Abstract

The Jones matrix calculus is applied to an electro-optic crystal with uniaxial symmetry when the light beam is incident nearly normally on the crystal face. The approach allows one to treat refracted waves and rays that diverge in the crystal and are modulated by an external low-frequency field. The effect of partial interference of overlapping refracted beams is allowed for and calculated for the case of uniform intensity of the beam over its cross section. The method is employed to analyze optical systems containing an imprecisely cut and aligned electro-optic crystal plate.

© 2004 Optical Society of America

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References

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    [CrossRef]
  6. M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
    [CrossRef]
  7. M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
    [CrossRef]
  8. J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially-stretched plastic films by generalized ellipsometry,” Thin Solid Films 313/314, 814–818 (1998).
    [CrossRef]
  9. W. Xu, L. T. Wood, T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
    [CrossRef]
  10. I. Ścierski, F. Ratajczyk, “The Jones matrix of the real dichroic elliptic object,” Optik (Stuttgart) 68, 121–125 (1984).
  11. T. A. Maldonado, T. K. Gaylord, “Electro-optic effect calculations: simplified procedure for arbitrary cases,” Appl. Opt. 27, 5051–5066 (1988).
    [CrossRef] [PubMed]
  12. M. Izdebski, W. Kucharczyk, R. E. Raab, “Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence,” J. Opt. Soc. Am. A 18, 1393–1398 (2001).
    [CrossRef]
  13. M. Izdebski, W. Kucharczyk, R. E. Raab, “Analysis of accuracy of measurement of quadratic electro-optic coefficients in uniaxial crystals: a case study of KDP,” J. Opt. Soc. Am. A 19, 1417–1421 (2002).
    [CrossRef]
  14. M. Izdebski, W. Kucharczyk, “Effect of divergence of light wave and alignment of crystal on the response of electro-optic modulators,” in International Conference on Solid State Crystals 2000: Growth, Characterization and Applications of Single Crystals, A. Rogalski, K. Adamiec, P. Madejczyk, eds., Proc. SPIE4412, 400–405 (2001).
    [CrossRef]
  15. 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]
  16. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
    [CrossRef]
  17. M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitia, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1668–1673 (2002).
    [CrossRef]
  18. M. Avendaño-Alejo, O. N. Stavroudis, “Huygens’s principle and rays in uniaxial anisotropic media. II. Crystal axis orientation arbitrary,” J. Opt. Soc. Am. A 19, 1674–1679 (2002).
    [CrossRef]
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
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    [CrossRef] [PubMed]
  21. M. C. Simon, R. M. Echarri, “Ray tracing formulas for monoaxial optical components: vectorial formulation,” Appl. Opt. 25, 1935–1939 (1986).
    [CrossRef] [PubMed]
  22. K.-H. Hellwege, A. M. Hellwege, eds., Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology—New Series (Springer, Berlin, 1984), Group III, Vols. 11 and 18.
  23. M. J. Gunning, R. Ledzion, P. Górski, W. Kucharczyk, “Studies of the quadratic electro-optic effect in KDP-type crystals,” in International Conference on Solid State Crystals ’98: Single Crystal Growth, Characterization, and Applications, A. Majchrowski, J. Zielinski, eds., Proc. SPIE3724, 249–255 (1999).
    [CrossRef]
  24. R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
    [CrossRef]

2002 (4)

2001 (2)

2000 (1)

W. Xu, L. T. Wood, T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
[CrossRef]

1999 (1)

R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
[CrossRef]

1998 (1)

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially-stretched plastic films by generalized ellipsometry,” Thin Solid Films 313/314, 814–818 (1998).
[CrossRef]

1996 (2)

1988 (1)

1986 (1)

1984 (1)

I. Ścierski, F. Ratajczyk, “The Jones matrix of the real dichroic elliptic object,” Optik (Stuttgart) 68, 121–125 (1984).

1983 (1)

1972 (2)

1956 (1)

1942 (1)

1941 (1)

Avendaño-Alejo, M.

Azzam, R. M. A.

Bashara, N. M.

Berreman, D. W.

Bondarczuk, K.

