
                     2 
                   
                 
               
               + 
               
                 
                   ( 
                   
                     λ 
                     - 
                     
                       
                         α 
                         2 
                       
                       ⁢ 
                       
                         τ 
                         2 
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   Ψ 
                   ⁡ 
                   
                     ( 
                     τ 
                     ) 
                   
                 
               
             
             = 
             0 
           
         
       
     
     which vanishes at infinity only if: 
             λ   =     α   ⁡     (       2   ⁢   n     +   1     )                     ψ   n     =         e       -     1   2       ⁢     ω   2     ⁢     τ   2         ⁢       d   n         d   ⁢           ⁢     τ   n       ⁢               ⁢     e       -     α   2       ⁢     τ   2           ∝       H   n     ⁡     (   τ   )               
Where H n (τ) is the Hermit functions and:
 
(Δ tΔf )=½(2 n+ 1)
 
     So Hermit functions H n (τ) occupy information blocks of 1/2, 3/2, 5/2, . . . with ½ as the minimum information quanta. 
     Squeezed States 
     Here we would derive the complete Eigen functions in the most generalized form using quantum mechanical approach of Dirac algebra. We start by defining the following operators: 
     
       
         
           
             b 
             = 
             
               
                 
                   
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ω 
                       ′ 
                     
                   
                   
                     2 
                     ⁢ 
                     ℏ 
                   
                 
               
               ⁢ 
               
                 ( 
                 
                   x 
                   + 
                   
                     ip 
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ω 
                         ′ 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               b 
               + 
             
             = 
             
               
                 
                   
                     
                       m 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ω 
                         ′ 
                       
                     
                     
                       2 
                       ⁢ 
                       ℏ 
                     
                   
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       x 
                       - 
                       
                         ip 
                         
                           m 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ω 
                             ′ 
                           
                         
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     
 
                   
                   [ 
                   
                     b 
                     , 
                     
                       b 
                       + 
                     
                   
                   ] 
                 
               
               = 
               
                 
                   1 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   a 
                 
                 = 
                 
                   
                     
                       λ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       b 
                     
                     - 
                     
                       μ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         b 
                         + 
                       
                       ⁢ 
                       
                         
 
                       
                       ⁢ 
                       
                         a 
                         + 
                       
                     
                   
                   = 
                   
                     
                       λ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         b 
                         + 
                       
                     
                     - 
                     
                       μ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       b 
                     
                   
                 
               
             
           
         
       
     
     Now we are ready to define Δx and Δp as: 
     
       
         
           
             
               
                 ( 
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   x 
                 
                 ) 
               
               2 
             
             = 
             
               
                 
                   ℏ 
                   
                     2 
                     ⁢ 
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     ω 
                     
                       ω 
                       ′ 
                     
                   
                   ) 
                 
               
               = 
               
                 
                   ℏ 
                   
                     2 
                     ⁢ 
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                   
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       λ 
                       - 
                       μ 
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   p 
                 
                 ) 
               
               2 
             
             = 
             
               
                 
                   
                     ℏ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                   
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       ω 
                       ′ 
                     
                     ω 
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ℏ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     ω 
                   
                   2 
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       λ 
                       + 
                       μ 
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   ( 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     x 
                   
                   ) 
                 
                 2 
               
               ⁢ 
               
                 
                   ( 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     p 
                   
                   ) 
                 
                 2 
               
             
             = 
             
               
                 ℏ 
                 4 
               
               ⁢ 
               
                 
                   ( 
                   
                     
                       λ 
                       2 
                     
                     - 
                     
                       μ 
                       2 
                     
                   
                   ) 
                 
                 2 
               
             
           
         
       
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               x 
               ⁢ 
               
                   
               
               ⁢ 
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               p 
             
             = 
             
               
                 
                   
                     ℏ 
                     2 
                   
                   4 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       λ 
                       2 
                     
                     - 
                     
                       μ 
                       2 
                     
                   
                   ) 
                 
               
               = 
               
                 ℏ 
                 2 
               
             
           
         
       
     
     Now let parameterize differently and instead of two variables λ, and μ, we would use only one variable ξ as follows:
 
λ=sin  hξ 
 
μ=cos  hξ 
 
λ+μ= e   ξ 
 
λ−μ=− e   −ξ 
 
     Now the Eigen states of the squeezed case are: 
     
       
         
           
             
               b 
               ⁢ 
               
                  
                 β 
                 〉 
               
             
             = 
             
               β 
               ⁢ 
               
                  
                 β 
                 〉 
               
             
           
         
       
       
         
           
             
               
                 ( 
                 
                   
                     λ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     a 
                   
                   + 
                   
                     μ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       a 
                       + 
                     
                   
                 
                 ) 
               
               ⁢ 
               
                  
                 β 
                 〉 
               
             
             = 
             
               β 
               ⁢ 
               
                  
                 β 
                 〉 
               
             
           
         
       
       
         
           
             b 
             = 
             
               UaU 
               + 
             
           
         
       
       
         
           
             U 
             = 
             
               e 
               
                 
                   ξ 
                   / 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       a 
                       2 
                     
                     - 
                     
                       a 
                       
                         + 
                         2 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   U 
                   + 
                 
                 ⁡ 
                 
                   ( 
                   ξ 
                   ) 
                 
               
               ⁢ 
               
                 aU 
                 ⁡ 
                 
                   ( 
                   ξ 
                   ) 
                 
               
             
             = 
             
               
                 a 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 cosh 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ξ 
               
               - 
               
                 
                   a 
                   + 
                 
                 ⁢ 
                 sinh 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ξ 
               
             
           
         
       
       
         
           
             
               
                 
                   U 
                   + 
                 
                 ⁡ 
                 
                   ( 
                   ξ 
                   ) 
                 
               
               ⁢ 
               
                 a 
                 + 
               
               ⁢ 
               
                 U 
                 ⁡ 
                 
                   ( 
                   ξ 
                   ) 
                 
               
             
             = 
             
               
                 
                   a 
                   + 
                 
                 ⁢ 
                 cosh 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ξ 
               
               - 
               
                 a 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 sinh 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 ξ 
               
             
           
         
       
     
     We can now consider the squeezed operator: 
     
       
         
           
             
                
               
                 α 
                 , 
                 ξ 
               
               〉 
             
             = 
             
               
                 U 
                 ⁡ 
                 
                   ( 
                   ξ 
                   ) 
                 
               
               ⁢ 
               
                 D 
                 ⁡ 
                 
                   ( 
                   α 
                   ) 
                 
               
               ⁢ 
               
                  
                 0 
                 〉 
               
             
           
         
       
       
         
           
             
               D 
               ⁡ 
               
                 ( 
                 α 
                 ) 
               
             
             = 
             
               
                 e 
                 
                   
                     - 
                     
                       
                          
                         α 
                          
                       
                       2 
                     
                   
                   2 
                 
               
               ⁢ 
               
                 e 
                 
                   α 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     a 
                     + 
                   
                 
               
               ⁢ 
               
                 e 
                 
                   
                     - 
                     
                       α 
                       * 
                     
                   
                   ⁢ 
                   a 
                 
               
             
           
         
       
       
         
           
             
                
               α 
               〉 
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 ∞ 
               
               ⁢ 
               
                 
                   
                     α 
                     n 
                   
                   
                     
                       n 
                       ! 
                     
                   
                 
                 ⁢ 
                 e 
                 ⁢ 
                 
                    
                   n 
                   〉 
                 
               
             
           
         
       
       
         
           
             
                
               α 
               〉 
             
             = 
             
               
                 e 
                 
                   
                     
                       - 
                       
                         
                            
                           α 
                            
                         
                         2 
                       
                     
                     2 
                   
                   + 
                   
                     α 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       a 
                       + 
                     
                   
                 
               
               ⁢ 
               
                  
                 0 
                 〉 
               
             
           
         
       
     
     For a distribution P(n) we would have: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             P 
                             ⁡ 
                             
                               ( 
                               n 
                               ) 
                             
                           
                           = 
                           
                              
                             
                               
                                 
                                   〈 
                                   n 
                                    
                                 
                                 ⁢ 
                                 β 
                               
                               , 
                               ξ 
                             
                             〉 
                           
                         
                          
                       
                       2 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       
                         
                           〈 
                           α 
                            
                         
                         ⁢ 
                         β 
                       
                       , 
                       ξ 
                     
                   
                   〉 
                 
                 = 
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     ∞ 
                   
                   ⁢ 
                   
                     
                       
                         α 
                         n 
                       
                       
                         
                           n 
                           ! 
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           
                             
                                
                               α 
                                
                             
                             2 
                           
                         
                         2 
                       
                     
                     ⁢ 
                     
                       〈 
                       n 
                        
                     
                     ⁢ 
                     β 
                   
                 
               
               , 
               ξ 
             
             〉 
           
         
       
       
         
           
             
               e 
               
                 
                   2 
                   ⁢ 
                   zt 
                 
                 - 
                 
                   t 
                   2 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 ∞ 
               
               ⁢ 
               
                 
                   
                     
                       H 
                       n 
                     
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                   ⁢ 
                   
                     t 
                     n 
                   
                 
                 
                   n 
                   ! 
                 
               
             
           
         
       
     
     Therefore the final result is: 
     
       
         
           
             〈 
             
               
                 n 
                 ⁢ 
                 
                    
                   
                     β 
                     , 
                     ξ 
                   
                   〉 
                 
               
               = 
               
                 
                   
                     
                       ( 
                       
                         tanh 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ξ 
                       
                       ) 
                     
                     
                       n 
                       / 
                       2 
                     
                   
                   
                     
                       2 
                       
                         n 
                         / 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             n 
                             ! 
                           
                           ⁢ 
                           cosh 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ξ 
                         
                         ) 
                       
                       2 
                     
                   
                 
                 ⁢ 
                 
                   e 
                   
                     
                       
                         - 
                         1 
                       
                       / 
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             β 
                              
                           
                           2 
                         
                         - 
                         
                           
                             β 
                             2 
                           
                           ⁢ 
                           tanh 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ξ 
                         
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   
                     H 
                     n 
                   
                   ⁡ 
                   
                     ( 
                     
                       β 
                       
                         2 
                         ⁢ 
                         sinh 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         ξcoshξ 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Another issue of concern with the use of QLO with QAM is a desire to improve bit error rate (BER) performance without impacting the information rate or bandwidth requirements of the queue a low signal. One manner for improving BER performance utilizes two separate oscillators that are separated by a known frequency Δf. Signals generated in this fashion will enable a determination of the BER. Referring now to  FIG. 64 , there is illustrated the generation of two bit streams b 1  and B 2  that are provided to a pair of QAM modulators  6402  and  6404  by a transmitter  6400 . Modulator  6402  receives a first carrier frequency F 1  and modulator  6404  receives a second carrier frequency F 2 . The frequencies F 1  and at two are separated by a known value Δf. The signals for each modulator are generated and combined at a summing circuit  6406  to provide the output s(t). The variables in the outputs of the QAM modulators are A i  (amplitude), f i  (frequency) and ϕ i  (phase). 
     Therefore, each constituent QAM modulation occupies a bandwidth: 
             BW   =       r   s     =         r   b         log   2     ⁢   m       ⁢           ⁢   symbols   ⁢     /     ⁢   sec             
where r s  equals the symbol rate of each constituent QAM signal.
 
     The total bandwidth of signal s(t) is: 
     
       
         
           
             W 
             = 
             
               
                 
                   r 
                   s 
                 
                 ⁡ 
                 
                   ( 
                   
                     1 
                     + 
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         f 
                       
                       
                         r 
                         s 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   r 
                   s 
                 
                 + 
                 
                   Δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   f 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     H 
                     z 
                   
                 
               
             
           
         
       
     
     Therefore, the spectral efficiency η of this two oscillator system is: 
             η   =       2   ⁢     r   b       W           
but
 
r b =r 2  log 2  m
 
     
       
         
           
             η 
             = 
             
               
                 
                   2 
                   ⁢ 
                   
                     r 
                     b 
                   
                 
                 W 
               
               = 
               
                 
                   
                     2 
                     ⁢ 
                     
                       r 
                       s 
                     
                     ⁢ 
                     
                       log 
                       2 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     m 
                   
                   
                     
                       r 
                       s 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         + 
                         
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             f 
                           
                           
                             r 
                             s 
                           
                         
                       
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       2 
                       ⁢ 
                       
                         log 
                         2 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       m 
                     
                     
                       1 
                       + 
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                         
                         
                           r 
                           s 
                         
                       
                     
                   
                   ⁢ 
                   
                     
                       bits 
                       ⁢ 
                       
                         / 
                       
                       ⁢ 
                       sec 
                     
                     Hz 
                   
                 
               
             
           
         
       
     
     The narrowband noise over the signal s(t) is:
 
 n ( t )= n   1 ( t )cos(2π f   0   t )− n   q ( t )sin(2π f   0   t )
 
Where: n I (t)=noise in I
 
     N q (t)=noise in Q 
     Each noise occupies a bandwidth of W [Hz] and the average power of each component is N 0 W. N 0  is the noise power spectral density in Watts/Hz. The value of f 0  is the mean value of f 1  and f 2 . 
     Referring now to  FIG. 65 , there is illustrated a receiver side block diagram for demodulating the signal generated with respect to  FIG. 65 . The received signal s(t)+n(t) is provided to a number of cosine filters  6502 - 6508 . Cosine filters  6502  and  6504  filter with respect to carrier frequency f 1  and cosine filters  6506  and  6508  filter the received signal for carrier frequency f 2 . Each of the filters  6502 - 6508  provide an output to a switch  6510  that provides a number of output to a transformation block  6512 . Transformation block  6512  provides two output signals having a real portion and an imaginary portion. Each of the real and imaginary portions associated with a signal are provided to an associated decoding circuit  6514 ,  6516  to provide the decoded signals b 1  and b 2 . 
                       [           a   ⁡     (     T   s     )                 b   ⁡     (     T   s     )                 c   ⁡     (     T   s     )                 d   ⁡     (     T   s     )             ]     =           T   s     ⁡     [         1       0         K   1           K   2             0       1         -     K   2             K   1               K   1           -     K   2           1       0             K   2           K   1         0       1         ]       ⁡     [             A   1     ⁡     (     cos   ⁢           ⁢     φ   1       )                   A   1     ⁡     (     sin   ⁢           ⁢     φ   1       )                   A   2     ⁡     (     cos   ⁢           ⁢     φ   2       )                   A   2     ⁡     (     sin   ⁢           ⁢     φ   2       )             ]       +     [             N     I   ⁢           ⁢   1       ⁡     (     T   s     )                   N     Q   ⁢           ⁢   1       ⁡     (     T   s     )                   N     I   ⁢           ⁢   2       ⁡     (     T   s     )                   N     Q   ⁢           ⁢   2       ⁡     (     T   s     )             ]         ⁢     
     ⁢           |     A   &gt;           𝕄         |     S   &gt;             |     N   &gt;             ⁢     
     ⁢     (     nonsingular   ⁢           ⁢   so   ⁢           ⁢   it   ⁢           ⁢   has   ⁢           ⁢     𝕄     -   1         )     ⁢     
     ⁢              A   &gt;=       T   s     ⁢   𝕄            ⁢   S     &gt;   +            ⁢   N     &gt;     
     ⁢   Where                 N     I     2   +       1   -         =           ∫   0     T   s       ⁢         η   s     ⁡     (   t   )       ⁢     cos   ⁡     (         2   ⁢   ηΔ   ⁢           ⁢   f     2     ⁢   t     )           ∓         η   G     ⁡     (   t   )       ⁢     sin   ⁡     (         2   ⁢   ηΔ   ⁢           ⁢   f     2     ⁢   t     )       ⁢   dt   ⁢     
     ⁢     N     Q     2   +       1   -             =             ∫   0     T   s       ⁢         η   I     ⁡     (   t   )       ⁢     din   ⁡     (         2   ⁢   ηΔ   ⁢           ⁢   f     2     ⁢   t     )           ∓         η   Q     ⁡     (   t   )       ⁢     cos   ⁡     (         2   ⁢   ηΔ   ⁢           ⁢   f     2     ⁢   t     )       ⁢   dt   ⁢     
     ⁢       |     ⁢   A         &gt;=       T   s     ⁢   𝕄   ⁢     |     ⁢   S     &gt;       +     |       ⁢   N     &gt;     
     ⁢     Multiply   ⁢           ⁢   by   ⁢             ⁢             ⁢     1     T   s       ⁢     𝕄     -   1       ⁢     
     ⁢       1     T   s       ⁢     𝕄     -   1       ⁢     |     ⁢   A       &gt;=       |     ⁢   S     &gt;       +     1     T   s         ⁢     𝕄     -   1       ⁢     |     ⁢   N     &gt;=       |     ⁢   S     &gt;       +     |       ⁢     N   ~     ⁢     
     ⁢           Output         |     𝕆   &gt;             |       N   ~     &gt;             ⁢     
     [             I   1     ⁡     (     T   s     )                   Q   1     ⁡     (     T   s     )                   I   2     ⁡     (     T   s     )                   Q   2     ⁡     (     T   s     )             ]         =       [             A   1     ⁡     (     cos   ⁢           ⁢     φ   1       )                   A   1     ⁡     (     sin   ⁢           ⁢     φ   1       )                   A   2     ⁡     (     cos   ⁢           ⁢     φ   2       )                   A   2     ⁡     (     sin   ⁢           ⁢     φ   2       )             ]     +       [               N   ~       I   ⁢           ⁢   1       ⁡     (     T   s     )                     N   ~       Q   ⁢           ⁢   1       ⁡     (     T   s     )                     N   ~       I   ⁢           ⁢   2       ⁡     (     T   s     )                     N   ~       Q   ⁢           ⁢   2       ⁡     (     T   s     )             ]     ⁢     
     ⁢                 |     𝕆   &gt;             |     S   &gt;                   |       N   ~     &gt;                           
Then the probability of correct decision P e  is
 
 P   e ≈(1− P   e ) 4 ≈1−4 P   e  for  P   e &lt;&lt;1
 
P e =well known error probability in one dimension for each consistuent m-QAM modulation.
 
Therefore, one can calculate BER.
 
