Abstract

We demonstrate a new method of detecting the presence of ferroelectric domains based on non-phase-matched second-harmonic generation. If a domain boundary is tilted relative to the input and output faces of the crystal, the far-field second-harmonic light consists of multiple beams, in contrast to the single beam generated in a single-domain crystal. The angular separation of the beams provides a measure of the tilt of the domain wall if the refractive-index difference n - n ω is known.

© 1998 Optical Society of America

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  1. D. J. Armstrong, W. J. Alford, T. D. Raymond, A. V. Smith, “Absolute measurement of the effective nonlinearities of KTP and BBO crystals by optical parametric amplification,” Appl. Opt. 35, 2032–2040 (1996).
    [CrossRef] [PubMed]
  2. F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
    [CrossRef]
  3. J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
    [CrossRef]
  4. G. M. Loiacono, R. A. Stolzenberger, “Observation of complex domain walls in KTiOPO4,” Appl. Phys. Lett. 53, 1498–1499 (1988).
    [CrossRef]
  5. Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
    [CrossRef]
  6. M. Houe, P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
    [CrossRef]
  7. G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
    [CrossRef]
  8. A. Reichert, K. Betzler, “Induced noncollinear frequency-doubling: a new characterization technique for electrooptic crystals,” J. Appl. Phys. 79, 2209–2212 (1996).
    [CrossRef]
  9. R. S. Cudney, V. Garces-Chavez, P. Negrete-Regagnon, “Analysis of ferroelectric 180°-domain structures in BaTiO3 by use of second-harmonic scattering,” Opt. Lett. 22, 439–441 (1997).
    [CrossRef] [PubMed]
  10. G. Dolino, “Effects of domain shapes on second-harmonic scattering in triglycine sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
    [CrossRef]
  11. S. Kurimura, Y. Uesu, “Application of the second harmonic generation microscope to nondestructive observation of periodically poled ferroelectric domains in quasi-phase-matched wavelength converters,” J. Appl. Phys. 81, 369–375 (1997).
    [CrossRef]
  12. Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, New York, 1984).
  13. R. L. Sutherland, Handbook of Nonlinear Optics, (Marcel Dekker, New York, 1996).
  14. J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
    [CrossRef]
  15. W. N. Herman, L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12, 416–427 (1995).
    [CrossRef]
  16. D. Chemla, P. Kupecek, “Analyse des experiences de generation de second harmonique,” Rev. Phys. Appl. 6, 31–50 (1971).
    [CrossRef]
  17. V. G. Dmitriev, G. G. Guradyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer-Verlag, New York, 1991).
  18. H. Vanherzeele, J. D. Bierlein, F. C. Zumsteg, “Index of refraction measurements and parametric generation in hydrothermally grown KTiOPO4,” Appl. Opt. 27, 3314–3316 (1988).
    [CrossRef] [PubMed]
  19. I. Tardjman, R. Masse, J. C. Gritel, “Structure cristalline du monophosphate KTiPO5,” Z. Kristallogr. 139, 103–115 (1974).
    [CrossRef]
  20. J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
    [CrossRef]

1997 (2)

R. S. Cudney, V. Garces-Chavez, P. Negrete-Regagnon, “Analysis of ferroelectric 180°-domain structures in BaTiO3 by use of second-harmonic scattering,” Opt. Lett. 22, 439–441 (1997).
[CrossRef] [PubMed]

S. Kurimura, Y. Uesu, “Application of the second harmonic generation microscope to nondestructive observation of periodically poled ferroelectric domains in quasi-phase-matched wavelength converters,” J. Appl. Phys. 81, 369–375 (1997).
[CrossRef]

1996 (3)

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

A. Reichert, K. Betzler, “Induced noncollinear frequency-doubling: a new characterization technique for electrooptic crystals,” J. Appl. Phys. 79, 2209–2212 (1996).
[CrossRef]

D. J. Armstrong, W. J. Alford, T. D. Raymond, A. V. Smith, “Absolute measurement of the effective nonlinearities of KTP and BBO crystals by optical parametric amplification,” Appl. Opt. 35, 2032–2040 (1996).
[CrossRef] [PubMed]

