We demonstrate a high-accuracy heterodyne measurement system for characterizing the magnitude of the frequency response of high-speed 1.55 µm photoreceivers from 2 MHz to greater than 50 GHz. At measurement frequencies below 2 GHz, we employ a phase-locked loop with a double-heterodyne detection scheme, which enables precise tuning of the heterodyne beat frequency with an RF synthesizer. At frequencies above 2 GHz the system is operated in free-run mode with thermal tuning of the laser beat frequency. We estimate the measurement uncertainties for the low frequency range and compare the measured high-frequency response of a photoreceiver to a measurement using electro-optic sampling.
©2011 Optical Society of America
Future optical networks for 100 Gb/s and beyond will continue to demand higher-accuracy characterization of signals to achieve optimum performance as signal margins become tighter. For example, eye-mask testing of transceivers relies critically on accurate time and amplitude calibration of sampling oscilloscopes in order to achieve and maintain economic production yields. NIST has supported the high-accuracy characterization of photodiode-based receivers as calibrated electrical sources to provide time- and frequency-domain transfer standards [1–5]. A consistent goal of this effort has been to reduce measurement uncertainties for frequency response calibrations while providing them at ever-higher bandwidths to support the ever-increasing bit rate of optical networks. However, as networks have moved toward complex modulation formats to achieve higher throughput at the same symbol rates, the signal margins have had to become even tighter. Network testing is also being done with longer data sequences that better resemble real data traffic. Doing so increasingly tests the low-frequency response or slow settling-time of high-bandwidth receivers. In addition, the cable TV (CATV) industry depends critically on being able to accurately characterize the optical response of transceivers from only a few megahertz to about 3 GHz. And finally, testing of microprocessors and interconnects depends critically on quantitative waveform metrology using high-bandwidth sampling oscilloscopes that have been characterized over long durations.
Likewise, high-bandwidth photoreceivers used as transfer standards for the calibration of waveform and frequency response measurement equipment should also be calibrated at high as well as low frequencies. With the response calibration of photoreceivers extending into the low-megahertz regime, quantitative measurements of optical and electrical waveforms approaching 1 µs in duration may become technically feasible. In this work we describe a technique for accurately achieving this calibration.
A number of techniques have been developed to characterize the frequency response of photoreceivers . The pulse excitation method usin g a sampling oscilloscope  is capable of providing the magnitude as well as the phase response, but depends on time-domain deconvolutions and some systems modeling. Electro-optic sampling (EOS) is another pulse excitation method that can provide the magnitude and phase response of a photoreceiver [8, 9]. Calibrations using EOS can be related to fundamental quantities; however, the method is not well-suited to measurements below about 600 MHz. The broadband noise excitation method [10, 11] is very simple to implement. However, like a number of techniques, it relies fundamentally on knowing the frequency response of another device, such as a transmitter-modulator, reference photoreceiver, or instrumentation such as a network analyzer. In effect, these techniques serve as a method of transferring a known response calibration to a photoreceiver under test. In the harmonic swept-frequency modulation methods [12–14], the frequency response of the modulator does not need to be known, but an assumption that the modulator follows a well-known or ideal process must be made, which may introduce measurement errors. The self-heterodyne methods [15, 16] offer the simplicity of a single laser source, but do not provide a way to control or compensate for linewidth noise on the resulting heterodyne signal. A stimulus with a broadened linewidth will make it difficult to operate at low frequencies or resolve sharp resonant features. Other methods for measuring response may not scale easily to high frequencies or may suffer from temporal instabilities. However, the heterodyne method involving two lasers [4, 5, 17–22] has proven to be of value to photoreceiver metrology for nearly 30 years, for a multitude of reasons.
The heterodyne method using two lasers is capable of generating an optical stimulus of very high accuracy, and also at very high frequencies without further difficulty. In fact, the uncertainties for a heterodyne response measurement are typically dominated by the uncertainty of the RF power sensor calibration [4, 5] required by the technique. As we will review, this technique has the advantage that the excitation of the photoreceiver can be calculated from first principles. In addition, the calibration of the measurement system is well-understood and well-behaved, remaining largely invariant during operation as the measurement frequency is swept.
