Abstract

We present a rigorous, but mathematically relatively simple and elegant, theory of first-order spatio-temporal distortions, that is, couplings between spatial (or spatial-frequency) and temporal (or frequency) coordinates, of Gaussian pulses and beams. These distortions include pulse-front tilt, spatial dispersion, angular dispersion, and a less well-known distortion that has been called “time vs. angle.” We write pulses in four possible domains, xt, xω, kω, and kt; and we identify the first-order couplings (distortions) in each domain. In addition to the above four “amplitude” couplings, we identify four new spatio-temporal “phase” couplings: “wave-front rotation,” “wave-front-tilt dispersion,” “angular temporal chirp,” and “angular frequency chirp.” While there are eight such couplings in all, only two independent couplings exist and are fundamental in each domain, and we derive simple expressions for each distortion in terms of the others. In addition, because the dimensions and magnitudes of these distortions are unintuitive, we provide normalized, dimensionless definitions for them, which range from -1 to 1. Finally, we discuss the definitions of such quantities as pulse length, bandwidth, angular divergence, and spot size in the presence of spatio-temporal distortions. We show that two separate definitions are required in each case, specifically, “local” and “global” quantities, which can differ significantly in the presence of spatio-temporal distortions.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Akturk , M. Kimmel , P. O’Shea , and R. Trebino , “ Measuring spatial chirp in ultrashort es using single-shot Frequency-Resolved Optical Gating ,” Opt. Express   11 , 68 – 78 ( 2003 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-68
    [CrossRef] [PubMed]
  2. X. Gu , S. Akturk , and R. Trebino , “ Spatial chirp in ultrafast optics ,” Opt. Commun.   242 , 599 – 604 ( 2004 ).
    [CrossRef]
  3. S. Akturk , M. Kimmel , P. O’Shea , and R. Trebino , “ Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE ,” Opt. Express   11 , 491 – 501 ( 2003 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-491
    [CrossRef] [PubMed]
  4. K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
    [CrossRef]
  5. K. Varju , A. P. Kovacs , and K. Osvay , “ Angular dispersion of femtosecond pulses in a Gaussian beam ,” Opt. Lett.   27 (22), 2034 – 2036 ( 2002 ).
    [CrossRef]
  6. O. E. Martinez , “ Pulse distortions in tilted pulse schemes for ultrashort pulses ,” Opt. Commun.   59 (3), 229 – 232 ( 1986 ).
    [CrossRef]
  7. C. Dorrer , E. M. Kosik , and I. A. Walmsley , “ Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry ,” Appl. Phys. B-Lasers and Optics   74 [suppl.] , 209 – 219 ( 2002 ).
    [CrossRef]
  8. Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
    [CrossRef]
  9. J. Hebling , “ Derivation of pulse-front tilt casued by angular dispersion ,” Opt. Quantum Eng.   28 , 1759 – 1763 ( 1996 ).
    [CrossRef]
  10. I. Z. Kozma , G. Almasi , and J. Hebling , “ Geometrical optical modeling of femtosecond setups having angular dispersion ,” Appl. Phys. B-Lasers and Optics B   76 , 257 – 261 ( 2003 ).
    [CrossRef]
  11. S. Akturk , X. Gu , E. Zeek , and R. Trebino , “ Pulse-front tilt caused by spatial and temporal chirp ,” Opt. Express   12 , 4399 – 4410 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4399
    [CrossRef] [PubMed]
  12. O. E. Martinez , “ Matrix formalism for pulse compressors ,” IEEE J. Quantum. Electron.   24 , 2530 – 2536 ( 1988 ).
    [CrossRef]
  13. O. E. Martinez , “ Matrix Formalism for Dispersive Laser Cavities ,” IEEE J. Quantum. Electron.   25 , 296 – 300 ( 1989 ).
    [CrossRef]
  14. S. P. Dijaili , A. Dienes , and J. S. Smith , “ ABCD Matrices for dispersive pulse propagation ,” IEEE J. Quantum. Electron.   26 , 1158 – 1164 ( 1990 ).
    [CrossRef]
  15. M. A. Larotonda and A. A. Hnilo , “ Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix ,” Opt. Commun.   183 , 207 – 213 ( 2000 ).
    [CrossRef]
  16. Q. Lin and S. Wang , “ Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics ,” Opt. Quantum Electron.   27 , 785 – 798 ( 1995 ).
    [CrossRef]
  17. A. G. Kostenbauder , “ Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems ,” IEEE J. Quantum. Electron.   26 , 1148 – 1157 ( 1990 ).
    [CrossRef]
  18. M. Born and E. Wolf , Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light ( Cambridge Univ Pr, 1999 ).
  19. A. E. Siegman , Lasers ( Univ Science Books, 1986 ).
  20. I. S. Gradshteyn and I. M. Ryzhik , Table of integrals, series and products ( Academic Press, 1994 ).
  21. K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).
  22. O. E. Martinez , “ Grating and prism compressors in the case of finite beam size ,” J. Opt. Soc. Am. B   3 , 929 – 934 ( 1986 ).
    [CrossRef]
  23. D. J. Kane and R. Trebino , “ Characterization of Arbitrary Femtosecond Pulses Using Frequency Resolved Optical Gating ,” IEEE J. Quantum Electron.   29 , 571 – 579 ( 1993 ).
    [CrossRef]
  24. P. O’Shea , M. Kimmel , X. Gu , and R. Trebino , “ Highly simplified device for ultrashort-pulse measurement ,” Opt. Lett.   26 (12), 932 – 934 ( 2001 ).
    [CrossRef]
  25. R. Trebino , Frequency-Resolved Optical Gating ( Kluwer Academic Publishers, Boston, 2002 ).
    [CrossRef]
  26. P. Gabolde and R. Trebino , “ Self-referenced measurement of the complete electric field of ultrashort pulses ,” Opt. Expr.ess   12 , 4423 – 4429 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4423
    [CrossRef]
  27. R. G. Lane and M. Tallon , “ Wave-front reconstruction using a Shack-Hartmann sensor ,” Appl. Opt.   31 , 6902 – 6908 ( 1992 ).
    [CrossRef] [PubMed]

