Abstract:
A laser radar, or “lidar” system, employs an asymmetric single-ended detector to detect received signals reflected back from targets. The asymmetric single-ended detector benefits from a reduced part count and fewer optical splices while nearly achieving a same gain as a symmetric differential detector.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 14/249,085, entitled “System and Method for Using Combining Couplers with Asymmetric Split Ratios in a Lidar System,” filed on Apr. 9, 2014, and granted as U.S. Pat. No. ______. The foregoing application is incorporated herein by reference in its entirety. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention is generally related to lidar system (i.e., laser radar systems), and more particularly, using asymmetric split ratio combining couplers and single-ended detection. 
       BACKGROUND OF THE INVENTION 
       [0003]    Various conventional lidar systems (i.e., laser rader systems) employ coherent detection, in which a received optical signal is combined with a mixing or reference optical signal, typically with a symmetric combining coupler, to produce an interference signal. A symmetric combining coupler is a combining coupler with a power transfer ratio of 50% between its two output ports (i.e., each port receives 50% of the output power from the combing coupler). 
         [0004]    In some conventional systems, the interference signal from a single port of the symmetric combining coupler is applied to a detector to produce a detected signal. This is referred to as single-ended detection. Because the two ports of the symmetric combining coupler each produce an interference signal with a 180-degree phase difference from one another, these two interference signals may be combined to produce a differential signal that can subsequently be applied to a detector to produce a detected signal. This is referred to as differential detection. 
         [0005]    Optical systems employing single-ended detection typically employ fewer detectors and require fewer optical fiber splices. However, such optical systems employing single-ended detection experience a 3 dB loss in signal-to-noise ratio (SNR) over those employing differential detection. Because of this, optical systems that require higher sensitivity (such as lidar systems) typically employ differential detection with symmetric combining couplers rather than single-ended detection. 
         [0006]    What is needed is a lidar system that employs single-ended detection that does not suffer the SNR losses of conventional systems. 
       SUMMARY OF THE INVENTION 
       [0007]    According to various implementations of the invention, a laser radar, or “lidar” system, employs an asymmetric single-ended detector to detect received signals reflected back from targets. The asymmetric single-ended detector benefits from a reduced part count (as compared with a differential detector) and fewer optical splices while achieving nearly a same gain as a symmetric differential detector. 
         [0008]    Various implementations of the invention are directed toward a lidar system that includes a laser source configured to generate a laser output; a splitter configured to split the laser output into a transmit signal and a mixing signal; and a single-ended detector comprising: an asymmetric combiner configured to combine a received signal and the mixing signal and output a combined signal, wherein the received signal is a reflected portion of the transmit signal reflected back from a target and received by the lidar, wherein the asymmetric combiner has a split ratio greater than 0.5, and a detector configured to detect the combined signal. 
         [0009]    Various implementations of the invention are directed toward a method for detecting a reflected signal from a target, the method comprising: receiving the reflected signal from the target; combining the received signal with a mixing signal using an asymmetric combiner having a split ratio greater than 0.5 to generate a combined signal; and detecting the combined signal using a single-ended detector. 
         [0010]    These implementations, their features and other aspects of the invention are described in further detail below. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]      FIG. 1  illustrates a conventional symmetric single-ended detector. 
           [0012]      FIG. 2  illustrates a conventional symmetric double-ended detector. 
           [0013]      FIG. 3  illustrates a lidar system (i.e., laser radar) according to various implementations of the invention. 
           [0014]      FIG. 4  illustrates a relationship between e-Field inputs and outputs for a combining coupler having a split ratio, σ. 
           [0015]      FIG. 5  illustrates respective scaling factors between e-Field inputs and outputs form a combining coupler having a split ratio, σ. 
           [0016]      FIG. 6  illustrates gain curves for a single-ended detector in comparison with a differential detector as a function of split ratio,  FIG. 4  illustrates a relationship between e-Field inputs and outputs from a combining coupler having a split ratio, σ. 
       