The first swept heterodyne demonstration using laser diodes to measure the response of an optical detector was performed with temperature tuning at 870 nm . Since that time, the heterodyne technique has expanded to include more wavelengths (1.3 and 1.5 µm), diode and solid state lasers, controlled frequency sweeping , frequency locking , and very high frequencies on wafer . Phase-locked-loop operation of a heterodyne system has been demonstrated by use of very low-noise Nd:YAG lasers at 1.3 µm, enabling Hertz-level frequency control with a sub-kilohertz heterodyne stimulus for high-accuracy and high-resolution response calibrations [5, 23]. However, at the 1.55 µm wavelength commonly used in telecommunications optical networks, a corresponding calibration system has not been realized, to our knowledge. The primary challenge has been the lack of low-noise tunable lasers at 1.55 µm. Most diode lasers, whether tuned by temperature or external gratings, are simply too noisy, with 1-second linewidths of 10 MHz or more. As a result, typical phase-locked-loop systems don’t have enough bandwidth to lock systems based on diode lasers,either because of limits to the electronic phase detection or the optical feedback element that corrects the phase. In the phase-locked-loop system we will describe, we avoid the linewidth problem of diode lasers by using 1.5 µm tunable fiber lasers having a linewidth specification of less than 1 kHz. In addition, we use an acousto-optic frequency modulator to externally provide high-bandwidth phase-locking.
The heterodyne technique combines the output of two narrow-linewidth lasers having a difference in frequency equal to the desired excitation frequency of the photoreceiver. If the two lasers are equal in power and have the same state of polarization, the desired condition of nearly 100% modulation depth will be achieved. The total optical power incident on the photoreceiver is then described byFig. 1 . Besides the photodiode, a typical module contains a load resistance RL and provides access to the bias current, which can be monitored by the external resistor Rb. The normalized frequency response of the photoreceiver, , is defined as the ratio R2(f)/R2(0), where R(f) is the responsivity (in A/W) of the photoreceiver at frequency f, and R(0) is the DC responsivity. As shown in Ref. 4 and 5, the normalized frequency response can be accurately approximated as the ratio of the RF power delivered to the load RL and designated as PRF, to the DC electrical power that would be delivered to load RL as related to the current idc by
The convenience of the normalized response, by needing only to measure idc, is that the individual optical powers delivered to the photoreceiver from each laser do not need to be known. Knowing these optical powers throughout a measurement with low uncertainty can be challenging, because the output powers of the lasers can vary with time or wavelength tuning and optical components such as splitters can have wavelength-dependent loss or create etalon effects. Because most high-speed photoreceiver modules are fiber-pigtailed and terminated with a connector, the normalized response includes the optical loss from this packaging. To report the absolute response, additional uncertainty contributions would need to be included to account for the optical power measurement and the optical fiber connector repeatability, both of which would dominate the total uncertainty and raise it to 1 dB or more. By contrast, the typical uncertainty for a normalized response measurement is 0.051 dB, as will be detailed later in Table 1 .
2. Heterodyne measurement system
Figure 1 illustrates the system we constructed to measure the magnitude of the frequency response of photoreceivers using the heterodyne technique. The block containing the heterodyne optics and phase-locked loop is described in detail later, and was used to produce a sinusoidally modulated optical stimulus at a frequency determined by the computer controlling the measurement system. The modulated signal was delivered to the photoreceiver under test by use of a singlemode optical fiber. In response to the optical stimulus, the photodiode generated an RF power across the load RL, which was then delivered to a diode-based power sensor by a coaxial waveguide. The coaxial waveguide contained an adaptor to match the connector on the photodiode (1.0 mm) to the connector on the power sensor (1.85 mm). In addition to the RF power, the computer simultaneously recorded the DC current idc. This photocurrent was measured by use of a digital volt meter (DVM) to monitor the voltage drop across the precision 1 kΩ resistor Rb. The computer also recorded the frequency of the heterodyne signal measured by either an electrical counter or a tracking electrical spectrum analyzer (ESA). At high heterodyne frequencies the system was operated without the phase-locked loop, in free-run mode, with thermal tuning of the difference frequency between the lasers.
To obtain the frequency response of just the receiver module, illustrated within the dotted circle of Fig. 1, the measurements by the power sensor must be corrected for the response of the coaxial connections to the power sensor, and the sensor itself. While such corrections may be negligible in some cases, they can become sizeable at low frequencies due to AC-coupling of the sensor, or at tens of gigahertz due to resonant features. Power sensors are characterized by a frequency-dependent calibration factor Kb, which accounts for both the effective efficiency and the refection coefficient of the sensor. The calibration factor of our high-frequency power sensor was measured at NIST by use of a direct power comparison technique relying upon transfer standards that have been calibrated against a calorimeter . To account for the coaxial connections, we used a vector network analyzer to measure the scattering parameters of the photodetector, coaxial adaptor, and the power sensor. The scattering information was used to calculate a frequency-dependent impedance-mismatch correction M.
