## Abstract

Sensitive, real-time chirp and spectral phase diagnostics along with full field reconstruction of femtosecond laser pulses are performed using a single rapid-scan interferometric autocorrelator. Through the use of phase retrieval error maps, ambiguities in pulse retrievals based on the pulse spectrum and various forms of MOSAIC traces are discussed. We show second-order autocorrelations can introduce significantly different amounts of chirp depending on the implementation. Examples are presented that illustrate the sensitivity and fidelity of the scheme even with low signal-to-noise.

©2008 Optical Society of America

## 1. Introduction

Characterization of the amplitude and phase of ultrashort laser pulses is vital to the controlled use of femtosecond laser systems [1, 2]. The diagnosis of such short pulses by means of direct electronic detection is limited by instrument bandwidth. Optical sampling techniques based on nonlinear autocorrelations or cross-correlations, as well as self-referencing spectral interferometric methods remain the most viable means of means for characterizing such short pulses. The first order or linear correlation contains spectral amplitude information, but provides no information on the phase of the ultrashort pulse. Nonlinear schemes such as second harmonic generation (SHG), two-photon fluorescence, two-photon conductivity and Kerr gating provide intensity autocorrelations that are routinely used to estimate the laser pulse width. No chirp (or phase) information is gained unless an interferometric setup such as a second-order interferometric auto-correlation (IAC) is used [2]. Furthermore, while the IAC contains amplitude and phase information, it will not yield a full characterization of the amplitude and phase of the pulse electric field. A number of elegant techniques [3-7] have been introduced that reconstruct the full electric field. The uniqueness of the retrieved spectral phase (or its ambiguity) typically varies with the degree of complexity in the implementation of these measurements. In many applications, however, the full field retrieval may not be necessary, and only a semi-quantitative yet sensitive measure of the phase distortion is of interest. The Modified Spectrum Auto-Interferometric Correlation (MOSAIC) algorithm detailed here achieves this by a very simple approach: an IAC trace is converted to a fringe free trace that provides a visual and unambiguous indication of the phase distortion (chirp) with very high sensitivity [8, 9]. The algorithm runs efficiently on a PC to allow precise experimental optimization in real-time.

The usefulness of MOSAIC has been extended with homodyne detection and signal averaging [9]. High fidelity traces are extracted using fringe-free averaging techniques in a high noise environment where the signal-to-noise ratio (SNR) approaches unity. Different pulses can be distinguished even when producing essentially identical IAC traces [10]. The fringe-free MOSAIC technique has recently been used to characterize ultrashort pulses in the mid-IR [11], because MOSAIC is an algorithm, it can be performed in any spectral region or experimental condition where an IAC or intensity autocorrelation and second harmonic spectrum can be measured. The MOSAIC envelope is not distorted by intensity imbalance in the autocorrelator, unlike an IAC. Additionally, MOSAIC traces are unaffected by residual linear absorption that may be present in two photon absorbing detectors. Other pulse characterization techniques depict temporal asymmetry using unbalanced interferometric correlation envelope (ICE) functions [12].

Although MOSAIC was originally intended for sensitive real-time analysis, we show here that the MOSAIC trace can also be used to retrieve the spectral phase and provide a full field reconstruction of an ultrashort pulse. In doing so, it is critical to eliminate potential ambiguities. The speed and simplicity of this 1-dimensional algorithm comes at the expense of time-direction ambiguity, but such precise knowledge is not necessary for many applications. Unique quantitative analysis gives information on the electric field, amplitude and phase in the presence of extreme noise; pulses with energy as low 60 pJ in 86 MHz pulse trains with an average power of 5 mW have been analyzed using averaged MOSAIC envelopes [9]. This is comparable to SHG FROG sensitivities [13]. A single autocorrelator is all that is required to implement this technique. A sketch of the experimental layout is shown in Fig. 1.

