Abstract

Maker fringe analysis was adapted to x-cut LiNbO3 wafers to examine variations in birefringence, thickness, and photoelastic strain. The pump beam was polarized parallel to the crystalline y axis and produced e- and o-polarized Maker fringes, owing to d31 and d22, respectively, by rotation of the sample about the y axis. Fitting our model to the o-polarized data enabled computation of the sample thickness to an uncertainty of approximately ±0.01 μm. The accuracy was limited by an implicit ±2×10-4 uncertainty in no that exists in the commonly used Sellmeier equation of G. J. Edwards and M. Lawrence, Opt. Quantum Electron . 16, 373 (1984). For a pump wavelength λp=1064 nm, fitting the model to the e-polarized fringes revealed that ne at 532 nm deviated from the Sellmeier result by typically -1.58×10-4. The uniformity of ne over a wafer 10 cm in diameter was approximately ±4×10-5. This result is consistent with that expected from compositional variations. Our model included multiple passes of the pump and second-harmonic waves. The effects of photoelastic strain in producing perturbations and mixing of the e- and o-polarized fringes was investigated. This was restricted to two experimentally motivated cases that suggested that strains produce rotations of the optic axis by typically ±0.05° about the x axis and y axis with the former assigned to an indeterminant combination of S1, S2, and S4 and the latter to an indeterminant combination of S5 and S6. In both cases the magnitude of the collective strains is of the order of 10-4. The birefringence variations that are due to strain are of the same magnitude as those expected from compositional variations. The formalism developed here is used in the subsequent mapping study of x-cut wafers.

© 1998 Optical Society of America

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References

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  1. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
    [CrossRef]
  2. J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
    [CrossRef]
  3. N. A. Sanford and J. A. Aust, “Maker fringe mapping of LiNbO3 wafers,” in Lasers and Optics for Manufacturing, Vol. 9 of 1997 OSA Trends in Optics and Photonics Series, Andrew C. Tam, ed. (Optical Society of America, Washington, D.C., 1997), pp. 23–32.
  4. P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
    [CrossRef]
  5. D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical properties of lithium-rich lithium niobate fabricated by vapor transport equilibration,” IEEE J. Quantum Electron. 26, 135 (1990).
    [CrossRef]
  6. S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
    [CrossRef]
  7. J. A. Aust, B. Steiner, N. A. Sanford, G. Fogarty, B. Yang, A. Roshko, J. Amin, and C. Evans, “examination of domain-reversed layers in z-cut LiNbO3 using Maker fringe analysis, atomic force microscopy, and high-resolution x-ray diffraction imaging,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 485.
  8. J. A. Aust, N. A. Sanford, and J. Amin, “Maker fringe analysis of z-cut lithium niobate,” in Proceedings of the 10th Annual Meeting of IEEE Lasers and Electro-Optics Society (IEEE Lasers and Electro-Optics Society, Piscataway, N.J., 1997), pp. 114–115.
  9. P. Bordui, Crystal Telchnology, Inc., Palo Alto, Calif. (personal communication, 1997).
  10. N. A. Sanford and J. A. Aust, Properties of Lithium Niobate and other Novel Ferroelectric Materials, K. K. Wong, ed., EMIS Data Review Series (Institution of Electrical Engineers, London, UK, to be published).
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  12. V. G. Dmitriev, G. G. Gurzadyan, and D. H. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).
  13. N. Bloembergen and J. Ducuing, “Experimental verification of the optical laws on non-linear reflection,” Phys. Lett. 6, 5 (1963).
    [CrossRef]
  14. G. J. Edwards and M. Lawerence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373 (1984).
    [CrossRef]
  15. D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
    [CrossRef]
  16. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
    [CrossRef]
  17. M. L. Bortz and M. M. Fejer, “Measurement of second-order nonlinear susceptibility of proton-exchanged LiNbO3,” Opt. Lett. 17, 704 (1992).
    [CrossRef] [PubMed]
  18. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, J. Opt. Soc. Am. B 14, 2268 (1997).
    [CrossRef]
  19. J. F. Lam and H. W. Yen, “Dynamics of optical TE to TM mode conversion in LiNbO3 channel waveguides,” Appl. Phys. Lett. 45, 1173 (1984).
    [CrossRef]
  20. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367 (1991).
    [CrossRef]
  21. Product data provided by Crystal Technology Inc, Palo Alto, Calif.
  22. R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A: Solids Surf. 37, 191 (1984). See also, J. F. Nye, Physical Properties of Crystals (Oxford U. Press, New York, 1995).
    [CrossRef]
  23. R. C. Miller and W. A. Nordland, J. Appl. Phys. 42, 4145 (1971). There is serious disagreement in the literature regarding the magnitudes of the nonlinear optical coefficients for LiNbO3. For example, our estimate of d22 is inferred from Miller’s Table I for congruent LiNbO3. Miller’s data is normalized to d36 for potassium dihydrogen phosphate. The magnitudes of d36 for potassium dihydrogen phosphate and d31 for LiNbO3 were taken from the recent work of Shoji (Ref. 18 above).
    [CrossRef]
  24. L. P. Avakyants, D. F. Kiselev, and N. N. Shchitov, “Photoelasticity of LiNbO3,” Sov. Phys. Solid State 18, 899 (1976).
  25. D. F. Nelson and M. Lax, “Theory of the photoelastic interaction,” Phys. Rev. B 3, 2778 (1971).
    [CrossRef]
  26. F. R. N. Nabarro, ed., Dislocations in Solids (Elsevier, Amsterdam, 1979).

