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In high-contrast (HC) photonic crystals (PC) slabs, the high-order coupling is so intense that it is indispensable for analyzing the guided mode resonance (GMR) effect. In this paper, a semi-analytical approach is proposed for analyzing GMR in HC PC slabs with TE-like polarization. The intense high-order coupling is included by using a convergent recursive procedure. The reflection of radiative waves at high-index-contrast interfaces is also considered by adopting a strict Green’s function for multi-layer structures. Modal properties of interest like band structure, radiation constant, field profile are calculated, agreeing well with numerical finite-difference time-domain simulations. This analysis is promising for the design and optimization of various HC PC devices.
© 2013 Optical Society of America
1. Introduction
Guided mode resonance (GMR) is an interesting phenomenon existing in photonic crystal (PC) slabs [1–7] that originates from the complex interaction between the in-plane guided waves and the external radiations. More specifically, although the in-plane guided waves are confined within the PC slab, the periodically modulated PC structure diffracts them into the leaky waves propagating in the vertical or oblique directions. Hence, GMR possesses a surface emission property with finite lifetime. It provides an efficient way to channel light from within the slab to the external environment, and allows subtle control over light at the wavelength scale.
Because of these characteristics, GMR has attracted wide interest in the research of active [8–16] and passive [17, 18] PC devices. By utilizing the complex resonant line shapes of the one-dimensional (1D) GMR, resonant grating waveguide structures are designed for many filter applications [3]. Besides, two-dimensional (2D) GMR effect has been adopted as mode controlling mechanism in photonic crystal surface-emitting lasers (PC-SELs) to produce a high-quality beam with single-mode coherent oscillation from the near-infrared to midinfrared [9], terahertz [10, 11], and blue-violet [12, 13]; and can be potentially employed in various applications, such as polarization control [14], pattern tailoring [15], and on-chip beam steering [16].
It is noticed that the GMR effect within high refractive dielectric slabs can result in a strong localization of energy and high-amplitude near fields, which are promising for numerous applications. Lately, a subwavelength high-index-contrast gratings (HCG) have been proposed by Chang-Hasnain et al. [19–22]. HCG creates much higher index contrast than conventional gratings and renders the Bloch modes strongly guided. Using the 1D GMR, HCG is a favorable broadband high-reflectivity mirror [20] with transverse-mode control [21] and large-error tolerance [22]. In the 2D case, GMR in macroscopic high-index-contrast (HC) PC slabs with ultralong lifetime has been observed at Γ point [23] and non-Γ points [24] by measuring the reflection spectrum. For applications, 2D HC GMR can be used for enabling low threshold lasing [25], enhancing and manipulating fluorescence of organic molecules [26], etc..
Increasing experimental advances of HC PC devices call for an efficient and accurate theoretical analysis. Conventionally, numerical algorithms such as FDTD [27] and RCWA [28] are widely used for the analysis of GMR within PC slabs. However, these methods require substantial amounts of computer resources. More importantly, they cannot depict the physics picture of mode resonances. Recently, we developed a 3D coupled-wave theory (CWT) [29–33] to realize accurate and efficient analysis of the GMR in PC-SELs. The 3D theory is able to analyze the GMR in finite [30] or infinite PC slabs with more crystalline geometries [31, 32] and arbitrary tilted sidewalls [33]. In fact, these models are essentially only valid for weakly guided PC slabs with relatively low-index-contrast (LC) interfaces. Thus, we refer to them as LC-CWT.
Compared with LC PC slabs, HC PC slabs have much stronger energy concentration and confinement of the Bloch modes within the PC layer, which significantly increases wave interactions and enhances the radiation of the GMR. Besides, considerable reflections occur at interfaces due to the high index contrast. To address these features, it is crucial to establish a rigorous model for HC PC slabs. In this paper, we extend the 3D CWT for LC PC slabs to HC PC slabs. The intense in-plane wave couplings and the reflections at dielectric interfaces are elucidated by our semi-analytical approach. We calculate the band structure and the radiation constants of the band-edge modes, validate with numerical results, and discuss their behaviors versus the filling factor.