This paper is organized as follows. The MOSAIC algorithm is presented followed by an example of spectral reconstruction using an iterative technique. Ambiguities in the reconstruction are discussed, demonstrated experimentally and finally a possible solution is presented. Algorithm structure and efficiency is discussed as well as a comparison to IAC based phase retrieval. MOSAIC reveals spectral phase distortion introduced by common two-photon photovoltaic detectors such as LEDs and planar metal-semiconductor-metal (MSM) photoconductive devices. Finally, pulse characterization is performed in high noise conditions where a stand-alone IAC is not possible.

## 2. Background

The simple principle of generating a MOSAIC trace can be described in the frequency domain as follows: The laser pulse is assumed to have an electric field given by
$\tilde{E}\left(t\right)=\sqrt{f\left(t\right)}{e}^{i\left[{\omega}_{o}t+\varphi \left(t\right)\right]}$, where *ϕ*(*t*) denotes the temporal phase. The well known second order IAC trace produced by a nonlinear autocorrelator is given by [1]:

$$\phantom{\rule{7.7em}{0ex}}+\int f\left(t\right)f\left(t+\tau \right)\mathrm{cos}\left(2\mathit{\omega \tau}+2\mathrm{\Delta}\varphi \right)\mathit{dt}\mathit{\text{}}$$

$$\phantom{\rule{8.2em}{0ex}}+2\int {f}^{\frac{1}{2}}\left(t\right){f}^{\frac{3}{2}}\left(t+\tau \right)\mathrm{cos}\left(\mathit{\omega \tau}+\mathrm{\Delta}\varphi \right)\mathit{dt}$$

$$\phantom{\rule{8.2em}{0ex}}+2\int {f}^{\frac{3}{2}}\left(t\right){f}^{\frac{1}{2}}\left(t+\tau \right)\mathrm{cos}\left(\mathrm{\omega \tau}+\mathrm{\Delta}\varphi \right)\mathit{dt}$$

where ∆*ϕ*(*t*, *τ*) = *ϕ*(*t* + *τ*) - *ϕ*(*t*) and ∫*f*(*t*)*dt* = 1. The spectrum (Fourier transform) of the above IAC contains three components at 0, *ω* and 2*ω*, where *ω* is the fringe frequency. The spectrum is then modified by retaining the dc term, removing the *ω* term and multiplying the 2*ω* component by 2[14]. The inverse Fourier transform produces a fringe-resolved MOSAIC trace [8]

where *g*(*τ*) = ∫*f*(*t*)*f*(*t* + *τ*)*dt* is the intensity autocorrelation, and

is the amplified 2*ω* component of the IAC, which is also the envelope of the second harmonic field autocorrelation. The term, **Φ**(*τ*) =-tan^{-1}{Im[*g*̃_{2ω}(*τ*)]/Re[*g*̃_{2ω}(*τ*)]}, is the fringe phase. The upper and lower envelopes of the MOSAIC trace are given by [15]

Note that the lower envelope, *S*
_{min}(*τ*) = *g*(*τ*)-|*g*̃_{2ω}(*τ*)|, exhibits a flat feature (i.e. equals zero for all *τ*) when no chirp is present (∆*ϕ* = 0) and thus provides a sensitive and background-free signal indicative of pulse chirp, see Fig. 2(a).

Examples of two different chirp conditions in MOSAIC traces are shown in Fig. 2(a, b) where the presence of shoulders on *S*
_{min} indicate chirp; the corresponding IAC traces appear indistinguishable, Fig. 2(c, d). The upper MOSAIC envelope is a measure of pulse duration but does not possess information about chirp information beyond *S*
_{min}. For comparison and later discussion the fringe resolved MOSAIC is also depicted in Fig. 2(a, b) insets. We find it more useful to replace the upper envelope by the intensity autocorrelation *g*(*τ*) so that *S*
_{max}(*τ*) = *g*(*τ*) and *S*
_{min}(*τ*) = *g*(*τ*)-|*g*̃(*τ*)| represent amplitude and phase profiles respectively [9].