     P e  comprises the known error probability in one dimension for each constituent member of the QAM modulation. Using the known probability error the bit error rate for the channel based upon the known difference between frequencies f 1  and f 2  may be calculated. 
     Adaptive Processing 
     The processing of signals using QLO may also be adaptively selected to combat channel impairments and interference. The process for adaptive QLO is generally illustrated in  FIG. 66 . First at step  6602  an analysis of the channel environment is made to determine the present operating environment. The level of QLO processing is selected at step  6604  based on the analysis and used to configure communications. Next, at step  6606 , the signals are transmitted at the selected level of QLO processing. Inquiry step  6608  determines if sufficient channel quality has been achieved. If so, the system continues to transmit and the selected QLO processing level at step  6606 . If not, control passes back to step  6602  to adjust the level of QLO processing to achieve better channel performance. In a further embodiment, the information provided by a mode crosstalk matrix as described herein below may also be used in conjunction with QLO processing of Hermite Gaussian modes, Laguerre Gaussian modes and Ince Gaussian modes to adaptively select the best modes for transmission to combat channel impairments and interference. 
     The processing of signals using mode division multiplexing (MDM) may also be adaptively selected to combat channel impairments and interference and maximize spectral efficiency. The process for adaptive MDM is generally illustrated in  FIG. 67 . First at step  6702  an analysis of the channel environment is made to determine the present operating environment. The level of MDM processing is selected at step  6704  based on the analysis and used to conFig. communications. Next, at step  6706 , the signals are transmitted at the selected level of MDM processing. Inquiry step  6708  determines if sufficient channel quality has been achieved. If so, the system continues to transmit and the selected MDM processing level at step  6706 . If not, control passes back to step  6702  to adjust the level of MDM processing to achieve better channel performance. In a further embodiment, the information provided by a mode crosstalk matrix as described herein below may also be used in conjunction with adaptive MDM processing of Hermite Gaussian modes, Laguerre Gaussian modes and Ince Gaussian modes to adaptively select the best modes for transmission to combat channel impairments and interference and maximize spectral efficiency. 
     The processing of signals using an optimal combination of QLO and MDM may also be adaptively selected to combat channel impairments and interference and maximize spectral efficiency. The process for adaptive QLO and MDM is generally illustrated in  FIG. 68 . First at step  6802  an analysis of the channel environment is made to determine the present operating environment. A selected combination of a level of QLO process and a level of MDM processing are selected at step  6804  based on the analysis and used to conFig. communications. Next, at step  6806 , the signals are transmitted at the selected level of QLO and MDM processing. Inquiry step  6808  determines if sufficient channel quality has been achieved. If so, the system continues to transmit and the selected combination of QLO and MDM processing levels at step  6806 . If not, control passes back to step  6802  to adjust the levels of QLO and MDM processing to achieve better channel performance. Adjustments through the steps continue until a most optimal combination of QLO and MDM processing is achieved to maximize spectral efficiency using a 2-dimensional optimization. In a further embodiment, the information provided by a mode crosstalk matrix as described herein below may also be used in conjunction with adaptive QLO and MDM processing of Hermite Gaussian modes, Laguerre Gaussian modes and Ince Gaussian modes to adaptively select the best modes for transmission to combat channel impairments and interference and maximize spectral efficiency using a 2-dimensional optimization. 
     The processing of signals using an optimal combination of QLO and QAM may also be adaptively selected to combat channel impairments and interference and maximize spectral efficiency. The process for adaptive QLO and QAM is generally illustrated in  FIG. 69 . First at step  6902  an analysis of the channel environment is made to determine the present operating environment. A selected combination of a level of QLO process and a level of QAM processing are selected at step  6904  based on the analysis and used to conFig. communications. Next, at step  6906 , the signals are transmitted at the selected level of QLO and QAM processing. Inquiry step  6908  determines if sufficient channel quality has been achieved. If so, the system continues to transmit and the selected combination of QLO and QAM processing levels at step  6906 . If not, control passes back to step  6902  to adjust the levels of QLO and QAM processing to achieve better channel performance. Adjustments through the steps continue until a most optimal combination of QLO and QAM processing is achieved to maximize spectral efficiency using a 2-dimensional optimization. In a further embodiment, the information provided by a mode crosstalk matrix as described herein below may also be used in conjunction with adaptive QLO and QAM processing of Hermite Gaussian modes, Laguerre Gaussian modes and Ince Gaussian modes to adaptively select the best modes and processing combination for transmissions to combat channel impairments and interference and maximize spectral efficiency using a 2-dimensional optimization. 
     The processing of signals using an optimal combination of QLO, MDM and QAM may also be adaptively selected to combat channel impairments and interference and maximize spectral efficiency. The process for adaptive QLO, MDM and QAM is generally illustrated in  FIG. 70 . First at step  7002  an analysis of the channel environment is made to determine the present operating environment. A selected combination of a level of QLO processing, a level of MDM processing and a level of QAM processing are selected at step  7004  based on the analysis and used to configure communications. Next, at step  7006 , the signals are transmitted at the selected level of QLO, MDM and QAM processing. Inquiry step  7008  determines if sufficient channel quality has been achieved. If so, the system continues to transmit and the selected combination of QLO, MDM and QAM processing levels at step  7006 . If not, control passes back to step  7002  to adjust the levels of QLO, MDM and QAM processing to achieve better channel performance. Adjustments through the steps continue until a most optimal combination of QLO, MDM and QAM processing is achieved to maximize spectral efficiency using a 3-dimensional optimization. In a further embodiment, the information provided by a mode crosstalk matrix as described herein below may also be used in conjunction with adaptive QLO, QAM and MDM modes using Laguerre Gaussian, Hermite Gaussian and Ince Gaussian orthogonal modes in fiber. The most optimum combination is selected based on channel impairments and interference environment to maximize spectral efficiency using a 3-dimensional optimization. 
     The adaptive approaches described herein above may be used with any combination of QLO, MDM and QAM processing in order to achieve optimal channel efficiency. In another application distinct modal combinations may also be utilized. 
     Improvement of Pilot Signal Modulation 
     The above described QLO, MDM and QAM processing techniques may also be used to improve the manner in which a system deals with noise, fading and other channel impairments by the use of pilot signal modulation techniques. As illustrated in  FIG. 71 , a pilot signal  7102  is transmitted between a transmitter  7104  to a receiver  7106 . The pilot signal includes an impulse signal that is received, detected and processed at the receiver  7106 . Using the information received from the pilot impulse signal, the channel  7108  between the transmitter  7104  and receiver  7106  may be processed to remove noise, fading and other channel impairment issues from the channel  7108 . The pilot signal assisted modulation may also be carried out using orthogonal modes in an elliptical core fiber as discussed herein below. 
     This process is generally described with respect to the flowchart of  FIG. 72 . The pilot impulse signal is transmitted at  7202  over the transmission channel. The impulse response is detected at step  7204  and processed to determine the impulse response over the transmission channel. Effects of channel impairments such as noise and fading may be countered by multiplying signals transmitted over the transmission channel by the inverse of the impulse response at step  7206  in order to correct for the various channel impairments that may be up on the transmission channel. In this way the channel impairments are counteracted and improved signal quality and reception may be provided over the transmission channel. 
     Power Control 
     Adaptive power control may be provided on systems utilizing QLO, MDM and QAM processing to also improve channel transmission. Amplifier nonlinearities within the transmission circuitry and the receiver circuitry will cause impairments in the channel response as more particularly illustrated in  FIG. 73 . As can be seen the channel impairments and frequency response increase and decrease over frequency as illustrated generally at  7302 . By adaptively controlling the power of a transmitting unit or a receiving unit and inverse frequency response such as that generated at  7304  may be generated. Thus, when the normal frequency response  7302  and the inverse frequency response  7304  are combined, a consistent response  7306  is provided by use of the adaptive power control. 
     Backward and Forward Channel Estimation 
     QLO techniques may also be used with forward and backward channel estimation processes when communications between a transmitter  7402  and a receiver  7404  do not have the same channel response over both the forward and backward channels. As shown in  FIG. 74 , the forward channel  7406  and backward channel  7408  between a transmitter  7402  and receiver  7404  me each be processed to determine their channel impulse responses. Separate forward channel estimation response and backward channel estimation response may be used for processing QLO signals transmitted over the forward channel  7406  and backward channel  7408 . The differences in the channel response between the forward channel  7406  and the backward channel  7408  may arise from differences in the topography or number of buildings located within the area of the transmitter  7402  and the receiver  7404 . By treating each of the forward channel  7406  and a backward channel  7408  differently better overall communications may be achieved. 
     Using MIMO Techniques with QLO 
     MIMO techniques may be used to improve the performance of QLO-based transmission systems. MIMO (multiple input and multiple output) is a method for multiplying the capacity of a radio link using multiple transmit and receive antennas to exploit multipath propagation. MIMO uses multiple antennas to transmit a signal instead of only a single antenna. The multiple antennas may transmit the same signal using modulation with the signals from each antenna modulated by different orthogonal signals such as that described with respect to the QLO modulation in order to provide an improved MIMO based system. 
     Diversions within OAM beams may also be reduced using phased arrays. By using multiple transmitting elements in a geometrical configuration and controlling the current and phase for each transmitting element, the electrical size of the antenna increases as does the performance of the antenna. The antenna system created by two or more individual intended elements is called an antenna array. Each transmitting element does not have to be identical but for simplification reasons the elements are often alike. To determine the properties of the electric field from an array the array factor (AF) is utilized. 
     The total field from an array can be calculated by a superposition of the fields from each element. However, with many elements this procedure is very unpractical and time consuming. By using different kinds of symmetries and identical elements within an array, a much simpler expression for the total field may be determined. This is achieved by calculating the so-called array factor (AF) which depends on the displacement (and shape of the array), phase, current amplitude and number of elements. After calculating the array factor, the total field is obtained by the pattern multiplication rule which is such that the total field is the product of the array factor in the field from one single element.
 
 E   total   =E   single element   ×AF  
 
     This formula is valid for all arrays consisting of identical elements. The array factor does not depend on the type of elements used, so for calculating AF it is preferred to use point sources instead of the actual antennas. After calculating the AF, the equation above is used to obtain the total field. Arrays can be 1D (linear), 2D (planar) or 3D. In a linear array, the elements are placed along the line and in a planar they are situated in a plane. 
     Referring now to  FIG. 75 , there is illustrated in the manner in which Hermite Gaussian beams and Laguerre Gaussian beams will diverge when transmitted from a phased array of antennas. For the generation of Laguerre Gaussian beams a circular symmetry over the cross-section of the phased antenna array is used, and thus, a circular grid will be utilized. For the generation of Hermite Gaussian beams  7502 , a rectangular array  7504  of array elements  7506  is utilized. As can be seen with respect to  FIG. 75 , the Hermite Gaussian waves  7508  provide a more focused beam front then the Laguerre Gaussian waves  7510 . 
     Reduced beam divergence may also be accomplished using a pair of lenses. As illustrated in  FIG. 76A , a Gaussian wave  7602  passing through a spiral phase plate  7604  generates an output Laguerre Gaussian wave  7606 . The Laguerre Gaussian wave  7606  when passing from a transmitter aperture  7608  to a receiver aperture  7610  diverges such that the entire Laguerre Gaussian beam does not intersect the receiver aperture  7610 . This issue may be addressed as illustrated in  FIG. 76B . As before the Gaussian waves  7602  pass through the spiral phase plate  7604  generating Laguerre Gaussian waves  7606 . Prior to passing through the transmitter aperture  7608  the Laguerre Gaussian waves  7606  pass through a pair of lenses  7614 . The pair of lenses  7614  have an effective focal length  7616  that focuses the beam  7618  passing through the transmitter aperture  7608 . Due to the focusing lenses  7614 , the focused beam  7618  fully intersects the receiver aperture  7612 . By providing the lenses  7614  separated by an effective focal length  7616 , a more focused beam  7618  may be provided at the receiver aperture  7612  preventing the loss of data within the transmission of the Laguerre Gaussian wave  7606 . 
     Application of OAM to Optical Communication 
     Utilization of OAM for optical communications is based on the fact that coaxially propagating light beams with different OAM states can be efficiently separated. This is certainly true for orthogonal modes such as the LG beam. Interestingly, it is also true for general OAM beams with cylindrical symmetry by relying only on the azimuthal phase. Considering any two OAM beams with an azimuthal index of l  1  and l  2 , respectively:
 
 U   1 ( r,θ,z )= A   1 ( r,z )exp( il   1 θ)  (12)
 
where r and z refers to the radial position and propagation distance respectively, one can quickly conclude that these two beams are orthogonal in the sense that:
 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       0 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         U 
                         1 
                       
                       ⁢ 
                       
                         U 
                         2 
                         * 
                       
                       ⁢ 
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       θ 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ℓ 
                                 1 
                               
                             
                             ≠ 
                             
                               ℓ 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             
                               A 
                               1 
                             
                             ⁢ 
                             
                               A 
                               2 
                               * 
                             
                           
                         
                         
                           
                             
                               if 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ℓ 
                                 1 
                               
                             
                             = 
                             
                               ℓ 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     There are two different ways to take advantage of the distinction between OAM beams with different l states in communications. In the first approach, N different OAM states can be encoded as N different data symbols representing “0”, “1”, . . . , “N−1”, respectively. A sequence of OAM states sent by the transmitter therefore represents data information. At the receiver, the data can be decoded by checking the received OAM state. This approach seems to be more favorable to the quantum communications community, since OAM could provide for the encoding of multiple bits (log 2(N)) per photon due to the infinitely countable possibilities of the OAM states, and so could potentially achieve a higher photon efficiency. The encoding/decoding of OAM states could also have some potential applications for on-chip interconnection to increase computing speed or data capacity. 
     The second approach is to use each OAM beam as a different data carrier in an SDM (Spatial Division Multiplexing) system. For an SDM system, one could use either a multi-core fiber/free space laser beam array so that the data channels in each core/laser beam are spatially separated, or use a group of orthogonal mode sets to carry different data channels in a multi-mode fiber (MMF) or in free space. Greater than 1 petabit/s data transmission in a multi-core fiber and up to 6 linearly polarized (LP) modes each with two polarizations in a single core multi-mode fiber has been reported. Similar to the SDM using orthogonal modes, OAM beams with different states can be spatially multiplexed and demultiplexed, thereby providing independent data carriers in addition to wavelength and polarization. Ideally, the orthogonality of OAM beams can be maintained in transmission, which allows all the data channels to be separated and recovered at the receiver. A typical embodiments of OAM multiplexing is conceptually depicted in  FIG. 27 . An obvious benefit of OAM multiplexing is the improvement in system spectral efficiency, since the same bandwidth can be reused for additional data channels. 
     Optical Fiber Communications 
     The use of orbital angular momentum and multiple layer overlay modulation processing techniques within an optical communications interface environment as described with respect to  FIG. 3  can provide a number of opportunities within the optical communications environment for enabling the use of the greater signal bandwidths provided by the use of optical orbital angular momentum processing, or multiple layer overlay modulation techniques alone.  FIG. 77  illustrates the general configuration of an optical fiber communication system. The optical fiber communication system  7700  includes an optical transmitter  7702  and an optical receiver  7704 . The transmitter  7702  and receiver  7704  communicate over an optical fiber  7706 . The transmitter  7702  includes information within a light wavelength or wavelengths that is propagated over the optical fiber  7706  to the optical receiver  7704 . 
     Optical communications network traffic has been steadily increasing by a factor of 100 every decade. The capacity of single mode optical fibers has increased 10,000 times within the last three decades. Historically, the growth in the bandwidth of optical fiber communications has been sustained by information multiplexing techniques using wavelength, amplitude, phase, and polarization of light as a means for encoding information. Several major discoveries within the fiber-optics domain have enabled today&#39;s optical networks. An additional discovery was led by Charles M. Kao&#39;s groundbreaking work that recognized glass impurities within an optical fiber as a major signal loss mechanism. Existing glass losses at the time of his discovery were approximately 200 dB per kilometer at 1 micrometer. 
     These discoveries gave birth to optical fibers and led to the first commercial optical fibers in the 1970s, having an attenuation low enough for communication purposes in the range of approximately 20 dBs per kilometer. Referring now to  FIGS. 78A-78C , there is more particularly illustrated the single mode fiber  7802 , multicore fibers  7808 , and multimode fibers  7810  described herein above. The multicore fibers  7808  consist of multiple cores  7812  included within the cladding  7813  of the fiber. As can be seen in  FIG. 78B , there are illustrated a 3 core fiber, 7 core fiber, and 19 core fiber. Multimode fibers  7810  comprise multimode fibers comprising a few mode fiber  7820  and a multimode fiber  7822 . Finally, there is illustrated a hollow core fiber  7815  including a hollow core  7814  within the center of the cladding  7816  and sheathing  7818 . The development of single mode fibers (SMF) such as that illustrated at  7802  ( FIG. 78A ) in the early 1980s reduced pulse dispersion and led to the first fiber-optic based trans-Atlantic telephone cable. This single mode fiber included a single transmission core  7804  within an outer sheathing  7806 . Development of indium gallium arsenide photodiodes in the early 1990s shifted the focus to near-infrared wavelengths (1550 NM), were silica had the lowest loss, enabling extended reach of the optical fibers. At roughly the same time, the invention of erbium-doped fiber amplifiers resulted in one of the biggest leaps in fiber capacity within the history of communication, a thousand fold increase in capacity occurred over a 10 year period. The development was mainly due to the removed need for expensive repeaters for signal regeneration, as well as efficient amplification of many wavelengths at the same time, enabling wave division multiplexing (WDM). 
     Throughout the 2000s, increases in bandwidth capacity came mainly from introduction of complex signal modulation formats and coherent detection, allowing information encoding using the phase of light. More recently, polarization division multiplexing (PDM) doubled channel capacity. Through fiber communication based on SMFs featured tremendous growth in the last three decades, recent research has indicated SMF limitations. Non-linear effects in silica play a significant role in long range transmission, mainly through the Kerr effect, where a presence of a channel at one wavelength can change the refractive index of a fiber, causing distortions of other wavelength channels. More recently, a spectral efficiency (SE) or bandwidth efficiency, referring to the transmitted information rate over a given bandwidth, has become theoretically analyzed assuming nonlinear effects in a noisy fiber channel. This research indicates a specific spectral efficiency limit that a fiber of a certain length can reach for any signal to noise (SNR). Recently achieved spectral efficiency results indeed show that the proximity to the spectral efficiency limit, indicating the need for new technologies to address the capacity issue in the future. 
     Among several possible directions for optical communications in the future, the introduction of new optical fibers  7706  other than single mode fibers  7802  has shown promising results. In particular, researchers have focused on spatial dimensions in new fibers, leading to so-called space division multiplexing (SDM) where information is transmitted using cores of multi-core fibers (MCF)  7808  ( FIG. 78B ) or mode division multiplexing (MDM) or information is transmitted using modes of multimode fibers (MMFs)  7810  ( FIG. 78C ). The latest results show spectral efficiency of 91 bits/S/Hz using 12 core multicore fiber  7808  for 52 kilometer long fibers and 12 bits/S/Hz using 6 mode multimode fiber  7810  and 112 kilometer long fibers. Somewhat unconventional transmissions at 2.08 micrometers have also been demonstrated in two 90 meter long photonic crystal fibers, though these fibers had high losses of 4.5 decibels per kilometer. 
     While offering promising results, these new types of fibers have their own limitations. Being noncircularly symmetric structures, multicore fibers are known to require more complex, expensive manufacturing. On the other hand, multimode fibers  7810  are easily created using existing technologies. However, conventional multimode fibers  7810  are known to suffer from mode coupling caused by both random perturbations in the fibers and in modal multiplexers/demultiplexers. 
     Several techniques have been used for mitigating mode coupling. In a strong coupling regime, modal cross talk can be compensated using computationally intensive multi-input multi-output (MIMO) digital signal processing (DSP). While MIMO DSP leverages the technique&#39;s current success in wireless networks, the wireless network data rates are several orders of magnitude lower than the ones required for optical networks. Furthermore, MIMO DSP complexity inevitably increases with an increasing number of modes and no MIMO based data transmission demonstrations have been demonstrated in real time thus far. Furthermore, unlike wireless communication systems, optical systems are further complicated because of fiber&#39;s nonlinear effects. In a weak coupling regime, where cross talk is smaller, methods that also use computationally intensive adapted optics, feedback algorithms have been demonstrated. These methods reverse the effects of mode coupling by sending a desired superposition of modes at the input, so that desired output modes can be obtained. This approach is limited, however, since mode coupling is a random process that can change on the order of a millisecond in conventional fibers. 
     Thus, the adaptation of multimode fibers  7810  can be problematic in long haul systems where the round trip signal propagation delay can be tens of milliseconds. Though 2×56 GB/S transmission at 8 kilometers length has been demonstrated in the case of two higher order modes, none of the adaptive optics MDM methods to date have demonstrated for more than two modes. Optical fibers act as wave guides for the information carrying light signals that are transmitted over the fiber. Within an ideal case, optical fibers are 2D, cylindrical wave guides comprising one or several cores surrounded by a cladding having a slightly lower refractive index as illustrated in  FIGS. 78A-78D . A fiber mode is a solution (an eigenstate) of a wave guide equation describing the field distribution that propagates within a fiber without changing except for the scaling factor. All fibers have a limit on the number of modes that they can propagate, and have both spatial and polarization degrees of freedom. 
     Single mode fibers (SMFs)  7802  is illustrated in  FIG. 78A  support propagation of two orthogonal polarizations of the fundamental mode only (N=2). For sufficiently large core radius and/or the core cladding difference, a fiber is multimoded for N&gt;2 as illustrated in  FIG. 78C . For optical signals having orbital angular momentums and multilayer modulation schemes applied thereto, multimode fibers  7810  that are weakly guided may be used. Weakly guided fibers have a core cladding refractive index difference that is very small. Most glass fibers manufactured today are weakly guided, with the exception of some photonic crystal fibers and air-core fibers. Fiber guide modes of multimode fibers  7810  may be associated in step indexed groups where, within each group, modes typically having similar effective indexes are grouped together. Within a group, the modes are degenerate. However, these degeneracies can be broken in a certain fiber profile design. 
     We start by describing translationally invariant waveguide with refractive index n=n(x, y), with n co  being maximum refractive index (“core” of a waveguide), and n ci  being refractive index of the uniform cladding, and ρ represents the maximum radius of the refractive index n. Due to translational invariance the solutions (or modes) for this waveguide can be written as:
 
 E   j ( x,y,z )= e   j ( x,y ) e   iβ     j     z ,
 
 H   j ( z,y,z )= h   j ( x,y ) e   iβ     j     z ,
 
where β j  is the propagation constant of the j-th mode. Vector wave equation for source free Maxwell&#39;s equation can be written in this case as:
 
(∇ 2   +n   2   k   2 −β j   2 ) e   j =−(∇ t   +iβ   j   {circumflex over (z)} )( e   tj ·∇ t  ln( n   2 ))
 
(∇ 2   +n   2   k   2 −β j   2 ) h   j =−(∇ t  ln( n   2 ))×( (∇   t   +iβ   j   {circumflex over (z)} )× h   j )
 
where k=2π/λ is the free-space wavenumber, λ is a free-space wavelength, e t =e x {circumflex over (x)}+e y ŷ is a transverse part of the electric field, ∇ 2  is a transverse Laplacian and ∇ t  transverse vector gradient operator. Waveguide polarization properties are built into the wave equation through the ∇ t  ln(n 2 ) terms and ignoring them would lead to the scalar wave equation, with linearly polarized modes. While previous equations satisfy arbitrary waveguide profile n(x, y), in most cases of interest, profile height parameter Δ can be considered small Δ«1, in which case waveguide is said to be weakly guided, or that weakly guided approximation (WGA) holds. If this is the case, a perturbation theory can be applied to approximate the solutions as:
 
 E ( x,y,z )= e ( x,y ) e   i(β+{tilde over (β)})z =( e   t   +{circumflex over (z)}e   z ) e   i(β+{tilde over (β)})z  
 
 H ( x,y,z )= h ( x,y ) e   i(β+{tilde over (β)})z =( h   t   +{circumflex over (z)}h   z ) e   i(β+{tilde over (β)})z  
 
where subscripts t and z denote transverse and longitudinal components respectively. Longitudinal components can be considered much smaller in WGA and we can approximate (but not neglect) them as:
 
               e   z     =           i   ⁡     (     2   ⁢   Δ     )         1   2       v     ⁢     (     ρ   ⁢       ∇   t     ⁢     ·     e   t           )                     h   z     =           i   ⁡     (     2   ⁢   Δ     )         1   2       V     ⁢     (     ρ   ⁢       ∇   t     ⁢     ·     h   t           )             
Where Δ and V are profile height and fiber parameters and transversal components satisfy the simplified wave equation.
 
