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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 10 — May. 9, 2011
  • pp: 9475–9491
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Analysis of two-dimensional photonic crystal with anisotropic gain

Shinichi Takigawa and Susumu Noda  »View Author Affiliations


Optics Express, Vol. 19, Issue 10, pp. 9475-9491 (2011)
http://dx.doi.org/10.1364/OE.19.009475


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Abstract

Photonic modes in a two-dimensional square-lattice photonic crystal (PC) with anisotropic gain are analyzed for the first time. A plane-wave expansion method is improved to include the gain, which depends on not only the position but also the propagation direction of each plane wave. The anisotropic gain varies the photonic band structure, the near-field distributions, and the gain dispersion curves through variation in PC symmetry. Low-threshold operation of a PC laser with anisotropic-gain material such as nonpolar InGaN requires that the direction of higher gain in the material aligns along the ΓX direction of the PC.

© 2011 OSA

1. Introduction

Two-dimensional (2D) photonic crystals (PC) have been actively developed for applications such as surface-emitting lasers [1

1. 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(5862), 445–447 (2008). [CrossRef]

7

7. O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 93(17), 171112 (2008). [CrossRef]

], light-emitting diodes [8

8. E. Matioli, B. Fleury, E. Rangel, T. Melo, E. Hu, J. Speck, and C. Weisbuch, “High extraction efficiency GaN-based photonic crystal light-emitting diodes: comparison of extraction lengths between surface and embedded photonic crystals,” Appl. Phys. Express 3(3), 032103 (2010). [CrossRef]

], and interferometers [9

9. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90(23), 231114 (2007). [CrossRef]

]. A PC is based on the periodic alignment of materials having different dielectric constants to form a photonic band structure (PBS). In general, the dielectric constants are in a complex form, whose imaginary parts provide the gain or loss factors of the electromagnetic (EM) wave propagating in the PC. Several reports show the importance of gain or loss characteristics in a PC. For example, Nojima addressed the 2D PC gain and numerically found an enhancement in optical gain for 2D PCs [10

10. S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]

,11

11. S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001). [CrossRef]

]. He showed that this enhancement results from zero velocity at the band edge. Sigalas et al. analyzed a PC made of dispersive and highly absorptive materials [12

12. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter 49(16), 11080–11087 (1994). [CrossRef] [PubMed]

]. The characteristics of a metallic PC with incident light absorption were demonstrated by Yanonpapas et al. [13

13. V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60(8), 5359–5365 (1999). [CrossRef]

] and El-Kady et al. [14

14. I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000). [CrossRef]

]. Brand et al. analyzed complex PCs to discuss effective plasma frequencies in a metal rod array [15

15. S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75(3), 035102 (2007). [CrossRef]

]. Sakai et al. demonstrated the threshold criteria of 2D PC lasers using the coupled wave theory [16

16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46(5), 788–795 (2010). [CrossRef]

,17

17. K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization,” Opt. Express 15(7), 3981–3990 (2007). [CrossRef] [PubMed]

]. These reports, however, only consider the gain/loss distribution that is dependent on the position in the PC. In other words, at each point of the PC medium, the gain does not depend on the propagation directions (wavevectors) of the EM wave, and thus the gain is isotropic.

Meanwhile, interest has been growing in InGaN-based materials as visible semiconductor light sources. For example, blue-violet InGaN-based laser diodes (LDs) have been commercialized for high-density optical data storage, and pure-blue and green LDs are candidates for mobile projectors. However, quantum-confined Stark (QCS) effects are a major issue of InGaN-based materials, since they decrease the recombination rate of electrons and holes, leading to a threshold increase in laser action. Epitaxial-grown layers on c-plane substrate cause the polarization at the interface of a layer (the so-called “polar” material), which induces a QCS effect. This effect is stronger in a higher indium-composition InGaN layer, i.e., the quantum well (QW) active layer for longer wavelength emission, such as green. Thus, in order to avoid this effect, nonpolar or semipolar InGaN-based materials that suppress this polarization have been developed. The nonpolar or semipolar InGaN-based materials are realized by epitaxial growths on, e. g., m-plane [18

18. K. Okamoto, J. Kashiwagi, T. Tanaka, and M. Kubota, “Nonpolar m-plane InGaN multiple quantum well laser diodes with a lasing wavelength of 499.8nm,” Appl. Phys. Lett. 94(7), 071105 (2009). [CrossRef]

], (11-22)-plane [19

19. A. Tyagi, Y.-D. Lin, D. A. Cohen, M. Saito, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Stimulated emission at blue-green (480nm) and green (514nm) wavelengths from nonpolar (m-plane) and semipolar (11-22) InGaN multiple quantum well laser diode structures,” Appl. Phys. Express 1, 091103 (2008). [CrossRef]

], (20-21)-plane [20

20. Y. Enya, Y. Yoshizumi, T. Kyono, K. Akita, M. Ueno, M. Adachi, T. Sumitomo, S. Tokuyama, T. Ikegami, K. Katayama, and T. Nakamura, “531 nm green lasing of InGaN based laser diodes on semipolar {20-21} free-standing GaN substrates,” Appl. Phys. Express 2, 082101 (2009). [CrossRef]

], and (30-31)-plane [21

21. P. S. Hsu, K. M. Kelchner, A. Tyagi, R. M. Farrell, D. A. Haeger, K. Fujito, H. Ohta, S. P. DenBaars, J. S. Speck, and S. Nakamura, “InGaN/GaN blue laser diode grown on semipolar (30-31) free-standing GaN substrates,” Appl. Phys. Express 3(5), 052702 (2010). [CrossRef]

] GaN substrates. Although this approach is useful in the suppression of QCS effects, the nonpolar or semipolar QW exhibits another feature, that is, anisotropic gain where the gain depends on the EM propagation direction. Ohtoshi et al. report the dependence of optical gain on the crystal orientation of GaN [22

