Abstract

We calculate the modal power distribution of a randomly and linearly polarized (LP) multimode beam inside a cylindrical fiber core from knowledge of spatial-intensity profiles of a beam emitted from the fiber. We provide an exact analysis with rigorous proofs that forms the basis for our calculations. The beam from the fiber end is collimated by a spherical lens with a specific focal length. The original LP-mode basis is transformed by the spherical lens and forms another orthogonal basis that describes the free-space beam. By using this basis, we calculate the modal power distribution from the mutual-intensity profile. This is acquired by adopting a well-known mutual-intensity-profile-retrieving technique based on measurements of the intensity patterns several times after two orthogonal cylindrical lenses with varying separation. The feasibility of our decomposition algorithm is demonstrated with simulations.

© 2004 Optical Society of America

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References

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  1. E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.
  2. L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11, 1435–1446 (1993).
    [CrossRef]
  3. T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
    [CrossRef]
  4. O. G. Leminger, G. K. Grau, “Nearfield-intensity and modal power distribution in multimode graded-index fibres,” Electron. Lett. 16, 678–679 (1980).
    [CrossRef]
  5. G. K. Grau, O. G. Leminger, “Relations between near-field and far-field intensities, radiance and modal power distribution of multimode graded-index fibres,” Appl. Opt. 20, 457–459 (1981).
    [CrossRef] [PubMed]
  6. A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
    [CrossRef]
  7. A. R. Mickelson, M. Eriksrud, “Mode-continuum approximation in optical fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).
  8. D. Rittich, “Practicability of determining the modal power distribution by measured near and far fields,” J. Lightwave Technol. LT-3, 652–661 (1985).
    [CrossRef]
  9. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis of partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
    [CrossRef]
  10. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  11. C. Elster, I. Weingartner, “Solution to the shearing problem,” Appl. Opt. 38, 5024–5031 (1999).
    [CrossRef]
  12. E. L. Lago, R. de la Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A Pure Appl. Opt. 4, 299–302 (2002).
    [CrossRef]
  13. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersory, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system—a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  14. W. Snyder, J. D. Love, Optical Waveguide Theory (Kluwer Academic, Boston, Mass., 1983).
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  16. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
    [CrossRef] [PubMed]
  17. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
    [CrossRef] [PubMed]
  18. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
    [CrossRef] [PubMed]
  19. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
    [CrossRef]
  20. G. E. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
    [CrossRef]
  21. J. Bertrand, P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
    [CrossRef]

2003

2002

E. L. Lago, R. de la Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

1999

1998

1997

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

1995

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

1994

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

G. E. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersory, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system—a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

1993

L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11, 1435–1446 (1993).
[CrossRef]

1992

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

1987

J. Bertrand, P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

1985

D. Rittich, “Practicability of determining the modal power distribution by measured near and far fields,” J. Lightwave Technol. LT-3, 652–661 (1985).
[CrossRef]

1984

A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
[CrossRef]

1981

1980

O. G. Leminger, G. K. Grau, “Nearfield-intensity and modal power distribution in multimode graded-index fibres,” Electron. Lett. 16, 678–679 (1980).
[CrossRef]

1973

A. R. Mickelson, M. Eriksrud, “Mode-continuum approximation in optical fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Aamlid, S.

A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
[CrossRef]

Beck, M.

Berthou, H.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Bertrand, J.

J. Bertrand, P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

Bertrand, P.

J. Bertrand, P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

Borghi, R.

Clarke, L.

Dan, J. P.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

de la Fuente, R.

E. L. Lago, R. de la Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

Dong, B. Z.

Elster, C.

Eriksrud, M.

A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
[CrossRef]

A. R. Mickelson, M. Eriksrud, “Mode-continuum approximation in optical fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Ersory, O. K.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gori, F.

Grau, G. K.

G. K. Grau, O. G. Leminger, “Relations between near-field and far-field intensities, radiance and modal power distribution of multimode graded-index fibres,” Appl. Opt. 20, 457–459 (1981).
[CrossRef] [PubMed]

O. G. Leminger, G. K. Grau, “Nearfield-intensity and modal power distribution in multimode graded-index fibres,” Electron. Lett. 16, 678–679 (1980).
[CrossRef]

Gu, B. Y.

Guattari, G.

Hakimi, F.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.

Healey, G. E.

G. E. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

Hintermann, H. E.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Kondepudy, R.

G. E. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

Kotrotsios, G.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Lago, E. L.

E. L. Lago, R. de la Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

Leminger, O. G.

G. K. Grau, O. G. Leminger, “Relations between near-field and far-field intensities, radiance and modal power distribution of multimode graded-index fibres,” Appl. Opt. 20, 457–459 (1981).
[CrossRef] [PubMed]

O. G. Leminger, G. K. Grau, “Nearfield-intensity and modal power distribution in multimode graded-index fibres,” Electron. Lett. 16, 678–679 (1980).
[CrossRef]

Love, J. D.

W. Snyder, J. D. Love, Optical Waveguide Theory (Kluwer Academic, Boston, Mass., 1983).

Luthy, W.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Mayer, A.

McAlister, D. F.

McCollum, B. C.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.

Mickelson, A. R.

A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
[CrossRef]

A. R. Mickelson, M. Eriksrud, “Mode-continuum approximation in optical fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Neuman, V.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Nugent, K. A.

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

Piquero, G.

Po, H.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.

Raymer, M. G.

Rittich, D.

D. Rittich, “Practicability of determining the modal power distribution by measured near and far fields,” J. Lightwave Technol. LT-3, 652–661 (1985).
[CrossRef]

Ryen, N.

A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
[CrossRef]

Santarsiero, M.

Simon, R.

Snitzer, E.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.

Snyder, W.

W. Snyder, J. D. Love, Optical Waveguide Theory (Kluwer Academic, Boston, Mass., 1983).

Tamura, S.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Tu, J.

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Tumminelli, R.

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.

Weber, H. P.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Weber, T.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

Weingartner, I.

Yang, G. Z.

Zenteno, L.

L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11, 1435–1446 (1993).
[CrossRef]

Zhuang, J. Y.

Appl. Opt.

Bell Syst. Tech. J.

A. R. Mickelson, M. Eriksrud, “Mode-continuum approximation in optical fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Electron. Lett.

O. G. Leminger, G. K. Grau, “Nearfield-intensity and modal power distribution in multimode graded-index fibres,” Electron. Lett. 16, 678–679 (1980).
[CrossRef]

Found. Phys.

J. Bertrand, P. Bertrand, “A tomographic approach to Wigner function,” Found. Phys. 17, 397–405 (1987).
[CrossRef]

IEEE J. Quantum Electron.

T. Weber, W. Luthy, H. P. Weber, V. Neuman, H. Berthou, G. Kotrotsios, J. P. Dan, H. E. Hintermann, “Cladding-pumped fiber laser,” IEEE J. Quantum Electron. 31, 326–329 (1995).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

G. E. Healey, R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Mach. Intell. 16, 267–276 (1994).
[CrossRef]

J. Lightwave Technol.

L. Zenteno, “High-power double-clad fiber lasers,” J. Lightwave Technol. 11, 1435–1446 (1993).
[CrossRef]

D. Rittich, “Practicability of determining the modal power distribution by measured near and far fields,” J. Lightwave Technol. LT-3, 652–661 (1985).
[CrossRef]

A. R. Mickelson, M. Eriksrud, S. Aamlid, N. Ryen, “Role of the fusion splice in the concatenation problem,” J. Lightwave Technol. LT-2, 126–138 (1984).
[CrossRef]

J. Opt. A Pure Appl. Opt.

E. L. Lago, R. de la Fuente, “Wavefront sensing by diffracted beam interferometry,” J. Opt. A Pure Appl. Opt. 4, 299–302 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. E

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Phys. Rev. Lett.

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1992).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Other

W. Snyder, J. D. Love, Optical Waveguide Theory (Kluwer Academic, Boston, Mass., 1983).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, B. C. McCollum, “Double-clad, offset core Nd fiber laser,” in Conference on Optical Fiber Sensors, New Orleans, January 25–28, 1988, Postdeadline paper PD5.

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Figures (6)

Fig. 1
Fig. 1

Measurement setup for determination of the mutual intensity profile.

Fig. 2
Fig. 2

Calculated intensity distribution of each mode: (a) modal solutions inside the fiber, (b) modal solutions after the spherical lens.

Fig. 3
Fig. 3

Test multimode beam with arbitrarily assigned initial modal power: (a) beam intensity inside fiber, (b) the free-space beam intensity at z=f.

Fig. 4
Fig. 4

Selected CCD images for different θ, ϕ, i.e., with different positions of the lenses and the CCD array.

Fig. 5
Fig. 5

Modal power distribution among modes. The test modal power distribution and the results with simulations A, B, and C are shown.

Fig. 6
Fig. 6

Comparison between the test multimode beam and reconstructed beam from calculated results. The figure shows the intensity pattern of the test multimode beam inside the fiber (solid curves), the reconstructed image from simulation A (dashed curves), the reconstructed image from simulation B (dotted curves), and the reconstructed image from simulation C (dashed–dotted curves). (b) is a magnified portion of (a) to clarify the small differences between results.

Tables (1)

Tables Icon

Table 1 Modes of the Step-Index Fiber Considered in the Simulations and Their Effective Indices a

Equations (61)

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E(x, y, z)=e(x, y)exp(iβz),
[t2+k2n(r)2-β2]e(x, y)=0,
n(R)=ncoif0R1nclif1<R,
elpij(x, y)=Flp(R)sinlϕ+π2 ixˆ+sinlϕ+π2 jyˆ,
d2dR2+1RddR+k2n2-l2R2-β2Flp(R)=0.
Flp(R)=Jl(UlpR)/Jl(Ulp),0R<1Kl(WlpR)/Kl(Wlp),otherwise,
UlpJl+1(Ulp)Jl(Ulp)=WlpKl+1(Wlp)Kl(Wlp).
e=i=1Ncie¯i.
P=1ηi=1N|ci|2,
ci=e, e¯i=--e(x, y)e¯i*(x, y)dxdy.
Γ(x1, y1; x2, y2)e(x1, y1)e*(x2, y2).
--Γ(x1, y1; x2, y2)e¯i(x2, y2)dx2dy2=cie(x1, y1).
|ci|2--e(x, y)e*(x, y)dxdy
=----Γ(x1, y1; x2, y2)×e¯i(x2, y2)dx2dy22dx1dy1.
--|e(x, y)|2dxdy=n=1N|cn|2.
|ci|2=Λi/n=1NΛn1/2,
Λi=----Γ(x1, y1; x2, y2)×e¯i(x2, y2)dx2dy22dx1dy1.
e(x1, y1; z=-f)
=1-jλfexp-j k2f (x12+y12)1-df
×--e(x1, y1; z=d)
×exp-j kf (x1x1+y1y1)dx1dy1,
|k(x12+y12)(1-d/f)/2f|k(x12+y12)d/2f21;
ρkdMfdm,
e(x1, y1; z=-f)
1-jλf--e(x1, y1; z=d)
×exp-j kf (x1x1+y1y1)dx1dy1.
e˜(x, y)[e(x, y; z=f)]e(x, y; z=d),
d,f<d<f2/ρ2k,
e¯Fi(x, y)=1-jλf--e¯i(x, y; z=-f)×expj kf (xx+yy)dxdy.
e¯Fi, e¯Fj=--e¯Fi(x, y)e¯Fj*(x, y)dxdy=1λ2f2------e¯i×(x, y)e¯j*(x, y)×expj kf [x(x-x)+y(y-y)]dxdydxdydxdy=1λ2f2----e¯i(x, y)e¯j*(x, y)×δ(x-x)δ(y-y)λ2f2dxdydxdy=--e¯i(x, y)e¯j*(x, y)dxdy=e¯i, e¯j.
e˜(x, y)=i=1Ncie¯Fi,
ΓF(x1, y1; x2, y2)e˜(x1, y1)e˜*(x2, y2),
--ΓF(x1, y1; x2, y2)e¯Fi(x2, y2)dx2dy2
=cie˜(x1, y1),
|ci|2=ΛFi/n=1NΛFn,
ΛFi=----ΓF(x1, y1; x2, y2)×e¯Fi(x2, y2)dx2dy22dx1dy1.
etlpi, etmqj=02π0Flp(R)Fmq(R)sin(lϕ+(π/2)i)×sin(mϕ+(π/2)j)RdRdϕ,
etlpi, etlqi=02π0Flp(R)Flq(R)sin2(lϕ+(π/2)i)RdRdϕ=2π0Flp(R)Flq(R)RdRifl=0π0Flp(R)Flq(R)RdRotherwise.
d2dR2+1RddR-l2R2+Up2-V2fn(R)Flp(R)=0,
d2dR2+1RddR-l2R2+Uq2-V2fn(R)Flq(R)=0,
fn(R)=0,if0R<11,otherwise,
Flq(R)R dFlp(R)dR0-Flp(R)R dFlq(R)dR0=(Uq2-Up2)0Flp(R)Flq(R)RdR.
W(x, kx, y, ky)=1/π2--exp[-2i(kxΔx+kyΔy)]Γ˜(x, y; Δx, Δy)dΔxdΔy,
Γ˜(x, y; Δx, Δy)=Γ(x+Δx, y+Δy; x-Δx, y-Δy).
W(xθ, kθ; yϕ, kϕ)=W(x, kx; y, ky),
xθkθ=cos θsin θ-sin θcos θ xkx,
yϕkϕ=cos θsin θ-sin θcos θ yky.
e(x, y; z=D)=--C expiβ˜xxL1+yyL2
-(x)22R1-(y)22R2]}e˜(x, y)dxdy,
Ri=Ri0+di,
Li=(D-di)(1+di/Ri0),
1Ri0=1D-di-1fi,(i=1, 2).
Pθϕ(xθ, yϕ)=--W(xθ, kθ; yϕ, kϕ)dkθdkϕ.
|e(x, y; z=D)|2=Pθϕ[(β˜ sin θ/L1)x, (β˜ sin ϕ/L2)y],
θ=-tan-1(R1/β˜),ϕ=-tan-1(R2/β˜).
R[ψ](x, θ)=-ψ(x cos θ-u sin θ, x sin θ+u cos θ)du.
MF(ϖ, θ)F{R[ψ](x, θ)}x
=-R[ψ](x, θ)exp(-iϖx)dx=--ψ(x cos θ-u sin θ, x sin θ+u cos θ)exp(-iϖx)dudx.
x1x2=cos θ-sin θsin θcos θ xu,ζξ=ϖ cos θϖ sin θ
MF(ζ, ξ)=--ψ(x1, x2)exp(-iζx1-iξx2)dx1dx2=F2[ψ(x1, x2)]x1,x2.
ψ(x1, x2)=F2-1[MF(ζ, ξ)]ζ,ξ=14π2--MF(ζ, ξ)exp(iζx1+iξx2)dζdξ=14π202π0MF(ν, φ)exp[iν(x1sin φ+x2cos φ)]|ν|dνdφ,

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