Abstract

We demonstrate how the Maker fringes that are observable in spontaneous parametric down-conversion (SPDC) give a direct visualization of the poling quality of a periodically-poled crystal. Identical Maker fringes are observed in the optical spectrum of collinear SPDC and the temperature dependence of second harmonic generation. We analyze these Maker fringes via a unified treatment of the tuning curve in crystals with small and slowly-varying deformations of the poling structure. Our theoretical model, based on a Fourier analysis of the poling deformations, distinguishes between duty-cycle variations and variations of the poling phase. The analysis indicates that the poling phase is approximately fixed, while the duty-cycle typically varies between 36% and 64%.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
    [CrossRef]
  2. M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second-harmonic generation," J. Phys. D: Appl. Phys. 28, 1747-1763 (1995).
    [CrossRef]
  3. C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006).
    [CrossRef]
  4. F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
    [CrossRef]
  5. H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
    [CrossRef]
  6. G. Rosenman, A. Skliar, I. Lareah, N. Angert, M. Tseitlin, and M. Roth, "Observation of ferroelectric domain structures by secondary-electron microscopy in as-grown KTiOPO4," Phys. Rev. B 54, 6222 (1996).
    [CrossRef]
  7. C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003).
    [CrossRef]
  8. S. K. Johansen and P. Baldi, "Characterization of quasi-phase-matching gratings in quadratic media through double-pass second-harmonic power measurements," J. Opt. Soc. Am. B 21, 1137 (2004).
    [CrossRef]
  9. I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
    [CrossRef]
  10. S. J. Holmgren, V. Pasiskevicius, S. Wang, and F. Laurell, "Three-dimensional characterization of the effective second-order nonlinearity in periodically poled crystals," Opt. Lett. 28, 1555 (2003).
    [CrossRef] [PubMed]
  11. G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
    [CrossRef]
  12. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, "Effects of dispersion and focussing on the production of optical harmonics," Phys. Rev. Lett. 8, 21 (1962).
    [CrossRef]
  13. S. Emanueli and A. Arie, "Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4," Appl. Opt. 42, 6661 (2003).
    [CrossRef] [PubMed]
  14. F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
    [CrossRef]
  15. T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOP04," Appl. Opt. 26, 2390 (1987).
    [CrossRef] [PubMed]
  16. Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
    [CrossRef]
  17. D. W. Anthon and C. D. Crowder, "Wavelength dependent phase matching in KTP," Appl. Opt. 27, 2650 (1988).
    [CrossRef] [PubMed]
  18. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
    [CrossRef]
  19. K. Kato and E. Takaoka, "Sellmeier and thermo-optic dispersion formulas for KTP," Appl. Opt. 41, 5040 (2002).
    [CrossRef] [PubMed]
  20. W. Wiechmann and S. Kubota, "Refractive-index temperature derivatives of potassium titanyl phosphate," Opt. Lett. 18, 1208-1210 (1993).
    [CrossRef] [PubMed]
  21. R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
    [CrossRef]
  22. M. V. Pack, D. J. Armstrong, and A. V. Smith, "Measurement of the |(2) tensors of KTiOPO4, KTiOAsO4, RbTiOPO4, and RbTiOAsO4 crystals," Appl. Opt. 43, 3319 (2004).
    [CrossRef] [PubMed]
  23. M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
    [CrossRef]
  24. G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
    [CrossRef]
  25. G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
    [CrossRef]

2007 (1)

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

2006 (1)

C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006).
[CrossRef]

2005 (1)

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

2004 (2)

2003 (3)

2002 (1)

2001 (1)

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

1999 (1)

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

1998 (2)

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

1997 (1)

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

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

1993 (2)

W. Wiechmann and S. Kubota, "Refractive-index temperature derivatives of potassium titanyl phosphate," Opt. Lett. 18, 1208-1210 (1993).
[CrossRef] [PubMed]

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

1992 (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (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-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

1988 (2)

D. W. Anthon and C. D. Crowder, "Wavelength dependent phase matching in KTP," Appl. Opt. 27, 2650 (1988).
[CrossRef] [PubMed]

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

1987 (1)

1962 (1)

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

Angert, N.

