Enhanced optical phase conjugation in nonlinear metamaterials Kihong Kim∗ Department of Energy Systems Research and Department of Physics, Ajou University, Suwon 443-749, South Korea ∗ [email protected]
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 OCIS codes: (190.5040) Phase conjugation; (190.4380) Nonlinear optics, four-wave mixing; (160.3918) Metamaterials.
References and links 1. N. I. Zheludev, “The road ahead for metamaterials,” Science 328, 582–583 (2010). 2. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of threedimensional 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). 4. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003). 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). 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). 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). 8. H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995). 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). 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). 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). 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). 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). 17. B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-H¨anchen effect in a phaseconjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1744
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). 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).
1.
Introduction
Photonic metamaterials have attracted a great deal of interest from researchers, due to their great potential in applications as well as their intriguing physical properties [1, 2]. A large amount of attention has been paid to the nonlinear photonic properties in various kinds of metamaterials, including those with very small dielectric permittivity [3]. It has been demonstrated that nonlinear optical phenomena such as second harmonic generation and optical bistability can be greatly enhanced in metamaterials [4-7]. In the present paper, I study optical phase conjugation by degenerate four-wave mixing in nonlinear metamaterials theoretically. My approach is based on a derivation of the coupled wave equations following the method of Arnoldus and George [8, 9] and uses a generalized version of the invariant imbedding method [10-12], which goes beyond the slowly-varying envelope approximation widely used in the analysis of optical phase conjugation [13]. I find that the efficiency of phase conjugation can be significantly enhanced in various metamaterial structures such as negative refractive index media, media with small dielectric permittivity and multilayered structures. It is interpreted that this is due to the enhancement of electromagnetic fields in those metamaterial structures. 2.
Optical phase conjugation in Kerr media through degenerate four-wave mixing
Fig. 1. Schematic of the situation studied in this paper.
I am interested in the propagation of electromagnetic waves in nonlinear phase-conjugating media. I consider the situation depicted in Fig. 1, where a Kerr nonlinear medium lying in 0 ≤ z ≤ L is illuminated by two strong monochromatic pump beams of frequency ω propagating perpendicularly to the z axis and oppositely to each other [8, 9]. Following Ref. 8, I assume that the electric fields associated with the pump beams are linearly polarized in the z direction and phase-matched. A weak probe wave with the frequency ω1 and the vacuum wave number k1 = ω1 /c is incident on the medium from the (vacuum) region where z > L with an incident angle θ . In addition to an ordinary wave of frequency ω1 reflected specularly from the boundary plane z = L, there exists a phase-conjugate reflected wave with the frequency ω2 (≈ ω1 ) and the vacuum wave number k2 = ω2 /c propagating almost oppositely to the incident wave. Similarly, there exist both ordinary and phase-conjugate transmitted waves in the (vacuum) region z < 0. In the current four-wave mixing geometry, the frequency of the pump beams, ω , is equal to
ω1 + ω2 . (1) 2 I assume that the medium is uniform along the x and y axes and all waves propagate in the xz plane. ω=
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1745
In nonlinear Kerr media, the electromagnetic fields satisfy the usual relationships B = μ H, D = ε E + 4π PNL ,
(2)
where ε is the linear part of the dielectric permittivity and μ is the magnetic permeability. The cgs units are used in this paper. The nonlinear polarization vector PNL describes the coupling between the ordinary and phase-conjugate waves. Starting from Maxwell’s equations, I derive the electromagnetic wave equations
ω2 ∇μ × (∇ × E) + 2 (ε μ E + 4π μ PNL ) = 0, μ c 2 ω ∇ε 4 π iω ∇ ε 2 ∇ H − ∇ (∇ · H) + × (∇ × H) + 2 ε μ H + × PNL − ∇ × PNL = 0. ε c c ε ∇2 E − ∇ (∇ · E) +
(3) (4)
For s-polarized waves, ∇ · E = 0 and it is convenient to use Eq. (3). Similarly for p-polarized waves, ∇ · H = 0 and it is convenient to use Eq. (4). In the present paper, I focus on the optical phase conjugation phenomenon due to the nonlinear polarization effect and ignore third harmonic generation. When the probe wave is much weaker than the pump beams, it is straightforward to show that PNL ≈
1 ˜ , P˜ NL ≈ 1 P 2γ0 E˜ + γ ∗ E , P 2γ0 E + γ E 4π 4π
(5)
˜ and where E and PNL are the field components associated with the prove wave, whereas E P˜ NL are those associated with the phase-conjugate wave [8, 9]. The assumption of undepleted pumps has been used in this derivation. In the present geometry, the operator P is defined by PE = Ex xˆ + Ey yˆ + 3Ez zˆ .
