Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls

Chao Peng; Yong Liang; Kyosuke Sakai; Seita Iwahashi; Susumu Noda

doi:10.1364/OE.19.024672

1. Introduction

Development of the band-edge-effect based two-dimensional (2D) photonic-crystal surface-emitting lasers (PC-SELs) has attracted considerable attention because they can show large-area and single-mode coherent oscillation [1]–[15]. High output power and lasing efficiency has been achieved in square-lattice photonic crystal structures at room temperature [1]–[7], and the performance of the PC structure has been optimized by tuning the air-hole shapes and filling factors. The lasing wavelengths have been extended from the near-infrared regime to the mid-infrared [9], tera-hertz [10, 11], and blue-violet regimes [12, 13]. Besides, control of the polarizations and beam patterns by appropriate design of the PC geometry [3, 8], on-chip beam steering by using a composite PC structure [14, 15], etc. have been reported.

The huge potential of using PC-SELs in many applications calls for the development of an accurate model to understand and analyze the characteristics of PC laser. However, despite numerous experimental efforts and advances, theoretical study on these types of lasers, which can depict the physics of wave coupling and enable further optimization, is limited. Conventionally, numerical approaches based on the finite-difference time-domain (FDTD) [16] and the plain-wave expansion (PWE) [4] methods are widely used for analyzing the characteristics of the PC structure. However, using these approaches to analyze the radiative properties of a large, finite-area PC structure is not feasible because the 2D PWE is applicable only to infinite structure and the FDTD method requires substantial amounts of computer resources for modeling realistic finite, large-area structures. Hence, several analytical methods [6][20] based on the one-dimensional (1D) coupled-wave theory (CWT) [17]–[19] have been reported. But in these over simplified methods, only four basic Bloch waves are considered, and hence, the important 2D optical coupling effects are neglected. As a result of these shortcomings, these initial studies cannot explain the experimental results [7]. Sakai et al. later derived the 2D CWT, which induces in-plane coupling by using an eight-wave model to describe 2D coherent lasing action [21][22]. This detailed formulation of the lasing process underscores the importance of accurately modeling complicated coupling effects in 2D PC-SELs.

Recently, we proposed a more rigorous formulation of the CWT for accurate and efficient analysis of PC-SELs [23]. The proposed CWT model is directly formulated from electric field components based Maxwell’s equation, and the explicit definition of coupling coefficients and the vertical field profiles of individual waves are included. Besides the four basic Bloch waves, radiative waves and a sufficient number of higher-order waves are consistently incorporated to depict all the important coupling effects. The mode frequencies and radiation constants are calculated by solving the corresponding eigen problem. Since our model can be applied to arbitrarily shaped holes, it is considered a simple, fast, and effective model for the quantitative analysis of PC-SELs.

However, because the previous derivation only assumes uniform refractive index distribution within the PC layer [23], the formulas are not rigorous for general cases in which the air holes possess arbitrary sidewalls. Engineering air holes in vertical direction is a promising approach for efficient tuning of PC-SEL properties to suit a variety of applications [24]. In-plane asymmetric air holes give rise to complicated diffraction effects and enhance vertical radiation [23]. Air holes with arbitrary sidewalls can also afford more design freedom, which influences the radiative characteristics. In this study, we extend the CWT model to an inhomogeneous vertical structure for comprehensively understanding the effects of air holes with arbitrary sidewalls.

The formulation of the general CWT analysis for air holes with arbitrary sidewall is derived by taking into account the vertically inhomogeneous permittivity. Air holes with two different in-plane geometries, circular (CC) and right-angled isosceles triangular (RIT), are investigated for tapered and tilted sidewalls, respectively. The results of CWT analysis agree well with those of the FDTD numerical simulation. Investigation of the symmetry properties of the band-edge modes shows that designing air holes with in-plane asymmetry is crucial for achieving high output power because partial constructive interference is caused by such air holes. Fabrication of air holes with in-plane asymmetry and vertically tilted sidewall help in inducing such asymmetries. By carefully breaking the symmetries of the band-edge modes, the two factors, air-hole shape and tilt angle, can be tuned independently, and this leads to enhanced radiation power and preservation of the mode selectivity performance. Finally, a top-down approach by reactive ion etching (RIE) is adopted for fabricating such a structure.

The rest of this paper is organized as follows. In Section2, derivation of the CWT formulation by assuming inhomogeneous vertical distribution of permittivity is presented. Section3 discusses the the CC and RIT air holes for “tapered” and “tilted” sidewall and confirmation of the results by FDTD simulation. The symmetries of the band-edge modes are discussed and a promising method for fabricating tilted air holes is described briefly. Section4 includes concluding remarks.

2. Coupled-wave theory for inhomogeneous vertical structure

In this section, we will present the derivation of the general CWT for inhomogeneous vertical structures. Major differences between the CWT formulation in our previous study and that in the present study will be highlighted. For a more detailed derivation and discussion, please refer to our previous paper [23].

