1. Introduction

A second harmonic autocorrelator is one of the most common pulse diagnostic tools in ultrafast laboratories. An autocorrelator can be aligned to produce interferometric fringe resolved traces [1, 2] which contain information about the phase and amplitude of the pulse. Such fringe resolved traces provide a direct and accurate self-calibration method. In addition, the fringes are also useful in the alignment of the autocorrelator [3]. Further, for few cycle pulses the interferometric autocorrelator (IAC) produces intuitive traces. Although the phase information of the pulse is contained in the IAC traces, this information is not visually evident. The MOSAIC [4] approach was originally introduced to visually display the temporal chirp information with high sensitivity.

Several elegant techniques have been invented to completely characterize ultrashort laser pulses. Some examples include; frequency resolved optical gating (FROG) and numerous variants [5, 6, 7], spectral phase interferometry for direct electrical field reconstruction [8], spectrally and temporally resolved upconversion technique [9], phase and intensity from correlation and spectrum only [10], and time resolved optical gating [11]. For many applications such as oscillator or amplifier alignment, exact knowledge of the ultrashort pulses is not necessary, as long as the pulse width and the chirp are minimized. In such cases, the optical system or pulse retrieval complexity of many of the complete characterization techniques is not required. Compared to complete characterization techniques, MOSAIC is very simple to implement and serves as a real time visualization tool for determining the temporal pulse width and the presence of chirp.

MOSAIC traces are derived from a conventional second order interferometric autocorrelation signal. Interferometric traces contain fringes at frequencies ω ₀ and 2ω ₀, where ω ₀ is the pulse center frequency. MOSAIC traces are obtained by filtering the ω ₀ fringe-band and multiplying the 2ω ₀ band by a factor of two. The shape of the resultant fringe resolved trace depends on the pulse chirp and one can easily distinguish a chirped pulse from a chirp-free pulse by examining their MOSAIC traces [4]. These traces were used to determine the pulse chirp parameters by assuming a shape for the pulse amplitude (for example sech) [4, 12]. More recently, fringe free MOSAIC envelopes were also obtained [13, 14]. A key advantage of this approach is that it allows averaging of several traces leading to very sensitive and noise free determination of MOSAIC envelopes, which otherwise would be difficult for fringe resolved traces due to fringe drifts. Bender et. al. [14] demonstrated a chirp measurement with remarkable sensitivity by producing average MOSAIC traces using IAC traces with signals just above the noise background.

Determination of the chirp by fitting to the MOSAIC traces is always based on an assumption; that is, a particular form of the electric field amplitude is usually assumed. Although very accurate MOSAIC traces can be obtained by averaging, due to the pulse amplitude assumption the accuracy of determined chirp remains unclear. For example, if a set of chirp parameters and an amplitude shape result in a very good fit to the MOSAIC trace then is it the correct pulse shape? Or can other pulse shapes have the same MOSAIC trace? In 1989, Naganuma et. al. [15] proved that magnitude of the pulse spectrum and the corresponding IAC trace are sufficient to uniquely determine the electric field completely. Since the proof only deals with a sufficiency condition, it does not clearly answer the above questions. However, it does suggest that there may be multiple electric fields that yield the same MOSAIC trace. The main focus of this article is to answer these questions. In particular, we describe an algorithm that allows the construction of multiple complex electric field shapes that have nearly identical MOSAIC traces which are not experimentally distinguishable. Our results show that there exist several pulse shapes with the same MOSAIC traces, implying that quantifying chirp using MOSAIC is not unique unless pulse amplitude information is known precisely. Finally we demonstrate an approach to remove this ambiguity in chirp characterization.

2. Chirp ambiguity from MOSAIC traces

2.1. MOSAIC traces

The symmetric second-order interferometric autocorrelation (S_IAC (τ)) between a pulse E(t) and its delayed (by τ) replica can be written as [2]

where

In these equations, we have used the normalization ${\int }_{-\infty }^{+\infty }$ I(t)dt=1. The MOSAIC trace (S_MOSAIC (τ)) is obtained by filtering A ₁(τ) and multiplying A ₂(τ) by a factor of two [4],

A typical IAC and the corresponding MOSAIC trace of a chirped pulse are shown in part (a) and (b) of Fig. 1. The constant DC shift in Fig. 1(b) is not useful and it is typically dropped. A key point is that when the pulse is chirped, it results in a nonzero symmetric lower trace (minima envelope), which is given by [4]

Fringe free MOSAIC envelopes ($S_{\text{MOSIAC}}^{\text{min}}$(τ) and 2A ₀(τ)) are plotted in part (c) and part (d) of Fig. 1.