(∇ 2   +n   2   k   2 −β j   2 ) e   j =0
 
Though WGA simplified the waveguide equation, further simplification can be obtained by assuming circularly symmetric waveguide (such as ideal fiber). If this is the case refractive index that can be written as:
 
 n ( r )= n   2   co (1−2 f ( R )Δ)
 
where f(R)≥0 is a small arbitrary profile variation.
 
     For a circularly symmetric waveguide, we would have propagation constants β lm  that are classified using azimuthal (l) and radial (m) numbers. Another classification uses effective indices n lm  (sometimes noted as n eff   lm , or simply n eff , that are related to propagation constant as: β lm =kn ef f ). For the case of l=0, the solutions can be separated into two classes that have either transverse electric (T E 0m ) or transverse magnetic (T M 0m ) fields (called meridional modes). In the case of l≠0, both electric and magnetic field have z-component, and depending on which one is more dominant, so-called hybrid modes are denoted as: HE lm  and EH lm . 
     Polarization correction δβ has different values within the same group of modes with the same orbital number (l), even in the circularly symmetric fiber. This is an important observation that led to development of a special type of fiber. 
     In case of a step refractive index, solutions are the Bessel functions of the first kind, J l (r), in the core region, and modified Bessel functions of the second kind, K l (r), in the cladding region. 
     In the case of step-index fiber the groups of modes are almost degenerate, also meaning that the polarization correction δβ can be considered very small. Unlike HE 11  modes, higher order modes (HOMs) can have elaborate polarizations. In the case of circularly symmetric fiber, the odd and even modes (for example He odd  and HE even  modes) are always degenerate (i.e. have equal n eff ), regardless of the index profile. These modes will be non-degenerate only in the case of circularly asymmetric index profiles. 
     Referring now to  FIG. 79 , there are illustrated the first six modes within a step indexed fiber for the groups L=0 and L=1. 
     When orbital angular momentums are applied to the light wavelength within an optical transmitter of an optical fiber communication system, the various orbital angular momentums applied to the light wavelength may transmit information and be determined within the fiber mode. 
     Angular momentum density (M) of light in a medium is defined as: 
     
       
         
           
             M 
             = 
             
               
                 
                   1 
                   
                     c 
                     2 
                   
                 
                 ⁢ 
                 r 
                 × 
                 
                   ( 
                   
                     E 
                     × 
                     H 
                   
                   ) 
                 
               
               = 
               
                 
                   r 
                   × 
                   P 
                 
                 = 
                 
                   
                     1 
                     
                       c 
                       2 
                     
                   
                   ⁢ 
                   r 
                   × 
                   S 
                 
               
             
           
         
       
     
     with r as position, E electric field, H magnetic field, P linear momentum density and S Poynting vector. 
     The total angular momentum (J), and angular momentum flux (Φ M ) can be defined as:
 
 J=∫∫∫M dV  
 
Φ=∫∫ M dA  
 
     In order to verify whether certain mode has an OAM let us look at the time averages of the angular momentum flux Φ M :
 
 Φ M     =∫∫     M     dA  
 
as well as the time average of the energy flux:
 
     
       
         
           
             
               〈 
               
                 Φ 
                 W 
               
               〉 
             
             = 
             
               ∫ 
               
                 ∫ 
                 
                   
                     
                       〈 
                       
                         S 
                         z 
                       
                       〉 
                     
                     c 
                   
                   ⁢ 
                   dA 
                 
               
             
           
         
       
     
     Because of the symmetry of radial and axial components about the fiber axis, we note that the integration in equation will leave only z-component of the angular momentum density non zero. Hence: 
     
       
         
           
             
               〈 
               M 
               〉 
             
             = 
             
               
                 
                   〈 
                   M 
                   〉 
                 
                 z 
               
               = 
               
                 
                   1 
                   
                     c 
                     2 
                   
                 
                 ⁢ 
                 r 
                 × 
                 
                   
                     〈 
                     
                       E 
                       × 
                       H 
                     
                     〉 
                   
                   z 
                 
               
             
           
         
       
     
     and knowing (S)=Re{S} and S=½E×H*leads to:
 
 S   Φ =½(− E   r   H   z   *   +E   z   H   r   * )
 
 S   z =½( E   x   H   y   *   −E   y   H   x   * )
 
     Let us now focus on a specific linear combination of the HE l+1,m   even  and HE l+1,m   odd  modes with π/2 phase shift among them:
 
 V   lm   +=HE   l+1,m   even   +iEH   l+1,m   odd  
 
     The idea for this linear combination comes from observing azimuthal dependence of the HE l+1,m   even  and modes comprising cos(φ) and sin (φ). If we denote the electric field of HE l+1,m   even  and HE l+1,m   odd  modes as e 1  and e 2 , respectively, and similarly, denote their magnetic fields as h 1  and h 2 , the expression for theis new mode can be written as:
 
 e=e   1   +ie   2 ,  (2.35)
 
 h=h   1   +ih   2 .  (2.36)
 
then we derive:
 
               e   r     =       e       i   ⁡     (     l   +   1     )       ⁢   φ       ⁢       F   l     ⁡     (   R   )                       h   z     =       e       i   ⁡     (     l   +   1     )       ⁢   φ       ⁢         n   co     ⁡     (       ϵ   0       μ   0       )         1   2       ⁢         (     2   ⁢   Δ     )       1   2       V     ⁢     G   l   -                     e   z     =       ie       i   ⁡     (     l   +   1     )       ⁢   φ       ⁢         (     2   ⁢   Δ     )       1   2       V     ⁢     G   l   -                     h   r     =       -     ie       i   ⁡     (     l   +   1     )       ⁢   φ         ⁢         n   co     ⁡     (       ϵ   0       μ   0       )         1   2       ⁢       F   l     ⁡     (   R   )               
Where F l (R) is the Bessel function and
 
     
       
         
           
             
               G 
               l 
               ± 
             
             = 
             
               
                 
                   dF 
                   l 
                 
                 dR 
               
               ± 
               
                 
                   l 
                   R 
                 
                 ⁢ 
                 
                   F 
                   l 
                 
               
             
           
         
       
     
     We note that all the quantities have e i(l+1)φ  dependence that indicates these modes might have OAM, similarly to the free space case. Therefore the azimuthal and the longitudinal component of the Poynting vector are: 
     
       
         
           
             
               S 
               φ 
             
             = 
             
               
                 - 
                 
                   
                     
                       n 
                       co 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           ϵ 
                           0 
                         
                         
                           μ 
                           0 
                         
                       
                       ) 
                     
                   
                   
                     1 
                     2 
                   
                 
               
               ⁢ 
               
                 
                   
                     ( 
                     
                       2 
                       ⁢ 
                       Δ 
                     
                     ) 
                   
                   
                     1 
                     2 
                   
                 
                 V 
               
               ⁢ 
               Re 
               ⁢ 
               
                 { 
                 
                   
                     F 
                     l 
                     * 
                   
                   ⁢ 
                   
                     G 
                     l 
                     - 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               S 
               z 
             
             = 
             
               
                 
                   
                     n 
                     co 
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         ϵ 
                         0 
                       
                       
                         μ 
                         0 
                       
                     
                     ) 
                   
                 
                 
                   1 
                   2 
                 
               
               ⁢ 
               
                 
                   ⌈ 
                   
                     F 
                     l 
                   
                   ⌉ 
                 
                 2 
               
             
           
         
       
     
     The ratio of the angular momentum flux to the energy flux therefore becomes: 
     
       
         
           
             
               
                 ϕ 
                 M 
               
               
                 ϕ 
                 W 
               
             
             = 
             
               
                 l 
                 + 
                 1 
               
               ω 
             
           
         
       
     
     We note that in the free-space case, this ratio is similar: 
     
       
         
           
             
               
                 ϕ 
                 M 
               
               
                 ϕ 
                 W 
               
             
             = 
             
               
                 σ 
                 + 
                 1 
               
               ω 
             
           
         
       
     
     where σ represents the polarization of the beam and is bounded to be −1&lt;σ&lt;1. In our case, it can be easily shown that SAM of the V +  state, is 1, leading to important conclusion that the OAM of the V +lm  state is l. Hence, this shows that, in an ideal fiber, OAM mode exists. 
     Thus, since an orbital angular momentum mode may be detected within the ideal fiber, it is possible to encode information using this OAM mode in order to transmit different types of information having different orbital angular momentums within the same optical wavelength. 
     The above description with respect to optical fiber assumed an ideal scenario of perfectly symmetrical fibers having no longitudinal changes within the fiber profile. Within real world fibers, random perturbations can induce coupling between spatial and/or polarization modes, causing propagating fields to evolve randomly through the fiber. The random perturbations can be divided into two classes, as illustrated in  FIG. 80 . Within the random perturbations  8002 , the first class comprises extrinsic perturbations  8004 . Extrinsic perturbations  8004  include static and dynamic fluctuations throughout the longitudinal direction of the fiber, such as the density and concentration fluctuations natural to random glassy polymer materials that are included within fibers. The second class includes extrinsic variations  8006  such as microscopic random bends caused by stress, diameter variations, and fiber core defects such as microvoids, cracks, or dust particles. 
     Mode coupling can be described by field coupling modes which account for complex valued modal electric field amplitudes, or by power coupling modes, which is a simplified description that accounts only for real value modal powers. Early multimode fiber systems used incoherent light emitting diode sources and power coupling models were widely used to describe several properties including steady state, modal power distributions, and fiber impulse responses. While recent multimode fiber systems use coherent sources, power coupling modes are still used to describe effects such as reduced differential group delays and plastic multimode fibers. 
     By contrast, single mode fiber systems have been using laser sources. The study of random birefringence and mode coupling in single mode fibers which leads to polarization mode dispersion (PMD), uses field coupling modes which predict the existence of principal states of polarization (PSPs). PSPs are polarization states shown to undergo minimal dispersion and are used for optical compensation of polarization mode dispersion in direct detection single mode fiber systems. In recent years, field coupling modes have been applied to multimode fibers, predicting principal mode which are the basis for optical compensation of modal dispersion in direct detection multimode fiber systems. 
     Mode coupling can be classified as weak or strong, depending on whether the total system length of the optical fiber is comparable to, or much longer than, a length scale over which propagating fields remain correlated. Depending on the detection format, communication systems can be divided into direct and coherent detection systems. In direct detection systems, mode coupling must either be avoided by careful design of fibers and modal D (multiplexers) and/or mitigated by adaptive optical signal processing. In systems using coherent detection, any linear cross talk between modes can be compensated by multiple input multiple output (MIMO) digital signal processing (DSP), as previously discussed, but DSP complexity increases with an increasing number of modes. 
     Referring now to  FIG. 81 , there were illustrated the intensity patterns of the first order mode group within a vortex fiber. Arrows  8102  within the illustration show the polarization of the electric field within the fiber. The top row illustrates vector modes that are the exact vector solutions, and the bottom row shows the resultant, unstable LP 11  modes commonly obtained at a fiber output. Specific linear combinations of pairs of top row modes resulting in the variety of LP 11  modes obtained at the fiber output. Coupled mode  8102  is provided by the coupled pair of mode  8104  and  8106 . Coupled mode  8104  is provided by the coupled pair of mode  8104  and mode  8108 . Coupled mode  8116  is provided by the coupled pair of mode  8106  and mode  8110 , and coupled mode  8118  is provided by the coupled pair of mode  8108  and mode  8110 . 
     Typically, index separation of two polarizations and single mode fibers is on the order of 10-7. While this small separation lowers the PMD of the fiber, external perturbations can easily couple one mode into another, and indeed in a single mode fiber, arbitrary polarizations are typically observed at the output. Simple fiber polarization controller that uses stress induced birefringence can be used to achieve any desired polarization at the output of the fiber. 
     By the origin, mode coupling can be classified as distributed (caused by random perturbations in fibers), or discrete (caused at the modal couplers and the multiplexers). Most importantly, it has been shown that small, effective index separation among higher order modes is the main reason for mode coupling and mode instabilities. In particular, the distributed mode coupling has been shown to be inversely proportional to Δ-P with P greater than 4, depending on coupling conditions. Modes within one group are degenerate. For this reason, in most multimode fiber modes that are observed in the fiber output are in fact the linear combinations of vector modes and are linearly polarized states. Hence, optical angular momentum modes that are the linear combination of the HE even, odd modes cannot coexist in these fibers due to coupling to degenerate TE 01  and TM 01  states. 
     Thus, the combination of the various OAM modes is not likely to generate modal coupling within the optical systems and by increasing the number of OAM modes, the reduction in mode coupling is further benefited. 
     Referring now to  FIGS. 82A and 82B , there is illustrated the benefit of effective index separation in first order modes.  FIG. 82A  illustrates a typical step index multimode fiber that does not exhibit effective index separation causing mode coupling. The mode TM 01  HE even   21 , mode HE odd   21 , and mode TE 01  have little effective index separation, and these modes would be coupled together. Mode HE x,1   11  has an effective index separation such that this mode is not coupled with these other modes. 
     This can be compared with the same modes in  FIG. 82B . In this case, there is an effective separation  8202  between the TM 01  mode and the HE even   21  mode and the TE 01  mode and the HE odd   21  mode. This effective separation causes no mode coupling between these mode levels in a similar manner that was done in the same modes in  FIG. 82A . 
     In addition to effective index separation, mode coupling also depends on the strength of perturbation. An increase in the cladding diameter of an optical fiber can reduce the bend induced perturbations in the fiber. Special fiber design that includes the trench region can achieve so-called bend insensitivity, which is predominant in fiber to the home. Fiber design that demonstrates reduced bends and sensitivity of higher order Bessel modes for high power lasers have been demonstrated. Most important, a special fiber design can remove the degeneracy of the first order mode, thus reducing the mode coupling and enabling the OAM modes to propagate within these fibers. 
     Topological charge may be multiplexed to the wave length for either linear or circular polarization. In the case of linear polarizations, topological charge would be multiplexed on vertical and horizontal polarization. In case of circular polarization, topological charge would be multiplexed on left hand and right hand circular polarization. 
     The topological charges can be created using Spiral Phase Plates (SPPs) such as that illustrated in  FIG. 11E , phase mask holograms or a Spatial Light Modulator (SLM) by adjusting the voltages on SLM which creates properly varying index of refraction resulting in twisting of the beam with a specific topological charge. Different topological charges can be created and muxed together and de-muxed to separate charges. When signals are muxed together, multiple signals having different orthogonal functions or helicities applied thereto are located in a same signal. The muxed signals are spatially combined in a same signal. 
     As Spiral Phase plates can transform a plane wave (l=0) to a twisted wave of a specific helicity (i.e. l=+1), Quarter Wave Plates (QWP) can transform a linear polarization (s=0) to circular polarization (i.e. s=+1). 
     Cross talk and multipath interference can be reduced using Multiple-Input-Multiple-Output (MIMO). 
     Most of the channel impairments can be detected using a control or pilot channel and be corrected using algorithmic techniques (closed loop control system). 
     Optical Fiber Communications Using OAM Multiplexing 
     OAM multiplexing may be implemented in fiber communications. OAM modes are essentially a group of higher order modes defined on a different basis as compared to other forms of modes in fiber, such as “linearly polarized” (LP) modes and fiber vector modes. In principle each of the mode sets form an orthogonal mode basis spanning the spatial domain, and may be used to transmit different data channels. Both LP modes and OAM modes face challenges of mode coupling when propagating in a fiber, and may also cause channel crosstalk problems. 
     In general, two approaches may be involved in fiber transmission using OAM multiplexing. The first approach is to implement OAM transmission in a regular few mode fiber such as that illustrated in  FIG. 78 . As is the case of SDM using LP modes, MIMO DSP is generally required to equalize the channel interface. The second approach is to utilize a specially designed vortex fiber that suffers from less mode coupling, and DSP equalization can therefore be saved for a certain distance of transmission. 
     OAM Transmission in Regular Few Mode Fiber 
     In a regular few mode fiber, each OAM mode represents approximately a linear combination of the true fiber modes (the solution to the wave equation in fiber). For example, as illustrated in  FIG. 83 , a linearly polarized OAM beam  8302  with l=+1 comprises the components of Eigen modes including TE 01 , TM 01  and HE 21 . Due to the perturbations or other non-idealities, OAM modes that are launched into a few mode fiber (FMF) may quickly coupled to each other, most likely manifesting in a group of LP modes at the fiber output. The mutual mode coupling in fiber may lead to inter-channel crosstalk and eventually failure of the transmission. One possible solution for the mode coupling effects is to use MIMO DSP in combination with coherent detection. 
     Referring now to  FIG. 84 , there is illustrated a demonstration of the transmission of four OAM beams (l=+1 and −1 each with 2 orthogonal polarization states), each carrying 20 Gbit/s QPSK data, in an approximately 5 kilometer regular FMF (few mode fiber)  8404 . Four data channels  8402  (2 with x-pol and 2 with y-pol) were converted to pol-muxed OAM beams with l=+1 and −1 using an inverse mode sorter  8406 . The pol-muxed to OAM beams  8408  (four in total) are coupled into the FMF  8404  for propagation. At the fiber output, the received modes were decomposed onto an OAM basis (l=+1 and −1) using a mode sorter  8410 . In each of the two OAM components of light were coupled onto a fiber-based PBS for polarization demultiplexing. Each output  8412  is detected by a photodiode, followed by ADC (analog-to-digital converter) and off-line processing. To mitigate the inter-channel interference, a constant modulus algorithm is used to blindly estimate the channel crosstalk and compensate for the inter-channel interference using linear equalization. Eventually, the QPSK data carried on each OAM beam is recovered with the assistance of a MIMO DSP as illustrated in  FIGS. 85A and 85B . 
     OAM Transmission in a Vortex Fiber 
     A key challenge for OAM multiplexing in conventional fibers is that different OAM modes tend to couple to each other during the transmission. The major reason for this is that in a conventional fiber OAM modes have a relatively small effective refractive index difference (Δ n eff ). Stably transmitting an OAM mode in fiber requires some modifications of the fiber. One manner for stably transmitting OAM modes uses a vortex fiber such as that illustrated in  FIG. 86 . A vortex fiber  8602  is a specially designed a few mode fiber including an additional high index ring  8604  around the fiber core  8606 . The design increases the effective index differences of modes and therefore reduces the mutual mode coupling. 
     Using this vortex fiber  8602 , two OAM modes with l=+1 and −1 and two polarizations multiplexed fundamental modes were transmitted together for 1.1 km. The measured mode cross talk between two OAM modes was approximately −20 dB. These four distinct modes were used to each carried a 100 Gbuad QPSK signal at the same wavelength and simultaneously propagate in the vortex fiber. After the mode demultiplexing, all data was recovered with a power penalty of approximately 4.1 dB, which could be attributed to the multipath effects and mode cross talk. In a further example, WDM was added to further extend the capacity of a vortex fiber transmission system. A 20 channel fiber link using to OAM modes and 10 WDM channels (from 1546.642 nm to 1553.88 nm), each channel sending 80 Gb/s 16-QAM signal was demonstrated, resulting in a total transmission capacity of 1.2 Tb/s under the FEC limit. 
     There are additional innovative efforts being made to design and fabricate fibers that are more suitable for OAM multiplexing. A recently reported air-core fiber has been demonstrated to further increase the refractive index difference of eigenmodes such that the fiber is able to stably transmit 12 OAM states (l=±7, ±8 and ±9, each with two orthogonal polarizations) for 2 m. A few mode fibers having an inverse parabolic graded index profile in which propagating 8 OAM orders (l=±1 and ±2, each with two orthogonal polarizations) has been demonstrated over 1.1 km. The same group recently presented a newer version of an air core fiber, whereby the supported OAM states was increased to 16. One possible design that can further increase the supported OAM modes and a fiber is to use multiple high contrast indexed ring core structure which is indicated a good potential for OAM multiplexing for fiber communications. 
     RF Communications with OAM 
     As a general property of electromagnetic waves, OAM can also be carried on other ways with either a shorter wavelength (e.g., x-ray), or a longer wavelength (millimeter waves and terahertz waves) than an optical beam. Focusing on the RF waves, OAM beams at 90 GHz were initially generated using a spiral phase plate made of Teflon. Different approaches, such as a phase array antenna and a helicoidal parabolic antenna have also been proposed. RF OAM beams have been used as data carriers for RF communications. A Gaussian beam and an OAM beam with l=+1 at approximately 2.4 GHz have been transmitted by a Yagi-Uda antenna and a spiral parabolic antenna, respectively, which are placed in parallel. These two beams were distinguished by the differential output of a pair of antennas at the receiver side. The number of channels was increased to three (carried on OAM beams with l=−1, 0 and +1) using a similar apparatus to send approximately 11 Mb/s signal at approximately 17 GHz carrier. Note that in these two demonstrations different OAM beams propagate along different spatial axes. There are some potential benefits if all of the OAM beams are actually multiplexed and propagated through the same aperture. In a recent demonstration eight polarization multiplexed (pol-muxed) RF OAM beams (for OAM beams on each of two orthogonal polarizations) our coaxially propagated through a 2.5 m link. 
     The herein described RF techniques have application in a wide variety of RF environments. These include RF Point to Point/Multipoint applications, RF Point to Point Backhaul applications, RF Point to Point Fronthaul applications (these provide higher throughput CPRI interface for cloudification and virtualization of RAN and future cloudified HetNet), RF Sattellite applications, RF Wifi (LAN) applications, RF Bluetooth (PAN) applications, RF personal device cable replacement applications, RF Radar applications and RF electromagnet tag applications. The techniques could also be used in a RF and FSO hybrid system that can provide communications in an RF mode or an FSO mode depending on which mode of operation is providing the most optimal or cost effective communications link at a particular point in time. 
     The four different OAM beams with l=−3, −1, +1 and +3 on each of 2 orthogonal polarizations are generated using customized spiral phase plates specifically for millimeter wave at 28 GHz. The observed intensity profile for each of the beams and their interferograms are shown in  FIG. 87 . These OAM beams were coaxially multiplexed using designed beam splitters. After propagation, the OAM channels were multiplexed using an inverse spiral phase plate and a spatial filter (the receiver antenna). The measured crosstalk it 28 GHz for each of the demultiplexed channels is shown in Table 8. It can be seen that the cross talk is low enough for 16-QAM data transmission without the assistance of extra DSPs to reduce the channel interference. 
     
