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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 1 — Jan. 14, 2013
  • pp: 565–580
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Three-dimensional coupled-wave analysis for triangular-lattice photonic-crystal surface-emitting lasers with transverse-electric polarization

Yong Liang, Chao Peng, Kenji Ishizaki, Seita Iwahashi, Kyosuke Sakai, Yoshinori Tanaka, Kyoko Kitamura, and Susumu Noda  »View Author Affiliations


Optics Express, Vol. 21, Issue 1, pp. 565-580 (2013)
http://dx.doi.org/10.1364/OE.21.000565


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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

Two-dimensional (2D) photonic-crystal surface-emitting lasers (PC-SELs) are becoming increasingly important owing to their promising functionality and improved performance compared to conventional semiconductor lasers. A number of successful demonstrations underpinning this promise have already been conducted [1

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

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]

]. By utilizing the band edge of the photonic band structure, single longitudinal and transverse mode oscillation in two dimensions has been achieved with a large lasing area, which enables high-power, single-mode operation [5

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

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

]. The output beam of such devices is generally emitted in the direction normal to the 2D PC plane and has a small beam divergence angle of less than 1° owing to the large area of coherent oscillation [5

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

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

]. Furthermore, both the polarization and beam pattern [2

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

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

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

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]

], as well as the beam direction [11

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]

], of the laser output can be controlled by appropriate design of the PC geometry. In addition, the lasing wavelengths of PC-SELs have spanned a wide range from near-infrared to blue-violet, mid-infrared, and terahertz regimes [6

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

10

10. L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

].

To further improve the device performance or exploit new functionalities of PC-SELs, it is important to develop an accurate first-principles model to describe the lasing properties in these structures. Realistic devices of PC-SELs require three-dimensional (3D) analysis, because both the surface emission in the vertical direction and the 2D optical feedback in the PC plane need to be taken into account. In addition, due to two important features of the PC laser cavities – a large area that is typically above 200 periods for one side length and increasingly complicated air-hole shape designs [2

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

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

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

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]

] – it is difficult to model this type of laser using brute-force computer simulation methods such as the finite-difference time-domain (FDTD) [14

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

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]

] or finite-element [10

10. L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

] methods.

The remainder of this paper is organized as follows. Section 2 describes the PC geometry and derivations of the coupled-wave equations. Section 3 presents numerical results, their comparison with experimental results, and discussions. Section 4 concludes this work by summarizing our findings.

2. Formulation

2.1. Device geometry and photonic-crystal design

Fig. 1 (a) Schematic structure of a photonic-crystal surface-emitting laser device with a triangular lattice. (b) Band structure of triangular-lattice PCs calculated by the 2D plane-wave expansion method for transverse-electric (TE) mode [3]. The red circle indicates the second-order Γ point. The inset shows the high-symmetry points at the corners of the irreducible Brillouin zone (shaded light blue).

Table 1. Structural parameters of the PC-SEL device.

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Figure 2(a) shows a schematic of a triangular-lattice PC. The primitive translation vectors of the triangular lattice, a1 and a2, can be expressed as
a1=(3a/2,a/2),a2=(3a/2,a/2).
(1)
Then, the corresponding reciprocal lattice vector shown in Fig. 2(b) is given by
Gmn=mb1+nb2=m+n2β0x^+3(nm)2β0y^m0x(m,n)β0x^+n0y(m,n)β0y^,
(2)
where the primitive reciprocal lattice vectors are b1=(β0/2,3β0/2) and b2=(β0/2,3β0/2), β0=4π/3a, and m and n are integers.

Fig. 2 (a) Schematic of a triangular-lattice PC in real space. The blue arrows denote the primitive translation vectors a1 and a2, and a is the lattice constant. (b) Reciprocal lattice space of a triangular-lattice PC. The colored arrows indicate the six basic waves: R1, S1, R2, S2, R3, and S3 at the second-order Γ point, whose wavenumber is equal to β0=4π/3a.

Light propagating inside a PC must obey Bloch’s theorem, and the Bloch wave state u(r) can be expressed as
u(r)=Gmnamn(z)ei(Δk+Gmn)r,
(3)
where amn represents the field amplitude of a wave of the order of (m, n) and Δk = Δxβ0 + Δyβ0ŷ 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]

]. Specifically, in the vicinity of the second-order Γ point (Δk ≃ 0), only the field amplitudes of the waves with |Gmn| = β0 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]

, 21

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

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]

]. We refer to these waves as basic waves: R1, S1, R2, S2, R3, and S3, as illustrated in Fig. 2(b).

