1. Introduction

Progress in optical pulse characterization has closely followed advances in optical pulse generation, and the temporal characterization of optical sources delivering attosecond pulses has now been performed [1]. While the concepts for characterizing pulses with attosecond, femtosecond, and picosecond duration are similar, the implementations differ widely because of the availability of components. Most optical pulse-characterization techniques use a nonlinear interaction to implement the time-nonstationary filter required to measure the temporal electric field [2]. Nonlinear media are available at a wide range of wavelengths, but they can hinder the practicality and decrease the sensitivity of the corresponding diagnostics. For pulses in the femtosecond and picosecond domain, high-bandwidth electrically driven temporal modulators can be used to implement the time-nonstationary filter [3]. Electrical modulation has greatly increased the sensitivity of pulse characterization. For example, 10-GHz sources can be characterized at an average power below 100 nW using linear spectrography [4], and single-shot characterization of 1-nJ pulses has been performed using spectral-shearing interferometry [5]. Electrically driven temporal modulators can also significantly simplify the implementation of the diagnostics, since temporal scanning can be performed in the RF domain, and optical spectrum measurements can be performed with standard test equipment [6]. These developments were triggered by the need for sensitive diagnostics in the optical telecommunication environment and were facilitated by the availability of high-bandwidth temporal modulators. Diagnostics at non-telecommunication wavelengths, such as 1053 nm [5] and 1060 nm [7], have also been reported.

Nonlinear pulse-characterization techniques generally generate ancillary replicas of the pulse under test used to modulate the pulse under test via a nonlinear interaction. The relative jitter between the pulse under test and the modulation is essentially set by the stability of the optical path between elements in the diagnostic, and is usually insignificant compared to the duration of the pulse under test. A train of identical pulses with jitter (which commonly occurs in practice) has, in most cases, an experimental trace that is not impacted by the pulse-to-pulse jitter, i.e., the electric field reconstructed from the experimental trace is the field of each pulse. For linear pulse-characterization techniques based on electrically driven modulators, the time-nonstationary filter is indirectly linked only to the pulse under test. For example, an RF clock is used to generate the pulse under test and to drive the modulator used for its characterization [4,8,9], a drive signal is triggered by the source under test [5,6], or a drive signal is reconstructed by a phase-lock loop [10]. There can be a random delay between the optical pulse under test and the modulation, even when the source under test is a periodic train of pulses. The electric drive signal may also correspond to an averaged period of the train of pulses under test, in which case pulse-to-pulse jitter can impact the measurement. The amplitude of the jitter between the pulse under test and the modulation is set by practical limitations that depend on both the optical source under test and the diagnostic. It is expected to be a fraction of the pulse duration. For example, the train of 2-ps optical pulses from the mode-locked laser diode used in Ref. [15] has a jitter specification of 150 fs, and since the same RF clock is used for pulse generation and temporal modulation, no significant jitter is added in the diagnostic.

It is of practical importance to understand the impact of this jitter on the measured experimental trace and on the electric field reconstructed from it. Three techniques belonging to different classes of pulse-characterization strategies—linear spectrography, spectral shearing interferometry, and simplified chronocyclic tomography—are considered in this article. These techniques can be implemented with an electrically driven modulator, but their interpretation and the role of the modulator are significantly different. In spectrography, the modulation corresponds to a generalized gating in the temporal domain (i.e., it can be amplitude and/or phase modulation). The combination of this modulation with measurements of the optical spectrum maps the spectrogram of the pulse in the time-frequency domain; since the spectrogram is the double convolution of the Wigner function of the pulse with the Wigner function of the gate, the modulator is used as a probe in the time-frequency space. In spectral-shearing interferometry, a spectral shear is generated by linear temporal-phase modulation, and spectral interference gives one slice of the two-frequency correlation function of the pulse. In chronocyclic tomography, a quadratic temporal-phase modulation is used to rotate the Wigner function of the pulse in the chronocyclic space, and projections of the Wigner function are used to reconstruct the electric field of the pulse using tomographic concepts. Details of these techniques and additional references are presented in Ref. [3]. A study of all techniques and their variants is beyond the scope of this work, and it is hoped that this work will trigger additional studies on this important issue. Simulations are presented for picosecond pulses characterized using temporal modulations with bandwidth of the order of 10 GHz. They could be scaled to the characterization of shorter pulses, provided that the modulator’s response is scaled accordingly and no practical limitation hinders the implementation. They could also be extended to nonlinear pulse-characterization techniques where temporal modulation of the pulse under test is not perfectly synchronized to the pulse under test. Derivations and simulations show that these three techniques tolerate jitter as high as several times the pulse duration with little noticeable decrease in performance.

