## Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser

Optics Express, Vol. 13, Issue 8, pp. 2869-2880 (2005)

http://dx.doi.org/10.1364/OPEX.13.002869

Acrobat PDF (630 KB)

### Abstract

Using the three-dimensional (3D) finite-difference time-domain (FDTD) method, we have investigated in detail the optical properties of a two-dimensional (2D) photonic crystal (PC) surface-emitting laser having a square-lattice structure. In this study we perform the 3D-FDTD calculation for the structure of an actual fabricated device. The device is based on band-edge resonance, and four band edges are present at the corresponding band edge point. For these band edges, we calculate the quality (*Q*) factor. The results show that the *Q* factor of a resonant mode labeled A_{1} is larger than that of other resonant modes; that is, lasing occurs easily in mode A_{1}. The device can thus achieve single-mode lasing oscillation. To increase the *Q* factor, we also consider the optimization of device parameters. The results provide important guidelines for device fabrication.

© 2004 Optical Society of America

## 1. Introduction

1. E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. **58**, 2059–2062 (1987). [CrossRef] [PubMed]

7. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. **75**, 316–318 (1999). [CrossRef]

16. K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D.M. Tennant, A.M. Sergent, D.L. Sivco, A.Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” App. Phys. Lett. **84**, 4164–4166 (2004). [CrossRef]

7. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. **75**, 316–318 (1999). [CrossRef]

9. 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]

10. M. Yokoyama and S. Noda, “Polarization mode control of two-dimensional photonic crystal laser having a square lattice structure,” IEEE J. Quantum Electron. **39**, 1074–1080 (2003). [CrossRef]

*Q*) factors that occur among the resonant modes originating in these band edge resonances for the case of a 2D square lattice PC slab, formed from a dielectric slab having circular air rods and with air cladding layers above and below. We discerned obvious differences in the mode profiles, far field patterns, and

*Q*factors among these resonant modes. However, the actual device that we fabricated is formed by an active layer sandwiched by p- and n-cladding layers. The 2D PC structure is embedded in the cladding layer near the active layer.

*Q*factor at each band edge of the Γ2 point for a structure similar to that of the actual device. Threshold gain is proportional to

*Q*factor, and high

*Q*factor is advantageous for achieving low-threshold lasing action. We use the 3D finite-difference time-domain (FDTD) method to calculate the mode profile and

*Q*factor. In section 2, we describe the method for calculating the resonant mode. In section 3, we describe in detail the results of resonant mode calculations. We prove that

*Q*factors differ depending on the band edges of the Γ2 point and especially on the unit cell structure size or air filling factor. Therefore, there exists an optimum unit cell structure size for achieving a low threshold current.

## 2. Calculation model and method

### 2.1 Calculation of realistic structure

*Q*factor. The FDTD calculation method is almost the same as that employed in our previous study of a PC slab[11]. However, unlike the PC slab structure discussed in the previous study, the structure discussed in this study leads to some problems in calculating the mode profile and

*Q*factor. One problem is that the effective refractive index difference of the PC of the structure discussed in this study is very small as compared with that of the PC slab. This is because the PC structure exists at the cladding layer, and only the evanescent component, which is leaked from the active layer, is influenced by modulation of the refractive index. To achieve a high

*Q*factor, we must sufficiently increase the period of the PC. Furthermore, for the structure discussed in this study, light confinement at the active layer is very weak as compared with the case of the PC slab, because the refractive index difference between the active layer and the cladding layer is very small. Therefore, in order to attenuate the evanescent component sufficiently, the cladding layer must be of sufficient thickness. As a result, FDTD calculation must be performed for a very large model. However, obtaining the solution for such a large model requires very high computation power, very large memory size, and a very long time. This is the first problem. The second problem is as follows. We consider the 2D square-lattice PC, which has a circular unit cell structure. As mentioned previously, in the case of the square-lattice PC, four band edges are present at the band edge of the Γ2 point. For the structure discussed in this study, resonant frequency differences among these resonant modes are very small, because, as described previously, the effective refractive index difference is very small. Therefore, multiple resonant modes are excited simultaneously, making it difficult to excite only a specific resonant mode. Also, as mentioned later, one of the resonant modes has a

