Abstract:
A system for suppressing even-order distortion in a photonic link includes a laser for providing laser light to a first input of a Mach-Zehnder modulator (MZM), where the MZM has a second input for receiving an RF input signal, a third input for applying a DC bias voltage to the MZM, and an optical signal output. A dc-voltage-biased photodiode has an input, coupled to the MZM optical signal output, and a modulated RF signal output. The MZM DC bias voltage is set at a value to generate an even-order distortion amplitude substantially equal to an even-order distortion amplitude from the photodiode and 180 degrees out of phase so as to substantially cancel the photodiode even-order distortion. The invention provides the cancellation of photodiode even-order distortion via predisortion linearization with a MZM biased slightly away from quadrature, employing a single fiber run and a single photodiode. The invention provides an improvement in carrier-to-intermodulation ratio (CIR) upwards of 40 dB.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application 61/787,392 filed on Mar. 15, 2013 and incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention is directed to a high-linearity photonic link, and in particular to a photonic link having an improved even-order distortion response. 
       BACKGROUND OF THE INVENTION 
       [0003]    High-linearity photodiodes are actively researched in the field of microwave photonics, with applications in the academic, industrial and military sectors. A recent survey collects reported state-of-the-art results from the component level (V. J. Urick, J. F. Diehl, M. N. Draa, J. D. McKinney, and K. J. Williams, “Wideband analog photonic links: some performance limits and considerations for multi-octave limitations,” Proc. SPIE 8259, 1-14 (2012) (“Urick et al. 1”). The concentration of high-linearity photodiode work is largely in terms of single-octave third-order-limited intermodulation distortion as quantified by a third-order output intercept point (OIP3). One of the inherent advantages of photonic solutions is the wide bandwidth available in the optical domain, making analog optical links attractive for multi-octave applications. However, even-order distortion generated by photodiodes can be inhibiting in such implementations as described in Urick et al. Previous works, e.g. Urick et al. 1 and V. J. Urick, A. S. Hastings, J. D. McKinney, P. S. Devgan, K. J. Williams, C. Sunderman, J. F. Diehl, and K. Colladay, “Photodiode linearity requirements for radio-frequency photonics and demonstration of increased performance using photodiode arrays,” in 2008  IEEE International Meeting on Microwave Photonics Digest , pp. 86-89 (“Urick et al. 2”), have described the photodiode requirements in high-linearity photonic links for single- and multi-octave applications in terms of OIP3 and second-order output intercept point (OIP2), respectively. Oftentimes the present photodiode technology falls short of the system requirements, particularly in multi-octave applications. Architectural techniques have been devised to mitigate the component limitations. For example, photodiode arrays have been shown to achieve better linearity than the individual photodiodes are capable of alone. Two- and four-photodiode arrays have been demonstrated (see, respectively, A. Joshi, “Highly linear dual photodiodes for Ku-Band applications,” in 2009  IEEE Avionics Fiber Optics and Photonics Conference Digest , pp. 9-10, and Y. Fu, H. Pan, and J. C. Campbell, “Photodiodes with monolithically integrated Wilkinson power combiner,” IEEE J. Quantum Electron. 46, 541-545 (2010); and S. Itakura, K. Sakai, T. Nagatsuka, E. Ishimura, M. Nakaji, H. Otsuka, K. Mori, and Y. Hirano, “High-current backside-illuminated photodiode array module for optical analog links,” J. Lightwave Technol. 28, 965-971 (2010) and Y. Fu, H. Pan, Z. Li, and J. Campbell, “High linearity photodiode array with monolithically integrated Wilkinson power combiner,” in 2010  IEEE International Meeting on Microwave Photonics Digest , pp. 111-113). This simple but quite effective technique is based on dividing the input signal between numerous non-linear devices and then linearly combining their outputs. The “array gain” scales with the number of elements for both even- and odd-order distortion, assuming that each element exhibits the same nonlinearity. Balanced photodiode arrays have been demonstrated that improve the OIP3 by the array gain but suppress photodiode-generated even-order distortion through the balanced detection process (Urick et al. 2, and A. S. Hastings, V. J. Urick, C. Sunderman, J. F. Diehl, J. D. McKinney, D. A. Tulchinsky, P. S. Devgan, and K. J. Williams, “Suppression of even-order photodiode nonlinearities in multioctave photonic links,” J. Lightwave Technol. 26, 2557-2562 (2008). This technique is attractive for multi-octave applications but requires two phase-matched fibers for the transmission span when implemented with a Mach-Zehnder modulator (MZM). 
       BRIEF SUMMARY OF THE INVENTION 
       [0004]    According to the invention, a system for suppressing even-order distortion in a photonic link includes a laser for providing laser light to a first input of a Mach-Zehnder modulator (MZM), where the MZM has a second input for receiving an RF input signal, a third input for applying a DC bias voltage to the MZM, and an optical signal output. A dc-voltage-biased photodiode has an input, coupled to the MZM optical signal output, and a modulated RF signal output. The MZM DC bias voltage is set at a value to generate an even-order distortion amplitude substantially equal to an even-order distortion amplitude from the photodiode and 180 degrees out of phase so as to substantially cancel the photodiode even-order distortion. 
