## Analysis of replica pulses in femtosecond pulse shaping with pixelated devices

Optics Express, Vol. 14, Issue 3, pp. 1314-1328 (2006)

http://dx.doi.org/10.1364/OE.14.001314

Acrobat PDF (1715 KB)

### 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

1. J.P. Heritage, R.N. Thurston, W.J. Tomlinson, A.M. Weiner, and R.H. Stolen, “Spectral windowing of frequency-modulated optical pulses in a grating compressor,” Appl. Phys. Lett. **47**, 87–89 (1985). [CrossRef]

2. A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. **71**, 1929–1969 (2000). [CrossRef]

3. M.M. Wefers and K.A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B **12**, 1343–1362 (1995). [CrossRef]

04. N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature **418**, 512–514 (2002). [CrossRef] [PubMed]

7. N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B **18**, 1742–1746 (2001). [CrossRef]

8. H.P. Saradesai, C.-C. Chang, and A.M. Weiner, “A Femtosecond Code-Division Multiple-Access Communication System Test Bed,” J. Lightwave Technol. **16**, 1953–1964 (1998). [CrossRef]

9. F. Huang, W. Yang, and W.S. Warren, “Quadrature spectral interferometric detection and pulse shaping,” Opt. Lett. **26**, 362–364 (2001). [CrossRef]

3. M.M. Wefers and K.A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B **12**, 1343–1362 (1995). [CrossRef]

10. A.M. Weiner, D.E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable Shaping of Femtosecond Optical Pulses by Use of 128-Element Liquid Crystal Phase Modulator,” IEEE J. Quantum Electron. **28**, 908–920 (1992). [CrossRef]

11. H. Wang, Z. Zheng, D.E. Leaird, A.M. Weiner, T.A. Dorschner, J.J. Fijol, L.J. Friedman, H.Q. Nguyen, and L.A. Palmaccio, “20-fs Pulse Shaping With a 512-Element Phase-Only Liquid Crystal Modulator,” IEEE J. Sel. Top. Quantum Electron. **7**, 718–727 (2001). [CrossRef]

12. G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B **72**, 627–630 (2001). [CrossRef]

13. M. Hacker, G. Stobrawa, R. Sauerbrey, T. Buckup, M. Motzkus, M. Wildenhain, and A. Gehner, “Micromirror SLM for femtosecond pulse shaping in the ultraviolet,” Appl. Phys. B **76**, 711–714 (2003). [CrossRef]

14. A. Monmayrant and B. Chatel, “New phase and amplitude high resolution pulse shaper,” Rev. Sci. Instrum. **75**, 2668–2671 (2004). [CrossRef]

15. J.C. Vaughan, T. Hornung, T. Feurer, and K.A. Nelson, “Diffraction-based femtosecond pulse shaping with a 2D SLM,” Opt. Lett. **30**, 323–325 (2005). [CrossRef] [PubMed]

17. R. Trebino, K.W. DeLong, D.N. Fittinghoff, J.N. Sweetser, M.A. Krumbugel, and B.A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. **68**, 3277–3295 (1997). [CrossRef]

18. T. Feurer, J.C. Vaughan, R.M. Koehl, and K. Nelson, “Multidimensional control of femtosecond pulses by use of a programmable liquid crystal matrix,” Opt. Lett. **27**, 652–654 (2002). [CrossRef]

19. J.C. Vaughan, T. Feurer, and K.A. Nelson, “Automated two-dimensional femtosecond pulse shaping,” J. Opt. Soc. Am. B **19**, 2489–2495 (2002). [CrossRef]

## 2. General Analysis

*E*(

_{in}*ν*) and the spectral modulation

*M*(

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

*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,

*x*is the position of the

_{n}*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

*= ΔΩ*

_{n}*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

*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

*B*is the spectral amplitude of the input laser pulse at the

_{n}*n*th pixel. Substitution of the above expressions for

*M*(Ω) and

*E*(Ω) into eq. 1 yields

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

_{out}*-*

_{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.

*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.