2004 (4)

X. Gu , S. Akturk , and R. Trebino , “ Spatial chirp in ultrafast optics ,” Opt. Commun.   242 , 599 – 604 ( 2004 ).
[CrossRef]

S. Akturk , X. Gu , E. Zeek , and R. Trebino , “ Pulse-front tilt caused by spatial and temporal chirp ,” Opt. Express   12 , 4399 – 4410 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4399
[CrossRef] [PubMed]

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

P. Gabolde and R. Trebino , “ Self-referenced measurement of the complete electric field of ultrashort pulses ,” Opt. Expr.ess   12 , 4423 – 4429 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4423
[CrossRef]

2003 (3)

2002 (3)

C. Dorrer , E. M. Kosik , and I. A. Walmsley , “ Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry ,” Appl. Phys. B-Lasers and Optics   74 [suppl.] , 209 – 219 ( 2002 ).
[CrossRef]

K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
[CrossRef]

K. Varju , A. P. Kovacs , and K. Osvay , “ Angular dispersion of femtosecond pulses in a Gaussian beam ,” Opt. Lett.   27 (22), 2034 – 2036 ( 2002 ).
[CrossRef]

2001 (1)

2000 (1)

M. A. Larotonda and A. A. Hnilo , “ Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix ,” Opt. Commun.   183 , 207 – 213 ( 2000 ).
[CrossRef]

1996 (1)

J. Hebling , “ Derivation of pulse-front tilt casued by angular dispersion ,” Opt. Quantum Eng.   28 , 1759 – 1763 ( 1996 ).
[CrossRef]

1995 (1)

Q. Lin and S. Wang , “ Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics ,” Opt. Quantum Electron.   27 , 785 – 798 ( 1995 ).
[CrossRef]

1993 (2)

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

D. J. Kane and R. Trebino , “ Characterization of Arbitrary Femtosecond Pulses Using Frequency Resolved Optical Gating ,” IEEE J. Quantum Electron.   29 , 571 – 579 ( 1993 ).
[CrossRef]

1992 (1)

1990 (2)

