Abstract

Electromagnetic beams in a uniaxial crystal are treated with emphasis on the extraordinary mode. A virtual source that generates a basic elliptical Gaussian wave propagating obliquely to the optic axis is identified. An exact expression is obtained for this basic elliptical Gaussian wave that simplifies to the corresponding basic elliptical Gaussian beam in the appropriate limit. In the direction of amplitude propagation, the paraxial result becomes identical to the exact result and the sum of all the nonparaxial contributions vanish. The characteristics of the basic elliptical Gaussian beam are illustrated with a numerical example. From the spectral representation of the basic Gaussian wave, the first three orders of nonparaxial corrections for the basic elliptical Gaussian beam are determined. The nonparaxial results reduce correctly to those of the fundamental Gaussian beam in an isotropic medium.

© 2003 Optical Society of America

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References

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  1. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 315–324.
  2. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 15.
  3. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994), pp. 740–820.
  4. S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
    [CrossRef]
  5. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  6. S. R. Seshadri, “Paraxial wave equation for the extraordinary-mode beam in a uniaxial crystal,” J. Opt. Soc. Am. A 18, 2628–2629 (2001).
    [CrossRef]
  7. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), Chap. 6.
  8. A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorialtheory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  9. G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
    [CrossRef]
  10. A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
    [CrossRef]
  11. G. Cincotti, A. Ciattoni, C. Palma, “Laguerre–Gauss and Bessel–Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680–1688 (2002).
    [CrossRef]
  12. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  13. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  14. L. B. Felsen, “On the use of refractive index diagrams for source-excited anisotropic regions,” J. Res. Natl. Bur. Stand. Sect. D 69, 155–169 (1965).
    [CrossRef]
  15. S. R. Seshadri, “Virtual source for a Laguerre–Gauss beam,” Opt. Lett. 27, 1872–1874 (2002).
    [CrossRef]
  16. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.
  17. M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  18. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]

2002 (4)

2001 (2)

1985 (1)

1983 (1)

1981 (1)

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1976 (1)

1974 (1)

S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1965 (1)

L. B. Felsen, “On the use of refractive index diagrams for source-excited anisotropic regions,” J. Res. Natl. Bur. Stand. Sect. D 69, 155–169 (1965).
[CrossRef]

Belanger, P. A.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 15.

Ciattoni, A.

Cincotti, G.

Couture, M.

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Crosignani, B.

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Di Porto, P.

Feit, M. D.

Felsen, L. B.

L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
[CrossRef]

S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

L. B. Felsen, “On the use of refractive index diagrams for source-excited anisotropic regions,” J. Res. Natl. Bur. Stand. Sect. D 69, 155–169 (1965).
[CrossRef]

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994), pp. 740–820.

Fleck, J. A.

Fukumitsu, O.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 315–324.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 315–324.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994), pp. 740–820.

Palma, C.

Seshadri, S. R.

Shin, S. Y.

S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

Takenaka, T.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 15.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), Chap. 6.

Yokota, M.

Appl. Phys. (1)

S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2002).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

J. Res. Natl. Bur. Stand. Sect. D (1)

L. B. Felsen, “On the use of refractive index diagrams for source-excited anisotropic regions,” J. Res. Natl. Bur. Stand. Sect. D 69, 155–169 (1965).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

M. Couture, P. A. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Other (5)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–287.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), Chap. 6.

L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, New York, 1960), pp. 315–324.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999), Chap. 15.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994), pp. 740–820.

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Figures (3)

Fig. 1
Fig. 1

Coordinate systems showing the directions of the optic axis (z^p), the wave vector (zˆ), and the normal (z^s) to the wave-vector surface. The angle between the wave vector and the optic axis is α and that between the normal to the wave-vector surface and the wave vector is β. For the Gaussian beam, the magnetic field, the electric field, and the electric flux density are in the x^s, y^s, and yˆ directions, respectively.

