Abstract

A general new 4(n+1)×4(n+1) matrix formulation of Maker fringes applicable to any anisotropic material containing n layers, convenient and straightforward for experimental data analyses, is proposed. The treatment of the transmitted and reflected harmonic waves includes the contribution of anisotropic one-photon absorption for the fundamental and harmonic waves under the assumption of no pump depletion and leads to a complete analysis of any linearly or elliptically polarized harmonic signal recorded under various incident polarization configurations. In the framework of the proposed model, we report detailed results of Maker fringes in various samples, for instance, in one and two z-cut quartz plates separated by a controlled air gap.

© 2002 Optical Society of America

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References

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  1. J. Zyss, in Molecular Nonlinear Optics. Materials, Physics and Devices (Academic, Boston, Mass., 1994).
  2. M. G. Kuzyk and C. W. Dirk, in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials (Marcel Dekker, New York, 1998).
  3. 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]
  4. 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]
  5. M. S. Wong, F. Pan, M. Bösch, R. Spreiter, C. Bosshard, P. Günter, and V. Gramlich, “Novel electro-optic molecular cocrystals with ideal chromophoric orientation and large second-order optical nonlinearities,” J. Opt. Soc. Am. B 15, 426–431 (1998).
    [CrossRef]
  6. C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
    [CrossRef]
  7. W. 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]
  8. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
    [CrossRef]
  9. 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]
  10. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742–752 (1989).
    [CrossRef]
  11. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989).
    [CrossRef]
  12. 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, 367–373 (1991).
    [CrossRef]
  13. M. Braun, F. Bauer, T. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4× 4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B 14, 1699–1706 (1997).
    [CrossRef]
  14. M. Braun, F. Bauer, T. Vogtmann, and M. Schwoerer, “Detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials by 4×4 matrix formulation,” J. Opt. Soc. Am. B 15, 2877–2884 (1998).
    [CrossRef]
  15. D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  16. R. M. A. Azzam and N. M. Bashara, in Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 340–355.
  17. F. Abéles, in Optical Properties of Solids (Academic, New York, 1972).
  18. D. Pureur, A. C. Liu, M. J. F. Digonnet, and G. S. Kino, “Absolute prism-assisted Maker fringe measurements of the nonlinear profile in thermally poled silica,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1977), pp. 299–301.
  19. D. Pureur, A. C. Liu, J. F. Digonnet, and G. S. Kino, “Absolute measurement of the second-order nonlinearity profile in poled silica,” Opt. Lett. 23, 588–590 (1998).
    [CrossRef]
  20. Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
    [CrossRef]
  21. D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
    [CrossRef]
  22. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, in Handbook of Nonlinear Optical Crystal. Optical Sciences 2nd ed. (Springer, Berlin, 1997), pp. 67–288.
  23. V. Rodriguez and C. Sourisseau, “SHG at near resonance and chromophore orientational distribution function in p(DR1M) thin films oriented by corona poling,” Nonlinear Opt. 25, 259–264 (2000).
  24. V. Rodriguez and F. Adamietz, “Time evolution and polar orientation of the orientational distribution function of DR1/PMMA polymer films wire poled under high field conditions,” presented at ICONO’6, Tucson, Ariz., December 16–20, 2001.
  25. G. A. Lindsay and K. D. Singer, in “Polymer for second-order nonlinear optics,” ACS Symp. Ser. 601, 130–238 (1995).
  26. U. Meier, M. Bösch, C. Bosshard, and P. Günter, “DAST a high optical nonlinearity organic crystal,” Synth. Met. 109, 19–22 (2000).
    [CrossRef]
  27. A. Yokoo, I. Yokohama, H. Kobayashi, and T. Kaino, “Linear and nonlinear optical properties of the organic nonlinear optical crystal 2-adamantylamino-5-nitropyridine,” J. Opt. Soc. Am. B 15, 432–437 (1998).
    [CrossRef]
  28. B. Ferreira, “Etude de la génération de seconde harmonique dans des verres polarisés thermiquement,” Ph.D. dissertation (University Bordeaux I, Bordeaux, France, 2002).

