Patent Publication Number: US-2011069969-A1

Title: Waveguides and devices for enhanced third order nonlinearities in polymer-silicon systems

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to and the benefit of co-pending U.S. provisional patent application Ser. No. 61/068,326, filed Mar. 5, 2008, which application is incorporated herein by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The invention relates to optical waveguides in general and particularly to optical waveguides, including split waveguides, which employ materials, such as polymers, having large nonlinear optical characteristics. 
     BACKGROUND OF THE INVENTION 
     The field of nonlinear optics is extremely rich in results, and has been around for many years. Basically the premise of nearly all measurements in the field is that one introduces a sufficiently high power flux (or “fluence,” a term of art) in an optical material, it is often possible to excite nonlinear behavior, meaning that the properties of the material change with the input optical power. This kind of effect is very often described through the use of, for instance. Chi 2  (χ 2 ) and Chi 3  (χ 3 ) which are material dependent constants that describe the strength of two of the relevant nonlinear optical activities of a material. Some nonlinearities, which are material dependent, will work at the full optical frequency, while others are slower. Recently, engineered organic materials have begun to be used for nonlinear optics, because they can be designed to have extremely large χ 2  and χ 3  moments. 
     There is a need for systems and methods that can fully exploit the optical properties of materials that exhibit large χ 2  and χ 3  moments without having to provide excessive amounts of optical power to do so. 
     SUMMARY OF THE INVENTION 
     In one aspect, the invention features an all-optical signal processing device. The all-optical signal processing device comprises an optical input of the all-optical signal processing device configured to receive an optical signal as input; an optical output of the all-optical signal processing device configured to provide a modulated optical signal as output; and an interaction region configured to permit the optical input signal to interact with at least another optical signal to produce an optical output signal. The interaction region comprises a high index contrast waveguide adjacent an insulating surface of a substrate, and a cladding adjacent the high index contrast waveguide, the cladding comprising a material that exhibits a third-order or higher odd order nonlinear optical coefficient. 
     In some embodiments, the interaction region configured to permit the optical input signal to interact with at least another optical signal is an interaction region configured to permit the optical signal to interact with a portion of the optical signal that is reintroduced so as to interact with itself. 
     In some embodiments, the high index contrast waveguide is a selected one of a ridge waveguide, a rib and a slot waveguide. In some embodiments, the high index contrast slot waveguide has at least two stripes defining the slot; and at least some of the cladding is situated within the slot. In some embodiments, the substrate is a silicon wafer. In some embodiments, the insulating surface is a layer comprising silicon and oxygen. In some embodiments, the substrate is selected from one of silicon-on-insulator (SOI) and silicon-on-sapphire (SOS). In some embodiments, the high index contrast slot waveguide adjacent the insulating surface comprises silicon stripes. In some embodiments, each of the silicon stripes is deliberately doped to attain a desired resistivity. In some embodiments, the slot is less than or equal to 100 nanometers in width. In some embodiments, the optical input comprises an input waveguide for coupling optical radiation into the high index contrast waveguide. 
     In some embodiments, the all-optical signal processing device is a logic gate. In some embodiments, the logic gate is a selected one of an AND gate, an OR gate, a NAND a NOR, and an XOR gate. 
     In some embodiments, the all-optical signal processing device is a selected one of an optical latch and an optical memory. In some embodiments, the all-optical signal processing device is a variable delay line. In some embodiments, the all-optical signal processing device is a self-oscillator. In some embodiments, the all-optical signal processing device is a multiplexer. In some embodiments, the all-optical signal processing device is a demultiplexer. In some embodiments, the all-optical signal processing device is a clock multiplier. 
     The foregoing and other objects, aspects, features, and advantages of the invention will become more apparent from the following description and from the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The objects and features of the invention can be better understood with reference to the drawings described below, and the claims. The drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the drawings, like numerals are used to indicate like parts throughout the various views. 
         FIG. 1  is a diagram showing dispersion plots for the fundamental mode (Ex polarized) of exemplary clad and unclad waveguides, shown as effective index vs. wavelength in μm. 
         FIG. 2  is a diagram showing an SEM image of an exemplary ring resonator. 
         FIG. 3  is a diagram showing the normalized transmission of light through the system (and past the ring) in dB, as a function of wavelength detuning in nm for both clad and unclad waveguides, shifted to overlay resonance peaks. 
         FIG. 4  is a diagram showing an exemplary slot waveguide mode profile. 
         FIG. 5  is a diagram showing the effective index vs. free space wavelength in microns for the slot waveguide of  FIG. 4 . 
         FIG. 6  is a diagram showing the device layout of an exemplary slot waveguide. 
         FIG. 7  is a diagram showing an SEM image of a portion of an oval slot waveguide. 
         FIG. 8  is a diagram showing a more detailed SEM image showing the coupling region of an exemplary slot waveguide and an input waveguide. 
         FIG. 9  is a diagram showing the measured transmission spectrum in dB vs. laser wavelength in nm past a high quality factor slot ring resonator. 
         FIG. 10  is a diagram showing the detail of the peak of the transmission spectrum near 1488 nm. 
         FIG. 11  is a diagram showing a shallow angle SEM view of a typical silicon-on-insulator ring resonator and waveguide having a sidewall roughness on the order of 10 nm. 
         FIG. 12  is a diagram of a slot ring resonator directional coupler region, and the associated input waveguide. 
         FIG. 13  is a diagram showing a slot waveguide structure that exhibits subfield stitching errors at the edge of an input waveguide. 
         FIG. 14  is a diagram showing yet another example of a rough wall that is likely to create problems in device fabrication and operation. 
         FIG. 15  is a diagram showing an exemplary high-index segmented waveguide structures, which in the embodiment shown comprises a central waveguide portion with fingers or ridges sticking out to the sides. 
         FIG. 16A  is a diagram that shows a dispersion diagram of both a segmented waveguide and the normal, unsegmented waveguide, taken on a plane parallel to the substrate that on a z plane that intersects the middle of a segment. 
         FIG. 16B  is a diagram that shows modal patterns of the Bloch mode, with contours of |E| plotted, starting at 10% of the max value and with contour increments of 10%. 
         FIG. 16C  is a diagram that shows a plot of modal patterns over four periods of a segmented waveguide on a horizontal plane that intersects the silicon layer halfway through. 
         FIG. 17  is a diagram that shows an exemplary electrical isolator that was constructed and tested, and which provided both a transition from a standard to a slotted waveguide and electrical isolation between the two sides of the slot waveguide. 
         FIG. 18  is a diagram showing the results of a baseline measurement of an EDFA and optical test system in the absence of a test sample. 
         FIG. 19  is a diagram showing the results for the measurement of a first exemplary material having a large value of χ 3 . 
         FIG. 20  is a diagram showing the results for the measurement of a second exemplary material having a large value of χ 3 . 
         FIG. 21  is a diagram that shows a plot of the numerically computed conversion efficiency for the second exemplary material having a large value of χ 3 , in dB vs 1 watt compared to length traveled in waveguide in μm. 
         FIG. 22  is a diagram showing a chemical reaction useful for the synthesis of a chromophore referred to as YLD  124 . 
         FIG. 23  is a four panel diagram that shows details of one embodiment of an optical modulator device, including the geometry of the photodetectors and filters, and including a cross section of the slotted waveguide. 
       Panel A of  FIG. 24  shows the transmission spectrum of detector device  1 , according to principles of the invention. 
       Panel B of  FIG. 24  shows the transmission spectrum of detector device  2 , according to principles of the invention. 
       Panel C of  FIG. 24  shows several curves of current vs. power for three measurement series. 
       Panel D of  FIG. 24  shows the output current as a function of wavelength, overlaid with the transmission spectrum. 
         FIG. 25  is a diagram showing the use of the structures embodying the invention as resonantly enhanced electro-optic modulators, and a result at approximately 6 MHz operating frequency. 
         FIG. 26  is a diagram showing a chemical formula for the chromophore referred to as JSC 1 . 
         FIG. 27  shows a diagram of a Mach-Zehnder modulator with a conventional electrode geometry in top-down view, including top contact, waveguide, and bottom contact layers. 
         FIG. 28  is an isometric three dimensional schematic of a conventional Mach-Zehnder polymer interferometer, showing top contact, waveguide, and bottom contact layers. 
         FIG. 29  is a three dimensional, isometric schematic of a slot-waveguide modulator, showing the slot waveguide, segmentation region and metal contacts. The device illustrated in  FIG. 29  functions by maintaining the two arms of the slot waveguide at differing voltages, creating a strong electric field in the slot. 
         FIG. 30  is a top-down view of a layout of a slot-waveguide based optical modulator of the device in  FIG. 29 . 
         FIG. 31A  shows the optical mode with |E| plotted in increments of 10%, for a mode with propagating power of 1 Watt. 
         FIG. 31B  shows a contour plot of the static electric field for the waveguide of  FIG. 31A  with the field of view slightly enlarged. 
         FIG. 31C  and  FIG. 31D  show analogous data to  FIG. 31A  and  FIG. 31B , respectively, for the most optimal slot waveguide geometry that is presently known to the inventors (corresponding to design #3 in Table 2). 
         FIG. 32A  shows the static voltage potential field distribution due to charging the two electrodes. 
         FIG. 32B  shows the electric field due to the potential distribution. |E| is plotted in increments of 10%. 
         FIG. 33  is a diagram that illustrates the dependence of susceptibility on gap size for several waveguide designs. 
         FIG. 34A  shows a cross section of the segmented, slotted waveguide, with the |E| field plotted in increments of 10% of max value. 
         FIG. 34B  shows a similar plot for the unsegmented waveguide. 
         FIG. 34C  shows a horizontal cross section of the segmented, slotted waveguide in which Re(Ex) is plotted in increments of 20% of max. 
         FIG. 35(   a ) is a diagram of the silicon slot waveguide used in the Mach-Zehnder modulator, according to principles of the invention. 
         FIG. 35(   b ) is an SEM micrograph of a slot waveguide, according to principles of the invention. 
         FIG. 36(   a ) is a diagram of the modulator layout, according to principles of the invention. 
         FIG. 36(   b ) and  FIG. 36(   c ) are two SEM micrographs of modulators constructed according to principles of the invention, that show the slotted, segmented region, as well as the location where the silicon makes contact with the electrical layer. 
         FIG. 37(   a ) is a diagram showing the transmission through the Mach-Zehnder device as a function of wavelength, for a modulator drive voltage of 0.2 V bias. 
         FIG. 37(   b ) is a diagram showing the transmission through the Mach-Zehnder device as a function of wavelength, for a modulator drive voltage of 0.4 V bias. 
         FIG. 38(   a ) and  FIG. 38(   b ) are diagrams illustrating the transmission through the device as a function of bias voltage, according to principles of the invention. 
         FIG. 38(   c ) is a diagram that shows the frequency response of a device, according to principles of the invention. 
         FIG. 39  is a diagram that shows a transmission spectrum of an electroded slot waveguide resonator with a gap of 70 nm. Fiber to fiber insertion loss is plotted in dB, against the test laser wavelength in nm. 
         FIG. 40  is a diagram that shows an SEM image of a portion of a typical slot waveguide with a sub-100 nm slot. The cursor width is 57 nm in this image. 
         FIG. 41  through  FIG. 45  illustrate additional options for waveguide designs, according to principles of the invention. 
         FIG. 46  is a schematic diagram of a lumped element design. 
         FIG. 47  is an illustration in elevation of the structure of a modulator using a slotted waveguide and an electro-optical polymer, according to principles of the invention. 
         FIG. 48  is an illustrative diagram of a variable delay line device based on all-optical switches. 
         FIG. 49  is a diagram showing an illustrative example of an all-optical multiplexer. 
         FIG. 50  is a diagram of an illustrative all-optical self-oscillator. 
         FIG. 51  is a diagram of an illustrative AND gate without a gain section when turned ON. 
         FIG. 52  is a diagram of an illustrative clock multiplier. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     High index contrast waveguides as described herein are useful to concentrate light in order to enhance nonlinear optical effects in various materials so that such effects can be employed to manipulate light (or more generally electromagnetic radiation) at low power levels, as compared to conventional systems and methods that employ nonlinear optical materials. The manipulation of electromagnetic radiation or light can be useful to provide a variety of components that perform operations on light such as rectification and logic operations in a manner analogous to the same operations which are provided using electronic devices operating on electrical signals. For example, an input a light wave to be processed is impressed onto the component. The light wave has at least one parameter characterizing the light wave, such as one of an intensity, a polarization, a frequency, a wavelength, and a duration (e.g., a pulse length, or in the case of continuous wave light, an effectively infinite duration). After the input light wave is processed (or interacts with the waveguide and the clad nonlinear optical material adjacent to the waveguide), an output signal is observed. In a circumstance where the input signal has been processed, the output signal has at least one parameter that is different from at least one parameter characterizing the input light wave, including possibly an electrical output signal when the input light wave had no electrical signal component (e.g., optical rectification). As used herein, the term “optical rectification” is intended to relate to input signals having frequencies ranging from of the order of 100s of gigahertz through terahertz, and also including IR, visible, UV, and x-ray input signals. 
     As described in greater detail herein, the present invention provides methods and structures that exhibit enhancement of the nonlinear effects in various electro-optical materials that is sufficient to make the nonlinear effects accessible with continuous-wave, low-power lasers. In some embodiments, pulsed lasers can be used in addition to or in place of CW lasers. As is described herein the waveguide is coated or clad with another material which provides or exhibits an enhanced nonlinear optical coefficient, such as certain kinds of organic electro-optical materials that can be specifically designed to operate in various regions of the electromagnetic spectrum. We have demonstrated that some designs of high index contrast waveguides are designed to concentrate light in the cladding. In some embodiments, the waveguide is a split waveguide. In some embodiments, the split waveguide is coated with a material which provides an enhanced nonlinear optical coefficient. In some embodiments, the waveguides of the invention, including slotted or split waveguides, can operate with a low optical index fluid as a cladding, for example, air, or with no cladding, for example, vacuum. In some embodiments, the coating or cladding can be a ferroelectric material. In some embodiments, the two sides of the split waveguide also comprise electrodes that are used for polling a χ 2  material introduced into the gap. As described herein, in some embodiments, the dispersion of a waveguide is engineered to enhance the optical power in the mode by slowing the propagation of the light. In some embodiments the waveguides are segmented waveguides. As discussed herein, the waveguide can provide optical field enhancement when the structure is arranged into a resonator, which in various embodiments can be either a ring resonator or a linear resonator. It is believes that appropriate claddings can comprise one or more of glass, semiconductor, quantum dots, saturable absorbers, quantum dots doped into an organic mains, electro-optic materials such as polymers and dendrimers, polymers or other organic materials providing large χ 3  coefficients, or other nonlinear optical material to provide large optical nonlinearities through field enhancement in the cladding. In some embodiments, the systems and methods of the invention can be used to provide a tunable infrared source. In some embodiments, by using a low power tunable laser and a high power fixed wavelength laser as the inputs, it is possible to produce a high power coherent tunable source. The tunable source can be a widely tunable coherent source. In addition, using systems and methods of the invention, the use of an incoherent input light source can result in an incoherent tunable source. With the provision of on-chip feedback, the systems and methods of the invention can be used to provide devices that exhibit optical self-oscillation. In some embodiments, the central high index waveguide comprises an amplifying medium, such as a gallium arsenide stripe laser. In some embodiments, where the cladding material exhibits nonlinearities, the laser can be operated as a pulsed source. In some embodiments, systems and methods of the invention can be constructed to provide optical logic functionality, such as optical AND or optical flip-flops. It is believed that systems and method according to the invention can be employed to create optical NAND, OR, NOR and XOR gates, and optical latches, or optical memory. In some embodiments, the systems of the invention can further comprise pump lasers integrated onto the same chip. In some embodiments, the systems of the invention can further comprise off-chip feedback or amplification for frequency conversion or pulse generation. In some embodiments, an additional electrical signal is coupled into the structure to provide active modelocking. 
     We have developed a set of tools for concentrating light to a high degree by using silicon or other high index contrast waveguides, and we have fabricated devices that demonstrate some of the many applications that can be contemplated when such nonlinear materials are exploited. While the description given will be expressed using single crystal silicon, it is believed that similar devices, systems and methods can be provided using polycrystalline silicon (“poly silicon”) or amorphous silicon (also referred to as “a-silicon” or “a-silicon”). In particular, by utilizing split waveguides (or slot waveguides), we are able to greatly enhance the optical fields in the cladding of a tightly confined waveguide, without greatly enhancing the optical losses of the same waveguide. Combining the high field concentrations available from the split waveguides with the high nonlinear activity of nonlinear optical polymers permits the development of nonlinear optical devices operating at much lower optical input power levels than are possible with conventional free space or chip based systems. We have demonstrated four-wave mixing (which is based upon χ 3 ), as well as optical rectification (based on χ 2 ), in such waveguides. Using these waveguides it is possible to decrease the power levels needed to observe significant nonlinearities to the point where, by contrast with conventional nonlinear optics, it can be done with non-pulsed, continuous wave lasers. 
     Chi2 (χ 2 ) and Chi3 (χ 3 ) based optical effects can be used in particular to build on-chip optical parametric oscillator (“OPO”) systems, where two input wavelengths can be mixed together to produce sum and difference frequencies. These frequencies can be either higher or lower than the input frequencies, and can be made tunable. These effects work for frequencies from the ultraviolet and X-ray regime all the way out into the far infrared and microwave, and in fact can work down to DC in some cases, particularly with optical rectification. 
     The material of which the high index waveguide is made can be any material having a high index that is reasonably transparent at the wavelengths of interest. This can include but is not limited to silicon, gallium nitride, indium phosphide, indium gallium nitride, gallium phosphide, diamond, sapphire, or the various quaternary II/V and II/VI materials such as aluminum gallium arsenide phosphide. III/V denotes materials having at least one element from column III of the periodic table of elements (or an element that is stable as a positive trivalent ion) and at least one element from column V (or an element that is stable as a negative trivalent ion). Examples of III/V compounds include BN, AlP, GaAs and InP. II/VI denotes materials having at least one element from column II of the periodic table of elements (or an element that is stable as a positive divalent ion) and at least one element from column VI (or an element that is stable as a negative divalent ion). Examples of II/VI compounds include MgO, CdS, ZnSe and HgTe. 
     We will now present a more detailed description of the systems and methods of the invention, including successively the mechanical structure of exemplary embodiments of high index waveguides, exemplary embodiments of cladding materials having large nonlinear constants χ 2  and χ 3  and their incorporation into devices having high index waveguides, exemplary results observed on some of the fabricated devices that are described, and some theoretical discussions about the devices and the underlying physics, as that theory is presently understood. Although the theoretical descriptions given herein are believed to be correct, the operation of the devices described and claimed herein does not depend upon the accuracy or validity of the theoretical description. That is, later theoretical developments that may explain the observed results on a basis different from the theory presented herein will not detract from the inventions described herein. 
     Exemplary High Index Waveguide Structures 
     Example 1 
     High-Q Ring Resonators in Thin Silicon-On-Insulator 
     Resonators comprising high-Q microrings were fabricated from thin silicon-on-insulator (SOI) layers. Measured Q values of 45 000 were observed in these rings, which were then improved to 57 000 by adding a PMMA cladding. Various waveguide designs were calculated, and the waveguide losses were analyzed. It is recognized that several forms of silicon on insulator, such as SOI comprising wafers having a silicon oxide layer fabricated on silicon, or such as silicon on sapphire (SOS) can be used in different embodiments. 
     Microring resonator structures as laser sources and as optical filter elements for dense wavelength division multiplexing systems have been studied in the past. The silicon-on-insulator (SOI) structure described here is particularly advantageous. It has low waveguide loss. One can extrapolate an uncoupled Q value of 94 000 and a waveguide loss of 7.1 dB/cm in the unclad case, and −6.6 dB/cm in the PMMA clad case, from the respective measured Q values of 45 000 and 57 000. Although higher Q values have been obtained for optical microcavities, we believe that our geometry has the highest Q for a resonator based on a single mode silicon waveguide. It is also noteworthy that a large amount of power appears outside the core silicon waveguide, which may be important in some applications. The modes that are described herein have approximately 57% of the power outside the waveguide, as compared to 20% for a single-mode 200-nm-thick silicon waveguide, and 10% for a single-mode 300-nm-thick silicon waveguide. 
     In the embodiment now under discussion, wafer geometries were selected that minimize the thickness of the SOI waveguiding layer as well as the buried oxide, but still yield low loss waveguides and bends. A number of different waveguide widths were compared by finite difference based mode solving. The geometry used in the exemplary embodiment comprises a 500-nm-wide waveguide formed in a 120-nm-thick silicon layer, atop a 1.4 μm oxide layer, which rests on a silicon handle, such as a silicon wafer as a substrate. Such a configuration supports only a single well-contained optical mode for near infrared wavelengths. The dispersion characteristics are shown in  FIG. 1  for both unclad and PMMA-clad waveguides. Our interest in unclad structures stems from the ease of fabrication, as detailed in the following, as well as the flexibility an open air waveguide may provide for certain applications. 
     These modes were determined by using a finite difference based Hermitian eigensolver, described further herein. It is possible to calculate the loss directly from the mode pattern with an analytic method valid in the low-loss limit. The waveguide loss at 1.55 μm calculated in such a fashion is approximately −4.5 dB. This loss figure was in agreement with the extrapolated results of FDTD simulation. 
     Because a loss of −4 dB/cm is attributed to substrate leakage, the waveguide loss can be improved by the addition of a cladding, which tends to pull the mode upwards. This notion is supported by the measured decrease in waveguide loss upon the addition of a PMMA cladding. It can be shown that the substrate leakage loss attenuation coefficient is nearly proportional to 
     
