Abstract

Methods for calculating the optical properties as a function of wavelength for second-order nonlinear optical polymers are presented. By combining the optical properties of the individual components, that is, those of the polymer and those of the chromophore, one can determine the real and the imaginary parts of the index of refraction and the second hyperpolarizability as a function of wavelength for a composite polymer, for either a guest–host or a side-chain system. These properties are important for characterizing and applying nonlinear optical polymers to devices. The methods allow for an accuracy in index of 3%, if the wavelength range includes the main absorption band (with absorption maximum at λmax) of the chromophore, and of 0.2% for wavelengths 400 nm and more to the long-wavelength side of λmax. In addition, if the index of the composite polymer is known at one specific wavelength on the long-wavelength side of λmax, the dispersion in the index in this wavelength range can be predicted to within 0.1%.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. J. Twieg and C. W. Dirk, “Design, properties and applications of nonlinear optical chromophores,” in Organic Thin Films for Waveguiding Nonlinear Optics, F. Kajzar and J. D. Swalen, eds. (Gordon & Breach, Amsterdam, 1996), Chap. 2, pp. 45–136.
  2. R. D. Miller, “Poled polymers for χ(2) applications,” in Organic Thin Films for Waveguiding Nonlinear Optics, F. Kajzar and J. D. Swalen, eds. (Gordon & Breach, Amsterdam, 1996), Chap. 8, pp. 329–456.
  3. L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
    [CrossRef]
  4. D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
    [CrossRef]
  5. C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
    [CrossRef]
  6. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 155.
  7. W. J. Welsh, “Densities of amorphous and crystalline polymers” in Physical Properties of Polymers Handbook, J. E. Mark, ed. (American Institute of Physics, New York, 1996), pp. 401–407.
  8. D. R. Lide, ed., Handbook of Chemistry and Physics (CRC, Cleveland, Ohio, 1990).
  9. D. W. van Krevelen and P. J. Hoftyzer, Properties of Polymers, Their Estimation and Correlation with Chemical Structures (Elsevier, Amsterdam, 1976).
  10. J. D. Swalen, R. Santo, M. Tacke, and J. Fischer, IBM J. Res. Dev. 21, 168 (1977).
    [CrossRef]
  11. J. D. Swalen, J. Molec. Electron. 2, 155 (1986).
  12. R. Azzam and W. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  13. S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communications Electronics (Wiley, New York, 1965), p. 335.
  14. F. W. King, J. Opt. Soc. Am. 68, 7 (1978).
  15. W. E. Press, B. R. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes (Cambridge U. Press, Cambridge, 1989).
  16. K. D. Singer, M. G. Kuzyk, and J. E. Sohn, J. Opt. Soc. Am. B 4, 968 (1987).
    [CrossRef]
  17. R. H. Page, M. C. Jurich, B. Reck, A. Den, R. J. Twieg, J. D. Swalen, G. C. Bjorklund, and C. G. Wilson, J. Opt. Soc. Am. B 7, 1239 (1990).
    [CrossRef]
  18. Data sheet available from Dan Dawson, IBM Almaden Research Center, 650 Harry Road, San Jose, Calif. 95120 [tel. (408) 927–1617].
  19. R. C. Raine, in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1950), pp. 6–93 and 6–94.
  20. T. Ishigure, E. Nihei, and Y. Koike, Appl. Opt. 35, 2048 (1996).
    [CrossRef] [PubMed]
  21. Ch. Bosshard, K. Sutter, Ph. Prêtre, J. Hulliger, M. Flörsheimer, P. Kaatz, and P. Günter, in Organic Nonlinear Optical Materials, A. F. Garito and F. Kajzar, eds., Vol. 1 of Advances in Nonlinear Optics (Gordon & Breach, Basel, 1995), p. 116; G. Knöpfle, Ch. Bosshard, R. Schlesser, and P. Günter, IEEE J. Quantum Electron. 30, 1303 (1994); F. Pan, G. Knöpfle, Ch. Bosshard, S. Follonier, R. Spreiter, M. S. Wong, and P. Günter, Appl. Phys. Lett. 69, 13 (1996).
    [CrossRef]
  22. This procedure is not strictly correct. It assumes that adding the chromophore results in no index change in the solution and consequently no change in transmission coefficients at the cell–solution boundaries. At the low concentrations used in these types of measurement, however, such a change (Δn≈10−5) will have no influence on the result.
  23. C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
    [CrossRef]

1996 (1)

1993 (2)

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

1992 (1)

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

1991 (1)

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

1990 (1)

1987 (1)

1978 (1)

F. W. King, J. Opt. Soc. Am. 68, 7 (1978).

1977 (1)

J. D. Swalen, R. Santo, M. Tacke, and J. Fischer, IBM J. Res. Dev. 21, 168 (1977).
[CrossRef]

Betterton, K. M.

