## Abstract

Photonic modes in a two-dimensional square-lattice photonic crystal (PC) with anisotropic gain are analyzed for the first time. A plane-wave expansion method is improved to include the gain, which depends on not only the position but also the propagation direction of each plane wave. The anisotropic gain varies the photonic band structure, the near-field distributions, and the gain dispersion curves through variation in PC symmetry. Low-threshold operation of a PC laser with anisotropic-gain material such as nonpolar InGaN requires that the direction of higher gain in the material aligns along the ΓX direction of the PC.

©2011 Optical Society of America

## 1. Introduction

Two-dimensional (2D) photonic crystals (PC) have been actively developed for applications such as surface-emitting lasers [1–7], light-emitting diodes [8], and interferometers [9]. A PC is based on the periodic alignment of materials having different dielectric constants to form a photonic band structure (PBS). In general, the dielectric constants are in a complex form, whose imaginary parts provide the gain or loss factors of the electromagnetic (EM) wave propagating in the PC. Several reports show the importance of gain or loss characteristics in a PC. For example, Nojima addressed the 2D PC gain and numerically found an enhancement in optical gain for 2D PCs [10,11]. He showed that this enhancement results from zero velocity at the band edge. Sigalas et al. analyzed a PC made of dispersive and highly absorptive materials [12]. The characteristics of a metallic PC with incident light absorption were demonstrated by Yanonpapas et al. [13] and El-Kady et al. [14]. Brand et al. analyzed complex PCs to discuss effective plasma frequencies in a metal rod array [15]. Sakai et al. demonstrated the threshold criteria of 2D PC lasers using the coupled wave theory [16,17]. These reports, however, only consider the gain/loss distribution that is dependent on the position in the PC. In other words, at each point of the PC medium, the gain does not depend on the propagation directions (wavevectors) of the EM wave, and thus the gain is isotropic.

Meanwhile, interest has been growing in InGaN-based materials as visible semiconductor light sources. For example, blue-violet InGaN-based laser diodes (LDs) have been commercialized for high-density optical data storage, and pure-blue and green LDs are candidates for mobile projectors. However, quantum-confined Stark (QCS) effects are a major issue of InGaN-based materials, since they decrease the recombination rate of electrons and holes, leading to a threshold increase in laser action. Epitaxial-grown layers on c-plane substrate cause the polarization at the interface of a layer (the so-called “polar” material), which induces a QCS effect. This effect is stronger in a higher indium-composition InGaN layer, i.e., the quantum well (QW) active layer for longer wavelength emission, such as green. Thus, in order to avoid this effect, nonpolar or semipolar InGaN-based materials that suppress this polarization have been developed. The nonpolar or semipolar InGaN-based materials are realized by epitaxial growths on, e. g., m-plane [18], (11-22)-plane [19], (20-21)-plane [20], and (30-31)-plane [21] GaN substrates. Although this approach is useful in the suppression of QCS effects, the nonpolar or semipolar QW exhibits another feature, that is, anisotropic gain where the gain depends on the EM propagation direction. Ohtoshi et al. report the dependence of optical gain on the crystal orientation of GaN [22]. Park demonstrates the crystal orientation effects of InGaN QWs [23], and Scheibenzuber et al. compare the optical gain of semipolar/nonpolar lasers with that of a polar laser [24]. For nonpolar m-plane InGaN QWs, several reports show experimentally that the propagation along the *c*-axis possesses larger gain than that along the *a*-axis [25,26]. As shown in [1,27], InGaN-based QW LDs integrated with 2D PCs are attractive for visible light sources with high optical output power and narrow beam divergence. Therefore, it is important to reveal the characteristics of 2D PCs with anisotropic gain, particularly for InGaN lasers operating in the visible wavelength.