2000 (4)

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[CrossRef]

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

V. Rodriguez and C. Sourisseau, “SHG at near resonance and chromophore orientational distribution function in p(DR1M) thin films oriented by corona poling,” Nonlinear Opt. 25, 259–264 (2000).

U. Meier, M. Bösch, C. Bosshard, and P. Günter, “DAST a high optical nonlinearity organic crystal,” Synth. Met. 109, 19–22 (2000).
[CrossRef]

1998 (4)

1997 (2)

1995 (2)

W. 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]

G. A. Lindsay and K. D. Singer, in “Polymer for second-order nonlinear optics,” ACS Symp. Ser. 601, 130–238 (1995).

1991 (1)

1989 (2)

1972 (1)

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, 1667–1681 (1970).
[CrossRef]

1962 (3)

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]

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

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Bauer, F.

Bernage, P.

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

Berreman, D. W.

Bethune, D. S.

Bloembergen, N.

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

Bösch, M.

Bosshard, C.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[CrossRef]

U. Meier, M. Bösch, C. Bosshard, and P. Günter, “DAST a high optical nonlinearity organic crystal,” Synth. Met. 109, 19–22 (2000).
[CrossRef]

M. S. Wong, F. Pan, M. Bösch, R. Spreiter, C. Bosshard, P. Günter, and V. Gramlich, “Novel electro-optic molecular cocrystals with ideal chromophoric orientation and large second-order optical nonlinearities,” J. Opt. Soc. Am. B 15, 426–431 (1998).
[CrossRef]

Braun, M.

Digonnet, J. F.

Douay, M.

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

Duthérage, P.

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

Gramlich, V.

Gubler, U.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[CrossRef]

Günter, P.

Hayden, L. M.

Herman, W.

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, 1667–1681 (1970).
[CrossRef]

Kaatz, P.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[CrossRef]

Kaino, T.

King, L. A.

Kino, G. S.

Kitamoto, A.

Kleinman, D. A.

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Kobayashi, H.

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, 1667–1681 (1970).
[CrossRef]

Kuzyk, M. G.

Lindsay, G. A.

G. A. Lindsay and K. D. Singer, in “Polymer for second-order nonlinear optics,” ACS Symp. Ser. 601, 130–238 (1995).

Liu, A. C.

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]

Martinelli, G.

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

Mazerant, W.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[CrossRef]

Meier, U.

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[CrossRef]

U. Meier, M. Bösch, C. Bosshard, and P. Günter, “DAST a high optical nonlinearity organic crystal,” Synth. Met. 109, 19–22 (2000).
[CrossRef]

Niay, P.

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[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]

Pan, F.

Pershan, P. S.

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

Pureur, D.

Quiquempois, Y.

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

Rodriguez, V.

V. Rodriguez and C. Sourisseau, “SHG at near resonance and chromophore orientational distribution function in p(DR1M) thin films oriented by corona poling,” Nonlinear Opt. 25, 259–264 (2000).

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]

Schwoerer, M.

Shirane, M.

Shoji, I.

Singer, K. D.

Sourisseau, C.

V. Rodriguez and C. Sourisseau, “SHG at near resonance and chromophore orientational distribution function in p(DR1M) thin films oriented by corona poling,” Nonlinear Opt. 25, 259–264 (2000).

Spreiter, R.

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]

Vogtmann, T.

Wong, M. S.

Yokohama, I.

Yokoo, A.

Zahn, H. E.

