Abstract

We describe a new method of modeling electro-optic (EO) devices, such as lanthanum-modified lead zirconate titanate polarization modulators, that resolves two deficiencies of current methods: (i) the inclusion of depolarization effects resulting from scattering and (ii) saturation of the EO response at strong electric-field strengths. Our approach to modeling depolarization is based on describing the transmitted optical field by superposition of a deterministic polarized wave and a scattered, randomly polarized, stochastic wave. Corresponding Jones matrices are used to derive a Mueller matrix to describe the wave propagation in scattering and depolarizing EO media accurately. A few simple optical measurements can be used to find the nonlinear behavior of the EO phase function, which is shown to describe accurately the material’s EO behavior for weak and strong applied electric fields.

© 1998 Optical Society of America

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References

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  1. G. Haertling, “Electro-optic ceramics and devices,” Electronic Ceramics: Properties, Devices, and Applications, L. Levinson, ed. (Marcel Dekker, New York, 1988), pp. 371–492.
  2. J. Thomas and Y. Fainman, “Programmable diffractive optical elements using a multichannel lanthanum-modified lead zirconate titanate phase modulator,” Opt. Lett. 20, 1510–1512 (1995).
    [CrossRef] [PubMed]
  3. Q. W. Song, X. M. Wang, and R. Bussjager, “Lanthanum-modified lead zirconate titanate ceramic wafer-based electro-optic dynamic diverging lens,” Opt. Lett. 21, 242–244 (1996).
    [CrossRef] [PubMed]
  4. T. Utsunomiya, “Optical switch using PLZT ceramics,” Ferroelectrics 109, 235–240 (1990).
    [CrossRef]
  5. J. T. Cutchen, J. O. Harris, and G. R. Laguna, “PLZT electrooptic shutters: applications,” Appl. Opt. 14, 1866–1873 (1975).
    [CrossRef] [PubMed]
  6. P. Shames, P. C. Sun, and Y. Fainman, “Modeling and optimization of electro-optic phase modulator,” in Physics and Simulation of Optoelectronic Devices IV, W. W. Chow and M. Osinski, eds., Proc. SPIE 2693, 787–796 (1996).
    [CrossRef]
  7. C. E. Land, “Variable birefringence, light scattering, and surface-deformation effects in PLZT ceramics,” Ferroelectrics 7, 45–51 (1974).
    [CrossRef]
  8. E. E. Klotin’sh, Yu. Ya. Kotleris, and Ya. A. Seglin’sh, “Geometrical optics of an electrically controlled phase plate made of PLZT-10 ferroelectric ceramic,” Avtometriya 6, 68–72 (1984).
  9. K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
    [CrossRef]
  10. M. Title and S. H. Lee, “Modeling and characterization of embedded electrode performance in transverse electro-optic modulators,” Appl. Opt. 29, 85–98 (1990).
    [CrossRef] [PubMed]
  11. Y. Wu, T. C. Chen, and H. Y. Chen, “Model of electro-optic effects by Green’s function and summary representation: applications to bulk and thin film PLZT displays and spatial light modulators,” in Proceedings of the Eighth IEEE International Symposium on Applications of Ferroelectrics, M. Liu, A. Safari, A. Kingon, and G. Haertling, eds. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 600–603.
    [CrossRef]
  12. A. E. Kapenieks and M. Ozolinsh, “Distortion of phase modulation by PLZT ceramics electrically controlled retardation plates,” Optik (Stuttgart) 63, 333–339 (1983).
  13. D. H. Goldstein, “PLZT modulator characterization,” Opt. Eng. 34, 1589–1592 (1995).
    [CrossRef]
  14. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.
  15. B. Mansoorian, “Design and characterization of flip-chip bonded Si/PLZT smart pixels,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1994), Chap. 2.
  16. E. T. Keve, “Structure–property relationships in PLZT ceramic materials,” Ferroelectrics 10, 169–174 (1976).
    [CrossRef]
  17. G. H. Haertling and C. E. Land, “Hot-pressed (Pb, La)(Zr, Ti)O3 ferroelectric ceramics for electrooptic applications,” J. Am. Ceram. Soc. 54, 1–10 (1971).
    [CrossRef]
  18. M. Ivey, “Birefringent light scattering in PLZT ceramics,” IEEE Trans. Ultrason. Ferroelec. Frequen. Contr. 38, 579–584 (1991).
    [CrossRef]
  19. R. A. Chipman, “The mechanics of polarization ray tracing,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 62–75 (1992).
    [CrossRef]
  20. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), pp. 554–555.
  21. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 2.
  22. A. E. Kapenieks, “Mueller matrix determination for the transparent ferroelectric PLZT ceramics in the transverse electric field,” Lat. PSR Zinat. Akad. Vestis Fiz. Teh. Zinat. Ser. 6, 61–65 (1980).
  23. F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Dover, New York, 1970), Chap. 3.
  24. K. Kim, L. Mandel, and E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  25. R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978).
    [CrossRef]
  26. J. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.
  27. S. M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 119–127 (1992).
    [CrossRef]
  28. R. A. Chipman, “Polarimetry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.
  29. C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
    [CrossRef]
  30. R. M. A. Azzam, “Ellipsometry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