The structure of PC-SELs is illustrated schematically in Fig. 1(a). It is a multiple-layer slab structure with the PC at the center. The average permittivity of the PC layer is given by ɛ_PC = fɛ_a + (1 – f)ɛ_b, where ɛ_a = 1 is the permittivity of air, ɛ_b = 12 is the permittivity of the dielectric material (GaAs), and f = 0.16 is the filling factor. The slab structure supports only a single waveguide mode propagating in the XY plane; and the wavevector is denoted by β. Let the PC lie in a square lattice (lattice constant a) with arbitrarily shaped holes; then, the reciprocal base vector is given by β₀ = 2π/a. For the PC-SELs where the side length is sufficiently large (exceeding 300a or 100μm for practical devices), the structure can be reasonably regarded as infinite in the x – y plane and the in-plane loss can be neglected [25]. We start from the electric components based Maxwell equation:

\nabla \times \nabla \times E (r) = ɛ (r) k_{0}^{2} E (r)

where k₀ is the wavenumber in vacuum. For the transverse-electric (TE) mode, the field should be E = (E_x, E_y, 0), and it can be expanded using the Bloch theorem, as E_x(z) = ΣE_x,mn(z)e^{–imβ₀x–inβ₀y}, E_y(z) = ΣE_y,mn(z)e^{–imβ₀x–inβ₀y}. Besides, the permittivity can similarly be expanded as

ɛ (z) = ɛ_{0} (z) + \sum_{m, n \neq 0} ξ_{m n} (z) e^{- im β_{0} x - in β_{0} y}

Here, ɛ₀(z) is the average permittivity and ξ_mn(z) = 0 for the no-periodicity case. In the case of a general inhomogeneous vertical structure, ɛ₀(z) and ξ_mn(z) are not necessarily constant in the z direction. Then, we get the wave equation for all E components,

[\frac{\partial^{2}}{\partial z^{2}} + ɛ_{0} (z) k_{0}^{2} - n^{2} β_{0}^{2}] E_{x, mn} + k_{0}^{2} \sum_{m^{'} \neq m, n^{'} \neq n} ξ_{m - m^{'}, n - n^{'}} (z) E_{x, m^{'} n^{'}} + mn β_{0}^{2} E_{y, mn} = 0

[\frac{\partial^{2}}{\partial z^{2}} + ɛ_{0} (z) 2 k_{0}^{2} - n^{2} β_{0}^{2}] E_{y, mn} + k_{0}^{2} \sum_{m^{'} \neq m, n^{'} \neq n} ξ_{m - m^{'}, n - n^{'}} (z) E_{y, m^{'} n^{'}} + mn β_{0}^{2} E_{x, mn} = 0

The Above mentioned wave equations include all possible waves whose amplitudes are explicitly dependent on the vertical position z; solving such equation is difficult because all the waves are coupled to one another. As we defined previously [23], the waves are classified into three groups according to their in-plane wavenumbers, $β_{mn} = \sqrt{m^{2} + n^{2}} β_{0}$ , as follows: basic waves (|m² + n²| = 1), high-order waves (|m² + n²| > 1), and radiative wave (|m² + n²| = 0). Most of energy is concentrated in the basic waves for the resonant case, and therefore, treating the four basic waves as sources to excite high-order waves and radiative waves, and the excited waves coupling back to the basic waves would be a good approximation. Coupled equations describing the interactions between the four basic waves can be obtained, and the four band-edge modes at the second-order Γ-point can be obtained as shown in Fig. 1(b).

Fig. 1 (a) Schematic structure of the surface-emitting PC laser with circular (CC) and right-angled isosceles triangular (RIT) air holes; (b) band structure of RIT PC laser near the Γ-point (using the PWE method)

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2.1. Basic waves

Assume that the four basic waves have in-plane amplitudes R_x, S_x, R_y, S_y with identical vertical profiles Θ₀(z). For a infinite periodical PC structure, the in-plane amplitudes are independent of x,y.

E_{x, 10} = 0 E_{y, 10} = R_{x} Θ_{0} (z)

E_{x, - 10} = 0 E_{y, - 10} = S_{x} Θ_{0} (z)

E_{x, 01} = R_{y} Θ_{0} (z) E_{y, 01} = 0

E_{x, 0 - 1} = S_{y} Θ_{0} (z) E_{y, 0 - 1} = 0

Without loss of generality, we focus on the case m = 1, n = 0: Eqs. 4 and 5 lead to

[\frac{\partial^{2} Θ_{0}}{\partial z^{2}} + (ɛ_{0} (z) k_{0}^{2} - β_{0}^{2}) Θ_{0}] R_{x} = - k_{0}^{2} \sum_{m^{'} \neq m, n^{'} \neq n} ξ_{1 - m^{'}, - n^{'}} (z) ɛ_{0} (z) E_{y, m^{'} n^{'}} (z)

Under resonant condition, the phase matching leads to β = β₀, and hence, the profile Θ₀(z) for basic waves can be determined by the waveguide mode profile; the basic wave profile satisfies following equation

\frac{\partial^{2} Θ_{0}}{\partial z^{2}} + [ɛ_{0} (z) k^{2} - β_{0}^{2}] Θ_{0} = 0

Θ₀(z) can be solved by using transfer matrix method (TMM) [23][26]. It is noteworthy that because the average permittivity may vary along z when the air hole possesses an arbitrary sidewall, the PC layer must be divided into multiple layers and constant permittivity must be assumed within every single layer, if necessary [26].