       
         
           
              
             
               
                 - 
                 2 
               
                
               
                 
                   
                     n 
                     eff 
                     2 
                   
                   - 
                   
                     n 
                     o 
                     2 
                   
                 
               
                
               
                 k 
                 0 
               
                
               A 
             
           
         
       
     
     if k 0  is the free space wave number, n eff  is the effective index of the mode, n 0  is the effective index of the oxide layer, and A is the thickness of the oxide. In the present case, the e-folding depth of the above-mentioned function turns out to be 180 nm, which explains why the substrate leakage is so high. 
     SOI material with a top silicon layer of approximately 120 nm and 1.4 μm bottom oxide was obtained in the form of 200 mm wafers, which were manually cleaved, and dehydrated for 5 min at 180° C. The wafers were then cleaned with a spin/rinse process in acetone and isopropanol, and air dried. HSQ electron beam resist from Dow Corning Corporation was spin coated at 1000 rpm and baked for 4 min at 180° C. The coated samples were exposed with a Leica EBPG-5000+electron beam writer at 100 kV. The devices were exposed at a dose of 4000 μc/cm 2 , and the samples were developed in MIF-300 TMAH developer and rinsed with water and isopropanol. The patterned SOI devices were subsequently etched by using an Oxford Plasmalab 100 ICP-RIE within 12 mTorr of chlorine, with 800 W of ICP power and 50 W of forward power applied for 33 s. Microfabricated devices such as the one shown in  FIG. 2  were tested by mounting the dies onto an optical stage system with a single-mode optical fiber array. A tunable laser was used first to align each device, and then swept in order to determine the frequency domain behavior of each of the devices. Light was coupled into the waveguides from a fiber mode by the use of grating couplers. Subsequently the devices were spin-coated with 11% 950 K PMMA in Anisole, at 2000 rpm, baked for 20 min at 180° C., and retested. 
     The theoretical development of the expected behavior of a ring resonator system has been described in the technical literature. In the present case the dispersion of the waveguide compels the addition of a dispersive term to the peak width. We take λ 0  to be the free space wavelength of a resonance frequency of the system, n 0  to be the index of refraction at this wavelength, (dn/δλ) 0 , the derivative of n with respect to λ taken at λ 0 , L to be the optical path length around the ring, a to be the optical amplitude attenuation factor due to loss in a single trip around the ring, and finally t to be the optical amplitude attenuation factor due to traveling past the coupling region. In the limit of a high Q, and thus (1−α)  and (1−t) 1, 
     we have 
     
       
         
           
             
               
                 
                   Q 
                   = 
                   
                     
                       
                         π 
                          
                         
                             
                         
                          
                         L 
                       
                       
                         λ 
                         0 
                       
                     
                      
                     
                       
                         
                           ( 
                           
                             
                               n 
                               0 
                             
                             - 
                             
                               
                                 
                                   λ 
                                   0 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       ∂ 
                                       n 
                                     
                                     
                                       ∂ 
                                       λ 
                                     
                                   
                                   ) 
                                 
                               
                               0 
                             
                           
                           ) 
                         
                         
                           ( 
                           
                             1 
                             - 
                             
                               α 
                                
                               
                                   
                               
                                
                               t 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The waveguide mode was coupled into a ring resonator from an adjacent waveguide. As shown in  FIG. 2 , the adjacent waveguide can in some embodiments be a linear waveguide. The strength of coupling can then be lithographically controlled by adjusting the distance between the waveguide and the ring. This ring was fabricated with a radius of 30 μm, a waveguide width of 500 nm, and a separation between ring and waveguide of 330 nm. For the clad ring presented, the measured Q is 45 000, and the extinction ratio is −22 dB, for the resonance peak at 1512.56 nm. The PMMA clad ring had a similar geometry, and achieved a Q of 57 000, but with an extinction ratio of −15.5 dB. Typical observed transmission spectra are shown in  FIG. 3 . The typical amount of optical power in the waveguide directly coupling into the resonator was about 0.03 mW. A dependence of the spectrum on this power was not observed, to within an order of magnitude. 
     From the mode-solving results for the unclad waveguides, we have (dn/δλ)(1.512)=−1.182 μm −1 , and n(λ=1.512)=1.688. Using this result and the earlier relations, the waveguide loss can be calculated from the measured Q value. Specifically, an extinction that is at least −22 dB indicates that a critically coupled Q in this geometry is greater than 38 500, which then implies a waveguide loss of less than −7.1 dB/cm. In similar fashion, the PMMA clad waveguide resonator with a Q of 57 000 but only −15.5 dB of extinction allows a worst case waveguide loss of −6.6 dB/cm. This also implies an intrinsic Q of 77 000 for the unclad resonator, and an intrinsic Q of 94 000 for the PMMA clad resonator. 
     These devices have a slight temperature dependence. Specifically, the resonance peak shifts correspondingly with the change in the refractive index of silicon with temperature, moving over 2 nm as temperature shifts from 18 to 65° C. The Q rises with higher temperatures slightly, from 33 k at 18° C. to 37 k on one device studied. This shift can probably be explained entirely by the dependence of Q on the effective index. 
     Example 2 
     High-Q Optical Resonators in Silicon-On-Insulator Based Slot Waveguides 
     We now describe the design, fabrication and characterization of high Q oval resonators based on slot waveguide geometries in thin silicon on insulator material. Optical quality factors of up to 27,000 were measured in such filters, and&#39;we estimate losses of −10 dB/cm in the slotted waveguides on the basis of our resonator measurements. Such waveguides enable the concentration of light to very high optical fields within nano-scale dimensions, and show promise for the confinement of light in low-index material with potential applications for optical modulation, nonlinear optics and optical sensing. As will be appreciated, the precise geometry of a resonator (or other kinds of devices) is frequently a matter of design, and the geometry can be varied based on such considerations as length of waveguide, area of a chip, and required interaction (or required non-interaction), such as coupling (or avoiding coupling) with other waveguide structures that are present in a device or on a chip. In some embodiments, the waveguide can be a closed loop, such as at least one ring or at least one oval shaped endless stripe. As has been explained, optical energy can be provided to such a closed loop, for example with an input waveguide. 
     One can form high quality factor ring or oval resonators in SOI. In these SOI waveguides, vertical confinement of light is obtained from the index contrast between the silicon core and the low index cladding and the buried silicon dioxide layer, whereas lateral confinement can be obtained by lithographically patterning the silicon. The majority of the light tends to be guided within the silicon core in such waveguide. Although the high refractive index contrast between silicon and its oxide provide excellent optical confinement, guiding within the silicon core can be problematic for some applications. In particular, at very high optical intensities, two-photon absorption in the silicon may lead to high optical losses. Moreover, it is often desirable to maximize the field intensity overlap between the optical waveguide mode and a lower index cladding material when that cladding is optically active and provides electro-optic modulation or chemical sensing. 
     One solution to these problems involves using a slot waveguide geometry. In a slot waveguide, two silicon stripes are formed by etching an SOI slab, and are separated by a small distance. In one embodiment, the separation is approximately 60 nm. The optical mode in such a structure tends to propagate mainly within the center of the waveguide. In the case of primarily horizontal polarization, the discontinuity condition at the cladding-silicon interface leads to a large concentration of the optical field in the slot or trench between the two stripes. One can predict that the electric field intensity would be approximately 10 8  vP V/m where P is the input power in watts.  FIG. 4  shows the approximate geometry used for the design in this embodiment, as well as the solved mode pattern for light at approximately 1.53 μm. As seen in  FIG. 4 , the mode profile comprises |E| contours, plotted in increments of 10% of the maximum field value. The E field is oriented primarily parallel to the wafer surface. This mode was obtained from a full vectoral eigensolver based on a finite difference time domain (FDTD) model. Some embodiments described herein use a 120 nm silicon on insulator layer and 300 nm wide by 200 nm thick silicon strips on top of a 1.4 μm thick buried oxide layer, which is in turn deposited on a silicon substrate. After the lithographic waveguide definition process, polymethylmethacrylate (PMMA) was deposited as the top cladding layer. Various widths for the central slot were fabricated to provide test devices with 50, 60 and 70 nm gaps. The mode profile shown in  FIG. 4  and the dispersion diagram shown in  FIG. 5  are for a 60 nm slot.  FIG. 5  is a diagram showing the effective index vs. free space wavelength in microns for the slot waveguide of  FIG. 4 . Slots larger than 70 nm have also been fabricated and were shown to work well. The slot waveguide with a 50 nm slot and 300×200 nm arms for enhancement of nonlinear moment enjoys an improvement by around a factor of 10 over the effective nonlinearity of simple ridge waveguides having a single 500×100 nm Si ridge geometry that are coated with a nonlinear polymer cladding. 
     In the 1.4-1.6 μm wavelength regime, the waveguide geometry is single node, and a well-contained optical mode is supported between the two silicon waveguide slabs. There is some loss that such an optical mode will experience even in the absence of any scattering loss or material absorption due to leakage of light into the silicon substrate. The substrate loss can be estimated semi-analytically via perturbation theory, and ranges from approximately −0.15 dB/cm at 1.49 μm to about −0.6 dB/cm at 1.55 μm for the SOI wafer geometry of the present embodiment. 
     Oval resonators were fabricated by patterning the slot waveguides into an oval shape. An oval resonator geometry was selected in preference to the more conventional circular shape to enable a longer coupling distance between the oval and the external coupling waveguide or input waveguide. See  FIG. 6 . Slots were introduced into both the oval and external coupling waveguides.  FIG. 7  and  FIG. 8  show scanning electron micrograph images of an exemplary resonator and the input coupler. 
     Predicting coupling strength and waveguide losses for such devices is not easy. Many different coupling lengths and ring to input waveguide separations were fabricated and tested. It is well known that the most distinct resonance behavior would be observed for critically coupled resonators, in which the coupling strength roughly matches the round trip loss in the ring. 
     An analytic expression for the quality factor of a ring resonator was presented in equation (1) hereinabove. 
     Also, the free spectral range can be calculated via: 
     
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     λ 
                   
                   = 
                   
                     
                       
                         λ 
                         2 
                       
                       / 
                       L 
                     
                     
                       n 
                       - 
                       
                         λ 
                          
                         
                           