(6)
The parameters γ0 and γ (= γ0 eiφ ) represent the coupling between the ordinary and phaseconjugate waves due to the nonlinear four-wave mixing process in the medium. γ0 is a small parameter proportional to the third-order nonlinear optical susceptibility χ (3) . Both γ0 and φ depend on the intensities, the polarizations and the frequency of the pump beams and the horizontal length of the medium in a quite complicated manner. In this paper, I will not show the expressions for γ0 and φ and treat them as phenomenological parameters. A detailed derivation of Eq. (5) and the expressions for γ0 and φ can be found in Ref. 9. ˜ and P˜ NL have only y components. Then, from Eqs. (3) In the s wave case, the fields E, PNL , E and (5), one can easily derive a set of two coupled ordinary differential equations for the electric field amplitudes for the prove and phase-conjugate waves, E1 and E2 , respectively, which has the form d 2 ψ dE −1 d ψ E (z) + E (z)K 2 M (z) − q2 I ψ = 0, − 2 dz dz dz where q is the x component of the wave vector and 1 0 k1 0 E1 (z) , I= , , K= ψ = ψs = 0 1 E2 (z) 0 k2 μ (z) 0 ε (z) + 2γ0 (z) γ (z) , M = Ms = . E = Es = μ (z) γ ∗ (z) ε (z) + 2γ0 (z) 0
(7)
(8)
˜ and P˜ NL have x and z components, while the magnetic In the p wave case, the fields E, PNL , E ˜ fields H and H have only y components. Starting from Eq. (4) and expressing PNL and P˜ NL in #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1746
terms of the magnetic fields, one can show that an equation of the same form as Eq. (7) is satisfied, except that now ψ , E and M are given by γ (z) H1 (z) ε (z) + 2γ0 (z) ρ , E = Ep = ψ = ψp = , H2 (z) ργ ∗ (z) ε (z) + 2γ0 (z) γ0 (z) γ (z) ε (z)−3γ0 (z) μ (z) + 2γ0 (z) εε(z)+3 ρ ε (z)+γ0 (z) Y (z)+γ0 (z) Y , (9) M = Mp = γ ∗ (z) ε (z)−3γ0 (z) γ0 (z) μ (z) + 2γρ02(z) εε(z)+3 ρ ε (z)+γ (z) Y (z)+γ (z) Y 0
0
where H1 and H2 are the magnetic field amplitudes for the prove and phase-conjugate waves respectively and
ρ =−
k2 2q2 , Y= . k1 [ε (z) + 3γ0 (z)][ε (z) + 9γ0 (z)]k12
(10)
When a prove wave is incident from a vacuum region with the incident angle θ , q is equal to k1 sin θ . I emphasize that the frequently-used slowly-varying envelope approximation has not been used in the present derivation. 3.
