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A stable optical frequency comb with 20-GHz spacing is shaped by a compact integrated silica arrayed waveguide grating (AWG) pair to produce optical waveforms with unprecedented fidelity. Complete characterization of both the intensity and phase of the crafted optical fields is accomplished with cross-correlation frequency resolved optical gating (XFROG) which has been optimized for periodic waveforms with resolvable modes. A new method is proposed to quantify, in a single number, the quality of the match in both the amplitude and phase between the measured optical waveform and the target waveform.
©2007 Optical Society of America
1. Introduction
Since its genesis over twenty years ago [1], optical pulse shaping on the sub-picosecond scale has evolved significantly. It began with the manipulation of groups of spectral lines [1–6] and it is now approaching a new level where there is complete control of the intensity and phase of the individual spectral modes that compose a periodic pulse train. This Fourier synthesis of optical waveforms via the manipulation of individual spectral modes is often referred to as line-by-line pulse shaping or optical arbitrary waveform generation (OAWG) [7, 8]. Enabling technologies such as high-stability optical frequency combs with large mode spacing and highresolution pulse shapers (both integrated [7, 9] and bulk-optic [10, 11]) have paved the way for OAWG. Ultimately, the crafted waveforms’ complexity is limited only by the number of optical lines available for shaping. Several groups, including ours, have recently presented line-by-line pulse shaping results that demonstrate rudimentary control of the amplitude and phase of the individual spectral components that make up the waveform [12–16]. So far, the accuracy of the shaped pulse is typically determined by studying its autocorrelation or cross-correlation. However, the autocorrelation measurements suffer from significant ambiguity [17] and crosscorrelations do not reveal any information concerning the temporal phase of the waveform. Techniques for measuring the relative phase differences via optical beating between adjacent lines have been used [18], but it can be difficult to implement the narrow zero-dispersion filters
required and time consuming to evaluate a large number of lines. Finally, very little effort has been made to explore the limits of waveform crafting accuracy.
As OAWG techniques and technologies mature and the ability to craft complicated optical waveforms evolves, many applications will demand high-fidelity waveforms [19]. Therefore, complete, accurate characterization of the optical waveforms will become necessary. Frequency resolved optical gating (FROG), and its variants, have become commonplace in ultrafast optics laboratories as a diagnostic tool [20]. FROG has been used to measure the intensity and phase of optical waveforms from the femtosecond regime to the nanosecond regime, single shot or periodic waveforms, and over an extremely large range of pulse energies. In this paper, we show how FROG measurement techniques can be adapted to measure optical waveforms when the individual spectral lines are to be resolved. Additionally, we suggest a means to quantify the quality of match between two complex waveforms which is based on the rms difference between the spectrograms of each pulse normalized to the pulse’s total energy. This quantity, called G′, simultaneously measures the match between two optical waveforms for both the real and imaginary parts (alternatively, intensity and phase). We demonstrate the sensitivity of G′ to show even small mismatches between waveforms. Finally, we show examples of the highestfidelity waveforms reported to date, each with a G′ well below 5%. In terms of the spectral domain accuracy, the spectral intensities of each optical line are within ±0.25 dB of their target values for lines greater than -10 dBc in amplitude and the spectral phases are within ±0.02 rad of their target values for lines greater than -20 dBc in amplitude, an order of magnitude more accurate than any OAWG result previously reported. required and time consuming to evaluate a large number of lines. Finally, very little effort has been made to explore the limits of waveform crafting accuracy. As OAWG techniques and technologies mature and the ability to craft complicated optical waveforms evolves, many applications will demand high-fidelity waveforms [19]. Therefore, complete, accurate characterization of the optical waveforms will become necessary. Frequency resolved optical gating (FROG), and its variants, have become commonplace in ultrafast optics laboratories as a diagnostic tool [20]. FROG has been used to measure the intensity and phase of optical waveforms from the femtosecond regime to the nanosecond regime, single shot or periodic waveforms, and over an extremely large range of pulse energies. In this paper, we show how FROG measurement techniques can be adapted to measure optical waveforms when the individual spectral lines are to be resolved. Additionally, we suggest a means to quantify the quality of match between two complex waveforms which is based on the rms difference between the spectrograms of each pulse normalized to the pulse’s total energy. This quantity, called G′, simultaneously measures the match between two optical waveforms for both the real and imaginary parts (alternatively, intensity and phase). We demonstrate the sensitivity of G′ to show even small mismatches between waveforms. Finally, we show examples of the highestfidelity waveforms reported to date, each with a G′ well below 5%. In terms of the spectral domain accuracy, the spectral intensities of each optical line are within ±0.25 dB of their target values for lines greater than -10 dBc in amplitude and the spectral phases are within ±0.02 rad of their target values for lines greater than -20 dBc in amplitude, an order of magnitude more accurate than any OAWG result previously reported.