       
         
           
               
             
               
                 TABLE 8 
               
             
            
               
                   
               
               
                 Crosstalk of the OAM channels measured at f = 28 GHz (CW) 
               
            
           
           
               
               
               
               
               
            
               
                   
                 l = −3 
                 l = −1 
                 l = +1 
                 l = +3 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 Single-pol (Y-pol) 
                 −25 dB 
                   −23 dB 
                   −25 dB 
                 −26 dB 
               
               
                 Dual-pol (X-pol) 
                 −17 dB 
                 −16.5 dB 
                 −18.1 dB 
                 −19 dB 
               
               
                 Dual-pol (Y-pol) 
                 −18 dB 
                 −16.5 dB 
                 −16.5 dB 
                 −24 dB 
               
               
                   
               
            
           
         
       
     
     Considering that each beam carries a 1 Gbaud 16-QAM signal, a total link capacity of 32 Gb/s at a single carrier frequency of 28 GHz and a spectral efficiency of 16 Gb/s/Hz may be achieved. In addition, an RF OAM beam demultiplexer (“mode sorter”) was also customize for a 28 GHz carrier and is implemented in such a link to simultaneously separate multiple OAM beams. Simultaneously demultiplexing for OAM beams at the single polarization has been demonstrated with a cross talk of less than −14 dB. The cross talk is likely to be further reduced by optimizing the design parameters. 
     Free Space Communications 
     An additional configuration in which the optical angular momentum processing and multi-layer overlay modulation technique described herein above may prove useful within the optical network framework is use with free-space optics communications. Free-space optics systems provide a number of advantages over traditional UHF RF based systems from improved isolation between the systems, the size and the cost of the receivers/transmitters, lack of RF licensing laws, and by combining space, lighting, and communication into the same system. Referring now to  FIG. 88  there is illustrated an example of the operation of a free-space communication system. The free-space communication system utilizes a free-space optics transmitter  8802  that transmits a light beam  8804  to a free-space optics receiver  8806 . The major difference between a fiber-optic network and a free-space optic network is that the information beam is transmitted through free space rather than over a fiber-optic cable. This causes a number of link difficulties, which will be more fully discussed herein below. Free-space optics is a line of sight technology that uses the invisible beams of light to provide optical bandwidth connections that can send and receive up to 2.5 Gbps of data, voice, and video communications between a transmitter  8802  and a receiver  8806 . Free-space optics uses the same concepts as fiber-optics, except without the use of a fiber-optic cable. Free-space optics systems provide the light beam  8804  within the infrared (IR) spectrum, which is at the low end of the light spectrum. Specifically, the optical signal is in the range of 300 Gigahertz to 1 Terahertz in terms of wavelength. 
     Presently existing free-space optics systems can provide data rates of up to 10 Gigabits per second at a distance of up to 2.5 kilometers. In outer space, the communications range of free space optical communications is currently on the order of several thousand kilometers, but has the potential to bridge interplanetary distances of millions of kilometers, using optical telescopes as beam expanders. In January of 2013, NASA used lasers to beam an image of the Mona Lisa to the Lunar Reconnaissance Orbiter roughly 240,000 miles away. To compensate for atmospheric interference, an error correction code algorithm, similar to that used within compact discs, was implemented. 
     The distance records for optical communications involve detection and emission of laser light by space probes. A two-way distance record for communication was established by the Mercury Laser Altimeter instrument aboard the MESSENGER spacecraft. This infrared diode neodymium laser, designed as a laser altimeter for a Mercury Orbiter mission, was able to communicate across a distance of roughly 15,000,000 miles (24,000,000 kilometers) as the craft neared Earth on a fly by in May of 2005. The previous record had been set with a one-way detection of laser light from Earth by the Galileo Probe as two ground based lasers were seen from 6,000,000 kilometers by the outbound probe in 1992. Researchers used a white LED based space lighting system for indoor local area network communications. 
     Referring now to  FIG. 89 , there is illustrated a block diagram of a free-space optics system using orbital angular momentum and multilevel overlay modulation according to the present disclosure. The OAM twisted signals, in addition to being transmitted over fiber, may also be transmitted using free optics. In this case, the transmission signals are generated within transmission circuitry  8902  at each of the FSO transceivers  8904 . Free-space optics technology is based on the connectivity between the FSO based optical wireless units, each consisting of an optical transceiver  8904  with a transmitter  8902  and a receiver  8906  to provide full duplex open pair and bidirectional closed pairing capability. Each optical wireless transceiver unit  8904  additionally includes an optical source  8908  plus a lens or telescope  8910  for transmitting light through the atmosphere to another lens  8910  receiving the information. At this point, the receiving lens or telescope  8910  connects to a high sensitivity receiver  8906  via optical fiber  8912 . The transmitting transceiver  8904   a  and the receiving transceiver  8904   b  have to have line of sight to each other. Trees, buildings, animals, and atmospheric conditions all can hinder the line of sight needed for this communications medium. Since line of sight is so critical, some systems make use of beam divergence or a diffused beam approach, which involves a large field of view that tolerates substantial line of sight interference without significant impact on overall signal quality. The system may also be equipped with auto tracking mechanism  8914  that maintains a tightly focused beam on the receiving transceiver  3404   b , even when the transceivers are mounted on tall buildings or other structures that sway. 
     The modulated light source used with optical source  8908  is typically a laser or light emitting diode (LED) providing the transmitted optical signal that determines all the transmitter capabilities of the system. Only the detector sensitivity within the receiver  8906  plays an equally important role in total system performance. For telecommunications purposes, only lasers that are capable of being modulated at 20 Megabits per second to 2.5 Gigabits per second can meet current marketplace demands. Additionally, how the device is modulated and how much modulated power is produced are both important to the selection of the device. Lasers in the 780-850 nm and 1520-1600 nm spectral bands meet frequency requirements. 
     Commercially available FSO systems operate in the near IR wavelength range between 750 and 1600 nm, with one or two systems being developed to operate at the IR wavelength of 10,000 nm. The physics and transmissions properties of optical energy as it travels through the atmosphere are similar throughout the visible and near IR wavelength range, but several factors that influence which wavelengths are chosen for a particular system. 
     The atmosphere is considered to be highly transparent in the visible and near IR wavelength. However, certain wavelengths or wavelength bands can experience severe absorption. In the near IR wavelength, absorption occurs primarily in response to water particles (i.e., moisture) which are an inherent part of the atmosphere, even under clear weather conditions. There are several transmission windows that are nearly transparent (i.e., have an attenuation of less than 0.2 dB per kilometer) within the 700-10,000 nm wavelength range. These wavelengths are located around specific center wavelengths, with the majority of free-space optics systems designed to operate in the windows of 780-850 nm and 1520-1600 nm. 
     Wavelengths in the 780-850 nm range are suitable for free-space optics operation and higher power laser sources may operate in this range. At 780 nm, inexpensive CD lasers may be used, but the average lifespan of these lasers can be an issue. These issues may be addressed by running the lasers at a fraction of their maximum rated output power which will greatly increase their lifespan. At around 850 nm, the optical source  8908  may comprise an inexpensive, high performance transmitter and detector components that are readily available and commonly used in network transmission equipment. Highly sensitive silicon (SI) avalanche photodiodes (APD) detector technology and advanced vertical cavity emitting laser may be utilized within the optical source  8908 . 
     VCSEL technology may be used for operation in the 780 to 850 nm range. Possible disadvantage of this technology include beam detection through the use of a night vision scope, although it is still not possible to demodulate a perceived light beam using this technique. 
     Wavelengths in the 1520-1600 nm range are well-suited for free-space transmission, and high quality transmitter and detector components are readily available for use within the optical source block  8908 . The combination of low attenuation and high component availability within this wavelength range makes the development of wavelength division multiplexing (WDM) free-space optics systems feasible. However, components are generally more expensive and detectors are typically less sensitive and have a smaller receive surface area when compared with silicon avalanche photodiode detectors that operator at the 850 nm wavelength. These wavelengths are compatible with erbium-doped fiber amplifier technology, which is important for high power (greater than 500 milliwatt) and high data rate (greater than 2.5 Gigabytes per second) systems. Fifty to 65 times as much power can be transmitted at the 1520-1600 nm wavelength than can be transmitted at the 780-850 nm wavelength for the same eye safety classification. Disadvantages of these wavelengths include the inability to detect a beam with a night vision scope. The night vision scope is one technique that may be used for aligning the beam through the alignment circuitry  8914 . Class  1  lasers are safe under reasonably foreseeable operating conditions including the use of optical instruments for intrabeam viewing. Class  1  systems can be installed at any location without restriction. 
     Another potential optical source  8908  comprised Class  1 M lasers. Class  1 M laser systems operate in the wavelength range from 302.5 to 4000 nm, which is safe under reasonably foreseeable conditions, but may be hazardous if the user employs optical instruments within some portion of the beam path. As a result, Class  1 M systems should only be installed in locations where the unsafe use of optical aids can be prevented. Examples of various characteristics of both Class  1  and Class  1 M lasers that may be used for the optical source  4708  are illustrated in Table 9 below. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 9 
               
               
                   
               
               
                 Laser 
                 Power 
                 Aperture Size 
                 Distance 
                 Power Density 
               
               
                 Classification 
                 (mW) 
                 (mm) 
                 (m) 
                 (mW/cm 2 ) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
            
               
                 850-nm Wavelength 
               
            
           
           
               
               
               
               
               
            
               
                 Class 1 
                 0.78 
                 7 
                 14 
                 2.03 
               
               
                   
                   
                 50 
                 2000 
                 0.04 
               
               
                 Class 1M 
                 0.78 
                 7 
                 100 
                 2.03 
               
               
                   
                 500 
                 7 
                 14 
                 1299.88 
               
               
                   
                   
                 50 
                 2000 
                 25.48 
               
            
           
           
               
            
               
                 1550-nm Wavelength 
               
            
           
           
               
               
               
               
               
            
               
                 Class 1 
                 10 
                 7 
                 14 
                 26.00 
               
               
                   
                   
                 25 
                 2000 
                 2.04 
               
               
                 Class 1M 
                 10 
                 3.5 
                 100 
                 103.99 
               
               
                   
                 500 
                 7 
                 14 
                 1299.88 
               
               
                   
                   
                 25 
                 2000 
                 101.91 
               
               
                   
               
            
           
         
       
     