1996 (2)

1995 (2)

1991 (1)

M. Ivey, “Birefringent light scattering in PLZT ceramics,” IEEE Trans. Ultrason. Ferroelec. Frequen. Contr. 38, 579–584 (1991).
[CrossRef]

1990 (2)

1987 (1)

1985 (1)

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

1984 (1)

E. E. Klotin’sh, Yu. Ya. Kotleris, and Ya. A. Seglin’sh, “Geometrical optics of an electrically controlled phase plate made of PLZT-10 ferroelectric ceramic,” Avtometriya 6, 68–72 (1984).

1983 (1)

A. E. Kapenieks and M. Ozolinsh, “Distortion of phase modulation by PLZT ceramics electrically controlled retardation plates,” Optik (Stuttgart) 63, 333–339 (1983).

1980 (1)

A. E. Kapenieks, “Mueller matrix determination for the transparent ferroelectric PLZT ceramics in the transverse electric field,” Lat. PSR Zinat. Akad. Vestis Fiz. Teh. Zinat. Ser. 6, 61–65 (1980).

1978 (1)

1976 (1)

E. T. Keve, “Structure–property relationships in PLZT ceramic materials,” Ferroelectrics 10, 169–174 (1976).
[CrossRef]

1975 (1)

1974 (1)

C. E. Land, “Variable birefringence, light scattering, and surface-deformation effects in PLZT ceramics,” Ferroelectrics 7, 45–51 (1974).
[CrossRef]

1971 (1)

G. H. Haertling and C. E. Land, “Hot-pressed (Pb, La)(Zr, Ti)O3 ferroelectric ceramics for electrooptic applications,” J. Am. Ceram. Soc. 54, 1–10 (1971).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, “Propagation of partially polarized light through anisotropic media with or without depolarization: a differential 4 × 4 matrix calculus,” J. Opt. Soc. Am. 68, 1756–1767 (1978).
[CrossRef]

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 2.

R. M. A. Azzam, “Ellipsometry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 2.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), pp. 554–555.

Brosseau, C.

C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

Bussjager, R.

Byron, F. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Dover, New York, 1970), Chap. 3.

Chen, H. Y.

Y. Wu, T. C. Chen, and H. Y. Chen, “Model of electro-optic effects by Green’s function and summary representation: applications to bulk and thin film PLZT displays and spatial light modulators,” in Proceedings of the Eighth IEEE International Symposium on Applications of Ferroelectrics, M. Liu, A. Safari, A. Kingon, and G. Haertling, eds. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 600–603.
[CrossRef]

Chen, T. C.

Y. Wu, T. C. Chen, and H. Y. Chen, “Model of electro-optic effects by Green’s function and summary representation: applications to bulk and thin film PLZT displays and spatial light modulators,” in Proceedings of the Eighth IEEE International Symposium on Applications of Ferroelectrics, M. Liu, A. Safari, A. Kingon, and G. Haertling, eds. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 600–603.
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.