1997 (1)

1992 (1)

1991 (2)

D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367 (1991).
[CrossRef]

P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
[CrossRef]

1990 (1)

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical properties of lithium-rich lithium niobate fabricated by vapor transport equilibration,” IEEE J. Quantum Electron. 26, 135 (1990).
[CrossRef]

1985 (1)

S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
[CrossRef]

1984 (2)

G. J. Edwards and M. Lawerence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

J. F. Lam and H. W. Yen, “Dynamics of optical TE to TM mode conversion in LiNbO3 channel waveguides,” Appl. Phys. Lett. 45, 1173 (1984).
[CrossRef]

1976 (1)

L. P. Avakyants, D. F. Kiselev, and N. N. Shchitov, “Photoelasticity of LiNbO3,” Sov. Phys. Solid State 18, 899 (1976).

1974 (1)

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

1971 (2)

D. F. Nelson and M. Lax, “Theory of the photoelastic interaction,” Phys. Rev. B 3, 2778 (1971).
[CrossRef]

R. C. Miller and W. A. Nordland, J. Appl. Phys. 42, 4145 (1971). There is serious disagreement in the literature regarding the magnitudes of the nonlinear optical coefficients for LiNbO3. For example, our estimate of d22 is inferred from Miller’s Table I for congruent LiNbO3. Miller’s data is normalized to d36 for potassium dihydrogen phosphate. The magnitudes of d36 for potassium dihydrogen phosphate and d31 for LiNbO3 were taken from the recent work of Shoji (Ref. 18 above).
[CrossRef]

1970 (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

1968 (1)

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

1963 (1)

N. Bloembergen and J. Ducuing, “Experimental verification of the optical laws on non-linear reflection,” Phys. Lett. 6, 5 (1963).
[CrossRef]

1962 (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[CrossRef]

Ashkin, A.

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

Avakyants, L. P.

L. P. Avakyants, D. F. Kiselev, and N. N. Shchitov, “Photoelasticity of LiNbO3,” Sov. Phys. Solid State 18, 899 (1976).

Ballman, A. A.

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

Bergman, J. G.

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

Bethune, D. S.

Bird, C. D.

P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
[CrossRef]

Bloembergen, N.

N. Bloembergen and J. Ducuing, “Experimental verification of the optical laws on non-linear reflection,” Phys. Lett. 6, 5 (1963).
[CrossRef]

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[CrossRef]

Bordui, P. F.

P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
[CrossRef]

Bortz, M. L.

Byer, R. L.

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical properties of lithium-rich lithium niobate fabricated by vapor transport equilibration,” IEEE J. Quantum Electron. 26, 135 (1990).
[CrossRef]

Calvert, G. D.

P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
[CrossRef]

Ducuing, J.

N. Bloembergen and J. Ducuing, “Experimental verification of the optical laws on non-linear reflection,” Phys. Lett. 6, 5 (1963).
[CrossRef]

Dziedzic, J. M.

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

Edwards, G. J.

G. J. Edwards and M. Lawerence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

Fejer, M. M.

M. L. Bortz and M. M. Fejer, “Measurement of second-order nonlinear susceptibility of proton-exchanged LiNbO3,” Opt. Lett. 17, 704 (1992).
[CrossRef] [PubMed]

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical properties of lithium-rich lithium niobate fabricated by vapor transport equilibration,” IEEE J. Quantum Electron. 26, 135 (1990).
[CrossRef]

Holmes, R. J.

S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
[CrossRef]

Ito, R.

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

Jundt, D. H.

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical properties of lithium-rich lithium niobate fabricated by vapor transport equilibration,” IEEE J. Quantum Electron. 26, 135 (1990).
[CrossRef]

Kim, Y. S.

S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
[CrossRef]

Kiselev, D. F.

L. P. Avakyants, D. F. Kiselev, and N. N. Shchitov, “Photoelasticity of LiNbO3,” Sov. Phys. Solid State 18, 899 (1976).

Kitamoto, A.

Kondo, T.

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

Lam, J. F.

J. F. Lam and H. W. Yen, “Dynamics of optical TE to TM mode conversion in LiNbO3 channel waveguides,” Appl. Phys. Lett. 45, 1173 (1984).
[CrossRef]

Lawerence, M.

G. J. Edwards and M. Lawerence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

Lax, M.

D. F. Nelson and M. Lax, “Theory of the photoelastic interaction,” Phys. Rev. B 3, 2778 (1971).
[CrossRef]

Levinstein, H. J.

