## Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization |

Optics Express, Vol. 21, Issue 1, pp. 565-580 (2013)

http://dx.doi.org/10.1364/OE.21.000565

Acrobat PDF (2471 KB)

### Abstract

Three-dimensional coupled-wave theory is extended to model triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization. A generalized coupled-wave equation is derived to describe the sixfold symmetry of the eigenmodes in a triangular lattice. The extended theory includes the effects of both surface radiation and in-plane losses in a finite-size laser structure. Modal properties of interest including the band structure, radiation constant, threshold gain, field intensity profile, and far-field pattern (FFP) are calculated. The calculated band structure and FFP, as well as the predicted lasing mode, agree well with experimental observations. The effect of air-hole size on mode selection is also studied and confirmed by experiment.

© 2013 OSA

## 1. Introduction

1. 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]

13. S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B **85**, 035304 (2012). [CrossRef]

5. 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]

6. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) **441**, 946 (2006). [CrossRef]

5. 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]

6. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) **441**, 946 (2006). [CrossRef]

2. 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]

6. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) **441**, 946 (2006). [CrossRef]

12. S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express **19**, 11963–11968 (2011). [CrossRef] [PubMed]

13. S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B **85**, 035304 (2012). [CrossRef]

11. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics **4**, 447–450 (2010). [CrossRef]

**441**, 946 (2006). [CrossRef]

2. 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]

11. Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics **4**, 447–450 (2010). [CrossRef]

21. 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]

22. 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]

14. H. Y. Ryu, M. Notomi, and Y. H. 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]

15. 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]

16. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. **43**, 2327–2335 (1972). [CrossRef]

4. I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. **39**, 689–700 (2003). [CrossRef]

17. M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. **28**, 1653–1662, (1992). [CrossRef]

20. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **46**, 788–795 (2010). [CrossRef]

21. 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]

23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express **20**, 15945–15961 (2012). [CrossRef] [PubMed]

24. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B **86**, 035108 (2012). [CrossRef]

13. S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B **85**, 035304 (2012). [CrossRef]

*C*

_{2v}or

*C*

_{4v}symmetry [25

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

*C*

_{6v}symmetry [19

19. 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]

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

21. 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]

24. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B **86**, 035108 (2012). [CrossRef]

## 2. Formulation

### 2.1. Device geometry and photonic-crystal design

**a**and

_{1}**a**, can be expressed as Then, the corresponding reciprocal lattice vector shown in Fig. 2(b) is given by where the primitive reciprocal lattice vectors are

_{2}*m*and

*n*are integers.

*u*(

**r**) can be expressed as where

*a*represents the field amplitude of a wave of the order of (

_{mn}*m*,

*n*) and Δ

**k**= Δ

_{x}β_{0}

*x̂*+ Δ

_{y}β_{0}

*ŷ*represents the wavenumber deviation from the Γ point [24

24. C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B **86**, 035108 (2012). [CrossRef]

**k**≃ 0), only the field amplitudes of the waves with |

**G**

*| =*

_{mn}*β*

_{0}are dominant [3

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

**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

*R*

_{1},

*S*

_{1},

*R*

_{2},

*S*

_{2},

*R*

_{3}, and

*S*

_{3}, as illustrated in Fig. 2(b).

### 2.2. Derivation of 3D coupled-wave equations

**84**, 195119 (2011). [CrossRef]

23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express **20**, 15945–15961 (2012). [CrossRef] [PubMed]

**86**, 035108 (2012). [CrossRef]

**E**(

**r**) is

*e*,

^{iωt}*k*

_{0}(=

*ω/c*) is the free-space wavenumber,

*ω*is the angular frequency,

*c*is the velocity of light in free space, and

*ñ*is the refractive index (a complex number) satisfying

23. Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express **20**, 15945–15961 (2012). [CrossRef] [PubMed]

*n*

_{0}(

*z*) represents the average refractive index of the material at position

*z*, and

*α̃*(

*z*) represents the gain (

*α̃*> 0) or loss (

*α̃*< 0) in each region. For TE polarization, the electric field can be assumed as (

*E*,

_{x}*E*, 0) and expanded as where

_{y}*m*≡

_{x,mn}*m*

_{0x}(

*m*,

*n*) + Δ

*and*

_{x}*n*=

_{y,mn}*n*

_{0y}(

*m*,

*n*) + Δ

*. Similarly, the periodic function of*

_{y}*n*

^{2}(

**r**) can be expanded as where

*ξ*(

_{m,n}*z*) = 0 outside the PC layer, and air holes within the PC region are assumed to have perfectly vertical sidewalls [22

22. 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]

*ξ*are constant within every region.

