Abstract

When a Dammann grating is used to split a beam of femtosecond laser pulses into multiple equal-intensity beams, chromatic dispersion will occur in beams of each order of diffraction and with different scale of angular dispersion because the incident ultrashort pulse contains a broad range of spectral bandwidths. We propose a novel method in which the angular dispersion can be compensated by positioning an m-time-density grating to collimate the mth-order beam that has been split, producing an array of beams that are free of angular dispersion. The increased width of the compensated output pulses and the spectral walk-off effect are discussed. We have verified this approach theoretically and validated it through experiments. It should be highly interesting in practical applications of splitting femtosecond laser pulses for pulse-width measurement, pump-probe measurement, and micromachining at multiple points.

© 2005 Optical Society of America

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References

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  3. R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  8. R. L. Morrison, “Symmetries that simplify the design of spot array phase gratings,” J. Opt. Soc. Am. A 9, 464–471 (1992).
    [CrossRef]
  9. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
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    [CrossRef]
  11. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  13. J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1996).
  14. C. Zhou, L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34, 5961–5969 (1995).
    [CrossRef] [PubMed]
  15. C. Zhou, J. Jia, L. Liu, “Circular Dammann grating,” Opt. Lett. 28, 2174–2176 (2003).
    [CrossRef] [PubMed]
  16. D. Strickland, G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
    [CrossRef]
  17. P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
    [CrossRef]

2003 (2)

N. H. Rizvi, “Femtosecond laser micromachining: Current status and applications,” RIKEN Rev. 50, 107–112 (2003).

C. Zhou, J. Jia, L. Liu, “Circular Dammann grating,” Opt. Lett. 28, 2174–2176 (2003).
[CrossRef] [PubMed]

2002 (1)

2000 (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

1995 (1)

1992 (1)

1991 (1)

1988 (1)

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

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]

1985 (1)

D. Strickland, G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Amako, J.

Bado, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

Chilla, J. L. A.

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

Goodman, J. W.

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1996).

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Iaconis, C.

Jia, J.

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Kazuhiro, N.

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Krumbügel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

Liu, L.

Maine, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

Martinez, O. E.

Morrison, R. L.

Mourou, G.

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

D. Strickland, G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

Nagasaka, K.

Pessot, M.

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

Rizvi, N. H.

N. H. Rizvi, “Femtosecond laser micromachining: Current status and applications,” RIKEN Rev. 50, 107–112 (2003).

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Strickland, D.

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

D. Strickland, G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Trebino, R.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, 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]

Walmsley, I. A.

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Zhou, C.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

P. Maine, D. Strickland, P. Bado, M. Pessot, G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24, 398–403 (1988).
[CrossRef]

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

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

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

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun. (2)

D. Strickland, G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219–221 (1985).
[CrossRef]

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, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

RIKEN Rev. (1)

N. H. Rizvi, “Femtosecond laser micromachining: Current status and applications,” RIKEN Rev. 50, 107–112 (2003).

Science (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Other (2)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

J. W. Goodman, Fourier Optics (McGraw-Hill, New York, 1996).

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Figures (7)

Fig. 1
Fig. 1

Geometrical arrangement of splitting and compensating femtosecond pulses by a Dammann grating and the corresponding compensation gratings.

Fig. 2
Fig. 2

(a) Output pulse width versus Dammann grating period; (b) broadening sources A and B versus Dammann grating period.

Fig. 3
Fig. 3

Comparison of the output pulse width of the Dammann splitter and the BK7 substrate.

Fig. 4
Fig. 4

Comparison of pulse widths of focused and unfocused beams.

Fig. 5
Fig. 5

(a) Femtosecond laser beams are split from a 1 × 8 Dammann grating ( d = 100   µ m ) without compensation. Note that the angular dispersion increases as the diffraction order m is increased, so the diffraction spots are becoming more elliptical. (b) Fifth diffraction order that has already been compensated by a compensation grating with period d c = 20   µ m .

Fig. 6
Fig. 6

Focused fifth-order diffraction spot of the split 1 × 8 array, (a) uncompensated and (b) compensated by a compensation grating. The focal length of the lens is f = 350   mm .

Fig. 7
Fig. 7

Output pulse widths of fifth order are measured by a single-shot autocorrelator by changing the distance L between the Dammann grating and the compensation grating.

Equations (16)

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d [ sin   γ + sin ( γ - θ 1 ) ] = m λ ,
d c = d / m ,
d c [ sin ( γ - θ 1 ) + sin ( γ - θ 1 - θ 2 ) ] = λ ,
S = L m d   λ 0 - m - 2 d   λ 0 ,
P = D + L   m 2 d   Δ λ + L   | m - 2 | 2 d   Δ λ ,
L min = 2 Dd / ( 4 λ 0 - ( m + | m - 2 | ) Δ λ ) .
E i ( t ) = exp [ - 2   ln   2 ( t 2 / τ FWHM 2 ) ] ,
τ out = τ FWHM 2 + 8   ln   2 β 2 L 2 σ 2 + ( 4   ln   2 k β 2 L ) 2 τ FWHM 2 + 8   ln   2 β 2 L 2 σ 2 1 / 2 = ( τ FWHM 2 + A + B ) 1 / 2 ,
n 2 ( λ ) = 1 + B 1 λ 2 λ 2 - C 1 + B 2 λ 2 λ 2 - C 2 + B 3 λ 2 λ 2 - C 3 ,
τ out = τ FWHM 2 + 4   ln   2   λ 0 3 L d 2 π C 2 d 2 n ( λ ) d λ 2 2 τ FWHM 2 1 / 2 ,
E ( x o ,   y o ,   ω ) = E i ( ω ) exp ( ik β 2 ω 2 L / 2 ) × exp { - [ ( x o + β ω L ) 2 + y o 2 ] / σ 2 } ,
E ( x f ,   y f ,   ω ) = A   exp [ i ( k / 2 f ) ( 1 - h / f ) ( x f 2 + y f 2 ) ] i λ f × E ( x o ,   y o ,   ω ) exp [ - i ( k / f ) ( x o x f + y o y f ) ] d x o d y o ,
E ( x f ,   y f ,   ω ) = exp ( - τ FWHM 2 ω 2 / 8   ln   2 ) × exp ( ik β 2 ω 2 L / 2 ) exp ik β ω L f   x f × exp - k 2 σ 2 4 f 2   ( x f 2 + y f 2 ) .
E ( x f ,   y f ,   t ) = E ( x o ,   y o ,   ω ) exp ( - i ω t ) d ω = exp - k 2 σ 2 4 f 2   ( x f 2 + y f 2 ) × exp - 2   ln   2 t - k β L f   x f 2 τ 2 × exp i 8 ( ln   2 ) 2 k β 2 L t - k β L f   x f 2 τ FWHM 4 + ( 4   ln   2 k β 2 L ) 2 ,
τ 2 = τ FWHM 2 + ( 4   ln   2 k β 2 L ) 2 τ FWHM 2 .
tan   ψ = k β LC f .

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