                             ∂ 
                             n 
                           
                           
                             ∂ 
                             λ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Here, L is the round trip length in the ring, and n 0  and λ 0  are the index of refraction, and the wavelength at resonance, respectively. The derivative of the effective index with respect to the wavelength at the resonance peak is given by (δn/δλ) 0 , and it can be shown that this term is roughly equal to −0.6 μm −1  from the 1.4-1.6 μm spectral range for the slot waveguides studied here. 
     We have observed a quality factor of 27,000 in a device fabricated with a slot size of 70 nm, a ring to input waveguide edge to edge separation of 650 nm, and a coupling distance of 1.6 μm. The radius of the circular part of the slotted oval was 50 μm. This resonance was observed near 1488 nm, and the resonance peak had an extinction ratio of 4.5 dB.  FIG. 9  shows the measured transmission spectrum past the ring, normalized for the input coupler baseline efficiency of our test system.  FIG. 10  shows the details of one peak in the vicinity of 1488 nm. Because the extinction ratio at the resonance peak was not very large in this case, it was not possible to accurately determine waveguide losses from this device. By measuring many devices with different geometries, we obtained data on resonators with higher extinction ratios that approached critical coupling. One such device was a 50 μm radius slotted ring resonator with a 60 nm waveguide gap, a ring to input waveguide spacing of 550 nm and coupling length of 1.6 μm. In this device, a Q of 23,400 was observed near 1523 nm, with an on-resonance extinction of 14.7 dB. 
     Since this resonance is nearly critically coupled, the waveguide loss can be estimated using equation (1) as −10 dB/cm. We can also use equation (2) to further validate our theoretical picture of the ring resonator. The observed free spectral range of this resonator was 2.74 nm, while equation (2) predicts 2.9 nm. This discrepancy is most likely due to small differences in the fabricated dimensions as compared to those for which the numerical solutions were obtained. 
     To further validate the waveguide loss result, several waveguide loss calibration loops were fabricated with varying lengths of the slot waveguide, ranging from 200 to 8200 μm in length. A total of five center slot waveguide devices were studied for each of the 50, 60 and 70 nm slot widths. Linear regression analysis on the peak transmission of each series yielded waveguide loss figures of 11.6±3.5 dB/cm for the 50 inn center waveguide, 7.7±2.3 dB/cm for the 60 nm center waveguide, and 8.1±1.1 dB/cm for the 70 nm center waveguide. These figures are in agreement with the loss estimated from the oval resonator. Since the theoretical loss due to substrate leakage is much lower than this, it is clear that a great deal of loss is due to surface roughness and possibly material absorption. It is believed that engineering improvements will decrease this loss further. For sensing and modulation applications as well as use in nonlinear optics, the high optical field concentration that can be supported in the cladding material of the slotted waveguide geometry should be very advantageous when compared to more conventional waveguides. 
       FIG. 11  is a diagram showing a shallow angle SEM view of a silicon-on-insulator ring resonator and waveguide having a sidewall roughness on the order of 10 nm. In the exemplary waveguide shown in  FIG. 11 , the silicon-insulator bond has been decorated with a brief buffered oxide etch.  FIG. 12  is a diagram of a slot ring resonator directional coupler region, and the associated input waveguide. 
     By comparison,  FIG. 13  is a diagram showing a slot waveguide structure that exhibits subfield stitching errors at the edge of the input waveguide in the example shown. Such errors can be devastating for waveguide loss. Because electric fields are known to concentrate at sharp corners or surface irregularities, it is expected that such sharp features occurring at undefined (or random) locations on the surface of a waveguide will have deleterious consequences for the desired electric field profiles.  FIG. 14  is yet another example of a rough wall that is likely to create problems in device fabrication and operation. It is therefore preferred that the walls of waveguides according to principles of the invention be constructed so as to minimize the occurrence of sharp features. 
     Other variations on the geometry of waveguides are possible.  FIG. 15  is a diagram showing an exemplary high-index segmented waveguide structures, which in the embodiment shown comprises a central waveguide portion with fingers or ridges sticking out to the sides. With the light localized in the center in a Bloch mode, electrical contact can be established using the fingers or ridges that stick off the sides of the waveguide. This structure provides a way to form both electrical contacts to waveguides and structures that would provide electrical isolation with low optical loss. Through an iterative process involving a combination of optical design using a Hermetian Bloch mode eigensolver and fabrication of actual structures, it was found that (non-slotted) segmented waveguide structures could be constructed in 120 nm thick SOL Waveguide losses as small as −16 dB per centimeter were observed, and insertion losses as small as −0.16 dB were shown from standard silicon waveguides. 
     The segmented waveguide structure can also be modeled as regards its expected properties, which can then be compared to actual results.  FIG. 16A  is a diagram that shows a dispersion diagram of both a segmented waveguide and the normal, unsegmented waveguide, taken on a plane parallel to the substrate that on a z plane that intersects the middle of a segment.  FIG. 16B  is a diagram that shows modal patterns of the Bloch mode, with contours of |E| plotted, starting at 10% of the max value and with contour increments of 10%.  FIG. 16C  is a diagram that shows a plot of modal patterns over four periods of a segmented waveguide on a horizontal plane that intersects the silicon layer halfway through. 
     By utilizing the same type of design methodology as was used for the segmented waveguides, one is able to able to construct structures that provide electrical isolation without substantial optical loss.  FIG. 17  is a diagram that shows an exemplary electrical isolator that was constructed and tested, and which provided both a transition from a standard to a slotted waveguide and electrical isolation between the two sides of the slot waveguide. Such structures were shown to have losses on the order of 0.5 dB. 
     Exemplary Results for Waveguides with Cladding Materials 
     Examples 1-4 
     Four-Wave Mixing in Silicon Waveguides with χ 3  Polymer Material 
     Two types of integrated nano-optical silicon waveguide structures were used for this demonstration. The first type of structure was a series of ring resonator structures, which allowed an estimation of the waveguide loss of the nonlinear material. The second type of structures used was long runout devices, which comprised a simple waveguide loop with distances on the order of 0.7 cm. Characterization of loss could be done passively. 
     For the actual nonlinear testing, a Keopsys EDFA was used to boost two lasers to a high power level, on the order of 30 dBm (1 Watt) or more. 
     The materials used for the demonstrations were clad on waveguides configured as previously described herein. The chromophore identified as JSC 1  is shown by its chemical structure in  FIG. 26 . The chromophores identified as JSC 1  and YLD  124  are two substances among many chromophores that were described in a paper by Alex Jen, et al., “Exceptional electro-optic properties through molecular design and controlled self-assembly,” Proceedings of SPIE—The International Society for Optical Engineering (2005), 5935 (Linear and Nonlinear Optics of Organic Materials V), 593506/1-593506/13. The paper describes at least five additional specific chromophores, and states in part that a “series of guest-host polymers furnished with high μβ chromophores have shown large electro-optic coefficients around 100˜160 pm/V @ 1.31 μm.” It is believed that the several examples given in the present description represent a few specific examples of many chromophores that can be used as materials having large nonlinear coefficients χ 2  and χ 3  according to principles of the systems and methods disclosed herein. Four types of claddings were applied to waveguides situated on silicon dies: 
     1. JSC 1 /APC: The chromophore JSC 1  is doped into amorphous polycarbonate (APC) with the loading of 35 wt %. The solvent we used is cyclohexanone, and concentration of overall solid in this solution is 14 wt %.
 
2. AJL 21 /PMMA: The chromophore AJL 21  is doped into PMMA with the loading of 40 wt %. The solvent used was 1,1,2-trichloroethane, and solution concentration was 10 wt %.
 
3. AJL 21  monolithic films: The chromophore AJL 21  is coated by itself monolithically. The solvent was 1,1,2 trichloroethane, and the concentration was 10 wt %.
 
4. AJC 212  monolithic films: The chromophore AJC 212  was coated by itself monolithically. The solvent was cyclopentanone, and concentration was 11 wt %. This film may have wetting problems, as evidenced by periphery shrinkage after baking.
 
     Passive Results 
     Waveguide loss was measured for each of the four die. Intrinsic waveguide loss with a cladding having an index of 1.46 is about 7 dB/cm. A cladding with n&gt;1.46 would lower this figure slightly. The total loss and the estimated loss due to the polymer are presented separately. This is based on subtracting 7 dB from the polymer, and then multiplying by three, because the polymer causes approximately as third as much loss as it would for the mode if it were in a bulk material, because not all of the optical energy interacts with the polymer. 
     Die  1 : 30 dB/cm; 69 dB/cm for bulk polymer
 
Die  2 : 5.7 dB/cm; &lt;1 dB/cm for bulk polymer
 
     Die  3 : the loss was too high to measure devices 
     Die  4 : 10 dB/cm; 12 dB/cm for hulk polymer 
     Active Results 
     The intrinsic nonlinear response of our EDFA and optical test system was measured to determine a baseline for measurements on devices.  FIG. 18  is a diagram showing the results of a baseline measurement of an EDFA and optical test system in the absence of a test sample. As can be seen in  FIG. 18 , there is a very small amount of four wave mixing that occurs. This test was performed with about 28 dBm of EDFA output. There is 40 dB of extinction from the peak to the sidebands. 
     A Die  1  loop device with 7000 μm of runlength produced about 29 dB of conversion efficiency (that is, sidebands were 29 dB down from peak at end of run). 
       FIG. 19  is a diagram showing the results for the measurement of a first exemplary material having a large value of χ 3 , namely Die  1  with a cladding. Even though the plot looks similar to that shown in  FIG. 18 , in fact there is an order of magnitude more nonlinear conversion that has occurred. The insertion loss is due to the grating couplers and the waveguide loss in the device. 
       FIG. 20  is a diagram showing the results for the measurement of a first exemplary material having a large value of χ 3 , namely Die  2  with a cladding, which showed better results than Die  1 . Here there is about 20 dB of extinction from the right peak to the left sideband, and 22 dB from the larger peak on the left to the left sideband. This is the result that represents a demonstration of 1% conversion efficiency. 
     The noise level on some of these scans is higher than others because some were taken with faster scan settings on the optical signal analyzer. 
     Semi-Analytic Results 
     The slowly varying approximation can be used to generate the characteristic equations to predict the conversion efficiency. Let a 0 ( z ), a 1 ( z ) and a 2 ( z ) be the amplitudes of the 3 wavelengths involved in a given four-wave mixing interaction. Let w 2 =2*w 0 −w 1 . Approximately, E is 10 8  V/m for 1 Watt of power. so if we take E=a 0 ( z )* 10   8  V/m then a 0  is power normalized to be 1 watt when |a 0 |=1. The characteristic equations are: 
     
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           ∂ 
                           
                             a 
                             0 
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       = 
                       
                         6 
                          
                         
                             
                         
                          
                         f 
                          
                         
                           
                              
                              
                             
                                 
                             
                              
                             
                               β 
                               0 
                             
                           
                           
                             neff 
                             0 
                             2 
                           
                         
                          
                         
                           exp 
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   
                                     
                                       - 
                                       2 
                                     
                                      
                                     
                                         
                                     
                                      
                                     
                                       β 
                                       0 
                                     
                                   
                                   + 
                                   
                                     β 
                                     2 
                                   
                                   + 
                                   
                                     β 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                                
                                
                               
                                   
                               
                                
                               z 
                             
                             ) 
                           
                         
                          
                         a 
                          
                         
                             
                         
                          
                         0 
                         * 
                         a 
                          
                         
                             
                         
                          
                         1 
                          
                         
                             
                         
                          
                         a 
                          
                         
                             
                         
                          
                         2 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           ∂ 
                           
                             a 
                             1 
                           
                         
                         
                           ∂ 
                           z 
                         
                       
                       = 
                       
                         
                           3 
                            
                           
                               
                           
                            
                           f 
                            
                           
                             
                                
                                
                               
                                   
                               
                                
                               
                                 β 
                                 1 
                               
                             
                             
                               neff 
                               1 
                               2 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ? 
                                       
                                     
                                     - 
                                     
                                       ? 
                                     
                                     - 
                                     
                                       ? 
                                     
                                   
                                   ) 
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 z 
                               
                               ) 
                             
                           
                            
                           a 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           a 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           a 
                            
                           
                               
                           
                            
                           2 
                           * 
                           
                             
 
                           
                            
                           
                               
                           
                            
                           
                             
                               ∂ 
                               
                                 a 
                                 2 
                               
                             
                             
                               ∂ 
                               z 
                             
                           
                         
                         = 
                         
                           3 
                            
                           
                               
                           
                            
                           f 
                            
                           
                             
                                
                                
                               
                                   
                               
                                
                               
                                 β 
                                 2 
                               
                             
                             
                               neff 
                               2 
                               2 
                             
                           
                            
                           
                             exp 
                              
                             
                               ( 
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         ? 
                                       
                                     
                                     - 
                                     
                                       ? 
                                     
                                     - 
                                     
                                       ? 
                                     
                                   
                                   ) 
                                 
                                  
                                  
                                  
                                 
                                     
                                 
                                  
                                 z 
                               
                               ) 
                             
                           
                            
                           a 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           a 
                            
                           
                               
                           
                            
                           0 
                            
                           
                               
                           
                            
                           a 
                            
                           
                               
                           
                            
                           1 
                           * 
                           
                             
 
                           
                            
                           
                             ? 
                           
                            
                           
                             indicates text missing or illegible when filed 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The quantity f is taken as an unknown fraction which reduces the effect of the nonlinear material due to the fact that some of the optical energy is not in the optical region, but in the waveguide core. It is estimated that f is about 0.1, with an uncertainty of perhaps a factor of 2. 
     The phasor factor turns out to have an oscillation period on the order of a meter for the waveguides under consideration, and can be ignored. Based on a numerical integration, one can then estimate the χ 3  coefficients for die  1  and die  2  as: 
     Die  1 : χ 3  is nearly 8×10 −22  (m/V) 2  
 