Invariant imbedding method
The invariant imbedding method is based on an exact transformation of the boundary value problem of second-order ordinary differential equations into an initial value problem of firstorder ordinary differential equations [10, 11, 14, 15]. In Ref. 10, a generalized version of the invariant imbedding method for studying the propagation of any number of coupled waves in arbitrarily-inhomogeneous stratified media has been developed. This method is suitable for solving current problems. I focus on the 2 × 2 matrix reflection coefficient r and transmission coefficient t. For the coupled wave equation, Eq. (7), the invariant imbedding equations satisfied by r and t take the forms dr = i [r(l)E (l)P + E (l)Pr(l)] dl i − [r(l) + 1] E (l)P − PM (l) − q2 P−1 M (l) + q2 P−1 E −1 (l) [r(l) + 1], 2 dt = it(l)E (l)P dl i − t(l) E (l)P − PM (l) − q2 P−1 M (l) + q2 P−1 E −1 (l) [r(l) + 1], (11) 2 where P is a diagonal 2 × 2 matrix such that Pi j = pi δi j [10]. The quantities p1 and p2 are the negative z components of the vacuum wave vectors for the incident ordinary and phaseconjugate waves respectively and are written as
(12) p1 = k1 cos θ , p2 = − k22 − q2 . The reflection coefficient component r21 is associated with the case where the incident wave is an ordinary wave and the reflected wave is a phase-conjugate wave. All other matrix components of r and t are defined in a similar manner. The invariant imbedding equations are integrated numerically from l = 0 to l = L using the matrix initial conditions r(0) = 0 and t(0) = I. In Eq. (11), the quantities E and M are 2 × 2 matrices representing the specific material properties of the structure. They take different forms, either Eq. (8) or Eq. (9), depending on the polarization of an incident wave. In layered or stratified media, the parameters ε , μ , γ0 and φ depend only on z. #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1747
4.
Numerical results
I have applied the theoretical method developed here to various structures consisting of phaseconjugating media. In Fig. 2, I compare the square root of the phase-conjugate reflectance, √ R21 (= |r21 |), for the positive refractive index case with √ ε = μ = 1 with that for the negative refractive index case with ε = μ = −1. The quantity R21 is plotted versus the frequency of the incident prove wave normalized with respect to the pump frequency, w (≡ ω1 /ω ), for both s and p waves, when γ0 = γ = 0.01, L/λ1 = k1 L/(2π ) = 105 and θ = 45◦ . It is noticed that the phase-conjugate reflectance can be greatly enhhanced when the frequency of the incident wave is close to the pump frequency. It is also observed that the curves are in general not perfectly symmetric with respect to w = 1. The results for ε = μ = 1 agree perfectly with those shown in Fig. 14 of Ref. 9. For the parameter values used in Fig. 2, the enhancement is substantially larger in the negative refractive index case. The energy of the enhanced phase-conjugate wave is provided by the pumps.
√ Fig. 2. Square root of the phase conjugate reflectance, R21 , plotted versus w ≡ ω1 /ω 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.
I have also calculated the Goos-H¨anchen-type lateral shift of a phase-conjugate wave beam, Δ, using the formula λ1 dΦ , (13) Δ= 2π cos θ dθ √ where Φ is the phase of the reflection coefficient r21 = R21 eiΦ [16-18]. In Fig. 3, the lateral shifts corresponding to the parameter values used in Fig. 2 are plotted versus w. They show a peculiar non-monotonic and oscillatory behavior as a function of the frequency. The lateral shift at w = 1 is zero and its sign for w > 1 is opposite to that for w < 1. The signs of Δ are also opposite in the positive and negative index cases. As |w − 1| (= |ω1 − ω |/ω ), which is the normalized detuning frequency, increases from zero, the magnitude of the lateral shift initially increases to large values and then decreases. In the s wave case with ε = μ = 1, the maximum of |Δ| occurs at |w − 1| ≈ 0.001. The nonmonotonic behavior near w = 1 is qualitatively similar to that reported in [17, 18]. In those references, however, the maximum of |Δ| occurs at |w − 1| ∼ 10−14 , which is greatly different from my result.
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1748
Fig. 3. Lateral shift of a phase-conjugate beam, Δ, normalized with respect to λ1 plotted versus w = ω1 /ω 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.
√ In Fig. 4, the square root of the phase conjugate reflectance, R21 , is plotted as a function of the incident angle for the parameter values used in Fig. 2, when w = 0.99975. A series of sharp resonances occur at well-defined incident angles. This result is due to a Fabry-Perot-type resonance and is fully consistent with Fig. 6 of Ref. 8.
√ Fig. 4. Square root of the phase conjugate reflectance, R21 , plotted versus incident angle for (a) s and (b) p waves, when γ0 = γ = 0.01, L/λ1 = 105 and w = ω1 /ω = 0.99975.
