Abstract

A new molecular engineering strategy is proposed that favors the packing of charge-transfer conjugated molecules to enhance their crystalline quadratic nonlinear efficiency. A highly polar substituent is grafted to an achiral molecule at a position remote from the donor–acceptor π-electron conjugated system. Dipolar interaction forces will act mainly toward the antiparallel coupling of local dipoles, while the remaining nonlinear portion of the molecule is freed, under other influences, to set up a noncentrosymmetric and possibly optimal structure. Among other 4-nitroanilinelike compounds, N-(4-nitrophenyl)-N-methylaminoacetonitrile (NPAN) exemplifies this new approach and is shown to have a powder second-harmonic generation efficiency of the same order as that of N-(4-nitrophenyl)-L-prolinol (NPP), i.e., more than 2 orders of magnitude above that of urea. The nonlinearity of both molecules (vector part of the β tensor projected along the dipole moment) has been measured by use of electric-field induced second-harmonic (EFISH) generation in solution at 1.06 μm. The nonlinearity of the NPAN molecule is roughly half that of NPP, but the transparency range of NPAN is significantly increased toward the UV compared with that of NPP. Two theoretical models, based, respectively, on a finite-field perturbation of the Hartree–Fock equations and on a sum-over-states expansion of tensor β both at a semiempirical level of approximation, are used to compute the coefficients of the first-order hyperpolarizability of NPP and NPAN. A two-level quantum model is used to account for frequency dispersion, and theoretical crystalline coefficients are obtained from an oriented-gas description of the crystal. Theoretical molecular polarizabilities are in satisfactory agreement with the EFISH experimental results. The experimental crystalline nonlinearity of NPP is also well accounted for by calculations, while the optimized nonlinear coefficient dZYY of crystalline NPAN is predicted to be of the order of 140 × 10−9 esu, coming close to that of NPP.

© 1987 Optical Society of America

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    [CrossRef]
  2. J. Zyss, “New organic molecular materials for nonlinear optics,” J. Non-Cryst. Solids 47, 211–225 (1982).
    [CrossRef]
  3. J. Zyss, D. S. Chemla, J. F. Nicoud, “Demonstration of efficient nonlinear optical crystals with vanishing molecular dipole moment: second-harmonic generation in 3-methyl-4-nitropyridine-1-oxyde,” J. Chem. Phys. 74, 4800–4811 (1981).
    [CrossRef]
  4. J. L. Oudar, R. Hierle, “An efficient organic material for non-linear optics: methyl-(2,4-dinitrophenyl)-aminopropanoate,” J. Appl. Phys. 48, 2699–2704 (1977).
    [CrossRef]
  5. K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
    [CrossRef]
  6. A. F. Garito, K. D. Singer, “Organic crystals and polymers, a new class of nonlinear optical materials,” Laser Focus 18(2), 59–68 (1982).
  7. I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
    [CrossRef]
  8. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  9. H. Rabin, C. L. Tang, Quantum Electronics (Academic, New York, 1965.
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    [CrossRef]
  11. P. V. Vidaković, M. Coquillay, F. Salin, “N-(4-nitrophenyl)-N-methylamino-aceto-nitrile: new organic material for efficient second-harmonic generation in bulk and waveguide configurations. I. Growth, crystal structure, and characterization of organic crystal-cored fibers,” J. Opt. Soc. Am. B 4, 998–1012 (1987).
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  13. J. F. Lipscomb, A. F. Garito, R. S. Narang, “An exceptionally large linear electrooptic effect in the organic solid MNA,” J. Chem. Phys. 75, 1509–1516 (1981).
    [CrossRef]
  14. A. D. Mighell, V. L. Himes, J. R. Rodgers, “Space group frequencies for organic compounds,” Acta Cryst. A39, 737–740 (1983).
  15. R. J. Twieg, IBM Almaden Research Center, San Jose, California 95102-6099 (personal communication).
  16. V. I. Minkin, O. A. Osipov, Yv. A. Zhdanov, Dipole Moments of Organic Chemistry (Plenum, New York, 1970).
    [CrossRef]
  17. N-Methylaminoacetonitrile is commercially available (from Tokyo Chemical Industry, Tokyo) or easily prepared: A. H. Cook, S. F. Cox, “The preparation of α-N-alkylamino-nitrile, -amide, and acids,” J. Am. Chem. Soc. 1949, 2334–2337 (1949).
  18. J. L. Oudar, H. Le Person, “Second order polarizabilities of some aromatic molecules,” Opt. Commun. 15, 258–262 (1976).
    [CrossRef]
  19. J. L. Oudar, “Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds,” J. Chem. Phys. 67, 446–457 (1977).
    [CrossRef]
  20. B. F. Levine, C. G. Bethea, “Ultraviolet dispersion of the donor–acceptor charge transfer contribution to the second order hyperpolarizability,” J. Chem Phys. 69, 5240–5245 (1978).
    [CrossRef]
  21. C. G. Bethea, “Experimental technique of dc induced SHG in liquids: measurement of the nonlinearity of CH2I2,” Appl. Opt. 14, 1447–1451 (1975).
    [CrossRef] [PubMed]
  22. B. F. Levine, C. G. Bethea, “Effects on hyperpolarizabilities of molecular interactions in associating liquid mixtures,” J. Chem. Phys. 65, 2429–2438 (1976).
    [CrossRef]
  23. B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516–520 (1976).
    [CrossRef]
  24. K. D. Singer, A. F. Garito, “Measurements of molecular second order optical susceptibilities using dc induced second harmonic generation,” J. Chem. Phys. 75, 3572–3580(1981).
    [CrossRef]
  25. I. Ledoux, J. Zyss, “Influence of the molecular environment in solution measurements of the second order optical susceptibility of urea and derivatives,” Chem. Phys. 73, 203–213 (1982).
    [CrossRef]
  26. P. D. Maker, R. W. Terhune, M. Nisenoff, C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
    [CrossRef]
  27. B. F. Levine, C. G. Bethea, “Second and third order hyper-polarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
    [CrossRef]
  28. J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. I. Perturbated INDO approach to monosubstituted benzene,” J. Chem. Phys. 70, 3333–3340 (1979).
    [CrossRef]
  29. J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. II. Substituent effects and respective σ–π contributions,” J. Chem. Phys. 70, 3341–3349 (1979).
    [CrossRef]
  30. J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. III. Study of a family of donor–acceptor disubstituted phenylpolynes,” J. Chem. Phys. 71, 909–916 (1979).
    [CrossRef]
  31. V. J. Dorcherty, D. Pugh, J. D. Morley, “Calculation of the second order electronic polarizabilities of some organic molecules,” J. Chem. Soc. Faraday Trans. 281, 1179–1192 (1985).
  32. S. Allen, J. D. Morley, D. Pugh, V. J. Docherty, “A cndovsb program for the calculation of second-order molecular polarizabilities and its application,” in Molecular and Polymeric Optoelectronic Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682 (to be published).
  33. J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mol. Phys. 37, 1–18 (1965).
    [CrossRef]
  34. J. L. Oudar, D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
    [CrossRef]
  35. I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
    [CrossRef]
  36. J. L. Oudar, J. Zyss, “Structural dependence of nonlinear optical properties of methyl-(2,4-dinitrophenyl)-aminopropanoate crystals,” Phys. Rev. A. 26, 2016–2027 (1982).
    [CrossRef]
  37. F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
    [CrossRef]
  38. J. Zyss, J. L. Oudar, “Relations between microscopic and macroscopic lowest order optical nonlinearities of molecular crystals with one or two dimensional units,” Phys. Rev. A 26, 2028–2048 (1982).
    [CrossRef]