The remainder of this paper is organized as follows. Section 2 describes the 3D high-index-contrast coupled-wave theory (HC-CWT). Section 3 presents semi-analytical results, their comparison with FDTD results, and discussions. Section 4 provides our conclusions.
2. High-index-contrast coupled-wave theory
In this section, we will present the derivation of the HC-CWT for analyzing the GMR effect in HC PC slabs. We will also demonstrate the major differences between the HC-CWT and the previous LC-CWT [29–33], which contains a more detailed derivation and discussion on their common formulation if needed to be referred to.
2.1. High-index-contrast PC slabs vs. Low-index-contrast PC slabs
Figures 1(a) and 1(b) compare the LC PC slabs and the HC PC slabs. The LC PC slabs are usually a multi-layer structure with a permittivity contrast around 1 at interfaces (permittivity of Si, GaxAs1−x, AlxGa1−xAs, etc. are all around 11∼12). In the LC PC slabs, Bloch waves are weakly guided, the electric field decays slowly in the cladding, and the vertical energy distribution is not restricted within the PC layer (the green curve in Fig. 1(a)). Hence, under the weakly guided condition, both the in-plane wave interactions and the vertical radiations in the PC layer are rather mild. Therefore, in the reciprocal space, treating the basic waves (blue arrows in Fig. 1(e)) as sources to excite high-order waves and radiative waves, and the excited waves coupling back to the basic waves is a good approximation, which we refer to as the LC approximation. Besides, also because of the low index contrast, the reflection between interfaces is negligibly small. Thus, a Green’s function for infinite homogeneous materials is able to provide accurate analyses on the band-edge modes and their radiation properties for LC PC slabs [29, 31].
Fig. 1 Schematic cross-sectional view of (a) a low-index-contrast (LC) photonic crystal (PC) multi-layer structure and (b) a high-index-contrast (HC) PC slab. Green curves indicate the profiles of basic Bloch waves. (c) Schematic view of an HC PC slab with circular air holes; (d) Examples of air-hole designs: (i) circular (CC) and (ii) equilateral triangular (ET); (e) Blochwave states represented by wave vectors (arrows) in reciprocal space. Basic waves (blue arrows, denoted by Rx, Sx, Ry, Sy) and high-order waves (green arrows and black arrows) are considered. A simplified model considers eight wave vectors (blue and green arrows) [34]. A large number of high-order waves are included in [29,31]. In HC PC slabs, wave coupling among high-order waves (dashed red arrows) should be depicted to ensure accuracy.
On the contrary, high permittivity contrast (For example, in Fig. 1(b), the contrast of Air-Silicon ≃ 12) appears at the interfaces of the HC PC slabs, in which Bloch waves are strongly guided (the green curve in Fig. 1(b)). The high index contrast confines most of the energy within the PC slab, and therefore, results in intense in-plane wave interactions. The high-order wave interactions (e.g. the dashed red arrow in Fig. 1(e)), which are negligible in the LC cases, become a considerable coupling effect due to the intense GMR. Consequently, the non-negligible high-order waves also become important sources of the vertical radiation. Hence, the LC approximation no longer applies for HC PC slabs. In addition, the high index contrast causes considerable reflections on interfaces and renders the homogenous Green’s function not accurate anymore. Therefore, for the analysis of strongly guided HC PC slabs, two critical problems should be addressed: first, the intense high-order coupling should be precisely depicted; second, a Green’s function that includes multi-layer reflections should be provided.