    
    
     DETAILED DESCRIPTION 
       [0017]    As discussed above, conventional lidar systems that employ single-ended detection use fewer receive detectors and require fewer fiber-optic splices than their double-ended, differential detection counterparts; however, detection in such conventional lidar system experiences a 3 dB loss in signal-to-noise ratio (SNR) in relation to conventional lidar systems that employ differential detection. According to various implementations of the invention, asymmetric combining couplers may be used in a lidar system employing single-ended detection to substantially reduce these losses in SNR. Moreover, in various implementations of the invention, use of the asymmetric combining couplers does not increase the cost of parts or labor, or increase package size in comparison with conventional single-ended detection systems. 
         [0018]    According to various implementations of the invention, an asymmetric combining coupler with an asymmetric split ratio between its output ports (e.g., 80%/20%, 90%/10%, etc.) may be used in the lidar system. In order to fully recognize the gains (i.e., increased sensitivity, improvement in SNR, etc.) of the asymmetric combining couplers in such systems, a power level of a reference signal (also referred to as a mixing signal) may be adjusted in relation to the asymmetric split ratio used. 
         [0019]    Before discussing various implementations of the invention, conventional detectors are first described.  FIG. 1  illustrates a conventional symmetric single-ended detector  100 . Symmetric single-ended detector  100  includes an optical mixing signal  110  (sometimes referred to herein as an “LO signal”) and a received optical signal  120  (typically reflected from a target as would be appreciated). Depending on the application, optical mixing signal  110  may be a time-shifted (i.e., time-delayed) version of an actual transmit signal as would be appreciated. Received optical signal  120  and optical mixing signal  110  may be mixed together, for example, via a coupler having a symmetric split ratio (e.g., 50/50) such as a coupler  130 . An output  140  of coupler  130  is applied to an electro-optical detector  150 , which as illustrated may be a PIN diode. Detector  150  converts optical power (e.g., from coupler  130 ) to an electrical current  160  as would be appreciated. Current  160  may be subsequently measured as would be appreciated. 
         [0020]      FIG. 2  illustrates a conventional symmetric double-ended detector  200 . Symmetric double-ended detector  200  includes an optical mixing signal  110  (which, again, may be a time-shifted version of the transmit signal) and a received optical  120 . Received optical signal  120  and optical mixing signal  110  may be mixed together, for example, via a coupler having a symmetric split ratio (e.g., 50/50) such as coupler  130 . With detector  200 , both outputs  240  (illustrated as an output  240 A and an output  240 B) of coupler  130  are applied to electro-optical detectors  250  (illustrated as an electro-optical detector  250 A and an electro-optical detector  250 B). Detectors  250 A,  250 B convert optical power to electrical currents  260 A,  260 B, respectively, which may be subsequently measured as would be appreciated. Creating a difference signal between these two currents typically realizes a sensitivity gain of 3 dB over the single-ended detector  100  as would be appreciated. 
         [0021]    A simple lidar system employing asymmetric detection is now described in accordance with various implementations of the invention.  FIG. 3  illustrates a lidar system  300  (i.e., laser radar) according to various implementations of the invention. Lidar system  300  includes a coherent laser source  310  that outputs an optical signal  315 , which in turn propagates to a splitting coupler  320 . Splitting coupler  320  splits output optical signal  315  into two components, a transmit signal  325  and a mixing signal  326  (i.e., an LO signal). 
         [0022]    Transmit signal  325  propagates through a optical separator  330  onto a fiber tip  340 , from which it transitions into free space, is focused by a lens system  350  onto a target  360  and reflected back as a reflected signal. The reflected signal follows a path back in the opposite direction through lens system  350  and fiber tip  340  and back to optical separator  330 . 
         [0023]    Optical separator  330  separates transmit signal  325  from the reflected signal and outputs a receive signal  336 . In various implementations of the invention, optical separator  330  may be a circulator, a splitter, or other suitable optical separator that separates transmit signal  325  from the reflected signal as would be appreciated. 
         [0024]    Receive signal  336  and mixing signal  326  are combined by a combining coupler  370  to output a combined signal  375 . In some implementations of the invention, a gain or attenuation stage (not otherwise illustrated) may be applied to mixing signal  326  prior to reaching combining coupler  370  in order to adjust mixing signal  326  to an anticipated power level of receive signal  336  as would be appreciated. According to various implementations of the invention, combining coupler  370  is single-ended and utilizes an asymmetric split ratio as discussed in further detail below. 
         [0025]    Combined signal  375  is provided to a detector  380  to convert the combined signal  375  to an electrical signal for measurement as would be appreciated. 
         [0026]    More complex lidar systems may benefit from various implementations of the invention as would be appreciated. For example, in some implementations of the invention, lidar system  300  may be incorporated into a dual laser source, chirped coherent laser radar system capable of unambiguously and simultaneously measuring both range and Doppler velocity of a point on target  360 . Such a laser radar system is described in U.S. Pat. No. 8,582,085 entitled “Chirped Coherent Laser Radar with Multiple Simultaneous Measurements,” which is incorporated herein by reference in its entirety. 
         [0027]    The amplitudes of noise power and signal power under certain conditions are derived for both conventional symmetric single-ended detector  100  and conventional symmetric differential detector  200  and are now described. Afterwards, an effect of varying the split ratio of combining coupler  370  is explored. 
         [0028]    For purposes of this discussion, the signal-to-noise ratio (SNR) of the electrical output of a detector is described as the power of the component with the frequency difference of LO and Rx signals relative to the noise power in a given receive bandwidth. To explore the effect of varying the split ratio of combining coupler  370  on the resulting SNR of the output of detector  380 , an analytical expression for the field strength at a detector surface and the resulting detector current is derived. Absolute values of noise levels or SNR are not of interest here, but rather the relative gain or loss of SNR relative to a symmetric coupler (i.e., a couple having a split ratio of 0.5 or “50/50”). Because relative gains are calculated, physical constants and their associated units are omitted. 
         [0029]    For illustrative purposes, this derivation of relative SNR is based on the following assumptions:
       The receive signal (i.e., Rx signal) is of substantially lower power than the transmit signal (i.e., LO signal). Therefore, the noise power density at the detector output is determined only by the fraction of optical power from the LO signal reaching the detector surface.   The detector is not saturated, i.e. the optical power on the detector surface is within the limits of linear operation of the detector.   Detector shot noise is the only significant contributor to the overall noise level, all other noise sources are negligible.   The detector converts optical power to a current within a bandwidth significantly lower than the optical frequency. All frequency components of the optical power at the optical frequency or its multiples are averaged by the detector; only components at differences of optical frequencies are propagated as the signal to be detected.       
 
