Abstract

We focus theoretically and experimentally on the fundamental limitations of spectral pulse shaping using an acousto-optic modulator. We analyze the role of the relative thickness of the interaction region as dictated by the acousto-optic Q parameter and show that varying Q allows flexibility in choosing between diffraction efficiency and pixels of resolution. We model and experimentally demonstrate the effects of potential nonidealities such as nonlinear acoustic attenuation. In addition, we derive a simple and intuitive expression to predict the magnitude of the distortions in the spatial profile generated by an acousto-optic-modulator spectral light modulator. Finally, we demonstrate amplification of acousto-optic-modulator-generated shaped pulses for the first time.

© 1997 Optical Society of America

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References

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  1. K. A. Nelson, “Coherent control: optics, molecules, and materials,” in Ultrafast Phenomena, A. H. Zewail, ed. (Springer-Verlag, New York, 1994), Vol. 9, p. 47.
  2. W. S. Warren, H. H. Rabitz, and M. Dahleh, “Coherent control: The dream is alive,” Science 259, 1581 (1993).
    [CrossRef] [PubMed]
  3. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” IEEE J. Quantum Electron. 19, 61 (1995).
  4. M. M. Wefers and K. A. Nelson, “Generation of high fidelity programmable ultrafast waveforms,” Opt. Lett. 20, 1047 (1995).
    [CrossRef]
  5. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
    [CrossRef]
  6. A. M. Weiner, D. E. Learid, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251 (1992); K. Ema and F. Shimuzu, “Optical pulse shaping using a Fourier transform hologram,” Jpn. J. Appl. Phys. 2, 631 (1990).
    [CrossRef]
  7. C. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, “Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses,” Opt. Lett. 19, 737 (1994).
    [CrossRef] [PubMed]
  8. J. X. Tull, M. A. Dugan, and W. S. Warren, “High resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Res. 20, 1 (1997).
  9. I. C. Chang, “Acousto-optic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
    [CrossRef]
  10. W. R. Klein and B. D. Cook, “Unified approaches to ultrasonic diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
    [CrossRef]
  11. G. Elston, M. Amano, and J. Lucero, “Material tradeoff for wideband Bragg cells,” Proc. SPIE 567, 150 (1985).
    [CrossRef]
  12. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 663.
  13. M. Wefers and K. A. Nelson, “Space-time profiles of shaped optical waveforms,” IEEE J. Quantum Electron. 32, 161 (1996); “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B 12, 1343 (1995).
    [CrossRef]
  14. J. Paye and A. Migus, “Analysis of a pulse shaper using space–time Wigner functions,” J. Opt. Soc. Am. B 12, 1480 (1995).
    [CrossRef]
  15. M. B. Danailov and I. P. Christov, “Time–space shaping of light pulses by Fourier optical processing,” J. Mod. Opt. 36, 725 (1989).
    [CrossRef]
  16. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1973).
  17. B. E. Lemoff and C. P. Barty, “Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses,” Opt. Lett. 18, 1651 (1993).
    [CrossRef] [PubMed]
  18. X. Liu, R. Wagner, A. Maksimchuk, E. Goodman, J. Workman, D. Umstadter, and A. Migus, “Nonlinear temporal diffraction and frequency shifts resulting from pulse shaping in chirped-pulse amplification systems,” Opt. Lett. 20, 1163 (1995).
    [CrossRef] [PubMed]

1995 (4)

1994 (1)

1993 (2)

1992 (1)

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

1989 (1)

M. B. Danailov and I. P. Christov, “Time–space shaping of light pulses by Fourier optical processing,” J. Mod. Opt. 36, 725 (1989).
[CrossRef]

1985 (1)

G. Elston, M. Amano, and J. Lucero, “Material tradeoff for wideband Bragg cells,” Proc. SPIE 567, 150 (1985).
[CrossRef]

1976 (1)

I. C. Chang, “Acousto-optic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
[CrossRef]

1967 (1)

W. R. Klein and B. D. Cook, “Unified approaches to ultrasonic diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

Amano, M.

G. Elston, M. Amano, and J. Lucero, “Material tradeoff for wideband Bragg cells,” Proc. SPIE 567, 150 (1985).
[CrossRef]

Barty, C. P.

Chang, I. C.

I. C. Chang, “Acousto-optic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
[CrossRef]

Christov, I. P.

M. B. Danailov and I. P. Christov, “Time–space shaping of light pulses by Fourier optical processing,” J. Mod. Opt. 36, 725 (1989).
[CrossRef]

Cook, B. D.

W. R. Klein and B. D. Cook, “Unified approaches to ultrasonic diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

Dahleh, M.

W. S. Warren, H. H. Rabitz, and M. Dahleh, “Coherent control: The dream is alive,” Science 259, 1581 (1993).
[CrossRef] [PubMed]

Danailov, M. B.

M. B. Danailov and I. P. Christov, “Time–space shaping of light pulses by Fourier optical processing,” J. Mod. Opt. 36, 725 (1989).
[CrossRef]

Elston, G.