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

Lunt, S. R.

S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
[CrossRef]

Mikulyak, R. M.

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

Miller, R. C.

R. C. Miller and W. A. Nordland, J. Appl. Phys. 42, 4145 (1971). There is serious disagreement in the literature regarding the magnitudes of the nonlinear optical coefficients for LiNbO3. For example, our estimate of d22 is inferred from Miller’s Table I for congruent LiNbO3. Miller’s data is normalized to d36 for potassium dihydrogen phosphate. The magnitudes of d36 for potassium dihydrogen phosphate and d31 for LiNbO3 were taken from the recent work of Shoji (Ref. 18 above).
[CrossRef]

Nelson, D. F.

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

D. F. Nelson and M. Lax, “Theory of the photoelastic interaction,” Phys. Rev. B 3, 2778 (1971).
[CrossRef]

Nordland, W. A.

R. C. Miller and W. A. Nordland, J. Appl. Phys. 42, 4145 (1971). There is serious disagreement in the literature regarding the magnitudes of the nonlinear optical coefficients for LiNbO3. For example, our estimate of d22 is inferred from Miller’s Table I for congruent LiNbO3. Miller’s data is normalized to d36 for potassium dihydrogen phosphate. The magnitudes of d36 for potassium dihydrogen phosphate and d31 for LiNbO3 were taken from the recent work of Shoji (Ref. 18 above).
[CrossRef]

Norwood, R. G.

P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[CrossRef]

Peterson, G. E.

S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
[CrossRef]

Shchitov, N. N.

L. P. Avakyants, D. F. Kiselev, and N. N. Shchitov, “Photoelasticity of LiNbO3,” Sov. Phys. Solid State 18, 899 (1976).

Shirane, M.

Shoji, I.

Smith, R. G.

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

Yen, H. W.

J. F. Lam and H. W. Yen, “Dynamics of optical TE to TM mode conversion in LiNbO3 channel waveguides,” Appl. Phys. Lett. 45, 1173 (1984).
[CrossRef]

Appl. Phys. Lett. (2)

J. G. Bergman, A. Ashkin, A. A. Ballman, J. M. Dziedzic, H. J. Levinstein, and R. G. Smith, “Curie temperature, birefringence, and phase-matching temperature variations in LiNbO3 as a function of melt stoichiometry,” Appl. Phys. Lett. 12, 92 (1968).
[CrossRef]

J. F. Lam and H. W. Yen, “Dynamics of optical TE to TM mode conversion in LiNbO3 channel waveguides,” Appl. Phys. Lett. 45, 1173 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical properties of lithium-rich lithium niobate fabricated by vapor transport equilibration,” IEEE J. Quantum Electron. 26, 135 (1990).
[CrossRef]

J. Appl. Phys. (3)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667 (1970).
[CrossRef]

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys. 45, 3688 (1974).
[CrossRef]

R. C. Miller and W. A. Nordland, J. Appl. Phys. 42, 4145 (1971). There is serious disagreement in the literature regarding the magnitudes of the nonlinear optical coefficients for LiNbO3. For example, our estimate of d22 is inferred from Miller’s Table I for congruent LiNbO3. Miller’s data is normalized to d36 for potassium dihydrogen phosphate. The magnitudes of d36 for potassium dihydrogen phosphate and d31 for LiNbO3 were taken from the recent work of Shoji (Ref. 18 above).
[CrossRef]

J. Cryst. Growth (1)

P. F. Bordui, R. G. Norwood, C. D. Bird, and G. D. Calvert, “Compositional uniformity in growth and poling of large-diameter lithium niobate crystals,” J. Cryst. Growth 113, 61 (1991).
[CrossRef]

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

G. J. Edwards and M. Lawerence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16, 373 (1984).
[CrossRef]

Phys. Lett. (1)

N. Bloembergen and J. Ducuing, “Experimental verification of the optical laws on non-linear reflection,” Phys. Lett. 6, 5 (1963).
[CrossRef]

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606 (1962).
[CrossRef]

Phys. Rev. B (1)

D. F. Nelson and M. Lax, “Theory of the photoelastic interaction,” Phys. Rev. B 3, 2778 (1971).
[CrossRef]

Proc. SPIE (1)

S. R. Lunt, G. E. Peterson, R. J. Holmes, and Y. S. Kim, “Maker fringe analysis of lithium niobate integrated optical substrates,” in Integrated Optical Circuit Engineering II, S. Sriram, ed., Proc. SPIE 578, 22 (1985).
[CrossRef]

Sov. Phys. Solid State (1)

L. P. Avakyants, D. F. Kiselev, and N. N. Shchitov, “Photoelasticity of LiNbO3,” Sov. Phys. Solid State 18, 899 (1976).

Other (10)

Product data provided by Crystal Technology Inc, Palo Alto, Calif.

