Abstract

The evolution of the frequency chirp of a laser pulse inside a classical pulse compressor is very different for plane waves and Gaussian beams, although after propagating through the last (4th) dispersive element, the two models give the same results. In this paper, we have analyzed the evolution of the frequency chirp of Gaussian pulses and beams using a method which directly obtains the spectral phase acquired by the compressor. We found the spatiotemporal couplings in the phase to be the fundamental reason for the difference in the frequency chirp acquired by a Gaussian beam and a plane wave. When the Gaussian beam propagates, an additional frequency chirp will be introduced if any spatiotemporal couplings (i.e. angular dispersion, spatial chirp or pulse front tilt) are present. However, if there are no couplings present, the chirp of the Gaussian beam is the same as that of a plane wave. When the Gaussian beam is well collimated, the introduced frequency chirp predicted by the plane wave and Gaussian beam models are in closer agreement. This work improves our understanding of pulse compressors and should be helpful for optimizing dispersion compensation schemes in many applications of femtosecond laser pulses.

©2009 Optical Society of America

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References

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  1. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
  2. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9(5), 150–152 (1984).
    [PubMed]
  3. J. Squier, F. Salin, G. Mourou, and D. Harter, “100-fs pulse generation and amplification in Ti:AI2O3,” Opt. Lett. 16(5), 324–326 (1991).
    [PubMed]
  4. C. L. Blanc, G. Grillon, J. P. Chambaret, A. Migus, and A. Antonetti, “Compact and efficient multipass Ti:sapphire system for femtosecond chirped-pulse amplification at the terawatt level,” Opt. Lett. 18(2), 140–142 (1993).
    [PubMed]
  5. O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3-1.6 um region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).
  6. S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, “Pulse broadening of the femtosecond pulses in a Gaussian beam passing an angular disperser,” Opt. Lett. 32(9), 1180–1182 (2007).
    [PubMed]
  7. D. Li, X. Li, S. Zeng, and Q. Luo, “A generalized analysis of femtosecond laser pulse broadening after angular dispersion,” Opt. Express 16(1), 237–247 (2008).
    [PubMed]
  8. D. Li, S. Zeng, Q. Luo, P. Bowlan, V. Chauahan, and R. Trebino, “Propagation dependence of chirp in Gaussian pulses and beams due to angular dispersion,” Opt. Lett. 34(7), 962–964 (2009).
    [PubMed]
  9. S. Szatmari, G. Kuhnle, and P. Simon, “Pulse compression and traveling wave excitation scheme using a single dispersive element,” Appl. Opt. 29(36), 5372–5379 (1990).
    [PubMed]
  10. S. Szatmári, P. Simon, and M. Feuerhake, “Group velocity dispersion compensated propagation of short pulses in dispersive media,” Opt. Lett. 21(15), 1156–1158 (1996).
    [PubMed]
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    [PubMed]
  12. M. Nakazawa, T. Nakashima, and H. Kubota, “Optical pulse compression using a TeO2 acousto-optical light deflector,” Opt. Lett. 13(2), 120–122 (1988).
    [PubMed]
  13. S. Zeng, X. Lv, C. Zhan, W. R. Chen, W. Xiong, S. L. Jacques, and Q. Luo, “Simultaneous compensation for spatial and temporal dispersion of acousto-optical deflectors for two-dimensional scanning with a single prism,” Opt. Lett. 31(8), 1091–1093 (2006).
    [PubMed]
  14. Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008).
    [PubMed]
  15. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986).
  16. O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59(3), 229–232 (1986).
  17. Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).
  18. K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27(22), 2034–2036 (2002).
  19. J. C. Diels, and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).
  20. C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).
  21. K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).
  22. K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).
  23. A. E. Siegman, Lasers, (University Science, Mill Valley, CA, 1986).
  24. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4-6), 599–604 (2004).
  25. S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating,” Opt. Express 11(1), 68–78 (2003).
    [PubMed]
  26. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12(19), 4399–4410 (2004).
    [PubMed]
  27. S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams,” Opt. Express 13(21), 8642–8661 (2005).
    [PubMed]
  28. P. Gabolde, D. Lee, S. Akturk, and R. Trebino, “Describing first-order spatio-temporal distortions in ultrashort pulses using normalized parameters,” Opt. Express 15(1), 242–251 (2007).
    [PubMed]

2009 (1)

2008 (2)

2007 (2)

2006 (2)

2005 (2)

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

S. Akturk, X. Gu, P. Gabolde, and R. Trebino, “The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams,” Opt. Express 13(21), 8642–8661 (2005).
[PubMed]

2004 (3)

S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Opt. Express 12(19), 4399–4410 (2004).
[PubMed]

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4-6), 599–604 (2004).