22. T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).

]. Park demonstrates the crystal orientation effects of InGaN QWs [23

23. S.-H. Park, “Crystal orientation effects on many-body optical gain of wurtzite InGaN/GaN quantum well lasers,” Jpn. J. Appl. Phys. 42(Part 2, No. 2B), L170–L172 (2003). [CrossRef]

], and Scheibenzuber et al. compare the optical gain of semipolar/nonpolar lasers with that of a polar laser [24

24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B 80(11), 115320 (2009). [CrossRef]

]. For nonpolar m-plane InGaN QWs, several reports show experimentally that the propagation along the c-axis possesses larger gain than that along the a-axis [25

25. K. Okamoto, H. Ohta, S. F. Chichibu, J. Ichihara, and H. Takasu, “Continuous-wave operation of m-plane InGaN multiple quantum well laser diodes,” Jpn. J. Appl. Phys. 46(9), L187–L189 (2007). [CrossRef]

,26

26. T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in m-plane InxGa1-xN/GaN multiple quantum well laser diode wafers fabricated on the low defect density freestanding GaN substrate,” Appl. Phys. Lett. 93(9), 091112 (2008). [CrossRef]

]. As shown in [1

1. 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(5862), 445–447 (2008). [CrossRef]

,27

27. H. Kitagawa, T. Suto, M. Fujita, Y. Tanaka, T. Asano, and S. Noda, “Green photoluminescence from GaInN photonic crystals,” Appl. Phys. Express 1, 032004 (2008). [CrossRef]

], InGaN-based QW LDs integrated with 2D PCs are attractive for visible light sources with high optical output power and narrow beam divergence. Therefore, it is important to reveal the characteristics of 2D PCs with anisotropic gain, particularly for InGaN lasers operating in the visible wavelength.

This paper analyzes a square-lattice PC with anisotropic gain, carried out here for the first time. In Sec. 2, we derive the calculation method for anisotropic-gain PCs. In Sec. 3, the calculation model for anisotropic-gain PCs is described. In Sec. 4, the numerical results are demonstrated. In Sec. 5, we discuss the importance of PC symmetry to anisotropic gain and the importance of radiation loss. Finally, Section 6 summarizes the conclusions.

2. Calculation method

In general, a 2D plane-wave expansion (PWE) method [28

28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991). [CrossRef] [PubMed]

] utilizes several plane waves with Fourier-expanded dielectric constants, both of which are based on reciprocal wavevectors of a PC. The distinctive feature of our improved PWE analysis is that, in order to include the anisotropic gain, expanded dielectric constants depend on the propagation direction of the plane waves. For simplicity, we look at the transverse electric (TE) mode waves in a nonpolar QW only, since a semipolar QW causes birefringence so that TE mode does not make sense [24

24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B 80(11), 115320 (2009). [CrossRef]

]. Figure 1
Fig. 1 Schematic diagram of a PC layer (period: a; radius:r). Here α and β are the orthogonal gain directions of the PC material; k' is the wavevector for an expanded plane wave. The dashed line is parallel to a primitive transformation vector of PC. HL and BG represent the regions within or outside a circular cross-section.
shows a schematic diagram of a PC. The PC has a circular cross-section region (denoted HL) in the background region (denoted BG). The dielectric constant difference between the HL and BG regions leads to the formation of the PC. Here we define coordinates x1 and x2 as being in the PC plane, while coordinate x3 is normal to the plane. The TE mode in the PC corresponds to so-called “H-polarization,” whose magnetic field is perpendicular to the PC planes. The magnetic field along the x3 axis, H3(x¯), is the sum of the expanded plane waves, h3(k,G,x¯), and can be expressed as
H3(x¯)=Gh3(k,G,x¯)exp(iωt),
(1)
h3(k,G,x¯)=B(k,G)exp{i(k+G)x¯},
(2)
where x¯=(x1,x2) is a position in the PC plane, k is the wavevector of the EM wave, G is a reciprocal vector of PC, and ω is an angular frequency of the propagating EM wave. Here,B(k,G) is a complex expansion coefficient. Since the dielectric constant in the PC plane is a periodic function of x¯, the inverse dielectric constant can be expanded in 2D Fourier series as
1/ε(k',x¯)=G'κ¯(k',G')exp(iG'x¯),
(3)
where G' is another reciprocal vector of PC, and κ¯(k',G') is a Fourier coefficient, through which the anisotropic gain results in dependence on the propagation direction of the plane wave (k'=k+G). Here, κ¯(k',G') can be expressed as [28

28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991). [CrossRef] [PubMed]

]
κ¯(k',G')={f/εa(k')+(1f)/εb(k')(G'=0),2f[1/εa(k')1/εb(k')]J1(|G'r|)/|G'r|(G'0),
(4)
where f, r, and J1(x) are the filling fraction, radius of circular region, and Bessel function, respectively. Since we here consider the anisotropic gain, εa(k') and εb(k') are set to be the k'-dependent relative dielectric constants of HL and BG regions, respectively. Since the PWE method assumes that the structure has infinite length with uniformity along the x3 direction, we use the effective relative dielectric constants for each region as εa(k') and εb(k'). Assuming that the Maxwell equations are satisfied for each expanded plane wave, the TE-modes Maxwell equations for h3(k,G,x¯) with the propagation direction of k+G is expressed as
x1{1ε(k+G,x¯)x1h3(k,G,x¯)}+x2{1ε(k+G,x¯)x2h3(k,G,x¯)}=ω2c2h3(k,G,x¯),
(5)
where c is velocity of light. By the summation of all reciprocal vectors, we can obtain
x1G{1ε(k+G,x¯)x1h3(k,G,x¯)}+x2G{1ε(k+G,x¯)x2h3(k,G,x¯)}=ω2c2Gh3(k,G,x¯).
(6)
By using Eqs. (2) and (3), Eq. (6) is modified as
GG'(k+G)(k+G')κ¯(k+G',GG')B(k,G')exp{i(k+G)x¯}=ω2c2Gh3(k,G,x¯).
(7)
In order to satisfy Eq. (7) with arbitrary x¯
G'(k+G)(k+G')κ¯(k+G',GG')B(k,G')=(ω2/c2)B(k,G)
(8)
is obtained. In the case of isotropic material, κ¯(GG') can be used instead of κ¯(k+G',GG'), so that Eq. (8) is equivalent to the eigenvalue equation of TE-mode PC [28