√ In Fig. 5, I plot R21 as a function of the dielectric permittivity ε , when γ0 = γ = 0.01, μ = 1, θ = 45◦ , L/λ1 = 105 and w = 0.99975. It is observed that a large number of strong resonances occur at small values of ε , where the phase-conjugate reflectance is strongly amplified. Media with small dielectric permittivity are generally expected to show strongly-enhanced optical phase conjugation. #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1749
√ Fig. 5. Square root of the phase conjugate reflectance, R21 , 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 /ω = 0.99975.
√ In Fig. 6, I plot R21 as a function of w for the three-layer system where a normal slab of thickness 15λ1 with ε = 1.5 and μ = 1 is sandwiched between two phase-conjugating slabs of equal thicknesses 45λ1 with γ0 = γ = 0.01 and ε = μ = 1. The incident angle is equal to θ = 45◦ . The three-layer structure supports waveguide modes, which cause an enhancement of electromagnetic fields. An extremely strong resonant enhancement of the phase-conjugate reflection, especially for p waves, occurs due to this effect.
√ Fig. 6. Square root of the phase conjugate reflectance, R21 , of a three-layer system plotted versus w ≡ ω1 /ω 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.
Finally, I consider the effects of absorption on the phase conjugation process. As is wellknown, it is difficult to suppress the absorption in metamaterials, especially in negative-index #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1750
media based on metal-dielectric composites. The attention is restricted to the s wave case, √ though similar results are obtained for p waves. In Fig. 7(a), the R21 curve corresponding to the negative index case in Fig. 2(a) is plotted versus w when ε = μ = −1, −1 + 0.001i and −1 + 0.002i. It is seen that the resonant enhancement of the phase conjugate reflection near w = 0 is suppressed as the imaginary part of ε and μ increases. As long as the size of the imaginary part is smaller than the nonlinearity parameter γ0 , however, the resonant effect is still manifested. The effect of absorption is not limited to suppressing the enhancement, as shown by the appearance of √ two peaks when ε = μ = −1 + 0.002i. In Fig. 7(b), the R21 curve corresponding to the negative index case in Fig. 4(a) is plotted versus θ , when ε = μ = −1 and −1 + 0.002i. For large incident angles, the resonant effect is strongly suppressed, whereas for some special incident angles, it is greatly amplified. Near the peak at θ ≈ 44◦ , a strong oscillatory behavior appears when ε = μ = −1 + 0.002i. It has been pointed out previously that four-wave mixing systems can exhibit new oscillatory behavior and strong enhancement√in phase-conjugate reflection due to linear absorption [19]. In Fig. 7(c), the R21 curve corresponding to Fig. 6(a) is plotted versus w, when the imaginary part of ε for both the defect layer and the surrounding nonlinear layers is nonzero. I find that the resonance peaks are suppressed in general, though new peaks may appear at different parameter values. A detailed understanding of the influence of absorption on resonant phase conjugation phenomena requires a further study and will be the subject of a future publication.
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) R21 versus w for s waves, when ε = μ = −1, −1 + 0.001i√ and −1 + 0.002i. (b) R21 versus θ for s waves, when ε = μ = −1 and −1 + 0.002i. (c) R21 versus w for s waves, when Im ε = 0 and 0.003i for both the defect and the nonlinear layers.
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1751
5.
Conclusion
I have developed a novel theoretical method for studying the optical phase conjugation in inhomogeneous metamaterials, using which I have studied the dependencies of optical phase conjugation on the frequency, the polarization, the incident angle, the material properties and the structure in detail. It has been found that the efficiency of phase conjugation can be significantly enhanced due to the enhancement of electromagnetic fields in various metamaterial structures. Acknowledgments This work has been supported by the National Research Foundation of Korea Grant (NRF2012R1A1A2044201) funded by the Korean Government.
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1752
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 OCIS codes: (190.5040) Phase conjugation; (190.4380) Nonlinear optics, four-wave mixing; (160.3918) Metamaterials.
