Abstract

The measurement of pulse-front tilt (PFT) in ultrashort pulse laser beams, without ambiguity of its sign, has been carried out using an in-house built single-shot tilted pulse-front autocorrelator (TPFAC) based on second harmonic generation in non-linear crystal. The experiment is carried out for the beam propagation through a single prism and a pair of prisms. Expressions of PFT angle have been derived to account for beam size, beam propagation effects and for arbitrary incidence of the laser beam. The measured PFT angles were found in accordance with theoretically calculated values. The calculated sensitivity of TPFAC is found to be larger than that offered by the other techniques.

© 2006 Optical Society of America

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Corrections

Avnish Kumar Sharma, Rajesh Kumar Patidar, Manchi Raghuramaiah, Prasad Anant Naik, and Parshotam Dass Gupta, "Measuring pulse-front tilt in ultrashort pulse laser beams without ambiguity of its sign using single-shot tilted pulse-front autocorrelator: Erratum," Opt. Express 15, 7537-7537 (2007)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-15-12-7537

References

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2006 (1)

J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006).
[Crossref]

2005 (4)

2004 (6)

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

T. Kanai, X. Zhou, T. Liu, A. Kosuge, T. Sekikawa, and S. Watanabe, “Generation of terawatt 10-fs blue pulses by compensation for pulse-front distortion in broadband frequency doubling,” Opt. Lett. 29, 2929–2931 (2004).
[Crossref]

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

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).
[Crossref] [PubMed]

2003 (4)

2002 (5)

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

J. Hebling, G. Almasi, I. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large area THz-pulse generation,” Opt. Express 10, 1161–1166 (2002).
[PubMed]

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002).
[Crossref]

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

T. Tajima and G. Mourou, “Zettawatt-exawatt lasers and their applications in ultrastrong-field physics,” Phys. Rev. Spl. topics accelerators and beams 5, 031301, 1–9 (2002).
[Crossref]

2001 (1)

2000 (2)

1998 (1)

1997 (2)

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

1996 (2)

C. Radzewicz, M. J. la Grone, and J. S. Krasinski, “Interferometric measurement of femtosecond pulse distortion by lenses,” Opt. Commun. 126, 185–190 (1996).
[Crossref]

J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Opt. Quantum. Electron. 28, 1759–1763 (1996).
[Crossref]

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, 1662–1670 (1994).
[Crossref]

1993 (1)

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[Crossref]

1986 (2)

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]

1985 (1)

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

Akahane, Y.

Akturk, S.

Almasi, G.

Almási, G.

Andreoni, A.

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

Aoyama, M.

Banfi, G. P.

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

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, 1662–1670 (1994).
[Crossref]

Bor, Z.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[Crossref]

Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

Brummitt, P. A.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Cardoso, L.

Carillon, A.

Chanteloup, J. C.

Clarke, R. J.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Collier, J. L.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Csatari, M.

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

Danielius, R.

Z. Sacks, G. Mourou, and R. Danielius, “Adjusting pulse front tilt and pulse duration by use of a single-shot autocorrelator,” Opt. Lett. 26, 462–464 (2001).
[Crossref]

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

Danson, C. N.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Di Trapani, P.

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

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, 201–209 (2005).
[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 74, 209–219 (2002).
[Crossref]

Dubietis, A.

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

Fell, G.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

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, 201–209 (2005).
[Crossref]

Figueira, G.

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, 1662–1670 (1994).
[Crossref]

Foggi, P.

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

Frackiewicz, A. J.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Fujita, H.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Gabolde, P.

Gu, X.

Gupta, P. D.

M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, “Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam,” Opt. Commun. 223, 163–168 (2003).
[Crossref]

Hancock, S.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Hawkes, S.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Hazim, H. A.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[Crossref]

Hebling, J.

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, 201–209 (2005).
[Crossref]

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

Hernandez-Gomez, C.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Hilbert, M.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[Crossref]

Holligan, P.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Hubert, P.

Hutchinson, M. H. R.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Imran, T.

J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006).
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Inoue, N.

Izawa, Y.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

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Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Kalachnikov, M.

Kanabe, T.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Kanai, T.

Kasper, A.

G. Pretzler, A. Kasper, and K. J. White, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

Kawasaki, T.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Kidd, A.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Kimmel, M.

Kiriyama, H.

Kitagawa, Y.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Kitamura, H.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

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, 201–209 (2005).
[Crossref]

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

Klisnick, A.

Kobayashi, T.

Kodama, R.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

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 74, 209–219 (2002).
[Crossref]

Kosuge, A.

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, 201–209 (2005).
[Crossref]

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

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002).
[Crossref]

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

Kozma, I.

Krasinski, J. S.

C. Radzewicz, M. J. la Grone, and J. S. Krasinski, “Interferometric measurement of femtosecond pulse distortion by lenses,” Opt. Commun. 126, 185–190 (1996).
[Crossref]

Kuhl, J.

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, 201–209 (2005).
[Crossref]

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

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002).
[Crossref]

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

la Grone, M. J.

C. Radzewicz, M. J. la Grone, and J. S. Krasinski, “Interferometric measurement of femtosecond pulse distortion by lenses,” Opt. Commun. 126, 185–190 (1996).
[Crossref]

Lee, Y. S.

J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006).
[Crossref]

Lester, W. J.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
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Liu, T.

Lopes, N.

Ma, J.

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O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986).
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O. E. Martinez, “Pulse distortions in tilted pulse schemes for ultrashort pulses,” Opt. Commun. 59, 229–232 (1986).
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Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Midorikawa, K.

Migus, A.

Miyanaga, N.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Mourou, G.

T. Tajima and G. Mourou, “Zettawatt-exawatt lasers and their applications in ultrastrong-field physics,” Phys. Rev. Spl. topics accelerators and beams 5, 031301, 1–9 (2002).
[Crossref]

Z. Sacks, G. Mourou, and R. Danielius, “Adjusting pulse front tilt and pulse duration by use of a single-shot autocorrelator,” Opt. Lett. 26, 462–464 (2001).
[Crossref]

Musgrave, I. O.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Nabekawa, Y.

Naik, P. A.

M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, “Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam,” Opt. Commun. 223, 163–168 (2003).
[Crossref]

Nam, C. H.

J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006).
[Crossref]

Neely, D.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Neville, D. R.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Nickles, P.

Norreys, P. A.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

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, 201–209 (2005).
[Crossref]

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

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (2002).
[Crossref]

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

Park, J. Y.

J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006).
[Crossref]

Pepler, D. A.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Piskarskas, A.

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

Pretzler, G.

G. Pretzler, A. Kasper, and K. J. White, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

Racz, B.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
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Z. Bor and B. Racz, “Group-velocity dispersion in prisms and its application to pulse-compression and traveling-wave excitation,” Opt. Commun. 54, 165–170 (1985).
[Crossref]

Radzewicz, C.

C. Radzewicz, M. J. la Grone, and J. S. Krasinski, “Interferometric measurement of femtosecond pulse distortion by lenses,” Opt. Commun. 126, 185–190 (1996).
[Crossref]

Raghuramaiah, M.

M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, “Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam,” Opt. Commun. 223, 163–168 (2003).
[Crossref]

Reason, C. J.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
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Riedle, E.

Ros, D.

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, 1662–1670 (1994).
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Sacks, Z.

Sakabe, S.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Sakane, I.

Salmon, E.

Sauteret, C.

Sekikawa, T.

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, 1662–1670 (1994).
[Crossref]

Shaikh, W.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Sharma, A. K.

M. Raghuramaiah, A. K. Sharma, P. A. Naik, and P. D. Gupta, “Simultaneous measurement of pulse front tilt and pulse duration of a femtosecond laser beam,” Opt. Commun. 223, 163–168 (2003).
[Crossref]

Shigemori, K.

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, “Prepulse-free petawatt laser for a fast ignitor,” IEEE J. Quantum. Electron. 40, 281–293 (2004).
[Crossref]

Shirakawa, A.

Solcia, C.

P. Di Trapani, A. Andreoni, C. Solcia, G. P. Banfi, R. Danielius, A. Piskarskas, and P. Foggi, “Powerful sub-100-fs pulses broadly tunable in the visible from a blue-pumped parametric generator and amplifier,” J. Opt. Soc. Am. B 14, 1245-(1997).
[Crossref]

Stepanov, A.

Sung, J. H.

J. H. Sung, J. Y. Park, T. Imran, Y. S. Lee, and C. H. Nam, “Generation of 0.2-TW5.5-fs optical pulses at 1 kHz using a differentially pumped hollow-fiber chirped-mirror compressor,” Appl. Phys. B Laser & Optics 82, 5–8 (2006).
[Crossref]

Szabo, G.

Z. Bor, B. Racz, G. Szabo, M. Hilbert, and H. A. Hazim, “Femtosecond pulse front tilt caused by angular-dispersion,” Opt. Eng. 32, 2501–2504 (1993).
[Crossref]

Tajima, T.

T. Tajima and G. Mourou, “Zettawatt-exawatt lasers and their applications in ultrastrong-field physics,” Phys. Rev. Spl. topics accelerators and beams 5, 031301, 1–9 (2002).
[Crossref]

TamoSauskas, G.

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

Trebino, R.

Ueda, H.

Valiulis, G.

A. Dubietis, G. Valiulis, G. TamoSauskas, R. Danielius, and A. Piskarskas, “Nonlinear pulse compression in the ultraviolet,” Opt. Commun. 141, 50–55 (1997).

Varjú, K.

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

K. Varjú, A. P. Kovács, G. Kurdi, and K. Osvay, “High precision measurement of angular dispersion in a CPA laser,” Appl. Phys. B 74, S259–S263 (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 74, 209–219 (2002).
[Crossref]

Watanabe, S.

Wemans, J.

White, K. J.

G. Pretzler, A. Kasper, and K. J. White, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70, 1–9 (2000).
[Crossref]

Winstone, T. B.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Wyatt, R. W. W.

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, G. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan petawatt - an ultra-high-intensity interaction facility,” Nucl. Fusion 44, S239–S249 (2004).
[Crossref]

Wyborn, B. E.

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Figures (8)

Fig. 1.
Fig. 1.

Diagram illustrating the various parameters used in an experiment using a single prism, for a plane wave. In the case of a beam with a curved wavefront, Δθ′≠Δθ [30]

Fig. 2.
Fig. 2.

Experimental setup of TPFAC: a) schematic drawing and b) pictorial view.

Fig. 3.
Fig. 3.

AC trace rotation angle (a) and its derivative (b) w. r. to the PFT angle at different cross over angles.

Fig. 4.
Fig. 4.

Typical AC trace (a) after prism insertion; and (b) without prism insertion in the 200fs laser beam directly from the GLX 200 laser oscillator.

Fig. 5.
Fig. 5.

The measured and theoretical values of PFT angle as a function of distance (z) from the SF10 prism. (AC trace corresponding to z=1m and z=3.6m are also depicted in the figure)

Fig. 6.
Fig. 6.

Typical AC trace at output of prism pair with a separation of (a) 3.6 m; and (b) 15 cm.

Fig. 7.
Fig. 7.

Typical AC trace corresponding to PFT angle of opposite sign due to mirror reflection compared to that shown in Fig. 4(a).

Fig. 8.
Fig. 8.

Experimental AC trace for a tilted pulse-front laser beam at beam cross-over angle of (a) 12°; and (b) 6°.

