Patent Publication Number: US-10326530-B2

Title: Bias control of nested mach-zehnder modulators for the generation of optical QAM signals

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This Application claims the benefit under 35 U.S.C. § 120 of U.S. application Ser. No. 14/615,350, entitled “BIAS CONTROL OF NESTED MACH-ZEHNDER MODULATORS FOR THE GENERATION OF OPTICAL QAM SIGNALS,” filed on Feb. 5, 2015, which is hereby incorporated herein by reference in its entirety. U.S. application Ser. No. 14/615,350 claims the benefit of U.S. Provisional Patent Application Ser. No. 61/936,061, filed Feb. 5, 2014 which is incorporated by reference in its entirety as if set forth at length herein. 
    
    
     TECHNICAL FIELD 
     This disclosure relates generally to optical communications structures systems and methods. More particularly, this disclosure pertains to techniques, methods and apparatus for the bias control of modulators for the generation of optical qam signals. 
     BACKGROUND 
     Contemporary optical communications systems oftentimes employ advanced modulation formats. Given their importance to these contemporary optical communications systems, techniques, methods and apparatus that facilitate the generation of such modulation formats would represent a welcome addition to the art. 
     SUMMARY 
     An advance in the art is made according to an aspect of the present disclosure directed to the generation of advanced modulation formats. More particularly, the present disclosure describes bias control methods of nested Mach-Zehnder modulators for the generation of optical QAM signals while ensuring correct I/Q polarity of the generated optical QAM signal. 
     Briefly, one illustrative method according to the present disclosure for ensuring correct I/Q polarity of the generated optical signal involves temporarily offsetting I and Q biases from ideal transmission null bias points. More specifically, control loops for I modulator bias, Q modulator bias, and I-Q phase delay are allowed to converge and V π /2 is added to the bias voltages applied to I and Q modulators. The phase delay bias is dithered around the above-converged value while measuring any variation in RF power. If the measured variation in RF power is 180 degrees out of phase with the bias dither, the converged bias&#39; determined above ensures proper operation. 
     Another aspect of the present disclosure is directed to a method for ensuring correct modulator I/Q polarity which temporarily makes I and Q data streams identical. More specifically, this alternative method according to the present disclosure drives the modulator with statistically independent I and Q data streams such control loops for the I modulator bias, Q modulator bias, and I-Q phase delay converge. A source driving the modulator is driven such that I and Q data streams become identical. The phase delay bias is dithered around the above-converged value while measuring any variation in RF power. If the measured RF power is 180 degrees out of phase with the bias dither, then the converged bias&#39; determined above ensures proper operation. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
       A more complete understanding of the present disclosure may be realized by reference to the accompanying drawing in which: 
         FIG. 1  shows a schematic of an illustrative nested Mach-Zehnder Interferometer (MZI); 
         FIG. 2  shows a schematic of an alternative illustrative nested MZI; 
         FIG. 3  is a graph depicting amplitude and power transfer as a function of the normalized drive voltage; 
         FIG. 4  is a graph depicting output power as a function of bias voltage; 
         FIG. 5  is a graph depicting the change in the polarity of the dither tone in the modulator output power changing from inverted to noninverted; 
         FIG. 6  is a graph showing bias voltage and drive waveform and amplitude; 
         FIG. 7  is a graph showing an illustrative data signal; 
         FIG. 8  is a graph showing optical field and power at a modulator output; 
         FIG. 9  is a graph showing optical field and power at a modulator output; 
         FIG. 10  is a graph showing optical power for ideal biasing; 
         FIG. 11  shows a shows a schematic of an illustrative nested MZI; 
         FIG. 12  is a graph showing dependency of RF power on R and the I-Q phase delay for a “realistic” I/Q drive waveform; 
         FIG. 13  is a graph showing normalized amplitude with respect to normalized electrical drive signals and time; 
         FIG. 14  is a graph showing normalized amplitude with respect to optical field after I/Q modulators and normalized time; 
         FIG. 15  is a graph showing normalized amplitude with respect to normalized electrical drive signals and time; 
         FIG. 16  is a graph showing normalized amplitude with respect to optical field after I/Q modulators and normalized time; 
         FIG. 17  is a graph showing relative change of RF power with respect to phase delay for a case where both modulators are noninverting; and 
         FIG. 18  is a graph showing relative change of RF power with respect to phase delay for a case where one of the modulators is noninverting and the other is inverting; 
     
