Abstract

A spatial Fourier transform approach is proposed to investigate the polarization changing and the beam profile deformation of light during the acousto-optic (AO) interaction in isotropic media. Two basic types of sound waves are considered, namely, longitudinal and shear waves interacting with the light in an AO bulk cell. The perturbation of the permittivity is then caused by these kinds of acoustic waves and can be expressed in tensor forms. The evolution of two different orders of scattered light under the Bragg condition can be properly described by a couple of solutions that are derived from the wave equation by using a spatial Fourier transform approach. The solutions explicitly comprise the effects of polarization changing, beam deformation, and propagational diffraction. It is shown that the spatial beam profiles of the scattered light are distorted during the process because of the effects of the AO interaction and the propagational diffraction. For both cases of longitudinal and shear sound waves, the degree of the profile deformation can be controlled by changing the amplitude and the frequency of the sound. It is also shown that the polarization states of the scattered light are different from those of the input light on account of the AO effect. The degree of difference of the polarization states, which depend on the propagation type, frequency, and amplitude of the sound wave, can be examined through the use of two polarization parameters, the ellipticity and the orientation of the major axis of the scattered light. Detailed numerical simulations, including Fourier-transforming the incident light profile to calculate the spectra and the on-axis values of ellipticity and orientation of the scattered light in space with use of the inverse Fourier transform, are presented.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See, for instance, the special issue on acousto-optics, Proc. IEEE 69, 48–120 (1981).
  2. A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1989), Chap. 1.
  3. N. J. Berg, J. N. Lee, eds., Acousto-Optic Signal Processing (Marcel Dekker, New York, 1983), Chap. 1.
  4. C. S. Tsai, Guided-Wave Acoustooptics (Springer-Verlag, Berlin, 1990), Chap. 1.
  5. See, for instance, the special section on acousto-optics, Opt. Eng. 31, 2047–2167 (1992).
  6. E. I. Gordon, “A review of acoustooptical deflection and modulation devices,” Proc. IEEE 54, 1391–1400 (1988).
    [CrossRef]
  7. V. I. Balakshy, J. A. Hassan, “Polarization effects in acoustooptic interaction,” Opt. Eng. 32, 746–751 (1993).
    [CrossRef]
  8. A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989), Chaps. 16–19.
  9. A. Alippi, “Half-wave plate behavior of ultrasonic waves light modulators,” Opt. Commun. 8, 397–400 (1973).
    [CrossRef]
  10. C. W. Tarn, “A spatio-temporal Fourier transform approach to acoustooptic interactions,” Ph.D. dissertation (Syracuse University, 1991).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.
  12. M. R. W. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
    [CrossRef]
  13. C. W. Tarn, C. N. Chang, “The polarization changing and beam profile deformation of light during the acoustooptic interaction in isotropic media,” Opt. Quantum Electron. 28, 1427–1442 (1996).
    [CrossRef]
  14. G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3.
  15. P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).
  16. A. Korpel, P. P. Banerjee, C. W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
    [CrossRef]
  17. C. W. Tarn, P. P. Banerjee, A. Korpel, “Two-dimensional strong, acousto-optic interaction between arbitrary light and sound profiles: a Fourier transform approach,” Opt. Commun. 104, 141–148 (1993).
    [CrossRef]
  18. P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Pacific Palisades, Calif., 1991), Chap. 4.
  19. R. F. Harrington, Time-Harmonic Electromagnetic Field (McGraw-Hill, New York, 1961), Chap. 2.
  20. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 9.
  21. M. J. P. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970), Chap. 1.
  22. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7.
  23. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.
  24. D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
    [CrossRef]
  25. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 22.1–22.36.

1996 (1)

C. W. Tarn, C. N. Chang, “The polarization changing and beam profile deformation of light during the acoustooptic interaction in isotropic media,” Opt. Quantum Electron. 28, 1427–1442 (1996).
[CrossRef]

1993 (3)

V. I. Balakshy, J. A. Hassan, “Polarization effects in acoustooptic interaction,” Opt. Eng. 32, 746–751 (1993).
[CrossRef]

A. Korpel, P. P. Banerjee, C. W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

C. W. Tarn, P. P. Banerjee, A. Korpel, “Two-dimensional strong, acousto-optic interaction between arbitrary light and sound profiles: a Fourier transform approach,” Opt. Commun. 104, 141–148 (1993).
[CrossRef]

1992 (1)

See, for instance, the special section on acousto-optics, Opt. Eng. 31, 2047–2167 (1992).