*Q*factor much smaller than those of the other resonant modes. The resonant mode with a small

*Q*factor is especially difficult to excite. This is the second problem. In order to avoid these two difficulties, we utilize symmetry of structure and resonant modes[15

15. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science **302**, 1374–1377 (2003). [CrossRef] [PubMed]

16. K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D.M. Tennant, A.M. Sergent, D.L. Sivco, A.Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” App. Phys. Lett. **84**, 4164–4166 (2004). [CrossRef]

18. M. Okano and S. Noda, “Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,” Phys. Rev. B **70**, 125105 (2004). [CrossRef]

### 2.2 Symmetry analysis

10. M. Yokoyama and S. Noda, “Polarization mode control of two-dimensional photonic crystal laser having a square lattice structure,” IEEE J. Quantum Electron. **39**, 1074–1080 (2003). [CrossRef]

19. M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Comm. **80**, 199–204 (1991). [CrossRef]

8. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B **65**, 195306 (2002). [CrossRef]

*x*and

*y*axes as shown in Fig. 2, the structure is invariant against a symmetry operation in which

*x*is changed to -

*x*; i.e., is symmetrical with respect to the

*y*axis. This symmetry operation is called σ

_{x}. Similarly, the symmetry operations in which

*y*is changed to -

*y*, (

*x*,

*y*) to (

*y*,

*x*), and (

*x*,

*y*) to (-

*y*, -

*x*) are called σ

_{y}, σ’

_{d}, σ”

_{d}, respectively. The structure is also invariant against rotations by 90, 180, and 270 degrees; such symmetry operations are called C

_{4},

_{4},

_{x}, σ

_{y}, σ’

_{d}, σ”

_{d}}, where E is the symmetry operation that maintains the structure as it is. The magnetic field (H

_{z}) distribution of the A

_{1}mode is invariant against all symmetry operations, including E, C

_{4},

_{z}distribution of the B

_{1}mode is invariant against symmetry operations E,

_{x}, and σ

_{y}, but not against symmetry operations C

_{4},

_{d}, σ”

_{d}, which change the sign of Hz. Moreover, even the E mode is relatively complicated. The important point is that one of the E modes is a replica of the other given by 90 degrees rotation; these two modes are essentially the same and cannot be distinguished from each other, and thus are doubly degenerated[20

20. K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B **52**, 7982–7986 (1995). [CrossRef]

### 2.3 FDTD calculation method

*x*-

*y*plane into four parts by symmetrical planes; i.e., the two symmetrical planes A and B. Then we introduce the PMC (Perfect Magnetic Conductor) or PEC (Perfect Electric Conductor) boundary condition to these two planes. By setting the PEC boundary condition for both the A and B planes, we can excite the A

_{1}mode or the B

_{1}mode. By setting the PEC and PMC boundary conditions for the A and B planes, respectively, we can excite one of the doubly degenerated E modes. Conversely, by setting the PMC and PEC boundary conditions for the A and B planes, respectively, we can excite the other E mode. However, these two E modes are equivalent and exhibit no differences. Therefore, it is sufficient to investigate either of these E modes. In order to distinguish the A

_{1}and B

_{1}modes, we must introduce a symmetrical plane at an angle of 45 degrees to the symmetrical planes A and B. However, introducing such a symmetrical plane in FDTD calculation is difficult. Accordingly, we distinguish the A

_{1}and B

_{1}modes by changing the excitation points and excitation frequency between the A

_{1}and B

_{1}modes[11]. As mentioned later, because the

*Q*factor is almost identical for the A

_{1}and B

_{1}modes and is sufficiently high, we can use this method to distinguish these two resonant modes. However, in order to investigate the characteristics of the E mode, which has a

*Q*factor drastically lower than those of the A

_{1}and B

_{1}modes, it is convenient to completely differentiate between the A

_{1}and B

_{1}modes and the E mode. The 2D-PC structure of the experimental device is embedded on either side; i.e., in the n- and p-cladding layers. Such an asymmetrical structure with respect to the

*z*direction generates an asymmetry mode along the

*z*direction and makes analysis difficult. In order to avoid this, we render the calculation model a symmetrical structure with respect to the

*z*direction and introduce the symmetrical boundary condition at the center of the active layer. As a result, the 2D-PC structure exists at both sides of the cladding layer. By adopting this model, we can distinguish a TE-like mode and a TM-like mode. Because the laser oscillation of the experimental device occurs in a TE-like mode, in this study we discuss only the TE-like mode. By introducing these symmetrical boundary conditions, we can reduce the calculation burden to one-eighth and can greatly reduce calculation time. In this study, we use these symmetry characteristics in calculating the resonant frequency, mode profile, and

*Q*factor by 3D-FDTD simulation.

