OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 9 — Apr. 28, 2008
  • pp: 6033–6040
« Show journal navigation

Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization

Kyosuke Sakai, Jianglin Yue, and Susumu Noda  »View Author Affiliations


Optics Express, Vol. 16, Issue 9, pp. 6033-6040 (2008)
http://dx.doi.org/10.1364/OE.16.006033


View Full Text Article

Acrobat PDF (520 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We present a coupled-wave model for a triangular-lattice two-dimensional (2D) photonic crystal (PC) with a transverse electric (TE) polarization and derive a set of coupled-wave equations. We use these equations to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. We calculate the resonant mode frequencies for a PC composed of circular holes. These agree well with the frequencies calculated using the 2D plane wave expansion method. We also evaluate the coupling constants of fabricated samples using their measured resonant mode frequencies. Our analytic expressions allow the design and evaluation of feedback strength in triangular-lattice 2D PC cavities.

© 2008 Optical Society of America

1. Introduction

Photonic crystal (PC) lasers based on the band-edge effect have attracted much attention due to the large-area coherent oscillation that they produce [1–13

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]

]. We have developed these PC lasers to realize continuous-wave operation at room temperature [7

7. 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]

] using a square-lattice PC, and have produced a range of beam patterns including circular single-lobed and doughnut-shaped beams [8

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

]. We have also extended coupled-wave theory [14

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

] to include square-lattice PCs and have derived a set of coupled-wave equations [15

15. 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]

,16

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

] that allows the basic resonant mode properties to be analyzed. For example, electromagnetic field distributions, radiation loss (output power) from the PC cavity and thus the threshold gains will be obtained. We have used these equations to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants [15

15. 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]

]. These analytic expressions allow the feedback strength in square-lattice PC cavities to be evaluated.

In order to further develop the PC laser, it is important to investigate different crystal geometries such as the triangular lattice; this has six-fold rotational symmetry and is thus expected to have resonant modes with stronger two-dimensionality and higher controllability than the square-lattice PC with four-fold rotational symmetry. A PC laser with triangular-lattice geometry has been studied [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]

,2

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

] and was found to have six resonant modes at the edge of the photonic band structure, as shown in Fig. 1. Although some of these resonant mode properties have been studied, such as the electromagnetic field [4

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

,6

6. 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]

], analytic expressions that allow the feedback strength in triangular-lattice PC cavities to be evaluated have not yet been obtained. In this paper, we present a coupled-wave model and derive analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants for triangular-lattice PC cavities without introduced defects. In section 2 of this paper, we derive a set of coupled-wave equations for a triangular-lattice PC with a transverse electric (TE) polarization and then derive the analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. In section 3, we use these expressions to calculate the resonant mode frequencies, which we compare with those obtained using the two-dimensional (2D) plane wave expansion method (PWEM) to assess the validity of our model. The coupling constants of previously fabricated devices are then evaluated using their measured resonant mode frequencies. Using our analytic expressions, we are thus able to evaluate the feedback strength of triangular-lattice PC cavities in fabricated samples.

Fig. 1. Photonic bandstructure for triangular lattice photonic crystal with a TE mode. The right-hand figure shows the detailed structure in the dotted circle, which has six dispersion curves. Each curve has a band edge that is indicative of the associated resonant mode (A – D). Modes B and D are doubly degenerate.

2. Coupled-wave model for triangular lattice

The PC structure investigated here consists of a triangular lattice of circular holes in the x-y plane with period a, as shown in Fig. 2(a). The structure is assumed to be uniform in the z-direction. The circular holes form a 2D Bravais lattice with sites given by the vectors:

X(l)=l1a1+l2a2.
(1)

Here a 1 and a 2 are the two primitive translation vectors of the lattice, while l 1 and l 2 are any two integers. The area enclosed by the primitive unit cell of this lattice is ac=|a 1 ×a 2|=√3a 2/2.

Fig. 2. (a). Triangular lattice photonic crystal. The black arrows indicate the primitive translation vectors of the lattice. (b). The reciprocal lattice of (a). The black arrows indicate the primitive reciprocal lattice vectors. The gray arrows indicate the wavevectors considered in this paper.

The corresponding reciprocal lattice is shown in Fig. 2(b) and the reciprocal lattice vectors G(h) are given by:

G(h)=h1b1+h2b2.
(2)

Here h 1 and h 2 are any two integers, denoted collectively by h, and the primitive translation vectors of this lattice are given by:

b1=2πac(a(2)y,a(2)x),
(3)

and

b2=2πac(a(1)y,a(1)x),
(4)

where a (i) j is the j th Cartesian component, x or y, of a i (i=1 or 2). If we express the primitive translation vectors as a 1=(√3a/2, a/2) and a 2=(0,a), as shown in Fig. 2(a), the primitive reciprocal lattice vectors are b 1=(4π/√3a, 0) and b 2=(-2π/√3a, 2π/a), as shown in Fig. 2(b).