1995 (3)

Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
[CrossRef]

M. Houe, P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
[CrossRef]

W. N. Herman, L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12, 416–427 (1995).
[CrossRef]

1992 (1)

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

1988 (2)

1987 (2)

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

1974 (1)

I. Tardjman, R. Masse, J. C. Gritel, “Structure cristalline du monophosphate KTiPO5,” Z. Kristallogr. 139, 103–115 (1974).
[CrossRef]

1972 (1)

G. Dolino, “Effects of domain shapes on second-harmonic scattering in triglycine sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
[CrossRef]

1971 (1)

D. Chemla, P. Kupecek, “Analyse des experiences de generation de second harmonique,” Rev. Phys. Appl. 6, 31–50 (1971).
[CrossRef]

1970 (1)

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Ahmed, F.

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

Alford, W. J.

Angert, N.

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Armstrong, D. J.

Betzler, K.

A. Reichert, K. Betzler, “Induced noncollinear frequency-doubling: a new characterization technique for electrooptic crystals,” J. Appl. Phys. 79, 2209–2212 (1996).
[CrossRef]

Bierlein, J. D.

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

H. Vanherzeele, J. D. Bierlein, F. C. Zumsteg, “Index of refraction measurements and parametric generation in hydrothermally grown KTiOPO4,” Appl. Opt. 27, 3314–3316 (1988).
[CrossRef] [PubMed]

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

Bindloss, M.

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

Chemla, D.

D. Chemla, P. Kupecek, “Analyse des experiences de generation de second harmonique,” Rev. Phys. Appl. 6, 31–50 (1971).
[CrossRef]

Cudney, R. S.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Guradyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer-Verlag, New York, 1991).

Dolino, G.

G. Dolino, “Effects of domain shapes on second-harmonic scattering in triglycine sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
[CrossRef]

Garces-Chavez, V.

Gritel, J. C.

I. Tardjman, R. Masse, J. C. Gritel, “Structure cristalline du monophosphate KTiPO5,” Z. Kristallogr. 139, 103–115 (1974).
[CrossRef]

Gupta, M. C.

Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
[CrossRef]

Guradyan, G. G.

V. G. Dmitriev, G. G. Guradyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer-Verlag, New York, 1991).

Hayden, L. M.

Herman, W. N.

Houe, M.

M. Houe, P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
[CrossRef]

Hsuing, H.

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

Hu, Z. W.

Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
[CrossRef]

Jerphagnon, J.

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Kupecek, P.

D. Chemla, P. Kupecek, “Analyse des experiences de generation de second harmonique,” Rev. Phys. Appl. 6, 31–50 (1971).
[CrossRef]

Kurimura, S.

S. Kurimura, Y. Uesu, “Application of the second harmonic generation microscope to nondestructive observation of periodically poled ferroelectric domains in quasi-phase-matched wavelength converters,” J. Appl. Phys. 81, 369–375 (1997).
[CrossRef]

Kurtz, S. K.

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Lareah, I.

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Laurell, F.

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

Loiacono, G. M.

G. M. Loiacono, R. A. Stolzenberger, “Observation of complex domain walls in KTiOPO4,” Appl. Phys. Lett. 53, 1498–1499 (1988).
[CrossRef]

Masse, R.

I. Tardjman, R. Masse, J. C. Gritel, “Structure cristalline du monophosphate KTiPO5,” Z. Kristallogr. 139, 103–115 (1974).
[CrossRef]

Negrete-Regagnon, P.

Nikogosyan, D. N.

V. G. Dmitriev, G. G. Guradyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer-Verlag, New York, 1991).

Raymond, T. D.

Reichert, A.

A. Reichert, K. Betzler, “Induced noncollinear frequency-doubling: a new characterization technique for electrooptic crystals,” J. Appl. Phys. 79, 2209–2212 (1996).
[CrossRef]

Risk, W. P.

Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
[CrossRef]

Roefols, M. G.

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

Rosenman, G.

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Roth, M.

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, New York, 1984).