         [0005]    The invention provides the cancellation of photodiode even-order distortion via predisortion linearization with a MZM biased slightly away from quadrature, employing a single fiber run and a single photodiode. The invention provides an improvement in carrier-to-intermodulation ratio (CIR) upwards of 40 dB. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0006]      FIG. 1  is a schematic diagram of an intensity-modulation direct-detection link employing an external Mach-Zehnder Modulator (MZM) according to the invention; 
           [0007]      FIG. 2  is a schematic diagram of an apparatus for characterization of photodiode linearity, where two lasers are both intensity modulated via an external Mach-Zehnder modulator (MZM); 
           [0008]      FIG. 3  is a graph of the measured OIP2 due to intermodulation distortion for the photodiode at 3 mA average photocurrent showing the measured fundamentals (circles), the measured IMD2 (squares), and the first- and second-order fits with slopes m=1 and m=2, respectively; 
           [0009]      FIG. 4  is a graph of the measured OIP2s for the link at quadrature and at the cancellation point showing the measured fundamentals (circles), the measured IMD2 at quadrature (squares), the measured IMD2 at the cancellation condition (triangles), and the first and second order fits with slopes m=1 and m=2, respectively, according to the invention; 
           [0010]      FIG. 5  is a graph of the CIR at quadrature (squares) and at the cancellation condition (triangles) for the data in  FIG. 4 , according to the invention; 
           [0011]      FIG. 6  is a graph of the measured fundamental output power (open circles), measured IMD2 (triangles) and measured DC photocurrent (gray circles) as a function of MZM bias for the link at −20 dBm input power to the fundamentals, and where the solid lines show the calculated fundamental power, IMD2 power and average photocurrent, according to the invention; 
           [0012]      FIG. 7  is a graph of the measured fundamental output power (open circles) and measured IMD2 (triangles) as a function of MZM bias for the link at −20 dB input power to the fundamentals, where the solid lines show the calculated fundamental power and IMD2 power, and the gray dotted line shows the calculator IMD2 with MZM only, according to the invention; 
           [0013]      FIG. 8  is a schematic diagram of a phase-modulation direct-detection link employing an external phase modulator and Mach-Zehnder interferometer (MZI) according to the invention; 
           [0014]      FIG. 9  is a graph of the measured fundamental output power (open circles), measured IMD2 (triangles) and measured DC photocurrent (gray circles) as a function of MZM bias for the link in  FIG. 8  at −25 dBm input power to the fundamentals, where the solid lines show the calculated fundamental power, IMD2 power and average photocurrent, according to the invention; and 
           [0015]      FIG. 10  is a schematic diagram of a phase modulation link employing a laser, polarization modulator with a beam splitter biased by a polarization controller with direct detection by a photodiode according to the invention. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0016]    A calculation demonstrating cancellation of photodiode even-order distortion with MZM-generated distortion is conducted assuming the architecture of the invention  100  shown in  FIG. 1 . A laser light source  102  outputs a laser light to a first input  104  of a Mach-Zehnder modulator (MZM)  106  having an RF signal input  108  and an input  110  for applying an adjustable DC bias voltage. MZM  106  generates an output optical signal  112  (a second MZM output  114  is coupled to an optical power meter, not illustrated, in order to monitor and adjust the DC bias voltage applied to the MZM  106 ) that is input to a dc-voltage-biased photodiode  116  that is biased at a selected level to generate an RF output signal  118  with photodiode-induced even-order distortion substantially cancelled. The bias voltage of the MZM  106  is adjusted so that the second-order MZM distortion matches the amplitude of the photodiode second-order distortion, which under certain conditions are out of phase with each other. The following analysis is conducted in three steps. First, the response for the intensity-modulation direct-detection (IMDD) link in  FIG. 1  is reviewed and cast in well-known closed-form expressions assuming an ideal linear photodiode. A Taylor series analysis is then employed to model the photodiode nonlinearity. Finally, these two sets of equations are combined to predict the composite response. 
         [0017]    Link with Ideal Photodiode 
         [0018]    The response for an IMDD link employing a MZM is well-known (see B. H. Kolner and D. W. Dolfi, “Intermodulation distortion and compression in an integrated electrooptic modulator,” Appl. Opt. 26, 3676-3680 (1987)). Here, the terms relevant to the cancellation technique are highlighted. We assume an ideal push-pull MZM with the following transfer function: 
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         [0019]    where E 1  and E 2  are the fields corresponding to the two MZM outputs, φ is the phase shift induced by the applied voltage, and E in  is the field at the MZM input. The frequency-dependent MZM half-wave voltage is V π (Ω). The input field is written as E in =κ√{square root over (2P o )}e iωt , where P o  is the average optical power at angular frequency ω and κ is a constant such that P o =E*E/(2κ 2 ). The input to the MZM comprises a DC bias voltage V dc  and a two-tone RF signal of the form V 1  sin(Ω 1 t)+V 2  sin(Ω 2 t), where Ω are the angular frequencies. With these input voltages the phase shift φ(t)=φ dc +φ 1  sin(Ω 1 t)+φ 2  sin(Ω 2 t) results, where φ dc =πV dc /V π  and φ 1,2 =πV 1,2 /V π . Assuming an ideal photodiode with responsivity          , the total photocurrent due to E 1  can be calculated and separated into three components: 
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                                  
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 p 
                                               