Then, the frequency k₀ is generalized by including a small deviation from the desired frequency k, as k₀ = k + δk. The radiative waves are denoted as E_x,00 = ΔE_x(z), E_y,00 = ΔE_y(z); hence, the coupled equation of the basic wave is obtained from Eq. 9 and Eq. 10, as

ɛ_{0} (z) (k_{0}^{2} - k^{2}) Θ_{0} (z) R_{x} = - k_{0}^{2} ξ_{20} (z) S_{x} Θ_{0} (z) - k_{0}^{2} ξ_{10} (z) Δ E_{y} (z) - k_{0}^{2} \sum_{| m^{2} + n^{2} | > 1} ξ_{\begin{array}{l} 1 - m^{'} \\ - n^{'} \end{array}} (z) E_{y, m^{'} n^{'}} (z)

Multiplication of the complex conjugate field

Θ_{0}^{*} (z)

on both sides and integration over (−∞, ∞) gives

\begin{array}{l} h δ k R_{x} = & - k_{0}^{2} S_{x} \int_{PC} ξ_{20} (z) {| Θ_{0} (z) |}^{2} d z - k_{0}^{2} \int_{PC} ξ_{10} (z) Δ E_{y} (z) Θ_{0}^{*} (z) d z \\ - k_{0}^{2} \sum_{| m^{2} + n^{2} | > 1} \int_{PC} ξ_{\begin{array}{l} 1 - m^{'} \\ - n^{'} \end{array}} (z) E_{y, m^{'} n^{'}} (z) Θ_{0}^{*} (z) d z \end{array}

with the normalization condition

\int_{- \infty}^{\infty} {| Θ_{0} (z) |}^{2} d z = 1

, and

h = 2 k \int_{- \infty}^{\infty} ɛ_{0} (z) Θ_{0} (z) Θ_{0}^{*} (z) d z

, which is determined by the basic wave profile. However, the radiative wave profile ΔE_y(z) and high-order profile E_y,m′n′ (z) are still unknown, and they will be determined in the subsequent sections.

Note that instead of the complex propagation constant (implying spatially decaying modes), the complex frequencies δk (implying temporally decaying modes) are used here. Simultaneous Bragg coupling and energy leakage perturb the bands corresponding to complex propagation constant, however, the two mechanisms are decoupled in the case of complex frequency [27]. Thus, using a complex frequency afford well defined, clear band gaps.

2.2. Radiative waves

For the radiative wave ΔE_y(z), the terms with (m, n) = (0, 0) are grouped from Eq. 4:

[\frac{\partial^{2}}{\partial z^{2}} + ɛ_{0} (z) k_{0}^{2}] Δ E_{y} (z) = - k_{0}^{2} \sum_{m^{'}, n^{'} \neq 0} ξ_{\begin{array}{l} - m^{'} \\ - n^{'} \end{array}} (z) E_{y, m^{'} n^{'}} (z)

Then, the approximation that only basic waves are important for exciting radiative waves is applied:

Δ E_{y} (z) ≃ k_{0}^{2} R_{x} \int_{PC} ξ_{- 1, 0} (z) G (z, z^{'}) Θ_{0} (z^{'}) d z^{'} - k_{0}^{2} S_{x} \int_{PC} ξ_{1, 0} (z) G (z, z^{'}) Θ_{0} (z^{'}) d z^{'}

The above equation is solved by employing the Green function approach. In our previous study, we found that using an approximation of the Green function G(z,z′) = −i/2β· e^{−iβ|z–z′|} for infinite homogenous space gives accurate results [23]. However, in this expression, reflections between individual layers are neglected, and hence, the equation is not very rigorous when the refractive index contrast between individual layers causes strong reflection. Therefore a generalized Green function for a multilayer inhomogeneous structure would be more accurate (see Appendix for details). The profile of the radiative waves is expressed in terms of the basic wave profiles and Green function. A similar approach can be adopted for the other radiative wave ΔE_x(z).

2.3. High-order waves

The high order waves shown by Eqs. 3 and 4 can be decoupled into two sets of equations by using a proper linear combination of E_x,mn(z) and E_y,mn(z); then we get

[\frac{\partial^{2}}{\partial z^{2}} + ɛ_{0} (z) k_{0}^{2} - (m^{2} + n^{2}) β_{0}^{2}] (n E_{x, mn} - m E_{y, m n}) = - k_{0}^{2} \sum ξ_{\begin{array}{l} m - m^{'} \\ n - n^{'} \end{array}} (z) (n E_{x, m^{'} n^{'}} - m E_{y, m^{'} n^{'}})

[\frac{\partial^{2}}{\partial z^{2}} + ɛ_{0} (z) k_{0}^{2}] (m E_{x, mn} + n E_{y, mn}) = - k_{0}^{2} \sum ξ_{\begin{array}{l} m - m^{'}, \\ n - n^{'} \end{array}} (z) (m E_{x, m^{'} n^{'}} + n E_{y, m^{'} n^{'}})