The ideal condition *S*
_{min}(0) = 0 can only be approached in practice. Small deviations in the order of quadratic nonlinearity, minor misalignment of the interferometric autocorrelator or insufficient bandwidth in the IAC acquisition may lead to a distorted trace for which *S*
_{min}(0) is nonzero. This can be addressed with a correction factor *η*:

that forces the trace to zero at zero delay. This correction takes the form:

and is needed to render a correct MOSAIC, separating pulse chirp information from distortion associated with autocorrelator misalignment and noise in the detection electronics. The second harmonic field autocorrelation is related to the second harmonic spectrum by

where the second harmonic power spectrum is |*E*̃(2*ω*) |^{2} and *F*
^{-1} denotes the inverse Fourier transform operation [15]. This property can be exploited to acquire MOSAIC traces in a non-interferometric setting suitable for single shot measurements and is termed Envelope-MOSAIC (or E-MOSAIC). The production of MOSAIC envelopes from the intensity autocorrelation and the second harmonic spectrum has been shown to provide the same information as first-generation MOSAIC generated from a fringe resolved IAC [15]. An additional rendering of MOSAIC in the form of a Hybrid-MOSAIC (H-MOSAIC) has been developed that distinguish between temporal and spectral phase [15]. The added capability of the H-MOSAIC requires the pulse spectrum in addition to the IAC. In rapid-scan IAC schemes, the spectrum can be obtained by splitting off a small amount of the autocorrelator output and directing it to a linear detector [9, 15 and 16], see Fig. 1. The pulse spectrum, |*E*̃(*ω*)|^{2}, is found from the Fourier transform of the resulting linear interferogram (or a spectrometer in a noninterferometric or single shot arrangement). The spectrum can be used to compute the complex transform limited amplified 2*ω* component of the IAC by employing the convolution integral:

The use of *g*̃^{TL}
_{2ω}(*τ*) is not important for visual appreciation of pulse distortion at this stage, but may play a role for phase retrieval as will be discussed in the next section.

## 2. Developments

#### 2.1 Retrieval MOSAIC: spectral phase reconstruction

Naganuma *et. al*. have shown that a combination of pulse spectrum and IAC is sufficient to uniquely reconstruct the complex electric field with only a time direction ambiguity [17]. Reconstruction results from SHG FROG or GRENOUILLE devices also have the direction of time ambiguity [3, 7]. An iterative, phase retrieval technique using IAC and pulse spectrum based on a population split genetic algorithm has recently shown promise for improved computational efficiency and accuracy [18]. The SNR required to uniquely reconstruct the phase, however, may not be experimentally practical [10]. By combining MOSAIC data with the first-order interferogram and performing additional analysis, the spectral phase of the electric field can be recovered [14].

Experimental reconstruction using MOSAIC and the pulse spectrum has been demonstrated using an iterative line minimization technique [16]. In this reconstruction method, all points in the spectral phase are optimized individually at the expense of processing time. It has been shown that processing time can be reduced about 7x by analyzing phase with a fourth-order Taylor-series expansion and adjusting the coefficients using the Retrieval (R)-MOSAIC algorithm [15]. The sequential R-MOSAIC algorithm accounts for the predominant contribution of the lowest order spectral phase coefficients encountered in realistic pulses. Our simulations have shown that simultaneous optimization of Taylor-series coefficients is less likely to converge on the optimal phase. Simultaneous optimization can weight higher order terms too heavily causing the algorithm to fall into a local well far from the correct solution in the search space. This was noted as a poorly reconstructed MOSAIC with a higher root mean square (RMS) error compared to the sequential routine.

The sequential R-MOSAIC algorithm minimizes the RMS error, ∆, between the measured and reconstructed MOSAIC traces. The function to be minimized is given by:

where *N* is the number of points used in the reconstruction, *S*
_{max/min} and *s*
_{max/min} represent measured and computed quantities, respectively. Minimization of the RMS defines convergence of the algorithm. An example of a fit (∆ = 0.0068) is shown in Fig. 3(b).

