Abstract:
In accordance with various aspects of the disclosure, a method, an apparatus and a system for characterizing and compensating for deterministic phase nonlinearities and distortion inherent in radio frequency and optical components utilized to synthesize a single sideband suppressed carrier optical waveform in the presence of random phase noise generated by an optical carrier source is disclosed. The method comprises mixing a modulated optical signal with a continuous wave optical signal in an optical coupler; optically heterodyning the mixed signal output from the optical coupler in a detector to produce a radio frequency waveform; and analyzing the produced radio frequency waveform in a processor based on a phase history of a preselected continuous wave signal to measure distortion characteristics of the radio frequency modulated optical signal.

Description:
BACKGROUND 
     This disclosure relates generally to the field of optics and, more specifically, to a method and apparatus for synthesizing and correcting phase distortions in ultra-wide bandwidth optical waveforms. 
     Conventionally, the process of synthesizing extremely high bandwidth, single-sideband, linear frequency modulated optical waveforms is complicated. Radio frequency (RF) and optical components can introduce gain, phase delay, nonlinearity and/or other distortion phenomena into the waveform synthesis procedure. 
     What is needed is a waveform synthesis method and apparatus capable of characterizing and at least minimizing, if not eliminating, deterministic waveform distortion arising from RF and optical components in the presence of laser phase noise. 
     SUMMARY 
     In accordance with various embodiments of this disclosure, a method for characterizing and compensating for deterministic phase nonlinearities and distortion inherent in radio frequency and optical components utilized to synthesize a single sideband suppressed carrier optical waveform in the presence of random phase noise generated by an optical carrier source is disclosed. The method comprises mixing a modulated optical signal with a constant frequency optical signal in an optical coupler; optically heterodyning the mixed signal output from the optical coupler in a detector to produce a radio frequency waveform; and comparing the produced radio frequency waveform in a processor based on a known, theoretical phase history of a preselected continuous wave signal to measure distortion characteristics of the radio frequency modulated optical signal. 
     In accordance with various embodiments of this disclosure, a system for characterizing and compensating for deterministic phase nonlinearities and distortion inherent in radio frequency and optical components utilized to synthesize a single sideband suppressed carrier optical waveform in the presence of random phase noise generated by an optical carrier source is disclosed. The system comprises an optical coupler configured to mix a modulated optical signal with a constant frequency optical signal; an optical detector configured to receive the mixed signal output from the optical coupler and optically heterodyne the mixed signal to produce a radio frequency waveform; and a processor in communication with a memory having instructions stored therein which, when executed compare the produced radio frequency waveform to a known, theoretical phase history of a preselected continuous wave signal and measure distortion characteristics of the radio frequency modulated optical signal. 
     In accordance with various embodiments of this disclosure, a computer-readable physical medium including instructions that, when executed by a processor, cause the processor to carry out functions related to compensating for deterministic phase nonlinearities and distortion in a radio frequency and optical components utilized to synthesize a single sideband suppressed carrier optical waveform is disclosed. The functions include generating a first waveform comprising a linear frequency modulated chirp; determining a distortion component from a received second waveform; and modifying the first waveform using the determined distortion component to compensate for the deterministic phase nonlinearities and distortion of the received second waveform produced by the radio frequency components and the one or more optical components. 
     These and other features and characteristics, as well as the methods of operation and functions of the related elements of structure and the combination of parts and economies of manufacture, will become more apparent upon consideration of the following description and the appended claims with reference to the accompanying drawings, all of which form a part of this specification, wherein like reference numerals designate corresponding parts in the various Figures. It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only and are not intended as a definition of the limits of claims. As used in the specification and in the claims, the singular form of “a”, “an”, and “the” include plural referents unless the context clearly dictates otherwise. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  shows a comparison between theoretical and synthesized waveforms. 
         FIG. 2  shows an example system architecture for simultaneous waveform synthesis and characterization in accordance with various aspects of the present disclosure. 
         FIG. 3  shows an example of a frequency spectrum of a Dual-Sideband, Suppressed-Carrier (DSB-SC) waveform in accordance with various aspects of the present disclosure. 
         FIG. 4  shows an example frequency spectrum of a Single-Sideband, Suppressed-Carrier (SSB-SC) waveform. 
         FIG. 5  shows an example of a frequency spectrum for optical heterodyne and RF waveform recovery in accordance with various aspects of the present disclosure. 
         FIG. 6  shows an example frequency spectrum for unmodified, generation  0  waveform (dotted line) compared with theoretical (line) in accordance with various aspects of the present disclosure. 
         FIG. 7  shows an example frequency spectrum for generation  1  waveform (dotted line) compared to theoretical (line) in accordance with various aspects of the present disclosure. 
         FIG. 8  shows an example frequency spectrum for generation  2  waveform (dotted line) compared to theoretical (line) in accordance with various aspects of the present disclosure. 
         FIG. 9  shows an example frequency spectrum for generation  3  waveform (dotted line) compared to theoretical (line) in accordance with various aspects of the present disclosure. 
         FIG. 10  is a block diagram illustrating an example computing device that is arranged to perform the various processes and/or methods in accordance with the various aspects of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     In the description that follows, like components have been given the same reference numerals, regardless of whether they are shown in different embodiments. To illustrate an embodiment(s) of the present disclosure in a clear and concise manner, the drawings may not necessarily be to scale and certain features may be shown in somewhat schematic form. Features that are described and/or illustrated with respect to one embodiment may be used in the same way or in a similar way in one or more other embodiments and/or in combination with or instead of the features of the other embodiments. 
       FIG. 1  depicts a theoretically perfect, linearly frequency modulated waveform ( 1 ) in comparison to a typical synthesized waveform ( 2 ) containing distortion created by real-world RF and optical components. The distortion is shown graphically as the solid curved line ( 2 ), deviating from the dashed theoretical waveform ( 1 ) and changing as a function of time during waveform synthesis. 
     The theoretical linear frequency modulated waveform ( 1 ) has a time-frequency relationship given by Equation 1 and a time-phase relationship given by Equation 2. Real-world linear frequency modulated waveforms ( 2 ) have a time-phase relationship similar to Equation 2, but possess a time-dependent deterministic phase distortion component, φ D (t), and a random noise phase component, φ N (t), shown in Equation 3. 
     