    
    
     DETAILED DESCRIPTION 
     The following merely illustrates the principles of the disclosure. It will thus be appreciated that those skilled in the art will be able to devise various arrangements which, although not explicitly described or shown herein, embody the principles of the disclosure and are included within its spirit and scope. More particularly, while numerous specific details are set forth, it is understood that embodiments of the disclosure may be practiced without these specific details and in other instances, well-known circuits, structures and techniques have not be shown in order not to obscure the understanding of this disclosure. 
     Furthermore, all examples and conditional language recited herein are principally intended expressly to be only for pedagogical purposes to aid the reader in understanding the principles of the disclosure and the concepts contributed by the inventor(s) to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. 
     Moreover, all statements herein reciting principles, aspects, and embodiments of the disclosure, as well as specific examples thereof, are intended to encompass both structural and functional equivalents thereof. Additionally, it is intended that such equivalents include both currently-known equivalents as well as equivalents developed in the future, i.e., any elements developed that perform the same function, regardless of structure. 
     Thus, for example, it will be appreciated by those skilled in the art that the diagrams herein represent conceptual views of illustrative structures embodying the principles of the disclosure. 
     In addition, it will be appreciated by those skilled in art that any flow charts, flow diagrams, state transition diagrams, pseudocode, and the like represent various processes which may be substantially represented in computer readable medium and so executed by a computer or processor, whether or not such computer or processor is explicitly shown. 
     In the claims hereof any element expressed as a means for performing a specified function is intended to encompass any way of performing that function including, for example, a) a combination of circuit elements which performs that function or b) software in any form, including, therefore, firmware, microcode or the like, combined with appropriate circuitry for executing that software to perform the function. The invention as defined by such claims resides in the fact that the functionalities provided by the various recited means are combined and brought together in the manner which the claims call for. Applicant thus regards any means which can provide those functionalities as equivalent as those shown herein. Finally, and unless otherwise explicitly specified herein, the drawings are not drawn to scale. 
     Thus, for example, it will be appreciated by those skilled in the art that the diagrams herein represent conceptual views of illustrative structures embodying the principles of the disclosure. 
     We begin by noting that advanced modulation formats are particularly attractive for application to optical communications systems that require a high spectral efficiency (bit/s/Hz). As may be appreciated by those skilled in the art, high spectral efficiencies are traditionally required in metro, long-haul, and submarine optical communications systems where optical link cost(s) are very high and increasing spectral efficiency is seen as an effective way of reducing cost per bit. Such high optical link costs drive the use of multiple, high-performance, state-of-the-art components in transceivers used in such links. 
       FIG. 1  shows in schematic form a “nested” Mach-Zehnder modulator for generation of polarization multiplex QAM signals according to an aspect of the present disclosure. As will be readily observed by those skilled in the art, such a nested structure includes an “outer” or “nested” modulator having an input, an output, and two arms connecting the input to the output. Each of the arms includes an “inner” modulator positioned therein. 
     These inner modulators—known as the I and Q modulators—are driven by electrical signals v i (t) and v q (t), respectively. 
     As may be appreciated and understood by those skilled in the art, bias points of the I and Q modulators may be locked based on an average optical power at their outputs. As shown in  FIG. 1 , the average optical power of the I and Q modulators may be detected by photodetectors, Q pow and I pow. This power is measured as the DC component of the current flowing in the photo detector and the modulator output. 
     The I and Q components from the inner modulators are combined in an additional section of the modulator with the relative phase e jφ  that is controlled by a phase delay bias voltage. Advantageously, this relative phase may be adjusted to a desired value based on the RF power of the current in a detector at the output of the nested modulator namely RF detect. 
     Turning now to  FIG. 2 , there is shown an alternative—somewhat simpler—structure nested structure according to an aspect of the present disclosure. 
     As may be observed from  FIG. 2 , a single photodetector DC and RF detect is used for control of I and Q bias as well as I-Q phase relation. More specifically, such control is performed in a round-robin fashion by observing the detector current (DC or RF) while manipulating one bias at a time keeping the two other biases constant. In particular: i) the detector current is monitored as the I modulator bias is being manipulated while Q bias and phase delay biases are kept constant, ii) the detector current is monitored as the Q bias is being manipulated while I bias and phase delay biases are kept constant, iii) the detector current is monitored as the phase delay bias is being manipulated while I and Q biases are kept constant, repeat i), ii), iii) . . . etc. 
     Bias Control for the I/Q (Inner) Modulators 
     We can express the complex envelope at the output of the I or Q modulator in terms of the complex envelope at the input, m in  (assumed to be 1 in the following for simplicity), and the drive signal v i (t) (or v q (t)) written as v b +v d (t) 
     
