Abstract

We propose a general method to calculate the dispersion of an arbitrary optical system. It is based on a nonlinear extension of the ABCD matrix model, where each optical element is described as an operator rather than a matrix. The deviation from a reference ray in terms of transverse position, angle, and phase as a function of wavelength is propagated through any optical system. This allows the calculation of all orders of dispersion, and also gives some insight in the space–time coupling phenomena such as spatial and angular chirp. This method is well-suited to compute the linear dispersive properties of complex and/or aberrated stretcher and compressor setups.

© 2008 Optical Society of America

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References

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  1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56, 219-221 (1985).
    [CrossRef]
  2. A. Chong, L. Kuznetsova, and F. Wise, “Theoretical optimization of nonlinear chirped-pulse fiber amplifiers,” J. Opt. Soc. Am. B 24, 1815-1823 (2007).
    [CrossRef]
  3. S. Kane, J. Squier, J. V. Rudd, and G. Mourou, “Hybrid grating-prism stretcher-compressor system with cubic phase and wavelength tenability and decreased alignment sensitivity,” Opt. Lett. 19, 1876-1878 (1994).
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  4. D. C. O'Shea, “Group-velocity dispersion using commercial optical design programs,” Appl. Opt. 45, 4740-4746 (2006).
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    [CrossRef] [PubMed]
  6. A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148-1157 (1990).
    [CrossRef]
  7. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530-2536 (1988).
    [CrossRef]
  8. Y. Nabekawa and K. Midorikawa, “High-order pulse front tilt caused by high-order angular dispersion,” Opt. Express 11, 3365-3376 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  13. E. A. Gibson, D. M. Gaudiosi, H. C. Kapteyn, R. Jimenez, S. Kane, R. Huff, C. Durfee, and J. Squier, “Efficient reflection grisms for pulse compression and dispersion compensation of femtosecond pulses,” Opt. Lett. 31, 3363-3365 (2006).
    [CrossRef] [PubMed]
  14. J. R. Buckley, S. W. Clark, and F. W. Wise, “Generation of ten-cycle pulses from an ytterbium fiber laser with cubic phase compensation,” Opt. Lett. 31, 1340-1342 (2006).
    [CrossRef] [PubMed]
  15. L. Kuznetsova, F. W. Wise, S. Kane, and J. Squier, “Chirped-pulse amplification of femtosecond pulses in a Yb-doped fiber amplifier near the gain narrowing limit using a reflection grism compressor,” presented at the Advanced Solid-State Photonics Topical Meeting and Tabletop Exhibit, Vancouver, Canada, January 28-31, 2007, paper TuB3.
  16. G. Cheriaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultrashort-pulse amplification,” Opt. Lett. 21, 414-416 (1996).
    [CrossRef] [PubMed]
  17. J. Jiang, Z. Zhang, and T. Hasama, “Evaluation of chirped-pulse-amplification systems with Offner triplet telescope stretchers,” J. Opt. Soc. Am. B 19, 678-683 (2002).
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  18. F. Verluise, V. Laude, Z. Cheng, C. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25, 575-577 (2000).
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2007

2006

2005

2003

2002

2000

1997

1996

1994

1990

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148-1157 (1990).
[CrossRef]

1988

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530-2536 (1988).
[CrossRef]

1985

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

Akturk, S.

Buckley, J. R.

Chambaret, J. P.

Cheng, Z.

Cheriaux, G.

Chong, A.

Clark, S. W.

Cormier, E.

Dimauro, L. F.

Druon, F.

Durfee, C.

Fuchs, U.

Gabolde, P.

Gaudiosi, D. M.

Georges, P.

Gibson, E. A.

Hanna, M.

Hasama, T.

Huff, R.

Jiang, J.

Jimenez, R.

Kane, S.

Kapteyn, H. C.

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148-1157 (1990).
[CrossRef]

Kuznetsova, L.

A. Chong, L. Kuznetsova, and F. Wise, “Theoretical optimization of nonlinear chirped-pulse fiber amplifiers,” J. Opt. Soc. Am. B 24, 1815-1823 (2007).
[CrossRef]

L. Kuznetsova, F. W. Wise, S. Kane, and J. Squier, “Chirped-pulse amplification of femtosecond pulses in a Yb-doped fiber amplifier near the gain narrowing limit using a reflection grism compressor,” presented at the Advanced Solid-State Photonics Topical Meeting and Tabletop Exhibit, Vancouver, Canada, January 28-31, 2007, paper TuB3.

Laude, V.

Lee, D.

Martinez, O. E.

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530-2536 (1988).
[CrossRef]

Midorikawa, K.