2.2. Structure description
In this paper we analyze an HC PC slab shown in Figs. 1(b) and 1(c). It is a single-layer structure: a PC slab surrounded by air. The thickness of the PC slab is 220 nm. The lattice constant a = 410 nm is designed to match the wavelength of light in the waveguide such that the photonic lattices serve to provide a 2D distributed feedback effect. Then, the reciprocal base vector is given by β0 = 2π/a. The average permittivity of the PC layer is given by εPC = fεa + (1 − f)εb, where εa = 1 is the permittivity of air, εb = 11.97 is the permittivity of the photonic crystal (Si), and f is the filling factor given by f = Sair−hole/a2 (i.e., the fraction of the area of a unit cell occupied by air holes). Let the PC lies in a square lattice with arbitrarily shaped holes. Figure 1(d) depicts examples of air-hole shapes that are considered later in this paper: (i) circles and (ii) equilateral triangles. For the PC slab where the side length is sufficiently large (exceeding 300a or 100 μm for practical devices), the structure can be reasonably regarded as infinite in the XY plane and the in-plane loss can be neglected [35].
2.3. High-order wave coupling
For the transverse-electric (TE) like mode, the electric fields are assumed as (Ex, Ey, 0). Thus, using the Bloch theorem and combining the master equation of electric components with the Fourier expansion of material permittivity, we obtain the GMR coupling equations in the reciprocal space [29, 31]:
Here n0 is the refractive index, k0 = ω/c (ω is the angular frequency and c is the light velocity in vacuum), mx and ny are the wave orders in the xy plane, and ξmn is the component of the Fourier expansion of permittivity with an index of (m,n). The above wave equations include all possible waves whose amplitudes are explicitly dependent on the vertical position z. As we defined previously [29], the waves are classified into three groups according to their in-plane wavenumbers, , as follows: basic waves (|m2 + n2| = 1), high-order waves (|m2 + n2| > 1), and radiative wave (|m2 + n2| = 0).In the GMR coupling equations Eqs. (1) and (2), the coupling intensity of respective waves (denoted by (m,n)) is determined by two factors: the intensity of source waves (denoted by (m′, n′)) and the Fourier component of the permittivity: ξm−m′,n−n′, which defines the coupling route of corresponding waves. Usually, the nearer the Bloch waves in the reciprocal space, the larger ξm−m′,n−n′ is, and vice versa. For HC PC slabs with a considerable filling factor, the intense energy confinement renders high-order waves significant and ξm−m′,n−n′ can be still large even if Bloch waves are far away from each other. So, the decisive factors of the high-order coupling, which are negligible in LC PC slabs, become significant in HC PC slabs, making the high-order coupling an important contribution in the whole wave interaction mechanism. That is the reason why the LC approximation no longer applies to the analysis of GMR in HC PC slabs.
To strictly solve Eqs. (1) and (2), the main difficulty is that all the waves are both sources and fields in the coupling effect. Consequently, we cannot obtain their accurate value instantly. In contrast, we need a self-consistent iteration procedure to find a convergent route of all the wave profiles (an early report [36] contains an iteration loop for the 1D DFB laser.). To start the iteration, an appropriate initialization is very important for a quick convergence. We adopt the LC approximation for initialization because the basic waves are still important sources (although not dominant any more) to excite all orders of waves in the HC case. In the initialization, the accurate coupling equations Eqs. (1) and (2) degenerate into simpler and soluble ones [29, 31]. However, the LC approximation is no longer self-consistent in the HC case because of the high energy concentration and intense wave couplings in the PC slab. Hence, after the initialization of radiative waves and high-order waves, we should trace back to the accurate equations to include the high-order wave coupling effect.
Figure 2 shows the flowchart of our iteration procedure, which is used to determine the value of radiative and high-order waves. First, the profiles of the four basic waves are obtained under the resonant condition using the TMM method [37]. Then, the values of radiative and high-order waves are initialized using the LC approximation. We initialize the coupling matrix of the structure as well. After that, we enter the main iteration process, where we use the initialized all orders of waves to excite themselves using the accurate coupling equation (Eqs. (1) and (2)). Thus, the renewed coupling matrix can also be calculated and is compared with the previous version of the coupling matrix. If the coupling matrix becomes convergent (for example, the relative error of the two versions of coupling matrix less than 0.001), the iteration stops and provides the complex eigenmodes and radiation properties of the structure. If not, the iteration continues using the renewed all orders of waves until the relative error is small enough.