     The 10,000 nm wavelength is relatively new to the commercial free space optic arena and is being developed because of better fog transmission capabilities. There is presently considerable debate regarding these characteristics because they are heavily dependent upon fog type and duration. Few components are available at the 10,000 nm wavelength, as it is normally not used within telecommunications equipment. Additionally, 10,000 nm energy does not penetrate glass, so it is ill-suited to behind window deployment. 
     Within these wavelength windows, FSO systems should have the following characteristics. The system should have the ability to operate at higher power levels, which is important for longer distance FSO system transmissions. The system should have the ability to provide high speed modulation, which is important for high speed FSO systems. The system should provide a small footprint and low power consumption, which is important for overall system design and maintenance. The system should have the ability to operate over a wide temperature range without major performance degradations such that the systems may prove useful for outdoor systems. Additionally, the mean time between failures should exceed 10 years. Presently existing FSO systems generally use VCSELS for operation in the shorter IR wavelength range, and Fabry-Pérot or distributed feedback lasers for operation in the longer IR wavelength range. Several other laser types are suitable for high performance FSO systems. 
     A free-space optics system using orbital angular momentum processing and multi-layer overlay modulation would provide a number of advantages. The system would be very convenient. Free-space optics provides a wireless solution to a last-mile connection, or a connection between two buildings. There is no necessity to dig or bury fiber cable. Free-space optics also requires no RF license. The system is upgradable and its open interfaces support equipment from a variety of vendors. The system can be deployed behind windows, eliminating the need for costly rooftop right. It is also immune to radiofrequency interference or saturation. The system is also fairly speedy. The system provides 2.5 Gigabits per second of data throughput. This provides ample bandwidth to transfer files between two sites. With the growth in the size of files, free-space optics provides the necessary bandwidth to transfer these files efficiently. 
     Free-space optics also provides a secure wireless solution. The laser beam cannot be detected with a spectral analyzer or RF meter. The beam is invisible, which makes it difficult to find. The laser beam that is used to transmit and receive the data is very narrow. This means that it is almost impossible to intercept the data being transmitted. One would have to be within the line of sight between the receiver and the transmitter in order to be able to accomplish this feat. If this occurs, this would alert the receiving site that a connection has been lost. Thus, minimal security upgrades would be required for a free-space optics system. 
     However, there are several weaknesses with free-space optics systems. The distance of a free-space optics system is very limited. Currently operating distances are approximately within 2 kilometers. Although this is a powerful system with great throughput, the limitation of distance is a big deterrent for full-scale implementation. Additionally, all systems require line of sight be maintained at all times during transmission. Any obstacle, be it environmental or animals can hinder the transmission. Free-space optic technology must be designed to combat changes in the atmosphere which can affect free-space optic system performance capacity. 
     Something that may affect a free-space optics system is fog. Dense fog is a primary challenge to the operation of free-space optics systems. Rain and snow have little effect on free-space optics technology, but fog is different. Fog is a vapor composed of water droplets which are only a few hundred microns in diameter, but can modify light characteristics or completely hinder the passage of light through a combination of absorption, scattering, and reflection. The primary answer to counter fog when deploying free-space optic based wireless products is through a network design that shortens FSO linked distances and adds network redundancies. 
     Absorption is another problem. Absorption occurs when suspended water molecules in the terrestrial atmosphere extinguish photons. This causes a decrease in the power density (attenuation) of the free space optics beam and directly affects the availability of the system. Absorption occurs more readily at some wavelengths than others. However, the use of appropriate power based on atmospheric conditions and the use of spatial diversity (multiple beams within an FSO based unit), helps maintain the required level of network availability. 
     Solar interference is also a problem. Free-space optics systems use a high sensitivity receiver in combination with a larger aperture lens. As a result, natural background light can potentially interfere with free-space optics signal reception. This is especially the case with the high levels of background radiation associated with intense sunlight. In some instances, direct sunlight may case link outages for periods of several minutes when the sun is within the receiver&#39;s field of vision. However, the times when the receiver is most susceptible to the effects of direct solar illumination can be easily predicted. When direct exposure of the equipment cannot be avoided, the narrowing of receiver field of vision and/or using narrow bandwidth light filters can improve system performance. Interference caused by sunlight reflecting off of a glass surface is also possible. 
     Scattering issues may also affect connection availability. Scattering is caused when the wavelength collides with the scatterer. The physical size of the scatterer determines the type of scattering. When the scatterer is smaller than the wavelength, this is known as Rayleigh scattering. When a scatterer is of comparable size to the wavelengths, this is known as Mie scattering. When the scattering is much larger than the wavelength, this is known as non-selective scattering. In scattering, unlike absorption, there is no loss of energy, only a directional redistribution of energy that may have significant reduction in beam intensity over longer distances. 
     Physical obstructions such as flying birds or construction cranes can also temporarily block a single beam free space optics system, but this tends to cause only short interruptions. Transmissions are easily and automatically resumed when the obstacle moves. Optical wireless products use multibeams (spatial diversity) to address temporary abstractions as well as other atmospheric conditions, to provide for greater availability. 
     The movement of buildings can upset receiver and transmitter alignment. Free-space optics based optical wireless offerings use divergent beams to maintain connectivity. When combined with tracking mechanisms, multiple beam FSO based systems provide even greater performance and enhanced installation simplicity. 
     Scintillation is caused by heated air rising from the Earth or man-made devices such as heating ducts that create temperature variations among different pockets of air. This can cause fluctuations in signal amplitude, which leads to “image dancing” at the free-space optics based receiver end. The effects of this scintillation are called “refractive turbulence.” This causes primarily two effects on the optical beams. Beam wander is caused by the turbulent eddies that are no larger than the beam. Beam spreading is the spread of an optical beam as it propagates through the atmosphere. 
     Referring now to  FIGS. 90A-90D , in order to achieve higher data capacity within optical links, an additional degree of freedom from multiplexing multiple data channels must be exploited. Moreover, the ability to use two different orthogonal multiplexing techniques together has the potential to dramatically enhance system performance and increased bandwidth. 
     One multiplexing technique which may exploit the possibilities is mode division multiplexing (MDM) using orbital angular momentum (OAM). OAM mode refers to laser beams within a free-space optical system or fiber-optic system that have a phase term of e ilφ  in their wave fronts, in which φ is the azimuth angle and l determines the OAM value (topological charge). In general, OAM modes have a “donut-like” ring shaped intensity distribution. Multiple spatial collocated laser beams, which carry different OAM values, are orthogonal to each other and can be used to transmit multiple independent data channels on the same wavelength. Consequently, the system capacity and spectral efficiency in terms of bits/S/Hz can be dramatically increased. Free-space communications links using OAM may support 100 Tbits/capacity. Various techniques for implementing this as illustrated in  FIGS. 90A-90D  include a combination of multiple beams  9002  having multiple different OAM values  9004  on each wavelength. Thus, beam  9002  includes OAM values, OAM 1  and OAM 4 . Beam  9006  includes OAM value  2  and OAM value  5 . Finally, beam  9008  includes OAM 3  value and OAM 6  value. Referring now to  FIG. 90B , there is illustrated a single beam wavelength  9010  using a first group of OAM values  9012  having both a positive OAM value  9012  and a negative OAM value  9014 . Similarly, OAM 2  value may have a positive value  9016  and a negative value  9018  on the same wavelength  9010 . While mode division multiplexing of OAM modes is described above, other orthogonal functions may be used with mode division multiplexing such as Laguerre Gaussian functions, Hermite Gaussian functions, Jacobi functions, Gegenbauer functions, Legendre functions and Chebyshev functions. 
       FIG. 90C  illustrates the use of a wavelength  9020  having polarization multiplexing of OAM value. The wavelength  9020  can have multiple OAM values  9022  multiplexed thereon. The number of available channels can be further increased by applying left or right handed polarization to the OAM values. Finally,  FIG. 90D  illustrates two groups of concentric rings  9060 ,  9062  for a wavelength having multiple OAM values. 
     Wavelength distribution multiplexing (WDM) has been widely used to improve the optical communication capacity within both fiber-optic systems and free-space communication system. OAM mode/mode division multiplexing and WDM are mutually orthogonal such that they can be combined to achieve a dramatic increase in system capacity. Referring now to  FIG. 91 , there is illustrated a scenario where each WDM channel  9102  contains many orthogonal OAM beam  9104 . Thus, using a combination of orbital angular momentum with wave division multiplexing, a significant enhancement in communication link to capacity may be achieved. By further combining polarization multiplexing with a combination of MDM and WDM even further increased in bandwidth capacity may be achieved from the +/−polarization values being added to the mode and wavelength multiplexing. The determination of the modes and number of modes for combination using combined polarization multiplex with a combination of MDM and WDM may be achieved using a mode crosstalk matrix such as that described herein below when transmitting Laguerre Gaussian modes, Hermite Gaussian modes and Ince Gaussian modes over a fiber. 
     Current optical communication architectures have considerable routing challenges. A routing protocol for use with free-space optic system must take into account the line of sight requirements for optical communications within a free-space optics system. Thus, a free-space optics network must be modeled as a directed hierarchical random sector geometric graph in which sensors route their data via multi-hop paths to a base station through a cluster head. This is a new efficient routing algorithm for local neighborhood discovery and a base station uplink and downlink discovery algorithm. The routing protocol requires order O log(n) storage at each node versus order O(n) used within current techniques and architectures. 
     Current routing protocols are based on link state, distance vectors, path vectors, or source routing, and they differ from the new routing technique in significant manners. First, current techniques assume that a fraction of the links are bidirectional. This is not true within a free-space optic network in which all links are unidirectional. Second, many current protocols are designed for ad hoc networks in which the routing protocol is designed to support multi-hop communications between any pair of nodes. The goal of the sensor network is to route sensor readings to the base station. Therefore, the dominant traffic patterns are different from those in an ad hoc network. In a sensor network, node to base stations, base station to nodes, and local neighborhood communication are mostly used. 
     Recent studies have considered the effect of unidirectional links and report that as many as 5 percent to 10 percent of links and wireless ad hoc networks are unidirectional due to various factors. Routing protocols such as DSDV and AODV use a reverse path technique, implicitly ignoring such unidirectional links and are therefore not relevant in this scenario. Other protocols such as DSR, ZRP, or ZRL have been designed or modified to accommodate unidirectionality by detecting unidirectional links and then providing bidirectional abstraction for such links. Referring now to  FIG. 92 , the simplest and most efficient solution for dealing with unidirectionality is tunneling, in which bidirectionality is emulated for a unidirectional link by using bidirectional links on a reverse back channel to establish the tunnel. Tunneling also prevents implosion of acknowledgement packets and looping by simply pressing link layer acknowledgements for tunneled packets received on a unidirectional link. Tunneling, however, works well in mostly bidirectional networks with few unidirectional links. 
     Within a network using only unidirectional links such as a free-space optical network, systems such as that illustrated in  FIGS. 92 and 93  would be more applicable. Nodes within a unidirectional network utilize a directional transmit  9202  transmitting from the node  9200  in a single, defined direction. Additionally, each node  9200  includes an omnidirectional receiver  9204  which can receive a signal coming to the node in any direction. Also, as discussed here and above, the node  9200  would also include a 0 log(n) storage  9206 . Thus, each node  9200  provide only unidirectional communications links. Thus, a series of nodes  9200  as illustrated in  FIG. 93  may unidirectionally communicate with any other node  9200  and forward communication from one desk location to another through a sequence of interconnected nodes. 
     Topological charge may be multiplexed to the wave length for either linear or circular polarization. In the case of linear polarizations, topological charge would be multiplexed on vertical and horizontal polarization. In case of circular polarization, topological charge would be multiplexed on left hand and right hand circular polarizations. 
     The topological charges can be created using Spiral Phase Plates (SPPs) such as that illustrated in  FIG. 12E , phase mask holograms or a Spatial Light Modulator (SLM) by adjusting the voltages on SLM which creates properly varying index of refraction resulting in twisting of the beam with a specific topological charge. Different topological charges can be created and muxed together and de-muxed to separate charges. 
     As Spiral Phase plates can transform a plane wave (l=0) to a twisted wave of a specific helicity (i.e. 1=+1), Quarter Wave Plates (QWP) can transform a linear polarization (s=0) to circular polarization (i.e. s=+1). 
     Cross talk and multipath interference can be reduced using Multiple-Input-Multiple-Output (MIMO). 
     Most of the channel impairments can be detected using a control or pilot channel and be corrected using algorithmic techniques (closed loop control system). 
     Multiplexing of the topological charge to the RF as well as free space optics in real time provides redundancy and better capacity. When channel impairments from atmospheric disturbances or scintillation impact the information signals, it is possible to toggle between free space optics to RF and back in real time. This approach still uses twisted waves on both the free space optics as well as the RF signal. Most of the channel impairments can be detected using a control or pilot channel and be corrected using algorithmic techniques (closed loop control system) or by toggling between the RF and free space optics. 
     In a further embodiment illustrated in  FIG. 94 , both RF signals and free space optics may be implemented within a dual RF and free space optics mechanism  9402 . The dual RF and free space optics mechanism  9402  include a free space optics projection portion  9404  that transmits a light wave having an orbital angular momentum applied thereto with multilevel overlay modulation and a RF portion  9406  including circuitry necessary for transmitting information with orbital angular momentum and multilayer overlay on an RF signal  9410 . The dual RF and free space optics mechanism  9402  may be multiplexed in real time between the free space optics signal  9408  and the RF signal  9410  depending upon operating conditions. In some situations, the free space optics signal  9408  would be most appropriate for transmitting the data. In other situations, the free space optics signal  9408  would not be available and the RF signal  9410  would be most appropriate for transmitting data. The dual RF and free space optics mechanism  9402  may multiplex in real time between these two signals based upon the available operating conditions. 
     Multiplexing of the topological charge to the RF as well as free space optics in real time provides redundancy and better capacity. When channel impairments from atmospheric disturbances or scintillation impact the information signals, it is possible to toggle between free space optics to RF and back in real time. This approach still uses twisted waves on both the free space optics as well as the RF signal. Most of the channel impairments can be detected using a control or pilot channel and be corrected using algorithmic techniques (closed loop control system) or by toggling between the RF and free space optics. 
     Quantum Communication using OAM 
     OAM has also received increasing interest for its potential role in the development of secure quantum communications that are based on the fundamental laws of quantum mechanics (i.e., quantum no cloning theorem). One of the examples is high dimensional quantum key distribution (QKD) QKD systems have conventionally utilized the polarization or phase of light for encoding. The original proposal for QKD (i.e., the BB 84 protocol of Bennett and Brassard) encodes information on the polarization states and so only allow one bit of information to be impressed onto each photon. The benefit of using OAM is that OAM states reside in an infinite dimensional Hilbert space, implying the possibility of encoding multiple bits of information on an individual photon. Similar to the use of OAM multiplexing in classical optical communications, the secure key rate can be further increased simultaneous encoding of information in different domains is implemented through making use of high dimensional entanglement. The addition to the advantages of a large alphabet for information encoding, the security of keys generated by an OAM-based QKD system have been shown to be improved due to the use of a large Hilbert space, which indicates increase robustness of the QKD system against eavesdropping. 
       FIG. 95  illustrates a seven dimensional QKD link based on OAM encoding.  FIG. 96  shows the two complementary seven dimensional bases used for information encoding. Recent QKD systems have been demonstrated to operate at a secure key rate of up to 1 Mb/s. However, in order to support an OAM-based QKD system with a higher secure key rate, the development of a OAM generation methods with speeds higher than MHz would be required. Another challenge arises from the efficiency inch sorting single photons in the OAM basis, although the current OAM sorting approach allows an OAM separation efficiency of greater than 92%. Additionally, adverse channel conditions pose a critical challenge. For a free space QKD system employing OAM states, atmospheric turbulence that distorts the phase front of an OAM state may significantly degrade the information content of the transmitted OAM light field. 
     Quantum Key Distribution 
     Referring now to  FIG. 97 , there is illustrated a further improvement of a system utilizing orbital angular momentum processing, Laguerre Gaussian processing, Hermite Gaussian processing or processing using any orthogonal functions. In the illustration of  FIG. 97 , a transmitter  9702  and receiver  9704  are interconnected over an optical link  9706 . The optical link  9706  may comprise a fiber-optic link or a free-space optic link as described herein above. The transmitter receives a data stream  9708  that is processed via orbital angular momentum processing circuitry  9710 . The orbital angular momentum processing circuitry  9710  provide orbital angular momentum twist to various signals on separate channels as described herein above. In some embodiments, the orbital angular momentum processing circuitry may further provide multi-layer overlay modulation to the signal channels in order to further increase system bandwidth. 
     The OAM processed signals are provided to quantum key distribution processing circuitry  9712 . The quantum key distribution processing circuitry  9712  utilizes the principals of quantum key distribution as will be more fully described herein below to enable encryption of the signal being transmitted over the optical link  9706  to the receiver  9704 . The received signals are processed within the receiver  9704  using the quantum key distribution processing circuitry  9714 . The quantum key distribution processing circuitry  9714  decrypts the received signals using the quantum key distribution processing as will be more fully described herein below. The decrypted signals are provided to orbital angular momentum processing circuitry  9716  which removes any orbital angular momentum twist from the signals to generate the plurality of output signals  9718 . As mentioned previously, the orbital angular momentum processing circuitry  9716  may also demodulate the signals using multilayer overlay modulation included within the received signals. 
     Orbital angular momentum in combination with optical polarization is exploited within the circuit of  FIG. 97  in order to encode information in rotation invariant photonic states, so as to guarantee full independence of the communication from the local reference frames of the transmitting unit  9702  and the receiving unit  9704 . There are various ways to implement quantum key distribution (QKD), a protocol that exploits the features of quantum mechanics to guarantee unconditional security in cryptographic communications with error rate performances that are fully compatible with real world application environments. 
     Encrypted communication requires the exchange of keys in a protected manner. This key exchanged is often done through a trusted authority. Quantum key distribution is an alternative solution to the key establishment problem. In contrast to, for example, public key cryptography, quantum key distribution has been proven to be unconditionally secure, i.e., secure against any attack, even in the future, irrespective of the computing power or in any other resources that may be used. Quantum key distribution security relies on the laws of quantum mechanics, and more specifically on the fact that it is impossible to gain information about non-orthogonal quantum states without perturbing these states. This property can be used to establish random keys between a transmitter and receiver, and guarantee that the key is perfectly secret from any third party eavesdropping on the line. 
     In parallel to the “full quantum proofs” mentioned above, the security of QKD systems has been put on stable information theoretic footing, thanks to the work on secret key agreements done in the framework of information theoretic cryptography and to its extensions, triggered by the new possibilities offered by quantum information. Referring now to  FIG. 98 , within a basic QKD system, a QKD link  9802  is a point to point connection between a transmitter  9804  and a receiver  9806  that want to share secret keys. The QKD link  9802  is constituted by the combination of a quantum channel  9808  and a classic channel  9810 . The transmitter  9804  generates a random stream of classical bits and encodes them into a sequence of non-orthogonal states of light that are transmitted over the quantum channel  9808 . Upon reception of these quantum states, the receiver  9806  performs some appropriate measurements leading the receiver to share some classical data over the classical link  9810  correlated with the transmitter bit stream. The classical channel  9810  is used to test these correlations. 
     If the correlations are high enough, this statistically implies that no significant eavesdropping has occurred on the quantum channel  9808  and thus, that has a very high probability, a perfectly secure, symmetric key can be distilled from the correlated data shared by the transmitter  9804  and the receiver  9806 . In the opposite case, the key generation process has to be aborted and started again. The quantum key distribution is a symmetric key distribution technique. Quantum key distribution requires, for authentication purposes, that the transmitter  9804  and receiver  9806  share in advance a short key whose length scales only logarithmically in the length of the secret key generated by an OKD session. 
     Quantum key distribution on a regional scale has already been demonstrated in a number of countries. However, free-space optical links are required for long distance communication among areas which are not suitable for fiber installation or for moving terminals, including the important case of satellite based links. The present approach exploits spatial transverse modes of the optical beam, in particular of the OAM degree of freedom, in order to acquire a significant technical advantage that is the insensitivity of the communication to relevant alignment of the user&#39;s reference frames. This advantage may be very relevant for quantum key distribution implementation to be upgraded from the regional scale to a national or continental one, or for links crossing hostile ground, and even for envisioning a quantum key distribution on a global scale by exploiting orbiting terminals on a network of satellites. 
     The OAM Eigen modes are characterized by a twisted wavefront composed of “l” intertwined helices, where “l” is an integer, and by photons carrying “±lh” of (orbital) angular momentum, in addition to the more usual spin angular momentum (SAM) associated with polarization. The potentially unlimited value of “l” opens the possibility to exploit OAM also for increasing the capacity of communication systems (although at the expense of increasing also the channel cross-section size), and terabit classical data transmission based on OAM multiplexing can be demonstrated both in free-space and optical fibers. Such a feature can also be exploited in the quantum domain, for example to expand the number of qubits per photon, or to achieve new functions, such as the rotational invariance of the qubits. 
     In a free-space QKD, two users (Alice and Bob) must establish a shared reference frame (SRF) in order to communicate with good fidelity. Indeed the lack of a SRF is equivalent to an unknown relative rotation which introduces noise into the quantum channel, disrupting the communication. When the information is encoded in photon polarization, such a reference frame can be defined by the orientations of Alice&#39;s and Bob&#39;s “horizontal” linear polarization directions. The alignment of these directions needs extra resources and can impose serious obstacles in long distance free space QKD and/or when the misalignment varies in time. As indicated, we can solve this by using rotation invariant states, which remove altogether the need for establishing a SRF. Such states are obtained as a particular combination of OAM and polarization modes (hybrid states), for which the transformation induced by the misalignment on polarization is exactly balanced by the effect of the same misalignment on spatial modes. These states exhibit a global symmetry under rotations of the beam around its axis and can be visualized as space-variant polarization states, generalizing the well-known azimuthal and radial vector beams, and forming a two-dimensional Hilbert space. Moreover, this rotation-invariant hybrid space can be also regarded as a decoherence-free subspace of the four-dimensional OAM-polarization product Hilbert space, insensitive to the noise associated with random rotations. 
     The hybrid states can be generated by a particular space-variant birefringent plate having topological charge “q” at its center, named “q-plate”. In particular, a polarized Gaussian beam (having zero OAM) passing through a q-plate with q=½ will undergo the following transformation:
 
(α| R     +β|R   ) π   |0   0   →α|L     π     |r     0   +β|R     π     |l     0  
 