       
         
           
             
               
                 
                   μ 
                   = 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       f 
                     
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       t 
                     
                   
                 
               
               
                 1 
               
             
             
               
                 
                   
                     
                       ϕ 
                       THEO 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         RF 
                       
                       ⁢ 
                       t 
                     
                     + 
                     
                       
                         μ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           t 
                           2 
                         
                       
                       2 
                     
                   
                 
               
               
                 2 
               
             
             
               
                 
                   
                     
                       ϕ 
                       SIG 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       π 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         f 
                         RF 
                       
                       ⁢ 
                       t 
                     
                     + 
                     
                       
                         μ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           t 
                           2 
                         
                       
                       2 
                     
                     + 
                     
                       
                         ϕ 
                         D 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     + 
                     
                       
                         ϕ 
                         N 
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                 
               
               
                 3 
               
             
           
         
       
     
     In order to characterize and eliminate the time-dependent, deterministic distortion from real-world waveforms, φ D (t), two criteria should be considered. First, a waveform synthesis technique and methodology that enables the simultaneous synthesis and characterization of linearly frequency modulated optical waveforms. Second, an algorithm capable of measuring and compensating for the distortion in linearly frequency modulated optical waveforms. 
       FIG. 2  shows an example system architecture block diagram that enables linearly frequency modulated optical waveforms (LFM) to be simultaneously synthesized and characterized. Waveform synthesis begins on a computer ( 10 ) where digital samples ( 50 ) of a LFM waveform are created using software such as MATLAB manufactured by The MathWorks headquartered in Natick, Mass.; however, other suitable software platform may be used. LFM waveform samples ( 50 ) are transferred from computer ( 10 ) to memory inside arbitrary waveform generator ( 11 ). Arbitrary waveform generator ( 11 ) can clock the waveform samples from memory to a digital-to-analog converter, producing baseband radio frequency (RF) signal ( 51 ). Baseband RF signal ( 51 ) undergoes a series of time-bandwidth modifications in RF upconversion ( 12 ) to produce final RF signal ( 52 ) that will be modulated onto an optical carrier. For example, RF upconversion ( 12 ) can be done by using several stages of RF doublers, where each stage can increase the time-bandwidth property of the RF waveform by 2×. 
     Laser source ( 13 ), such as a narrow-linewidth, nominal 1.5 μm laser source, is configured to provide a low level, for example +13 dBm, optical carrier ( 55 ) that is amplified by amplifier ( 14 ), such as a Erbium-Doped Fiber Amplifier (EDFA), to increase optical carrier ( 56 ) power. Amplified optical carrier ( 56 ) can be split into two signals of equal power ( 57 ,  58 ) by splitter ( 15 ), such as a fiber-optic 50/50 splitter. Optical carrier on PATH 1  ( 57 ) can be modulated by modulator ( 27 ), such as a Mach-Zehnder modulator, driven with final RF signal ( 52 ) to produce a dual-sideband, suppressed carrier (DSB-SC), linear frequency modulated optical waveform ( 59 ). 