Die  2 : χ 3  is approximately 1.5 −22  (m/V) 2  
 
       FIG. 21  is a diagram that shows a plot of the numerically computed conversion efficiency for Die  2 , in dB vs 1 watt compared to length traveled in waveguide in μm. 
     The devices that were tested were observed in all cases to eventually fail, either when ramping up the power levels or after extended testing. It is believed that the problem is caused by heating damage. Fortunately the damage seems not to extend to the silicon waveguides. This means that devices that fail in this way can be recovered by stripping the polymers, and then being recoated. With additional experience, solutions for the problem of this damage problem may be identified and solved. 
     It is unfortunate that the waveguide loss in the die  1  material is so high, because it is a material that exhibits extremely high χ 3 . Nevertheless, reasonable efficiencies were demonstrated with material exhibiting a lower χ 3 . It would be advantageous to identify a material with a value of χ 3  that is larger by a factor of 10 or so. It would also be advantageous to lower the waveguide loss slightly. With these two adjustments, it would be possible to enter the “strong coupling” regime, so that one might observe 100% conversion in lengths &lt;0.5 cm. One likely possibility would be to lower the optical loss of the Die  1  material, JSC 1 . 
     Example 5 
     Optical Modulation and Detection in Slotted Silicon Waveguides 
     In some embodiments, an optical input signal can be directly converted to an electrical output signal via a process known as optical rectification. This process occurs when a particularly intense optical beam is incident on a χ 2  material, and induces a low frequency electric field as a result. The large magnitude of this electric field is due to the enhancement of the optical field in a slot waveguide. This process has many advantages over conventional detection schemes, such as photodiodes. In particular, there will be nearly no speed limit for this type of detector, because the mechanism is ultrafast and functions at the optical frequency. 
     In this example, we describe a system and process that provide low power optical detection and modulation in a slotted waveguide geometry filled with nonlinear electro-optic polymers and present examples that demonstrate such methods. The nanoscale confinement of the optical mode, combined with its close proximity to electrical contacts, enables the direct conversion of optical energy to electrical energy, without external bias, via optical rectification, and also enhances electro-optic modulation. We demonstrate this process for power levels in the sub-milliwatt regime, as compared to the kilowatt regime in which optical nonlinear effects are typically observed at short length scales. The results presented show that a new class of detectors based on nonlinear optics can be fabricated and operated. 
     Waveguide-based integrated optics in silicon provide systems and methods for concentrating and guiding light at the nanoscale. The high index contrast between silicon and common cladding materials enables extremely compact waveguides with very high mode field concentrations, and allows the use of established CMOS fabrication techniques to define photonic integrated circuits. As we have already explained hereinabove, by using slotted waveguides, it is possible to further concentrate a large fraction of the guided mode into a gap within the center of a silicon waveguide. This geometry greatly magnifies the electric field associated with the optical mode, resulting in electric fields of at least (or in excess of)  10   6  V/m for continuous-wave, sub-milliwatt optical signals. Moreover, since the slotted geometry comprises two silicon strips which can be electrically isolated, a convenient mechanism for electro-optic interaction is provided. Such waveguides can be fabricated with low loss. We have previously described systems that provide losses below −10 dB/cm. 
     In the present example, we exploit both the high intensity of the optical field and the close proximity of the electrodes for several purposes. First, we demonstrate detection of optical signals via direct conversion to electrical energy by means of nonlinear optical rectification. An exemplary device comprises a ring resonator with an electro-optic polymer based χ 2  material deposited as a cladding. Inside the slot, the high optical field intensity creates a standing DC field, which creates a virtual voltage source between the two silicon electrodes, resulting in a measurable current flow, in the absence of any external electrical bias. Though optical rectification has been observed in electro-optic polymers, typically instantaneous optical powers on the order of 1 kW are needed for observable conversion efficiencies, often achieved with pulsed lasers. The exemplary embodiment provides measurable conversion with less than 1 mW of non-pulsed input, obtained from a standard, low power tunable laser operating near 1500 nm. 
     In one embodiment, systems and methods of the invention provide standard Pockels effect based modulation, which is similarly enhanced by means of the very small scale of our device. The close proximity of the electrodes, and ready overlap with the optical mode, causes an external voltage to produce a far larger effective electric modulation field, and therefore refractive index shift, than would be obtained through conventional waveguide designs. In one embodiment, the modulation and refractive index shift is provided by tuning the resonance frequencies of a slot waveguide ring resonator. 
     Device Fabrication 
     Waveguide Fabrication 
     The devices described in this example were fabricated in electronic grade silicon-on-insulator (SOI) with a top layer thickness of 110 nm and an oxide thickness of 1.3 microns. The silicon layer is subsequently doped to approximately 10 19  Phosphorous atoms/cm 3 , yielding resistivities after dopant activation of about 0.025 ohm-cm. Electro-optic (“EO”) polymers were then spin-deposited onto the waveguide structures and subsequently poled by using a high field applied across the slot in the waveguide. 
     Lithography was performed using a Leica EBPG 5000+ electron beam system at 100 kv. Prior to lithography, the samples were manually cleaved, cleaned in acetone and isopropanol, baked for 20 minutes at 180 C, coated with 2 percent HSQ resist from Dow Corning Corporation, spun for two minutes at 1000 rpm, and baked for an additional 20 minutes. The samples were exposed at 5 nm step size, at 3500 μC/cm 2 . The samples were developed in AZ 300 TMAH developer for 3 minutes, and etched on an Oxford Instruments PLC Plasmalab 100 with chlorine at 80 scorn, forward power at 50 W, ICP power at 800 W, 12 mTorr pressure, and 33 seconds of etch time. The samples were then implanted with phosphorous at normal incidence, 30 keV energy, and 1×10 14  ions/cm 2  density. The sample was annealed under a vacuum at 950 C in a Jipilec Jetstar rapid thermal annealer. The samples were dipped in buffered hydrofluoric acid in order to remove the remnants of electron beam resist from the surface. 
     After initial optical testing, the samples were coated with YLD  124  electro-optic polymer, and in one case with dendrimer-based electro-optic material. The samples were stored under a vacuum at all times when they were not being tested, in order to reduce the chances of any degradation. 
     Synthesis of YLD  124  Coating Solution 
       FIG. 22  is a diagram showing a chemical reaction useful for the synthesis of a chromophore referred to as YLD  124 . The compound denoted in  FIG. 22  by 1 is discussed in the paper by C. Zhang, L. R. Dalton, M. C. Oh, H. Zhang, W. H. Steier, entitled “Low V-pi electro-optic modulators from CLD-1: Chromophore design and synthesis, material processing, and characterization,” which was published in Chem. Mater., volume 13, pages 3043-3050 (2001). 
     To a solution of 0.56 g (0.96 mmol) of 1 and 0.36 g of 2 (1.1 mmol) in 1.5 mL of THF was added 6 mL of absolute ethanol. The mixture was stirred for 6 h at room temperature. The precipitate was collected by filtration and washed by ethanol and methanol. The crude product was dissolved in minimum amount of CH 2 Cl 2 . The resultant solution was added dropwisely to 100 mL of methanol. The product (0.76 g) was collected as dark green precipitate. Yield was 90%.  1 H NMR (CDCl 3 ): 8.05 (t, J=13.6 Hz, 1H), 7.45-7.58 (m, 5H), 7.38 (d, J=8.9 Hz, 2H) 6.93 (d, J=15.9 Hz, 1H) 6.79 (d, J=15.9 Hz, 1H), 6.70 (d, J=8.9 Hz, 2H), 6.40-6.25 (m, 3H), 3.80 (t, J=5.8 Hz, 4H), 3.59 (t, J=5.8 Hz, 4H), 2.42 (s, 2H), 2.40 (s, 2H), 1.04 (s, 3H), 0.98 (s, 3H), 0.90 (s, 18H), 0.04 (5, 12H). MS (ESP): 879.48 (M+H). UV-Vis (THF): 765 nm. m.p. 173° C. 
     One part of YLD  124  was mixed with three parts of APC (PoIy[Bisphenol. A carbonate-co-4,4′-(3,3,5-trimethylcyclohexylidene)diphenol carbonate]). The mixture was dissolved in cyclopentanone. The total solid content (YLD  124  and APC) is about 12%. The resultant solution was filtered through a 0.2 pm filter before being used on the device to provide a cladding layer comprising the chromophore YLD  124 . 
     Measurement Results 
     Optical Rectification Based Detection 
       FIG. 23  is a four panel diagram that shows details of one embodiment of an optical modulator device, including the geometry of the photodetectors and filters, and including a cross section of the slotted waveguide. Panel A of  FIG. 23  shows a cross section of the device geometry with optical mode superimposed on a waveguide. In  FIG. 23(A) , the optical mode was solved using a finite-difference based Hermetian Eigensolver, such as that described by A. Taflove,  Computational Electrodynamics , (Artech House, Boston. MA, 1995), and has an effective index of approximately 1.85 at 1500 nm. Most of the electric field is parallel to the plane of the chip, and it is possible to contact both sides of the slot in a slotted ring resonator, as shown in  FIG. 23(B) . Panel B of  FIG. 23  shows a SEM image of the resonator electrical contacts. Electrically isolated contacts between the silicon rails defining the slotted waveguide introduce only about 0.1 dB of optical loss. Panel C of  FIG. 23  shows the logical layout of device, superimposed on a SEM image of a device.  FIG. 23(C)  details the layout of a complete slotted ring resonator, with two contact pads connected to the outer half of the ring, and two pads electrically connected to the inner half of the ring. A shunt resistor provides a means of confirming electrical contact, and typical pad-to-pad and pad-to-ring resistances range from 1 MO to 5 MO.  FIG. 23(D)  displays a typical electrically contacted slotted ring described in this study. Panel D of  FIG. 23  is an image of the ring and the electrical contact structures. 
     Measurements were performed with single-mode polarization maintaining input and output fibers, grating coupled to slotted waveguides with an insertion loss of approximately 8 dB. Optical signal was provided from an Agilent 81680a tunable laser and in some cases an erbium doped fiber amplifier (“EDFA”) from Keopsys Corporation. A continuous optical signal inserted into a poled polymer ring results in a measurable current established between the two pads, which are electrically connected through a pico-Ammeter. In the most sensitive device, a DC current of ˜1.3 nA was observed, indicating an electrical output power of ˜10 −9  of the optical input power (5×10 −12  W of output for approximately 0.5 mW coupled into the chip). Control devices, in which PMMA or un-poled EO material was substituted, show no photocurrent. 
     The fact that there is no external bias (or indeed any energy source) other than the optical signal applied to the system of this embodiment demonstrates conclusively that power is being converted from the optical signal. To establish that the conversion mechanism is actually optical rectification, we performed a number of additional measurements. A steady bias was applied to the chip for several minutes, as shown in Table IA. A substantial change in the photoresponse of the device was observed. This change depends on the polarity of the bias voltage, consistent with the expected influence of repoling of the device in-place at room temperature. Specifically, if the external bias was applied opposing the original poling direction, conversion efficiency generally decreased, while an external bias in the direction of the original poling field increased conversion efficiency. 
     In the present invention, we understand that an optical material can be subject to spatially periodic repoling of the electrooptic material, for example to provide a particular functionality, such as a nonlinear or exponential functionality or behavior. 
     
       
         
           
               
             
               
                 TABLE I 
               
               
                   
               
               
                 Poling Results 
               
               
                   
               
             
            
               
                 Part A: 
               
            
           
           
               
               
               
            
               
                   
                 Action 
                 New Steady State Current (6 dBm input) 
               
               
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Initial State 
                 −5.7 pA 
                   
               
               
                   
                 +10 V for 2 minutes 
                 0 pA 
               
               
                   
                 −10 V for 2 minutes 
                 −7.1 pA 
               
               
                   
                 +10 V for 2 minutes 
                 −4.4 pA 
               
               
                   
                 +10 V for 4 minutes 
                 −6.1 pA 
               
               
                   
                 −10 V for 4 minutes 
                 −4.5 pA 
               
               
                   
                 −10 V for 2 minutes 
                 −14.8 pA 
               
               
                   
                   
               
            
           
           
               
            
               
                 Part B: 
               
            
           
           
               
               
               
            
               
                   
                   
                 Current Polarity of 
               
               
                 Device 
                 Action 
                 Optical Rectification 
               
               
                   
               
               
                 1 
                 Positive Poling 
                 Positive 
               
               
                 1 
                 Thermal Cycling to 
                 Rapid fluctuation, did 
               
               
                   
                 poling temperature 
                 not settle 
               
               
                   
                 with no voltage 
               
               
                 1 
                 Negative Poling 
                 Negative 
               
               
                 2 
                 Negative Poling 
                 Negative 
               
               
                 2 
                 Thermal Cycling to 
                 None observable 
               
               
                   
                 Poling temperature 
               
               
                   
                 with no voltage 
               
               
                 2 
                 Positive Poling 
                 Negative 
               
               
                 3 
                 Negative Poling 
                 Negative 
               
               
                 4 
                 Positive Poling 
                 Positive 
               
               
                 5 
                 Negative Poling 
                 Negative 
               
               
                   
               
            
           
         
       
     