       
         
           
             
               
                 m 
                 out 
               
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             = 
             
               
                 m 
                 
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                   ⁢ 
                   
                       
                   
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                 sin 
                 ⁡ 
                 
                   ( 
                   
                     
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                           V 
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                     ⁢ 
                     
                       ( 
                       
                         
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                           b 
                         
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     where V π  is a characteristic voltage for the modulator that specifies how much the drive voltage must be changed to induce a 180 deg change of the relative phase of the optical fields traveling in the upper and the lower arms of the inner modulator (which varies from device to device), v 0  is a characteristic voltage for the modulator that specifies a drive voltage where the output is 0 and the slope of the sin( ) function is positive (varies from device to device and drifts over time and temperature), v b  is the DC component of the drive signal (the bias voltage), and v d (t) is the AC component that carries the data. 
     Notably, if the modulator is biased with v b =v 0 , the output amplitude is 0 for v d (t)= . . . −2V π ,0,2V π  . . . and maximum for v d (t)= . . . −V π ,V π ,3V π  . . . . Specifically, if v d (t)=V π i(t) where i(t)∈{−1,1} as an example is the I component of an optical QPSK signal we wish to generate, we find that m out (t)=i(t). However, if the modulator is instead biased at v b =v 0 +2V π , we find that m out (t)=−i(t), i.e. the modulator is inverting the I component. 
     In general, the bias points v 0 ,v 0 ±4V π ,v 0 ±8V π  . . . ensure m out (t)=i(t) whereas the bias points v 0 +2V π ±4V π ,v 0 +2V π ±8V π  . . . invert the signal: m out (t)=−i(t). For any other bias points, the transformation from i(t)/q(t) to the I/Q component of the optical signal is distorted. 
     Those skilled in the art will readily appreciate that one function of the bias control is to ensure that the bias voltage v b  continuously tracks the v 0  of the modulator. 
     According to an aspect of the present disclosure, bias control of the I/Q modulators is based on the power at the output of the modulator according to the following relationship: 
     
       
         
           
             
               
                 p 
                 out 
               
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                       out 
                     
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                 2 
               
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                 ⁡ 
                 
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       FIG. 3  is a graph which shows amplitude and power transfer as a function of the normalized drive voltage namely, 
     
       
         
           
             
               
                 
                   v 
                   b 
                 
                 ⁢ 
                 
                     
                 