       FIG. 3  shows the frequency spectrum of DSB-SC linear frequency modulated waveform ( 59 ). DSB-SC waveform has lower sideband ( 100 ), upper sideband ( 102 ) and some residual carrier ( 101 ) after modulation. The optical carrier ( 101 ;  57  of  FIG. 2 ) has frequency f λ , while final RF waveform ( 52 ) begins at frequency f RF  and has bandwidth of f BW  GHz. 
     Returning to  FIG. 2 , DSB-SC LFM optical waveform ( 59 ) can pass through optical circulator ( 19 ) to grating ( 20 ), such as a Fiber Bragg Grating, operating in reflection mode ( 69 ). Grating ( 20 ) can filter one sideband of DSB-SC LFM optical waveform to produce single-sideband, suppressed-carrier (SSB-SC) LFM optical waveform ( 62 ) shown in  FIG. 4 . 
     Grating ( 20 ;  113  of  FIG. 4 ) can pass upper sideband ( 102  of  FIG. 2 ;  112  of  FIG. 4 ) and filter lower sideband ( 100  of  FIG. 3 ) and optical carrier ( 101  of  FIG. 3 ) from DSB-SC waveform ( 50 ). Some residual lower sideband signal ( 110 ) may remain along with some residual optical carrier ( 111 ), but these residual signal components are below the signal power of the upper sideband ( 112 ), for example 30-45 dB below the signal power. 
     Optical carrier on PATH 2  ( 58 ) can be modulated by modulator ( 28 ), for example a 2 nd  Mach-Zehnder modulator, driven by constant frequency RF signal ( 53 ) synthesized by RF source ( 17 ) to produce DSB-SC optical carrier ( 60 ) having upper sideband ( 128  of  FIG. 5 ) and lower sideband ( 129  of  FIG. 5 ). DSB-SC optical carrier ( 60 ) can be amplified by amplifier ( 18 ), for example an EDFA optical amplifier, to produce local oscillator optical carrier ( 61 ). 
     Local oscillator optical carrier ( 61 ) signal power and polarization can be controlled via a variable optical attenuator (VOA) ( 21   a ) and a polarization controller ( 22   a ) to produce the final local oscillator optical carrier ( 63 ). In similar fashion, SSB-SC LFM optical waveform ( 62 ) signal power and polarization can be controlled via 2 nd  VOA ( 21   b ) and polarization controller ( 22   b ) to prepare signal ( 64 ) for optical mixing. 
     Attenuated, polarized local oscillator optical carrier ( 63 ) and attenuated, polarized, SSB-SC LFM optical waveform ( 64 ) can be mixed in optical combiner/splitter ( 23 ), for example a  99 / 1  optical combiner/splitter. One percent of the power can be split off ( 66 ) to be measured on power meter ( 25 ), for example an IR power meter. Ninety nine percent of the mixed optical signals ( 65 ) can illuminate detector ( 24 ), for example a high-bandwidth InGaAs detector, where optical heterodyne can occur.  FIG. 5  shows the frequency spectrum of the mixed optical signals ( 65 ) that are optically heterodyned on InGaAs detector ( 24 ). 
     Heterodyning of optical carrier ( 128 ) and SSB-SC LFM optical waveform ( 112  of  FIG. 4 ;  127  of  FIG. 5 ) can produce RF signal ( 67 ) which is the difference in optical frequency between the two signals; the RF waveform at Reference  67  has identical bandwidth to the RF waveform at Reference  52 , but now contains the distortion created by the RF and optical components during waveform synthesis. RF waveform ( 67 ) from detector ( 24 ) can be digitized by oscilloscope ( 26 ), such as a high-bandwidth oscilloscope, and the digitized oscilloscope data ( 68 ) can be transferred to computer ( 10 ) for distortion analysis. 
     The algorithms for distortion analysis are now discussed. Legendre Polynomials are a mathematically orthogonal basis set over the interval of [−1, 1]. Equations 4, 5 and 6 provide the definition of Legendre Polynomials and the principle of orthogonality; i.e., if any two Legendre Polynomials P m (x) and P n (x) are integrated over the interval [−1, 1], the result will be zero. Finally, if the same order Legendre Polynomials are integrated over the interval of [−1, 1], the result is a constant that depends on the polynomial order, n. 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                     
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           2 
                           n 
                         