28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991). [CrossRef] [PubMed]

].Next, we discuss κ¯(k',G'). For nonpolar QW, the optic axis exists in the PC plane. In this case, the index ellipsoid method [29

29. A. Yariv, Introduction to Optical Electronics (Holts, Rinehart and Winston. Inc, 1974).

] gives us that the refractive index, nγ, of the EM waves propagating along the γ direction (k') in the PC plane, is expressed as
1/nγ2=cos2φ/nα2+sin2φ/nβ2,
(9)
where nα and nβ are refractive indices of the EM waves propagating along the orthogonal directions, α and β, respectively; φ is the angle between the α and γ directions (see Fig. 1). The refractive indices are expanded as:
1/nγ2=1/ε(k',x¯)=G'κ¯(k',G')exp(iG'x¯),
(10)
1/nα2=1/εα(x¯)=G'κ¯α(G')exp(iG'x¯),
(11)
1/nβ2=1/εβ(x¯)=G'κ¯β(G')exp(iG'x¯),
(12)
where, εα(x¯) and εβ(x¯) are the relative dielectric constants for the EM wave propagating along the orthogonal α and β directions, respectively, and κ¯α(G') and κ¯β(G') are the expanded Fourier coefficients of εα(x¯) and εβ(x¯), respectively (see Eq. (3)). Substitution of Eqs. (10) to (12) into Eq. (9) gives
κ¯(k',G')=κ¯α(G')cos2φ+κ¯β(G')sin2φ.
(13)
Here, as shown in Fig. 1, θ is defined as the angle between the α direction and a primitive transformation vector of PC along the ΓX direction. The cosφ is expressed ascosφ=k'Iα/|k'|, where Iα=(cosθ,sinθ) is a unit vector along the α direction.

Finally, we discuss the gain or loss of the EM wave. Since the EM wave experiences gain or loss during propagation in the PC, the wave will grow or decay temporally. Thus, in order to express this variation, complex frequency is introduced in Eq. (8) as
ω=ωr+iωi.
(14)
Substituting of Eq. (14) into Eq. (1) shows that positive and negative ωi indicate the growth and decay of an EM wave, respectively. The gain, g, is calculated as g=2εeff0.5ωi/c, where εeff is the relative effective dielectric constant for a guided EM wave.

Solving Eq. (8) with Eqs. (13) and (14) gives the ωrk dispersion characteristics (the PBS) and the ωik dispersion characteristics (the gain dispersion characteristics). Both characteristics include the anisotropic gain of the PC material.

3. Calculation model

Figure 2
Fig. 2 Cross-section of calculated model. AROG denotes “air holes retained over growth,” whose layers provide the PC configuration. HL and BG correspond the regions within or outside a circular cross-section in Fig. 1.
shows the calculation model. This model is based on the AROG (air holes retained over growth) structure [1

1. 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(5862), 445–447 (2008). [CrossRef]

], which is a useful method for realizing InGaN-based PC lasers. The model consists of an Al0.04Ga0.96N lower cladding layer, a GaN (0.1 μm) or SiO2 (0.01 μm)/Air (0.09 μm) AROG layer, a GaN lower optical confinement layer (0.15 μm), a double QW active layer (In0.26Ga0.74N well (0.003 μm) and In0.16Ga0.84N barrier (0.015 μm)), a GaN upper optical confinement layer (0.06 μm), an Al0.18Ga0.82N electron blocking layer (0.02 μm), an Al0.06Ga0.94N upper cladding layer (0.6 μm), a GaN contact layer (0.05 μm), and a gold electrode. The air/SiO2 region in the AROG layer forms a square-aligned circular hole so that it corresponds to HL in Fig. 1, while the other region corresponds to BG. The period, a, and r/a ratio of the PC is 207 nm and 0.2, respectively. The PC cavity is set to be infinity in plane.

We assume that the structure above is formed in a nonpolar m-plane. For the anisotropic-gain spectrum of a nonpolar QW, we use the calculation results given by Scheibenzuber et al. [24

24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B 80(11), 115320 (2009). [CrossRef]

] with carrier density of 6 × 1012 cm−2, although the gain peak wavelength is shifted to 504 nm in this study. The propagation direction with larger gain (maximum material gain of approximate 3500 cm−1) is set to the α direction, while that with the smaller (approximate 680cm−1) is set to the β direction. The dielectric constant of GaN is the square of refractive indices in [30

30. T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82(7), 3528–3535 (1997). [CrossRef]

]. The dielectric constants of AlGaN and InGaN are obtained by shifting that of GaN, the method for which is based on [31

31. M. J. Bergmann and H. C. Casey Jr., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998). [CrossRef]

]. The dielectric constants of SiO2 and gold are from [31

31. M. J. Bergmann and H. C. Casey Jr., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998). [CrossRef]

,32

32. L. Q. Zhang, D. S. Jiang, J. J. Zhu, D. G. Zhao, Z. S. Liu, S. M. Zhang, and H. Yang, “Confinement factor and absorption loss of AlInGaN based laser diodes emitting from ultraviolet to green,” J. Appl. Phys. 105(2), 023104 (2009). [CrossRef]

], respectively. The gain is expressed through the imaginary parts of the dielectric constants of InGaN well layer. The effective relative dielectric constants of guided EM waves in the HL and BG regions are obtained by solving the Maxwell equations for the waves. In the following sections, “isotropic gain” indicates the averaged gain of α and β directions and has no dependence on the propagation direction.

4. Calculation results

4.1 Photonic band structure

Figure 3
Fig. 3 Photonic band structure with four photonic modes, A, B, C, and D. Although calculation is undertaken for isotropic gain and anisotropic gain with θ = 0, 45, and 90 deg, no significant difference is observed in this axis range.
shows the PBS around the second symmetric Γ point (Γ2). Although we calculate the PBS for isotropic gain and anisotropic gain with different θ directions, no significant difference is observed in the axis range in Fig. 3. The left ordinate of Fig. 3 represents the normalized frequency (ωra/2πc), while the right one represents the corresponding wavelength. The photonic band gap is approximately 2 × 10−4 in ωra/2πc at the Γ2 point. The PBS shows four photonic modes, designated A, B, C, and D. Four similar modes can be seen in the conventional square-lattice PC [1

1. 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(5862), 445–447 (2008). [CrossRef]

].

4.2 Near- field distributions

4.3 Gain dispersion characteristics

Gain dependence on θ is summarized in Fig. 7
Fig. 7 Gains of photonic modes at the Γ2 point as a function of θ. Solid and dotted lines are anisotropic and isotropic gains, respectively.
. For anisotropic gain, as θ increases, the gain of A and D modes decreases, while the B and C modes obtain higher gain. This is because the resonant direction of A and D (B and C) modes become far from (close to) the direction of higher gain in the PC material. At θ = 45 deg, the gain contribution to each mode is the same, so that the crossover between A-D and B-C modes occurs. Over the θ = 45 deg, the gains of B and C modes are larger than those of A and D modes. Therefore, when θ varies from 0 deg to 90 deg, the PC laser with anisotropic gain should function as follows: first, strong emission with the resonant direction parallel to the ΓX direction, then weak emission around θ = 45 deg, and finally strong emission with the resonant direction perpendicular to the ΓX direction.

5. Discussions

5.1 PC symmetry change by anisotropic gain

As shown in Secs. 3 and 4, anisotropic gain induces variation in the resonance frequencies, field distributions, and gain. In order to clarify this, we investigate the field distributions and the expanded waves of H3(x¯) around θ = 45 deg in more detail. Here, Δθ is defined as 45 minus θ in units of degree.

Near-field distribution at the Γ2 point is shown in Fig. 8
Fig. 8 Near-field distribution of the anisotropic-gain photonic modes at Γ2 point. Δθ is defined as Δθ = 45 deg - θ. Black circles represent the HL shape of the PC.
as a function of Δθ. The fields ofC and D modes are very sensitive to the Δθ. At Δθ = 10−4 deg, the near-field distributions of the C and D modes are almost identical to those at θ = 45 deg. At Δθ = 10−3 deg, the strong field portions of C and D modes affect the horizontal and vertical neighboring strong portions, respectively. At Δθ = 10−2 deg, the field distribution is similar to that at θ = 0 or 90 deg. In contrast, such small Δθ results in little variation for the A and B modes. At Δθ = 0.8 deg, the near-field distributions of the A and B modes remain almost identical to those at θ = 45 deg. At Δθ = 1 deg, the strong field regions of A and B modes affect the vertical and horizontal neighboring strong regions, respectively. At Δθ = 2 deg, the field distributions of A and B modes are similar to those at θ = 0 or 90 deg. In addition, Fig. 8 clearly indicates that the A, B, C, and D photonic modes originate from the monopole, quadrupole, symmetric dipole, and asymmetric dipole modes, respectively.

These variations can be shown by using the expanded waves of H3(x¯). According to Eq. (1), an EM wave consists of the summation of many reciprocal vectors; our calculation shows that four reciprocal vectors, G1=(1,0)2π/a, G2=(0,1)2π/a, G3=(0,1)2π/a, and G4=(1,0)2π/a, contribute mainly at the Γ2 point, i.e.,
H3(x¯)n=14B(k,Gn,ω)expi[(k+Gn)x¯ωt].
(15)
Figure 9
Fig. 9 Real and imaginary parts of B coefficients of expanded plane waves as a function of Δθ. (a), (b), (c), and (d) are A, B, C, and D modes, respectively. The parameters are reciprocal vectors, (G)1 to (G)4.
shows the real and imaginary parts of the B-coefficients, B(k,Gn,ω), of expanded plane waves of Eq. (15) at the Γ2 point, as a function of Δθ. When Δθ is less than 10−4 deg, each mode consists of four reciprocal vectors, G 1 to G 4. These four vectors cause 2D resonances, as shown in Fig. 8. When Δθ increases from 10−4 to 10−2 deg, the amplitudes of the B-coefficients of G 1 and G 4 (G 2 and G 3) reduce in C (D) modes monotonically, leading to 1D resonances. When Δθ is around 0.8 deg, the real part of the B-coefficients of G 2 and G 3 (G 1 and G 4) in A (B) modes decrease, while, in turn, the imaginary parts appear and fade away. This indicates that, when the resonances of A and B modes change from 2D to 1D, the phase of the B-coefficients of G 2 and G 3 (G 1 and G 4) vary toward π/2.

The anisotropic gain changes the symmetry of the square-lattice PC. That is, although square-lattice PCs with isotropic gain possess 4-fold rotational symmetry, square-lattice PCs with anisotropic gain with θ ≠ 45 deg have 2-fold rotational symmetry. Note that anisotropic gain with θ = 45 deg produces 4-fold rotational symmetry despite there being geometrical 2-fold symmetry, because H3(x¯) does not possess a reciprocal vector consisting of both x1 and x2 components such as (1,1)2π/a. In 2-fold rotational symmetry, the resonance occurs in 1D only. Therefore the resonance dimension changes around θ = 45 deg, which causes an abrupt change in the resonance frequency and the near-field distribution. In addition, since the A and B modes inherently possess a symmetric field distribution with respect to the hole shape as shown in Fig. 5, these modes are relatively robust to disturbance by the anisotropic gain. On the other hand, the C and D modes have an inherently asymmetric distribution with respect to the hole shape, and thus they obtain an asymmetric nature easily caused by the anisotropic gain. Such difference explains the sensitivity of the field or B-coefficient variation to Δθ variation, as shown in Figs. 8 and 9.

