## Abstract

We present a detailed analysis of commonly encountered waveform distortions in femtosecond pulse shaping with pixelated devices, including the effects of discrete sampling, pixel gaps, smooth pixel boundaries, and nonlinear dispersion of the laser spectrum. Experimental and simulated measurements are used to illustrate the effects. The results suggest strategies for reduction of some classes of distortions.

© 2006 Optical Society of America

## 1. Introduction

Since its development approximately 20 years ago [1], femtosecond pulse shaping [2] has been applied to a wide variety of problems in, for example, spectroscopy [3], microscopy [4], laser control of matter [5, 6], laser pulse compression [7], telecommunications [8], and optical metrology [9]. Although the quality of experimental results obtained using shaped femtosecond pulses is typically a strong function of the quality of the shaped pulses themselves, commonly encountered pulse distortions such as replica pulses have received only a little attention in the literature [3, 10, 11, 12, 13, 14] and without comprehensive treatment. In this article, we will analyze in detail the factors responsible for various waveform distortions, including the effects of discrete sampling, imperfections in modulator elements such as gaps or blurred-out pixel regions, and nonlinear dispersion of the laser spectrum.

The typical 4-f apparatus for pulse shaping is shown in fig. 1. A short (typically less than 1 ps) laser pulse incident from the right is spectrally dispersed by a grating-lens pair. At the spectral plane, a computer-controlled liquid-crystal (LC) spatial light modulator (SLM) modulates the phase and/or amplitude of the dispersed laser spectrum. The modulated spectral components are then recombined by another grating-lens pair, producing an output waveform. Note that the output waveform shown in fig. 1 was produced with a pulse shaping device based on a two-dimensional LC SLM [15, 16] and consists of 10 unevenly spaced pulses spread out over 6 ps.

Consider now the relatively simple goal of delaying a pulse in time. This may be achieved through the application of a linear spectral phase with the LC SLM. Even for a perfectly aligned and calibrated pulse shaping apparatus, the obtained waveforms depart dramatically from a simple delayed pulse, as can be seen in the XFROG (cross-correlation frequency-resolved optical gating [17]) measurements shown in fig. 2. In (a) through (i), the pulse has been delayed by varying amounts, from 0 to 14 ps, in steps of 2 ps. The “desired” pulse can be seen in all but the last plot as a short pulse at the specified time. Additional undesired pulses are also observed at other times. Specifically, a strongly chirped pulse is observed at negative times in (b)-(i), where the temporal separation between the chirped pulse and the pulse at the desired time is constant at about 14 ps. Several unchirped pulses are also observed in (b) and (c), where the spacing between each is the same as the delay of the desired pulse. All plots show a pulse at time zero.

This article will analyze in detail the factors governing these waveform distortions. Numerous experimental and simulated measurements will be used to illustrate the effects discussed. Other than the data in fig. 2 from the group of professor R. Sauerbrey of Jena, Germany, which were obtained through the use of a 1D LC SLM, all measurements were performed using our reflective-mode Hamamatsu 2D LC SLM, described in detail in [18, 19]. Note that the analysis presented here applies for all LC SLMs, although the relative magnitudes of the different effects depends upon the specific properties of the device under consideration. The organization of this paper is as follows. In section 2, we will provide a general analysis of femtosecond pulse shaping with pixelated LC SLMs. Section 3 will analyze a particular class of waveform distortions called sampling replica pulses that result from the discrete pixelation of the modulator elements used in femtosecond pulse shaping. Sections 4 and 5 will describe a second class of waveform distortions, which we call modulator replica pulses, that arises from pixel smoothing in the LC SLM. We will conclude in section 6 with a summary of the replica pulse effects and some discussion on methods for reducing their severity.