R. A. Chipman, “The mechanics of polarization ray tracing,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 62–75 (1992).
[CrossRef]

Cutchen, J. T.

Fainman, Y.

J. Thomas and Y. Fainman, “Programmable diffractive optical elements using a multichannel lanthanum-modified lead zirconate titanate phase modulator,” Opt. Lett. 20, 1510–1512 (1995).
[CrossRef] [PubMed]

P. Shames, P. C. Sun, and Y. Fainman, “Modeling and optimization of electro-optic phase modulator,” in Physics and Simulation of Optoelectronic Devices IV, W. W. Chow and M. Osinski, eds., Proc. SPIE 2693, 787–796 (1996).
[CrossRef]

Fuller, R. W.

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Dover, New York, 1970), Chap. 3.

Goldstein, D. H.

D. H. Goldstein, “PLZT modulator characterization,” Opt. Eng. 34, 1589–1592 (1995).
[CrossRef]

Goodman, J.

J. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.

Haertling, G.

G. Haertling, “Electro-optic ceramics and devices,” Electronic Ceramics: Properties, Devices, and Applications, L. Levinson, ed. (Marcel Dekker, New York, 1988), pp. 371–492.

Haertling, G. H.

G. H. Haertling and C. E. Land, “Hot-pressed (Pb, La)(Zr, Ti)O3 ferroelectric ceramics for electrooptic applications,” J. Am. Ceram. Soc. 54, 1–10 (1971).
[CrossRef]

Harris, J. O.

Ivey, M.

M. Ivey, “Birefringent light scattering in PLZT ceramics,” IEEE Trans. Ultrason. Ferroelec. Frequen. Contr. 38, 579–584 (1991).
[CrossRef]

Kapenieks, A. E.

A. E. Kapenieks and M. Ozolinsh, “Distortion of phase modulation by PLZT ceramics electrically controlled retardation plates,” Optik (Stuttgart) 63, 333–339 (1983).

A. E. Kapenieks, “Mueller matrix determination for the transparent ferroelectric PLZT ceramics in the transverse electric field,” Lat. PSR Zinat. Akad. Vestis Fiz. Teh. Zinat. Ser. 6, 61–65 (1980).

Keve, E. T.

E. T. Keve, “Structure–property relationships in PLZT ceramic materials,” Ferroelectrics 10, 169–174 (1976).
[CrossRef]

Kim, K.

Klotin’sh, E. E.

E. E. Klotin’sh, Yu. Ya. Kotleris, and Ya. A. Seglin’sh, “Geometrical optics of an electrically controlled phase plate made of PLZT-10 ferroelectric ceramic,” Avtometriya 6, 68–72 (1984).

Kotleris, Yu. Ya.

E. E. Klotin’sh, Yu. Ya. Kotleris, and Ya. A. Seglin’sh, “Geometrical optics of an electrically controlled phase plate made of PLZT-10 ferroelectric ceramic,” Avtometriya 6, 68–72 (1984).

Laguna, G. R.

Land, C. E.

C. E. Land, “Variable birefringence, light scattering, and surface-deformation effects in PLZT ceramics,” Ferroelectrics 7, 45–51 (1974).
[CrossRef]

G. H. Haertling and C. E. Land, “Hot-pressed (Pb, La)(Zr, Ti)O3 ferroelectric ceramics for electrooptic applications,” J. Am. Ceram. Soc. 54, 1–10 (1971).
[CrossRef]

Lee, S. H.

Mandel, L.

Murata, M.

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

Nee, S. M. F.

S. M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 119–127 (1992).
[CrossRef]

Ozolinsh, M.

A. E. Kapenieks and M. Ozolinsh, “Distortion of phase modulation by PLZT ceramics electrically controlled retardation plates,” Optik (Stuttgart) 63, 333–339 (1983).

Seglin’sh, Ya. A.

E. E. Klotin’sh, Yu. Ya. Kotleris, and Ya. A. Seglin’sh, “Geometrical optics of an electrically controlled phase plate made of PLZT-10 ferroelectric ceramic,” Avtometriya 6, 68–72 (1984).

Seto, H.

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

Shames, P.