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

Bjorklund, G. C.

Burland, D. M.

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

Cheng, L.-T.

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Den, A.

Fischer, J.

J. D. Swalen, R. Santo, M. Tacke, and J. Fischer, IBM J. Res. Dev. 21, 168 (1977).
[CrossRef]

Ishigure, T.

Jurich, M. C.

King, F. W.

F. W. King, J. Opt. Soc. Am. 68, 7 (1978).

Koike, Y.

Kuzyk, M. G.

Lee, V. Y.

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

Marder, S. R.

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Meredith, G. R.

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Miller, R. D.

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

Moylan, C. R.

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

Nihei, E.

Page, R. H.

Reck, B.

Reiser, O.

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

Rikken, G.

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Santo, R.

J. D. Swalen, R. Santo, M. Tacke, and J. Fischer, IBM J. Res. Dev. 21, 168 (1977).
[CrossRef]

Singer, K. D.

Sohn, J. E.

Stevenson, S. H.

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Swalen, J. D.

Swanson, S. A.

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

Tacke, M.

J. D. Swalen, R. Santo, M. Tacke, and J. Fischer, IBM J. Res. Dev. 21, 168 (1977).
[CrossRef]

Tam, W.

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Thackara, J. I.

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

Twieg, R. J.

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

R. H. Page, M. C. Jurich, B. Reck, A. Den, R. J. Twieg, J. D. Swalen, G. C. Bjorklund, and C. G. Wilson, J. Opt. Soc. Am. B 7, 1239 (1990).
[CrossRef]

Walsh, C. A.

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

Wilson, C. G.

Appl. Opt. (1)

IBM J. Res. Dev. (1)

J. D. Swalen, R. Santo, M. Tacke, and J. Fischer, IBM J. Res. Dev. 21, 168 (1977).
[CrossRef]

J. Am. Chem. Soc. (1)

C. R. Moylan, R. J. Twieg, V. Y. Lee, S. A. Swanson, K. M. Betterton, and R. D. Miller, J. Am. Chem. Soc. 115, 12599 (1993).
[CrossRef]

J. Appl. Phys. (1)

D. M. Burland, R. D. Miller, O. Reiser, R. J. Twieg, and C. A. Walsh, J. Appl. Phys. 71, 410 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

F. W. King, J. Opt. Soc. Am. 68, 7 (1978).

J. Opt. Soc. Am. B (2)

J. Phys. Chem. (1)

L.-T. Cheng, W. Tam, S. H. Stevenson, G. R. Meredith, G. Rikken, and S. R. Marder, J. Phys. Chem. 95, 10,631 (1991).
[CrossRef]

Proc. SPIE (1)

C. R. Moylan, S. A. Swanson, C. A. Walsh, J. I. Thackara, R. J. Twieg, R. D. Miller, V. Y. Lee, “From electric field-induced second harmonic generation (EFISH) to electro-optic measurements of nonlinear chromophores,” in Nonlinear Optical Properties of Organic Materials VI, G. R. Moehlmann, ed., Proc. SPIE 2025, 192 (1993).
[CrossRef]

Other (14)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 155.

W. J. Welsh, “Densities of amorphous and crystalline polymers” in Physical Properties of Polymers Handbook, J. E. Mark, ed. (American Institute of Physics, New York, 1996), pp. 401–407.

D. R. Lide, ed., Handbook of Chemistry and Physics (CRC, Cleveland, Ohio, 1990).

D. W. van Krevelen and P. J. Hoftyzer, Properties of Polymers, Their Estimation and Correlation with Chemical Structures (Elsevier, Amsterdam, 1976).

J. D. Swalen, J. Molec. Electron. 2, 155 (1986).

R. Azzam and W. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

S. Ramo, J. R. Whinnery, and T. van Duzer, Fields and Waves in Communications Electronics (Wiley, New York, 1965), p. 335.