R. S. Weis and T. K. Gaylord, “Lithium niobate: summary of physical properties and crystal structure,” Appl. Phys. A: Solids Surf. 37, 191 (1984). See also, J. F. Nye, Physical Properties of Crystals (Oxford U. Press, New York, 1995).
[CrossRef]

F. R. N. Nabarro, ed., Dislocations in Solids (Elsevier, Amsterdam, 1979).

J. A. Aust, B. Steiner, N. A. Sanford, G. Fogarty, B. Yang, A. Roshko, J. Amin, and C. Evans, “examination of domain-reversed layers in z-cut LiNbO3 using Maker fringe analysis, atomic force microscopy, and high-resolution x-ray diffraction imaging,” in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 485.

J. A. Aust, N. A. Sanford, and J. Amin, “Maker fringe analysis of z-cut lithium niobate,” in Proceedings of the 10th Annual Meeting of IEEE Lasers and Electro-Optics Society (IEEE Lasers and Electro-Optics Society, Piscataway, N.J., 1997), pp. 114–115.

P. Bordui, Crystal Telchnology, Inc., Palo Alto, Calif. (personal communication, 1997).

N. A. Sanford and J. A. Aust, Properties of Lithium Niobate and other Novel Ferroelectric Materials, K. K. Wong, ed., EMIS Data Review Series (Institution of Electrical Engineers, London, UK, to be published).

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

V. G. Dmitriev, G. G. Gurzadyan, and D. H. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1991).

N. A. Sanford and J. A. Aust, “Maker fringe mapping of LiNbO3 wafers,” in Lasers and Optics for Manufacturing, Vol. 9 of 1997 OSA Trends in Optics and Photonics Series, Andrew C. Tam, ed. (Optical Society of America, Washington, D.C., 1997), pp. 23–32.

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Figures (14)

Fig. 1
Fig. 1

A schematic showing the wave vectors γ, k, κ, ks, κs, kp, and the associated angles of incidence, reflection, and refraction for the input pump field, the reflected e- and o-polarized SHG fields, the transmitted e-polarized SHG field, the transmitted o-polarized SHG field, and the nonlinear source polarization, respectively.

Fig. 2
Fig. 2

The normalized e- and o-polarized SHG generated on reflection of the pump field at the first surface (x=0) as a function of pump angle of incidence θi.

Fig. 3
Fig. 3

A schematic showing the fields retained when we consider multiple reflections of the SHG. At x=0 the y-polarized pump field EyP,i produces y-polarized reflected field EyR1, x-polarized reflected field ExR1, z-polarized reflected field EzR1, y-polarized transmitted field Ey, x-polarized transmitted field Ex, and z-polarized transmitted field Ez. These fields reflected and transmitted at x=L produce the associated fields superscripted by R2 and T2, respectively. These latter fields, when reflected and transmitted from the boundary at x=0, produce the fields superscripted by R3 and T3, respectively. Finally, with reflection and transmission at the boundary x=L, the R3 and T3 superscripted fields produce the R4 and T4 superscripted fields, respectively.

Fig. 4
Fig. 4

(a) The normalized e-polarized SHG power P¯e(T2) as a function of pump angle of incidence θi for λp=1064 nm, Ts=25 °C and L=1 mm. (b) Graphs showing normalized e-polarized SHG P¯e(T2) as a function of pump angle of incidence θi for λp=1064 nm, which compare the sensitivity of the fringe patterns with small changes in birefringence and thickness. The result shows that a deviation Δsne=5×10-6 of ne from the Sellmeier result may be resolved but that variations in sample thickness of ±1 μm near L=1 mm are not resolved. Ts=25 °C for all graphs shown. (c) P¯e(T2) for λp=1319 nm, Ts=25 °C and L=1 mm. (d) Graph illustrating the sensitivity of P¯e(T2) to a variation Δsne=2×10-5 for λp=1319 nm, Ts=25 °C, and L=1 mm. (e) Graph illustrating the critical phase matching peak for λp=1090 nm, Ts=25 °C, and L=1 mm. (f) Graphs illustrating the shift in the critical phase matching angle θpm that is due to a variation Δsne=2×10-5 for λp=1090 nm, Ts=25 °C, and L=1 mm.

Fig. 5
Fig. 5

(a) Normalized o-polarized SHG P¯o(T2) as a function of θi for λp=1064 nm, Ts=25 °C and L=1 mm. (b) P¯o(T2) illustrating the resolution of thickness variations where L=1 mm±0.1 μm. (c) Normalized o-polarized SHG P¯o(T2) as a function of θi for λp=1319 nm, Ts=25 °C, and L=1 mm. (d) Normalized o-polarized SHG P¯o(T2) as a function of θi for λp=1090 nm, Ts=25 °C and L=1 mm.

Fig. 6
Fig. 6

Representation of overlapping multiple reflections of the Gaussian pump beam. Elliptical distortions of the pump-beam profile are ignored.

