Abstract

We describe propagation in a uniaxially anisotropic medium by relying on a suitable plane-wave angular-spectrum representation of the electromagnetic field. We obtain paraxial expressions for both ordinary and extraordinary components that satisfy two decoupled parabolic equations. As an application, we obtain, for a particular input beam (a quasi-Gaussian beam), analytical results that allow us to identify some relevant features of propagation in uniaxial crystals.

© 2001 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
  2. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  3. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).
  4. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  5. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
    [CrossRef]
  6. A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
    [CrossRef]
  7. A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
    [CrossRef]
  8. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  9. See, for example, G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1984), Sec. 16.6.

2000 (2)

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
[CrossRef]

1976 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Arfken, G.

See, for example, G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1984), Sec. 16.6.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Ciattoni, A.

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
[CrossRef]

Di Porto, P.

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Sherman, G. C.

Stamnes, J. J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

Yariv, A.

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
[CrossRef]

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

J. Opt. Soc. Am. (2)

A. Ciattoni, P. Di Porto, B. Crosignani, A. Yariv, “Vectorial nonparaxial propagation equation in the presence of a tensorial refractive-index perturbation,” J. Opt. Soc. Am. B17, 809–819 (2000).
[CrossRef]

J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
[CrossRef]

Opt. Commun. (1)

A. Ciattoni, B. Crosignani, P. Di Porto, “Vectorial free-space optical propagation: a simple approach for generating all-order nonparaxial corrections,” Opt. Commun. 177, 9–13 (2000).
[CrossRef]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Other (5)

See, for example, G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1984), Sec. 16.6.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, 1985).

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

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Figures (2)

Fig. 1
Fig. 1

Plot of the intensity |Ax|2+|Ay|2 normalized to E02 for (a) z=0 μm, (b) z=1500 μm, (c) z=3000 μm. The wavelength is λ=0.6328 μm, s=5 μm, and the crystal is rutile, i.e., no=2.616 and ne=2.903.

Fig. 2
Fig. 2

Plot of the moduli of Ax and Ay normalized to |E0| for (a) z=0 μm, (b) z=1500 μm, (c) z=3000 μm.

Equations (34)

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2E-(  E)+k02  E=0,
=no2000no2000ne2,
Eo(±)=Uo(±)exp[ik  r±iz(k02no2-k2)1/2]Uo(±)exp(ik  r±ikozz),
Ee(±)=Ue(±)exp[ik  r±iz none (k02ne2-k2)1/2]Ue(±)exp(ik  r±ikezz),
Ue(±)=Ue(±)[kxe^x+kye^y-nonek2(ko2ne2-k2)1/2e^x]
E(r, z)=d2k[Uo(+)(k)exp(ik  r+ikozz)+Ue(+)(k)exp(ik  r+ikezz)]Eo(r, z)+Ee(r, z).
E˜(k)=1(2π)2d2rexp(-ik  r)E(r, 0),
Uo(+)(k)=-1k2 [kyE˜x(k)-kxE˜y(k)],
Ue(+)(k)=1k2 [kxE˜x(k)+kyE˜y(k)],
Eo(r, z)=d2kexp[ik  r+iz(k02no2-k2)1/2]Po  E(k),
Ezo(k, z)=0,
Ee(r, z)=d2kexp[ik  r+iz none (k02ne2-k2)1/2]Pe  E(k),
Eze(r, z)=-noned2kexp[ik  r+iz none (k02ne2-k2)1/2] k  E(k)(k02ne2-k2)1/2 ,
Po=1k2ky2-kxky-kxkykx2,Pe=1k2kx2kxkykxkyky2.
Eo(r, z)=exp(ik0noz)d2k×expik  r-ik22k0no zPo  E˜(k)exp(ik0noz)Ao(r, z),
Ezo(k, z)=0,
Ee(r, z)=exp(ik0noz)d2k×expik  r-inok22k0ne2 zPe  E(k)exp(ik0noz)Ae(r, z),
Eze(r, z)=exp(ik0noz)-nok0ne2d2k×expik  r-inok22k0ne2 zk  E˜(k)exp(ik0noz)Aze(r, z),
i z+12k0no 2Ao=0,
i z+no2k0ne2 2Ae=0,
Ao(r, 0)=12πd2rlog|r-r|×y2-xy-xyx2E(r, 0),
Ae(r, 0)=12πd2rlog|r-r|×x2xyxyx2E(r, 0),
Aze(r, z)=inok0ne2   Aor, no2ne2 z+Ae(r, z).
E(r, 0)=E0r2s2-2exp-r22s2e^x,
E˜(k)=-E0s42π k2exp-s2k22e^x.
Ao(r, z)=E0s4s2+izk0no3y2-s2+izk0noe^x+(-xy)e^yexp-r22s2+izk0no,
Ae(r, z)=E0s4s2+inozk0ne23x2-s2+inozk0ne2e^x+(xy)e^yexp-r22s2+inozk0ne2.
Aze(r, z)=E0s4inok0ne2xs2+inozk0ne244s2+inozk0ne2-r2exp-r22s2+inozk0ne2.
Ao(r, 0)=d2kexp(ik  r) 1k2ky2-kxky-kxkykx2×E˜x(k)E˜y(k).
2Ao(r, 0)=-d2kexp(ik  r)ky2-kxky-kxkykx2×E˜x(k)E˜y(k)=y2-xy-xyx2d2kexp(ik  r)×E˜x(k)E˜y(k)=y2-xy-xyx2E(r, 0).
Ao(r, 0)=12πd2rlog|r-r|×y2-xy-xyx2E(r, 0),
Aze(r, z)=-nok0ne2d2kexpik  r-inok22k0ne2 zk  E˜(k),
Aze(r, z)=inok0ne2   d2k×expik  r-inok22k0ne2 zE˜(k),
Aze(r, z)=inok0ne2 d2k×expik  r-ik22k0nono2ne2 zPo  E˜(k)+inok0ne2 d2k×expik  r-inok22k0ne2 zPe  E˜(k)=inok0ne2 Aor, no2ne2 z+Ae(r, z).

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