Abstract

We demonstrate a variation of the wedge method of Maker-fringe measurement in which the fundamental beam diameter is large enough to contain several second-harmonic coherence fringes. In the far field the second harmonic forms spatially separated beams from which both Δks and deffs can be deduced on a single laser pulse. Analysis is simple because no fringe analysis is required and because the method is immune to multiple surface reflections, birefringent walk-off, group-velocity walk-off, and surface effects such as longitudinal polarization. Example measurements on KDP and LiIO3 are presented.

© 1998 Optical Society of America

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References

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  1. R. L. Sutherland, Handbook of Nonlinear Optics (Dekker, New York, 1996).
  2. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
    [CrossRef]
  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, 1667–1681 (1970).
    [CrossRef]
  4. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12, 416–427 (1995).
    [CrossRef]
  5. D. Chemla and P. Kupecek, “Analyse des experiences de generation de second harmonique,” Rev. Phys. Appl. 6, 31–50 (1971).
    [CrossRef]
  6. M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693–1706 (1976).
    [CrossRef]
  7. R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
    [CrossRef]
  8. I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
    [CrossRef]
  9. Y. Yamamoto, T. Ashida, S. Kurimura, and Y. Uesu, “Two-dimensional observation of the Maker fringe and its application to the poling state evaluation of ferroelectric domains,” Appl. Opt. 36, 602–605 (1997).
    [CrossRef] [PubMed]
  10. S. Kurimura and 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]
  11. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
    [CrossRef]
  12. N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for nonlinear films with C∞V symmetry,” J. Opt. Soc. Am. B 9, 2083–2087 (1992).
    [CrossRef]
  13. A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995); A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2276 (1995). SNLO freeware for modeling χ2 nonlinear optics is available from A. V. Smith.
    [CrossRef]
  14. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, New York, 1997).
  15. K. Kato, “High-power difference-frequency generation at 4.4–5.7 μm in LiIO3,” IEEE J. Quantum Electron. QE-21, 119–120 (1985).
    [CrossRef]
  16. M. Okada and S. Ieiri, “Kleinman’s symmetry relation in non-linear optical coefficient of LiIO3,” Phys. Lett. A 34, 63–64 (1971).
    [CrossRef]
  17. J. Jerphagnon, “Optical nonlinear susceptibilities of lithium iodate,” Appl. Phys. Lett. 16, 298–299 (1970).
    [CrossRef]
  18. D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
    [CrossRef]
  19. R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
    [CrossRef]
  20. G. Nath and S. Haussühl, “Large nonlinear optical coefficient and phase matched second harmonic generation in LiIO3,” Appl. Phys. Lett. 14, 154–156 (1969).
    [CrossRef]
  21. A. J. Campillo and C. L. Tang, “Spontaneous parametric scattering of light in LiIO3,” Appl. Phys. Lett. 16, 242–244 (1970).
    [CrossRef]

1997 (3)

1995 (1)

1992 (2)

N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for nonlinear films with C∞V symmetry,” J. Opt. Soc. Am. B 9, 2083–2087 (1992).
[CrossRef]

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

1990 (1)

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

1988 (1)

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

1985 (1)

K. Kato, “High-power difference-frequency generation at 4.4–5.7 μm in LiIO3,” IEEE J. Quantum Electron. QE-21, 119–120 (1985).
[CrossRef]

1976 (1)

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693–1706 (1976).
[CrossRef]

1971 (2)

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

M. Okada and S. Ieiri, “Kleinman’s symmetry relation in non-linear optical coefficient of LiIO3,” Phys. Lett. A 34, 63–64 (1971).
[CrossRef]

1970 (3)

J. Jerphagnon, “Optical nonlinear susceptibilities of lithium iodate,” Appl. Phys. Lett. 16, 298–299 (1970).
[CrossRef]

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

A. J. Campillo and C. L. Tang, “Spontaneous parametric scattering of light in LiIO3,” Appl. Phys. Lett. 16, 242–244 (1970).
[CrossRef]

1969 (1)

G. Nath and S. Haussühl, “Large nonlinear optical coefficient and phase matched second harmonic generation in LiIO3,” Appl. Phys. Lett. 14, 154–156 (1969).
[CrossRef]

1962 (2)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Ashida, T.

