## Low-threshold laser oscillation due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals

Optics Express, Vol. 4, Issue 12, pp. 481-489 (1999)

http://dx.doi.org/10.1364/OE.4.000481

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### Abstract

An analytical expression of the lasing threshold for arbitrary photonic crystals was derived, which showed their reduction due to small group velocities of electromagnetic eigenmodes. The lasing threshold was also evaluated numerically for a two-dimensional photonic crystal by examining the divergence of its transmission and reflection coefficients numerically. A large reduction of lasing threshold caused by a group-velocity anomaly that is peculiar to two- and three-dimensional photonic crystals was found.

© Optical Society of America

## 1. Introduction

5. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal,” Opt. Express **4**, 167–176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm [CrossRef] [PubMed]

5. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal,” Opt. Express **4**, 167–176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm [CrossRef] [PubMed]

6. K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B **54**, 5732–5741 (1996). [CrossRef]

7. K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B **54**, 5742–5749 (1996). [CrossRef]

5. K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal,” Opt. Express **4**, 167–176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm [CrossRef] [PubMed]

8. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. **75**, 1896–1899 (1994). [CrossRef]

8. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. **75**, 1896–1899 (1994). [CrossRef]

9. S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. 2, Lett. **37**L565–L567 (1998). [CrossRef]

## 2. Theory

*ω*

_{kμ}is the angular frequency of the eigenmode and

*υ*

_{g}(

**k**

_{μ}) is the component the group velocity parallel to

**k**.

*F*(

**k**

*μ*) is the averaged density of the impurity weighted with the distribution of the eigenfunction,

**E**

_{kμ}:

*V*

_{0}denotes the volume of the unit cell.

**E**

_{kμ}is normalized as follows.

*∊*(

**r**) denotes the position-dependent dielectric constant. Note that

**E**

_{kμ}is dimensionless by this definition. Also note that the real part of

*β*(

**k**

*μ*) is positive since the imaginary part of

*α*(

*ω*

_{kμ}) is negative due to the population inversion. Because the amplification factor is inversely proportional to the group velocity, we can expect the enhancement of stimulated emission at photonic band edges and other frequency regions where the eigenmodes have small group velocities. Since the energy velocity is equal to the group velocity in the photonic crystal [16

16. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. **69**, 742–756 (1979). [CrossRef]

8. J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. **75**, 1896–1899 (1994). [CrossRef]

*L*and take a simple model. That is, we assume that the wave function inside the specimen is the same as that of an infinite crystal. Although this is a rough assumption and the field distribution near the surface of the specimen may be considerably different from that in the infinite system, this assumption leads to a qualitatively correct estimation as will be shown below. When we denote the amplitude reflection coefficient of the relevant eigenmode at each surface by

*R*(

**k**

*μ*), the lasing threshold is given by the balance between the loss at both surfaces and the optical gain in the pass of 2

*L*:

*k*= |

**k**|. In this equation, we took into consideration the phase shift of 2

*kL*, which is consistent with Bloch’s theorem. On the other hand, we have neglected additional loss mechanisms such as spontaneous emission and light scattering by defects.

17. K. Sakoda, “Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice,” J. Opt. Soc. Am. B **14**, 1961–1966 (1997). [CrossRef]

*η*

_{eff}=

*c*/

*υ*

_{g}, where

*c*is the light velocity in vacuum. Therefore, we assume that the reflection coefficient can also be approximated by that of a uniform material with a refractive index

*η*

_{eff}:

*m*is an integer. Here we neglect the (

**k**,

*μ*) dependence of

*F*for simplicity and approximate it by the ratio of the mean value of the density of the impurity atoms to that of the real part of the dielectric constant,

*n̅*/

*∊̅*′. When we denote the imaginary part of the dielectric constant and that of the polarizability by

*∊*″ and

*α*″, the real part of Eq. (6) gives the lasing threshold as follows.