     |L&gt; π   _  and |R&gt; π  denote the left and right circular polarization states (eigenstates of SAM with eigenvalues “±h”), |0&gt; O  represents the transverse Gaussian mode with zero OAM and the |L O   _  and |R&gt; O  eigenstates of OAM with |l|=1 and with eigenvalues “±lh”). The states appearing on the right hand side of equation are rotation-invariant states. The reverse operation to this can be realized by a second q-plate with the same q. In practice, the q-plate operates as an interface between the polarization space and the hybrid one, converting qubits from one space to the other and vice versa in a universal (qubit invariant) way. This in turn means that the initial encoding and final decoding of information in our QKD implementation protocol can be conveniently performed in the polarization space, while the transmission is done in the rotation-invariant hybrid space. 
     OAM is a conserved quantity for light propagation in vacuum, which is obviously important for communication applications. However, OAM is also highly sensitive to atmospheric turbulence, a feature which limits its potential usefulness in many practical cases unless new techniques are developed to deal with such issues. 
     Quantum cryptography describes the use of quantum mechanical effects (in particular quantum communication and quantum computation) to perform cryptographic tasks or to break cryptographic systems. Well-known examples of quantum cryptography are the use of quantum communication to exchange a key securely (quantum key distribution) and the hypothetical use of quantum computers that would allow the breaking of various popular public-key encryption and signature schemes (e.g., RSA). 
     The advantage of quantum cryptography lies in the fact that it allows the completion of various cryptographic tasks that are proven to be impossible using only classical (i.e. non-quantum) communication. For example, quantum mechanics guarantees that measuring quantum data disturbs that data; this can be used to detect eavesdropping in quantum key distribution. 
     Quantum key distribution (QKD) uses quantum mechanics to guarantee secure communication. It enables two parties to produce a shared random secret key known only to them, which can then be used to encrypt and decrypt messages. 
     An important and unique property of quantum distribution is the ability of the two communicating users to detect the presence of any third party trying to gain knowledge of the key. This results from a fundamental aspect of quantum mechanics: the process of measuring a quantum system in general disturbs the system. A third party trying to eavesdrop on the key must in some way measure it, thus introducing detectable anomalies. By using quantum superposition or quantum entanglement and transmitting information in quantum states, a communication system can be implemented which detects eavesdropping. If the level of eavesdropping is below a certain threshold, a key can be produced that is guaranteed to be secure (i.e. the eavesdropper has no information about it), otherwise no secure key is possible and communication is aborted. 
     The security of quantum key distribution relies on the foundations of quantum mechanics, in contrast to traditional key distribution protocol which relies on the computational difficulty of certain mathematical functions, and cannot provide any indication of eavesdropping or guarantee of key security. 
     Quantum key distribution is only used to reduce and distribute a key, not to transmit any message data. This key can then be used with any chosen encryption algorithm to encrypt (and decrypt) a message, which is transmitted over a standard communications channel. The algorithm most commonly associated with QKD is the one-time pad, as it is provably secure when used with a secret, random key. 
     Quantum communication involves encoding information in quantum states, or qubits, as opposed to classical communication&#39;s use of bits. Usually, photons are used for these quantum states and thus is applicable within optical communication systems. Quantum key distribution exploits certain properties of these quantum states to ensure its security. There are several approaches to quantum key distribution, but they can be divided into two main categories, depending on which property they exploit. The first of these are prepare and measure protocol. In contrast to classical physics, the act of measurement is an integral part of quantum mechanics. In general, measuring an unknown quantum state changes that state in some way. This is known as quantum indeterminacy, and underlies results such as the Heisenberg uncertainty principle, information distribution theorem, and no cloning theorem. This can be exploited in order to detect any eavesdropping on communication (which necessarily involves measurement) and, more importantly, to calculate the amount of information that has been intercepted. Thus, by detecting the change within the signal, the amount of eavesdropping or information that has been intercepted may be determined by the receiving party. 
     The second category involves the use of entanglement based protocols. The quantum states of two or more separate objects can become linked together in such a way that they must be described by a combined quantum state, not as individual objects. This is known as entanglement, and means that, for example, performing a measurement on one object affects the other object. If an entanglement pair of objects is shared between two parties, anyone intercepting either object alters the overall system, revealing the presence of a third party (and the amount of information that they have gained). Thus, again, undesired reception of information may be determined by change in the entangled pair of objects that is shared between the parties when intercepted by an unauthorized third party. 
     One example of a quantum key distribution (QKD) protocol is the BB84 protocol. The BB84 protocol was originally described using photon polarization states to transmit information. However, any two pairs of conjugate states can be used for the protocol, and optical fiber-based implementations described as BB84 can use phase-encoded states. The transmitter (traditionally referred to as Alice) and the receiver (traditionally referred to as Bob) are connected by a quantum communication channel which allows quantum states to be transmitted. In the case of photons, this channel is generally either an optical fiber, or simply free-space, as described previously with respect to  FIG. 97 . In addition, the transmitter and receiver communicate via a public classical channel, for example using broadcast radio or the Internet. Neither of these channels needs to be secure. The protocol is designed with the assumption that an eavesdropper (referred to as Eve) can interfere in any way with both the transmitter and receiver. 
     Referring now to  FIG. 99 , the security of the protocol comes from encoding the information in non-orthogonal states. Quantum indeterminacy means that these states cannot generally be measured without disturbing the original state. BB84 uses two pair of states  9902 , each pair conjugate to the other pair to form a conjugate pair  9904 . The two states  9902  within a pair  9904  are orthogonal to each other. Pairs of orthogonal states are referred to as a basis. The usual polarization state pairs used are either the rectilinear basis of vertical (0 degrees) and horizontal (90 degrees), the diagonal basis of 45 degrees and 135 degrees, or the circular basis of left handedness and/or right handedness. Any two of these basis are conjugate to each other, and so any two can be used in the protocol. In the example of  FIG. 100 , rectilinear basis are used at  10002  and  10004 , respectively, and diagonal basis are used at  10006  and  10008 . 
     The first step in BB84 protocol is quantum transmission. Referring now to  FIG. 101  wherein there is illustrated a flow diagram describing the process, wherein the transmitter creates a random bit (0 or 1) at step  10102 , and randomly selects at  10104  one of the two basis, either rectilinear or diagonal, to transmit the random bit. The transmitter prepares at step  10106  a photon polarization state depending both on the bit value and the selected basis, as shown in  FIG. 55 . So, for example, a 0 is encoded in the rectilinear basis (+) as a vertical polarization state and a 1 is encoded in a diagonal basis (X) as a 135 degree state. The transmitter transmits at step  10108  a single proton in the state specified to the receiver using the quantum channel. This process is repeated from the random bit stage at step  10102  with the transmitter recording the state, basis, and time of each photon that is sent over the optical link. 
     According to quantum mechanics, no possible measurement distinguishes between the four different polarization states  10002  through  10008  of  FIG. 100 , as they are not all orthogonal. The only possible measurement is between any two orthogonal states (and orthonormal basis). So, for example, measuring in the rectilinear basis gives a result of horizontal or vertical. If the photo was created as horizontal or vertical (as a rectilinear eigenstate), then this measures the correct state, but if it was created as 45 degrees or 135 degrees (diagonal eigenstate), the rectilinear measurement instead returns either horizontal or vertical at random. Furthermore, after this measurement, the proton is polarized in the state it was measured in (horizontal or vertical), with all of the information about its initial polarization lost. 
     Referring now to  FIG. 102 , as the receiver does not know the basis the photons were encoded in, the receiver can only select a basis at random to measure in, either rectilinear or diagonal. At step  10202 , the transmitter does this for each received photon, recording the time measurement basis used and measurement result at step  10204 . At step  10206 , a determination is made if there are further protons present and, if so, control passes back to step  10202 . Once inquiry step  10206  determines the receiver had measured all of the protons, the transceiver communicates at step  10208  with the transmitter over the public communications channel. The transmitter broadcast the basis for each photon that was sent at step  10210  and the receiver broadcasts the basis each photon was measured in at step  10212 . Each of the transmitter and receiver discard photon measurements where the receiver used a different basis at step  10214  which, on average, is one-half, leaving half of the bits as a shared key, at step  10216 . This process is more fully illustrated in  FIG. 103 . 
     The transmitter transmits the random bit  01101001 . For each of these bits respectively, the transmitter selects the sending basis of rectilinear, rectilinear, diagonal, rectilinear, diagonal, diagonal, diagonal, and rectilinear. Thus, based upon the associated random bits selected and the random sending basis associated with the signal, the polarization indicated in line  10202  is provided. Upon receiving the photon, the receiver selects the random measuring basis as indicated in line  10304 . The photon polarization measurements from these basis will then be as indicated in line  10306 . A public discussion of the transmitted basis and the measurement basis are discussed at  10308  and the secret key is determined to be  0101  at  10310  based upon the matching bases for transmitted photons  1 ,  3 ,  6 , and  8 . 
     Referring now to  FIG. 104 , there is illustrated the process for determining whether to keep or abort the determined key based upon errors detected within the determined bit string. To check for the presence of eavesdropping, the transmitter and receiver compare a certain subset of their remaining bit strings at step  10402 . If a third party has gained any information about the photon&#39;s polarization, this introduces errors within the receiver&#39;s measurements. If more than P bits differ at inquiry step  10404 , the key is aborted at step  10406 , and the transmitter and receiver try again, possibly with a different quantum channel, as the security of the key cannot be guaranteed. P is chosen so that if the number of bits that is known to the eavesdropper is less than this, privacy amplification can be used to reduce the eavesdropper&#39;s knowledge of the key to an arbitrarily small amount by reducing the length of the key. If inquiry step  10404  determines that the number of bits is not greater than P, then the key may be used at step  10408 . 
     The E91 protocol comprises another quantum key distribution scheme that uses entangled pairs of protons. This protocol may also be used with entangled pairs of protons using orbital angular momentum processing, Laguerre Gaussian processing, Hermite Gaussian processing or processing using any orthogonal functions for Q-bits. The entangled pairs can be created by the transmitter, by the receiver, or by some other source separate from both of the transmitter and receiver, including an eavesdropper. The photons are distributed so that the transmitter and receiver each end up with one photon from each pair. The scheme relies on two properties of entanglement. First, the entangled states are perfectly correlated in the sense that if the transmitter and receiver both measure whether their particles have vertical or horizontal polarizations, they always get the same answer with 100 percent probability. The same is true if they both measure any other pair of complementary (orthogonal) polarizations. However, the particular results are not completely random. It is impossible for the transmitter to predict if the transmitter, and thus the receiver, will get vertical polarizations or horizontal polarizations. Second, any attempt at eavesdropping by a third party destroys these correlations in a way that the transmitter and receiver can detect. The original Ekert protocol (E91) consists of three possible states and testing Bell inequality violation for detecting eavesdropping. 
     Presently, the highest bit rate systems currently using quantum key distribution demonstrate the exchange of secure keys at 1 Megabit per second over a 20 kilometer optical fiber and 10 Kilobits per second over a 100 kilometer fiber. 
     The longest distance over which quantum key distribution has been demonstrated using optical fiber is 148 kilometers. The distance is long enough for almost all of the spans found in today&#39;s fiber-optic networks. The distance record for free-space quantum key distribution is 144 kilometers using BB84 enhanced with decoy states. 
     Referring now to  FIG. 105 , there is illustrated a functional block diagram of a transmitter  10502  and receiver  10504  that can implement alignment of free-space quantum key distribution. The system can implement the BB84 protocol with decoy states. The controller  10506  enables the bits to be encoded in two mutually unbiased bases Z={|0&gt;, |1&gt;} and X={|+&gt;, |−&gt;}, where |0&gt; and |1&gt; are two orthogonal states spanning the qubit space and |± =1/√2 (|0 ±|1 ). The transmitter controller  10506  randomly chooses between the Z and X basis to send the classical bits  0  and  1 . Within hybrid encoding, the Z basis corresponds to {|L   π   |r   O , |R   π   |l   O } while the X basis states correspond to 1/√2 (|L   π   |r   O ±|R   90    |l   O ). The transmitter  10502  uses four different polarized attenuated lasers  10508  to generate quantum bits through the quantum bit generator  10510 . Photons from the quantum bit generator  41050  are delivered via a single mode fiber  10512  to a telescope  10514 . Polarization states |H&gt;, |V&gt;, |R&gt;, |L&gt; are transformed into rotation invariant hybrid states by means of a q-plate  10516  with q=½. The photons can then be transmitted to the receiving station  10504  where a second q-plate transform  10518  transforms the signals back into the original polarization states |H&gt;, |V&gt;, |R&gt;, |L&gt;, as defined by the receiver reference frame. Qubits can then be analyzed by polarizers  10520  and single photon detectors  10522 . The information from the polarizers  10520  and photo detectors  10522  may then be provided to the receiver controller  10524  such that the shifted keys can be obtained by keeping only the bits corresponding to the same basis on the transmitter and receiver side as determined by communications over a classic channel between the transceivers  10526 ,  10528  in the transmitter  10502  and receiver  10504 . 
     Referring now to  FIG. 106 , there is illustrated a network cloud based quantum key distribution system including a central server  10602  and various attached nodes  10604  in a hub and spoke configuration. Trends in networking are presenting new security concerns that are challenging to meet with conventional cryptography, owing to constrained computational resources or the difficulty of providing suitable key management. In principle, quantum cryptography, with its forward security and lightweight computational footprint, could meet these challenges, provided it could evolve from the current point to point architecture to a form compatible with multimode network architecture. Trusted quantum key distribution networks based on a mesh of point to point links lacks scalability, require dedicated optical fibers, are expensive and not amenable to mass production since they only provide one of the cryptographic functions, namely key distribution needed for secure communications. Thus, they have limited practical interest. 
     A new, scalable approach such as that illustrated in  FIG. 106  provides quantum information assurance that is network based quantum communications which can solve new network security challenges. In this approach, a BB84 type quantum communication between each of N client nodes  10604  and a central sever  10602  at the physical layer support a quantum key management layer, which in turn enables secure communication functions (confidentiality, authentication, and nonrepudiation) at the application layer between approximately N 2  client pairs. This network based communication “hub and spoke” topology can be implemented in a network setting, and permits a hierarchical trust architecture that allows the server  10602  to act as a trusted authority in cryptographic protocols for quantum authenticated key establishment. This avoids the poor scaling of previous approaches that required a pre-existing trust relationship between every pair of nodes. By making a server  10602 , a single multiplex QC (quantum communications) receiver and the client nodes  10604  QC transmitters, this network can simplify complexity across multiple network nodes. In this way, the network based quantum key distribution architecture is scalable in terms of both quantum physical resources and trust. One can at time multiplex the server  10602  with three transmitters  10604  over a single mode fiber, larger number of clients could be accommodated with a combination of temporal and wavelength multiplexing as well as orbital angular momentum multiplexed with wave division multiplexing to support much higher clients. 
     Referring now to  FIGS. 107 and 108 , there are illustrated various components of multi-user orbital angular momentum based quantum key distribution multi-access network.  FIG. 107  illustrates a high speed single photon detector  10702  positioned at a network node that can be shared between multiple users  10704  using conventional network architectures, thereby significantly reducing the hardware requirements for each user added to the network. In an embodiment, the single photon detector  10702  may share up to 64 users. This shared receiver architecture removes one of the main obstacles restricting the widespread application of quantum key distribution. The embodiment presents a viable method for realizing multi-user quantum key distribution networks with resource efficiency. 
     Referring now also to  FIG. 108 , in a nodal quantum key distribution network, multiple trusted repeaters  10802  are connected via point to point links  10804  between node  10806 . The repeaters are connected via point to point links between a quantum transmitter and a quantum receiver. These point to point links  10804  can be realized using long distance optical fiber lengths and may even utilize ground to satellite quantum key distribution communication. While point to point connections  10804  are suitable to form a backbone quantum core network, they are less suitable to provide the last-mile service needed to give a multitude of users access to the quantum key distribution infrastructure. Reconfigurable optical networks based on optical switches or wavelength division multiplexing may achieve more flexible network structures, however, they also require the installation of a full quantum key distribution system per user which is prohibitively expensive for many applications. 
     The quantum key signals used in quantum key distribution need only travel in one direction along a fiber to establish a secure key between the transmitter and the receiver. Single photon quantum key distribution with the sender positioned at the network node  10806  and the receiver at the user premises therefore lends itself to a passive multi-user network approach. However, this downstream implementation has two major shortcomings. Firstly, every user in the network requires a single photon detector, which is often expensive and difficult to operate. Additionally, it is not possible to deterministically address a user. All detectors, therefore, have to operate at the same speed as a transmitter in order not to miss photons, which means that most of the detector bandwidth is unused. 
     Most systems associated with a downstream implementation can be overcome. The most valuable resource should be shared by all users and should operate at full capacity. One can build an upstream quantum access network in which the transmitters are placed at the end user location and a common receiver is placed at the network node. This way, an operation with up to 64 users is feasible, which can be done with multi-user quantum key distribution over a 1×64 passive optical splitter. 
     The above described QKD scheme is applicable to twisted pair, coaxial cable, fiber optic, RF satellite, RF broadcast, RF point-to point, RF point-to-multipoint, RF point-to-point (backhaul), RF point-to-point (fronthaul to provide higher throughput CPRI interface for cloudification and virtualization of RAN and cloudified HetNet), free-space optics (FSO), Internet of Things (IOT), Wifi, Bluetooth, as a personal device cable replacement, RF and FSO hybrid system, Radar, electromagnetic tags and all types of wireless access. The method and system are compatible with many current and future multiple access systems, including EV-DO, UMB, WIMAX, WCDMA (with or without), multimedia broadcast multicast service (MBMS)/multiple input multiple output (MIMO), HSPA evolution, and LTE. The techniques would be useful for combating denial of service attacks by routing communications via alternate links in case of disruption, as a technique to combat Trojan Horse attacks which does not require physical access to the endpoints and as a technique to combat faked-state attacks, phase remapping attacks and time-shift attacks. 
     Thus, using various configurations of the above described orbital angular momentum processing, multi-layer overlay modulation, and quantum key distribution within various types of communication networks and more particularly optical fiber networks and free-space optic communication network, a variety of benefits and improvements in system bandwidth and capacity maybe achieved. 
     OAM Based Networking Functions 
     In addition to the potential applications for static point to point data transmission, the unique way front structure of OAM beams may also enable some networking functions by manipulating the phase using reconfigurable spatial light modulators (SLMs) or other light projecting technologies. 
     Data Swapping 
     Data exchange is a useful function in an OAM-based communication system. A pair of data channels on different OAM states can exchange their data in a simple manner with the assistance of a reflective phase hologram as illustrated in  FIG. 109 . If two OAM beams  10902 ,  10904 , e.g., OAM beams with l=+L 1  and +L 2 , which carry two independent data streams  10906 ,  10908 , are launched onto a reflective SLM  10910  loaded with a spiral phase pattern with an order of −(L 1 +L 2 ), the data streams will swap between the two OAM channels. The phase profile of the SLM will change these two OAM beams to l=−l 2  and l=−l 1 , respectively. In addition, each OAM beam will change to its opposite charge under the reflection effect. As a result, the channel on l=+l 1  is switched to l=+l 2  and vice versa, which indicates that the data on the two OAM channels is exchanged.  FIG. 109  shows the data exchange between l=+6  10912  and l=8  10914  using a phase pattern on the order of l=−14 on a reflective SLM  10910 . A power penalty of approximately 0.9 dB is observed when demonstrating this in the experiment. 
     An experiment further demonstrated that the selected data swapping function can handle more than two channels. Among multiple multiplexed OAM beams, any two OAM beams can be selected to swap their data without affecting the other channels. In general, reconfigurable optical add/drop multiplexers (ROADM) are important function blocks in WDM networks. A WDM RODAM is able to selectively drop a given wavelength channel and add in a different channel at the same wavelength without having to detect all pass-through channels. A similar scheme can be implemented in an OAM multiplexed system to selectively drop and add a data channel carried on a given OAM beam. One approach to achieve this function is based on the fact that OAM beams generally have a distinct intensity profile when compared to a fundamental Gaussian beam. 
     Referring now to  FIGS. 110 and 111  there is illustrated the manner for using a ROADM for exchanging data channels. The example of  FIG. 111  illustrates SLM&#39;s and spatial filters. The principle of an OAM-based ROADM uses three stages: down conversion, add/drop and up conversion. The down conversion stage transforms at step  11002  the input multiplexed OAM modes  11102  (donut like transverse intensity profiles  11104 ) into a Gaussian light beam with l=0 (a spotlight transverse intensity profile  11106 ). After the down conversion at step  11002 , the selected OAM beam becomes a Gaussian beam while the other beams remain OAM but have a different l state. The down converted beams  11106  are reflected at step  11004  by a specially designed phase pattern  11108  that has different gratings in the center and in the outer ring region. The central and outer regions are used to redirect the Gaussian beam  11106  in the center (containing the drop channel  11110 ) and the OAM beams with a ring-shaped (containing the pass-through channels) in different directions. Meanwhile, another Gaussian beam  11112  carrying a new data stream can be added to the pass-through OAM beams (i.e., add channel). Following the selective manipulation, an up conversion process is used at step  11006  for transforming the Gaussian beam back to an OAM beam. This process recovers the l states of all of the beams.  FIG. 112  illustrates the images of each step in the add/drop of a channel carried by an OAM beam with l=+2. Some other networking functions in OAM based systems have also been demonstrated including multicasting, 2 by 2 switching, polarization switching and mode filtering. 
     In its fundamental form, a beam carrying OAM has a helical phase front that creates orthogonality and hence is distinguishable from other OAM states. Although other mode groups (e.g., Hermite-Gaussian modes, etc.) also have orthogonality and can be used for mode multiplexing, OAM has the convenient advantage of its circular symmetry which is matched to the geometry of most optical systems. Indeed, many free-space data link demonstrations attempt to use OAM-carrying modes since such modes have circular symmetry and tend to be compatible with commercially available optical components. Therefore, one can consider that OAM is used more as a technical convenience for efficient multiplexing than as a necessarily “better” type of modal set. 
     The use of OAM multiplexing in fiber is potentially attractive. In a regular few mode fiber, hybrid polarized OAM modes can be considered as fiber eigenmodes. Therefore, OAM modes normally have less temporal spreading as compared to LP mode basis, which comprise two eigenmode components each with a different propagation constant. As for the specially designed novel fiber that can stably propagate multiple OAM states, potential benefits could include lower receiver complexity since the MIMO DSP is not required. Progress can be found in developing various types of fiber that are suitable for OAM mode transmission. Recently demonstrated novel fibers can support up to 16 OAM states. Although they are still in the early stages, there is the possibility that further improvement of performance (i.e., larger number of “maintained” modes and lower power loss) will be achieved. 
     OAM multiplexing can be useful for communications in RF communications in a different way than the traditional spatial multiplexing. For a traditional spatial multiplexing system, multiple spatially separated transmitter and receiver aperture pairs are adopted for the transmission of multiple data streams. As each of the antenna elements receives a different superposition of the different transmitted signals, each of the original channels can be demultiplexed through the use of electronic digital signal processing. The distinction of each channel relies on the spatial position of each antenna pair. However, OAM multiplexing is implemented such that the multiplexed beams are completely coaxial throughout the transmission medium, and only one transmitter and receiver aperture (although with certain minimum aperture sizes) is used. Due to the OAM beam orthogonality provided by the helical phase front, efficient demultiplexing can be achieved without the assist of further digital signal post-processing to cancel channel interference. 
     Many of the demonstrated communication systems with OAM multiplexing use bulky and expensive components that are not necessarily optimized for OAM operation. As was the case for many previous advances in optical communications, the future of OAM would greatly benefit from advances in the enabling devices and subsystems (e.g., transmitters, (de)multiplexers and receivers). Particularly with regard to integration, this represents significant opportunity to reduce cost and size and to also increase performance. 
     Orthogonal beams using for example OAM, Hermite Gaussian, Laguerre Gaussian, spatial Bessel, Prolate spheroidal or other types of orthogonal functions may be multiplexed together to increase the amount of information transmitted over a single communications link. The structure for multiplexing the beams together may use a number of different components. Examples of these include spatial light modulators (SLMs); micro electromechanical systems (MEMs); digital light processers (DLPs); amplitude masks; phase masks; spiral phase plates; Fresnel zone plates; spiral zone plates; spiral phase plates and phase plates. 
     Multiplexing Using Holograms 
     Referring now to  FIG. 113 , there is illustrated a configuration of generation circuitry for the generation of an OAM twisted beam using a hologram within a micro-electrical mechanical device. Configurations such as this may be used for multiplexing multiple OAM twisted beams together. A laser  11302  generates a beam having a wavelength of approximately 543 nm. This beam is focused through a telescope  11304  and lens  11306  onto a mirror/system of mirrors  11308 . The beam is reflected from the mirrors  11308  into a DMD  11310 . The DMD  11310  has programmed in to its memory a one or more forked holograms  11312  that generate a desired OAM twisted beam  11313  having any desired information encoded into the OAM modes of the beam that is detected by a CCD  11314 . The holograms  11312  are loaded into the memory of the DMD  11310  and displayed as a static image. In the case of 1024×768 DMD array, the images must comprise 1024 by 768 images. The control software of the DMD  11310  converts the holograms into .bmp files. The holograms may be displayed singly or as multiple holograms displayed together in order to multiplex particular OAM modes onto a single beam. The manner of generating the hologram  11312  within the DMD  11310  may be implemented in a number of fashions that provide qualitative differences between the generated OAM beam  11313 . Phase and amplitude information may be encoded into a beam by modulating the position and width of a binary amplitude grating used as a hologram. By realizing such holograms on a DMD the creation of HG modes, LG modes, OAM vortex mode or any angular mode may be realized. Furthermore, by performing switching of the generated modes at a very high speed, information may be encoded within the helicity&#39;s that are dynamically changing to provide a new type of helicity modulation. Spatial modes may be generated by loading computer-generated holograms onto a DMD. These holograms can be created by modulating a grating function with 20 micro mirrors per each period. 
     Rather than just generating an OAM beam  11313  having only a single OAM value included therein, multiple OAM values may be multiplexed into the OAM beam in a variety of manners as described herein below. The use of multiple OAM values allows for the incorporation of different information into the light beam. Programmable structured light provided by the DLP allows for the projection of custom and adaptable patterns. These patterns may be programmed into the memory of the DLP and used for imparting different information through the light beam. Furthermore, if these patterns are clocked dynamically a modulation scheme may be created where the information is encoded in the helicities of the structured beams. 
     Referring now to  FIG. 114 , rather than just having the laser beam  11402  shine on a single hologram multiple holograms  11404  may be generated by the DMD  4410 .  FIG. 114  illustrates an implementation wherein a 4×3 array of holograms  11404  are generated by the DMD  4410 . The holograms  11404  are square and each edge of a hologram lines up with an edge of an adjacent hologram to create the 4×3 array. The OAM values provided by each of the holograms  11404  are multiplexed together by shining the beam  11402  onto the array of holograms  11404 . Several configurations of the holograms  11404  may be used in order to provide differing qualities of the OAM beam  11313  and associated modes generated by passing a light beam through the array of holograms  11404 . 
     Referring now to  FIG. 115  there is illustrated an alternative way of multiplexing various OAM modes together. An X by Y array of holograms  11502  has each of the hologram  11502  placed upon a black (dark) background  11504  in order to segregate the various modes from each other. In another configuration illustrated in  FIG. 116 , the holograms  11602  are placed in a hexagonal configuration with the background in the off (black) state in order to better segregate the modes. 
       FIG. 117  illustrates yet another technique for multiplexing multiple OAM modes together wherein the holograms  11702  are cycled through in a loop sequence by the DMD  11310 . In this example modes T 0 -T 11  are cycled through and the process repeats by returning back to mode T 0  This process repeats in a continuous loop in order to provide an OAM twisted beam with each of the modes multiplex therein. 
     In addition to providing integer OAM modes using holograms within the DMD, fractional OAM modes may also be presented by the DMD using fractional binary forks as illustrated in  FIG. 118 .  FIG. 118  illustrates fractional binary forks for generating fractional OAM modes of 0.25, 0.50, 0.75, 1.25, 1.50 and 1.75 with a light beam. 
     Referring now to  FIGS. 119-132 , there are illustrated the results achieved from various configurations of holograms program within the memory of a DMD.  FIG. 119  illustrates the configuration at  11902  having no hologram separation on a white background producing the OAM mode image  11904 .  FIG. 120  uses a configuration  12002  consisting of circular holograms  11904  having separation on a white background. The OAM mode image  11906  that is provided therefrom is also illustrated. Bright mode separation yields less light and better mode separation. 
       FIG. 121  illustrates a configuration  12102  having square holograms with no separation on a black background. The configuration  12102  generates the OAM mode image  12104 .  FIG. 122  illustrates the configuration of circular holograms (radius˜256 pixels) that are separated on a black background. This yields the OAM mode image  12204 . Dark mode separation yields more light in the OAM image  12204  and has slightly better mode separation. 
       FIG. 123  illustrates a configuration  12302  having a bright background and circular hologram (radius˜256 pixels) separation yielding an OAM mode image  12304 .  FIG. 124  illustrates a configuration  12402  using circular holograms (radius˜256 pixels) having separation on a black background to yield the OAM mode image  12404 . The dark mode separation yields more light and has a slightly worse mode separation within the OAM mode images. 
       FIG. 125  illustrates a configuration  12502  including circular holograms (radius˜256 pixels) in a hexagonal distribution on a bright background yielding an OAM mode image  12504 .  FIG. 126  illustrates at  12602  small circular holograms (radius˜256 pixels) in a hexagonal distribution on a bright background that yields and OAM mode image  12604 . The larger holograms with brighter backgrounds yield better OAM mode separation images. 
     Referring now to  FIG. 127 , there is illustrated a configuration  12702  of circular holograms (radius˜256 pixels) in a hexagonal distribution on a dark background with each of the holograms having a radius of approximately 256 pixels. This configuration  12702  yields the OAM mode image  12704 .  FIG. 128  illustrates the use of small holograms (radius˜256 pixels) having a radius of approximately 190 pixels arranged in a hexagonal distribution on a black background that yields the OAM mode image  12804 . Larger holograms (radius of approximately 256 pixels) having a dark background yields worse OAM mode separation within the OAM mode images. 
       FIG. 129  illustrates a configuration  12902  of small holograms (radius of approximately 190 pixels) in a hexagonal separated distribution on a dark background that yields the OAM mode image  12904 .  FIG. 130  illustrates a configuration  13002  of small holograms (radius˜256 pixels) in a hexagonal distribution that are close together on a dark background that yields the OAM mode image  13004 . The larger dark boundaries ( FIG. 129 ) yield worse OAM mode image separation than a smaller dark boundary. 
       FIG. 131  illustrates a configuration  13102  of small holograms (radius˜256 pixels) in a separated hexagonal configuration on a bright background yielding OAM mode image  13104 .  FIG. 63  illustrates a configuration  6302  of small holograms (radius˜256 pixels) more closely spaced in a hexagonal configuration on a bright background yielding OAM mode image  6304 . The larger bright boundaries ( FIG. 131 ) yield a better OAM mode separation. 
     Additional illustrations of holograms, namely reduced binary holograms are illustrated in  FIGS. 133-136 .  FIG. 133  illustrates reduced binary holograms having a radius equal to 100 micro mirrors and a period of 50 for various OAM modes. Similarly, OAM modes are illustrated for reduced binary for holograms having a radius of 50 micro mirrors and a period of 50 ( FIG. 134 ); a radius of 100 micro mirrors and a period of 100 ( FIG. 135 ) and a radius of 50 micro mirrors and a period of 50 ( FIG. 136 ). 
     The illustrated data with respect to the holograms of  FIGS. 119-136  demonstrates that full forked gratings yield a great deal of scattered light. Finer forked gratings yield better define modes within OAM images. By removing unnecessary light from the hologram (white regions) there is a reduction in scatter. Holograms that are larger and have fewer features (more dark zones) having a hologram diameter of 200 micro mirrors provide overlapping modes and strong intensity. Similar configurations using 100 micro mirrors also demonstrate overlapping modes and strong intensity. Smaller holograms having smaller radii between 100-200 micro mirrors and periods between 50 and 100 generated by a DLP produce better defined modes and have stronger intensity than larger holograms with larger radii in periods. Smaller holograms having more features (dark zones with hologram diameters of 200 micro mirrors provide well-defined modes with strong intensity. However, hundred micro mirror diameter holograms while providing well-defined modes provide weaker intensity. Thus, good, compact hologram sizes are between 100-200 micro mirrors with zone periods of between 50 and 100. Larger holograms have been shown to provide a richer OAM topology. 
     Referring now also to  FIG. 137  there are illustrated additional methods of multimode OAM generation by implementing multiple holograms within a MEMs device.  FIG. 138  illustrates binary spiral holograms. 
     Referring now to  FIGS. 139 and 140 , there are illustrated a block diagram of a circuit for generating a muxed and multiplexed data stream containing multiple new Eigen channels ( FIG. 139 ) for transmission over a communications link (free space, fiber, RF, etc.), and a flow diagram of the operation of the circuit ( FIG. 140 ). Multiple data streams  13902  are received at step  14002  and input to a modulator circuit  13904 . The modulator circuit  13904  modulates a signal with the data stream at step  14004  and outputs these signals to the orthogonal function circuit  13906 . The orthogonal function circuit  13906  applies a different orthogonal function to each of the data streams at step  14006 . These orthogonal functions may comprise orbital angular momentum functions, Hermite Gaussian functions, Laguerre Gaussian functions, prolate spheroidal functions, Bessel functions or any other types of orthogonal functions. Each of the data streams having an orthogonal function applied thereto are applied to the mux circuit  13098 . The mux circuit  13098  performs a spatial combination of multiple orthogonal signals onto a same physical bandwidth at step  14008 . Thus, a single signal will include multiple orthogonal data streams that are all located within the same physical bandwidth. A plurality of these muxed signals are applied to the multiplexing circuit  13910 . The multiplexing circuit  13910  multiplexes multiple muxed signals onto a same frequency or wavelength at step  14010 . Thus, the multiplexing circuit  13910  temporally multiplexes multiple signals onto the same frequency or wavelength. The muxed and multiplexed signal is provided to a transmitter  13912  such that the signal  13914  may be transmitted at step  14012  over a communications link (Fiber, FSO, RF, etc.). 
     Referring now to  FIGS. 141 and 142 , there is illustrated a block diagram ( FIG. 141 ) of the receiver side circuitry and a flow diagram ( FIG. 142 ) of the operation of the receiver side circuitry associated with the circuit of  FIG. 139 . A received signal  14102  is input to the receiver  14104  at step  14202 . The receiver  14014  provides the received signal  14102  to the de-multiplexer circuit  14106 . The de-multiplexer circuit  14106  separates the temporally multiplexed received signal  14102  into multiple muxed signals at step  14204  and provides them to the de-mux circuit  14108 . As discussed previously with respect to  FIGS. 139 and 140 , the de-multiplexer circuit  14106  separates the muxed signals that are temporally multiplexed onto a same frequency or wavelength. The de-mux circuit  14108  separates (de-muxes) the multiple orthogonal data streams at step  14206  from the same physical bandwidth. The multiple orthogonal data streams are provided to the orthogonal function circuit  14110  that removes the orthogonal function at step  14208 . The individual data streams may then be demodulated within the demodulator circuit  14112  at step  14210  and the multiple data streams  14114  provided for use. 
     Thus, the above described process enables multiple data streams to be first placed within a same physical bandwidth to create a muxed signal of orthogonal data streams. Multiple of these muxed signals may then be multiplexed onto a same frequency or wavelength in order to provide more information on a same communications link. Each orthogonal function within the muxed signals that are then multiplexed together represents a new Eigen channels that may carry a unique information stream thus greatly increasing the amount of data which may be transmitted over the communications link. As described here and above the communications link may comprise free space optical, fiber, RF or any other communication structure. The manner for muxing and multiplexing the data may also use any of the processing techniques described herein above. 
     Optical communications may be carried out over an optical fiber using optical signals processed with orthogonal functions such as Hermite-Gaussian functions and Laguerre-Gaussian functions as described hereinabove in order to improve system bandwidth. Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) signals have three important properties and advantages and one major disadvantage. The advantages include the ability to form two complete families of exact and orthogonal solutions of the paraxial wave equations. Another advantage is that the HG and LG signals are transverse eigenmodes of stable resonators. Finally, the HG and LG signals do not change shape on propagation and provide stable modes of propagation for signals. The disadvantage of HG and LG signals is that the eigenmodes coupled to one another after a long distance of propagation. However, for short distance applications such as intra-data and inter-data center connectivity, and front haul applications or back haul applications, both LG and HG signals provide good solutions within these applications. 
     Referring now to  FIG. 143 , there are illustrated various types of orthogonal functions that may be utilized in optical fiber transmissions. Hermite-Gaussian functions  14302  use Hermite-Gaussian (HG) signals that are derived from the expansion of Laplacian equations in a rectangular coordinate system  14304  with paraxial approximation in the z-direction. Laguerre-Gaussian functions  14306  use Laguerre-Gaussian (LG) signals that are derived from the expansion of Laplacian equations in a cylindrical coordinate system with a paraxial approximation in the Z-direction. Depending on the geometrical symmetries of the problem, for example, one mode would propagate better than the others if they are the natural symmetry of the chosen geometry. Both HG and LG signals have been discussed in corresponding U.S. patent application Ser. No. 14/882,085, filed Oct. 13, 2015, entitled APPLICATION OF ORBITAL ANGULAR MOMENTUM TO FIBER, FSO AND RF, which is incorporated herein by reference in its entirety. The theoretical connection between the two modes has also been studied on a Poincare sphere in U.S. patent application Ser. No. 14/818,050, entitled MODULATION AND MULTIPLE ACCESS TECHNIQUE USING ORBITAL ANGULAR MOMENTUM, which is incorporated herein by reference in its entirety. 
     A technique for further improving system bandwidth in optical fiber transmissions involves the use of input signals processed using an elliptical coordinate system that is transmitted over an elliptical core fiber. Elliptical core fibers may be used with Ince-Gaussian spatial orthogonal wave fronts for spatial modulation. Ince-Gaussian functions  14310  use Ince-Gaussian (IG) signals that are derived from the expansion of Laplacian equation in an elliptical coordinate system with paraxial approximation in the Z-direction. The Ince-Gaussian functions  14310  may be applied in each of the same manners and applications as the Hermite-Gaussian functions, Laguerre-Gaussian functions and other orthogonal functions describe previously herein. 
     Ince-Gaussian signals comprise a third family of exacting orthogonal solutions of the paraxial wave equation in the elliptical coordinate system. These new Eigen functions have natural resonating modes in stable resonators and constitute the exact and continuous transition modes between HG and LG eigenmodes. The transverse distribution of these fields is described by new Eigen modes and HG and LG have a complete set of solutions of the paraxial wave equations, such that any paraxial field can be expressed as a superposition of these eigenmodes. The generation of multiple modes of these Eigen functions, the muxing of them, their demuxing, and detection characteristics of the eigenmodes have potential uses of such modes in a commercial setting. These new eigenmodes can approximate LG or HG as a parameter of the Wigner transform going to zero or infinity. 
     Referring now to  FIG. 144 , there is illustrated a transmitter  14402  and receiver  14404  transmitting signals over an elliptical core fiber  14406 . In a fiber  14406  with an elliptical core, such elliptical eigenmodes, as provided by Ince-Gaussian functions, can propagate further before coupling with one another and more of these modes can be muxed together within an elliptical core fiber as compared to other modes such as LG modes or HG modes. A passively specially designed lens  14408  having a diamond cut such that plane waves  14410  incoming at different angles can be converted to elliptical modes  14412  and muxed together before leaving the lens  14408 . The elliptical modes  14412  can be demuxed with another specially designed lens  14414  that demultiplexes and converts the elliptical modes  14412  back into plane waves  14410 . Thus, each orthogonal eigenmode acts as an independent channel on one wavelength over the elliptical fiber  14406 . Therefore, techniques such as CWDM (course wavelength division multiplexing) or DWDM (dense wavelength division multiplexing) are still applicable as such eigenmodes are muxed on one particular wavelength and therefore, can create multiple independent channels on one specific wavelength and more wavelengths can be aggregated together to increase throughput or capacity of the system. 
     Referring now more particularly to  FIG. 145 , there is generally illustrated the process for generating an Ince-Gaussian process signal for transmission on elliptical fiber. An input signal/signals  14502  comprises the data or information that is to be transmitted to the optical processing circuitry  14504  for processing. The optical processing circuitry  14504 , in one example, comprises the circuitry described in corresponding U.S. patent application Ser. No. 14/882,085. While the disclosure of the &#39;085 application describes applying either rectangular coordinates that yield Hermite-Gaussian signals or cylindrical coordinates providing Laguerre-Gaussian signals, the optical processing circuitry  1504  of the present system processes the input signals  15402  into an optical signal using an elliptical coordinate system. 
     The optical processing circuitry  14504  expands the Laplacian of the wave equation in elliptical contexts and solves them as separable solutions for each coordinate. The angular solutions produce new functions called Mathieu and modified Mathieu functions. The optical processing circuitry  14504  performs a paraxial approximation to produce new differential equations and solutions for the paraxial equations. The process signals may be generated using spatial light modulators (SLM&#39;s) or other techniques such as those described in U.S. patent application Ser. No. 14/882,085 to transmit the information over an elliptical core fiber  14508 . The optical processing circuitry  14504  may perform muxing and multiplexing process to combine IG processed signals. A multiplexing technique implemented by the optical processing circuitry  14504  can sequentially combine different Ince Gaussian orthogonal modes in an elliptical core fiber with a higher multiplier effect that achieves a higher number of orthogonal Eigen channels with distinct modal combinations. 
     Ince-Gaussian (IG) beams generated within the optical processing circuitry  14504  are the solutions of paraxial beams in an elliptical coordinate system. IG beams are the third kind of orthogonal Eigen states and can probe the chirality structures of samples. Since IG modes have a preferred symmetry (long axis versus short axis) this enables it to probe chirality better than Laguerre-Gaussian or Hermite-Gaussian modes. This enables the propagation of more IG modes within an elliptical core fiber than Laguerre-Gaussian modes or Hermite-Gaussian modes. 
     The solution of the Ince-Gaussian equations is performed in an elliptical coordinate system. The wave equation can be represented as a Helmholtz equation in Cartesian coordinates as follows:
 