                         ⁢ 
                         
                           n 
                           ! 
                         
                       
                     
                     ⁢ 
                     
                       
                         ⅆ 
                         n 
                       
                       
                         ⅆ 
                         
                           x 
                           n 
                         
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             x 
                             2 
                           
                           - 
                           1 
                         
                         ) 
                       
                       n 
                     
                   
                 
               
               
                 4 
               
             
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         1 
                       
                       1 
                     
                     ⁢ 
                     
                       
                         
                           P 
                           m 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           P 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 5 
               
             
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         1 
                       
                       1 
                     
                     ⁢ 
                     
                       
                         
                           P 
                           n 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         
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                           n 
                         
                         ⁡ 
                         
                           ( 
                           x 
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                       ⁢ 
                       
                           
                       
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                         x 
                       
                     
                   
                   = 
                   
                     2 
                     
                       
                         2 
                         ⁢ 
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
               
                 6 
               
             
           
         
       
     
     The definition of Legendre Polynomials presented in Equations 4, 5 and 6 has one minor problem for implementation in algorithms and software which is that it requires repeated derivatives which are not easy to program. An alternative definition, based upon a recursive definition known as “Rodriguez&#39; formula”, is presented as Equations 7, 8 and 9. Unlike differentiation, recursion is readily adaptable to software programming; hence Equations 7, 8 and 9 better suited for synthesizing Legendre Polynomials in software. 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       n 
                     
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
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                               ⁢ 
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                             - 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         
                           
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                               n 
                               - 
                               1 
                             
                           
                           ⁡ 
                           
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                             ( 
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                 7 
               
             
             
               
                 
                   
                     
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                       0 
                     
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                 8 
               
             
             
               
                 
                   
                     
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                       1 
                     
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                       ( 
                       x 
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                   = 
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                 9 
               
             
           
         
       
     
     The recovered signal phase history from the digitized waveform ( 67 ) can be defined as φ ADC (t), and the theoretical phase history for a perfect CW signal can be defined as φ THEO (t). The recovered signal and theoretical CW phase histories can be defined as a sum of N Legendre Polynomials with scaling coefficients as shown in Equations 10 &amp; 11; P n (x) defines the n th  order Legendre Polynomial basis vector. A collection of scaling coefficients, A 0 -A n , represent the recovered phase history, while a similar group of scaling coefficients, B 0 -B n , represent the theoretical phase history. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         
                           ϕ 
                           ADC 
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       ≈ 
                       
                         
                           
