Abstract

Acousto-optic programmable dispersive filters (AOPDF) can compensate in real time for large amounts of group-delay dispersion. This feature can be used in chirped-pulse amplification femtosecond laser chains to compensate adaptively for dispersion. An analytical expression relating the group delay at the output of the AOPDF to the input acoustic signal is obtained with coupled-wave theory in the case of collinear and quasi-collinear bulk acousto-optic interactions and also in the case of planar waveguides and optical fibers. With this relation, the acoustic signal that will induce an arbitrary group-delay variation with frequency can be easily obtained. Numerical simulations are shown to support the principle of arbitrary group-delay control with an AOPDF.

© 2000 Optical Society of America

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References

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  1. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
    [CrossRef]
  2. M. A. Wefers and K. A. Nelson, “Programmable phase and amplitude femtosecond pulse shaping,” Opt. Lett. 18, 2032–2034 (1993).
    [CrossRef] [PubMed]
  3. M. A. 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]
  4. K. M. Mahoney and A. M. Weiner, “A femtosecond pulse-shaping apparatus containing microlens arrays for use with pixellated spatial light modulators,” IEEE J. Quantum Electron. 32, 2071–2077 (1996).
    [CrossRef]
  5. C. Dorrer, F. Salin, F. Verluise, and J.-P. Huignard, “Programmable phase control of femtosecond pulses by use of a nonpixelated spatial light modulator,” Opt. Lett. 23, 709–711 (1998).
    [CrossRef]
  6. A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, “Adaptive control of pulse phase in a chirped-pulse amplifier,” Opt. Lett. 23, 1915–1917 (1998).
    [CrossRef]
  7. M. E. Fermann, V. da Silva, D. A. Smith, Y. Silberberg, and A. M. Weiner, “Shaping of ultrashort optical pulses by using an integrated acousto-optic tunable filter,” Opt. Lett. 18, 1505–1507 (1993).
    [CrossRef] [PubMed]
  8. P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140, 245–249 (1997).
    [CrossRef]
  9. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  10. F. Verluise, V. Laude, J.-P. Huignard, and P. Tournois, “Design of an improved acousto-optic programmable dispersive filter,” in Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), p. 99.
  11. M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995).
  12. D. F. Elliot, ed., Handbook of Digital Signal Processing (Academic, San Diego, 1986).

1998 (2)

1997 (1)

P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140, 245–249 (1997).
[CrossRef]

1996 (1)

K. M. Mahoney and A. M. Weiner, “A femtosecond pulse-shaping apparatus containing microlens arrays for use with pixellated spatial light modulators,” IEEE J. Quantum Electron. 32, 2071–2077 (1996).
[CrossRef]

1995 (1)

1993 (2)

1992 (1)

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
[CrossRef]

Beach, N. M.

da Silva, V.

Dorrer, C.

Efimov, A.

Fermann, M. E.

Huignard, J.-P.

Krause, J. L.

Leaird, D. E.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
[CrossRef]

Mahoney, K. M.

K. M. Mahoney and A. M. Weiner, “A femtosecond pulse-shaping apparatus containing microlens arrays for use with pixellated spatial light modulators,” IEEE J. Quantum Electron. 32, 2071–2077 (1996).
[CrossRef]

Moores, M. D.

Nelson, K. A.

Patel, J. S.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
[CrossRef]

Reitze, D. H.

Salin, F.

Silberberg, Y.

Smith, D. A.

Tournois, P.

P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140, 245–249 (1997).
[CrossRef]

Verluise, F.

Wefers, M. A.

Weiner, A. M.

K. M. Mahoney and A. M. Weiner, “A femtosecond pulse-shaping apparatus containing microlens arrays for use with pixellated spatial light modulators,” IEEE J. Quantum Electron. 32, 2071–2077 (1996).
[CrossRef]

M. E. Fermann, V. da Silva, D. A. Smith, Y. Silberberg, and A. M. Weiner, “Shaping of ultrashort optical pulses by using an integrated acousto-optic tunable filter,” Opt. Lett. 18, 1505–1507 (1993).
[CrossRef] [PubMed]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
[CrossRef]

Wullert II, J. R.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. M. Mahoney and A. M. Weiner, “A femtosecond pulse-shaping apparatus containing microlens arrays for use with pixellated spatial light modulators,” IEEE J. Quantum Electron. 32, 2071–2077 (1996).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert II, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid-crystal phase modulator,” IEEE J. Quantum Electron. 28, 908–919 (1992).
[CrossRef]

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

Opt. Commun. (1)

P. Tournois, “Acousto-optic programmable dispersive filter for adaptive compensation of group delay time dispersion in laser systems,” Opt. Commun. 140, 245–249 (1997).
[CrossRef]

Opt. Lett. (4)

Other (4)

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

F. Verluise, V. Laude, J.-P. Huignard, and P. Tournois, “Design of an improved acousto-optic programmable dispersive filter,” in Conference on Lasers and Electro-Optics (CLEO/U.S.), Vol. 6 of 1998 OSA Technical Digest Series (Optical Society of America, Washington D.C., 1998), p. 99.