An alternative to characterizing the isolated response of a module, when intended to be used as a modulation reference, is to characterize the module in conjunction with a particular power sensor [4, 5]. This combined device requires no corrections and can have uncertainties for the measured response that are more than two times lower than those of the isolated module. This combined reference receiver is illustrated within the dotted rectangle in Fig. 1. If the power sensor inside a combined receiver cannot be made available externally, it will not be possible to calibrate the sensor to an external power source, and the measured frequency response curve will be relative rather than absolute.
3. Heterodyne optics and phase-locked loop
Our heterodyne system can operate in closed-loop mode at measurement frequencies up to 1.9 GHz, and in open-loop mode at higher frequencies extending beyond 50 GHz. Figure 2 illustrates in detail the heterodyne optics and phase-locked loop system we constructed. The upper portion of the figure is composed mainly of optical paths (red dotted lines) that aloneconstitute the open-loop system, while the lower portion adds electrical paths (blue solid lines) that enable phase-locked operation.
The single-mode lasers we used were commercial 10 mW fiber lasers having up to 1 nm of continuously single-mode wavelength tuning and a linewidth specification of <1 kHz. Broad tuning of the heterodyne beat frequency was performed by controlling the operating temperature of the lasers, while fine tuning and frequency locking were accomplished with piezo-electric transducer control of the fiber laser cavity length. The output of laser 2 was passed though a fiber-pigtailed acousto-optic modulator (AOM), which shifted the frequency of the laser by 100 MHz and enabled fast feedback control for phase-locking.
The outputs of both lasers were launched into a beam coupling system composed of free-space optics, as detailed in the inset box of Fig. 2. The use of free-space optics rather than fiber components reduced the wavelength and polarization dependence of the coupled beams. The lasers were collimated and combined in a plate beam-splitter, and the combined beams were passed through polarizing isolators. Fiber-pigtailed collimators were used again to collect and deliver the polarized light to the fiber-coupled device under test (DUT) and the photoreceivers used for frequency measurement and locking. Polarizing and mode-matching the beams in a single-mode fiber was critical to ensuring that the modulation of the light incident on the DUT was precisely calculable. Adjustable rotary attenuators were used to equalize the power from each laser delivered to the DUT, as measured by monitoring the bias current flowing through resistor Rb.
The frequency of the heterodyne signal was detected by use of two amplified photoreceivers, one covering between dc and 500 MHz and another covering up to 50 GHz. The heterodyne frequency was monitored with an electrical counter up to 1.9 GHz in closed-loop operation. At heterodyne frequencies extending up to 50 GHz, an ESA was used to track and measure the frequency, with the assistance of harmonic mixers beyond 26.5 GHz. The frequency range beyond 50 GHz can be covered with higher-bandwidth photoreceivers and additional harmonic mixers, but with increasing difficulty.
The lower portion of Fig. 2 illustrates phase-locked loop operation at a heterodyne frequency of 200 MHz. This signal frequency is detected with an amplified photoreceiver having a 3 dB bandwidth of 1 GHz (labeled LF PD, followed by an amplifier with gain G). The signal is electrically mixed (the second heterodyne) with a tunable frequency synthesizer at 280 MHz, and the resulting intermediate frequency at 80 MHz is isolated with a band-pass filter. We used a divide-by-eight frequency divider to compress the deviations in phase, thereby increasing the phase capture range of the feedback loop by a factor of eight.
The phase of the mixed and divided heterodyne frequency, now at 10 MHz, was detected relative to a 10 MHz synchronization frequency provided by the tunable oscillator. This same reference frequency was distributed to the ESA and the counter. We used a custom NIST phase-locked loop circuit based on an emitter-coupled logic (ECL) comparator and an ECL phase-detector chip to create a voltage signal proportional to the phase difference . This voltage signal represents the error in the phase of the heterodyne frequency, which we then fed back through a second-order, active loop filter before driving a voltage-controlled oscillator. The oscillator has a center frequency of 100 MHz, which matches the resonance frequency of the AOM. The AOM was used to provide high-bandwidth, external corrections to the phase of laser 2, such that its output tracked the phase of laser 1 at a difference frequency 80 MHz below the frequency of the tunable oscillator.
