Abstract

We propose a novel scheme for directly determining the intensity and phase of ultrashort pulses. The technique involves time-to-space conversion of an input pulse by a diffraction grating, followed by the detection of its spectral intensity distributions in the Fourier transform plane with and without an exponential filter placed in front of the grating. By use of an efficient noniterative phase-retrieval algorithm, the spectral phase of the pulse is retrieved from these spectral intensities. The intensity and phase of the pulse in the temporal domain can be reconstructed by the inverse Fourier transformation of the retrieved spectral amplitude. Computer simulation results confirm that robust reconstructions of a linearly chirped pulse and a pulse with quadratic and cubic spectral phases are possible with high fidelity.

© 1999 Optical Society of America

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  1. R. L. Fork, C. H. Brito-Cruz, P. C. Becker, C. V. Shank, “Compression of optical pulses to six femtoseconds by using cubic phase compensation,” Opt. Lett. 12, 483–485 (1987).
    [CrossRef] [PubMed]
  2. H. P. Weber, “Method for pulse width measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
    [CrossRef]
  3. A. M. Levine, E. Özizmir, R. Trebino, C. C. Hayden, A. M. Johnson, K. L. Tokuda, “Induced-grating autocorrelation of ultrashort pulses in a slowly responding medium,” J. Opt. Soc. Am. B 11, 1609–1618 (1994), and references therein.
    [CrossRef]
  4. Y. Tomita, M. Shibata, J. Bergquist, “Pulsewidth dependence of time-resolved two-photon absorption with picosecond pump-probe excitation,” J. Appl. Phys. 71, 2102–2105 (1992).
    [CrossRef]
  5. R. Trebino, D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency-resolved optical gating,” J. Opt. Soc. Am. A 10, 1101–1111 (1993).
    [CrossRef]
  6. D. J. Kane, R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823–825 (1993).
    [CrossRef] [PubMed]
  7. A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.
  8. N. Nakajima, “Phase retrieval from two intensity measurements using the Fourier series expansion,” J. Opt. Soc. Am. A 4, 154–158 (1987).
    [CrossRef]
  9. N. Nakajima, “Phase retrieval using the logarithmic Hilbert transform and the Fourier-series expansion,” J. Opt. Soc. Am. A 5, 257–262 (1988).
    [CrossRef]
  10. N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.
  11. C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.
  12. A. M. Weiner, D. E. Leaird, J. S. Patel, J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. 15, 326–328 (1990).
    [CrossRef] [PubMed]
  13. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
    [CrossRef]
  14. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
    [CrossRef]
  15. M. C. Nuss, M. Li, T. H. Chiu, A. M. Weiner, A. Partovi, “Time-to-space mapping of femtosecond pulses,” Opt. Lett. 19, 664–666 (1994).
    [CrossRef] [PubMed]
  16. P. C. Sun, Y. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
    [CrossRef]
  17. K. W. DeLong, R. Trebino, J. Hunter, W. E. White, “Frequency-resolved optical gating using second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
    [CrossRef]
  18. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [CrossRef]
  19. In the noniterative phase-retrieval experiment for image reconstruction a photographic film was used as the exponential filter with the 1/e2 width of about 1.5 mm [see N. Nakajima, “Reconstruction of phase objects from experimental far field intensities by exponential filtering,” Appl. Opt. 29, 3369–3374 (1990)]. The 1/e2 width of ∼0.4 mm can also be easily prepared by use of a holographic film or plate whose spatial resolution is on the order of thousands of lines per millimeters.
    [CrossRef] [PubMed]
  20. A. M. Kan’an, A. M. Weiner, “Efficient time-to-space conversion of femtosecond optical pulses,” J. Opt. Soc. Am. B 15, 1242–1245 (1998).
    [CrossRef]
  21. A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
    [CrossRef]
  22. Note that the purpose of preparing two replicas of a test pulse in our system is merely to realize the temporal gating of the filtered test pulse in the Fourier plane at an appropriate delay, not to obtain the spectral interferogram.

1998 (1)

1997 (2)

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

P. C. Sun, Y. T. Mazurenko, Y. Fainman, “Femtosecond pulse imaging: ultrafast optical oscilloscope,” J. Opt. Soc. Am. A 14, 1159–1170 (1997).
[CrossRef]

1994 (3)

1993 (2)

1992 (2)

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Y. Tomita, M. Shibata, J. Bergquist, “Pulsewidth dependence of time-resolved two-photon absorption with picosecond pump-probe excitation,” J. Appl. Phys. 71, 2102–2105 (1992).
[CrossRef]

1990 (2)

1988 (1)

1987 (2)

1986 (2)

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

1967 (1)

H. P. Weber, “Method for pulse width measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
[CrossRef]

Becker, P. C.