2.4. Green’s function for vertical structure with considerable reflection
In our previous study, we found that using an approximation of Green’s function G(z,z′) = −i/2β · e−iβ|z−z′| for an infinite homogenous space gives accurate results for the analysis of GMR in LC PC slabs [29, 31]. Due to the low-index contrast between layers, the reflection at interfaces can be neglected in the LC case. However, in the HC case, as we have demonstrated in section 2.1, the reflection becomes non-negligible when Bloch waves are strongly guided.
Under the strongly guided condition, a reflection-included Green’s function for the radiative waves should be adopted [33]:
where hPC is the thickness of the PC slab, and Δ is the normalized deviation from the Γ point in the x or y direction [31]. In the LC case, κ can be very small. Using the Silicon-GaAs interface for example, the corresponding amplitude of κ at Γ point is about 0.012, which leads to a 0.014% reflection. On the contrary, κ becomes much larger in the HC case. At the air-silicon interface, the corresponding amplitude of κ is about 0.5, which leads to a considerable 20% reflection. This validates the necessity of adopting a strict reflection-included Green’s function for HC PC slabs.2.5. Coupled wave equations
Since the high-order coupling is accurately depicted using the self-consistent iteration process and the reflection-included Green’s function is derived, we solve the two critical problems in the GMR analysis of the strongly guided HC PC slabs. Now, all orders of waves have reached convergence after the analytical iteration and they only depend on the PC geometry and the multilayer waveguide structure. Thus the coupling matrix C also becomes self-consistent and the eigenvalue equation is given by [31]:
where V = (Rx, Sx, Ry, Sy)T. The complex frequencies ω can be obtained from the eigenvalues by solving Eq. (4) as ω = ck. The Q factors of the band-edge modes can be determined from the real and imaginary parts of ω, as Re(ω)/|2Im(ω)|, without using any ambiguous definition of the effective refractive index. The radiation constant αr can be obtained from the Q factor [3, 29, 31]: αr = β0/Q = (2π/a)/Q.3. Results and discussion
In this section, we solve the coupled-wave Eq. (4) as an eigenvalue problem, and directly evaluate the two most important GMR properties of the band-edge modes, i.e., the mode frequency ω and the radiation constant αr: the modal power loss due to the surface emission. A truncation of the summation terms with an appropriate order of m and n is required to obtain numerical results. Here, we define a quantity D such that |m, n| ≤ D. As discussed in our previous study [29], LC-CWT converges well when D is larger than 10. We also find that D = 10 is a large enough truncation number for HC-CWT. Hence, we use D = 10 in all the following calculations.
In order to confirm the accuracy of the HC-CWT analysis, we also performed the LC-CWT analysis and 3D-FDTD simulations [27, 38] for the structure shown in Fig. 1(b). The FDTD simulations were accomplished on our super-computer system with modest grid-size. We used a computational cell of 64 × 64 × 1024 pixels (x × y × z), corresponding to 1 × 1 × 16 lattice periods, with absorbing boundary layers in the z direction and periodic boundary conditions in the x and y directions. The Q factors and radiation constants were obtained from the numerical simulation, and compared directly with CWT analysis results.
The processing speed of HC-CWT is an obvious advantage. Since HC-CWT is a semi-analytical algorithm, the analysis requires a much shorter calculation time than the 3D-FDTD simulation. To depict a detailed band structure, the FDTD simulation takes more than one day using a parallelized super-computer system, whereas the HC-CWT analysis takes several minutes with a single-core personal computer. The proposed CWT is superior to the numerical simulation in terms of computation time and resource consumption.
3.1. In-plane and vertical field profiles
After obtaining the eigenmodes, we are able to show the in-plane field distribution by combining all the Bloch waves, which is shown in Fig. 3. The band-edge modes are characterized by different field patterns distributed in two dimensions. Modes C and D correspond to the symmetric mode (leaky mode) of 1D-DFB lasers, and thus have significant loss. In contrast, modes A and B correspond to the antisymmetric mode (nonleaky mode) of 1D-DFB lasers, and lase more easily than modes C and D.