As in the case of the basic wave case, Eq. 15 can be solved by using the Green function approach with the corresponding Green function G_mn(z, z′). In addition, only basic waves are assumed to excite high-order waves and the generalized Green function is used for better accuracy (see Appendix):

\int_{PC} (n E_{x, mn} - m E_{y, mn}) Θ_{0}^{*} (z) dz ≃ - m μ_{m, n}^{1, 0} R_{x} - m μ_{m, n}^{- 1, 0} S_{x} + n μ_{m, n}^{0, 1} R_{y} + n μ_{m, n}^{0, - 1} S_{y} ≜ E^{-}

where

μ_{m, n}^{r, s} = k_{0}^{2} \iint_{PC} ξ_{m - r, n - s} (z) G_{mn} (z, z^{'}) Θ_{0}^{*} (z) Θ_{0} (z^{'}) dz d z^{'}

On the other hand, in the TE mode (no E_z component) and under the transversity condition ∇ · [ɛ(r)E(r)] = 0, Eq 16 gives

\int_{PC} (m E_{x, mn} + n E_{y, mn}) Θ_{0}^{*} (z) dz ≃ n ν_{m, n}^{1, 0} R_{x} + n ν_{m, n}^{- 1, 0} S_{x} + m ν_{m, n}^{0, 1} R_{y} + m ν_{m, n}^{0, - 1} S_{y} ≜ E^{+}

where

ν_{m, n}^{r, s} = - \int_{PC} \frac{1}{ɛ_{0} (z)} ξ_{m - r, n - s} (z) {| Θ_{0} (z) |}^{2} d z

The integrals in Eq. 17 and 18 take into account the z dependency of ɛ₀(z) and ξ_mn(z). Finally, the overlap integrals in Eq. 12 are obtained in the form of Θ₀(z) and the Green function:

\int_{P C} E_{x, mn} (z) Θ_{0}^{*} (z) dz = \frac{m E^{+} + n E^{-}}{m^{2} + n^{2}}, \int_{PC} E_{y, mn} (z) Θ_{0}^{*} (z) d z = \frac{n E^{+} - m E^{-}}{m^{2} + n^{2}}

2.4. Coupled wave equations

Now, all the terms in Eq. 12 have been solved analytically and they only depend on the PC geometry and the multilayer waveguide structure. Similar treatment to other basic waves gives the eigen value problem equations of the basic wave vector V = (R_x, S_x, R_y, S_y) with the coupling matrix C:

δ k V = C V

The complex frequencies ω can be obtained from the eigen values by solving Eq. 20 as ω = (k + δk)/c. In this case, the Q factors of the band-edge modes can be determined from the real and imaginary parts of ω, as Re(ω)/|2Im(ω)|, and directly compared with the FDTD simulation results without using any ambiguous definition of the effective refractive index. Nevertheless, for the infinite PC structure, only the radiative wave causes radiation loss in the spatial domain and afford a corresponding finite Q factor in the time domain. The Q factor describes the same radiative characteristics of PC-SELs as do the radiative constant derived from the complex propagating constant, but from another point of view. Thus, the radiation constant can be obtained from the Q factor using the following equation [23][28]:

α_{r} = \frac{β_{0}}{Q} = \frac{2 π / a}{Q}

3. Radiative characteristics of air holes with arbitrary sidewalls

By extending the CWT to a general vertical structure, we can analyze the PC-SEL structure with arbitrary air-hole sidewalls. We are interested in the structure shown in Fig. 2(a–b). When the sizes of the holes vary from top to bottom but the center position remains the same, we refer the structure as “tapered case.” In-plane asymmetry of air hole also has a significant impact on the performance of PC-SELs [23]. As a result, we adopt the “tilted case”; here, the in-plane asymmetry is induced by the tilting of the air hole to the given direction (Fig. 2(c–d)). In this case, the shapes and sizes of the air holes are the same along the PC slab, but the center position shift to given tilt direction.

Fig. 2 Schematic of two types of air-hole sidewalls for two in-plane shapes. (a),(c): CC air holes; (b),(d): RIT air holes. Figs. (a)(b) show the “tapered case” for tilt angle θ. Figs. (c)(d) show the “tilted case” for tilt angle θ and direction ϕ.

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3.1. Air-hole shape and calculation parameters

The parameters for the vertical structure used in numerical calculations are presented in Table. 1. A vertically symmetric structure is assumed to minimize the influence of TM-like modes. As shown in Fig. 2, for the “tapered case,” the sidewall is depicted by tilt angle θ, and θ = 0° indicates an ideal vertical sidewall. For the “tilted case,” the direction of shift of the center position is defined as the tilt direction ϕ; ϕ = 0° is in the X = 0 direction, which matches with the square lattice of the PC. Two air-hole shapes, CC and RIT, will be discussed.

