Abstract

The influence of the aberrations on the characteristic parameters of a laser beam is analyzed. An analytical treatment is proposed that uses the decomposition of the phase of the beam in terms of the Zernike polynomials. It is shown that the width remains unaffected by the aberrations. Changes in the divergence, the radius of curvature, and the quality factor are analyzed, and some numerical examples are shown.

© 1997 Optical Society of America

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References

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  1. H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [Crossref]
  2. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 15, pp. 581–625.
  3. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [Crossref] [PubMed]
  4. A. E. Siegman, “Defining the effective radius of curvature for a non-ideal opticalbeam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [Crossref]
  5. M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussianand nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
    [Crossref] [PubMed]
  6. M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
    [Crossref]
  7. C. B. Hogge, R. R. Butts, M. Burlakoff, “Characteristics of phase-aberrated non-diffraction-limited laser beams,” Appl. Opt. 13, 1065–1070 (1974).
    [Crossref] [PubMed]
  8. J. T. Hunt, P. A. Renard, R. G. Nelson, “Focusing properties of an aberrated laser beam,” Appl. Opt. 15, 1458–1464 (1976).
    [Crossref] [PubMed]
  9. V. N. Mahajan, “Uniform versus Gaussian beam: a comparison of the effects of diffracted,obscuration, and aberrations,” J. Opt. Soc. Am. A 3, 470–485 (1986).
    [Crossref]
  10. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 9, p. 204.
  11. J. Arnaud, H. Kogelnik, “Gaussian light beam with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [Crossref] [PubMed]
  12. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phaseaberrations,” Appl. Opt. 32, 5893–5901 (1993).
    [Crossref] [PubMed]
  13. A. E. Siegman, J. A. Ruff, “Effects of spherical aberration on laser beam quality, in Laser Energy Distribution Profiles: Measurement and Applications, J.M. Darchuk, ed., Proc. SPIE1834, 130–139 (1993).
    [Crossref]
  14. J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberrationin a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
    [Crossref]
  15. G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherical aberratedlenses,” Opt. Commun. 107, 179–183 (1994).
    [Crossref]
  16. R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
    [Crossref]
  17. R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “Beam quality changes produced by quartic phase transmittances,” Opt. Quantum Electron. 27, 173–183 (1995).
    [Crossref]
  18. M. Born, E. Wolf, Principles of Optics, 6th. ed. (Pergamon, Oxford, 1987), Chap. 9, p. 464.
  19. D. Malacara, Optical Shop Testing, 2nd. ed. (Wiley/Interscience, New York, 1992), Chap. 13, p. 461.
  20. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [Crossref] [PubMed]
  21. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  22. D. E. Novoseller, “Zernike-ordered adaptive correction of thermal blooming,” J. Opt. Soc. Am. A 5, 1937–1942 (1988).
    [Crossref]
  23. J. Alda, G. D. Boreman, “Zernike-based matrix model of deformable mirror: optimization of aperturesize,” Appl. Opt. 32, 2431–2438 (1993).
    [Crossref] [PubMed]
  24. M. A. Porras, J. Alda, E. Bernabeu, “Nonlinear propagation and transformation of arbitrary laser beams bymeans of the generalized ABCD law,” Appl. Opt. 32, 5885–5892 (1993).
    [Crossref] [PubMed]
  25. W. Swantner, W. W. Chow, “Gram–Schmidt orthonormalization of Zernike polynomials for generalaperture shapes,” Appl. Opt. 34, 1832–1837 (1994).
    [Crossref]
  26. M. A. Porras, J. Alda, E. Bernabeu, “Gaussian beams passing through apertured ABCDoptical systems,” Optik 94, 23–32 (1993).
  27. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 7, pp. 212–217.
  28. M. A. Porras, “Leyes de propagación y transformación de haces láser por sistemas ópticos ABCD,” Ph.D. dissertation (Universidad Complutense de Madrid, Madrid, 1992).
  29. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagatingthrough ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [Crossref]
  30. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first orderoptical systems,” Optik 88, 163–168 (1991).

1996 (1)

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[Crossref]

1995 (2)

R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[Crossref]

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “Beam quality changes produced by quartic phase transmittances,” Opt. Quantum Electron. 27, 173–183 (1995).
[Crossref]

1994 (3)

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberrationin a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[Crossref]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherical aberratedlenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

W. Swantner, W. W. Chow, “Gram–Schmidt orthonormalization of Zernike polynomials for generalaperture shapes,” Appl. Opt. 34, 1832–1837 (1994).
[Crossref]

1993 (4)

1992 (1)

1991 (4)

P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[Crossref] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature for a non-ideal opticalbeam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagatingthrough ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first orderoptical systems,” Optik 88, 163–168 (1991).

1988 (1)

1986 (1)

1980 (1)

1976 (2)

1974 (1)

1969 (1)

1966 (1)

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Alda, J.

Arnaud, J.

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first orderoptical systems,” Optik 88, 163–168 (1991).

Bélanger, P. A.

Bernabeu, E.

Boreman, G. D.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 9, p. 204.

M. Born, E. Wolf, Principles of Optics, 6th. ed. (Pergamon, Oxford, 1987), Chap. 9, p. 464.

Burlakoff, M.

Butts, R. R.

Chow, W. W.

W. Swantner, W. W. Chow, “Gram–Schmidt orthonormalization of Zernike polynomials for generalaperture shapes,” Appl. Opt. 34, 1832–1837 (1994).
[Crossref]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 7, pp. 212–217.

Hodgson, N.

R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[Crossref]

Hogge, C. B.

Hunt, J. T.

Kogelnik, H.

Mahajan, V. N.

Malacara, D.

D. Malacara, Optical Shop Testing, 2nd. ed. (Wiley/Interscience, New York, 1992), Chap. 13, p. 461.

Marti´nez-Herrero, R.

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “Beam quality changes produced by quartic phase transmittances,” Opt. Quantum Electron. 27, 173–183 (1995).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[Crossref]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherical aberratedlenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagatingthrough ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Meji´as, P. M.

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “Beam quality changes produced by quartic phase transmittances,” Opt. Quantum Electron. 27, 173–183 (1995).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[Crossref]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherical aberratedlenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagatingthrough ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[Crossref]

Nelson, R. G.

Noll, R. J.

Novoseller, D. E.

Piquero, G.

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “Beam quality changes produced by quartic phase transmittances,” Opt. Quantum Electron. 27, 173–183 (1995).
[Crossref]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherical aberratedlenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

Porras, M. A.

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[Crossref]

M. A. Porras, J. Alda, E. Bernabeu, “Nonlinear propagation and transformation of arbitrary laser beams bymeans of the generalized ABCD law,” Appl. Opt. 32, 5885–5892 (1993).
[Crossref] [PubMed]

M. A. Porras, J. Alda, E. Bernabeu, “Gaussian beams passing through apertured ABCDoptical systems,” Optik 94, 23–32 (1993).

M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussianand nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
[Crossref] [PubMed]

M. A. Porras, “Leyes de propagación y transformación de haces láser por sistemas ópticos ABCD,” Ph.D. dissertation (Universidad Complutense de Madrid, Madrid, 1992).

Renard, P. A.

Ruff, J. A.

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberrationin a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[Crossref]

A. E. Siegman, J. A. Ruff, “Effects of spherical aberration on laser beam quality, in Laser Energy Distribution Profiles: Measurement and Applications, J.M. Darchuk, ed., Proc. SPIE1834, 130–139 (1993).
[Crossref]

Serna, J.

Siegman, A. E.

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberrationin a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[Crossref]

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phaseaberrations,” Appl. Opt. 32, 5893–5901 (1993).
[Crossref] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature for a non-ideal opticalbeam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 15, pp. 581–625.

A. E. Siegman, J. A. Ruff, “Effects of spherical aberration on laser beam quality, in Laser Energy Distribution Profiles: Measurement and Applications, J.M. Darchuk, ed., Proc. SPIE1834, 130–139 (1993).
[Crossref]

Silva, D. E.

Swantner, W.

W. Swantner, W. W. Chow, “Gram–Schmidt orthonormalization of Zernike polynomials for generalaperture shapes,” Appl. Opt. 34, 1832–1837 (1994).
[Crossref]

Wang, J. Y.

Weber, H.

R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th. ed. (Pergamon, Oxford, 1987), Chap. 9, p. 464.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 9, p. 204.

Appl. Opt. (9)

IEEE J. Quantum Electron. (2)

R. Martı́nez-Herrero, P. M. Mejı́as, N. Hodgson, H. Weber, “Beam-quality changes generated by thermally-induced spherical aberrationin laser cavities,” IEEE J. Quantum Electron. 31, 2173–2176 (1995).
[Crossref]

A. E. Siegman, “Defining the effective radius of curvature for a non-ideal opticalbeam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

M. A. Porras, “Non-paraxial vectorial moment theory of light beam propagation,” Opt. Commun. 127, 79–95 (1996).
[Crossref]

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherical aberratedlenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

R. Martı́nez-Herrero, G. Piquero, P. M. Mejı́as, “Beam quality changes produced by quartic phase transmittances,” Opt. Quantum Electron. 27, 173–183 (1995).
[Crossref]

J. A. Ruff, A. E. Siegman, “Measurement of beam quality degradation due to spherical aberrationin a simple lens,” Opt. Quantum Electron. 26, 629–632 (1994).
[Crossref]

Optik (2)

M. A. Porras, J. Alda, E. Bernabeu, “Gaussian beams passing through apertured ABCDoptical systems,” Optik 94, 23–32 (1993).

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first orderoptical systems,” Optik 88, 163–168 (1991).

Proc. IEEE (1)

H. Kogelnik, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[Crossref]

Other (7)

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 15, pp. 581–625.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1987), Chap. 9, p. 204.

M. Born, E. Wolf, Principles of Optics, 6th. ed. (Pergamon, Oxford, 1987), Chap. 9, p. 464.

D. Malacara, Optical Shop Testing, 2nd. ed. (Wiley/Interscience, New York, 1992), Chap. 13, p. 461.

A. E. Siegman, J. A. Ruff, “Effects of spherical aberration on laser beam quality, in Laser Energy Distribution Profiles: Measurement and Applications, J.M. Darchuk, ed., Proc. SPIE1834, 130–139 (1993).
[Crossref]

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 7, pp. 212–217.

M. A. Porras, “Leyes de propagación y transformación de haces láser por sistemas ópticos ABCD,” Ph.D. dissertation (Universidad Complutense de Madrid, Madrid, 1992).

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Figures (5)

Fig. 1
Fig. 1

Variation of the trace of the divergence tensor as a function of the coefficients of defocus c4 and spherical aberration c11 for an elliptical Gaussian beam with wx=0.2, wy=0.3, x0 =0.3, y0=0.2, and φ=0. The axes representing the a2 and a4 coefficients of pure quadratic and quartic aberrations are plotted by the dashed lines. The dotted line represents the points where the change in the divergence is minimum. The axis ranges are typical in a nonoptimized biconvex lens. In all the examples, the selected wavelength relative to the aperture is λ/D=6×10-5.

Fig. 2
Fig. 2

Dependence of the element (R-1)xx on spherical and defocus aberrations for a centered elliptical Gaussian beam. The dependence is linear with respect to these aberrations, and therefore the curvature values form a plane. The intersection of this plane with the plane (R-1)xx=0 gives the aberration combination that produces an average plane wave in the xx direction represented by the thick straight line.

Fig. 3
Fig. 3

Value of the invariant quality parameter J as a function of the individual aberrations considered one at a time. The parameters of the laser beam are wx=0.2, wy=0.4, x0=0.2, y0 =0.2, and φ=π/6. (a) Spherical aberration; (b) coma 1; (c) coma 2; (d) t. astigmatism 1; (e) t. astigmatism 2.

Fig. 4
Fig. 4

Elliptical Gaussian beam located on an aperture composed of coma c7 and a constant amount of spherical aberration c11=1. The gray levels represent the phase function that is due only to coma, and the elliptical section is plotted with its center at the location x0=y0=0.3 and rotated by an angle φ.

Fig. 5
Fig. 5

Changes in J parameters when the beam rotates, showing different orientations φ of the beam's principal widths with respect to the coordinate system. The beam here has a fixed amount of spherical aberration c11=λ plus a variable contribution of coma.

Tables (3)

Tables Icon

Table 1 Zernike Polynomials and Third-Order Aberrations

Tables Icon

Table 2 Dependence of Variation of the Trace of the Divergence Tensor (Θ2)xx+(Θ2)yy=(Θ2)r for One Individual Aberration n on Decentering (x0, y0), Size of the Elliptical Gaussian Beam (wx, wy), and Orientation φ

Tables Icon

Table 3 Dependence of Quality Parameter J for One Individual Aberration n on Decentering (x0, y0), Size of the Elliptical Gaussian Beam (wx, wy), and Orientation ϕ

Equations (57)

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

Ψ=|Ψ|exp(iΦ).
Φ(x, y)=2πλ W(x, y).
W(x, y)=n=0N cnZn(x, y),
W2=4D2x2xyxyy2-xy[xy].
xmyn=μmnμ00=xmyn|Ψ(x, y)|2dxdy|Ψ(x, y)|2dxdy,
Θ2=4λ2D2 ξ2ξηξηη2-ξη[ξη],
ξmηn= ξmηn|FT[Ψ]|2dξdη|FT[Ψ]|2dξdη.
Θ2=4k2D2I(Ψ) dxdy|Ψ|x2|Ψ|x |Ψ|y|Ψ|x |Ψ|y|Ψ|y2+|Ψ|2Φx2Φx ΦyΦx ΦyΦy2-4k2D2I2(Ψ) |Ψ|2 Φx dxdy|Ψ|2 Φy dxdy×|Ψ|2 Φx dxdy|Ψ|2 Φy dxdy,
R-1=(W2)-1S+1Tr(W2) [ST-(W2)-1SW2],
S=-2λπI(Ψ) ×dxdy|Ψ|2×(x-x) Φx(x-x) Φy(y-y) Φx(y-y) Φy.
Znx=jγnjxZj,Zny=jγnjyZj,
Φx=2πC ΓxZ,Φy=2πC ΓyZ.
ZjZk= ZjZk|Ψ|2dxdy|Ψ|2dxdy.
ZjZk=m,nMmnjkxmyn,
|Ψ|2 Φx Φx dxdy
=4π2C Γx|Ψ|2ZZTdxdyΓxTC T,
|Ψ|2 Φx Φy dxdy
=4π2C Γx|Ψ|2ZZTdxdyΓyTC T,
|Ψ|2 Φy Φy dxdy
=4π2C Γy|Ψ|2ZZTdxdyΓyTC T.
|Ψ|2 Φx dxdy
=2πC1|Ψ|2ZZTdxdyΓxTC T,
|Ψ|2 Φy dxdy
=2πC1|Ψ|2ZZTdxdyΓyTC T,
Θ2=Θ02+4λ2π2D2 C ΓxZZTΓxTC TC ΓxZZTΓyTC TC ΓxZZTΓyTC TC ΓyZZTΓyTC T-4λ2π2D2 C1ZZTΓxTC TC1ZZTΓyTC T×[C1ZZTΓxTC TC1ZZTΓyTC T].
W(r)=a2r2+a4r4,
Csph=a22+a43, 0, 0, a2+a423, 0, 0, 0, 0, 0, 0, a465.
a2=-2a4 r4/r2,
r2n=k=0nnkx2ky2(n-k).
c4=151-2r4r2c11.
Δ(Θ2)r=16a42λ2π2D2 r2r6-(r4)2(r2)4.
|Ψ|2x Φx dxdy
=2πCx|Ψ|2ZZTdxdyΓxTC T,
|Ψ|2x Φy dxdy
=2πCx|Ψ|2ZZTdxdyΓyTC T,
|Ψ|2y Φx dxdy
=2πCy|Ψ|2ZZTdxdyΓxTC T,
|Ψ|2y Φy dxdy
=2πCy|Ψ|2ZZTdxdyΓyTC T,
|Ψ|2xdxdy=C1|Ψ|2ZZTdxdyCxT,
|Ψ|2ydxdy=C1|Ψ|2ZZTdxdyCyT.
S=-4λCxZZTΓxTC TCxZZTΓyTC TCyZZTΓxTC TCyZZTΓyTC T-C1ZZTCxTC1ZZTCyT×[C1ZZTΓxTC TC1ZZTΓyTC T].
Ψ(x, y)=exp-(x-x0)2wxx2-(y-y0)2wyy2-2(x-x0)(y-y0)wxy2×expi2πn=111cnZn(x, y),
[Δ(Θ2)r]n=θn(wx, wy, x0, y0, ϕ)cn2,
c4=523 (6-9wx2-3wy2)c11.
M4=π2λ2 (W2Θ2-S2).
J=1+54(c72+c82)w6+18(c92+c102)w6+360c112w8.
Jn=1+αn(wx, wy, x0, y0, φ)cn2,n7,
μmn=j=0mk=0nmjnkx0m-jy0n-kμjkc,
ΔJ=16a42[r2r6-(r4)2].
Φπλ [x2(R-1)xx+xy(R-1)xy+yx(R-1)yx+yy(R-1)yy]+σxx+σyy+t,
Φxk[x(R-1)xx+y(R-1)xy]+σx,
Φyk[x(R-1)yx+y(R-1)yy]+σy.
|Ψ(x, y)|2
×Φx-k[x(R-1)xx+y(R-1)xy]+σx2+Φy-k[x(R-1)yx+y(R-1)yy]+σy2dxdy-mk[(R-1)xy-(R-1)yx],
R-1=(W2)-1S+1Tr(W2) [ST-(W2)-1SW2],
S=iλπI(Ψ)  dxdy(x-x)Ψx Ψ*-Ψ Ψ*x(x-x)ψy Ψ*-Ψ Ψ*y(y-y)Ψx Ψ*-Ψ Ψ*x(y-y)Ψy Ψ*-Ψ Ψ*y,

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