2. FROG characterization of optical waveforms with resolvable lines
The various forms of FROG allow for retrieval of the intensity and phase of optical waveforms and, through the Fourier transform, the ability to display this information in the spectral or temporal domain. This paper specifically addresses issues as they relate to second harmonic generation (SHG) FROG and cross-correlation FROG (XFROG), but the concepts are easily transferred to other types of FROG. If E(t) is the optical waveform in the time domain, then its spectrally-resolved autocorrelation, or FROG trace, is defined as
where τ is the relative delay between waveforms [20, 21]. Making FROG measurements of optical waveforms where the individual lines will be resolved in the retrieved fields (i.e., lineby- line characterization) requires a few modifications to the FROG algorithm and the binning process. Although it isn’t necessarily obvious, it is not mandatory to fully resolve the spectral lines within the measured FROG trace. This is due to the oversampling inherent in the FROG trace and how the optical waveform’s information is stored in both the temporal and spectral domains. Generally, the FROG algorithm processes an N ×N binned FROG trace where the frequency step size of one axis is the inverse of the total time window on the other axis and N only increments by powers of two. Line-by-line characterization (i.e., mode resolved FROG) also requires that the frequency step size be equal to, or a sub-harmonic of, the mode spacing of the waveform to be retrieved. Thus, restricting the temporal window size to an integer number times the inverse of the waveform repetition rate. Although it is possible to have the frequency step size be equal to sub-harmonics of the mode spacing (i.e., longer time window), it is generally not worth the processing effort since it requires a larger N and the bins between modes do not have any energy for stable, unmodulated pulse trains.
Fig. 1. Comparison of two different FROG traces. (a) Sample waveform used to create the traces. (b) SHG FROG trace of (a). (c) XFROG trace of (a) using a 5-ps transform-limited gate pulse. Complexity of XFROG traces can be minimized by the proper choice of the gate pulse.
When binning a measured FROG trace, the frequency bins must be centered on the spectral modes of the waveform. Otherwise, the spectral lines are incorrectly retrieved (easily confirmed by independent spectral power measurements). Although not necessary, spectrally resolving the modes within the measured FROG trace can make it easier to center the optical modes within the frequency bins during the binning process.
The FROG algorithm also implements boundary conditions to include the energy at the edges of the FROG trace’s temporal window. We do not enforce the standard requirement that the normalized FROG trace must be below ~10-4 at the edges of the temporal window [21], but this requirement still applies to the frequency window. Essentially, the boundary condition presumes that the waveform is periodic and that the trace is repeated temporally.
The resolution of the FROG spectrometer does not need to be significantly less than the mode spacing. This is especially true for XFROG measurements, where a simple, well behaved gate pulse can be used to keep the features of the XFROG trace from becoming complicated by the features of the gate pulse. For example, we use a relatively long transform-limited pulse for the gate in our measurements. In contrast, SHG FROG uses the waveform itself for the gate pulse and as the waveform becomes more and more complicated, the SHG FROG trace will have exceedingly more complicated features that must be resolved. Figure 1 shows two example FROG traces of the same optical waveform, Fig. 1(a). The SHG FROG trace [Fig. 1(b)] has greater complexity and therefore resolution requirements, while the XFROG trace [Fig. 1(c)] is much simpler and has lower resolution demands. The gate pulse used in Fig. 1(c) is a 5-ps transform limited sech2 pulse. Generally, the choice of gate pulse should try to minimize the complexity of the trace. Note the FROG traces in Fig. 1 have expanded frequency axes to more easily show the details and the traces are plotted on a logarithmic intensity scale to enhance the features. The algorithmic method used for waveform retrieval from the FROG traces was generalized projections (GP), a method discussed at length in [20–22]. Although binning to a 32 × 32 array would have been sufficient, all experimental FROG traces in this paper were binned to a 128 × 128 trace, improving the immunity to noise. The FROG error, G, was below 0.0026 for all retrieved fields.