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

Anthon, D. W.

Arie, A.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

S. Emanueli and A. Arie, "Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4," Appl. Opt. 42, 6661 (2003).
[CrossRef] [PubMed]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Armstrong, D. J.

Aust, J. A.

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

Baldi, P.

Bassi, P.

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

Bierlein, J. D.

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

Bindloss, W.

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

Bluhm, H.

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOP04," Appl. Opt. 26, 2390 (1987).
[CrossRef] [PubMed]

Canalias, C.

C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006).
[CrossRef]

C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003).
[CrossRef]

Cristiani, I.

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

Crowder, C. D.

Degiorgio, V.

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

Dyakov, V. A.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[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-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOP04," Appl. Opt. 26, 2390 (1987).
[CrossRef] [PubMed]

Eger, D.

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

Emanueli, S.

Enger, D.

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

Fan, T. Y.

Fan, Y. X.

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, "Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOP04," Appl. Opt. 26, 2390 (1987).
[CrossRef] [PubMed]

Feigelson, R. S.

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Ferraro, P.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Fradkin, K.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Fragemann, A.

C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003).
[CrossRef]

Garb, K.

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

Grilli, S.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Holmgren, S. J.

Hsiung, H.

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

Hu, B. Q.

Huang, C. E.

Johansen, S. K.

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Kang, C. H.

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Kato, K.

Katz, M.

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

Kholodnykh, A. I.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Kitaeva, G. K. H.

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Krasnikov, V. V.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Kubota, S.

Lareah, I.

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

Laurell, F.

C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006).
[CrossRef]

S. J. Holmgren, V. Pasiskevicius, S. Wang, and F. Laurell, "Three-dimensional characterization of the effective second-order nonlinearity in periodically poled crystals," Opt. Lett. 28, 1555 (2003).
[CrossRef] [PubMed]

C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003).
[CrossRef]

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

Liberale, C.

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, "Effects of dispersion and focussing on the production of optical harmonics," Phys. Rev. Lett. 8, 21 (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-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

Nada, N.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Nam, D.

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

Natale, P. D.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Naumova, I. I.

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Nicola, S. D.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Nisenoff, M.

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

Oron, M.

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

Pack, M. V.

Pasiskevicius, V.

C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006).
[CrossRef]

S. J. Holmgren, V. Pasiskevicius, S. Wang, and F. Laurell, "Three-dimensional characterization of the effective second-order nonlinearity in periodically poled crystals," Opt. Lett. 28, 1555 (2003).
[CrossRef] [PubMed]

C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003).
[CrossRef]

Penin, A. N.

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Pignatiello, F.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Pryalkin, V. I.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Pshenichnikov, M. S.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Razumikhina, T. B.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Roelofs, M. G.

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

Rosa, M. D.

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Rosenman, G.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

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

Roshko, A.

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

Roth, M.

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

Saitoh, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Savage, C. M.

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

Skliar, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

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

Smith, A. V.

Solomatin, V. S.

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Suna, A.

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

Takaoka, E.

Tang, S. H.

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Tartarini, G.

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

Terhune, R. W.

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

Tishkova, V. V.

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Tseitlin, M.

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

Wadas, A.

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

Wang, S.

Watanabe, K.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Wiechmann, W.

Wiesendanger, R.

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

Yamada, M.

