Abstract

The Gaussian beam propagation in the direction of the optic axis of a uniaxial crystal is treated by the complex-source-point technique. At the input plane the electric field is linearly polarized. A particular superposition of the ordinary-mode and the extraordinary-mode beams is generated. The electrodynamics of the composite beam has features that are different from those of the two constituent beams. As a result of the anisotropy, on propagation, the cross-polarized component of the electric field is generated except along the beam axis; the cross section of the beam, which is circular at the input plane, becomes elliptical; and the mean squared width of the beam departs from the usual quadratic dependence on the distance from the waist in the direction of propagation.

© 2005 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).
  3. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994), pp. 740–820.
  4. H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), Chaps. 4, 5, and 11.
  5. S. Y. Shin, L. B. Felsen, “Gaussian beams in anisotropic media,” Appl. Phys. 5, 239–250 (1974).
    [CrossRef]
  6. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  7. M. Nazarathy, J. W. Goodman, “Diffraction transforms in homogeneous birefringent media,” J. Opt. Soc. Am. A 3, 523–531 (1986).
    [CrossRef]
  8. L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).
  9. J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001). See also references cited in this paper.
    [CrossRef]
  10. A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
    [CrossRef]
  11. G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2001).
    [CrossRef]
  12. 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]
  13. A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
    [CrossRef]
  14. A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  15. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  16. L. B. Felsen, “Evanescent waves,” J. Opt. Soc. Am. 66, 751–760 (1976).
    [CrossRef]
  17. E. Heyman, L. B. Felsen, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A 18, 1588–1611 (2001).
    [CrossRef]
  18. H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pure Appl. Math. 3, 355–391 (1950). See Eqs. (3.1).
    [CrossRef]
  19. S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, Reading, Mass., 1971). See Eqs. (9.2.1)–(9.2.6) and (9.10.1)–(9.10.5).
  20. S. R. Seshadri, “Basic elliptical Gaussian wave and beam in a uniaxial crystal,” J. Opt. Soc. Am. A 20, 1818–1826 (2003).
    [CrossRef]
  21. S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
    [CrossRef]
  22. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]

2003 (1)

2002 (2)

2001 (5)

2000 (1)

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

1999 (1)

1986 (1)

1983 (1)

1977 (1)

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]

1950 (1)

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pure Appl. Math. 3, 355–391 (1950). See Eqs. (3.1).
[CrossRef]

Born, M.

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

Ciattoni, A.

Cincotti, G.

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

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]

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2001).
[CrossRef]

Crosignani, B.

Deschamps, G. A.

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

Dhayalan, V.

Di Porto, P.

Feit, M. D.

Felsen, L. B.

Fleck, J. A.

Garea, M. T.

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

Goodman, J. W.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), Chaps. 4, 5, and 11.

Heyman, E.

Landau, L. D.

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

Levine, H.

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pure Appl. Math. 3, 355–391 (1950). See Eqs. (3.1).
[CrossRef]

Lifshitz, E. M.

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

Marcuvitz, N.

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

Nazarathy, M.

Palma, C.

A. Ciattoni, G. Cincotti, C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792–796 (2002).
[CrossRef]

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]

G. Cincotti, A. Ciattoni, C. Palma, “Hermite–Gauss beams in uniaxially anisotropic crystals,” IEEE J. Quantum Electron. 37, 1517–1524 (2001).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

Perez, L. I.

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

Schwinger, J.

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pure Appl. Math. 3, 355–391 (1950). See Eqs. (3.1).
[CrossRef]

Seshadri, S. R.

Shin, S. Y.

S. Y. Shin, L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977).
[CrossRef]

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

Stamnes, J. J.

Wolf, E.

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

Appl. Phys. (1)

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

Commun. Pure Appl. Math. (1)

H. Levine, J. Schwinger, “On the theory of electromagnetic wave diffraction by an aperture in an infinite plane conducting screen,” Commun. Pure Appl. Math. 3, 355–391 (1950). See Eqs. (3.1).
[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 (2001).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (8)

Opt. Commun. (1)

A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
[CrossRef]

Optik (1)

L. I. Perez, M. T. Garea, “Propagation of 2D and 3D Gaussian beams in an anisotropic uniaxial medium: vectorial and scalar treatment,” Optik 111, 297–306 (2000).

Other (5)

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).

