OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 8 — Apr. 18, 2005
  • pp: 2869–2880
« Show journal navigation

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

Mitsuru Yokoyama and Susumu Noda  »View Author Affiliations


Optics Express, Vol. 13, Issue 8, pp. 2869-2880 (2005)
http://dx.doi.org/10.1364/OPEX.13.002869


View Full Text Article

Acrobat PDF (630 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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 A1 is larger than that of other resonant modes; that is, lasing occurs easily in mode A1. 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

In this study, we report characteristics of resonant modes, focusing mainly on the 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

2.2 Symmetry analysis

Figure 1 depicts the magnetic field distribution normal to the PC plane, obtained by the 2D plane wave (PW) expansion method using the effective refractive index[10

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

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

]. The method for determining the effective index follows Ref.[8

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

]. Thick black circles indicate the locations and shapes of lattice points. We can classify the resonant modes originating from the four band edges at the Γ2 point into A1, B1, and E modes. This nomenclature is based on the representation under group theory, which indicates the symmetry of the eigen mode at each band edge when the period of the PC is infinite. The characteristics of each mode are as follows. When we take the 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 σx. 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 C4, C42, and C41, respectively. Thus, the square lattice structure with a circular unit cell structure is invariant against symmetry operations {E, C4, C41, C42, σ x , σ y , σ’ d , σ” d }, where E is the symmetry operation that maintains the structure as it is. The magnetic field (Hz) distribution of the A1 mode is invariant against all symmetry operations, including E, C4, C41, and the others. On the other hand, the Hz distribution of the B1 mode is invariant against symmetry operations E, C42, σ x , and σ y , but not against symmetry operations C4, C41, σ’ 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]

,21

21. K. Sakoda, Optical Properties of Photonic Crystals, (Springer Verlag, Berlin, 2001).

].

Fig. 1. (a), (b), (c), (d) Magnetic field distributions at band edges A1, B1, E, and E, respectively. The amplitudes of the magnetic fields in the direction perpendicular to the plane are indicated by white and black areas, which denote positive and negative regions, respectively. The thick black circles indicate the shapes and locations of lattice points.
Fig. 2. Coordinates and symmetry of the square lattice structure having a circular unit cell structure

2.3 FDTD calculation method

Figure 3 shows the calculation model. As shown in Fig. 3, we divide the 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 A1 mode or the B1 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 A1 and B1 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 A1 and B1 modes by changing the excitation points and excitation frequency between the A1 and B1 modes[11

11. M. Yokoyama and S. Noda, “Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser having a Square-Lattice Slab Structure,” IEICE Trans. Electron. E87-C, 386–392 (2004).

]. As mentioned later, because the Q factor is almost identical for the A1 and B1 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 A1 and B1 modes, it is convenient to completely differentiate between the A1 and B1 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.

Here we introduce two important parameters concerning PC size. One is the number of air rods in the Γ-X direction (N), as shown in Fig. 3. Because we introduce symmetrical boundary conditions, the total number of air rods is equal to 2N×2N. 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 0.6a, 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

22. G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” in Proceedings of IEEE Conference on Electromagn. Compat. EMC-23 (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 377–382.

]. 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 xyz =1/10a, and Δt=0.5Δx/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

11. M. Yokoyama and S. Noda, “Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser having a Square-Lattice Slab Structure,” IEICE Trans. Electron. E87-C, 386–392 (2004).

] 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]

].

Fig. 3. Schematic of 3D-FDTD calculations. The device is composed of an n-cladding layer, a p-cladding layer, and an active layer. The 2D PC is embedded at the cladding layer close to the active layer.

3. Calculation results

Fig. 4. (a) Magnetic field distributions of A1 mode normal to the photonic crystal plane at the center of the active layer. The PC size is 100×100. (b) Wave number space (k-space) patterns of (a). The figure is obtained by Fourier transformation of the magnetic field of the photonic crystal area in Fig 4(a). The white dotted square represents the first Brillouin zone.