The minima envelope, $S_{\text{MOSIAC}}^{\text{min}}$(τ), is only present when the pulse is chirped. It is easy to see this as follows; when there is no chirp, the second harmonic field and intensity envelopes are the same (E ²(t)=I(t)) implying that the envelopes A ₀(τ) and A ₂(τ) are identical and hence the minima envelope $S_{\text{MOSIAC}}^{\text{min}}$(τ) is zero. It is this property of the minima envelope that makes it a powerful qualitative visualization tool for distinguishing chirped pulses from chirp-free pulses.

 

Fig. 1. Illustration of an IAC and its corresponding MOSAIC traces. (a) Second order interferometric autocorrelation trace. (b) Fringe resolved MOSAIC trace. (c) and (d) Envelopes of MOSAIC trace on linear and log scales.

Download Full Size | PDF

 

Fig. 2. Gerchberg-Saxton like algorithm for the construction of an electric field E(t) that has a nearly identical MOSAIC trace as the input field E_i (t).

Download Full Size | PDF

2.2. Pulse construction algorithm

In order to construct pulses that have the same MOSAIC traces, we use the following approach. The MOSAIC envelopes are constructed using the intensity correlation, A ₀(τ) and second harmonic correlation A ₂(τ) envelopes. The Fourier transformed counter-parts of these correlations, Ã _0,2(Ω), are real functions [15]; that is

Here ~ represents the Fourier transform with respect to τ. Therefore the MOSAIC traces are entirely determined by the magnitudes of intensity spectrum (|Ĩ(Ω)|) and second harmonic spectrum (|Ẽ ²(Ω)|). In general, it is known [15] that these two spectral magnitudes in addition to the magnitude of field spectrum are sufficient to completely determine the field E(τ) (except for the time direction ambiguity). Therefore, it is likely that multiple field shapes are constrained to have same magnitude of the intensity spectrum and same magnitude of the second harmonic spectrum. Our algorithm is based on finding a field E(τ) given the intensity and second harmonic spectral magnitudes.

We first start with an input pulse shape, E_i (τ), and compute the magnitudes |Ĩ_i (Ω) | and |${\stackrel{\mathit{~}}{E}}_i^2$ (Ω)|. The Gerchberg-Saxton [16] like algorithm shown in Fig. 2 iteratively computes a new field, E(τ), that satisfies the spectral magnitude constraints. This algorithm is very similar to Naganuma’s [15] pulse retrieval method except that the field spectrum magnitude constraint is absent. During any epoch of the algorithm, the field envelope E(τ) is used to compute the intensity, I(τ) and second harmonic envelopes, E ²(τ). These time domain fields are Fourier transformed and their Fourier magnitudes are replaced by |Ĩ_i (Ω)| and |${\stackrel{\mathit{~}}{E}}_i^2$ (Ω)|, while retaining the phase. The resultant fields are Fourier transformed back to the time domain to obtain a new estimate of I(τ) and E ²(τ). The magnitude of second harmonic field is replaced by |I(τ)| while retaining its phase. From this updated E ²(τ), a new estimate for the field envelope E(τ) is obtained which is used in the next iteration of the algorithm. After nearly 100 iterations, the adapted field E(τ) has the same magnitudes for its intensity and second harmonic spectra as the input field E_i (τ). By varying the initial condition, namely the guess field E_g (τ) in the first epoch, the algorithm can be expected to result in different output pulse shapes.

2.3. Reconstruction results

We tried the reconstruction approach on various input shapes and initial fields. During reconstruction, the error between the adapted |Ĩ(Ω)| and the input spectral magnitude |Ĩ_i (Ω) | decreased monotonically, whereas the error for the second harmonic spectral magnitude did not decrease monotonically. After adaptation using this approach, output fields have two trivial ambiguities, namely temporal shifts, and time direction ambiguity (E(τ) vs E*(-τ)). We plot the reconstruction results after choosing a time direction and correcting for the time shift such that the resultant output field is a better comparison to the input field.