References and links 1. N. I. Zheludev, “The road ahead for metamaterials,” Science 328, 582–583 (2010). 2. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of threedimensional 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). 4. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91(3), 037401 (2003). 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). 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). 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). 8. H. F. Arnoldus and T. F. George, “Theory of optical phase conjugation in Kerr media,” Phys. Rev. A 51(5), 4250–4263 (1995). 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). 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). 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). 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). 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). 17. B. M. Jost, A.-A. R. Al-Rashed, and B. E. A. Saleh, “Observation of the Goos-H¨anchen effect in a phaseconjugate mirror,” Phys. Rev. Lett. 81(11), 2233–2235 (1998).
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1744
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). 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).
1.
Introduction
Photonic metamaterials have attracted a great deal of interest from researchers, due to their great potential in applications as well as their intriguing physical properties [1, 2]. A large amount of attention has been paid to the nonlinear photonic properties in various kinds of metamaterials, including those with very small dielectric permittivity [3]. It has been demonstrated that nonlinear optical phenomena such as second harmonic generation and optical bistability can be greatly enhanced in metamaterials [4-7]. In the present paper, I study optical phase conjugation by degenerate four-wave mixing in nonlinear metamaterials theoretically. My approach is based on a derivation of the coupled wave equations following the method of Arnoldus and George [8, 9] and uses a generalized version of the invariant imbedding method [10-12], which goes beyond the slowly-varying envelope approximation widely used in the analysis of optical phase conjugation [13]. I find that the efficiency of phase conjugation can be significantly enhanced in various metamaterial structures such as negative refractive index media, media with small dielectric permittivity and multilayered structures. It is interpreted that this is due to the enhancement of electromagnetic fields in those metamaterial structures. 2.
Optical phase conjugation in Kerr media through degenerate four-wave mixing
Fig. 1. Schematic of the situation studied in this paper.
I am interested in the propagation of electromagnetic waves in nonlinear phase-conjugating media. I consider the situation depicted in Fig. 1, where a Kerr nonlinear medium lying in 0 ≤ z ≤ L is illuminated by two strong monochromatic pump beams of frequency ω propagating perpendicularly to the z axis and oppositely to each other [8, 9]. Following Ref. 8, I assume that the electric fields associated with the pump beams are linearly polarized in the z direction and phase-matched. A weak probe wave with the frequency ω1 and the vacuum wave number k1 = ω1 /c is incident on the medium from the (vacuum) region where z > L with an incident angle θ . In addition to an ordinary wave of frequency ω1 reflected specularly from the boundary plane z = L, there exists a phase-conjugate reflected wave with the frequency ω2 (≈ ω1 ) and the vacuum wave number k2 = ω2 /c propagating almost oppositely to the incident wave. Similarly, there exist both ordinary and phase-conjugate transmitted waves in the (vacuum) region z < 0. In the current four-wave mixing geometry, the frequency of the pump beams, ω , is equal to
ω1 + ω2 . (1) 2 I assume that the medium is uniform along the x and y axes and all waves propagate in the xz plane. ω=
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1745
In nonlinear Kerr media, the electromagnetic fields satisfy the usual relationships B = μ H, D = ε E + 4π PNL ,
(2)
where ε is the linear part of the dielectric permittivity and μ is the magnetic permeability. The cgs units are used in this paper. The nonlinear polarization vector PNL describes the coupling between the ordinary and phase-conjugate waves. Starting from Maxwell’s equations, I derive the electromagnetic wave equations
ω2 ∇μ × (∇ × E) + 2 (ε μ E + 4π μ PNL ) = 0, μ c 2 ω ∇ε 4 π iω ∇ ε 2 ∇ H − ∇ (∇ · H) + × (∇ × H) + 2 ε μ H + × PNL − ∇ × PNL = 0. ε c c ε ∇2 E − ∇ (∇ · E) +
(3) (4)
For s-polarized waves, ∇ · E = 0 and it is convenient to use Eq. (3). Similarly for p-polarized waves, ∇ · H = 0 and it is convenient to use Eq. (4). In the present paper, I focus on the optical phase conjugation phenomenon due to the nonlinear polarization effect and ignore third harmonic generation. When the probe wave is much weaker than the pump beams, it is straightforward to show that PNL ≈
1 ˜ , P˜ NL ≈ 1 P 2γ0 E˜ + γ ∗ E , P 2γ0 E + γ E 4π 4π
(5)
˜ and where E and PNL are the field components associated with the prove wave, whereas E P˜ NL are those associated with the phase-conjugate wave [8, 9]. The assumption of undepleted pumps has been used in this derivation. In the present geometry, the operator P is defined by PE = Ex xˆ + Ey yˆ + 3Ez zˆ .