Fig. 2
Fig. 2

Inclination angle β in degrees as a function of α in degrees, where α is the angle between the wave vector and the optic axis, and β is the angle between the normal to the wave-vector surface and the wave vector for the (positive) uniaxial crystal n=2.616 and n=2.903.

Fig. 3
Fig. 3

Ratio R of the rms width of the beam in the x direction to that in the y direction as a function of the angle of inclination α of the wave vector to the optic axis for the positive uniaxial crystal with n=2.616 and n=2.903. The ordinate also represents the Rayleigh distance normalized by its value corresponding to the beam propagation along the optic axis.

Equations (94)

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=(x^px^p+y^py^p)+ z^pz^p.
Ep=(0)1/2E˜p,Hp=(μ0)1/2H˜p,Jmp=(0)1/2J˜mp,
Jep=(μ0)1/2J˜ep,ρm=(μ0)-1/2ρ˜m,ρe=(0)-1/2ρe
p×Ep=ik0Hp-Jmp,
p×Hp=-ik0Ep+Jep,
pHp=ρm,
p(Ep)=ρe,
p=x^pxp+y^pyp+z^pzp
Jmp=z^pJmzp,Jep=z^pJezp.
Ep=-p×(z^pFzp)=z^p×pFzp.
Hp=ikz^pFzp-pψp
ψp=1ikFzpzp
2xp2+2yp2+2zp2+k02Fzp=-Jmzp.
Hp=-1ikpzp+z^pk2Fzp.
Hp=p×(z^pAzp)=-z^p×pAzp.
Ep=ikz^pAzp-pϕp,
ϕp=1ikAzpzp
2xp2+2yp2+2zp2+k02Azp=-Jezp.
Ep=-1ikpzp+z^pk2Azp.
(a)x=xp,(b)y=ypcos α+zpsin α,
(c)z=-ypsin α+zpcos α.
2x2+eyy2y2+ezz2z2+eyz2yz+k2Azp=-Jzp,
eyy=cos2 α+(/)sin2 α,
ezz=sin2 α+(/)cos2 α,
eyz=-(1-/)2 cos α sin α.
f(kx, ky, kz)=kx2+eyyky2+ezzkz2+eyzkykz-k2=0,
kf(kx, ky, kz)=xˆfkx+yˆfky+zˆfkz.
tan β=yz=fky/fkzkx=ky=0=eyz2ezz.
(a)xs=x,(b)ys=y cos β-z sin β,
(c)zs=y sin β+z cos β.
2xs2+cos2 βezz2ys2+ezz2z2+k2Azp(xs, ys, z)
=-Sexδ(xs)δ(ys)δ(zs-zs,ex).
yss=(ezz/cos2 β)1/2ys.
xs=ρscos θs;yss=ρssin θs.
2ρs2+1ρsρs+ezz2z2+k2Azp(ρs, z)
=-Sexezz1/2δ(ρs)2πρs δ(z-zs,excos β).
Azp(ρs, z)=iSex4πezz1/20dηηJ0(ηρs)ζ-1×exp[iζ(z-zs,excos β)]
ζ=(ezz)-1/2(k2-η2)1/2.
Azp,p(ρs, z)=Sex4π(z-zs,excos β)ezz1/2×expikezz1/2(z-zs,excos β)×expik(ezz)1/2ρs22(z-zs,excos β).
Azp,p(ρs, 0)=exp(-x2/w0x2-y2/w0y2),
ρs2=x2+(ezz/)(y-z tan β)2
zs,excos β=ib=i 12 kw0x2(ezz)1/2
zs,excos β=i 12 kw0y2(ezz)1/2ezz.
