## Abstract

We investigate the propagation of electromagnetic waves in a one-dimensional photonic crystal containing a defect layer made of an isotropic chiral medium. Using the invariant imbedding method, we calculate the transmission spectrum for both linearly- and circularly-polarized incident waves. In the normal incidence case, there is one defect mode, which does not depend on the chiral index and the polarization of the incident wave. When the waves are incident obliquely, however, we find that there appear double defect modes regardless of the polarization. The interval between the two defect frequencies increases monotonically as the chiral index or the incident angle increases. We argue that this phenomenon occurs due to the coupling and conversion between *s* and *p* waves inside the chiral defect layer.

© 2014 Optical Society of America

## 1. Introduction

Photonic crystals have been intensively studied over the last 25 years due to their ability to control electromagnetic waves efficiently and the possibility for various applications including optical switches, tunable filters, and waveguides, among many others. The key property of photonic crystals is to prohibit the propagation of electromagnetic waves the frequency of which lies within a frequency band called the photonic bandgap. If a defect is introduced into such a structure, a narrow defect mode appears in the bandgap. It is crucial to understand the nature of the defect modes for applications such as filters, resonators, splitters, mode converters, cavities, detectors, and low-threshold lasers [1–6].

During the past decade, considerable research interest has been directed at photonic crystals made of complex media such as cholesteric liquid crystals [7, 8], nonlinear optical materials [9, 10] and chiral media [11–13]. The study of the electromagnetic wave propagation in chiral media has a long-standing history [14]. Natural chiral media are composed of chiral molecules with no mirror symmetry, the examples of which include sugars, amino acids and DNA, or chiral arrangements of achiral molecules. Crystalline solids such as quartz, AgGaS_{2} and *α*-LiIO_{3} also display optical activity due to microscopic chirality. Chiral media are characterized by a cross-coupling between the electric and magnetic fields, the strength of which is measured by a dimensionless parameter *γ* termed chiral index [14, 15]. In the uniform case, right-circularly-polarized (RCP) and left-circularly-polarized (LCP) waves are the eigenmodes of isotropic chiral media [15].

In natural chiral media, the value of the chiral index *γ* is typically small, ranging approximately from 10^{−4} to 10^{−3}. Motivated by recent rapid developments in electromagnetic meta-materials, much renewed interest has been given to developing artificial chiral metamaterials which show a greatly-enhanced effect of chirality [16, 17]. These metamaterials are constructed typically by a periodic arrangement of a large number of identical micro- or nano-structures having the shapes of helices or other chiral shapes. For electromagnetic waves whose wavelength is substantially larger than the size of these structures, chiral metamaterials behave as uniform chiral media. They are interesting due to their capability to affect the polarization state of electromagnetic waves in an enhanced manner. Many interesting phenomena, such as giant optical activity, strong circular dichroism and chirality-induced negative refractive index can occur in these systems [18–20].

A large number of experiments have been performed on one-dimensional photonic crystals with structural chirality, which are not directly relevant to our work [7, 8, 21]. Experimental studies on chiral media with dielectric chirality, where the length scale of chiral inclusions is much smaller than the wavelength, have been usually limited to uniform cases. Recent experiments on chiral metamaterials have shown that the effective value of the chiral index can be as large as O(1) or larger than the refractive index of the same system, which may lead to very strong chirality-induced effects [22]. It will be very interesting to do experiments combining strongly chiral metamaterials with photonic crystals.

In this paper, we study theoretically the defect mode in one-dimensional photonic crystals containing a defect layer made of an isotropic chiral medium. Using a generalized version of the invariant imbedding method, we calculate the transmission spectrum for linearly- and circularly-polarized incident waves. When the waves are incident obliquely and the chirality is sufficiently strong, we find an interesting phenomenon that double defect modes are formed in the presence of one defect layer, regardless of the polarization state of incident waves. The interval between the two defect frequencies is found to increase monotonically as the chiral index or the incident angle increases. This phenomenon is due to the coupling and conversion between *s* and *p* waves inside the chiral defect layer. We will present a detailed analysis of the unusual behavior of the defect mode.

## 2. Theoretical method

The appropriate constitutive relations of isotropic chiral media are given by

*ε*,

*μ*and

*γ*are the dielectric permittivity, the magnetic permeability and the chiral index respectively [15]. When these parameters are real-valued, the constitutive relations (1) describe chiral media which are lossless and reciprocal [14, 23]. It is possible to deduce Eq. (1) starting from microscopic equations, if one assumes weak spatial dispersion, which means that the inhomogeneity scale is considerably smaller than the characteristic wavelength [23].

We assume that the chiral medium is stratified in the *z* direction and *ε*, *μ* and *γ* are functions of *z* only. The wave is assumed to propagate in the *xz* plane. From the Maxwell’s equations and Eq. (1), we derive the coupled wave equations satisfied by the *y* components of the electric and magnetic fields, *E _{y}* =

*E*(

_{y}*z*) and

*H*=

_{y}*H*(

_{y}*z*):

*k*(=

*ω/c*) is the vacuum wave number and

*q*is the

*x*component of the wave vector. The prime denotes a differentiation with respect to

*z*. In uniform chiral media, RCP and LCP waves are the eigenmodes of Eq. (2) with the effective refractive indices $\sqrt{\epsilon \mu}+\gamma $ and $\sqrt{\epsilon \mu}-\gamma $ respectively.