Skliar, A.

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Smith, A. V.

Stolzenberger, R. A.

G. M. Loiacono, R. A. Stolzenberger, “Observation of complex domain walls in KTiOPO4,” Appl. Phys. Lett. 53, 1498–1499 (1988).
[CrossRef]

Suma, A.

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

Sutherland, R. L.

R. L. Sutherland, Handbook of Nonlinear Optics, (Marcel Dekker, New York, 1996).

Tardjman, I.

I. Tardjman, R. Masse, J. C. Gritel, “Structure cristalline du monophosphate KTiPO5,” Z. Kristallogr. 139, 103–115 (1974).
[CrossRef]

Thomas, P. A.

Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
[CrossRef]

Townsend, P. D.

M. Houe, P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
[CrossRef]

Tseitlin, M.

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Uesu, Y.

S. Kurimura, Y. Uesu, “Application of the second harmonic generation microscope to nondestructive observation of periodically poled ferroelectric domains in quasi-phase-matched wavelength converters,” J. Appl. Phys. 81, 369–375 (1997).
[CrossRef]

Vanherzeele, H.

Zumsteg, F. C.

Appl. Opt. (2)

Appl. Phys. Lett. (4)

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

G. M. Loiacono, R. A. Stolzenberger, “Observation of complex domain walls in KTiOPO4,” Appl. Phys. Lett. 53, 1498–1499 (1988).
[CrossRef]

Z. W. Hu, P. A. Thomas, M. C. Gupta, W. P. Risk, “Multiple-crystal x-ray topographic characterization of periodically domain-inverted KTiOPO4 crystal,” Appl. Phys. Lett. 66, 13–15 (1995).
[CrossRef]

J. D. Bierlein, F. Ahmed, “Observation and poling of ferroelectric domains in KTiOPO4,” Appl. Phys. Lett. 51, 1322–1324 (1987).
[CrossRef]

J. Appl. Phys. (4)

F. Laurell, M. G. Roefols, M. Bindloss, H. Hsuing, A. Suma, J. D. Bierlein, “Detection of ferroelectric domain reversal in KTiOPO4 waveguides,” J. Appl. Phys. 71, 4664–4670 (1992).
[CrossRef]

A. Reichert, K. Betzler, “Induced noncollinear frequency-doubling: a new characterization technique for electrooptic crystals,” J. Appl. Phys. 79, 2209–2212 (1996).
[CrossRef]

S. Kurimura, Y. Uesu, “Application of the second harmonic generation microscope to nondestructive observation of periodically poled ferroelectric domains in quasi-phase-matched wavelength converters,” J. Appl. Phys. 81, 369–375 (1997).
[CrossRef]

J. Jerphagnon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

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

J. Phys. D (1)

M. Houe, P. D. Townsend, “An introduction to methods of periodic poling for second-harmonic generation,” J. Phys. D 28, 1747–1763 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

G. Dolino, “Effects of domain shapes on second-harmonic scattering in triglycine sulfate,” Phys. Rev. B 6, 4025–4035 (1972).
[CrossRef]

G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, M. Roth, “Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4 crystals,” Phys. Rev. B 54, 6222–6226 (1996).
[CrossRef]

Rev. Phys. Appl. (1)

D. Chemla, P. Kupecek, “Analyse des experiences de generation de second harmonique,” Rev. Phys. Appl. 6, 31–50 (1971).
[CrossRef]

Z. Kristallogr. (1)

I. Tardjman, R. Masse, J. C. Gritel, “Structure cristalline du monophosphate KTiPO5,” Z. Kristallogr. 139, 103–115 (1974).
[CrossRef]

Other (3)

Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, New York, 1984).

R. L. Sutherland, Handbook of Nonlinear Optics, (Marcel Dekker, New York, 1996).

V. G. Dmitriev, G. G. Guradyan, D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, (Springer-Verlag, New York, 1991).