(∇ 2   +k   2 ) E ( x,y,z )=0
 
E(x, y, z) is complex field amplitude which can be expressed in terms of its slowly varying envelope and fast varying part in z-direction.
 
 E ( x,y,z )=ψ( x,y,z ) e   jkz  
 
     A Paraxial Wave approximation may be determined by substituting our assumption in the Helmholtz Equation.
 
(∇ 2   +k   2 )ψ, e   jkz =0
 
     
       
         
           
             
               
                 
                   
                     δ 
                     2 
                   
                   ⁢ 
                   ψ 
                 
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     x 
                     2 
                   
                 
               
               + 
               
                 
                   
                     δ 
                     2 
                   
                   ⁢ 
                   ψ 
                 
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     y 
                     2 
                   
                 
               
               + 
               
                 
                   
                     δ 
                     2 
                   
                   ⁢ 
                   ψ 
                 
                 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     z 
                     2 
                   
                 
               
               - 
               
                 j 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 k 
                 ⁢ 
                 
                   δψ 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     z 
                   
                 
               
             
             = 
             0 
           
         
       
     
     A slowly varying envelope approximation is made as follows: 
                          δ   2     ⁢   ψ       δ   ⁢           ⁢     z   2              ⪡              δ   2     ⁢   ψ       δ   ⁢           ⁢     x   2                ,              δ   2     ⁢   ψ       δ   ⁢           ⁢     y   2              ,     2   ⁢           ⁢   k   ⁢          δψ     δ   ⁢           ⁢   z                                ∇   t   2     ⁢   ψ     +     j   ⁢           ⁢   2   ⁢           ⁢   k   ⁢     δψ     δ   ⁢           ⁢   z           =   0         
This comprises a Paraxial wave equation.
 
