Abstract

We report numerical results of second-harmonic generation in a type II potassium dihydrogen phosphate crystal with a time predelay for picosecond and/or femtosecond Yb-doped solid-state lasers, and clarify the dependence of the self compression in the second-harmonic laser pulse on the initial frequency chirp, fundamental duration and intensity, and phase-mismatching angle. We also show numerically the generation possibility of a self-compressed second-harmonic laser pulse near 20 fs.

©2007 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Numerical study of fourth-harmonic generation of a picosecond laser pulse with time predelay

Tiejun Zhang, Yoshiaki Kato, and Hiroyuki Daido
J. Opt. Soc. Am. B 13(6) 1166-1178 (1996)

Nonlinear second-harmonic pulse compression with tilted pulses

A. Dubietis, G. Valiulis, G. Tamošauskas, R. Danielius, and A. Piskarskas
Opt. Lett. 22(14) 1071-1073 (1997)

References

  • View by:
  • |
  • |
  • |

  1. Y. Zaouter, J. Diderjean, F. Balembois, G. Lucas-Leclin, F. Druon, P. Georges, J. Petit, P. Golner, and B. Viana, “47-fs diode pumped Yb3+:CaGdAlO4 laser,” Opt. Lett. 31, 119–121 (2006).
    [Crossref] [PubMed]
  2. K. Yamakawa, M. Aoyama, Y. Akahane, K. Ogawa, K. Tsuji, A. Sugiyama, T. Harimoto, J. Kawanaka, H. Nishioka, and M. Fujita, “Ultra-broadband optical parametric chirped-pulse amplification using an Yb:LiYF4 chirped-pulse amplification pump laser,” Opt. Express 15, 5018–5023 (2007).
    [Crossref] [PubMed]
  3. A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
    [Crossref]
  4. Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990)
    [Crossref] [PubMed]
  5. Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
    [Crossref] [PubMed]
  6. R. Danielius, A. Dubietis, G. Valiulis, and A. Piskaraskas, “Femotosecond high-contrast pulses from a parametric generator pumped by the self-compressed second harmonic of a Nd:glass laser,” Opt. Lett. 20, 2225–2227 (1995).
    [Crossref] [PubMed]
  7. R. Danielius, A. Dubietis, A. Piskaraskas, G. Valiulis, and A. Varanavicius, “Generation of compressed 600–720-nm tunable femotosecond pulses by transient frequency mixing in a β-barium borate crystal,” Opt. Lett. 21, 216–218 (1996).
    [Crossref] [PubMed]
  8. T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
    [Crossref]
  9. T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
    [Crossref]
  10. C. Y. Chien, G. Korn, J. S. Coe, J. Squier, G. Mourou, and R. S. Craxton, “Highly efficient second-harmonic generation of ultraintense Nd:glass laser pulses,” Opt. Lett. 20, 353–355 (1995).
    [Crossref] [PubMed]
  11. L. Zheng and D. D. Meyerhofer, “Self- and cross-phase-modulation coefficients in KDP crystals measured by a Z-scan technique,” LLE Review 74, 125–130 (1998).
  12. R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
    [Crossref]
  13. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, (Springer-Verlag, 1991).
  14. G. Duchateau and A. Dyan, “Coupling statistics and heat transfer to study laser-induced crystal damage by nanosecond pulses,” Opt. Express 15, 4557–4576 (2007).
    [Crossref] [PubMed]
  15. P. DeMange, C. W. Carr, R. A. Negres, H. B. Radousky, and S. G. Demos “Multiwavelength investigation of laser-damage performance in potassium dihydrogen phosphate after laser annealing,” Opt. Lett. 30, 221–223 (2005).
    [Crossref] [PubMed]
  16. H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
    [Crossref]
  17. T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. 39, 1146–1150 (2000).
    [Crossref]

2007 (2)

2006 (2)

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Y. Zaouter, J. Diderjean, F. Balembois, G. Lucas-Leclin, F. Druon, P. Georges, J. Petit, P. Golner, and B. Viana, “47-fs diode pumped Yb3+:CaGdAlO4 laser,” Opt. Lett. 31, 119–121 (2006).
[Crossref] [PubMed]

2005 (1)

2004 (1)

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

2001 (1)

T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
[Crossref]

2000 (1)

T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. 39, 1146–1150 (2000).
[Crossref]

1998 (1)

L. Zheng and D. D. Meyerhofer, “Self- and cross-phase-modulation coefficients in KDP crystals measured by a Z-scan technique,” LLE Review 74, 125–130 (1998).

1996 (1)

1995 (3)

1992 (1)

1991 (1)

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[Crossref]

1990 (1)

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990)
[Crossref] [PubMed]

Akahane, Y.

Aoyama, M.

K. Yamakawa, M. Aoyama, Y. Akahane, K. Ogawa, K. Tsuji, A. Sugiyama, T. Harimoto, J. Kawanaka, H. Nishioka, and M. Fujita, “Ultra-broadband optical parametric chirped-pulse amplification using an Yb:LiYF4 chirped-pulse amplification pump laser,” Opt. Express 15, 5018–5023 (2007).
[Crossref] [PubMed]

T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
[Crossref]

T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. 39, 1146–1150 (2000).
[Crossref]

Balembois, F.

Carr, C. W.

Chien, C. Y.

Coe, J. S.

Craxton, R. S.

Daido, H.

T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
[Crossref]

Danielius, R.

DeMange, P.

Demos, S. G.

Diderjean, J.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, (Springer-Verlag, 1991).

Dragila, R.

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990)
[Crossref] [PubMed]

Druon, F.

Dubietis, A.

Duchateau, G.

Dyan, A.

Fujita, H.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Fujita, M.

Ganeev, R. A.

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Georges, P.

Golner, P.

Gurzadyan, G. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, (Springer-Verlag, 1991).

Harimoto, T.

Ibragimov, E. A.

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[Crossref]

Izawa, Y.

T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
[Crossref]

Kamimura, T.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Kato, Y.

T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
[Crossref]

Kawanaka, J.

Korn, G.

Kulagin, I. A.

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Lucas-Leclin, G.

Luther-Davies, B.

Meyerhofer, D. D.

L. Zheng and D. D. Meyerhofer, “Self- and cross-phase-modulation coefficients in KDP crystals measured by a Z-scan technique,” LLE Review 74, 125–130 (1998).

Mourou, G.

Nakatsuka, M.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Negres, R. A.

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, (Springer-Verlag, 1991).

Nishioka, H.

Ogawa, K.

Petit, J.

Piskaraskas, A.

Radousky, H. B.

Ryasnyansky, A. I.

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Sasaki, T.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Squier, J.

Stabinis, A.

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[Crossref]

Sugiyama, A.

Tsuji, K.

Tugushev, R. I.

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Usmanov, T.

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

Valiulis, G.

Varanavicius, A.

Viana, B.

Wang, Y.

Y. Wang and B. Luther-Davies, “Frequency-doubling pulse compressor for picosecond high-power neodymium laser pulses,” Opt. Lett. 17, 1459–1461 (1992).
[Crossref] [PubMed]

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990)
[Crossref] [PubMed]

Yamakawa, K.

K. Yamakawa, M. Aoyama, Y. Akahane, K. Ogawa, K. Tsuji, A. Sugiyama, T. Harimoto, J. Kawanaka, H. Nishioka, and M. Fujita, “Ultra-broadband optical parametric chirped-pulse amplification using an Yb:LiYF4 chirped-pulse amplification pump laser,” Opt. Express 15, 5018–5023 (2007).
[Crossref] [PubMed]

T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
[Crossref]

T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. 39, 1146–1150 (2000).
[Crossref]

T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
[Crossref]

Yonemura, M.

T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
[Crossref]

Yoshida, H.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Yoshida, K.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Yoshimura, M.

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

Zaouter, Y.

Zhang, T.

T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
[Crossref]

T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. 39, 1146–1150 (2000).
[Crossref]

T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
[Crossref]

Zheng, L.

L. Zheng and D. D. Meyerhofer, “Self- and cross-phase-modulation coefficients in KDP crystals measured by a Z-scan technique,” LLE Review 74, 125–130 (1998).

Jpn. J. Appl. Phys. (4)

H. Yoshida, H. Fujita, M. Nakatsuka, M. Yoshimura, T. Sasaki, T. Kamimura, and K. Yoshida, “Dependences of laser-induced bulk damage threshold and crack patterns in several nonlinear crystals on irradiation direction,” Jpn. J. Appl. Phys. 45, 766–769 (2006).
[Crossref]

T. Zhang, M. Aoyama, and K. Yamakawa: “Noncollinear chirp-compensated second-harmonic generation with subpicosecond laser pulses,” Jpn. J. Appl. Phys. 39, 1146–1150 (2000).
[Crossref]

T. Zhang, Y. Kato, K. Yamakawa, H. Daido, and Y. Izawa : “Peak intensity enhancement and pulse compression of a picosecond laser pulse by frequency doubling with a predelay,” Jpn. J. Appl. Phys. 34, 3552–3561 (1995).
[Crossref]

T. Zhang, M. Yonemura, M. Aoyama, and K. Yamakawa : “A simulation code for tempo-spatial analysis of three-wave interaction with ultra-short and ultra-high intensity laser pulses,” Jpn. J. Appl. Phys. 40, 6455–6456 (2001).
[Crossref]

LLE Review (1)

L. Zheng and D. D. Meyerhofer, “Self- and cross-phase-modulation coefficients in KDP crystals measured by a Z-scan technique,” LLE Review 74, 125–130 (1998).

Opt. Commun. (2)

R. A. Ganeev, I. A. Kulagin, A. I. Ryasnyansky, R. I. Tugushev, and T. Usmanov, “Characterization of nonlinear optical parameters of KDP, LiNbO3 and BBO crystals,” Opt. Commun. 229, 403–412 (2004).
[Crossref]

A. Stabinis, G. Valiulis, and E. A. Ibragimov, “Effective sum frequency pulse compression in nonlinear crystals,” Opt. Commun. 86, 301–306 (1991).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. A (1)

Y. Wang and R. Dragila, “Efficient conversion of picosecond laser pulses into second-harmonic frequency using group-velocity dispersion,” Phys. Rev. A 41, 5645–5649 (1990)
[Crossref] [PubMed]

Other (1)

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of nonlinear optical crystals, (Springer-Verlag, 1991).

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. (a). Dependence of the optimized thickness on the chirp-rate coefficient Φ at fundamental intensities of 5, 10, and 20 GW/cm2. (b) SH energy conversion efficiency with respect to Φ in the type II KDP crystal with the same thickness used in Fig.1 (a). (c) The SH pulse duration.
Fig. 2.
Fig. 2. (a). Optimized thickness versus the fundamental duration for intensities of 5, 10, and 20 GW/cm2. (b) The SH energy conversion efficiency at the same thickness used in Fig. 2(a). (c) The SH pulse duration. The time delay of two chirp-free fundamental pulses is twice as large as the pulse duration. The phase-matching condition is used in the calculation.
Fig.3.
Fig.3. a). Optimized thickness as a function of the fundamental intensity for durations of 50, 75, and 125 fs. (b) The SH energy conversion efficiency at the same thickness used in Fig. 3(a). (c) The SH pulse duration. The time delay of two chirp-free fundamental pulses is twofold of the fundamental duration. The phase-matching condition is assumed.
Fig.4.
Fig.4. a). Optimized thickness as a function of the phase-mismatching angle Δθ0 for fundamental intensities of 0.5, 0.75, and 1 TW/cm2. (b) The SH energy conversion efficiency at the same thickness used in Fig. 4(a). (c) The SH pulse duration. The fundamental pulse duration is 50 fs and the time delay is 150 fs.

Equations (8)

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

A 1 o z + 1 ν 1 o A 1 o t + j g 1 o 2 2 A 1 o t 2 β 1 o 6 3 A 1 o t 3 + δ x 1 o A 1 o x + δ y 1 o A 1 o y j 2 k 1 o 2 A 1 o + α 1 o 2 A 1 o
= j ω 0 n 1 o c d eff A 2 e A 1 e * exp ( j Δ k z ) + j k 1 o ( γ 1 o 1 o I 1 o + γ 1 e 1 o I 1 e + γ 2 e 1 o I 2 e ) A 1 o ,
A 1 e z + 1 ν 1 e A 1 e t + j g 1 e 2 2 A 1 e t 2 β 1 e 6 3 A 1 e t 3 + δ x 1 e A 1 e x + ( δ y 1 e + ρ 1 e ) A 1 e y j 2 k 1 e 2 A 1 e + α 1 e 2 A 1 e
= j ω 0 n 1 e c d eff A 2 e A 1 o * exp ( j Δ k z ) + j k 1 e ( γ 1 o 1 e I 1 o + γ 1 e 1 e I 1 e + γ 2 e 1 e I 2 e ) A 1 e ,
A 2 e z + 1 ν 2 e A 2 e t + j g 2 e 2 2 A 2 e t 2 β 2 e 6 3 A 2 e t 3 + δ x 2 e A 2 e x + ( δ y 2 e + ρ 2 e ) A 2 e y j 2 k 2 e 2 A 2 e + α 2 e 2 A 2 e
= j ω 0 n 2 e c d eff A 1 e A 1 o exp ( j Δ k z ) + j k 2 e ( γ 1 o 2 e I 1 o + γ 1 e 2 e I 1 e + γ 2 e 2 e I 2 e ) A 2 e ,
A m = A 0 m exp { 2 2 G 1 ln 2 [ ( x x 0 m ) 2 + ( y y 0 m ) 2 D m 2 ] G 2 ln 2 ( t t 0 m τ m ) 2 } exp [ j b m ( t t 0 m ) 2 ] ,
τ m Δ ν m = ( 2 ln 2 π ) [ 1 + ( b τ m 2 2 ln 2 ) 2 ] 1 2 = 0.44 Φ ,

Metrics