Next, in Fig. 5 we show the resonant frequency of each resonant mode as a function of 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 A1, B1, 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 A1, B1, and E. We call this region A. When F is around 30%, the resonant frequency increases in the order of A1, E, and B1. 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, A1, and B1. We call this region C.

Fig. 5. The resonant frequency of each resonant mode versus air filling factor. We can classify air filling factor into three regions depending on the by the relative height the resonant frequency. The horizontal line at a frequency of 0.3116 is the cladding light line, above which light leaks to the cladding layer.

We can obtain the same results by the 2D PW expansion method using the effective refractive index. In Fig. 6, we show the typical band structure of each region around the Γ2 point, as calculated by the 2D PW expansion method. From Fig. 6, we can see changes in not only the relative height of the resonant frequency, but also the band shape. For example, the shape of the band around the band edge A1 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 B1 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.

Fig. 6. (a) (b) (c) Typical band structures of regions A, B, and C around the Γ2 point, respectively. These band structures are calculated by the 2D PW expansion method. The dielectric constants inside the unit cell are 9.26, 9.86, and 10.01, respectively. The dielectric constants outside the unit cell are 10.60, 10.47, and 10.38 respectively. The air filling factors are 8.50%, 30.0%, and 50.0%, respectively.

Next, we explain the relationship between the stability of the guided mode and 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 A1 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, 2π/a) and/or k=(0,-2π/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, 2π/a) and/or k=(0, -2π/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, 2π/a) and/or k=(0, -2π/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.

Fig. 7. (a)(b) The wave number space (k-space) patterns of the electric field (E x ) of the A1 mode, when the air filling factors are 10% and 60%, respectively. 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.

To consider the 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 A1 and B1 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 A1 and B1 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

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

]. The trend of 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]

]. It is also similar to the results derived from Fig. 5 or Fig. 6.

Fig. 8. Q factors as functions of air filling factor for A1, B1, and E modes. Size of the PC is 100×100. The Q factor of the A1 mode is larger than those of the other resonant modes. This result indicates that, among the resonant modes, the A1 mode allows relatively easy lasing.
Fig. 9. (a)(b) Horizontal Q factor (Q) and vertical Q factor (Q ) as functions of the air filling factor for A1, B1, and E modes. Q and Q represent losses of guided mode and out-of-plane radiation, respectively, in the direction normal to the photonic crystal plane.

Fig. 10. Frequency differences between maximum and minimum resonant frequencies for each of the A1, B1, and E resonant modes. The solid line is obtained by the 3D-FDTD calculation and the dotted line is obtained by the 2D PW expansion method using the effective refractive index. The characteristic trend is similar to that for Q.

4. Summary

Next, we investigate the Q factors of the resonant modes. Q calculations indicate that Q total of the A1 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 A1 mode has the highest Q total; that is, among the resonant modes, the A1 mode allows relatively easy lasing. The A1 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 A1 and B1 modes, but that of the E mode is almost entirely determined by Q . The A1 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.

Finally, the existence of dual peaks in the graph of 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.

References and links

1.

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

2.

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-Dimensional Photonic Band-Gap Defect Mode Laser,” Science 284, 1819–1821 (1999). [CrossRef] [PubMed]

3.

S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full Three-Dimensional Photonic Crystals at Near-Infrared Wavelengths,” Science 289, 604–606 (2000). [CrossRef] [PubMed]

4.

S. Noda, M. Imada, and A. Chutinan, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000). [CrossRef] [PubMed]

5.

B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science 300, 1537 (2003). [CrossRef] [PubMed]

6.

S. Noda and T. Baba, Eds., Roadmap on Photonic Crystals, (Kluwer Academic, New York, 2003).

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]

8.

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

11.

M. Yokoyama and S. Noda, “Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser having a Square-Lattice Slab Structure,” IEICE Trans. Electron. E87-C, 386–392 (2004).

12.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, “Laser action from two-dimensional distributed feedback in photonic crystals,” Appl. Phys. Lett. 74, 7–9 (1999). [CrossRef]

13.