*N*), as shown in Fig. 3. Because we introduce symmetrical boundary conditions, the total number of air rods is equal to 2

*N*×2

*N*. The other is the air filling factor (

*F*), which is defined as the area fraction occupied by the air rod per unit cell. The parameters used in the calculation model are a dielectric constant of 11.5, and an active layer thickness of 0.24 µm (or 0.6

*a*, where

*a*is the lattice constant and is equal to 0.4 µm.). The cladding layer has a dielectric constant of 10.3 and a thickness of 1.44 µm. The thickness of the air layer (

*ε*=1) outside the cladding layer is 1.08 µm. The PC layer is placed 0.04 µm from the active layer, and has a thickness of 0.4 µm. The PC layer contains circular air rods (

*ε*=1). The air rods are arranged in a square region as shown in Fig. 3. Mur’s second-order absorbing boundary condition is employed[22]. To estimate the resonant frequency at the corresponding band edge, the photonic band structure is also calculated by the 3D-FDTD method. For this calculation, Bloch’s and Mur’s second-order absorbing boundary conditions are employed for the horizontal and vertical boundary conditions, respectively. The parameters used in the FDTD calculation are Δ

_{x}=Δ

_{y}=Δ

_{z}=1/10

*a*, and Δ

*t*=0.5Δ

*x*/

*c*, where

*c*is the speed of light in a vacuum. The excitation method and the calculation method of the

*Q*factor described in Ref[11] are employed. To separate the loss of the guided mode in the PC plane and the out-of-plane radiation loss in the direction normal to the PC plane, we decompose the

*Q*factor (

*Q*

_{total}) into a horizontal

*Q*factor (

*Q*) and a vertical

_{‖}*Q*factor (

*Q*

_{⊥})[23

23. O. J. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B **16**, 275–285 (1999). [CrossRef]

## 3. Calculation results

*Q*factors of the resonant modes. First, we show the resonant field distributions in Fig. 4. Figure 4(a) shows the magnetic field distributions normal to the PC plane at the center of the active layer in the A

_{1}mode. The number of air rods in the Γ-X direction (

*N*) is 50, and the air filling factor (

*F*) is 10%. The rectangular region shown by the black dotted line in Fig. 4(a) indicates the PC region. Figure 4(a) shows that the magnetic field at the active layer is subject to strong feedback by the PC embedded at the cladding layer. The top-right inset of Fig. 4(a) shows a magnification of the field distribution near the center of the PC region. The field distribution indicated by the inset is consistent with that calculated by the 2D PW expansion method as shown in Fig. 1(a), indicating that the resonant field distributions are caused by band edge resonance. Figure 4(b) is obtained by the Fourier transform of the magnetic field of the PC area in Fig. 4(a). We can consider Fig. 4(b) as representing the

*k*(wave number)-space pattern of the A

_{1}mode. The white dotted square in Fig. 4(b) represents the first Brillouin zone. The

*k*-space pattern of the A

_{1}mode, as shown in Fig. 4(b), indicates that this mode is caused by the resonance at the Γ point. This is caused by finiteness of the structure and influence of the window function; the wave number is not fixed securely at the Γ point and spreads over a certain region around the Γ point. These results are in agreement with those obtained in a previous study using the PC slab[11]. We also confirm that other resonant modes such as the B

_{1}and E modes, which are not shown here, are consistent with results obtained by study of PC slab and/or the 2D PW expansion method.

*F*when

*N*=50. The maximum value of

*F*is 70% in Fig. 5, because the filling factor determined by the area of a circle inscribed in a square is 79.6%. Figure 5 shows the height of the resonant among the A

_{1}, B

_{1}, and E modes changes as a function of

*F*. We can classify air filling factor into three regions depending on the relative height of the resonant frequency. When

*F*is less than about 30%, the resonant frequency increases in the order of A

_{1}, B

_{1}, and E. We call this region A. When

*F*is around 30%, the resonant frequency increases in the order of A

_{1}, E, and B

_{1}. We call this region B. In this region, the differences in resonant frequency among the resonant modes are very small. When

*F*is more than about 30%, the resonant frequency increases in the order of E, A

_{1}, and B

_{1}. We call this region C.

_{1}is upwards convex when

*F*is less then 30%; that is, in region A or B, but becomes flatter toward Γ-X when

*F*is more than 30%; that is, region C. Meanwhile, the shape of band around the band edge B

_{1}is flat toward the Γ-X direction in region A, but is downwards convex in regions B and C. An interesting observation is that the boundary points between region A and region B and between region B and region C are triply degenerated.

*F*. A large air filling factor is equivalent to a low equivalent refractive index for the PC layer. In the example discussed in this study, the PC layer exists at the cladding layer on both sides of the active layer and is located very close to the active layer. When the refractive index of the PC layer becomes low, light cannot be stably confined to the active layer and light leaks into the cladding layer. However, because an air layer exists outside the cladding layer, the guided mode is maintained by total reflection between the air and cladding layers and the mode cannot operate as an anti-guide. We explain this phenomenon by reference to Fig. 7, which shows the

*k*-space patterns of the A

_{1}mode when

*F*is 10% or 60%. Considering the propagation of light normal to the PC plane, these

*k*-space patterns are obtained by Fourier transformation of the electric field (E

_{x}) at the center of the active layer, unlike the case in Fig. 4(b). In Fig. 7, the circles indicated by a solid line, a broken line, and a dotted line are the light cones determined by the refractive index of the active layer, the cladding layer, and air, respectively. Figure 7(a) reveals that when

*F*is 10%; that is, when the resonant frequency is 0.304, only the wave number components around

*k*=(0, 0) exist inside the air light cone. These wave number components are the origin of the surface-emitting component. The wave number components around

*k*=(0, 2

*π*/

*a*) and/or

*k*=(0,-2

*π*/

*a*) exist between the active layer light cone and the cladding layer light cone. This means that these wave number components are confined to the active layer by total reflection between the active and cladding layers and that they generate a guided mode. On the other hand, Fig. 7(b) shows that when

*F*is 60%; that is, when the resonant frequency is 0.318, the wave number components around

*k*=(0, 2

*π*/

*a*) and/or

*k*=(0, -2

*π*/

*a*) exist inside the cladding light cone but outside the air light cone. Thus, these wave number components are confined by total reflection between the cladding and air layers. Even in this case, the wave number components existing inside the air light cone are components around

*k*=(0, 0). Therefore, the wave number components around

*k*=(0, 2

*π*/

*a*) and/or

*k*=(0, -2

*π*/

*a*) cannot be emitted to air. However, this causes a decrease in light confinement factor at the active layer and an accompanying drop in

*Q*factor. As a result, the threshold current is expected to increase. The horizontal line at the frequency of 0.3116 in Fig. 5 indicates the threshold above which light leaks to the cladding layer. Thus, the air filling factor greatly influences the stability of the guided mode and the light confinement factor. This phenomenon is caused by the 2D-PC layer at both sides of the cladding layer, with the effective refractive index becoming low. When the PC layer is embedded on either side; i.e., in the n- and p-cladding layers as in our experimental device or in a PC slab composed of three simple layers, this phenomenon does not occur.