S. P. Dijaili , A. Dienes , and J. S. Smith , “ ABCD Matrices for dispersive pulse propagation ,” IEEE J. Quantum. Electron.   26 , 1158 – 1164 ( 1990 ).
[CrossRef]

A. G. Kostenbauder , “ Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems ,” IEEE J. Quantum. Electron.   26 , 1148 – 1157 ( 1990 ).
[CrossRef]

1989 (1)

O. E. Martinez , “ Matrix Formalism for Dispersive Laser Cavities ,” IEEE J. Quantum. Electron.   25 , 296 – 300 ( 1989 ).
[CrossRef]

1988 (1)

O. E. Martinez , “ Matrix formalism for pulse compressors ,” IEEE J. Quantum. Electron.   24 , 2530 – 2536 ( 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 (3), 229 – 232 ( 1986 ).
[CrossRef]

Akturk, S.

Almasi, G.

I. Z. Kozma , G. Almasi , and J. Hebling , “ Geometrical optical modeling of femtosecond setups having angular dispersion ,” Appl. Phys. B-Lasers and Optics B   76 , 257 – 261 ( 2003 ).
[CrossRef]

Bor, Z.

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

Born, M.

M. Born and E. Wolf , Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light ( Cambridge Univ Pr, 1999 ).

Csatari, M.

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

Dienes, A.

S. P. Dijaili , A. Dienes , and J. S. Smith , “ ABCD Matrices for dispersive pulse propagation ,” IEEE J. Quantum. Electron.   26 , 1158 – 1164 ( 1990 ).
[CrossRef]

Dijaili, S. P.

S. P. Dijaili , A. Dienes , and J. S. Smith , “ ABCD Matrices for dispersive pulse propagation ,” IEEE J. Quantum. Electron.   26 , 1158 – 1164 ( 1990 ).
[CrossRef]

Dorrer, C.

C. Dorrer , E. M. Kosik , and I. A. Walmsley , “ Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry ,” Appl. Phys. B-Lasers and Optics   74 [suppl.] , 209 – 219 ( 2002 ).
[CrossRef]

Gabolde, P.

P. Gabolde and R. Trebino , “ Self-referenced measurement of the complete electric field of ultrashort pulses ,” Opt. Expr.ess   12 , 4423 – 4429 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4423
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik , Table of integrals, series and products ( Academic Press, 1994 ).

Gu, X.

Hazim, H. A.

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

Hebling, J.

I. Z. Kozma , G. Almasi , and J. Hebling , “ Geometrical optical modeling of femtosecond setups having angular dispersion ,” Appl. Phys. B-Lasers and Optics B   76 , 257 – 261 ( 2003 ).
[CrossRef]

J. Hebling , “ Derivation of pulse-front tilt casued by angular dispersion ,” Opt. Quantum Eng.   28 , 1759 – 1763 ( 1996 ).
[CrossRef]

Heiner, Z.

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

Hilbert, M.

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

Hnilo, A. A.

M. A. Larotonda and A. A. Hnilo , “ Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix ,” Opt. Commun.   183 , 207 – 213 ( 2000 ).
[CrossRef]

Kane, D. J.

D. J. Kane and R. Trebino , “ Characterization of Arbitrary Femtosecond Pulses Using Frequency Resolved Optical Gating ,” IEEE J. Quantum Electron.   29 , 571 – 579 ( 1993 ).
[CrossRef]

Kimmel, M.

Klebniczki, J.

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

Kosik, E. M.

C. Dorrer , E. M. Kosik , and I. A. Walmsley , “ Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry ,” Appl. Phys. B-Lasers and Optics   74 [suppl.] , 209 – 219 ( 2002 ).
[CrossRef]

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]

Kovacs, A.

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

Kovacs, A. P.

K. Varju , A. P. Kovacs , and K. Osvay , “ Angular dispersion of femtosecond pulses in a Gaussian beam ,” Opt. Lett.   27 (22), 2034 – 2036 ( 2002 ).
[CrossRef]

K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
[CrossRef]

Kozma, I. Z.

I. Z. Kozma , G. Almasi , and J. Hebling , “ Geometrical optical modeling of femtosecond setups having angular dispersion ,” Appl. Phys. B-Lasers and Optics B   76 , 257 – 261 ( 2003 ).
[CrossRef]

Kurdi, G.

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
[CrossRef]

Lane, R. G.

Larotonda, M. A.

M. A. Larotonda and A. A. Hnilo , “ Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix ,” Opt. Commun.   183 , 207 – 213 ( 2000 ).
[CrossRef]

Lin, Q.

Q. Lin and S. Wang , “ Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics ,” Opt. Quantum Electron.   27 , 785 – 798 ( 1995 ).
[CrossRef]

Martinez, O. E.

O. E. Martinez , “ Matrix Formalism for Dispersive Laser Cavities ,” IEEE J. Quantum. Electron.   25 , 296 – 300 ( 1989 ).
[CrossRef]

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

O. E. Martinez , “ Pulse distortions in tilted pulse schemes for ultrashort pulses ,” Opt. Commun.   59 (3), 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]

O’Shea, P.

Osvay, K.

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
[CrossRef]

K. Varju , A. P. Kovacs , and K. Osvay , “ Angular dispersion of femtosecond pulses in a Gaussian beam ,” Opt. Lett.   27 (22), 2034 – 2036 ( 2002 ).
[CrossRef]

Racz, B.

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik , Table of integrals, series and products ( Academic Press, 1994 ).

Siegman, A. E.

A. E. Siegman , Lasers ( Univ Science Books, 1986 ).

Smith, J. S.

S. P. Dijaili , A. Dienes , and J. S. Smith , “ ABCD Matrices for dispersive pulse propagation ,” IEEE J. Quantum. Electron.   26 , 1158 – 1164 ( 1990 ).
[CrossRef]

Szabo, G.

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

Tallon, M.

Trebino, R.

P. Gabolde and R. Trebino , “ Self-referenced measurement of the complete electric field of ultrashort pulses ,” Opt. Expr.ess   12 , 4423 – 4429 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4423
[CrossRef]

S. Akturk , X. Gu , E. Zeek , and R. Trebino , “ Pulse-front tilt caused by spatial and temporal chirp ,” Opt. Express   12 , 4399 – 4410 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4399
[CrossRef] [PubMed]

X. Gu , S. Akturk , and R. Trebino , “ Spatial chirp in ultrafast optics ,” Opt. Commun.   242 , 599 – 604 ( 2004 ).
[CrossRef]

S. Akturk , M. Kimmel , P. O’Shea , and R. Trebino , “ Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE ,” Opt. Express   11 , 491 – 501 ( 2003 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-5-491
[CrossRef] [PubMed]

S. Akturk , M. Kimmel , P. O’Shea , and R. Trebino , “ Measuring spatial chirp in ultrashort es using single-shot Frequency-Resolved Optical Gating ,” Opt. Express   11 , 68 – 78 ( 2003 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-68
[CrossRef] [PubMed]

P. O’Shea , M. Kimmel , X. Gu , and R. Trebino , “ Highly simplified device for ultrashort-pulse measurement ,” Opt. Lett.   26 (12), 932 – 934 ( 2001 ).
[CrossRef]

D. J. Kane and R. Trebino , “ Characterization of Arbitrary Femtosecond Pulses Using Frequency Resolved Optical Gating ,” IEEE J. Quantum Electron.   29 , 571 – 579 ( 1993 ).
[CrossRef]

R. Trebino , Frequency-Resolved Optical Gating ( Kluwer Academic Publishers, Boston, 2002 ).
[CrossRef]

Varju, K.

K. Varju , A. P. Kovacs , and K. Osvay , “ Angular dispersion of femtosecond pulses in a Gaussian beam ,” Opt. Lett.   27 (22), 2034 – 2036 ( 2002 ).
[CrossRef]

K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
[CrossRef]

Walmsley, I. A.

C. Dorrer , E. M. Kosik , and I. A. Walmsley , “ Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry ,” Appl. Phys. B-Lasers and Optics   74 [suppl.] , 209 – 219 ( 2002 ).
[CrossRef]

Wang, S.

Q. Lin and S. Wang , “ Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics ,” Opt. Quantum Electron.   27 , 785 – 798 ( 1995 ).
[CrossRef]

Wolf, E.