## 2. General Analysis

Most temporal pulse shaping schemes demonstrated to date manipulate the phase and/or amplitude of a broad-bandwidth input laser pulse in order to create a desired time-dependent optical waveform, as shown in fig. 1. Mathematically, the modulated laser spectrum can be described as the product of the input laser pulse *E _{in}*(

*ν*) and the spectral modulation

*M*(

*ν*) applied by the pulse shaping apparatus, giving

An ideal LC SLM consists of *N* sharply defined pixels separated by Δ*x*, with no gaps present between the pixels. The LC SLM may independently modulate the phase (for a single mask LC SLM) or the amplitude and phase (for a dual-mask LC SLM) of the spectrum of the laser pulse. The modulating function *M*(*x*) is then simply the convolution of the spatial profile *S*(*x*) of a given spectral component with the phase and amplitude modulation applied by the LC SLM,

where *x _{n}* is the position of the

*n*th pixel,

*A*and

_{n}*ϕ*are the amplitude and phase modulation applied by the

_{n}*n*th pixel, Δ

*x*is the separation of adjacent pixels, and the top-hat function squ(

*x*) is defined as

For the moment, we will assume a linear spectral dispersion given by Ω* _{n}* = ΔΩ

*n*, where the frequency Ω

*of the*

_{n}*n*th pixel is defined relative to the center frequency

*ν*by Ω

_{o}*n*=

*ν*-

_{n}*ν*, and where ΔΩ is the frequency separation of adjacent pixels corresponding to Δ

_{o}*x*. Whether position is linearly or nonlinearly mapped to frequency depends on the details of the optics used to disperse the spectrum (grating or prism and lens, etc.). For a sufficiently small spectral range, the linear approximation is valid. Assuming also that the spatial field profile of a given spectral component is a Gaussian function

*S*(

*x*) = exp(-

*x*

^{2}/

*δx*

^{2}), the modulation function may be written as

Here, the width of the Gaussian function has been expressed in terms of *δ*Ω, the spectral resolution of the grating-lens pair, where *δ*Ω = *δx*ΔΩ/*δx*. The spot size *δx* (measured as full-width at 1/*e* of the intensity maximum, assuming a Gaussian input beam profile) is dependent upon the input beam diameter *D* and lens focal length *F* according to *δx* = 4*πF*/*λD*. If we assume that the input laser pulse is bandwidth-limited (that the spectral phase is flat), we can then approximate the input laser pulse as

where *B _{n}* is the spectral amplitude of the input laser pulse at the

*n*th pixel. Substitution of the above expressions for

*M*(Ω) and

*E*(Ω) into eq. 1 yields

_{in}Finally, Fourier transformation of *E _{out}*(Ω) yields an expression for the output of the pulse shaping apparatus,

The summation term in eq. 7 describes the basic properties of the output pulse, such as would be obtained by modulating the amplitude and/or phase of the input pulse at the points Ω* _{n}* -

*n*ΔΩ with a grating-lens apparatus that has perfect spectral resolution. The sinc term is the Fourier transformation of the top-hat pixel shape, where the width of the sinc function is inversely proportional to the pixel separation Δ

*x*, or equivalently, ΔΩ. The Gaussian term results from the finite spectral resolution of the grating-lens pair, where the width of the Gaussian function is inversely proportional to the spectral resolution

*δ*Ω. Collectively, the product of the Gaussian and sinc terms is known as the time window. Use of a LC SLM with a larger number of pixels over the same distance decreases ΔΩ and therefore increases the width of the sinc function. As the pixel separation decreases to be less than the spot size of a frequency component at the spectral plane,

*δx*, the additional pixels do not result in a larger temporal range over which pulses may be shaped due to the rolloff of the Gaussian at times far from zero delay. In such a scenario, a greater number of pixels will only result in a larger time window if it is also accompanied by an improvement in the spectral resolution.

While eq. 7 provides a compact and useful analytical result, it neglects some important limitations of LC SLMs. First, LC SLMs typically have a phase range that is only slightly in excess of 2*π*. Fortunately, since phases that differ by 2*π* are mathematically equivalent, the phase modulation may be applied modulo 2*π*. Thus, whenever the phase would otherwise exceed integer multiples of 2*π*, it is “wrapped” back to be within the range of 0 – 2*π*. Second, the pixels of the LC SLM are not perfectly sharp, and there are gap regions between the pixels whose properties are somewhat intermediate between those of the adjacent pixels. Although smoothing of the pixelated phase and/or amplitude pattern might in general sound desirable, when it is combined with the phase-wraps, distortions in the spectral phase and/or amplitude modulation are introduced at phase-wrap points, as will be shown in section 4. Third, while the pixels are evenly distributed in space, the frequency components of the dispersed spectrum are not. We will refer to this nonlinear mapping of pixel number to frequency as nonlinear spectral dispersion. These three considerations make the determination of an exact analytical expression for *M*(Ω) difficult. Instead, we will formulate a general expression for *M*(*x*) and then specify a procedure for numerical computation of the generated output pulse. At the same time we will attempt to glean a physical understanding of the trends that are observed.