Table 1. Parameters for the vertical structure

View Table

In CWT formulas, the inhomogeneous vertical structure is described by the z-dependent permittivity, i.e., ɛ₀(z) and ξ_mn(z). For the “tapered case” shown in Fig. 2(a–b), the sizes of the air holes vary along the PC layer, and hence the sidewall can be represented by the linear change in the filling factor. Because the air-hole shape and center positions are unchanged in z-direction, ξ_mn(z) only changes with the filling factor. The average permittivity ɛ₀(z) also varies in the z-direction, and hence the PC layer must be divided into multiple layers when applying the TMM to basic wave profile calculation [26].

On the other hand, for the “tilted case” shown in Fig. 2(c–d), the air-hole size is constant, but the center position shifts to the given direction. The shift of center position Δr induces an extra phase shift e^{−iβ₀·Δr} to ξ_mn(z). As a result, ɛ₀(z) is constant within the PC layer, whereas ξ_mn(z) varies with the linear change in the phase shift in the z-direction. In this case, the PC slab can be treated as a single homogeneous layer in the TMM calculation.

To confirm the accuracy of the proposed CWT analysis, we performed 3D-FDTD simulations for the structures described above. We used a computational domain x × y × z of 40 × 40 × 640 pixels, corresponding to 1 × 1 × 16 lattice periods, with the PML in the z direction and the periodic boundary in the xy plane [29]. The Q factors and radiation constant were obtained from the numerical simulation, and compared directly with CWT analysis results.

It is noteworthy that since the CWT algorithm is a semi-analytical algorithm, the analysis requires a much shorter calculation time than does the 3D-FDTD simulation. For a specific data point, the FDTD simulation takes around 4 hours because a parallelized super-computer system (4 × 16 cores) is used, while the CWT analysis only takes a few minutes since a single-core personal computer is used. Hence, the proposed CWT is more advantageous in terms of computation time and resource consumption than is the numerical simulation.

3.2. Calculation results and discussion

The results of CWT analysis and the FDTD numerical data are presented in Figs. 4–6, which show that the two sets of results agree with each other, the thus, confirming the validity of the proposed CWT model. These results also reveal the complex phenomena involving changes in the radiative characteristics of PC-SELs caused by the tilted air-hole sidewalls. In this section, we will analyze these results by investigating the wave interaction effects in PC-SELs.

Fig. 4 Radiation constant vs. tilt angle θ for modes A and B in the “tapered case” shown in Figs. 2(a–b): (a) CC air holes; (b) RIT air holes. The inset solid and dashed shapes show the up and down facets of the air holes, respectively; both CWT and FDTD results are plotted.

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Fig. 5 Radiation constant vs. tilt angle θ for modes A and B in the “tilt case” with the CC air holes shown in Fig. 2(c) in tilt directions indicated by red arrows: (a) ϕ = 0°; (b) ϕ = 135°. The inset solid and dashed shapes show the up and down facets of the air holes, respectively; both CWT and FDTD results are plotted.

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Fig. 6 Radiation constant vs. tilt angle θ for modes A and B in the “tilt case” with the RIT air holes shown in Fig. 2(d) in tilt directions indicated by red arrows indicated: (a) ϕ = 0°; (b) ϕ = 135°. The inset solid and dashed shapes show the up and down facets of the air holes, respectively; both CWT and FDTD results are plotted.

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As explained in the earlier paper [23], the vertical radiation is caused by the Bragg diffraction of in-plane waves (|m² + n²| = 1) to radiative waves (|m² + n²| = 0). The main contribution to the radiation power is from the basic waves (|m² + n²| = 1), where most of the energy is concentrated. As shown in Eq. 14, the radiation can be approximated as the superposition of these four basic waves. On the basis of the eigenvectors solved by the CWT, we first explain the in-plane symmetries of resonant modes A and B, assuming ideal vertical sidewalls, referred to “uniform case,” and then analyze the “tapered case” and “tilted case” using the insights from the symmetry.

Uniform case

Fig. 3 shows the field patterns and eigenvectors of modes A and B for the CC and RIT air holes in real space. The eigenvectors solved from Eq. 20 are complex values that represent the amplitudes and phases of the four basic waves, with their polarizations given by Eqs. 5–8. For convenience, only the amplitudes are plotted in Fig. 3; the lengths of the red and blue arrows indicate the amplitudes, and the arrow directions indicate the electric field polarization of the waves. Because of the perfect symmetry of the CC air holes, the four basic waves have equal amplitudes, while the asymmetric RIT air-hole shape results in different amplitudes between Rx, Ry and Sx, Sy, as shown in Fig. 3.

Fig. 3 Field distribution and eigenvectors of modes A and B, for the: (a) CC air holes, and (b) RIT air holes. The color pattern in the field distribution shows the magnetic field distribution, and the black arrows show the electric field distribution. The eigenvectors of modes A and B are illustrated in coordinate space, with the dashed arrows showing the propagating directions of four basic waves; the red and blue arrows show the amplitudes and polarizations of their electric fields, and the gray arrows show the amplitudes and polarizations of the radiative waves determined by Eq 14.