         [0034]    With these assumptions, the electrical component of the optical LO signal electromagnetic field (i.e., E-field) may be described as: 
         [0000]        LO ( t )= E   LO *cos(ω LO ( t )* t+φ   LO )
 
         [0035]    where
       E LO  denotes the field strength;   ω LO (t) is the time-varying optical frequency of the LO signal; and   φ LO  is the LO signal phase at t=0.
 
Similarly, the receive signal may be described as:
       
 
         [0000]        Rx ( t )= E   Rx *cos(ω Rx ( t )* t+φ   Rx )
 
         [0039]    where
       E Rx  denotes the field strength;   ω Rx (t) is the time-varying optical frequency of the Rx signal; and   φ Rx  is the Rx signal phase at t=0.       
 
         [0043]    Symmetric coupler  130  is a reciprocal and symmetric four-port device which exhibits a power split ratio, σ, between its two inputs E 1 , E 2  and its two outputs E 3 , E 4  as illustrated in  FIG. 4 . The E-fields of the optical inputs are propagated according to the following transfer matrix: 
         [0000]    
       
         
           
             
                
               
                 
                   
                     
                       E 
                       3 
                     
                   
                 
                 
                   
                     
                       E 
                       4 
                     
                   
                 
               
                
             
             = 
             
               
                  
                 
                   
                     
                       
                         
                           1 
                           - 
                           σ 
                         
                       
                     
                     
                       
                         
                           - 
                           j 
                         
                         * 
                         
                           σ 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           j 
                         
                         * 
                         
                           σ 
                         
                       
                     
                     
                       
                         
                           1 
                           - 
                           σ 
                         
                       
                     
                   
                 
                  
               
               * 
               
                  
                 
                   
                     
                       
                         E 
                         1 
                       
                     
                   
                   
                     
                       
                         E 
                         2 
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    In many applications of these couplers, only three of the four ports are used and the unconnected port is terminated to eliminate back reflection. In such applications the split ratio of the ports on one side relative to the one port on the other side is indicated as a percentage such as illustrated in  FIG. 5 . 
         [0044]    For single-ended detectors, the power split ratio of a given coupler is σ, where 0&lt;σ&lt;1. For symmetric couplers, σ=0.5; whereas for asymmetric couplers, (e.g., σ&lt; &gt;0.5), the power from the Rx input reaching the detector is scaled by σ; and the power from the LO input reaching the detector is scaled by 1−σ. The starting phases φ LO  and φ Rx  are arbitrary because the phase at t=0 depends on the location of the measurement. 
         [0045]    The electrical field strength of the optical signal at the output of the coupler connected to the detector may be expressed as: 
         [0000]        E   1   =√{square root over (σ)}*E   Rx *sin(ω Rx ( t )* t+φ   Rx )+√{square root over (1−σ)}* E   LO *cos(ω LO ( t )* t+φ   LO )
 