This paper analyzes a square-lattice PC with anisotropic gain, carried out here for the first time. In Sec. 2, we derive the calculation method for anisotropic-gain PCs. In Sec. 3, the calculation model for anisotropic-gain PCs is described. In Sec. 4, the numerical results are demonstrated. In Sec. 5, we discuss the importance of PC symmetry to anisotropic gain and the importance of radiation loss. Finally, Section 6 summarizes the conclusions.

## 2. Calculation method

In general, a 2D plane-wave expansion (PWE) method [28] utilizes several plane waves with Fourier-expanded dielectric constants, both of which are based on reciprocal wavevectors of a PC. The distinctive feature of our improved PWE analysis is that, in order to include the anisotropic gain, expanded dielectric constants depend on the propagation direction of the plane waves. For simplicity, we look at the transverse electric (TE) mode waves in a nonpolar QW only, since a semipolar QW causes birefringence so that TE mode does not make sense [24]. Figure 1
shows a schematic diagram of a PC. The PC has a circular cross-section region (denoted HL) in the background region (denoted BG). The dielectric constant difference between the HL and BG regions leads to the formation of the PC. Here we define coordinates *x _{1}* and

*x*as being in the PC plane, while coordinate

_{2}*x*is normal to the plane. The TE mode in the PC corresponds to so-called “H-polarization,” whose magnetic field is perpendicular to the PC planes. The magnetic field along the

_{3}*x*axis, ${H}_{3}(\overline{x})$, is the sum of the expanded plane waves, ${h}_{3}(k,G,\overline{x})$, and can be expressed as

_{3}**k**is the wavevector of the EM wave,

**G**is a reciprocal vector of PC, and

*ω*is an angular frequency of the propagating EM wave. Here,$B(k,G)$ is a complex expansion coefficient. Since the dielectric constant in the PC plane is a periodic function of $\overline{x}$, the inverse dielectric constant can be expanded in 2D Fourier series as

**G'**is another reciprocal vector of PC, and $\overline{\kappa}(k\text{'},G\text{'})$ is a Fourier coefficient, through which the anisotropic gain results in dependence on the propagation direction of the plane wave ($k\text{'}=k+G$). Here, $\overline{\kappa}(k\text{'},G\text{'})$ can be expressed as [28]

*f*,

*r*, and

*J*are the filling fraction, radius of circular region, and Bessel function, respectively. Since we here consider the anisotropic gain, ${\epsilon}_{a}(k\text{'})$ and ${\epsilon}_{b}(k\text{'})$ are set to be the

_{1}(x)**k'**-dependent relative dielectric constants of HL and BG regions, respectively. Since the PWE method assumes that the structure has infinite length with uniformity along the

*x*direction, we use the effective relative dielectric constants for each region as ${\epsilon}_{a}(k\text{'})$ and ${\epsilon}_{b}(k\text{'})$. Assuming that the Maxwell equations are satisfied for each expanded plane wave, the TE-modes Maxwell equations for ${h}_{3}(k,G,\overline{x})$ with the propagation direction of $k+G$ is expressed as

_{3}*c*is velocity of light. By the summation of all reciprocal vectors, we can obtain

*n*, of the EM waves propagating along the γ direction (

_{γ}**k'**) in the PC plane, is expressed as

*n*and

_{α}*n*are refractive indices of the EM waves propagating along the orthogonal directions, α and β, respectively; φ is the angle between the α and γ directions (see Fig. 1). The refractive indices are expanded as:

_{β}*θ*is defined as the angle between the α direction and a primitive transformation vector of PC along the ΓX direction. The $\mathrm{cos}\phi $ is expressed as$\mathrm{cos}\phi =k\text{'}\cdot {I}_{\alpha}/\left|k\text{'}\right|$, where ${I}_{\alpha}=(\mathrm{cos}\theta ,\mathrm{sin}\theta )$ is a unit vector along the α direction.