P. Shames, P. C. Sun, and Y. Fainman, “Modeling and optimization of electro-optic phase modulator,” in Physics and Simulation of Optoelectronic Devices IV, W. W. Chow and M. Osinski, eds., Proc. SPIE 2693, 787–796 (1996).
[CrossRef]

Song, Q. W.

Sun, P. C.

P. Shames, P. C. Sun, and Y. Fainman, “Modeling and optimization of electro-optic phase modulator,” in Physics and Simulation of Optoelectronic Devices IV, W. W. Chow and M. Osinski, eds., Proc. SPIE 2693, 787–796 (1996).
[CrossRef]

Tanaka, K.

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

Thomas, J.

Title, M.

Utsunomiya, T.

T. Utsunomiya, “Optical switch using PLZT ceramics,” Ferroelectrics 109, 235–240 (1990).
[CrossRef]

Wakino, K.

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

Wang, X. M.

Wolf, E.

Wu, Y.

Y. Wu, T. C. Chen, and H. Y. Chen, “Model of electro-optic effects by Green’s function and summary representation: applications to bulk and thin film PLZT displays and spatial light modulators,” in Proceedings of the Eighth IEEE International Symposium on Applications of Ferroelectrics, M. Liu, A. Safari, A. Kingon, and G. Haertling, eds. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 600–603.
[CrossRef]

Yamaguchi, M.

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.

Appl. Opt. (2)

Avtometriya (1)

E. E. Klotin’sh, Yu. Ya. Kotleris, and Ya. A. Seglin’sh, “Geometrical optics of an electrically controlled phase plate made of PLZT-10 ferroelectric ceramic,” Avtometriya 6, 68–72 (1984).

Ferroelectrics (3)

T. Utsunomiya, “Optical switch using PLZT ceramics,” Ferroelectrics 109, 235–240 (1990).
[CrossRef]

C. E. Land, “Variable birefringence, light scattering, and surface-deformation effects in PLZT ceramics,” Ferroelectrics 7, 45–51 (1974).
[CrossRef]

E. T. Keve, “Structure–property relationships in PLZT ceramic materials,” Ferroelectrics 10, 169–174 (1976).
[CrossRef]

IEEE Trans. Ultrason. Ferroelec. Frequen. Contr. (1)

M. Ivey, “Birefringent light scattering in PLZT ceramics,” IEEE Trans. Ultrason. Ferroelec. Frequen. Contr. 38, 579–584 (1991).
[CrossRef]

J. Am. Ceram. Soc. (1)

G. H. Haertling and C. E. Land, “Hot-pressed (Pb, La)(Zr, Ti)O3 ferroelectric ceramics for electrooptic applications,” J. Am. Ceram. Soc. 54, 1–10 (1971).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jap. J. Appl. Phys. (1)

K. Tanaka, M. Yamaguchi, H. Seto, M. Murata, and K. Wakino, “Analyses of PLZT electrooptic shutter and shutter array,” Jap. J. Appl. Phys. 24, 177–182 (1985).
[CrossRef]

Lat. PSR Zinat. Akad. Vestis Fiz. Teh. Zinat. Ser. (1)

A. E. Kapenieks, “Mueller matrix determination for the transparent ferroelectric PLZT ceramics in the transverse electric field,” Lat. PSR Zinat. Akad. Vestis Fiz. Teh. Zinat. Ser. 6, 61–65 (1980).

Opt. Commun. (1)

C. Brosseau, “Mueller matrix analysis of light depolarization by a linear optical medium,” Opt. Commun. 131, 229–235 (1996).
[CrossRef]

Opt. Eng. (1)

D. H. Goldstein, “PLZT modulator characterization,” Opt. Eng. 34, 1589–1592 (1995).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

A. E. Kapenieks and M. Ozolinsh, “Distortion of phase modulation by PLZT ceramics electrically controlled retardation plates,” Optik (Stuttgart) 63, 333–339 (1983).

Other (13)

R. M. A. Azzam, “Ellipsometry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 27.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5.

B. Mansoorian, “Design and characterization of flip-chip bonded Si/PLZT smart pixels,” Ph.D. dissertation (University of California, San Diego, La Jolla, Calif., 1994), Chap. 2.