The smoothed pixels of the LC SLM may be accounted for by using a smoothed applied phase in the analysis or simulations. Although the exact nature of the smooth pixel boundaries is expected to be highly dependent upon the specific device that is being considered, it can be approximated by convolving a spatial response function *L*(*x*) with an idealized phase modulation function that would result in the case of sharply defined pixel and gap regions. We will now consider the phase response of a phase-only LC SLM with smooth pixels. Similar effects are observed for both phase and amplitude LC SLMs, although analytical results are less tractable. For a phase-only LC SLM with pixels separated by Δ*x* and gaps of width *w*, the applied phase modulation is given by

where *ϕ _{o}* is the phase applied in the gap region. Note that in eq. 8 the phase values have been indicated modulo 2

*π*, although 4

*π*or 6

*π*, etc., could be substituted depending on the properties of the device being used.

Next, the spatial phase modulation function will be converted to a spectral phase modulation function in the case of nonlinear spectral dispersion. This can be written generically as

Since *x*(Ω) represents the position of the frequency component Ω, eq. 9 performs a convolution of the phase applied by the mask as a function of position, *ϕ*(*x*), with a Gaussian function representing the spot size of the spectral component Ω. Unfortunately, due to the nonlinear dependence of *x*(Ω) on Ω, as well as the convolution *contained within ϕ*(*x*), eq. 9 is not easily evaluated. Instead, *M*′(Ω) may be calculated numerically. To do this, we first calculate *ϕ*(*x*) with about 10 grid points per pixel. Then *ϕ*(*x*) may be resampled on a grid of points evenly spaced in frequency, before evaluating eq. 9. Finally, the output pulse is calculated by fast Fourier transformation (FFT) of the product *E _{in}*(Ω)

*M*′(Ω). It is important to use the evenly-spaced frequency grid in order to make use of the computationally efficient (Cooley-Tukey) FFT algorithm.

## 3. Sampling Replica Pulses

The expression for a shaped output pulse with an idealized pulse shaping apparatus (eq. 7) contains a summed term that is a complex Fourier series,

A property of Fourier series (with evenly-spaced frequency samples) is that they repeat themselves with a period given by the reciprocal of the frequency increment. The present case is no exception, and the general waveform described by expression 10 is therefore repeated infinitely in time with a period of 1/ΔΩ, although the Gaussian-sinc window (see eq. 7) suppresses repetitions that are at very long or very early times. Nonetheless, these pulse repetitions are a cause for concern since they can degrade the quality of a desired output waveform. Although these pulse repetitions have received some attention in the literature [3, 10, 11], it is typically without quantitative analysis, especially in the case of nonlinear spectral dispersion.

We refer to the undesired repetitions of the waveform mentioned above as *sampling replica* pulses since they are a direct consequence of the discrete sampling of the LC SLM. In a sense, the distinction between a “desired pulse” and a “sampling replica pulse” is an arbitrary one since both are part of a coherent optical waveform. Nonetheless, the distinction is useful in that it exposes the limitations inherent to pixelated modulators. One must also be careful when using the term “pulse.” Although the input to a pulse shaper is usually expected to be a single pulse, the desired output may consist of multiple pulses or some general output waveform. By “sampling replica pulses” we actually mean copies of the desired output waveform, whatever its temporal profile. It should be noted that nonpixelated devices, such as acousto-optic modulators (AOMs) used in either a 4-f configuration [20] or in an in-line configuration [21] allow for smoothly varying spectral phase and/or amplitude modulations free of sampling replica pulses (and modulator replica pulses, the subject of the next section). At present, pulse shaping with 4-f AOMs is not widely used due to a low efficiency and a significantly higher degree of complexity to operate, especially for very short pulses. See reference [2] for a detailed comparison of pulse shaping with 4-f AOMs versus LC SLMs. In-line AOMs, better known as acousto-optic programmable dispersive filters (AOPDFs), have better efficiencies and are simpler to operate, but cannot be used for input pulses at oscillator repetition rates (successive pulses would be shaped differently due to acoustic wave propagation) and cannot withstand the output powers of amplified femtosecond laser pulses used in many coherent control experiments. AOPDFs are typically used with frequency down-counted femtosecond oscillator pulses prior to amplification in order to precompensate for dispersion and/or gain narrowing. To the best of our knowledge, AOPDFs have not been used to generate highly structured output pulses in a well-controlled manner.