The scalar wave equations for the magnetic field Hz in the TE mode are written in the form [17

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

]:

x{1k2Hzx}+y{1k2Hzy}+Hz=0.
(5)

Here the constant k is given by [18

18. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

]:

k2=(2πλ)2GεGexp(iG·r)+i2πε012λGαGexp(iG·r).
(6)

In the above expression λ is the wavelength of the light in free space, ε G is the Fourier coefficient of the modulated dielectric constant ε(r), ε 0 (=ε G=0) is the averaged dielectric constant, and α G is the Fourier coefficient of the modulated gain constant α(r). In PC lasers, it is assumed that the gain is small over distances of the order of a wavelength, and that the modulations of the dielectric constant and gain constant are small, such that:

αβ2πε012λ,εG0ε0,αGβ.
(7)

These assumptions allow us to express the constant k in the form:

1k2=1β4(β2i2αβ+2βG0κGexp(iG·r)).
(8)

Here, α(=α G=0/2) is the averaged gain constant and κ G is the coupling constant defined as:

κG=πλε012εGiαG2.
(9)

It should be noted that we consider the resonance at the Γ-point with the smallest non-zero frequency in the photonic bandstructure [4

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

], in which the coupling constants for |G|=4π/√3a, 4π/a, 8π/√3 a contribute significantly. We list the corresponding coupling constants as:

κ1=κG|G=β0,
(10)
κ2=κG|G=3β0,
(11)
κ3=κG|G=2β0,
(12)

where β 0=4π/√3a. In a periodic structure, the magnetic field is given by the Bloch mode [17

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

]:

Hz(r)=GHGexp[i(k+G)·r],
(13)

where H G is the amplitude of each plane wave, and k is a wavevector in the first Brillouin zone that becomes zero at the Γ-point. In principle, a periodic perturbation of the medium generates an infinite set of diffraction orders. However, at the specific Γ-point discussed in this paper, only the amplitudes H G with |G|=β 0 are significant. All other amplitudes are small and can be neglected. Therefore, six waves with |G|=β 0, as indicated by the gray arrows in Fig. 2(b), are considered in our model. These six waves propagate in the PC structure and interfere with each other due to diffraction by the circular holes. As a result, the amplitudes of the waves become position-dependent. We describe these waves using the complex amplitude Hi (r) (i=1 to ~6), and rewrite the expression for the magnetic field as the sum:

Hz(r)=H1exp(iβ0x)+H2exp(iβ02xi3β02y)+H3exp(iβ02xi3β02y)
+H4exp(iβ0x)+H5exp(iβ02x+i3β02y)+H6exp(iβ02x+i3β02y).
(14)

In view of Eq. (7), these amplitudes vary slowly and their second derivatives can thus be neglected. By substituting Eqs. (8), (9) and (14) into Eq. (5), using expressions (10)–(12), and comparing the exponential terms, we obtain six equations of the form:

xH1+(αiδ)H1=iκ12(H2+H6)+iκ22(H3+H5)+iκ3H4,
(15-1)
12xH232yH2+(αiδ)H2=iκ12(H1+H3)+iκ22(H4+H6)+iκ3H5,
(15-2)
12xH332yH3+(αiδ)H3=iκ12(H2+H4)+iκ22(H1+H5)+iκ3H6,
(15-3)
xH4+(αiδ)H4=iκ12(H3+H5)+iκ22(H2+H6)+iκ3H1,
(15-4)
12xH5+32yH5+(αiδ)H5=iκ12(H4+H6)+iκ22(H1+H3)+iκ3H2,
(15-5)
12xH6+32yH6+(αiδ)H6=iκ12(H1+H5)+iκ22(H2+H4)+iκ3H3.
(15-6)

The parameter δ is a normalized frequency defined by:

δβ2β202β0ββ0=n(ωω0)c,
(16)

where n is the averaged refractive index which is equal to ε 0 1/2 and c is the speed of light in free space. The parameter δ is a measure of the deviation of the oscillation frequency ω from the Bragg frequency ω 0. Because this frequency deviation is assumed to be small, we have set β/β 0≈1 in the above derivation.

The above set of equations expresses the coupling of waves propagating in the triangular-lattice PC structure. For example, Eq. (15-1) describes the coupling of waves H 1 and H 4 that travel in opposite directions; the coupling constant is κ 3. The same equation also describes the coupling of waves that propagate in oblique directions. That is, wave H 1 propagating along the x-axis couples to waves H 2 and H 6 with the coupling constant κ 1, and it also couples to waves H 3 and H 5 with the coupling constant κ 2. These oblique couplings with constants κ 1 and κ 2 provide 2D optical feedback, which gives rise to coherent 2D oscillation. By numerically solving the set of equations (15) under some boundary conditions, the eigenvalues α and δ provide the threshold gain and frequency of the resonant mode in triangular-lattice PC cavities, respectively. Note that, for other crystal geometries such as square-lattice PCs, different set of equations, as shown in Ref. 15

15. 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]

, are required.