Y. Wu, T. C. Chen, and H. Y. Chen, “Model of electro-optic effects by Green’s function and summary representation: applications to bulk and thin film PLZT displays and spatial light modulators,” in Proceedings of the Eighth IEEE International Symposium on Applications of Ferroelectrics, M. Liu, A. Safari, A. Kingon, and G. Haertling, eds. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 600–603.
[CrossRef]

G. Haertling, “Electro-optic ceramics and devices,” Electronic Ceramics: Properties, Devices, and Applications, L. Levinson, ed. (Marcel Dekker, New York, 1988), pp. 371–492.

P. Shames, P. C. Sun, and Y. Fainman, “Modeling and optimization of electro-optic phase modulator,” in Physics and Simulation of Optoelectronic Devices IV, W. W. Chow and M. Osinski, eds., Proc. SPIE 2693, 787–796 (1996).
[CrossRef]

F. W. Byron and R. W. Fuller, Mathematics of Classical and Quantum Physics (Dover, New York, 1970), Chap. 3.

J. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 8.

S. M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 119–127 (1992).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook in Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 2, Chap. 22.

R. A. Chipman, “The mechanics of polarization ray tracing,” in Polarization Analysis and Measurement, D. H. Goldstein and R. A. Chipman, eds., Proc. SPIE 1746, 62–75 (1992).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), pp. 554–555.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 2.

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Figures (5)

Fig. 1
Fig. 1

(a) PLZT sample with transverse electrodes oriented to generate an electric field along the y axis. A crossed polarizer–analyzer pair is oriented at 45° and -45°, respectively, to the y axis. Linear polarized light is rotated 90° if the induced birefringence results in a phase difference of ϕ = π. (b) Experimental measurement results (filled circles) of the transmission intensity of light passing through the system as a function of the electric-field strength (for a PLZT sample of L = 388 μm). The solid curve shows the simulated performance by use of Eq. (1) and a best-fit quadratic EO model. To account for the weak EO response at low electric fields, a constant field of E 0 = 2 × 105 V/m, opposite to the applied field direction, is included in the model.

Fig. 2
Fig. 2

Experimental setup for the characterization and modeling of PLZT samples. The PLZT sample is fully illuminated to avoid localized photorefractive effects. The low-pass filter was only used for experimental measurements for which the scattered fields needed to be filtered out.

Fig. 3
Fig. 3

Using a low-pass filter (see Fig. 2), we remove the scattered light from the detected signal. The remaining dc component contains no optical bias, allowing us to conclude that the dc light is polarized. In this measurement we used parallel polarizers, which resulted in a maximum intensity at zero voltage. The PLZT sample used was 1 mm thick, which accounts for the increase in phase variation compared with Fig. 1(b).

Fig. 4
Fig. 4

Seven optical intensity measurements normalized to illustrate the envelope functionality of the vertical and horizontal components to the I 45°,45° data.

Fig. 5
Fig. 5

Change in the relative phase calculated from measured data for PLZT 8.9/65/35 (circles). For comparison we also show the calculated change in phase obtained by use of a shifted (E 0 = 1.5 × 105 V/m) electric field in the standard quadratic model (solid curve) with the quadratic EO coefficient R = 17.0 × 10-16 (m/V)2. The change in the index of refraction is calculated by use of the linear relation Δϕ/L = 2πΔn/λ.

Equations (43)