R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Boyain y Goitia, A. R.

Chen, C. G.

Ciattoni, A.

Crosignani, B.

Di Porto, P.

Echarri, R. M.

Elman, J. F.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially-stretched plastic films by generalized ellipsometry,” Thin Solid Films 313/314, 814–818 (1998).
[CrossRef]

Ferrera, J.

Gaylord, T. K.

Golding, T. D.

W. Xu, L. T. Wood, T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
[CrossRef]

Górski, P.

R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
[CrossRef]

M. J. Gunning, R. Ledzion, P. Górski, W. Kucharczyk, “Studies of the quadratic electro-optic effect in KDP-type crystals,” in International Conference on Solid State Crystals ’98: Single Crystal Growth, Characterization, and Applications, A. Majchrowski, J. Zielinski, eds., Proc. SPIE3724, 249–255 (1999).
[CrossRef]

Greener, J.

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially-stretched plastic films by generalized ellipsometry,” Thin Solid Films 313/314, 814–818 (1998).
[CrossRef]

Gunning, M. J.

M. J. Gunning, R. Ledzion, P. Górski, W. Kucharczyk, “Studies of the quadratic electro-optic effect in KDP-type crystals,” in International Conference on Solid State Crystals ’98: Single Crystal Growth, Characterization, and Applications, A. Majchrowski, J. Zielinski, eds., Proc. SPIE3724, 249–255 (1999).
[CrossRef]

Heilmann, R. K.

Herzinger, C. M.

Izdebski, M.

M. Izdebski, W. Kucharczyk, R. E. Raab, “Analysis of accuracy of measurement of quadratic electro-optic coefficients in uniaxial crystals: a case study of KDP,” J. Opt. Soc. Am. A 19, 1417–1421 (2002).
[CrossRef]

M. Izdebski, W. Kucharczyk, R. E. Raab, “Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence,” J. Opt. Soc. Am. A 18, 1393–1398 (2001).
[CrossRef]

M. Izdebski, W. Kucharczyk, “Effect of divergence of light wave and alignment of crystal on the response of electro-optic modulators,” in International Conference on Solid State Crystals 2000: Growth, Characterization and Applications of Single Crystals, A. Rogalski, K. Adamiec, P. Madejczyk, eds., Proc. SPIE4412, 400–405 (2001).
[CrossRef]

Johs, B.

Jones, R. C.

Konkola, P. T.

Kucharczyk, W.

M. Izdebski, W. Kucharczyk, R. E. Raab, “Analysis of accuracy of measurement of quadratic electro-optic coefficients in uniaxial crystals: a case study of KDP,” J. Opt. Soc. Am. A 19, 1417–1421 (2002).
[CrossRef]

M. Izdebski, W. Kucharczyk, R. E. Raab, “Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence,” J. Opt. Soc. Am. A 18, 1393–1398 (2001).
[CrossRef]

R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
[CrossRef]

M. J. Gunning, R. Ledzion, P. Górski, W. Kucharczyk, “Studies of the quadratic electro-optic effect in KDP-type crystals,” in International Conference on Solid State Crystals ’98: Single Crystal Growth, Characterization, and Applications, A. Majchrowski, J. Zielinski, eds., Proc. SPIE3724, 249–255 (1999).
[CrossRef]

M. Izdebski, W. Kucharczyk, “Effect of divergence of light wave and alignment of crystal on the response of electro-optic modulators,” in International Conference on Solid State Crystals 2000: Growth, Characterization and Applications of Single Crystals, A. Rogalski, K. Adamiec, P. Madejczyk, eds., Proc. SPIE4412, 400–405 (2001).
[CrossRef]

Ledzion, R.

R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
[CrossRef]

M. J. Gunning, R. Ledzion, P. Górski, W. Kucharczyk, “Studies of the quadratic electro-optic effect in KDP-type crystals,” in International Conference on Solid State Crystals ’98: Single Crystal Growth, Characterization, and Applications, A. Majchrowski, J. Zielinski, eds., Proc. SPIE3724, 249–255 (1999).
[CrossRef]

Maldonado, T. A.