ACS Symp. Ser. (1)

G. A. Lindsay and K. D. Singer, in “Polymer for second-order nonlinear optics,” ACS Symp. Ser. 601, 130–238 (1995).

J. Appl. Phys. (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, 1667–1681 (1970).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (9)

M. Braun, F. Bauer, T. Vogtmann, and M. Schwoerer, “Detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials by 4×4 matrix formulation,” J. Opt. Soc. Am. B 15, 2877–2884 (1998).
[CrossRef]

M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742–752 (1989).
[CrossRef]

D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989).
[CrossRef]

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, 367–373 (1991).
[CrossRef]

W. 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]

M. Braun, F. Bauer, T. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4× 4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B 14, 1699–1706 (1997).
[CrossRef]

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]

M. S. Wong, F. Pan, M. Bösch, R. Spreiter, C. Bosshard, P. Günter, and V. Gramlich, “Novel electro-optic molecular cocrystals with ideal chromophoric orientation and large second-order optical nonlinearities,” J. Opt. Soc. Am. B 15, 426–431 (1998).
[CrossRef]

A. Yokoo, I. Yokohama, H. Kobayashi, and T. Kaino, “Linear and nonlinear optical properties of the organic nonlinear optical crystal 2-adamantylamino-5-nitropyridine,” J. Opt. Soc. Am. B 15, 432–437 (1998).
[CrossRef]

Nonlinear Opt. (1)

V. Rodriguez and C. Sourisseau, “SHG at near resonance and chromophore orientational distribution function in p(DR1M) thin films oriented by corona poling,” Nonlinear Opt. 25, 259–264 (2000).

Opt. Commun. (1)

Y. Quiquempois, G. Martinelli, P. Duthérage, P. Bernage, P. Niay, and M. Douay, “Localization of the induced second-order nonlinearity within Infrasil and Suprasil thermally poled glasses,” Opt. Commun. 176, 479–487 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (2)

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

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)

C. Bosshard, U. Gubler, P. Kaatz, W. Mazerant, and U. Meier, “Non-phase-matched optical third-harmonic generation in noncentrosymmetric media: cascaded second-order contributions for the calibration of third-order nonlinearities,” Phys. Rev. B 61, 10688–10701 (2000).
[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]

Synth. Met. (1)

U. Meier, M. Bösch, C. Bosshard, and P. Günter, “DAST a high optical nonlinearity organic crystal,” Synth. Met. 109, 19–22 (2000).
[CrossRef]

Other (8)

R. M. A. Azzam and N. M. Bashara, in Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 340–355.

F. Abéles, in Optical Properties of Solids (Academic, New York, 1972).

D. Pureur, A. C. Liu, M. J. F. Digonnet, and G. S. Kino, “Absolute prism-assisted Maker fringe measurements of the nonlinear profile in thermally poled silica,” in Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals, Vol. 17 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1977), pp. 299–301.

B. Ferreira, “Etude de la génération de seconde harmonique dans des verres polarisés thermiquement,” Ph.D. dissertation (University Bordeaux I, Bordeaux, France, 2002).

V. Rodriguez and F. Adamietz, “Time evolution and polar orientation of the orientational distribution function of DR1/PMMA polymer films wire poled under high field conditions,” presented at ICONO’6, Tucson, Ariz., December 16–20, 2001.

J. Zyss, in Molecular Nonlinear Optics. Materials, Physics and Devices (Academic, Boston, Mass., 1994).

M. G. Kuzyk and C. W. Dirk, in Characterization Techniques and Tabulations for Organic Nonlinear Optical Materials (Marcel Dekker, New York, 1998).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, in Handbook of Nonlinear Optical Crystal. Optical Sciences 2nd ed. (Springer, Berlin, 1997), pp. 67–288.

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Figures (14)

Fig. 1
Fig. 1

Schematic representation of a multilayer planar system.

Fig. 2
Fig. 2

Representation of the four generalized field vectors in layer i at the two boundaries’ interfaces +(zi) and -(zi+di). In the semi-infinite approximation, the two reflected solutions, denoted 3 and 4, at zi are eliminated from the boundary conditions for interface i1/i.

