Abstract

The Jones matrix of a quarter-wave plate is studied theoretically and experimentally, taking into account internal reflections, the ellipsoid of the indices, geometric defects, the tilt angle, and the characteristics of the incident Gaussian beam. The influence of the different parameters is isolated, and large discrepancies are observed with respect to results obtained from the Jones matrix that are usually given in textbooks. It is shown that the effective Jones matrix of the plate does not depend on the longitudinal position of the plate on the Gaussian beam but only on the beam-waist size. This leads to a method of characterization of the defects of a quarter-wave plate that is more precise than the usual methods. Different procedures to optimize the efficiency of a given plate are discussed, taking the plate defects into account. In all cases, a good agreement between experiments and theory is obtained.

© 1995 Optical Society of America

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References

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  1. V. Evtuhov, A. E. Siegman, “A ‘twisted-mode’ technique for obtaining axially uniform energy density in a laser cavity,” Appl. Opt. 4, 142–143 (1965); A. Kastler, “Champ lumineux stationnaire à structure hélicoïdale dans une cavité laser: possibilité d’imprimer cette structure hélicoïdale à un milieu matériel transparent isotrope,” C. R. Acad. Sci. B 271, 999–1001 (1970).
    [CrossRef]
  2. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  3. E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
    [CrossRef] [PubMed]
  4. M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
    [CrossRef] [PubMed]
  5. M. Bass, ed., Handbook of Optics, 2nd ed., McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.
  6. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).
  7. F. Heismann, “Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems,” J. Lightwave Technol. 12, 690–699 (1994).
    [CrossRef]
  8. M. Suzuki, K. Midorikawa, H. Tashino, “Direct generation of circularly polarized pulse from a transversally excited atmospheric CO2 laser by injection locking,” Appl. Opt. 31, 1210–1212 (1992); J.-C. Cotteverte, G. Ropars, M. Brunel, F. Bretenaker, A. Le Floch, “Angular momentum transfer between quantum oscillators,” Phys. Rev. Lett. 74, 1966–1969 (1995).
    [CrossRef] [PubMed]
  9. M.-A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
    [CrossRef]
  10. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  11. W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964), p. 89; E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), p. 325.
  12. K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik 70, 6–13 (1985).
  13. X. Zhu, “Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate surface,” Appl. Opt. 33, 3502–3506 (1994).
    [CrossRef] [PubMed]
  14. I. T. Bodnar, A. K. Soika, “Polarization effects in quartz from laser-beam transmission,” Opt. Spectrosc. 69, 120–121 (1990).
  15. D. Clarke, “Nomenclature of polarized light: linear polarization,” Appl. Opt. 13, 3–5 (1974); “Nomenclature of polarized light: elliptical polarization,” Appl. Opt. 13, 222–224 (1974).
    [CrossRef] [PubMed]
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).
  17. J.-C. Cotteverte, F. Bretenaker, A. Le Floch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30, 305–311 (1991).
    [CrossRef] [PubMed]
  18. S. Huard, Polarisation de la Lumière (Masson, Paris, 1994).
  19. Y. Le Grand, A. Le Floch, “Measurement of residual reflectivities using the two eigenstates of a passive cavity,” Appl. Opt. 27, 4925–4930 (1988).
    [CrossRef]
  20. G. Bruhat, Optique (Masson, Paris, 1954).
  21. R. B. Sosman, The Properties of Silica (Chemical catalog, New York, 1927).
  22. B. R. Grunstra, H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585–587 (1966); L. Yao, Z. Zhiyao, W. Runwen, “Optical heterodyne measurement of the phase retardation of a quarter-wave plate,” Opt. Lett. 13, 553–555 (1988); L.-H. Shyu, C.-L. Chen, D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993).
    [CrossRef] [PubMed]
  23. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

1994

F. Heismann, “Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems,” J. Lightwave Technol. 12, 690–699 (1994).
[CrossRef]

X. Zhu, “Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate surface,” Appl. Opt. 33, 3502–3506 (1994).
[CrossRef] [PubMed]

1993

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

1992

1991

1990

I. T. Bodnar, A. K. Soika, “Polarization effects in quartz from laser-beam transmission,” Opt. Spectrosc. 69, 120–121 (1990).

1988

1987

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

1985

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik 70, 6–13 (1985).