2. Description of simulations

2.1. rms field error

The discrepancy between the electric field of a test pulse and the electric field reconstructed by a pulse-characterization device technique can be quantified using the rms field error [11]. The rms field error between two electric fields E ₁(t) and E ₂(t) with energy equal to 1 $[i.e.,\sqrt{{\int \mid E_1\left(t\right)\mid }^2}=\sqrt{\int {\mid E_2\left(t\right)\mid }^{2}dt}=1]$ is defined as

where τ and φ are chosen to minimize the difference between the two fields. This minimization is consistent with the fact that in most cases, pulses with temporal electric fields that differ only by a delay and a constant phase are considered identical. It is also consistent with the practical limitation of most pulse-characterization techniques, which cannot reconstruct the delay and phase relative to a reference. This minimization can be performed efficiently using an algebraic non-iterative simple algorithm [11]. Advantages of such an error metrics are that it can be calculated identically in the time and frequency domain; it naturally weights the amplitude and phase fluctuation in these two domains; and it can be interpreted easily. For example, two fields with identical constant amplitude over their temporal support and a phase difference having a standard deviation of 0.1 rad have an rms field error equal to 0.1. Similarly, two fields differing by an amplitude with a standard deviation equal to 0.1 have an rms field error equal to 0.1.

2.2. Pulses under test

A set of three pulses under test was considered for illustration purposes

The respective second- and third-order dispersion were arbitrarily chosen to yield a group delay of 2.5 ps at a frequency offset from the center frequency by half the FWHM in the frequency domain. The spectrum and spectral phase of these pulses are shown in Fig. 1(a) and the corresponding temporal intensities in Fig. 1(b).

 

Fig. 1. (a) Spectral intensity (black line) and phase of the Fourier transform (FT)-limited test pulse (blue line), pulse with second-order dispersion (green line), and pulse with third-order dispersion (red line). (b) Temporal intensity of the FT-limited pulse (blue line), pulse with second-order dispersion (green line), and pulse with third-order dispersion (red line).

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3. Linear spectrography

3.1. Linear spectrography in the presence of jitter

Linear spectrography is based on the measurement of a spectrogram of the pulse under test using a temporal modulator [4,12]. This spectrogram calculated from the electric field of the pulse E(t) and transfer function of the modulator R(t) is the two-dimensional trace

where ω is the optical frequency and τ is the relative delay between the pulse under test and the modulation. This experimental trace can be inverted using the Principal Component Generalized Projection Algorithm (PCGPA) [13]. In most applications of linear spectrograms, an estimate of the modulator transfer function (which is essentially independent of the pulse under test) is available, and this information can be used during the inversion.

The three pulses under test described in the previous section have been considered. Two different real gates, a Gaussian gate with a 10-ps FWHM and a Gaussian gate with a 30-ps FWHM, were considered. The sampling of the spectrogram was done on a 128×128 grid, with a sampling rate of 10 GHz in the spectral domain, and with the associated Nyquist rate in the temporal domain. The jitter was specified as a random Gaussian variable with zero mean and given standard deviation σ. For each value of the jitter standard deviation, 10,000 spectrograms were calculated according to Eq. (2), where each spectrum of Eq. (2) is calculated for the delay τ+δτ, δτ being a randomly drawn variable corresponding to a Gaussian distribution with the prescribed standard deviation. This procedure corresponds to an experimental setup where optical spectra are measured with an array detector. It was checked that the average of the spectrograms calculated with a random draw of δτ at each delay τ and each frequency ω is identical to the average of the spectrograms calculated with a random draw of δτ at each delay τ and spectrum. The averaged spectrogram S _simulated (ω,τ) was inverted using the PCPGA. The rms error between the reconstructed field and the input field was calculated. The “FROG error” [14], i.e., the quantity

where S _calculated is the spectrogram calculated from the retrieved pulse and retrieved gate and N=128, was calculated to test the consistency of the retrieval. Figure 2 displays the spectrograms of the three test pulses calculated with the 10-ps Gaussian gate without jitter and with a jitter of standard deviation equal to 5 ps. As expected, the jitter significantly blurs the spectrogram along the temporal axis.