*Q*

_{total}) and

*F*for each resonant mode at

*N*=50. In our previous study, we reported that

*Q*

_{total}of the A

_{1}mode achieves over 5000 at a PC size of 50×50, but in the structure discussed in this study

*Q*

_{total}of the A

_{1}mode is about 1400 at a PC size of 100×100 (

*N*=50). Thus, the effect of the PC structure can be considered smaller for this structure than for the PC slab structure. Consequently, to achieve a sufficiently high

*Q*factor to lase, we need a larger PC. In fact, our experimental device has a PC size of 1000×1000. The structure discussed in this study has a PC size of 100×100 at the largest and is considerably smaller than the experimental device, but we still consider the model to be useful for analyzing the

*Q*factor tendency with different resonant modes and filling factors. From Fig. 8, the height of

*Q*

_{total}between the A

_{1}and B

_{1}modes is inverted when

*F*is over 50%, but when

*F*is less than about 50%,

*Q*

_{total}decreases in the order of A

_{1}, B

_{1}, and E. This result shows that among the corresponding resonant modes, the A

_{1}mode allows lasing to occur relatively easily. Meanwhile,

*Q*

_{total}of the E mode is considerably smaller than those of the other resonant modes. These results are in agreement with a previous study of a PC slab. We can obtain the maximum

*Q*

_{toal}at

*F*=10%. To achieve a low threshold current, a device should be fabricated with this filling factor. Recently, we experimentally confirmed that the threshold current decreases at an air filling factor of around 10%[24

24. D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express **12**, 1562–1568 (2004). [CrossRef] [PubMed]

*Q*factor characteristics as a function of

*F*, we decompose

*Q*

_{total}into

*Q*and

_{‖}*Q*

_{⊥}, as shown in Fig. 9, which shows

*Q*and

_{‖}*Q*

_{⊥}for each resonant mode plotted against

*F*. Figure 9 shows that, in the A

_{1}and B

_{1}modes,

*Q*

_{total}is almost entirely determined by

*Q*;

_{‖}*Q*is much smaller than

_{‖}*Q*

_{⊥}. On the other hand, in the E mode,

*Q*

_{total}is almost entirely determined by

*Q*

_{⊥};

*Q*

_{⊥}is much more smaller than

*Q*. The reason is that in a PC of infinite size, the A

_{‖}_{1}and B

_{1}modes do not couple to free space normal to the PC plane, because of a symmetry mismatch, but the E mode does couple to free space normal to the PC plane[25

25. T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B **63**, 125107 (2001). [CrossRef]

26. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B **65**, 235112 (2002). [CrossRef]

*Q*with

_{‖}*F*is notable. For all modes,

*Q*exhibits local maximums at around

_{‖}*F*=10% and

*F*=45%, and a minimum at around

*F*=30%. As a result, the graph of

*Q*against

_{‖}*F*shows dual peaks. This characteristic is similar to that of the coupling coefficient

*κ*of a 2nd order 1D DFB laser, which shows dual peaks with a node at

*F*=50%[27

27. M. Imada, S. Noda, H. Kobayashi, and G. Sasaki, “Characterization of a Distributed Feedback Laser with Air/Semiconductor Gratings Embedded by the Wafer Fusion Technique,” IEEE J. Quantum Electron. **35**, 1277–1283 (1999). [CrossRef]

*F*. The solid line is obtained by the 3D-FDTD calculation shown in Fig. 5 and the dotted line is obtained by the 2D PW expansion method using the effective refractive index shown in Fig. 6. The absolute values are not consistent between the 3D-FDTD and 2D PW expansions, but the tendencies are consistent. An important point is that the trends observed in Fig. 10 are similar to those for

*Q*. If we consider this frequency difference to be the stopband width, light receives a large feedback when the frequency difference is large; that is,

_{‖}*Q*becomes large. This result indicates that we can easily estimate the

_{‖}*Q*characteristics from the resonant frequency difference. As shown in Fig. 6, we can estimate the resonant frequency difference from the 2D PW expansion method using the effective refractive index. The 3D-FDTD calculation, which requires a very long time and very high computation power, is not required. Even more convenient is that the 2D-FDTD calculation is also unnecessary, and we can estimate the

_{‖}*Q*characteristics simply by using the 2D PW expansion method. As mentioned before,

_{‖}*Q*

_{total}is almost entirely determined by

*Q*for the A

_{‖}_{1}mode; therefore, we can regard the characteristics of

*Q*to be those of

_{‖}*Q*

_{total}. Because

*Q*

_{total}of the A

_{1}mode is larger than that of other modes, the A

_{1}mode allows relatively easy lasing. As a result, we can obtain the optimum filling factor of the device by this estimation. However, it is also important to consider application of this estimation method. For example, in the case of a PC slab which has a strong refractive index modulation, estimating the effective refractive index and effective index difference is very difficult, leading to difficulty in determining the resonant frequency by the 2D PW expansion method. In summary, a significant point to consider is whether it is reasonable to perform the 2D PW expansion method using the effective refractive index. We consider this estimation to be valid, as long as it is possible to replace the 3D model with the 2D model. This may prove very difficult and the results of future investigations in this area will be reported elsewhere.