         [0000]    As indicated, both components E Rx  and E LO  are scaled with the square root of the respective split ratio (i.e., either the square root of σ or the square root of 1−σ). The detector current is proportional to the square of this field strength. 
         [0000]    
       
      
       I 
       1 
       =η*E 
       1 
       2  
      
     
         [0046]    The proportionality factor η contains the detector efficiency, among other constants. Only the SNR change relative to split ratio is of interest; the detector efficiency does not affect this ratio and is omitted in the following equations. 
         [0047]    After squaring the expression for E 1  and rearranging with trigonometric identities, the detector current can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   I 
                   1 
                 
                 ~ 
                 
                   σ 
                   2 
                 
               
               * 
               
                 E 
                 Rx 
                 2 
               
               * 
               
                 ( 
                 
                   1 
                   - 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                           * 
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           * 
                           t 
                         
                         + 
                         
                           2 
                           * 
                           
                             ϕ 
                             Rx 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
             + 
             
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
               * 
               
                 ( 
                 
                   1 
                   + 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                           * 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           * 
                           t 
                         
                         + 
                         
                           2 
                           * 
                           
                             ϕ 
                             LO 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     + 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The first term is the contribution of the receive power and is negligible due to the assumption of low receive signal power. The second term is the main contribution to the total detector power from the LO signal; the shot noise power density of the resulting electrical current scales with the square root of this power. The third term is the mixing product with the frequency difference between Rx and LO inputs. This term is in a frequency range detectable by the detector and is converted into the electrical receive signal. The fourth term describes a signal at twice the optical frequency; since it does not produce an average current and is proportional to the receive field strength, its contribution to the total detector power is negligible. 
         [0048]    The detector will average all frequency components at optical frequency or above and the detector current then may be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   I 
                   1 
                 
                 ~ 
                 
                   σ 
                   2 
                 
               
               * 
               
                 E 
                 Rx 
                 2 
               
             
             + 
             
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the resulting contribution to the detector current by average power on the detector surface is therefore given by: 
         [0000]    
       
         
           
             
               
                 I 
                 avg 
               
               ~ 
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
             
             * 
             
               E 
               LO 
               2 
             
           
         
       
     
         [0049]    To arrive at an SNR expression relative to the case of the symmetric coupler (σ=0.5), the shot noise power and therefore the average detector current are held constant by adjusting the power of the LO signal according to the split ratio. The square of the field strength of the LO signal for symmetric split ratio is denoted as N LO   2  and given by: 
         [0000]    
       
         
           
             
               
                 
                   I 
                   avg 
                 
                 ~ 
                 0.25 
               
               * 
               
                 N 
                 LO 
                 2 
               
             
             = 
             
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
             
           
         
       
     
         [0000]    Expressing the LO power supplied to the coupler in terms of split ratio while keeping the average detector current constant yields: 
         [0000]    
       
         
           
             
               E 
               LO 
             
             = 
             
               
                 1 
                 
                   
                     2 
                     * 
                     
                       ( 
                       
                         1 
                         - 
                         σ 
                       
                       ) 
                     
                   
                 
               
               * 
               
                 N 
                 LO 
               
             
           
         
       
     
         [0050]    The receive signal of interest is given by the third term in the equation for detector current: 
         [0000]        I   sig ˜+√{square root over (σ−σ 2 )}* E   Rx   *E   LO *sin((ω Rx ( t )−ω LO ( t ))* t+φ   Rx −φ LO )
 
         [0000]    Accounting for the adjustment of LO power depending on split ratio, the signal current will then become 
         [0000]    
       
         
           
             
               
                 I 
                 sig 
               
                
               
                 ∼ 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 1 
                 
                   
                     2 
                     * 
                     
                       ( 
                       
                         1 
                         - 
                         σ 
                       
                       ) 
                     
                   
                 
               
               * 
               
                 N 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and after some simplification: 
         [0000]    
       
         
           
             
               
                 I 
                 sig 
               
                
               
                 ∼ 
               
             
             + 
             
               
                 
                   σ 
                   2 
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 N 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    with the RMS amplitude 
         [0000]    
       
         
           
             
               I 
               sig_RMS 
             
              
             