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

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, Englewood Cliffs, N.J., 1984), Chaps. 4, 5, and 11.

S. R. Seshadri, Fundamentals of Transmission Lines and Electromagnetic Fields (Addison-Wesley, Reading, Mass., 1971). See Eqs. (9.2.1)–(9.2.6) and (9.10.1)–(9.10.5).

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Figures (2)

Fig. 1
Fig. 1

Ratio r of the minor axis to the major axis of the elliptical cross section of the beam as a function of the normalized distance zp/b0 in the propagation direction (b0=k0s2).

Fig. 2
Fig. 2

Fx(zp) and Fy(zp) as functions of zp/b0. (a) Fx(zp) and (b) Fy(zp).

Equations (94)

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=(xˆpxˆp+yˆpyˆp)+zˆpzˆp.
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)-1/2ρ˜e
p×Ep=ik0Hp-Jmp,
p×Hp=-ik0Ep+Jep,
pHp=ρm,
p(Ep)=ρe,
p=xˆpxp+yˆpyp+zˆpzp
Jmp=zˆpJmzp,Jep=zˆpJezp.
Exp=-1Fzpyp,
Eyp=1Fzpxp,
Ezp=0,
Hxp=-1ik02Fzpxpzp,
Hyp=-1ik02Fzpypzp,
Hzp=-1ik02zp2+k02Fzp,
2xp2+2yp2+2zp2+k02Fzp=-Jmzp.
Hxp=Azpyp,
Hyp=-Azpxp,
Hzp=0,
Exp=-1ik02Azpxpzp,
Eyp=-1ik02Azpypzp,
Ezp=-1ik02zp2+k02Azp,
2xp2+2yp2+2zp2+k02Azp=-Jezp.
Jmzp(xp, yp, zp)=Smypδ(xp)δ(yp)δ(zp),
Jezp(xp, yp, zp)=Sexpδ(xp)δ(yp)δ(zp),
Fzp(xp, yp, zp)=Smypexp[ik0()1/2RF,ph]4πRF,am,
Azp(xp, yp, zp)=Se1/2xpexp[ik0()1/2RA,ph]4πRA,am,
RF,ph=RF,am=(xp2+yp2+zp2)1/2,
RA,ph=RA,am=xp2+yp2+zp21/2.
Fzp(xp, yp, zp)=F0(xp, yp, zp)exp[ik0()1/2zp],
Azp(xp, yp, zp)=A0(xp, yp, zp)exp[ik0()1/2zp],
F0(xp, yp, zp)=Sm4πEmyp,
A0(xp, yp, zp)=Se4π 1/2Eexp,
Em=exp[-ik0()1/2zp]exp[ik0()1/2RF,ph]RF,am,
Ee=exp[-ik0()1/2zp]exp[ik0()1/2RA,ph]RA,am.
fm(kxp, kyp, kzp)=kxp2+kyp2+kzp2-k02=0,
fe(kxp, kyp, kzp)=kxp2+kyp2+kzp2-k02=0,
kpfm(kxp, kyp, kzp)=2(xˆpkxp+yˆpkyp+zˆpkzp),
kpfe(kxp, kyp, kzp)=2xˆpkxp+yˆpkyp+zˆpkzp.
zpm=zp-ibm=-ibmαm,
zpe=zp-ibe=-ibeαe.
F0(xp, yp, zp)=Smi4πbmαmypexp-k0()1/2(xp2+yp2)2bmαm,
A0(xp, yp, zp)=Sei4πbeαexpexp-k0(xp2+yp2)2be()1/2αe.
[Eyp(xp, yp, 0)]m=Smi4πbm2xpypexp-k0()1/2(xp2+yp2)2bm,
[Eyp(xp, yp, 0)]e=-Sei4πbe()1/22xpypexp-k0(xp2+yp2)2be()1/2.
bm=be.
Sm=Se()1/2.
[Exp(xp, yp, 0)]m=Smik0()1/24πbm21-k0()1/2yp2bm×exp-k0()1/2(xp2+yp2)2bm,