1991 (1)

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

1988 (1)

E. I. Gordon, “A review of acoustooptical deflection and modulation devices,” Proc. IEEE 54, 1391–1400 (1988).
[CrossRef]

1981 (1)

See, for instance, the special issue on acousto-optics, Proc. IEEE 69, 48–120 (1981).

1974 (1)

M. R. W. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

1973 (1)

A. Alippi, “Half-wave plate behavior of ultrasonic waves light modulators,” Opt. Commun. 8, 397–400 (1973).
[CrossRef]

1970 (1)

D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
[CrossRef]

Alippi, A.

A. Alippi, “Half-wave plate behavior of ultrasonic waves light modulators,” Opt. Commun. 8, 397–400 (1973).
[CrossRef]

Balakshy, V. I.

V. I. Balakshy, J. A. Hassan, “Polarization effects in acoustooptic interaction,” Opt. Eng. 32, 746–751 (1993).
[CrossRef]

Banerjee, P. P.

C. W. Tarn, P. P. Banerjee, A. Korpel, “Two-dimensional strong, acousto-optic interaction between arbitrary light and sound profiles: a Fourier transform approach,” Opt. Commun. 104, 141–148 (1993).
[CrossRef]

A. Korpel, P. P. Banerjee, C. W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Pacific Palisades, Calif., 1991), Chap. 4.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7.

Chang, C. N.

C. W. Tarn, C. N. Chang, “The polarization changing and beam profile deformation of light during the acoustooptic interaction in isotropic media,” Opt. Quantum Electron. 28, 1427–1442 (1996).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 22.1–22.36.

Forshaw, M. R. W.

M. R. W. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

Ghatak, A.

A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989), Chaps. 16–19.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

Gordon, E. I.

E. I. Gordon, “A review of acoustooptical deflection and modulation devices,” Proc. IEEE 54, 1391–1400 (1988).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Field (McGraw-Hill, New York, 1961), Chap. 2.

Hassan, J. A.

V. I. Balakshy, J. A. Hassan, “Polarization effects in acoustooptic interaction,” Opt. Eng. 32, 746–751 (1993).
[CrossRef]

Kino, G. S.

G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3.

Korpel, A.

A. Korpel, P. P. Banerjee, C. W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

C. W. Tarn, P. P. Banerjee, A. Korpel, “Two-dimensional strong, acousto-optic interaction between arbitrary light and sound profiles: a Fourier transform approach,” Opt. Commun. 104, 141–148 (1993).
[CrossRef]

A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1989), Chap. 1.

Musgrave, M. J. P.

M. J. P. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970), Chap. 1.

Pinnow, D. A.

D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
[CrossRef]

Poon, T. C.

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Pacific Palisades, Calif., 1991), Chap. 4.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

Tarn, C. W.

C. W. Tarn, C. N. Chang, “The polarization changing and beam profile deformation of light during the acoustooptic interaction in isotropic media,” Opt. Quantum Electron. 28, 1427–1442 (1996).
[CrossRef]

C. W. Tarn, P. P. Banerjee, A. Korpel, “Two-dimensional strong, acousto-optic interaction between arbitrary light and sound profiles: a Fourier transform approach,” Opt. Commun. 104, 141–148 (1993).
[CrossRef]

A. Korpel, P. P. Banerjee, C. W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

C. W. Tarn, “A spatio-temporal Fourier transform approach to acoustooptic interactions,” Ph.D. dissertation (Syracuse University, 1991).

Thyagarajan, K.

A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989), Chaps. 16–19.

Tsai, C. S.

C. S. Tsai, Guided-Wave Acoustooptics (Springer-Verlag, Berlin, 1990), Chap. 1.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7.

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 9.

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 9.

Acustica (1)

P. P. Banerjee, C. W. Tarn, “A Fourier transform approach to acoustooptic interactions in the presence of propagational diffraction,” Acustica 74, 181–191 (1991).

IEEE J. Quantum Electron. (1)

D. A. Pinnow, “Guide lines for the selection of acoustooptic materials,” IEEE J. Quantum Electron. QE-6, 223–238 (1970).
[CrossRef]

Opt. Commun. (4)

A. Alippi, “Half-wave plate behavior of ultrasonic waves light modulators,” Opt. Commun. 8, 397–400 (1973).
[CrossRef]

M. R. W. Forshaw, “Diffraction of a narrow laser beam by a thick hologram: experimental results,” Opt. Commun. 12, 279–281 (1974).
[CrossRef]

A. Korpel, P. P. Banerjee, C. W. Tarn, “A unified treatment of spectral formalisms of light propagation and their application to acousto-optics,” Opt. Commun. 97, 250–258 (1993).
[CrossRef]

C. W. Tarn, P. P. Banerjee, A. Korpel, “Two-dimensional strong, acousto-optic interaction between arbitrary light and sound profiles: a Fourier transform approach,” Opt. Commun. 104, 141–148 (1993).
[CrossRef]

Opt. Eng. (2)

See, for instance, the special section on acousto-optics, Opt. Eng. 31, 2047–2167 (1992).