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Figures (6)

Fig. 1
Fig. 1

Diagram of the fundamental wave (dashed line) plus the free (F) and driven (D) second-harmonic waves in a crystal with a tilted planar ferroelectric domain boundary. The sign of the nonlinear coefficient d is reversed in the two domains. This diagram is for Δk > 0; for Δk < 0, the deflection is in the opposite direction.

Fig. 2
Fig. 2

Simulated Maker fringes for a crystal with (a) a single domain and an untilted boundary separating (b) two equal length domains, (c) two domains with a length ratio of 3:1, and (d) two domains with a length ratio of 200:1.

Fig. 3
Fig. 3

Experimental setups for (a) non-phase-matched second-harmonic generation and (b) phase-matched second-harmonic generation. PBS, polarizing beam splitter.

Fig. 4
Fig. 4

Beam patterns for the second-harmonic light generated by multiple-domain KTP crystals. (a) Crystal 4, one array of beams indicating a single-domain boundary. (b) Crystal 5, one array of beams believed to be due to two closely spaced parallel domain boundaries. (c) Crystal 6, two arrays of beams indicating two nonparallel domain boundaries. (d) Crystal 7, five arrays of beams indicating five domain boundaries. In (a) and (b), the untilted beam is indicated by the vertical line; in (c) and (d) the untilted beam is at the intersection of the arrays. In (c) and (d) the third beam of each array is outside the camera’s field of view.

Fig. 5
Fig. 5

Measured (filled circles) and simulated (curves) phase-matching curves for second-harmonic generation of 1.32 μm for multiple-domain KTP crystals. (a) Crystal 4, data display two peaks typical of a two-domain crystal. The simulated curve is for two domains with a length ratio of 62.5:37.5. (b) Crystal 5, data appear similar to that for a single-domain crystal, except for slightly larger than usual secondary peaks. The simulated curve is for three domains with a length ratio of 28:1:19. (c) Crystal 6, data show a peak that is significantly broader than for single domain. Simulated curve is for three domains with a length ratio of 3:42:3. (d) Crystal 7, data indicate more than two domains. Simulated curve is for four equal length domains.

Fig. 6
Fig. 6

Measured (open triangles) tilt angles 〈ϕ zyy 〉 and computed tilt angles (filled squares) for best-fit low-order crystallographic planes identified by their Miller indices.

Tables (2)

Tables Icon

Table 1 Comparison of Measured (ϕ’s) and Calculated (Δn’s) Second-Harmonic Beam Tilt Ratios

Tables Icon

Table 2 Comparison of Measured and Calculated Beam Tilts

Equations (12)

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

E 2 ω z = E ω 2 d eff Kz   sin Δ kz 2 Δ kz 2 exp i Δ kz / 2 .
ϕ = arcsin n 2 ω sin γ - arcsin n ω n 2 ω sin   γ .
ϕ = n 2 ω - n ω   tan   γ = Δ k 2 k 0 tan   γ ,
E 2 ω L = K   L 1 sin Δ kL 1 / 2 Δ kL 1 / 2   exp i Δ kL 1 / 2 + i Δ kL 2 - L 2 sin Δ kL 2 / 2 Δ kL 2 / 2   exp i Δ kL 2 / 2 ,
E 2 ω L = - iK Δ k exp i Δ kL + 1 - 2   exp i Δ kL 2 .
E 2 ω x = iK Δ k exp i Δ kL + 1 - 2   exp i Δ kL 0   exp i Δ kx   tan   γ .
d ˆ = a   x ˆ + m b   y ˆ + n c   z ˆ a 2 + m b 2 + n c 2 1 / 2 ,
k ˆ = sin   θ x ˆ + cos   θ z ˆ ,
p ˆ = cos   θ x ˆ - sin   θ z ˆ ,
γ = arccos k ˆ · d ˆ = arccos   a   sin   θ + n c   cos   θ a 2 + m b 2 + n c 2 1 / 2 ,
ϕ y = ϕ   d ˆ · y ˆ d ˆ · y ˆ 2 + d ˆ · p ˆ 2 1 / 2 ,
ϕ xz = ϕ   d ˆ · p ˆ d ˆ · y ˆ 2 + d ˆ · p ˆ 2 1 / 2 .

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