     To further understand the photo-conversion mechanism, 5 EO detection devices were poled with both positive and negative polarities, thus reversing the direction of the relative χ 2  tensors. For these materials, the direction of χ 2  is known to align with the polling E field direction, and we have verified this through Pockels&#39; effect measurements. In all but one case, we observe that the polarity of the generated potential is the same as that used in poling, and the +V terminal during poling acts as the −V terminal in spontaneous current generation, as shown in Table 1B. Furthermore, the polarity of the current is consistent with a virtual voltage source induced through optical rectification. It was observed that these devices decay significantly over the course of testing, and that in one case the polarity of the output current was even observed to spontaneously switch after extensive testing. However, the initial behavior of the devices after polling seems largely correlated to the χ 2  direction. 
     Part A of Table I shows the dependence of the steady state observed current after room temperature biasing with various voltage polarities for one device. The device was originally polled with a ˜12 V bias, though at 110 C. With one exception, applying a voltage in the direction of the original polling voltage enhances current conversion efficiencies, while applying a voltage against the direction of the polling voltage reduces the current conversion efficiencies. It should be noted that the power coupled on-chip in these measurements was less than 1 mW due to coupler loss. 
     Part B of Table I shows the behavior of several different devices immediately after thermal polling or cycling without voltage. Measurements were taken sequentially from top to bottom for a given device. The only anomaly is the third measurement on device  2 ; this was after significant testing, and the current observed was substantially less than was observed in previous tests on the same device. We suspect that the polymer was degraded by repeated testing in this case. 
     A number of measurements were performed to attempt to produce negative results, and to exclude the possibility of a mistaken measurement of photocurrent. The power input to the chip was turned on and off by simply moving the fiber array away from the chip mechanically, without changing the circuit electrically, and the expected change in the electrical output signal of our detector was observed. A chip was coated in polymethylmethacrylate and tested, resulting in no observed photocurrents. Also, when some of the devices shown in Table I were tested before any polling had been performed; no current was observed. 
     We used a lock-in amplifier to establish a quantitative relationship between the laser power in the EQ material and the photo-current, and achieved a noise floor of about 0.2 pA. This resulted in a reasonable dynamic range for the 10-200 pA photocurrent readings.  FIG. 24(A)  and  FIG. 24(B)  show optical transmission curves for typical devices.  FIG. 24(C)  shows several traces of output current versus input laser power, and a fairly linear relationship is observed. The relationship I=cP, where I is the output current, P is the input laser power, and c is a proportionality constant ranging from 88+/−10 pA/mW at a 1 kHz lock-in measurement and when the wavelength is on resonance, changing to a lower value of 58+/−8 pA/mW off resonance for the best device. It is important to note that current was easily observed with only a pico-ammeter, or by simply connecting an oscilloscope to the output terminal and observing the voltage deflection. 
     Panel A of  FIG. 24  shows the transmission spectrum of detector device I. Panel B of  FIG. 24  shows the transmission spectrum of detector device  2 . Panel C of  FIG. 24  shows several curves of current vs. power for three measurement series. Series  1  is of the first device with the wavelength at 1549.26 nm, on a resonance peak. Series  2  is the first device with the wavelength at 1550.5 nm off resonance. Series  3  is for device  2 , with the wavelength at 1551.3 nm, on resonance. Finally, panel D of  FIG. 24  shows the output current as a function of wavelength, overlaid with the transmission spectrum. The transmission spectrum has been arbitrarily resealed to show the contrast. 
     As another demonstration of the dependence of the output current on the amount of light coupled into the resonator, we also tuned the laser frequency and measured the output current. As can be seen in  FIG. 24(D) , the amount of output current increases as the laser is tuned onto a resonance peak. This again indicates that the overlap between the EO polymer in the resonator and the optical mode is responsible for the photo-current. We have overlaid a photocurrent vs. wavelength response scan to show the resonance peaks for comparison. It should not be surprising that a small photocurrent is still measured when the laser is off resonance, since the amount of radiation in a low-Q ring resonator is non-negligible oven off resonance. We have successfully observed this detector function at speeds up to 1 MHz, without significant observable rolloff. This is again consistent with optical rectification. Unfortunately, our devices could not be measured at higher speeds, due to substantial output impedance. 
     The conversion efficiency from our first measurements is thought to be several orders of magnitude below the ultimate limit, and can be explained by the high insertion losses in our system. In the present embodiment, 75% of the input power in the fiber is not coupled onto the chip. Our low-Q resonators only provide a limited path length within which light can interact with the electro-optic material. Furthermore, by design a great deal of the light in the resonator will be dumped to an output port, and not absorbed. It is expected that with further design and higher Q resonators, the efficiency of these devices can be greatly increased. It is, however, important to note that nothing about this effect depends on the presence of rings. The rings provide a convenient and compact device for observing these effects, but one could just as easily observe optical rectification by using other geometries, such as a long linear, polymer coated, split waveguide, with each side connected to an electrical pad. 
     Pockels&#39; Effect Modulation 
     At DC, the Pockels effect was measured by applying varying voltages to the device and observing the device transmission as a function of wavelength. For devices having operative modulation, the resonance peaks were shifted, often to a noticeable degree. To counter the systemic drift due to temperature fluctuations, a series of random voltages were applied to a device under test and the wavelength responses noted. The intersection of a resonance peak and a certain extinction, chosen to be at least 10 dB above the noise floor, was followed across multiple scans. A 2d linear regression was performed, resulting in two coefficients, one relating drift to time, and one relating drift to voltage. 
     At AC, a square wave input voltage was applied across the device. The input wavelength was tuned until the output signal had the maximum extinction. It was determined what power levels were implied by the output voltage, and then the observed power levels were fit to a wavelength sweep of the resonance peak. This readily allowed the tuning range to be calculated. We successfully measured AC tuning up to the low MHz regime. The limitation at these frequencies was noise in our electrical driving signal path, and not, as far as we can tell, any rolloff in the modulation process itself. 
       FIG. 25  is a diagram showing the use of the structures embodying the invention as resonantly enhanced electro-optic modulators, and a result at approximately 6 MHz operating frequency, representing a bit pattern generated by Pockels&#39; Effect modulation of 5 dB. The vertical axis represents input voltage and output power, both in arbitrary units. The horizontal axis represents time in units of microseconds. Voltage swing on the input signal is 20 volts. These measurements clearly demonstrate that low-voltage electro-optic tuning and modulation can be achieved in the same geometries as have been described for photodetection. It should be emphasized that these devices are not optimized as modulators. By increasing the Q of the resonators to exceed 20,000, which has been described hereinabove, it will be possible to achieve much larger extinction values per applied voltage. 
     By utilizing new dendrimer-based electro-optic materials, we have achieved 0.042±008 nm/V, or 5.2±1 GHz/V for these rings. This implies an r 33  of 79±15 pm/V. This result is better than those obtained for rings of 750 micron radius, which we believe to be the best tuning figure published to date. By contrast, our rings have radii of 40 microns. We credit our improvement over the previous results mainly to the field enhancement properties of our waveguide geometry. 
     Additional Results 
     Optical modulators are a fundamental component of optical data transmission systems. They are used to convert electrical voltage into amplitude modulation of an optical carrier frequency, and they can serve as the gateway from the electrical to the optical domain. High-bandwidth optical signals can be transmitted through optical fibers with low loss and low latency. All practical high-speed modulators that are in use today require input voltage shifts on the order of 1V to obtain full extinction. However it is extremely advantageous in terms of noise performance for modulators to operate at lower drive voltages. Many sensors and antennas generate only millivolts or less. As a result it is often necessary to include an amplifier in optical transmission systems, which often limits system performance. By using silicon nano-slot waveguide designs and optical polymers, it is possible today to construct millivolt-scale, broadband modulators. In some embodiments, a millivolt-scale signal is one having a magnitude of hundreds of millivolts. In some embodiments, a millivolt-scale signal is one having a magnitude of tens of millivolts. In some embodiments, a millivolt-scale signal is one having a magnitude of units of millivolts. Using novel nanostructured waveguide designs, we have demonstrated a 100× improvement in Vp over conventional electro-optic polymer modulators. 
     A variety of physical effects are available to produce optical modulation, including the acousto-optic effect, the Pockels effect either in hard materials, such as lithium niobate or in electro-optic polymers, free-carrier or plasma effects, electro-absorption, and thermal modulation. For many types of optical modulation, the basic design of a modulator is similar; a region of waveguide on one arm of a Mach-Zehnder interferometer is made to include an active optical material that changes index in response to an external signal. This might be, for instance, a waveguide of lithium niobate, or a semiconductor waveguide in silicon. In both cases, a voltage is introduced to the waveguide region by means of external electrodes. This causes the active region to shift in index slightly, causing a phase delay on the light traveling down one arm of the modulator. When the light in that arm is recombined with light that traveled down a reference arm, the phase difference between the two signals causes the combined signal to change in amplitude, with this change depending on the amount of phase delay induced on the phase modulation arm. Other schemes, where both arms are modulated in order to improve performance, are also common. 
     The measure of the strength of a modulation effect is how much phase shift is obtained for a given input voltage. Typical conventional modulators obtain effective index shifts on the order of 0.004% for 1 V. This implies that a Mach-Zehnder 1 cm in length, meant to modulate radiation near 1550 nm, would require 1 V of external input for the arms to accumulate a relative phase shift of p radians. The half wave voltage V p  (or V pi ) is the voltage needed for an interarm phase shift of p radians (or 180 degrees). Lower values for V p  imply that less power is needed to operate the modulator. Often, the responsivity, a length-independent product V p −L is reported. Typical V p −L values are in the range of 8 Vcm in silicon, or 6 V-cm for lithium niobate modulators. This voltage-length product, or responsivity, is an important figure of merit for examining a novel modulator design. Making a modulator physically longer generally trades lower halfwave voltage against reduced operating frequency and higher loss. Because generating high-speed and high-power signals requires specialized amplifiers, particularly if broadband performance is required, lowering the operating voltage of modulators is extremely desirable, particularly for on-chip integrated electronic/photonic applications, (including chip-to-chip interconnects) where on-chip voltages are limited to levels available in CMOS.  FIG. 27  shows a diagram of a Mach-Zehnder modulator with a conventional electrode geometry. 
       FIG. 27  is a top-down view of a simple conventional Mach-Zehnder polymer interferometer, showing top contact, waveguide, and bottom contact layers. Such a device is usually operated in ‘push/pull’ mode, where either opposite voltages are applied to the different arms, or where the two arms are poled in opposite directions to achieve the same effect. 
     In the past several years, silicon has gained attention as an ideal optical material for integrated optics, in particular at telecommunications wavelengths. Low loss optical devices have been built, and modulation obtained through free carrier effects. One of the waveguides that can be supported by silicon is the so-called slot waveguide geometry. This involves two ridges of silicon placed close to each other, with a small gap between them. As shown above with regard to  FIGS. 23 ,  24  and  25 , we have demonstrated modulation regions based on filling this gap with a nonlinear material, and using the two waveguide halves as electrodes. In such a geometry, the silicon is doped to a level that allows electrical conductivity without causing substantial optical losses. This allows the two wires or ridges to serve both as transparent electrical contacts and as an optical waveguide. 
     Using slot waveguides, we previously obtained an improvement in modulation strength of nearly 5× when compared to the best contemporary conventional waveguide geometries with electrodes separated from the waveguide, with the initial, non-optimized designs. This improvement was based on the remarkably small width of the gap across which the driving voltage drops. It is expected that smaller gaps translate into higher field per Volt, and the Pockels Effect depends on the local strength of the electric field. The smaller the gap, the larger the index shift. A unique property of slot waveguides is that, even as these gaps become nanoscale, the divergence conditions on the electric field require that much of the optical mode remains within the central gap. As a result, changing the index within a nanoscale gap can give a remarkably large change in the waveguide effective index. Because of these divergence conditions, the optical mode&#39;s effective index is largely determined by the shift found even in very small gaps. 
     Low V p  Modulators 
     Several major approaches toward achieving low V p  modulation have recently been pursued. The free-carrier dispersion effect in silicon waveguides has been used. Green et al. achieved a V p  of 1.8 V with this effect. Modulators based on lithium niobate are also frequently used. Typical commercially obtained V p  values are 4 V. Recently, Mathine and co-workers have demonstrated a nonlinear polymer based modulator with a V p  of 0.65 V. For the devices produced by others, the attained values of V p  are large. 
     A number of approaches have been proposed for developing low V p  modulators. Different proposed approaches rely on the development of new electrooptic materials, or on optical designs that trade bandwidth for sensitivity, either through the use of resonant enhancement, or through dispersion engineering. The designs presented herein are based upon conventional, high-bandwidth Mach-Zehnder traveling wave approaches, but achieve appreciable benefits from using nano-slot waveguides. Of course, these designs can also take advantage of the newest and best electrooptic polymers. In principle, any material that can be coated conformally onto the surface of the silicon waveguides and that is reasonably resistive could be used to provide modulation in these systems, making the system extremely general. 
       FIG. 28  is an isometric three dimensional schematic of a conventional Mach-Zehnder polymer interferometer, showing top contact, waveguide, and bottom contact layers. Such a device is usually operated in ‘push/pull’ mode, where either opposite voltages are applied to the different arms, or where the two arms are poled in opposite directions to achieve the same effect. 
       FIG. 29  is a three dimensional, isometric schematic of a slot-waveguide modulator, showing the slot waveguide, segmentation region and metal contacts. The device illustrated in  FIG. 29  functions by maintaining the two arms of the slot waveguide at differing voltages, creating a strong electric field in the slot. 
       FIG. 30  is a top-down view of a layout of a slot-waveguide based optical modulator of the device in  FIG. 29 . 
     The nonlinear polymers that have been used with slot waveguides exhibit a local anisotropic shift in their dielectric constant when they are exposed to an electric field. This is characterized by r 33 , which is a component of the electro-optic tensor. A simplification is appropriate to the case of slot waveguides, where the poling field, the modulation field, and the optical electric field are all nearly parallel. In this case, r 33  is defined as: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         
                           ( 
                           
                             n 
                             + 
                             
                               δ 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                           ) 
                         
                         2 
                       
                     
                     - 
                     
                       1 
                       
                         n 
                         2 
                       
                     
                   
                   = 
                   
                     
                       r 
                       33 
                     
                      
                     
                       E 
                       
                         d 
                          
                         
                             
                         
                          
                         c 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     4 
                      
                     
                         
                     
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     That is, a shift in the bulk index for this particular polarization is defined as a product of r 33  and the modulating electric field. 
     We have developed an analytic model to express the modulation strength that will be observed in a given slot waveguide geometry. Assuming that a nonlinear electro-optic polymer is used, the local shift in dielectric constant can be expressed as Eqn. (4b): 
       δ∈=| E   dc   |vv′ ( n   4   r   33 )  (4b)
 
     Here v is the unit vector of the direction of the dc electric field, and n is the bulk refractive index of the nonlinear polymer. Note that de is a second rank tensor in Eqn. (4b). It has been assumed that the poling dc field is identical to the modulation dc field. Nonlinear polymers have become increasingly strong in recent years, with some of the most recently developed material having an r 33  of 500 pm/V. This corresponds to an on axis χ 2  moment of 4.2×10 −9  m/V. 
     With the optical mode known, the shift in effective index is given by Eqn. (5): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       n 
                     
                     
                       ∂ 
                       V 
                     
                   
                   = 
                   
                     γ 
                      
                     
                       ( 
                       
                         
                           n 
                           4 
                         
                          
                         
                           r 
                           33 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The key parameter for any waveguide involving a nonlinear electro-optic material is γ, which we term the effective index susceptibility. γ is independent of the nonlinear material properties, and depends only on the waveguide geometry, and is given by Eqn. (6): 
     
       
         
           
             
               
                 
                   γ 
                   = 
                   
                     
                       
                         ∫ 
                         
                           
                             
                                
                               
                                 
                                   E 
                                   opt 
                                 
                                 · 
                                 v 
                               
                                
                             
                             2 
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             
                               w 
                                
                               
                                 ( 
                                 
                                   
                                     E 
                                     
                                       d 
                                        
                                       
                                           
                                       
                                        
                                       c 
                                     
                                   
                                   · 
                                   v 
                                 
                                 ) 
                               
                             
                             / 
                             V 
                           
                            
                           
                              
                             A 
                           
                         
                       
                       
                         ∫ 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             Re 
                              
                             
                               ( 
                               
                                 
                                   
                                     Ex 
                                     opt 
                                     * 
                                   
                                    
                                   
                                     Hy 
                                     opt 
                                   
                                 
                                 - 
                                 
                                   
                                     Ey 
                                     opt 
                                     * 
                                   
                                    
                                   
                                     Hx 
                                     opt 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             A 
                           
                         
                       
                     
                      
                     
                       1 
                       
                         k 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The ultimate V p  that can be obtained is inversely proportional to γ. It is noteworthy that this model accurately predicts Steier et al.&#39;s results, as shown described below. 
     For a conventional all-polymer geometry with electrodes external to the waveguide, γ is 0.026 μm −1 . For the slot waveguide that we used in our previous experiments, γ was 0.4 μm −1 . Finally, for a more optimal design, shown in  FIG. 33 , has a γ of 2.3 μm −1 . This design comprises a 200 nm thick silicon-on-insulator layer on a silicon dioxide substrate that is etched to create arms with widths of 200 nm with a 20 nm gap between them, which is described in more detail as design #3 presented in Table 2 below. This geometry enjoys an increase of about a factor of 100 in the tuning sensitivity compared to the conventional electrode geometry; this corresponds to a decrease by a factor of 100 in the Vp needed for modulation. These numbers assume a minimum lithographic linewidth of 20 nm, which is easily achievable today with electron beam lithography. Narrower linewidths are expected to further improve the achievable performance. 
       FIG. 31A  and  FIG. 31B  show a conventional electrode geometry for a nonlinear polymer waveguide described by Tazawa and Steier (H. Tazawa, Y. Kuo, I. Dunayevskiy, J. Luo, A. K. Y. Jen, H. Fetterman and W. Steier, “Ring resonator based electrooptic polymer traveling-wave modulator,” IEEE J. Lightwave Technol. 24, 3514-3519 (2006) and Tazawa, H. &amp; Steier, W. H., “Analysis of ring resonator-based traveling-wave modulators,”  IEEE Photonics Technology Letters  18, 211-213 (2006)).  FIG. 31A  shows the optical mode with |E| plotted in increments of 10%, for a mode with propagating power of 1 Watt.  FIG. 31B  shows a contour plot of the static electric field, with the field of view slightly enlarged.  FIG. 31C  and  FIG. 31D  show analogous data for an improved slot waveguide geometry according to the present invention. In the slot waveguide, the silicon provides both the optical guiding layer and the electrical contacts. 
     The most recent nonlinear polymers achieve a high nonlinear coefficient, expressed as an r 33  of 500 pm/V. Using this in combination with the high susceptibilities described above, it is believed that it is possible today to construct a 1 cm Mach-Zehnder modulator with a V p  of 8 mV. This corresponds to a ring resonator with a tuning sensitivity of 795 GHz/V. Both of these values are two orders of magnitude better than the performance obtained by current approaches. Current commercially available modulators typically have Vp&#39;s from 1 to 9 V, and current tunable electro-optic polymer based resonators achieve 1 GHz/V of tunability. If the r 33  value of 33 pm/V demonstrated by Tazawa and Steier for conventional polymer designs is used, then a V, of 64 mV and a resonator tunability of 50 GHz/V are obtained. 
     Segmented waveguide contact structures can be formed that allow very low resistance electrical contact to slot waveguides. We have described above, in similar circumstances, electrical contact to waveguides can be established via segmented waveguides. See  FIG. 23B  and  FIG. 23D  and the discussion related thereto. When the RC circuits implied by the segmentation geometry and the gap are examined, it is found that RC turn on times on the order of 200 GHz or more are achievable. Because the nonlinear polymers exhibit an ultrafast nonlinearity, these waveguide geometries present a path to making Terahertz scale optical modulators. Because the modulation is so strong, it is also possible to trade the length of the modulator against V. For example, our optimal geometry is expected obtain a Vp of 0.6 V with a 100 μm long Mach-Zehnder modulator. This device is expected be exceptionally simple to design for 10 GHz operation, as it could likely be treated as a lumped element. We have shown above that lateral contact structures with low loss and low resistance can be constructed with these slot waveguides. See  FIG. 23B  and  FIG. 23D  and the discussion related thereto. 
     We believe these nano-slot waveguide designs present a path to realizing very high speed, low voltage modulators. It is advantageous to be able to attain a responsivity V p −L of less than 1 V-cm. The physical principles involved in such devices are based on employing a nonlinear material of at least moderate resistivity, and a high index contrast waveguide with tight lithographic tolerances. Therefore, it is expected that nano-slot waveguides, either as Mach-Zehnder or ring-based devices, are likely an advantageous geometry for optical modulation with nonlinear materials in many situations. In addition, materials compatibility and processing issues are greatly reduced for such devices compared to conventional multilayer patterned polymer modulator structures. 
     These high index contrast devices have (or are expected to have) extremely small bend radii, which are often orders of magnitude smaller than corresponding all-polymer designs with low loss and high Q. These geometric features translate into extremely high free spectral ranges for ring modulators, compact devices, and wide process latitudes for their fabrication. Given the inexpensive and readily available foundry SOI and silicon processes available today, and the commercial availability of electron beam lithography at sub-10 nm line resolution, it is expected that slot-waveguide based modulators are likely to replace conventional modulators in many applications in the coming years. 
     Waveguide Susceptibility 
     The primary design goal of any electro-optic waveguide geometry is to maximize the amount of shift in effective index that can occur due to an external voltage. The exact modal patterns for these waveguides can be calculated using a Hermetian eigensolver on the FDTD grid. Once the modal patterns are known, the shift in effective index due to an index shift in part of the waveguide can be readily calculated. The static electric field due to the two waveguide arms acting as electrodes can be calculated by simply solving the Poisson equation. 
     The use of nonlinear polymers with slot waveguides provides an anisotropic effect on the local dielectric constant of the material when exposed to an electric field. The local shift in relative dielectric constant for the optical frequency can be expressed as in Eqn. (4) above. 
     Consider the x-y plane to be the plane of the waveguide, while the z direction is the direction of propagation. In this case, the total shift in effective index for the optical mode can be calculated to be that given in Eqn (7): 
     