G. Elston, M. Amano, and J. Lucero, “Material tradeoff for wideband Bragg cells,” Proc. SPIE 567, 150 (1985).
[CrossRef]

Goodman, E.

Goswami, D.

Hillegas, C.

Klein, W. R.

W. R. Klein and B. D. Cook, “Unified approaches to ultrasonic diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

Leaird, D. E.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

Lemoff, B. E.

Liu, X.

Lucero, J.

G. Elston, M. Amano, and J. Lucero, “Material tradeoff for wideband Bragg cells,” Proc. SPIE 567, 150 (1985).
[CrossRef]

Maksimchuk, A.

Migus, A.

Nelson, K. A.

Patel, J. S.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

Paye, J.

Rabitz, H. H.

W. S. Warren, H. H. Rabitz, and M. Dahleh, “Coherent control: The dream is alive,” Science 259, 1581 (1993).
[CrossRef] [PubMed]

Strickland, D.

Tull, J. X.

Umstadter, D.

Wagner, R.

Warren, W. S.

Wefers, M. M.

Weiner, A. M.

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” IEEE J. Quantum Electron. 19, 61 (1995).

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

Workman, J.

Wullert, J. R.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” IEEE J. Quantum Electron. 19, 61 (1995).

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable pulse shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. 28, 908 (1992).
[CrossRef]

IEEE Trans. Sonics Ultrason. (2)

I. C. Chang, “Acousto-optic devices and applications,” IEEE Trans. Sonics Ultrason. SU-23, 2 (1976).
[CrossRef]

W. R. Klein and B. D. Cook, “Unified approaches to ultrasonic diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123 (1967).
[CrossRef]

J. Mod. Opt. (1)

M. B. Danailov and I. P. Christov, “Time–space shaping of light pulses by Fourier optical processing,” J. Mod. Opt. 36, 725 (1989).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (4)

Proc. SPIE (1)

G. Elston, M. Amano, and J. Lucero, “Material tradeoff for wideband Bragg cells,” Proc. SPIE 567, 150 (1985).
[CrossRef]

Science (1)

W. S. Warren, H. H. Rabitz, and M. Dahleh, “Coherent control: The dream is alive,” Science 259, 1581 (1993).
[CrossRef] [PubMed]

Other (6)

K. A. Nelson, “Coherent control: optics, molecules, and materials,” in Ultrafast Phenomena, A. H. Zewail, ed. (Springer-Verlag, New York, 1994), Vol. 9, p. 47.

J. X. Tull, M. A. Dugan, and W. S. Warren, “High resolution, ultrafast laser pulse shaping and its applications,” Adv. Magn. Opt. Res. 20, 1 (1997).

A. M. Weiner, D. E. Learid, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251 (1992); K. Ema and F. Shimuzu, “Optical pulse shaping using a Fourier transform hologram,” Jpn. J. Appl. Phys. 2, 631 (1990).
[CrossRef]

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 663.

M. Wefers and K. A. Nelson, “Space-time profiles of shaped optical waveforms,” IEEE J. Quantum Electron. 32, 161 (1996); “Analysis of programmable ultrashort waveform generation using liquid-crystal spatial light modulators,” J. Opt. Soc. Am. B 12, 1343 (1995).
[CrossRef]

E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1973).

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Figures (13)

Fig. 1
Fig. 1

(a) Generic spectral pulse-shaping apparatus. A multielement modulator (typically a microlithographically fabricated mask or liquid-crystal array) shapes the spatially separated frequency components inside a zero-dispersion line. (b) Zero-dispersion line modified to use an AOM as a SLM. Compared with other SLM’s, AOM’s offer the advantages of a higher resolution with existing devices, faster waveform update rates, and a continuous phase and amplitude mask.

Fig. 2
Fig. 2

Experimental (dashed curve) and theoretical (solid curve) cross-correlation scans corresponding to a highly structured pulse train, combining linear, quadratic, cubic, and quartic spectral phase (see text).

Fig. 3
Fig. 3

Real (top) and imaginary (bottom) parts of the RF waveform corresponding to the train of chirped pulses in Fig. 2. The full-time aperture of the crystal is 10.2 µs (corresponding to a length of 43 mm); hence this highly structured RF waveform is simple to generate with commercially available waveform generators.

Fig. 4
Fig. 4

Theoretical falloff of diffraction efficiency (defined in terms of amplitude) as a function of normalized frequency (ν/ν0) is plotted for a rectangular transducer AOM with Q=4π. Equation (2) accurately predicts this falloff in the region from ν/ν0=0.61.2.

Fig. 5
Fig. 5

Fractional bandwidth, defined in terms of the 1-dB rolloff points of the square root of the third function of Eq. (2), is plotted as a function of Q.

Fig. 6
Fig. 6

Effects of intensity-dependent acoustic attenuation. Dashed curve: undistorted sinc function. Solid curve: nonlinearly attenuated sinc function for 1-W RF power, calculated by the measured acoustic attenuation with frequency and amplitude for our modulator.