                             A 
                             0 
                           
                           ⁢ 
                           
                             
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                               0 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
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                             1 
                           
                           ⁢ 
                           
                             
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                               1 
                             
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                               ( 
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                               ) 
                             
                           
                         
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                             2 
                           
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                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
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                       ⁢ 
                       
                         
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                             n 
                           
                           ⁡ 
                           
                             ( 
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                 10 
               
             
             
               
                 
                   
                     
                       ϕ 
                       THEO 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           ω 
                           O 
                         
                         ⁢ 
                         t 
                       
                       ≈ 
                       
                         
                           
                             B 
                             0 
                           
                           ⁢ 
                           
                             
                               P 
                               0 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                         + 
                         
                           
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                             1 
                           
                           ⁢ 
                           
                             
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                               1 
                             
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                               ( 
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                               ) 
                             
                           
                         
                         + 
                         
                           
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                             2 
                           
                           ⁢ 
                           
                             
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                               2 
                             
                             ⁡ 
                             
                               ( 
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                               ) 
                             
                           
                         
                         + 
                         … 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           n 
                           = 
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                           - 
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                       ⁢ 
                       
                         
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                             n 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 11 
               
             
           
         
       
     
     Using the orthogonality principle, the individual scaling coefficients, A n  and B n  can be calculated, from their respective phase data, φ ADC (t) and φ THEO (t). This is analogous to computing Fourier Frequency-Domain Coefficients from time-domain data. Combining Equations 10 and 11 with the orthogonality principle defined in Equation 6, Equations 12 and 13 can be obtained to define how to compute the n th  order Legendre scaling coefficients, A n  and B n , from the recovered phase history and the theoretical phase history respectively. 
     
       
         
           
             
               
                 
                   
                     A 
                     n 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           n 
                         
                         + 
                         1 
                       
                       2 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             ϕ 
                             ADC 
                           
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             n 
                           
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 12 
               
             
             
               
                 
                   
                     B 
                     n 
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           n 
                         
                         + 
                         1 
                       
                       2 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         
                           
                             ϕ 
                             THEO 
                           
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             n 
                           
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 13 
               
             
           
         
       
     
     Similar to Fourier analysis, the signal properties can be compared in Legendre space and the distortion measured in the recovered signal phase history, φ ADC (t). Equation 14 defines how the distortion, D n , is computed for the n th  order Legendre basis vector; N defines the number of Legendre basis vectors required to accurately model the signals. Each difference coefficient, D n , is computed by subtracting the measured phase coefficient, A n , from the theoretical phase coefficient, B n . The distortion for the 0 th  and 1 st  order terms, which represent DC phase and linear phase offset need not be determined since they are not required for phase compensation.
 
 D   n   =B   n   −A   n ,2 ≦k≦N− 1  14
 
     Each Legendre difference coefficient, D n , is defined for the full-bandwidth signal. In order to perform phase compensation, the distortion coefficient is transformed or input-referred to baseband signal bandwidth ( 51 ) initially produced by digital-to-analog converter (DAC) in AWG ( 11 ). Equation 15 defines a bandwidth based scaling coefficient for this input-referred mapping; L defines the total number of bandwidth scaling operations that occur in the RF and optical hardware ( 12 ).
 
 R   BW =10 LOG 10 (2 L )  15
 
     Equation 16 defines the initial or 0 th  generation baseband waveform ( 50 ) synthesized in the software platform, such as Matlab, and loaded into the AWG; φ BB0 (t) defines a theoretical linear FM chirp and is identical to Equation 2. The j th  generation baseband waveform ( 11 ), defined in Equation 17, is iteratively generated by adding the input-referred distortion, D j,k  for 1&lt;k&lt;N−1, to the previous generation baseband waveform. It has been found by the inventors through hardware experiments that typically 4 to 5 iterations are needed for proper convergence and ultra-high fidelity phase compensation, however, more or less can be used depending on the application. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       
                         ϕ 
                         
                           BB 
                           0 
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ω 
                           O 
                         