For pulses not having a Taylor-series expandable phase it becomes necessary to use an individual point line search method to adequately reconstruct MOSAIC traces. Such algorithms can be seeded with the output of the resulting spectral phase from R-MOSAIC to allow for more rapid convergence. An experimental example of a phase recovered from individual point line search is shown across a measured spectrum in Fig. 3(c). Simultaneous recording of both linear and second-order interferometric autocorrelation traces is done on a two-channel digital oscilloscope controlled with National Instruments LabVIEW software. The associated IAC and 100x averaged MOSAIC trace can be seen in Fig. 3(a) and (b) with the retrieved time domain pulse displayed in Fig. 3(d). While a low error was achieved in the pulse retrieval, we show in the next section that ambiguities in the phase retrieval error map require additional information to be used to correctly identify the spectral phase.

#### 2.2 Error mapping

Spectral phase retrieval using the three envelope dataset outlined by Naganuma *et. al*. can be improved with the application of the MOSAIC algorithm. Here we show that preprocessing of the IAC to a MOSAIC trace leads to better localization of the error minima in the parameter space explored by iterative retrieval algorithms. To demonstrate this we simulate an ultrashort pulse having a symmetric spectrum given by |*E*̃(*ω*)| =exp[-*ω*
^{2}/(∆*ω*)^{2}]. The spectrum is centered at 800 nm and has a FWHM ≈ 57nm. A spectral phase is assigned to the pulse having 50 fs^{2} of GVD and 175 fs^{3} of TOD. From this pulse a target IAC, Eq. (1), and a fringe resolved MOSAIC, Eq. (2), are computed. To visualize ambiguities in the retrieval we use the trial pulses to produce an error map in a manner similar to Ref. [6]. An error map as a function of GVD and TOD is produced by computing the RMS error between the target IAC and the trial IAC traces, Fig. 4(a).

Trial reconstruction pulses are produced by taking the target spectral amplitude and assigning a trial spectral phase for all values of GVD between -100 and 100 fs^{2} and TOD between -600 and 600 fs^{3}. Regions of dark blue indicate low error. The corresponding error map for the target fringe resolved MOSAIC and trial fringe resolved MOSAIC trace is shown in Fig. 4(b). It is important to note that the region of low error surrounding the solution shrinks in the case of the fringe resolved MOSAIC. This highly localized solution suggests faster convergence and better accuracy for iterative phase retrieval schemes. In the case of an intensity imbalanced autocorrelator or residual linear absorption on the detector the IAC error map would be affected. Because the *ω* term is removed and it is background free, MOSAIC error maps are insensitive to these distortions.

We further compute the error map for the case of the E-MOSAIC, Eq. (4). Results are presented in Fig. 4(c). Here it can be seen that regions of low error become broader and less localized relative to the fringe resolved case. In addition, there is also four-fold degeneracy for the region of lowest error. This degeneracy results in ambiguity on the sign of GVD and TOD coefficients. The ambiguity and poor localization is due to neglecting the fringe phase, **Φ**(*τ*) , from Eq. (2). For visual interpretation of the E-MOSAIC trace, **Φ**(*τ*) is not needed, however, for phase retrieval its inclusion is important particularly when the pulse spectrum is symmetric. Figure 4(c) suggests that E-MOSAIC depends weakly on TOD. Incorporation of the fringe phase is needed to better define the solution in the error map. Experimentally, **Φ**(*τ*) can be signal averaged in the same way as a MOSAIC envelope. Spectral dependence on the fringe phase can be removed by making use of the phase on *g*̃^{TL}
_{2ω}(*τ*) presented in Eq. (8) in section 1. The spectral dependence present on **Φ**(*τ*) is also present on **Φ**
^{TL}(*τ*) since they are both produced from the same spectrum and the same rapid scan autocorrelator. By subtracting the two we can have a background free measure of a differential fringe phase (DFP)