Finally, we discuss the gain or loss of the EM wave. Since the EM wave experiences gain or loss during propagation in the PC, the wave will grow or decay temporally. Thus, in order to express this variation, complex frequency is introduced in Eq. (8) as

Substituting of Eq. (14) into Eq. (1) shows that positive and negative ${\omega}_{i}$ indicate the growth and decay of an EM wave, respectively. The gain,*g,*is calculated as $g=2{\epsilon}_{eff}{}^{0.5}{\omega}_{i}/c$, where ${\epsilon}_{eff}$ is the relative effective dielectric constant for a guided EM wave.

Solving Eq. (8) with Eqs. (13) and (14) gives the ${\omega}_{r}-k$ dispersion characteristics (the PBS) and the ${\omega}_{i}-k$ dispersion characteristics (the gain dispersion characteristics). Both characteristics include the anisotropic gain of the PC material.

## 3. Calculation model

Figure 2
shows the calculation model. This model is based on the AROG (air holes retained over growth) structure [1], which is a useful method for realizing InGaN-based PC lasers. The model consists of an Al_{0.04}Ga_{0.96}N lower cladding layer, a GaN (0.1 μm) or SiO_{2} (0.01 μm)/Air (0.09 μm) AROG layer, a GaN lower optical confinement layer (0.15 μm), a double QW active layer (In_{0.26}Ga_{0.74}N well (0.003 μm) and In_{0.16}Ga_{0.84}N barrier (0.015 μm)), a GaN upper optical confinement layer (0.06 μm), an Al_{0.18}Ga_{0.82}N electron blocking layer (0.02 μm), an Al_{0.06}Ga_{0.94}N upper cladding layer (0.6 μm), a GaN contact layer (0.05 μm), and a gold electrode. The air/SiO_{2} region in the AROG layer forms a square-aligned circular hole so that it corresponds to HL in Fig. 1, while the other region corresponds to BG. The period, *a,* and *r/a* ratio of the PC is 207 nm and 0.2, respectively. The PC cavity is set to be infinity in plane.

We assume that the structure above is formed in a nonpolar m-plane. For the anisotropic-gain spectrum of a nonpolar QW, we use the calculation results given by Scheibenzuber et al. [24] with carrier density of 6 × 10^{12} cm^{−2}, although the gain peak wavelength is shifted to 504 nm in this study. The propagation direction with larger gain (maximum material gain of approximate 3500 cm^{−1}) is set to the α direction, while that with the smaller (approximate 680cm^{−1}) is set to the β direction. The dielectric constant of GaN is the square of refractive indices in [30]. The dielectric constants of AlGaN and InGaN are obtained by shifting that of GaN, the method for which is based on [31]. The dielectric constants of SiO_{2} and gold are from [31,32], respectively. The gain is expressed through the imaginary parts of the dielectric constants of InGaN well layer. The effective relative dielectric constants of guided EM waves in the HL and BG regions are obtained by solving the Maxwell equations for the waves. In the following sections, “isotropic gain” indicates the averaged gain of α and β directions and has no dependence on the propagation direction.

## 4. Calculation results

#### 4.1 Photonic band structure

Figure 3
shows the PBS around the second symmetric Γ point (Γ_{2}). Although we calculate the PBS for isotropic gain and anisotropic gain with different *θ* directions, no significant difference is observed in the axis range in Fig. 3. The left ordinate of Fig. 3 represents the normalized frequency (*ω _{r}a/2πc*), while the right one represents the corresponding wavelength. The photonic band gap is approximately 2 × 10

^{−4}in

*ω*at the Γ

_{r}a/2πc_{2}point. The PBS shows four photonic modes, designated A, B, C, and D. Four similar modes can be seen in the conventional square-lattice PC [1].