1987

1986

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
[CrossRef]

1985

V. J. Dorcherty, D. Pugh, J. D. Morley, “Calculation of the second order electronic polarizabilities of some organic molecules,” J. Chem. Soc. Faraday Trans. 281, 1179–1192 (1985).

1984

J. Zyss, J. F. Nicoud, M. Coquillay, “Chirality and hydrogene bonding in molecular crystals for phase-matched second harmonic generation: N-(4-nitrophenyl)-L-prolinol (NPP),” J. Chem. Phys. 81, 4160–4167 (1984).
[CrossRef]

1983

A. D. Mighell, V. L. Himes, J. R. Rodgers, “Space group frequencies for organic compounds,” Acta Cryst. A39, 737–740 (1983).

1982

J. Zyss, “New organic molecular materials for nonlinear optics,” J. Non-Cryst. Solids 47, 211–225 (1982).
[CrossRef]

A. F. Garito, K. D. Singer, “Organic crystals and polymers, a new class of nonlinear optical materials,” Laser Focus 18(2), 59–68 (1982).

I. Ledoux, J. Zyss, “Influence of the molecular environment in solution measurements of the second order optical susceptibility of urea and derivatives,” Chem. Phys. 73, 203–213 (1982).
[CrossRef]

J. L. Oudar, J. Zyss, “Structural dependence of nonlinear optical properties of methyl-(2,4-dinitrophenyl)-aminopropanoate crystals,” Phys. Rev. A. 26, 2016–2027 (1982).
[CrossRef]

J. Zyss, J. L. Oudar, “Relations between microscopic and macroscopic lowest order optical nonlinearities of molecular crystals with one or two dimensional units,” Phys. Rev. A 26, 2028–2048 (1982).
[CrossRef]

1981

K. D. Singer, A. F. Garito, “Measurements of molecular second order optical susceptibilities using dc induced second harmonic generation,” J. Chem. Phys. 75, 3572–3580(1981).
[CrossRef]

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

J. Zyss, D. S. Chemla, J. F. Nicoud, “Demonstration of efficient nonlinear optical crystals with vanishing molecular dipole moment: second-harmonic generation in 3-methyl-4-nitropyridine-1-oxyde,” J. Chem. Phys. 74, 4800–4811 (1981).
[CrossRef]

J. F. Lipscomb, A. F. Garito, R. S. Narang, “An exceptionally large linear electrooptic effect in the organic solid MNA,” J. Chem. Phys. 75, 1509–1516 (1981).
[CrossRef]

1979

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. I. Perturbated INDO approach to monosubstituted benzene,” J. Chem. Phys. 70, 3333–3340 (1979).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. II. Substituent effects and respective σ–π contributions,” J. Chem. Phys. 70, 3341–3349 (1979).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. III. Study of a family of donor–acceptor disubstituted phenylpolynes,” J. Chem. Phys. 71, 909–916 (1979).
[CrossRef]