Fig. 7
Fig. 7

(a) Normalized o-polarized SHG P¯o as a function of θi that includes the effect of pump and SHG overlap. Note that the influence of multiple-pump reflections is most evident near normal incidence, whereas the effects of multiple SHG reflections are stronger at the extreme range of θi. Here λp=1064 nm, Ts=25 °C, L=1 mm, and 2δ=70 μm. (b) P¯o (including pump and SHG reflections) compared with P¯o(T2) (neglecting such reflections) over the range 0°<θi<18°. Here λp=1064 nm, Ts=25 °C, and L=1 mm. (c) P¯o (including pump and SHG reflections) compared with P¯o(T2) (neglecting such reflections) over the range 60°<θi<80°. Here λp=1064 nm, Ts=25 °C, L=1 mm, and 2δ=70 μm. (d) P¯o over the range 0°<θi<4° illustrating variations in resolution of thickness where L=1 mm±0.01 μm. Over this range of θi with 2δ=70 μm, the sensitivity of P¯o to small changes in thickness is dominated by the multiple reflections of the pump field. Here λp=1064 nm and Ts=25 °C. (e) Graphs illustrating the approximate periodic nature of P¯o where L=1.002 mm±2lc, lc=5.846 μm, λp=1064, and Ts=25°C. For 0°<θi<18° all three graphs are nearly identical whereas for wider angles, 70°<θi<90°, for example, the graphs are clearly distinct and do not overlap. (f) Fitting of Eq. (51a) to an o-polarized Maker fringe pattern that was collected at Ts=25.7 °C. The fit shown was obtained with L=0.98621 mm±0.01 μm, λp=1064, and 2δ=70 μm. (g) Fit of Eq. (51a) to an o-polarized Maker fringe data set is shown for a sample nominally 0.2 mm in thickness. The pump beam was polarized parallel to the y axis, and the sample was a z-cut LiNbO3 plate rotated about the y-axis during the collection of the SHG data. With L varied as the fitting parameter, the result shown illustrates the fit when L=0.19559 mm±0.01 μm.

Fig. 8
Fig. 8

(a) Comparison of the normalized e-polarized SHG P¯e(T2) (neglecting pump and SHG reflections) with the normalized e-polarized SHG P¯e (including such reflections). Here λp=1064 nm, Ts=25 °C, L=0.94 mm, and 2δ=70 μm. (b) Graphs illustrating the same parameters and conditions as Fig. 8(a), except for L=0.2 mm. (c) Fitting Eq. (58a) to an e-polarized Maker fringe pattern that was collected at Ts=25.7 °C. The value of L computed in regard to Fig. 7(f) was used as an input parameter. With Δsne as the only adjustable parameter, the fit shown was obtained with Δsne=-1.58×10-4±5×10-6, λp=1064, and 2δ=70 μm. The inset shows the residual deviation of the model from the data for θi near 40°. Figure 12(b) shows how the inclusion of strain effects can correct this deviation. (d) Graphs illustrating the approximate periodic nature of Eq. (58a) with respect to shifts in the extraordinary index where Δsne=-1.58×10-4-5.32×10-4 and Δsne=-1.58×10-4+5.48×10-4. The sample temperature and thickness are the same as used for Fig. 8(c). The two graphs show reasonable overlap with the same e-polarized SHG data set appearing in Fig. 8(c) with regard to locating the fringe nulls. However, neither graph shows an acceptable fit of the fringe envelope in comparison with Fig. 8(c).

Fig. 9
Fig. 9

Data showing the corruption of the weaker o-polarized Maker fringe data by the dominant e-polarized SHG signal. Note that the extra fringes in the o-polarized data align exactly with the corresponding e-polarized data taken at the same point on the sample as indicated by the arrows. This effect is often highly localized such that an o-polarized scan performed at a position less than 1 mm away may show substantially greater or lesser corruption.

Fig. 10
Fig. 10

(a) Schematic of the strain-induced rotation of the optic axis about x by an angle ζ, leading to leakage of the e-polarized SHG signal into the o-polarized data as shown in Fig. 9. The transformed axis z is associated with the transformed extraordinary index ne and the transformed normalized e-polarized SHG field E¯z(T2). The transformed axis y is associated with the transformed ordinary indices nyp and ny at the pump and SHG wavelengths, respectively. The y axis conforms with the orientation of the output polarizer used to separate the e- and o-polarized SHG signals. The projection of E¯z(T2) onto y is approximately ζE¯z(T2). (b) Normalized opolarized SHG ℘ in the presence of strain-induced leakage of ζE¯z(T2). The case in which strain-induced rotation of the optic axis ζ=0 is compared with ζ=-0.002 and ζ=-0.004. (c) Improved fit of the o-polarized SHG data shown above in Fig. 7(f) when a strain induced rotation ζ=-0.004 is included. (d) Appearance of extra features in the o-polarized SHG fringes that conform to the corresponding e-polarized fringes are indicated by the vertical arrows. The angle ζ=-0.01.