               ∼ 
             
              
             
               
                 σ 
               
               2 
             
             * 
             
               E 
               Rx 
             
             * 
             
               N 
               LO 
             
           
         
       
     
         [0051]    Shot noise current density in the electrical signal is proportional to the square root of the average detector current; multiplying with the square root of some fixed receive bandwidth yields the noise current magnitude: 
         [0000]        I   noise   _   RMS =√{square root over (2* q*I   avg   *Bw )}
 
         [0000]    where q is the elementary charge and Bw is the receive bandwidth. 
         [0052]    Inserting the equation for average current and ignoring the constant 2*q gives: 
         [0000]    
       
         
           
             
               
                 I 
                 noise_RMS 
               
                
               
                 ( 
                 σ 
                 ) 
               
             
             ∼ 
             
               
                 
                   N 
                   LO 
                 
                 2 
               
               * 
               
                 Bw 
               
             
           
         
       
     
         [0000]    The RMS noise current is not depending on split ratio which is expected, because LO power level was adjusted for average noise current to be independent of split ratio. 
         [0000]    
       
         
           
             
               S 
                
               
                   
               
                
               N 
                
               
                   
               
                
               
                 R 
                  
                 
                   ( 
                   σ 
                   ) 
                 
               
             
             = 
             
               10 
               * 
               
                 
                   log 
                   10 
                 
                  
                 
                   ( 
                   
                     
                       I 
                       sig_RMS 
                       2 
                     
                     
                       I 
                       noise_RMS 
                       2 
                     
                   
                   ) 
                 
               
                
               
                 ∼ 
               
                
               10 
               * 
               
                 
                   
                     log 
                     10 
                   
                    
                   
                     ( 
                     
                       σ 
                       * 
                       
                         E 
                         Rx 
                         2 
                       
                       * 
                       
                         1 
                         Bw 
                       
                     
                     ) 
                   
                 
                  
                 
                     
                 
                 [ 
                 dB 
                 ] 
               
             
           
         
       
     
         [0053]    The ratio of signal amplitudes over split ratio given constant noise power translates directly to the SNR dependency in dB on split ratio relative to σ=0.5: 
         [0000]      SNR rel (σ)=SNR(σ)−SNR(0.5)=10*log 10 (2*σ)[dB]
 
         [0054]    For differential detectors, the power split ratio of the coupler is σ, where 0&lt;σ&lt;1; for symmetric couplers σ=0.5. For asymmetric couplers, the power from the Rx input reaching a first detector is scaled by σ and the power from the LO input reaching the first detector is scaled by 1−-σ. These scaling factors are reversed for the second detector, i.e., the power from the Rx input reaching the second detector is scaled by 1−σ and the power from the LO input reaching the second detector is scaled by σ. Again, the starting phases are arbitrary. 
         [0055]    The electrical field strength of the optical signal at the output of the coupler connected to the detector generating current I 1  (e.g., current  260 B in  FIG. 2 ) may be expressed as: 
         [0000]    
       
         
           
             
               E 
               1 
             
             = 
             
               
                 
                   σ 
                 
                 * 
                 
                   E 
                   Rx 
                 
                 * 
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         
                           
                             ω 
                             Rx 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         * 
                         t 
                       
                       + 
                       
                         ϕ 
                         Rx 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   
                     1 
                     - 
                     σ 
                   
                 
                 * 
                 
                   E 
                   LO 
                 
                 * 
                 
                   cos 
                    
                   
                     ( 
                     
                       
                         
                           
                             ω 
                             LO 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         * 
                         t 
                       
                       + 
                       
                         ϕ 
                         LO 
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    and the optical field strength contained in the output of the coupler connected to the detector generating current I 2  (e.g., current  260 A in  FIG. 2 ) may be expressed as: 
         [0000]        E   2 =√{square root over (1 −σ)}* E   Rx *cos(ω Rx ( t )* t+φ   Rx )+√{square root over (σ)}* E   LO *sin(ω LO ( t )* t+φ   LO )
 
         [0056]    The dependency of I 1  on E LO  and E Rx  is identical to the result derived for the single-ended case and given by: 
         [0000]    
       
         
           
             
               
                 I 
                 1 
               
                
               
                 ∼ 
               
                
               
                 σ 
                 2 
               
               * 
               
                 E 
                 Rx 
                 2 
               
               * 
               
                 ( 
                 
                   1 
                   - 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                           * 
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           * 
                           t 
                         
                         + 
                         
                           2 
                           * 
                           
                             ϕ 
                             Rx 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
             + 
             
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
               * 
               
                 ( 
                 
                   1 
                   + 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                           * 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           * 
                           t 
                         
                         + 
                         
                           2 
                           * 
                           
                             ϕ 
                             LO 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     + 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The dependency of I 2  on E LO  and E Rx  is arrived at by squaring the expression for E 2 : 
         [0000]    
       
         
           
             
               
                 I 
                 2 
               
                
               
                 ∼ 
               
                
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 Rx 
                 2 
               
               * 
               
                 ( 
                 
                   1 
                   + 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                           * 
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           * 
                           t 
                         
                         + 
                         
                           2 
                           * 
                           
                             ϕ 
                             Rx 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
             + 
             
               
                 σ 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
               * 
               
                 ( 
                 
                   1 
                   - 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           2 
                           * 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           * 
                           t 
                         
                         + 
                         
                           2 
                           * 
                           
                             ϕ 
                             LO 
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
             - 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     + 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The first terms are the contribution of the receive power and are negligible due to the assumption of low receive signal power. The second terms are the main contribution to the total detector power from the LO signal; the shot noise power densities of the resulting electrical signals scale with the square root of these powers. The third terms are the mixing products with the frequency difference between Rx and LO inputs. These terms are in a frequency range detectable by the PIN diode and their difference is converted into the electrical receive signal. The fourth terms describe signals at twice the optical frequency. Because these fourth terms do not produce an average current and are proportional to the receive field strength, their contributions to the total detector power are negligible. 
         [0057]    The detectors will average all frequency components at optical frequency or above and the detector currents may be expressed as: 
         [0000]    
       
         
           
             
               
                 I 
                 1 
               
                
               
                 ∼ 
               
                
               
                 σ 
                 2 
               
               * 
               
                 E 
                 Rx 
                 2 
               
             
             + 
             
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
             
             + 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 I 
                 2 
               
                
               
                 ∼ 
               
                
               
                 
                   1 
                   - 
                   σ 
                 
                 2 
               
               * 
               
                 E 
                 Rx 
                 2 
               
             
             + 
             
               
                 σ 
                 2 
               
               * 
               
                 E 
                 LO 
                 2 
               
             
             - 
             
               
                 
                   σ 
                   - 
                   
                     σ 
                     2 
                   
                 
               
               * 
               
                 E 
                 Rx 
               
               * 
               
                 E 
                 LO 
               
               * 
               
                 sin 
                  
                 
                   ( 
                   
                     
                       
                         ( 
                         
                           
                             
                               ω 
                               Rx 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           - 
                           
                             
                               ω 
                               LO 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                         
                         ) 
                       
                       * 
                       t 
                     
                     + 
                     
                       ϕ 
                       Rx 
                     
                     - 
                     
                       ϕ 
                       LO 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and the resulting contribution to the detector currents by average power on the detector surface may therefore be expressed as: 
         [0000]    
       
         
           
             
               I 
               
                 avg 
                  
                 
                     
                 
                  
                 1 
               
             
              
             
               ∼ 
             
              
             
               
                 1 
                 - 
                 σ 
               
               2 
             
             * 
             
               E 
               LO 
               2 
             
           
         
       
       
         
           
             
               I 
               
                 avg 
                  
                 
                     
                 
                  
                 2 
               
             
              
             
               ∼ 
             
              
             
               σ 
               2 
             
             * 
             
               E 
               LO 
               2 
             
           
         
       
     
         [0058]    Shot noise current density in the electrical signals is proportional to the square root of the average detector currents: 
         [0000]        I   noise   _   RMS =√{square root over (2* q*I   avg   *Bw )}
 
         [0000]    where q is the elementary charge and Bw is the receive bandwidth. Because the shot noise contributions of both detectors are mutually uncorrelated, the noise power of both detectors may be added to derive the resulting noise current of the differential arrangement for a given bandwidth. This yields the following dependence on split ratio: 
         [0000]        I   noise   _   RMS   _   DIFF (σ)˜√{square root over (1−2*σ2*σ 2 )}* E   LO   *√{square root over (Bw)} 
 
         [0059]    The amplitudes of the receive signal of interested are given by the third term in the equations for detector current: 
         [0000]        I   sig1 ˜+√{square root over (σ−σ 2 )}* E   Rx   *E   LO *sin((ω Rx ( t )−ω LO ( t ))* t+φ   Rx −φ LO )
 