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     No I/Q Data Modulation 
     In the absence of data modulation, the electrical drive signal equals the bias voltage e.g. v i (t)=v b .  FIG. 4  shows the output power as a function of the bias voltage for a case where v 0 =0.5V π . 
     As may now be appreciated, the bias control will try to set v b  to v 0  by minimizing the power. However, having only access to the power of the signal, the bias control will not have any preference for one power null over the other meaning that it&#39;s throughout possible that the bias point ends up at e.g. v b =v 0 +2V π , i.e. a bias point where the optical field is inverted relative to the data. 
     To minimize the power, a small dither tone is impressed on the bias v b =v b0 +ε sin(ω b t). As illustrated in  FIG. 5  below, the polarity of the sin(ω b t) dither tone in the modulator output power will change from inverted (v b0 &lt;v 0 ) to noninverted (v b0 &gt;v 0 ) because the slope of the bias-to-power transfer function is negative below the minimum and positive above. When v b0 =v 0 , the dither tone will disappear in the optical power. Consequently, the sign and amplitude of the dither tone in the optical power can be used in e.g. a PID control loop to set v b  to v 0  (or v b =v 0 +2V π  etc). 
     I/Q Data Modulation Present 
     As will also be appreciated by those skilled in the art, the optical power at a modulator&#39;s output will vary when the electrical drive signal is connected however, averaging the output optical power over “many” (thousands) symbols provides a good feed-back signal for the bias control. The detailed relationship between the bias voltage and the optical average power depends on the drive waveform and amplitude. If the drive amplitude is small (significantly smaller than 2V π ), the average power will have a minimum at v b =v 0  similar to the case without data modulation. However, for a “reasonably ideal” waveform and a drive amplitude around 2V π , it can be seen from  FIG. 6  that the average power will have a maximum at v b =v 0 . 
     As an example, consider the data signal i(t) shown in  FIG. 7 . With a drive signal peak-peak amplitude of 2V π (v d (t)=V π i(t)) and “ideal” biasing v b =v 0 , we have an optical field and power at the modulator&#39;s output such as that shown in  FIG. 8  and  FIG. 9 . 
     The average optical power for this ideal biasing v b =v 0  as well as for biasing offset from the ideal setting are shown in  FIG. 10 . As may be readily understood, the average power is maximum at the ideal biasing v b =v 0 . If we dither the bias and look for the dither tone in the modulator output power, the polarity of the sin(ω b t) dither tone in the modulator output power will change from noninverted (v b0 &lt;v 0 ) to inverted (v b0 &gt;v 0 ) because the slope of the bias-to-power transfer function is positive below the minimum and negative above. Notably, this behavior is the opposite of the case without data modulation—which has to be taken into account by an appropriate sign change in the bias control algorithm. 
     Control of the Relative Phase of I and Q Components 
     With reference now to  FIG. 11  which depicts in schematic form a nested Mach-Zehnder modulator, we now let i(t) and q(t) denote the complex envelope of the optical field at the output of the inner, I and Q modulators shown in that figure. As may be appreciated, i(t) and q(t) are encoded with statistically independent bit sequences and it is assumed that the drive signals to the I and the Q modulators are symmetrical around the 0 level so that E{i(t)}=E{i 3 (t)}=E{q(t)}=E{q 3 (t)}=0 when the modulators are biased at a transmission null. 
     As may be understood by those skilled in the art—with reference to  FIG. 11 , it may be observed that there exists a phase shifting portion or section of the modulator situated after the inner Q modulator. This phase shifting section after the Q modulator delays the phase of the Q component by an amount φ controlled by the phase delay bias voltage so total complex envelope at the output of the “nested” Mach-Zehnder modulator becomes:
 
 i ( t )+ e   jφ   q ( t );
 
Wherein the instantaneous power of this output signal is represented by:
 
 |i ( t )+ e   jφ   q ( t )| 2 =( i   2 ( t )+ q   2 ( t )+2 i ( t ) q ( t )cos φ);
 
     which, apart from a constant expressing the detector response (responsivity), is also the current in an ideal photo detector at the output of the nested modulator. 
     The RF power R of this photo current is 
     
       
         
           
             
               
                 
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     As noted above, E{i(t)}=E{i 3 (t)}=E{q(t)}=E{q 3 (t)}=0 when the inner modulators are biased at zero transmission points. Accordingly, a relationship between RF power and the relative phase between the I and Q components may be represented by:
 
 R=E{i   4 ( t )}+ E{q   4 ( t )}+2 E{i   2 ( t )} E{q   2 ( t )}(1+2 cos 2  φ);
 
     By adjusting the bias voltage to minimize the RF power (the cos 2  φ terms vanishes), the phase delay is set to 
                 π   2     +     k   ⁢           ⁢   π       ,         
k integer, which, apart from uncertainty of the sign, is the desired relative phase between the I and Q components of the optical QAM signal. This minimization can be done as described in the end of the section “no I/Q data modulation” by adding a dither tone sin(ω b t) to the phase delay bias voltage and use the phase and amplitude of the corresponding dither tone in the RF power detector signal as a feed-back signal for the bias control loop.
 