Bloembergen, N.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Byer, R. L.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693–1706 (1976).
[CrossRef]

Campillo, A. J.

A. J. Campillo and C. L. Tang, “Spontaneous parametric scattering of light in LiIO3,” Appl. Phys. Lett. 16, 242–244 (1970).
[CrossRef]

Chemla, D.

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

Choy, M. M.

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693–1706 (1976).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

Fan, Y. X.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

Haussühl, S.

G. Nath and S. Haussühl, “Large nonlinear optical coefficient and phase matched second harmonic generation in LiIO3,” Appl. Phys. Lett. 14, 154–156 (1969).
[CrossRef]

Hayden, L. M.

Herman, W. N.

Hirano, Y.

Ieiri, S.

M. Okada and S. Ieiri, “Kleinman’s symmetry relation in non-linear optical coefficient of LiIO3,” Phys. Lett. A 34, 63–64 (1971).
[CrossRef]

Ito, R.

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
[CrossRef]

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

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, 1667–1681 (1970).
[CrossRef]

J. Jerphagnon, “Optical nonlinear susceptibilities of lithium iodate,” Appl. Phys. Lett. 16, 298–299 (1970).
[CrossRef]

Kaneda, Y.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Kato, K.

K. Kato, “High-power difference-frequency generation at 4.4–5.7 μm in LiIO3,” IEEE J. Quantum Electron. QE-21, 119–120 (1985).
[CrossRef]

Kitamoto, A.

Kondo, T.

I. Shoji, T. Kondo, A. Kitamoto, M. Shirane, and R. Ito, “Absolute scale of second-order nonlinear-optical coefficients,” J. Opt. Soc. Am. B 14, 2268–2294 (1997).
[CrossRef]

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Kupecek, P.

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

Kurimura, S.

Y. Yamamoto, T. Ashida, S. Kurimura, and Y. Uesu, “Two-dimensional observation of the Maker fringe and its application to the poling state evaluation of ferroelectric domains,” Appl. Opt. 36, 602–605 (1997).
[CrossRef] [PubMed]

S. Kurimura and 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 and 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]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Masuda, H.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

Morita, R.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Nath, G.

G. Nath and S. Haussühl, “Large nonlinear optical coefficient and phase matched second harmonic generation in LiIO3,” Appl. Phys. Lett. 14, 154–156 (1969).
[CrossRef]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Ogasawara, N.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Okada, M.

M. Okada and S. Ieiri, “Kleinman’s symmetry relation in non-linear optical coefficient of LiIO3,” Phys. Lett. A 34, 63–64 (1971).
[CrossRef]

Okamoto, N.

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Roberts, D. A.

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Shirane, M.

Shoji, I.

Sugihara, O.

Sugihashi, A.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Tang, C. L.

A. J. Campillo and C. L. Tang, “Spontaneous parametric scattering of light in LiIO3,” Appl. Phys. Lett. 16, 242–244 (1970).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Uesu, Y.

Y. Yamamoto, T. Ashida, S. Kurimura, and Y. Uesu, “Two-dimensional observation of the Maker fringe and its application to the poling state evaluation of ferroelectric domains,” Appl. Opt. 36, 602–605 (1997).
[CrossRef] [PubMed]

S. Kurimura and 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]

Umegaki, S.

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Yamamoto, Y.