1981

M.-A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

1974

1966

1965

1941

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Ballard, S. S.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964), p. 89; E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), p. 325.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

Bodnar, I. T.

I. T. Bodnar, A. K. Soika, “Polarization effects in quartz from laser-beam transmission,” Opt. Spectrosc. 69, 120–121 (1990).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Bouchiat, M.-A.

M.-A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

Bretenaker, F.

Bruhat, G.

G. Bruhat, Optique (Masson, Paris, 1954).

Cable, A.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

Chu, S.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

Clarke, D.

Cotteverte, J.-C.

de Boer, M. P.

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

Evtuhov, V.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

Grunstra, B. R.

Heismann, F.

F. Heismann, “Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems,” J. Lightwave Technol. 12, 690–699 (1994).
[CrossRef]

Hoogenraad, J. H.

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

Huard, S.

S. Huard, Polarisation de la Lumière (Masson, Paris, 1994).

Jones, R. C.

Le Floch, A.

Le Grand, Y.

Melle, H.

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik 70, 6–13 (1985).

Midorikawa, K.

Moser, J.

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik 70, 6–13 (1985).

Muller, H. G.

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

Noordam, L. D.

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

Perkins, H. B.

Pottier, L.

M.-A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

Prentiss, M.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

Pritchard, D. E.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

Raab, E. L.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

Shurcliff, W. A.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964), p. 89; E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), p. 325.

Siegman, A. E.

Soika, A. K.

I. T. Bodnar, A. K. Soika, “Polarization effects in quartz from laser-beam transmission,” Opt. Spectrosc. 69, 120–121 (1990).

Sosman, R. B.

R. B. Sosman, The Properties of Silica (Chemical catalog, New York, 1927).

Suzuki, M.

Tashino, H.

Vrijen, R. B.

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

Zander, K.

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik 70, 6–13 (1985).

Zhu, X.

Appl. Opt.

B. R. Grunstra, H. B. Perkins, “A method for the measurement of optical retardation angles near 90 degrees,” Appl. Opt. 5, 585–587 (1966); L. Yao, Z. Zhiyao, W. Runwen, “Optical heterodyne measurement of the phase retardation of a quarter-wave plate,” Opt. Lett. 13, 553–555 (1988); L.-H. Shyu, C.-L. Chen, D.-C. Su, “Method for measuring the retardation of a wave plate,” Appl. Opt. 32, 4228–4230 (1993).
[CrossRef] [PubMed]

D. Clarke, “Nomenclature of polarized light: linear polarization,” Appl. Opt. 13, 3–5 (1974); “Nomenclature of polarized light: elliptical polarization,” Appl. Opt. 13, 222–224 (1974).
[CrossRef] [PubMed]

Y. Le Grand, A. Le Floch, “Measurement of residual reflectivities using the two eigenstates of a passive cavity,” Appl. Opt. 27, 4925–4930 (1988).
[CrossRef]

J.-C. Cotteverte, F. Bretenaker, A. Le Floch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30, 305–311 (1991).
[CrossRef] [PubMed]

M. Suzuki, K. Midorikawa, H. Tashino, “Direct generation of circularly polarized pulse from a transversally excited atmospheric CO2 laser by injection locking,” Appl. Opt. 31, 1210–1212 (1992); J.-C. Cotteverte, G. Ropars, M. Brunel, F. Bretenaker, A. Le Floch, “Angular momentum transfer between quantum oscillators,” Phys. Rev. Lett. 74, 1966–1969 (1995).
[CrossRef] [PubMed]

X. Zhu, “Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate surface,” Appl. Opt. 33, 3502–3506 (1994).
[CrossRef] [PubMed]

V. Evtuhov, A. E. Siegman, “A ‘twisted-mode’ technique for obtaining axially uniform energy density in a laser cavity,” Appl. Opt. 4, 142–143 (1965); A. Kastler, “Champ lumineux stationnaire à structure hélicoïdale dans une cavité laser: possibilité d’imprimer cette structure hélicoïdale à un milieu matériel transparent isotrope,” C. R. Acad. Sci. B 271, 999–1001 (1970).
[CrossRef]

J. Lightwave Technol.

F. Heismann, “Analysis of a reset-free polarization controller for fast automatic polarization stabilization in fiber-optic transmission systems,” J. Lightwave Technol. 12, 690–699 (1994).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

M.-A. Bouchiat, L. Pottier, “A high-purity circular polarization modulator: application to birefringence and circular dichroism measurements on multidielectric mirrors,” Opt. Commun. 37, 229–233 (1981).
[CrossRef]

Opt. Spectrosc.

