Patent Publication Number: US-2022231760-A1

Title: Timing measurement apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This patent document is a continuation application of U.S. Non-Provisional patent application Ser. No. 16/898,218, entitled “TIMING MEASUREMENT APPARATUS,” filed on Jun. 10, 2020, which claims priority to and benefits of U.S. Provisional Patent Application No. 62/875,429, entitled “TIMING MEASUREMENT APPARATUS,” filed on Jul. 17, 2019. The entire contents of the before-mentioned patent applications are incorporated by reference as part of the disclosure of this patent document. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     The United States Government has rights in this invention pursuant to Contract No. DE-AC52-07NA27344 between the U.S. Department of Energy and Lawrence Livermore National Security, LLC, for the operation of Lawrence Livermore National Laboratory. 
    
    
     BACKGROUND 
     Generating microwave signals with high spectral purity and stability is crucial in communication systems, radars, signal processing, radio astronomy, satellites, GPS navigation, spectroscopy, and in time and frequency metrology. Optical frequency combs can achieve a phase noise that is orders of magnitude lower than what is available from commercial microwave references. Thus, they have become revolutionary tools in high-precision applications, such as low phase noise microwave oscillators and generators, low sample timing error of high frequency microwaves and millimeter waves, photonic analog-to-digital converters, photonics-based radars, dual-comb ranging, timing synchronization and distribution, and alike. However, all free-running optical frequency combs exhibit high phase noise at various timescales. 
     SUMMARY 
     The disclosed techniques can be implemented in various embodiments to obtain an accurate measurement of timing errors to generate a frequency agile radio-frequency (RF) signal. The disclosed embodiments, among other features and benefits, allow the timing information to be recorded in digital form for subsequent compensation or processing and allow timing error information to be obtained and utilized in real time. 
     One aspect of the disclosed embodiments relates to a timing measurement device that includes an optical hybrid configured to receive two optical pulse trains as inputs and produce two or more optical outputs that are each phase shifted with respect to one another. The timing measurement device further includes two or more optical filters each coupled to the optical hybrid to receive an output from the optical hybrid, the two or more optical filters configured to produce multiple pulse signals with distinctive frequency bands. The timing measurement device also includes one or more photodetectors positioned to receive and convert each of the multiple optical signals produced by the two or more optical filters to an associated electrical signal. The timing measurement device additionally includes one or more analog-to-digital converters coupled to the one or more photodetectors to convert the plurality of electrical signals into a plurality of digital signals corresponding to the outputs of the two or more optical filters, where processing of the plurality of digital signals enables a determination of a timing error associated with the two optical pulse trains based on a computed phase difference between a first frequency band signal and a second frequency band signal and a computed frequency difference between the first frequency band signal and the second frequency band. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  illustrates a perfect pulse train and its associated optical spectrum. 
         FIG. 1B  illustrates an example phase offset caused by jitter in the pulse train. 
         FIG. 1C  illustrates an example phase offset caused by both jitter and carrier envelope offset. 
         FIG. 2A  illustrates an example timing measurement apparatus in accordance with the present technology. 
         FIG. 2B  illustrates another representation of the timing measurement apparatus in accordance with an example embodiment. 
         FIG. 3A  illustrates another timing measurement apparatus in accordance with the present technology. 
         FIG. 3B  illustrates another representation of a timing measurement apparatus in accordance with an example embodiment. 
         FIG. 4A  illustrates another example timing measurement apparatus in accordance with the present technology. 
         FIG. 4B  illustrates another representation of a timing measurement apparatus in accordance with the present technology. 
         FIG. 5  illustrates yet another example timing measurement apparatus in accordance with the present technology. 
         FIG. 6  illustrates an example architecture of a timing measurement system in accordance with the present technology. 
         FIG. 7  illustrates another example architecture of a timing measurement system in accordance with the present technology. 
         FIG. 8  illustrates another example architecture of a timing measurement system in accordance with the present technology. 
         FIG. 9  illustrates an example architecture of a timing measurement system that reduces environmental dependency in accordance with the present technology. 
         FIG. 10  illustrates another example architecture of a timing measurement system in accordance with the present technology. 
         FIG. 11  illustrates another example architecture of a timing measurement system in accordance with the present technology. 
         FIG. 12  illustrates an example architecture of a real-time calibration system in accordance with the present technology. 
         FIG. 13  illustrates an example architecture of an error-compensated Photonic Analog-to-Digital converter that uses a timing measurement device in accordance with the present technology. 
     
    
    
     DETAILED DESCRIPTION 
     Timing variations or irregularity of the optical pulse trains, such as jitter, become important properties for optical frequency comb applications. For example, the general approach to remove or reduce jitter from an optical pulse train in an RF signal generator is to measure the jitter of an optical pulse train as accurately as possible, use that jitter information to correct its effects as precisely as possible, and use that jitter corrected signal to make a very low jitter RF signal generator. However, existing techniques that use an electronic reference can be limited by the poor high frequency offset phase noise of electronic references. Techniques that use an optical reference can be limited by the poor low frequency offset phase noise of optical references. Techniques that use a stabilized continuous-wave reference require ultrahigh quadrature cavities which are extremely fragile and temperature and/or vibration sensitive. Some of the conventional techniques also require multi-staged phase-locked loops for both the repetition rate and the carrier-envelope offset, or complicated subsystems that perform spectral broadening and carrier-envelope offset control. 
     Many conventional techniques are limited by environmental sensitivity, such as temperature or vibrations through various physical mechanisms. In rougher environments, the amount of requisite isolation increases, posing more challenges to the size and weight of the system. Furthermore, many conventional techniques measures timing irregularities (such as jitter) using an averaging instrument (e.g., an RF spectrum analyzer). However, temporal dependence of the timing error information is lost during averaging, so this information cannot be used to correct the error in real-time. It is thus desirable to obtain a real-time timing error measurement so as to create a signal generator that is more stable. 
     The techniques disclosed herein, among other features and benefits, overcome the above limitations and rely in-part on a digitally corrected optical delay reference. The disclosed embodiments enable precise measurements of timing errors in signals using optical techniques. The example timing error measurements described herein, by the way of example and not by limitation, sometimes refer to jitter measurements. It should be understood, however, that the disclosed embodiments are applicable to measuring all types of timing errors, and can be specially beneficial in applications where real-time timing error measurements on a pulse-by-pulse basis is needed for in-situ correction or optimization of signal generation or processing systems. 
     The concept of the digitally corrected delay reference is illustrated using jitter measurements as an example.  FIGS. 1A-1C  illustrate schematic diagrams of performing jitter measurements. When the pulse has a specific timing component, the pulse demonstrates a phase slope associated with the timing.  FIG. 1A  illustrates a perfect pulse train and its associated optical spectrum. The initial phase slope in this particular example is 0. In addition to the phase slope, there is also a phase offset caused by any irregularities in the timing component.  FIG. 1B  illustrates an example phase offset caused by jitter in the pulse train. Furthermore, the phase offset is also related to the carrier envelope offset (CEO) phase. This is the phase of the optical pulse carrier, which can vary independently from the timing of the pulse itself.  FIG. 1C  illustrates an example phase offset caused by both jitter and carrier envelope offset. When two pulses interfere with each other, the spectral interference pattern encodes the phase differences in the intensity pattern. Given two phase offsets ΔØ high  and ΔØ low  measured at different frequencies, ν high  and ν low , respectively, the phase offset caused by carrier envelope offset ΔØ CEO  can be eliminated ac follows: 
     
       
         
           
             
               
                 
                   
                     
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     The different frequency values ν high  and ν low  can be selected using different optical filters. An optical filter is a device that takes in an optical wave and outputs that optical wave with some wavelengths of the spectrum with lower output power than others. For example, an optical filter can pass a contiguous fraction of the input bandwidth, with each of the optical filters passing a different band of wavelengths, e.g., arrayed waveguide gratings, thin-film filters, or fiber Bragg gratings. Many optical filters have multiple outputs, each corresponding to a distinct frequency band of the input. In this document, optical filters are also referred as wavelength division multiplexers (WDMs). In addition to arrayed waveguide gratings and thin-film filters, they can be constructed in multiple ways, including but not limited to: a coupler followed by single output filters on each output, a set of reflective filters can be combined with circulator, such that the reflection from one filter is circulated to the next filter. The filters can be selected so that they can cover the optical spectrum of the pulse train while maintaining sufficient separation of the frequencies. Phase offsets caused by other types of timing variations or irregularities can be determined in a similar fashion. 
       FIG. 2A  illustrates an example timing measurement apparatus  200  in accordance with the present technology. The apparatus  200  accepts two inputs  210 ,  202  into an optical hybrid  203  device. The optical hybrid  203  is a device that can include a number of beam splitters and one or more quarter-wave plates. The optical hybrid  203  includes at least two inputs and at least two outputs such that the two inputs are interfered at each output, with a phase difference between the inputs that is different for each output. For example, the device can produce 0 and 90-degree phase difference outputs, or 0, 90, 180, and 270-degree outputs. In some embodiments, the phase difference between the outputs can have values other than 90°. 
     In this embodiment, the optical hybrid  203  generates two outputs whose phases are shifted 90 degrees from each other (e.g., 0° and 90° outputs). The two outputs of the optical hybrid  203  are fed into wavelength division multiplexers  204   a  to  204   k  (e.g., optical filters). The outputs of each wavelength division multiplexer (e.g., ν high  and v low ) are fed into photodetectors  206   a  to  206   n  to convert the optical signals into radio-frequency (RF) signals. The RF outputs from the photodetectors  206   a  to  206   n  are then digitized using analog-to-digital converters  208   a  to  208   n.  The digital signals output from the timing measurement apparatus can be fed into a digital processor  210  to calculate a pulse pair phase difference at each wavelength. In some embodiments, the digital processor  210  is a part of the timing measurement apparatus  200 . It should be noted that in  FIG. 2A , and other figures herein, the depicted ellipses indicate the capability of processing two or more wavelength channels using similar configurations. 
     In general, at least some of the components in  FIG. 2A  (as well as other figures in this patent document) can be implemented as part of the timing measurement device, or as separate components and/or at remote locations with respect to other components of the system. For example, in some embodiments, the digital processor is a separate component implemented outside of the timing measurement apparatus. Similarly, the analog-to-digital converters or even, in some instances, the photodetectors can be implemented as separated components. 
       FIG. 2B  illustrates another representation of the timing measurement apparatus in accordance with the present technology to provide better understanding of the phase offset calculations. The phase offset ΔØ high  can be obtained as: 
       ΔØ high =α tan 2( P   0°,high   ,P   90°,high )   Eq. (4)
 
     The phase offset ΔØ low  can be obtained as: 
       ΔØ low =α tan 2( P   0°,low   ,P   90°,low )   Eq. (5)
 
     The phase offset caused by timing error (e.g., jitter) can then be determined according to Eq. (3). As shown in Eq. (3), the pulse pair phase differences at each wavelength for the same original pulse are subtracted to eliminate the carrier envelope offset phase. The result can be scaled by the optical frequency difference between phases from any pair of wavelengths to yield the time difference between the pulses. 
       FIG. 3A  illustrates another example timing measurement apparatus  300  in accordance with the present technology. The apparatus  300  accepts two inputs  301 ,  302  into an optical hybrid  303  device. The optical hybrid  303  generates four outputs whose phases are shifted 90 degrees from each other (e.g., 0°, 90°, 180°, and 270° outputs). The four outputs of the optical hybrid  303  are fed into wavelength division multiplexers  304   a  to  304   k.  The outputs of each wavelength division multiplexer  304   a  to  304   k  (e.g., ν high  and ν low ) are fed into photodetectors  306   a  to  306   n  to convert optical signals into radio-frequency (RF) signals. The RF outputs from the photodetectors  306   a  to  306   n  are digitized using analog-to-digital converters  308   a  to  308   n.    
     The digital signals output from the timing measurement apparatus are fed into a digital processor  310 . Similar to the embodiment shown in  FIG. 2 , some of the apparatus components, such as the digital processor, can be a part of the timing measurement apparatus or a separate component implemented outside of the timing measurement apparatus. The digital processor  310  calculates the difference between the 0° and 180° pulses from each frequency band as well as the 90° and 270° outputs from each frequency band. The two differences are fed as inputs into the pulse pair phase difference algorithm. 
       FIG. 3B  illustrates another representation of the timing measurement apparatus in accordance with the present technology to provide better understanding of the phase offset calculations. The phase offset ΔØ high  can be obtained as: 
       ΔØ high =α tan 2( P   0°,high   −P   180°,high   ,P   90°,high   −P   270°,high )   Eq. (6)
 
     The phase offset ΔØ low  can be obtained as: 
       ΔØ low =α tan 2( P   0°,low   −P   180°,low   ,P   90°,low   −P   270°,low )   Eq. (7)
 
     The phase offset caused by timing error (e.g., jitter) can then be determined according to Eq. (3). 
       FIG. 4A  illustrates another example timing measurement apparatus  400  in accordance with the present technology. The apparatus  400  accepts two inputs  401 ,  402  into an optical hybrid device  403 . The optical hybrid  403  generates four outputs whose phases are shifted 90 degrees from each other (e.g., 0°, 90°, 180°, and 270° outputs). The 0° and 180° pulses from the optical hybrid are directed to two WDMs  404   a,    404   b.  Outputs from the same frequency band of the 0° and 180° pulses are directed into one set of balanced photodetectors  406   a  to  406   n.  Likewise, outputs from the same frequency band of the 90° and 270° pulses are directed into another set of balanced photodetectors. 
       FIG. 4B  illustrates another representation of the timing measurement apparatus in accordance with the present technology. The phase offset ΔØ high  can be obtained as: 
       ΔØ high =α tan 2( P   0°-180°,high   ,P   90°-270°,high )   Eq. (8)
 
     The phase offset ΔØ low  can be obtained as: 
       ΔØ=α tan 2( P   0°-180°,low   ,P   90°-270°,low )   Eq. (9)
 
     The phase offset caused by timing error (e.g., jitter) can then be determined according to Eq. (3). 
       FIG. 5  illustrates yet another example timing measurement apparatus  500  in accordance with the present technology. In this embodiment, the timing measurement apparatus  500  is similar to the apparatus  200  in  FIG. 5  but is illustrated as including the digital processor  510  as part of the apparatus. In particular, in  FIG. 5 , two optical inputs  501 ,  503  are provided to an optical hybrid  503 . The outputs of the optical hybrid  503  are provided to the wavelength division multiplexers  504   a  to  504   k;  the outputs of the wavelength division multiplexers  504   a  to  504   k  are provided to the photodetectors  506   a  to  506   n.  The ADCs  508   a  to  506   n  receive the electrical signals from the photodetectors  506   a  to  506   n  and provide the digitized signals to the digital processor  510 . In addition, the digital processor  510  receives not only the digital signals from the analog-to-digital converters  508   a  to  508   n  but also one or more control signals  511 . The one or more control signals  511  can be used to account for timing offsets or timing error compensation, can be signals associated with environmental and external factors, such as temperature, vibrations or other channel information. The digital processor  510  can use such control signal to correct or compensate for such timing or environmental factors. 
       FIG. 6  illustrates an example architecture of a timing measurement system  600  in accordance with the present technology. As shown in  FIG. 6 , a pulse source generates an optical pulse train  601 . The optical pulse train  601  is then provided to a coupler  602 . The optical coupler  602  is a device that splits the input optical wave (or combines two input optical waves from two ports) into two output optical waves at ports Out 1  and Out 2 , each with a fraction of the power of the input(s). The coupler  602  can maintain the polarization state of the input for optimal operation. In this embodiment, one of the outputs of the coupler  602  is directed into a timing measurement device  604  (such as those shown in example configurations of  FIGS. 2 to 5 ) directly. The other output of the coupler  602  is fed into the timing measurement device  604  via an optical delay component  603 . The optical delay component  603  can be a polarization maintaining optical fiber, such as an integrated photonic optical delay line (e.g. silicon photonic, planar lightwave circuit, InP, GaAs, etc.). The length of the optical delay can be chosen to overlap pulses that are N≥1 periods apart. 
       FIG. 7  illustrates another example architecture of a timing measurement system  700  in accordance with the present technology. In this embodiment, similar to the configuration in  FIG. 6 , an optical pulse train  701  from an optical source is provide to the optical coupler  702 ; one output of the coupler  702  is directly provided to the timing measurement device  704 , while the other output of the coupler is provided to an optical delay component  703 , whose output is provided to the timing measurement device  704 . The timing measurement system  700  further accepts one or more control signals  705  so that a timing offset can be added to the timing error signal. In both embodiments shown in  FIG. 6  and  FIG. 7 , the optical hybrid, which is a part of the timing measurement device  604 ,  704 , eliminates the need for a phase locked loop and repetition rate tunable laser as used in some of the conventional techniques. The timing error information can be recorded in the digital form so that it can be used for compensation or combined with other subsequent processing. In some embodiments, the average peak of the pulses from the photodetectors can be aligned to the sample time of the digitizer, whose sample rate is equal to the pulse repetition rate. 
       FIG. 8  illustrates another example architecture of a timing measurement system  800  in accordance with the present technology. This embodiment can be implemented when the digitizer sample rate is higher than the pulse repetition rate, and the multiple samples on each pulse correspond to the multiple optical frequencies of the pulse. The timing measurement system  800  includes a dispersive device  804  that receives the optical pulse train  801  and separates each pulse&#39;s optical frequencies temporally that are provided to the coupler  802 . The system  800  then aligns the average peak of the pulses from the photodetectors  806 ,  807  to the sample time of the digitizers. The outputs from the optical hybrid  802  are fed directly to the photodetectors  806 ,  807 , thereby removing the need for additional photodetectors and digitizers. The analog to digital converters  808 ,  809  digitize the analog signals produced by the photodetectors  806 ,  807 . Alternatively, multiple dispersive devices can be used to connect the outputs from the optical hybrid to the photodetectors. The pulses are temporally aligned either in the analog domain (e.g., using optical delay lines) or in the digital processor with a shift (e.g., using circshift). The digital processor  810  then processes the samples corresponding to the same pulse. The phase offset ΔØ(n, m) as a function of pulse number n and sample point within a single pulse m can be obtained as: 
       ΔØ( n,m )=α tan 2( P   0° ( n,m ), P   90° ( n,m ))   Eq. (10)
 
     In some embodiments, a line can be fit to the independent variable optical frequency ω, which is related to sample number m by the frequency-time mapping, and dependent variable ΔØ(n,m), for a single pulse, yielding an intercept and slope for each pulse. The timing error can be deemed as being proportional to the slope of this line. 
     Typically, optical delay lines have temperature and vibration dependency.  FIG. 9  illustrates an example architecture of a timing measurement system  900  that reduces environmental dependency of the timing measurement in accordance with the present technology. Similar to the configuration of  FIG. 6 , the optical pulse train  901  is provided to the coupler  902 . One output of the coupler is provided to the timing measurement device  904 , and the other output of the coupler  902  is provided to the optical delay component  903 . The timing measurement system  900  adds a temperature and/or vibration sensor  905  onto the optical delay component  903  so that information from the sensor  905  can be used to digitally remove or reduce optical delay variations caused by environmental fluctuations on the timing error measurement. 
       FIG. 10  illustrates another example architecture of a timing measurement system  1000  in accordance with the present technology. In this embodiment, the optical pulse train  1001  is coupled into a first coupler  1002  to create two copies: Out 1  and Out 2 . The first copy, Out 1 , is transmitted directly into a first timing measurement device  1004 . The second copy, Out 2 , is fed into both polarizations of a birefringent optical delay line that maintains polarization. In some embodiments, the polarizations are created by a polarization rotator  1003 , such as a 45-degree splice that sends linearly polarized light half into each polarization, that is placed between the coupler  1002  and the birefringent optical delay line  1005 . The optical signal from the optical delay line  1005  is then fed into a polarizing beam splitter  1006  that separates light that travels through the slow and fast axes of the birefringent optical delay line. Light from one axis (e.g. slow) is sent to a second coupler  1007 . One output of the second coupler  1007  is fed into the first timing measurement device  1004 , whose other input receives the first copy of the first optical pulse train Out 1 . The other output of the second coupler  1007  and light from the other axis (e.g. fast) from the beam splitter  1006  are fed into a second timing measurement device  1008 . The digital processor  1009  uses the information obtained from the timing measurement devices  1004 ,  1008  to produce the timing error signal. 
     The first timing error signal t TMD1 (n) as a function of measurement number n from the first timing measurement device  1004  is between pulses that are many pulse periods (M) apart, and the delay is proportional to temperature changes ΔT(n) that vary with measurement number n and with coefficient K 1  that is independent of n due to changes in the delay line&#39;s length and group index from temperature changes. The second timing error t TMD2 (n) from the second timing measurement device  1008  is between pulses that are a few pulse periods (N) apart, and the delay is proportional to a temperature change ΔT(n) with coefficient K 2  that is independent of n due to changes in the delay line&#39;s length and birefringence. Both signals are sensitive and linearly proportional to temperature, yielding a linear system with two equations (relationship between temperature change and single or dual polarization delay) and two unknowns (temperature change and delay between each far apart pulse pair). 
         t   TMD1 ( n )= MΔt ( n )+ t   1   +K   1   ΔT ( n )   Eq. (11)
 
         t   TMD2 ( n )= NΔt ( n )+ t   2   +K   2   ΔT ( n )   Eq. (12)
 