Detailed investigation at the Γ_{2} point clearly reveals the *ω _{r}a/2πc* difference between photonic modes, as shown in Fig. 4
, where the normalized frequencies of each mode at the Γ

_{2}point is calculated as a function of

*θ*. This has two features:

- (1) As
*θ*increases, the normalized frequencies of anisotropic-gain A and D modes decrease, while those of anisotropic-gain B and C modes increase. At*θ*= 45 deg, the normalized frequencies of all modes are close to those with isotropic gain. - (2) When
*θ*is far from 45 deg, the normalized frequencies of A and B modes with anisotropic gain abruptly shift from those with isotropic gain. At the other angle, the normalized frequencies of A and B modes are around the mean value of those with isotropic gain. C and D modes are not degenerate in anisotropic gain except when*θ*= 45 deg, while they are degenerate in the case of isotropic gain.

*θ*= 0 and 90 deg equal those of the α and β directions, respectively. Our calculation shows that the effective refractive indices for the α direction is smaller than that for the β directions so that the refractive indices of A and D modes at

*θ*= 0 deg are smaller than those at

*θ*= 90 deg. In general, the normalized frequency of the Γ

_{2}point can approximate 1/

*n*from a simple calculation using the empty lattice method [33], where

_{a}*n*denotes the refractive index of the averaged PC material. Therefore, the frequencies of the Γ

_{a}_{2}point for the A and D modes become small as

*θ*increases. Concerning B and C modes, since these modes propagate perpendicular to the ΓX direction, the frequencies for B and C modes at the Γ

_{2}point become large as

*θ*increases. Feature (2) is related to the symmetry change and is discussed in Sec. 5.

#### 4.2 Near- field distributions

Figure 5
shows the near-field distribution of ${H}_{3}(\overline{x})$ at the Γ_{2} point. The near field is calculated in the plane of −0.3 μm ≤ *x _{1}*,

*x*≤ 0.3 μm. In the case of isotropic gain, A and B modes show a 2D resonant ${H}_{3}(\overline{x})$ distribution, while C and D modes are degenerate and display an orthogonal one-dimensional (1D) resonant distribution. These are the same as the reported properties in the square-lattice PC [16]. The A and B modes are classified as the monopole and quadrupole modes, respectively. On the other hand, in case of anisotropic gain with

_{2}*θ*= 45 deg, A and B modes show the same 2D resonance as the isotropic gain. However, unlike the isotropic gain, C and D modes show unique 2D resonances, categorized as symmetric dipole and asymmetric dipole modes. We suppose that these dipole modes result from the interference arising through the dependence of gain on directions, which does not occur in isotropic gain. In the case of anisotropic gain with

*θ*= 0 or 90 deg, the 2D resonance does not occur for any modes. Resonance of the A and D (B and C) modes occurs parallel (perpendicular) to the ΓX direction in 1D. The A and B modes show the maximum intensity at the center of the HL region (that is, are symmetric), while the C and D modes do so at the edge of HL (that is, are asymmetric). Since the frequency of the A and B modes are lower than that of the C and D mode as shown in Fig. 3, the A and B modes are categorized as having “dielectric” band characteristics, while C and D modes have “air band” characteristics [33]. Such variation in the near-field distribution is related to symmetry change, as shown in Sec. 5.

#### 4.3 Gain dispersion characteristics

As well as a PBS where the real part (*ω _{r}*) of the frequency of a photonic mode provides the dispersion curve as a function of wavevectors, the imaginary part (

*ω*) gives the dispersion curve of the gain. Here,

_{i}*ω*is normalized as

_{i}*ω*Fig. 6 shows gain dispersion characteristics for isotropic gain and anisotropic gain with

_{i}a/2πc.*θ*= 0, 45, and 90 deg. Each graph has four lines, which are the A, B, C, and D photonic modes. The left axis is the normalized imaginary frequency (

*ω*) and the right axis is the corresponding gain. This gain is not the threshold gain, but the net modal gain. In anisotropic gain, at

_{i}a/2πc*θ*= 0 deg (Fig. 6(b)), the A and D modes possess the highest gain of

*ω*= 4.8 × 10

_{i}a/2πc^{−5}, or 71 cm

^{−1}. On the other hand, the value of

*ω*is the smallest for the B and C modes, giving a gain of 11 cm

_{i}a/2πc^{−1}. Thus the A and D modes, unlike the B and C modes, are able to oscillate. At

*θ*= 90 deg (Fig. 6(d)), the larger gain in the B and C modes lead to their lasing operation. At

*θ*= 45 deg (Fig. 6(c)), the gain is not high compared with other value of

*θ*and has the same value as in the case of isotropic gain (Fig. 6(a)). This is because photonic modes consist of plane waves whose directions are parallel or perpendicular only to the ΓX direction (see Sec. 5.1). Thus, when the direction of larger gain in the PC material is parallel or perpendicular to the ΓX direction, the directions of material gain and the plane waves coincide, so that the gains of corresponding photonic modes become large. When the direction of larger gain in the material is parallel to the ΓM direction, they do not coincide, so that the gains of photonic modes become small. This means that the lasing action is difficult, when

*θ*is 45 deg.

Gain dependence on *θ* is summarized in Fig. 7
. For anisotropic gain, as *θ* increases, the gain of A and D modes decreases, while the B and C modes obtain higher gain. This is because the resonant direction of A and D (B and C) modes become far from (close to) the direction of higher gain in the PC material. At *θ* = 45 deg, the gain contribution to each mode is the same, so that the crossover between A-D and B-C modes occurs. Over the *θ* = 45 deg, the gains of B and C modes are larger than those of A and D modes. Therefore, when *θ* varies from 0 deg to 90 deg, the PC laser with anisotropic gain should function as follows: first, strong emission with the resonant direction parallel to the ΓX direction, then weak emission around *θ* = 45 deg, and finally strong emission with the resonant direction perpendicular to the ΓX direction.

## 5. Discussions

#### 5.1 PC symmetry change by anisotropic gain

As shown in Secs. 3 and 4, anisotropic gain induces variation in the resonance frequencies, field distributions, and gain. In order to clarify this, we investigate the field distributions and the expanded waves of ${H}_{3}(\overline{x})$ around *θ* = 45 deg in more detail. Here, *Δθ* is defined as 45 minus *θ* in units of degree.

Near-field distribution at the Γ_{2} point is shown in Fig. 8
as a function of *Δθ*. The fields ofC and D modes are very sensitive to the *Δθ*. At *Δθ* = 10^{−4} deg, the near-field distributions of the C and D modes are almost identical to those at *θ* = 45 deg. At *Δθ* = 10^{−3} deg, the strong field portions of C and D modes affect the horizontal and vertical neighboring strong portions, respectively. At *Δθ* = 10^{−2} deg, the field distribution is similar to that at *θ* = 0 or 90 deg. In contrast, such small *Δθ* results in little variation for the A and B modes. At *Δθ* = 0.8 deg, the near-field distributions of the A and B modes remain almost identical to those at *θ* = 45 deg. At *Δθ* = 1 deg, the strong field regions of A and B modes affect the vertical and horizontal neighboring strong regions, respectively. At *Δθ* = 2 deg, the field distributions of A and B modes are similar to those at *θ* = 0 or 90 deg. In addition, Fig. 8 clearly indicates that the A, B, C, and D photonic modes originate from the monopole, quadrupole, symmetric dipole, and asymmetric dipole modes, respectively.

These variations can be shown by using the expanded waves of ${H}_{3}(\overline{x})$. According to Eq. (1), an EM wave consists of the summation of many reciprocal vectors; our calculation shows that four reciprocal vectors, ${G}_{1}=(-1,0)2\pi /a$, ${G}_{2}=(0,-1)2\pi /a$, ${G}_{3}=(0,1)2\pi /a$, and ${G}_{4}=(1,0)2\pi /a$, contribute mainly at the Γ_{2} point, i.e.,

_{2}point, as a function of

*Δθ*. When

*Δθ*is less than 10

^{−4}deg, each mode consists of four reciprocal vectors,

**G**

_{1}to

**G**

_{4}. These four vectors cause 2D resonances, as shown in Fig. 8. When

*Δθ*increases from 10

^{−4}to 10

^{−2}deg, the amplitudes of the B-coefficients of

**G**

_{1}and

**G**

_{4}(

**G**

_{2}and

**G**

_{3}) reduce in C (D) modes monotonically, leading to 1D resonances. When

*Δθ*is around 0.8 deg, the real part of the B-coefficients of

**G**

_{2}and

**G**

_{3}(

**G**

_{1}and

**G**

_{4}) in A (B) modes decrease, while, in turn, the imaginary parts appear and fade away. This indicates that, when the resonances of A and B modes change from 2D to 1D, the phase of the B-coefficients of

**G**

_{2}and

**G**

_{3}(

**G**

_{1}and

**G**

_{4}) vary toward π/2.

The anisotropic gain changes the symmetry of the square-lattice PC. That is, although square-lattice PCs with isotropic gain possess 4-fold rotational symmetry, square-lattice PCs with anisotropic gain with *θ* ≠ 45 deg have 2-fold rotational symmetry. Note that anisotropic gain with *θ* = 45 deg produces 4-fold rotational symmetry despite there being geometrical 2-fold symmetry, because ${H}_{3}(\overline{x})$ does not possess a reciprocal vector consisting of both *x _{1}* and

*x*components such as (1,1)2π/

_{2}*a*. In 2-fold rotational symmetry, the resonance occurs in 1D only. Therefore the resonance dimension changes around

*θ*= 45 deg, which causes an abrupt change in the resonance frequency and the near-field distribution. In addition, since the A and B modes inherently possess a symmetric field distribution with respect to the hole shape as shown in Fig. 5, these modes are relatively robust to disturbance by the anisotropic gain. On the other hand, the C and D modes have an inherently asymmetric distribution with respect to the hole shape, and thus they obtain an asymmetric nature easily caused by the anisotropic gain. Such difference explains the sensitivity of the field or B-coefficient variation to

*Δθ*variation, as shown in Figs. 8 and 9.

#### 5.2 Radiation loss effects

2D PC suffers from radiation loss (see the Appendix). For the model shown in Sec. 3, the radiation losses for photonic modes are calculated as 45.9 cm^{−1} for C and D modes, while there is no loss for A and B modes. Taking these losses into consideration, the dependence of gain on *θ* is calculated as shown in Fig. 10
. Negative gain indicates loss. Compared to Fig. 7, gains of C and D modes decrease drastically, since they suffer from the radiation loss. At *θ* = 0 (90) deg, the gain of A and D (B and C) modes separates so that the PC laser will oscillate in a single photonic mode of A (B). In contrast, the no gain difference between A and B modes is observed for the isotropic gain, e.g., c-plane case, which leads to dual mode operation. Thus, Fig. 10 indicates that it is important to consider anisotropic gain when characterizing surface-emitting performance.

The reason that the C and D modes experience the radiation loss is as follows. Since the electric field in the PC plane is given by the derivative of the magnetic field, the asymmetric *H _{3}* fields of C and D give symmetric electric fields. Since the symmetric electric fields interfere constructively in the PC plane, the radiation becomes strong. On the other hand, symmetric

*H*fields of A and B forms an asymmetric electric field in the PC plane, leading to destructive interference and weak radiation. Sakai et al. studied a square-lattice PC laser and showed that each PC mode suffers from this radiation loss [16]. A, B, and E modes in [16]. correspond to A, B, and C (D) modes, respectively, in this work. Sakai et al. indicate that, compared to A and B modes, the C (D) mode suffers from radiation loss to a great extent, which is consistent with our results.