(6)
The parameters γ0 and γ (= γ0 eiφ ) represent the coupling between the ordinary and phaseconjugate waves due to the nonlinear four-wave mixing process in the medium. γ0 is a small parameter proportional to the third-order nonlinear optical susceptibility χ (3) . Both γ0 and φ depend on the intensities, the polarizations and the frequency of the pump beams and the horizontal length of the medium in a quite complicated manner. In this paper, I will not show the expressions for γ0 and φ and treat them as phenomenological parameters. A detailed derivation of Eq. (5) and the expressions for γ0 and φ can be found in Ref. 9. ˜ and P˜ NL have only y components. Then, from Eqs. (3) In the s wave case, the fields E, PNL , E and (5), one can easily derive a set of two coupled ordinary differential equations for the electric field amplitudes for the prove and phase-conjugate waves, E1 and E2 , respectively, which has the form d 2 ψ dE −1 d ψ E (z) + E (z)K 2 M (z) − q2 I ψ = 0, − 2 dz dz dz where q is the x component of the wave vector and 1 0 k1 0 E1 (z) , I= , , K= ψ = ψs = 0 1 E2 (z) 0 k2 μ (z) 0 ε (z) + 2γ0 (z) γ (z) , M = Ms = . E = Es = μ (z) γ ∗ (z) ε (z) + 2γ0 (z) 0
(7)
(8)
˜ and P˜ NL have x and z components, while the magnetic In the p wave case, the fields E, PNL , E ˜ fields H and H have only y components. Starting from Eq. (4) and expressing PNL and P˜ NL in #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1746
terms of the magnetic fields, one can show that an equation of the same form as Eq. (7) is satisfied, except that now ψ , E and M are given by γ (z) H1 (z) ε (z) + 2γ0 (z) ρ , E = Ep = ψ = ψp = , H2 (z) ργ ∗ (z) ε (z) + 2γ0 (z) γ0 (z) γ (z) ε (z)−3γ0 (z) μ (z) + 2γ0 (z) εε(z)+3 ρ ε (z)+γ0 (z) Y (z)+γ0 (z) Y , (9) M = Mp = γ ∗ (z) ε (z)−3γ0 (z) γ0 (z) μ (z) + 2γρ02(z) εε(z)+3 ρ ε (z)+γ (z) Y (z)+γ (z) Y 0
0
where H1 and H2 are the magnetic field amplitudes for the prove and phase-conjugate waves respectively and
ρ =−
k2 2q2 , Y= . k1 [ε (z) + 3γ0 (z)][ε (z) + 9γ0 (z)]k12
(10)
When a prove wave is incident from a vacuum region with the incident angle θ , q is equal to k1 sin θ . I emphasize that the frequently-used slowly-varying envelope approximation has not been used in the present derivation. 3.