Appl. Opt. (1)

Appl. Phys. Lett. (3)

G. Nath and S. Haussühl, “Large nonlinear optical coefficient and phase matched second harmonic generation in LiIO3,” Appl. Phys. Lett. 14, 154–156 (1969).
[CrossRef]

A. J. Campillo and C. L. Tang, “Spontaneous parametric scattering of light in LiIO3,” Appl. Phys. Lett. 16, 242–244 (1970).
[CrossRef]

J. Jerphagnon, “Optical nonlinear susceptibilities of lithium iodate,” Appl. Phys. Lett. 16, 298–299 (1970).
[CrossRef]

IEEE J. Quantum Electron. (3)

D. A. Roberts, “Simplified characterization of uniaxial and biaxial nonlinear optical crystals: a plea for standardization of nomenclature and conventions,” IEEE J. Quantum Electron. 28, 2057–2074 (1992).
[CrossRef]

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, “Absolute and relative nonlinear optical coefficients of KDP, KD*P, BaB2O4, LiIO3, MgO:LiNbO3, and KTP measured by phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 26, 922–933 (1990).
[CrossRef]

K. Kato, “High-power difference-frequency generation at 4.4–5.7 μm in LiIO3,” IEEE J. Quantum Electron. QE-21, 119–120 (1985).
[CrossRef]

J. Appl. Phys. (2)

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

S. Kurimura and 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. Opt. Soc. Am. B (3)

Jpn. J. Appl. Phys. (1)

R. Morita, T. Kondo, Y. Kaneda, A. Sugihashi, N. Ogasawara, S. Umegaki, and R. Ito, “Multiple-reflection effects in optical second-harmonic generation,” Jpn. J. Appl. Phys. 27, L1134–L1136 (1988).
[CrossRef]

Phys. Lett. A (1)

M. Okada and S. Ieiri, “Kleinman’s symmetry relation in non-linear optical coefficient of LiIO3,” Phys. Lett. A 34, 63–64 (1971).
[CrossRef]

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Phys. Rev. B (1)

M. M. Choy and R. L. Byer, “Accurate second-order susceptibility measurements of visible and infrared nonlinear crystals,” Phys. Rev. B 14, 1693–1706 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Rev. Phys. Appl. (1)

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

Other (3)

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

A. V. Smith and M. S. Bowers, “Phase distortions in sum- and difference-frequency mixing in crystals,” J. Opt. Soc. Am. B 12, 49–57 (1995); A. V. Smith, W. J. Alford, T. D. Raymond, and M. S. Bowers, “Comparison of a numerical model with measured performance of a seeded, nanosecond KTP optical parametric oscillator,” J. Opt. Soc. Am. B 12, 2253–2276 (1995). SNLO freeware for modeling χ2 nonlinear optics is available from A. V. Smith.
[CrossRef]

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, New York, 1997).

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Figures (4)

Fig. 1
Fig. 1

Diagram of nonlinear crystal and the fundamental beam plus the free and driven harmonic beams.

Fig. 2
Fig. 2

Diagram of propagation in a birefringent crystal with its optic axis tilted in the same plane as the exit face. Poynting vectors of the incident and reflected waves are labeled Si and Sr respectively. The ρ’s are walk-off angles.

Fig. 3
Fig. 3

Contour plots of second-harmonic fluence at the exit face of 19-mm-long LiIO3 crystal (θ=23°, α=12.55°). The fundamental beam diameter is 0.2 mm, and the calculated e-wave birefringent walk-off is 1.2 mm. Plot (a) is for (eo, o) with a coherence length of 38.5 μm. The left beam is the driven wave created at the crystal exit face by the undisplaced o-polarized fundamental, and the right beam is the free wave created at the crystal input face and displaced by walk-off. Plot (b) is for (ee, e) with a coherence length of 14.3 μm. Both the free and driven waves are displaced by walk-off and overlap at the crystal exit face to create an interference pattern owing to the variation of crystal length across the beam plus the differing wavelengths within the crystal. Plot (c) is for (oe, o) with a coherence length of 12.2 μm. The undeviated o-polarized free wave generated at the crystal input face is evident, but the driven wave is absent because the e- and o-polarized fundamental beams do not overlap at the exit face.