Fig. 11
Fig. 11

(a) Asymmetric angular shift of e-polarized Maker fringe patterns with respect to pump angle of incidence θi. The two e-polarized data scans shown were collected from locations separated by 60 mm on the sample. Note that the relative difference between these data sets is characterized by a symmetric breathing of the fringes with respect to θi as well as an asymmetric displacement of the two sets with respect to θi. The breathing effect may be described by composition-induced variations in Δsne, and the asymmetric displacement is described by a strain-induced rotation of the optic axis about y. (b) Schematic illustrating the strain-induced rotation of the optic axis about y by an angle ϑ with respect to the incident pump field, the e-polarized SHG field, and the original crystal axes.  

Fig. 12
Fig. 12

(a) Asymmetric shift in the e-polarized fringes that results with the inclusion of aggregate shear strains that are due to S5 and S6, resulting in ϑ=-3×10-3 with Ts=25 °C, λp=1064, and 2δ=70 μm. Also shown is a graph with Δsne=2×10-5. Comparison of these graphs shows that variations in the fringes that are due to Δsne are more heavily weighted near normal incidence, whereas while shifts in the fringes that are due to rotations ϑ appear throughout the full range of θi. (b) Improvement of the fit shown in Fig. 8(c) by the inclusion of a small strain-induced rotation ϑ of the optic axis about y. Δsne=-1.58×10-4, ϑ=-8×10-4 rad, λp=1064, and 2δ=70 μm. The inset details the improved fit for θi near 40°.

Fig. 13
Fig. 13

P¯e(T2) as a function of temperature with L=1 cm. The FWHM of P¯e(T2) is ∼1.2 °C. Also shown is the shift of graph associated with Δsne=-5×10-5.

Fig. 14
Fig. 14

P¯e(T2) is illustrated with Δsne varied from -4.8×10-4 to -8×10-4. This range of Δsne produces graphs of P¯e(T2) that bracket the excursion of the phase-matching temperature Tpm that is expected for congruently grown material, as discussed in Ref. 4.

Equations (177)