_{3}## 6. Conclusions

We have presented the analysis of photonic modes with anisotropic gain for a 2D PC structure. We developed a novel PWE method to investigate the anisotropic gain, where the gain factor is expressed as a function of complex frequency. Numerical calculations were undertaken for a nonpolar InGaN-based PC laser. The results indicate that anisotropic gain induces variation not only in PBS but also in the gain characteristics and in the near-field distribution. Small variation in the direction of gain from the ΓM direction changes the symmetry of the square-lattice PC from 4-fold rotational symmetry to 2-fold symmetry. Higher gain induced by the PC material requires the alignment in which the direction of larger gain in the material coincides with the ΓX direction of the square-lattice PC.

## Appendix

Radiation loss necessitates the three-dimensional (3D) treatment of the radiation wave. Similar to [34], the magnetic field *H _{3}* in a square-lattice PC can be expressed as:

where *x _{3}* is the axis normal to the PC plane; $\widehat{x}$ is a position in space, i.e., $\widehat{x}=({x}_{1},{x}_{2},{x}_{3})$; and $\mathrm{\Delta}H(\widehat{x})$ is the radiation perturbation. Here we consider four reciprocal vectors based on Eq. (15). Note that, unlike Eq. (2), the Fourier expansion coefficient,${B}_{\overline{x}}(G,\omega )$, is a function of $\overline{x}$. We assume that $\mathrm{\Delta}H(\widehat{x})$ and ${B}_{\overline{x}}(G,\omega )$ vary slowly in $\overline{x}$, so that

where *i*, *j* = 1, 2. In Eq. (a.1), $\phi ({x}_{3})$ is a near-field distribution along *x _{3}* and is given by:

where $\mathrm{\Delta}\epsilon ({x}_{3})={\epsilon}_{eff}-\epsilon ({x}_{3})$; ${\epsilon}_{eff}$ denotes the averaged effective dielectric constant of HL and BG region, i.e., ${\epsilon}_{eff}=f{\epsilon}_{a}+(1-f){\epsilon}_{b}$; and $\epsilon ({x}_{3})$ represents the dielectric constant along *x _{3}*, which is in-plane averaged for the PC layer. The inverse of the dielectric constant, $\epsilon (\widehat{x})$, in 3D form is expanded as

where

and

Here, ${\epsilon}_{pa}$ and ${\epsilon}_{pb}$ represent the dielectric constants of the HL and the BG regions, respectively, in the PC layer (in our model, the PC layer corresponds to the AROG layer); *g _{1}* and

*g*are defined as the

_{2}*x*range of the PC layer. From Eqs. (a.5) and (a.6), we can derive $\widehat{\kappa}(G,{x}_{3})$=$\widehat{\kappa}(-G,{x}_{3})$ and ${\overline{\kappa}}_{p}(G)=$ ${\overline{\kappa}}_{p}(-G)$. The 3D Maxwell equations for H-polarization give:

_{3}By substituting Eqs. (a.1), (a,3), and (a.4) into Eq.(a.7), the following equation is obtained at the ${\mathrm{\Gamma}}_{2}$ point:

where $\overline{\nabla}=(\partial /\partial {x}_{1},\partial /\partial {x}_{2})$. Constant terms with respect to the exponential function in Eq. (a.8) give

Equation (a.9) can be solved by the Green function method [34] and gives

where $\mathrm{\Delta}{\epsilon}_{p}={\epsilon}_{eff}-\left[f{\epsilon}_{pa}+(1-f){\epsilon}_{pb}\right]$; $\tilde{G}({x}_{3},t)=\mathrm{exp}\left[\pm i{K}_{3}({x}_{3}-t)\right]/2i{K}_{3}$ (positive sign for *x _{3}* >

*t*, negative sign for

*x*<

_{3}*t*); and ${K}_{3}=\omega /a{\overline{\kappa}}_{p}{(0)}^{0.5}$. Coefficient terms with respect to the exponential function in Eq. (a.8) for a

**G**

_{m}give

By multiplying Eq. (a.11) with $\phi ({x}_{3})$, integrating it in the PC layer, i.e., ${g}_{1}\le {x}_{3}\le {g}_{2}$, and substituting Eq. (a.10) into it, we can obtain the following equation:

where

${\mathrm{\Gamma}}_{g}$ is a confinement factor in PC layer, defined as:

When $\phi ({x}_{3})$ is regarded as almost constant in the PC layer, Eqs. (a.15) and (a.16) are approximated as

where *d _{g}*=|

*g*-

_{2}*g*|.