While feedback to the AOM provided for fast phase-locking, lower-bandwidth feedback was necessary to correct for larger frequency deviations and maintain the lock during tuning of the heterodyne frequency. Therefore, the error signal from the phase-locked-loop circuit was low-pass-filtered and passed through a NIST-built loop filter composed of adjustable integrator, proportional, and gain feedback stages. A voltage amplifier with a gain of 20 was used to create a high-voltage drive signal (180 V maximum) for the piezo-electric tuning element of laser 2. We used a passive 3.3 kHz low-pass filter placed between the high-voltage amplifier and the piezo-electric element of laser 2 to reduce resonant oscillations. A second feedback path to the piezo element, with a bandwidth of just 2 Hz, was used to counteract drift of the dc operating point caused by frequency tuning of the heterodyne frequency.
4. Phase-locked loop characterization
We characterized the effectiveness of the phase lock by observing the spectrum of the heterodyne beatnote seen by the electrical spectrum analyzer and frequency counter shown in Fig. 2. The system was locked and held at an output frequency of 1 GHz under typical operating conditions. Figure 3(a) shows a spectral measurement with a 1 MHz span and 10 kHz resolution, and formed from the average of 10 traces. The total electrical power in the carrier defined by the beat frequency is estimated to be −22.3 dBm. A noise pedestal centered about the carrier is visible above the instrumentation noise floor of −93 dBm. We estimate that the noise level at the carrier frequency is about −53 dBm, or about 30 dB down from the carrier. Even with a phase-lock, a finite amount of residual noise remains in the system. The lack of broad side-lobes to define the pedestal is probably indicative of a weak locking condition. While it is possible that some of the pedestal noise could have been reduced by use of a narrower 80 MHz band-pass filter (see Fig. 2), doing so would have also made it much more likely that the system would not automatically re-lock each time the tunable oscillator tried to move the beat frequency. Figure 3(b) shows a single spectral trace with a 100 Hz span and 1 Hz resolution, and demonstrates that the system is in fact phase-locked because the width of the apparent beat frequency is limited by the 1 Hz resolution of the spectrum analyzer. The electrical power level of the noise pedestal is about 40 dB lower than Fig. 3(a), which is consistent with the 104 change in the frequency span between the two spectra.
At an operating frequency of 1 GHz, the system could maintain a robust lock for manyhours at a time. However, at frequencies above 1.8 GHz the lock did become noticeably less stable, and the measured beat frequency typically deviated by up to 1 kHz from the desired operating frequency. This deviation represents an error of 2 parts in 106. The high-frequency error was caused by the bandwidth limitation of the low-frequency photodiode used to detect the phase-lock signal (again, labeled LF PD in Fig. 2). This commercial photoreceiver is DC-coupled and has a 3 dB bandwidth specification of 1 GHz stated by the manufacturer.
5. Experimental results
We used the phase-locked operation of our heterodyne system to measure the low-frequency response of a reference receiver composed of a combined photodiode and power sensor. The reference receiver is represented by the components inside the dotted rectangle of Fig. 1. The photocurrent flowing through the bias resistor Rb was approximately 100 µA throughout the measurements, and the detected RF power was about −39 dBm. The normalized response from 2 MHz to 1.9 GHz was scanned five times in 1 MHz steps, with a wait time of 1 second between steps to allow the system to settle. To check measurement repeatability, two of the five measurements were performed four months after the first three. The largest difference between any two response scans was less than 0.02 dB. The five scans were combined with a “kernel” smoothing technique [5, 26] to produce the red magnitude response curve shown in Fig. 4 . Because of the sealed packaging of this combined receiver, we were unable to reference calibrate the RF power sensor before each scan. Instead, we arbitrarily set the response at 50 MHz to 0 dB in Fig. 4.