     The elliptical-cylindrical coordinate system is defined as shown in  FIG. 146 . 
     
       
         
           
             x 
             = 
             
               acosh 
               ⁢ 
               
                   
               
               ⁢ 
               ξ 
               ⁢ 
               
                   
               
               ⁢ 
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               η 
             
           
         
       
       
         
           
             y 
             = 
             
               a 
               ⁢ 
               
                   
               
               ⁢ 
               sinh 
               ⁢ 
               
                   
               
               ⁢ 
               ξ 
               ⁢ 
               
                   
               
               ⁢ 
               sin 
               ⁢ 
               
                   
               
               ⁢ 
               η 
             
           
         
       
       
         
           
             
               ξ 
               ∈ 
               
                 ( 
                 
                   0 
                   , 
                   ∞ 
                 
                 ) 
               
             
             , 
             
               η 
               ∈ 
               
                 ( 
                 
                   0 
                   , 
                   
                     2 
                     ⁢ 
                     π 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             a 
             = 
             
               
                 
                   f 
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   f 
                   ⁡ 
                   
                     ( 
                     z 
                     ) 
                   
                 
               
               = 
               
                 
                   
                     f 
                     0 
                   
                   ⁢ 
                   
                     w 
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
                 
                   w 
                   0 
                 
               
             
           
         
       
     
     Curves of constant value of ξ trace confocal ellipses as shown in  FIG. 147 . 
     
       
         
           
             
               
                 
                   x 
                   2 
                 
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     cosh 
                     2 
                   
                   ⁢ 
                   ξ 
                 
               
               + 
               
                 
                   y 
                   2 
                 
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     sinh 
                     2 
                   
                   ⁢ 
                   ξ 
                 
               
             
             = 
             
               1 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 Ellipse 
                 ) 
               
             
           
         
       
     
     A constant value of η give confocal hyperbolas as shown in  FIG. 148 . 
     
       
         
           
             
               
                 
                   x 
                   2 
                 
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     cos 
                     2 
                   
                   ⁢ 
                   η 
                 
               
               - 
               
                 
                   y 
                   2 
                 
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     sin 
                     2 
                   
                   ⁢ 
                   η 
                 
               
             
             = 
             
               1 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 hyperbola 
                 ) 
               
             
           
         
       
     
     An elliptical-cylindrical coordinate system may then be defined in the following manner: 
               ∇   t   2     ⁢     =         1     h   ξ   2       ⁢       δ   2       δξ   2         +       1     h   η   2       ⁢       δ   2       δη   2                   
where h ξ , h η  are scale factors
 
     
       
         
           
             
               h 
               ξ 
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       δξ 
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         y 
                       
                       δξ 
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               h 
               η 
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       δη 
                     
                     ) 
                   
                   2 
                 
                 + 
                 
                   
                     ( 
                     
                       
                         δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         y 
                       
                       δη 
                     
                     ) 
                   
                   2 
                 
               
             
           
         
       
       
         
           
             
               h 
               ξ 
             
             = 
             
               
                 h 
                 η 
               
               = 
               
                 a 
                 ⁢ 
                 
                   
                     
                       
                         sinh 
                         2 
                       
                       ⁢ 
                       ξ 
                     
                     + 
                     
                       
                         sin 
                         2 
                       
                       ⁢ 
                       η 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               ∇ 
               t 
               2 
             
             ⁢ 
             
               = 
               
                 
                   1 
                   
                     
                       a 
                       2 
                     
                     ⁢ 
                     
                       sinh 
                       2 
                     
                     ⁢ 
                     ξ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     η 
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         δ 
                         2 
                       
                       
                         δξ 
                         2 
                       
                     
                     + 
                     
                       
                         δ 
                         2 
                       
                       
                         δη 
                         2 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     The solution to the paraxial wave equations may then be made in elliptical coordinates. Paraxial Wave Equation in Elliptic Cylindrical co-ordinates are defined as: 
     
       
         
           
             
               
                 
                   1 
                   
                     
                       a 
                       2 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           sinh 
                           2 
                         
                         ⁢ 
                         ξ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         η 
                       
                       ) 
                     
                   
                 
                 ⁢ 
                 
                   ( 
                   
                     
                       
                         
                           δ 
                           2 
                         
                         ⁢ 
                         ψ 
                       
                       
                         δξ 
                         2 
                       
                     
                     + 
                     
                       
                         
                           δ 
                           2 
                         
                         ⁢ 
                         ψ 
                       
                       
                         δη 
                         2 
                       
                     
                   
                   ) 
                 
               
               - 
               
                 j 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 k 
                 ⁢ 
                 
                   δψ 
                   
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     z 
                   
                 
               
             
             = 
             0 
           
         
       
     
     Assuming separable solution as modulated version of fundamental Gaussian beam. 
               IG   ⁡     (     r   ~     )       =       E   ⁡     (   ξ   )       ⁢     N   ⁡     (   η   )       ⁢     exp   ⁡     (     jZ   ⁡     (   z   )       )       ⁢       ψ   GB     ⁡     (     r   ~     )                       where   ⁢           ⁢       ψ   GB     ⁡     (     r   ~     )         =         w   0       w   ⁡     (   z   )         ⁢     exp   ⁡     [       -       r   2         w   2     ⁡     (   z   )           +     j   ⁢       kr   2       2   ⁢           ⁢     R   ⁡     (   z   )             -     j   ⁢           ⁢       ψ   GS     ⁡     (   z   )           ]               
E, N &amp; Z are real functions. They have the same wave-fronts as ψ GB  but different intensity distribution.
 
     Separated differential equations are defined as: 
                     d   2     ⁢   E       d   ⁢           ⁢     ξ   2         -     ϵ   ⁢           ⁢   sinh   ⁢           ⁢   2   ⁢   ξ   ⁢     dE     d   ⁢           ⁢   ξ         -       (     a   -     p   ⁢           ⁢   ϵcosh2ξ       )     ⁢   E       =   0                       d   2     ⁢   N       d   ⁢           ⁢     η   2         -     ϵ   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢   η   ⁢     dN     d   ⁢           ⁢   η         -     (     a   =     p   ⁢           ⁢   ϵ   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢   η       )       =       0   ⁢     
     -       (         z   2     +     z   r   2         z   r       )     ⁢     dZ   dz         =   p           
where a and p are separation constants
 
     
       
         
           
             ϵ 
             = 
             
               
                 
                   f 
                   0 
                 
                 ⁢ 
                 
                   w 
                   0 
                 
               
               
                 w 
                 ⁡ 
                 
                   ( 
                   z 
                   ) 
                 
               
             
           
         
       
     
     The even solutions for the Ince-Gaussian equations are: 
                 IG   pm   e     ⁡     (       r   ~     ,   ϵ     )       =         Cw   0       w   ⁡     (   z   )         ⁢       C   p   m     ⁡     (       j   ⁢           ⁢   ξ     ,   ϵ     )       ⁢       C   p   m     ⁡     (     η   ,   ϵ     )       ⁢     exp   ⁡     (     -       r   2         w   2     ⁡     (   z   )           )       ×   exp   ⁢           ⁢     j   ⁡     (     kz   +       kr   2       2   ⁢           ⁢     R   ⁡     (   z   )           -       (     p   +   1     )     ⁢       ψ   GS     ⁡     (   z   )           )               
The frequency of the even Ince-Polynomials are illustrated in  FIGS. 149A and 149B  and the modes and their phases are illustrated in  FIG. 150 .
 
     The odd solutions for the Ince-Gaussian equations are: 
                 IG   pm   o     ⁡     (       r   ~     ,   ϵ     )       =         sw   0       w   ⁡     (   z   )         ⁢       S   p   m     ⁡     (       j   ⁢           ⁢   ξ     ,   ϵ     )       ⁢       S   p   m     ⁡     (     η   ,   ϵ     )       ⁢     exp   ⁡     (     -       r   2         w   2     ⁡     (   z   )           )       ×   exp   ⁢           ⁢     j   ⁡     (     kz   +       kr   2       2   ⁢           ⁢     R   ⁡     (   z   )           -       (     p   +   1     )     ⁢       ψ   GS     ⁡     (   z   )           )               
The frequency of the odd Ince Polynomials are illustrated in  FIGS. 151A and 151B  and the modes and their phases are illustrated in  FIG. 152 . Processing of the input signal  14502  by the optical processing circuitry  14504  using an elliptical coordinate system creates a unique item function that provides improved throughput characteristics when utilized upon an elliptical core optical fiber  14508  when transmitted from an optical transmitter  14506  utilizing the processed signal.
 
     Referring now back to  FIG. 145 , the optical transmitter  14506  transmits the generated Ince-Gaussian beam over the elliptical fiber  14508 . The signals processed according to the elliptical coordinate system will provide better transmission characteristics over the elliptical core optical fiber  14508  allowing greater data throughput. The input signal  14502  processed according to the elliptical core system will have a distinct index of refraction profile arising from processing using the elliptical coordinate system. The signals transmitted over the elliptical fiber  14508  are processed and demodulated using a receiver optical processing circuitry  14510 . The receiver optical processing circuitry  14510  removes the Eigen function applied using the elliptical coordinate system to reconstruct an output signal  14512  that comprises the originally received input signals. 
     In an alternative embodiment, the transmitter optical processing circuitry  14504  and the receiver optical processing circuitry  14510  may utilize the techniques discussed in U.S. patent application Ser. No. 14/882,085 to generate the Laguerre-Gaussian, Hermite-Gaussian or Ince-Gaussian processed signals that are transmitted over an elliptical core fiber to improve data bandwidth. 
     Referring now to  FIG. 153 , the elliptical core fiber  14508  has a parabolic index profile and is weakly guided. These are boundary conditions that are input to the wave equation (Laplacian). Also, the elliptical fiber  14508  includes stress rods  15304  on either size of the semi-major axis (the longer elliptical axis)  15305  along the semi-minor axis (the shorter elliptical axis)  15306  that may be taken into account in the Laplacian wave equation. The elliptical fiber  14508  may be produced using the preform techniques similar to those disclosed in U.S. patent application Ser. No. 15/430,981, filed Feb. 13, 2017, entitled SYSTEM AND METHOD FOR PRODUCING VORTEX FIBER, which is incorporated herein by reference in its entirety or using the fiber forming techniques disclosed in U.S. patent application Ser. No. 14/882,085, filed Oct. 13, 2015, entitled APPLICATION OF ORBITAL ANGULAR MOMENTUM TO FIBER, FSO AND RF, which is incorporated herein by reference in its entirety. 
     An elliptical core fiber may comprise a few mode fiber as illustrated in  FIG. 154 . The elliptical core-few mode fiber (EC-FMF)  15402  includes an elliptical core  15404  within the center of the fiber that has a major axis of approximately 15 μm and a minor axis of approximately 10 μm. The core shape is described by the ovality parameter: 
     
       
         
           
             o 
             = 
             
               
                 2 
                 ⁢ 
                 
                   ( 
                   
                     a 
                     - 
                     b 
                   
                   ) 
                 
               
               
                 a 
                 + 
                 b 
               
             
           
         
       
     
     Here,
 
 a= 15 μm
 
 b= 10 μm
 
     Giving ovality,
 
 o= 40%
 
     The two dark regions  15406  along the minor axis are stress-applying parts (stress and rods) for creating asymmetric stress during the fiber draw process to make the core elliptical. 
     Ovality may also be written o→X=2(R a −R b )/(R a +R b ) and is a measure of how “out of round” cylindrical part is. 
     Referring now to  FIG. 155 , there are illustrated intensity diagrams for various different types of beam topologies within elliptical core fibers. Row  15502  illustrates linearly polarized signals transmitted through an elliptical core fiber for various modes. Row  15504  illustrates intensity diagrams for Laguerre-Gaussian signals for LG 01  mode, LG 10  mode and LG 01  mode. Row  15506  illustrates the intensity diagrams for Hermite-Gaussian signals for HG 01  mode, HG 10  mode and HG 20  mode. Finally, row  15508  illustrates the intensity diagrams for Ince-Gaussian beams in IG 11  mode, IG 20  mode and IG 22  mode. 
     A mode crosstalk matrix provides an illustration of the relative amount of light scattered into adjacent modes that were not launched into a fiber. Referring now to  FIG. 156  there is illustrated a measurement technique for generating a mode crosstalk matrix.  FIG. 157  is a flow diagram illustrating the steps of the process. A 980 nm laser  15602  generates a laser signal that is transmitted through a first lens  15604  A first spatial light modulator  15606  utilizes a hologram  15608  to generate a first Ince-Gaussian, Hermite-Gaussian, LaGuerre-Gaussian or other type of orthogonal function processed beam of a first mode having a particular intensity diagram  15610 . The generated beam from the SLM  15606  passes through a series of lenses  15612  that focuses and launches the generated beam having a particular mode onto an elliptical core few mode fiber  15614  at step  15702  (see  FIG. 157 ). While the present embodiment discloses transmitting the signal through an elliptical core few mode fiber, other types of fibers and communications links may also be tested using the described process to populate a mode crosstalk matrix. 
     After passing through the fiber  15614 , the output of the fiber is monitored at step  15704 . The output will comprise a superposition of the various output mode components. Within the monitoring process, the output signal passes through a lens  15616  to a second spatial light modulator (SLM)  15618 . For every output mode component to be measured, the SLM  15618  will imprint at step  15706  an inverse of the modes face front onto the output beam from the fiber  15614 . The output of the SLM  15618  next passes through a further series of lenses  15620  before being input to a spatial filter  15622 . The output mode components from the SLM  15618  to be measured will have a planar face front after the imprinting with the inverse of the Ince-Gaussian, Hermite-Gaussian, LaGuerre-Gaussian or other type of orthogonal function processed beam. The spatial filter  15622  passes the plane wave component and blocks other output mode components within the signal from the SLM  15618  at step  15710 . The relative amount of power through the spatial filter  15622  is the amount of the output mode component to be measured at step  15712  by the mode power measurement circuitry  15644 . By interrogating each of the adjacent mode components, the mode power measurement circuitry  15624  populates the columns of the mode crosstalk matrix for the fiber or communications link under test for a particular row associated with the mode being transmitted. 
     Initially, the mode output power for a first of the adjacent modes measured at step  15712  is used to populate a first column of the row associated with the row associated with the mode under test. Each of the columns is associated with one of the adjacent mode components of the mode being transmitted. After storage of the power value for the adjacent mode in the mode crosstalk matrix, inquiry step  15714  determines if there are adjacent modes for test. If so, the measurement circuitry  15624  proceeds to the next adjacent mode at step  15716  and measures at step  15712  the output power for the next adjacent mode at step  15712 . After population of the matrix row associated with the current mode being transmitted through the fiber or communications link is determined to be complete at inquiry step  15714 , inquiry step  15718  determines if a new mode for test exists. If so, a next mode may be launched at step  15720  into the fiber or communications link under test to complete a next row of the crosstalk matrix. The output associated with this mode is then monitored at step  15704 . The above process is repeated for each mode to be tested. 
       FIG. 158  illustrates a generated single row of the mode crosstalk matrix. The mode HG 11  has been launched into the fiber  15614  (or communications link). The power output is measured for various other adjacent modes to the transmitted mode and the values associated with the adjacent modes are stored in the columns within the row. Thus, the selectively measured output for mode HG 00  is −23.7, for mode HG 01  is −21.2, for mode HG 10  is −18.8, for mode HG 02  is −17.1 and for mode HG 20  is −20.9. 
     Referring now to  FIG. 159 , there is illustrated the results for the comparison of each of the selectively excited modes  15902  that are calculated from the first SLM  15606 . Each of the modes after passing through the fiber  15614  is measured by set of masks (holograms)  15904  using the second SLM  15618  to yield an associated set of intensity images  15906 . For each of the intensity images  15906 , the optical power of the light coupled into the fiber is measured using the mode power measurement circuitry  15644 . These are represented for the various modes as indicated generally at  15908  from a level of −28.1 dB to −14.5 dB. 
     Each of the selectively measured outputs  15908  that are measured may be used to populate a mode crosstalk matrix using Hermite-Gaussian modes as illustrated in  FIG. 160 . The mode crosstalk matrix  16002  generates the power values for adjacent modes to a transmitted mode for provided input values according to dB→10 log 10  P ij /P ii  in each row and are measured for each selectively measured output by the spatial light modulator  15618  for a given selectively excited input to the fiber  15614  by the SLM  15606 . The rows of the mode crosstalk matrix  16002  comprise the modes selectively excited by the spatial light modulator  15606 . The columns of the mode crosstalk matrix  16002  comprise the modes selectively excited by the spatial light modulator  15618 . A mode crosstalk matrix for Laguerre-Gaussian modes, linearly polarized modes and Ince-Gaussian modes are illustrated in  FIGS. 161, 162 and 163 , respectively. 
     Once a mode crosstalk matrix such as that described herein above has been determined, the results illustrated in the matrix can be used to determine whether particular modes within a matrix can be multiplexed together. The mode crosstalk matrix is a correlation matrix. Thus, any entries having a low value means that there is a low correlation and thus the associated adjacent mode may be multiplexed with the transmitted mode. Entries with high values have high correlation and may not be multiplexed due to high interference. The diagonal elements have a high correlation of 100%. Since the signals are being represented in dB, the matrix is logarithmic and 100% correlation is represented as 0 dB. Thus, entries have a low, predetermined dB value are suitable for muxing. 
     When an adjacent mode is illustrated to fall below a particular predetermined level (for example, 20 dB, however other predetermined thresholds may be utilized), the mode or modes that fall below this predetermined level may be selected as options for being multiplexed with the transmitted mode. The reason being that if there is not a too high level of crosstalk between a launched mode and an adjacent mode than multiplex, then multiplexing the launched and adjacent modes together is feasible. If the crosstalk exceeds the predetermined level, then multiplexing the launched and adjacent modes is not feasible to the excessive crosstalk that would cause channel interference. The lower the value of the entries within the mode crosstalk matrix, the more modes that can be multiplexed together. Thus, a greater number of lower valued modes may be multiplexed together than the number of higher valued entries associated with modes. 
     Referring now to  FIG. 164 , there is illustrated a flow diagram describing the process for selecting modes for multiplexing together utilizing a mode crosstalk matrix. Initially, the mode crosstalk matrix is created at step  16402  in the manner described hereinabove. Once the matrix is generated for all of the transmitted and adjacent modes, a determination is made at step  16404  of the modes falling below the predetermined threshold level. The selected modes may then be multiplexed together at step  16406  to create the signals for transmission over the communication links between transmitting and receiving units. 
     In addition to those various orthogonal function mode techniques discussed hereinabove to which Ince Gaussian functions may be applied in a manner similar to that of Hermite Gaussian, Laguerre Gaussian or other types of orthogonal functions, Ince Gaussian orthogonal modes/functions may be applied in a number of other manners. The mode crosstalk matrix with Hermite-Gaussian, Laguerre-Gaussian and Ince-Gaussian spatial orthogonal modes may also be used with WDM and DWDM to determine a number of modes that can be multiplexed together based upon the determined crosstalk as indicated by the mode crosstalk matrix. These further include the use of Ince Gaussian orthogonal modes in elliptical core fibers to perform phase estimation in carer recovery, to perform symbol. Estimation and clock recovery, to perform decision directed carrier recovery where decoded signals are compared with the closest constellation points or for dealing with amplifier nonlinearity. Ince Gaussian orthogonal modes within an elliptical core fiber may also be used for adaptive power control, adaptive variable symbol rate and within adaptive equalization techniques. 
     It will be appreciated by those skilled in the art having the benefit of this disclosure that this system and method for transmissions using elliptical core fibers provides an improved method for optical signal transmission within a fiber. It should be understood that the drawings and detailed description herein are to be regarded in an illustrative rather than a restrictive manner, and are not intended to be limiting to the particular forms and examples disclosed. On the contrary, included are any further modifications, changes, rearrangements, substitutions, alternatives, design choices, and embodiments apparent to those of ordinary skill in the art, without departing from the spirit and scope hereof, as defined by the following claims. Thus, it is intended that the following claims be interpreted to embrace all such further modifications, changes, rearrangements, substitutions, alternatives, design choices, and embodiments.