1978

B. F. Levine, C. G. Bethea, “Ultraviolet dispersion of the donor–acceptor charge transfer contribution to the second order hyperpolarizability,” J. Chem Phys. 69, 5240–5245 (1978).
[CrossRef]

1977

J. L. Oudar, “Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds,” J. Chem. Phys. 67, 446–457 (1977).
[CrossRef]

J. L. Oudar, R. Hierle, “An efficient organic material for non-linear optics: methyl-(2,4-dinitrophenyl)-aminopropanoate,” J. Appl. Phys. 48, 2699–2704 (1977).
[CrossRef]

J. L. Oudar, D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
[CrossRef]

F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
[CrossRef]

1976

J. L. Oudar, H. Le Person, “Second order polarizabilities of some aromatic molecules,” Opt. Commun. 15, 258–262 (1976).
[CrossRef]

B. F. Levine, C. G. Bethea, “Effects on hyperpolarizabilities of molecular interactions in associating liquid mixtures,” J. Chem. Phys. 65, 2429–2438 (1976).
[CrossRef]

B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516–520 (1976).
[CrossRef]

1975

C. G. Bethea, “Experimental technique of dc induced SHG in liquids: measurement of the nonlinearity of CH2I2,” Appl. Opt. 14, 1447–1451 (1975).
[CrossRef] [PubMed]

B. F. Levine, C. G. Bethea, “Second and third order hyper-polarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

1965

J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mol. Phys. 37, 1–18 (1965).
[CrossRef]

1962

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

1949

N-Methylaminoacetonitrile is commercially available (from Tokyo Chemical Industry, Tokyo) or easily prepared: A. H. Cook, S. F. Cox, “The preparation of α-N-alkylamino-nitrile, -amide, and acids,” J. Am. Chem. Soc. 1949, 2334–2337 (1949).

Allen, S.

S. Allen, J. D. Morley, D. Pugh, V. J. Docherty, “A cndovsb program for the calculation of second-order molecular polarizabilities and its application,” in Molecular and Polymeric Optoelectronic Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682 (to be published).

Antonetti, A.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

Bertinelli, F.

F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
[CrossRef]

Bethea, C. G.

B. F. Levine, C. G. Bethea, “Ultraviolet dispersion of the donor–acceptor charge transfer contribution to the second order hyperpolarizability,” J. Chem Phys. 69, 5240–5245 (1978).
[CrossRef]

B. F. Levine, C. G. Bethea, “Effects on hyperpolarizabilities of molecular interactions in associating liquid mixtures,” J. Chem. Phys. 65, 2429–2438 (1976).
[CrossRef]

C. G. Bethea, “Experimental technique of dc induced SHG in liquids: measurement of the nonlinearity of CH2I2,” Appl. Opt. 14, 1447–1451 (1975).
[CrossRef] [PubMed]

B. F. Levine, C. G. Bethea, “Second and third order hyper-polarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Brillante, A.

F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
[CrossRef]

Chemla, D. S.

J. Zyss, D. S. Chemla, J. F. Nicoud, “Demonstration of efficient nonlinear optical crystals with vanishing molecular dipole moment: second-harmonic generation in 3-methyl-4-nitropyridine-1-oxyde,” J. Chem. Phys. 74, 4800–4811 (1981).
[CrossRef]

J. L. Oudar, D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
[CrossRef]

Cheng, Y. Y.

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

Cook, A. H.

N-Methylaminoacetonitrile is commercially available (from Tokyo Chemical Industry, Tokyo) or easily prepared: A. H. Cook, S. F. Cox, “The preparation of α-N-alkylamino-nitrile, -amide, and acids,” J. Am. Chem. Soc. 1949, 2334–2337 (1949).

Coquillay, M.

Cox, S. F.

N-Methylaminoacetonitrile is commercially available (from Tokyo Chemical Industry, Tokyo) or easily prepared: A. H. Cook, S. F. Cox, “The preparation of α-N-alkylamino-nitrile, -amide, and acids,” J. Am. Chem. Soc. 1949, 2334–2337 (1949).

Crowley, J. I.

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

Docherty, V. J.

S. Allen, J. D. Morley, D. Pugh, V. J. Docherty, “A cndovsb program for the calculation of second-order molecular polarizabilities and its application,” in Molecular and Polymeric Optoelectronic Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682 (to be published).

Dorcherty, V. J.

V. J. Dorcherty, D. Pugh, J. D. Morley, “Calculation of the second order electronic polarizabilities of some organic molecules,” J. Chem. Soc. Faraday Trans. 281, 1179–1192 (1985).

Etchepare, J.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

Garito, A. F.

A. F. Garito, K. D. Singer, “Organic crystals and polymers, a new class of nonlinear optical materials,” Laser Focus 18(2), 59–68 (1982).