_{1}In case of the 1D resonance, the near-field distribution can be considered to consist of **G**
_{1} and **G**
_{2} reciprocal vectors, where ${G}_{m}=(2\pi /a,0)$, ${G}_{1}={G}_{m}$, and ${G}_{2}=-{G}_{m}$, which does not lose generality. In this case ${B}_{\overline{x}}({G}_{n})$ is a function of *x _{1}* only, so that we can use ${B}_{{x}_{1}}({G}_{n})$ instead of ${B}_{\overline{x}}({G}_{n})$. For symmetric and asymmetric

*H*fields, we can set ${B}_{{x}_{1}}({G}_{1})={B}_{{x}_{1}}({G}_{2})$ and${B}_{{x}_{1}}({G}_{1})=-{B}_{{x}_{1}}({G}_{2})$, respectively. Then Eq. (a.12) can be simplified as

_{3}where plus and minus signs denote symmetric and asymmetric *H _{3}* fields, respectively;

*a*and

_{m}*b*are

_{m}*x*components of $F({G}_{m},{G}_{m})$ and $F({G}_{m},-{G}_{m})$, respectively; ${c}_{m}=A({G}_{m},{G}_{m})+{(\omega /c)}^{2}$; and ${d}_{m}=A({G}_{m},-{G}_{m})$. Equations (a.20) and (a.21) indicate that the imaginary part of $\left({c}_{m}\pm {d}_{m}\right)/\left({a}_{m}\pm {b}_{m}\right)$ represents the radiation loss, since the expanded wave of the

_{1}**G**

_{1}(

**G**

_{2}) component in Eq. (a.1) at the ${\mathrm{\Gamma}}_{2}$ point propagates in the +

*x*(−

_{1}*x*) direction. Note that $\left({c}_{m}\pm {d}_{m}\right)/\left({a}_{m}\pm {b}_{m}\right)$ represents the loss only in the PC layer, because the integration of Eq. (a.12) is undertaken within the PC layer. Thus, the radiation loss for symmetric and asymmetric

_{1}*H*fields,

_{3}*Q*and

_{s}*Q*, respectively, can be obtained as

_{a}where ${P}_{1}={\overline{\kappa}}_{p}({G}_{m})/{\overline{\kappa}}_{p}(0)$, ${P}_{2}={\overline{\kappa}}_{p}(2{G}_{m})/{\overline{\kappa}}_{p}(0)$, and $x=\omega a/2\pi c$.

In the case of 2D resonance, we can treat Eq. (a.12) in the same manner as 1D resonance. The radiation loss for A and B modes, *Q _{A}* and

*Q*, respectively, are as follows;

_{B}where ${P}_{3}={\overline{\kappa}}_{p}\left({G}_{1}+{G}_{2}\right)/{\overline{\kappa}}_{p}(0)$. The radiation loss for both C and D modes, *Q _{C}*, and

*Q*, respectively, is equal to the radiation loss of the 1D asymmetric resonance mode,

_{D}*Q*. Note that the radiation loss, Eqs. (a.22) to (a.25), should be zero in the case of a negative value, because the radiation does not result in gain.

_{a}## Acknowledgments

S. Takigawa is grateful to Dr. Daisuke Ueda and Dr. Tsuyoshi Tanaka in Panasonic Corporation for their encouragements.

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