I. T. Bodnar, A. K. Soika, “Polarization effects in quartz from laser-beam transmission,” Opt. Spectrosc. 69, 120–121 (1990).

Optik

K. Zander, J. Moser, H. Melle, “Change of polarization of linearly polarized, coherent light transmitted through plane-parallel anisotropic plates,” Optik 70, 6–13 (1985).

Phys. Rev. Lett.

E. L. Raab, M. Prentiss, A. Cable, S. Chu, D. E. Pritchard, “Trapping of neutral sodium atoms with radiation pressure,” Phys. Rev. Lett. 59, 2631–2634 (1987); D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, G. Grynberg, “Raman spectroscopy of caesium atoms in a laser trap,” Europhys. Lett. 15, 149–154 (1991).
[CrossRef] [PubMed]

M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, H. G. Muller, “Indications of high-intensity adiabatic stabilization in neon,” Phys. Rev. Lett. 71, 3263–3266 (1993).
[CrossRef] [PubMed]

Other

M. Bass, ed., Handbook of Optics, 2nd ed., McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987).

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

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964), p. 89; E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), p. 325.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965).

S. Huard, Polarisation de la Lumière (Masson, Paris, 1994).

G. Bruhat, Optique (Masson, Paris, 1954).

R. B. Sosman, The Properties of Silica (Chemical catalog, New York, 1927).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, San Diego, Calif., 1980).

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Figures (10)

Fig. 1
Fig. 1

Double refraction inside the QWP. θ i is the incidence angle, and θ ro and θ re are the refraction angles for the ordinary and the extraordinary polarizations, respectively.

Fig. 2
Fig. 2

Successive internal reflections inside the QWP. θ i is the incidence angle, θ r is the mean refraction angle inside the plate, and Δx is the transverse walk-off between two successive beams.

Fig. 3
Fig. 3

Theoretical evolution of ∑ r versus the angle of incidence θ i for an uncoated QWP with no thickness defect (k = 10, ɛ = 0, n o = 1.542637, n e = 1.551646, Θ = π/4, λ = 6328 Å) for two different Gaussian beams: (a), (b) w 0 = 234 μm; (c), (d) w 0 = 110 μm. (a) and (c) are computed with y parallel to the extraordinary axis, and (b) and (d) are computed with y parallel to the ordinary axis. The arrows show that the oscillations are in opposition when the rotation axis is changed. Notice that even if the thickness of the plate is perfect (ɛ = 0), ∑ r is far from zero at normal incidence.

Fig. 4
Fig. 4

Theoretical evolution of ∑ r versus angle of incidence for an uncoated QWP (k = 10, n o = 1.542637, n e = 1.551646, θ = π/4) for a given Gaussian beam (w 0 = 234 μm) with two opposite values of the thickness defect: (a), (b) ɛ = −5%; (c), (d) ɛ = +5%. (a) and (c) are computed with y parallel to the extraordinary axis, and (b) and (d) are computed with y parallel to the ordinary axis. The arrows indicate the positions of the pinches.

Fig. 5
Fig. 5

Experimental arrangement: P1 and P2 are linear polarizers, L is a lens, and D is a detector.

Fig. 6
Fig. 6

Experimental [(a), (b), (e), (f)] and theoretical [(c), (d)] evolutions of ∑ r versus angle of incidence for an uncoated QWP (k = 10, n o = 1.542637, n e = 1.551646, Θ = π/4, ɛ = −7.36%) for a given Gaussian beam (w 0 = 110 μm). (a), (c), and (e) are obtained with y parallel to the extraordinary axis, and (b), (d), and (f) are obtained with y parallel to the ordinary axis. The difference between (a), (b) and (e), (f) lies in the longitudinal position z of the plate with respect to the beam waist.

Fig. 7
Fig. 7

Same as Figs. 6(a)–6(d) for another Gaussian beam (w 0 = 234 μm).

Fig. 8
Fig. 8

Experimental [(a), (b)] and theoretical [(c), (d)] evolutions of ∑ r versus angle of incidence for an uncoated QWP (k = 1, n o = 1.542637, n e = 1.551646, Θ = π/4, ɛ = −5.58%) for a given Gaussian beam (w 0 = 234 μm). (e), (f) are obtained with an AR-coated first-order plate. (a), (c), and (e) are obtained with y parallel to the extraordinary axis, and (b), (d), and (f) are obtained with y parallel to the ordinary axis.

Fig. 9
Fig. 9

Evolution of ∑ r at normal incidence (θ i = 0) versus polarization angle Θ of the incident beam. The experimental measurements (filled circles) are performed with the same plate and in the same conditions as in Fig. 7. The corresponding theoretical curve computed with the effective Jones matrix is shown by the solid curve. The dashed curve is obtained from the textbook matrix of Eq. (1), corresponding to a hypothetical perfect QWP without any internal reflections.

Fig. 10
Fig. 10

Theoretical evolution of ∑ r versus Θ for different values of the angle of incidence θ i , in order to optimize the purity of the circular polarization produced by the plate. (a) ɛ = −7.36%, (b) ɛ = −4.00%. The rotation axis is parallel to the optic axis.

Equations (43)