V. I. Balakshy, J. A. Hassan, “Polarization effects in acoustooptic interaction,” Opt. Eng. 32, 746–751 (1993).
[CrossRef]

Opt. Quantum Electron. (1)

C. W. Tarn, C. N. Chang, “The polarization changing and beam profile deformation of light during the acoustooptic interaction in isotropic media,” Opt. Quantum Electron. 28, 1427–1442 (1996).
[CrossRef]

Proc. IEEE (2)

E. I. Gordon, “A review of acoustooptical deflection and modulation devices,” Proc. IEEE 54, 1391–1400 (1988).
[CrossRef]

See, for instance, the special issue on acousto-optics, Proc. IEEE 69, 48–120 (1981).

Other (14)

A. Korpel, Acousto-Optics (Marcel Dekker, New York, 1989), Chap. 1.

N. J. Berg, J. N. Lee, eds., Acousto-Optic Signal Processing (Marcel Dekker, New York, 1983), Chap. 1.

C. S. Tsai, Guided-Wave Acoustooptics (Springer-Verlag, Berlin, 1990), Chap. 1.

A. Ghatak, K. Thyagarajan, Optical Electronics (Cambridge U. Press, Cambridge, 1989), Chaps. 16–19.

P. P. Banerjee, T. C. Poon, Principles of Applied Optics (Aksen, Pacific Palisades, Calif., 1991), Chap. 4.

R. F. Harrington, Time-Harmonic Electromagnetic Field (McGraw-Hill, New York, 1961), Chap. 2.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1983), Chap. 9.

M. J. P. Musgrave, Crystal Acoustics (Holden-Day, San Francisco, 1970), Chap. 1.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 7.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

G. S. Kino, Acoustic Waves: Devices, Imaging, and Analog Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1987), Chap. 3.

C. W. Tarn, “A spatio-temporal Fourier transform approach to acoustooptic interactions,” Ph.D. dissertation (Syracuse University, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds., 2nd ed. (McGraw-Hill, New York, 1995), Vol. 2, pp. 22.1–22.36.

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 (15)

Fig. 1
Fig. 1

Basic acousto-optic interaction configuration.

Fig. 2
Fig. 2

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, zero-order light as functions of x/σ and the sound intensity I for the longitudinal sound frequency equal to 50 MHz.

Fig. 3
Fig. 3

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, minus-one-order light as functions of x/σ and the sound intensity I for the longitudinal sound frequency equal to 50 MHz.

Fig. 4
Fig. 4

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, zero-order light as functions of x/σ and the sound intensity I for the shear sound frequency equal to 50 MHz.

Fig. 5
Fig. 5

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, minus-one-order light as functions of x/σ and the sound intensity I for the shear sound frequency equal to 50 MHz.

Fig. 6
Fig. 6

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, zero-order light as functions of x/σ and the sound intensity I for the longitudinal sound frequency equal to 250 MHz.

Fig. 7
Fig. 7

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, minus-one-order light as functions of x/σ and the sound intensity I for the longitudinal sound frequency equal to 250 MHz.

Fig. 8
Fig. 8

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, zero-order light as functions of x/σ and the sound intensity I for the shear sound frequency equal to 250 MHz.

Fig. 9
Fig. 9

Three-dimensional plots showing variation of the normalized intensity of the (a) x-polarized and (b) y-polarized, minus-one-order light as functions of x/σ and the sound intensity I for the shear sound frequency equal to 250 MHz.

Fig. 10
Fig. 10

(a) On-axis variation of (a) the magnitudes of the ellipticity of the zero- and minus-one-order and (b) the degrees of the azimuthal orientation major axis of the zero- and minus-one-order light as functions of the sound intensity I for the longitudinal sound frequency equal to 50 MHz and for linearly polarized incident light; Ex=cos 60°, and Ey=sin 60°. For Figs. 1015, the solid line represents zero-order and the dotted line represents minus-one-order-light.  

