Abstract

We describe monochromatic light propagation in uniaxial crystals by means of an exact solution of Maxwell’s equations. We subsequently develop a paraxial scheme for describing a beam traveling orthogonal to the optical axis. We show that the Cartesian field components parallel and orthogonal to the optical axis are extraordinary and ordinary, respectively, and hence uncoupled. The ordinary component exhibits a standard Fresnel behavior, whereas the extraordinary one exhibits interesting anisotropic diffraction dynamics. We interpret the anisotropic diffraction as a composition of two spatial geometrical affinities and a single Fresnel propagation step. As an application, we obtain the analytical expression of the extraordinary Gaussian beam. We then derive the first nonparaxial correction to the paraxial beam, thus giving a scheme for describing slightly nonparaxial fields. We find that nonparaxiality couples the Cartesian components of the field and that the resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. Finally, we derive the analytical expression for the nonparaxial correction to the paraxial Gaussian beam.

© 2003 Optical Society of America

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References

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  1. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  2. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
  4. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic media,” J. Opt. Soc. Am. 66, 780–788 (1976).
    [CrossRef]
  5. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  6. A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
    [CrossRef]
  7. 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]
  8. M. Lax, W. H. Luoisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  9. A. Ciattoni, G. Cincotti, C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55–61 (2001).
    [CrossRef]
  10. A. Ciattoni, G. Cincotti, C. Palma, H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1894–1900 (2002).
    [CrossRef]
  11. A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
    [CrossRef]
  12. To be more precise, Eq. (23) describes an harmonic oscillator only if k02ne2-(ne2/no2)kx2-ky2>0 and the corresponding plane waves are homogeneous. For k02ne2-(ne2/no2)kx2-ky2<0, Eq. (A3) permits exponential but not sinusoidal solutions, and the corresponding plane waves are the well-known evanescent ones.
  13. Analogously to the case of Eq. (A3), Eq. (A5) is the case of a forced harmonic oscillator only if k02no2-k⊥2>0(homogeneous waves); for k02no2-k⊥2<0 its solutions exhibit exponential behavior (evanescent waves).

2002 (2)

A. Ciattoni, G. Cincotti, C. Palma, H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1894–1900 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

2001 (2)

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

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[CrossRef]

2000 (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]

1983 (1)

1976 (1)

1975 (1)

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

Born, M.

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

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

Ciattoni, A.

A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1894–1900 (2002).
[CrossRef]

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[CrossRef]

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

Cincotti, G.

A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1894–1900 (2002).
[CrossRef]

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

Crosignani, B.

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[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]

Di Porto, P.

A. Ciattoni, B. Crosignani, P. Di Porto, “Paraxial vector theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656–1661 (2001).
[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]

Feit, M. D.

Fleck, J. A.

Lax, M.

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

Luoisell, W. H.

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

McKnight, W. B.

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

Palma, C.

A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, C. Palma, H. Weber, “Energy exchange between the Cartesian components of a paraxial beam in a uniaxial crystal,” J. Opt. Soc. Am. A 19, 1894–1900 (2002).
[CrossRef]

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

Provenziani, D.

A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

Sherman, G. C.

Stamnes, J. J.

Weber, H.

Wolf, E.

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

Yariv, A.

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).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

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

Phys. Rev. A (1)

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

Phys. Rev. E (1)

A. Ciattoni, G. Cincotti, D. Provenziani, C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phys. Rev. E 66, 036614 (2002).
[CrossRef]

Other (5)

To be more precise, Eq. (23) describes an harmonic oscillator only if k02ne2-(ne2/no2)kx2-ky2>0 and the corresponding plane waves are homogeneous. For k02ne2-(ne2/no2)kx2-ky2<0, Eq. (A3) permits exponential but not sinusoidal solutions, and the corresponding plane waves are the well-known evanescent ones.

Analogously to the case of Eq. (A3), Eq. (A5) is the case of a forced harmonic oscillator only if k02no2-k⊥2>0(homogeneous waves); for k02no2-k⊥2<0 its solutions exhibit exponential behavior (evanescent waves).

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

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, New York, 1983).

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

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Figures (3)

Fig. 1
Fig. 1

The light beam propagates along the z axis, whereas the optical axis of the crystal coincides with the x axis.

Fig. 2
Fig. 2

Level plot of the normalized intensity |Ax|2/|E0|2 of an extraordinary Gaussian beam of waist s=8 µm (vacuum wavelength λ=2π/k0=0.5 µm) propagating in a crystal with no=2 and ne=2.5 on the planes (a) z=0, (b) z=L, (c) z=2L. Here L=1600 µm is the average diffraction length. Note that the boundary circularly symmetric field becomes more and more elliptical during propagation because of the anisotropic diffraction.

Fig. 3
Fig. 3

Level plot of the normalized moduli (a) |Ex|/|E0|, (b) |Ey|/|E0|, and (c) |Ez|/|E0| of a nonparaxial Gaussian beam with sx=0.12 µm, sy=0.2 µm (vacuum wavelength λ=2π/k0=0.5 µm) propagating in a crystal with no=2 and ne=2.5. The field is evaluated at z=1.28 µm (about four diffraction lengths). The paraxiality degree of the beam is f=2/[k0(sx+sy)]=0.47 (slightly nonparaxial beam).

Equations (44)

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2E-(·E)+k02E=0,
=ne2000no2000no2,
E(r, z)=d2k exp(ik·r)E˜(k, z),
E(r, z)=d2k exp(ik·r)exp(ikezz)×E˜x(k)-kxkyk02no2-kx2E˜x(k)-kezkxk02no2-kx2E˜x(k)+d2k exp(ik·r)exp(ikozz)×0kxkyk02no2-kx2E˜x(k)+E˜y(k)-kykozkxkyk02no2-kx2E˜x(k)+E˜y(k),
E˜(k)=1(2π)2d2r exp(-ik·r)E(r, 0)
koz(k)=(k02no2-k2)1/2,
kez(k)=[k02ne2-(ne2/no2)kx2-ky2]1/2.
Ex(r, z)=exp(ik0nez)d2k exp(ik·r)×exp-ine2kx2+no2ky22k0neno2zE˜x(k)exp(ik0nez)Ax(r, z),
Ey(r, z)=exp(ik0noz)d2k exp(ik·r)×exp-ikx2+ky22k0nozE˜y(k)exp(ik0noz)Ay(r, z),
iz+12k0no2x2+2y2Ax=0,
iz+12k0neno2ne22x2+no22y2Ay=0,
Ax(r, z)=k0no2πizd2rexp-k02izne[no2(x-x)2+ne2(y-y)2]Ex(r, 0),
Ay(r, z)=k0no2πizd2rexp-k0no2iz[(x-x)2+(y-y)2]Ey(r, 0),
Ax(r, z)=k0no2πizdx×exp-k02izne[no2(x-x)2]F(x)×k0no2πizdy×exp-k02izne[ne2(y-y)2]G(y),
ξ=nonex,ξ=nonex,
η=ney,η=ney,
Ax(ξ, η, z)=k02πizdξdη×exp-k02iz[(ξ-ξ)2+(η-η)2]×Ex(ξ, η, 0),
Ax(ξ, η, z)=Ax(ne/noξ, neη, z),
Ex(ξ, η, 0)=Ex(ne/noξ, neη, 0).
E(r, 0)=E0 exp-x22sx2-y22sy2eˆx.
Ax(r, z)=E0exp-x22sx21+inezk0no2sx2-y22sy21+izk0nesy21+inezk0no2sx21+izk0nesy21/2.
E(r, z)=exp(ik0nez)d2k exp(ik·r)×exp-ine2kx2+no2ky22k0neno2z×E˜x(k)-kxkyk02no2E˜x(k)-nekxk0no2E˜x(k)+exp(ik0noz)d2k exp(ik·r)×exp-ikx2+ky22k0noz
×0kxkyk02no2E˜x(k)+E˜y(k)-kyk0noE˜y(k).
E(r, z)=exp(ik0nez)Ax(r, z)eˆx+exp(ik0noz)Ay(r, z)eˆy+Enp(r, z),
Enp(r, z)=exp(ik0nez)01k02no22xyAx(r, z)inek0no2xAx(r, z)+exp(ik0noz)0-1k02no22xyFx(r, z)ik0noyAy(r, z)
Fx(r, z)=d2k exp(ik·r)exp-ikx2+ky22k0nozE˜x(k).
Enp(r, z)=exp(ik0nez)×0xyk0no2sx2sy2Ax(r, z)1+inezk0no2sx21+izk0nesy2-inexk0no2Ax(r, z)1+inezk0no2sx2-exp(ik0noz)×0xyk0no2sx2sy2Fx(r, z)1+izk0nosx21+izk0nosy20,
Fx(r, z)=E0exp-x22sx21+izk0nosx2-y22sy21+izk0nosy21+izk0nosx21+izk0nosy21/2.
2E˜xz2-ikxE˜zz+(k02ne2-ky2)E˜x+kxkyE˜y=0,
2E˜yz2-ikyE˜zz+(k02no2-kx2)E˜y+kxkyE˜x=0,
E˜z=i(k02no2-k2)kxE˜xz+kyE˜yz,
k02no2-ky2k02no2-k22E˜xz2+kxkyk02no2-k22E˜yz2+(k02ne2-ky2)E˜x+kxkyE˜y=0,
k02no2-kx2k02no2-k22E˜yz2+kxkyk02no2-k22E˜xz2+(k02no2-kx2)E˜y+kxkyE˜x=0.
2E˜xz2+kez2E˜x=0,
E˜x(k, z)=F˜x(k)exp(ikezz).
2E˜yz2+koz2E˜y=ne2-no2no2kxkyF˜x exp(ikezz),
E˜y(k, z)=F˜y(k)exp(ikozz)-kxkyk02no2-kx2F˜x(k)exp(ikezz).
E(r, z)=d2k exp(ik·r)exp(ikezz)F˜x(k)-exp(ikezz)kxkyk02no2-kx2F˜x(k)+exp(ikozz)F˜y(k)-exp(ikezz)kezkxk02no2-kx2F˜x(k)-exp(ikozz)kykozF˜y(k).
F˜x(k)=E˜x(k),
F˜y(k)=kxkyk02no2-kx2E˜x(k)+E˜y(k),
Ax(r, z)=d2rGr, r; neno2, 1neEx(r, 0),
Ay(r, z)=d2rGr, r; 1no, 1noEy(r, 0),
G(r, r; qx, qy)=1(2π)2d2k expik·(r-r)-iz2k0(qxkx2+qyky2).
G(r, r; qx, qy)=k02πizqxqyexpk02iz(x-x)2qx+(y-y)2qy,

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