An inhomogeneous chiral medium of thickness *L* is assumed to lie in the range 0 ≤ *z* ≤ *L*. Then the coupled wave equation takes the form

*ψ*= (

*E*,

_{y}*H*)

_{y}*,*

^{T}*I*is a 2 × 2 unit matrix, and

*ℰ*and

*ℳ*are 2 × 2 matrix functions that depend on

*z*in an arbitrary manner. The waves are incident from the vacuum region where

*z*>

*L*and transmitted to another vacuum region where

*z*< 0. By comparing Eq. (2) with Eq. (3), we obtain the expressions for

*ℰ*and

*ℳ*:

We are mainly interested in calculating the 2 × 2 matrix reflection and transmission coefficients *r* = *r*(*L*) and *t* = *t*(*L*). Using the invariant imbedding method [24, 25], we derive exact differential equations satisfied by *r*(*l*) and *t*(*l*):

*θ*is the incident angle. We integrate the coupled differential equations (5) and (6) numerically from

*l*= 0 to

*l*=

*L*using the initial conditions

*r*(0) = 0 and

*t*(0) =

*I*and obtain

*r*and

*t*as functions of

*L*. In previous works, these equations have been applied successfully to the study of the propagation of electromagnetic waves in chiral media [24, 26] and in magnetized plasmas [25].

We define *t _{ss}* (

*t*) to be the transmission coefficient when the incident wave is

_{ps}*s*-polarized and the transmitted wave is

*s*-polarized (

*p*-polarized). Similar definitions are applied to

*t*and

_{pp}*t*and to the reflection coefficients. We obtain a new set of the transmission coefficients

_{sp}*t*, where

_{ij}*i*and

*j*are either + or −, from

*t*

_{++}(

*t*

_{−+}) represents the transmission coefficient when the incident wave is RCP and the transmitted wave is RCP (LCP).

*t*

_{−−}and

*t*

_{+−}are defined similarly. The transmittances are defined by

*T*= |

_{ij}*t*|

_{ij}^{2}. We also define

*T*(≡

_{s}*T*+

_{ss}*T*) and

_{ps}*T*(≡

_{p}*T*+

_{pp}*T*) as the total transmittances for

_{sp}*s*- and

*p*-polarized incident waves. Similarly, we define the total transmittances

*T*

_{+}(≡

*T*

_{++}+

*T*

_{−+}) and

*T*

_{−}(≡

*T*

_{−−}+

*T*

_{+−}) for RCP and LCP incident waves.

## 3. Numerical results

We consider a multilayered structure represented as (AB)* ^{N}*C(BA)

*, where C denotes a defect layer made of an isotropic chiral medium. The A and B layers are considered to be made of SiO*

^{N}_{2}and TiO

_{2}, the refractive indices of which are 1.46 and 2.3 respectively. The refractive index of the C layer is assumed to be 1.5 and its chiral index is denoted by

*γ*. The thicknesses of the A, B and C layers are

*d*= 61.2 nm,

_{A}*d*= 38.8 nm and

_{B}*d*= 119 nm respectively. With these definitions, the optical thickness of the C layer is twice those of the A and B layers. We define Λ (≡

_{C}*d*+

_{A}*d*= 100 nm) to be the period of the photonic crystal. The total number of the periods in our structure is 2

_{B}*N*. In Fig. 1, we show the schematic of our structure.

We illustrate our main result in Fig. 2, where we compare the transmittance spectra of a photonic crystal with a chiral defect with those of a photonic crystal with no defect or an achiral defect. From the comparison of Figs. 2(a) and 2(b) with the rest of Fig. 2, it is obvious that the appearance of defect modes is due to the presence of a defect layer, whether it is chiral or not. In the case of ordinary achiral defects with *γ* = 0, the number of defect modes is equal to that of defects, therefore we observe one defect mode at slightly different frequencies *ω*Λ/*c* = 1.842 and 1.841 for *s*- and *p*-polarized incident waves in Figs. 2(c) and 2(d). When the defect is chiral, however, *two* defect modes are observed regardless of the polarization of incident waves because the *s* and *p* wave defect modes are coupled to each other due to the chirality. In Figs. 2(e), 2(f), 2(g) and 2(h), we plot the total transmittances *T _{s}*,

*T*,

_{p}*T*

_{+}and

*T*

_{−}versus normalized frequency

*ω*Λ/

*c*, when

*θ*= 30°,

*γ*= 0.5 and

*N*= 8. Two split defect modes with

*ω*Λ/

*c*= 1.837 and 1.859 are clearly seen inside the photonic band gap, even though there is only one defect layer in our structure. We observe that

*T*(

_{s}*T*) is larger (smaller) at

_{p}*ω*Λ/

*c*= 1.859 than at

*ω*Λ/

*c*= 1.837. Both

*T*

_{+}and

*T*

_{−}are approximately 0.5 at either frequency.

