## Abstract

A conventional one-dimensional photonic crystal with a conjugated pair of *ε*-negative and *μ*-negative defects has been presented, and only the defects are presumed to possess quadratic nonlinearity. Large enhancement of second-harmonic generation is predicted in numerical simulation. Interface and volume nonlinearity are both utilized in the process of second-harmonic generation due to the strong localization of the fundamental wave.

© 2009 Optical Society of America

In recent decades, much attention has been paid on the research of photonic crystals (PCs),[1, 2] which are also called photonic band gap (PBG) materials proposed in early 70’s both theoretically and experimentally. Most of the researchers are focused on the investigation of insulating properties of PCs, such as omnidirectional gap, [3] high reflection properties [4] and transmission properties. [5] The nonlinear properties of PCs, for example, the second [6, 7] and third-order [8] nonlinear optical process, and optical parametric oscillation (OPO) [9] though less studied, are also interesting and worth of attention. One remarkable advantage of PCs is that the density of modes (DOMs) at band gap edges or defective modes are increased notably, leading to strong interaction of photons. It has been reported by *Ren* that simultaneous localization of the fundamental wave (FW) and second harmonic wave (SHW) in the same defective layer could bring giant enhancement of the conversion efficiency by more than eight orders. [7] It is well known that only the structure which lacks of centrosymmetry can lead to quadratic nonlinearity, while the interface nonlinearity does not need the structure lacking of centrosymmetry. In the PCs composed of layered negative index material (NIM) and positive index material (PIM), the electromagnetic wave tunneling at the high frequency band edge of the zero-average band gap can lead to localization at each of the interface of NIM and PIM strictly. This localization is very helpful to the interface second-harmonic generation which has been discussed in Ref. 10.

When one of the permittivity and permeability is negative,[11] we call these metamaterials single negative (SNG) materials including epsilon-negative (ENG) materials with *ε* < 0 and *μ* > 0 and mu-negative (MNG) materials with *μ* < 0 and *ε* > 0. The conventional PC with a pair of ENG-MNG defects have higher localization at the interface of ENG and MNG than the PC made of layered ENG-MNG materials,[13] and the defects under some special conditions do not influence the transmittance property of the structure. This unique property can be used in the surface second-harmonic generation to enhance the nonlinear conversion efficiency. Regrettably, up to now, there is not much work done concerning the contribution of the interface nonlinearity in nonlinear process using these SNG materials.

In this paper, our purpose is to study the interface and volume second-harmonic generation (SHG) from the conventional PC with a pair of ENG-MNG defects. The structure we proposed here is a PIM/PIM photonic crystal with a pair of ENG-MNG defects as we show in Fig. 1. The multi-layers are alternating along the z direction, which is denoted as (AB)_{PN}CD(AB)_{PN-1}A. Here, A and B are PIMs, C and D represent mu-negative (MNG) material and epsilon-negative (ENG) material respectively. PN represents the period number of AB layers. Here we choose PN = 9. *ε*
_{A,B,C,D} and *μ*
_{A,B,C,D} are the permittivity and permeability of A, B, C, D layers respectively. *d*
_{A,B,C,D} is the thickness of the corresponding layers respectively. In this paper, *ε _{A}* = 9,

*μ*= 1,

_{A}*ε*= 2,

_{B}*μ*= 1, the structure is surrounded with air. μ

_{B}_{C}and

*ε*are described by lossy Drude model, and the parameters are chosen as

_{D}in a mu-negative (MNG) material and

in an epsilon-negative (ENG) material, where *ω* = 2*πf* is angular frequency in unit of gigahertz, *γ _{m}* (

*γ*) is the damping coefficient. In the calculation, the normally incident electromagnetic wave is a transverse magnetic (TM) wave (the magnetic field

_{e}**H**is in the y direction) propagating along the z direction as shown in Fig. 1. The transverse electric (TE) wave can be treated in a similar way. The parameters of the pair of SNG materials should be chosen specially. Only if the pair of the SNG materials are conjugatedly matched pair of MNG-ENG layers,

*i.e*. the parameters of MNG-ENG layers must satisfy the relationship described as

*ε*= −

_{C}*ε*,

_{D}*μ*= −

_{C}*μ*, and

_{D}*d*=

_{C}*d*, so that the defects will not influence the transmission property of the structure. It is actually a particular case of zero-reflection condition. Using transmission-line model, the zero-reflection condition can be written as

_{D}*Z*= −

_{C}*Z*,

_{D}*k*=

_{MNG}d_{C}*k*, [14] where

_{ENG}d_{D}*Z*(

_{C}*D*) and

*k*

_{MNG(ENG)}indicate the impedance and wave number in MNG and ENG layers respectively.