                                               + 
                                               1 
                                             
                                             ) 
                                           
                                            
                                           
                                             Ω 
                                             2 
                                           
                                         
                                         - 
                                         
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 n 
                                               
                                               + 
                                               1 
                                             
                                             ) 
                                           
                                            
                                           
                                             Ω 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     t 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         - 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             ∞ 
                           
                            
                           
                             
                               ∑ 
                               
                                 p 
                                 = 
                                 0 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     1 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   J 
                                   
                                     
                                       2 
                                        
                                       p 
                                     
                                     + 
                                     1 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   [ 
                                   
                                     
                                       ( 
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 p 
                                               
                                               + 
                                               1 
                                             
                                             ) 
                                           
                                            
                                           
                                             Ω 
                                             2 
                                           
                                         
                                         + 
                                         
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 n 
                                               
                                               + 
                                               1 
                                             
                                             ) 
                                           
                                            
                                           
                                             Ω 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     t 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         - 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     2 
                                      
                                     k 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     1 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   J 
                                   
                                     2 
                                      
                                     m 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   [ 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           m 
                                            
                                           
                                               
                                           
                                            
                                           
                                             Ω 
                                             2 
                                           
                                         
                                         - 
                                         
                                           k 
                                            
                                           
                                               
                                           
                                            
                                           
                                             Ω 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     t 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                         - 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             ∞ 
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 1 
                               
                               ∞ 
                             
                              
                             
                               
                                 
                                   J 
                                   
                                     2 
                                      
                                     k 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     1 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   J 
                                   
                                     2 
                                      
                                     m 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   [ 
                                   
                                     2 
                                      
                                     
                                       ( 
                                       
                                         
                                           m 
                                            
                                           
                                               
                                           
                                            
                                           
                                             Ω 
                                             2 
                                           
                                         
                                         + 
                                         
                                           k 
                                            
                                           
                                               
                                           
                                            
                                           
                                             Ω 
                                             1 
                                           
                                         
                                       
                                       ) 
                                     
                                      
                                     t 
                                   
                                   ] 
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   
                     2 
                      
                     c 
                   
                   ) 
                 
               
             
           
         
       
     