2.2. Derivation of 3D coupled-wave equations

By following Refs. [21

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

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

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]

], we seek solutions to
××E(r)=k02n˜2(r)E(r),
(4)
where the time dependence of the electric field E(r) is eiωt, k0(= 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 k02n˜2(r)k02n2(r)+2ik0n0(z)α˜(z)[23

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]

]; n0(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 (Ex, Ey, 0) and expanded as
Ej(z)=m,nEj,m,n(z)eimx,mnβ0xiny,mnβ0y,j=x,y,
(5)
where mx,mnm0x(m, n) + Δx and ny,mn = n0y(m, n) + Δy. Similarly, the periodic function of n2(r) can be expanded as
n2(r)=n02(z)+m0,n0ξm,n(z)eim0xβ0xin0yβ0y,
(6)
where ξm,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]

]; thus, n02(z) and ξm,n are constant within every region.

By substituting Eqs. (5) and (6) into Eq. (4) and collecting all terms that are multiplied by the factor eimxβ0xinyβ0y, we obtain
[2z2+k02n02(z)+2ik0n0(z)α˜(z)ny2β02]Ex,m,n+mxnyβ02Ey,m,n2inyβ0Ex,m,ny+iβ0(mxEy,m,ny+nyEy,m,nx)=k02mm,nnξmm,nnEx,m,n,
(7)
[2z2+k02n02(z)+2ik0n0(z)α˜(z)mx2β02]Ey,m,n+mxnyβ02Ex,m,n2imxβ0Ey,m,nx+iβ0(mxEx,m,ny+nyEx,m,nx)=k02mm,nnξmm,nnEy,m,n,
(8)
z[Ex,m,nx+Ey,m,nyiβ0(mxEx,m,n+nyEy,m,n)]=0.
(9)
Here, we have neglected the second-order spatial derivatives [23

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]

].

By introducing a proper linear combination of Ex,m,n(z) and Ey,m,n(z), i.e., (nyEx,m,nmxEy,m,n) as described in [21

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

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]

], we obtain
[2z2+k02n02+2ik0n0(z)α˜(z)(mx2+ny2)β02](nyEx,m,nmxEy,m,n)+iβ0[mxnyEy,m,ny(mx2+2ny2)Ex,m,ny]+iβ0[mxnyEx,m,nx+(ny2+2mx2)Ey,m,nx]=k02mm,nnξmm,nn(nyEx,m,nmxEy,m,n).
(10)

3. Numerical results and discussion

In this section, we present numerical results of the coupled-wave equations derived above. By solving the coupled-wave Eq. (15), we can obtain various properties of interest, including the band structure, radiation constant, threshold gain, field intensity envelope profile within the device, and FFPs. To validate our CWT analysis, we also compare some of the calculated results with experimental observations. Unless otherwise noted in the following examples, the lattice constant 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

Fig. 3 Calculated (a) band structure and (b) radiation constant of the eigenmodes in the vicinity of the second-order Γ point for a triangular-lattice PC-SEL with circular air holes. Six modes near the Γ point are referred to as modes: A, B1, B2, C, D1, and D2, in the order of increasing frequency. Modes B1 and B2, as well as D1 and D2, are doubly degenerate at the Γ point. (c) E-field vector distribution (arrows) and H-field patterns (in color) of the individual band-edge modes. The black circles indicate the air holes. Band-edge modes A and C are known as hexapole and monopole modes, respectively [3]. In the calculations, we use the structural parameters listed in Table 1, the lattice constant a = 341 nm, and the air-hole filling factor f = 0.15.

A noteworthy fact is that as the in-plane wavevector is slightly detuned from the band edge, most of the antisymmetric modes start to become significantly more lossy, regardless of the directions, thereby ensuring that band-edge modes keep lasing stably. However, the only exception is mode 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]

], where a multiple-mode lasing action occurring at both the band edge and the flat-band modes in the Γ–J direction was observed.