Fig. 3
Fig. 3

Propagation of the bound wave vector, ki,b2ω, resulting from the sum of the two fundamental wave vectors ki,kω and ki,lω. This harmonic bound source term can interfere with the two free harmonic rays with wave vector ki,12ω and ki,22ω.

Fig. 4
Fig. 4

Sample quasi-index-matching setup using hemicylindrical lenses, corresponding to media 1 and n. The NLO optical path of layer i, which is typically twice that with suprasil hemicylindrical lenses, enhances the Maker-fringe resolution when the thickness of layer i is well beyond its coherence length.

Fig. 5
Fig. 5

Scheme of the experimental setup for SHG acquisitions: BA, beam attenuator; BS, beam splitter; D, near-infrared diode; F, 1064-nm filter or LiNbO3 crystal and 532-nm filter; PM1, 532-nm filter, analyzer, and photomultiplier tube; PM2, 532-nm or 1064-nm filter, analyzer, and photomultiplier tube. The incident laser beam is focused on the sample, which is mounted upon a double goniometer. A first unit detection (PM1) permits the transmitted SHG signal to be detected, and a coupled rotation (2θ) of the second detection unit (PM2), with respect to the sample angle (θ), allows reflection measurements.

Fig. 6
Fig. 6

Scheme of the SHG Maker-fringe experiments on a z cut of an α-quartz plate with thickness ∼500 µm. The angle ϕ1 between the rotation axis and the x axis of the crystal was set at ∼105°.

Fig. 7
Fig. 7

Experimental (diamonds), calculated (curves), and difference (crosses) Maker fringes on a z-cut α-quartz plate with thickness 500 µm (ϕ1105°). The patterns were recorded over the [-80°; +80°] angular range with various polarizations for the incident and transmitted beams, namely, s (vertical or 90°), p (horizontal or 0°), 30°, 45°, or 60°.

Fig. 8
Fig. 8

Ellipsometric analysis of experimental (diamonds) and calculated (curves) Maker fringes of the α-quartz plate with Φ1105° and thickness 500 µm. The incident fundamental polarization was p, and the transmitted second-harmonic signal was recorded with a linear analyzer varying from p (0°) to s (90°) position in steps of 15°.

Fig. 9
Fig. 9

Scheme of the SHG Maker fringes of two successive z-cut α-quartz plates, with a respective thickness of ∼500 µm (plate 1) and ∼340 µm (plate 2), separated by a 50-µm air gap. The angles between the horizontal axis X and the x1 and x2 axes of each crystal were set at ϕ1199° and ϕ1274°, respectively. The z1 axis of the first plate points toward a Z-negative value (ϕ31=180°), where the z2 axis of the second one points toward a Z-positive value (ϕ32=0°).

Fig. 10
Fig. 10

Experimental [diamonds: (a), (b)], calculated [curves: (a), (b)], and difference [crosses: (b)] Maker fringes of two successive z-cut α-quartz plates, with a respective thickness of 500 µm and 340 µm, separated by an air gap of 50 µm. The SHG patterns were calculated (a) under the assumption that the net output field is only the sum of contributions from individual layers and (b) with the model described in this paper. The polarization configurations were sp, pp, 30°–30°, and 60°–60° (see comments in Fig. 7 caption).

Fig. 11
Fig. 11

Effect of the poling conditions and of the d31/d33 ratio value on the normalized Maker-fringe pp patterns (over the +80°, -80° angular range) for a representative isotropically absorbing NLO poled polymer film [thickness=1 µm, α532 nm=5.7 µm-1, n(1064 nm)=1.62, n(532 nm)=1.70, coherence length lc(θ=0)=3.48 µm]: (curve) d31/d33=1/3, i.e., low poling field; (crosses) d31/d33=1/5, i.e., medium poling field; (filled circles) d31/d33=1/6, i.e., high poling field.