M. Born and E. Wolf , Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light ( Cambridge Univ Pr, 1999 ).

Zeek, E.

Appl. Opt. (1)

Appl. Phys. B-Lasers and Optics (2)

K. Varju , A. P. Kovacs , G. Kurdi , and K. Osvay , “ High-precision measurement of angular dispersion in a CPA laser ,” Appl. Phys. B-Lasers and Optics   B74[Suppl] , 259 – 263 ( 2002 ).
[CrossRef]

C. Dorrer , E. M. Kosik , and I. A. Walmsley , “ Spatio-temporal characterization of ultrashort optical pulses using two-dimensional shearing interferometry ,” Appl. Phys. B-Lasers and Optics   74 [suppl.] , 209 – 219 ( 2002 ).
[CrossRef]

Appl. Phys. B-Lasers and Optics B (1)

I. Z. Kozma , G. Almasi , and J. Hebling , “ Geometrical optical modeling of femtosecond setups having angular dispersion ,” Appl. Phys. B-Lasers and Optics B   76 , 257 – 261 ( 2003 ).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. J. Kane and R. Trebino , “ Characterization of Arbitrary Femtosecond Pulses Using Frequency Resolved Optical Gating ,” IEEE J. Quantum Electron.   29 , 571 – 579 ( 1993 ).
[CrossRef]

IEEE J. Quantum. Electron. (4)

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

O. E. Martinez , “ Matrix Formalism for Dispersive Laser Cavities ,” IEEE J. Quantum. Electron.   25 , 296 – 300 ( 1989 ).
[CrossRef]

S. P. Dijaili , A. Dienes , and J. S. Smith , “ ABCD Matrices for dispersive pulse propagation ,” IEEE J. Quantum. Electron.   26 , 1158 – 1164 ( 1990 ).
[CrossRef]

A. G. Kostenbauder , “ Ray-Pulse Matrices: A Rational Treatment for Dispersive Optical Systems ,” IEEE J. Quantum. Electron.   26 , 1148 – 1157 ( 1990 ).
[CrossRef]

IEEE JSTQE (1)

K. Osvay , A. Kovacs , Z. Heiner , G. Kurdi , J. Klebniczki , and M. Csatari , “ Angular Dispersion and Temporal Change of Femtosecond Pulses From Misaligned Pulse Compressors ,” IEEE JSTQE   10 (1), 213 – 220 ( 2004 ).

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

Opt. Commun. (3)

M. A. Larotonda and A. A. Hnilo , “ Short laser pulse parameters in a nonlinear medium: different approximations of the ray-pulse matrix ,” Opt. Commun.   183 , 207 – 213 ( 2000 ).
[CrossRef]

O. E. Martinez , “ Pulse distortions in tilted pulse schemes for ultrashort pulses ,” Opt. Commun.   59 (3), 229 – 232 ( 1986 ).
[CrossRef]

X. Gu , S. Akturk , and R. Trebino , “ Spatial chirp in ultrafast optics ,” Opt. Commun.   242 , 599 – 604 ( 2004 ).
[CrossRef]

Opt. Engineering (1)

Z. Bor , B. Racz , G. Szabo , M. Hilbert , and H. A. Hazim , “ Femtosecond pulse front tilt caused by angular dispersion ,” Opt. Engineering   32 (10), 2501 – 2503 ( 1993 ).
[CrossRef]

Opt. Expr.ess (1)

P. Gabolde and R. Trebino , “ Self-referenced measurement of the complete electric field of ultrashort pulses ,” Opt. Expr.ess   12 , 4423 – 4429 ( 2004 ). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4423
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

Q. Lin and S. Wang , “ Spatial-temporal coupling in a grating-pair pulse compression system analysed by matrix optics ,” Opt. Quantum Electron.   27 , 785 – 798 ( 1995 ).
[CrossRef]

Opt. Quantum Eng. (1)

J. Hebling , “ Derivation of pulse-front tilt casued by angular dispersion ,” Opt. Quantum Eng.   28 , 1759 – 1763 ( 1996 ).
[CrossRef]

Other (4)

M. Born and E. Wolf , Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light ( Cambridge Univ Pr, 1999 ).

A. E. Siegman , Lasers ( Univ Science Books, 1986 ).

I. S. Gradshteyn and I. M. Ryzhik , Table of integrals, series and products ( Academic Press, 1994 ).

R. Trebino , Frequency-Resolved Optical Gating ( Kluwer Academic Publishers, Boston, 2002 ).
[CrossRef]

Supplementary Material (2)

» Media 1: AVI (2320 KB)     
» Media 2: AVI (2096 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

The effect of the WFR. The movie shows the wave fronts in (x,t) domain as a function of time and position (file size 2.26 MB).