[Exp(xp, yp, 0)]e=Seik04πbe21-k0xp2be()1/2×exp-k0(xp2+yp2)2be()1/2.
E(xp, yp, 0)=-xˆpE02-xp2+yp2s2×exp-xp2+yp22s2,
bm=b0()1/2=k0s2()1/2,
be=b0()-1/2=k0s2()-1/2.
[Exp(xp, yp, 0)]m=Smi4πk0()1/2s41-yp2s2×exp-xp2+yp22s2,
[Exp(xp, yp, 0)]e=Smi4πk0()1/2s41-xp2s2×exp-xp2+yp22s2.
Sm=i4πk0()1/2s4E0,
Se=i4πk0s4E0.
F0(xp, yp, zp)=E0ypαm2exp-(xp2+yp2)2s2αm,
A0(xp, yp, zp)=E0()1/2xpαe2exp-(xp2+yp2)2s2αe.
Expm=()-1/2Hypm=-E0αm21-yp2s2αm×exp-(xp2+yp2)2s2αm×exp[ik0()1/2zp],
Eypm=-()-1/2Hxpm=-E0xpyps2αm3×exp-(xp2+yp2)2s2αm×exp[ik0()1/2zp].
Expe=()-1/2Hype=-E0αe21-xp2s2αe×exp-(xp2+yp2)2s2αe×exp[ik0()1/2zp],
Eype=-()-1/2Hxpe=E0xpyps2αe3×exp-(xp2+yp2)2s2αe×exp[ik0()1/2zp].
Exp=Expm+Expe
S=zˆpSzp=zˆpc2Re(ExpHyp*-EypHxp*).
Szp=Szpm+Szpe+Szpc+Szpc*,
Szpm=c()1/22(ExpmExpm*+EypmEypm*)=S0|αm|41-2yp2s2|αm|2+yp4s4|αm|2+xp2yp2s4|αm|2×exp-(xp2+yp2)s2|αm|2,
Szpe=c()1/22 (ExpeExpe*+EypeEype*)=S0|αe|41-2xp2s2|αe|2+xp4s4|αe|2+xp2yp2s4|αe|2×exp-(xp2+yp2)s2|αe|2,
Szpc=c()1/22 (ExpmExpe*+EypmEype*)=S0αm2(αe*)21-xp2s2αe*-yp2s2αm×exp-(xp2+yp2)2s2βme.
S0=c2()1/2|E0|2,
1βme=1αm+1αe*.
P=-dxp-dypSzp(xp, yp, zp)=Pm+Pe+Pc+Pc*,
Pm=-dxp-dypSzpm(xp, yp, zp)=S0πs2,
Pe=-dxp-dypSzpe(xp, yp, zp)=S0πs2,
Pc=-dxp-dypSzpc(xp, yp, zp)=0.
P=c()1/2|E0|2πs2.
E0=[c()1/2πs2]-1/2,
(u)av,zp=-dxp-dypuSzp(xp, yp, zp)foru=xpandyp.
(xp)av,zp=(yp)av,zp=0.
(xpyp)av,zp=-dxp-dypxpypSzp(xp, yp, zp)=0.
(xp2)av,zp=-dxp-dypxp2Szp(xp, yp, zp).
(xp2)av,zp=[(xp2)av,zp]m+[(xp2)av,zp]e+[(xp2)av,zp]c+[(xp2)av,zp]c*,
[(xp2)av,zp]m=-dxp-dypxp2Szpm(xp, yp, zp),
[(xp2)av,zp]e=-dxp-dypxp2Szpe(xp, yp, zp),
[(xp2)av,zp]c=-dxp-dypxp2Szpc(xp, yp, zp).
[(xp2)av,zp]m=38s21+zp2bm2,
[(xp2)av,zp]e=58s21+9zp25be2.
[(xp2)av,zp]c+[(xp2)av,zp]c*=-12s2-41bm-1bes2zp2×4+zp21bm-1be2-3f(bm, be),
f(bm, be)=12be-zp221bm-1be25bm-3be-zp481bm-1be5.
(xp2)av,zp=12s2+38s2zp21bm2+3be2-41bm-1bes2zp2×4+zp21bm-1be2-3f(bm, be).
(yp2)av,zp=-dxp-dypyp2Szp(xp, yp, zp).
(σφ2)zp=(xp2)av,zp cos2φ0+(yp2)av,zp sin2φ0.
[(yp2)av,zp]m=58s21+9zp25bm2,
[(yp2)av,zp]e=38s21+zp2be2.
Fx(zp)=[(xp2)av,zps-2-1/2](zp/b0)-2.
Fy(zp)=[(yp2)av,zps-2-1/2](zp/b0)-2.

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