*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

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

_{o}*π*, although 4

*π*or 6

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

*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*(Ω)

_{in}*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

3. M.M. Wefers and K.A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B **12**, 1343–1362 (1995). [CrossRef]

10. A.M. Weiner, D.E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable Shaping of Femtosecond Optical Pulses by Use of 128-Element Liquid Crystal Phase Modulator,” IEEE J. Quantum Electron. **28**, 908–920 (1992). [CrossRef]

11. H. Wang, Z. Zheng, D.E. Leaird, A.M. Weiner, T.A. Dorschner, J.J. Fijol, L.J. Friedman, H.Q. Nguyen, and L.A. Palmaccio, “20-fs Pulse Shaping With a 512-Element Phase-Only Liquid Crystal Modulator,” IEEE J. Sel. Top. Quantum Electron. **7**, 718–727 (2001). [CrossRef]

*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

20. J.X. Tull, M.A. Dugan, and W.S. Warren, “High resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. **20**, 1–56 (1997). [CrossRef]

21. F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. **25**, 575–577 (2000). [CrossRef]

2. A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. **71**, 1929–1969 (2000). [CrossRef]

*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

*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.

*n*in a power-series expansion:

*ϕ*between the replica pulse phase

_{n}*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

*n*by substitution of Ω

*given in eq. 12. Exact expressions for the coefficients*

_{n}*α*,

*β*,

*γ*,

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

*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.

*π*ΔΩ/4 (for linear spectral dispersion) or

*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).

*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.

*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

*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.

13. M. Hacker, G. Stobrawa, R. Sauerbrey, T. Buckup, M. Motzkus, M. Wildenhain, and A. Gehner, “Micromirror SLM for femtosecond pulse shaping in the ultraviolet,” Appl. Phys. B **76**, 711–714 (2003). [CrossRef]

*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

*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:

*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

*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

15. J.C. Vaughan, T. Hornung, T. Feurer, and K.A. Nelson, “Diffraction-based femtosecond pulse shaping with a 2D SLM,” Opt. Lett. **30**, 323–325 (2005). [CrossRef] [PubMed]

22. T. Hornung, J.C. Vaughan, T. Feurer, and K.A. Nelson, “Degenerate four-wave mixing spectroscopy based on two-dimensional femtosecond pulse shaping,” Opt. Lett. **29**, 2052–2054 (2004). [CrossRef] [PubMed]

*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.

*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

23. L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B **12**, 2467–2474 (1995). [CrossRef]

19. J.C. Vaughan, T. Feurer, and K.A. Nelson, “Automated two-dimensional femtosecond pulse shaping,” J. Opt. Soc. Am. B **19**, 2489–2495 (2002). [CrossRef]

*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.

11. H. Wang, Z. Zheng, D.E. Leaird, A.M. Weiner, T.A. Dorschner, J.J. Fijol, L.J. Friedman, H.Q. Nguyen, and L.A. Palmaccio, “20-fs Pulse Shaping With a 512-Element Phase-Only Liquid Crystal Modulator,” IEEE J. Sel. Top. Quantum Electron. **7**, 718–727 (2001). [CrossRef]

24. M.M. Wefers and K.A. Nelson, “Space-Time Profiles of Shaped Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. **32**, 161–172 (1996). [CrossRef]

*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.

*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

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

^{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

15. J.C. Vaughan, T. Hornung, T. Feurer, and K.A. Nelson, “Diffraction-based femtosecond pulse shaping with a 2D SLM,” Opt. Lett. **30**, 323–325 (2005). [CrossRef] [PubMed]

04. N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature **418**, 512–514 (2002). [CrossRef] [PubMed]

25. D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature **396**, 239–242 (1998). [CrossRef]

## 6. Conclusions

**30**, 323–325 (2005). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J.P. Heritage, R.N. Thurston, W.J. Tomlinson, A.M. Weiner, and R.H. Stolen, “Spectral windowing of frequency-modulated optical pulses in a grating compressor,” Appl. Phys. Lett. |

2. | A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. |

3. | M.M. Wefers and K.A. Nelson, “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B |