Fig. 3 In-plane E-field vector distribution (arrows) and Hz patterns (in color) of the four band-edge modes (A–D). The thick black circles indicate the shapes and locations of the air holes.
In addition to the in-plane fields, our 3D formulation also allows us to calculate the field profile in the vertical direction for each wave vector. In Fig. 4, vertical field profiles before and after the convergent iteration are both plotted for comparison. The FDTD profiles, obtained from the vertical distributions of the electromagnetic field’s Fourier components, are also plotted.
Fig. 4 Vertical field profile comparison of wave vectors with different in-plane wave index: (a) basic, (m,n)=(1,0) (b) radiative, (m,n)=(0,0) and high-order waves with (c) (m,n) = (2,1) and (d) (m,n) = (4,3). The colored regions indicate the PC layer. The vertical field profile of basic waves is the same as that of the fundamental waveguide mode. The FDTD profiles are obtained by solving (Ex, Ey, Hz) in whole space considering the vertical multilayer structure and the horizontal permittivity modulation. The vertical field profiles of radiative and high-order waves are plotted for mode B (similar features are observed for other modes) and are calculated based on Eqs. (1) and (2) (only Ey field is plotted and field outside the PC layer is calculated by imposing continuity conditions on E field at the layer interfaces).
Here, we show the results for circular air-hole shapes with a filling factor of 0.12 (similar results are obtained for other asymmetric shapes and higher filling factors). The basic waves (Fig. 4(a)) calculated by CWT agrees well with the numerical result. The amplitude of basic waves reaches maximum at the center of the PC layer and decays symmetrically upward and downward. Due to the large index contrast at the air-silicon interface, the structure more resembles a planar waveguide than other multilayer structures (e.g. Fig. 1(a)), which leads to the high accuracy of the TMM method. Also, the amplitude of basic waves is still much larger than that of the high-order and radiative waves. Thus, we can treat the basic waves as accurate sources for initialization. The radiative wave (Fig. 4(b)) possesses an oscillating field and emanates in the vertical direction. Although the normalized profile of the radiative wave is the same for LC-CWT and HC-CWT, their calculated radiation constants (reflecting absolute radiation properties) are quite different, which will be discussed later.
The field profiles of the high-order waves are more complicated. As has been stated above, the high-order coupling becomes non-negligible for analyzing GMR in HC PC slabs. For instance, we show the profiles of high-order waves with (m,n) = (2,1) and (m,n) = (4,3) in Figs. 4(c) and 4(d), respectively. Similar to basic waves, high-order waves decay evanescently outside the PC layer, but they are more strongly confined within the PC layer: the higher the index (m,n), the more strongly the field is confined in the PC layer. More importantly, the huge difference between the LC-CWT result and the numerical result in the PC layer shows that the LC approximation is no longer accurate in this HC case. The high-order coupling becomes a significant contribution in the absolute value, as well as the shape of the high-order wave profiles. After taking the high-order coupling into account using the iteration process, we obtain the HC-CWT profiles, which agree well with the FDTD results inside and outside the PC layer.
In sum, with regard to the depiction of field profiles in a 3D HC structure, each field profile must be treated individually to accurately quantify their coupling effects. Besides, the high-order wave coupling must be incorporated to correctly reflect the coupling mechanism in the structure. In previous CWT analyses [34, 39], an approximation based on the effective refractive index was used in the 2D calculations to compensate for effects of the 3D nature. This approximation assumes that all the individual wave vectors have the same profile in the vertical direction as the basic waveguide mode. Failing to reflect these complicated field profile changes induces significant inaccuracy. In the 3D LC-CWT analyses [29, 31, 33], field profiles are precisely depicted when permittivity varies slowly in the vertical direction. However, when analyzing GMR in structures with large index-contrast interfaces, the LC approximation no longer applies, and also induces considerable error.