3.2. Threshold gain and field intensity envelope

In this section, we solve the coupled-wave Eq. (15) in a finite-size system to evaluate the threshold gain of the eigenmodes. We discretize Eq. (15) by using the staggered-grid finite-difference method [23

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]

]. It is important to note here that, for the triangular-lattice PC configuration, it is crucial to both discretize the computational domain on a hexagonal grid [29

29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

] instead of the commonly used square grid and use appropriate boundary conditions (Appendix B). We focus our analysis on the antisymmetric band-edge modes mentioned above (i.e., modes A, B1, B2, 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 μm (Appendix B).

For finite-size structures, threshold gain and mode frequency correspond to the imaginary and real parts of the normalized eigenvalue, (δ + )L[23

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]

]. As an example, we show in Fig. 4(a) a plot of the normalized threshold gain (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]

, 23

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]

]. We note that there exists an additional mode that has a low threshold comparable to that of the fundamental band-edge modes. We refer to this mode as the 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 B1 and B2). This fact reveals that stable single-mode operation at mode C can be possibly achieved for triangular-lattice PC-SELs.

Fig. 4 (a) Normalized threshold gain (αL) as a function of normalized mode frequency deviation (δL). The fundamental band-edge modes (A, B1, B2, and C) and an additional mode W are indicated by arrows. (b) Field intensity envelopes of the modes indicated by arrows in (a). The data are calculated by using the same parameters as specified in the caption of Fig. 3. Note that the field intensity envelopes are plotted on hexagonal grids with a circular-shape computational domain (dashed circle). The radius of the circular domain, L = 30 μm, is discretized to span seven grid cells for which the eigenvalues converge well (see Appendix B for details).

Table 2. Normalized threshold gain (αL) of the low-threshold modes indicated by arrows in Fig. 4(a).

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3.3. Comparison of experimental and theoretical results and discussion

3.3.1. Comparison of band structures and FFPs

Figure 5(a) shows the measured band structure of a fabricated device with the same structural parameters as specified above. The band structure was mapped out around the Γ point by measuring the angle-dependent spontaneous emission spectra well below the lasing threshold [26

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]

]. The threshold current (Ith) 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 0.9Ith 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 D1 and C near the Γ point) were also observed. These bands are considered to be TM modes as discussed in Ref. [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]

]. As the laser structure described in Table 1 is not completely symmetric in the vertical direction, TM mode usually coexists with TE mode and may derive some optical gain from the multiple quantum-well active layer. However, since the optical gain is greater for the TE mode than for the TM mode [4

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]

], the existence of the TM bands rarely becomes an issue for our laser devices. It is also interesting to note that, especially near the Γ point, the intensity of spontaneous emission from the antisymmetric modes AC is extremely weak compared to that from modes D1 and D2. 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 1.2Ith 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.

Fig. 5 (a) Comparison of the measured (in color) and calculated (white dashed curves) band structures. (b) Lasing spectrum measured above the lasing threshold in the direction normal to the PC plane. The threshold current (Ith) at room temperature CW operation was 25 mA. Band structure and lasing spectrum were measured at CW current levels of 0.9Ith and 1.2Ith, respectively. The frequency of the lasing peak in (b) is 0.3434 (a/λ), indicating that the lasing mode is band-edge mode C (yellow dashed line).

Next, we examine the FFP and polarization profile of the lasing mode. (See Ref. [23

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]

] for details of the calculation method for FFP.) Figure 6(a) shows the calculated FFP of mode 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).

Fig. 6 (a) Calculated and (b) measured FFPs of mode C. A scanning microscope image of the fabricated PC with a = 341 nm and f = 0.15 (where the r/a ratio is 0.20 and r is the air-hole radius) is shown in the left inset of (b). Ex (Ey) displayed in the right insets represents the x(y) component of the FFP. Parameters used for the calculations are the same as those shown in the caption of Fig. 4. The yellow arrows in (b) indicate the directions of the measured beam polarization. The beam divergence angle of the FFPs for both cases is around 1°, reflecting the large area of coherent oscillation.

3.3.2. Lasing mode control by tuning air-hole size

Fig. 7 (a) Calculated and (b) measured FFPs of mode A. A scanning microscope image of the fabricated PC with a = 341 nm and f = 0.26 (where r/a = 0.27) is shown in the left inset of (b). Ex (Ey) displayed in the right insets represents the x (y) component of the FFP. Parameters used for the calculations are specified in the caption of Table 3. The yellow arrows in (b) indicate the directions of the measured beam polarization.

Table 3. Normalized threshold gain (αL) of the low-threshold modes AC. Parameters used for the calculations are the same as those shown in Fig. 4 except that a larger air-hole filling factor f = 0.26 is used.

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To further demonstrate the predictive power of our CWT analysis, we fabricated another device with 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

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

Radiative waves

The radiative waves (for which m = n = 0) can be expressed as
(ΔExΔEy)=k02PCG(z,z)Θ0(z)dz(ξ1,0ρ1,0ξ1,0ρ1,0ξ0,1ρ0,1ξ0,1ρ0,1ξ1,1ρ1,1ξ1,1ρ1,1ξ1,0η1,0ξ1,0η1,0ξ0,1η0,1ξ0,1η01ξ1,1η1,1ξ1,1η1,1)V,
(A1)
where G(z,z′) = −i/2βz · ez|zz′| is an approximated Green’s function [21

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

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]

], βz = k0n0(z), and V = (R1, S1, R2, S2, R3, S3)t.

High-order waves

The integrals of high-order waves (for which |m2 + n2| > 1) included in the second term of the right-hand side of the coupled-wave Eq. (15) can be expressed as
(PCEx,m,n(z)Θ0*(z)dzPCEy,m,n(z)Θ0*(z)dz)=1mx2+ny2(nymxmxny)(μm,n(1,0)μm,n(1,0)μm,n(0,1)μm,n(0,1)μm,n(1,1)μm,n(1,1)νm,n(1,0)νm,n(1,0)νm,n(0,1)νm,n(0,1)νm,n(1,1)νm,n(1,1))V(ςx,m,n(1,0)ςx,m,n(1,0)ςx,m,n(0,1)ςx,m,n(0,1)ςx,m,n(1,1)ςx,m,n(1,1)ςy,m,n(1,0)ςy,m,n(1,0)ςy,m,n(0,1)ςy,m,n(0,1)ςy,m,n(1,1)ςy,m,n(1,1))V,
(A2)
where
μmn(rs)=k02PCξmr,ns(nyρrsmxηrs)Gm,n(z,z)Θ0(z)Θ0*(z)dzdz,
(A3)
νmn(rs)=PC1n02ξmr,ns(mxρrs+nyηrs)|Θ0(z)|2dz,
(A4)
Gm,n(z,z)=12βz,m,neβz,m,n|zz|,βz,m,n=(mx2+ny2)β02k02n02(z).
(A5)

As a consequence, the overlap integrals appearing on the right-hand side of Eq. (15) can be replaced by terms only associated with basic waves. Finally, a coupled-wave equation for infinite periodic structures [for which the derivative terms in Eq. (15) are neglected] can be written in the matrix form as
(δ+iα)V=CV,
(A6)
where C is a 6 × 6 matrix. The matrix elements of C can be written as
C=dk0,mn+Cb+Cr+Ch,
(A7)
where
dk0,mn=(dk0;1,0000000dk0;1,0000000dk0;0,1000000dk0;0,1000000dk0;1,1000000dk0;1,1),
(A8)
Cb=(0κ1,0(1,0)κ1,0(0,1)κ1,0(0,1)κ1,0(1,1)κ1,0(1,1)κ1,0(1,0)0κ1,0(0,1)κ1,0(0,1)κ1,0(1,1)κ1,0(1,1)κ0,1(1,0)κ0,1(1,0)0κ0,1(0,1)κ0,1(1,1)κ0,1(1,1)κ0,1(1,0)κ0,1(1,0)κ0,1(0,1)0κ0,1(1,1)κ0,1(1,1)κ1,1(1,0)κ1,1(1,0)κ1,1(0,1)κ1,1(0,1)0κ1,1(1,1)κ1,1(1,0)κ1,1(1,0)κ1,1(0,1)κ1,1(0,1)κ1,1(1,1)0),
(A9)
Cr=(ζ1,0(1,0)ζ1,0(1,0)ζ1,0(0,1)ζ1,0(0,1)ζ1,0(1,1)ζ1,0(1,1)ζ1,0(1,0)ζ1,0(1,0)ζ1,0(0,1)ζ1,0(0,1)ζ1,0(1,1)ζ1,0(1,1)ζ0,1(1,0)ζ0,1(1,0)ζ0,1(0,1)ζ0,1(0,1)ζ0,1(1,1)ζ0,1(1,1)ζ0,1(1,0)ζ0,1(1,0)ζ0,1(0,1)ζ0,1(0,1)ζ0,1(1,1)ζ0,1(1,1)ζ1,1(1,0)ζ1,1(1,0)ζ1,1(0,1)ζ1,1(0,1)ζ1,1(1,1)ζ1,1(1,1)ζ1,1(1,0)ζ1,1(1,0)ζ1,1(0,1)ζ1,1(0,1)ζ1,1(1,1)ζ1,1(1,1)),
(A10)
Ch=(χ1,0(1,0)χ1,0(1,0)χ1,0(0,1)χ1,0(0,1)χ1,0(1,1)χ1,0(1,1)χ1,0(1,0)χ1,0(1,0)χ1,0(0,1)χ1,0(0,1)χ1,0(1,1)χ1,0(1,1)χ0,1(1,0)χ0,1(1,0)χ0,1(0,1)χ0,1(0,1)χ0,1(1,1)χ0,1(1,1)χ0,1(1,0)χ0,1(1,0)χ0,1(0,1)χ0,1(0,1)χ0,1(1,1)χ0,1(1,1)χ1,1(1,0)χ1,1(1,0)χ1,1(0,1)χ1,1(0,1)χ1,1(1,1)χ1,1(1,1)χ1,1(1,0)χ1,1(1,0)χ1,1(0,1)χ1,1(0,1)χ1,1(1,1)χ1,1(1,1)),
(A11)
and
dk0;m,n=(mx(m,n)2+ny(m,n)21)β0,
(A12)
ζmn(rs)=k042β0PCξm,nξr,s(ρmnρrs+ηmnηrs)G(z,z)Θ0(z)Θ0*(z)dzdz,
(A13)
χmn(rs)=k022β0m2+n2>1ξmm,nn(ρmnςx,m,n(rs)+ηmnςy,m,n(rs)).
(A14)
Here, a large number of high-order waves are included by truncating the summation terms in Eq. (A14) at |m′,n′| ≤ 10 [21

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]

]. Note that the matrix dk0,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 Cb, Cr, and Ch correspond to the coupling effects of basic, radiative, and high-order waves, respectively.

Appendix B: Staggered-grid finite-difference method on hexagonal grids

The coupled-wave equation for finite systems can be solved by using a staggered-grid finite-difference method. We rewrite the coupled-wave Eq. (15) for the band-edge modes (for which Δx = Δy = 0) in the following form:
(δ+iα)[R1S1R2S2R3S3]=[C][R1S1R2S2R3S3]+i[12R1x32R1y12S1x+32S1y12R2x+32R2y12S2x32S2yR3xS3x],
(B1)
where C is a 6 × 6 matrix composed of coupling coefficients defined by Eqs. (A7)(A11).

To model a triangular-lattice PC having a sixfold symmetry, we discretize the computational domain on a hexagonal grid (see, e.g., Sec. 3.7.2 of Ref. [29

29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

]). The location of the six basic-wave components and the coupling coefficients in the vicinity of the (j,k)th hexagonal cell of the grid is illustrated in Fig. 8(a). Then, Eq. (B1) becomes
12(δ+iα)[R1j+1,k+R1j,kS1j+1,k+S1j,kR2j,k+1+R2j,kS2j,k+1+S2j,kR3j+1,k+1+R3j,kS3j+1,k+1+S3j,k]=12[C][R1j+1,k+R1j,kS1j+1,k+S1j,kR2j,k+1+R2j,kS2j,k+1+S2j,kR3j+1,k+1+R3j,kS3j+1,k+1+S3j,k]+i1h[R1j+1,kR1j,kS1j+1,k+S1j,kR2j,k+1R2j,kS2j,k+1+S2j,kR3j+1,k+1R3j,kS3j+1,k+1+S3i,k],
(B2)
where h is the distance between two adjacent grid points and the coupling coefficients Cmn (1 ≤ 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

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]

] for details), we consider a circular computational domain. We define the radius of the circular-shape gain region as L = Nh (N: integer). Schematic depicted in Fig. 8(b) corresponds to the case when N = 2. By multiplying 2L on both sides of Eq. (B2), we obtain
L(δ+iα)[R1j+1,k+R1j,kS1j+1,k+S1j,kR2j,k+1+R2j,kS2j,k+1+S2j,kR3j+1,k+1+R3j,kS3j+1,k+1+S3j,k]=L[C][R1j+1,k+R1j,kS1j+1,k+S1j,kR2j,k+1+R2j,kS2j,k+1+S2j,kR3j+1,k+1+R3j,kS3j+1,k+1+S3j,k]+i2N[R1j+1,kR1j,kS1j+1,k+S1j,kR2j,k+1R2j,kS2j,k+1+S2j,kR3j+1,k+1R3j,kS3j+1,k+1+S3j,k],
(B3)
which can be treated as a generalized eigenvalue problem with spatially dependent eigenvectors (R1j,k,S1j,k,R2j,k,S2j,k,R3j,k,S3j,k,R1j+1,k,S1j+1,k,R2j+1,k,S2j+1,k,R3j+1,k,S3j+1,k,)t and normalized eigenvalues (δ + )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]

]. The boundary consists of the set of points lying immediately outside but closest to one half of the circular domain on the side opposite to the wave’s propagation. As an example, the boundary of R3 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).

Fig. 8 (a) Location of field components (basic waves) and coupling coefficients in the vicinity of the (j,k)th hexagonal cell of the grid. Positions of the unknown field components (colored hollow dots) are staggered from the positions of the known coupling coefficients (solid dots). The colored hollow dots correspond to the points that are updated using the finite-difference scheme, while the solid points are points that are not solved. (b) Schematic of a circular computational domain (yellow shaded region) discretized on the hexagonal grids with L = 2h (L is the radius of the circular shape and h is the distance between two adjacent grid points). The green hollow dots inside the black squares indicate the boundary of R3; these are set to be zero in the calculations.

Acknowledgments

The authors are grateful to Dr. T. Asano, Dr. A. Oskooi, Dr. Y. Kurosaka, T. Nakamura, and T. Okino for their helpful discussions and valuable suggestions. This work was partly supported by the Core Research for Evolution Science and Technology (CREST) and Consortium for Photon Science and Technology (C-PhoST) programs of the Japan Science and Technology Agency. One of the authors (Y. Liang) is supported by research fellowships from the Japan Society for the Promotion of Science (JSPS).

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. 75, 316–318 (1999). [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]

3.

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

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]

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 319, 445–447 (2008). [CrossRef]

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. 88, 191105 (2006). [CrossRef]

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]

10.

L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev. 5, 647–658 (2011).

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]

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]

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]

17.

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

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. 89, 021101 (2006). [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]

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]

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]

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]

25.

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

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]

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]

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]

29.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

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

  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]
  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,” Science293, 1123–1125 (2001). [CrossRef] [PubMed]
  3. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B65, 195306 (2002). [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]
  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. Express12, 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]
  7. 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]
  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.88, 191105 (2006). [CrossRef]
  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]
  10. L. Mahler and A. Tredicucci, “Photonic engineering of surface-emitting terahertz quantum cascade lasers,” Laser Photonics Rev.5, 647–658 (2011).
  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. Photonics4, 447–450 (2010). [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. Express19, 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. B85, 035304 (2012). [CrossRef]
  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. B68, 045209 (2003). [CrossRef]
  15. 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]
  16. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys.43, 2327–2335 (1972). [CrossRef]
  17. M. Toda, “Proposed cross grating single-mode DFB laser,” IEEE J. Quantum Electron.28, 1653–1662, (1992). [CrossRef]
  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.89, 021101 (2006). [CrossRef]
  19. 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]
  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. B84, 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. Express19, 24672–24686 (2011). [CrossRef] [PubMed]
  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. Express20, 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. B86, 035108 (2012). [CrossRef]
  25. K. Sakoda, “Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices,” Phys. Rev. B52, 7982 (1995). [CrossRef]
  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]
  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]
  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. B82, 539–541 (2006). [CrossRef]
  29. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, Norwood, 2005).

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