J. F. Lipscomb, A. F. Garito, R. S. Narang, “An exceptionally large linear electrooptic effect in the organic solid MNA,” J. Chem. Phys. 75, 1509–1516 (1981).
[CrossRef]

K. D. Singer, A. F. Garito, “Measurements of molecular second order optical susceptibilities using dc induced second harmonic generation,” J. Chem. Phys. 75, 3572–3580(1981).
[CrossRef]

Grillon, G.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

Hewig, G. H.

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

Hierle, R.

J. L. Oudar, R. Hierle, “An efficient organic material for non-linear optics: methyl-(2,4-dinitrophenyl)-aminopropanoate,” J. Appl. Phys. 48, 2699–2704 (1977).
[CrossRef]

Himes, V. L.

A. D. Mighell, V. L. Himes, J. R. Rodgers, “Space group frequencies for organic compounds,” Acta Cryst. A39, 737–740 (1983).

Jain, K.

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

Josse, D.

I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
[CrossRef]

Le Person, H.

J. L. Oudar, H. Le Person, “Second order polarizabilities of some aromatic molecules,” Opt. Commun. 15, 258–262 (1976).
[CrossRef]

Ledoux, I.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
[CrossRef]

I. Ledoux, J. Zyss, “Influence of the molecular environment in solution measurements of the second order optical susceptibility of urea and derivatives,” Chem. Phys. 73, 203–213 (1982).
[CrossRef]

Levine, B. F.

B. F. Levine, C. G. Bethea, “Ultraviolet dispersion of the donor–acceptor charge transfer contribution to the second order hyperpolarizability,” J. Chem Phys. 69, 5240–5245 (1978).
[CrossRef]

B. F. Levine, C. G. Bethea, “Effects on hyperpolarizabilities of molecular interactions in associating liquid mixtures,” J. Chem. Phys. 65, 2429–2438 (1976).
[CrossRef]

B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516–520 (1976).
[CrossRef]

B. F. Levine, C. G. Bethea, “Second and third order hyper-polarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

Lipscomb, J. F.

J. F. Lipscomb, A. F. Garito, R. S. Narang, “An exceptionally large linear electrooptic effect in the organic solid MNA,” J. Chem. Phys. 75, 1509–1516 (1981).
[CrossRef]

Maker, P. D.

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

Mighell, A. D.

A. D. Mighell, V. L. Himes, J. R. Rodgers, “Space group frequencies for organic compounds,” Acta Cryst. A39, 737–740 (1983).

Migus, A.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

Minkin, V. I.

V. I. Minkin, O. A. Osipov, Yv. A. Zhdanov, Dipole Moments of Organic Chemistry (Plenum, New York, 1970).
[CrossRef]

Morley, J. D.

V. J. Dorcherty, D. Pugh, J. D. Morley, “Calculation of the second order electronic polarizabilities of some organic molecules,” J. Chem. Soc. Faraday Trans. 281, 1179–1192 (1985).

S. Allen, J. D. Morley, D. Pugh, V. J. Docherty, “A cndovsb program for the calculation of second-order molecular polarizabilities and its application,” in Molecular and Polymeric Optoelectronic Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682 (to be published).

Narang, R. S.

J. F. Lipscomb, A. F. Garito, R. S. Narang, “An exceptionally large linear electrooptic effect in the organic solid MNA,” J. Chem. Phys. 75, 1509–1516 (1981).
[CrossRef]

Nicoud, J. F.

J. Zyss, J. F. Nicoud, M. Coquillay, “Chirality and hydrogene bonding in molecular crystals for phase-matched second harmonic generation: N-(4-nitrophenyl)-L-prolinol (NPP),” J. Chem. Phys. 81, 4160–4167 (1984).
[CrossRef]

J. Zyss, D. S. Chemla, J. F. Nicoud, “Demonstration of efficient nonlinear optical crystals with vanishing molecular dipole moment: second-harmonic generation in 3-methyl-4-nitropyridine-1-oxyde,” J. Chem. Phys. 74, 4800–4811 (1981).
[CrossRef]

J. F. Nicoud, R. J. Twieg, “Design and synthesis of organic molecular compounds for efficienty second harmonic generation,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1986), Vol. 1.

Nisenoff, M.

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

Osipov, O. A.

V. I. Minkin, O. A. Osipov, Yv. A. Zhdanov, Dipole Moments of Organic Chemistry (Plenum, New York, 1970).
[CrossRef]

Oudar, J. L.

J. L. Oudar, J. Zyss, “Structural dependence of nonlinear optical properties of methyl-(2,4-dinitrophenyl)-aminopropanoate crystals,” Phys. Rev. A. 26, 2016–2027 (1982).
[CrossRef]

J. Zyss, J. L. Oudar, “Relations between microscopic and macroscopic lowest order optical nonlinearities of molecular crystals with one or two dimensional units,” Phys. Rev. A 26, 2028–2048 (1982).
[CrossRef]

J. L. Oudar, “Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds,” J. Chem. Phys. 67, 446–457 (1977).
[CrossRef]

J. L. Oudar, D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
[CrossRef]

J. L. Oudar, R. Hierle, “An efficient organic material for non-linear optics: methyl-(2,4-dinitrophenyl)-aminopropanoate,” J. Appl. Phys. 48, 2699–2704 (1977).
[CrossRef]

J. L. Oudar, H. Le Person, “Second order polarizabilities of some aromatic molecules,” Opt. Commun. 15, 258–262 (1976).
[CrossRef]

Palmieri, P.