Firstly, we give the transmission spectrum of the structure of (AB)_{17}A and (AB)_{9}CD(AB)_{8}A respectively, using transfer-matrix method.[12, 15] In order to make the SNG defects satisfy the zero-reflection condition at the high frequency band edge of the first band gap shown in Fig. 2(a), the parameters of the MNG and ENG defects are chosen as follows: *ε*
_{1} = *μ*
_{1} = 1, *ε*
_{2} = *μ*
_{2} = 1, *α* = *β* = 400, *γ _{m}* =

*γ*= 2

_{e}*π*×3×10

^{-6}. The values of the damping chosen here are small enough, so that their influence on

*μ*,

_{C}*ε*can be neglected.

_{D}In Fig. 2(a), the red solid line shows the transmission spectrum of the conventional PC, the green dashed and blue dotted lines represent the transmission spectrum of the photonic crystals with different thickness of *d _{C}* =

*d*= 48

_{D}*mm*,

*d*=

_{C}*d*= 18

_{D}*mm*, respectively, while

*d*=

_{A}*d*= 18 mm are invariable in these three cases. It is noted that the transmission function within the bands is periodically modulated. This periodic modulation is mainly due to the reflections from end facets of the structure yielding Fabry-Pérot resonances. We can see from Fig. 2(a) that the higher frequency edge of band gap is always at 2.25 GHz, which means the pair of the defects do not change the transmittance at FW because the zero-reflection condition is satisfied. However, the transmittance of SHW changes slightly. This change is owing to the consideration of the lossy Drude model, and does not have much effect on the conversion efficiency of SHG. We will illustrate this phenomenon clearly in the following calculation.

_{B}As well known, the relative electric field intensity distribution at the band edge in conventional PC is a standing wave. The energy of EM is almost concentrated in the low refractive index or the high refractive material, and the amplitude of the energy is generally not high (Fig. 2(b)). But, as another unique property of the PC with a conjugated pair of MNG-ENG
Fig. 2(b)). The structural parameters used here are *d _{A}* =

*d*= 18

_{B}*mm*,

*d*=

_{C}*d*= 48

_{D}*mm*. A high localization of the electric field is found in the MNG and ENG layers, and the intensity of relative electric field is amplified by 600 times. Obviously, the amplitude of intensity of localized electric field is improved greatly in the MNG-ENG defects due to the confinement of the two conventional PCs outside of the defects. In this defective PC, the dramatically localized field will greatly contribute to the SHG.

The relative electric field intensity distribution is amplified in the defects, and the maximum is at the interface of the MNG-ENG layers. Under the precondition of conjugation matching, an interesting phenomenon is the influence of *d _{C}* (=

*d*) on the maximum of the relative electric field intensity distribution which was called

_{D}*localization peak*here. The thickness of the MNG and ENG materials have to be changed simultaneously, as the thickness of the MNG and ENG materials affects the localization peak obviously. In Fig. 3(a), we plot the relationship between the localization peak and

*d*by red solid line. The localization peak increases exponentially when

_{C}*d*becomes thicker, until to a maximum value about 2250 a.u. with

_{C}*d*= 55

_{C}*mm*. Then the localization peak decreases if

*d*grows continuously , because the zero-reflection condition does not exactly satisfied at this frequency (2.25 GHz), otherwise, the localization peak would keep increasing when dC strides over 55 mm.[16] When dC increases, the mismatch of the zero-reflection condition leads to the decrement of the localization peak.

_{C}In such a defective PC, we give a quantitative calculation for the total conversion efficiency including the forward and backward SHG in the undepleted pump wave approximation. For normal incidence, the coupled-wave equations governing quadratic nonlinear interaction of two monochromatic plane waves with different frequencies at *ω* and 2*ω* can be written as:[17]

where *ε*
_{ω,2ω,μω,2ω} are susceptibility and permeability respectively of the FW or SHW. *d*(*z*) is the quadratic coupling coefficient.