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

2003 (1)

2002 (1)

1996 (1)

1994 (1)

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

1993 (2)

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

C. L. Blanc, G. Grillon, J. P. Chambaret, A. Migus, and A. Antonetti, “Compact and efficient multipass Ti:sapphire system for femtosecond chirped-pulse amplification at the terawatt level,” Opt. Lett. 18(2), 140–142 (1993).
[PubMed]

1991 (1)

1990 (1)

1988 (1)

1987 (1)

O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3-1.6 um region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).

1986 (2)

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986).

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

1984 (1)

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).

Akturk, S.

Antonetti, A.

Benkö, Z.

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

Blanc, C. L.

Blanchot, N.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

Bor, Z.

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

Bourdieu, L.

Bowlan, P.

Candela, Y.

Chambaret, J. P.

Chauahan, V.

Chen, W. R.

Csatári, M.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

Dieudonné, S.

Divall, M.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

Ferincz, I. E.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

Feuerhake, M.

Fiorini, C.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

Fork, R. L.

Gabolde, P.

Gordon, J. P.

Grillon, G.

Gu, X.

Harter, D.

Hazim, H. A.

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

Heiner, Z.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

Honnorat, N.

Horváth, Z. L.

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

Jacques, S. L.

Kimmel, M.

Klebniczki, J.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

Kovacs, A. P.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

Kovács, A. P.

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27(22), 2034–2036 (2002).

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

Kremer, Y.

Kubota, H.

Kuhnle, G.

Kurdi, G.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27(22), 2034–2036 (2002).

Lapole, R.

Lee, D.

Léger, J. F.

Li, D.

Li, X.

Liu, J.

Luo, Q.

Lv, X.

Martinez, O. E.

O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3-1.6 um region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).

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

O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986).

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9(5), 150–152 (1984).
[PubMed]

Migus, A.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

C. L. Blanc, G. Grillon, J. P. Chambaret, A. Migus, and A. Antonetti, “Compact and efficient multipass Ti:sapphire system for femtosecond chirped-pulse amplification at the terawatt level,” Opt. Lett. 18(2), 140–142 (1993).
[PubMed]

Mourou, G.

Nakashima, T.

Nakazawa, M.

O’Shea, P.

Osvay, K.

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27(22), 2034–2036 (2002).

Rouyer, C.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

Salin, F.

Sauteret, C.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

Seznec, S.

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

Simon, P.

Squier, J.

Szatmari, S.

Szatmári, S.

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).

Trebino, R.

Varjú, K.

Xiong, W.

Zeek, E.

Zeng, S.

Zhan, C.

Appl. Opt. (1)

IEEE J. Quantum Electron. (3)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).

O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in 1.3-1.6 um region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).

C. Fiorini, C. Sauteret, C. Rouyer, N. Blanchot, S. Seznec, and A. Migus, “Temporal aberrations due to misalignments of a stretcher-compressor system and compensation,” IEEE J. Quantum Electron. 30(7), 1662–1670 (1994).

IEEE J. Sel. Top. Quantum Electron. (1)

K. Osvay, A. P. Kovács, Z. Heiner, G. Kurdi, J. Klebniczki, and M. Csatári, “Angular dispersion and temporal change of femtosecond pulses from misaligned pulse compressors,” IEEE J. Sel. Top. Quantum Electron. 10(1), 213–220 (2004).

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

Opt. Commun. (3)

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

K. Osvay, A. P. Kovacs, G. Kurdi, Z. Heiner, M. Divall, J. Klebniczki, and I. E. Ferincz, “Measurement of non-compensated angular dispersion and the subsequent temporal lengthening of femtosecond pulses in a CPA laser,” Opt. Commun. 248(1-3), 201–209 (2005).