5.2 Radiation loss effects

2D PC suffers from radiation loss (see the Appendix). For the model shown in Sec. 3, the radiation losses for photonic modes are calculated as 45.9 cm−1 for C and D modes, while there is no loss for A and B modes. Taking these losses into consideration, the dependence of gain on θ is calculated as shown in Fig. 10
Fig. 10 Gains of photonic modes at the Γ2 point as a function of θ with radiation loss. Solid and dotted lines are anisotropic and isotropic gains, respectively.
. Negative gain indicates loss. Compared to Fig. 7, gains of C and D modes decrease drastically, since they suffer from the radiation loss. At θ = 0 (90) deg, the gain of A and D (B and C) modes separates so that the PC laser will oscillate in a single photonic mode of A (B). In contrast, the no gain difference between A and B modes is observed for the isotropic gain, e.g., c-plane case, which leads to dual mode operation. Thus, Fig. 10 indicates that it is important to consider anisotropic gain when characterizing surface-emitting performance.

The reason that the C and D modes experience the radiation loss is as follows. Since the electric field in the PC plane is given by the derivative of the magnetic field, the asymmetric H3 fields of C and D give symmetric electric fields. Since the symmetric electric fields interfere constructively in the PC plane, the radiation becomes strong. On the other hand, symmetric H3 fields of A and B forms an asymmetric electric field in the PC plane, leading to destructive interference and weak radiation. Sakai et al. studied a square-lattice PC laser and showed that each PC mode suffers from this radiation loss [16

16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46(5), 788–795 (2010). [CrossRef]

]. A, B, and E modes in [16

16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46(5), 788–795 (2010). [CrossRef]

]. correspond to A, B, and C (D) modes, respectively, in this work. Sakai et al. indicate that, compared to A and B modes, the C (D) mode suffers from radiation loss to a great extent, which is consistent with our results.

6. Conclusions

We have presented the analysis of photonic modes with anisotropic gain for a 2D PC structure. We developed a novel PWE method to investigate the anisotropic gain, where the gain factor is expressed as a function of complex frequency. Numerical calculations were undertaken for a nonpolar InGaN-based PC laser. The results indicate that anisotropic gain induces variation not only in PBS but also in the gain characteristics and in the near-field distribution. Small variation in the direction of gain from the ΓM direction changes the symmetry of the square-lattice PC from 4-fold rotational symmetry to 2-fold symmetry. Higher gain induced by the PC material requires the alignment in which the direction of larger gain in the material coincides with the ΓX direction of the square-lattice PC.

Appendix

Radiation loss necessitates the three-dimensional (3D) treatment of the radiation wave. Similar to [34

34. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21(2), 144–150 (1985). [CrossRef]

], the magnetic field H3 in a square-lattice PC can be expressed as:

H3(x^)=n=14Bx¯(Gn)exp[i(k+Gn)x¯]φ(x3)+ΔH(x^), (a.1)

where x3 is the axis normal to the PC plane; x^ is a position in space, i.e., x^=(x1,x2,x3); and ΔH(x^) is the radiation perturbation. Here we consider four reciprocal vectors based on Eq. (15). Note that, unlike Eq. (2), the Fourier expansion coefficient,Bx¯(G,ω), is a function of x¯. We assume that ΔH(x^) and Bx¯(G,ω) vary slowly in x¯, so that

2ΔH(x^)/xixj=2Bx¯(G,ω)/xixj=0, (a.2)

where i, j = 1, 2. In Eq. (a.1), φ(x3) is a near-field distribution along x3 and is given by:

2φ(x3)/x32=(ω/c)2Δε(x3)φ(x3), (a.3)

where Δε(x3)=εeffε(x3); εeff denotes the averaged effective dielectric constant of HL and BG region, i.e., εeff=fεa+(1f)εb; and ε(x3) represents the dielectric constant along x3, which is in-plane averaged for the PC layer. The inverse of the dielectric constant, ε(x^), in 3D form is expanded as

1/ε(x^)=Gκ^(G,x3)exp(iGx¯), (a.4)

where

κ^(G,x3)={κ¯p(G)(g1x3g2),1/ε(x3)(x3<g1,g2x3,G=0),0(x3<g1,g2x3,G0), (a.5)

and

κ¯p(G)={f/εpa+(1f)/εpb(G=0),(1/εpa1/εpb)2fJ1(|Gr|)/|Gr|(G0). (a.6)

Here, εpa and εpb represent the dielectric constants of the HL and the BG regions, respectively, in the PC layer (in our model, the PC layer corresponds to the AROG layer); g1 and g2 are defined as the x3 range of the PC layer. From Eqs. (a.5) and (a.6), we can derive κ^(G,x3)=κ^(G,x3) and κ¯p(G)= κ¯p(G). The 3D Maxwell equations for H-polarization give:

x1[1ε(x^)H3(x^)x1]+x2[1ε(x^)H3(x^)x2]+1ε(x^)2H3(x^)x32=ω2c2H3(x^). (a.7)

By substituting Eqs. (a.1), (a,3), and (a.4) into Eq.(a.7), the following equation is obtained at the Γ2 point:

G[n=14κ^(GGn,x3){i(G+Gn)¯GGn+(ω/c)2Δε(x3)}Bx¯(Gn)φ(x3)+κ^(G,x3)(iG¯+2/x32)ΔH(x^)]exp(iGx¯)+(ω/c)2[n=14Bx¯(Gn)φ(x3)exp(iGnx¯)+ΔH(x^)]=0, (a.8)

where ¯=(/x1,/x2). Constant terms with respect to the exponential function in Eq. (a.8) give

[κ^(0,x3)2/x32+(ω/c)2]ΔH(x^)=n=14κ^(Gn,x3)[iGn¯+(ω/c)2Δε(x3)]Bx¯(Gn)φ(x3). (a.9)

Equation (a.9) can be solved by the Green function method [34

34. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21(2), 144–150 (1985). [CrossRef]