Mourou, G.

Nabekawa, Y.

O'Shea, D. C.

Papadopoulos, D. N.

Rousseau, P.

Rudd, J. V.

Salin, F.

Sherriff, R. E.

Spielmann, C.

Squier, J.

Strickland, D.

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

Tournois, P.

Trebino, R.

Tünnerman, A.

Verluise, F.

Walker, B.

Wise, F.

Wise, F. W.

J. R. Buckley, S. W. Clark, and F. W. Wise, “Generation of ten-cycle pulses from an ytterbium fiber laser with cubic phase compensation,” Opt. Lett. 31, 1340-1342 (2006).
[CrossRef] [PubMed]

L. Kuznetsova, F. W. Wise, S. Kane, and J. Squier, “Chirped-pulse amplification of femtosecond pulses in a Yb-doped fiber amplifier near the gain narrowing limit using a reflection grism compressor,” presented at the Advanced Solid-State Photonics Topical Meeting and Tabletop Exhibit, Vancouver, Canada, January 28-31, 2007, paper TuB3.

Zaouter, Y.

Zeitner, U. D.

Zhang, Z.

Appl. Opt.

IEEE J. Quantum Electron.

A. G. Kostenbauder, “Ray-pulse matrices: a rational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148-1157 (1990).
[CrossRef]

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530-2536 (1988).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

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

Opt. Express

Opt. Lett.

Other

L. Kuznetsova, F. W. Wise, S. Kane, and J. Squier, “Chirped-pulse amplification of femtosecond pulses in a Yb-doped fiber amplifier near the gain narrowing limit using a reflection grism compressor,” presented at the Advanced Solid-State Photonics Topical Meeting and Tabletop Exhibit, Vancouver, Canada, January 28-31, 2007, paper TuB3.

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Figures (10)

Fig. 1
Fig. 1

Geometry describing a single optical system.

Fig. 2
Fig. 2

Geometry of the interface.

Fig. 3
Fig. 3

Transmission gratings compressor setup.

Fig. 4
Fig. 4

(a) Second-order spectral phase introduced by a gratings compressor as a function of wavelength. Black crosses, our method; red circles, analytic. (b) Third-order order spectral phase introduced by a gratings compressor as a function of wavelength. Black crosses, our method; red circles, analytic.

Fig. 5
Fig. 5

Second-order spectral phase introduced by a prism compressor as a function of wavelength. Circles: our method, no matter. Crosses: our method, 2 mm matter in prisms. Red dotted line: analytic approximation, no matter. Red solid line: analytic approximation, 2 mm matter in prisms.

Fig. 6
Fig. 6

Hybrid prism–gratings compressor setup.

Fig. 7
Fig. 7

Tuning of the third- to second-order spectral phase ratio as a function of the distance L2 (cf. Fig. 6).

Fig. 8
Fig. 8

Two stretcher configurations.

Fig. 9
Fig. 9

(a) Case of parabolic mirrors: second- (top left) and third-order spectral phase (top right), angular (bottom left) and spatial chirp (bottom right) introduced by a classical (black circles) and Öffner (blue crosses) stretcher. (b) Case of spherical mirrors: second- (top left) and third-order spectral phase (top right), angular (bottom left) and spatial chirp (bottom right) introduced by a classical (black circles) and Öffner (blue crosses) stretcher.

Fig. 10
Fig. 10

Output pulse width and relative peak power as a function of input pulse width in a classical stretcher with f = 1 m optics, 1100 lines mm gratings, and spherical mirrors.

Equations (48)