As described in section 2, the phase applied by an LC SLM to the *n*th pixel is of the form *ϕ*
_{applied,n} = mod[*ϕ*
_{desired,n}, 2*π*]. Due to the mathematical equivalence of phase values that differ by integer multiples of 2*π*, there are an infinite number of ways to “unwrap” the applied phase. Sampling replica pulses constitute an important class of these equivalent phase functions, and their phase as a function of pixel, *ϕ*
_{replica,n}, may be described by

where *R* is the sampling replica order and may be any nonzero integer (*R* = 0 corresponds to the desired pulse) and where *ϕ*
_{applied,n} is the applied phase. Note that the analysis in this section will assume that the LC SLM has well defined pixels [*L*(*x*) = *δ*(*x*)] without gaps. In the case of linear spectral dispersion, *ϕ*
_{replica,n} for different values of R differ by the linear spectral phase 2*πRν*/ΔΩ, which by virtue of the Fourier-shift theorem precisely corresponds to a temporal shift of *R*/ΔΩ. Therefore, many sampling replica pulses are produced, where each is temporally separated from the next by 1/ΔΩ.

In the case of nonlinear spectral dispersion, the sampling replica pulses gain additional spectral phase. The nonlinear spectral dispersion is given below as a function of pixel number *n* in a power-series expansion:

The variable
$\overline{\mathrm{\Delta \Omega}}$ is used here as the coefficient for the linear spectral dispersion to convey its physical meaning (it is approximately the average frequency span per pixel), but should not be confused with the variable ΔΩ that was used earlier, when the spectral dispersion was explicitly assumed to be linear. The phase difference Δ*ϕ _{n}* between the replica pulse phase

*ϕ*

_{replica,n}and the applied phase

*ϕ*

_{applied,n}can now be equated to a power-series expansion of the phase difference in terms of frequency

$$\phantom{\rule{2.em}{0ex}}=\alpha {\Omega}_{n}+\beta {\Omega}_{n}^{2}+\gamma {\Omega}_{n}^{3}+\dots ,$$

which can then be solved in powers of *n* by substitution of Ω* _{n}* given in eq. 12. Exact expressions for the coefficients

*α*,

*β*,

*γ*,

*δ*… may thus be obtained (only the first four are shown):

$$\frac{\beta}{2\mathrm{\pi R}}=-\frac{K}{{\overline{\mathrm{\Delta \Omega}}}^{3}}$$

$$\frac{\gamma}{2\mathrm{\pi R}}=-\frac{L}{{\overline{\mathrm{\Delta \Omega}}}^{4}}+\frac{2{K}^{2}}{{\overline{\mathrm{\Delta \Omega}}}^{5}}$$

$$\frac{\delta}{2\mathrm{\pi R}}=-\frac{M}{{\overline{\mathrm{\Delta \Omega}}}^{5}}+\frac{5KL}{{\overline{\mathrm{\Delta \Omega}}}^{6}}-\frac{5{K}^{3}}{{\overline{\mathrm{\Delta \Omega}}}^{7}}.$$

The term *α* describes the expected linear delay of the sampling replica pulse of order *R* and is not dependent upon the nonlinear dispersion coefficients *K*, *L*, *M*, etc. The quadratic, cubic, and quartic spectral phases do, however, depend in varying degrees on the higher order spectral dispersion terms. All coefficients of the spectral phase are proportional to the replica order. Note that the above coefficients are completely general for a pixelated modulator, and apply regardless of whether phase and/or amplitude shaping of the pulse is used.