_{m,n}*e*

^{−imxβ0x−inyβ0y}, we obtain

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*E*(

_{x,m,n}*z*) and

*E*(

_{y,m,n}*z*), i.e., (

*n*−

_{y}E_{x,m,n}*m*) as described in [21

_{x}E_{y,m,n}**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

*, Δ*

_{x}*≪ 1. We assume that all of the six basic waves have an identical vertical field profile Θ*

_{y}_{0}(

*z*) [24

**86**, 035108 (2012). [CrossRef]

*β*(

*β*≃

*β*

_{0}) and the vertical field profile Θ

_{0}(

*z*) can be calculated by employing the transfer matrix method (TMM) [21

**84**, 195119 (2011). [CrossRef]

_{0}is normalized as

*E*and

_{x,m,n}*E*as with where (

_{y,m,n}*m,n*) ∈ {(1, 0), (−1, 0), (0, 1), (0, −1), (1, 1), (−1, −1)} which correspond, in order, to {

*R*

_{1},

*S*

_{1},

*R*

_{2},

*S*

_{2},

*R*

_{3},

*S*

_{3}} for

*A*(

_{m,n}*x*,

*y*), as shown in Fig. 2(b);

*ρ*and

_{mn}*η*denote the perturbation of the basic waves’ polarizations [24

_{mn}**86**, 035108 (2012). [CrossRef]

*δ*=

*β*−

*β*

_{0}=

*n*−

_{eff}ω/c*β*

_{0}is the deviation from the Bragg condition,

*n*is the effective refractive index for the fundamental guided mode,

_{eff}*α*is the modal loss defined by

*ξ*= 0 outside that range. The field profiles [i.e.,

_{m,n}*E*(

_{x,m′,n′}*z*) and

*E*(

_{y,m′,n′}*z*)] of radiative (

*m′*=

*n′*= 0) and high-order (|

*m′*

^{2}+

*n′*

^{2}| > 1) waves can be expressed in terms of basic waves as described in Appendix A.

4. I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. **39**, 689–700 (2003). [CrossRef]

19. 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]

27. M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. **125**, 365–389 (2012). [CrossRef]

**85**, 035304 (2012). [CrossRef]

**86**, 035108 (2012). [CrossRef]

## 3. Numerical results and discussion

*a*= 341 nm, the air-hole shape is circular with air-hole filling factor

*f*= 0.15 (corresponding to an

*r/a*ratio of 0.20, where

*r*is the air-hole radius), and the structural parameters listed in Table 1 are used.

### 3.1. Band structure and radiation constant

*C*, which has a slow-group-velocity region in the Γ–

*J*directions. (We refer to modes in this region as flat-band modes.) Because the radiation constant becomes noticeably small within this region, the flat-band modes may also participate in lasing mode competition. This type of mode competition was reported in Ref. [28

28. K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B **82**, 539–541 (2006). [CrossRef]

*J*direction was observed.

### 3.2. Threshold gain and field intensity envelope

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*A*,

*B*

_{1},

*B*

_{2}

*,*and

*C*), which are most likely to be lasing modes. In the following calculations, we assume an absorbing boundary condition and the radius of the circular-shape computational region

*L*is set to be

*L*= 30

*μ*m (Appendix B).

*δ*+

*iα*)

*L*[23

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*αL*) as a function of deviation from the Bragg condition (

*δL*). Though a large number of modes are found to exist as a result of the numerical calculation, we identify the fundamental modes (indicated by arrows) using the techniques described in Refs. [19

19. 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]

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*W*mode. The normalized threshold gains of the low-threshold modes indicated by arrows in Fig. 4(a) are listed in Table 2. As the lasing action occurs at the mode with the lowest threshold gain (loss), Table 2 indicates that mode

*C*[indicated by red arrow in Fig. 4(a)] is favored for lasing with a large threshold-gain discrimination of over 20 cm

^{−1}against the competing side modes (i.e., modes

*B*

_{1}and

*B*

_{2}). This fact reveals that stable single-mode operation at mode

*C*can be possibly achieved for triangular-lattice PC-SELs.

20. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. **46**, 788–795 (2010). [CrossRef]

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*A*,

*B*

_{1},

*B*

_{2}, and

*C*) is relatively confined to the central part of the laser cavity, whereas mode

*W*exhibits a null intensity in the center. The exact physical interpretation of the origin of mode

*W*is not yet clear. From the character of the doughnut-shape intensity envelope, we suggest that mode

*W*might be a whispering-gallery-like mode that was discussed in Ref. [9

9. 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]

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*C*possesses the smallest amount of energy that escapes from the boundary (i.e., the in-plane losses). In contrast, mode

*A*exhibits a much higher threshold because of the relatively larger in-plane losses. This intuitive explanation is confirmed by a quantitative study of the in-plane losses based on the formula derived in Ref. [23

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*A*are a factor of 2.7 larger than those of mode

*C*.

### 3.3. Comparison of experimental and theoretical results and discussion

#### 3.3.1. Comparison of band structures and FFPs

26. K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. **23**, 1335–1340 (2005). [CrossRef]

*I*

_{th}) at room temperature continuous wave (CW) operation was 25 mA. The measurements were performed in the Γ–

*X*and Γ–

*J*directions under a CW current level of 0.9

*I*

_{th}and in the measurements the sample was attached on a heat sink. The dashed curves in Fig. 5(a) replot the data shown in Fig. 3(a) in order to enable a direct comparison with the experimental results. It can be seen that the measured and calculated band structures are in satisfactory agreement. The slight discrepancy may be mainly attributed to the experimental details, such as the change in refractive index owing to carrier injection and thermal effects. We note that in the measured band structure, several additional bands (e.g., bands existing between modes

*D*

_{1}and

*C*near the Γ point) were also observed. These bands are considered to be TM modes as discussed in Ref. [24

**86**, 035108 (2012). [CrossRef]

4. I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. **39**, 689–700 (2003). [CrossRef]

*A*–

*C*is extremely weak compared to that from modes

*D*

_{1}and

*D*

_{2}. This is due to the difference in their radiative nature, as presented in Fig. 3(b). The lasing spectrum at the Γ point measured at 1.2

*I*

_{th}current level is shown in Fig. 5(b), which evidently suggests that single-mode lasing indeed occurs at band-edge mode

*C*(yellow dashed line). This observation is consistent with the lasing mode predicted above.

**20**, 15945–15961 (2012). [CrossRef] [PubMed]

*C*. A doughnut-shape beam with a divergence angle of around 1° is obtained; this closely matches the measured FFP shown in Fig. 6(b). The polarization profiles shown in the insets indicate that mode

*C*exhibits an azimuthal polarization, which stems from the monopolar nature of its field pattern, as depicted in Fig. 3(c).

#### 3.3.2. Lasing mode control by tuning air-hole size

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

12. S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express **19**, 11963–11968 (2011). [CrossRef] [PubMed]

*A*(i.e., the hexapole mode). Based on our CWT analysis, we find that by tuning some structural parameters, e.g., by varying the air-hole size, mode

*A*may be favored for lasing. As an example, we list in Table 3 the normalized threshold gain of the low-threshold modes

*A*–

*C*for air holes with a larger filling factor of

*f*= 0.26. The data given in Table 3 suggest that mode

*A*is the lasing mode with a threshold-gain discrimination of around 20 cm

^{−1}against the competing modes (i.e., modes

*B*

_{1}and

*B*

_{2}). The calculated FFP and polarization profile of mode

*A*are shown in Fig. 7(a). It is seen that mode

*A*exhibits a doughnut-shape pattern with hexagonal symmetry and possesses a characteristic polarization: The electric-field vector is rotated 4

*π*around the circumference of the doughnut beam (instead of 2

*π*for the monopole mode

*C*). The weak side lobes of the calculated FFP result from the radiation field being terminated abruptly in the numerical calculation.

*f*= 0.26. By measuring the band structure and the lasing spectrum in the same manner as described for Fig. 5, we confirmed that lasing indeed occurs at band-edge mode

*A*. The measured FFP and polarization profile are shown in Fig. 7(b), and they are in good agreement with the calculated FFP shown in Fig. 7(a). The slight asymmetry in the measured FFP (e.g., the weak streak in the central part of the doughnut-shape beam) can be attributed to the nonuniform electrical pumping or the asymmetric geometry of the gain region determined by carrier diffusion in practical experimental situations.