     In the above equations, Δt(n) is the timing difference between adjacent pulses, averaged over M pulses; t 1  and t 2  can be freely chosen, e.g., such that for the first measurement point n=1, Δt(1)=0 and ΔTM=0 for both equations; regardless, t 1  and t 2  do not vary with n. As long as the ratio of the timing changes to temperature is different from the ratio of number of pulse periods apart (which is the case for standard polarization maintaining fiber), the system of equations Eq. (11) and Eq. (12) is invertible. Solving that system yields the delay between pulses from the optical pulse train. 
       FIG. 11  illustrates another example architecture of a timing measurement system  1100  in accordance with the present technology. In this embodiment, the optical pulse train  1101  is receive by the coupler  1102 ; one output of the coupler is provided to the timing measurement device  1104  and the other output of the coupler  1102  is provided to the optical delay line  1103 . The timing measurement system  1100  can stabilizes the optical delay line  1103  using a temperature and/or vibration sensor  1105 . Information from the temperature and/or vibration sensor  1105  can create a compensational signal along the optical delay line  1103 . The compensation signal can be sent to a delay line adjustment device, e.g. a fiber stretcher, a heater, or a tunable optical delay line placed before or after the optical delay line  1103  to perform the stabilization. 
       FIG. 12  illustrates an example architecture of a real-time calibration system  1200  in accordance with the present technology. The calibration system  1200  uses a digital-to-analog converter  1203  and a calibrated tunable delay line. The digital processor included in the timing measurement device  1204  can adopt a calibration algorithm to determine calibration coefficients to enable real-time calibration. The digital-to-analog converter  1203  sends a calibration signal to the calibrated tunable delay line, and the timing measurement device records the digital inputs. For example, the calibration signal can be a rectangular wave, with period given by the laser period or an integer fraction of that, multiplied by a ramp that ranges from 0% to 100%. The calibration signal can cause timing differences between pulses to be distributed uniformly across the range of possible calibration values. In some embodiments, for fast calibration, the calibrated tunable delay line can an electro-optic phase modulator  1202  with known V π , and the calibration signal can be used immediately before and/or after one or more acquisitions in the field. 
       FIG. 13  illustrates an example architecture of an error-compensated photonic Analog-to-Digital converter  1300  that uses a timing measurement device in accordance with the present technology. When the timing error of the optical pulse train source in the photonic analog-to-digital converter is larger than the measurement noise of the timing measurement device, the negative impact of the timing errors on the RF signal measurement process can be compensated in a digital processor. In this particular embodiment, the converter is arranged such that: (1) an optical pulse train  1301  source is directed into a coupler  1032 ; (2) one output of the coupler  1302  is fed into a timing measurement device  1304 , generating a timing error signal; (3) the second output of the coupler  1302  is fed into an electro-optic intensity modulator  1303 , generating an intensity modulated optical pulse train in accordance with the RF signal  1305 ; (4) the intensity modulated optical pulse train is directed to a photodetector  1306  to form an intensity modulated RF pulse train; (5) an analog-to-digital converter  1307  converts the intensity modulated RF pulse train into intensity modulated digital samples; (6) the digital processor  1308  takes the intensity modulated digital samples and timing error signal, and compensates the timing error on the intensity modulated digital samples, thereby obtaining the electro-optic intensity modulator transfer function to generate a digitized RF signal  1309 . 
     In all of the embodiments described herein, the digital processor can use a calibration algorithm that removes nonidealities such as: unequal modulation amplitudes between 0 degree/90 degree/other channels and/or phase differences between channels besides 90 degrees. In addition, 0° and 90° pulses (as well as the 180° and 270° pulses) are ideally proportional to the sine and cosine of the modulated phase at the time of sampling. Thus, when plotted on the axes of a graph, the 0° and 90° pulses (similarly, the 180° and 270° pulses) from all possible modulated phases should form a circle. Any deviation from a true circle is a result of distortion in the system. To remove such distortions, the digital processor can sample the received data to fit an ellipse to the shape (e.g., using a least-squares fit). From the ellipse coefficients, the digital processor can calculate coefficients to transform that ellipse to a circle to removes biases and/or offsets in the components such as the modulators, digitizers, etc. 
     One aspect of the disclosed embodiments relates to a timing measurement apparatus that includes an optical hybrid configured to receive two optical pulse trains as inputs and produce two or more optical outputs that are each phase shifted with respect to one another. The timing measurement apparatus further includes two or more optical filters each coupled to the optical hybrid to receive an output from the optical hybrid, the two or more optical filters configured to produce multiple pulse signals with distinctive frequency bands. The timing measurement apparatus also includes one or more photodetectors positioned to receive and convert each of the multiple optical signals produced by the two or more optical filters to an associated electrical signal. The above timing measurement apparatus additionally includes one or more analog-to-digital converters coupled to the one or more photodetectors to convert the plurality of electrical signals into a plurality of digital signals corresponding to the outputs of the two or more optical filters, where processing of the plurality of digital signals enables a determination of a timing error associated with the two optical pulse trains based on a computed phase difference between a first frequency band signal and a second frequency band signal and a computed frequency difference between the first frequency band signal and the second frequency band. 