3. Quantifying the quality of match
The measure of quality of optical waveform crafting has generally been a fit to an autocorrelation or cross-correlation trace. However, as will be shown in a later section, this can easily result in overly optimistic assessments of how well the measured and target waveforms truly match. Ideally, a single value would describe the quality of match between waveforms. It would repre- sent mismatches in the amplitude and phase of the waveform whether the errors are described in the spectral or temporal domains. Therefore, it makes sense to look at time-frequency distributions (i.e., chronocyclic distributions) of an optical waveform which can contain complete information about the waveform. There are an infinite number of possible time-frequency distributions [23], but some have become quite common in the characterization of optical pulses. Specifically, the SHG FROG trace [Eq. (1)] is widely known and we investigate its use to quantify the quality of match below.
The quality of measurement retrieval in FROG is quantified by the “FROG error”, G. This is defined as the rms difference between the normalized measured trace I FROG(ωi,τj) and the trace I(k) FROG(ωi,τj) calculated from the retrieved field E(k)(ti),
where k is the latest iteration number and α is a scaling constant that is chosen to minimize G [20]. To highlight G’s normalization, the peak value in the FROG trace I FROGPeak is factored out of Eq. (2) and shown explicitly in Eq. (3), where I Raw FROG is the uncorrected or raw FROG trace. In addition to being normalized to the peak of the trace, G is also dependent on the binned trace size. This definition works well for minimizing error within the FROG retrieval algorithm, but it has limitations for general application due to its normalization (e.g., G-numbers that have been binned differently cannot be directly compared) especially when comparing the quality of FROG traces of optical pulses with very different temporal or spectral widths.
We propose a newly modified version of the FROG error value which is still based on the well known SHG FROG trace, but instead of normalizing the error to the binned trace size (N 2) the integrated error energy is normalized to the total FROG trace energy which, in turn, is related to the waveform’s total energy. This way, the normalized ratio of the integrated error energy is quantified. Since complete information about the waveform’s intensity and phase (alternatively its real and imaginary parts) are contained within the FROG trace, we can use it to quantify the match between two waveforms in a single number, called G′. Thus, the rms difference between a target waveform and a measured waveform is given by,
where the respective I FROG(ω,τ) has been calculated from the target and measured waveforms according to Eq. (1). In practice, the FROG traces are not continuous functions, but are discrete since they have been binned to a particular size and therefore the integrals in Eq. (4) become summations over the binned traces.
Using the SHG FROG trace as a basis for quantifying the match is a good choice for several reasons. First, it does not require external signals or quantities other than the optical waveform itself for the calculation. Second, and this could be an advantage or disadvantage depending on the application, the SHG FROG trace is time-shift invariant. Put differently, it will be unaffected by a linear spectral phase slope. Thus, the trace is always temporally centered; leaving one less parameter to be adjusted when comparingwaveforms. Finally, if SHG FROG is used to measure the crafted waveform, the measured FROG trace can be used directly in the calculation of G′. Otherwise, measurement of the crafted waveform can be made with any technique that yields complete intensity and phase information. Above all, the most effective technique should be used, one that yields the highest fidelity measurement of the waveform and then the SHG FROG trace can be computed from the measured intensity and phase of the waveform. The waveforms shown later in this paper were all measured using XFROG and then the SHG FROG trace was calculated from the retrieved fields. There are a couple of disadvantages to using SHG FROG traces, one is time ambiguity, and the other is phase ambiguity (φ ↔ φ +π) for widely separated pulses. Neither of these are a serious problem when determining the quality of match, and many techniques exist to remove such ambiguities [20].
Fig. 2. Comparison of the target and measured (simulated) Gaussian pulse with cubic spectral phase. (a) Target spectral intensity (black dashes) and spectral phase (solid red line) shown with a simulated measurement of the spectral intensity (bars) and phase (circles). (b) Target temporal intensity (black dotted) and phase (red dotted) with simulated measured intensity (solid blue) and phase (solid red). (c) SHG FROG trace of target waveform. (d) SHG FROG trace of simulated measured waveform. (e) Absolute difference between (c) and (d). Calculated G′=0.124 from this example.