## 4. Conclusions

**86**, 035108 (2012). [CrossRef]

*C*

_{6v}symmetry of the eigenmodes in the vicinity of the second-order Γ point; additionally, both surface radiation and in-plane losses, which are critical for analyzing the finite-size laser structure, were included. By solving the extended coupled-wave equation, modal properties, including the band structure, radiation constant, threshold gain, field intensity profile, and FFP were studied. Comparison of the calculated band structure and FFP of the lasing mode with experimental results showed good agreement. Single-mode lasing at the monopole mode

*C*predicted by our threshold gain analysis was confirmed by experimental data. Furthermore, it was shown that the hexapole mode

*A*can be favored for lasing by using a larger air-hole size; this was also confirmed by the experiments.

**84**, 195119 (2011). [CrossRef]

*C*

_{2v}or

*C*

_{4v}symmetry, thereby unifying the analysis of the modal properties of PC-SELs. Our future work will focus on a more systematic study of the modal properties of PC-SELs by varying structural parameters of PCs, such as the air-hole shape and the lattice angle, to obtain a comprehensive understanding the physics of PC-SELs and exploit their potential applications.

*linear*theory, and it gives reliable solutions only at or near the threshold current. When the current level is far above the threshold, modal properties of the practical laser devices may deviate from predictions based on the linear analysis. We will leave the above-threshold 3D-CWT analysis that incorporates the optical nonlinearities such as hole burning, nonuniform gain distribution, and thermal effects to future work.

## Appendix A: Solutions of radiative and high-order waves for triangular-lattice PCs

*E*and

_{x,m′,n′}*E*in Eq. (15)] can be solved in a manner similar to that described in our previous works [21

_{y,m′,n′}**84**, 195119 (2011). [CrossRef]

**86**, 035108 (2012). [CrossRef]

## Radiative waves

*m*=

*n*= 0) can be expressed as

*G*(

*z,z′*) = −

*i*/2

*β*·

_{z}*e*

^{−iβz|z−z′|}is an approximated Green’s function [21

**84**, 195119 (2011). [CrossRef]

22. 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]

*β*=

_{z}*k*

_{0}

*n*

_{0}(

*z*), and

**V**= (

*R*

_{1},

*S*

_{1},

*R*

_{2},

*S*

_{2},

*R*

_{3},

*S*

_{3})

*.*

^{t}## High-order waves

*m*

^{2}+

*n*

^{2}| > 1) included in the second term of the right-hand side of the coupled-wave Eq. (15) can be expressed as

**C**is a 6 × 6 matrix. The matrix elements of

**C**can be written as where

*m′*,

*n′*| ≤ 10 [21

**84**, 195119 (2011). [CrossRef]

**dk**

_{0,mn}represents the variation of the in-plane wavenumbers resulting from deviation of the individual basic wave vectors from the second-order Γ point, and

**C**

*,*

_{b}**C**

*, and*

_{r}**C**

*correspond to the coupling effects of basic, radiative, and high-order waves, respectively.*

_{h}## Appendix B: Staggered-grid finite-difference method on hexagonal grids

*= Δ*

_{x}*= 0) in the following form:*

_{y}**C**is a 6 × 6 matrix composed of coupling coefficients defined by Eqs. (A7)–(A11).

*j,k*)th hexagonal cell of the grid is illustrated in Fig. 8(a). Then, Eq. (B1) becomes

*h*is the distance between two adjacent grid points and the coupling coefficients

*C*(1 ≤

_{mn}*m,n*≤ 6) are assumed to be constant throughout the laser cavity. Figure 8(b) shows a schematic of the circular-shape computational domain (yellow shaded region) considered in this work. The computational domain physically corresponds to the gain region determined by the electrode geometry. Because the shape of the electrode at the bottom of the fabricated device is circular (see Ref. [13

**85**, 035304 (2012). [CrossRef]

*L*=

*Nh*(

*N*: integer). Schematic depicted in Fig. 8(b) corresponds to the case when

*N*= 2. By multiplying 2

*L*on both sides of Eq. (B2), we obtain

*δ*+

*iα*)

*L*. In this work, we assume the following absorbing boundary condition: The field amplitude of the basic waves always starts from zero at the boundaries. This generalizes a similar concept for the 1D distributed feedback laser structure originally proposed by Kogelnik and Shank [16

16. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. **43**, 2327–2335 (1972). [CrossRef]

*R*

_{3}is indicated by black squares in Fig. 8(b). We investigated the impact of discretization on the solutions of Eq. (B3) by varying

*N*and confirmed that well-converged solutions can be obtained at

*N*= 7 with a short calculation time (∼1 minute with a personal computer).