If we assume an infinite periodic structure, we can use Eq. (15) to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. Neglecting the derivatives and the threshold gain α in Eq. (15), the normalized frequency δ can be derived from the condition that the determinant of the matrix formed by the amplitude coefficients is zero. Using Eq. (16), the analytic expressions for the resonant modes (A - D) are obtained in the form:

ωD=cn(β0+12κ1+12κ2+κ3),(doublydegenerate),
(17-1)
ωC=cn(β0+κ1κ2κ3),
(17-2)
ωB=cn(β012κ1+12κ2κ3),(doublydegenerate),
(17-3)
ωA=cn(β0κ1κ2+κ3).
(17-4)

Here, modes B and D are doubly degenerate. Given the coupling constants of the specific PC structure under consideration, one can then calculate the resonant mode frequencies using the set of equations (17).

Equation (17) can also be derived by considering the point group C6. By using the array of group characters for an irreducible representation, we can obtain the same results, (e.g. ϕ=(1, 1, 1, 1, 1, 1) [19

19. T. Rowland and E. W. Weisstein, “Character Table,” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CharacterTable.html

] leads to Eqs. (17-2)).

3. Resonant mode frequencies and coupling constants

In order to assess the validity of the analytic expressions (17), we now obtain the theoretical coupling constants and calculate the resonant mode frequencies for a PC with circular holes. We then compare the resultant mode frequencies with those obtained using the 2D PWEM. As shown in Eq. (9), the coupling constants are found using the Fourier coefficients of both the modulated dielectric constant and the gain constant. Using the dielectric constant εa and gain constant αa of the circular hole, and the dielectric constant εb and gain constant αb of the surrounding material, as shown in Fig. 2(a), the Fourier coefficients are obtained in the form [17

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

]:

εG={εaf+εb(1f),G=0,(εaεb)1acAexp(iG·r)dr,G0,
(18)
αG={αaf+αb(1f),G=0,(αaαb)1acAexp(iG·r)dr,G0,
(19)

Here f is the filling factor (the fraction of the area occupied by the circular hole) and A is the region inside the circular hole. For a circular hole with radius R, Eqs. (18) and (19) can be expressed for G≠0 as:

εG=(εaεb)2fJ1(GR)GR,
(20)
αG=(αaαb)2fJ1(GR)GR.
(21)

Here, J 1(x) is a Bessel function of the first kind for integer order 1 and the filling factor is given by f=(2π/√3)R 2/a 2. By substituting Eqs. (20) and (21) into Eq. (9) and using λ 0= 0 1/2, the theoretical coupling constant for a circular hole is given by:

κG={πaε0(εaεb)i12(αaαb)}2fJ1(GR)GR.
(22)

We calculate the coupling constant using the parameters in Ref. 4

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

, where εa=10.5625, εb=10.6834, a=462 nm, αa=0, and αb=0. The calculated coupling constants are plotted as a function of the filling factor in Fig. 3. At f≈0.9, neighboring holes are in contact with each other. The coupling constant κ 3 becomes zero at f≈0.25 and the coupling constant κ 2 becomes zero at f≈0.33, where the feedback effect induced by each type of coupling vanishes. Even if these coupling constants vanish, the coupling constant κ 1 still provides the 2D optical feedback. Therefore the 2D lasing oscillation can be still maintained.

Fig. 3. Coupling constants as a function of the filling factor for a circular hole with εa=10.5625, εb=10.6834, and a=462 nm.
Fig. 4. Normalized resonant mode frequencies as a function of the filling factor, calculated using (a) Eq. (17), and (b) the 2D PWEM (225 waves).

In Fig. 4(a) we present the resonant mode frequencies as a function of the filling factor, calculated using Eq. (17) and the coupling constants from Fig. 3. The corresponding frequencies calculated using the 2D PWEM are shown in Fig. 4(b). The two sets of results are in good agreement; the minor deviations are due to the different number of waves used in each method. We used 6 waves to determine the resonant mode frequencies from Eq. (17), while 225 waves were used in the 2D PWEM.