w0x2=w0y2(ezz/).
Sex=-i2πkw0x2()-1/2exp[-kb(/ezz)1/2].
Azp,p(ρs, z)=exp[ik(/ezz)1/2z]q2(z)exp(-ν),
q2(z)=(1+iz/b)-1,
ν=q2(z)[x2w0x-2+(y-z tan β)2w0y-2],
Azp(ρs, z)=bezzexp[-kb(/ezz)1/2]×0dηηJ0(ηρs)ζ-1×exp[iζ(z-ib)].
Hx=cos α y-sin α zAzp,
Hy=-cos α x Azp,
Hz=sin α x Azp,
Ex=-1ikxsin α y+cos α zAzp,
Ey=-1ikysin α y+cos α z+k2sin αAzp,
Ez=-1ikzsin α y+cos α z+k2cos αAzp.
Hx=-ik sin α(/ezz)1/2Azp,p,
Hy=Hz=Ex=0,
Ey=ik sin α Azp,p,
Ez=-ik sin α(eyz/2ezz)Azp,p.
E=y^sik sin αcos β Azp,p.
=xˆxˆ+yˆyˆeyy+(yˆzˆ+zˆyˆ) 12 eyz+zˆzˆezz,
Dx=(0)-1/2D˜x=Ex=0,
Dy=(0)-1/.2D˜y=(eyyEy+12eyzEz)=ik sin α(/ezz)Azp,p,
Dz=(0)-1/.2D˜z=(12eyzEy+ezzEz)=0.
S=c2Re(E×H*)=zsc2k2sin2 αcos βezz1/2|Azp,p(ρs, z)|2,
|Azp,p(ρs, z)|2=1+z2b2-1exp-21+z2b2-1×x2w0x2+(y-z tan β)2w0y2.
P=--dxdyzˆS=c4 k2sin2 αezz1/2πw0xw0y.
(x)aν=1P--dxdyxzˆS=0.
(y)aν=1P--dxdyyzˆS=z tan β.
(σx2)z=1P--dxdy[x-(x)aν]2zˆS=w0x241+z2b2.
(σy2)z=1P--dxdy[y-(y)aν]2zˆS=w0y241+z2b2=w0y2w0x2 (σx2)z.
R=[(σx2)z]1/2[(σy2)z]1/2=w0xw0y=ezz1/2.
b0=12 kw0x221/2 forα=0.
bn=bb0=ezz1/2,
zss=(ezz)-1/2z;kss=()1/2k.
2xs2+2yss2+2zss2+kss2Azp(xs, yss, zss)=0.
Azp(xs, yss, zss)=A(xs, yss, zss)exp(iksszss)=A(xs, yss, zss)exp[ik(/ezz)1/2z],
2xs2+2yss2+2iksszssA(xs, yss, zss)=0.
S=δ(x)δ(y)=δ(xs)δ(ys/cos β).
S=δ(xs)ezz1/2δ(yss),
G(xs, ys, z)=ezz1/2-ikss2πzssexpikss2zss (xs2+yss2).
G(xs, ys, z)=-ikezz()1/22πzexpik(ezz)1/22z×xs2+ezzyscos β2,
Pzp,p(ρs, z)=1πw0xw0y Azp,p(ρs, z)exp[-ik(/ezz)1/2z]=-ikezz()1/22π(z-ib)expik(ezz)1/22(z-ib)xs2+ezzyscos β2.
2ρs2+1ρsρs+2zss2+kss2Azp(ρs, z)
=--i4πb(ezz)1/2exp[-kb(/ezz)1/2]δ(ρs)2πρs× δ[zss-ib(ezz)-1/2].
Azp(ρs, z)=-i4πb(ezz)1/2exp[-kb(/ezz)1/2] exp(ikssRss)4πRss,
Rss2=ρs2+[zss-ib(ezz)-1/2]2,
Azp(ρs, z)=-ib(ezz)1/2exp[-kb(/ezz)1/2]×exp[ik()1/2Rss]Rss,
Rss2=x2+ezz (y-z tan β)2+1ezz (z-ib)2.
Azp(0, z)=(1+iz/b)-1exp[ik(/ezz)1/2z].
Azp(ρs, z)
=Azp,p(ρs, z)×1+s=1s=(kw0x)-2s-sq2s(z)Azp,2s(ρs, z).
Azp,2s(ρs, z)=n=0n=s(2s)!(-1)nn!(s-n)! Ls+n(ν)
fors=1, 2, 3,

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