M. Bass, ed., Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995).

D. F. Elliot, ed., Handbook of Digital Signal Processing (Academic, San Diego, 1986).

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of the AOPDF principle. The acoustic wave and the incident and diffracted optical waves are collinear and propagating along the z axis. See text for details.

Fig. 2
Fig. 2

Graphical representation of the computation of the acoustic signal given a desired group-delay function τ(ω). The group-delay equation (20) gives z(ω); i.e., the position at which phase matching should occur for each frequency (arrow 1). The phase-matching condition [Eq. (21)] then gives the acoustic frequency K(z) (arrow 2).

Fig. 3
Fig. 3

Determination of the compensation bandwidth. vg1(ω) and vg2(ω) are the optical group velocities on modes 1 and 2, respectively; vg(ω) is the group delay of the diffracted pulse.

Fig. 4
Fig. 4

Possible group-delay trajectories. The AOPDF can impose an arbitrary group-delay variation τ(ω) on the diffracted optical pulse, provided that the corresponding group velocity vg(ω)=L/τ(ω) remains between vg1(ω) and vg2(ω).

Fig. 5
Fig. 5

Representation of the quasi-collinear acousto-optic interaction geometry.

Fig. 6
Fig. 6

Spectral intensity and phase induced on mode 2 when the acoustic signal is computed from Eq. (24) for a 3-cm-long collinear AOPDF. The inset shows a magnified portion of the spectral phase versus frequency curve.

Fig. 7
Fig. 7

Group delay computed from the spectral phase of Fig. 6.

Fig. 8
Fig. 8

Numerical simulation of the propagation of an incident 15-fs Gaussian pulse through a 3-cm-long collinear AOPDF. The spectrum of the incident pulse is centered on 380 THz and is 29.4-THz FWHM large. The inset shows the magnified central portion of the curves.

Fig. 9
Fig. 9

Five examples of group-delay control. Table 1 lists the designed versus the obtained dispersion values.

Tables (1)

Tables Icon

Table 1 Designed Versus Obtained Dispersion Values for the Five Examples of Fig. 9

Equations (37)

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(z)=+21(z)cos[ψ(z)],
Ψ(z)=0zK(z)dz.
E(t, r)=E(ω, r)exp(iωt)dω.
Em(ω, r)=S(ω)em exp[-ikm(ω)z],
em=2μ0ωkm(ω)1/2pm.
d2dz2E(ω, r)+ω2μ0(z)E(ω, r)=0.
E(ω, r)=S(ω){A1(z)e1 exp[-ik1(ω)z]+A2(z)e2 exp[-ik2(ω)z]},
ddzA1(z)=-iκ(z)A2(z){exp[-iϕ+(z)]+exp[-iϕ-(z)]},
ddzA2(z)=-iκ(z)A1(z){exp[iϕ+(z)]+exp[iϕ-(z)]},
ϕ±(z)=[k2(ω)-k1(ω)]z±ψ(z),
κ(z)=ω2μ02[k1(ω)k2(ω)]1/2p1·1(z)·p2.
ddzϕ±(z)=0=k2(ω)-k1(ω)±K(z).
K[z(ω)]=k1(ω)-k2(ω)=ωc[n1(ω)-n2(ω)].
A2(z)a(ω)exp{iϕ[z(ω)]},z>z(ω),
φ(ω)=k2(ω)L-ϕ[z(ω)]=k1(ω)z(ω)-ψ[z(ω)]+k2(ω)[L-z(ω)].
τ(ω)=dz(ω)dω{k1(ω)-k2(ω)-K[z(ω)]}+dk1(ω)dωz(ω)+dk2(ω)dω[L-z(ω)].
τ(ω)=z(ω)vg1(ω)+L-z(ω)vg2(ω).
Δvg(ω)=[1/vg1(ω)-1/vg2(ω)]-1,
τ(ω)=z(ω)Δvg(ω)+Lvg2(ω)
z(ω)=Δvg(ω)τ(ω)-Lvg2(ω).
K[z(ω)]=ωΔvp(ω),
τ(ω)=Dω+τ0,
K[z(ω)]=1DΔvp(ω)z(ω)Δvg(ω)+Lvg2(ω)-τ0.
z(ω)=Δvg(ω)T-Lvg2(ω).
Lvg2(ω)TLvg1(ω),
maxω[vg1(ω)]minω[vg2(ω)].
vg(ω)=z(ω)L1vg1(ω)+L-z(ω)L1vg2(ω)-1.
s(t)=cos0tΩ(t)dt,
Ω(t)=VK(Vt).
Ω(t)=VΔvpVΔvgtD+Ω0,
τz0(ω)=τ(ω)+z0Δvg(ω),
Em(ω, r)=S(ω)em exp[-ikm(ω)·r]=S(ω)em exp{-i[αm(ω)x+βm(ω)z]},
ϕ(z)=[β2(ω)-β1(ω)]z-ψ(z)
α2(ω)=α1(ω),
K(z)=β2(ω)-β1(ω).
φ(ω)=k2(ω)·r-ϕ[z(ω)].
τ(ω)=z(ω)vg1z(ω)+L-z(ω)vg2z(ω)+xvgx(ω).

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