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

M = [ exp ( i π 4 ) 0 0 exp ( - i π 4 ) ] .
e r = 1 2 ( 1 i ) ,
e l = 1 2 ( 1 - i ) .
E i ( x , y ) = A exp ( - x 2 + y 2 w 2 ) exp [ i π λ ρ ( x 2 + y 2 ) ] ,
A = a ( cos Θ sin Θ )
n e - n o e = k λ + λ 4 ( 1 + ɛ ) .
Δ ϕ x y = π 2 ( 1 + ɛ ) .
n x = n e = [ n e 2 + ( 1 - n e 2 n o 2 ) sin 2 θ i ] 1 / 2 .
φ x = 2 π e λ n x cos θ r x ,
φ y = 2 π e λ n y cos θ r y .
ψ x = 2 φ x ,
ψ y = 2 φ y .
Δ x = 2 e tan θ r cos θ i ,
E t ( x , y ) = a p = 0 B p exp [ - ( x - p Δ x ) 2 + y 2 w 2 ] × exp { i π λ ρ [ ( x - p Δ x ) 2 + y 2 ] } ,
B p = ( t x r x p cos Θ exp [ i ( φ x + p ψ x ) ] t y r y p sin Θ exp [ i ( φ y + p ψ y ) ] )
B p = ( t x r x p cos Θ exp [ i ( 2 p + 1 ) φ x ] t y r y p sin Θ exp [ i ( 2 p + 1 ) φ y ] ) .
M e = [ [ 2 a 1 ( w 0 ) ] 1 / 2 0 0 [ 2 a 2 ( w 0 ) ] 1 / 2 exp { - i arcsin a 3 ( w 0 ) [ a 1 ( w 0 ) a 2 ( w 0 ) ] 1 / 2 } ] ,
1 2 k = 0 { t x 2 r x 2 k + t y 2 r y 2 k 2 - t x t y ( r x r y ) k × sin [ ( 2 k + 1 ) ( φ x - φ y ) ] }
E r ( x , y ) = a 2 p = 0 ( t x r x p cos Θ exp [ i ( 2 p + 1 ) φ x ] + t y r y p sin Θ exp { i [ ( 2 p + 1 ) φ y - π 2 ] } ) × exp [ - ( x - p Δ x ) 2 + y 2 w 2 ] × exp { i π λ ρ [ ( x - p Δ x ) 2 + y 2 ] } ,
E l ( x , y ) = a 2 p = 0 ( t x r x p cos Θ exp { i [ ( 2 p + 1 ) φ x ] } - t y r y p sin Θ exp { i [ ( 2 p + 1 ) φ y - π 2 ] } ) × exp [ - ( x - p Δ x ) 2 + y 2 w 2 ] × exp { i π λ ρ [ ( x - p Δ x ) 2 + y 2 ] } .
I r ( x , y ) = E r ( x , y ) E r * ( x , y ) .
α p a ( t x r x p cos Θ exp { i [ ( 2 p + 1 ) φ x ] } + t y r y p sin Θ exp { i [ ( 2 p + 1 ) φ y - π 2 ] } ) × exp [ - ( x - p Δ x ) 2 + y 2 w 2 ] × exp { i π λ ρ [ ( x - p Δ x ) 2 + y 2 ] } .
E r ( x , y ) = 1 2 p = 0 α p ,
I r ( x , y ) = 1 2 p = 0 q = 0 α p α q * .
I r ( x , y ) = 1 2 u = 0 v = 0 u α u - v α v * .
I r ( x , y ) = 1 2 k = 0 ( v = 0 2 k α 2 k - v α v * + v = 0 2 k + 1 α 2 k + 1 - v α v * ) .
I r ( x , y ) = 1 2 k = 0 ( α k α k * + v = 0 v k 2 k α 2 k - v a v * + v = 0 k α 2 k + 1 - v α v * + α v α 2 k + 1 - v * ) .
I r ( x , y ) = 1 2 k = 0 [ α k α k * + p = 1 k ( α k - p α k + p * + α k + p α k - p * ) + p = 0 k ( α k + 1 + p α k - p * + α k - p α k + 1 + p * ) ] .
α k α k * = a 2 { t x 2 r x 2 k cos 2 Θ + t y 2 r y 2 k sin 2 Θ + t x t y ( r x r y ) k sin 2 Θ cos [ ( 2 k + 1 ) ( φ x - φ y ) + π 2 ] } × exp [ - 2 ( x - p Δ x ) 2 + y 2 w 2 ] ,
α k - p α k + p * + α k + p α k - p * = a 2 exp { - [ x - ( k - p ) Δ x ] 2 + [ x - ( k + p ) Δ x ] 2 + 2 y 2 w 2 } × [ 2 t x 2 r x 2 k cos 2 Θ × cos ( 4 p φ x + π λ ρ { [ x - ( k - p ) Δ x ] 2 - [ x - ( k + p ) Δ x ] 2 } ) + 2 t y 2 r y 2 k sin 2 Θ × cos ( 4 p φ y + π λ ρ { [ x - ( k - p ) Δ x ] 2 - [ x - ( k + p ) Δ x ] 2 } ) + t x t y r x k - p r y k + p sin 2 Θ × sin ( ( 2 k + 2 p + 1 ) φ y - ( 2 k - 2 p + 1 ) φ x + π λ ρ { [ x - ( k + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) - t x t y r x k + p r y k - p sin 2 Θ sin ( ( 2 k + 2 p + 1 ) φ x - ( 2 k - 2 p + 1 ) φ y + π λ ρ { [ x - ( k + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) ] ,
α k + 1 + p α k - p * + α k - p α k + 1 + p * = a 2 exp { - [ x - ( k + 1 + p ) Δ x ] 2 + [ x - ( k - p ) Δ x ] 2 + 2 y 2 w 2 } × [ 2 t x 2 r x 2 k + 1 cos 2 Θ × cos ( 2 ( 2 p + 1 ) φ x + π λ ρ { [ x - ( k + 1 + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) + 2 t y 2 r y 2 k + 1 sin 2 Θ cos ( 2 ( 2 p + 1 ) φ y + π λ ρ { [ x - ( k + 1 + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) + t x t y r x k - p r y k + 1 + p sin 2 Θ sin ( ( 2 k + 2 p + 3 ) φ y - ( 2 k - 2 p + 1 ) φ x + π λ ρ { [ x - ( k + 1 + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) - t x t y r x k + 1 + p r y k - p sin 2 Θ × sin ( ( 2 k + 2 p + 3 ) φ x - ( 2 k - 2 p + 1 ) φ y + π λ ρ { [ x - ( k + 1 + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) ] .
I r = - + - + I r ( x , y ) d x d y .
I r = a 2 2 k = 0 ( I 1 { t x 2 r x 2 k cos 2 Θ + t y 2 r y 2 k sin 2 Θ - t x t y ( r x r y ) k × sin 2 Θ sin [ ( 2 k + 1 ) ( φ x - φ y ) ] } + p = 1 k { 2 t x 2 r x 2 k cos 2 Θ I 2 ( 4 p φ x ) + 2 t y 2 r y 2 k × sin 2 Θ I 2 ( 4 p φ y ) + t x t y r x k - p r y k + p × sin 2 Θ I 2 [ ( 2 k + 2 p + 1 ) φ y - ( 2 k - 2 p + 1 ) φ x ] - t x t y r x k + p r y k - p sin 2 Θ I 2 [ ( 2 k + 2 p + 1 ) φ x - ( 2 k - 2 p + 1 ) φ y ] } + p = 0 k { 2 t x 2 r x 2 k + 1 cos 2 Θ I 3 [ 2 ( 2 p + 1 ) φ x ] + 2 t y 2 r y 2 k + 1 sin 2 Θ I 3 [ 2 ( 2 p + 1 ) φ y ] + t x t y r x k - p r y k + 1 + p sin 2 Θ I 3 [ ( 2 k + 2 p + 3 ) φ y - ( 2 k - 2 p + 1 ) φ x ] - t x t y r x k + 1 + p r y k - p × sin 2 Θ I 3 [ ( 2 k + 2 p + 3 ) φ x - ( 2 k - 2 p + 1 ) φ y ] } )
I 1 = - + - + exp [ - 2 ( x - k Δ x ) 2 + y 2 w 2 ] d x d y = w 2 π 2 ,
I 2 ( ψ ) = - + - + exp { - [ x - ( k - p ) Δ x ] 2 + [ x - ( k + p ) Δ x ] 2 + 2 y 2 w 2 } × sin ( ψ + π λ ρ { [ x - ( k + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) d x d y = w 2 π 2 sin ψ exp [ - ( 2 p ) 2 Δ x 2 2 w 0 2 ] ,
I 3 ( ψ ) = - + - + exp { - [ x - ( k + 1 + p ) Δ x ] 2 + [ x - ( k - p ) Δ x ] 2 + 2 y 2 w 2 } × sin ( ψ + π λ ρ { [ x - ( k + 1 + p ) Δ x ] 2 - [ x - ( k - p ) Δ x ] 2 } ) d x d y = w 2 π 2 sin ψ exp [ - ( 2 p + 1 ) 2 Δ x 2 2 w 0 2 ] .
I i = w 2 π 2 a 2 .
Σ r ( Θ ) = a 1 ( w 0 ) cos 2 Θ + a 2 ( w 0 ) sin 2 Θ + a 3 ( w 0 ) sin 2 Θ ,
a 1 ( w 0 ) = t x 2 2 k = 0 r x 2 k { 1 + 2 p = 1 k cos ( 4 p φ x ) × exp [ - ( 2 p ) 2 Δ x 2 2 w 0 2 ] + 2 r x p = 0 k cos [ 2 ( 2 p + 1 ) φ x ] × exp [ - ( 2 p + 1 ) 2 Δ x 2 2 w 0 2 ] } ,
a 2 ( w 0 ) = t y 2 2 k = 0 r y 2 k { 1 + 2 p = 1 k cos ( 4 p φ y ) × exp [ - ( 2 p ) 2 Δ x 2 2 w 0 2 ] + 2 r y p = 0 k cos [ 2 ( 2 p + 1 ) φ y ] × exp [ - ( 2 p + 1 ) 2 Δ x 2 2 w 0 2 ] } ,
a 3 ( w 0 ) = t x t y 2 k = 0 ( r x r y ) k [ - sin [ ( 2 k + 1 ) ( φ x - φ y ) ] + p = 1 k ( exp [ - ( 2 p ) 2 Δ x 2 2 w 0 2 ] × { r x - p r y p sin [ ( 2 k + 2 p + 1 ) φ y - ( 2 k - 2 p + 1 ) φ x ] - r x p r y - p sin [ ( 2 k + 2 p + 1 ) φ x - ( 2 k - 2 p + 1 ) φ y ] } ) + p = 0 k ( exp [ - ( 2 p + 1 ) 2 Δ x 2 2 w 0 2 ] × { r x - p r y 1 + p sin [ ( 2 k + 2 p + 3 ) φ y - ( 2 k - 2 p + 1 ) φ x ] - r x 1 + p r y - p sin [ ( 2 k + 2 p + 3 ) φ x - ( 2 k - 2 p + 1 ) φ y ] } ) ] .
Σ l ( Θ ) = a 1 ( w 0 ) cos 2 Θ + a 2 ( w 0 ) sin 2 Θ - a 3 ( w 0 ) sin 2 Θ .
M e = [ [ 2 a 1 ( w 0 ) ] 1 / 2 0 0 [ 2 a 2 ( w 0 ) ] 1 / 2 exp { - i arcsin a 3 ( w 0 ) [ a 1 ( w 0 ) a 2 ( w 0 ) ] } ] .

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