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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References

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  1. N. I. Zheludev, “The road ahead for metamaterials,” Science 328, 582–583 (2010).
    [Crossref] [PubMed]
  2. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).
  3. M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014).
    [Crossref]
  4. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003).
    [Crossref] [PubMed]
  5. A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. B 81(4), 043839 (2010).
    [Crossref]
  6. A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011).
    [Crossref] [PubMed]
  7. M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
    [Crossref]
  8. H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995).
    [Crossref] [PubMed]
  9. H. F. Arnoldus and T. F. George, Phase Conjugation in a Layer of Nonlinear Material (Nova Publishers, 2005).
  10. K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. 69(2), 207–213 (2006).
    [Crossref]
  11. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses,” Opt. Express 16(2), 1150–1164 (2008).
    [Crossref] [PubMed]
  12. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, “Strong influence of nonlinearity and surface plasmon excitations on the lateral shift,” Opt. Express 16(20), 15506–15513 (2008).
    [Crossref] [PubMed]
  13. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
  14. R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).
  15. V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. 33, 1–127 (1994).
    [Crossref]
  16. G. C. Papen and B. E. A. Saleh, “Lateral and focal shifts of phase-conjugated beams in photorefractive materials,” Opt. Lett. 14(14), 745–747 (1989).
    [Crossref] [PubMed]
  17. B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
    [Crossref]
  18. A.-A. R. Al-Rashed, B. M. Jost, and B. E. A. Saleh, “Spatial shifts of the conjugate beam generated by a nondegenerate photorefractive phase-conjugate mirror,” Appl. Opt. 37(33), 7821–7826 (1998).
    [Crossref]
  19. B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
    [Crossref]

2014 (1)

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014).
[Crossref]

2012 (1)

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
[Crossref]

2011 (2)

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011).
[Crossref] [PubMed]

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).

2010 (2)

N. I. Zheludev, “The road ahead for metamaterials,” Science 328, 582–583 (2010).
[Crossref] [PubMed]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. B 81(4), 043839 (2010).
[Crossref]

2008 (2)

2006 (1)

K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. 69(2), 207–213 (2006).
[Crossref]

2003 (1)

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003).
[Crossref] [PubMed]

1998 (2)

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

A.-A. R. Al-Rashed, B. M. Jost, and B. E. A. Saleh, “Spatial shifts of the conjugate beam generated by a nondegenerate photorefractive phase-conjugate mirror,” Appl. Opt. 37(33), 7821–7826 (1998).
[Crossref]

1997 (1)

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[Crossref]

1995 (1)

H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995).
[Crossref] [PubMed]

1994 (1)

V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. 33, 1–127 (1994).
[Crossref]

1989 (1)

Al-Rashed, A.-A. R.

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

A.-A. R. Al-Rashed, B. M. Jost, and B. E. A. Saleh, “Spatial shifts of the conjugate beam generated by a nondegenerate photorefractive phase-conjugate mirror,” Appl. Opt. 37(33), 7821–7826 (1998).
[Crossref]

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[Crossref]

Arnoldus, H. F.

H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995).
[Crossref] [PubMed]

H. F. Arnoldus and T. F. George, Phase Conjugation in a Layer of Nonlinear Material (Nova Publishers, 2005).

Bellman, R.

R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Ciattoni, A.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
[Crossref]

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. B 81(4), 043839 (2010).
[Crossref]

de Ceglia, D.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
[Crossref]

George, T. F.

H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995).
[Crossref] [PubMed]

H. F. Arnoldus and T. F. George, Phase Conjugation in a Layer of Nonlinear Material (Nova Publishers, 2005).

Huang, D.

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011).
[Crossref] [PubMed]

Jost, B. M.

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

A.-A. R. Al-Rashed, B. M. Jost, and B. E. A. Saleh, “Spatial shifts of the conjugate beam generated by a nondegenerate photorefractive phase-conjugate mirror,” Appl. Opt. 37(33), 7821–7826 (1998).
[Crossref]

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[Crossref]

Kim, K.

Kivshar, Y. S.

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014).
[Crossref]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003).
[Crossref] [PubMed]

Klyatskin, V. I.

V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. 33, 1–127 (1994).
[Crossref]

Lapine, M.

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014).
[Crossref]

Lee, D.-H.

K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. 69(2), 207–213 (2006).
[Crossref]

Lim, H.