Also shown in Fig. 4 is the expanded uncertainty for the measurement of the normalized response curve in blue, where the expansion factor k is 2 for an approximate 95% confidence interval . As shown, the expanded uncertainty is largely independent of frequency, with a value of about 0.05 dB. Table 1 lists the components that contribute to the total uncertainty, given for a measurement frequency of 1 GHz. The largest component of uncertainty is the power-meter range scaling, which is the estimated uncertainty in the gain of the power meter across its measurement range. The power meter manufacturer specifies the range scaling uncertainty at 0.5%. The second largest component of uncertainty is the measurement of the bias current, which is limited by the temperature dependence of the bias resistor Rb. The temperature during each response scan was stable to ±1 C. The combined standard uncertaintyis the square root of the summed squared uncertainties, and the expanded uncertainty includes the coverage factor k = 2.
Used in open-loop mode, our heterodyne system is capable of measuring the normalized magnitude response of a receiver at frequencies approaching 100 GHz. To demonstrate some of this capability, we measured a reference receiver module used at NIST as a check-standard for the NIST EOS system [8, 9]. The module has the topology illustrated in the dotted circle of Fig. 1, with integral bias circuitry and a waveguide-integrated photodiode chip that provides a 3 dB bandwidth in excess of 50 GHz. In this measurement, the power sensor was external to the receiver, which allowed the sensor to be calibrated against a reference sourcebefore each heterodyne scan. For this reason, the normalized response measured is absolute rather than relative, and no constant vertical offsets are applied.
During our open-loop measurements, the total average photocurrent from the receiver was about 100 µA, and the detected RF power within the bandwidth of the detector was about −42 dBm. Scanning of the system was accomplished by thermally tuning one of the lasers at a constant rate while periodically recording the normalized response, as well as the heterodyne frequency on a tracking ESA. As described above in Section 2, the isolated frequency response of the receiver module was obtained by normalizing to the measured calibration factor Kb of the power sensor head and the mismatch correction M of the system . The red line in the plot of Fig. 5 shows the frequency response of the module obtained with open-loop scanning between 40 MHz and 50 GHz. This result was obtained from separate measurements covering the spans 40 MHz to 26 GHz, 25.5 GHz to 40 GHz, and 39 GHz to 50 GHz. In total, each frequency span was measured a total of five times before the results were combined with kernel smoothing.
As a time-domain method [8, 9], EOS can provide independent verification of a frequency-domain heterodyne response measurement. The frequency response of the same NIST check standard module measured with the EOS technique is shown for comparison as the blue line in Fig. 5. The EOS curve was aligned vertically with respect to the heterodyne curve by minimizing the mean difference between the responses across the entire frequency range. By use of this criterion, the largest differences between the curves occur at frequencies above 45 GHz, and are no larger than 0.4 dB. Differences of this magnitude are consistent with the EOS uncertainty and our preliminary analysis of the heterodyne uncertainty. The EOS uncertainty is dominated by the mismatch correction, whereas the heterodyne uncertainty is dominated by the mismatch correction and the RF power sensor calibration. A complete uncertainty analysis of the heterodyne system, which includes the elements of Table 1 as well as contributions from the power sensor calibration and mismatch correction, will be presented elsewhere.
6. Discussion and conclusion
In open-loop mode, our heterodyne system is capable of generating a sinusoidal optical stimulus in excess of 100 GHz, limited only by the thermal tuning range of the fiber lasers. However, in practice there are a couple of technical challenges to measuring the response of a photoreceiver with a coaxial connector at such high frequencies, all of which are electrical in nature. Measuring the heterodyne frequency becomes increasingly difficult above 50 GHz because the necessary microwave amplification, harmonic mixing, and measurement equipment are both expensive and lossy in coaxial form. Above 110 GHz, the cut-off frequency of 1.0 mm coaxial connectors, signal measurements must presently be done with waveguide-based equipment. An alternative approach is to use a high-precision optical wavelength meter to independently measure the two laser wavelengths and calculate the heterodyne frequency from the wavelength difference. With a commercially available wavelength meter having sub-picometer resolution it is possible to measure nearly unlimited heterodyne frequencies with better than 50 MHz resolution.
The other measurement challenge at high frequencies is the lack of coaxial-based power sensors above 65 GHz that have accurately measured calibration factors Kb. Sensors available in this range may also incur the high loss associated with an integral waveguide mixer and require waveguide-to-coaxial adapters. Both the calibration factor and any mismatch corrections for adapters must be accurately characterized in order to isolate the response of a receiver from the measurement system. At high frequencies such characterization of the measurement equipment is difficult, translating into increased measurement uncertainties for the frequency response of photodiodes. Our response measurements were conducted at low power to avoid photodiode nonlinearities, whereas some high-frequency sensors require high power to operate. While an ESA could be used to measure the RF power, this approach involves lossy internal or external mixers and high noise levels which may require the photodiode to operate in a nonlinear regime. In addition, the ESA may introduce large calibration factor and mismatch correction uncertainties.