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

I = 1 2 sin 2 ϕ 2 ,
E x t = E 1 t exp i α 1 t - 2 π ν ¯ t , E y t = E 2 t exp i α 2 t - 2 π ν ¯ t ,
S S 0 S 1 S 2 S 3 E 1 2 + E 2 2 E 1 2 - E 2 2 2 E 1 E 2   cos   δ t 2 E 1 E 2   sin   δ t ,
M system = M N M N - 1     M n     M 1 ,
S out = M system S in .
M = T J J * T - 1 ,
J ϕ = exp i ϕ 2 0 0 exp - i   ϕ 2 ,
M PLZT ideal ϕ = 1 0 0 0 0 1 0 0 0 0 cos ϕ - sin ϕ 0 0 sin ϕ cos ϕ .
J e = a   exp i   ϕ 2 0 0 b   exp - i   ϕ 2 + u 1 u 4 u 3 u 2 e = a   exp i   ϕ 2 + u 1 e u 3 e u 4 e b   exp - i   ϕ 2 + u 2 e ,
u i u k * e = 0 if   i k U i if   i = k ,
u i exp ± j   ϕ 2 e = 0 ,
M = T J e J e * e T - 1 .
M PLZT = 1 2 A + B + U 1 + U 2 + U 3 + U 4 1 2 A - B + U 1 - U 2 - U 3 + U 4 0 0 1 2 A - B + U 1 - U 2 + U 3 - U 4 1 2 A + B + U 1 + U 2 - U 3 - U 4 0 0 0 0 AB cos ϕ - AB sin ϕ 0 0 AB sin ϕ AB cos ϕ ,
A E = A ,     B E = B , U i E = U i ,     ϕ E = ϕ ,
S in = 1 0 1 0 .
M Pol θ = 1 2 1 cos 2 θ sin 2 θ 0 cos 2 θ cos 2 2 θ cos 2 θ sin 2 θ 0 sin 2 θ cos 2 θ sin 2 θ sin 2 2 θ 0 0 0 0 0 ,
S out = M Pol θ a M PLZT M Pol θ p S in ,
I 45 ° , 45 ° = 1 4 A + B + U 1 + U 2 + U 3 + U 4 + 2 AB cos ϕ ,
I 0 ° , 0 ° = 1 2 A + U 1 ,
I 90 ° , 90 ° = 1 2 B + U 2 ,
I 0 ° , 90 ° = 1 2   U 3 ,
I 90 ° , 0 ° = 1 2   U 4 ,
I 0 ° , 0 ° - Filter = 1 2   A ,
I 90 ° , 90 ° - Filter = 1 2   B .
U = U i ,     i = 1 ,   2 ,   3 ,   4 .
M PLZT = A + 2 U 0 0 0 0 A 0 0 0 0 A   cos ϕ - A   sin ϕ 0 0 A   sin ϕ A   cos ϕ .
I 45 ° , 45 ° = 1 2 A + 2 U + A   cos ϕ ,
I 0 ° , 0 ° = I 90 ° , 90 ° = 1 2 A + U ,
I 0 ° , 90 ° = I 90 ° , 0 ° = 1 2   U .
M PLZT = A 0 0 0 0 A 0 0 0 0 A   cos ϕ - A   sin ϕ 0 0 A   sin ϕ A   cos ϕ + 2 U 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .
I 0 ° , 0 ° I 90 ° , 90 ° ,     I 0 ° , 90 ° I 90 ° , 0 ° .
I 45 ° , 45 ° max = A E + U E = 2 I 0 ° , 0 ° = 2 I 90 ° , 90 ° .
I 45 ° , 45 ° min = U E = 2 I 0 ° , 90 ° = 2 I 90 ° , 0 ° ,
ϕ = cos - 1 I 45 ° , 45 ° - I 0 ° , 0 ° - I 0 ° , 90 ° I 0 ° , 0 ° - I 0 ° , 90 ° .
E x = J 11 E x + J 12 E y ,
E y = J 21 E x + J 22 E y ,
E x E y = J 11 J 12 J 21 J 22 E x E y ,
J = J 11 J 12 J 21 J 22 ,
J J = J 11 J 11 J 11 J 12 J 12 J 11 J 12 J 12 J 11 J 21 J 11 J 22 J 12 J 21 J 12 J 22 J 21 J 11 J 21 J 12 J 22 J 11 J 22 J 12 J 21 J 21 J 21 J 22 J 22 J 21 J 22 J 22 .
C = E x E x * E x E y * E y E x * E y E y * ,
S 0 = C xx + C yy , S 1 = C xx - C yy , S 2 = C xy + C yx , S 3 = j C xy - C yx .
C = C xx C xy C yx C yy ;
T = 1 0 0 1 1 0 0 - 1 0 1 1 0 0 j - j 0 .

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