Data sheet available from Dan Dawson, IBM Almaden Research Center, 650 Harry Road, San Jose, Calif. 95120 [tel. (408) 927–1617].

R. C. Raine, in American Institute of Physics Handbook, D. E. Gray, ed. (McGraw-Hill, New York, 1950), pp. 6–93 and 6–94.

W. E. Press, B. R. Flannery, S. A. Teukolsky, and W. T. Vettering, Numerical Recipes (Cambridge U. Press, Cambridge, 1989).

R. J. Twieg and C. W. Dirk, “Design, properties and applications of nonlinear optical chromophores,” in Organic Thin Films for Waveguiding Nonlinear Optics, F. Kajzar and J. D. Swalen, eds. (Gordon & Breach, Amsterdam, 1996), Chap. 2, pp. 45–136.

R. D. Miller, “Poled polymers for χ(2) applications,” in Organic Thin Films for Waveguiding Nonlinear Optics, F. Kajzar and J. D. Swalen, eds. (Gordon & Breach, Amsterdam, 1996), Chap. 8, pp. 329–456.

Ch. Bosshard, K. Sutter, Ph. Prêtre, J. Hulliger, M. Flörsheimer, P. Kaatz, and P. Günter, in Organic Nonlinear Optical Materials, A. F. Garito and F. Kajzar, eds., Vol. 1 of Advances in Nonlinear Optics (Gordon & Breach, Basel, 1995), p. 116; G. Knöpfle, Ch. Bosshard, R. Schlesser, and P. Günter, IEEE J. Quantum Electron. 30, 1303 (1994); F. Pan, G. Knöpfle, Ch. Bosshard, S. Follonier, R. Spreiter, M. S. Wong, and P. Günter, Appl. Phys. Lett. 69, 13 (1996).
[CrossRef]

This procedure is not strictly correct. It assumes that adding the chromophore results in no index change in the solution and consequently no change in transmission coefficients at the cell–solution boundaries. At the low concentrations used in these types of measurement, however, such a change (Δn≈10−5) will have no influence on the result.

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 (8)

Fig. 1
Fig. 1

Structural schematic of the polymer PMMA with the chromophore DR1 chemically attached. For our materials n =1 and m=9. The dashed line shows the separation between the chromophore and the oligomer in our calculations.

Fig. 2
Fig. 2

Concentration of DR1 as a function of absorbance for four cases. The inverse of the slope is equal to the extinction coefficient times the unit thickness. Measurements in solution exhibit a linear behavior, and therefore it is sufficient to perform a single measurement at a known concentration for an accurate determination of the extinction coefficient.

Fig. 3
Fig. 3

Dispersion in the refractive index of PMMA. Data in Ref. 20 are calculated with a three-term Sellmeier equation. Sellmeier parameters for the waveguide and ellipsometer measurements are given in Table 1.

Fig. 4
Fig. 4

Lorentz equation and effective-index approach calculation: Dispersion in the refractive index of 10 mol.% DR1–PMMA from absorption measurements of DR1 in several solvents: solid curve, chloroform; dashed curve, cyclohexanone; dotted curve, tert-butyl acetate. Crosses, ellipsometric data for a thin film of this NLO polymer. The insets show the curves after an additional background subtraction for a least-squares deviation from measured data.

Fig. 5
Fig. 5

Conjugate Fourier series and effective-index approach calculation: Dispersion in the refractive index of 10 mol.% DR1–PMMA from absorption measurements of DR1 in several solvents: solid curve, chloroform; dashed curve, cyclohexanone; dotted curve, tert-butyl acetate. Dashed–dotted curve, side-chain polymer; crosses, ellipsometric data for a thin film of this NLO polymer. The insets show the same curves on a enlarged ordinate scale.

Fig. 6
Fig. 6

Direct conjugate Fourier series calculation: Dispersion in the refractive index of 10 mol.% DR1–PMMA from absorption measurements of DR1 in several solvents: solid curve, chloroform; dashed curve, cyclohexanone; dotted curve, tert-butyl acetate. Dashed–dotted curve, side-chain polymer; crosses, ellipsometric data for a thin film of this NLO polymer. The insets show the same curves on a enlarged ordinate scale.

