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Enhanced optical phase conjugation in nonlinear metamaterials

Kihong Kim

Open Access Open Access

Abstract

Optical phase conjugation by degenerate four-wave mixing in nonlinear metamaterials is studied theoretically by solving the coupled wave equations using a generalized version of the invariant imbedding method. The phase-conjugate reflectance and the lateral shift of the phase-conjugate reflected beams are calculated and their dependencies on the frequency, the polarization, the incident angle, the material properties and the structure are investigated in detail. It is found that the efficiency of phase conjugation can be significantly enhanced due to the enhancement of electromagnetic fields in various metamaterial structures.

© 2014 Optical Society of America

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Figures (7)
Fig. 1
Fig. 1 Schematic of the situation studied in this paper.

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Fig. 2
Fig. 2 Square root of the phase conjugate reflectance, R 21, plotted versus w ≡ ω1/ω̄ for both (a) s and (b) p waves, when γ0 = γ = 0.01, L/λ1 = 105 and θ = 45°. The negative index case with ε = μ = −1 is compared with the positive index case with ε = μ = 1.

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Fig. 3
Fig. 3 Lateral shift of a phase-conjugate beam, Δ, normalized with respect to λ1 plotted versus w = ω1/ω̄ for (a) s and (b) p waves, when γ0 = γ = 0.01, L/λ1 = 105 and θ = 45°. The case with ε = μ = −1 is compared with that with ε = μ = 1.

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Fig. 4
Fig. 4 Square root of the phase conjugate reflectance, R 21, plotted versus incident angle for (a) s and (b) p waves, when γ0 = γ = 0.01, L/λ1 = 105 and w = ω1/ω̄ = 0.99975.

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Fig. 5
Fig. 5 Square root of the phase conjugate reflectance, R 21 , plotted versus ε for (a) s and (b) p waves, when γ0 = γ = 0.01, μ = 1, θ = 45°, L/λ1 = 105 and w = ω1/ω̄ = 0.99975.

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Fig. 6
Fig. 6 Square root of the phase conjugate reflectance, R 21 , of a three-layer system plotted versus w ≡ ω1/ω̄ for (a) s and (b) p waves when θ = 45°. A linear layer of thickness 15λ1 with ε = 1.5 and μ = 1 is surrounded by nonlinear layers of equal thicknesses 45λ1 with γ0 = γ = 0.01 and ε = μ = 1.

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Fig. 7
Fig. 7 Influence of absorption on the phase conjugate reflectance curves corresponding to (a) Fig. 2(a), (b) Fig. 4(a) and (c) Fig. 6(a). (a) R 21 versus w for s waves, when ε = μ = −1, −1 + 0.001i and −1 + 0.002i. (b) R 21 versus θ for s waves, when ε = μ = −1 and −1 + 0.002i. (c) R 21 versus w for s waves, when Im ε = 0 and 0.003i for both the defect and the nonlinear layers.

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Equations (13)

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ω ¯ = ω 1 + ω 2 2 .
B = μ H ,                 D = ε E + 4 π P NL ,
∇ 2 E − ∇ ( ∇ ⋅ E ) + ∇ μ μ × ( ∇ × E ) + ω 2 c 2 ( ε μ E + 4 π μ P NL ) = 0 ,
∇ 2 H − ∇ ( ∇ ⋅ H ) + ∇ ε ε × ( ∇ × H ) + ω 2 c 2 ε μ H + 4 π i ω c ( ∇ ε ε × P NL − ∇ × P NL ) = 0 .
P NL ≈ 1 4 π 𝒫 ( 2 γ 0 E + γ E ˜ ) ,                 P ˜ NL ≈ 1 4 π 𝒫 ( 2 γ 0 E ˜ + γ * E ) ,
𝒫 E = E x x ^ + E y y ^ + 3 E z z ^ .
d 2 ψ d z 2 − d ℰ d z ℰ − 1 ( z ) d ψ d z + [ ℰ ( z ) K 2 ℳ ( z ) − q 2 I ] ψ = 0 ,
ψ = ψ s = ( E 1 ( z ) E 2 ( z ) ) ,                   K = ( k 1 0 0 k 2 ) ,                 I = ( 1 0 0 1 ) , ℰ = ℰ s = ( μ ( z ) 0 0 μ ( z ) ) ,                 ℳ = ℳ s = ( ε ( z ) + 2 γ 0 ( z ) γ ( z ) γ * ( z ) ε ( z ) + 2 γ 0 ( z ) ) .
ψ = ψ p = ( H 1 ( z ) H 2 ( z ) ) ,                 ℰ = ℰ p = ( ε ( z ) + 2 γ 0 ( z ) γ ( z ) ρ ρ γ * ( z ) ε ( z ) + 2 γ 0 ( z ) ) , ℳ = ℳ p = ( μ ( z ) + 2 γ 0 ( z ) ε ( z ) + 3 γ 0 ( z ) ε ( z ) + γ 0 ( z ) Y γ ( z ) ρ ε ( z ) − 3 γ 0 ( z ) ε ( z ) + γ 0 ( z ) Y γ * ( z ) ρ ε ( z ) − 3 γ 0 ( z ) ε ( z ) + γ 0 ( z ) Y μ ( z ) + 2 γ 0 ( z ) ρ 2 ε ( z ) + 3 γ 0 ( z ) ε ( z ) + γ 0 ( z ) Y ) ,
ρ = − k 2 k 1 ,                   Y = 2 q 2 [ ε ( z ) + 3 γ 0 ( z ) ] [ ε ( z ) + 9 γ 0 ( z ) ] k 1 2 .
d r d l = i [ r ( l ) ℰ ( l ) P + ℰ ( l ) Pr ( l ) ] − i 2 [ r ( l ) + 1 ] [ ℰ ( l ) P − P ℳ ( l ) − q 2 P − 1 ℳ ( l ) + q 2 P − 1 ℰ − 1 ( l ) ] [ r ( l ) + 1 ] , d t d l = i t ( l ) ℰ ( l ) P − i 2 t ( l ) [ ℰ ( l ) P − P ℳ ( l ) − q 2 P − 1 ℳ ( l ) + q 2 P − 1 ℰ − 1 ( l ) ] [ r ( l ) + 1 ] ,
p 1 = k 1 cos θ ,               p 2 = − k 2 2 − q 2 .
Δ = λ 1 2 π cos θ d Φ d θ ,
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