Fig. 4
Fig. 4

Second-harmonic pulse energy in the e-polarized free beam versus the polarization angle of the fundamental light. The solid curve taken without the quarter-wave plate shows that deff has the same sign for (ee, e) and (eo, o). The dashed curve taken with the quarter-wave plate in simulates oppositely signed deff’s.

Tables (8)

Tables Icon

Table 1 |deff| for Crystals Belonging to Symmetry Group 6 (LiIO3) and Group 6mm (CdSe) a

Tables Icon

Table 2 |deff| for Crystals Belonging to Symmetry Group 4¯2m (KDP, AgGaS2, AgGaSe2, ZnGeP2, CdGeAs2) and Group 3m (β-BaB2O4, LiNbO3)

Tables Icon

Table 3 |deff| for KTP Family in the Optical Frame (x, y, z)

Tables Icon

Table 4 |deff| for KNbO3 in the Optical Frame (x, y, z)

Tables Icon

Table 5 |deff| for LiB3O5 in the Optical Frame (x, y, z)

Tables Icon

Table 6 Comparison of Measured and Calculated Δk’s and deffs for θ=75° KDP

Tables Icon

Table 7 Comparison of Measured and Calculated n’s and Δk’s for θ=23° LiIO3

Tables Icon

Table 8 Comparison of Measured d’s for LiIO3

Equations (31)

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

Lc=2π/Δk=2π/|(k2ω-2kω)|.
β=arcsin(n2ω sin α),
2k¯ω=(nω+nω)ω/c=2n¯ωω/c,
γ=arcsin(n¯ω sin α).
δ=β-γ=arcsin(n2ω sin α)-arcsin(n¯ω sin α).
δ=(n2ω-n¯ω) sin α1-n¯ω2 sin2 α=Δk2k0 sin α1-n¯ω2 sin2 α,
Edriven=-Ppar0(n2ω2-n¯ω2),
Efree=Ppar0(n2ω2-n¯ω2) n¯ω+1n2ω+1=Ppar2k00(n2ω+n¯ω)(Δk) n¯ω+1n2ω+1.
Ppar=0deffEω2,
Efree(z)=Efree(0)exp(-2k0κ2ωz).
Edriven(z)=Edriven(0)exp[-(κω+κω)k0z].
ΔkρR,
Δkστ,
Eω(r, t)=E0 exp(-t2/τ2)exp(-r2/R2),
Ufree=Uω2 2deff2π3/2τR20c n¯ω+1n2ω+122k0(n2ω+n¯ω)(Δk)2|tω|2|tω|2|t2ω|2 exp(-2κ2ωωL/c),
tω=21+nω,
tω=21+nω,
t2ω=2n2ω cos αn2ω cos α+cos β
t2ω=2n2ω cos αn2ω cos β+cos α
tω=2cos ρ+nω cos ρ.
t2ω=nr cos(α+ρi)cos ρr+ni cos(σ+ρr)cos ρinr cos β cos ρr+cos(σ+ρr),
××E2ω-4k02˜E2ω=4 k020 PNL exp(i2kωr)
Edriven=-n2ω20(n2ω2-n¯ω2) ˜-1PNL-k¯ω(k¯ωPNL)(k0none)2,
Efree=A˜-1tˆ,
Efree=Ppar0(n2ω2-n¯ω2) n¯ω+1n2ω+1×n2ω4cos2 θno4+sin2 θne4,
Ppar=0deffEω2,
d=000dxyzdxxz0000dxxz-dxyz0dzxxdzxxdzzz000.
d=0000dxxz-dyyy-dyyydyyy0dxxz00dxxzdxxzdzzz000.
Efree=dzyyEω2 sin2 ψn2ω,z2-nω,y2 nω,y+1n2ω,z+12nω,y+12+dzzzEω2 cos2 ψn2ω,z2-nω,z2 nω,z+1n2ω,z+12nω,z+12t2ω.
12 dXYZS2θ
12 (dXYZ+dZXY)S2θC2ϕ

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