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EyP,T1=yˆTEyP,iexp[i(γxpx+γzpz-ωpt)].
P=(yˆd22+zˆd31)(TEyP,i)2exp[i(kxPx+kzPz-ωt)].
×H=o˜ Et+Pt,
×E=-μoHt.
-zHy=-iωono2Ex,
-xHz+zHx=-iωono2Ey-iωPy,
xHy=-iωone2Ez-iωPz,
-zEy=iωμoHx,
-xEz+zEx=iωμoHy,
xEy=iωμoHz.
xx2 Ey+zz2 Ey+k2no2 Ey=-k2o Py.
D=o(xˆno2 Ex+yˆno2 Ey+zˆne2 Ez)+yˆPy+zˆPz.
x Ex=-1ono2 (one2zEz+zPz),
zEz=-1one2 (ono2xEx+zPz),
zx2Ex=-1ono2 (one2zz2Ez+zz2Pz),
xz2Ez=-1one2 (ono2xx2Ex+xz2Pz).
none2xx2Ex+zz2Ex+k2no2Ex=-1one2 xz2Pz,
xx2Ez+neno2zz2Ez+k2ne2Ez=-1ono2 zz2Pz
-k20 Pz.
Exh=Exoexp[i(kxsx+kzsz-iωt)],
Eyh=Eyoexp[i(κxsx+κzsz-iωt)],
Ezh=Ezoexp[i(kxsx+kzsz-iωt)].
n(θs)=ne2no2ne2sin2 θs+no2cos2 θs1/2 or
1n2(θs)=sin2 θsno2+cos2 θsne2,
Ex={Exoexp[i(kxsx+kzsz)]+Axexp[i(kxpx+kzpz)]}exp(-iωt),
Ey={Eyoexp[i(κxsx+κzsz)]+Ayexp[i(kxpx+kzpz)]}exp(-iωt),
Ez={Ezoexp[i(kxsx+kzsz)]+Azexp[i(kxpx+kzpz)]}exp(-iωt).
Ax=d31(EyP,iT)2kxpkzpone2k2no2-none kxp2-(kzp)2,
Ay=d22(EyP,iT)2o[(nop)2-no2],
Az=d31(EyP,iT)2[(kzp)2-(nok)2]no2ok2ne2-neno kzp2-(kxp)2.
ExR1=Ex(R1)oexp[i(-kxx+kzz-ωt)],
EyR1=Ey(R1)oexp[i(-κxx+κzz-ωt)],
EzR1=Ez(R1)oexp[i(-kxx+kzz-ωt)].
Exo=-ne2kzsno2kxs Ezo,
Ex(R1)o=kzkx Ez(R1)o.
HyR1|(x=0)=Hy|(x=0),
HzR1|(x=0)=Hz|(x=0),
EyR1|(x=0)=Ey|(x=0),
EzR1|(x=0)=Ez|(x=0).
exp(ikzpz),exp(ikzsz),exp(ikzz),
exp(iκzsz),exp(iκzz)
Ezo=Ax-[kxp/kz+kx/kz+kz/kx]Az[(ne/no)2kz/kxs+kxs/kz+kx/kz+kz/kx],
Ez(R1)o=Ezo+Az,
Eyo=-Ay[κx+kxp][κx+κxs],
Ey(R1)o=Ay[κxs-kxp][kxs+κx].
θR=θi,
sin θP=sin θinop,
sin θS=nosin θi[(none)2+(no2-ne2)sin2 θi]1/2,
sin ϕs=sin θino.
N=d31o (EyP,i)2,
M=d22o (EyP,i)2.
T=2 cos θinopcos θp+cos θi.
A¯x=T2sin 2 θp2ne2nonop2-nocos θpne2-sin2 θp,
A¯y=T2[(nop)2-no2],
A¯z=T2sin2 θp-nonop2no2nenop2-nesin θpno2-cos2 θp.
E¯zo=A¯x-A¯z[cot θp+2 csc 2θi]neno2tan θs+cot θs+2 csc 2θi.
P¯eR1= |E¯z(R1)o|2sec2 θi.
P¯oR1=(A¯y)2nocos ϕs-nopcos θpnocos ϕs+cos θi2.
ExR2=(Ex(R2)oexp{i[kxs(-x+L)+kzsz]}+Ax(2)exp{i[kxp(-x+L)+kzpz]})exp(-iωt),
EzR2=(Ez(R2)oexp{i[kxs(-x+L)+kzsz]}+Az(2)exp{i[kxp(-x+L)+kzpz]})exp(-iωt).
ExT2=Ex(T2)oexp{i[kx(x-L)+kzz]}exp(-iωt),
EzT2=Ez(T2)oexp{i[kx(x-L)+kzz]}exp(-iωt).
Ax(2)=-AxR2,
Az(2)=AzR2.
R2=nopcos θp-cos θinopcos θp+cos θi2.
Ez|(x=L)+EzR2|(x=L)=EzT2|(x=L),
Hy|(x=L)+HyR2|(x=L)=HyT2|(x=L).
Ez(T2)o=b1Ezoexp(ikxsL)+(Azb2+Axb3)exp(ikxpL)+[Az(2)b4+Ax(2)b3].
b1=2a1a1+a2,
b2=a1+kxpa1+a2,
b3=-kzpa1+a2,
b4=a1-kxpa1+a2
a1=kxs+neno2(kzs)2kxs,
a2=kx+kz2kx.
P¯eT2={H12+H22+H32+2H1H3cos(kxsL)+2H2H3cos(kxpL)+2H1H2cos[(kxs-kxp)L]}sec2 θi.
H1=E¯zob1,
H2=(A¯zb2+A¯xb3),
H3=[A¯z(2)b4+A¯x(2)b3].
EyR2=(Ey(R2)oexp{i[κxs(-x+L)+κzsz]+Ay(2)exp{i[kxp(-x+L)+kzpz]})exp(-iωt),
EyT2=Ey(T2)oexp{i[κx(x-L)+κzz]}exp(-iωt).
Ey|(x=L)+EyR2|(x=L)=EyT2|(x=L),
Hz|(x=L)+HzR2|(x=L)=HzT2|(x=L).
E¯y(T2)o=G1exp(iκxsL)+G2exp(ikxpL)+G3.
P¯oT2=G12+G22+G32+2G1G3cos(κxsL)+2G2G3cos(kxpL)+2G1G2cos[(κxs-kxp)L].
G1=2κxsκxs+kxE¯yo,
G2=kxp+κxskx+κxsA¯y,