       
         
           
             
               
                 
                   
                     δ 
                      
                     
                         
                     
                      
                     n 
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           
                             
                                
                               
                                 
                                   E 
                                   opt 
                                 
                                 · 
                                 v 
                               
                                
                             
                             2 
                           
                            
                           
                             ɛ 
                             0 
                           
                            
                           
                             w 
                              
                             
                               ( 
                               
                                 
                                   E 
                                   
                                     d 
                                      
                                     
                                         
                                     
                                      
                                     c 
                                   
                                 
                                 · 
                                 v 
                               
                               ) 
                             
                           
                            
                           
                              
                             A 
                           
                         
                       
                       
                         ∫ 
                         
                           2 
                            
                           
                               
                           
                            
                           
                             Re 
                              
                             
                               ( 
                               
                                 
                                   
                                     Ex 
                                     opt 
                                     * 
                                   
                                    
                                   
                                     Hy 
                                     opt 
                                   
                                 
                                 - 
                                 
                                   
                                     Ey 
                                     opt 
                                     * 
                                   
                                    
                                   
                                     Hx 
                                     opt 
                                   
                                 
                               
                               ) 
                             
                           
                            
                           
                              
                             A 
                           
                         
                       
                     
                      
                     
                       1 
                       
                         k 
                         0 
                       
                     
                      
                     δ 
                      
                     
                         
                     
                      
                     
                       ɛ 
                        
                       
                         ( 
                         
                           
                             n 
                             4 
                           
                            
                           
                             r 
                             33 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     The integral in the numerator is taken over only regions where the nonlinear polymer has been deposited, while the integral in the denominator should be taken over all space. Note that Eqn. (7) presumes that in the poling process, any region where the nonlinear polymer is exposed to a dc field is poled to the maximal extent; that is, the maximal r 33  will be demonstrated in the resulting material. In regions where the dc field is very small, this is unlikely to be the case, but these regions already do not contribute to Eqn. (7) much anyway, so this approximation is unlikely to cause substantial error. 
     The particular strength of the polymer, however, is not directly relevant to the configuration of the waveguide geometry. It is convenient to factor the last term out of Eqn (7), leaving what we will define as the effective index susceptibility as shown above in Eqn (6). 
     Here V has been introduced, the external voltage that corresponds to E dc . The units of the effective index susceptibility are m −1 . The derivative of the effective index with respect to applied voltage is then as shown above in Eqn. (5). 
     This relationship expresses how much the effective index of the waveguide shifts in response to a change in index in one of its constituent parts. Before continuing, it is useful to note an approximate maximum value for Eqn (5). In the case that the mode were contained entirely inside a material of a given index, we would have n+δn+√{square root over (∈+δ∈)}. It is in this situation that the mode is maximally sensitive to a shift in the waveguide index. Thus, in the most sensitive case we would have the value of γ as given by Eqn. (8): 
       γ=1/(2 n )( E   d   ·v )/ V   (8)
 
     Here it has been assumed that the dc field is uniform over the entire waveguide region. This provides a useful approximate upper bound on the effective index susceptibility that we can expect to obtain from any waveguide design. 
     Before proceeding, however, we must consider how the performance of various active devices depend on the effective index susceptibility. A Mach-Zehnder modulator can be formed by having both arms made of a slot waveguide with infiltrated nonlinear polymer. Note that in Eqn. (6), there is no constraint on the sign of the shift in index. Therefore, a change in the sign of the voltage will change the sense of the shift in index shift. Modulator performance is often characterized by V p , the amount of voltage needed to obtain a relative p of phase shift between the two arms. The optimal modulator design, with one arm positively biased and one arm negatively biased, has a V p  given by Eqn. (9): 
     
       
         
           
             
               
                 
                   
                     V 
                     π 
                   
                   = 
                   
                     π 
                     
                       2 
                        
                       
                           
                       
                        
                       
                         k 
                         0 
                       
                        
                       
                         L 
                          
                         
                           ( 
                           
                             
                               ∂ 
                               n 
                             
                             
                               ∂ 
                               V 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     Multiplying both sides by L, we have: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       π 
                     
                      
                     L 
                   
                   = 
                   
                     π 
                     
                       2 
                        
                       
                           
                       
                        
                       
                         
                           k 
                           0 
                         
                          
                         
                           ( 
                           
                             
                               ∂ 
                               n 
                             
                             
                               ∂ 
                               V 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     9 
                      
                     
                         
                     
                      
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     Here L is the length of the Mach-Zehnder, and k 0  is the free space wavenumber of the optical signal under modulation. State of the art results for Vp&#39;s for optical modulators are currently on the order of 1-5 V. The tunability of a resonator and the value 1/(V p −L) for a Mach-Zehnder modulator are both proportional to the figure of merit, y. Thus, increasing the figure of merit will lead to better device performance for both ring and MZI geometries. 
     Ring resonators have also been used to enable optical signals to be modulated or switched based on a nonlinear polymer being modulated by an external voltage. In this case, the performance of the tunable ring resonator is usually reported in the frequency shift of a resonance peak due to an externally applied voltage. This can be expressed as shown in Eqn. (10): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       f 
                     
                     
                       ∂ 
                       V 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           c 
                           λ 
                         
                       
                        
                       
                         
                           ∂ 
                           n 
                         
                         
                           ∂ 
                           V 
                         
                       
                     
                     
                       ( 
                       
                         n 
                         - 
                         
                           λ 
                            
                           
                             
                               ∂ 
                               n 
                             
                             
                               ∂ 
                               λ 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Results of 1 GHz/V have recently been reported for ring resonators based on large electrodes. We have observed 5.2 GHz/V of tuning. 
     Waveguide Geometries 
     We now describe several different waveguide geometries, and show the effective index susceptibility as a function of the slot sizes of the waveguide. In all cases, the modes have been solved using the aforementioned Hermetian eigensolver, and Eqn. (5). The susceptibilities are calculated near a 1550 nm free space wavelength. However, the values obtained will not vary much from 1480 nm to 1600 nm as the modal pattern does not change significantly. In the embodiments described, the waveguides are composed of silicon, and assumed to rest on a layer of silicon dioxide. The top cladding is a nonlinear polymer with an index of 1.7. This is similar to the waveguide geometry that we have used in our modulation work described hereinabove.  FIG. 32  shows the static electric fields solved as part of analyzing waveguide design  1  with a gap of 40 nm, as described in Table 2. As one would expect, the field is nearly entirely concentrated inside the slot area. The field shown was calculated assuming a voltage difference of 1 Volt. It is slightly larger than simply the reciprocal of the gap size due to the singular nature of the solution to Poisson&#39;s equation near the corners of the waveguide. 
       FIGS. 32A and 32B  illustrate solved field patterns for the analysis of waveguide  1  at a 40 nm gap.  FIG. 32A  shows the static voltage potential field distribution due to charging the two electrodes.  FIG. 32B  shows the electric field due to the potential distribution. |E| is plotted in increments of 10%. 
     We have constrained ourselves to use waveguide geometries that have minimum feature sizes of at least 20 nm. These are near the minimum feature sizes that can be reliably fabricated using e-beam lithography. Table 2 lists a description of each type of waveguide studied. Each waveguide was studied for a number of different gap sizes. In all cases, the maximum susceptibility was obtained at the minimum gap size. The maximum gap size studied and the susceptibility at this point are also listed. In some cases, the study was terminated because at larger gap sizes, the mode is not supported; this is noted in Table 2. For multislot waveguide designs where there are N arms, there are N−1 gaps; the design presumes that alternating arms will be biased either at the input potential or ground. 
     Table 2 shows the effective index susceptibility for various waveguide designs. 
     The dependence of susceptibility on gap size is presented in  FIG. 33  for several waveguides. The susceptibility is approximately inversely proportional to gap size. 
     It is clear that within the regime of slotted waveguides, it is always advantageous to make the slot size smaller, at least down to the 20 nm gap we have studied. This causes the DC electric field to increase, while the optical mode tends to migrate into the slot region, preventing any falloff due to the optical mode failing to overlap the modulation region. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                 Waveguide 
                 Waveguide 
                 Arm Sizes 
                   
                   
               
               
                 Design 
                 Height (nm) 
                 (nm) 
                 Maximum γ (μm −1 ) 
                 Minimum γ (μm −1 ) 
               
               
                   
               
             
            
               
                 1 
                 100 
                 300, 300 
                 1.3, 20 nm gap 
                 .40, 140 nm gap 
               
               
                 2 
                 150 
                 300, 300 
                 1.6, 20 nm gap 
                 .68, 120 nm gap 
               
               
                 3 
                 200 
                 300, 300 
                 2.3, 20 nm gap 
                 .74, 120 nm gap 
               
               
                 4 
                 100 
                 400, 400 
                 1.1, 20 nm gap 
                 .67, 60 nm gap, 
               
               
                   
                   
                   
                   
                 modal limit 
               
               
                 5 
                 100 
                 250, 250 
                 1.2, 20 nm gap 
                 .56, 60 nm gap, 
               
               
                   
                   
                   
                   
                 modal limit 
               
               
                 6 
                 100 
                 300, 40, 300 
                 1.6, 20 nm gap 
                 .53, 80 nm gap, 
               
               
                   
                   
                   
                   
                 modal limit 
               
               
                 7 
                 100 
                 300, 40, 40, 
                 1.9, 20 nm gap 
                 .76, 60 nm gap, 
               
               
                   
                   
                 300 
                   
                 modal limit 
               
               
                 8 
                 200 
                 200, 40, 200 
                     3, 20 nm gap 
                 1.4, 60 nm gap, 
               
               
                   
                   
                   
                   
                 modal limit 
               
               
                 9 
                 300 
                 300, 300 
                 2.5, 20 nm gap 
                 2.5, 20 nm gap, 
               
               
                   
                   
                   
                   
                 modal limit 
               
               
                 Steier et al. 
                 N/A 
                 N/A 
                 .026, 10 μm gap  
                 N/A 
               
               
                   
               
            
           
         
       
     