3.2. Frequency and radiation analysis
In order to understand the effects of asymmetric air-hole shapes on GMR, we present numerical results for two air-hole shapes shown in Fig. 1(d): circular (CC) and equilateral triangular (ET). In the LC PC slabs, it is usually very difficult to evaluate all the band-edge modes using the FDTD method because of the overlap of the TE-like and TM-like modes. On the contrary, in the HC PC slabs, the intense energy concentration widens the bandgap and thus reduces the TE-TM overlap. Hence, we are able to assess the accuracy of the HC-CWT using the FDTD method with all the four band-edge modes.
Figures 5 and 6 show the mode frequencies and radiation constants as a function of filling factor for circular air holes, obtained by the LC-CWT, HC-CWT and 3D-FDTD methods. All the three methods agree well except the mode frequency of mode A. With large filling factors, mode A’s frequency obtained by LC-CWT deviates from numerical results. In contrast, HC-CWT correctly provides mode A’s frequency after including the high-order coupling. For other modes, the acceptable accuracy of LC-CWT is an artificial phenomenon induced by the destructive interference in the perfectly symmetric circular slab [29]. The high-order coupling, although indispensable for analyzing GMR in this HC structure (Fig. 4), still cancels each other in the calculation of the radiation properties for circular air holes. However, when replacing circular air holes with equilateral triangular ones, the symmetry is broken and LC-CWT demonstrates its inaccuracy. Corresponding results are shown in Figs. 7 and 8, in which HC-CWT still agrees quite well with the numerical results.
Figures 5 and 8 also show that the filling factor is an important parameter that relates to the amplitude of high-order coupling. With small filling factors, the in-plane variation of the permittivity is also low, which results in rather small high-order permittivity components in the Fourier expansion (i.e. ξmn is very small for large (m, n)). A small ξmn for high-order (m, n) restricts the amplitude of high-order coupling (Eq. (1) and (2)). Thus, the LC approximation applies again to small filling factors. Nevertheless, with large filling factors, high-order coupling is no longer negligible. The necessity of including high-order coupling is validated by the significant errors of LC-CWT in Figs. 7 and 8.
3.3. Band structure
Treating the non-Γ points as perturbations to all the wave vectors, we are able to depict the band structure of a given multi-layer structure [31]. Therefore, the impact of high-order coupling can also be evaluated from the comparisons of LC-CWT, HC-CWT, and FDTD.
Figures 9 and 10 show the band structure of the HC PC slabs with circular air holes and a filling factor of 0.25. For an infinite periodic structure with circular air holes, the radiation constants of both mode A and B are zero at Γ point because the perfect symmetry leads to complete destructive interference [2, 29]. Thus, the accuracy of LC-CWT is generally acceptable at Γ point. Again, at non-Γ points where symmetry no longer exists, LC-CWT becomes inaccurate. On the contrary, HC-CWT maintains high accuracy in the whole band structure by correctly including the important high-order wave coupling effect.
When the air hole shape is equilateral triangular, the band structure becomes more complicated, as is shown in Figs. 11 and 12 (the filling factor is still 0.25). As expected, LC-CWT is inaccurate in the whole band structure because of the double asymmetry induced by the air hole shape and the wave vector deviation. On the other hand, the accuracy of HC-CWT is much higher than that of LC-CWT but there still exists some divergence between the results of FDTD and HC-CWT, especially when the deviation from Γ point becomes large. In the following we explain the divergence. There is a mutual assumption in LC-CWT and HC-CWT: the vertical profiles of the basic waves are identical, which is strict at Γ point. However, at non-Γ points, the larger the deviation from the Γ point, the more different the basic wave profiles are, and thus, the less accurate the assumption is. So, high accuracy can be achieved by HC-CWT in the vicinity of Γ point but we can only provide qualitative results for large wave vector deviations.
It is noteworthy that HC-CWT is not restricted for HC PC slabs. It can also provide accurate calculations for the mode frequencies and the band structure of LC PC slabs with various air hole shapes, tilted angles, and lattice angles. The main difference is that the calculation of the LC PC slabs has faster convergence while that of the HC PC slabs needs more iteration loops.
