Abstract

We study a statistical ensemble of multimode laser beams. Each beam is made up of an incoherent superposition of off-axis polychromatic Hermite–Gaussian modes. We obtain analytic expressions for the squared beam radius, the waist position, the Rayleigh range, the skewness parameter, the kurtosis parameter, and the squared beam-propagation factor. We demonstrate that the squared beam radius has a quadratic dependence on the distance from the waist plane. The skewness parameter may be different from zero in the near-field zone, but it tends to zero in the far-field zone. The kurtosis parameter in the far-field zone coincides with the kurtosis parameter of the incoherent superposition of on-axis modes.

© 2002 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1560 (1966).
    [CrossRef] [PubMed]
  2. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  3. A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
    [CrossRef]
  4. C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
    [CrossRef]
  5. S. Wang, Q. Lin, X. Jiang, “Axial superposition of Gaussian spherical beams,” Opt. Laser Technol. 31, 151–155 (1999).
    [CrossRef]
  6. Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
    [CrossRef]
  7. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  8. F. Encinas-Sanz, J. M. Guerra, I. Pastor, “Transverse pattern morphogenesis in a CO2 laser,” Opt. Lett. 21, 1153–1155 (1996).
    [CrossRef] [PubMed]
  9. J. Merlin, C. Oliveira, J. Dietz, “Energy distribution analysis of high-power laser beam from spots on paper,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. SPIE952, 731–735 (1988).
  10. E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
    [CrossRef] [PubMed]
  11. B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
    [CrossRef]
  12. Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
    [CrossRef]
  13. O. E. Martı́nez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988).
    [CrossRef]
  14. O. E. Martı́nez, “Matrix formalism for disperse laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
    [CrossRef]
  15. A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
    [CrossRef]
  16. C. J. R. Sheppard, X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
    [CrossRef]
  17. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [CrossRef]
  18. M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
    [CrossRef]
  19. L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
    [CrossRef]
  20. L. Martı́-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
    [CrossRef]
  21. L. Martı́-López, O. Mendoza-Yero, J. A. Ramos-de-Campo, “Propagation of polychromatic Gaussian beams through thin lenses,” J. Opt. Soc. Am. A 18, 1348–1356 (2001).
    [CrossRef]
  22. M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).
  23. R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
    [CrossRef]

2001 (3)

L. Martı́-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, J. A. Ramos-de-Campo, “Propagation of polychromatic Gaussian beams through thin lenses,” J. Opt. Soc. Am. A 18, 1348–1356 (2001).
[CrossRef]

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

2000 (2)

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
[CrossRef]

1999 (4)

S. Wang, Q. Lin, X. Jiang, “Axial superposition of Gaussian spherical beams,” Opt. Laser Technol. 31, 151–155 (1999).
[CrossRef]

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

1998 (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

1997 (2)

C. J. R. Sheppard, X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

1996 (2)

F. Encinas-Sanz, J. M. Guerra, I. Pastor, “Transverse pattern morphogenesis in a CO2 laser,” Opt. Lett. 21, 1153–1155 (1996).
[CrossRef] [PubMed]

E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
[CrossRef] [PubMed]

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1993 (1)

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

1990 (1)

A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

1989 (1)

O. E. Martı́nez, “Matrix formalism for disperse laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

1988 (1)

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

1966 (1)

Beran, M. J.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Borghi, R.

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Cao, Q.

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Chen, T.

Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
[CrossRef]

Dangoisse, D.

E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
[CrossRef] [PubMed]

Deng, X.

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Dietz, J.

J. Merlin, C. Oliveira, J. Dietz, “Energy distribution analysis of high-power laser beam from spots on paper,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. SPIE952, 731–735 (1988).

Encinas-Sanz, F.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

F. Encinas-Sanz, J. M. Guerra, I. Pastor, “Transverse pattern morphogenesis in a CO2 laser,” Opt. Lett. 21, 1153–1155 (1996).
[CrossRef] [PubMed]

Gan, X.

C. J. R. Sheppard, X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Glorieux, P.

E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
[CrossRef] [PubMed]

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Guerra, J. M.

