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

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References

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

Adv. Magn. Opt. Reson.

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]

Appl. Phys. B

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]

Appl. Phys. B

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]

Appl. Phys. Lett.

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]

IEEE J. Quantum Electron

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

IEEE J. Quantum Electron.

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]

IEEE J. Sel. Top. Quantum Electron.

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]

J. Lightwave Technol.

J. Opt. Soc. Am. B

J. Opt. Soc. Am. B

J. Opt. Soc. Am. B

J. Opt. Soc. Am. B

Nature

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

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

Opt. Lett.

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

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]

Opt. Lett.

Opt. Lett.

Rev. Sci. Instrum.

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

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

Rev. Sci.Instrum.

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 grating.” Rev. Sci.Instrum 68, 3277–3295 (1997).
[CrossRef]

Other

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).

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

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

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Figures (8)

Fig. 1.
Fig. 1.

Schematic illustration of experimental apparatus used for temporal-only pulse shaping and representative input and output pulse shapes.

Fig. 2.
Fig. 2.

Experimental XFROG measurements of waveforms resulting from the application of a linear spectral phase, illustrating various waveform distortions. The y-axis of the plots is wavelength, increasing from top (378 nm) to bottom (432 nm), and the color map is logarithmic. Each plot is rescaled so that the maximum intensity within each is the same color. The “desired” waveform in each case is a single pulse with a temporal delay between 0 and 14 ps. These measurements are courtesy of the group of professor Roland Sauerbrey of FSU in Jena, Germany.

Fig. 3.
Fig. 3.

Illustration of sampling replica pulses. Applied phase (black boxes), desired phase (blue line for R = 0), and sampling replica pulse phases (red line for R = 1 and green line for R = -1) in the case of (a) linear and (b) nonlinear spectral dispersion. The grey vertical lines represent pixel boundaries. Simulated XFROG and cross-correlation measurements (on a logarithmic scale) of the corresponding output waveforms for linear (c) and nonlinear (d) spectral dispersion. To make the curvature of the R = ±1 replica pulses more apparent, the extent of nonlinear spectral dispersion was exaggerated in (b), but the actual nonlinear spectral dispersion of our apparatus was used for the simulation in (d).

Fig. 4.
Fig. 4.

(a) Simulated spectral phases with slope 2π × 0.4 ps: desired phase (red); applied phase (black); unwrapped applied phase (blue). (b) simulated output pulse at -0.4 ps with weak modulator replica pulses separated by 0.4 ps. (c) Spectral interferogram of a pulse shifted to -0.4 ps. (d) Extracted (blue) and desired (red) spectral phase from (c) showing smoothed out pixel wraps.

Fig. 5.
Fig. 5.

(a) Experimental cross-correlation measurement of a pulse shifted to negative 3 ps with modulator replica pulses and (b) a simulation of the cross-correlation measurement. (c) Delayed pulse peak intensity (dots) with the simulated time window including the effects of modulator replica pulses (solid) compared to Gaussian-sinc time window for the pulse shaping apparatus.

Fig. 6.
Fig. 6.

Simulations illustrating the dependence of modulator replica on the periodicity of pixel wraps in the cases of linear spectral dispersion (a) and nonlinear spectral dispersion (b).

Fig. 7.
Fig. 7.

Experimental cross-correlation measurements of chirped pulses with (a) and without (d) modulator replica pulse features. For comparison, the blue curve in (d) shows the unshaped pulse. Spectral interferogram (b) and the extracted spectral phase (c) of a chirped pulse with modulator replica features. Spectral interferogram (e) and extracted spectral phase (f) of a chirped pulse where modulator replica pulses have been eliminated using a diffraction-based pulse shaping scheme. The measured spectral phases in (c) and (f) are shown in blue, while the desired spectral phase is shown in red.

Fig. 8.
Fig. 8.

Simulated XFROG and cross-correlation measurements (on a logarithmic scale) of waveforms generated by application of a sinusoidal spectral phase that does not exceed 2π with (a) a LC SLM that has sharp pixels and (b) one that has smooth pixels.

Equations (23)

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E out ( ν ) = M ( ν ) E in ( ν ) .
M ( x ) = S ( x ) n = N 2 N 2 1 squ ( x x n Δ x ) A n exp ( i ϕ n ) ,
squ ( x ) = { 1 x 1 2 0 x > 1 2 .
M ( Ω ) = exp ( Ω 2 δ Ω 2 ) n = N 2 N 2 1 squ ( Ω Ω n ΔΩ ) A n exp ( i ϕ n ) .
E in ( Ω ) = n = N 2 N 2 1 squ ( Ω Ω n ΔΩ ) B n ,
E out ( Ω ) = exp ( Ω 2 δ Ω 2 ) n = N 2 N 2 1 squ ( Ω Ω n ΔΩ ) A n B n exp ( i ϕ n ) .
e out ( t ) exp ( π 2 δ Ω 2 t 2 ) sin c ( π ΔΩ t ) n = N 2 N 2 1 A n B n exp [ i ( 2 π ν n t + ϕ n ) ] .
ϕ ( x ) = L ( x ) n = N 2 N 2 1 { mod [ ϕ n , 2 π ] squ ( x x n Δ x w ) + mod [ ϕ o , 2 π ] squ ( x x n + Δx 2 w ) } ,
M ( Ω ) exp [ ( x ) ] exp [ ( x x ( Ω ) ) 2 δ x 2 ] dx .
n = N 2 N 2 1 A n B n exp [ i ( 2 π ν n t + ϕ n ) ] .
ϕ replica , n = ϕ applied , n + 2 πRn ,
Ω n = ΔΩ ¯ n + K n 2 + L n 3 + M n 4 +
Δ ϕ n = 2 πRn
= α Ω n + β Ω n 2 + γ Ω n 3 + ,
α 2 πR = 1 ΔΩ ¯
β 2 πR = K ΔΩ ¯ 3
γ 2 πR = L ΔΩ ¯ 4 + 2 K 2 ΔΩ ¯ 5
δ 2 πR = M ΔΩ ¯ 5 + 5 K L ΔΩ ¯ 6 5 K 3 ΔΩ ¯ 7 .
M ( Ω ) = squ [ Ω N ΔΩ ] { comb [ Ω ΔΩ ] squ [ Ω ΔΩ 2 w ΔΩ Δ x ] exp [ i ϕ o ] +
( exp [ i ϕ applied ( Ω ) ] comb [ Ω ΔΩ ] ) squ [ Ω ΔΩ ( 1 w Δ x ) ] } .
comb ( Ω ) = n = n = δ ( Ω n ) ,
m ( t ) sin c [ πN ΔΩ t ] { comb [ ΔΩ t ] w ΔΩ Δ x sin c [ π w ΔΩ Δ x t ] exp [ i ϕ o ] +
comb [ ΔΩ t ] ΔΩ ( 1 w Δ x ) sin c [ π ΔΩ ( 1 w Δ x ) t ] ( n = N 2 N 2 1 exp [ i ( 2 π ν n t + ϕ n ) ] ) } .

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