     In examining the results of our calculations, it is useful to calculate the maximum susceptibilities that can be obtained. For an effective index of about 2, which is approximately correct for these waveguides, and a gap size of 20 nm, the maximum achievable γ is approximately 12.5 μm −1 . Thus, for a gap size of 20 nm, waveguide design  8  is already within 25% of the theoretical maximum value. 
     It is also worth noting the corresponding y value that can be obtained by calculation using our methods for the separated electrode approach of Steier. The effective index of the mode is expected to be about 1.8, and the gap distance for the dc field is 10 μm. Under the most optimistic assumptions about mode overlap with the active polymer region (that is, assuming complete overlap), this corresponds to a γ of about 0.03 μm −1 . 
     It is useful to calculate, given the current r 33  values that are available, the index tuning that might be achieved with these designs. The most advanced polymers now yield r 33  values of 500 pm/V, If a bulk refractive index of 1.7 is used, then a ∂n/∂V of 0.006 V −1  is obtained with the best design given above. Using a waveguide with an effective index of 2 and a group index of 3, which are typical of silicon-polymer nano-slot waveguides, the V p  for a Mach-Zehnder with a length of 1 cm is expected to be about 6 mV, The resonance shift that is expected to be obtained in a ring resonator configuration would be 380 GHz per volt. Both of these values represent orders of magnitude improvement in the performance of these devices compared to current designs. 
     Segmented Contacting 
     As we have shown empirically, silicon can be doped to about 0.025 Ω-cm of resistivity with a n-type dopant without substantially increasing losses. Other dopants or perhaps other high index waveguiding materials may have even higher conductivities that can be induced, without significantly degrading optical performance. However, it is known that the conductivity cannot be increased endlessly without impacting optical loss. 
     This naturally presents a serious challenge for the issue of driving a slot waveguide of any substantial length. Consider a slot waveguide arm of length 1 mm, formed of our optimal design. The capacitor formed by the gap between the two electrodes is about 0.25 pF. The ‘down the arm’ resistance of the structure, however, is 4 MΩ. Therefore, the turn on time of an active waveguide based on this is about 0.1 μS, implying a 10 MHz bandwidth. 
     A solution to this problem is presented by continuously contacting the waveguide via a segmented waveguide. This comprises contacting the two silicon ridges with a series of silicon arms. Even though the silicon aims destroy the continuous symmetry of the waveguide, for the proper choice of periodicity no loss occurs, and the mode is minimally distorted. This is because a Bloch mode is formed on the discrete lattice periodicity, with no added theoretical loss. Of course the performance of fabricated devices will be different from that of conventional slot waveguides due to fabrication process differences. We have previously demonstrated empirically that continuous electrical contact can be formed for non-slotted waveguide via segmentation with relatively low optical losses. 
     Here we present a simulation of a particular segmentation geometry for our optimal slot waveguide design, that with 200 nm tall and 300 nm wide arms and a gap of 20 nm. We have found that a segmentation with 40 nm arms, and a periodicity of 100 nm, appears to induce no loss or significant mode distortion in the waveguide. Around 2 um of clearance appears to be needed from the edge of the segmented waveguide to the end of the arms.  FIGS. 34A ,  34 B and  34 C show plots of several cross sections of the segmented slot waveguide with a plot of the modal pattern overlaid. For comparison, a cross section of the unsegmented slot waveguide is presented as well. Simulations were also performed to confirm that the index shift formula continued to apply to the segmented slotted waveguide. It was found that the index shift was in approximate agreement with the value predicted for the non-segmented case. Non-segmented modesolvers were used for the rest of the simulations in this work, because simulation of the segmented designs is radically more computationally burdensome than solving for the unsegmented case, as they require solving for the modes of a 3d structure. Since the index shifts for the unsegmented and segmented cases are extremely similar, solving for the modes in the unsegmented cases is adequate for purposes of design and proof-of-concept. 
       FIG. 34  A shows a cross section of the segmented, slotted waveguide, with the |E| field plotted in increments of 10% of max value.  FIG. 34B  shows a similar plot for the unsegmented waveguide.  FIG. 34C  shows a horizontal cross section of the segmented, slotted waveguide; Re(Ex) is plotted in increments of 20% of max. In an actual device, some sort of metal based transmission line would undoubtedly provide the driving voltage for the waveguide. The metal electrodes that would likely form part of this transmission line have been noted in  FIG. 34C . In all cases the mode has been normalized to have 1 Watt of propagating power.  FIG. 34A  and  FIG. 34C  show the location of the other respective cross section as a line denoted C in  FIG. 34A  and A in  FIG. 34C . 
     Assuming a 0.025 Ω-cm resistivity, one can calculate the outer arm resistance as 63 kΩ per side per period, while the inner arm resistance is 25 kΩ per side per period. The gap capacitance per period is 2.5×10 −17  Farads. This implies a bandwidth on the order of 200 GHz. 
     We now describe an electro-optic modulator fabricated from a silicon slot waveguide and clad in a nonlinear polymer. In this geometry, the electrodes form parts of the waveguide, and the modulator driving voltage drops across a 120 nm slot. As a result, a half wave voltage of 0.25 V is achieved near 1550 nm. This is one of the lowest values for any modulator obtained to date. As the nonlinear polymers are extremely resistive, our device also has the advantage of drawing almost no current. It is believed that this type of modulator could operate at exceedingly low power. 
     A unique advantage with nonlinear polymers is that an integrated optical circuit can be conformally coated by a nonlinear polymer. This property, when combined with a slot waveguide, enables the construction of a uniquely responsive modulator. We describe the use of a push-pull Mach-Zehnder modulator configuration in which each arm has an opposing bias, leading to an opposing phase shift. 
       FIG. 35(   a ) shows the slot waveguide used for the Mach-Zehnder modulator. The modal pattern near 1550 nm is plotted, and contours of |E| are shown.  FIG. 35(   b ) is an SEM micrograph of a slot waveguide. In this case, the slot waveguide is being coupled to with a ridge waveguide; this mode converter involves tiny gaps which ensure electrical isolation between the two arms. Contacting arms are also present around 3 μm from the ridge/slot junction. The dimensions are two 300×100 nm arms separated by a 120 nm slot. 
     Nonlinear polymers typically have very high resistivity of 10 11  Ωcm. As a result, the two silicon arms are electrically isolated and can be used as modulator electrodes. The voltage drop between the arms occurs across a 120 nm electrode spacing, as opposed to the 5-10 μm that is typically required for modulators involving a nonlinear polymer and metallic contacts. This is a fundamental advantage that slot waveguide geometries have for electro-optic modulation. 
     It is advantageous to contact the silicon arms with an external electrode throughout the length of the Mach-Zehnder device to minimize parasitic resistances. We use a segmented waveguide in which a periodic set of small arms touches both waveguide arms. We use a segmentation with a periodicity of 0.3 μm and arm size of 0.1 μm that is largely transparent to the optical mode. 
     Because the polymer has a second order nonlinearity, a Mach-Zehnder modulator can be operated in push-pull mode, even with no de bias, effectively doubling the modulator response.  FIG. 36(   a ) is a diagram of the modulator layout. Contacts A, B, and C are shown.  FIG. 36(   b ) and  FIG. 36(   c ) are two SEM micrographs that show the slotted, segmented region, as well as the location where the silicon makes contact with the electrical layer. 
     Devices were fabricated with electron beam lithography and dry etching. The second order nonlinear polymer YLD  124  doped 25% by weight into an inert host polymer (APC), was used as a coating. Mixing and poling were done in the standard fashion, and a poling field of 150 V/μm was used. Coupling on and off the chip was accomplished via grating couplers, which had a bandwidth of around 40 nm. Total device insertion losses were approximately −40 dB fiber to fiber. 
     Referring to  FIG. 36(   a ), there are three regions in the modulator that are capable of maintaining distinct voltages. During poling operation, contact A is given a voltage of 2V pole , contact B a voltage of V pole , and contact C is held at ground. To achieve a poling field of 150 V/μm, V pole  was 18 V. This has the effect of symmetrically orienting the polymer in the two Mach-Zehnder arms. During device operation, contact B is driven at the desired voltage, while contacts A and C are both held at ground, leading to asymmetric electric fields in the two arms for a single bias voltage. This is the source of the asymmetric phase response. Electrical regions A and C cross the waveguide by means of a slotted ridged waveguide. At the ridge to slot mode converter, a small gap is left that maintains electrical isolation but is optically transparent. This enables the device to be built without requiring any via layers. 
     A driving voltage from a DC voltage source was applied to contact B, while contacts A and C were held at ground.  FIG. 37(   a ) and  FIG. 37(   b ) show device transmission spectra for various drive voltages.  FIG. 37(   a ) is a diagram showing the transmission through the Mach-Zehnder device as a function of wavelength, for a modulator drive voltage of 0.2 V bias, and  FIG. 37(   b ) is a diagram showing the transmission through the Mach-Zehnder device as a function of wavelength, for a modulator drive voltage of 0.4 V bias. As can be seen, a 0.2 V bias is just short of the V p  voltage, while a 0.4 V bias is substantially past this point. 
     This data is for an unbalanced Mach-Zehnder with arm lengths of 2 and 2.01 cm. This difference in length is the source of the variation in transmission as a function of wavelength with approximately 10 nm periodicity. The more gradual variation is from the grating coupler bandwidth. 
       FIG. 37(   a ) and  FIG. 37(   b ) are consistent with a V p  voltage somewhere from 0.2 to 0.3 V. At exactly the V, value, the minima of the spectrum would coincide with the maxima of the 0 V bias spectrum. The slight ripple visible on the various spectra is probably due to back reflections from the grating coupler and scattering noise from the segmented regions. To more accurately measure the V p  value for the device, the drive voltage was varied for a constant laser wavelength at 1574 nm and the transmission observed. The V p  varied from 0.25 to 0.28 V in two examples, possibly due to thermal drift.  FIG. 38(   a ) and  FIG. 38(   b ) are diagrams that show traces of the transmission plotted against the bias voltage.  FIG. 38(   a ) and  FIG. 38(   b ) are diagrams illustrating the transmission through the device as a function of bias voltage. V p  values of 0.25 and 0.28 V are observed, respectively. 
       FIG. 38(   c ) is a diagram that shows the frequency response of the device that was also characterized. This was done by using a sinusoidal function generator and a lock-in amplifier on the output of the modulator. The modulator was biased at a p/2 bias point, corresponding to 3 dB of extinction, by setting the signal wavelength to the appropriate value, and a 0.2 V peak to peak signal was used. The slight variation in the response below 1 kHz is possibly due to slight glitches in the lock-in and function generator. 
     The device does begin to show severe falloff around 1 kHz. This is possibly due to an RC time constant implied by the capacitor formed by the slot waveguide in each Mach-Zehnder arm, Electrical testing with control structures revealed that the resistance of these small silicon regions is much higher than expected. Typical resistances across a 400 μm length of ridged, segmented waveguide were well in excess of 10 9  O, often so high as to be immeasurable. The capacitance of a 400 μm long slot waveguide is 12 fF, and so an RC time constant could easily approach a millisecond. Further fabrication work is expected to remedy this deficiency. In particular, it is believed that one can build segmented and slotted waveguides with intrinsic speed limitations of 70 GHz. It is expected that the ultrafast response of YLD  124  combined with the observed behavior will allow one to build modulators with exceptionally high bandwidth. The V p L figure of merit for this modulator can be calculated as 0.5 V cm. From this value, one can calculate the r 33  value achieved in the polymer  12  to be 30 pm/V. This is lower than the optimal r 33  of 100 pm/V for YLD  124 . It is likely that the polymer in the slot was not fully poled. 
     The small amount of cross-slot current in the device causes the power consumption to be negligible for steady drive voltage. It is expected that constructing the slot waveguide modulator on a silicon platform will allow the driving circuitry to be placed next to the modulator, obviating the need for a transmission line. In this case, an impedance matching resistor would not be needed and a sufficiently short Mach-Zehnder could have exceptionally low power consumption. An r 33  of only 30 pm/V was obtained for the devices as tested. As a result, if the r 33  values of 170 pm/V that have been demonstrated previously are obtained here, the V p  value should decrease by nearly a factor of 6. It is believed that slot waveguide-based nonlinear polymer modulators will prove to be an attractive approach for integrated electro-optic modulators. 
     One might doubt that a nonlinear polymer, however active, could change its effective index by as much as is implied by this figure. We stress that the actual driving voltage would probably be less than a volt in such an instance, and so the actual modulating electric field experienced by the polymer would not be much different from what is present in today&#39;s larger devices. The maximum drive voltage will be determined by the breakdown field in the polymer, which is typically approximately 300 V/μm. This implies for a 20 nm gap a maximum drive voltage of 6 V which is well within the range of reasonable voltages to be applied from an external source. 
     We have recently demonstrated empirically that slot sizes of around 70 nm can be fabricated in 110 nm SOI as ring resonators with electrical contacts, as shown in  FIG. 39 .  FIG. 39  is a diagram that shows a transmission spectrum of an electroded slot waveguide resonator with a gap of 70 nm. Fiber to fiber insertion loss is plotted in dB, against the test laser wavelength in nm.  FIG. 40  is a diagram that shows an SEM image of a portion of a typical slot waveguide with a sub-100 nm slot. The cursor width is 57 nm in this image. These waveguides would have figures of merit about a factor of 2 higher than we have previously achieved with 140 nm slots. A Q of around 8,000 has been obtained, which, when combined with the massive amounts of tuning we expect, should prove sufficient for many applications. The Q does not approach the value we have obtained with non-electroded slotted ring resonators due to excess loss from the electrical contacts. The same electrode geometry was used as in an earlier approach. We have also confirmed through electrical measurements that the two halves of the slots are largely electrically isolated. 
     We believe that there is the possibility of constructing even narrower slot waveguides, on the scale of 1-5 nm in thickness. For example, one could use epitaxial techniques to grow a horizontal slot structure (rather than the vertical structures we have explored thus far) with an active, insulating material, with silicon beneath and above. This could be done in a layer form analogous to SOI wafer technology, in which a very thin layer of electroactive material such as the polymers we have described herein could be introduced. Such structures offer the possibility of yet another order of magnitude of improvement in the low-voltage performance of modulators. Here we should also mention that we anticipate our slot structures to be fairly robust even in the presence of fabrication errors. Fabrication imperfections may cause some of the narrower slots to have tiny amounts of residual silicon or oxide in their centers, or to even be partially fused in places. As long as electrical isolation is obtained, and the optical loss is acceptable, we would expect the slot performance to decrease only in a linear proportion to the amount of the slot volume that is no longer available to the nonlinear polymer cladding. 
     Still other designs for waveguides have been considered.  FIG. 41  through  FIG. 45  illustrate additional options for waveguide designs, which are shown in elevation.  FIG. 41  through  FIG. 45  are illustrations showing waveguides in vertical section.  FIGS. 41 ,  44  and  45  include a thin (50 nm) silicon “wing” that extends horizontally on each side of the respective waveguides. The wing is continuous if viewed in plan, e.g., top down. The reader should understand that the waveguide is constructed with a length dimension perpendicular to the plane of the illustration. The dimensions of the elements of the waveguides shown in  FIG. 41  through  FIG. 45  are described in Table 3. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                   
                   
                   
                   
                   
                 Effective 
               
               
                   
                 Arm 
                 Slot 
                 Strip load 
                 γ 
                 index (near 
               
               
                 Figure 
                 dimensions 
                 dimension 
                 dimension 
                 μm −1   
                 1550 nm) 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 41 
                 300 by 200 nm 
                 100 nm 
                 50 nm 
                 0.66 
                 2.17 
               
               
                 42 
                 300 by 200 nm 
                 100 nm 
                 none 
                 0.88 
                 2.095 
               
               
                 43 
                 300 by 200 nm 
                  50 nm 
                 none 
                 1.7 
                 2.21 
               
               
                 44 
                 300 by 200 nm 
                  50 nm 
                 50 nm 
                 1.42 
                 2.27 
               
               
                 45 
                 150 by 220 nm 
                 100 nm 
                 50 
                 0.59 
                 1.89 
               
               
                   
               
            
           
         
       
     
     As can be seen from Table 3, the addition of a 50 nm strip load, which is seen in each of  FIGS. 41 ,  44 , and  45 , can provide some benefit in certain cases. The design with the strip load is also referred to as a slot plus winged waveguide, or as a rib waveguide. For the waveguide of  FIG. 45 , the mode is not tightly contained, and dn eff /de=0.059. 
     There are several potential optical loss sources in such system. The losses can include waveguide losses and coupler losses. The waveguide losses can arise from scattering from geometric imperfections, from substrate leakage (which would be expected to be negligible with a 3 μm oxide layer), and from surface state absorption. It is believed that these losses can be reduced with the proper surface treatments. In its expected that slot waveguide which typically exhibit ˜8 dB/cm of loss can be improved dramatically with good lithography and surface treatments, including etching or other smoothing methods, and deposition of layers to control optical properties, such as materials deposited using atomic layer deposition methods. 
     The expected coupler losses can be contained or reduced by using edge coupling, in which looses of &lt;2 dB have been demonstrated. Another possibility is the use of grating couplers to appreciably reduce the footprint of our devices. If one expects that there will be of the order of 3 dB of total coupler loss and about 3 dB of total waveguide loss, one needs to maintain the devices as short length dimension devices with as high activity per Volt-mm as can be obtained. 
     Yet additional designs are envisioned to provide improved modulator waveguides. In one embodiment, a lumped element design can be used. In this design, the arms of the Mach-Zehnder interferometer are treated as capacitors, and the use of a shunt resistor is envisioned with an off-chip driver, as shown in  FIG. 46 . The use of an on-chip driver would allow the use of less power at lower speeds. As used herein, the term “driver” is intended to denote a selected one or more of a power supply, an amplifier, and a control circuit. The resistance of silicon strip loaded contacts can be reduced via doping to less than 50 O levels. In addition, one or more detectors and associated control, detection, conversion and analysis hardware may be provided on- or off-chip. 
     Table 4 presents some estimated performance parameters for a number of strip loaded lumped element waveguides. In all instances, a value of V p  of 0.25 volts is used as a design parameter. 
     
       
         
           
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 4 
               
               
                   
               
               
                   
                   
                   
                   
                   
                 Device Internal 
                 Drive 
                   
                   
               
               
                   
                   
                   
                 Polymer 
                   
                 Power 
                 Power @ 
               
               
                 Length 
                 Gap 
                 C gap   
                 Activity 
                 Vp 
                 Consumption @ 
                 50 O 
                 f 3dB   
               
               
                 (mm) 
                 (nm) 
                 (fF) 
                 (pm/V) 
                 (V) 
                 20 GHz (μW) 
                 (μW) 
                 (GHz) 
                 Optimizing 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
            
               
                 1.65 
                 30 
                 500 
                 73 
                 0.25 
                 312 
                 625 
                 22 
                 Polymer 
               
               
                 0.4 
                 30 
                 120 
                 300 
                 0.25 
                 75 
                 625 
                 90 
                 Length 
               
               
                 2.75 
                 50 
                 500 
                 60 
                 0.25 
                 312 
                 625 
                 22 
                 Polymer 
               
               
                 0.55 
                 50 
                 100 
                 300 
                 0.25 
                 63 
                 625 
                 110 
                 Length 
               
               
                 5.5 
                 100 
                 500 
                 68 
                 0.25 
                 312 
                 625 
                 22 
                 Polymer 
               
               
                 1.2 
                 100 
                 108 
                 300 
                 0.25 
                 68 
                 625 
                 100 
                 Length 
               
               
                   
               
            
           
         
       
     