F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
[CrossRef]

Pugh, D.

V. J. Dorcherty, D. Pugh, J. D. Morley, “Calculation of the second order electronic polarizabilities of some organic molecules,” J. Chem. Soc. Faraday Trans. 281, 1179–1192 (1985).

S. Allen, J. D. Morley, D. Pugh, V. J. Docherty, “A cndovsb program for the calculation of second-order molecular polarizabilities and its application,” in Molecular and Polymeric Optoelectronic Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682 (to be published).

Rabin, H.

H. Rabin, C. L. Tang, Quantum Electronics (Academic, New York, 1965.

Rodgers, J. R.

A. D. Mighell, V. L. Himes, J. R. Rodgers, “Space group frequencies for organic compounds,” Acta Cryst. A39, 737–740 (1983).

Salin, F.

Savage, C. M.

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

Singer, K. D.

A. F. Garito, K. D. Singer, “Organic crystals and polymers, a new class of nonlinear optical materials,” Laser Focus 18(2), 59–68 (1982).

K. D. Singer, A. F. Garito, “Measurements of molecular second order optical susceptibilities using dc induced second harmonic generation,” J. Chem. Phys. 75, 3572–3580(1981).
[CrossRef]

Taliani, C.

F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
[CrossRef]

Tang, C. L.

H. Rabin, C. L. Tang, Quantum Electronics (Academic, New York, 1965.

Terhune, R. W.

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

Tweig, R. J.

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

Twieg, R. J.

J. F. Nicoud, R. J. Twieg, “Design and synthesis of organic molecular compounds for efficienty second harmonic generation,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1986), Vol. 1.

R. J. Twieg, IBM Almaden Research Center, San Jose, California 95102-6099 (personal communication).

Vidakovic, P.

I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
[CrossRef]

Vidakovic, P. V.

Ward, J. F.

J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mol. Phys. 37, 1–18 (1965).
[CrossRef]

Zhdanov, Yv. A.

V. I. Minkin, O. A. Osipov, Yv. A. Zhdanov, Dipole Moments of Organic Chemistry (Plenum, New York, 1970).
[CrossRef]

Zyss, J.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
[CrossRef]

J. Zyss, J. F. Nicoud, M. Coquillay, “Chirality and hydrogene bonding in molecular crystals for phase-matched second harmonic generation: N-(4-nitrophenyl)-L-prolinol (NPP),” J. Chem. Phys. 81, 4160–4167 (1984).
[CrossRef]

J. Zyss, “New organic molecular materials for nonlinear optics,” J. Non-Cryst. Solids 47, 211–225 (1982).
[CrossRef]

I. Ledoux, J. Zyss, “Influence of the molecular environment in solution measurements of the second order optical susceptibility of urea and derivatives,” Chem. Phys. 73, 203–213 (1982).
[CrossRef]

J. L. Oudar, J. Zyss, “Structural dependence of nonlinear optical properties of methyl-(2,4-dinitrophenyl)-aminopropanoate crystals,” Phys. Rev. A. 26, 2016–2027 (1982).
[CrossRef]

J. Zyss, J. L. Oudar, “Relations between microscopic and macroscopic lowest order optical nonlinearities of molecular crystals with one or two dimensional units,” Phys. Rev. A 26, 2028–2048 (1982).
[CrossRef]

J. Zyss, D. S. Chemla, J. F. Nicoud, “Demonstration of efficient nonlinear optical crystals with vanishing molecular dipole moment: second-harmonic generation in 3-methyl-4-nitropyridine-1-oxyde,” J. Chem. Phys. 74, 4800–4811 (1981).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. I. Perturbated INDO approach to monosubstituted benzene,” J. Chem. Phys. 70, 3333–3340 (1979).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. II. Substituent effects and respective σ–π contributions,” J. Chem. Phys. 70, 3341–3349 (1979).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. III. Study of a family of donor–acceptor disubstituted phenylpolynes,” J. Chem. Phys. 71, 909–916 (1979).
[CrossRef]

Acta Cryst.

A. D. Mighell, V. L. Himes, J. R. Rodgers, “Space group frequencies for organic compounds,” Acta Cryst. A39, 737–740 (1983).

Appl. Opt.

Appl. Phys. Lett.

I. Ledoux, J. Zyss, A. Migus, J. Etchepare, G. Grillon, A. Antonetti, “Generation of high peak power subpicosecond pulses in the 1.0–1.6-μ m range by parametric amplification in an organic crystal,” Appl. Phys. Lett. 48, 1564–1566 (1986).
[CrossRef]

Chem. Phys.

I. Ledoux, J. Zyss, “Influence of the molecular environment in solution measurements of the second order optical susceptibility of urea and derivatives,” Chem. Phys. 73, 203–213 (1982).
[CrossRef]

F. Bertinelli, P. Palmieri, A. Brillante, C. Taliani, “Electronic excited states of nitroanilines. II. A configuration interaction study and U.V. Spectrum of the paranitroaniline single crystal,” Chem. Phys. 25, 333–341 (1977).
[CrossRef]

Chem. Phys. Lett.