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. B (1)

G. K. H. Kitaeva, V. V. Tishkova, I. I. Naumova, A. N. Penin, C. H. Kang, and S. H. Tang, "Mapping of periodically poled crystals via spontaneous parametric down conversion," Appl. Phys. B 81, 645-650 (2005).
[CrossRef]

Appl. Phys. Lett. (6)

M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, "First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation," Appl. Phys. Lett. 62, 435 (1993).
[CrossRef]

G. Rosenman, K. Garb, A. Skliar, M. Oron, D. Eger, and M. Katz, "Domain broadening in quasi-phase-matched nonlinear optical devices," Appl. Phys. Lett. 73, 865 (1998).
[CrossRef]

G. Rosenman, A. Skliar, D. Enger, M. Oron, and M. Katz, "Low temperature periodic electrical poling of fluxgrown KTiOPO4 and isomorphic crystals," Appl. Phys. Lett. 73, 3650 (1998).
[CrossRef]

H. Bluhm, A. Wadas, R. Wiesendanger, A. Roshko, J. A. Aust, and D. Nam, "Imaging of domain-inverted gratings in LiNbO3 by electrostatic force microscopy." Appl. Phys. Lett. 71, 146 (1997).
[CrossRef]

C. Canalias, V. Pasiskevicius, A. Fragemann, and F. Laurell, "High-resolution domain imaging on the nonpolar y-face of periodically poled KTiOPO4 by means of atomic force microscopy." Appl. Phys. Lett. 83, 734 (2003).
[CrossRef]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914 (1999).
[CrossRef]

Ferroelectrics (1)

C. Canalias, V. Pasiskevicius, and F. Laurell, "Periodic poling of KTiOPO4: from micrometer to sub-micrometer domain gratings," Ferroelectrics 340, 27-47 (2006).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, "Quasi-phase-matched second harmonic generation: tuning and tolerances," IEEE J. Quantum Electron. 28, 2631 (1992).
[CrossRef]

R. C. Eckardt, H. Masuda, Y. X. Fan, and R. L. Byer, "Absolute and relative nonlinear optical coefficients of KDP, KD-STAR-P, BaB2O3, LiIO3, MgO-LiNbO3, and KTP measured by phase-matched 2nd harmonic generation," IEEE J. Quantum Electron. 26, 922 (1990).
[CrossRef]

J. Appl. Phys. (1)

F. Laurell, M. G. Roelofs, W. Bindloss, H. Hsiung, A. Suna, and J. D. Bierlein, "Detection of ferroelectric domain reversal in KTiOPO4 waveguides," J. Appl. Phys. 71, 15 (1992).
[CrossRef]

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

J. Phys. D: Appl. Phys. (1)

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

Opt. Commun. (2)

I. Cristiani, C. Liberale, V. Degiorgio, G. Tartarini, and P. Bassi, "Nonlinear characterization and modeling of periodically poled lithium niobate waveguides for 1.5-�??m-band cascaded wavelength conversion," Opt. Commun. 187, 263 (2001).
[CrossRef]

F. Pignatiello, M. D. Rosa, P. Ferraro, S. Grilli, P. D. Natale, A. Arie, and S. D. Nicola, "Measurement of the thermal expansion coefficients of ferroelectric crystals by a moir�??e interferometer," Opt. Commun. 227, 14 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (1)

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

Phys. Rev. Lett. (1)

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

Sov. J. Quant. Electron. (1)

Q1. V. A. Dyakov, V. V. Krasnikov, V. I. Pryalkin, M. S. Pshenichnikov, T. B. Razumikhina, V. S. Solomatin, and A. I. Kholodnykh, "Sellmeier equation and tuning characteristics of PPKTP crystal frequency converters in the 0.4-4.0 ?m range," Sov. J. Quant. Electron. 18, 1059-1060 (1988).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Three different experimental setups that are used to measure Maker fringes of a periodically-poled KTiOPO4 crystal (PPKTP). Setup A and B utilize the nonlinear processes of spontaneous parametric down conversion, while setup C uses second harmonic generation. The indicated coordinate system corresponds to the crystallographic axes of the PPKTP crystal. Setup A: The far field of a narrow spectral band of the down-converted light is projected onto an intensified CCD camera (ICCD). Setup B: The spectral decomposition within a small angle (selected by the aperture) of the down-converted light is measured by placing an ICCD camera behind a spectrometer. Setup C: The power of the up-converted light is measured with a photodiode, while the crystal temperature is varied.

Fig. 2.
Fig. 2.

Measured angular intensity pattern of light generated via spontaneous parametric down conversion in a 5-mm-long periodically-poled KTP crystal. The detection bandwidth is much smaller than the inverse angular dispersion of the Maker fringes. The color scale represents the logarithm (log10) of the intensity divided by the maximum intensity of the three images. The full width and full height of each image is 96 mrad. The three images are taken at different crystal temperatures: (a) 53.5 °C, (b) 60.7 °C, and (c) 72.0 °C. Phase matching in forward direction is achieved in situation (b). The sixth and ninth fringe stand out and are marked with triangles in each image. The same PPKTP crystal is used in Figs. 3, 4, 5, and 6. The fringe pattern is slightly elliptical.

Fig. 3.
Fig. 3.

Composite plot of the measured angular intensity pattern of light generated via spontaneous parametric down conversion in a 5-mm-long periodically-poled KTP crystal. The plot is obtained by combining 25 images of Maker fringes, like the ones shown in Fig. 2, photographed at different crystal temperatures ranging from 45 °C to 89 °C. The horizontal axis represents the square of the far field angle at a crystal temperature of 60.7 °C. The intensity values of the nodes have limited quantitative meaning (see text for details). The sixth and ninth fringe stand out and are marked with arrows. The same PPKTP crystal is used in Figs. 2, 4, 5, and 6.

Fig. 4.
Fig. 4.

Measured SPDC ring pattern observed for pumping at two different positions on the crystal. The conditions are similar to those described in Fig. 2. The crystal temperature is now 69.7 °C, the image size is 90×90 mrad2, and the false color scale has been adjusted. The pump positions on the crystal are separated by 1.4 mm in the y direction. The images are very much alike. For instance, the sixth and ninth fringe stand out and are marked with triangles in both images. Only minor differences are visible; these require a good printer. The patterns do not change upon a pump displacement in z direction. The same PPKTP crystal is used in Figs. 2, 3, 5, and 6.

Fig. 5.
Fig. 5.

Spectrum of light generated via spontaneous parametric down conversion in a 5-mm-long periodically-poled KTP crystal. The crystal is pumped by a mildly focussed beam at a wavelength of 413.1 nm. This spectrum is measured within a cone-shaped solid angle of 42 mrad2 in forward direction. The crystal temperature is 60.7 °C at which phase matching is achieved in forward direction [see Fig. 2(b)]. The sixth fringe stands out and is marked with an arrow. The same PPKTP crystal is used in Figs. 2, 3, 4, and 6.

Fig. 6.
Fig. 6.

Red curve: measured temperature dependence of the conversion efficiency in SHG in a 5-mm-long periodically-poled KTP crystal that is pumped by a weakly focussed beam (w 0=61±4 µm) at a wavelength of 825.9 nm and a pump power of 270 mW. The nonlinear temperature dependence of the mismatch parameter ϕ is indicated by the nonequidistant markers on the ϕ axis on top of the figure. The sixth and ninth fringe stand out and are marked with arrows. The same PPKTP crystal is used in Figs. 2, 3, 4, and 5. Black curve: plot of the ideal non-deformed tuning curve of Eq. (2).

Fig. 7.
Fig. 7.

Red curve: measured temperature dependence of the conversion efficiency in SHG in a (different) 5-mm-long periodically-poled KTP crystal that is pumped by a weakly focussed beam (w 0=61±4 µm) at a wavelength of 826.4 nm and a pump power of 270 mW. The nonlinear temperature dependence of the mismatch parameter ϕ is indicated by the non-equidistant markers on the ϕ axis on top of the figure. Black curve: plot of the ideal non-deformed tuning curve of Eq. (2).