Invariant imbedding method
The invariant imbedding method is based on an exact transformation of the boundary value problem of second-order ordinary differential equations into an initial value problem of firstorder ordinary differential equations [10, 11, 14, 15]. In Ref. 10, a generalized version of the invariant imbedding method for studying the propagation of any number of coupled waves in arbitrarily-inhomogeneous stratified media has been developed. This method is suitable for solving current problems. I focus on the 2 × 2 matrix reflection coefficient r and transmission coefficient t. For the coupled wave equation, Eq. (7), the invariant imbedding equations satisfied by r and t take the forms dr = i [r(l)E (l)P + E (l)Pr(l)] dl i − [r(l) + 1] E (l)P − PM (l) − q2 P−1 M (l) + q2 P−1 E −1 (l) [r(l) + 1], 2 dt = it(l)E (l)P dl i − t(l) E (l)P − PM (l) − q2 P−1 M (l) + q2 P−1 E −1 (l) [r(l) + 1], (11) 2 where P is a diagonal 2 × 2 matrix such that Pi j = pi δi j [10]. The quantities p1 and p2 are the negative z components of the vacuum wave vectors for the incident ordinary and phaseconjugate waves respectively and are written as
(12) p1 = k1 cos θ , p2 = − k22 − q2 . The reflection coefficient component r21 is associated with the case where the incident wave is an ordinary wave and the reflected wave is a phase-conjugate wave. All other matrix components of r and t are defined in a similar manner. The invariant imbedding equations are integrated numerically from l = 0 to l = L using the matrix initial conditions r(0) = 0 and t(0) = I. In Eq. (11), the quantities E and M are 2 × 2 matrices representing the specific material properties of the structure. They take different forms, either Eq. (8) or Eq. (9), depending on the polarization of an incident wave. In layered or stratified media, the parameters ε , μ , γ0 and φ depend only on z. #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1747
4.
Numerical results
I have applied the theoretical method developed here to various structures consisting of phaseconjugating media. In Fig. 2, I compare the square root of the phase-conjugate reflectance, √ R21 (= |r21 |), for the positive refractive index case with √ ε = μ = 1 with that for the negative refractive index case with ε = μ = −1. The quantity R21 is plotted versus the frequency of the incident prove wave normalized with respect to the pump frequency, w (≡ ω1 /ω ), for both s and p waves, when γ0 = γ = 0.01, L/λ1 = k1 L/(2π ) = 105 and θ = 45◦ . It is noticed that the phase-conjugate reflectance can be greatly enhhanced when the frequency of the incident wave is close to the pump frequency. It is also observed that the curves are in general not perfectly symmetric with respect to w = 1. The results for ε = μ = 1 agree perfectly with those shown in Fig. 14 of Ref. 9. For the parameter values used in Fig. 2, the enhancement is substantially larger in the negative refractive index case. The energy of the enhanced phase-conjugate wave is provided by the pumps.
√ Fig. 2. Square root of the phase conjugate reflectance, R21 , plotted versus w ≡ ω1 /ω 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.
I have also calculated the Goos-H¨anchen-type lateral shift of a phase-conjugate wave beam, Δ, using the formula λ1 dΦ , (13) Δ= 2π cos θ dθ √ where Φ is the phase of the reflection coefficient r21 = R21 eiΦ [16-18]. In Fig. 3, the lateral shifts corresponding to the parameter values used in Fig. 2 are plotted versus w. They show a peculiar non-monotonic and oscillatory behavior as a function of the frequency. The lateral shift at w = 1 is zero and its sign for w > 1 is opposite to that for w < 1. The signs of Δ are also opposite in the positive and negative index cases. As |w − 1| (= |ω1 − ω |/ω ), which is the normalized detuning frequency, increases from zero, the magnitude of the lateral shift initially increases to large values and then decreases. In the s wave case with ε = μ = 1, the maximum of |Δ| occurs at |w − 1| ≈ 0.001. The nonmonotonic behavior near w = 1 is qualitatively similar to that reported in [17, 18]. In those references, however, the maximum of |Δ| occurs at |w − 1| ∼ 10−14 , which is greatly different from my result.
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1748
Fig. 3. Lateral shift of a phase-conjugate beam, Δ, normalized with respect to λ1 plotted versus w = ω1 /ω 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.
√ In Fig. 4, the square root of the phase conjugate reflectance, R21 , is plotted as a function of the incident angle for the parameter values used in Fig. 2, when w = 0.99975. A series of sharp resonances occur at well-defined incident angles. This result is due to a Fabry-Perot-type resonance and is fully consistent with Fig. 6 of Ref. 8.
√ Fig. 4. Square root of the phase conjugate reflectance, R21 , plotted versus incident angle for (a) s and (b) p waves, when γ0 = γ = 0.01, L/λ1 = 105 and w = ω1 /ω = 0.99975.