B. F. Levine, “Donor-acceptor charge transfer contributions to the second order hyperpolarizability,” Chem. Phys. Lett. 37, 516–520 (1976).
[CrossRef]

J. Am. Chem. Soc.

N-Methylaminoacetonitrile is commercially available (from Tokyo Chemical Industry, Tokyo) or easily prepared: A. H. Cook, S. F. Cox, “The preparation of α-N-alkylamino-nitrile, -amide, and acids,” J. Am. Chem. Soc. 1949, 2334–2337 (1949).

J. Appl. Phys.

J. L. Oudar, R. Hierle, “An efficient organic material for non-linear optics: methyl-(2,4-dinitrophenyl)-aminopropanoate,” J. Appl. Phys. 48, 2699–2704 (1977).
[CrossRef]

J. Chem Phys.

B. F. Levine, C. G. Bethea, “Ultraviolet dispersion of the donor–acceptor charge transfer contribution to the second order hyperpolarizability,” J. Chem Phys. 69, 5240–5245 (1978).
[CrossRef]

J. Chem. Phys.

J. L. Oudar, D. S. Chemla, “Hyperpolarizabilities of the nitroanilines and their relations to the excited state dipole moment,” J. Chem. Phys. 66, 2664–2668 (1977).
[CrossRef]

K. D. Singer, A. F. Garito, “Measurements of molecular second order optical susceptibilities using dc induced second harmonic generation,” J. Chem. Phys. 75, 3572–3580(1981).
[CrossRef]

B. F. Levine, C. G. Bethea, “Effects on hyperpolarizabilities of molecular interactions in associating liquid mixtures,” J. Chem. Phys. 65, 2429–2438 (1976).
[CrossRef]

B. F. Levine, C. G. Bethea, “Second and third order hyper-polarizabilities of organic molecules,” J. Chem. Phys. 63, 2666–2682 (1975).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. I. Perturbated INDO approach to monosubstituted benzene,” J. Chem. Phys. 70, 3333–3340 (1979).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. II. Substituent effects and respective σ–π contributions,” J. Chem. Phys. 70, 3341–3349 (1979).
[CrossRef]

J. Zyss, “Hyperpolarizabilities of substituted conjugated molecules. III. Study of a family of donor–acceptor disubstituted phenylpolynes,” J. Chem. Phys. 71, 909–916 (1979).
[CrossRef]

J. Zyss, D. S. Chemla, J. F. Nicoud, “Demonstration of efficient nonlinear optical crystals with vanishing molecular dipole moment: second-harmonic generation in 3-methyl-4-nitropyridine-1-oxyde,” J. Chem. Phys. 74, 4800–4811 (1981).
[CrossRef]

J. F. Lipscomb, A. F. Garito, R. S. Narang, “An exceptionally large linear electrooptic effect in the organic solid MNA,” J. Chem. Phys. 75, 1509–1516 (1981).
[CrossRef]

J. L. Oudar, “Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds,” J. Chem. Phys. 67, 446–457 (1977).
[CrossRef]

J. Zyss, J. F. Nicoud, M. Coquillay, “Chirality and hydrogene bonding in molecular crystals for phase-matched second harmonic generation: N-(4-nitrophenyl)-L-prolinol (NPP),” J. Chem. Phys. 81, 4160–4167 (1984).
[CrossRef]

J. Chem. Soc. Faraday Trans.

V. J. Dorcherty, D. Pugh, J. D. Morley, “Calculation of the second order electronic polarizabilities of some organic molecules,” J. Chem. Soc. Faraday Trans. 281, 1179–1192 (1985).

J. Non-Cryst. Solids

J. Zyss, “New organic molecular materials for nonlinear optics,” J. Non-Cryst. Solids 47, 211–225 (1982).
[CrossRef]

J. Opt. Soc. Am. B

Laser Focus

A. F. Garito, K. D. Singer, “Organic crystals and polymers, a new class of nonlinear optical materials,” Laser Focus 18(2), 59–68 (1982).

Opt. Commun.

J. L. Oudar, H. Le Person, “Second order polarizabilities of some aromatic molecules,” Opt. Commun. 15, 258–262 (1976).
[CrossRef]

Opt. Eng.

I. Ledoux, D. Josse, P. Vidakovic, J. Zyss, “Highly efficient single crystalline organic thin films for quadratic nonlinear optics,” Opt. Eng. 25, 202–210 (1986).
[CrossRef]

Opt. Laser Technol.

K. Jain, J. I. Crowley, G. H. Hewig, Y. Y. Cheng, R. J. Tweig, “Optically nonlinear organic materials,” Opt. Laser Technol. 13, 297–301 (1981).
[CrossRef]

Phys. Rev. A

J. Zyss, J. L. Oudar, “Relations between microscopic and macroscopic lowest order optical nonlinearities of molecular crystals with one or two dimensional units,” Phys. Rev. A 26, 2028–2048 (1982).
[CrossRef]

Phys. Rev. A.

J. L. Oudar, J. Zyss, “Structural dependence of nonlinear optical properties of methyl-(2,4-dinitrophenyl)-aminopropanoate crystals,” Phys. Rev. A. 26, 2016–2027 (1982).
[CrossRef]

Phys. Rev. Lett.

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

Rev. Mol. Phys.