As an example of how G′ reflects the quality of match for a particular optical waveform, Fig. 2 shows simulated target and measured waveforms. Figure 2(a) shows the target waveform with a Gaussian spectral intensity (black dashes) and purely cubic spectral phase (solid red) with the simulated measured spectral intensity (bars) and phase (circles). Here, the “measured” waveform is the target with some spectral intensity noise and spectral phase noise included. Figure 2(b) shows the time domain match between target and “measured” and is simply the Fourier transform of Fig. 2(a). The corresponding SHG FROG traces are shown in Figs. 2(c,d) and Fig. 2(e) is the absolute difference between the two traces. The temporal intensity traces in Fig. 2(b) show relatively good agreement between the measured and target. However as evidenced by their respective SHG FROG traces, there are significant differences in their temporal phases. This mismatch is easily apparent in the G′ for this target-measured pair which is 0.124, or 12.4%. As will be shown by examples of real measurements later in this paper, G′ values below 5% indicate a very close match between the measured and target waveforms.
4. Optical comb source and optical arbitrary waveform generation
When crafting high-fidelity optical waveforms it is critical to start with with a stable optical frequency comb; any amplitude, frequency or phase shifts within the comb will be propagated to the final waveform. Even though the repetition rate is locked to a stable microwave source, the frequency comb produced by many modelocked lasers, especially harmonically modelocked fiber lasers, is not suitable because the optical frequency is not actively locked and can wander [24]. An alternative to the modelocked laser is based on a stable, narrow-linewidth singlefrequency laser which is then heavily modulated to create sidebands around the optical carrier. This technique is often referred to as an optical frequency comb generator (OFCG) [25]. Of course the sideband creation can occur via either amplitude modulation (AM) or phase modulation (PM). However, AM alone is very lossy and pure PM produces an uneven frequency comb since sideband amplitudes are determined by the Bessel function. Several years ago, it was shown that an AM and a PM modulator in series could create a combination of AM and PM that resulted in a flatten frequency comb [26]. More recently Sakamoto, et al. demonstrated that the combination of amplitude and phase modulation necessary for a flattened frequency comb could be created using a single modulator [27]. The optical comb source used to craft waveforms shown in this paper is based on a single dual-electrode Mach-Zehnder modulator (DEMZM) as shown in Fig. 3. The DEMZM is simultaneously capable of both amplitude and phase modulation depending on the relative optical phase difference between the two arms of the interferometer (i.e., bias) and the relative amplitude and phase differences of the driving RF fields. Since the DEMZM is a broadband traveling-wave modulator (40 GHz bandwidth) it is inherently repetition-rate (comb spacing) tunable [28]. A typical output spectral intensity and phase, as retrieved by SHG FROG, is shown in the inset of Fig. 3. There are nine comb lines above 1% (-20 dBc) of the peak sideband amplitude. Note, the spectral phase is sinusoidal in shape, indicating a nonlinear frequency chirp across the pulse which is intrinsic to comb generators based on large phase modulation of CW laser light. The average powers at RF1 and RF2 are approximately 0.50 W and 0.32 W, respectively and the bias has been adjusted to approximately π/2 rad. The optical loss through the DEMZM is typically 9 dB in this configuration and the output power is -1 dBm when using an Agilent 81682B as the tunable single-frequency laser source.
Fig. 3. Experimental arrangement for the production and characterization of optical arbitrary waveforms. The 20-GHz optical frequency comb is generated via a dual-electrode Mach-Zehnder modulator (DEMZM). A silica AWG pair sets the amplitude and phase of each optical mode and XFROG is used to retrieve the amplitude and phase of the shaped waveform. The inset plot shows the retrieved spectral intensity and phase of the waveform at the output of the DEMZM.
Fig. 4. (a) Spectral transmission of the 20-GHz pulse shaper showing one full FSR (10.23 nm). (b) Close-in view of the center portion used for this experiment. (c) The relative phase response to heater power for three typical channels.
The output of the OFCG is then amplified by an erbium-doped fiber amplifier (EDFA) before entering the pulse shaper, sometimes referred to as an encoder. The pulse shaper is a 64-channel silica AWG pair with 20.03-GHz channel spacing and a free spectral range (FSR) of 10.23 nm. The first AWG acts as a wavelength demultiplexer, separating the wavelengths and putting them onto their own channel. The second AWG acts as a multiplexer and combines the 64 wavelength channels into one output. The pulse shaper is completely integrated and mounted on a temperature controlled substrate and the center wavelength of the transmission spectrum can be adjusted by tuning the temperature. Figure 4(a) shows the spectral transmission of the pulse shaper with a magnified view of the center wavelengths shown in Fig. 4(b). Each spectral channel of the pulse shaper is equipped with both an amplitude and a phase modulator placed in series. The amplitude modulator utilizes a Mach-Zehnder structure with a thermo-optic phase shifter on one arm resulting in a nominal 0–100% transmission change for a zero-to-π induced phase shift. Phase shifts are achieved through individual resistive heaters which alter the local temperature of the waveguides to change their index of refraction. Typical calibration curves for the phase modulators are shown in Fig. 4(c) with approximately 1.8π of phase shift available for 500 mW of electrical power to the heater. For this experiment, the electrical power delivered to the heaters is manually controlled via multi-turn potentiometers connected to a DC voltage source. However, in future work we expect to utilize commercially available computer controlled digital-to-analog converters (DAC) which will significantly increase the accuracy and update rate.
The optical waveforms from the pulse shaper are characterized using XFROG with a gate pulse produced by a harmonically modelocked fiber laser (PriTel, UOC-3) as shown in Fig. 3. To synchronize the gate pulses to the optical waveforms, the laser’s 10.015-GHz repetition rate is locked to exactly one-half the microwave synthesizer’s frequency (20.03 GHz). The XFROG apparatus consists of a fiber coupled, spectrally resolved cross-correlator which employs a stepper-motor-controlled variable optical delay. The sum frequency generation (SFG) crystal is a 2-mm thick piece of LiNbO3. The FROG spectrometer has a spectral resolution of ~15 GHz at 775 nm and uses a low-noise, high efficiency, cooled (-10 °C) charge-coupled device (CCD) for detection. A typical XFROG measurement scans a 50-ps delay and takes ~60 s to acquire the FROG trace. Retrieval of the waveform intensity and phase from the measured XFROG trace using the GP algorithm typically takes 1–2 s when using a previously characterized gate pulse. By using XFROG and a relatively high power gate pulse at a lower repetition rate, we are able to increase the sensitivity of the FROG measurements (signal levels were typically -3 to 0 dBm) and lower the relative noise of the FROG traces. Although a fiber laser was used for the gate pulse in the measurements, the use of an amplified and compressed sample of the comb source output would be a more ideal solution for the gate pulse.
5. Crafting optical arbitrary waveforms
Depending on the application, a specific optical waveform is defined in the time or frequency domain and is referred to as the target waveform. Then, if necessary, the target waveform is Fourier transformed into the frequency domain where the spectral intensity and phase is determined. Based on this, and a measurement of the output waveform of the un-driven pulse shaper (control inputs set to zero), the amplitude and phase controls are adjusted to set the output spectral intensity and phase to that of the target waveform. Several iterations are generally required due to crosstalk among adjacent channels and crosstalk between amplitude and phase adjustments within a channel. One of the first test waveforms investigated is a transform-limited pulse (i.e., the spectral phase is set to zero) with a simple spectral envelope. The retrieved target waveform is shown in Fig. 5. The target values for the spectral intensity and phase are shown in Fig. 5(a) by black dashes and a solid red line, respectively. Note, the phase of the field becomes undefined when the amplitude approaches zero and this is particularly clear in the time domain plots. The quality of match between the target and measured waveforms, G′, is 1.49%.
The second target waveform is a pulse with the same spectral envelope as in Fig. 5(a), but a purely quadratic spectral phase (i.e., linear frequency chirp) is applied. The retrieved waveform is shown in Fig. 6. The quality of match between the target and measured waveforms, G′, is 1.55%. For both waveforms (Figs. 5, 6), the spectral phase is within ±0.02 rad of the target phase and the spectral amplitude is within ±0.25 dB of the target values for lines that are greater than 10% of peak.
Another interesting test waveform is the zero-π pulse. Figure 7 shows the retrieved waveforms and corresponding XFROG traces for two different versions of a zero-π pulse. In the first example, Fig. 7(a–c), the spectral phase transition occurs just below the center frequency.
Fig. 5. The retrieved intensity and phase in both the spectral and temporal domains for the measured transform-limited optical waveform. (a) Target spectral intensity values are shown as black dashes and the target spectral phase is shown as a solid red line (note expanded phase axis). (b) Measured time domain data (solid lines) and target time domain data (dotted lines). Calculated G′=0.0149.
Fig. 6. The retrieved intensity and phase in both the spectral and temporal domains for the linearly chirped optical waveform. (a) Measured spectral intensity and phase data (bars and circles, respectively) and target spectral intensity (black dashes) and spectral phase (solid line). (b) Measured time domain data (solid lines) and target time domain data (dotted lines). Calculated G′=0.0155.