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4-6), 599–604 (2004).

Opt. Eng. (1)

Z. L. Horváth, Z. Benkö, A. P. Kovács, H. A. Hazim, and Z. Bor, “Propagation of femtosecond pulses through lenses, gratings, and slits,” Opt. Eng. 32(10), 2491–2500 (1993).

Opt. Express (7)

Opt. Lett. (9)

D. Li, S. Zeng, Q. Luo, P. Bowlan, V. Chauahan, and R. Trebino, “Propagation dependence of chirp in Gaussian pulses and beams due to angular dispersion,” Opt. Lett. 34(7), 962–964 (2009).
[PubMed]

S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, “Pulse broadening of the femtosecond pulses in a Gaussian beam passing an angular disperser,” Opt. Lett. 32(9), 1180–1182 (2007).
[PubMed]

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9(5), 150–152 (1984).
[PubMed]

J. Squier, F. Salin, G. Mourou, and D. Harter, “100-fs pulse generation and amplification in Ti:AI2O3,” Opt. Lett. 16(5), 324–326 (1991).
[PubMed]

C. L. Blanc, G. Grillon, J. P. Chambaret, A. Migus, and A. Antonetti, “Compact and efficient multipass Ti:sapphire system for femtosecond chirped-pulse amplification at the terawatt level,” Opt. Lett. 18(2), 140–142 (1993).
[PubMed]

M. Nakazawa, T. Nakashima, and H. Kubota, “Optical pulse compression using a TeO2 acousto-optical light deflector,” Opt. Lett. 13(2), 120–122 (1988).
[PubMed]

S. Zeng, X. Lv, C. Zhan, W. R. Chen, W. Xiong, S. L. Jacques, and Q. Luo, “Simultaneous compensation for spatial and temporal dispersion of acousto-optical deflectors for two-dimensional scanning with a single prism,” Opt. Lett. 31(8), 1091–1093 (2006).
[PubMed]

S. Szatmári, P. Simon, and M. Feuerhake, “Group velocity dispersion compensated propagation of short pulses in dispersive media,” Opt. Lett. 21(15), 1156–1158 (1996).
[PubMed]

K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27(22), 2034–2036 (2002).

Other (2)

J. C. Diels, and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

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

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Figures (3)

Fig. 1
Fig. 1 (a) A classical pulse compressor, which consists of four identical prisms (or other angular dispersers, such as gratings). (b) Optical path diagram of a femtosecond laser pulse passing through a classical pulse compressor. For two adjacent elements, the spacings are L 1, L 2, L 3, respectively. Due to angular dispersion, in the system, different spectral components have different paths. Taking the entrance vertices (O 1 ~O 4) of each element as the original point, four reference distances can be established: they are x 1-z 1 through x 4-z 4. The distances for any spectral component are defined as x 1 ω -z 1 ω through x 4 ω -z 4 ω . In this figure, the dashed line between the second and third element is the wave front of the pulse. When the alignment is perfect, θ 1 = θ 3 and L 1 = L 3. BW is the beam waist, and d is the distance between the beam waist and the first dispersive element.

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Fig. 2
Fig. 2 Comparison of the chirp evolution of a Gaussian beam (green) and plane wave (black) when passing through a 4-dispersive element pulse compressor. The red arrows in the figure show the difference in the two models.

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Fig. 3
Fig. 3 The chirp evolution of a femtosecond laser pulse when passing through the pulse compressor. The Gaussian beam (green line) compared with a plane wave (black line). Each of the subfigures has different Rayleigh range: from (a) to (d), they are 1m, 3m, 5m, and 10m, respectively. The other parameters are the same as those of Fig. 2. When the Rayleigh range increases, the chirp evolution of the Gaussian beam becomes closer to that of a plane wave.