                         ⁢ 
                         t 
                       
                       + 
                       
                         
                           u 
                           2 
                         
                         ⁢ 
                         
                           t 
                           2 
                         
                       
                     
                   
                 
               
               
                 16 
               
             
             
               
                 
                   
                     
                       
                         ϕ 
                         
                           BB 
                           J 
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     ≈ 
                     
                       
                         
                           ϕ 
                           
                             
                               BB 
                               J 
                             
                             - 
                             1 
                           
                         
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       + 
                       
                         
                           
                             D 
                             
                               j 
                               , 
                               2 
                             
                           
                           
                             R 
                             BW 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           
                             D 
                             
                               j 
                               , 
                               3 
                             
                           
                           
                             R 
                             BW 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             3 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                       + 
                       … 
                       + 
                       
                         
                           
                             D 
                             
                               j 
                               , 
                               
                                 N 
                                 - 
                                 1 
                               
                             
                           
                           
                             R 
                             BW 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ϕ 
                         
                           
                             BB 
                             J 
                           
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         t 
                         ) 
                       
                     
                     + 
                     
                       
                         1 
                         
                           R 
                           BW 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             2 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             D 
                             
                               j 
                               , 
                               k 
                             
                           
                           ⁢ 
                           
                             
                               P 
                               k 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 17 
               
             
           
         
       
     
     Phase compensation via Legendre Polynomial decomposition is based upon the idea that the master oscillator is a low phase-noise device. While such oscillators are readily obtainable in the RF domain, lasers do not generally possess this characteristic. The optical phase of a laser can be described as a function of three terms, shown in Equation 18.
 