         [0020]    where I dc,q  is the photocurrent at quadrature and J is a Bessel function of the first kind. The quadrature condition is given by φ dc =(2k+1)π/2 where k is an integer. Equation (2a) is the average (DC) current, Eq. (2b) are the odd-order RF terms, and Eq. (2c) are the even-order RF terms. Thus, the total photocurrent for this output is I 1 (t)=I dc,mzm +I odd,mzm +I even,mzm . The photocurrent associated with E 2  is I 2 (t)=2I dc,q −I 1 (t). The treatment here will assume a small-signal two-tone test with equal amplitude tones, thus φ 1 =φ 2 =φ&lt;&lt;1. A small-signal approximation allows for the Bessel functions to be written as J n (φ)≈φ n /(2 n n!). These conditions can be applied to Eq. (2) to yield the fundamental photocurrents 
         [0000]        I   fund,mzm   =φI   dc,q  sin(φ dc )[ sin(Ω 1   t )+ sin(Ω 2   t )]  (3)
 
         [0021]    Assuming all of the current is delivered to a load with resistance R, the average output power for both the fundamentals is P fund,mzm =φ 2 I dc,q   2  sin 2 (φ dc )R/2. The work here concentrates on even-order distortion. The largest small-signal distortion in Eq. (2) is second-order intermodulation distortion (IMD2) at frequencies |f 1 ±f 2 | given by the first two double summations in Eq. (2c) with n=p=0. The small-signal photocurrent for these two terms is 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     
                       
                         imd 
                          
                         
                             
                         
                          
                         2 
                       
                       , 
                       mzm 
                     
                   
                   = 
                   
                     
                       ± 
                       
                         
                           
                             φ 
                             2 
                           
                            
                           
                             I 
                             
                               dc 
                               , 
                               q 
                             
                           
                            
                           
                             cos 
                              
                             
                               ( 
                               
                                 φ 
                                 dc 
                               
                               ) 
                             
                           
                         
                         2 
                       
                     
                      
                     
                       
                         cos 
                          
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   Ω 
                                   2 
                                 
                                 ∓ 
                                 
                                   Ω 
                                   11 
                                 
                               
                               ) 
                             
                              
                             t 
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0022]    The average power associated with Eq. (4) is P imd2,mzm =φ 4 I dc,q   2  cos 2 (φ dc )R/8. Finally, the OIP2 due to MZM-generated IMD2 is 
         [0000]    
       
         
           
             
               
                 
                   
                     O 
                      
                     
                         
                     
                      
                     I 
                      
                     
                         
                     
                      
                     P 
                      
                     
                         
                     
                      
                     
                       2 
                       mzm 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           
                             sin 
                             4 
                           
                            
                           
                             ( 
                             
                               φ 
                               dc 
                             
                             ) 
                           
                         
                       
                       
                         
                           cos 
                           2 
                         
                          
                         
                           ( 
                           
                             φ 
                             dc 
                           
                           ) 
                         
                       
                     
                      
                     
                       I 
                       
                         dc 
                         , 
                         q 
                       
                       2 
                     
                      
                     
                       R 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0023]    As given by Eq. (5) and detailed previously in Urick et al. 1, small deviations from quadrature bias can significantly degrade the OIP2. In fact, the tolerance on MZM bias can be quite stringent to maintain third-order-limited performance in multi-octave links. 
         [0024]    Photodiode Distortion 
         [0025]    Numerous models have been developed to describe photodiode distortions in microwave photonics applications (see K. J. Williams and R. D. Esman, “Design considerations for high-current photodetectors,” J. Lightwave Technol. 17, 1443-1454 (1999), and Y. Fu, H. Pan, Z. Li, A. Beling, and J. C. Campbell, “Characterizing and modeling nonlinear intermodulation distortions in modified uni-traveling carrier photodiodes,” IEEE J. Quantum Electron. 47, 1312-1319 (2011)). Here, we assume that the fundamentals from the MZM drive the photodiode, which can be described by a Taylor series expansion. We apply the following definition for a Taylor series expansion 
         [0000]        I   pd   =a   0   +a   1 ( I   in   −I   dc )+ a   2 ( I   in   −I   dc ) 2   +a   3 ( I   in   −I   dc ) 3 + . . .  (6)
 
         [0026]    where I pd  is the output current of the photodiode with an injection current of I in  and an average current I dc . The Taylor coefficients are defined as usual 
         [0000]    
       
         
           
             
               
                 
                   
                     a 
                     m 
                   
                   = 
                   
                     
                       
                         1 
                         
                           m 
                           ! 
                         