     Optimizing to reduce length also serves to reduce losses. For lengths greater than about 2 mm, the lumped element analysis is expected to break down. As previously indicated, there exist polymers with activity of approximately 600 pm/volt. It is expected that surface treatment will improve the optical losses as discussed hereinabove. 
     While the present description has been presented with regard to electro-optic materials that require poling, the inventors also contemplate the use of use of self-assembled EO materials in the slot to eliminate the need for poling. 
     We now present illustrative embodiments for devices that employ waveguides that enhance the nonlinearity that an optical mode would experience from a nonlinear polymer cladding, such as the slot waveguides in silicon-on-insulator substrates which concentrate the field sharply in the center of a slot, such as shown in  FIG. 4 . Third-order nonlinear polymers such as those described hereinabove, for example with regard to  FIG. 22  and  FIG. 26 , can be deposited over the chip and in particular in this slot, thus enabling the field enhancement to greatly increase the effective nonlinearity of a nano-scale SOI waveguide clad with nonlinear polymer as compared to, for example, a simple ridge waveguide design. In some embodiments, polymers exhibiting higher odd order nonlinear optical coefficients can also be used as the optically responsive medium. 
     Several illustrative designs are provided for devices that are believed to be practical with the higher available nonlinearities, and which could be integrated on chip in a silicon-polymer system. Using a series of Mach-Zehnder all-optical switches based on the Kerr effect, which become practical with higher nonlinearities, one can construct a variable delay line of particularly high switching speed. These devices are expected to be capable of multiplexing bitstreams into speed regimes that have heretofore been inaccessible. An illustrative design for a self-oscillator device is provided, which can employ several CW optical signals and generate a pulsed signal, by virtue of having the output of a Mach-Zehnder all-optical switch turn the switch on or off, depending on the state of the switch. An illustrative design for a clock signal generator operating at extremely high frequencies is provided in which a square wave clock signal at a first frequency is used as input, and the frequency is increased by use of an AND gate and a series of delay lines. 
     Each of the designs includes at least one input port that accepts an optical input signal, an output port at which an output optical signal appears or is provided, and an interaction region having at least one optical input signal that defines an interaction between the at least one input signal and another optical signal (including possibly a copy or a portion of the input signal itself). In general, the interaction region includes an input port (for example, a gate input port, a clock input port, a pump input port or the like) if there is only one input port for an input signal, but may not have an input port if there are a plurality of input ports for input signals (such as in an AND gate, which has two input ports, and in which the interaction region is lacking a gate input port). 
     Variable Delay Line 
       FIG. 48  is an illustrative diagram  4800  of a variable delay line device based on all-optical switches. An input signal  4810  is introduced at an input port located at a first end of the device. 
     Using a series of Mach-Zehnder all-optical switches, which become practical with higher nonlinearities, one can construct a variable delay line having a particularly high switching speed. A Mach-Zehnder all-optical switch relies on a Kerr induced phase shift to allow a gate optical mode to induce a phase shift on a signal optical mode. This phase shift then directs the signal to one of two possible output ports, resulting in an all-optical switch. Note that even if the switch is as much as a cm long or more, the effective speed can still be quite high, in the terahertz frequency range. 
     As illustrated in  FIG. 48 , three delay elements having respective delays ΔT 1 , ΔT 2 , and ΔT 3  are present in series in the delay line. Each delay element has a respective gate terminal. In this device, a series of optical clocking signals on gates  1 - 3  determine which, if any, of ΔT 1 , ΔT 2 , and ΔT 3  are added to the delay of the signal. The delayed signal  4820  is provided as output at an output port at a second terminal of the device. As indicated in  FIG. 48  schematically, the delay leg of delay element  3  is longer than the delay leg of delay element  2 , which is in turn longer than the delay leg of delay element  1 . Accordingly, for this device, it is expected that the delays ΔT 1 , ΔT 2 , and ΔT 3  have magnitudes in the relation ΔT 1 &lt;ΔT 2 &lt;ΔT 3 , with the precise values of the delay determined by the design parameters. 
     Multiplexer 
     A multiplexer is a device which can take a number of bitstreams at slower speeds and merge them into a fast bitstream.  FIG. 49  is a diagram  4900  showing an illustrative example of an all-optical multiplexer. As shown in  FIG. 49 , the multiplexer accepts two 100 GBit/sec streams (Bitstream  0  and Bitstream  1 ) as input at respective input ports and produces a single 200 GBit/sec stream (Muxed Bitstream) as output at an output port. A 200 GBit clock signal is provided at a gate port to control which signals are active. This could be built with a single 2×2 switch based on the all-optical Mach-Zehnder switching geometry. In this illustrative embodiment, two input bitstreams are switched via an optical clocking signal. Such a device is expected to be capable of multiplexing bitstreams of speeds up to 1 THz or more, because an ultrafast switching mechanism is used. 
     Self-Oscillator 
       FIG. 50  is a diagram  5000  of an illustrative all-optical self-oscillator. A self-oscillator is an optical device which takes CW inputs and produces oscillations. In the illustrative embodiment, an input signal is introduced at an input port of the device. 
     A wavelength converter and amplifier is provided which is based on the well-known third order nonlinear process that occurs between a powerful pump beam (Pump) and a small signal (for example, a portion of some input signal). As a result of the interaction, an idler signal is created at 2*wp−ws, and both signal and idler are amplified. 
     In the illustrative embodiment, the output from a Mach-Zehnder switch is passed through a wavelength converter and amplifier, and the output from the converter and amplifier is then used to switch the Mach-Zehnder switch. This causes the oscillator to switch itself on and off repeatedly, thereby creating an integrated oscillator. The output signal that appears at an output port is a modulated signal. 
     AND Gate 
       FIG. 51  is a diagram  5100  of an illustrative AND gate without a gain section when turned ON (that is, both inputs are on). One good way to construct an AND gate is in two stages. First, using two signals as input, four wave mixing is used in a length of waveguide to produce an output signal. The four wave mixing will be performed if and only if the two input signals are present, thus implementing a logical AND. In the course of four wave mixing, wavelengths at frequencies different from the frequencies of the two input signals are generated. One can then filter out at least one of the new wavelengths from the input beams to generate a resultant signal, which is at a new frequency. As desired, one can then amplify or convert the resultant signal to a new wavelength as needed, and the signal so created can be provided at an output port as an output signal. The output signal will be present (or will have a first defined value) if and only if both input signals were present, and will be absent (or will have a different value) if either or both of the input signals are absent. It is important to note that the bandwidth limitation is dispersion, and therefore operation up to at least 1 THz or more can still be obtained even if the device is several cm long. 
     One can of course provide a gate having a larger number than two of ANDed inputs by cascading AND gates. For example, to AND three inputs, one could use a first AND gate to generate a signal indicative of the AND of a first signal, and a second signal. One would provide the output of the first AND gate as one input of a second AND gate, and provide the third input signal at the second input port of the second AND gate. The output of the cascaded gates would be “true” (or “one” or “ON”) if and only if all three of the input signals were simultaneously “true” and would be “false” (or “zero” or “OFF”) otherwise. 
     Clock Multiplier 
       FIG. 52  is a diagram  5200  of an illustrative clock multiplier. Once an AND gate can be realized, a clock multiplier can be constructed as is now described. Two input clock signals  5210  and  5220 , each at frequency f and period T, are introduced at respective input ports. The input signals are square waves. One signal is delayed relative to the other by a delay given by T/4. The two waves are passed into an AND gate. 
     The result of this operation is a partial clock signal  5230  at frequency 2/T as indicated in  FIG. 52 . This signal can be converted to a complete clock signal  5240  by splitting the signal and adding a relative delay of T/2 to one side, and then recombining. The resulting signal  5240  can be amplified to account for any lost strength, and then a full clock signal at frequency 2/T results. 
     Note that the input to such a gate can be a square wave intensity modulation on two different frequencies of optical radiation, which can be readily obtained from simply wavelength converting a single square wave intensity modulated frequency. 
     This frequency doubling procedure can be repeated (iterated) to generate ever higher clock rates. Another possibility is to obtain a speed multiplier of N by using an initial relative delay of T/(2*N)−T/2, splitting the signal into N components, and then employing N relative delay lines of delay  0 , T/N, 2*T/N, (N−1)*T/N. and recombining the resulting signals. 
     Applications 
     Contemplated applications include: the use of nano-scale ridge, rib or slot waveguides combined with a nonlinear polymer cladding to enhance the nonlinearity of a waveguide; the use of nonlinear polymer-clad slot and ridge nano-scale waveguides to construct a variable delay line with all-optical switches; the use of nonlinear polymer-clad slot and ridge nano-scale waveguides to construct a multiplexer based on a high speed all-optical switch; the use of nonlinear polymer-clad slot and ridge nano-scale waveguides to construct a self-oscillator by feeding the output of an all-optical switch through an amplification and conversion waveguide and then used as the gate optical mode of the self-oscillator; the use of nonlinear polymer-clad slot and ridge nano-scale waveguides to construct a logic gate, such as an AND gate, an OR gate, a NAND a NOR, and an XOR gate; and the use of nonlinear polymer-clad slot and ridge nano-scale waveguides to construct an ultrafast clock multiplier based on a combination of an all-optical AND gate and a series of delay lines used to recombine the pulses that can be created in this fashion. 
     Theoretical Discussion 
     Optical Rectification Theory 
     The general governing equation of nonlinear optics is known to be: 
         D   i =∈ 0 (∈ r   E   i +χ ijk   2   E   j   E   k + . . . )  (11)
 
     Our EO polymers are designed to exhibit a relatively strong χ 2  moment, ranging from 10-100 pm/V. In most χ 2  EO polymer systems, the Pockel&#39;s effect is used to allow the electric field (due to a DC or RF modulation signal) to modify the index of refraction. In such systems the modulating electric field is typically far in excess of the electric field from the optical signal and the term that produces the material birefringence is the only term of importance in the above equation. 
     Our waveguides, however, have a very large electric field as most of the radiation is confined to a 0.01 square micron cross section. It can be shown that the electric field is approximately uniform in the transverse direction, with a magnitude of 
     
       
         
           
             
               
                 
                   
                     10 
                     8 
                   
                    
                   
                     P 
                   
                    
                   
                     V 
                     m 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where P is the optical power in Watts. At large optical fields, the non-Pockels terms involved in the governing nonlinear equation cannot be neglected. For coherent input power, at a given location in the waveguide, the optical field is: 
         E   optical ( t )= A cos ( wt +θ)  (13)
 
     The term 
     
       
         
           
             
               
                 
                   
                     E 
                     optical 
                     2 
                   
                   = 
                   
                     
                       
                         
                           A 
                           2 
                         
                         2 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               ( 
                               
                                 wt 
                                 + 
                                 θ 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         A 
                         2 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     will therefore contain not only frequency doubled components, but also a “DC” component. This phenomenon is known as optical rectification. We believe that this DC component provides a likely explanation for the photo-current that we observe. Because we have positioned electrodes (the two sides of the slot waveguide) at precisely the bounds of the induced field, the effect of optical rectification takes a small slice of the optical power and converts it into a virtual voltage source between the two arms. This in turn induces a current that we can measure and is linearly proportional to the input power E optical   2 . 
     Now let us consider the solution to Maxwell&#39;s equation in more detail. Our system can be approximated for this discussion as having two dimensions, with both the optical and DC electric field in the x direction and propagation in the z direction, for instance. Let us imagine that the χ 2  is nonzero and small for a tiny region from 0 to w in the x dimension. χ 2  is sufficiently small that the electric field due to the optical mode is still uniform. Let us imagine the system has no charge anywhere. The optical electric field can be written as E=Ae (ikz-iwt) +c.c. where c.c. indicates a complex conjugate. Let us further assume that the rectified DC field is of real amplitude C and uniformly directed in the x dimension on (0, w), and 0 elsewhere. 
     Other than the divergence condition, Maxwell&#39;s equations are still satisfied by this system. But at the edge of an interface on the interior, the DC frequency component of D x , the displacement electric field, is discontinuous. At x 0 , we have: 
       D x   − =0  (15)
 
         D   x   + =∈ 0 (∈ r   C+χ   2   C   2 +2χ 2   |A|   2 )  (16)
 
     We neglect χ 2 C 2  because we expect the amplitude of the rectified field to be far smaller than that of the optical field. Clearly, the boundary condition of zero divergence can only be satisfied if D x   +  is 0. Then, 
     
       
         
           
             
               
                 
                   C 
                   = 
                   
                     
                       
                         2 
                          
                         
                             
                         
                          
                         
                           χ 
                           2 
                         
                       
                       
                         ɛ 
                         r 
                       
                     
                      
                     
                       
                          
                         A 
                          
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Thus the direction of the rectified field is reversed compared to the direction of χ 2 . Note that there is no particular direction associated with the optical field as it is continually oscillating. As we have seen, this rectified DC field would then, if acting as a virtual voltage source, create an effective positive terminal on the positive polling terminal. 
     Analysis of Data for Optical Rectification 
     To derive the magnitude of the expected photocurrent, we assume that the χ 2  magnitude relating to the Pockels&#39; effect is similar to that for optical rectification. A measurement of χ 2  can then be obtained from the direct observation of the electro-optic coefficient by the standard measurements described earlier. The typical measured tuning value of 2 GHz/V yields approximately 50 pm/V. 
     In the best case, devices with 6 dBm of input power returned approximately 1.4 nA of current. With Qs ranging from 3 k to 5 k, and assuming approximately 7 dB of insertion loss in the input grating coupler on one of our chips, in the best case as much as 0 dBm might be circulating in a resonator on resonance. This implies a peak electric field due to the optical signal of approximately 3.1×10 6  V/m. The induced static nonlinear polarization field is then nearly 1000 V/m, which amounts to a voltage drop of 14×10 −5  V across a 140 nm gap. If this voltage is assumed to be perfectly maintained, and the load resistance is assumed to be 5 MO, then 28 pA would be generated, about a factor of 100 less than is observed in the largest measurement made, but within a factor of 20 of the typical measurement of 352 pA for 6 dBm of input. Significantly, because the generated current is quadratic in E, it is clear that the current will be linearly proportional to the input intensity. This is in accordance with our observations. The best results for optical rectification were obtained with YLD  124 /APC polymer, whereas our best Pockels&#39; Effect results were obtained with the dendrimer materials. It is believed that acceptable performance can be attained for peak electric fields of the order of 1×10 5  V/m (that is in the range of 1×10 5  V/m to 9×10 5  V/m) that are generated due to optical signals. It is further believed that acceptable performance can be attained for peak electric fields of the order of 1×10 4  V/m (that is in the range of 1×10 4  V/m to 9×10 4  V/m) that are generated due to optical signals. 
     Significantly, the sign of the output current matches that which would be predicted by nonlinear optical rectification, as discussed above. Specifically, since positive current emanates from the positive terminal, the rectified E field has a sign reversed from the χ 2  and the polling E field. It is well established that the χ 2  direction tends to align with the direction of the polling E field. Because of this, the rectified field acting as a voltage source will produce an effective positive terminal at the terminal that had the positive polling voltage. 
     We do not yet fully understand the current generation mechanism. In particular, it is not clear what provides the mechanism for charge transport across the gap. The APC material in which the nonlinear polymer is hosted is insulating, and though it does exhibit the photoconductivity effect due to visible light, it is unclear whether it can for near-infrared radiation. Photoconductivity due to second harmonic generation may play a role in this effect. It is certainly the case, however, that current flows through this gap; that is the only region in the entire system where an electromotive force exists. Also, photoconductivity alone is not adequate to explain the reversal of the current coming from the detector devices when the poling direction is reversed, nor the conversion of the optical input into directed current in general. The only mechanism to our knowledge that adequately explains this data is optical rectification. 
     If we assume that it will be possible to achieve a 10-fold improvement in the Q&#39;s of the resonators, while still getting more than 10 dB of extinction, then the intensity circulating in such a ring would be about 13 dB up from the intensity of the input wave. By comparison, with a Q of about 1000 and high extinction, the peak circulating intensity is about the same as the intensity in the input waveguide. Therefore, it is reasonable to expect that it will be possible to get at least 10 dB of improvement in the circulating intensity, and thus in the conversion efficiency, by fabricating higher Q rings. 
     By combining the nano-scale slotted waveguide geometry with electro-optical polymers having high nonlinear constants, we have obtained massive enhancement of the optical field. That has in turn enabled us to exploit nonlinear optical processes that are typically only available in the kW regime in the sub-mW regime. This difference is so considerable that we believe it represents a change in kind for the function of nonlinear optical devices. In addition, it is believed that this hybrid material system provides systems and methods for creating compact devices that exploit other nonlinear phenomena on-chip. 
     Optical rectification based detectors can have many advantages over currently available technology. In particular, such detectors are expected to function at a higher intrinsic rate than the typical photodiode in use, as the optical rectification process occurs at the optical frequency itself, on the order of 100 THz in WDM systems. The absence of an external bias, and the generation of a voltage rather than a change in current flow, both provide certain advantages in electronic operation. We also believe that a device based on nonlinear optical rectification will not suffer from the limitation of a dark current. This in turn can provide WDM systems that will function with lower optical power, providing numerous benefits. Similarly, our demonstration of enhanced modulation using these waveguide geometries provides useful components for future communications systems. 
     We conclude by stressing advantageous economic aspects of our invention in various embodiments. Because our devices can be fabricated in planar electronics grade silicon-on-insulator, using processes compatible with advanced CMOS processing, it is expected that devices embodying these principles will be less expensive to fabricate. 
     While the present invention has been particularly shown and described with reference to the structure and methods disclosed herein and as illustrated in the drawings, it is not confined to the details set forth and this invention is intended to cover any modifications and changes as may come within the scope and spirit of the following claims.