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V ( λ ) = ( δ θ ( λ ) δ x ( λ ) δ φ ( λ ) ) .
δ x int = δ x in [ 1 cos θ 0 , in + tan θ 0 , in sin δ θ in cos ( θ 0 , in + δ θ in ) ] .
δ θ out = f ( θ 0 , in + δ θ in ) f ( θ 0 , in ) ,
δ x out = δ x in ( 1 cos θ 0 , in + tan θ 0 , in sin δ θ in cos ( θ 0 , in + δ θ in ) ) ( 1 cos θ 0 , out + tan θ 0 , out sin δ θ out cos ( θ 0 , out + δ θ out ) ) ,
δ φ out = δ φ in ,
θ out = arcsin ( n 1 n 2 sin θ in ) plane interface ,
θ out = arcsin [ n sin ( A arcsin ( sin θ in n ) ) ] very thin prism ,
θ out = arcsin [ ± 1 n 2 ( m λ d + n 1 sin θ in ) ] grating ,
δ θ out = δ θ in ,
δ x out = δ x in + L tan δ θ in ,
δ φ out = δ φ in + 2 π n ( λ ) L λ cos δ θ in .
δ θ out = a sin ( n 1 n 2 sin ( sgn ( n 2 ) . δ θ in a tan ( δ x in R ) ) ) + a tan ( δ x in R ) ,
δ x out = δ x in ,
δ φ out = δ φ in + 2 π λ ( n 2 n 1 ) n 1 R ( 1 sin 2 ( a tan ( δ x in R ) ) 1 ) ,
δ θ out = δ θ in + 2 a tan ( δ x in R ) ,
δ x out = δ x in ,
δ φ out = δ φ in + 2 π λ 2 R 1 sin 2 ( a tan ( δ x in R ) ) .
δ θ out δ θ in δ x in f + δ x 3 f 3 ( n 1 ) 3 ( 5 6 1 2 n n 2 2 + n 3 6 ) ,
δ x out = δ x in ,
δ φ out δ φ in + 2 π λ [ δ x in 2 2 f + 3 δ x in 4 32 f 3 ( n 1 ) 2 ] .
δ θ out = a sin ( n 1 n 2 sin ( sgn ( n 2 ) . δ θ in a tan ( δ x in + h 0 R ) ) ) + a tan ( δ x in + h 0 R ) θ 0 , out ,
δ x out = δ x in 1 cos θ 0 , out + tan θ 0 , out sin δ θ out cos ( θ 0 , out + δ θ out ) ,
δ φ out = δ φ in + 2 π λ ( n 2 n 1 ) n 1 R ( 1 sin 2 ( a tan ( δ x in + h 0 R ) ) 1 ) ,
θ 0 , out = a sin ( n 1 n 2 sin ( a tan ( h 0 R ) ) + a tan ( h 0 R ) ) .
δ θ out δ θ in δ x in f + δ x in 3 f 3 ( 1 3 ( 1 + ε a ) 4 ) ,
δ x out = δ x in ,
δ φ out δ φ in 2 π n λ [ δ x in 2 2 f + ( 1 + ε a 4 1 ) δ x in 4 8 f 3 ] ,
V out = G ¯ θ F ¯ L G ¯ θ F ¯ L G ¯ θ F ¯ L G ¯ θ ( V in ) ,
ϕ 2 = λ 3 Z π c 2 d 2 [ 1 ( λ d sin θ ) 2 ] 3 2 ,
ϕ 3 = ϕ 2 3 λ 2 π c [ 1 + λ d λ d sin θ 1 ( λ d sin θ ) 2 ] ,
ϕ 2 4 L λ 0 3 π c 2 ( d n d λ λ 0 ) 2 + L prism λ 0 3 2 π c 2 d 2 n d λ 2 λ 0 ,
φ a ( ω ) = ( 1 + ε a ) ω c f tan 4 δ θ ( ω ) ,
δ θ ( ω ) = a sin ( 2 π c d ω + sin θ 0 ) ,
δ θ out = δ θ in + a sin ( n 1 n 2 sin ( sgn ( n 2 ) . δ θ in a tan ( δ x in R δ x in R 2 δ x in 2 + Z [ δ x in , δ y in ] δ x in ) ) ) + a tan ( δ x in R δ x in R 2 δ x in 2 + Z [ δ x in , δ y in ] δ x in ) ,
δ γ out = δ γ in + a sin ( n 1 n 2 sin ( sgn ( n 2 ) . δ γ in a tan ( δ y in R δ y in R 2 δ y in 2 + Z [ δ x in , δ y in ] δ y in ) ) ) + a tan ( δ y in R δ y in R 2 δ y in 2 + Z [ δ x in , δ y in ] δ y in ) ,
δ x out = δ x in ,
δ y out = δ y in ,
δ φ out = δ φ in + 2 π λ n 2 n 1 n 1 . Z [ R sin ( a tan ( δ x in R ) ) , R sin ( a tan ( δ y in R ) ) ] ,
V ( λ ) = ( δ θ ( λ ) δ x ( λ ) δ ϕ ( λ ) I ( λ ) ) .
I out ( λ ) = H ( δ x in h 0 Φ 2 ) . ( 1 H ( δ x in h 0 + Φ 2 ) ) . T ( λ ) . I in ( λ ) ,
δ θ out = δ θ in ,
δ x out = δ x in ,
δ φ out = δ φ in + ψ ( δ x in ) ,
I out = T ( δ x in ) . I in ,
δ θ out = δ θ in ,
δ x out = δ x in ,
δ φ out = δ φ in + ψ ( λ ) ,
I out = T ( λ ) . I in ,

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