     An example of the dependency of the RF power R on the I-Q phase delay pp is shown in  FIG. 12  for a case with “realistic” I/Q drive waveforms and a peak-peak drive amplitude of 2V π  and I/Q modulators biased at v b −v 0 =0 (the first 10 bits of the electrical drive waveforms and the corresponding complex envelopes at the output of the I and Q modulators are shown in  FIG. 13  and  FIG. 14  below). 
     Control of I/Q Polarity 
     If the inner modulators are both noninverting i.e. biased so that the I and Q components of the optical field have the same polarity as the respective drive signals (v b −v 0 =0, ±4V π , ±8V π  . . . ) and the I-Q phase delay is 
               φ   =       π   2     +     k   ⁢           ⁢   2   ⁢   π         ,         
k integer, demodulation of the optical QAM signal is straightforward. The same is the case if both inner modulators invert (v b −v 0 =2V π ,2V π ±4V π ,2V π ±8V π  . . . ) and
 
             φ   =       π   2     +     k   ⁢           ⁢   2   ⁢   π             
(corresponding to a fixed 180 deg phase rotation of the QAM signal) or if one of the inner modulators invert and the other doesn&#39;t and
 
             φ   =       3   ⁢           ⁢     π   2       +     k   ⁢           ⁢   2   ⁢     π   .               
For other conditions, either the I or the Q component of the optical QAM signal at the output of the modulator is inverted relative to the drive signal and the optical QAM signal cannot be demodulated correctly without special processing at the receiver. Two methods for detecting possible I/Q inversion in the modulator may not be described.
 
Method 1—Temporarily Offsetting I and Q Biases from Ideal Transmission Null Bias Points.
 
     In this method, we assume that all three bias control loops have converged such that the I and the Q modulators are biased at a transmission null and the RF power is minimized i.e. 
                 π   2     +     k   ⁢           ⁢   π       ,         
where k is an integer. Now, if the bias voltage for the I modulator is increased by a “small” offset relative to the transmission null, E{i(t)} and E{i 3 (t)} will become positive if the slope of the optical field vs drive voltage transfer curve at the transmission null is positive corresponding to noninverting operation. This is illustrated in  FIG. 15  and  FIG. 16  for a bias voltage increase of V π /2 (compare to the  FIG. 13  and  FIG. 14 ).
 
     Similarly, if the slope of the transfer curve is negative at the transmission null (inverting operation), E{i(t)} and E{i 3 (t)} will become negative. The same holds for the Q modulator. This means that the terms E{t 3 (t)}E{q(t)} and E{i(t)}E{q 3 (t)} in the expression for the RF power will become positive when the I and Q bias voltages are increased if both the I modulator and the Q modulator are noninverting or if both modulators are inverting. However, if one modulator is inverting and the other is noninverting, the two terms will become negative. 
     As may be appreciated by those skilled in the art, the sign of the terms E{i 3 (t)}E{q(t)} and E{i(t)}E{q 3 (t)} influences the derivative of the RF power R with respect to the I-Q phase delay φ. If we look at this derivative near the phase delay found by minimizing the RF power (before adding the offset to the I and Q modulator biases) i.e. 
               φ   ≈       π   2     +     k   ⁢           ⁢   π         ,         
we find that
 
     
       
         
           
             
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                 ⁢ 
                 sin 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 φ 
               
             
           
         
       
     
     In the cases where both modulators are noninverting or both are inverting, adding a “small” positive offset to the I and Q modulator biases makes the parenthesis (E{i 3 (t)}E{q(t)}+E{i(t)}E{q 3 (t))}) positive and the slope of the derivative of the RF power with respect to φ will have the opposite sign of sin φ. In these cases, the desired optical field is generated when 
               φ   =       π   2     +     k   ⁢           ⁢   2   ⁢   π         ,         
k integer, as discussed above, showing that the desired operation of the modulator is characterized by the RF power being a decreasing function of the phase delay φ. This is illustrated in  FIG. 17  for the case where both modulators are noninverting. The graph shows the relative change of RF power with respect to φ in radians i.e.
 
               1   R     ⁢     dR     d   ⁢           ⁢   φ             
for the drive waveforms and amplitude considered above.
 
     If, on the other hand, we consider the cases where one of the two modulators is inverting and the other is noninverting, the parenthesis (E{i 3 (t)}E{q(t)}+E{i(t)}E{q 3 (t)}) becomes negative when a “small” positive offset is added to the I and Q modulator biases so that the derivative of the RF power with respect to φ will have the same sign as sin φ. In these cases, the desired optical field is generated when 
               φ   ≈         3   ⁢   π     2     +     k   ⁢           ⁢   2   ⁢   π         ,         
k integer, showing again that the desired operation of the modulator is characterized by the RF power being a decreasing function of the phase delay φ.  FIG. 18  illustrates the situation where the I modulator is inverting and the Q modulator is noninverting.
 
     This leads to the following method according to the present disclosure that ensures correct I/Q polarity of a generated optical QAM signal:
         1. Let the control loops for I bias, Q bias, and I-Q phase delay converge.   2. Freeze the I and Q bias control loops and add e.g. V π /2 to the bias voltages for the I and Q modulators   3. Open the phase delay loop and dither the phase delay bias (and consequently φ) around the value determined in step 1 and look for the corresponding variation of the RF power.   4. If the variation of the RF power is 180 deg out of phase with the impressed bias dither, the slope of the RF power vs bias voltage curve is negative and the bias points determined in step 1. Ensure the desired modulator operation. Set all bias points to the values found in step 1.   5. However, if the variation of the RF power is in-phase with the impressed bias dither, the slope of the RF power vs bias voltage curve is positive and the bias points determined in step 1 causes inversion of the I or Q component of the optical field relative to the drive signals. To correct this I/Q inversion one of the following actions can be taken: i) increase/decrease the I modulator bias by 2v π  compared to the value found in step 1, and set the Q modulator bias to the value found in step 1, ii) increase/decrease the Q modulator bias by 2V π  relative to the value found in step 1 and set the I modulator bias to the value found in step 1, or iii) increase/decrease the I-Q phase delay bias by V π  and set the I and Q modulator biases to the values found in step 1, or iv) invert either the I or the Q data driving the modulator and set the I and Q biases to the values found in step 1.   6. Close and unfreeze all three control loops to enable continuous tracking of the correct bias points.       

     Instead of detecting the phase relation between the dither tone impressed on the phase delay bias and the resulting dither tone in the RF power, we can modify step 3. and keep the phase delay control loop closed while we observe the direction in which the loop drives the phase delay bias when the “small” offset is added to the I and Q biases. Since the phase delay loop attempts to minimize the RF power, it will increase the bias voltage if the derivative of the RF power with respect to the phase delay bias is negative (negative slope slightly below of the point of minimum RF power), and it will decrease the bias voltage if the derivative is positive (positive slope slightly above the point of minimum RF power). This means that the negative slope condition in step 4. is detected if the phase delay loop increases the phase delay bias in response to adding “small” positive offsets to the I and Q biases. The condition is step 5. is detected if the phase delay loop decreases the phase bias. 
     As an alternative to monitoring at the RF power of the detector current when we add “small” offsets to the I and Q modulator biases, we can monitor the DC component which is proportional to the optical average power expressed by:
 
 E{|i ( t )+ e   jφ   q ( t )| 2   }=E {( t )}+ E{q   2 ( t )}+2 E{i ( t )} E{q ( t )}cos φ.
 
     The derivative of the DC component with respect to the phase delay I
 
−2 E{i ( t )} E{q ( t )}sin φ.
 
     This quantity has the same sign properties as the derivative of the RF power, 
               dR     d   ⁢           ⁢   φ       ,         
and can be used for determination of possible I/Q inversion.
 
Method 2—Temporarily Make I and Q Data Streams Identical.
 
     If we program a data source driving the modulator to output identical bit sequences, we get i(t)=q(t) if both modulators are noninverting or both modulators are inverting. In these cases the instantaneous power of the modulator output signal is defined by:
 
| i ( t )+ e   jφ   q ( t )| 2   =i   2 ( t )|1+ e   jφ | 2 =2 i   2 ( t )(1+cos(φ))
 
and the RF component of the photo current will be proportional to
 
4 E{i   4 ( t )}(1+cos(φ)) 2  
 
so that the derivative of the RF power with respect to the I-Q phase difference φ equals
 
−8 E{i   4 ( t )}(1+cos(φ))sin(φ)
 
which is negative for
 
             φ   =       π   2     +     k   ⁢           ⁢   2   ⁢   π             
corresponding to the correct I/Q polarity and positive for
 
             φ   =         3   ⁢   π     2     +     k   ⁢           ⁢   2   ⁢   π             
corresponding to incorrect I/Q polarity.
 