     In one example embodiment, the two or more optical outputs include two optical outputs that are phase shifted by 0 and 90 degrees, respectively. In another example embodiment, the two or more optical outputs include four optical outputs that are each phase shifted with respect to one another. In yet another example embodiment, the four optical outputs are phase shifted by 0, 90, 180 and 270 degrees, respectively. According to another example embodiment, the one or more photodetectors include one or more balanced photodetectors, each configured to receive inputs from two of the two or more optical outputs, wherein the two optical outputs feeding each balanced photodetector have the same frequency band. In one example embodiment, a first of the one or more balanced photodetectors is configured to receive a first input that is phase shifted by 0 degree and a second input that is phase shifted by 180 degrees, and wherein a second of the one or more balanced photodetectors is configured to receive a third input that is phase shifted by 90 degree and a fourth input that is phase shifted by 270 degrees. 
     In another example embodiment, the timing measurement system a digital processor configured to determine timing information associated with the two optical pulse trains based on the plurality of digital signals. In yet another example embodiment, the digital processor is configured to receive one or more control signals and to adjust the determination of the timing error according to the one or more control signals. 
     According to an example embodiment, the timing measurement apparatus is implemented as part of a timing measurement system, where the timing measurement system includes a coupler configured to receive an input optical pulse train and to produce a first pulse train that is provided to the timing measurement apparatus as one of the two optical pulse trains, the coupler further configured to produce a second pulse train. The timing measurement system further includes an optical delay component coupled to the coupler to allow transmission of the second pulse train to the timing measurement apparatus as another of the two optical pulse trains. In another example embodiment, the optical delay component allows an interference to occur between the first and the second pulse trains. In yet another example embodiment, the timing measurement system further comprises a sensor that is configured to produce the one or more control signals to a delay line adjustment component to stabilize a length of the optical delay component. In still another example embodiment, the timing measurement apparatus is further configured to receive one or more control signals to enable further adjustments to the determination of the timing error. In one example embodiment, the timing measurement system further comprises a sensor that is configured to produce the one or more control signals for provision to the timing measurement apparatus to enable additional adjustments to the determination of the timing error. In an example embodiment, the sensor includes a temperature or vibration sensor. 
     In another example embodiment, the timing measurement apparatus is a first timing measurement apparatus that is implemented as part of a timing measurement system, and the timing measurement system further includes a first coupler configured to receive an input optical pulse train and produce a first optical pulse train that is provided to the first timing measurement apparatus as one of the two optical pulse trains, the coupler further configured to produce a second optical pulse train. The timing measurement system also includes an optical rotator coupled to the first coupler to receive the second optical pulse train, an optical delay component coupled to the optical rotator, a polarizing beam splitter coupled to the optical rotator via the optical delay component to receive and separate an optical pulse train that is output from the optical delay component into a slow pulse train and a fast pulse train, a second coupler coupled to the polarizing beam splitter to receive the slow pulse train and produce a first intermediate pulse train and a second intermediate pulse train, wherein the first intermediate pulse train is directed to the timing measurement apparatus as another of the two optical pulse trains, and a second timing measurement apparatus coupled to the second coupler and to the polarizing beam splitter to receive the second intermediate pulse train and the fast pulse train as two optical pulse trains, wherein computation of timing error information associated with the input pulse train is enabled based on processing of signals representative of outputs of the first timing measurement apparatus and the second timing measurement apparatus. 
     In another example embodiment, the timing measurement apparatus is a first timing measurement apparatus that is implemented as part of a timing measurement system, and the timing measurement system further includes one or more couplers, a polarization rotator, an optical delay component and a polarizing beam splitter to produce a slow and a fast pulse train based on an input optical pulse train; and a second timing measurement apparatus, wherein the first timing measurement apparatus is configured to receive the input optical pulse train and the slow pulse train, and second timing measurement apparatus is configured to receive the slow pulse train and the fast pulse train to enable a determination of a timing error in the input optical pulse train. 
     In yet another example embodiment, the optical delay component is a birefringent optical delay line. In still another example embodiment, the timing measurement system is implemented as part of a real-time calibration system, where the real-time calibration system includes a digital-to-analog converter configured to convert a digital signal to an electrical analog signal, and an electro-optic (EO) modulator coupled to the digital-to-analog converter to receive the electrical analog signal, the EO modulator further configured to receive a train of pulses and to produce a phase modulated signal based on the received train of pulse and the electrical analog signal, wherein the timing measurement system is coupled to the EO modulator to receive the phase modulated signal as the input pulse train. 