As we have described, the use of a modulation transfer standard based on a combined photodiode and power sensor will effectively eliminate the uncertainties associated with a coaxial mismatch and a power sensor calibration factor. Conveniently, there are high-frequency, diode-based waveguide sensors that have sufficient sensitivity yet calibration factors that have not been well-characterized that could be considered for a combined standard that extends well beyond 50 GHz.
We have demonstrated a heterodyne system for the calibration of photoreceivers at 1.5 µm that covers the frequency range from 2 MHz to 50 GHz, with the capacity to extend to much higher frequencies. At frequencies below 1.9 GHz, the system can operate with a phase-locked loop for precise frequency tuning and high repeatability, which enables the averaging of repeated measurements and the potential to resolve sharp resonant features. Operated in open-loop mode, we used the same heterodyne system to measure the response of a photoreceiver up to 50 GHz. After making adapter mismatch and calibration factor corrections, the result was compared to an EOS measurement for independent verification. The two results showed excellent agreement, with a difference of no more than 0.4 dB occurring at the highest frequencies.
The heterodyne measurement performance demonstrated by this work makes possible the calibration of photoreceivers at the very high and low limits of their operating frequency range. In turn, this performance enables the quantitative measurement of high-bandwidth optical and electrical waveforms of durations approaching 1 µs, with implications for optical complex modulation formats, CATV applications, and the characterization of signals in microprocessors and interconnects.
We acknowledge Dr. Nathan Newbury for extensive assistance with the design of the phase-locked loop system, and Dr. Thomas Mitchell Wallis for high-accuracy measurements of the calibration factor of the power sensor. We also thank Dr. Jeffrey Jargon for measurements of the coaxial mismatch correction factor and Dr. Dylan Williams for supplying the EOS measurements for the comparison in Fig. 5.
1. T. S. Clement, P. D. Hale, D. F. Williams, J. C. M. Wang, A. Dienstfrey, and D. A. Keenan, “Calibration of sampling oscilloscopes with high-speed photodiodes,” IEEE Trans. Microw. Theory Tech. 54(8), 3173–3181 (2006). [CrossRef]
2. A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, and D. F. Williams, “Minimum-phase calibration of sampling-oscilloscopes,” IEEE Trans. Microw. Theory Tech. 54(8), 3197–3208 (2006). [CrossRef]
3. P. D. Hale, A. Dienstfrey, J. C. M. Wang, D. F. Williams, A. Lewandowski, D. A. Keenan, and T. S. Clement, “Traceable waveform calibration with a covariance-based uncertainty analysis,” IEEE Trans. Instrum. Meas. 58(10), 3554–3568 (2009). [CrossRef]
4. P. D. Hale, C. M. Wang, R. Park, and W. Y. Lau, “A transfer standard for measuring photoreceiver frequency response,” J. Lightwave Technol. 14(11), 2457–2466 (1996). [CrossRef]
5. P. D. Hale and C. M. Wang, “Calibration service of optoelectronic frequency response at 1319 nm for combined photodiode/RF power sensor transfer standards,” NIST Special Publication 250–51 (1999).
6. J. E. Bowers and C. A. Burrus, “Ultrawide-band long-wavelength p-i-n photodetectors,” J. Lightwave Technol. 5(10), 1339–1350 (1987). [CrossRef]
7. R. T. Hawkins, M. D. Jones, S. H. Pepper, J. H. Goll, and M. K. Ravel, “Vector characterization of photodetectors, photoreceivers, and optical pulse sources by time-domain pulse response measurements,” IEEE Trans. Instrum. Meas. 41(4), 467–475 (1992). [CrossRef]
8. D. F. Williams, P. D. Hale, T. S. Clement, and J. M. Morgan, “Calibrating electro-optic sampling systems,” in IEEE MTT-S Int. Microw. Symp. Dig., vol. 3 (2001), pp. 1527–1530.
