Abstract

The propagation of electromagnetic beams through uniaxial anisotropic media is investigated. The Maxwell equations are solved in the paraxial limit in terms of the plane-wave spectrum associated with each Cartesian field component. Attention is focused on the global changes in the spatial structure of the beam, which are described by means of the second-order intensity moment formalism. In particular, the propagation law for the intensity moments through this kind of media is obtained. As a consequence it is inferred that it is possible to improve the beam-quality parameter by using these media.

© 2001 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
    [CrossRef]
  4. T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1990).
    [CrossRef]
  5. G. D. Landry, T. A. Maldonado, “Gaussian beam transmission and reflection from a general anisotropic multilayer structure,” Appl. Opt. 35, 5870–5879 (1996).
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    [CrossRef]
  8. P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [CrossRef]
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    [CrossRef]
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  12. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
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    [CrossRef]
  14. J. Serna, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  15. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [CrossRef]
  16. R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spa-tial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
    [CrossRef]
  17. J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
    [CrossRef]
  18. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1993).
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    [CrossRef]
  20. J. Serna, J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 504–507 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2001 (1)

J. Serna, J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 504–507 (2001).
[CrossRef]

2000 (1)

1998 (3)

S. R. Seshadri, “Electromagnetic Gaussian beam,” J. Opt. Soc. Am. A 15, 2712–2719 (1998).
[CrossRef]

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

1997 (1)

1996 (2)

1994 (1)

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

1991 (1)

1990 (1)

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1990).
[CrossRef]

1989 (2)

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1988 (1)

1983 (1)

1979 (1)

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

1975 (1)

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

1966 (1)

Bastiaans, M. J.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1993).

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

Chrostowski, J.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Hall, D. G.

Hauss, H. A.

H. A. Hauss, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).

Huang, W. P.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

Keren, E.

Kogelnik, H.

Kozaki, S.

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1990).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

Landry, G. D.

Lavi, S.

Lax, M.

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

Li, T.

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]

Maldonado, T. A.

Marti´nez-Herrero, R.

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]

Meji´as, P. M.

Movilla, J. M.

J. Serna, J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26, 504–507 (2001).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spa-tial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

Piquero, G.

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Prochaska, R.

Serna, J.

Seshadri, S. R.

Sheppard, C. J. R.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Sonoda, T.

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1990).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

Török, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Varga, P.

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1993).

Xu, C. L.

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

Appl. Opt. (3)

Electron. Commun. Jpn. (2)

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam of a uniaxially anisotropic medium,” Electron. Commun. Jpn. 72, 95–103 (1989).
[CrossRef]

T. Sonoda, S. Kozaki, “Reflection and transmission of Gaussian beam from an anisotropic dielectric slab,” Electron. Commun. Jpn. 73, 85–92 (1990).
[CrossRef]

J. Lightwave Technol. (1)

C. L. Xu, W. P. Huang, J. Chrostowski, S. K. Chaudhuri, “A full-vectorial beam propagation method for anisotropic waveguides,” J. Lightwave Technol. 12, 1926–1931 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[CrossRef]

J. M. Movilla, G. Piquero, R. Martı́nez-Herrero, P. M. Mejı́as, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun. 149, 230–234 (1998).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Optik (Stuttgart) (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

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 (4)

H. A. Hauss, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1993).

“Optics and optical instruments—lasers and laser related equipment—test methods for laser beam parameters:Beam widths, divergence angle and beam propagation factor,” ISO/DIS 11146 (International Organization for Standardization, Geneva, 1995).

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Equations (72)

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×H-1cDt=0,
×E+1cHt=0,
D=0,
H=0,
××E+1c22Dt2=0,
2E-(E)+ω2c2D=0,
D=E.
=e000o000o.
oE+oeo-1Esx=0,
E=(1-γ2) Esx,
γ=neno=eo1/2.
2Es+k2ne2Es=(1-γ2) 2Esx2,
2Ep+k2no2Ep=(1-γ2) 2Esxy,
2Ez+k2no2Ez=(1-γ2) 2Esxz,
h(r, η, z)=Γr+s2, r-s2, zexp(ikηs)ds,
Γ(r1, r2)=E(r1)E(r2)¯
xmynupvq=k24π21Ixmynupvqh(r, η, z)drdη,
I=k24π2h(r, η, z)drdη
P=x2xyxuxvxyy2yuyvxuyuu2uvxvyvuvv2,
Po=MPiMt,
Q3D=x2+y2u2+v2-xu+yv2,
M2=2kQ3D.
Es(x, y, z)=As(x, y, z)exp(iknez),
γ22Asx2+2Asy2+2Asz2+2ikneAsz=0.
x0=x/W,
y0=y/W,
z0=z/d,
W2=x2+y2z=0,
d=kW2.
γ22Asx02+2Asy02+δ22Asz02+2ineAsz0=0,
δWd=1kW.
δ2u2+v2.
As=As0+δ2As2+δ4As4+ ,
γ22As0x02+2As0y02+2ineAs0z0=0,
Es0(x, y, z)=As0(x, y, z)exp(iknez),
As0(x, y, z)=A˜s0(u, v, z)exp[ikne(xu+yv)]dudv,
A˜s0(u, v, z)=A˜s0(u, v)exp-iknez2(u2γ2+v2).
 
Ep,z(x, y, z)=Ap,z(x, y, z)exp(iknoz)+Bp,z(x, y, z)exp(iknez),
2Ap,z+2iknoAp,zz=0,
2Bp+2ikneBpz+k2ne21γ2-1Bp
=(1-γ2) 2Asxy,
2Bz+2ikneBzz+k2ne21γ2-1Bz
=(1-γ2) 2Asxz.
Bp0,z0=0,
02Ap0,z0+2inoAp0,z0z0=0,
Ep(x, y, z)=Ap0(x, y ,z)exp(iknoz),
Ap0(x, y, z)=A˜p0(u, v, z)exp[ikno(xu+yv)]dudv,
A˜p0(u, v, z)=A˜p0(u, v)exp-i knoz2 (u2+v2).
eEsx+oEpy+oEzz=0,
Es(x, y, z)=A˜s(u, v, z)exp[ikne(xu+yv)],
Ep(x, y, z)=A˜p(u, v, z)exp[ikno(xu+yv)],
Ez(x, y, z)=0,
A˜s(u, v, z)=A˜s(u, v)expiknez1-γ2u2+v22,
A˜p(u, v, z)=A˜p(u, v)expiknoz1-u2+v22.
Ep(x, y, z)=E˜p(u, v, z)exp[ikno(xu+yv)]dudv,
E˜p(u, v, z)=E˜p(u, v)expiknoz1-u2+v22.
Pz=z0=MoPz=0Mot,
Mo=Iz0noI0I,
Es(x, y, z)=E˜s(u, v, z)exp[ikne(xu+yv)]dudv,
E˜(u, v, z)=E˜s(u, v)expiknez1-γ2u2+v22.
Pz=z0=MePz=0Met,
Me=Iz0neT0I,
T=γ2001.
Q3D,z=z0=Q3D,z=0+2z0ne(γ2-1)(xuv2-yvu2)+z02ne2(γ2-1)2u2v2,
Sxuu2-yvv2=0
(zw)x=-xu/u2,
(zw)y=-yv/v2,
2|S|>|z0ne(γ2-1)|
sign Ssign[nez0(γ2-1)].
z=S/ne(1-γ2),
ΔQ3DQ3D,z=z0-Q3D,z=0=-u2v2S2.

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