                       
                       · 
                       
                         
                           
                             d 
                             m 
                           
                            
                           
                             I 
                             pd 
                           
                         
                         
                           dI 
                           in 
                           m 
                         
                       
                     
                      
                     
                        
                       
                         
                           I 
                           in 
                         
                         = 
                         
                           I 
                           dc 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0027]    Now, if we assume that I in =I fund,mzm  as given by Eq. (3), that is, the ideal IMDD link provides the injection current to a nonlinear photodiode described by Eq. (6), then 
         [0000]    
       
         
           
             
               
                 
                   
                     I 
                     pd 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           a 
                           0 
                         
                         + 
                         
                           
                             a 
                             2 
                           
                            
                           
                             I 
                             2 
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         a 
                         1 
                       
                        
                       I 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               Ω 
                               1 
                             
                              
                             t 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         a 
                         1 
                       
                        
                       I 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               Ω 
                               2 
                             
                              
                             t 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         
                           
                             a 
                             2 
                           
                            
                           
                             I 
                             2 
                           
                         
                         2 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               Ω 
                               1 
                             
                              
                             t 
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         
                           
                             a 
                             2 
                           
                            
                           
                             I 
                             2 
                           
                         
                         2 
                       
                        
                       
                         cos 
                          
                         
                           ( 
                           
                             2 
                              
                             
                               Ω 
                               2 
                             
                              
                             t 
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                        
                       
                         I 
                         2 
                       
                        
                       
                         cos 
                          
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   Ω 
                                   1 
                                 
                                 - 
                                 
                                   Ω 
                                   2 
                                 
                               
                               ) 
                             
                              
                             t 
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         a 
                         2 
                       
                        
                       
                         I 
                         2 
                       
                        
                       
                         cos 
                          
                         
                           [ 
                           
                             
                               ( 
                               
                                 
                                   Ω 
                                   1 
                                 
                                 + 
                                 
                                   Ω 
                                   2 
                                 
                               
                               ) 
                             
                              
                             t 
                           
                           ] 
                         
                       
                     
                     + 
                     … 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0028]    where I=φI dc,q  sin(φ dc ) and the expansion has been carried out to terms of second order. The currents for the IMD2 terms in Eq. (8) are 
         [0000]        I   imd2,pd   =±a   2 φ 2   I   dc,q   2  sin 2 (φ dc )cos [(Ω 2 ∓Ω 1 ) t]   (9)
 
         [0000]    The OIP2 for the photodiode can be determined by the expression OIP2 pd =P fund,pd   2 /P imd2,pd , where P fund,pd =a 1   2 I 2 R/2 and P imd2,pd =a 2   2 I 4 R/2 are the average powers for the fundamental and IMD2, respectively. Thus, 
         [0000]    
       
         
           
             
               
                 
                   
                     O 
                      
                     
                         
                     
                      
                     I 
                      
                     
                         
                     
                      
                     P 
                      
                     
                         
                     
                      
                     
                       2 
                       pd 
                     
                   
                   = 
                   
                     
                       
                         
                           a 
                           1 
                           4 
                         
                          
                         R 
                       
                       
                         2 
                          
                         
                           a 
                           2 
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0029]    Combined Response 
         [0030]    Our proposition is that the MZM biased can be adjusted to generate even-order distortion matching the amplitude of that arising from the photodiode. The forms of Eqs. (4) and (9) predict that two sources of distortion can be out of phase as well. The treatments of MZM- and photodiode-generated distortion above can be combined to derive the cancellation condition. The peak current at both IMD2 terms is obtained by addition of Eqs. (4) and (9): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       
                         
                           imd 
                            
                           
                               
                           
                            
                           2 
                         
                         , 
                         peak 
                       
                     
                     = 
                     
                       
                         ± 
                         
                           φ 
                           2 
                         
                       
                        
                       
                         
                           I 
                           
                             dc 
                             , 
                             q 
                           
                         
                          
                         
                           [ 
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     dc 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 a 
                                 2 
                               
                                
                               
                                 I 
                                 
                                   dc 
                                   , 
                                   q 
                                 
                               
                                
                               
                                 
                                   sin 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     dc 
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0031]    where the “+” and “−” signs correspond to the terms at (Ω 2 −Ω 1 ) and (Ω 2 +Ω 1 ), respectively. Setting Eq. (11) to zero yields the cancellation condition as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       cos 
                        
                       
                         ( 
                         
                           φ 
                           dc 
                         
                         ) 
                       
                     
                     
                       
                         sin 
                         2 
                       
                        
                       
                         ( 
                         
                           φ 
                           dc 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       - 
                       2 
                     
                      
                     
                       a 
                       2 
                     
                      
                     