Palange, E.

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. B 81(4), 043839 (2010).
[Crossref]

Papen, G. C.

Phung, D. K.

Rizza, C.

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. B 81(4), 043839 (2010).
[Crossref]

Rose, A.

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011).
[Crossref] [PubMed]

Rotermund, F.

Saleh, B. E. A.

A.-A. R. Al-Rashed, B. M. Jost, and B. E. A. Saleh, “Spatial shifts of the conjugate beam generated by a nondegenerate photorefractive phase-conjugate mirror,” Appl. Opt. 37(33), 7821–7826 (1998).
[Crossref]

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[Crossref]

G. C. Papen and B. E. A. Saleh, “Lateral and focal shifts of phase-conjugated beams in photorefractive materials,” Opt. Lett. 14(14), 745–747 (1989).
[Crossref] [PubMed]

Scalora, M.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
[Crossref]

Shadrivov, I. V.

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014).
[Crossref]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003).
[Crossref] [PubMed]

Smith, D. R.

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011).
[Crossref] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).

Tataronis, J. A.

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[Crossref]

Vincenti, M. A.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
[Crossref]

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).

Wing, G. M.

R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

Zharov, A. A.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003).
[Crossref] [PubMed]

Zheludev, N. I.

N. I. Zheludev, “The road ahead for metamaterials,” Science 328, 582–583 (2010).
[Crossref] [PubMed]

Appl. Opt. (1)

Europhys. Lett. (1)

K. Kim, D.-H. Lee, and H. Lim, “Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media,” Europhys. Lett. 69(2), 207–213 (2006).
[Crossref]

Nat. Photon. (1)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photon. 5, 523–530 (2011).

Opt. Commun. (1)

B. M. Jost, A.-A. R. Al-Rashed, J. A. Tataronis, and B. E. A. Saleh, “Enhancement of phase-conjugate reflectivity with linear absorption in four-wave mixing systems,” Opt. Commun. 144, 222–226 (1997).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (2)

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84(6), 063826 (2012).
[Crossref]

H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995).
[Crossref] [PubMed]

Phys. Rev. B (1)

A. Ciattoni, C. Rizza, and E. Palange, “Extreme nonlinear electrodynamics in metamaterials with very small linear dielectric permittivity,” Phys. Rev. B 81(4), 043839 (2010).
[Crossref]

Phys. Rev. Lett. (3)

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett. 107(6), 063902 (2011).
[Crossref] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003).
[Crossref] [PubMed]

B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-Hänchen effect in a phase-conjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
[Crossref]

Prog. Opt. (1)

V. I. Klyatskin, “The imbedding method in statistical boundary-value wave problems,” Prog. Opt. 33, 1–127 (1994).
[Crossref]

Rev. Mod. Phys. (1)

M. Lapine, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear metamaterials,” Rev. Mod. Phys. 86(3), 1093 (2014).
[Crossref]

Science (1)

N. I. Zheludev, “The road ahead for metamaterials,” Science 328, 582–583 (2010).
[Crossref] [PubMed]

Other (3)

H. F. Arnoldus and T. F. George, Phase Conjugation in a Layer of Nonlinear Material (Nova Publishers, 2005).

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

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Figures (7)

Fig. 1
Fig. 1 Schematic of the situation studied in this paper.
Fig. 2
Fig. 2 Square root of the phase conjugate reflectance, R 21, plotted versus wω1/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.
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.
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.
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.
Fig. 6
Fig. 6 Square root of the phase conjugate reflectance, R 21 , of a three-layer system plotted versus wω1/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.
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.

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 d z 1 ( z ) d ψ d z + [ ( z ) K 2 ( 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 ) , = s = ( μ ( z ) 0 0 μ ( z ) ) , = 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 ) ) , = p = ( ε ( z ) + 2 γ 0 ( z ) γ ( z ) ρ ρ γ * ( z ) ε ( z ) + 2 γ 0 ( z ) ) , = 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 ) ( l ) P + ( l ) Pr ( l ) ] i 2 [ r ( l ) + 1 ] [ ( l ) P P ( l ) q 2 P 1 ( l ) + q 2 P 1 1 ( l ) ] [ r ( l ) + 1 ] , d t d l = i t ( l ) ( l ) P i 2 t ( l ) [ ( l ) P P ( l ) q 2 P 1 ( l ) + q 2 P 1 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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