Figure 3 illustrates the principle of sampling replica pulses in the simple case when the desired spectral phase is linear with respect to frequency, with a slope of either 2*π*ΔΩ/4 (for linear spectral dispersion) or
$2\pi \frac{\overline{\mathrm{\Delta \Omega}}}{4}$ (for nonlinear spectral dispersion). The desired phase, applied phase, and *R* = ±1 sampling replica phases, are shown in (a) for the case of linear spectral dispersion. A simulated XFROG measurement and cross-correlation measurement of the output waveform (both plotted on logarithmic scales) are shown in (c). The desired pulse occurs at about 2.6 ps and is accompanied by three weaker replica pulses, each separated by about 11 ps, where 11 ps is equal to 1/ΔΩ. Note in (c) that the relative intensities of the sampling replica pulses are completely determined by the Gaussian-sinc time window. Additional, but much weaker sampling replica pulses, are present beyond the time range shown. All simulations shown in this article used the experimental parameters for our reflective-mode pulse shaper (1200 lp/mm grating, 23.8° grating input angle, 15 cm focal length lens, 790 nm center wavelength, and 480-pixel phase-only LC SLM).

When the spectral dispersion is nonlinear, the spectral phase of the sampling replica pulses is no longer linear as can be seen in fig. 3(b). The extent of nonlinear spectral dispersion has been exaggerated in (b) in order to make the nonlinear characteristics of the *R* = ±1 sampling replica pulse phases more apparent. There, the *R* = 1 sampling replica pulse phase (red line) has a negative curvature while the *R* = -1 sampling replica pulse (green line) has a positive curvature. The actual nonlinear spectral dispersion for our pulse shaping apparatus was used in the simulation in (d), where simulated XFROG and cross-correlation measurements show that nonlinear spectral dispersion causes the replica pulses to become chirped. As expected from equations 14, the *R* = ±1 sampling replica pulses have opposite chirps and the weak *R* = -2 sampling replica pulse near *τ* = 19 ps has a chirp twice that of the *R* = -1 sampling replica pulse near *τ* = 8 ps. The slight nonuniform tilt (or curvature) of the *R* = ±1 replica pulses in the XFROG simulations is the result of non-negligible cubic spectral phase. In general, the presence of higher order spectral phase (quadratic, cubic, etc.) on sampling replica pulses in addition to the desired spectral phase has the effect of reducing their intensity by temporal spreading, as can be seen in the simulated cross-correlation plot in (d). One obvious exception is when the desired pulse itself is chirped, in which case one of the replica pulses may be partly (or even completely) compressed.

Looking back to the measurements in fig. 2, we may now interpret the chirped pulses at negative times in (a) through (i) as sampling replica pulses. The *R* = -1 sampling replica pulses are not observed in fig. 2 since they would appear at times > 14 ps, beyond the range of the measured data. In each of the plots, the *R* = 1 sampling replica pulse precedes the desired pulse by 14 ps, where this delay corresponds to
$\frac{1}{\overline{\mathrm{\Delta \Omega}}}$ for the Jena pulse shaper. For successively larger delays of the desired pulse, the *R* = 1 sampling replica pulses grow in intensity relative to the desired pulse, overtaking it for delays greater than 8 ps. The slope of the chirp of the sampling replica pulses is approximately -13 THz/ps. For the experimental parameters of the Jena pulse shaper (1800 lp/mm grating, 56.4° grating input angle, 28.9 cm focal length lens, and 804 nm center wavelength), we obtain *β* = -0.33 ps^{2}, which corresponds to an expected quadratic chirp of -9.5 THz/ps. Careful numerical simulations (not shown) exactly reproduced the slope of -13 THz/ps and revealed that the discrepancy was due to cubic and higher order spectral phase terms that contributed to the overall slope of the chirped pulse.