Bergquist, J.

Y. Tomita, M. Shibata, J. Bergquist, “Pulsewidth dependence of time-resolved two-photon absorption with picosecond pump-probe excitation,” J. Appl. Phys. 71, 2102–2105 (1992).
[CrossRef]

Brito-Cruz, C. H.

Chiu, T. H.

Colombeau, B.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

DeLong, K. W.

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

K. W. DeLong, R. Trebino, J. Hunter, W. E. White, “Frequency-resolved optical gating using second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
[CrossRef]

Fainman, Y.

Fittinghoff, D. N.

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

Fork, R. L.

Froehly, C.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

Hayden, C. C.

Hunter, J.

Johnson, A. M.

Kan’an, A. M.

Kane, D. J.

Krumbügel, M. A.

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

Leaird, D. E.

Levi, A.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Levine, A. M.

Li, M.

Martinez, O. E.

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
[CrossRef]

Mazurenko, Y. T.

Nakajima, N.

Nuss, M. C.

Özizmir, E.

Paek, E. G.

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Partovi, A.

Patel, J. S.

Reitze, D. H.

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

Richman, B. A.

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

Shank, C. V.

Shibata, M.

Y. Tomita, M. Shibata, J. Bergquist, “Pulsewidth dependence of time-resolved two-photon absorption with picosecond pump-probe excitation,” J. Appl. Phys. 71, 2102–2105 (1992).
[CrossRef]

Stark, H.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

Sun, P. C.

Sweetser, J. N.

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

Tokuda, K. L.

Tomita, Y.

Y. Tomita, M. Shibata, J. Bergquist, “Pulsewidth dependence of time-resolved two-photon absorption with picosecond pump-probe excitation,” J. Appl. Phys. 71, 2102–2105 (1992).
[CrossRef]

Trebino, R.

Vampouille, M.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

Weber, H. P.

H. P. Weber, “Method for pulse width measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
[CrossRef]

Weiner, A. M.

White, W. E.

Wullert, J. R.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

A. M. Weiner, D. E. Leaird, D. H. Reitze, E. G. Paek, “Femtosecond spectral holography,” IEEE J. Quantum Electron. 28, 2251–2261 (1992).
[CrossRef]

J. Appl. Phys. (2)

H. P. Weber, “Method for pulse width measurement of ultrashort light pulses generated by phase-locked lasers using nonlinear optics,” J. Appl. Phys. 38, 2231–2234 (1967).
[CrossRef]

Y. Tomita, M. Shibata, J. Bergquist, “Pulsewidth dependence of time-resolved two-photon absorption with picosecond pump-probe excitation,” J. Appl. Phys. 71, 2102–2105 (1992).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
[CrossRef]

Opt. Lett. (4)

Rev. Sci. Instrum. (1)

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

Other (4)

Note that the purpose of preparing two replicas of a test pulse in our system is merely to realize the temporal gating of the filtered test pulse in the Fourier plane at an appropriate delay, not to obtain the spectral interferogram.

A. Levi, H. Stark, “Restoration from phase and magnitude by generalized projections,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), pp. 277–320.

N. Nakajima, “Phase retrieval using the properties of entire functions,” in Advances in Imaging and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1995), Vol. 93, pp. 109–171.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. XX, pp. 65–153.

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the pulse reconstruction system based on time-to-space conversion by a grating and on the noniterative phase-retrieval algorithm by use of an exponential filter; the SHG crystal is used to gate pulse 2 by pulse 1 in the Fourier plane. Linear detector arrays (D1 and D2) are used to measure intensity distributions frequency 2ω and ω, respectively, in the Fourier plane.

Fig. 2
Fig. 2

Top view of the pulse reconstruction system used to analyze optical fields in the grating and Fourier planes. The beam splitter shown in Fig. 1 is not illustrated here for simplicity. The plane for detector arrays D2 shown in Fig. 1 is equivalent to the u plane.

Fig. 3
Fig. 3

Illustrations that describe the pulse spread relative to the spatial beam profile and the size of the exponential filter in the x2 plane for (a) pulse 1 and (b) pulse 2. Positions for pulse 1 and pulse 2 are at the initial gating time immediately after pulse 2 has passed across the unity transmittance position of the filter at x2=0. Position xa denotes the tail position of pulse 2 at the initial gating time, while position xb denotes the 1/e2 position of the filter’s amplitude transmittance.