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Equations (50)

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

ϕ(r,z)=kz+kr22R(z)−tan−1(zzR).
Δθ(γ,ω)=αΔγ+βΔω=∂θ∂γΔγ+∂θ∂ωΔω.
α2=1α1,β2=−β1α1,
α3=α1,β3=−β1,
α4=1α3,β4=−β3α3.
ϕ1(x1ω,z1ω,ω)=k(d/α2+z1ω)+kx1ω22R(d/α2+z1ω)−tan−1(d/α2+z1ωzR1).
ϕ2(x2ω,z2ω,ω)=k(d+α2z1ω+z2ω)+kx2ω22R(d+α2z1ω+z2ω)−tan−1(d+α2z1ω+z2ωzR2),
ϕ3(x3ω,z3ω,ω)=k(d/α2+z1ω+z2ω/α2+z3ω)+kx3ω22R(d/α2+z1ω+z2ω/α2+z3ω)−tan−1(d/α2+z1ω+z2ω/α2+z3ωzR3),
ϕ4(x4ω,z4ω,ω)=k(d+α2z1ω+z2ω+α2z3ω+z4ω)+kx4ω22R(d+α2z1ω+z2ω+α2z3ω+z4ω)−tan−1(d+α2z1ω+z2ω+α2z3ω+z4ωzR4).
z1ω=z1cosθ1+x1sinθ1,
x1ω=−z1sinθ1+x1cosθ1.
z1ω'=0,
z1ω''=−z1(dθ1dω)2,
x1ω'=−z1dθ1dω,
x1ω''=−z1d2θ1dω2.
ϕG''(z1)=−kβ2z1+kβ2z12R(d/α2+z1)=−kβ2z1[1−α2z1(d+α2z1)(d+α2z1)2+zR2].
z2ω=z2+const,
x2ω=x2+ξω+const.
z2ω'=0,
z2ω''=0,
x2ω'=αβL1,
x2ω''=0.
ϕG''(z2)=−kβ2L1+kα2β2L121R(d+α2L1+z2)=−kβ2L1[1−α2L1(d+α2L1+z2)(d+α2L1+z2)2+zR2].
z3ω=z3cosθ3−x3sinθ3,
x3ω=−(L3−z3)sinθ3+x3cosθ3.
z3ω'=0,
z3ω''=−z3(dθ3dω)2,
x3ω'=−(L3−z3)dθ3dω,
x1ω''=−(L3−z3)d2θ3dω2.
ϕG''(z3)=−kβ2L1−kβ2z3+kβ2(L3−z3)21R(d/a2+L1+L2/α2+z3)=−kβ2L1−kβ2z3+kβ2(L3−z3)2α2(d+α2L1+L2+α2z3)(d+α2L1+L2+α2z3)2+zR2.
z4ω=z4,
x4ω=x4.
ϕG''(z4)=−kβ2L1−kβ2L3=−2kβ2L1.
ϕP''(z1)=−kβ2z1,
ϕP''(z2)=−kβ2L1,
ϕP''(z3)=−kβ2L1−kβ2z3,
ϕP''(z4)=−2kβ2L1.
ΔϕP''(z1)=−kβ2z1,
ΔϕP''(z2)=0,
ΔϕP''(z3)=−kβ2z3,
ΔϕP''(z4)=0.
ΔϕG''(z1)=−kβ2z1[1−α2z1(d+α2z1)(d+α2z1)2+zR2],
ΔϕG''(z2)=kα2β2L12[(d+α2L1+z2)(d+α2L1+z2)2+zR2−(d+α2L1)(d+α2L1)2+zR2]=kα2β2L12[1R(d+α2L1+z2)−1R(d+α2L1)],
ΔϕG''(z3)=−kβ2z3+kβ2(L3−z3)2α2(d+α2L1+L2+α2z3)(d+α2L1+L2+α2z3)2+zR2−kβ2L1α2L1(d+α2L1+L2)(d+α2L1+L2)2+zR2=−kβ2z3−kα2β2L12[1R(d+α2L1+L2)−(L3−z3L3)21R(d+α2L1+L2+α2z3)],
ΔϕG''(z4)=0.
ΔϕPG''(z1)=kα2β2z121R(d+α2z1),
ΔϕPG''(z2)=kα2β2L12[1R(d+α2L1+z2)−1R(d+α2L1)],
ΔϕPG''(z3)=−kα2β2L12[1R(d+α2L1+L2)−(L3−z3L3)21R(d+α2L1+L2+α2z3)],
ΔϕPG''(z4)=0.
ΔϕPG''(L1)+ΔϕPG''(L2)+ΔϕPG''(L3)=0.

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