## 4. Summary

*Q*factors of the resonant modes.

*Q*calculations indicate that

*Q*

_{total}of the A

_{1}mode is about 1400 at a PC size of 100×100 (

*N*=50). Compared with the result for the PC slab,

*Q*

_{total}obtained by this study is considerably smaller, indicating that, in order to achieve a sufficiently high

*Q*factor to lase, our PC must be larger than the PC slab. Among the resonant modes, the A

_{1}mode has the highest

*Q*

_{total}; that is, among the resonant modes, the A

_{1}mode allows relatively easy lasing. The A

_{1}mode is shown to attain a maximum

*Q*

_{total}when the air filling factor is 10%. Results of our recent experiments indicate that the threshold current decreases at an air filling factor of around 10%. The characteristics of

*Q*

_{total}are clarified by decomposing

*Q*

_{total}into

*Q*and

_{‖}*Q*

_{⊥}. This decomposition shows that

*Q*

_{total}is almost entirely determined by

*Q*for the A

_{‖}_{1}and B

_{1}modes, but that of the E mode is almost entirely determined by

*Q*

_{⊥}. The A

_{1}and B1 modes do not easily couple to free space normal to the PC plane because of the symmetry mismatch, but the E mode can couple to free space normal to the PC plane.

*Q*against F is considered for all resonant modes. This result is similar to that for the resonant frequency difference between the maximum and minimum values of frequency at each resonant mode or bandgap width at the Γ2 point. Thus, the band gap width at the Γ2 point can be regarded to be the stopband width. The influence of bandgap width at the Γ2 point on

_{‖}*Q*factor may be estimated by the 2D PW expansion method, rather than using the complex 3D-FDTD calculation, allowing the estimation to be performed simply and conveniently.

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14. | M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Mekis, A. Timko, C. A. Murray, R. Ruel, and O. Nalamasu, “Emission characteristics of two-dimensional organic photonic crystal lasers fabricated by replica molding,” J. Appl. Phys. |

15. | R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. Gmachl, D. M. Tennant, A. M. Sergent, D. L. Sivco, A. Y. Cho, and F. Capasso, “Quantum cascade surface-emitting photonic crystal laser,” Science |

16. | K. Srinivasan, O. Painter, R. Colombelli, C. Gmachl, D.M. Tennant, A.M. Sergent, D.L. Sivco, A.Y. Cho, M. Troccoli, and F. Capasso, “Lasing mode pattern of a quantum cascade photonic crystal surface-emitting microcavity laser,” App. Phys. Lett. |

17. | K. S. Yee, “Numerical Solution of Initial Boundary Value Problem Involving Maxwell’s Equations in Isotropic Media,” in |

18. | M. Okano and S. Noda, “Analysis of multimode point-defect cavities in three-dimensional photonic crystals using group theory in frequency and time domains,” Phys. Rev. B |

19. | M. Plihal, A. Shambrook, and A. A. Maradudin, “Two-dimensional photonic band structures,” Opt. Comm. |

20. | K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B |

21. | K. Sakoda, |

22. | G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” in |

23. | O. J. Painter, J. Vuckovic, and A. Scherer, “Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab,” J. Opt. Soc. Am. B |

24. | D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express |

25. | T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B |

26. | S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B |

27. | M. Imada, S. Noda, H. Kobayashi, and G. Sasaki, “Characterization of a Distributed Feedback Laser with Air/Semiconductor Gratings Embedded by the Wafer Fusion Technique,” IEEE J. Quantum Electron. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(250.7270) Optoelectronics : Vertical emitting lasers

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 4, 2004

Revised Manuscript: March 27, 2005

Published: April 18, 2005

**Citation**

Mitsuru Yokoyama and Susumu Noda, "Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser," Opt. Express **13**, 2869-2880 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-2869

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### References

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