G3=κxs-kxpkx+κxsA¯y(2),
Ey(R2)o=Ey(T2)o-Eyoexp(iκxsL)-Ayexp(ikxpL)-Ay(2).
Ey(T3)=Ey(T3)oexp[i(-kxx+kzz-ωt)],
Ey(R3)=Ey(R3)oexp[i(κxsx+κzsz-ωt)].
Ey(T4)=Ey(T4)oexp{i[kx(x-L)+kzz-ωt]},
Ey(R4)=Ey(R4)oexp{i[κxs(L-x)+κzsz-ωt]}.
Ey(T4)o=ToRoEy(R2)oexp(2iκxsL).
To=2nocos ϕs(nocos ϕs+cos θi),
Ro=nocos ϕs-cos θinocos ϕs+cos θi.
E¯y(T4)o=g1exp(i3κxsL)+g2exp[i(2κxs+kxp)L]+g3exp(i2κxsL),
g1=ToRo(G1-E¯yo),
g2=ToRo(G2-A¯y),
g3=-ToRoA¯y(2).
EyP,cexp-r2δ2=TEyP,iexp-r2δ2+R2exp-r12δ2exp(i2kxpL),
r12=r2+ro2-2rrocos α,
ro=2L tan θp,
r2=y2+z2.
Pyp=nop2 [TEyP,i]2oμo1/202π0exp-2r2δ2+2[cos(2kxpL)]R2exp-r2+r12δ2rdrdα.
|EyP,c|2=(TEyP,i)2[1+2R2ϕ cos(2kxpL)],
ϕ=exp[-(2L tan θp/δ)2].
EyP,c=TEyP,iη,
η=[1+R2ϕ exp(i2kxpL)].
Po=12oμo1/202π0[|Ey(T2)o|2+|Ey(T4)o|2]×exp-2r2δs2rdrdα+2 Re02π0Ey(T2)o(Ey(T4)o)*×exp-(r2+r12)δs2rdrdα.
P¯o=(|E¯y(T2)o|2+2 Re{E¯y(T2)o[E¯y(T4)o]*}ϕo+|E¯y(T4)o|2)|(η)2|2,
P¯o=(Po)/M2δs2π4oμo1/2,
ϕo=exp-2L tan ϕsδs2.
Ez(R2)o=Ez(T2)o-Ezoexp(ikxsL)-Azexp(ikxpL)-Az(2).
Ex(T3)=Ex(T3)oexp[i(-kxx+kzz-ωt)],
Ez(T3)=Ez(T3)oexp[i(-kxx+kzz-ωt)],
Ex(R3)=Ex(R3)oexp[i(kxsx+kzsz-ωt)],
Ez(R3)=Ez(R3)oexp[i(kxsx+kzsz-ωt)],
Ex(T4)=Ex(T4)oexp{i[kx(x-L)+kzz-ωt]},
Ez(T4)=Ez(T4)oexp{i[kx(x-L)+kzz-ωt]},
Ex(R4)=Ex(R4)oexp{i[kxs(L-x)+kzsz-ωt]},
Ez(R4)=Ez(R4)oexp{i[kxs(L-x)+kzsz-ωt]}.
Ez(T4)o=P1exp(i3kxsL)+P2exp[i(kxp+2kxs)L]+P3exp(i2kxsL)+P4exp[i(kxp+kxs)L].
P1=T4c1(b1-1)Ezo,
P2=T4c1[Az(b2-1)+b3Ax],
P3=-T4c1Az(2),
P4=T4(c2Az(2)+c3Ax(2)),
c1=a1-a2a1+a2,
c2=kxp-a2a1+a2,
c3=kzp(a1+a2),
T4=2a1a1+a2.
Pe=12oμo1/202π0[|Ex(T2)o|2+|Ex(T4)o|2+|Ez(T2)o|2+|Ez(T4)o|2]exp-2r2δs2rdrdα+2 Re02π0{Ex(T2)o[Ex(T4)o]*+Ez(T2)o[Ez(T4)o]*}×exp-(r2+r12)δs2rdrdα.
P¯e=(|E¯z(T2)o|2+2 Re{E¯z(T2)o[E¯z(T4)o]*}ϕe+|E¯z(T4)o|2)|(η)2|2sec2(θi),
=PeN2δs2π4oμo1/2,
ϕe=exp-2L tan θsδs2.
Δηij=PijklSkl.
Δη11=P11S1+P12S2+P13S3+2P14S4,
Δη12=2P14S1+(P11-P12)S6,
Δη13=2P44S5+2P41S6,
Δη22=P12S1+P11S2+P13S3-2P14S4,
Δη23=P41S1-P41S2+2P44S4,
Δη33=P31S1+P31S2+P33S3.
Δij=-imΔηmnnj.
Δ11=-(11)2Δη11,
Δ12=-1122Δη12,
Δ13=-1133Δη13,
Δ22=-(22)2Δη22,
Δ23=-2233Δη23,
Δ33=-(33)2Δη33.
ij=no2000no2ξ0ξne2.
ξ=-(neno)2[P41(S1-S2)+2P44S4].
kl=akialjij,
aki=1000cos ζsin ζ0-sin ζcos ζ.
11=11,
22=22cos2 ζ+33sin2 ζ+ξ sin 2ζ,
23=[33-22] sin 2ζ2+ξ cos 2ζ,
33=22sin2 ζ+33cos2 ζ-ξ sin 2ζ.
ζξ(22-33).
Po= |E¯y(T2)o|2(d22)2+|E¯z(T2)o|2(ζd31)2+2ζd31d22Re{E¯y(T2)o[E¯z(T2)o]*}.
ζ=-66.49[P41(S1-S2)+2P44S4].
nx=no,
nyp=nop=[(nop)2-(nepζ)2+2(ζnop)2]1/2,
nz=ne=[(ne)2-(noζ)2+2(ζne)2]1/2.
nop-nenop-ne+7×10-7,
nop-nonop-no-2×10-7.
ij=no20ψ0no20ψ0ne2.
kl=akialjij,
aki=cos ϑ0-sin ϑ010sin ϑ0cos ϑ.
ϑ=ψne2-no2,
ψ=ne2no2[P44S5+P41S6].
nx=[no2-(ϑne)2+2(ϑno)2]1/2,
ny=no,
nz=[ne2-(ϑno)2+2(ϑne)2]1/2.
sin θi=n(θs-ϑ)sin θs.
sin θs=-(2A1A2+4A32ϑ2)+[(2A1A2+4A32ϑ2)2-4A12(A22-4A32ϑ2)]1/2(2A12)1/2.
A1=(ne2-no2)sin2 θi-(neno)2,
A2=(nosin2 θi)2,
A3=(no2-ne2)sin2 θi.

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