Abstract

We analyze the spatiotemporal intensity of pulses with durations of 20fs and shorter and a carrier wavelength of 810nm at the paraxial focal plane of an achromatic doublet lens. The incident pulse is well-collimated, and we use the Seidel aberration theory for thin lenses to evaluate the phase change due to the aberrations of the lens. In a set of cemented thin lenses with the stop at the lens, there is only spherical aberration, coma, astigmatism and field curvature, whereas the distortion aberration in the phase front is zero. We analyze the effect of these aberrations in the focusing of ultrashort pulses for homogenous illumination. We will show that the temporal spreading introduced by these aberrations in pulses shorter than 20fs at 810nm is very small but the spatial spreading is not, which reduces the intensity of the pulse considerably.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Z. Bor, “Distortion of femtosecond laser pulse in lenses and lens systems,” J. Mod. Opt. 35, 1907-1918 (1988).
    [CrossRef]
  2. Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119-121 (1989).
    [CrossRef] [PubMed]
  3. Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249-258(1992).
    [CrossRef]
  4. Z. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, J.L.Martin, A.Migus, G.A.Mourou, and A.H.Zewail, eds., Springer Series in Chemical Physics (Springer-Verlag, 1993), Vol. 55.
    [CrossRef]
  5. W. T. Welford, Aberration of Optical Systems (Adam Hilger, 1986).
  6. M. J. Kidger, Fundamental Optical Design (SPIE, 2002).
  7. Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220-222 (2007).
    [CrossRef]
  8. M. Kempe and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158-1165 (1992).
    [CrossRef]
  9. V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775-777 (2004).
    [CrossRef] [PubMed]
  10. P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
    [CrossRef] [PubMed]
  11. F. C. Estrada Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825-834 (2011).
    [CrossRef]
  12. P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
    [CrossRef]
  13. P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
    [CrossRef] [PubMed]
  14. D. Pestov, B. Xu, H. Li, and M. Dantus, “Nonlinear optical imaging with sub-8 fs laser pulses,” in Novel Techniques in Microscopy, OSA Technical Digest (CD) (Optical Society of America, 2011), paper NMD2.
  15. X. Zhu, T. C. Gunaratne, V. V. Lozovoy, and M. Dantus, “In-situ femtosecond laser pulse characterization and compression during micromachining,” Opt. Express 15, 16061-16066(2007).
    [CrossRef] [PubMed]
  16. J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
    [CrossRef]

2011

F. C. Estrada Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825-834 (2011).
[CrossRef]

2010

J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
[CrossRef]

2009

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

2008

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

2007

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220-222 (2007).
[CrossRef]

X. Zhu, T. C. Gunaratne, V. V. Lozovoy, and M. Dantus, “In-situ femtosecond laser pulse characterization and compression during micromachining,” Opt. Express 15, 16061-16066(2007).
[CrossRef] [PubMed]

2004

1992

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249-258(1992).
[CrossRef]

M. Kempe and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158-1165 (1992).
[CrossRef]

1989

1988

Z. Bor, “Distortion of femtosecond laser pulse in lenses and lens systems,” J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

Andegeko, Y.

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

Bor, Z.

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220-222 (2007).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249-258(1992).
[CrossRef]

Z. Bor, “Distortion of femtosecond laser pulses in lenses,” Opt. Lett. 14, 119-121 (1989).
[CrossRef] [PubMed]

Z. Bor, “Distortion of femtosecond laser pulse in lenses and lens systems,” J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, J.L.Martin, A.Migus, G.A.Mourou, and A.H.Zewail, eds., Springer Series in Chemical Physics (Springer-Verlag, 1993), Vol. 55.
[CrossRef]

Dantus, M.