Raab, R. E.

Ratajczyk, F.

I. Ścierski, F. Ratajczyk, “The Jones matrix of the real dichroic elliptic object,” Optik (Stuttgart) 68, 121–125 (1984).

Rheinländer, B.

Schattenburg, M. L.

Schubert, M.

Scierski, I.

I. Ścierski, F. Ratajczyk, “The Jones matrix of the real dichroic elliptic object,” Optik (Stuttgart) 68, 121–125 (1984).

Simon, M. C.

Stavroudis, O. N.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Wood, L. T.

W. Xu, L. T. Wood, T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
[CrossRef]

Woollam, J. A.

Xu, W.

W. Xu, L. T. Wood, T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
[CrossRef]

Appl. Opt. (3)

Cryst. Res. Technol. (1)

R. Ledzion, K. Bondarczuk, P. Górski, W. Kucharczyk, “Effect of deuteration on the quadratic electrooptic properties of KDP,” Cryst. Res. Technol. 34, 745–749 (1999).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (7)

M. Schubert, B. Rheinländer, J. A. Woollam, B. Johs, C. M. Herzinger, “Extension of rotating-analyzer ellipsometry to generalized ellipsometry: determination of the dielectric function tensor from uniaxial TiO2,” J. Opt. Soc. Am. A 13, 875–883 (1996).
[CrossRef]

M. Izdebski, W. Kucharczyk, R. E. Raab, “Effect of beam divergence from the optic axis in an electro-optic experiment to measure an induced Jones birefringence,” J. Opt. Soc. Am. A 18, 1393–1398 (2001).
[CrossRef]

M. Izdebski, W. Kucharczyk, R. E. Raab, “Analysis of accuracy of measurement of quadratic electro-optic coefficients in uniaxial crystals: a case study of KDP,” J. Opt. Soc. Am. A 19, 1417–1421 (2002).
[CrossRef]

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]

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
[CrossRef]

M. Avendaño-Alejo, O. N. Stavroudis, A. R. Boyain y Goitia, “Huygens’s principle and rays in uniaxial anisotropic media. I. Crystal axis normal to refracting surface,” J. Opt. Soc. Am. A 19, 1668–1673 (2002).
[CrossRef]

M. Avendaño-Alejo, O. N. Stavroudis, “Huygens’s principle and rays in uniaxial anisotropic media. II. Crystal axis orientation arbitrary,” J. Opt. Soc. Am. A 19, 1674–1679 (2002).
[CrossRef]

Optik (Stuttgart) (1)

I. Ścierski, F. Ratajczyk, “The Jones matrix of the real dichroic elliptic object,” Optik (Stuttgart) 68, 121–125 (1984).

Phys. Rev. B (2)

W. Xu, L. T. Wood, T. D. Golding, “Optical degeneracies in anisotropic layered media: treatment of singularities in a 4×4 matrix formalism,” Phys. Rev. B 61, 1740–1743 (2000).
[CrossRef]

M. Schubert, “Polarization-dependent optical parameters of arbitrarily anisotropic homogeneous layered systems,” Phys. Rev. B 53, 4265–4274 (1996).
[CrossRef]

Thin Solid Films (1)

J. F. Elman, J. Greener, C. M. Herzinger, B. Johs, “Characterization of biaxially-stretched plastic films by generalized ellipsometry,” Thin Solid Films 313/314, 814–818 (1998).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

M. Izdebski, W. Kucharczyk, “Effect of divergence of light wave and alignment of crystal on the response of electro-optic modulators,” in International Conference on Solid State Crystals 2000: Growth, Characterization and Applications of Single Crystals, A. Rogalski, K. Adamiec, P. Madejczyk, eds., Proc. SPIE4412, 400–405 (2001).
[CrossRef]

K.-H. Hellwege, A. M. Hellwege, eds., Landolt-Börnstein: Numerical Data and Functional Relationships in Science and Technology—New Series (Springer, Berlin, 1984), Group III, Vols. 11 and 18.

M. J. Gunning, R. Ledzion, P. Górski, W. Kucharczyk, “Studies of the quadratic electro-optic effect in KDP-type crystals,” in International Conference on Solid State Crystals ’98: Single Crystal Growth, Characterization, and Applications, A. Majchrowski, J. Zielinski, eds., Proc. SPIE3724, 249–255 (1999).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Relative error Δω in the measure of the modulation index caused by inaccuracies in alignment of a LiNbO3 crystal in the absence of inaccuracies in the crystal cutting. The intended parameters are E=(0, E, 0), σ=(0, 0, 1), amplitude of modulating field E0=103 V/m, crystal length l=1 cm, and wavelength λ=630 nm. The angles βa and γa are defined by Eq. (19). (a) Extended calculus, (b) original Jones calculus.

Fig. 2
Fig. 2

Relative error Δω in the measure of the modulation index caused by inaccuracies in cutting of a LiNbO3 crystal in the absence of inaccuracies in the crystal alignment. The intended parameters are E=(E, 0, 0), σ=(1, 0, 1)/2, amplitude of modulating field E0=103 V/m, crystal length l=1 cm, and wavelength λ=630 nm. The angles βc and γc are defined by Eq. (18). (a) Extended calculus, (b) original Jones calculus.

Fig. 3
Fig. 3

Relative error Δω in the measure of the modulation index caused by inaccuracies in alignment of a LiNbO3 crystal in the absence of inaccuracies in the crystal cutting. The intended parameters are E=(E, 0, 0), σ=(1, 0, 1)/2, amplitude of modulating field E0=103 V/m, crystal length l=1 cm, and wavelength λ=630 nm. The angles βa and γa are defined by Eq. (19). (a) Extended calculus, (b) original Jones calculus.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

E=JmJm-1J1E0.
J=Tf cos2 βf+Ts sin2 βf exp(-iΓ)(sin βf cos βf)[Tf-Ts exp(-iΓ)]exp(-iδ)(sin βf cos βf)[Tf-Ts exp(-iΓ)]exp(iδ)Tf sin2 βf+Ts cos2 βf exp(-iΓ),
Γ=2πlλ(ns-nf),
βfβf,lfls,
Tf0,Ts=0,
Ts0,Tf=0.
J=j11j12j21j22,
j11=Tf cos2 βf+Ts sin2 βf exp(-iΓ),
j12=[Tf sin βf cos βf-Ts sin βf cos βf exp(-iΓ)]exp(-iδ),
j21=[Tf sin βf cos βf-Ts sin βf cos βf exp(-iΓ)]exp(iδ),
j22=Tf sin2 βf+Ts cos2 βf exp(-iΓ),
Γ=2πλ(nsls-nf lf).
Srel=Sπr2.
Srel=1-uπr 1-u24r21/2-2π sin-1 u2rforu2r0foru>2r.
Φ=(πr2-S)If+(πr2-S)Is+SIm,
Φ=πr2[If+Is+(Im-If-Is)Srel],
I=If+Is+(Im-If-Is)Srel.
a=cos γσ0-sin γσ010sin γσ0cos γσ1000cos βσsin βσ0-sin βσcos βσ,
b=cos γc0-sin γc010sin γc0cos γc1000cos βcsin βc0-sin βccos βc.
c=cos γa0-sin γa010sin γa0cos γa1000cos βasin βa0-sin βacos βa.
d=cos γE0-sin γE010sin γE0cos γE1000cos βEsin βE0-sin βEcos βE.
EXEYEZ=aTbTadT00E,
Bij=1nij2+krijkEk,
pX2c2/v2-nX2+pY2c2/v2-nY2+pZ2c2/v2-nZ2=0,
ni=1/Biidiag,i=X,Y,Z,
sin ξ=sy/sin α,
cos ξ=sx/sin α,
sxsysz=cT001,
sin α=sx2+sy2.
pXpYpZ=eaTbTcos ξ-sin ξ0sin ξcos ξ0001pXpYpZ.
X=visot/sin α,YR,Z=0,
AvX+BvZ+Aviso/sin α=0,vYR.
nXu=nYu=12(nX+nY),nZu=nZ.
Δω=(Aω-Aωid)/Aωid,

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