Fig. 2.
Fig. 2.

The effect of the WFD. The movie shows the wave fronts in (x,ω) as a function of frequency and position (file size 2.06 MB).

Fig. 3.
Fig. 3.

Intensity profile of a pulse expressed in the four different domains. This pulse simultaneously has all four spatio-temporal amplitude couplings: PFT, SPC, AGD and TVA, as can be seen from the tilted images.

Fig. 4.
Fig. 4.

Intensity profile of a pulse expressed in the four different domains. This pulse has significant PFT and SPC but very small AGD and TVA, as can be seen from the tilt of the traces.

Tables (3)

Tables Icon

Table 1. Summary of the relations of spatio-temporal distortions in all four domains.

Tables Icon

Table 2 The correlation coefficients of the pulse field shown in Figure 3.

Tables Icon

Table 3 The correlation coefficients of the pulse field shown in Fig. 4.

Equations (86)

Equations on this page are rendered with MathJax. Learn more.

E ( x , t ) = exp { i π λ 0 ( x t ) T Q 1 ( x t ) } =
exp [ i π λ 0 ( ( Q 1 ) 11 x 2 + ( Q 1 ) 12 xt ( Q 1 ) 21 xt ( Q 1 ) 22 t 2 ) ]
K = [ x out x in x out θ in 0 x out v in x out x in θ out θ in 0 θ out v in t out x in t out θ in 1 t out v in 0 0 0 1 ] = [ A B 0 E C D 0 F G H 1 I 0 0 0 1 ]
Q out = { [ A 0 G 1 ] Q in + [ B E λ 0 H I λ 0 ] } · { [ C 0 0 0 ] Q in + [ D F λ 0 0 1 ] } 1
E ( x , t ) exp { Q ˜ xx x 2 + 2 Q ˜ xt xt Q ˜ tt t 2 }
Q in = [ Q 11 Q 12 Q 12 Q 22 ] = i λ 0 π [ Q ˜ xx Q ˜ xt Q ˜ xt Q ˜ tt ] 1
Q ˜ xx = i π λ 0 R ( z ) 1 w 2 ( z )
Q ˜ tt = + 1 τ 2
E ( x , t ) exp { Q ˜ xx x 2 + 2 Q ˜ xt xt Q ˜ tt t 2 }
Re { Q ˜ xx } beam spot size ( BSS )
Im { Q ˜ xx } wave front curvature ( WFC )
Re { Q ˜ tt } temporal pulse width ( TPW )
Im { Q ˜ tt } temporal chirp ( TCH )
Re { Q ˜ xt } pulse front tilt ( PFT )
Im { Q ˜ xt } wave front rotation ( WER )
E ( x , ω ) = 1 2 π E ( x , t ) e iωt dt
exp ( p 2 x 2 ± qx ) dx = exp ( q 2 4 p ) π p
E ( x , ω ) exp { R xx x 2 + 2 R R ωω ω 2 }
R xx = Q ˜ xx + Q ˜ xt Q ˜ tt
R = i 2 Q ˜ xt Q ˜ tt
R ωω = 1 4 Q ˜ tt
Re { R xx } beam spot size ( BSS )
Im { R xx } wave front curvature ( WFC )
Re { R ωω } band width ( BDW )
Im { R ωω } frequency chirp ( FCH )
Re { R } spatial chirp ( SPC )
Im { R } wave front tilt dispersion ( WFD )