The DFP is no longer coupled with spectral dependence if a single interferometer is used. With a spectrometer (as in a single shot) it is important to have a well calibrated frequency axis as uncertainties in determining *ω* and 2*ω* can affect retrieval. The DFP is only sensitive to spectral phase. The value of the DFP can become large far from zero delay where *g*(*τ*) goes to zero. To accentuate the relevant features of the DFP we weight it by the amplitude of *g*(*τ*). The normalized DFP is then *g*(*τ*)[**Φ**(*τ*) - **Φ**
^{TL}(*τ*)]. Simulation for different dispersion conditions is shown in Fig. 5.

It is important to note that the DFP is sensitive to the relative sign of GVD and TOD coefficients across a symmetric spectrum while E-MOSAIC is not. If both GVD and TOD have the same sign, the DFP will show a peak followed by a valley, Fig. 5(blue line), while an opposite sign between the coefficients is seen as a valley followed by a peak, Fig. 5(red line).

The inclusion of the DFP to E-MOSAIC is shown in the error map of Fig. 4(d). The error function in this case includes equal weighting for the upper and lower envelopes of MOSAIC as well as the normalized DFP. Spurious solutions are eliminated and highly localized solutions are restored, indicating the DFP is an important contribution to E-MOSAIC based phase retrieval for pulses having a symmetric spectrum.

To demonstrate the experimental relevance of the DFP we consider the retrieved pulse from Fig. 3. The highly symmetric spectrum allows for the ambiguity displayed in Fig. 4(c). Identically reconstructed MOSAIC traces were produced with different retrieved pulses by changing the sign of the starting point in the iterative line search algorithm. The retrieved pulses from the different starting points are shown in Fig. 6(a, b). The red line is a polynomial fit across the FWHM of the pulse spectrum to the retrieved spectral phase. The fit in Fig. 6(a) is *ϕ*_{1}(*ω*) = -466*fs*
^{2}
*ω*
^{2} - 11700*fs*
^{3}
*ω*
^{3} while the fit in Fig. 6(b) is *ϕ*
_{2}(*ω*) = -1000*fs*
^{2}
*ω*
^{2} + 11900*fs*
^{3}
*ω*
^{3}. The sign difference between the GVD and TOD coefficients in the two fits is a nontrivial ambiguity that can be resolved with the DFP. The DFP for each reconstructed pulse is shown in the insets. The correlation of the peaks and valleys of the reconstructed DFP with the measured DFP indicates correct sign for the pulse in Fig. 6(a). Similarly, the anti-correlation of the peaks and valleys of the DFP shown in Fig. 6(b) indicate that it is not the correct pulse; despite the fact that its reconstructed E-MOSAIC is the same.

Note that the advantage of the E-MOSAIC trace is that it can be averaged for pulse reconstruction at very low pulse energies. If sufficient pulse energy is available, however, the fringe-resolved MOSAIC trace can be used for full pulse reconstruction and issues of ambiguities in the E-MOSAIC error map can be avoided. The DFP can be included with E-MOSAIC in the reconstruction algorithm. This will automatically resolve sign ambiguities. Because the DFP can become distorted from asymmetries in the IAC and noise, the autocorrelator must be aligned to obtain a symmetric IAC and averaging is needed to suppress noise.

Several optimization algorithms have been investigated for effectiveness in phase retrieval using the procedure outlined above. Performance is defined by the minimum achievable RMS error and processing time. We tested line search, simplex, Levenberg-Marquardt, genetic and pattern search algorithms. Each of these algorithms was unmodified from the standard code available in the MATLAB software platform. Analysis of mode-locked Ti:sapphire laser pulses are presented in Fig. 7.

Best phase reconstruction is obtained with either a simplex or line search routine. The genetic and pattern search algorithms also produce satisfactory fits, but do so at the expense of computation time.

## 3. Additional examples

#### 3.1 Two-photon conductivity induced dispersion

By applying the MOSAIC algorithm to IAC traces generated in different ways but originating from the same pulse we are able to perform detector characterization. We generate IAC traces of near-infrared mode-locked Ti:sapphire laser pulses by (i) frequency doubling with a BBO crystal into a linear detector, Fig. 2(c) and (ii) two-photon absorption-induced photocurrent in a light-emitting diode [19], Fig. 2(d). The MOSAIC algorithm is applied to both IAC waveforms to generate the envelopes shown in Figs. 2(a) and 2(b), respectively. While the IACs appear essentially the same, the MOSAIC waveforms reveal a striking difference.

The pronounced shoulders in Fig. 2(b) indicate the LED detection scheme introduces chirp that is nearly five times greater than the same pulse detected via second-harmonic generation with BBO. We measure LEDs from four different manufacturers and find in all cases higher chirp and longer pulse duration compared to second-order autocorrelation with a BBO crystal. The chirp introduced by the LED epoxy dome is negligible for our 60 fs pulses. Removal of the dome is desirable, however, for enhanced SNR and ease of alignment. MOSAIC also reveals significant variation in induced chirp for identical wavelength LEDs produced by the same manufacturer. A detailed comparison of LED and BBO detection can be found in Ref. [20].

The metal-semiconductor-metal (MSM) structure can be used as a two-photon detector in second order autocorrelations; we examine a ZnSe MSM device for induced pulse chirp [21]. The structure is a single crystal ZnSe substrate with interleaved titanium electrodes and a gold cap layer. Titanium provides high adhesion to the ZnSe [22]. A 3.5x microscope objective focuses the autocorrelator output on the region between the electrodes (bias: 30 V). A 500 average MOSAIC is obtained for different lens positions; shoulder height (chirp) is plotted in Fig. 8(a). Error bars are due to slight asymmetry in MOSAIC traces. The SNR of the second order IAC is shown in Fig. 8(b). Lowest chirp occurs with the focusing geometry that produces the highest SNR, but this alignment optimization must be made with caution. Changing the lens position shifts the region of two-photon absorption from the surface to deeper in the bulk ZnSe. Significant chirp is introduced as the interaction length in the material increases, while the SNR decreases by < 10%.

#### 3.2 Characterization using MOSAIC in low signal-to-noise

MOSAIC exhibits high fidelity even when SNR is poor. Figure 9(a) shows an autocorrelation of a frequency doubled Ti:sapphire at *λ* = 415 nm obtained with an ultraviolet Michelson interferometer and suitably cut BBO crystal to produce 208 nm light that is detected with a photo-multiplier tube. An averaged MOSAIC waveform generated from 1000 low SNR IAC traces is presented in Fig. 9(b). Pulse duration and chirp can be determined with excellent accuracy.

## 4. Summary

We have presented the principle of the MOSAIC algorithm for ultrashort pulse characterization. Ambiguities in MOSAIC based phase retrieval were presented and experimentally demonstrated. A solution for avoiding ambiguities was developed. Error maps showing high localization of phase retrieval were presented. Second-order autocorrelations can introduce significantly different amounts of chirp depending on the implementation. Second harmonic generation and linear detection produces less distortion compared to two-photon absorbing LEDs. The metal-semiconductor-metal structure introduces minimal chirp provided the light is focused near the surface to achieve maximum SNR. MOSAIC is helpful for characterizing very weak ultrashort pulses such as from frequency doubled Ti:sapphire lasers. The efficiency of different optimization algorithms was also discussed. Open source MOSAIC software is available for free download at http://www.optics.unm.edu/sbahae/.

## Acknowledgment

The authors wish to acknowledge helpful discussions with B. Yellampalle and M. Hasselbeck. Support provided through National Science Foundation awards ECS-0100636 and DGE-0114319 is gratefully acknowledged.

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