Before moving on to a discussion of modulator replica pulses, in which the effects of pixel smoothing are discussed in detail, we will first examine the effects of LC SLM pixel gaps. For example, pulses due to pixel gaps were reported in [13] when using a MEMS-based modulator that possessed noticeable gaps. Equation 8 described the general phase pattern resulting from an LC SLM with pixels separated by Δ*x* and gaps of width *w*. If we assume linear spectral dispersion, phase-only modulation, and sharply-defined pixels, the spectral modulation applied by the LC SLM is given by

The first term within the brackets describes the spectral phase applied by the gap regions of the LC SLM, while the second term describes the spectral phase applied by the pixel regions of the LC SLM. The function comb(Ω) is defined as

requiring the use of the aperture function squ(Ω/*N*ΔΩ) such that *M*(Ω) is defined to be zero outside the range of the LC SLM. The mask’s temporal response may then be determined by Fourier transformation:

The first term describes the temporal response due to gap regions within the LC SLM and the second term describes the temporal response due to the pixel regions within the LC SLM.

Equation 17 illustrates two effects resulting from the presence of the gap regions. First, the gaps create “gap” replica pulses centered about time *t* = 0 with a period of 1/ΔΩ. The amplitude of the gap replica pulses is governed by a sinc envelope with a temporal width determined by the reciprocal of the spectral width of the gap. As the gap width *w* goes to zero, the gap replica pulses decrease in intensity. In the case of nonlinear spectral dispersion, the modulator replica pulses due to the gaps (except the one at time *t* = 0) become chirped as they gain additional spectral phase according to the arguments laid out above. The summation in the second term represents the desired phase-modulated output pulse, where the convolution of the desired output pulse with the function comb[ΔΩ*t*] creates sampling replica pulses separated by 1/ΔΩ as described above. As the pixel gap width increases from zero, the width of the term sinc[*π*ΔΩ(1 - *w*/Δ*x*)*t*] grows, with the result that the sampling replica pulses are somewhat less suppressed than otherwise. In the case when no phase modulation is applied, the gap replica pulses in the first term and the sampling replica pulses in the second term cancel out such that the output pulse is a single unshaped pulse as expected. In practice, however, it turns out that pixel-smoothing effects tend to dominate the gap regions that would otherwise be expected for LC SLMs, as will be shown in the next section.

## 4. Modulator Replica Pulses

As opposed to the sampling replica pulses discussed in the previous section, there is an entirely different class of output waveform distortions that result from smoothed-out pixel regions [i.e. finite spatial response *L*(*x*)] in combination with abrupt jumps or phase wraps. Somewhat loosely, we refer to these distortions as *modulator replica* pulses since discrete (and usually unwanted) pulses are often produced [15]. In this section we will focus on the case in which the objective is to delay a pulse in time. The implications for some other types of shaped pulses will be briefly addressed in section 5.

Delaying a pulse in time, although it is a relatively simple type of “pulse shaping,” is an especially important capability for applications of two-dimensional femtosecond pulse shaping, such as phase-stable degenerate four-wave mixing [22]. Additionally, the output waveforms are sufficiently simple that they readily illustrate the origins of the distortions. As mentioned above, all that is required to delay a pulse by *τ* is to apply a spectral phase with the slope - 2*πτν*. Since LC SLMs typically have the ability to apply a maximum spectral phase of only slightly in excess of 2*π*, the phase is applied modulo 2*π*. The presence of these phase-wraps in combination with a finite spatial response *L*(*x*) creates periodic distortions in the applied phase.

A simulated example of periodic distortions in the applied phase and the corresponding simulated output pulse intensity are shown in fig. 4(a) and fig. 4(b), respectively. In (a), the desired spectral phase is a line (red line) with a slope that corresponds to a delay of -0.4 ps. Only a small portion of the applied phase has been shown. The applied spectral phase (black line) appears as a smoothed-out sawtooth function, due to convolution with *L*(*x*) which is in this case sufficiently broad that it blurs the distinctions between separate pixels. The periodic deviations in the applied phase become clear when it is “unwrapped” (blue curve). The simulated output pulse intensity (b) shows numerous weak modulator replica pulses at both positive and negative times. A spectral interferometry [23] measurement of a pulse shifted to -0.4 ps and the extracted spectral phase of the pulse are shown in (c) and (d), respectively. In (d), the smoothed out pixel boundaries are clearly visible in the extracted spectral phase (blue line) as compared to the desired spectral phase (red line).

