M. G. Kuzyk, J. E. Sohn, and C. W. Dirk, "Mechanisms of quadratic electro-optic modulation of dye-doped polymer systems," J. Opt. Soc. Am. B 7, 842-858 (1990)

We present the experimental and theoretical considerations needed to determine the mechanisms of quadratic electro-optic modulation in dye-doped polymer systems. The modulation is observed in thin dye-doped polymer films and the quadratic Kerr coefficient is determined by using a modified Mach–Zehnder interferometric technique. The theory of several nonlinear mechanisms is developed and applied to representative systems to determine the various contributions. The fast, virtual electronic mechanism is shown to be the largest contribution, as inferred from the measured frequency dispersion of the quadratic electro-optic effect. This quick interferometric method is also shown to be useful for determining nonlinear-optical structure–property relationships.

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Typical experimental values of refractive index n, sample thickness d, modulating frequency Ω, resistance R, and coefficient of expansion (1/n)/(∂n/∂T).

Table 3

Molecular Values Used to Calculate Orientational Contributions in DRl/PMMA^{a}

Calculated orientational contribution
${s}_{1133}^{\text{or}}$, second-order induced orientational contribution
${s}_{1133}^{\text{sor}}$ and measured quadratic electro-optic coefficient
${s}_{1133}^{\text{expt}}$ for isotropic DR1/PMMA with N = 1.8 × 10^{20} cm^{−3} and n_{11} = 1.53. Note that the doped sample had a refractive index of ∼1.52.

Table 5

Third-Order Susceptibility Comparison between Electro-Optic Measurement and Third-Harmonic Measurement^{a}

Note that the electro-optic measurement was at λ = 633 nm with a number density of N = 1.8 × 10^{20} cm^{−3} and that the third-harmonic was at a fundamental wavelength of λ = 2050 with number density N = 1.9 × 10^{21} cm^{−3}.

Table 6

Third-Order Susceptibility of a Series of Molecules as Determined with Quadratic Electro-Optic Modulation^{a}

All measurements are at λ = 633 nm except for ISQ, which was measured at 799 nm. Orientational effects are not subtracted. DNBA, a diaminonitrobenzaniline dye; DNTA, a diaminonitrothiophene aniline dye; NFAI, a nitrofuran-substituted dye; NPCV, a nitrophenylcyanovinylazo dye.
The refractive index n was purposely overestimated owing to resonance enhancement, yielding as upper limits values of 〈γ_{zzzz}*〉 and χ^{(3)}. Also, the orientational effects have not been taken into account, so the dressed molecular third-order susceptibility could be systematically high by ∼30% for the noncentrosymmetric molecules.

Table 7

Experimental and Theoretical Isotropic Averages of Molecular Nonlinear-Optical Susceptibilities of Small Molecules^{a}

Molecule

N(10^{21} cm^{−3})

〈γ〉_{exp}(10^{−35} esu)

〈γ〉_{the}(10^{−35} esu)

m-Dinitrobenzene

1.24

0.6(± 0.3)

1.0

m-Dicyanobenzene

1.09

0.8 (±0.4)

1.0–3.0

The range in calculated values represents the range of results over several parameter sets.

Tables (7)

Table 1

Some Physical Properties of PMMA at Room Temperature^{a}

Thermal refractive-index coefficient

$\frac{1}{n}\frac{\partial n}{\partial T}$

1.1 × 10^{−4} K^{−1}

Thermal-expansion coefficient

$\frac{1}{d}\frac{\partial d}{\partial T}$

7 × 10^{−5} K^{−1}

Density

ρ

1.19 × 10^{3} kg/m^{2}

Specific heat

c

1.42 × 10^{3} J/kg-K

Dielectric constant

∊(ω = 1kHz)

3.0

Sound velocity

ν(2 MHz)

2700 m/sec

Elastic modulus

E′

6.6 × 10^{10} dyn/cm^{2}

Shear modulus

G′

2.0 × 10^{10} dyn/cm^{2}

Thermal conductivity

σ

0.193 W/m-K

See Ref. 71. The elastic moduli at 4 kHz were extrapolated from lower-frequency measurements.55

Table 2

Heating Contribution to the Third-Order Susceptibility in PMMA^{a}

Typical experimental values of refractive index n, sample thickness d, modulating frequency Ω, resistance R, and coefficient of expansion (1/n)/(∂n/∂T).

Table 3

Molecular Values Used to Calculate Orientational Contributions in DRl/PMMA^{a}

Calculated orientational contribution
${s}_{1133}^{\text{or}}$, second-order induced orientational contribution
${s}_{1133}^{\text{sor}}$ and measured quadratic electro-optic coefficient
${s}_{1133}^{\text{expt}}$ for isotropic DR1/PMMA with N = 1.8 × 10^{20} cm^{−3} and n_{11} = 1.53. Note that the doped sample had a refractive index of ∼1.52.

Table 5

Third-Order Susceptibility Comparison between Electro-Optic Measurement and Third-Harmonic Measurement^{a}

Note that the electro-optic measurement was at λ = 633 nm with a number density of N = 1.8 × 10^{20} cm^{−3} and that the third-harmonic was at a fundamental wavelength of λ = 2050 with number density N = 1.9 × 10^{21} cm^{−3}.

Table 6

Third-Order Susceptibility of a Series of Molecules as Determined with Quadratic Electro-Optic Modulation^{a}

All measurements are at λ = 633 nm except for ISQ, which was measured at 799 nm. Orientational effects are not subtracted. DNBA, a diaminonitrobenzaniline dye; DNTA, a diaminonitrothiophene aniline dye; NFAI, a nitrofuran-substituted dye; NPCV, a nitrophenylcyanovinylazo dye.
The refractive index n was purposely overestimated owing to resonance enhancement, yielding as upper limits values of 〈γ_{zzzz}*〉 and χ^{(3)}. Also, the orientational effects have not been taken into account, so the dressed molecular third-order susceptibility could be systematically high by ∼30% for the noncentrosymmetric molecules.

Table 7

Experimental and Theoretical Isotropic Averages of Molecular Nonlinear-Optical Susceptibilities of Small Molecules^{a}

Molecule

N(10^{21} cm^{−3})

〈γ〉_{exp}(10^{−35} esu)

〈γ〉_{the}(10^{−35} esu)

m-Dinitrobenzene

1.24

0.6(± 0.3)

1.0

m-Dicyanobenzene

1.09

0.8 (±0.4)

1.0–3.0

The range in calculated values represents the range of results over several parameter sets.