Fig. 11
Fig. 11

On-axis variation of (a) the magnitudes of the ellipticity of the zero- and minus-one-order light and (b) the degrees of the azimuthal orientation major axis of the zero- and minus-one-order light as functions of the sound intensity I for the shear sound frequency equal to 50 MHz and for linearly polarized incident light; Ex=cos 60°, and Ey=sin 60°.

Fig. 12
Fig. 12

On-axis variation of (a) the magnitudes of the ellipticity of the zero- and minus-one-order light and (b) the degrees of the azimuthal orientation major axis of the zero- and minus-one-order light as functions of the sound intensity I for the longitudinal sound frequency equal to 50 MHz and for elliptically polarized incident light; Ex=1, and Ey=exp(jπ/3).

Fig. 13
Fig. 13

On-axis variation of (a) the magnitudes of the ellipticity of the zero- and minus-one-order light and (b) the degrees of the azimuthal orientation major axis of the zero- and minus-one-order light as functions of the sound intensity I for the shear sound frequency equal to 50 MHz and for elliptically polarized incident light; Ex=1, and Ey=exp(jπ/3).

Fig. 14
Fig. 14

On-axis variation of (a) the magnitudes of the ellipticity of the zero- and minus-one-order light and (b) the degrees of the azimuthal orientation major axis of the zero- and minus-one-order light as functions of the sound intensity I for the longitudinal sound frequency equal to 50 MHz and for circularly polarized incident light; Ex=1, and Ey=exp(-jπ/2).

Fig. 15
Fig. 15

On-axis variation of (a) the magnitudes of the ellipticity of the zero- and minus-one-order light and (b) the degrees of the azimuthal orientation major axis of the zero- and minus-one-order light as functions of the sound intensity I for the shear sound frequency equal to 50 MHz and for circularly polarized incident light; Ex=1, and Ey=exp(-jπ/2).

Equations (40)

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

Einc(r, t)=12 Exax+Eyay(|Ex|2+|Ey|2)1/2ψinc(r)exp(jω0t-jk0x sin ϕinc-jk0z cos ϕinc)+c.c.,
2E(r, t)-μ0¯¯ 2E(r, t)t2μ0Δ¯¯ 2E(r, t)t2.
Δ¯¯L(r, t)=-0n4p11Sx+p12Sy+p12Sz000p12Sx+p11Sy+p12Sz000p12Sx+p12Sy+p11Sz,
Sx=12A0 exp(jΩt-Kx)+c.c.,
Sy=0,
Sz=0,
Δ¯¯L(r, t)=-0n4A0 cos(Ωt-Kx) p11000p12000p12=-0n4A0 cos(Ωt-Kx)P¯¯L.
Δ¯¯S(r, t)=-120n40(p11-p12)Sxy(p11-p12)Sxz(p11-p12)Sxy0(p11-p12)Syz(p11-p12)Sxz(p11-p12)Syz0,
Sxy=12A0 exp(jΩt-Kx)+c.c.,
Sxz=0,
Syz=0.
Δ¯¯S(r, t)=-12 0n4A0 cos(Ωt-Kx)×0p11-p120p11-p1200000=-0n4A0 cos(Ωt-Kx)P¯¯S.
E(r, t)=12 m=- emψm(r)exp(jω0t+mΩt-jnk0x sin ϕm-jnk0z cos ϕm)+c.c.,
sin ϕm=sin ϕinc+m Knk0.
2ψmx2+2ψmy2+2ψmz2-2jnk0(sin ϕm) ψmx-2jnk0(cos ϕm) ψmzem
=12k02n4A0ψm-1exp[-jnk0z(cos ϕm-1-cos ϕm)]×P¯¯em-1+12k02n4A0*ψm+1exp[-jnk0z×(cos ϕm+1-cos ϕm)]P¯¯em+1.
ψ0z e0=-j2nk0 cos ϕB 2ψ0x2+2ψ0y2e0-(tan ϕB)ψ0x e0+jk0n3A04 cos ϕB ψ-1P¯¯e-1,
ψ0z e-1=-j2nk0 cos ϕB 2ψ-1x2+2ψ-1y2e-1+(tan ϕB) ψ-1x e-1+jk0n3A0*4 cos ϕB ψ0P¯¯e0,
Ψ(kx, ky; z)=F[ψ(x, y, z)]=- ψ(x, y, z)exp(jkxx+jkyy)dxdy,
ψ(x, y, z)=F-1[Ψ(kx, ky; z)]=14π2 - Ψ(kx, ky; z)×exp(-jkxx-jkyy)dkxdky,
Ψ0(kx, ky; z)z e0
=jkx2+ky22nk0 cos ϕB+kx tan ϕBΨ0(kx, ky; z)e0+j k0n3A04 cos ϕB Ψ-1(kx, ky; z)P¯¯e-1,
Ψ-1(kx, ky; z)z e-1
=jkx2+ky22nk0 cos ϕB-kx tan ϕBΨ-1(kx, ky; z)e-1+j k0n3A0*4 cos ϕB Ψ0(kx, ky; z)P¯¯e0,
Ψm=Ψincδm0atz0,
E0(r, t)=14π2 - Ψinc(kx, ky; 0)H¯¯0(kx, ky; z)×exp[j(-kxx-kyy+ω0t-nk0x sin ϕB-nk0z cos ϕB)]dkxdky,
E-1(r, t)=14π2 -Ψinc(kx, ky; 0)H¯¯-1(kx, ky; z)×exp[j(-kxx-kyy+ω0t-Ωt+nk0x ×sin ϕB-nk0z cos ϕB)]dkxdky,
H¯¯0,L(kx, ky; z)=expj(kx2+ky2)z2nk0 cos ϕB Exax(|Ex|2+|Ey|2)1/2 cos(kx tan ϕB)2+k0n3p11A04 cos ϕB21/2z+j kx tan ϕB sin(kx tan ϕB)2+k0n3p11A04 cos ϕB21/2z(kx tan ϕB)2+k0n3p11A04 cos ϕB21/2+Eyay(|Ex|2+|Ey|2)1/2 cos(kx tan ϕB)2+k0n3p12A04 cos ϕB21/2z+j kx tan ϕB sin(kx tan ϕB)2+k0n3p12A04 cos ϕB21/2z(kx tan ϕB)2+k0n3p12A04 cos ϕB21/2,
 