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As is evident in Fig. 3 and the fact that E is a vector while H is a pseudovector, mode A with CC air holes is antisymmetric with respect to the Y = X and Y = −X axes, but mode B is symmetric with respect to the same axes. As a result of these symmetries and the equal amplitudes of the eigenvectors, the combination of these four basic waves according to Eq. 14 leads to complete destructive interference, which minimizes the radiative power and yields a large Q factor for modes A and B, as confirmed by the FDTD simulation [23]. The RIT air holes, on the other hand, break the symmetry in one of the two axes, and as a result, complete destructive interference is no longer applicable but partial constitutive interference appears. Therefore, a high radiation power and a low Q factor are obtained for modes A and B. The symmetries also determine the polarization of the radiative waves. As shown in Fig. 3, cancellation of the four basic waves results in the the polarization of radiative wave in the Y = −X direction for mode A but in the Y = X direction for mode B.

From this discussion we confirmed that the in-plane asymmetries of the air holes play a crucial role in achieving high-power PC-SELs since the asymmetry determines the interference of the individual modal eigenvectors. Partial constructive interference is enhanced by such asymmetries, which can result from in-plane asymmetric air-hole shapes or vertically tilted sidewalls.

Tapered case

The results of the “tapered case” are shown in Fig. 4. For the CC air holes (Fig. 4(a)), the radiation constant is almost zero when the tilt angle varies. For the RIT air holes (Fig. 4(b)), the radiation constant increases with the tilt angle, and the radiation discrimination, defined as the gap between the two curves, also increases. Theses results are very similar to our earlier results ([23], Fig. 7), which show the radiation constant as a function of the filling factor when an ideal vertical sidewall is assumed. This indicates the effect of the “tapered case,” in which the given tilt angle would be equal to the effective filling factor of an air hole if the sidewall has perfect verticality.

We can understand these results from the in-plane symmetries for the “tapered case.” Although the amplitudes of the modal eigenvectors change owing to the tilted sidewall, the in-plane symmetries of air holes are preserved. Thus, the interference scheme of individual modal eigenvectors is also valid, and the overall effect is similar to that observed in the case of ideal air holes with an effective filling factor and modal eigenvectors with equal amplitudes. As a result, the performance of the PC-SELs may not degrade significantly for small tilt angles.

Moreover, because the tilted sidewalls break the symmetry in vertical direction, the radiative waves propagating in the +z and −z directions are not identical and can be calculated from the generalized Green function (see Appendix). To improve the laser efficiency, the radiative power can be maximized in one direction by imposing a vertically asymmetric design. Detail discussion on this topic is beyond the scope of this paper, but it indicates that the arbitrary sidewall provides more freedom to optimize the performance of PC-SELs.

Tilt case

In the “tilt case,” the in-plane symmetries of the air holes are modified because of the tilted sidewall in given direction. The partial constructive interference is enhanced by the in-plane asymmetries induced by tilted sidewalls. With an increase in the tilt angle, the radiation constant increases steadily as shown in Figs. 5 and 6.

For the CC and RIT air holes with ϕ = 0° (Figs. 5(a) and 6(a)), the radiation constant of mode B increases more rapidly than that for mode A, and the radiation discrimination significantly changes with the tilt angle. As we have explained, the symmetric axes of modes A and B are Y = X and Y = −X and the tilt ϕ = 0° breaks the in-plane symmetry for both the symmetric axes. The symmetry breaking contributes radiations to modes A and B simultaneously, but the contribution differs with the tilt angle. As a result, the radiation constant curves diverge strongly for the CC case shown in Fig. 5(a) where the radiation is contributed entirely by the tilted-sidewall-induced asymmetries. One the other hand, in the RIT case in Fig. 6(a), the in-plane asymmetries induced by the asymmetric air holes shape and the tilted sidewall in the ϕ = 0° direction cannot be combined linearly, and hence the radiation discrimination changes significantly with the tilt angles

Tilting along one of the symmetric axes of modes A and B would break only one in-plane symmetry and maintain the other. Fig. 5(b) and Fig. 6(b) illustrate the CC and RIT air holes along ϕ = 135°, i.e., the Y = −X direction. Fig. 5(b) shows that the curves are degenerate, and Fig. 6(b) shows that, the gap between two curves is maintained at all tilt angles, which indicates that radiation discrimination is determined only by the in-plane air-hole geometry and not by the tilted sidewall. This can be explained as follows. When asymmetric air holes and tilted sidewall break the symmetries along the two orthogonal symmetric axes of modes A and B, radiative waves with two normal polarizations are generated. Hence, radiative waves with normal polarization contribute to the total radiation power orthogonally and independently, and can be linearly combined. As a result, the use of in-plane asymmetric air holes and vertically tilted sidewalls can help enhance the radiation power without affecting the mode selectivity performance. As shown in Fig. 6(b), the radiation at a moderate tilt angle can be much higher than that in the ideal vertical case, which is a very promising feature for high-power PC-SELs devices.

Discussions

Arbitrarily shaped sidewalls may interweave the “tapered case” and “tilted case” and thus can be more complicated. It might be difficult to present a simple picture for depicting such complex phenomena. Nevertheless, the proposed CWT analysis can be used to evaluate the radiative characteristics of general PC-SELs structures quantitatively.