     In the case that one of the modulators invert and the other doesn&#39;t i(t)=−q(t) and the RF component of the photo current becomes
 
4 E{i   4 ( t )}(1−cos(φ)) 2  
 
with the following derivative of the RF power with respect to the I-Q phase delay
 
+8 E{i   4 ( t )}(1−cos(φ))sin(φ)
 
     This expression is negative for 
             φ   =         3   ⁢   π     2     +     k   ⁢           ⁢   2   ⁢   π             
corresponding to the correct I/Q polarity and positive for
 
             φ   =       π   2     +     k   ⁢           ⁢   2   ⁢   π             
corresponding to incorrect I/Q polarity.
 
     Thus, it is found that correct I/Q polarity is characterized by a negative derivative of the RF power with respect to the I-Q phase delay φ when the I and Q data streams are identical. To detect the slope of the RF power vs phase delay bias curve, we can open the phase delay bias loop and compare the phase of a dither tone impressed on the phase bias and the resulting dither tone in the RF power as discussed above (negative slope means that the two dither tones are 180 deg out of phase). This leads to the following method for ensuring correct I/Q polarity:
         1. Driving the modulator with statistically independent I and Q data streams, let the control loops for I bias, Q bias, and I-Q phase delay converge;   2. Program the source driving the modulator so the I and Q data streams become identical. Optionally freeze the I and Q biases.   3. Open the phase delay loop and dither the phase delay bias (and consequently φ) around the value determined in step 1 and look for the corresponding variation of the RF power.   4. If the variation of the RF power is 180 deg out of phase with the impressed bias dither, the slope of the RF power vs bias voltage curve is negative and the bias points determined in step 1. ensures the desired modulator operation.   5. However, if the variation of the RF power is in-phase with the impressed bias dither, the slope of the RF power vs bias voltage curve is positive and the bias points determined in step 1 causes inversion of the I or Q component of the optical field relative to drive signals. To correct this I/Q inversion one of the following actions can be taken: i) increase/decrease the I modulator bias by 2V π  compared to the value found in step 1, and set the Q modulator bias to the value found in step 1, ii) increase/decrease the Q modulator bias by 2V π  relative to the value found in step 1 and set the I modulator bias to the value found in step 1, or iii) increase/decrease the I-Q phase delay bias by V π  and set the I and Q modulator biases to the values found in step 1, or iv) invert either the I or the Q data driving the modulator and set the I and Q biases to the values found in step 1.       

     6. Revert to normal operation with statistically independent I and Q data (using the same drive signal polarity as in step 1. with the possible inversion added in step 6.) and close and unfreeze all three control loops to enable continuous tracking of the correct bias points. 
     Similar to the discussion for Method 1, we can determine the slope of the RF power vs phase bias curve by keeping the phase delay control loop closed while we look at the direction in which the loop drives the phase delay bias when the source driving the modulator is programmed to output identical I and Q sequences: since the phase delay loop attempts to minimize the RF power, it will increase the bias voltage if the derivative of the RF power with respect to the phase delay bias is negative (negative slope slightly below of the point of minimum RF power), and it will decrease the bias voltage if the derivative is positive (positive slope slightly above the point of minimum RF power). This means that the negative slope condition in step 4. is detected if the phase delay loop increases the phase delay bias in response to identical I and Q data streams. The condition is step 5. is detected if the phase delay loop decreases the phase bias. 
     As an alternative to looking at the RF power of the detector current, we can look at its DC component which is proportional to the optical average power
 
 E{i   2 ( t )}(1±cos(φ))
 
     and its derivative with respect to the phase delay φ
 
∓2 E{i   2 ( t )}sin(φ)
 
     Notably, this quantity has the same sign properties as the derivative of the RF power and can be used for determination of possible I/Q inversion. 
     At this point, those skilled in the art will readily appreciate that while the methods, techniques and structures according to the present disclosure have been described with respect to particular implementations and/or embodiments, those skilled in the art will recognize that the disclosure is not so limited. More specifically and as generally described, alternative modulator structures than those shown may be employed with methods according to the present disclosure. Accordingly, the scope of the disclosure should only be limited by the claims appended hereto.