4. Conclusion
In this study, we propose a semi-analytical approach, the HC-CWT, for analyzing the GMR effect in high-index-contrast photonic crystal slabs. The intense high-order coupling effect is accurately depicted using a convergent recursive procedure: high-order waves are initialized by adopting the LC approximation and all orders of waves are combined to construct the strict self-consistent coupling equation. The reflection at interfaces is also incorporated by utilizing a strict Green’s function. Important properties of the GMR effect like the mode frequencies and the radiation constants of the band-edge modes are calculated and validated with the FDTD results.
The comparison of the results of HC-CWT, LC-CWT and FDTD clearly demonstrates that the high index contrast indeed has significant influence on the GMR effect: the stronger localization of energy and higher-amplitude near fields. Field vertical profiles are compared before and after the iteration procedure. The inaccuracy of LC-CWT confirms the significant contribution of the high-order coupling effect in HC PC slabs. In addition, band structure of the four band-edge modes are calculated separately using LC-CWT and HC-CWT with two different air hole shapes. LC-CWT is inaccurate for nearly all the modes, especially with the equilateral triangular air holes that break the structural symmetry. In contrast, HC-CWT ensures its precision and agrees well with the FDTD result for both of the air hole shapes. Besides, aside from the vertical index contrast, the in-plane filling factor is another important parameter that affects the GMR effect in HC PC slabs. As the filling factor is closely related to the high-order component of the in-plane permittivity, it restricts the amplitude of coupling between individual waves, and consequently, significantly influences the mode frequencies and radiation constants.
As a semi-analytical approach, the HC-CWT realizes more precise and efficient quantitative analysis of the GMR effect in HC PC slabs than does numerical methods. The approach also has wide compatibility: LC PC devices can still be precisely analyzed using the approach with very fast convergence. The insights obtained in this work are essential for understanding the performances and physics of various HC PC devices.
Acknowledgments
This work was supported by the National Key Basic Research Program of China (973 Program of China) 2013CB329205 and 2010CB328203, the National Natural Science Foundation of China (NSFC) under grant No. 61071084, and State Key Laboratory of Advanced Optical Communication Systems & Networks, China.
References and links
1. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]
2. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002). [CrossRef]
3. D. Rosenblatt, A. Sharon, and A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997). [CrossRef]
4. M. Kanskar, P. Paddon, V. Pacradouni, R. Morin, A. Busch, J. Young, S. Johnson, J. MacKenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with a two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997). [CrossRef]
5. M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and purcell enhancement from thin-film 2-d photonic crystals,” J. Lightwave Technol. 17, 2096–2112 (1999). [CrossRef]
6. P. Paddon and J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000). [CrossRef]
7. V. Pacradouni, W. J. Mandeville, A. R. Cowan, P. Paddon, J. F. Young, and S. R. Johnson, “Photonic band structure of dielectric membranes periodically textured in two dimensions,” Phys. Rev. B 62, 4204–4207 (2000). [CrossRef]
8. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. Slusher, J. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7–9 (1999). [CrossRef]
9. M. Kim, C. S. Kim, W. W. Bewley, J. R. Lindle, C. L. Canedy, I. Vurgaftman, and J. R. Meyer, “Surface-emitting photonic-crystal distributed-feedback laser for the midinfrared,” Appl. Phys. Lett. 88, 191105 (2006). [CrossRef]
10. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasersbased on two-dimensional photoniccrystal resonators,” Opt. Express 16, 5206–5217 (2008). [CrossRef] [PubMed]
11. Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, H. E. Beere, D. A. Ritchie, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Electrically pumped photonic-crystal terahertz lasers controlled by boundary conditions,” Nature (London) 457, 174–178 (2009). [CrossRef]