                       
                         I 
                         
                           dc 
                           , 
                           q 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0032]    The analysis above also predicts that second-harmonic distortion will cancel as well; expanding the Taylor series to higher order shows that all even-order distortion is suppressed with this technique. The parameters in Eq. (12) are readily determined. The bias phase and photocurrent at quadrature are easily measured. The small-signal gain of the link will allow for the magnitude of a 1  to be calculated. With this information, a measurement of the photodiode OIP2 will give the magnitude for a 2  by way of Eq. (10). Cancellation of the IMD2 is then predicted by Eq. (12) to be cyclic as a function of φ dc . 
         [0033]    The experimental investigation involves two apparatuses, a single-MZM link  100  ( FIG. 1 ) to demonstrate the cancellation and a two-MZM setup to isolate photodiode IMD2  200 . The two-MZM setup  200  is shown in  FIG. 2 . Two lasers  102  and  202  respectively at different wavelengths output laser lights  104  and  204  that are respectively modulated via two quadrature-biased MZMs  106  and  206  each having a respective RF signal input  108  and  208 . The RF signal generator power is adjusted to yield the same modulation depth on each laser. Variable optical attenuators (VOAs)  120  and  214  are then employed to equalize the average optical power from each MZM output  112  and  212  before coupling the two channels onto the photodiode  116  being evaluated to generate the RF output signal  118 . If the two lasers are spaced at a frequency difference much larger than the modulation frequencies, then the IMD2 from the apparatus should be dominated by the photodiode nonlinearity. As compared to three-tone architectures employing three of more MZMs for odd-order distortion measurements (see A. Ramaswamy, N. Numoya, K. J. Williams, J. Klamkin, M. Piels, L. A. Johansson, A. S. Hastings, L. A. Coldren, and J. E. Bowers, “Measurement of intermodulation distortion in high-linearity photodiodes,” Opt. Exp. 18, 2317-2324 (2010), and M. N. Draa, A. S. Hastings, and K. J. Williams, “Comparison of photodiode nonlinearity measurement systems,” Opt. Exp. 19, 12635-12645 (2011)), this relatively simple two-tone setup is adequate for IMD2. 
         [0034]    The structure shown in  FIG. 2  was constructed using two 100-mW semiconductor lasers (EM4, Inc.) at 1548 nm and 1560 nm. The two MZMs (EOSPACE) exhibited 20 GHz of analog bandwidth and had nearly equal V π (Ω). The two signal generators were set at f 1 =0.9 GHz and f 2 =1.1 GHz. With the variable optical attenuators set to output the same average optical power, the signal generators were tuned to establish the same modulation depth on each laser. The photodiode being examined was an Applied Optoelectronics PD3000 with about a 3-GHz 3-dB bandwidth. The photodiode was reversed biased with 1 V and the OIP2 was measured by sweeping the input power at f 1  and f 2 . The measured OIP2=13.5 dBm due to IMD2 at 2.0 GHz for an average photocurrent of I dc =3.0 mA as shown in  FIG. 3 . Also measured were OIP2=13.8 and 13.5 dBm at I dc =2.5 and 3.5 mA, respectively. 
         [0035]    The same photodiode was used in a link such as shown in  FIG. 1 . An 80-mW semiconductor laser near 1550 nm (EM4, Inc.) was used as the source at the input to a dual-output 20-GHz MZM (EOSPACE) with V π =4.5 V at 1 GHz. One output of the MZM was fed to an optical power meter to monitor the bias, while the other was attenuated and connected to the photodiode. The quadrature photocurrent was set at I dc,q =3.0 mA. A network analyzer was employed to measure the single-tone small-signal gain with an input power of −20 dBm. The response was 4-dB less than that predicted by Eq. (3) with R=50Ω at 1 GHz. With this result, Eq. (8) allows for a 1 =0.631 to be calculated. Equation (10) can then be solved for a 2 =13.3 A −1 . 
         [0036]    A two-tone test was applied to the link with frequencies f 1 =0.9 GHz and f 2 =1.1 GHz, the results of which are shown in  FIG. 4 . The fundamentals both exhibit the same gain. The IMD2 at 2.0 GHz resulted in an OIP2=12.5 dBm when the MZM was biased at quadrature. This level is very close to that measured for the photodiode alone with the setup in  FIG. 2 , indicating that the photodiode is limiting the IMD2. The MZM bias was adjusted to determine the minimum IMD2, which was observed well away from quadrature at an average photocurrent of 2.5 mA. The IMD2 at this cancellation point is also plotted in  FIG. 4 . Very strong suppression was measured at input powers below −15 dBm; the suppression is good but much less at higher input powers. A second-order function is fit to the data at higher input powers resulting in OIP2=30 dBm. This limiting OIP2 is attributed to reflections in the apparatus but is still 16.5 dB better than the OIP2 of the photodiode. As opposed to other linearization techniques, the fundamental is not affected by this method. The carrier-to-intermodulation ratio (CIR) is plotted in  FIG. 5  to demonstrate this point. At high input powers, the CIR is 17.5 dB higher at 2.5 mA than that at 3.0 mA (MZM quadrature). In the strong cancellation region, the CIR difference is upwards of 40 dB. 