J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
[CrossRef]

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

X. Zhu, T. C. Gunaratne, V. V. Lozovoy, and M. Dantus, “In-situ femtosecond laser pulse characterization and compression during micromachining,” Opt. Express 15, 16061-16066(2007).
[CrossRef] [PubMed]

V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775-777 (2004).
[CrossRef] [PubMed]

D. Pestov, B. Xu, H. Li, and M. Dantus, “Nonlinear optical imaging with sub-8 fs laser pulses,” in Novel Techniques in Microscopy, OSA Technical Digest (CD) (Optical Society of America, 2011), paper NMD2.

Estrada Silva, F. C.

F. C. Estrada Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825-834 (2011).
[CrossRef]

Garduño-Mejía, J.

F. C. Estrada Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825-834 (2011).
[CrossRef]

Gunaratne, T. C.

Gunn, J. M.

J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
[CrossRef]

High, S. H.

J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
[CrossRef]

Horváth, Z. L.

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220-222 (2007).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249-258(1992).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, J.L.Martin, A.Migus, G.A.Mourou, and A.H.Zewail, eds., Springer Series in Chemical Physics (Springer-Verlag, 1993), Vol. 55.
[CrossRef]

Kempe, M.

Kidger, M. J.

M. J. Kidger, Fundamental Optical Design (SPIE, 2002).

Kovács, A. P.

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220-222 (2007).
[CrossRef]

Li, H.

D. Pestov, B. Xu, H. Li, and M. Dantus, “Nonlinear optical imaging with sub-8 fs laser pulses,” in Novel Techniques in Microscopy, OSA Technical Digest (CD) (Optical Society of America, 2011), paper NMD2.

Lovozoy, V. V.

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

Lozovoy, V. V.

J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
[CrossRef]

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

X. Zhu, T. C. Gunaratne, V. V. Lozovoy, and M. Dantus, “In-situ femtosecond laser pulse characterization and compression during micromachining,” Opt. Express 15, 16061-16066(2007).
[CrossRef] [PubMed]

V. V. Lozovoy, I. Pastirk, and M. Dantus, “Multiphoton intrapulse interference. IV. Ultrashort laser pulse spectral phase characterization and compensation,” Opt. Lett. 29, 775-777 (2004).
[CrossRef] [PubMed]

Pastirk, I.

Pestov, D.

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

D. Pestov, B. Xu, H. Li, and M. Dantus, “Nonlinear optical imaging with sub-8 fs laser pulses,” in Novel Techniques in Microscopy, OSA Technical Digest (CD) (Optical Society of America, 2011), paper NMD2.

Rosete-Aguilar, M.

F. C. Estrada Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825-834 (2011).
[CrossRef]

Rudolph, W.

Weisel, L. R.

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberration of Optical Systems (Adam Hilger, 1986).

Xi, P.

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

Xu, B.

D. Pestov, B. Xu, H. Li, and M. Dantus, “Nonlinear optical imaging with sub-8 fs laser pulses,” in Novel Techniques in Microscopy, OSA Technical Digest (CD) (Optical Society of America, 2011), paper NMD2.

Zhu, X.

J. Biomed. Opt.

P. Xi, L. R. Weisel, Y. Andegeko, D. Pestov, V. V. Lozovoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002 (2009).
[CrossRef] [PubMed]

P. Xi, Y. Andegeko, D. Pestov, V. V. Lovozoy, and M. Dantus, “Two-photon imaging using adaptive phase compensated ultrashort laser pulses,” J. Biomed. Opt. 14, 014002(2009).
[CrossRef] [PubMed]

J. Mod. Opt.

F. C. Estrada Silva, J. Garduño-Mejía, and M. Rosete-Aguilar, “Third order dispersion effects generated by non-ideal achromatic doublets on sub-20 femtosecond pulses,” J. Mod. Opt. 58, 825-834 (2011).
[CrossRef]

Z. Bor, “Distortion of femtosecond laser pulse in lenses and lens systems,” J. Mod. Opt. 35, 1907-1918 (1988).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. C

J. M. Gunn, S. H. High, V. V. Lozovoy, and M. Dantus, “Measurement and control of ultrashort optical pulse propagation in metal nanoparticle-covered dielectric surfaces,” J. Phys. Chem. C 114, 12375-12381 (2010).
[CrossRef]

Opt. Commun.

P. Xi, Y. Andegeko, L. R. Weisel, V. V. Lovozoy, and M. Dantus, “Greater signal, increased depth, and less photobleaching in two-photon microscopy with 10 fs pulses,” Opt. Commun. 281, 1841-1849 (2008).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249-258(1992).
[CrossRef]

Opt. Express

Opt. Lett.

Springer Ser. Chem. Phys.

Z. L. Horváth, A. P. Kovács, and Z. Bor, “Distortion of ultrashort pulses caused by aberrations,” Springer Ser. Chem. Phys. 88, 220-222 (2007).
[CrossRef]

Other

Z. Bor and Z. L. Horváth, “Distortion of a 6 fs pulse in the focus of a BK7 lens,” in Ultrafast Phenomena VIII, J.L.Martin, A.Migus, G.A.Mourou, and A.H.Zewail, eds., Springer Series in Chemical Physics (Springer-Verlag, 1993), Vol. 55.
[CrossRef]

W. T. Welford, Aberration of Optical Systems (Adam Hilger, 1986).

M. J. Kidger, Fundamental Optical Design (SPIE, 2002).

D. Pestov, B. Xu, H. Li, and M. Dantus, “Nonlinear optical imaging with sub-8 fs laser pulses,” in Novel Techniques in Microscopy, OSA Technical Digest (CD) (Optical Society of America, 2011), paper NMD2.

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 (9)

Fig. 1
Fig. 1

System coordinates.

Fig. 2
Fig. 2

Lens and illumination parameters.

Fig. 3
Fig. 3

Spatiotemporal pulse intensities, | U | 2 , at the paraxial focal plane for normal incidence angle, u ¯ = 0 . The numbers 0 or 1 indicate that, in the numerical model, the corresponding aberration contribution, S I , S I I , S I I I , and S I V , has been multiplied by 0 or 1. The spatial and temporal axis are normalized quantities given by v = ρ k 0 r 2 f 0 and t / T int , respectively.

Fig. 4
Fig. 4

Spatiotemporal pulse intensities, | U | 2 for u ¯ = 5 ° and u ¯ = 8 ° at the paraxial focal plane. Each row corresponds to the individual contribution of S I , S I I , S I I I , and S I V (spherical aberration, coma, astigmatism, and field curvature, respectively). The last row shows the pulse when all aberrations are taken into account. The spatial and temporal axis are normalized quantities given by v = ρ k 0 r 2 f 0 and t / T int , respectively.

Fig. 5
Fig. 5

Normalized pulse intensity, I ( t ) and focused beam spot at the paraxial focal plane, I ( v ) for u ¯ = 0 ° , u ¯ = 5 ° , and u ¯ = 8 ° . The pulse has an initial duration of 20 fs at 810 nm .

Fig. 6
Fig. 6

Normalized pulse intensity, I ( t ) , and focused beam profile, I ( v ) , as a function of pupil aperture r with respect to the lens semidiameter ρ, for an incident pulse beam making an angle of 5 ° with the optical axis. The pulse has an initial duration of 20 fs at 810 nm .

Fig. 7
Fig. 7

Temporal and spatial aberration effects for input pulses of 6, 10, and 20 fs at 810 nm .

Fig. 8
Fig. 8

Aberration effects for an input pulse of 6 fs at 810 nm and at u ¯ = 5 ° incidence angle. For each case, 0 or 1 indicates that, in the numerical model, the corresponding aberration contribution, S I , S I I , S I I I , and S I V , has been multiplied by 0 or 1.

Fig. 9
Fig. 9

Temporal aberration effect compensation by pupil aperture reduction corresponding to r / ρ = 1 / 8 for the 6 fs pulse at 810 nm .

Equations (53)

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

U ( x 2 , y 2 , z , u ¯ , Δ ω ) d x 1 d y 1 U 0 ( x 1 , y 1 , u ¯ ) P ( x 1 , y 1 ) A ( Δ ω ) exp { i Θ ( x 1 , y 1 ; η ) } exp { i Φ ( x 1 , y 1 ) } × exp { i k a 2 z [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ] } ,
P ( x 1 , y 1 ) = { 1 , if     x 1 2 + y 1 2 = r 1 2 = ( ρ r ) 2 0 , otherwise r [ 0 , 1 ]
A ( Δ ω ) = A 0 × exp [ ( T Δ ω / 2 ) 2 ]
Φ ( x 1 , y 1 ) = ( k 1 d 1 + k 2 d 2 ) ( x 1 2 + y 1 2 ) 2 [ k 0 R 1 ( n 1 1 ) β 1 k 0 R 2 ( n 1 1 ) β 1 + k 0 R 2 ( n 2 1 ) β 2 k 0 R 3 ( n 2 1 ) β 2 ] ,
β 1 = 1 + b 1 1 Δ ω + b 2 1 ( Δ ω ) 2 + b 3 1 ( Δ ω ) 3 ,
β 2 = 1 + b 1 2 Δ ω + b 2 2 ( Δ ω ) 2 + b 3 1 ( Δ ω ) 3 ,
k j = ω c n j ( ω ) k 0 n 0 j [ 1 + a 1 j Δ ω + a 2 j ( Δ ω ) 2 + a 3 j ( Δ ω ) 3 ] ,
a 1 j = 1 ω 0 + 1 n j d n j d ω | ω 0 ,
a 2 j = 1 ω 0 n j d n j d ω | ω 0 + 1 2 n j d 2 n j d ω 2 | ω 0 ,
a 3 j = 1 2 ω 0 n j d 2 n j d ω 2 | ω 0 + 1 6 n j d 3 n j d ω 3 | ω 0 ,
Θ ( x 1 , y 1 ; η ) = k a W ( x 1 , y 1 ; η ) ,
W ( x 1 , y 1 ; η ) = 1 8 S I ( x 1 2 + y 1 2 ) 2 ρ 4 + 1 2 S I I y 1 ( x 1 2 + y 1 2 ) ρ 3 η η max + 1 2 S I I I y 1 2 ρ 2 η 2 η max 2 + 1 4 ( S I I I + S I V ) ( x 1 2 + y 1 2 ) ρ 2 η 2 η max 2 .
x 1 = r 1 sin θ , y 1 = r 1 cos θ ,
W ( r , η , θ ) = S I 8 r 4 + S I I 2 r 3 cos θ ( η η max ) + 1 2 S I I I ( r 2 cos 2 θ ) ( η η max ) 2 + 1 4 ( S I I I + S I V ) ( r 2 ) ( η η max ) 2 ,
S I , i = ρ 4 K i 3 4 ( n i 2 ( n i 1 ) 2 + ( n i + 2 ) n i ( n i 1 ) 2 ( B i + 2 ( n i 2 1 ) C i n i + 2 ) 2 n i C i 2 n i + 2 ) ,
S I I , i = ρ 2 K i 2 H 2 { ( n i + 1 ) B i n i ( n i 1 ) + ( 2 n i + 1 ) C i n i } ,
S I I I , i = H 2 K i ,
S I V , i = H 2 K i n i ,
B 1 = ( ς 1 + ς 2 ) ( ς 1 ς 2 ) ,
B 2 = ( ς 2 + ς 3 ) ( ς 2 ς 3 ) ,
C 1 = 1 ,
C 2 = μ 1 + μ 2 μ 1 μ 2 ,
S I = i = 1 2 S I , i ; S I I = i = 1 2 S I I , i ; S I I I = i = 1 2 S I I I , i ; S I V = i = 1 2 S I V , i .
U ( x 2 , y 2 , z , u ¯ , Δ ω ) exp { i [ ( k 1 d 1 + k 2 d 2 ) ] } d x 1 d y 1 U 0 ( x 1 , y 1 , u ¯ ) P ( x 1 , y 1 ) A ( Δ ω ) exp { i Θ ( x 1 , y 1 ) } exp [ i k 0 ( x 1 2 + y 1 2 2 ) ( 1 f 0 1 z ) ] × exp { i k 0 ( x 1 2 + y 1 2 ) 2 Δ ω [ ( n 1 1 ) R 1 ( β 1 1 ) Δ ω ( n 1 1 ) R 2 ( β 1 1 ) Δ ω + ( n 2 1 ) R 2 ( β 2 1 ) Δ ω ( n 2 1 ) R 3 ( β 2 1 ) Δ ω 1 z ω 0 ] } × exp { i k 0 ( 1 + Δ ω ω 0 ) 2 f 0 ( x 2 2 + y 2 2 ) } exp { i k 0 ( 1 + Δ ω ω 0 ) f 0 ( x 1 x 2 + y 1 y 2 ) } ,
x 2 = r 2 sin ϕ , y 2 = r 2 cos ϕ ,
U ( r 2 , ϕ , z , u ¯ , Δ ω ) exp { i [ ( k 1 d 1 + k 2 d 2 ) ] } 0 2 π 0 r 1 r 1 d r 1 d θ U 0 ( r 1 , θ , u ¯ ) P ( r 1 ) A ( Δ ω ) exp { i Θ ( r 1 ; θ ) } exp [ i k 0 ( r 1 2 2 ) ( 1 f 0 1 z ) ] × exp { i k 0 ( r 1 2 ) 2 Δ ω [ ( n 1 1 ) R 1 ( β 1 1 ) Δ ω ( n 1 1 ) R 2 ( β 1 1 ) Δ ω + ( n 2 1 ) R 2 ( β 2 1 ) Δ ω ( n 2 1 ) R 3 ( β 2 1 ) Δ ω 1 z ω 0 ] } × exp { i k 0 ( 1 + Δ ω ω 0 ) 2 f 0 ( r 2 2 ) } exp { i k 0 ( 1 + Δ ω ω 0 ) f 0 ( r 1 sin θ r 2 sin ϕ + r 1 cos θ r 2 sin ϕ ) } .
U ( u , v , ϕ , u ¯ , Δ ω ) exp { i [ ( k 1 d 1 + k 2 d 2 ) ] } 0 1 r d r A ( Δ ω ) exp { i ( u 2 ) r 2 } × exp { i r 2 ( τ Δ ω + δ Δ ω 2 ) } × exp { i ( τ Δ ω + δ Δ ω 2 ) } × exp { i v 2 4 N ( 1 + Δ ω ω 0 ) } 0 2 π d θ exp [ i ( 1 + Δ ω ω 0 ) v r cos ( θ ϕ ) ] × exp { i Θ ( r 1 ; θ ) } exp { i k 0 r 1 cos θ sin u ¯ } ,
τ = k 0 ρ 2 2 ( ( n 1 1 ) b 1 1 R 1 ( n 1 1 ) b 1 1 R 2 + ( n 2 1 ) b 1 2 R 2 ( n 2 1 ) b 1 2 R 3 ) ( k 0 ρ 2 2 f 0 ω 0 u 2 ω 0 ) ,
δ = ρ 2 k 0 2 [ ( n 1 1 ) b 2 1 R 1 ( n 1 1 ) b 2 1 R 2 + ( n 2 1 ) b 2 2 R 2 ( n 2 1 ) b 2 2 R 3 ] ,
τ = k 0 ( n 1 d 1 a 1 1 + n 2 d 2 a 1 2 ) ,
δ = k 0 ( n 1 d 1 a 2 1 + n 2 d 2 a 2 2 ) .
exp { i v 2 4 N ( 1 + Δ ω ω 0 ) } exp ( i v 2 4 N ) ,
exp [ i ( 1 + Δ ω ω 0 ) v r cos ( θ ϕ ) ] exp [ i v r cos ( θ ϕ ) ] ,
Θ ( r 1 ; θ ) k 0 W ( r 1 ; θ ) .
0 2 π d θ exp [ i v r cos ( θ ϕ ) ] × exp { i Θ ( r 1 ; θ ) exp { i k 0 ρ r cos θ sin u ¯ } } = exp [ i k 0 ( S I 8 r 4 + 1 4 ( S I I I + S I V ) r 2 ) ] 0 2 π d θ exp [ i v r cos ( θ ϕ ) ] exp { i k 0 ρ r cos θ sin u ¯ } × exp { i k 0 ( S I I 2 r 3 cos θ + S I I I 2 r 2 cos 2 θ ) } = exp [ i k 0 ( S I 8 r 4 + 1 4 ( S I I I + S I V ) r 2 ) ] × G ( ν r , ϕ , u ¯ ) ,
G ( v r , ϕ , u ¯ ) = 0 2 π d θ exp [ i v r cos ( θ ϕ ) ] × exp { i k 0 ( S I I 2 r 3 cos θ + S I I I 2 r 2 cos 2 θ ) } × exp { i k 0 ρ r cos θ sin u ¯ } .
U ( u , v , ϕ , u ¯ , t ) d ( Δ ω ) exp { i ( Δ ω ) t } U ( u , v , ϕ , u ¯ , Δ ω ) .
U ( u , v , ϕ , u ¯ , t ) K d ( Δ ω ) exp { ( Δ ω ) 2 p 2 } exp { ( Δ ω ) q } × 0 1 r d r exp { i ( u 2 ) r 2 } × exp [ i k 0 ( S I 8 r 4 + 1 4 ( S I I I + S I V ) r 2 ) ] × G ( ν r , ϕ , u ¯ ) ,
p = T 2 4 i ( δ r 2 δ ) ,
q = i ( t τ + r 2 τ ) ,
K = exp { i [ ( k 1 d 1 + k 2 d 2 ) ] } exp ( i ν 2 4 N ) .
exp ( p 2 x 2 ± q x ) d x = π p 2 exp ( q 2 4 p 2 ) .
U ( u , v , ϕ , u ¯ , t ) K 0 1 r d r × ( 4 π T 2 [ 1 + i ξ ] [ 1 + ξ 2 ] ) 1 / 2 exp { ( t τ + r 2 τ ) 2 [ 1 + i ξ ] T 2 [ 1 + ξ 2 ] } exp { i ( u 2 ) r 2 } × exp [ i k 0 ( S I 8 r 4 + 1 4 ( S I I I + S I V ) r 2 ) ] × G ( ν r , ϕ , u ¯ ) ,
i ξ = 4 i ( δ r 2 δ ) T 2 .
U ( u , v , ϕ , u ¯ , t ) K d ( Δ ω ) exp { ( Δ ω ) q } exp { ( Δ ω ) 2 p 2 } exp { ( Δ ω ) 3 s } 0 r r d r P ( r ) exp [ i k 0 ( S I 8 r 4 + 1 4 ( S I I I + S I V ) r 2 ) ] × G ( ν r , ϕ , u ¯ ) ,
K = exp { i [ k 0 ( n 1 d 1 + n 2 d 2 ) ] } exp { i ( v 2 4 N ) } ,
p = T 2 4 i ( δ r 2 δ ) ,
q = i ( t τ + r 2 τ ( u ) ) ,
s = i ( γ r 2 γ ) ,
γ = ρ 2 k 0 2 [ ( n 1 1 ) b 3 1 R 1 ( n 1 1 ) b 3 1 R 2 + ( n 2 1 ) b 3 2 R 2 ( n 2 1 ) b 3 2 R 3 ] ,
γ = k 0 ( n 1 d 1 a 3 1 + n 2 d 2 a 3 2 ) .
I ( t ) 0 d v v | U ( u , v , ϕ , u ¯ , t ) | 2 ,
I ( v ) d t | U ( u , v , ϕ , u ¯ , t ) | 2 .

Metrics