Fig. 12
Fig. 12

Effect of the poling conditions and of the d31/d33 ratio value on the Maker-fringe pp patterns (over the +80°, -80° angular range) for differently absorbing NLO poled polymer films with either an isotropic absorption (curves: P2=0), a weakly anisotropic absorption (crosses: P2=0.2], or a strongly anisotropic absorption (circles: P2=0.4). The plots reported in (a), (b), and (c) correspond to low poling field (d31/d33=1/3), medium poling field (d31/d33=1/5), and high poling field (d31/d33=1/6) conditions, respectively. For a better comparison, the SHG intensities are normalized in each set of plots to the maximum reached in the isotropic case (Δα(532 nm)=0).

Fig. 13
Fig. 13

Experimental (diamonds) and calculated (curves) Maker-fringe patterns under the 45°–45° polarization configuration, for an α-quartz plate as a function of the fundamental intensity I(ω) for a fringe maximum occurring at θ=38°. The parabolic curve gives the quadratic dependence of SHG intensity.

Fig. 14
Fig. 14

Experimental (diamonds) and calculated (curves) Maker fringes in the ps configuration from a y-cut plate of KDP of 1530-µm thickness. The vertical x crystal axis was the rotation axis, and the z crystal axis was in the plane of incidence.

Tables (1)

Tables Icon

Table 1 Best Set of Data Obtained from the Fit of the Experimental Maker Fringes, within the Proposed Model, for a z-Cut α-Quartz Plate

Equations (45)