## Acknowledgments

## References and links

1. | 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. |

2. | 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 |

3. | M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B |

4. | I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron. |

5. | 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 |

6. | E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London) |

7. | H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science |

8. | 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. |

9. | 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) |

10. | L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. |

11. | Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics |

12. | S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express |

13. | S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B |

14. | H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B |

15. | M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express |

16. | H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. |

17. | M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron. |

18. | 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. |

19. | K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express |

20. | K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. |

21. | 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 |

22. | 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 |

23. | Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express |

24. | C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B |

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

26. | K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun. |

27. | M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res. |

28. | K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B |

29. | A. Taflove and S. C. Hagness, |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(160.5298) Materials : Photonic crystals

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: October 15, 2012

Revised Manuscript: December 7, 2012

Manuscript Accepted: December 19, 2012

Published: January 7, 2013

**Citation**

Yong Liang, Chao Peng, Kenji Ishizaki, Seita Iwahashi, Kyosuke Sakai, Yoshinori Tanaka, Kyoko Kitamura, and Susumu Noda, "Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization," Opt. Express **21**, 565-580 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-565

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### References

- 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]
- 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,” Science293, 1123–1125 (2001). [CrossRef] [PubMed]
- M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002). [CrossRef]
- I. Vurgaftman and J. R. Meyer, “Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers,” IEEE J. Quantum Electron.39, 689–700 (2003). [CrossRef]
- 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. Express12, 1562–1568 (2004). [CrossRef] [PubMed]
- E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, “Lasers producing tailored beams,” Nature (London)441, 946 (2006). [CrossRef]
- H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, and S. Noda, “GaN photonic-crystal surface-emitting laser at blue-violet wavelengths,” Science319, 445–447 (2008). [CrossRef]
- 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]
- 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]
- L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev.5, 647–658 (2011).
- Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, and S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics4, 447–450 (2010). [CrossRef]
- S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, “Higher-order vector beams produced by photonic-crystal lasers,” Opt. Express19, 11963–11968 (2011). [CrossRef] [PubMed]
- S. Iwahashi, K. Sakai, Y. Kurosaka, and S. Noda, “Centered-rectangular lattice photonic-crystal surface-emitting lasers,” Phys. Rev. B85, 035304 (2012). [CrossRef]
- H. Y. Ryu, M. Notomi, and Y. H. Lee, “Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,” Phys. Rev. B68, 045209 (2003). [CrossRef]
- M. Yokoyama and S. Noda, “Finite-difference time-domain simulation of two-dimensional photonic crystal surface-emitting laser,” Opt. Express13, 2869–2880 (2005). [CrossRef] [PubMed]
- H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972). [CrossRef]
- M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662, (1992). [CrossRef]
- 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]
- K. Sakai, J. Yue, and S. Noda, “Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization,” Opt. Express16, 6033–6040 (2008). [CrossRef] [PubMed]
- K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron.46, 788–795 (2010). [CrossRef]
- 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. B84, 195119 (2011). [CrossRef]
- 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. Express19, 24672–24686 (2011). [CrossRef] [PubMed]
- Y. Liang, C. Peng, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave analysis for square-lattice photonic-crystal lasers with transverse electric polarization: Finite-size effects,” Opt. Express20, 15945–15961 (2012). [CrossRef] [PubMed]
- C. Peng, Y. Liang, K. Sakai, S. Iwahashi, and S. Noda, “Three-dimensional coupled-wave theory analysis of centered-rectangular lattice photonic crystal with transverse-electric-like mode,” Phys. Rev. B86, 035108 (2012). [CrossRef]
- K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B52, 7982 (1995). [CrossRef]
- K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, “Lasing band-edge identification for a surface-emitting photonic crystal laser,” IEEE J. Sel. Areas Commun.23, 1335–1340 (2005). [CrossRef]
- M. Koba and P. Szczepanski, “The threshold mode structure analysis of the two-dimensional photonic crystal lasers,” Prog. Electromagn. Res.125, 365–389 (2012). [CrossRef]
- K. Forberich, M. Diem, J. Crewett, U. Lemmer, A. Gombert, and K. Busch, “Lasing action in two-dimensional organic photonic crystal lasers with hexagonal symmetry,” Appl. Phys. B82, 539–541 (2006). [CrossRef]
- A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

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