Equations (31)

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

ϕ m = L 0 2 [ Δ k f ( T ) 2 π m Λ 0 ] ,
η m ( ϕ m ) ( 2 π m ) 2 sinc 2 ( ϕ m ) ,
Δ k SHG = k ( 2 ω , T ) 2 k ( ω , T ) ,
Δ k SHG = k ( 2 ω , T 0 ) 2 k ( ω , T 0 )
+ 2 ω c [ Δ n z ( 2 ω , T ) Δ n z ( ω , T ) ] ,
Δ k SPDC = k ( 2 ω , T ) [ k s , x + k i , x ] ,
Δ k SPDC = k ( 2 ω , T ) 2 k ( ω , T )
+ k ( ω , T ) [ θ y 2 + ( n z n x ) 2 θ z 2 ]
2 k ω 2 ω , T ( Ω 2 ) 2 ,
Δ n z ( 2 ω , T ) Δ n z ( ω , T ) = c 1 [ T T 0 ] + c 2 [ T T 0 ] 2 ,
Φ n Δ k 0 δ x n .
E ̂ ( δ Δ k ) E ( δ Δ k ) E perfect = 1 N n = 1 N e i δ Δ k x e i Φ n .
Φ ( x ¯ n , 0 ) 1 2 ( Φ n + Φ n + 1 ) ,
A ( x ¯ n , 0 ) cos [ 1 2 ( Φ n Φ n + 1 ) ] ,
E ̂ ( δ Δ k ) 1 L L 2 L 2 A ( x ) e i Φ ( x ) e i δ Δ k x d x .
η ̂ ( δ Δ k ) = E ̂ ( δ Δ k ) 2 .
Φ ( x ) a 0 2 = n = 1 a n cos ( 2 π n x L ) + b n sin ( 2 π n x L ) ,
A ( x ) A 0 1 = n = 1 c n cos ( 2 π n x L ) + d n sin ( 2 π n x L ) ,
E ̂ ( ϕ ) A 0 sinc ( ϕ ) + n = 1 c n i a n + b n + i d n 2 sinc ( ϕ + n π )
+ n = 1 c n i a n b n i d n 2 sinc ( ϕ n π ) ,
η ̂ ( ϕ = ± n π ) A 0 2 α ~ n β ~ n 2 2 ,
η ̂ ( ϕ = ± [ s 1 2 ] π ) A 0 2 1 ( s 1 2 ) π + ζ ~ s ξ ~ s 2 2 ,
α ~ n ( 1 ) n [ c n i a n ] ,
β ~ n ( 1 ) n [ b n + i d n ] ,
ζ ~ s 2 π n = 1 α ~ n [ 1 2 ( s n ) 1 + 1 2 ( s + n ) 1 ] ,
ξ ~ s 2 π n = 1 β ~ n [ 1 2 ( s n ) 1 1 2 ( s + n ) 1 ] .
α ~ n = 2 π s = 1 ζ ~ s [ 1 2 ( s n ) 1 + 1 2 ( s + n ) 1 ] ,
β ~ n = 2 π s = 1 ξ ~ s [ 1 2 ( s n ) 1 1 2 ( s + n ) 1 ] ,
s = 1 ζ ~ s 2 s 1 = 0 ,
Φ ( x ) a 0 2 = s = 1 ( 1 ) s Im ( ζ ~ s ) cos ( 2 π ( s 1 2 ) x L ) + ( 1 ) s + 1 Re ( ξ ~ s ) sin ( 2 π ( s 1 2 ) x L ) ,
A ( x ) A 0 1 = s = 1 ( 1 ) s + 1 Re ( ζ ~ s ) cos ( 2 π ( s 1 2 ) x L ) + ( 1 ) s + 1 Im ( ξ ~ s ) sin ( 2 π ( s 1 2 ) x L ) .

Metrics