Fig. 7
Fig. 7

Calculated dispersion for the second-harmonic coefficient d33 of 10 mol.% DR1–PMMA for a poling-field strength of 130 V/μm. Solid curve, worst case for the index dispersion, i.e., the index for DR1 in tert-butyl acetate from Fig. 4. Any other index set will be closer to the data points calculated with measured index values.

Fig. 8
Fig. 8

Calculated dispersion for the electro-optic coefficient r33 of 10 mol.% DR1–PMMA for a poling-field strength of 130 V/μm. Solid curve, worst case for the index dispersion, i.e., the index for DR1 in tert-butyl acetate from Fig. 4. Any other index set will be closer to the data points calculated with measured index values.

Tables (5)

Tables Icon

Table 1 Sellmeier Parameters According to Eq. (12) for PMMA and 10 mol.% DR1–PMMA a

Tables Icon

Table 2 Extinction Coefficient at the Wavelength of Maximum Absorption. FWHM Absorption, and Calculated Absorbance per Unit Length of a Neat DR1 Sample

Tables Icon

Table 3 Lorentz Equation and Effective-Index Approach Calculations at Selected Wavelengths for 10 mol.% DR1–PMMA from Absorption Measurements of DR1 in Chloroform

Tables Icon

Table 4 Conjugate Fourier Series Method and Effective-Index Approach Calculations at Selected Wavelengths for 10 mol.% DR1–PMMA from Absorption Measurements of DR1 in Chloroform

Tables Icon

Table 5 Direct Conjugate Fourier Series Calculations at Selected Wavelengths for 10 mol.% DR1–PMMA from Absorption Measurements of DR1 in Chloroform

Equations (45)

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

n2-1n2+2=4π3 Nα.
Ntαeff=Ntαc+Ntαp,
Np=Aρp/Mp.
Nc=Aρc/Mc.
neff2-1neff2+2=fρtρc nc2-1nc2+2+(1-f )ρtρp np2-1np2+2.
mtρt=mcρc+mpρp.
ρt=ρp/(1-fΔ),
neff2-np2neff2+2=f nc2-np2nc2+2.
weight/molecule
=M/A,where A is Avogadros number,
volume/molecule=M/Aρ,where ρ is the density,
numberofmolecules/1000cm3=1000Aρ/M,
c=1000ρ/Mmol/1.
ν=cMc/1000ρc,
f=νρcνρc+(1-ν)ρp.
A=εmaxct,
n2-1=B+q1/λmax2-1λ2.
ε=εb,neat+ωp2ω02-ω2-iγ.
ε=εb,near+ωp2(ω02-ω2)(ω02-ω2)2+ω2γ2,
ε=ωp2ωγ(ω02-ω2)2+ω2γ2.
ε=εb,neat,
ε=ωp2/ω0γ.
k=k0(n-iκ)=ω[μ(ε-iε)]1/2=-iα+β.
α2=k02ε2 1+εε21/2-1.
ωp2=ω0εb,neatγη=η24ω02,
η=4α2k02εb,neat=A/t ln 102πν¯0nb, neat2.
Z=(ω-i)/(ω+i),
n˜(ω)-1n˜[-i(Z+1)/(Z-1)]-1=m=0cmZm,
|Z| 1.
Z=exp(iθ)
n˜-cot θ2-1=m=0cm cos mθ+im=1cm sin mθn(θ)-in(θ),
n(θ)=a02+m=1am cos mθ,
n(θ)=-m=1am sin mθ,
limω[n(ω)-1]=limθ0 n(θ)=a02+m=1am=0
a0=-2m=1am.
χzzz(-2ω; ω, ω)=Nf2ωfω2βcos3 θ,
χzxx(-2ω; ω, ω)=1/2Nf2ωfω2βcos θ sin2 θ.
β(-2ω; ω, ω)=β0/[(1-ζ)(1-4ζ)],
β(-ω; ω, 0)=β0 (3-ζ)3(1-ζ)2,
fω=n2+23
f0=ε(0)(n2+2)n2+2ε(0).
cos3 θ=1+6u2coth u-3u 1+2u2u5,u1,
cos θ=coth u-1uu3,u1,
cos θ sin2 θ=1/2(cos θ-cos3 θ)u15,u1,
nb=np+f(nb, neat-np).

Metrics