] and gives

ΔH(x^)=n=14[κ^(Gn,t)/κ^(0,t)][iGn¯+(ω/c)2Δε(t)]Bx¯(Gn)φ(t)G˜(x3,t)dt=n=14[κ¯p(Gn)/κ¯p(0)][iGn¯+(ω/c)2Δεp]Bx¯(Gn)g1g2φ(t)G˜(x3,t)dt, (a.10)

where Δεp=εeff[fεpa+(1f)εpb]; G˜(x3,t)=exp[±iK3(x3t)]/2iK3 (positive sign for x3 > t, negative sign for x3 < t); and K3=ω/aκ¯p(0)0.5. Coefficient terms with respect to the exponential function in Eq. (a.8) for a G m give

n=14κ^(GmGn,x3){i(Gm+Gn)¯GmGn+(ω/c)2Δε(x3)}Bx¯(Gn)φ(x3)+κ^(Gm,x3)(iGm¯+2/x32)ΔH(x^)+(ω/c)2Bx¯(Gm)φ(x3)=0. (a.11)

By multiplying Eq. (a.11) with φ(x3), integrating it in the PC layer, i.e., g1x3g2, and substituting Eq. (a.10) into it, we can obtain the following equation:

n=14[i(F(Gn,Gm)¯+A(Gn,Gm)]Bx¯(Gn)+(ω/c)2Bx¯(Gm)=0, (a.12)

where

F(Gn,Gm)=κ¯p(GmGn)(Gm+Gn)[κ¯p(Gm)κ¯p(Gn)/κ¯p(0)][ς(ω/c)2ΔεpGm+ξGn], (a.13)
A(Gn,Gm)=κ¯p(GmGn)[GmGn+(ω/c)2Δεp][κ¯p(Gm)κ¯p(Gn)/κ¯p(0)]ξ(ω/c)2Δεp, (a.14)
ς=(1/Γg)g1g2g1g2φ(x3)φ(t)G˜(x3,t)dtdx3, (a.15)
ξ=(1/Γg)g1g2φ(x3)[2x32g1g2φ(t)G˜(x3,t)dt]dx3. (a.16)

Γg is a confinement factor in PC layer, defined as:

Γg=g1g2φ(x3)2dx3. (a.17)

When φ(x3) is regarded as almost constant in the PC layer, Eqs. (a.15) and (a.16) are approximated as

ς=[iK3dg+1exp(iK3dg)]/iK33dg, (a.18)
ξ=i[1exp(iK3dg)]/K3dg. (a.19)

where dg=|g2-g1|.

In case of the 1D resonance, the near-field distribution can be considered to consist of G 1 and G 2 reciprocal vectors, where Gm=(2π/a,0), G1=Gm, and G2=Gm, which does not lose generality. In this case Bx¯(Gn) is a function of x1 only, so that we can use Bx1(Gn) instead of Bx¯(Gn). For symmetric and asymmetric H3 fields, we can set Bx1(G1)=Bx1(G2) andBx1(G1)=Bx1(G2), respectively. Then Eq. (a.12) can be simplified as

i[Bx1(G1)/x1]=[(cm±dm)/(am±bm)]Bx1(G1), (a.20)
i[Bx1(G2)/x1]=[(cm±dm)/(am±bm)]Bx1(G2), (a.21)

where plus and minus signs denote symmetric and asymmetric H3 fields, respectively; am and bm are x1 components of F(Gm,Gm) and F(Gm,Gm), respectively; cm=A(Gm,Gm)+(ω/c)2; and dm=A(Gm,Gm). Equations (a.20) and (a.21) indicate that the imaginary part of (cm±dm)/(am±bm) represents the radiation loss, since the expanded wave of the G 1 (G 2) component in Eq. (a.1) at the Γ2 point propagates in the +x1 (−x1) direction. Note that (cm±dm)/(am±bm) represents the loss only in the PC layer, because the integration of Eq. (a.12) is undertaken within the PC layer. Thus, the radiation loss for symmetric and asymmetric H3 fields, Qs and Qa, respectively, can be obtained as

Qs=Im[Γg(cm+dm)/(am+bm)]=Im{Γg|Gm|2x2[(1+P2)Δε+1/κ¯p(0)2ξP12Δε](1P2)1ξP12}, (a.22)
Qa=Im[Γg(cmdm)/(ambm)]=Im{Γg|Gm|2x2[(1P2)Δε+1/κ¯p(0)](1+P2)1ςΔε(x|Gm|P1)2}, (a.23)

where P1=κ¯p(Gm)/κ¯p(0), P2=κ¯p(2Gm)/κ¯p(0), and x=ωa/2πc.

In the case of 2D resonance, we can treat Eq. (a.12) in the same manner as 1D resonance. The radiation loss for A and B modes, QA and QB, respectively, are as follows;

QA=Im{Γg|Gm|2x2[(1+P2+2P3)Δε+1/κ¯p(0)4ξP12Δε](1P2)1+P3ξP12}, (a.24)
QB=Im{Γg|Gm|2x2[(1+P22P3)Δε1/κ¯p(0)](1P2)1+P3ξP12}, (a.25)

where P3=κ¯p(G1+G2)/κ¯p(0). The radiation loss for both C and D modes, QC, and QD, respectively, is equal to the radiation loss of the 1D asymmetric resonance mode, Qa . Note that the radiation loss, Eqs. (a.22) to (a.25), should be zero in the case of a negative value, because the radiation does not result in gain.

Acknowledgments

S. Takigawa is grateful to Dr. Daisuke Ueda and Dr. Tsuyoshi Tanaka in Panasonic Corporation for their encouragements.

References and links

1.

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(5862), 445–447 (2008). [CrossRef]

2.

M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002). [CrossRef]

3.

R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. 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(5649), 1374–1377 (2003). [CrossRef] [PubMed]

4.

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 Comm. 23(7), 1335–1340 (2005). [CrossRef]

5.

G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101(8), 081726 (2007). [CrossRef]

6.

L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]

7.

O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 93(17), 171112 (2008). [CrossRef]

8.