     In one example embodiment, the timing measurement system is implemented as part of a photonic analog-to-digital converter, where the photonic analog-to-digital converter includes a second coupler configured to receive a train of optical pulses, wherein the timing measurement system is coupled to a first output of the second coupler to receive the train of optical pulses from the first output of the second coupler, and wherein the second coupler further configured to produce a separate optical pulse train at a second output thereof. The photonic analog-to-digital converter further includes an electro-optic (EO) modulator coupled to the second output of the second coupler to receive the separate optical pulse train, the EO modulator further configured to receive a radio-frequency (RF) signal and to produce a modulated optical signal based on the separate optical pulse train and the RF signal. The photonic analog-to-digital converter also includes a photodetector coupled to the EO modulator to convert the modulated optical signal into an associated electrical signal, and a digitizer to convert the electrical signals produced by the one or more photodetectors into digital signals. In another example embodiment, the photonic analog-to-digital converter includes a digital processor coupled to the digitizer and to the timing measurement systems to obtain an adjusted digital signal corresponding to the train of optical pulses. 
     Another aspect of the disclosed embodiments relates to a timing measurement system that includes a dispersive device configured to receive an input optical pulse train and to produce a train of pulses that are spectrally dispersed in time domain, a coupler coupled to the dispersive device to receive the train of spectrally dispersed pulses and to produce two outputs, an optical hybrid coupled to a first output of the coupler to receive a first pulse train from the coupler, an optical delay component coupled to a second output of the coupler to provide a second pulse train from the coupler to the optical hybrid, wherein the optical hybrid is configured to produce two or more optical outputs that are each phase shifted with respect to one another based on the first pulse train and the second pulse train, one or more photodetectors coupled to the optical hybrid to convert each of the two or more optical outputs into an associated electrical signal, and one or more analog-to-digital converters coupled to the one or more photodetectors to convert the electrical signals into digital signals, wherein processing of the digital signals enables a determination of timing information of the input optical pulse train and adjustment of the digital signals according to the timing information. 
     At least parts of the disclosed embodiments (e.g., the digital processor) can be implemented in digital electronic circuitry, or in computer software, firmware, or hardware. For example, electronic circuits can be used to control the operation of the detector arrays and/or to process electronic signals that are produced by the detectors. At least some of those embodiments or operations can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a computer-readable medium for execution by, or to control the operation of, data processing apparatus. The computer-readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter effecting a machine-readable propagated signal, or a combination of one or more of them. The term “data processing apparatus” encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers. The apparatus can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them. A propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal, that is generated to encode information for transmission to suitable receiver apparatus. 
     A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment. A computer program does not necessarily correspond to a file in a file system. A program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code). A computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network. 
     The processes and logic flows described in this document can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output. The processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). 
     Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer. Generally, a processor will receive instructions and data from a read-only memory or a random-access memory or both. The essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data. Generally, a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto-optical disks, or optical disks. However, a computer need not have such devices. Computer-readable media suitable for storing computer program instructions and data include all forms of non-volatile memory, media and memory devices, including, by way of example, semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry. 
     While this patent document contains many specifics, these should not be construed as limitations on the scope of any invention or of what may be claimed, but rather as descriptions of features that may be specific to particular embodiments of particular inventions. Certain features that are described in this patent document in the context of separate embodiments can also be implemented in combination in a single embodiment. Conversely, various features that are described in the context of a single embodiment can also be implemented in multiple embodiments separately or in any suitable subcombination. Moreover, although features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be excised from the combination, and the claimed combination may be directed to a subcombination or variation of a subcombination. 
     Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. Moreover, the separation of various system components in the embodiments described in this patent document should not be understood as requiring such separation in all embodiments. 
     Only a few implementations and examples are described, and other implementations, enhancements and variations can be made based on what is described and illustrated in this patent document.