J. F. Ward, “Calculation of nonlinear optical susceptibilities using diagrammatic perturbation theory,” Rev. Mol. Phys. 37, 1–18 (1965).
[CrossRef]

Other

S. Allen, J. D. Morley, D. Pugh, V. J. Docherty, “A cndovsb program for the calculation of second-order molecular polarizabilities and its application,” in Molecular and Polymeric Optoelectronic Materials, L. G. DeShazer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.682 (to be published).

D. J. Williams, ed., “Nonlinear optical properties of organic and polymeric materials,” ACS Symp. Ser. 233 (American Chemical Society, Washington, D.C., 1983).
[CrossRef]

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

H. Rabin, C. L. Tang, Quantum Electronics (Academic, New York, 1965.

J. F. Nicoud, R. J. Twieg, “Design and synthesis of organic molecular compounds for efficienty second harmonic generation,” in Nonlinear Optical Properties of Organic Molecules and Crystals, D. S. Chemla, J. Zyss, eds. (Academic, Orlando, Fla., 1986), Vol. 1.

R. J. Twieg, IBM Almaden Research Center, San Jose, California 95102-6099 (personal communication).

V. I. Minkin, O. A. Osipov, Yv. A. Zhdanov, Dipole Moments of Organic Chemistry (Plenum, New York, 1970).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Highly efficient ONL material of the 4-nitroaniline family.

Fig. 2
Fig. 2

Electronic absorption spectra of NPAN and NPP in ETOH.

Fig. 3
Fig. 3

Experimental setup. The Q-switched Nd:YAG laser is operating at 1.064 μm, the pulse duration is 20 nsec, and the repetition rate 10 Hz. The high-voltage pulses, up to 10-kV amplitude and 2-μsec duration, are applied synchronously to the laser pulses.

Fig. 4
Fig. 4

Coherence length of NPP and NPAN in acetone as a function of the mass ratio x. The accuracy is 2%.

Fig. 5
Fig. 5

DC SHG macroscopic susceptibility ΓL(x) of NPP and NPAN solutions in acetone. ΓL(x) is given in 10−14 esu. The accuracy is 10%.

Fig. 6
Fig. 6

NPAN and NPP molecules. The x1 axis lies along the CT axis; the z1 axis is orthogonal to the mean plane containing the six aromatic carbons and the two nitrogen atoms. The fictitious molecule NPAM is derived from NPAN by substitution of a hydrogen atom in the cyano group, keeping the same orientation.

Fig. 7
Fig. 7

Crystal structure of NPP (space group P21, Z = 2 molecules). a = 5.261 Å, b = 14.908 Å, c = 7.185 Å, β = 105.18°.

Fig. 8
Fig. 8

Crystal structure of NPAN (orthorhombic Fdd2, Z = 16 molecules). a = 25.9222 Å, b = 33.956 Å, c = 4.319 Å. The angle between the z1 axis and the (011) crystallographic plane is less than 60.

Tables (12)

Tables Icon

Table 1 Structural Formula and SHG Powder Tests (λ = 1.06 μm) of the New 4-NA Derivatives

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Table 2 UV Absorption Spectra Data for NPAN, NPPN, and DMNAa

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Table 3 1H-NMR Data at 90 MHza

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Table 4 13C-NMR data at 20 MHza

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Table 5 Mean Microscopic Hyperpolarizabilities γ 2 0, and z Component of the Vector Part of the Quadratic Hyperpolarizability Tensor β z 2 ω along the Permanent Dipole Momenta

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Table 6 Permanent Dipole Moment in the Molecular Frame x1y1z1a

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Table 7 β Components of NPAN in the Molecular frame x1y1z1a

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Table 8 β Components of NPP (in 10−30 esu) in the Molecular Frame x1y1z

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Table 9 Comparison between NPAN and NPAM by the FF Methoda

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Table 10 Calculated and Experimental Values of the βz Component (z along the Permanent Dipole) at the Fundamental Wavelength ħω = 1.7 eVa

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Table 11 Calculated and Experimental Values of d Coefficients for NPP Crystal at the Fundamental Wavelength ħω = 1.17 eVa

Tables Icon

Table 12 Calculated Values of d Coefficients for NPAN Crystal at the Fundamental Wavelength ħω = 1.17 eVa

Equations (36)