Equations (29)

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E ( x , z = 0 , ω ) = E ( α x , ω ) exp ( i k β ω x )
Δ θ = α Δ η + β ω ; where α = d θ d η ω o and β = d θ d ω η
α = cos η cos θ cos η cos θ ; and β = sin A cos η cos θ ( d n d ω )
β = β ( 1 + α 2 z d ) + ( z R d ) 2 ( 1 + α 2 z d ) 2 + ( z R d ) 2
E ( x , y , z , ω ) = i λ z E ( x , y , z = 0 , ω ) exp [ i π λ z { ( x x ) 2 ( y y ) 2 } ] d x d y
E ( x , ω ) = E ( ω ) exp { i k ( x ξ 0 ω ) 2 2 q ( d ) }
E ( ω ) = E 0 exp { ω 2 τ 0 2 4 } exp ( i φ 2 ω 2 2 }
E ( x , z , ω ) = b 1 i k q ( d ) 2 π z ( d + α 2 z ) E ( ω ) exp [ i k z q ( d ) 2 q ( d + α 2 z ) { β ω + ( x ξ 0 ω α ) z } 2 ]
exp [ i k { β ξ 0 ω 2 α + ( x ξ 0 ω α ) 2 2 z } ]
q ( d ) q ( d + α 2 z ) = d + i π σ 2 λ d + α 2 z + i π σ 2 λ = a 1 ( 1 + i 2 α 2 z a 2 k σ 2 )
where a 1 = ( 1 + α 2 z d ) + ( z R d ) 2 ( 1 + α 2 z d ) 2 + ( z R d ) 2 and a 2 = 1 + d 2 z R 2 + α 2 z d z R 2
E ( x , z , ω ) = b 2 c 1 E 0 i k a 1 2 π z ( 1 + i 2 α 2 z a 2 k σ 2 ) exp ( ω 2 τ o 2 4 ) exp ( i ω 2 φ 2 ( z ) 2 )
exp { α 2 a 1 σ 2 a 2 ( x ξ ( z ) ω ) 2 } exp ( i k β " ω x )
where b 2 is a constant multiplier and c 1 = exp ( i k ( a 1 1 ) x 2 2 z )
β " = a 1 β + ξ o ( a 1 1 ) α z
φ 2 ( z ) = φ 2 a 1 k β 2 z 2 k β ξ o ( a 1 1 ) α k ξ o 2 ( a 1 1 ) α 2 z
ξ ( z ) = ξ 0 α + β z ;
E ( x , z , t ) = b 3 c 1 E 0 i k a 1 2 π z ( 1 + i 2 α 2 z a 2 k σ 2 ) 4 π τ 2 + i 2 φ 2 ( z ) exp ( x 2 σ 2 ) exp { ( t t o τ ) 2 }
exp ( i φ 2 ( z ) υ 2 ( z ) x 2 2 ) exp { i ( ϕ 1 ( t t 0 ) + ϕ 2 ( t t 0 ) 2 2 ) }
t 0 = ( k β " + φ 2 ( z ) ν ( z ) x
ϕ 1 = ν ( z ) x ; ν ( z ) = ξ ( z ) ξ 2 ( z ) + a 2 σ 2 τ 0 2 4 a 1 α 2
ϕ 2 = φ 2 ( z ) τ 4 4 + φ 2 2 ( z ) ; τ 2 = τ 0 2 + 4 a 1 α 2 ξ 2 ( z ) a 2 σ 2
1 σ 2 = a 1 α 2 a 2 σ 2 υ 2 ( z ) τ 0 2 4
tan γ = k β " c + c φ 2 ( z ) ν ( z )
tan γ = k β c a 1 a 2 a 1 ( σ τ o 2 α β z ) 2 [ 1 + a 2 a 1 ( σ τ o 2 α β z ) 2 ]
tan γ = k β c ( σ τ o 2 α β z ) 2 [ 1 + ( σ τ o 2 α β z ) 2 ]
tan γ = k β ( a 1 1 ) α c c k a 1 β α [ 1 + a 2 a 1 ( σ τ o 2 α β z ) 2 ]
tan γ = k β c α [ 1 + ( σ τ o 2 α β z ) 2 ]
γ = tan 1 [ { 1 ( λ n ) ( d n d λ ) } tan δ sin ψ ]

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