Typical reconstruction results from the algorithm are shown in Fig. 3. The input field, E_i (τ), for this simulation is derived by applying a phase, ϕ(ω)=10.4ω ²+50ω ³-0.4ω ⁴(ω is in units of pHz), to the Fourier transform of sech(τ/τ_p ), where τ_p =5 fs. The initial condition for the algorithm is a secant field E_g (τ)=sech(τ/τ_p ). The intensity and phase of the reconstruction E(τ), shown in blue, in Fig. 3(b) are obtained after 150 iterations and they differ considerably from the input. The field spectrum is also quite different from the input field spectrum (Fig. 3(c)). However, the adapted spectra |Ĩ(Ω)|, | Ẽ ²(Ω) | match very closely with the corresponding input spectra as seen from the error plots shown in Fig. 3(d) and (e). It is clear from Fig. 3(a) that output field E(τ) has a MOSAIC trace that closely matches the original to one part in 10³, which is similar to a typical experimental accuracy of an average MOSAIC envelope measurement [14]. In order to see if the reconstructed and the original pulse can be distinguished by other methods, we generated second harmonic (SHG) FROG traces as shown in parts (c) and (d) of Fig. 3. We notice that the SHG FROG traces are sensitive to the differences between the two fields. The reconstructed field does not resemble a pulse that is broadened by the usual linear dispersive propagation like the input pulse. However, both the pulses are chirped and their MOSAIC traces reveal a non-vanishing minima envelope.

 

Fig. 3. Numerical results of the reconstruction algorithm. Red curves and image correspond to the input field, E_i (τ) and blue represents the reconstructed field, E(τ). (a) MOSAIC traces. (b) Intensity and phase of the electric fields. (c) Spectrum of E(τ) and E_i (τ). (d) and (e) Adaptation errors, $\Delta \mid \tilde{I}\left(\Omega \right)\mid =\frac{\Sigma \parallel \tilde{I}\left(\Omega \right)\mid -\mid {\tilde{I}}_i\left(\Omega \right)\parallel }{\Sigma \mid {\tilde{I}}_i\left(\Omega \right)\mid }$ and $\Delta \mid {\tilde{E}}^2\left(\Omega \right)\mid =\frac{\Sigma \parallel {\tilde{E}}^2\left(\Omega \right)\mid -\mid {\tilde{E}}_i^2\left(\Omega \right)\parallel }{\Sigma \mid {\tilde{E}}_i^2\left(\Omega \right)\mid }$ as a function iteration number. (f) and (g) Numerical second harmonic FROG traces for E_i (τ) and E(τ).

Download Full Size | PDF

 

Fig. 4. Reconstruction results corresponding to a secant hyperbolic initial field (red), E_i (τ), with higher order temporal phase similar to Ref. [14]. Green and blue represent reconstructions for two different initial conditions. (a), (b) MOSAIC traces, (c), (d) Temporal intensity and phase, (e), (f) and (g) comparison of SHG FROG traces.

Download Full Size | PDF

A second reconstruction example is illustrated in Fig. 4. In this example the input pulse shape is assumed to be E_i (τ)=sech(τ/τ_p )exp[i(0.18(τ/τ_p )²+0.2(τ/τ_p )³-0.82(τ/τ_p )⁴)], where τ_p =100 fs. This expression is an experimental fit obtained to a MOSAIC trace in [14]. Starting from the MOSAIC traces and the assumption that the pulse amplitude is sech(τ/τ_p ), a fit can be obtained for the chirp. However, we want to see if there are any other pulse shapes that have nearly the same MOSAIC trace. We use the algorithm with an initial condition E_g (τ)=sech(τ/τ^p )exp[πiαϕ_r ], where ϕ_r is a vector of uniformly distributed random numbers between zero and one, and α is a real constant. Two different reconstructions, color coded green and blue, are shown in Fig. 4. The two reconstructions differ in the values of α and the random vector ϕ_r . The values of α for the blue and green reconstructions are 0.1 and 0.8 respectively. Both reconstructions have very similar MOSAIC traces as the input trace and they are nearly identical over four orders of magnitude. Such MOSAIC traces would be indiscernible experimentally.

The green reconstruction shows an asymmetric intensity profile (although input intensity is symmetric) and the phase is considerably different from the input phase. The blue reconstruction is particularly interesting. The intensity profile of the reconstruction only differs from the input intensity (red) by three percent and however the phase is not similar in the central region of the pulse. The phase profile is also very different in the wings due to a difference in higher order chirp. We found that by varying the random vector ϕ_r (and keeping α=0.1), several pulses with different higher order chirp but nearly the same intensity profiles can be generated. This example suggests that there can be significant errors in the chirp measurement even when the pulse amplitude used to fit the MOSAIC is incorrect by only a few percent. We wish to point out that the differences in chirp between the original and the reconstructions are significant compared to the phase accuracy (typically 0.02 rad within the intensity full width half maximum of the pulse) of techniques like FROG.