E. Matioli, B. Fleury, E. Rangel, T. Melo, E. Hu, J. Speck, and C. Weisbuch, “High extraction efficiency GaN-based photonic crystal light-emitting diodes: comparison of extraction lengths between surface and embedded photonic crystals,” Appl. Phys. Express 3(3), 032103 (2010). [CrossRef]

9.

D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90(23), 231114 (2007). [CrossRef]

10.

S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]

11.

S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001). [CrossRef]

12.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter 49(16), 11080–11087 (1994). [CrossRef] [PubMed]

13.

V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60(8), 5359–5365 (1999). [CrossRef]

14.

I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000). [CrossRef]

15.

S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75(3), 035102 (2007). [CrossRef]

16.

K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46(5), 788–795 (2010). [CrossRef]

17.

K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization,” Opt. Express 15(7), 3981–3990 (2007). [CrossRef] [PubMed]

18.

K. Okamoto, J. Kashiwagi, T. Tanaka, and M. Kubota, “Nonpolar m-plane InGaN multiple quantum well laser diodes with a lasing wavelength of 499.8nm,” Appl. Phys. Lett. 94(7), 071105 (2009). [CrossRef]

19.

A. Tyagi, Y.-D. Lin, D. A. Cohen, M. Saito, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Stimulated emission at blue-green (480nm) and green (514nm) wavelengths from nonpolar (m-plane) and semipolar (11-22) InGaN multiple quantum well laser diode structures,” Appl. Phys. Express 1, 091103 (2008). [CrossRef]

20.

Y. Enya, Y. Yoshizumi, T. Kyono, K. Akita, M. Ueno, M. Adachi, T. Sumitomo, S. Tokuyama, T. Ikegami, K. Katayama, and T. Nakamura, “531 nm green lasing of InGaN based laser diodes on semipolar {20-21} free-standing GaN substrates,” Appl. Phys. Express 2, 082101 (2009). [CrossRef]

21.

P. S. Hsu, K. M. Kelchner, A. Tyagi, R. M. Farrell, D. A. Haeger, K. Fujito, H. Ohta, S. P. DenBaars, J. S. Speck, and S. Nakamura, “InGaN/GaN blue laser diode grown on semipolar (30-31) free-standing GaN substrates,” Appl. Phys. Express 3(5), 052702 (2010). [CrossRef]

22.

T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).

23.

S.-H. Park, “Crystal orientation effects on many-body optical gain of wurtzite InGaN/GaN quantum well lasers,” Jpn. J. Appl. Phys. 42(Part 2, No. 2B), L170–L172 (2003). [CrossRef]

24.

W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B 80(11), 115320 (2009). [CrossRef]

25.

K. Okamoto, H. Ohta, S. F. Chichibu, J. Ichihara, and H. Takasu, “Continuous-wave operation of m-plane InGaN multiple quantum well laser diodes,” Jpn. J. Appl. Phys. 46(9), L187–L189 (2007). [CrossRef]

26.

T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in m-plane InxGa1-xN/GaN multiple quantum well laser diode wafers fabricated on the low defect density freestanding GaN substrate,” Appl. Phys. Lett. 93(9), 091112 (2008). [CrossRef]

27.

H. Kitagawa, T. Suto, M. Fujita, Y. Tanaka, T. Asano, and S. Noda, “Green photoluminescence from GaInN photonic crystals,” Appl. Phys. Express 1, 032004 (2008). [CrossRef]

28.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991). [CrossRef] [PubMed]

29.

A. Yariv, Introduction to Optical Electronics (Holts, Rinehart and Winston. Inc, 1974).

30.

T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82(7), 3528–3535 (1997). [CrossRef]

31.

M. J. Bergmann and H. C. Casey Jr., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998). [CrossRef]

32.

L. Q. Zhang, D. S. Jiang, J. J. Zhu, D. G. Zhao, Z. S. Liu, S. M. Zhang, and H. Yang, “Confinement factor and absorption loss of AlInGaN based laser diodes emitting from ultraviolet to green,” J. Appl. Phys. 105(2), 023104 (2009). [CrossRef]

33.

K. Inoue and K. Ohtaka, Photonic Crystals (Springer-Verlag, 2004).

34.

R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21(2), 144–150 (1985). [CrossRef]

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: April 4, 2011
Manuscript Accepted: April 20, 2011
Published: April 29, 2011

Citation
Shinichi Takigawa and Susumu Noda, "Analysis of two-dimensional photonic crystal with anisotropic gain," Opt. Express 19, 9475-9491 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9475