9. D. F. Williams, A. Lewandowski, T. S. Clement, J. C. M. Wang, P. D. Hale, J. M. Morgan, D. A. Keenan, and A. Dienstfrey, “Covariance-based uncertainty analysis of the NIST electrooptic sampling system,” IEEE Trans. Microw. Theory Tech. 54(1), 481–491 (2006). [CrossRef]
10. F. Z. Xie, D. Kuhl, E. H. Böttcher, S. Y. Ren, and D. Bimberg, “Wide-band frequency response measurements of photodetectors using low-level photocurrent noise detection,” J. Appl. Phys. 73(12), 8641–8646 (1993). [CrossRef]
11. D. M. Baney, W. V. Sorin, and S. A. Newton, “High-frequency photodiode characterization using a filtered intensity noise technique,” IEEE Photon. Technol. Lett. 6(10), 1258–1260 (1994). [CrossRef]
13. D. A. Humphreys, M. R. Harper, A. J. A. Smith, and I. M. Smith, “Vector calibration of optical reference receivers using a frequency-domain method,” IEEE Trans. Instrum. Meas. 54(2), 894–897 (2005). [CrossRef]
14. B. H. Zhang, N. H. Zhu, W. Han, J. H. Ke, H. G. Zhang, M. Ren, W. Li, and L. Xie, “Development of swept frequency method for measuring frequency response of photodetectors based on harmonic analysis,” IEEE Photon. Technol. Lett. 21(7), 459–461 (2009). [CrossRef]
15. J. Wang, U. Krüger, B. Schwarz, and K. Petermann, “Measurement of frequency response of photoreceivers using self-homodyne method,” Electron. Lett. 25(11), 722–723 (1989). [CrossRef]
16. N. H. Zhu, J. M. Wen, H. S. San, H. P. Huang, L. J. Zhao, and W. Wang, “Improved optical heterodyne methods for measuring frequency response of photodetectors,” IEEE J. Quantum Electron. 42(3), 241–248 (2006). [CrossRef]
17. L. Piccari and P. Spano, “New method for measuring ultrawide frequency response of optical detectors,” Electron. Lett. 18(3), 116–118 (1982). [CrossRef]
18. S. Kawanishi and M. Saruwatari, “A very wide-band frequency response measurement system using optical heterodyne detection,” IEEE Trans. Instrum. Meas. 38(2), 569–573 (1989). [CrossRef]
19. O. Ishida, H. Toba, and F. Kano, “Optical sweeper with double-heterodyne frequency-locked loop,” Electron. Lett. 25(22), 1495–1496 (1989). [CrossRef]
20. A. Beling, H.-G. Bach, G. G. Mekonnen, R. Kunkel, and D. Schmidt, “High-speed miniaturized photodiode and parallel-fed traveling-wave photodetectors based on InP,” IEEE J. Sel. Top. Quantum Electron. 13(1), 15–21 (2007). [CrossRef]
21. T. S. Tan, R. L. Jungerman, and S. S. Elliott, “Optical receiver and modulator frequency response measurement with a Nd:YAG ring laser heterodyne technique,” IEEE Trans. Microw. Theory Tech. 37(8), 1217–1222 (1989). [CrossRef]
22. D. A. Humphreys, “Measurement of high-speed photodiodes using DFB heterodyne system with microwave reflectometer,” in High-Speed Electronics and Optoelectronics, Proc. SPIE 1680–15, 138—152 (1992).
23. K. J. Williams, L. Goldberg, R. D. Esman, M. Dagenais, and J. F. Weller, “6-34 GHz offset phase-locking of Nd:YAG 1319 nm nonplanar ring lasers,” Electron. Lett. 25(18), 1242–1243 (1989). [CrossRef]
24. M. Weidman, “Direct comparison transfer of microwave power sensor calibration,” NIST Technical Note 1379 (1996).
25. L. D’Evelyn, L. Hollberg, and Z. B. Popovic, “A CPW phase-locked loop for diode-laser stabilization,” in Microwave Symposium Digest 1994, IEEE MTT-S International (1994), pp. 65–68.
26. B. W. Silverman, Density estimation for statistics and data analysis (Chapman and Hill, London, England, 1986).
27. B. N. Taylor and C. E. Kuyatt, “Guidelines for evaluating and expressing the uncertainty of NIST measurement results,” NIST Technical Note 1297 (1994).
28. P. D. Hale, T. S. Clement, and D. F. Williams, “Frequency response metrology for high-speed optical receivers, in Optical Fiber Communication Conference and Exhibit, OFC Technical Digest Series (Optical Society of America, 2001), paper WQ1.