Fig. 7. The retrieved intensity and phase in both the spectral and temporal domains for two different zero-π pulses (phase shift above or below center frequency) and their corresponding XFROG traces. (a) Measured spectral data (bars and circles) and target spectral intensity (black dashes) and spectral phase (solid line). (b) Measured temporal waveform (solid lines) and target waveform (dotted lines). (a–c) Calculated G′=0.0301, (d–e) calculated G′=0.0209. XFROG traces have the same scaling as Fig. 1.
This π phase shift in the spectral domain causes a dip to appear in the time domain. The quality of match between the target and measured waveforms, G′, is 3.01%. In the second example, the phase transition occurs just above the center frequency and G′ is 2.09%. The temporal intensities of the two different waveforms appear the same, but their respective phases in time and frequency are different. In fact, the only way to distinguish these waveforms apart is to use a phase sensitive measurement technique like FROG.
A final example of how important phase sensitive measurement techniques are when crafting optical waveforms is shown in Fig. 8. In this simulated waveform, significant spectral phase errors have been added to the zero-π pulse of Fig. 7(d) and they are shown as bars and circles in Fig. 8(a) while the original target spectral intensity and phase are shown as black dashes and a solid red line, respectively. Although a cross-correlation would show some of the intensity distortions, as evidenced by the time domain intensity of Fig. 8(b), the autocorrelation [Fig. 8(c)] of the distorted waveform is very similar to an autocorrelation of the target pulse and shows how inadequate autocorrelation is as a diagnostic when trying to craft waveforms. However, as we have shown earlier, typically a G′ which exceeds 5% signals a poor match as evidenced by the calculated G′ of 18.45% for the waveform in Fig. 8.
Fig. 8. Simulation showing the limitations of using autocorrelations as a diagnostic. (a) Significant spectral phase errors are added to the zero-π pulse of Fig. 7(d). (b) Distortions appear in the temporal intensity and phase (simulation = solid lines; target waveform = dotted lines). (c) Autocorrelation of distorted (solid) zero-π pulse is compared to the autocorrelation of the target waveform (dotted). Calculated G′=0.1845
The OAWG waveforms shown in this section are the most accurately crafted waveforms presented to date. Although they are based on just nine modes, we have presented a group of technologies and tools necessary to craft waveforms of nearly unlimited complexity and fidelity. The ultimate limit is determined by the capabilities of the three constituent elements that make up a high-fidelity OAWGsystem; the optical frequency comb source, the characterization technique, and the pulse shaper. Wide-bandwidth, stable optical frequency combs have been demonstrated that utilize specialty fiber in combination with an OFCG [28] and it may soon be possible to utilize octave spanning OFCG’s based on high-repetition rate, self-referenced Kerrlens modelocked Ti:sapphire lasers [29]. Characterization techniques based on FROG are able to measure extremely complicated, octave-spanning waveforms [20]. Currently, it is the lineby- line pulse shaping technology that is the determining element and as pulse shapers based on integrated AWG-pairs or bulk-optic elements evolve, individual control of hundreds of modes should be feasible.
6. Conclusion
We have demonstrated high-fidelity optical waveform crafting using a completely integrated approach. By utilizing a very stable OFCG and an integrated silica AWG pair we show that it is possible to create optical arbitrary waveforms with exceptionally high fidelity (G′ < 5%). Critical to the creation of these waveforms is their complete characterization through mode-resolved FROG. Proper binning and satisfaction of boundary conditions are necessary to utilizing FROG on waveforms with resolvable lines, but it is not necessary to fully spectrally resolve the lines when measuring the FROG trace. We have also proposed a new method, and a corresponding parameter, to quantify the quality of match between a target waveform and the measured waveform which simultaneously accounts for variations in the real and imaginary parts of the waveform. This parameter (G′) has been shown to be a very sensitive measure of the match between waveforms and its use could provide transparency in future high-fidelity OAWG work.
Acknowledgement
The authors would like to thank T. Sakamoto of NICT for insightful discussions about the OFCG, and E. Ippen of MIT and J. Lowell of DARPA for their encouragements. This work was supported in part by the Defense Sciences Office of the Defense Advanced Research Projects Agency (DARPA DSO) and SPAWAR under OAWG contract HR0011-05-C-0155.
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