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

Optics Express, Vol. 16, Issue 9, pp. 6033-6040 (2008)

http://dx.doi.org/10.1364/OE.16.006033

Acrobat PDF (520 KB)

### 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

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]

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]

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]

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]

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]

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

## 2. Coupled-wave model for triangular lattice

*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:

**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

*a*=|

_{c}**a**

_{1}

**×a**

_{2}|=√3

*a*

^{2}/2.

**G**(

*h*) are given by:

*h*

_{1}and

*h*

_{2}are any two integers, denoted collectively by

*h*, and the primitive translation vectors of this lattice are given by:

*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}=(√3

*a*/2,

*a*/2) and

**a**

_{2}=(0,

*a*), as shown in Fig. 2(a), the primitive reciprocal lattice vectors are

**b**

_{1}=(4π/√3

*a*, 0) and

**b**

_{2}=(-2π/√3

*a*, 2π/

*a*), as shown in Fig. 2(b).

*H*in the TE mode are written in the form [17

_{z}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]

*k*is given by [18]:

*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:

*k*in the form:

*α*(=

*α*

_{G=0}/2) is the averaged gain constant and

*κ*

_{G}is the coupling constant defined as:

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

**G**|=4π/√3

*a*, 4π/

*a*, 8π/√3

*a*contribute significantly. We list the corresponding coupling constants as:

*β*

_{0}=4π/√3

*a*. 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]

*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

*H*(

_{i}**r**) (

*i*=1 to ~6), and rewrite the expression for the magnetic field as the sum:

*δ*is a normalized frequency defined by:

*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.

*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]

*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:

_{6}. 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

## 3. Resonant mode frequencies and coupling constants

*ε*and gain constant

_{a}*α*of the circular hole, and the dielectric constant

_{a}*ε*and gain constant

_{b}*α*of the surrounding material, as shown in Fig. 2(a), the Fourier coefficients are obtained in the form [17

_{b}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]

*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:

*J*

_{1}(

*x*) is a Bessel function of the first kind for integer order 1 and the filling factor is given by

*f*=(2

*π*/√3)

*R*

^{2}/

*a*

^{2}. By substituting Eqs. (20) and (21) into Eq. (9) and using

*λ*

_{0}=

*aε*

_{0}

^{1/2}, the theoretical coupling constant for a circular hole is given by:

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

*ε*=10.5625,

_{a}*ε*=10.6834,

_{b}*a*=462 nm,

*α*=0, and

_{a}*α*=0. The calculated coupling constants are plotted as a function of the filling factor in Fig. 3. At

_{b}*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.

*ω*=

*a*/

*λ*) calculated from the lasing wavelength in Ref. 4

**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*

_{3}L≈1.87, while the strength of the oblique couplings are found to be

*κ*

_{1}L≈6.24 and

*κ*

_{2}L≈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]

*κ*

_{3}L 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*

_{1}L and

*κ*

_{2}L have larger values than the 1D optical coupling strength

*κ*

_{3}L.

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. |

2. | 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. |

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 |

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

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. |

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

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 |

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

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. |

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

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. |

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 |

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. |

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

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. |

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 |

17. | M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B |

18. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. |

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: Year | Journal | Reset

### References

- M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, "Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure," Appl. Phys. Lett. 75, 316-318 (1999). [CrossRef]
- 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]
- 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]
- M. Imada, A. Chutinan, S. Noda, and M. Mochizuki, "Multidirectionally distributed feedback photonic crystal lasers," Phys. Rev. B 65, 195306 (2002). [CrossRef]
- 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]
- 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]
- 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]
- E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, "Lasers producing tailored beams," Nature 441, 946 (2006). [CrossRef] [PubMed]
- K. Sakai, E. Miyai, T. Sakaguchi, D. Ohnishi, T. Okano, and S. Noda, "Lasing band-edge identification for a surface-emitting photonic crystal laser," IEEE J. Sel. Areas Commun. 23, 1335-1340 (2005). [CrossRef]
- M. 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]
- 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]
- 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]
- 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]
- H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," J. Appl. Phys. 43, 2327-2335 (1972). [CrossRef]
- K. Sakai, E. Miyai, and S. Noda, "Coupled-wave model for square-lattice two-dimensional photonic crystal with transverse-electric-like mode," Appl. Phys. Lett. 89, 021101 (2006). [CrossRef]
- K. Sakai, 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]
- M. Plihal and A. A. Maradudin, "Photonic band structure of two-dimensional systems: The triangular lattice," Phys. Rev. B 44, 8565-8571 (1991). [CrossRef]
- H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).
- 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.

« Previous Article | Next Article »

OSA is a member of CrossRef.