φ λ ( t )=2 πf   λ   t+φ   ST ( t )+φ RW ( t )  18
 
     The first term of Equation 18 defines phase from a nominal optical frequency, f λ . The second term, φ ST (t), is a stationary white noise process, implying it has zero-mean and constant variance over time. The third term, φ RW (t), is a random walk process. 
     During phase compensation, the stationary white noise process, φ ST (t), should converge to zero-mean if enough pulses are captured per waveform generation. It can be argued that during the short waveform duration, a narrow-linewidth laser&#39;s random walk, φ RW (t), behaves as a similar white noise source that can be averaged to zero with enough waveform observations per generation. 
     Phase compensation algorithm performance and convergence are graphically depicted in  FIG. 6  through  FIG. 9 . For these Figures, black is the theoretically best achievable pulse compression via FFT derived from the theoretical phase history of Equation 2. The dotted lines are of four individual pulse compressions acquired over a five minute interval for each waveform generation. As the phase compensation algorithm calculates and refines the distortion measurements of the RF and optical hardware, one observes the synthesized signal pulse compression graphed in a dotted line converge and match the theoretical compression graphed black in the presence of phase noise from the laser. 
       FIG. 6  through  FIG. 9  shows the synthesis, iterative measurement and subsequent phase compensation of an ultra-high bandwidth optical waveform over multiple waveform iterations with actual hardware.  FIG. 6  shows an unmodified, generation  0  waveform (dotted line) compared with theoretical (line) in accordance with various aspects of the present disclosure.  FIGS. 7-9  show generation  1 ,  2  and  3  waveform, respectively, compared with the theoretical (line). 
       FIG. 10  is a block diagram illustrating an example computing device  200  that is arranged to perform the various processes and/or methods in accordance with the various aspects of the present disclosure. In a very basic configuration  201 , computing device  200  typically includes one or more processors  210  and a system memory  220 . A memory bus  230  may be used for communicating between processor  210  and system memory  220 . 
     Depending on the desired configuration, processor  210  may be of any type including but not limited to a microprocessor (μP), a microcontroller (μC), a digital signal processor (DSP), or any combination thereof. Processor  210  may include one more levels of caching, such as a level one cache  211  and a level two cache  212 , a processor core  213 , and registers  214 . An example processor core  213  may include an arithmetic logic unit (ALU), a floating point unit (FPU), a digital signal processing core (DSP Core), or any combination thereof. An example memory controller  215  may also be used with processor  210 , or in some implementations memory controller  215  may be an internal part of processor  210 . 
     Depending on the desired configuration, system memory  220  may be of any type including but not limited to volatile memory (such as RAM), non-volatile memory (such as ROM, flash memory, etc.), or any combination thereof. System memory  220  may include an operating system  221 , one or more applications  222 , and program data  224 . Application  222  may include one or more of the various algorithms, processes or methods  223 , as discussed above, that is arranged to perform the functions as described with respect to processes of  FIGS. 1-9 . Program data  224  may include data  225  that may be useful for one or more of the various algorithms, methods or processes as described herein. In some embodiments, application  222  may be arranged to operate with program data  224  on operating system  221  such that implementations of the various algorithms, processes or methods may be provided as described herein. This described basic configuration  201  is illustrated in  FIG. 10  by those components within the inner dashed line. 
     Computing device  200  may have additional features or functionality, and additional interfaces to facilitate communications between basic configuration  201  and any required devices and interfaces. For example, a bus/interface controller  242  may be used to facilitate communications between basic configuration  201  and one or more data storage devices  250  via a storage interface bus  241 . Data storage devices  250  may be removable storage devices  251 , non-removable storage devices  252 , or a combination thereof. Examples of removable storage and non-removable storage devices include magnetic disk devices such as flexible disk drives and hard-disk drives (HDD), optical disk drives such as compact disk (CD) drives or digital versatile disk (DVD) drives, solid state drives (SSD), and tape drives to name a few. Example computer storage media may include volatile and nonvolatile, removable and non-removable media implemented in any method or technology for storage of information, such as computer readable instructions, data structures, program modules, or other data. 
     System memory  220 , removable storage devices  251  and non-removable storage devices  252  are examples of computer storage media. Computer storage media includes, but is not limited to, RAM, ROM, EEPROM, flash memory or other memory technology, CD-ROM, digital versatile disks (DVD) or other optical storage, magnetic cassettes, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other medium which may be used to store the desired information and which may be accessed by computing device  200 . Any such computer storage media may be part of computing device  200 . 
     Computing device  200  may also include an interface bus  291  for facilitating communication from various interface devices (e.g., output devices  260 , peripheral interfaces  270 , and communication devices  280 ) to basic configuration  201  via bus/interface controller  242 . Example output devices  260  include graphics processing unit  261  and audio processing unit  262 , which may be configured to communicate to various external devices such as a display or speakers via one or more A/V ports  263 . Example peripheral interfaces  270  include serial interface controller  271  or parallel interface controller  272 , which may be configured to communicate with external devices such as input devices (e.g., keyboard, mouse, pen, voice input device, touch input device, etc.) or other peripheral devices (e.g., printer, scanner, etc.) via one or more I/O ports  273 . An example communication device  280  includes network controller  281 , which may be arranged to facilitate communications with one or more other computing devices  290  over a network communication link via one or more communication ports  282 . 
     The network communication link may be one example of a communication media. Communication media may typically be embodied by computer readable instructions, data structures, program modules, or other data in a modulated data signal, such as a carrier wave or other transport mechanism, and may include any information delivery media. A “modulated data signal” may be a signal that has one or more of its characteristics set or changed in such a manner as to encode information in the signal. By way of example, and not limitation, communication media may include wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, radio frequency (RF), microwave, infrared (IR) and other wireless media. The term computer readable media as used herein may include both physical storage media and communication media. 
     Computing device  200  may be implemented as a portion of a small-form factor portable (or mobile) electronic device such as a cell phone, a personal data assistant (PDA), a personal media player device, a wireless web-watch device, a personal headset device, an application specific device, or a hybrid device that include any of the above functions. Computing device  200  may also be implemented as a personal computer including both laptop computer and non-laptop computer configurations. 
     Although the above disclosure discusses what is currently considered to be a variety of useful embodiments, it is to be understood that such detail is solely for that purpose, and that the appended claims are not limited to the disclosed embodiments, but, on the contrary, are intended to cover modifications and equivalent arrangements that are within the spirit and scope of the appended claims.