Using the set of analytic expressions (17), we can extract the coupling constants from the resonant mode frequencies:

κ1=2(ωAωB+ωC+ωD)ωA+2ωB+ωC+2ωDβ0,
(23-1)
κ2=2(ωA+ωBωC+ωD)ωA+2ωB+ωC+2ωDβ0,
(23-2)
κ3=ωA2ωBωC+2ωDωA+2ωB+ωC+2ωDβ0.
(23-3)

Using the resonant mode frequencies (ω=a/λ) calculated from the lasing wavelength in Ref. 4

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

(ω A=0.35906, ω B=0.35942, ω C=0.35965, and ω D=0.35987), the following coupling constants are obtained: κ 1~130 cm-1, κ 2~74 cm-1, and κ 3~39 cm-1. Taking into account the diameter of the lasing area, L=480 µm, the strength of the coupling between waves propagating in opposite directions is found to be κ 3L≈1.87, while the strength of the oblique couplings are found to be κ 1L≈6.24 and κ 2L≈3.55. According to the 1D coupled wave theory [14

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

], if the coupling strength in the opposite direction κ 3L is about 1~2, then the electromagnetic field intensity uniformly distributes throughout the cavity. In addition, sufficient 2D optical coupling is considered to be achieved since the oblique coupling strength κ 1L and κ 2L have larger values than the 1D optical coupling strength κ 3L.

4. Conclusion

We have presented a coupled-wave model for a triangular-lattice 2D PC with a TE polarization and derived a set of coupled-wave equations. We have used these equations to obtain analytic expressions that describe the relations between the resonant mode frequencies and the coupling constants. The resonant mode frequencies were then calculated as a function of the filling factor for a PC with circular holes. Our calculated frequencies are in good agreement with those obtained using the plane wave expansion method. The coupling constants in previously fabricated devices were evaluated using our set of analytic expressions. The method presented in this paper allows the feedback strength of PC cavities in fabricated devices to be evaluated.

Acknowledgments

The authors thank Y. Funato for discussion and help at the beginning of this work. This work was partly supported by Special Coordination Funds for Promoting Science and Technology (SCF), commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

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.

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

3.

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]

4.

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

5.

G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel, “Operating characteristics of a semiconducting polymer laser pumped by a microchip laser,” Appl. Phys. Lett. 82, 313–315 (2003). [CrossRef]

6.

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]

7.

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]

8.

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

9.

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]

10.

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]

11.

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]

12.

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]

13.

T. C. Lu, S. W. Chen, L. F. Lin, T. T. Kao, C. C. Kao, P. Yu, H. C. Kuo, and S. C. Wang, “GaN-based two-dimensional surface-emitting photonic crystal lasers with AlN/GaN distributed Bragg reflector,” Appl. Phys. Lett. 92, 011129 (2008). [CrossRef]

14.

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

15.

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]

16.

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

17.

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

18.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

19.

T. Rowland and E. W. Weisstein, “Character Table,” From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CharacterTable.html

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(160.5298) Materials : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: February 25, 2008
Revised Manuscript: April 5, 2008
Manuscript Accepted: April 7, 2008
Published: April 14, 2008

Citation
Kyosuke Sakai, Jianglin Yue, and Susumu Noda, "Coupled-wave model for triangular-lattice photonic crystal with transverse electric polarization," Opt. Express 16, 6033-6040 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6033


Sort:  Author  |  Year  |  Journal  |  Reset  

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. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, "Laser action from two-dimensional distributed feedback in photonic crystals," Appl. Phys. Lett. 74, 7-9 (1999). [CrossRef]
  3. 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]
  4. M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002). [CrossRef]
  5. G. A. Turnbull, P. Andrew, W. L. Barnes, and I. D. W. Samuel, "Operating characteristics of a semiconducting polymer laser pumped by a microchip laser," Appl. Phys. Lett. 82, 313-315 (2003). [CrossRef]
  6. I. Vurgaftman and J. Meyer, "Design optimization for high-brightness surface-emitting photonic-crystal distributed-feedback lasers," IEEE J. Quantum Electron. 39, 689-700 (2003). [CrossRef]
  7. 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]
  8. E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006). [CrossRef] [PubMed]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. T. C. Lu, S. W. Chen, L. F. Lin, T. T. Kao, C. C. Kao, P. Yu, H. C. Kuo, and S. C. Wang, "GaN-based two-dimensional surface-emitting photonic crystal lasers with AlN/GaN distributed Bragg reflector," Appl. Phys. Lett. 92,011129 (2008). [CrossRef]
  14. H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972). [CrossRef]
  15. 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]
  16. K. Sakai, E. Miyai, and S. Noda, "Two-dimensional coupled wave theory for square-lattice photonic-crystal lasers with TM-polarization," Opt. Express 15, 3981-3990 (2007). [CrossRef]
  17. M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991). [CrossRef]
  18. H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
  19. T. Rowland and E. W. Weisstein, "Character Table," From MathWorld--A Wolfram Web Resource, http://mathworld.wolfram.com/CharacterTable.html.

Cited By

Alert me when this paper is cited

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

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited