Abstract

We investigate the paraxial propagation along the optical axis of a uniaxially anisotropic crystal of a general paraxial beam whose boundary Cartesian components possess cylindrical symmetry. This property allows us to obtain expressions whose dependence on the azimuth angle ϕ (in cylindrical coordinates) is fully described and very simple. We also find that the beam loses its boundary cylindrical symmetry during propagation, as a consequence of medium anisotropy. Further, these expressions elucidate the way in which the anisotropy changes the state of polarization. As an example, we discuss the case of a Gaussian beam focused into the crystal by a thin spherical lens.

© 2002 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. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1999).
  3. T. Feng, Y.-Z. Wu, P.-D. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 25, 531–536 (1989).
    [CrossRef]
  4. S. Selleri, L. Vincetti, M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
    [CrossRef]
  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. J. J. Stamnes, G. C. Sherman, “Radiation of electromagnetic fields in biaxially anisotropic media,” J. Opt. Soc. Am. 68, 502–508 (1978).
    [CrossRef]
  7. J. A. Fleck, M. D. Feit, “Beam propagation in uniaxial anisotropic media,” J. Opt. Soc. Am. 73, 920–926 (1983).
    [CrossRef]
  8. 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]
  9. J. J. Stamnes, G. C. Sherman, “Reflection and refraction of an arbitrary wave at the plane interface separating two uniaxial crystals,” J. Opt. Soc. Am. 67, 683–695 (1977).
    [CrossRef]
  10. D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in unaixial crystals,” Opt. Commun. 163, 55–71 (1999).
    [CrossRef]
  11. J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
    [CrossRef]
  12. D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves in uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
    [CrossRef]
  13. G. Sithambaranathan, J. J. Stamnes, “Transmission of a Gaussian beam into a biaxial crystal,” J. Opt. Soc. Am. A 18, 1662–1669 (2001).
    [CrossRef]
  14. J. J. Stamnes, V. Dhayalan, “Transmission of a two-dimensional Gaussian beam into a uniaxial crystal,” J. Opt. Soc. Am. A 18, 1670–1677 (2001).
    [CrossRef]
  15. See, e.g., R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., rev. (McGraw-Hill, New York, 1986).
  16. See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1956), Chap. 3.
  17. See, e.g., A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 2, formula no. 3., p. 186 (Gordon & Breach, Amsterdam, 1986).
  18. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

2001 (3)

2000 (2)

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves in uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

S. Selleri, L. Vincetti, M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[CrossRef]

1999 (1)

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in unaixial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

1998 (1)

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

1989 (1)

T. Feng, Y.-Z. Wu, P.-D. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 25, 531–536 (1989).
[CrossRef]

1983 (1)

1978 (1)

1977 (1)

1976 (1)

Born, M.

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

Bracewell, R. N.

See, e.g., R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., rev. (McGraw-Hill, New York, 1986).

Brychkov, Yu. A.

See, e.g., A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 2, formula no. 3., p. 186 (Gordon & Breach, Amsterdam, 1986).

Ciattoni, A.

Crosignani, B.

Dhayalan, V.

Di Porto, P.

Feit, M. D.

Feng, T.

T. Feng, Y.-Z. Wu, P.-D. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 25, 531–536 (1989).
[CrossRef]

Fleck, J. A.

Goodman, J. W.

See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1956), Chap. 3.

Jiang, D.

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves in uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in unaixial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

Marichev, O. I.

See, e.g., A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 2, formula no. 3., p. 186 (Gordon & Breach, Amsterdam, 1986).

Prudnikov, A. P.

See, e.g., A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 2, formula no. 3., p. 186 (Gordon & Breach, Amsterdam, 1986).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Selleri, S.

S. Selleri, L. Vincetti, M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[CrossRef]

Sherman, G. C.

Sithambaranathan, G.

Stamnes, J. J.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Vincetti, L.

S. Selleri, L. Vincetti, M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[CrossRef]

Wolf, E.

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

Wu, Y.-Z.

T. Feng, Y.-Z. Wu, P.-D. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 25, 531–536 (1989).
[CrossRef]

Yariv, A.

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

Ye, P.-D.

T. Feng, Y.-Z. Wu, P.-D. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 25, 531–536 (1989).
[CrossRef]

Yeh, P.

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

Zoboli, M.

S. Selleri, L. Vincetti, M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[CrossRef]

IEEE J. Quantum Electron. (2)

T. Feng, Y.-Z. Wu, P.-D. Ye, “Improved coupled-mode theory for anisotropic waveguide modulators,” IEEE J. Quantum Electron. 25, 531–536 (1989).
[CrossRef]

S. Selleri, L. Vincetti, M. Zoboli, “Full-vector finite-element beams propagation method for anisotropic optical device analysis,” IEEE J. Quantum Electron. 36, 1392–1401 (2000).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Opt. Commun. (3)

D. Jiang, J. J. Stamnes, “Numerical and asymptotic results for focusing of two-dimensional waves in unaixial crystals,” Opt. Commun. 163, 55–71 (1999).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of electromagnetic waves into a uniaxial crystal,” Opt. Commun. 150, 251–262 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Numerical and experimental results for focusing of two-dimensional electromagnetic waves in uniaxial crystals,” Opt. Commun. 174, 321–334 (2000).
[CrossRef]

Other (6)

See, e.g., R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., rev. (McGraw-Hill, New York, 1986).

See, e.g., J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1956), Chap. 3.

See, e.g., A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, Vol. 2, formula no. 3., p. 186 (Gordon & Breach, Amsterdam, 1986).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

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

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

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Figures (3)

Fig. 1
Fig. 1

Plot of the vector field nˆ(ϕ, α) for α=0.

Fig. 2
Fig. 2

Plot of the vector field nˆ(ϕ, α) for α=π/3.

Fig. 3
Fig. 3

Plot of the ordinary and extraordinary beam widths in a LiNbO3 crystal. We assumed λ=0.63 μm, w=100 μm and f=30 cm.

Equations (36)

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ϵ=no2000no2000ne2,
E(r, z)=exp(ik0noz)[Ao(r, z)+Ae(r, z)],
Ao(r, z)= d2kk2 ky2-kxky-kxkykx2·E˜(k)×expik·r-ik22k0noz,
Ae(r, z)= d2kk2 kx2kxkykxkyky2·E˜(k)×expik·r-inok22k0ne2z,
E˜=1(2π)2 d2r exp(-ik·r)E(r, 0).
E(r, 0)=E(r, 0),
E˜(k)=12π 0drrJ0(kr)E(r, 0),
r=r(cos ϕeˆx+sin ϕeˆy),
k=k(cos θeˆx+sin θeˆy).
Ao(r, ϕ, z)=0dkk exp-iz2k0nok2×Mo(r, k, ϕ)·E˜(k),
Ae(r, ϕ, z)=0dkk exp-inoz2k0ne2k2×Me(r, k, ϕ)·E˜(k),
Mo(r, k, ϕ)=02πdθ exp[ikr cos(ϕ-θ)]×sin2 θ-cos θ sin θ-cos θ sin θcos2 θ,
Me(r, k, ϕ)=02πdθ exp[ikr cos(ϕ-θ)]×cos2 θcos θ sin θcos θ sin θsin2 θ.
Mo(r, k, ϕ)=πJ0(kr)I+πJ2(kr)R(ϕ),
Me(r, k, ϕ)=πJ0(kr)I-πJ2(kr)R(ϕ),
R(ϕ)=cos 2ϕsin 2ϕsin 2ϕ-cos 2ϕ.
Ao(r, ϕ, z)=Ao(0)(r, z)+R(ϕ)Ao(2)(r, z),
Ae(r, ϕ, z)=Ae(0)(r, z)-R(ϕ)Ae(2)(r, z),
Ao(n)(r, z)=π0dkk exp-iz2k0nok2Jn(kr)E˜(k),
Ae(n)(r, z)=π0dkk exp-inoz2k0ne2k2Jn(kr)E˜(k),n=0,2.
E(r, ϕ, z)=exp(ik0noz)[A(0)(r, z)+R(ϕ)A(2)(r, z)],
A(0)(r, z)=π0 dkkexp-iz2k0nok2+exp-inoz2k0ne2k2J0(kr)E˜(k),and
A(2)(r, z)=π0dkkexp-iz2k0nok2-exp-inoz2k0ne2k2J2(kr)E˜(k).
E(r, 0)=E(r, 0)tˆ(α),
E(r, ϕ, z)=exp(ik0noz)[A(0)(r, z)tˆ(α)+A(2)(r, z)R(ϕ)·tˆ(α)],
A(0)(r, z)=π0 dkkexp-iz2k0nok2+exp-inoz2k0ne2k2J0(kr)E˜(k),
A(2)(r, z)=π0 dkkexp-iz2k0nok2-exp-inoz2k0ne2k2J2(kr)E˜(k).
nˆ(ϕ, α)=R(ϕ)·tˆ(α)=cos(2ϕ-α)eˆx+sin(2ϕ-α)eˆy.
E(r, 0)=E¯ exp-1w2+ik02 fr2tˆ(α),
E˜(k)=E¯4π 1w2+ik02 f-1 exp-k24 1w2+ik02 f-1tˆ(α)E˜(k)tˆ(α).
A(0)(r, z)=E¯21w2+ik02 f×exp-r2Qo(z)Qo(z)+exp-r2Qe(z)Qe(z),
A(2)(r, z)=-E¯21w2+ik02 f×exp-r2Qo(z)Qo(z)-exp-r2Qe(z)Qe(z)+exp-r2Qo(z)-exp-r2Qe(z)r2,
Qo(z)=w21+k02w44f2+i2zk0no-k02 f w41+k02w44f2,
Qe(z)=w21+k02w44f2+i2nozk0ne2-k02 f w41+k02w44f2.
Γo(z)=|Qo(z)|[Re{Qo(z)}]1/2,
Γe(z)=|Qe(z)|[Re{Qe(z)}]1/2,

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