Looking back to the measurements shown in fig. 2, several weak (<2% of the desired pulse in all cases) modulator replica pulses are clearly visible in (b) and (c), although the t=0 peak in the plots (d)-(i) may be attributed to some combination of modulator replica pulses and gap replica pulses. The modulator replica pulses in fig. 2, for the Jena pulse shaper, are weaker than those of our reflective-mode pulse shaper since the Jena LC SLM has more well-defined pixels. In fact, the pixels of the optically-addressed two-dimensional LC SLM used in our reflective-mode pulse shaper have been deliberately smoothed in order to reduce gap effects [19].

Note that the temporal separation of the modulator replica pulses is inversely proportional to the spectral separation of the phase wraps. If the LC SLM applied phase shifts modulo 4*π*, then the modulator replica pulses would be separated by only 0.2 ps. Furthermore, a lower number of phase wrap points would reduce the intensity of the modulator replica pulses. In the limit of no phase wrap points, the modulator replica pulses would disappear.

As the desired spectral phase becomes steeper and steeper, more and more phase wrap points are introduced, causing the energy content of the modulator replica pulses to grow relative to that of the desired pulse. Figure 5 shows an experimental measurement (a) and a simulation (b) of a pulse shifted to -3 ps along with several modulator replica pulses. Due to the modulator replica pulses, the peak intensity of the desired pulse is lower than that predicted by the Gaussian-sinc time window of the pulse shaping apparatus (eq. 7). Figure 5(c) plots measured (dots) and simulated (black curve) peak intensities as a function of delay. For reference, the dashed line shows the Gaussian-sinc time window. Note that the results of simulations that account for modulator replica effects are in good agreement with experimental measurements of variably delayed pulses. Wang et al. [11] observed a similar delay-dependent intensity rolloff for variably-delayed pulses performed with a different LC SLM, although they did not identify the origin of the unexpectedly fast rolloff as a function of delay. The noticeable asymmetry in the measured peak intensities in (c) at positive times compared to negative times is due to space-time coupling effects [24] which have not been implemented in the simulations.

Modulator replica pulses can also be strongly influenced by nonlinear spectral dispersion. As mentioned above, it is the periodicity of the phase wraps that determines the temporal separation of modulator replica pulses. In the case of linear dispersion, then, it follows that the phase wraps will only be evenly spaced when the slope of the spectral phase is 2*π*/*A*ΔΩ, where *A* may be any nonzero integer. For instance, a linear spectral phase with slope of 2*π*/4ΔΩ produces a phase wrap every 4 pixels with replica pulses separated in time by 1/4ΔΩ. Correspondingly, a linear spectral phase with a slope of 2*π*/4.5ΔΩ produces phase wraps in alternating 4- and 5-pixel groups that repeats every 9 pixels. The resulting replica pulses therefore have a periodicity of 1/9ΔΩ.

Nonlinear spectral dispersion in general destroys the periodicity of the phase wraps as they tend to be closer together on one side of the spectrum and further apart on the other side of the spectrum. In this case, modulator replica pulses occur with a temporal separation corresponding to the average phase wrap period 2*π*/*τ*, where the slope of the desired phase is 2*πτ*, and where other possible modulator replica pulses are chirped to a much lower intensity. The above effects are illustrated in the simulations shown in fig. 6, where the slope of the desired spectral phase is 2*π*/4.5ΔΩ in the case of linear spectral dispersion (a) and
$2\pi \u20444.5\overline{\mathrm{\Delta \Omega}}$ ps in the case of nonlinear spectral dispersion (b). In (b), only 4 modulator replica pulses are observed, while in (a), additional modulator replica at intermediate times are observed. The modulator replica pulses at intermediate times in (a) have become chirped to a nearly negligible intensity in (b), although careful examination of the baseline in (b) near 1.5 ps and 4 ps reveals two “noisy” regions corresponding to the chirped modulator replica pulses.