12. T. Lu, S. Chen, L. Lin, T. Kao, C. Kao, P. Yu, H. Kuo, S. Wang, and S. Fan, “GaN-based two-dimensional surface-emitting photonic crystal lasers with AlN/GaN distributed bragg reflector,” Appl. Phys. Lett. 92, 011129 (2008). [CrossRef]
13. H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008). [CrossRef]
14. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001). [CrossRef] [PubMed]
15. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Photonics: Lasers producing tailored beams,” Nature (London) 441, 946 (2006). [CrossRef]
16. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nature Photon. 4, 447–450 (2010). [CrossRef]
17. A. Mekis, A. Dodabalapur, R. E. Slusher, and J. D. Joannopoulos, “Two-dimensional photonic crystal couplers for unidirectional light output,” Opt. Lett. 25, 942–944 (2000). [CrossRef]
18. S. Peng and G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996). [CrossRef]
19. Y. Zhou, M. Huang, C. Chase, V. Karagodsky, M. Moewe, B. Pesala, F. Sedgwick, and C. Chang-Hasnain, “High-index-contrast grating (HCG) and its applications in optoelectronic devices,” IEEE J. Sel. Top. Quantum Electron. 15, 1485–1499 (2009). [CrossRef]
20. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high-index-contrast subwavelength grating,” Nat. Photonics 1, 119–122 (2007). [CrossRef]
21. M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “Single mode high-contrast subwavelength grating vertical cavity surface emitting lasers,” Appl. Phys. Lett. 92, 171108 (2008). [CrossRef]
22. Y. Zhou, M. C. Y. Huang, and C. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high-contrast grating,” IEEE Photon. Technol. Lett. 20, 434–436 (2008). [CrossRef]
23. J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and differentiation of unique high-Q optical resonances near zero wave vector in macroscopic photonic crystal slabs,” Phys. Rev. Lett. 109, 067401 (2012). [CrossRef] [PubMed]
24. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature (London) 499, 188–191 (2013). [CrossRef]
25. N. Ganesh, W. Zhang, P. C. Mathias, E. Chow, J. a. N. T. Soares, V. Malyarchuk, A. D. Smith, and B. T. Cunningham, “Enhanced fluorescence emission from quantum dots on a photonic crystal surface,” Nature Nano. 2, 515–520 (2007). [CrossRef]
26. B. Zhen, S. L. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Enabling enhanced emission and low-threshold lasing of organic molecules using special fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. USA (2013). In press. [CrossRef] [PubMed]
27. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Commun. 181, 687–702 (2010). [CrossRef]
28. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]
29. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave model for square-lattice photonic crystal lasers with transverse electric polarization: A general approach,” Phys. Rev. B 84, 195119 (2011). [CrossRef]
30. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic crystal surface emitting lasers with transverse-electric polarization: finite-size effects,” Opt. Express 20, 15945–15961 (2012). [CrossRef] [PubMed]
31. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of a centered-rectangular lattice photonic crystal laser with a transverse-electric-like mode,” Phys. Rev. B 86, 035108 (2012). [CrossRef]
32. Y. Liang, C. Peng, K. Ishizaki, S. Iwahashi, K. Sakai, Y. Tanaka, K. Kitamura, and S. Noda, “Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization,” Opt. Express 21, 565–580 (2013). [CrossRef] [PubMed]
33. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Coupled-wave analysis for photonic-crystal surface-emitting lasers on air holes with arbitrary sidewalls,” Opt. Express 19, 24672–24686 (2011). [CrossRef] [PubMed]
34. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode,” Appl. Phys. Lett. 89, 021101 (2006). [CrossRef]
35. H. Ryu, M. Notomi, and Y. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B 68, 045209 (2003). [CrossRef]
36. W. Streifer, D. R. Scifres, and R. Burnham, “Analysis of grating-coupled radiation in GaAs:GaAlAs lasers and waveguides - I,” IEEE J. Quantum Electron. 12, 422–428 (1976). [CrossRef]
37. M. J. Bergmann and H. C. Casey, “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84, 1196–1203 (1998). [CrossRef]
38. M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express 13, 2869–2880 (2005). [CrossRef] [PubMed]
39. K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express 16, 6033–6040 (2008). [CrossRef] [PubMed]