It is noteworthy that, the in-plane asymmetries also significantly affect the far-field radiation patterns of PC-SELs. For infinite PC structure, the far-field is determined by the radiative waves because only these waves are propagating to infinity. The in-plane asymmetry affects the partial constructive interference that contributes radiative wave as Eq. 14 shown, so that, determines the polarizations of radiative waves, namely, the far-fields. Moreover, for finite PC structure, the amplitudes of basic waves possess slow-varying envelopes on xy plane due to the boundary conditions of finite PC cavity. As a result, the finite structure gives rise to more complicate near-filed distributions, and hence, leads to a variety of far-field patterns through Fraunhofer diffraction [8]. In this case, CWT analysis for finite structure is required and we will present the details elsewhere [30].

3.3. Fabrication method of tilted air-hole

An we discussed, tilting the air-hole with given direction and angle provides an extra freedom to tune the radiation characteristics of the PC-SELs. However, fabricating such structure still remains as a huge challenge. Recently, a top-down approach of reactive ion etching (RIE) process was developed in our group [31]. With placing a metal control plate on the sample surface, the ion sheath was modified and the ion trajectory is now at an oblique angle to the surface. As a result, etching within arbitrarily controlled directions and wall angles is enabled. This process was designed for the direct creation of 3D PC structures originally, but also can be applied to the fabrication of tilted air-hole PC structure we proposed here.

4. Conclusion

In this study, we extend the CWT model to analyze inhomogeneous vertical refractive index distributions. The formulations are derived from the electric field components based Maxwell equations, considering the permittivity change in the vertical direction. Sufficient numbers of waves are incorporated with their vertical field profiles, to depict the complicated 2D coupling effect with good accuracy. As a result, explicit definitions of the wave profiles are obtained and the complex mode frequencies are calculated by solving the corresponding eigen value problem. To confirm the validity of the proposed CWT model, the air holes with two geometries, CC and RIT, are investigated for two types of sidewalls, i.e., “tapered case” and “tilted case,” and the results are compared with the FDTD simulation results. The CWT analysis results agree well with the FDTD data, thus confirming the validity of the proposed model.

The proposed CWT model is different from numerical calculations because it provides a semianalytic means of the physics of PC-SELs. On the basis of understanding the eigenvectors of the basic waves obtained from the CWT, the symmetric properties of the band-edge modes are investigated. It is found that the band-edge modes themselves possess intrinsic symmetries with respect to the given axes. In 2D PC-SELs, the radiation is determined by the superposition of contributions from the four basic waves. The in-plane asymmetry of air hole leads to partial constructive interference, which generates radiative power and is crucial for achieving high-output power PC-SELs. Asymmetric air holes and tilted sidewalls help in inducing in-plane asymmetries, and the two factors can be tuned independently when symmetry breaking occurs in the two orthogonal symmetric axes of the band-edge modes. As a result, the radiation power can be enhanced while preserving the mode selectivity performance. Moreover, a top-down reactive ion etching (RIE) approach is suggested to fabricate such structure.

As a semianalytical method, the proposed CWT realizes faster and more efficient modeling of 2D PC-SELs than does the numerical method which requires substantial amounts of computational resources. Thus, the CWT model becomes a simple, fast, and accurate tool to evaluate the characteristics of PC-SELs quantitatively. The discussion in this paper is restricted to infinite structures and TE polarization, but it can be extended to finite structure [22][30] with more complex geometries and polarization. Development of analysis based on the CWT enables us to further optimize the laser performance efficiently for a number of applications.

Appendix: Analytical solution for the Green function

The solution for the vertical field profiles of in-plane waves can be obtained by the Green function approach:

[\frac{\partial^{2}}{\partial z^{2}} + (ɛ_{0} (z) k_{0}^{2} - β_{mn}^{2})] G (z, z^{'}) = - δ (z - z^{'})

For a multilayer structure h_−N, h_−N+1,..., h₋₁, h₁, h₂, h_N, where h_i is the interface between the layers, and the PC layer ∈ [h₁, h₋₁]. Assume that z′ ∈ [h₁, h₋₁] lies in the PC layer; for z > z′, the Green function G(z,z′) can be written as

G = A_{i} e^{- i k_{i, z} (z - h_{i})} + B_{i} e^{i k_{i, z} (z - h_{i})}

G = A_{i + 1} e^{- i k_{i + 1, z} (z - h_{i + 1})} + B_{i + 1} e^{i k_{i + 1, z} (z - h_{i + 1})}

where

k_{i} = \sqrt{ɛ_{0, i} (z) k_{0}^{2} - β_{mn}^{2}}

. G(z,z′)∂G(z, z′)/∂z should be continuous across the interface, so that

\begin{array}{l} (\begin{matrix} A_{i} \\ B_{i} \end{matrix}) & = & [\begin{matrix} \frac{1}{2} (1 + \frac{k_{i + 1, z}}{k_{i, z}}) e^{i k_{i, z} (h_{i + 1} - h_{i})} & \frac{1}{2} (1 - \frac{k_{i + 1, z}}{k_{i, z}}) e^{i k_{i, z} (h_{i + 1} - h_{i})} \\ \frac{1}{2} (1 - \frac{k_{i + 1, z}}{k_{i, z}}) e^{- i k_{i, z} (h_{i + 1} - h_{i})} & \frac{1}{2} (1 + \frac{k_{i + 1, z}}{k_{i, z}}) e^{- i k_{i, z} (h_{i + 1} - h_{i})} \end{matrix}] (\begin{matrix} A_{i + 1} \\ B_{i + 1} \end{matrix}) \\ = & T_{i + 1} (\begin{matrix} A_{i + 1} \\ B_{i + 1} \end{matrix}) \end{array}