He, P.

Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
[CrossRef]

Hennequin, D.

E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
[CrossRef] [PubMed]

Jiang, X.

S. Wang, Q. Lin, X. Jiang, “Axial superposition of Gaussian spherical beams,” Opt. Laser Technol. 31, 151–155 (1999).
[CrossRef]

Kogelnik, H.

Kostenbauder, A. G.

A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

Li, T.

Lin, Q.

S. Wang, Q. Lin, X. Jiang, “Axial superposition of Gaussian spherical beams,” Opt. Laser Technol. 31, 151–155 (1999).
[CrossRef]

Louvergneaux, E.

E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
[CrossRef] [PubMed]

Lü, B.

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

Ma, H.

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

Marti´-López, L.

L. Martı́-López, O. Mendoza-Yero, J. A. Ramos-de-Campo, “Propagation of polychromatic Gaussian beams through thin lenses,” J. Opt. Soc. Am. A 18, 1348–1356 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

Marti´nez, C.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Marti´nez, O. E.

O. E. Martı́nez, “Matrix formalism for disperse laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

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

Marti´nez-Herrero, R.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Meji´as, P. M.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Mendoza-Yero, O.

L. Martı́-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, J. A. Ramos-de-Campo, “Propagation of polychromatic Gaussian beams through thin lenses,” J. Opt. Soc. Am. A 18, 1348–1356 (2001).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

Merlin, J.

J. Merlin, C. Oliveira, J. Dietz, “Energy distribution analysis of high-power laser beam from spots on paper,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. SPIE952, 731–735 (1988).

Oliveira, C.

J. Merlin, C. Oliveira, J. Dietz, “Energy distribution analysis of high-power laser beam from spots on paper,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. SPIE952, 731–735 (1988).

Parrent, G. B.

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

Pastor, I.

Piquero, G.

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Porras, M. A.

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Ramos-de-Campo, J. A.

Santarsiero, M.

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Serna, J.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

Siegman, A. E.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

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

Townsend, S. W.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

Wang, S.

S. Wang, Q. Lin, X. Jiang, “Axial superposition of Gaussian spherical beams,” Opt. Laser Technol. 31, 151–155 (1999).
[CrossRef]

Wang, Z. Y.

Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
[CrossRef]

Zuo, T. C.

Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (4)

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

O. E. Martı́nez, “Matrix formalism for disperse laser cavities,” IEEE J. Quantum Electron. 25, 296–300 (1989).
[CrossRef]

A. G. Kostenbauder, “Ray-pulse matrices: a rotational treatment for dispersive optical systems,” IEEE J. Quantum Electron. 26, 1148–1157 (1990).
[CrossRef]

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

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

Opt. Commun. (6)

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Z. Y. Wang, T. Chen, P. He, T. C. Zuo, “Calculation of mode contents of high-power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (2000).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

C. J. R. Sheppard, X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[CrossRef]

B. Lü, H. Ma, “Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry,” Opt. Commun. 171, 185–194 (1999).
[CrossRef]

Q. Cao, X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Opt. Laser Technol. (3)

S. Wang, Q. Lin, X. Jiang, “Axial superposition of Gaussian spherical beams,” Opt. Laser Technol. 31, 151–155 (1999).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Effect of chromatic aberration on Gaussian beams: non-dispersive laser resonators,” Opt. Laser Technol. 31, 239–245 (1999).
[CrossRef]

L. Martı́-López, O. Mendoza-Yero, “Polychromatic Gaussian beams emitted by dispersive laser resonators,” Opt. Laser Technol. 33, 1–5 (2001).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 17–30 (2000).
[CrossRef]

Phys. Rev. A (2)

M. A. Porras, “Propagation of single-cycle pulsed light beams in dispersive media,” Phys. Rev. A 60, 5069–5073 (1999).
[CrossRef]

E. Louvergneaux, D. Hennequin, D. Dangoisse, P. Glorieux, “Transverse mode competition in a CO2 laser,” Phys. Rev. A 53, 4435–4438 (1996).
[CrossRef] [PubMed]

Phys. Rev. E (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Other (3)

J. Merlin, C. Oliveira, J. Dietz, “Energy distribution analysis of high-power laser beam from spots on paper,” in Laser Technologies in Industry, S. P. Almeida, L. M. Bernardo, O. D. Soares, eds., Proc. SPIE952, 731–735 (1988).

M. J. Beran, G. B. Parrent, Theory of Partial Coherence (Prentice-Hall, Englewood Cliffs, N.J., 1964).

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

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Figures (1)

Fig. 1
Fig. 1

Off-axis monochromatic TEMmn mode of a multimode polychromatic beam emitted by a dispersive optical resonator. x¯mn, y¯mn denote the centroid’s coordinates.

Equations (126)

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

2x2u˜(x, y, z, λ)+2y2u˜(x, y, z, λ)-j4πλzu˜(x, y, z, λ)=0,
u˜(x, y, z, λ)=n=0m=0Bmn(λ)δ(x-Δxmn)δ(y-Δymn)×δ(z-Δzmn)*x,y,zu˜mn(x, y, z, λ),
I(x, y, z)=l=1Lqln=0m=0Imnl(x, y, z)=0l=1Lqln=0m=0Imnl(x, y, z, λ)dλ,
Imnl(x, y, z, λ)=ρmnl(λ)δ(x-x¯mnl)δ(y-y¯mnl)×δ(z-z¯mnl)*x,y,z|u˜mn(x, y, z, λ)|2,
ρmnl(λ)=|Bmnl(λ)|2.
Imnl(x, y, z, λ)=ρmnl(λ)δ(x-x¯mn)δ(y-y¯mn)×δ[z-z0(λ)]*x,y,z|u˜mn(x, y, z, λ)|2.
I(x, y, z)=0n=0m=0ρmn(λ)δ(x-x¯mn)δ(y-y¯mn)×δ(z-z0(λ))*x,y,z|u˜mn(x, y, z, λ)|2dλ,
ρmn(λ)=l=1Lqlρmnl(λ),
Imn(x, y, z, λ)=l=1LqlImnl(x, y, z, λ)=ρmn(λ)δ(x-x¯mn)δ(y-y¯mn)×δ[z-z0(λ)]*x,y,z|u˜mn(x, y, z, λ)|2.
Imn(x, y, z)=0Imn(x, y, z, λ)dλ.
x¯(z)=1P--I(x, y, z)xdxdy,
y¯(z)=1P--I(x, y, z)ydxdy,
P=--I(x, y, z)dxdy=n=0m=0Pmn,
Pmn=--Imn(x, y, z)dxdy.
x¯(z)=n=0m=0pmnx¯mn(z),
y¯(z)=n=0m=0pmny¯mn(z),
x¯mn(z)=1Pmn--Imn(x, y, z)xdxdy,
y¯mn(z)=1Pmn--Imn(x, y, z)ydxdy,
pmn=PmnP.
σ2(z)=σx2(z)+σy2(z),
σx2(z)=1P--I(x, y, z)[x-x¯(z)]2dxdy,
σy2(z)=1P--I(x, y, z)[y-y¯(z)]2dxdy.
σx2(z)=n=0m=0pmnσxmn2(z)+n=0m=0pmn[x¯mn(z)-x¯(z)]2,
σy2(z)=n=0m=0pmnσymn2(z)+n=0m=0pmn[y¯mn(z)-y¯(z)]2,
σ2(z)=n=0m=0pmnσmn2(z)+n=0m=0pmnr¯mn2(z),
σxmn2(z)=1Pmn--Imn(x, y, z)[x-x¯mn(z)]2dxdy,
σymn2(z)=1Pmn--Imn(x, y, z)[y-y¯mn(z)]2dxdy,
σmn2(z)=σxmn2(z)+σymn2(z),
r¯mn2(z)=[x¯mn(z)-x¯(z)]2+[y¯mn(z)-y¯(z)]2.
θx2=limzσx2(z)(z-zw)2=n=0m=0pmnθxmn2+n=0m=0pmnϕxmn2,
θxmn2=limzσxmn2(z)(z-zw)2,
ϕxmn2=limz[x¯mn(z)-x¯(z)]2(z-zw)2,
θ2=limzσ2(z)(z-zw)2=n=0m=0pmnθmn2+n=0m=0pmnϕmn2,
θmn2=limzσmn2(z)(z-zw)2=θxmn2+θymn2,
ϕmn2=ϕxmn2+ϕymn2.
|u˜mn(x, y, z, λ)|2=12m+n-1πm!n!w2(z, λ)Hm221/2xw(z, λ)×Hn221/2yw(z, λ)exp-2x2+y2w2(z, λ),
w2(z, λ)=w02(λ)1+z2zR2(λ);
σxmn2(z)=1Pmn0--ρmn(λ)x2×|u˜mn[x, y, z-z0(λ), λ]|2dxdydλ=1Pmn0ρmn(λ)σxλm2(z, λ)dλ,
σxλm2(z, λ)=--x2|u˜mn[x, y, z-z0(λ), λ]|2dxdy=σxλwm2(λ)1+[z-z0(λ)]2zR2(λ),
σxmn2(z)=θxmn2(z-z¯0,mn)2-2Gx0,mn(z-z¯0,mn)+σx1,mn2+σx0,mn2,
z¯0,mn=1Pmn0ρmn(λ)z0(λ)dλ,
θxmn2=1Pmn0ρmn(λ)θxλm2(λ)dλ,
Gx0,mn=1Pmn0ρmn(λ)θxλm2(λ)δmn(λ)dλ,
σx1,mn2=1Pmn0ρmn(λ)θxλm2(λ)δmn2(λ)dλ,
σx0,mn2=1Pmn0ρmn(λ)σxλwm2(λ)dλ,
θxλm2(λ)=limzσxλm2(z, λ)(z-zw)2,
δmn(z)=z0(λ)-z¯0,mn.
zxwmn=z¯0,mn+Gx0,mnθxmn2,
σxwmn2=σxmn2(zxwmn)=σx0,mn2+σx1,mn2-Gx0,mn2θxmn2.
σxmn2(z)=θxmn2(z-zxwmn)2+σxwmn2.
zxRmn2=σxwmn2θxmn2.
σmn2(z)=θmn2(z-zwmn)2+σwmn2,
θmn2=θxmn2+θymn2,
zwmn=z¯0,mn+G0,mnθmn2,
σwmn2=σmn2(zwmn)=σ0,mn2+σ1,mn2-G0,mn2θmn2,
G0,mn=Gx0,mn+Gy0,mn,
σ1,mn2=σx1,mn2+σy1,mn2,
σ0,mn2=σx0,mn2+σy0,mn2.
zRmn2=σwmn2θmn2.
σx2(z)=θx2(z-zxw)2+σxw2,
θx2=n=0m=0pmnθxmn2,
zxw=z¯0+Gx1θx2,
σxw2=σx02+σx12+σx22+σx32-Gx12/θx2,
Gx1=Gx0-n=0m=0pmnθxmn2(z¯0-z¯0,mn),
Gx0=n=0m=0pmnGx0,mn,
σx02=n=0m=0pmnσx0,mn2,
σx12=n=0m=0pmnσx1,mn2,
σx22=2n=0m=0pmnx¯mn2,
σx32=n=0m=0pmnθxmn2(z¯0-z¯0,mn)2-2n=0m=0pmnGx0,mn(z¯0-z¯0,mn).
z¯0=n=0m=0pmnz¯0,mn,
zRx2=σxw2/θx2.
σ2(z)=σx2(z)+σy2(z)=θ2(z-zw)2+σw2,
θ2=θx2+θy2=n=0m=0pmnθmn2.
zw=θx2zxw+θy2zywθ2=z¯0+G1θ2,
G1=Gx1+Gy1.
σw2=σxw2+σyw2+σ42=σ02+σ12+σ22+σ32+σ42-Gx12θx2-Gy12θy2,
σ02=σx02+σy02,
σ12=σx12+σy12,
σ22=σx22+σy22,
σ32=σx32+σy32,
σ42=θx2(zw-zxw)2+θy2(zw-zyw)2.
zRMP2=σw2/θ2.
σw2=σ02=n=0m=0pmnσ0,mn2.
Sx(z)=[σx2(z)]-3/2P--I(x, y, z)x3dxdy=[σx2(z)]-3/2n=0m=0[σxmn2(z)]3/2pmnSxmn(z)+3[σx2(z)]-3/2n=0m=0pmnσxmn2(z)x¯mn+[σx2(z)]-3/2n=0m=0pmnx¯mn3,
Sxmn(z)=[σxmn2(z)]-3/2Pmn--Imn(x, y, z)×(x-x¯mn)3dxdy.
Sx(z)=3[σx2(z)]-3/2n=0m=0pmnσxmn2(z)x¯mn+[σx2(z)]-3/2n=0m=0pmnx¯mn3.
Sx(z)=3[θx2(z-zxw)2+σxw2]-3/2(z-zxw)2×n=0m=0pmnx¯mnθxmn2+6[θx2(z-zxw)2+σxw2]-3/2(z-zxw)×n=0m=0pmnx¯mnθxmn2(zxw-zxwmn)+3[θx2(z-zxw)2+σxw2]-3/2×n=0m=0pmnx¯mnθxmn2(zxw-zxwmn)2+3[θx2(z-zxw)2+σxw2]-3/2×n=0m=0pmnσxwmn2(z)x¯mn+[θx2(z-zxw)2+σxw2]-3/2n=0m=0pmnx¯mn3.
Sx(zxw)=3(σxw2)-3/2n=0m=0pmnx¯mnθxmn2(zw-zxwmn)2+3(σxw2)-3/2n=0m=0pmnσxwmn2x¯mn+(σxw2)-3/2n=0m=0pmnx¯mn3.
limz Sx(z)=0.
K(z)=σ4(z)[σ2(z)]2,
σ4(z)=1P--I(x, y, z)r4dxdy,
K(z)=[σ2(z)]-2n=0m=0pmn[σmn2(z)]2Kmn(z)+4[σ2(z)]-2n=0m=0pmn[σxmn2(z)x¯mn2+σymn2(z)y¯mn2]+2[σ2(z)]-2n=0m=0pmnσmn2(z)r¯mn2+[σ2(z)]-2n=0m=0pmnr¯mn4+4n=0m=0pmn[σxmn2(z)]3/2Sxmn(z)x¯mn+4n=0m=0pmn[σymn2(z)]3/2Symn(z)y¯mn,
Kmn(z)=σmn4(z)[σmn2(z)]2,
σmn4(z)=1Pmn--Imn(x, y, z)[(x-x¯mn)2+(y-y¯mn)2]2dxdy.
K(z)=[σ2(z)]-2n=0m=0pmn[σmn2(z)]2Kmn(z).
K(zw)=[σw2(z)]-2n=0m=0pmn[θmn2(zw-zwmn)2+σwmn2]2Kmn(zw)+4[σw2(z)]-2n=0m=0pmn{[θxmn2(zw-zxwmn)2+σxwmn2]x¯mn2+[θymn2(zw-zywmn)2+σywmn2]y¯mn2}+2[σw2(z)]-2n=0m=0pmn[θmn2(zw-zwmn)2+σwmn2]r¯mn2+2[σw2(z)]-2n=0m=0pmnr¯mn4.
limz K(z)=(θ2)-2n=0m=0pmn(θmn2)2limz Kmn(z).
limz Kmn(z)=mnγmn21pmn0ρmn(λ)λ2zR-2(λ)dλ1pmn0ρmn(λ)λzR-1(λ)dλ2,
limz K(z)=(θ2)-2n=0m=0pmn(θmn2)2mnγmn2×1pmn0ρmn(λ)λ2zR-2(λ)dλ1pmn0ρmn(λ)λzR-1(λ)dλ2.
limz K(z)=(θ2)-2n=0m=0pmn(θmn2)2mnγmn2+(θ2)-2n=0m=0pmn(θmn2)2mnγmn2(λ-λ¯mn)2¯(λ¯mn)2,
(λ-λ¯mn)2¯=1pmn0ρmn(λ)(λ-λ¯mn)dλ,
λ¯mn=1pmn0ρmn(λ)λdλ.
SMP2=4π2θ2σw2λ¯2.
SMP2=SMPCSM2+4π2θ2λ¯2σ12+σ22+σ32+σ42-Gx12θx2-Gy12θy2
SMPCSM2=4π2θ2σ02λ¯2,
λ¯=n=0m=0pmnλ¯mn.
SMPCSM2=4π2λ¯2n=0m=0pmnθmn2n=0m=0pmnσ0,mn2=1λ¯2n=0m=0pmn2λ¯mn2SMmn2+4π2λ¯2nnmmn=0m=0pmnpmn×θmn2σ0,mn2,
SMmn2=4π2λ¯mn2θmn2σ0,mn2.
SMP2=1λ¯2n=0m=0pmn2λ¯mn2SMmn2+4π2λ¯2nnmmn=0m=0pmnpmnθmn2σ0,mn2+4π2θ2λ¯2σ12+σ22+σ32+σ42-Gx12θx2-Gy12θy2.
I(x, y, z)=0l=1Lql[I10l(x, y, z, λ)+I01l(x, y, z, λ)]dλ.
I10l(x, y, z, λ)=ρ10l(λ)δ(x-x¯10)δ(y-y¯10)×δ[z-z¯0(λ)]*x,y,z|u˜10(x, y, z, λ)|2,
I01l(x, y, z, λ)=ρ10l(λ)δ(x-x¯01)δ(y-y¯01)×δ[z-z¯0(λ)]*x,y,z|u˜01(x, y, z, λ)|2.
σx2(z)=1Pl=1Lql0--x2[I10l(x, y, z, λ)+I01l(x, y, z, λ)]dxdydλ=p10σx102(z)+p01σx012(z)+p10x¯102+p01x¯012=σ102(z)2+x¯102+x¯0122,
σy2(z)=1Pl=1Lql0--y2[I10l(x, y, z, λ)+I01l(x, y, z, λ)]dxdydλ=p10σy102(z)+p01σy012(z)+p10y¯102+p01y¯012=σ102(z)2+y¯102+y¯0122,
σ2(z)=σx2(z)+σy2(z)=p10σ102(z)+p01σ012(z)+p10r¯102+p01r¯012=σ102(z)+r¯102+r¯0122,
σ102(z)=1P100ρ10(λ)w02(λ)1+[z-z0(λ)]2zR2(λ)dλ,
P=0l=1Lqlρ10l(λ)dλ+0l=1Lqlρ01l(λ)dλ=0ρ10(λ)dλ+0ρ01(λ)dλ=P10+P01.
p10=p01=12.
θ2=p10θ102+p01θ012=θ102,
θ102=1P100ρ10(λ)λπzR(λ)dλ.
Sx(z)=38(3x¯10+x¯01)σ102(z)σ102(z)2+x¯102+x¯01223/2+12x¯103+x¯013σ102(z)2+x¯102+x¯01223/2.
K(z)=3σ104(z)+[3(x¯012+y¯102)+5(x¯102+y¯012)]σ102(z)+r¯104+r¯0142σ102(z)+r¯102+r¯01222,
σ104(z)=1P100ρ10(λ)w04(λ)1+[z-z0(λ)]2zR2(λ)2dλ.
limz K(z)=K101P100ρ10(λ)λ2zR-2(λ)dλ1P100ρ10(λ)λzR-1(λ)dλ2,
limz K(z)=K10+K10(λ-λ¯10)2¯λ¯102.
SMP2=4π2θ102λ¯21P100ρ10(λ)w02(λ)dλ+4π2θ102λ¯2r¯102+r¯0122.

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