In Fig. 3, we show the total transmittance versus normalized frequency, when *θ* = 0° and *N* = 8. There is only one defect mode, regardless of the value of *γ* and the polarization of the incident wave. In the normal incidence case, there is no distinction between the *s* and *p* waves, and it is straightforward to show that *t _{ss}* =

*t*and

_{pp}*t*= −

_{sp}*t*. Then from Eq. (7), we obtain

_{ps}*T*

_{+−}=

*T*

_{−+}= 0 and

*T*

_{+}=

*T*

_{−}=

*T*=

_{s}*T*for all values of

_{p}*γ*.

In the oblique incidence case, the defect modes for *s* and *p* waves appear at different frequencies. When the chirality index of the defect is sufficiently large, these two modes are strongly coupled and manifest as double defect modes. In Fig. 4, we plot *T _{s}*,

*T*,

_{p}*T*

_{+}and

*T*

_{−}versus normalized frequency, when

*θ*= 30°,

*N*= 8 and

*γ*= 0.1, 0.3, 0.5. When

*γ*is sufficiently small,

*T*and

_{s}*T*show a single peak at slightly different frequencies. The defect frequency associated with

_{p}*p*waves is lower than that associated with

*s*waves. As

*γ*increases, both

*T*and

_{s}*T*begin to show double peaks at the same frequencies. The peak at the smaller (larger) frequency is dominantly associated with

_{p}*p*(

*s*) waves, therefore the value of

*T*(

_{p}*T*) is larger at the lower (upper) peak. One of these two peaks is formed by the conversion between

_{s}*s*and

*p*waves in the chiral defect layer, which becomes stronger as

*γ*increases. We observe that the interval between the two defect frequencies increases monotonically as

*γ*increases at a rate faster than the linear rate. Since RCP and LCP waves are linear superpositions of

*s*and

*p*waves, we obtain two peaks even for small values of

*γ*. The shapes of the

*T*

_{+}and

*T*

_{−}spectra are similar and the peak values are approximately 0.5.

In Fig. 5, we plot *T _{ss}*,

*T*,

_{pp}*T*and

_{sp}*T*versus normalized frequency, when

_{ps}*θ*= 30°,

*N*= 8 and

*γ*= 0.5. We find that

*T*coincides perfectly with

_{ps}*T*.

_{sp}*T*and

_{sp}*T*, which represent the conversion between

_{ps}*s*and

*p*waves, show two peaks at the same frequencies as those associated with

*T*and

_{ss}*T*. This shows clearly that

_{pp}*the formation of double defect modes is due to the conversion between s and p waves in the chiral defect layer*.

Next, we show how the defect mode spectrum changes as the incident angle *θ* increases. In Fig. 6, we plot *T _{p}* versus normalized frequency for incident angles

*θ*= 15°, 30° and 45°, when

*γ*= 0.5 and

*N*= 8. We find that the two defect-mode peaks shift to higher frequencies and the interval between them increases monotonically as

*θ*increases, at a rate faster than the linear rate. This occurs because the difference between the defect frequencies for

*s*and

*p*waves increases as

*θ*increases.

Finally, we show the dependence of the defect mode spectrum on the number of periods of the photonic crystal *N*. In Fig. 7, *T _{p}* is plotted versus normalized frequency for

*N*= 6, 8 and 10, when

*θ*= 30° and

*γ*= 0.5. As is expected, the peaks become sharper and sharper as

*N*increases, while the peak positions are unchanged.

## 4. Conclusion and discussion

In this paper, we have studied the nature of the defect mode in a one-dimensional photonic crystal containing one defect layer made of an isotropic chiral medium. Using the invariant imbedding method, we have analyzed the transmission spectra for both linearly- and circularly-polarized incident waves. In the normal incidence case, there is one defect mode, which does not depend on the chiral index and the polarization of the incident wave. When the waves are incident obliquely, however, we have found that there appear double defect modes regardless of the polarization. The interval between the two defect frequencies increases monotonically as the chiral index or the incident angle increases. We have argued that this phenomenon occurs due to the coupling and conversion between *s* and *p* waves inside the chiral defect layer.

In order to observe the effect discussed here, the chiral index *γ* needs to be sufficiently large. In natural chiral media, the value of *γ* is rather small. Recently, there has been much interest in artificially fabricated metamaterials with unusual optical properties. It has been demonstrated that it is possible to fabricate a chiral metamaterial with a very large chiral index, where *γ* is of order one [20, 22, 27]. It will be interesting to verify our theory using a photonic crystal containing a metamaterial defect layer with strong chirality.

## Acknowledgments

This work has been supported by the National Research Foundation of Korea Grant ( NRF-2012R1A1A2044201) funded by the Korean Government and by CNRS-Ewha International Research Center Program.

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