3. Ambiguity free reconstruction

Here we describe an approach to reconstruct the pulses without the chirp ambiguity mentioned in the previous section. According to Naganuma’s proof [15], the MOSAIC traces and spectrum are sufficient to reconstruct the pulse with only trivial ambiguities. Our approach to remove ambiguities is based on an additional measurement of pulse spectral amplitude, |Ẽ(Ω)|. When the pulse spectrum is known, reconstructing the unknown phase of the pulse spectrum completely characterizes the pulse. A Gerchberg-Saxton like approach to recover the phase was already presented in Ref. [15]. However, this method did not converge to the correct field in several test cases. Therefore, we reconstruct the spectral phase using a different method based on an iterative line-minimization technique. We start with a guess phase, ϕ_g (Ω), in place of the missing phase of |Ẽ(Ω)| and obtain the time domain electric field E_g (τ). Using this field, we obtain estimates of intensity, | Ĩ_g (Ω)|, and second harmonic, | Ẽ ²(Ω)|, spectra. The guess phase, ϕ_g (Ω), is updated using Powell’s minimization [17] approach such that the error,

is reduced. The error metric Δ represents the mean squared error between the measured and adapted spectral intensity profiles.

 

Fig. 5. Reconstruction of pulse intensity and chirp using MOSAIC envelopes and pulse spectral magnitude. Complete and unambiguous pulse characterization is possible with this approach. Solid lines correspond to the original pulse (E_i (τ)) in Fig. 3 and the symbols represent ambiguity free reconstruction.

Download Full Size | PDF

 

Fig. 6. (a) Solid red lines represent experimentally measured MOSAIC envelopes and blue symbols represent reconstruction results. (b) Red curve is an experimentally measured pulse spectral intensity and the blue curve is the reconstructed phase. (c) Reconstructed pulse intensity and phase in time.

Download Full Size | PDF

We applied this approach to several pulse shapes and found that it converged in all test cases, unlike the method mentioned earlier. Reconstruction results corresponding to the example in Fig. 3 are shown in Fig. 5. For this reconstruction, magnitude of the pulse spectrum and the MOSAIC envelopes were used. We can see that both the temporal amplitude and chirp are in good agreement with the original pulse (E_i (τ)). This reconstruction was performed using 128 element vectors for |Ẽ(Ω)|, |Ĩ(Ω)|, and | Ẽ ²(Ω)|.

Using the line-minimization approach, we reconstructed pulse shapes for experimentally measured MOSAIC envelopes. Results for this reconstruction are presented in Fig. 6. The MOSAIC traces are obtained by averaging envelopes from fringe-resolved interferometric autocorrelation measurements. The MOSAIC envelopes plotted in Fig. 6(a) are corrected for distortions using a small correction factor, η, similar to Ref. [14]. The pulse spectrum is obtained in the same setup using a first-order autocorrelation with a linear detector. For these experiments, the two photon detector used about a milli Watt of power from a 75MHz Ti:Sapphire oscillator and the first order detector used a small fraction of a milli Watt. The reconstructed frequency phase [18] is shown in Fig. 6(b) and the time temporal pulse obtained by Fourier transforming Ẽ(Ω) is plotted in Fig. 6(c). The reconstruction does not resolve the time direction and hence Ẽ*(Ω) (that is E*(-τ)) is also a possible solution.

4. Conclusions

We have investigated ambiguities in quantifying chirp from the MOSAIC envelope traces. Although the MOSAIC minima envelope trace is very sensitive to chirp, there can be severe ambiguities in fitting a chirp to the measurement due to the uncertainty in the assumed pulse (or spectral) amplitude. In general, for any given MOSAIC trace, we found there are several pulse shapes that have nearly identical (experimentally indistinguishable) MOSAIC traces. This suggests that unless the pulse (or spectral) amplitude is well known, chirp determination from MOSAIC traces may not be correct. Furthermore, we found that a pulse amplitude uncertainty of a few percent can lead to considerable phase errors. This chirp ambiguity is removed when the spectral amplitude is known in addition to the MOSAIC envelopes. We present an algorithm based on line-minimization which extracts the complete pulse information (amplitude and phase) unambiguously. Without such an additional spectral measurement, MOSAIC is still very useful as a qualitative visual tool for pulse chirp.