         [0000]        I   sig2 ˜−√{square root over (σ−σ 2 )}* E   Rx   *E   LO *sin((ω Rx ( t )−ω LO ( t ))* t+φ   Rx −φ LO )
 
         [0060]    As noted above, the detection circuit determines the difference of the two currents, effectively doubling the amplitude of the signal current which may be expressed as: 
         [0000]        I   sig   _   DIFF   =I   sig1   −I   sig2 ˜√{square root over (2*σ−σ 2 )}* E   Rx   *E   LO *sin((ω Rx ( t )−ω LO ( t ))* t+φ   Rx −φ LO )
 
         [0000]    with the RMS amplitude 
         [0000]        I   sig   _   RMS   _   DIFF ˜√{square root over (2)}*√{square root over (σ−σ 2 )}* E   Rx   *E   LO  
 
         [0061]    Unlike the scenario of single-ended detection, the LO power is kept constant. For asymmetric split ratios, the total shot noise power level is taken into account. The SNR dependence on split ratio may be expressed as: 
         [0000]    
       
         
           
             
               S 
                
               
                   
               
                
               N 
                
               
                   
               
                
               
                 R 
                  
                 
                   ( 
                   σ 
                   ) 
                 
               
             
             = 
             
               10 
               * 
               
                 
                   log 
                   10 
                 
                  
                 
                   ( 
                   
                     
                       I 
                       
                         sig_RMS 
                          
                         _DIFF 
                       
                       2 
                     
                     
                       I 
                       
                         noise_RMS 
                          
                         _DIFF 
                       
                       2 
                     
                   
                   ) 
                 
               
                
               
                 ∼ 
               
                
               10 
               * 
               
                 
                   log 
                   10 
                 
                  
                 
                   ( 
                   
                     
                       
                         2 
                         * 
                         
                           ( 
                           
                             σ 
                             - 
                             
                               σ 
                               2 
                             
                           
                           ) 
                         
                       
                       
                         1 
                         - 
                         
                           2 
                           * 
                           σ 
                         
                         + 
                         
                           2 
                           * 
                           
                             σ 
                             2 
                           
                         
                       
                     
                     * 
                     
                       E 
                       Rx 
                       2 
                     
                     * 
                     
                       1 
                       Bw 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0062]    SNR dependency on split ratio relative to σ=0.5 for differential detection may then be expressed as: 
         [0000]    
       
         
           
             
               S 
                
               
                   
               
                
               N 
                
               
                   
               
                
               
                 
                   R 
                   rel 
                 
                  
                 
                   ( 
                   σ 
                   ) 
                 
               
             
             = 
             
               
                 
                   S 
                    
                   
                       
                   
                    
                   N 
                    
                   
                       
                   
                    
                   
                     R 
                      
                     
                       ( 
                       σ 
                       ) 
                     
                   
                 
                 - 
                 
                   S 
                    
                   
                       
                   
                    
                   N 
                    
                   
                       
                   
                    
                   
                     R 
                      
                     
                       ( 
                       0.5 
                       ) 
                     
                   
                 
               
               = 
               
                 10 
                 * 
                 
                   log 
                   10 
                 
                  
                 
                   
                     
                       2 
                       * 
                       
                         ( 
                         
                           σ 
                           - 
                           
                             σ 
                             2 
                           
                         
                         ) 
                       
                     
                     
                       1 
                       - 
                       
                         2 
                         * 
                         σ 
                       
                       + 
                       
                         2 
                         * 
                         
                           σ 
                           2 
                         
                       
                     
                   
                    
                   
                       
                   
                   [ 
                   dB 
                   ] 
                 
               
             
           
         
       
     
         [0063]    For purposes of comparing detector efficiency in both architectures, the SNR dependence on split ratio for a given receive bandwidth may derived. As the calculations above show, SNR does not depend on LO power level as long as the assumptions stated at the beginning are valid. 
         [0064]    The derivations of SNR for both architectures above gave the following results under identical neglect of proportionality constants (and physical units): 
         [0000]    
       
         
           
             Single 
              
             
               - 
             
              
             Ended 
              
             
               : 
             
              
             
                 
             
              
             S 
              
             
                 
             
              
             N 
              
             
                 
             
              
             
               R 
                
               
                 ( 
                 σ 
                 ) 
               
             
              
             
               ∼ 
             
              
             10 
             * 
             
               
                 log 
                 10 
               
                
               