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References

  1. 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(5862), 445–447 (2008). [CrossRef]
  2. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65(19), 195306 (2002). [CrossRef]
  3. R. Colombelli, K. Srinivasan, M. Troccoli, O. Painter, C. F. 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(5649), 1374–1377 (2003). [CrossRef] [PubMed]
  4. 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 Comm. 23(7), 1335–1340 (2005). [CrossRef]
  5. G. Scalari, L. Sirigu, R. Terazzi, C. Walther, M. I. Amanti, M. Giovannini, N. Hoyler, J. Faist, M. L. Sadowski, H. Beere, D. Ritchie, L. A. Dunbar, and R. Houdre, “Multi-wavelength operation and vertical emission in THz quantum-cascade lasers,” J. Appl. Phys. 101(8), 081726 (2007). [CrossRef]
  6. L. Sirigu, R. Terazzi, M. I. Amanti, M. Giovannini, J. Faist, L. A. Dunbar, and R. Houdré, “Terahertz quantum cascade lasers based on two-dimensional photonic crystal resonators,” Opt. Express 16(8), 5206–5217 (2008). [CrossRef] [PubMed]
  7. O. P. Marshall, V. Apostolopoulos, J. R. Freeman, R. Rungsawang, H. E. Beere, and D. A. Ritchie, “Surface-emitting photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 93(17), 171112 (2008). [CrossRef]
  8. E. Matioli, B. Fleury, E. Rangel, T. Melo, E. Hu, J. Speck, and C. Weisbuch, “High extraction efficiency GaN-based photonic crystal light-emitting diodes: comparison of extraction lengths between surface and embedded photonic crystals,” Appl. Phys. Express 3(3), 032103 (2010). [CrossRef]
  9. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90(23), 231114 (2007). [CrossRef]
  10. S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998). [CrossRef]
  11. S. Nojima, “Optical-gain enhancement in two-dimensional active photonic crystals,” J. Appl. Phys. 90(2), 545–551 (2001). [CrossRef]
  12. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Ho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B Condens. Matter 49(16), 11080–11087 (1994). [CrossRef] [PubMed]
  13. V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B 60(8), 5359–5365 (1999). [CrossRef]
  14. I. El-Kady, M. M. Sigalas, R. Biswas, K. H. Ho, and C. M. Soukoulis, “Metallic photonic crystals at optical wavelengths,” Phys. Rev. B 62(23), 15299–15302 (2000). [CrossRef]
  15. S. Brand, R. A. Abram, and M. A. Kaliteevski, “Complex photonic band structure and effective plasma frequency of a two-dimensional array of metal rods,” Phys. Rev. B 75(3), 035102 (2007). [CrossRef]
  16. K. Sakai, E. Miyai, and S. Noda, “Coupled-wave theory for square-lattice photonic crystal lasers with TE polarization,” IEEE J. Quantum Electron. 46(5), 788–795 (2010). [CrossRef]
  17. K. Sakai, E. Miyai, and S. Noda, “Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization,” Opt. Express 15(7), 3981–3990 (2007). [CrossRef] [PubMed]
  18. K. Okamoto, J. Kashiwagi, T. Tanaka, and M. Kubota, “Nonpolar m-plane InGaN multiple quantum well laser diodes with a lasing wavelength of 499.8nm,” Appl. Phys. Lett. 94(7), 071105 (2009). [CrossRef]
  19. A. Tyagi, Y.-D. Lin, D. A. Cohen, M. Saito, K. Fujito, J. S. Speck, S. P. DenBaars, and S. Nakamura, “Stimulated emission at blue-green (480nm) and green (514nm) wavelengths from nonpolar (m-plane) and semipolar (11-22) InGaN multiple quantum well laser diode structures,” Appl. Phys. Express 1, 091103 (2008). [CrossRef]
  20. Y. Enya, Y. Yoshizumi, T. Kyono, K. Akita, M. Ueno, M. Adachi, T. Sumitomo, S. Tokuyama, T. Ikegami, K. Katayama, and T. Nakamura, “531 nm green lasing of InGaN based laser diodes on semipolar {20-21} free-standing GaN substrates,” Appl. Phys. Express 2, 082101 (2009). [CrossRef]
  21. P. S. Hsu, K. M. Kelchner, A. Tyagi, R. M. Farrell, D. A. Haeger, K. Fujito, H. Ohta, S. P. DenBaars, J. S. Speck, and S. Nakamura, “InGaN/GaN blue laser diode grown on semipolar (30-31) free-standing GaN substrates,” Appl. Phys. Express 3(5), 052702 (2010). [CrossRef]
  22. T. Ohtoshi and T. Kuroda, “Dependence of optical gain on crystal orientation in wurtzite-GaN strained quantum-well lasers,” Appl. Phys. Lett. 82, 1518–1520 (1997).
  23. S.-H. Park, “Crystal orientation effects on many-body optical gain of wurtzite InGaN/GaN quantum well lasers,” Jpn. J. Appl. Phys. 42(Part 2, No. 2B), L170–L172 (2003). [CrossRef]
  24. W. Scheibenzuber, U. Schwarz, R. Veprek, B. Witzigmann, and A. Hangleiter, “Calculation of optical eigenmodes and gain in semipolar and nonpolar InGaN/GaN laser diodes,” Phys. Rev. B 80(11), 115320 (2009). [CrossRef]
  25. K. Okamoto, H. Ohta, S. F. Chichibu, J. Ichihara, and H. Takasu, “Continuous-wave operation of m-plane InGaN multiple quantum well laser diodes,” Jpn. J. Appl. Phys. 46(9), L187–L189 (2007). [CrossRef]
  26. T. Onuma, K. Okamoto, H. Ohta, and S. F. Chichibu, “Anisotropic optical gain in m-plane InxGa1-xN/GaN multiple quantum well laser diode wafers fabricated on the low defect density freestanding GaN substrate,” Appl. Phys. Lett. 93(9), 091112 (2008). [CrossRef]
  27. H. Kitagawa, T. Suto, M. Fujita, Y. Tanaka, T. Asano, and S. Noda, “Green photoluminescence from GaInN photonic crystals,” Appl. Phys. Express 1, 032004 (2008). [CrossRef]
  28. M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991). [CrossRef] [PubMed]
  29. A. Yariv, Introduction to Optical Electronics (Holts, Rinehart and Winston. Inc, 1974).
  30. T. Kawashima, H. Yoshikawa, S. Adachi, S. Fuke, and K. Ohtsuka, “Optical properties of hexagonal GaN,” J. Appl. Phys. 82(7), 3528–3535 (1997). [CrossRef]
  31. M. J. Bergmann and H. C. Casey., “Optical-field calculations for lossy multiple-layer AlxGa1-xN/InxGa1-xN laser diodes,” J. Appl. Phys. 84(3), 1196–1203 (1998). [CrossRef]
  32. L. Q. Zhang, D. S. Jiang, J. J. Zhu, D. G. Zhao, Z. S. Liu, S. M. Zhang, and H. Yang, “Confinement factor and absorption loss of AlInGaN based laser diodes emitting from ultraviolet to green,” J. Appl. Phys. 105(2), 023104 (2009). [CrossRef]
  33. K. Inoue and K. Ohtaka, Photonic Crystals (Springer-Verlag, 2004).
  34. R. F. Kazarinov and C. H. Henry, “Second-order distributed feedback lasers with mode selection provided by first-order radiation losses,” IEEE J. Quantum Electron. 21(2), 144–150 (1985). [CrossRef]

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