 

Fig. 2. Simulated spectrograms for a 10-ps gate and the Fourier transform–limited pulse, the pulse with second-order dispersion, and the pulse with third-order dispersion (from left to right) without jitter (first line) and with jitter with a standard deviation of 5 ps (second line).

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3.2. Influence of jitter on short pulse characterization using linear spectrography

Figure 3 presents the simulation results corresponding to linear spectrography in the presence of jitter. For the 10-ps gate (first line), the rms field error on the retrieved electric field is lower than 0.1 for jitter standard deviations lower than 6 ps for the three test pulses [Fig. 3(a)]. The consistency of the retrieval is very good, and ε_S is smaller than 10⁻³ for jitters smaller than 6 ps [Fig. 3(b)]. Figure 3(c) displays ε_E versus ε_S. There is a correlation between these two quantities, which demonstrates in this particular case a correlation between the consistency of the measurement and the accuracy of the measurement. One should note that such correlation is mostly fortuitous, as an iterative algorithm can generally converge to a solution that can be highly consistent, but completely inaccurate (for example, in the case of ambiguities, such as the time-direction ambiguity in SHG-FROG). The influence of the jitter is even smaller for the 30-ps gate: the field error is smaller than 0.1 for all jitters with standard deviation smaller than 10 ps [Fig. 3(d)], and the trace error is smaller than 10⁻³ in the same range. For jitters smaller than 6 ps, ε_E and ε_S are, respectively, smaller than 0.01 and 10⁻⁴, which indicates very good retrieval and consistency. In this case, the correlation between accuracy and consistency is not as obvious [Fig. 3(f)], but this is not critical since both quantities indicate very good retrieval. These simulations indicate that jitters with standard deviation several times the pulse duration can be tolerated in linear spectrography. The influence of jitter is smaller as the gate duration increases. Gates with bandwidth much smaller than the bandwidth of the pulse under test can make the retrieval more difficult because the spectrogram is dominated by the spectrum of the pulse under test. However, one should note that a 900-fs pulse has been characterized with a 30-ps gate [15]).

 

Fig. 3. Simulation results for accuracy and consistency of linear spectrography in the presence of jitter. (a)–(c) correspond to the field error ε_E versus rms jitter σ, the trace error ε_S versus rms jitter σ, and the field error ε_E versus trace error ε_S, for a 10-ps gate and the three test pulses (FT-limited pulse, pulse with second-order dispersion, and pulse with third-order dispersion; blue, green, and red lines, respectively). (d)–(f) correspond to the field error ε_E versus rms jitter σ, the trace error ε_S versus rms jitter σ, and the field error ε_E versus trace error ε_S, for a 30-ps gate and the three test pulses (FT-limited pulse, pulse with second-order dispersion, and pulse with third-order dispersion; blue, green, and red lines, respectively).

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4. Spectral-shearing interferometry

4.1. Principle of spectral-shearing interferometry

Spectral-shearing interferometry reconstructs the spectral phase of an optical pulse using interference between two optical pulses, with at least one of them being spectrally sheared [16,17]. For a relative delay τ between two pulses with respective shears Ω₁ and Ω₂, the optical spectrum is

The interferometric component Ẽ(ω−Ω₁)Ẽ*(ω−Ω₂)exp(−iωτ) can be extracted by Fourier processing of the interferogram, and its phase is concatenated after removal of the linear component ωτ due to the delay to obtain the spectral phase of the pulse under test. In the linear implementation of spectral-shearing interferometry, the spectral shear is implemented by linear temporal-phase modulation, and a temporal phase Ωt is equivalent to a spectral shear Ω [5,8]. For a relative delay δt between one replica of the pulse under test and the temporal modulation, the field of the modulated pulse in the time domain is

For relative jitters δt ₁ and δt ₂, the interferogram is

The corresponding interferometric component is

which can be decomposed for jitters that are small compared to the inverse of Ω₁ and Ω₂ as

When the interferogram is averaged with jitters of zero-mean, one has

which shows that jitters between the optical pulses and the temporal modulations decrease only the relative amplitude of the interferometric component and, equivalently, reduce the contrast of the fringes. Since the phase of the pulse under test is encoded on the phase of the interferometric component, this amplitude decrease does not introduce any error on the recovered phase, but reduces only the signal-to-noise ratio of the measurement. The relative decrease 〈(Ω₁ δt ₁−Ω₂ δt ₂)²〉/2 depends on the shears Ω₁ and Ω₂, on the variances 〈δt ₁ ²〉, 〈δt ₂ ²〉, and on the correlation 〈δt ₁ δt ₂〉, but it does not depend on the spectral phase of the pulse under test. For example, the decrease is Ω²〈δt ²〉 for two opposite shears Ω₁=−Ω₂=Ω and uncorrelated jitters with 〈δt ₁ ²〉=〈δt ₂ ²〉=〈δt ²〉. The relative decrease is 2Ω²〈δt ²〉 for two opposite shears and identical jitters. For a single modulation corresponding to a shear Ω and jitter 〈δt ²〉, the decrease is only Ω²〈δt ²〉/2. Since the signal-to-noise ratio in spectral-shearing interferometry is essentially proportional to the relative shear as long as the shear satisfies the Nyquist criterion for the representation of the electric field of the pulse under test, the influence of the jitter on the signal-to-noise ratio is identical in the single-modulation implementation and double-modulation implementation with opposite shears and uncorrelated jitters.

4.2. Simulations of the effect of jitter on spectral-shearing interferometry

The simulations shown in Fig. 4 demonstrate the decrease of the amplitude of the interferometric component for jitters with 5-ps standard deviations. This decrease is not significant for shears equal to 2% of the FWHM of the pulse, and even in the case of an identical jitter with standard deviation of 5 ps and two opposite shears equal to 10% of the FWHM of the pulse, the amplitude of the interferometric component is approximately divided by 2. In all the simulated cases, the rms field error ε_E calculated with the extracted electric field was of the order of 10⁻⁵. Jitter between the optical pulse under test and the temporal modulation has no effect for spectral-shearing interferometry for a wide range of jitter values.

 

Fig. 4. Simulated interferograms for jitter with standard deviation of 5 ps. (a)–(c) correspond to one or two shears equal to 2% of the FWHM of the pulse. (d)–(f) correspond to one or two shears equal to 10% of the FWHM of the pulse. The plots correspond from left to right to: a single shear, two opposite shears with uncorrelated jitters, and two opposite shears with identical jitter.

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5. Simplified chronocyclic tomography

5.1 Principle of simplified chronocyclic tomography

Simplified chronocyclic tomography (SCT) is based on the measurement of the optical spectrum of the pulse after a small rotation of its Wigner function in the chronocyclic space [9,18]. The implementation is simple, since such rotation can be performed using a temporalphase modulator inducing a quadratic temporal phase on the pulse under test, i.e., multiplication by exp(iψt ²/2). One should note that temporal- and spectral-quadratic-phase modulations are required for an arbitrary rotation, but the projection of the Wigner after a small rotation in the chronocyclic space can be obtained without spectral-phase modulation. The variation of the corresponding optical spectrum S_ψ when ψ converges to zero is directly linked to the spectral phase of the pulse φ and spectrum S via

This derivative can be obtained as a finite difference by considering the spectrum of one phase-modulated pulse and the spectrum of the pulse under test. It can also be obtained from the spectrum of two phase-modulated pulses with quadratic phase modulation of opposite signs, which in practice has been done. From the measured finite difference, the spectral phase can be reconstructed by integration of Eq. (10).

For a delay δt between the pulse under test and the modulation, one has

which can be simplified into

Equation (12) shows that

When the spectrum for a finite ψ is averaged over a jitter with zero mean, one has

Calculation of the derivative of Eq. (14) shows that

i.e., at first order, the experimental trace of simplified chronocyclic tomography is not modified by averaging over a large number of realizations of the jitter with zero mean. Since the experimental trace is not modified, the recovery of the electric field is accurate, and one expects a low influence of the jitter on simplified chronocyclic tomography.

5.2. Simulation of simplified chronocyclic tomography in the presence of jitter

Figure 5 shows the differential signal S_ψ−S _−ψ for ψ=2×10²¹s⁻² in the absence of jitter and with a jitter having a standard deviation of 5 ps. It can be seen that there is no modification of the experimental trace. The spectral phase error, i.e., the difference between the spectral phase reconstructed by SCT and the spectral phase of the three test pulses is plotted in Fig. 6. The spectral-phase error is less than 0.01 rad over most of the bandwidth of the pulses under test. The corresponding field errors ε_E are 1.4×10⁻⁴, 3.3×10⁻⁴, and 3.9×10⁻³, which demonstrate a very accurate reconstruction.

 

Fig. 5. (a)–(c) Spectrum (black line), differential signal in the absence of jitter (red line), and differential signal with a jitter having 5-ps standard deviation (red markers) for the characterization of the Fourier transform–limited pulse, the pulse with second-order dispersion, and the pulse with third-order dispersion, respectively.

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Fig. 6. Spectrum (black line) and phase error (red line) for the characterization of (a) the Fourier transform–limited pulse, (b) the pulse with second-order dispersion, and (c) the pulse with third-order dispersion with an experimental trace averaged over a 5-ps jitter.

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Figure 7 shows the evolution of the field error as a function of the jitter for the three test pulses. The accuracy of simplified chronocyclic tomography without jitter is intrinsically limited by the amplitude of the temporal phase modulation, which must be kept significantly smaller than one over the temporal support of the pulse under test. As can be seen in this figure, longer pulses (i.e., the pulse with second- and third-order dispersion), and a larger value of the amplitude of the temporal phase modulation correspond to higher values of the field error. The field error increases with the amount of jitter, but the resistance to jitter is very strong. The field error stays below 0.1 even for a 10-ps jitter and the higher-than-specification value of the temporal phase modulation of ψ=10²² s⁻². For an amplitude of the temporal phase modulation of 2×10²¹ s⁻², the field error is, for the most part, smaller than 0.01 even for jitters of 20 ps. From this point of view, simplified chronocyclic tomography is superior to linear spectrography.

 

Fig. 7. Field error ε_E versus standard deviation of the jitter for the FT-limited pulse, the pulse with second-order dispersion, and the pulse with third-order dispersion, (blue, green, and red lines, respectively) for an amplitude of the temporal phase modulation 4×10²⁰ s⁻², 2×10²¹ s⁻², and 10²² s⁻² [(a), (b), and (c), respectively].

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6. Conclusion

The accuracy of three linear pulse-characterization techniques using electrically driven temporal modulators has been studied in the presence of jitter between the pulse under test and the modulation. Linear spectrography, spectral-shearing interferometry, and simplified chronocyclic tomography are highly resistant to jitter, and can tolerate jitters as high as several times the duration of the pulse under test. Jitter modifies only the contrast of the fringes of the interferogram used in spectral-shearing interferometry, which reduces the signal-to-noise ratio but does not introduce an intrinsic error. For simplified chronocyclic tomography, jitter does not modify the experimental trace at first order; this technique is also highly accurate in the presence of jitter. The spectrograms used in linear spectrography are blurred along the temporal axis in the presence of jitter, but simulated retrievals are accurate in a range covering several times the duration of the pulse under test. From the perspective of jitter sensitivity, simplified chronocyclic tomography and spectral-shearing interferometry are advantageous since jitter does not significantly modify the corresponding measured experimental trace. Jitter insensitivity for these three techniques is a significant result from a practical point of view, since jitter is somewhat inherent to any optical source.

This work was supported by the U.S. Department of Energy Office of Inertial Confinement Fusion under Cooperative Agreement No. DE-FC52-08NA28302, the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article.