K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “A Two-Dimensional Photonic Crystal Laser,” Jpn. J. Appl. Phys. 38, L157–L159 (1999). [CrossRef]

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. 86, 3502–3507 (1999). [CrossRef]

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]

17.

K. S. Yee, “Numerical Solution of Initial Boundary Value Problem Involving Maxwell’s Equations in Isotropic Media,” in Proceedings of IEEE Conference on Antennas and Propagat. AP-14 (Institute of Electrical and Electronics Engineers, New York, 1966), pp. 302–307.

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]

19.

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

20.

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

21.

K. Sakoda, Optical Properties of Photonic Crystals, (Springer Verlag, Berlin, 2001).

22.

G. Mur, “Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,” in Proceedings of IEEE Conference on Electromagn. Compat. EMC-23 (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 377–382.

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]

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]

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]

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]

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


Sort:  Journal  |  Reset  

References

  1. E. Yablonovitch, �??Inhibited Spontaneous Emission in Solid-State Physics and Electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  2. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, �??Two-Dimensional Photonic Band-Gap Defect Mode Laser,�?? Science 284, 1819-1821 (1999). [CrossRef] [PubMed]
  3. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, �??Full Three-Dimensional Photonic Crystals at Near-Infrared Wavelengths,�?? Science 289, 604-606 (2000). [CrossRef] [PubMed]
  4. S. Noda, M. Imada, and A. Chutinan, �??Trapping and emission of photons by a single defect in a photonic bandgap structure,�?? Nature 407, 608-610 (2000). [CrossRef] [PubMed]
  5. B. S. Song, S. Noda and T. Asano, �??Photonic devices based on in-plane hetero photonic crystals,�?? Science 300, 1537 (2003). [CrossRef] [PubMed]
  6. S. Noda and T. Baba, Eds., Roadmap on Photonic Crystals, (Kluwer Academic, New York, 2003).
  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]
  8. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, �??Multidirectionally distributed feedback photonic crystal lasers,�?? Phys. Rev. B 65, 195306 (2002). [CrossRef]
  9. S. Noda, M. Yokoyama, M. Imada, A. Chutinan, and M. Mochizuki, �??Polarization mode control of twodimensional 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]
  11. M. Yokoyama and S. Noda, �??Finite-Difference Time-Domain Simulation of Two-Dimensional Photonic Crystal Surface-Emitting Laser having a Square-Lattice Slab Structure,�?? IEICE Trans. Electron. E87-C, 386-392 (2004).
  12. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos and O. Nalamasu, �??Laser action from two-dimensional distributed feedback in photonic crystals,�?? Appl. Phys. Lett. 74, 7-9 (1999). [CrossRef]
  13. K. Inoue, M. Sasada, J. Kawamata, K. Sakoda and J. W. Haus, �??A Two-Dimensional Photonic Crystal Laser,�?? Jpn. J. Appl. Phys. 38, L157-L159 (1999). [CrossRef]
  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. 86, 3502-3507 (1999). [CrossRef]
  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]
  17. K. S. Yee, �??Numerical Solution of Initial Boundary Value Problem Involving Maxwell�??s Equations in Isotropic Media,�?? in Proceedings of IEEE Conference on Antennas and Propagat. AP-14 (Institute of Electrical and Electronics Engineers, New York, 1966), pp. 302-307.
  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]
  19. M. Plihal, A. Shambrook, and A. A. Maradudin, �??Two-dimensional photonic band structures,�?? Opt. Comm. 80, 199-204 (1991). [CrossRef]
  20. K. Sakoda, �??Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,�?? Phys. Rev. B 52, 7982-7986 (1995). [CrossRef]
  21. K. Sakoda, Optical Properties of Photonic Crystals, (Springer Verlag, Berlin, 2001).
  22. G. Mur, �??Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations,�?? in Proceedings of IEEE Conference on Electromagn. Compat. EMC-23 (Institute of Electrical and Electronics Engineers, New York, 1981), pp. 377-382.
  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]
  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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1562">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1562</a>. [CrossRef] [PubMed]
  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]
  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited