Abstract

Sinusoidal-Gaussian beam solutions are derived for the propagation of electromagnetic waves in free space and in media having at most quadratic transverse variations of the index of refraction and the gain or loss. The resulting expressions are also valid for propagation through other real and complex lens elements and systems that can be represented in terms of complex beam matrices. The solutions are in the form of sinusoidal functions of complex argument times a conventional Gaussian beam factor. In the limit of large Gaussian beam size, the sine and cosine factors of the beams are dominant and reduce to the conventional modes of a rectangular waveguide. In the opposite limit the beams reduce to the familiar fundamental Gaussian form. Alternate hyperbolic-sinusoidal-Gaussian beam solutions are also found.

© 1997 Optical Society of America

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References

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  1. G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
    [Crossref]
  2. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
    [Crossref]
  3. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain profile,” Appl. Opt. 4, 1562–1569 (1965).
    [Crossref]
  4. L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
    [Crossref]
  5. N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).
  6. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
    [Crossref]
  7. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
    [Crossref]
  8. J. A. Arnaud, “Mode coupling in first order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
    [Crossref]
  9. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [Crossref]
  10. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.
  11. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [Crossref] [PubMed]
  12. L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
    [Crossref]
  13. R. Pratesi, L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. 67, 1274–1276 (1977).
    [Crossref]
  14. M. Nazarathy, A. Hardy, J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,” J. Opt. Soc. Am. 72, 1409–1420 (1982).
    [Crossref]
  15. A. E. Siegman, Lasers (University Science, Mill Valley, California, 1986), especially Subsec. 20.5.
  16. A. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
    [Crossref]
  17. C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
  18. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [Crossref]
  19. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [Crossref] [PubMed]
  20. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [Crossref] [PubMed]
  21. Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. Ser. 5 43, 125–132 (1897).
    [Crossref]
  22. J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  23. J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref] [PubMed]
  24. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [Crossref]
  25. A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
    [Crossref]
  26. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [Crossref]
  27. L. W. Casperson, “Mode stability in lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
    [Crossref]
  28. See, for example, E. Sklar, “Fourier-transform ring laser,” J. Opt. Soc. Am. A 1, 537–540 (1984).
    [Crossref]
  29. L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
    [Crossref]

1996 (1)

1995 (2)

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
[Crossref]

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
[Crossref]

1994 (1)

1991 (1)

1987 (3)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

1984 (1)

1982 (1)

1978 (2)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[Crossref]

1977 (1)

1976 (1)

1974 (1)

L. W. Casperson, “Mode stability in lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[Crossref]

1973 (2)

1971 (1)

1970 (2)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[Crossref]

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[Crossref]

1968 (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[Crossref]

1965 (3)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain profile,” Appl. Opt. 4, 1562–1569 (1965).
[Crossref]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[Crossref]

1961 (2)

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[Crossref]

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[Crossref]

1897 (1)

Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. Ser. 5 43, 125–132 (1897).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, “Mode coupling in first order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
[Crossref]

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[Crossref]

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[Crossref]

Casperson, L. W.

A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
[Crossref]

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
[Crossref]

A. A. Tovar, L. W. Casperson, “Off-axis complex-argument polynomial-Gaussian beams in optical systems,” J. Opt. Soc. Am. A 8, 60–68 (1991).
[Crossref]

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[Crossref]

L. W. Casperson, “Beam modes in complex lenslike media and resonators,” J. Opt. Soc. Am. 66, 1373–1379 (1976).
[Crossref]

L. W. Casperson, “Mode stability in lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[Crossref]

L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
[Crossref] [PubMed]

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[Crossref]

Durnin, J. E.

J. E. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[Crossref]

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Goubau, G.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Hall, D. G.

Hardy, A.

Jordan, R. H.

Kogelnik, H.

Miceli, J. J.

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Nazarathy, M.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Pratesi, R.

Rayleigh, Lord

Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. Ser. 5 43, 125–132 (1897).
[Crossref]

Ronchi, L.

Schwering, F.

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[Crossref]

Shamir, J.

Sheppard, C. J. R.

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).

Siegman, A. E.

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[Crossref]

A. E. Siegman, Lasers (University Science, Mill Valley, California, 1986), especially Subsec. 20.5.

Sklar, E.

Tovar, A. A.

Vakhimov, N. G.

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).

Yariv, A.

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[Crossref]

Zucker, H.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

L. W. Casperson, A. Yariv, “The Gaussian mode in optical resonators with a radial gain profile,” Appl. Phys. Lett. 12, 355–357 (1968).
[Crossref]

Bell Syst. Tech. J. (4)

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[Crossref]

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[Crossref]

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[Crossref]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[Crossref]

IEEE J. Quantum Electron. (1)

L. W. Casperson, “Mode stability in lasers and periodic optical systems,” IEEE J. Quantum Electron. QE-10, 629–634 (1974).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

A. A. Tovar, L. W. Casperson, “Gaussian beam optical systems with high gain or high loss media,” IEEE Trans. Microwave Theory Tech. 43, 1857–1862 (1995).
[Crossref]

IRE Trans. Antennas Propag. (1)

G. Goubau, F. Schwering, “On the guided propagation of electromagnetic wave beams,” IRE Trans. Antennas Propag. AP-9, 248–256 (1961).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (4)

Microwaves Opt. Acoust. (1)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).

Opt. Commun. (1)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[Crossref]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[Crossref]

Philos. Mag. Ser. 5 (1)

Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Philos. Mag. Ser. 5 43, 125–132 (1897).
[Crossref]

Phys. Rev. Lett. (1)

J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref] [PubMed]

Radio Eng. Electron. Phys. (USSR) (1)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

Other (2)

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), especially Subsecs. 2.16, 2.17, and 4.18–4.20.

A. E. Siegman, Lasers (University Science, Mill Valley, California, 1986), especially Subsec. 20.5.

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Figures (3)

Fig. 1
Fig. 1

Transverse intensity profiles of a sine-Gaussian beam at the reference plane z=0 for the normalized modal parameter values a=1, 3, and 5. For small values of this parameter the intensity distribution approaches that of a first-order Hermite–Gaussian beam mode.

Fig. 2
Fig. 2

Transverse intensity distribution of a sine-Gaussian beam with a=5 for the normalized propagation distance values z=0.0, 0.5, 1.0. For large values of the propagation distance the intensity distribution evolves into that of a sinh-Gaussian beam mode.

Fig. 3
Fig. 3

Amplitude profiles from Eq. (55) for (a) a fundamental Gaussian beam and (b) a cosh-Gaussian beam. The horizontal variable for these plots is x/w, and the vertical variable is normalized to unity. The cosh scale width for part (b) is given by a×w=5.

Equations (60)

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2E(x, y, z)+k2(x, y, z)E(x, y, z)=0,
k2(x, y, z)=k0(z)[k0(z)-k1x(z)x-k1y(z)y-k2x(z)x2-k2y(z)y2].
Ex(x, y, z)=A(x, y, z) exp-i0zk0(z)dz,
2Ax2+2Ay2-2ik0 Az-i dk0dzA-k0(k1xx+k1yy
+k2xx2+k2yy2)A=0,
A(x, y, z)=B(x, y, z)exp -iQx(z)x22+Qy(z)y22+Sx(z)x+Sy(z)y.
Qx2+k0 dQxdz+k0k2x=0,
Qy2+k0 dQydz+k0k2y=0,
QxSx+k0 dSxdz+k0k1x2=0,
QySy+k0 dSydz+k0k1y2=0,
2Bx2-2i(Sx+Qxx) Bx+2By2-2i(Sy+Qyy) By-(Sx2+Sy2)B-i(Qx+Qy)B-2ik0 Bz
-i dk0dzB=0.
Qx(z)=k0(z)Rx(z)-i 2wx2(z),
x=ax(z)x+bx(z),
y=ay(z)y+by(z),
z=z,
ax22Bx2-2iaxSx+i(x-bx)Qx+ik0 (x-bx)axdaxdz+ik0 dbxdz Bx+ay2 2By2-2iaySy+i(y-by)Qy+ik0 (y-by)aydaydz+ik0 dbydz By-(Sx2+Sy2)B-i(Qx+Qy)B
-2ik0 Bz-i dk0dzB=0.
B(x, y, z)=C(x, y, z)exp[-iP(z)]
dPdz=-12k0(Sx2+Sy2)+i(Qx+Qy)+(γx2ax2+γy2ay2)+i dk0dz,
ax2 2Cx2-2iaxSx+i(x-bx)Qx+ik0 (x-bx)axdaxdz+ik0 dbxdz Cx+ay2 2Cy2-2iaySy+i(y-by)Qy+ik0 (y-by)aydaydz+ik0 dbydz Cy-2ik0 Cz
+(γx2ax2+γy2ay2)C=0,
ax22Cx2+γx2C+ay22Cy2+γy2C
-2ik0 Cz=0.
Qx+k0axdaxdz=0,
axSx+k0 dbxdz=0,
C(x, y, z)=X(x)Y(y)
2Xx2+γx2X=0,
2Yy2+γy2Y=0.
Ex(x, y, z)=Ex0 exp-i0zk0(z)dz+Qx(z)x22+Qy(z)y22+Sx(z)x+Sy(z)y+P(z)×sin{γx[ax(z)x+bx(z)]}cos{γx[ax(z)x+bx(z)]}sinh{γx[ax(z)x+bx(z)]}cosh{γx[ax(z)x+bx(z)]}sin{γy[ay(z)y+by(z)]}cos{γy[ay(z)y+by(z)]}sinh{γy[ay(z)y+by(z)]}cosh{γy[ay(z)y+by(z)]},
Qx2k0=1qx2=Cx+Dx/qx1Ax+Bx/qx1,
1qx=1Rx-iλn0πwx2.
AxBxCxDx=cos[(k2x/k0)1/2z](k0/k2x)1/2 sin[(k2x/k0)1/2z]-(k2x/k0)1/2 sin[(k2x/k0)1/2z]cos[(k2x/k0)1/2z].
Sx2=Sx1Ax+Bx/qx1,
ax2=ax1Ax+Bx/qx1,
dbxdz=-1k0ax(z)Sx(z)=-1k0ax1Sx1{cos[(k2x/k0)1/2z]+(k0/k2x)1/2 sin[(k2x/k0)1/2z]/qx1}2.
bx(z)=bx1-ax1Sx1k0(k0/k2x)1/2 sin[(k2x/k0)1/2z]cos[(k2x/k0)1/2z]+(k0/k2x)1/2 sin[(k2x/k0)1/2z]/qx1,
bx2=bx1-ax1Sx1k0BxAx+Bx/qx1.
dP(z)dz=-12k0[(Sx2+Sy2)+i(Qx+Qy)+(γx2ax2+γy2ay2)].
P2=P1-i2[ln(Ax+Bx/qx1)+ln(Ay+By/qy1)]-12k0(Sx12+γx2ax12)BxAx+Bx/qx1+(Sy12+γy2ay12)ByAy+By/qy1,
AxBxCxDx=1z01,
1qx2=1/qx11+z/qx1,
Sx2=Sx11+z/qx1,
ax2=ax11+z/qx1,
bx2=bx1-ax1Sx1z/k01+z/qx1,
P2=P1-i2[ln(1+z/qx1)+ln(1+z/qy1)]-z2k0Sx12+γx2ax121+z/qx1+Sy12+γy2ay121+z/qy1.
Ex(x, z)=Ex0 exp-ik0z+k0x22qx(z)+P(z)×sin[γxax(z)x],
P2=P1-i2ln(1+z/qx1)-z2k0γx2ax121+z/qx1.
wx2=wx0[1+(z/z0)2]1/2,
Rx2=z[1+(z0/z)2],
ax2r=ax01+(z/z0)2,
ax2i=ax0(z/z0)1+(z/z0)2.
P2r=-12tan-1(z/z0)-zγx2ax02/(2k0)1+(z/z0)2,
P2i=-14ln[1+(z/z0)2]-zγx2ax02(z/z0)/(2k0)1+(z/z0)2,
I(x, z)=I0[1+(z/z0)2]1/2exp-2(x/wx0)2+(z0γx2ax02/k0)(z/z0)1+(z/z0)2cosh2γxxax0(z/z0)1+(z/z0)2-cos2γxxax01+(z/z0)22.
I(x, z)=I0(1+z2)1/2exp-2x2+a2z2/21+z2×cosh2axz1+z2-cos2ax1+z22.
I(x, z=0)=I0 exp(-2x2)[1-cos(2ax)]/2=I0 exp(-2x2)sin2(ax).
I(x, z=0)I0a2x2 exp(-2x2),
I(x, z)=I0z-1 exp[-(2x2/z2+a2/2)]×[cosh(2ax/z)-1]/2=I0z-1 exp[-(2x2/z2+a2/2)]×sinh2(ax/z).
Ex(x, z)cosh[a(z)x]exp[-x2/w(z)2].

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