                 ( 
                 
                   σ 
                   * 
                   
                     E 
                     Rx 
                     2 
                   
                   * 
                   
                     1 
                     Bw 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               Differential 
                
               
                 : 
               
                
               
                   
               
                
               S 
                
               
                   
               
                
               N 
                
               
                   
               
                
               
                 R 
                  
                 
                   ( 
                   σ 
                   ) 
                 
               
             
             ∼ 
             
               10 
               * 
               
                 
                   log 
                   10 
                 
                  
                 
                   ( 
                   
                     
                       
                         2 
                         * 
                         
                           ( 
                           
                             σ 
                             - 
                             
                               σ 
                               2 
                             
                           
                           ) 
                         
                       
                       
                         1 
                         - 
                         
                           2 
                           * 
                           σ 
                         
                         + 
                         
                           2 
                           * 
                           
                             σ 
                             2 
                           
                         
                       
                     
                     * 
                     
                       E 
                       Rx 
                       2 
                     
                     * 
                     
                       1 
                       Bw 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The single-ended case with σ=0.5 yields: 
         [0000]    
       
         
           
             S 
              
             
                 
             
              
             N 
              
             
                 
             
              
             
               R 
                
               
                 ( 
                 0.5 
                 ) 
               
             
              
             
               ∼ 
             
              
             10 
             * 
             
               
                 log 
                 10 
               
                
               
                 ( 
                 
                   0.5 
                   * 
                   
                     
                       E 
                       Rx 
                       2 
                     
                     Bw 
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    Expressing both SNR equations relative to the single-ended case with σ=0.5 yields: 
         [0000]    
       
         
           
             
               Single 
                
               
                 - 
               
                
               Ended 
                
               
                 : 
               
                
               
                   
               
                
               S 
                
               
                   
               
                
               N 
                
               
                   
               
                
               
                 
                   R 
                   rel_SE 
                 
                  
                 
                   ( 
                   σ 
                   ) 
                 
               
             
             = 
             
               10 
               * 
               log 
                
               
                   
               
                
               10 
                
               
                 ( 
                 
                   2 
                   * 
                   σ 
                 
                 ) 
               
             
           
         
       
       
         
           
             
               Differential 
                
               
                 : 
               
                
               
                   
               
                
               
                 
                   SNR 
                   rel_DE 
                 
                  
                 
                   ( 
                   σ 
                   ) 
                 
               
             
             = 
             
               10 
               * 
               log 
                
               
                   
               
                
               10 
                
               
                 ( 
                 
                   
                     4 
                     * 
                     
                       ( 
                       
                         σ 
                         - 
                         
                           σ 
                           2 
                         
                       
                       ) 
                     
                   
                   
                     1 
                     - 
                     
                       2 
                       * 
                       σ 
                     
                     + 
                     
                       2 
                       * 
                       
                         σ 
                         2 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0065]      FIG. 6  illustrates the relative SNR for both a single-ended detector and a differential detector over varying split ratios, σ. As illustrated, the relative SNR gain of differential detection over single-ended detection with a symmetric coupler is 3 dB. This is expected as differential detection adds signal amplitudes, equivalent to quadrupling the signal power, while the noise contributions of both coupler outputs are added in power, not amplitude. As also illustrated, the relative SNR gain of single-ended detection continues to increase with the increase of the split ratio until at an approximate split ratio, σ=0.8, the difference in gain between such a single-ended asymmetric detector and a symmetric differential detector is only roughly 1 dB; and at an approximate split ratio, σ=0.9, this difference in gain is even further reduced. Experimental results confirmed this analysis and the attendant assumptions. 
         [0066]    Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having a split ratio, σ&gt;0.5. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having a split ratio, σ&gt;0.6. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having a split ratio, σ&gt;0.7. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having a split ratio, σ&gt;0.8. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having a split ratio, σ&gt;0.9. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having an empirical split ratio, σ=0.7. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having an empirical split ratio, σ=0.8. Various implementations of the invention utilize asymmetric single-ended detection with a combining coupler  370  having an empirical split ratio, σ=0.9. 
         [0067]    While the invention has been described herein in terms of various implementations, it is not so limited and is limited only by the scope of the following claims, as would be apparent to one skilled in the art. These and other implementations of the invention will become apparent upon consideration of the disclosure provided above and the accompanying figures. In addition, various components and features described with respect to one implementation of the invention may be used in other implementations as well.