In the undepleted regime, the conversion efficiency can be written as[17]

$$+{\mid \frac{1}{L}{\int}_{0}^{L}{d}^{\left(2\right)}\left(z\right){\left[{\Phi}_{\omega}^{+}\left(z\right)\right]}^{2}{\Phi}_{2\omega}^{+}\left(z\right)\mathrm{dz}\mid}^{2}){I}_{\omega}^{\mathrm{Input}}$$

Φ^{±}
_{ω,2ω} is the right-to-left (RTL) (+) or left-to-right (LTR) (-) linear modes of the structure at *ω* or 2*ω* as described in Ref. 18. The propagating modes of RTL and LTR can be calculated independently using transfer-matrix method.[12, 15] L is the total length of the structure. If we only calculate the interface nonlinear influence, we need to write *d*(*z*) as a Dirac *δ* function. But in this paper, we consider both of the interface and volume nonlinearity contribution, and for convenience we chose the nonlinear coefficients of them as the same. *I ^{Input}_{ω}* is the power density of the pump beam.

The nonlinear conversion efficiency can be largely enhanced due to the high localization of the FW. In Fig. 3(b), we plot the relationship between input power and the conversion efficiency of a defective PC and a conventional PC respectively. For comparison, the parameters of the structure here are chosen as those in Fig. 2(b), and the nonlinear coefficient *d _{eff}* of the MNG and ENG is chosen to be 43.9 pm/V, the same as in Ref. 7. The unit of the x-axis is KW/cm

^{2}. The conversion efficiency of the conventional PC with a conjugated pair of MNG-ENG defects (represented by blue solid line) is largely improved by more than five orders of magnitude comparing with that of the conventional PCs (represented by red dashed line, which has been amplified by a factor of 10

^{5}for clarity). We should summarize that there are two reasons resulting in the large improvement of the conversion efficiency: 1) FW is strongly localized in the MNG-ENG layers and the amplification is much higher than that in the conventional PC, although the FW lies in the band edge similarly. In conventional PC, the field distribution corresponding to the edge of band gap is very small, which is ascribed to the low quality factor of F-P cavity. While in our SNG defective PC, the EM is decaying from the center of the defect to both sides, which represents a localized state. 2) The maximum of the field distribution of FW is exactly at the interface of MNG-ENG defects, which makes the surface nonlinearity to be usable in SHG. It is necessary to point out that the improvement of the conversion efficiency of second harmonic generation is mainly due to the introduction of the meta-material defects and the contribution of the surrounded PC is negligible. Our calculation shows the conversion efficiency due to the surrounded PC is only in the order of 10

^{-7}, which is much smaller than the total conversion efficiency. The main reason of the conversion efficiency enhancement is due to field localization in the defects of the meta-material as we discussed above.

We also plot the dependence of conversion efficiency on *d _{C}* by blue dashed line in Fig. 3(a) with the input power equaling to 1MW/cm

^{2}, and the parameters used here are the same as Fig. 2(b). The conversion efficiency also depends on

*d*and nearly coincides with the curve of localization peak, which implies that the localization peak of FW of the structure mainly determines the conversion efficiency. So, we can conclude that the SHW does not influence the conversion efficiency significantly. When

_{C}*d*= 55 mm, the conversion efficiency reaches the maximum (3×10

_{C}^{-2}).

In Fig. 4(a), we show the dependence of the localization peak (in red solid line) or the conversion efficiency (in blue dashed line) on the period number (PN) of the surrounded conventional PC. The input power remains 1MW/cm^{2} in the calculation of conversion efficiency as in Fig. 3(a). Since the transmittance and the localization of the FW are affected by PN periodically, correspondently, the conversion efficiencies will also be periodically influenced by PN similarly. Each localization peak means the FW is one of transmission resonance modes. When the FW is close to the second transmission resonance, the conversion efficiency is larger than the others.[18] The transmittance of the structure when the FW is close to the second transmission resonance is shown in Fig. 4(b). The higher localization of the FW near the second transmission resonance also contributes to the higher conversion efficiency. The oscillations in Fig. 4(b) have the same origin as that in Fig. 2(a).

In conclusion, we have shown that the FW, which lies in the band edge of the conventional PC with a conjugated pair of MNG-ENG defects, can be localized in MNG-ENG defects with dramatic amplification. The SHG is enhanced greatly resulted from two reasons: FW represents a localized state in the MNG-ENG layers, and both volume and surface quadratic nonlinearity has been sufficiently utilized. The influence of the thickness of defect and the period numbers of the surrounded conventional PC on the localization and conversion efficiency are investigated in detail.

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