E ( k , ω ) exp { S kk k 2 + 2 S S ωω ω 2 }
S kk = 1 4 R xx
S = i 2 R R xx
S ωω = R ωω + R 2 R xx
[ Q ˜ ] = [ Q ˜ xx Q ˜ xt Q ˜ xt Q ˜ tt ]
[ S ] = [ S kk S S S ωω ]
[ S ] = 1 4 [ Q ˜ T ] 1
Re { S kk } angular div ergence ( ADV )
Im { S kk } angular phase front curvature ( APC )
Re { S ωω } band width ( BDW )
Im { S ωω } frequency chirp ( FCH )
Re { S } angular dispersion ( AGD )
Im { S } angular spectral chirp ( ASC )
E ( k , t ) exp { P kk k 2 + 2 P kt kt P tt t 2 }
P kk = S kk + S 2 S ωω
P kt = i 2 S S ωω
P tt = 1 4 S ωω
[ P ] = 1 4 [ R T ] 1
[ R ] = [ R xx R R R ωω ] [ P ] = [ P kk P kt P kt P tt ]
Re { P kk } angular div ergence ( ADV )
Im { P kk } angular phase front curvature ( APC )
Re { P tt } temporal pulse width ( TPW )
Im { P tt } temporal chirp ( TCH )
Re { P kt } time vs . angle ( TVA )
Im { P kt } angular temporal chirp ( ATC )
Δ x = [ x 2 I ( x , t ) dx ( xI ( x , t ) dx ) 2 I ( x , t ) dx ] 1 2
Δ x L ( t ) = [ x 2 I ( x , t ) dx ( xI ( x , t ) dx ) 2 I ( x , t ) dx ] 1 2
Δ x G = [ x 2 I ( x , t ) dxdt I ( x , t ) dxdt ] 1 2
Δ x L = 1 2 [ 1 Q ˜ xx R ] 1 2
Δ x G = 1 2 [ Q ˜ tt R Q ˜ xx R Q ˜ tt R + Q ˜ xt R 2 ] 1 2
Δ t L = 1 2 [ 1 Q ˜ tt R ] 1 2
Δ t G = 1 2 [ Q ˜ xx R Q ˜ xx R Q ˜ tt R + Q ˜ xt R 2 ] 1 2
E ( x , ω ) exp [ ( x x 0 ω ω 4 Δ x L ) 2 ω 2 4 Δ ω G 2 ]
E ( x , ω ) exp [ R xx R x 2 + 2 R R R ωω R ω 2 ]
= exp [ R xx R ( x + R R R xx R ω ) 2 ( R R 2 R xx R + R ωω R ) ω 2 ]
Δ t L = 1 2 [ 1 R xx R ] = w 2
x 0 ω = R R R xx R
Δ ω G = 1 2 ( R xx R R xx R 2 + R ωω R R xx R ) 1 2
E ( x , ω ) = exp [ x 2 4 Δ x G 2 ( ω ω 0 x x 4 Δ ω L ) 2 ]
= exp [ ( R R 2 R ωω R + R xx R ) x 2 R ωω R ( ω R R R ωω R x ) 2 ]
Δ x G = 1 2 ( R ωω R R R 2 + R ωω R R xx R ) 1 2
ω 0 x = R R R ωω R
Δ ω L = 1 2 [ 1 R ωω R ] 1 2
ω 0 x = x 0 ω ( x 0 ω ) 2 + Δ x L 2 Δ ω G 2
ρ ∫∫ xωI ( x , ω ) dxdω ∫∫ I ( x , ω ) dxdω 1 Δ x G Δ ω G
ρ = R R R xx R R ωω R
Δ x L ( ω ) = Δ x G 1 ρ 2
Δ ω L ( x ) = Δ ω G 1 ρ 2
Q ˜ xt R = 1 2 R ωω 2 [ R ωω R R I + R ωω I R R ]
PET = 2 WFD + 2 FCH × FRG
PET = AGD + 2 FCH × FRG
SPD = 2 ASC 2 AGD × APC
AGD = 2 ATC + 2 TVA × TCH
TVA = 2 WFR + 2 PFT × WFC
Q ˜ xt = i 2 FRG + Q ˜ tt PFT
WFR = FRG 2 + TCH × PET
WFD = AGD 2 + SPD × WFC
ASC = TVA 2 + AGD × FCH
ATC = PFT 2 + TVA × APC

Metrics