Fig. 7
Fig. 7

Experimental shaped-pulse cross correlations (thin solid curve) resulting from an RF sinc function, which would give a rectangular laser pulse if there were no acoustic attenuation (thick curve). The dashed curve shows the theoretical cross correlation predicted after incorporating the power-dependent attenuation calibration measurements.

Fig. 8
Fig. 8

Spatial resolution criteria for AO pulse shaping are illustrated. The optical spectrum is linearly dispersed along the x axis, and the input beam propagates along the z axis. To maintain spatial resolution across the full interaction region defined by the transducer thickness L, the average beam diameter for a monochromatic frequency component, illustrated on the left side of the figure, must be less than the minimum acoustic-feature size. Since the modulator is aligned at the Bragg angle to the Fourier plane, only the frequency component corresponding to the center of the interaction region can be focused in the middle of the interaction region. A reasonable criteria to maintain spatial resolution is to require that the maximum variation in focal position be kept less than the Rayleigh range. In addition, the x displacement of the diffracted beam (wave vector kdif) in the interaction region, illustrated in the middle of the figure, must be made less than the minimum acoustic-feature size.

Fig. 9
Fig. 9

Ratio of the average beam diameter to the minimum possible acoustic-feature size as a function of Q. In order to obtain the maximum resolution from the modulator (given by the time-aperture bandwidth product) this ratio must be significantly less than 1.

Fig. 10
Fig. 10

SLM’s send different temporal slices of the shaped waveform in different directions. This is best seen by considering the effects of an AOM as a time-delay generator, which would imply a simple linear phase shift with frequency (generated by a linear phase shift with position). For the LCM case the linear phase shift is created by varying the index of refraction and hence the optical path length of the individual pixels, and the LCM array acts as a wedged optic. In the AOM case the linear phase shift is determined by the center RF frequency, which determines the Bragg angle. In both cases the time-delayed components of the shaped pulse are diffracted at different angles from the modulator; in the paraxial limit the second lens and grating pair [Fig. 1(b)] converts this into a spatial translation.

Fig. 11
Fig. 11

Time-dependent displacement of the shaped pulse is illustrated for a seven-pulse phase-locked pulse sequence by showing the relationship between the time-domain laser pulse and the RF spectrum. The shaped laser pulse is the convolution of the input laser pulse ein(tL) with SRF(tL), which has the same functional form as the RF spectrum in the limit of perfect spatial resolution. SRF(tL) is plotted both as a function of laser time tL and RF frequency ωRF, where these variables are related by the scaling factor tL=(0.91 ps/MHz)ωRF. Since the features of SRF(tL) are generally narrower than those of ein(tL), the amplitude of the shaped pulse at a given time is roughly proportional to the amplitude of a particular RF frequency. As discussed in Section 2, the diffraction angle is proportional to RF frequency, so that different time-delayed components are diffracted in different directions. The second lens and grating combination converts this variation of diffraction angles into a time-dependent displacement of the shaped pulse along the pulse-shaping axis, as illustrated in the lowest graph. The center spatial position of each pulse in the phase-locked train is displaced approximately 0.1 mm for our experimental conditions.

Fig. 12
Fig. 12

Amplified pulse shaping, with an AOM pulse shaper placed before the stretcher of a regenerative amplifier. Phase-modulated RF pulses can be used to compensate for spectral phase dispersion created by propagation effects and/or misalignment in the stretcher or compressor. In the figure above, successive iterations of third- and fourth-order spectral phase modulation reduce the autocorrelation signal of the amplified pulse to nearly the transform limit.

Fig. 13
Fig. 13

Autocorrelations of a series of amplified two-pulse phase-locked sequences are illustrated. The middle trace corresponds to the RF waveform that does not incorporate compensation for spectral phase dispersion generated by the amplifier. The bottom trace corresponds to a RF waveform that incorporates cubic and quartic spectral phase modulation of the correct magnitude, but incorrect sign. Finally, in the top trace, the RF waveform incorporates cubic and quartic modulation of the correct magnitude and sign necessary to produce a distortion-free phase-locked pulse sequence.

Equations (14)

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Q=2πλmLν02vac2=2πλmLΛ2.
II0=σ(Q)sin2π LλM2Ia21/2sinc2Q4νν01-νν0.
Eshaped(ωL)=Ein(ωL)M(ωL).
x=α(ωL-ω0L),
(x-x)=exp-2(x-x)2w02.
M(x)=-+sac(x)(x-x)dx.
FT[Eshaped(ωL)]=FT[Ein(ωL)M(ωL)]=Ein(tL)M(tL),
L=Q[F(Q)]2Δx022πλm.
w(z)Δx0=γ(Q)={0.038Q[F(Q)2]n}1/2.
ν0max<vac4πγ(Q)2λ2L0[F(Q)]21/3=vac0.327Qnλ2L01/3.
Npmax=0.69F(Q)(nQ)1/3L0λ2/3.
f(t)=-+G(ω)exp(iωt)dω;
f(t+τ)=-+[G(ω)exp(iωt)]exp(iωτ)dω,
eshaped(τL)=-+ein(tL-τL)SRF(tL)dtL.

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