04. | N. Dudovich, D. Oron, and Y. Silberberg, “Single-pulse coherently controlled nonlinear Raman spectroscopy and microscopy,” Nature |

5. | S.A. Rice and M. Zhao, Optical Control of Molecular Dynamics (John Wiley and Sons, New York, 2000). |

6. | M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes (Wiley-Interscience, New Jersey, 2003). |

7. | N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, “Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation,” J. Opt. Soc. Am. B |

8. | H.P. Saradesai, C.-C. Chang, and A.M. Weiner, “A Femtosecond Code-Division Multiple-Access Communication System Test Bed,” J. Lightwave Technol. |

9. | F. Huang, W. Yang, and W.S. Warren, “Quadrature spectral interferometric detection and pulse shaping,” Opt. Lett. |

10. | A.M. Weiner, D.E. Leaird, J.S. Patel, and J.R. Wullert, “Programmable Shaping of Femtosecond Optical Pulses by Use of 128-Element Liquid Crystal Phase Modulator,” IEEE J. Quantum Electron. |

11. | H. Wang, Z. Zheng, D.E. Leaird, A.M. Weiner, T.A. Dorschner, J.J. Fijol, L.J. Friedman, H.Q. Nguyen, and L.A. Palmaccio, “20-fs Pulse Shaping With a 512-Element Phase-Only Liquid Crystal Modulator,” IEEE J. Sel. Top. Quantum Electron. |

12. | G. Stobrawa, M. Hacker, T. Feurer, D. Zeidler, M. Motzkus, and F. Reichel, “A new high-resolution femtosecond pulse shaper,” Appl. Phys. B |

13. | M. Hacker, G. Stobrawa, R. Sauerbrey, T. Buckup, M. Motzkus, M. Wildenhain, and A. Gehner, “Micromirror SLM for femtosecond pulse shaping in the ultraviolet,” Appl. Phys. B |

14. | A. Monmayrant and B. Chatel, “New phase and amplitude high resolution pulse shaper,” Rev. Sci. Instrum. |

15. | J.C. Vaughan, T. Hornung, T. Feurer, and K.A. Nelson, “Diffraction-based femtosecond pulse shaping with a 2D SLM,” Opt. Lett. |

16. | J.C. Vaughan, T. Feurer, T. Hornung, and K.A. Nelson, “Spatial, Temporal, and Spectral Properties of Two-Dimensional Femtosecond Pulse Shaping,” In preparation (2006). |

17. | R. Trebino, K.W. DeLong, D.N. Fittinghoff, J.N. Sweetser, M.A. Krumbugel, and B.A. Richman, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. |

18. | T. Feurer, J.C. Vaughan, R.M. Koehl, and K. Nelson, “Multidimensional control of femtosecond pulses by use of a programmable liquid crystal matrix,” Opt. Lett. |

19. | J.C. Vaughan, T. Feurer, and K.A. Nelson, “Automated two-dimensional femtosecond pulse shaping,” J. Opt. Soc. Am. B |

20. | J.X. Tull, M.A. Dugan, and W.S. Warren, “High resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Reson. |

21. | F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. |

22. | T. Hornung, J.C. Vaughan, T. Feurer, and K.A. Nelson, “Degenerate four-wave mixing spectroscopy based on two-dimensional femtosecond pulse shaping,” Opt. Lett. |

23. | L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B |

24. | M.M. Wefers and K.A. Nelson, “Space-Time Profiles of Shaped Ultrafast Optical Waveforms,” IEEE J. Quantum Electron. |

25. | D. Meshulach and Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature |

**OCIS Codes**

(320.0320) Ultrafast optics : Ultrafast optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: October 19, 2005

Revised Manuscript: January 21, 2006

Manuscript Accepted: January 22, 2006

Published: February 6, 2006

**Citation**

Joshua Vaughan, T. Feurer, Katherine Stone, and Keith Nelson, "Analysis of replica pulses in femtosecond pulse shaping with pixelated devices," Opt. Express **14**, 1314-1328 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1314

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### References

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