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

P I 2 ω = Γ I J K L E J ω E K ω E L 0 .
Γ L = Γ X X X X .
Γ L = N f γ X X X X 0 = N f γ 0 .
f = ( n 2 ω 2 + 2 3 ) ( n ω 2 + 2 3 ) 2 ( ω + 2 ) ω + 2 ,
γ 0 = γ e + μ z β z 2 ω 5 k T ,
γ e = [ γ x x x x + γ y y y y + γ z z z z + ( γ x y y x + γ x x y y + γ x z z x + γ z x x z + γ z y y z + γ y z z y + 2 γ x y x y + 2 γ y x y x + 2 γ x z x z + 2 γ z x z x + 2 γ y z y z + 2 γ z y z y ) ] .
β z 2 ω = ( 3 β z z z 2 ω + β z x x 2 ω + β z y y 2 ω + 2 β x x z 2 ω + 2 β y y z 2 ω ) .
γ 0 μ z × β z 2 ω / 5 K T .
Γ L mix = f ( N 1 γ 1 0 + N 2 γ 2 0 ) .
Γ L mix = ρ ( x ) f ( x ) N 1 + x ( γ 1 0 M 1 + x γ 2 0 M 2 ) .
γ 2 0 = M 2 x ρ N f [ Γ L ( x ) - Γ L ( o ) ] .
Γ L ( x ) = { ± [ I L ( x ) I q ] 1 / 2 T q E q t 2 ω S + T S E S } 1 T L E L .
T q = n ω q + n 2 w q 1 + n 2 ω q ,             T s = n ω S + n 2 ω L n 2 ω S + n 2 ω L ,             T L = n ω L + n 2 ω L n 2 ω S + n 2 ω L .
E q = - 4 π ( n 2 ω q ) 2 - ( n ω q ) 2 d 11 q ( 2 1 + n ω q ) 2 10 - D q / 2 ( E ω ) 2 ,
E S = - 4 π ( n 2 ω S ) 2 - ( n ω S ) 2 Γ s E 0 ( 2 1 + n ω S ) 2 10 - D L / 2 ( E ω ) 2 ,
E L = - 16 π n 2 ω L + n ω L l c L λ E 0 ( 2 1 + n ω S 2 n ω S n ω S + n ω L ) 2 10 - D L / 2 ( E ω ) 2 ;
t 2 ω S = 2 n 2 ω S / ( 1 + n 2 ω S ) .
l c L = ½ × δ × tan α ,
n 2 ω L = 1.3596 ,             n ω L = n 2 ω L - λ 4 l c L = 1.3494.
μ i = μ i 0 + α i j 0 E j + β i j k 0 E j E k + .
β x x x 0 = 1 2 E 2 x { p x ( E x , 0 , 0 ) + p x ( - E x , 0 , 0 ) - 2 p x ( 0 , 0 , 0 ) } ,
β x y y 0 = 1 2 E 2 y { p x ( 0 , E y , 0 ) + p x ( 0 , - E y , 0 ) - 2 p x ( 0 , 0 , 0 ) } ,
β x x y 0 = 1 8 E x E y { p x ( E x , E y , 0 ) + p x ( - E x - E y , 0 ) - p x ( - E x , E y , 0 ) - p x ( E x , - E y , 0 ) } .
Δ β x x x 0 = 1 2 E x 2 { Δ p x ( E x , 0 , 0 ) + Δ p x ( - E x , 0 , 0 ) + 2 Δ p x ( 0 , 0 , 0 ) } .
β i j k 2 ω + β i k j 2 ω = - e 3 4 2 × n , n ( ( Γ g n i Γ n n i Γ n g k + Γ g n k Γ n n i Γ n g j ) × { [ ( ω n g - ω ) ( ω n g + ω ) ] - 1 + [ ( ω n g + ω ) ( ω n g - ω ) ] - 1 } + ( Γ g n i Γ n n j Γ n g k + Γ g n i Γ n n k Γ n g i ) { [ ( ω n g + 2 ω ) ( ω n g + ω ) ] - 1 + [ ( ω n g - 2 ω ) ( ω n g - ω ) ] - 1 } + ( Γ g n j Γ n n k Γ n g i + Γ g n k Γ n n j Γ n g i ) { [ ( ω n g - ω ) ( ω n g - 2 ω ) ] - 1 + [ ( ω n g + ω ) ( ω n g + 2 ω ) ] - 1 } ) .
β x x x 111 2 ω = 3 e 2 2 W f Δ μ / { 2 m [ W 2 - ( 2 ω ) 2 ] [ W 2 - ( ω ) 2 ] } ,
β x 1 x 1 x 1 2 ω = F ( W , ω ) × β x 1 x 1 x 1 0 ,
F ( W , ω ) = 1 / { [ 1 - ( 2 ω / W ) 2 ] [ 1 - ( ω / W ) 2 ] } .
β x 1 x 1 x 1 0 ( NPAN , CNDOVSB ) / β x 1 x 1 x 1 0 ( NPP , CNDOVSB ) = 0.75
β x 1 x 1 x 1 0 ( NPAN , FF ) / β x 1 x 1 x 1 0 ( NPP , FF ) = 0.70.
d Y Y Y 2 ω = N f Y Y Y cos 3 θ β x 1 x 1 x 1 2 ω ,
d Y X X 2 ω = N f Y X X cos θ sin 2 θ β x 1 x 1 x 1 2 ω ,
d Z X X 2 ω = N f Z X X cos ( Z , x 1 ) cos 2 ( X , x 1 ) β x 1 x 1 x 1 2 ω ,
d Z Y Y 2 ω = N f Z Y Y cos ( Z , x 1 ) cos 2 ( Y , x 1 ) β x 1 x 1 x 1 2 ω ,
d Z Z Z 2 ω = N f Z Z Z cos 3 ( Z , x 1 ) β x 1 x 1 x 1 2 ω ,
d Z Y Y 2 ω ( NPAN ) / d Z X X 2 ω ( NPP ) = 0.56 , d Z Z Z 2 ω ( NPAN ) / d Y Y Y 2 ω ( NPP ) = 0.66

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