         [0037]    To confirm the cancellation condition, equations were plotted with measured data as a function of φ dc . As shown in  FIG. 6 , the calculated fundamental power [Eq. 8], the IMD2 power [Eq. (11)] and the DC photocurrent [Eq. (2a)] follow the experimental results. The measured and calculated DC photocurrents agree precisely. The calculated and experimental fundamental powers agree well at quadrature (I dc,q =3.0 mA). However, the measured values diverge above the calculation below 3.0 mA and go below the calculation photocurrents above 3.0 mA. The IMD2 curves follow the same trend with a larger divergence but do agree quite well at the cancellation condition. The reason for the divergences is attributed to photodiode compression, which is worse at higher photocurrents and at higher frequencies. The three measured OIP2 values using the setup in  FIG. 2  also support this attribution. With this caveat, the measured results very much support the theory in predicting the cancellation of photodiode IMD2 with MZM-generated IMD2. 
         [0038]    To demonstrate the utility at high frequencies and under conditions where the photodiode is uncompressed, a second device was measured. A u 2 t Photonics (Model: u 2 t S/N 46167) commercial device was measured as above biased at 3V with I dc,q =2.5 mA, at center frequency 35.5 GHz with 500 MHz separation. Shown in  FIG. 7 , the calculated fundamental power [Eq. 8] and IMD2 power [Eq. 11] precisely agree with the experimental result over the bias conditions, as the uncompressed diode behaves according to the theoretical assumption. The calculated MZM generated IMD2 (gray dotted line) is also plotted to distinguish between the measured results and what would be obtained with an ideal linear photodiode. 
         [0039]    A phase- or polarization-modulation link can be employed to cancel photodiode even-order distortion in much the same way as the IMDD link. A similar theoretical treatment can be applied to the phase- or polarization-modulation architecture. The phase-modulation link  300  is shown in  FIG. 8 . A laser  302  outputs a laser light  304  to a phase modulator  306  having an RF signal input  308  and an output connected to a Mach-Zehnder interferometer (MZI)  312 . MZI  312  has an input  314  connected to a DC voltage source for applying an adjustable DC bias voltage to MZI  312  where the bias is adjusted on the MZI to achieve the distortion cancellation, and an optical signal output  316  that is input to a photodiode  318 , producing RF output signal  320 . In contrast to the IMDD link, the phase-modulation link allows for precise bias control of the MZI in a point-to-point link at the receiver end, with the trade-off of the additional difficulty in stabilizing the MZI. A preliminary result is plotted in  FIG. 9 , similar to  FIGS. 6 and 7 . For the link measured, the MZI had a 375 ps differential delay. The same PD300 was measured under the same conditions, 1 V bias and I dc,q =3.0 mA where the photodiode is compressed. The calculated fundamental power, the IMD2 power and the DC photocurrent power again match well at quadrature but similar to  FIG. 6  are not in good agreement at the cancellation condition due to the photodiode compression. However the general trend is followed for both the measurement and theoretical analysis. 
         [0040]    The polarization-modulation link architecture  400  is shown in  FIG. 10 . A laser  402  provides a laser light input  404  to a polarization modulator  406  with an RF input  408  and an output  410  where the modulation axis is adjusted by a polarization controller  412  and acts as the DC bias control with its output  414  input to a beam splitter  416 , with an output  418  that is input to a photodiode  420 , producing an RF output signal  422 . The theoretical analysis for the architecture follows the same method as previously. 
         [0041]    Obviously many modifications and variations of the present invention are possible in the light of the above teachings. It is therefore to be understood that the scope of the invention should be determined by referring to the following appended claims.