H¯¯-1,L(kx, ky; z)=j expj(kx2+ky2)z2nk0 cos ϕB ax Ex k0n3p11A04 cos ϕB(|Ex|2+|Ey|2)1/2 sin(kx tan ϕB)2+k0n3p11A04 cos ϕB21/2z(kx tan ϕB)2+k0n3p11 A04 cos ϕB21/2+ay Ey k0n3p12A04 cos ϕB(|Ex|2+|Ey|2)1/2 sin(kx tan ϕB)2+k0n3p12A04 cos ϕB21/2z(kx tan ϕB)2+k0n3p12A04 cos ϕB21/2.
H¯¯0,S(kx, ky; z)=expj(kx2+ky2)z2nk0 cos ϕB Exax+Eyay(|Ex|2+|Ey|2)1/2 cos(kx tan ϕB)2+k0n3(p11-p12)A08 cos ϕB21/2z+j kx tan ϕB sin(kx tan ϕB)2+k0n3(p11-p12)A08 cos ϕB21/2z(kx tan ϕB)2+k0n3(p11-p12)A08 cos ϕB21/2,
H¯¯-1,S(kx, ky; z)=j expj(kx2+ky2)z2nk0 cos ϕB Eyax+Exay(|Ex|2+|Ey|2)1/2 k0n3(p11-p12)A08 cos ϕB(kx tan ϕB)2+k0n3(p11-p12)A08 cos ϕB21/2×sin(kx tan ϕB)2+k0n3(p11-p12)A08 cos ϕB21/2z.
ψ0,L(z)=ψincExax(|Ex|2+|Ey|2)1/2 cosk0n3p11A04 cos ϕBz+Eyay(|Ex|2+|Ey|2)1/2 cosk0n3p12A04 cos ϕBz,
ψ-1,L(z)=jψincExax(|Ex|2+|Ey|2)1/2 (p11A0)*|p11A0| ×sink0n3p11A04 cos ϕBz+Eyay(|Ex|2+|Ey|2)1/2 ×(p12A0)*|p12A0| sink0n3p12A04 cos ϕBz.
ψ0,S(z)=ψinc Exax+Eyay(|Ex|2+|Ey|2)1/2 ×cosk0n3(p11-p12)A08 cos ϕBz,
ψ-1,S(z)=jψinc Eyax+Exay(|Ex|2+|Ey|2)1/2 [(p11-p12)A0]*|(p11-p12)A0| ×sink0n3(p11-p12)A08 cos ϕBz.
tan-1Ey cosk0n3p12A04 cos ϕB/Ex cosk0n3p11A04 cos ϕB,
tan-1Ey sink0n3p12A04 cos ϕB/Ex sink0n3p11A04 cos ϕB
ψinc(x, y, 0)=exp-x2+y2σ2,
Ψinc(kx, ky; 0)=πσ2 exp-kx2+ky24 σ2,

Metrics