## 5. Effects of Smooth Pixels in More Complex Waveforms

Although smooth pixels create phase distortions that can lead to extra pulses, there are other possible consequences of having smooth pixels, two of which we will address in this section. First, in some cases, smooth pixels may create modulator replica pulse features which overlap temporally with the desired pulse shape itself, typically creating heavily distorted output waveforms due to interference effects. Figure 7(a) shows a cross-correlation measurement of the output waveform resulting from a large quadratic spectral phase modulation (0.06 ps ^{2}) applied by our 2D LC SLM. The cross-correlation measurement is clearly a temporally broadened waveform compared to the approximately 40 fs input pulse, but consists of several short pulses spread out over 1 ps rather than a single, smooth, temporally broad pulse. The distorted chirped pulse was characterized by spectral interferometry (b), revealing numerous jumps in the extracted spectral phase [blue curve in (c)] when compared to the desired spectral phase [red curve in (c)]. As before, these jumps correspond to the phase wrap locations on the 2D LC SLM. When the modulator replica pulse features are removed using our diffractive pulse shaping scheme (see references [15, 16]) with the identical 2D LC SLM, the resulting smooth output waveform (green curve) shown in (d) is obtained. A cross-correlation measurement (blue curve) of the unshaped pulse is shown in comparison to the chirped pulse. Note that the slight structure present in the shaped pulse (green curve) shown in (d) is primarily determined by structure in the spectrum of the laser pulse itself. A spectral interferogram of the undistorted chirped pulse is shown in (e), where the extracted spectral phase [blue curve in (f)] of the undistorted pulse nearly exactly matches the desired spectral phase [red curve in (f)].

A second notable effect of pixel smoothing on output waveforms may actually help to clean up output pulses, when the phase pattern used to generate the output waveform does not require the use of phase wraps. One class of output waveforms commonly used in coherent control experiments (for instance, references [4, 25]) is a pulse train, or a series of evenly spaced pulses, generated by the application of a sinusoidal spectral phase. Typically the amplitude of the sinusoidal spectral phase is smaller than the maximum phase modulation range of the LC SLM being used. Since the phase modulation falls within the range of the device, no phase wraps are needed, eliminating phase distortions at the wrap points. This means that the otherwise pixelated phase response of the LC SLM is smoothed out, resulting in greatly diminished sampling replica pulses. Figure 8(a) shows simulated XFROG and cross-correlation measurements of a pulse train resulting from the application of a sinusoidal spectral phase by a device with well-defined (not smooth, but sharp) pixels. Chirped sampling replica pulses are observed. When the simulation is repeated for the case of smooth pixels, the sampling replica pulses disappear from the plot in (b). Note that the amount of pixel smoothing used in the simulation in (b) was comparable to that of our LC SLM.

## 6. Conclusions

To summarize, we have presented a detailed analysis of two categories of replica waveforms commonly encountered in pulse shaping when pixelated devices are used. While the first category is related to the pixelated nature of the device itself, the second category is caused by imperfections at the pixel boundaries.

We have not treated replica waveforms or artifacts which are evoked by miscalibration of the device. For example, an erroneous phase calibration of the LC SLM can create what sometimes appear to be modulator replica pulses but are not. Furthermore, if the wavelength-pixel calibration is incorrect, distorted pulse shapes will be produced. For instance, in the case where a single delayed pulse is desired, if the linear dispersion term is incorrect, the pulse will not be delayed to the correct times. Furthermore, if the nonlinear spectral dispersion term is incorrect or even neglected, the pulse will become increasingly chirped as it is delayed to larger times.

While neither sampling nor modulator replica pulses can be completely avoided, a detailed knowledge of their origin and their time-frequency distribution may be helpful to optimally suppress replica pulses for specific waveforms. For instance, one could take efforts to deliberately introduce nonlinear spectral dispersion in order to reduce sampling replica pulses. The analysis may also help in the design of next generation modulators and pulse shaping strategies. For example, devices with a larger overall phase modulation range would reduce the need for phase wraps and thereby reduce the appearance of modulator replica pulses. Use of a diffraction-based approach to pulse shaping [15, 16] permits suppression of modulator replica pulse features.

## Acknowledgments

This work was supported in part by National Science Foundation Grant No. CHE-0212375. We also would like to acknowledge the group of R. Sauerbrey at the University of Jena for providing the previously unpublished data shown in figure 2.

## References and links

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