Fig. 4
Fig. 4

Original test pulse with a linear chirp. The modulus and phase of the test pulse are represented by the solid and the short-dashed curves, respectively. The long-dashed curve denotes the round-trip amplitude transmittance distribution of the exponential filter converted into the time domain, which is evaluated by use of the time-to-space scaling factor of 2πβ/λ0.

Fig. 5
Fig. 5

Reconstruction of the test pulse amplitude shown in Fig. 4 from noiseless moduli: (a) the spectral modulus (solid curve) of the test pulse and the modulus (dashed curve) of the complex amplitude corresponding to the second-harmonic (2ω) field, and (b) the moduli and phases of the reconstructed pulse (solid curve) and the test pulse (dashed curve).

Fig. 6
Fig. 6

Same as Fig. 5, except that background noises were added to the spectral intensities.

Fig. 7
Fig. 7

Original pulse with quadratic and cubic spectral phases. The modulus and the phase of the test pulse in a temporal domain are represented by the solid and the short-dashed curves, respectively. The long-dashed curve denotes the same round-trip amplitude transmittance distribution of the exponential filter converted into the time domain as shown in Fig. 4.

Fig. 8
Fig. 8

Reconstruction of the test pulse amplitude shown in Fig. 7 from noiseless moduli: (a) the spectral modulus (solid curve) of the test pulse and the modulus (dashed curve) of the complex amplitude corresponding to the second-harmonic (2ω) field, and (b) the moduli and phases of the reconstructed pulse (solid curve) and the test pulse (dashed curve).

Fig. 9
Fig. 9

Same as Fig. 8, except that background noises were added to the spectral intensities.

Equations (27)

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p1(x1; ω)=A(ω)w(x1),
p1(x2; ω)=expi2πλ0β(ω-ω0)x2A(ω)w(αx2),
p1(x2; t)=at+2πλ0βx2w(αx2)exp-i2πλ0βω0x2,
a(t)=12π0A(ω)exp(iωt)dω.
P1(u; t)=(T1)1/2-at+2πλ0βx2w(αx2)×exp-i2πλ0βω0x2exp-i2πux2λ0fdx2=(T1)1/22πα0A(ω)Wβ(ω0-ω)λ0α+uλ0αf×exp(iωt)dω,
P1(u; t)Aω0+uβf(T1)1/22πα0Wβ(ω0-ω)λ0α+uλ0αfexp(iωt)dω=Aω0+uβfλ0(T1)1/22πβ×expiω0+uβftw-λ0αt2πβ,
p2(x2; t)=at-2πλ0βx2exp(-2πqx2)w(αx2)×expi2πλ0βω0x2.
P2(u; t)Aω0-u-iλ0fqβfλ02πβ×expiω0-u-iλ0fqβftwλ0αt2πβ.
S(u; t)ηP1(u; t-τ)P2(u; t),
I2(u)=|S(u; t)|2,
=K1Aω0+uβf2Aω0-u-iλ0fqβf2,
K1=ηλ04T1(2πβ)4w-λ0α2πβ(t-τ)2×wλ0αt2πβ2 exp-2λ0qtβ.
I1(u)=K2|A(ω0+u/βf)|2,
K2=R1λ02(2πβ)2w-λ0α2πβt2,
A(ω0-u/βf)=M(u)exp[iϕ(u)].
I2(u)=K1M2(-u)|M(u-iv)|2 exp[-2 Im ϕ(u-iv)],
I1(u)=K2M2(-u),
12lnI2(u)I1(u)|I1(-u+iv)|=-Im ϕ(u-iv)-12lnK22K1.
I1(-u+iv)=--I1(u)exp(2πixu)du×exp(2πvx)exp(2πixu)dx.
D(u)=-Im ψ(u-iv),
ψ(u)n=1Nan cosnπlu+bn sinnπlu,
D(u)n=1N-an sinnπlu+bn cosnπlusinhnπvl.
M[(ω-ω0)/2π]=exp[-(ω-ω0)2/4π2(5.422/2 ln 2)2]
ϕ[(ω-ω0)/2π]=[0.011(ω-ω0)2+0.001579(ω-ω0)3]/4π2.
0.02σspacemm-1<q<0.25σspacemm-1.
0.05mm<σspace<0.625mm.
29fs<τFWHM<367fs.

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