Therefore,

(\begin{matrix} A_{0} \\ B_{0} \end{matrix}) = T_{1} (\begin{matrix} A_{1} \\ B_{1} \end{matrix}) = \dots = T_{1} T_{2} \dots T_{N} (\begin{matrix} A_{N} \\ B_{N} \end{matrix}) = T (\begin{matrix} A_{N} \\ B_{N} \end{matrix})

According to the Sommerfeld radiation condition, waves never reflect back from infinity, and hence, A_N = 0. This would lead to

B_{0} = \frac{T_{21}}{T_{11}} A_{0} = κ_{p 1} A_{0}

For z < z′, we notate the amplitude as C, D instead of A,B, and the Green function G(z,z′) becomes

G = C_{i} e^{- i k_{i, z} (z - h_{i})} + D_{i} e^{i k_{i, z} (z - h_{i})}

G = C_{i + 1} e^{- i k_{i + 1, z} (z - h_{i + 1})} + D_{i + 1} e^{i k_{i + 1, z} (z - h_{i + 1})}

Similarly, we have

(\begin{matrix} C_{0} \\ D_{0} \end{matrix}) = T_{0}^{- 1} (\begin{matrix} C_{- 1} \\ D_{- 1} \end{matrix}) = \dots = T_{0}^{- 1} T_{- 1}^{- 1} \dots T_{- N + 1}^{- 1} (\begin{matrix} C_{- N} \\ D_{- N} \end{matrix}) = T^{*} (\begin{matrix} C_{- N} \\ D_{- N} \end{matrix})

Consider D_−N = 0, and thus,

C_{0} = \frac{T_{12}^{*}}{T_{22}^{*}} D_{0} = κ_{p 2} D_{0}

G(z,z′) should ensure continuity at z = z′ and the “jump condition” G′(z′₊₀, z′) − G′(z′₋₀, z′) = 1

A_{0} + B_{0} = C_{0} + D_{0}

A_{0} - B_{0} - C_{0} + D_{0} = - i / k_{0, z}

Finally,

A_{0} = - \frac{i}{2 k_{0, Z}} \frac{1 + κ_{p 2}}{1 - κ_{p 1} κ_{p 2}} B_{0} = - \frac{i}{2 k_{0, Z}} \frac{κ_{p 1} (1 + κ_{p 2})}{1 - κ_{p 1} κ_{p 2}}

C_{0} = - \frac{i}{2 k_{0, z}} \frac{κ_{p 2} (1 + κ_{p 1})}{1 - κ_{p 1} κ_{p 2}} D_{0} = - \frac{i}{2 k_{0, z}} \frac{1 + κ_{p 1}}{1 - κ_{p 1} κ_{p 2}}

and

G (z, z^{'}) = {\begin{matrix} A_{0} e^{- i k_{0, z} (z - z^{'})} + B_{0} e^{i k_{0, z} (z - z^{'})} & z > z^{'} \\ C_{0} e^{- i k_{0, z} (z - z^{'})} + D_{0} e^{i k_{0, z} (z - z^{'})} & z < z^{'} \end{matrix}

Here κ_p1 and κ_p2 present the reflection induced by the refractive index contrast between individual layers. When the reflections are negligible, i.e., κ_p₁, κ_p₂ → 0, above Green function reduces to the infinite homogeneous space form

G (z, z^{'}) = - \frac{i}{2 k_{0}} e^{- i k_{0} | z - z^{'} |}

.

Acknowledgments

The authors are grateful to Dr. A. Oskooi for helpful discussions and valuable suggestions. This study was partly supported by the Core Research for Evolution Science and Technology program of the Japan Science and Technology Agency (CREST-JST), Consortium for Photon Science and Technology (C-PhoST), and by the Global COE Program. Three of the authors (C. Peng, Y. Liang, and S. Iwahashi) are supported by Research Fellowships of the Japan Society for Promotion of Science.

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Layer name	Thickness (a)	Dielectric constant
n-clad (AlGaAs)	∞	11.0224
GaAs	0.2a	12.8603
PC	0.4a	ɛ_PC
GaAs	0.2a	12.8603
p-clad (AlGaAs)	∞	11.0224

Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls

Abstract

1. Introduction

2. Coupled-wave theory for inhomogeneous vertical structure

2.1. Basic waves

2.2. Radiative waves

2.3. High-order waves

2.4. Coupled wave equations

3. Radiative characteristics of air holes with arbitrary sidewalls

3.1. Air-hole shape and calculation parameters

3.2. Calculation results and discussion

Uniform case

Tapered case

Tilt case

Discussions

3.3. Fabrication method of tilted air-hole

4. Conclusion

Appendix: Analytical solution for the Green function

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (36)

Optics Express