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(εilω)(x, y, z)=n˜i,xlω2000n˜i,ylω2000n˜i,zlω2,
(εilω)(X, Y, Z)=Ui(εilω)(x, y, z)Ui-1,
Ui=cos ϕ1 cos ϕ3-sin ϕ1 sin ϕ2 sin ϕ3sin ϕ1 cos ϕ2cos ϕ1 sin ϕ3+sin ϕ1 sin ϕ2 cos ϕ3-sin ϕ1 cos ϕ3-cos ϕ1 sin ϕ2 sin ϕ3cos ϕ1 cos ϕ2-sin ϕ1 sin ϕ3+cos ϕ1 sin ϕ2 cos ϕ3-cos ϕ2 sin ϕ3-sin ϕ2cos ϕ2 cos ϕ3,
(X, Y, Z)RZ(ϕ1)(X, Y, Z)RX(ϕ2)(X, Y, Z)RY(ϕ3)(x, y, z).
dψilω(z)=-jk1lωΔilωψilω(z)dz,
ψiω(z)=Ei,xlωHi,ylωEi,ylω-Hi,xlω
ψilω(zi+di)=Lilω(di)×ψilω(zi),
Ψilω=QilωZilω.
Qilω=(ψi,1lωψi,2lωψi,3lωψi,4lω),
Ψilω(zi)=-Ψilω(zi)=-Qilω(zi)×Zilω.
Ψilω(zi+di)=+Ψilω(zi+di)=+Qilω(zi+di)×Zilω.
layer(i-1)layer(i)layer(i+1) +Qi-1lωZi-1lω= -QilωZilω +QilωZilω= -Qi+1lωZi+1lω,
Ψilω=QilωZilω=cos θilω0-cos θilω0nilω0nilω001010nilω cos θilω0-nilω cos θilω×Ei,1lωEi,2lωEi,3lωEi,4lω,
 rΨ1lω=rQ1lωrZ1lω=-cos θ1lω0n1lω0010-n1lω cos θ1lω×E1,3lωE1,4lω.
 tΨnlω=tQnlω tZnlω=cos θnlω0nnlω0010nnlω cos θnlω×En,1lωEn,2lω.
MlωZlω=Ωlω,
Mlω=(1)(2)(i-1)(i)(i+1)(n-1)(n)+rQ1lω--Q2lω00++Q2lω··000++Qi-1lω--Qilω00++Qilω--Qi+1lω000··--Qn-1lω00++Qn-1lω-tQnlω,
Zlω= rZ1lωZ2lωZi-1lωZilωZi+1lωZn-1lω tZnlω.
Zlω=(Mlω)-1Ωlω.
 -Qilω=(-Qi,1lω-Qi,2lω00),
+Qilω=(+Qi,1lω+Qi,2lω+Qi,3lω+Qi,4lω).
 tΨ1ω=tQ1ω tZ1ω=cos θ1ω0n1ω0010n1ω cos θ1ω× tE1,1ω tE1,2ω= tΨ1,pω tΨ1,sω,
Ωω=-tΨ1ω00.
Pi,j2ω=12k=14l=14χjkl(2)(-2ω; ω, ω)Ei,kωEi,lω=12k=14l=14djmi(-2ω; ω, ω)Ei,kωEi,lω=12k=14l=14d11id12id13id14id15id16id21id22id23id24id25id26id31id32id33id34id35id36i×Ei,k,xωEi,l,xωEi,k,yωEi,l,yωEi,k,zωEi,l,zωEi,k,yωEi,l,zω+Ei,k,zωEi,l,yωEi,k,xωEi,l,zω+Ei,k,zωEi,l,xωEi,k,xωEi,l,yω+Ei,k,zωEi,l,yω,
Ei,kω(z)=Qi,kω(z)Zi,kω=|Qi,kω(z)Zi,kω|exp(-jki,k,zωz),
EtoωEtoωEtoωEteωEteωEtoωEteωEteωEtoωEroωEtoωEreωEteωEroωEteωEreω/EroωEtoωEroωEteωEreωEtoωEreωEteωEroωEroωEroωEreωEreωEroωEreωEreω.
(c/2ω)2××Ei,b2ω(z)-(εi2ω)Ei,b2ω(z)=4πPi,b2ω,
Ei,b2ω(z)=ei,b2ωexp(-jki,b,z2ωz),
ki,b2ω=ki,kω+ki,lω.
1/4π(c/2ω)2Bi,b2ωEi,b2ω(z)=Pi,b2ω(z),
Bi,b2ω=(ki,b2ω)y2+(ki,b2ω)z2-bi,x2ω-(ki,b2ω)x(ki,b2ω)y-(ki,b2ω)x(ki,b2ω)z-(ki,b2ω)x(ki,b2ω)y(ki,b2ω)x2+(ki,b2ω)z2-bi,y2ω-(ki,b2ω)y(ki,b2ω)z-(ki,b2ω)x(ki,b2ω)z-(ki,b2ω)y(ki,b2ω)z(ki,b2ω)x2+(ki,b2ω)y2-bi,z2ω,
bi,τ2ω=(2ω/c)2ε˜i,τ2ω,τ=x, y,orz.
Ei,b2ω(z)=4π(2ω/c)2(Bi,b2ω)-1Pi,b2ω(z).
ψi,b2ω(z)=Ei,b,x2ωHi,b,y2ωEi,b,y2ω-Hi,b,x2ω.
 bΨi2ω(z)=b=18ψi,b2ω(z).
Ω2ω=--bΨ22ω +bΨ22ω--bΨ32ω  +bΨi-12ω--bΨi2ω+bΨi2ω--bΨi+12ω +bΨn-22ω--bΨn-12ω +bΨn-12ψ .
tan ψ exp(jΔ)=E˜y2ω/E˜x2ω,
 tE1,jω=2tE0,jω/(1+n1ω) fortheincomingfundamentalwave,
 rE0,j2ω=2n12ωrE1,j2ω/(1+n12ω) forthereflectedharmonicwave,
 tEn+1,j2ω=2nn2ω  tEn,j2ω/(1+nn2ω) forthetransmittedharmonicwave,
d=d11-d11000000000-d11000000.
d=0000d310000d310d31d31d33000.
Rp=j|Ijexp(2ω)-Ijcalc(2ω)|2j|Ijexp(2ω)|2,
Rwp
=jwj|Ijexp(2ω)-Ijcalc(2ω)|2jwj|Ijexp(2ω)|2,

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