√ In Fig. 5, I plot R21 as a function of the dielectric permittivity ε , when γ0 = γ = 0.01, μ = 1, θ = 45◦ , L/λ1 = 105 and w = 0.99975. It is observed that a large number of strong resonances occur at small values of ε , where the phase-conjugate reflectance is strongly amplified. Media with small dielectric permittivity are generally expected to show strongly-enhanced optical phase conjugation. #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1749
√ Fig. 5. Square root of the phase conjugate reflectance, R21 , 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 /ω = 0.99975.
√ In Fig. 6, I plot R21 as a function of w for the three-layer system where a normal slab of thickness 15λ1 with ε = 1.5 and μ = 1 is sandwiched between two phase-conjugating slabs of equal thicknesses 45λ1 with γ0 = γ = 0.01 and ε = μ = 1. The incident angle is equal to θ = 45◦ . The three-layer structure supports waveguide modes, which cause an enhancement of electromagnetic fields. An extremely strong resonant enhancement of the phase-conjugate reflection, especially for p waves, occurs due to this effect.
√ Fig. 6. Square root of the phase conjugate reflectance, R21 , of a three-layer system plotted versus w ≡ ω1 /ω 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.
Finally, I consider the effects of absorption on the phase conjugation process. As is wellknown, it is difficult to suppress the absorption in metamaterials, especially in negative-index #224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1750
media based on metal-dielectric composites. The attention is restricted to the s wave case, √ though similar results are obtained for p waves. In Fig. 7(a), the R21 curve corresponding to the negative index case in Fig. 2(a) is plotted versus w when ε = μ = −1, −1 + 0.001i and −1 + 0.002i. It is seen that the resonant enhancement of the phase conjugate reflection near w = 0 is suppressed as the imaginary part of ε and μ increases. As long as the size of the imaginary part is smaller than the nonlinearity parameter γ0 , however, the resonant effect is still manifested. The effect of absorption is not limited to suppressing the enhancement, as shown by the appearance of √ two peaks when ε = μ = −1 + 0.002i. In Fig. 7(b), the R21 curve corresponding to the negative index case in Fig. 4(a) is plotted versus θ , when ε = μ = −1 and −1 + 0.002i. For large incident angles, the resonant effect is strongly suppressed, whereas for some special incident angles, it is greatly amplified. Near the peak at θ ≈ 44◦ , a strong oscillatory behavior appears when ε = μ = −1 + 0.002i. It has been pointed out previously that four-wave mixing systems can exhibit new oscillatory behavior and strong enhancement√in phase-conjugate reflection due to linear absorption [19]. In Fig. 7(c), the R21 curve corresponding to Fig. 6(a) is plotted versus w, when the imaginary part of ε for both the defect layer and the surrounding nonlinear layers is nonzero. I find that the resonance peaks are suppressed in general, though new peaks may appear at different parameter values. A detailed understanding of the influence of absorption on resonant phase conjugation phenomena requires a further study and will be the subject of a future publication.
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) R21 versus w for s waves, when ε = μ = −1, −1 + 0.001i√ and −1 + 0.002i. (b) R21 versus θ for s waves, when ε = μ = −1 and −1 + 0.002i. (c) R21 versus w for s waves, when Im ε = 0 and 0.003i for both the defect and the nonlinear layers.
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1751
5.
Conclusion
I have developed a novel theoretical method for studying the optical phase conjugation in inhomogeneous metamaterials, using which I have studied the dependencies of optical phase conjugation on the frequency, the polarization, the incident angle, the material properties and the structure in detail. It has been found that the efficiency of phase conjugation can be significantly enhanced due to the enhancement of electromagnetic fields in various metamaterial structures. Acknowledgments This work has been supported by the National Research Foundation of Korea Grant (NRF2012R1A1A2044201) funded by the Korean Government.
#224207 - $15.00 USD Received 1 Oct 2014; revised 21 Oct 2014; accepted 21 Oct 2014; published 28 Oct 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. S7 | DOI:10.1364/OE.22.0A1744 | OPTICS EXPRESS A1752