*f*denotes the proportion, or the filling factor, of the dielectric material in which the impurity atoms with population inversion are doped. If

*υ*

_{g}≪

*c*, the threshold is proportional to

**4**, 167–176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm [CrossRef] [PubMed]

*Γ*-

*X*direction, or (1, 0) direction. The front and rear surfaces of the crystal were perpendicular to the propagation direction and that the distance between each surface and the center of the first air-cylinder was half a lattice constant. The transmittance and the reflectance were calculated by means of the plane-wave expansion method formulated previously [17

17. K. Sakoda, “Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice,” J. Opt. Soc. Am. B **14**, 1961–1966 (1997). [CrossRef]

19. K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B **51**, 4672–4675 (1995). [CrossRef]

20. K. Sakoda, “Transmittance and Bragg reflectivity of two-dimensional photonic lattices,” Phys. Rev. B **52**, 8992–9002 (1995). [CrossRef]

## 3. Numerical Results and Discussion

### 3.1 E polarization

*E*polarization with electric field parallel to the cylinder axis. On the other hand, the right-hand side shows the threshold of laser oscillation, which will be explained later. In Fig. 1, the ordinate denotes the normalized frequency where

*a*stands for the lattice constant of the two-dimensional photonic crystal. The dispersion relations were calculated by the plane-wave expansion method [21

21. 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*points in the two-dimensional Brillouin zone, i.e., from (0, 0) to (

*π*/

*a*, 0). Solid lines denote symmetric modes, whereas a dashed line denotes an antisymmetric (uncoupled) mode that does not contribute to light transmission because of the mismatching of a symmetry property [22–25

22. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. **68**, 2023–2026 (1992). [CrossRef] [PubMed]

*ω*,

*∊*″). The maximum of the calculated values was about 10

^{5}in Fig. 2. We obtained a larger maximum when we examined the region around the peak with a finer mesh of

*ω*and

*∊*″. Therefore, we could judge that the transmittance and/or reflectance were divergent, and hence, the threshold of laser oscillation was attained. We found six other divergent points in the frequency range of the third symmetric band. These points correspond to longitudinal modes with different

*m*in Eq. (6). Figures 3 and 4 show other examples of the divergence found in the frequency ranges of the first and the second bands, respectively. Seven other divergent points were found for each band.

*∊*″

_{th}smaller by two orders of magnitude than that necessary for the first and the second bands. Because -

*∊*″

_{th}is proportional to the pumping rate to create the inverted population, we can conclude that the group-velocity anomaly brings about the reduction of the threshold by two orders of magnitude for the present example. We should also note that the threshold is somewhat small at the upper edges of the first and the second bands compared with that in the middle of both bands. This decrease is caused by the small group velocity at the band edges, which should be equal to zero for a system with infinite thickness. However, for a system with finite thickness, the group-velocity anomaly is much more efficient for the reduction of the lasing threshold even though the relevant group velocity is not exactly equal to zero.

*ωa*/2

*πc*= 0.789 at the upper edge of the third symmetric band is an exception. The incident light excites the third and the fourth symmetric bands simultaneously at this frequency, since the lower edge of the latter is

*ωa*/2

*πc*= 0.784. Because the average group velocity of the fourth symmetric band is comparable with that of the second band, a lasing threshold of the same order is expected. Therefore, the lasing at

*ωa*/2

*πc*= 0.789 should be attributed to the fourth symmetric band. On the other hand, a low-threshold lasing with

*∊*″

_{th}= -2.5 × 10

^{-3}, which should be attributed to the third symmetric band, is observed at

*ωa*/2

*πc*= 0.788 just below the lasing frequency for the fourth symmetric band mentioned above. As this example shows, the lasing with different origins can coexist in the frequency ranges where more than one band overlap each other.

*ωa*/2

*πc*= 0.191 (1st band), 0.558 (2nd band), and 0.757 (3rd band), which are obtained from the slope of each band, are 1.35, 1.4, and 7.0. Then, the lasing threshold predicted by Eq. (7) are

*∊*″

_{th}= -2.9,-8.9×10

^{-1}, and-2.1×10

^{-2}, respectively. These values coincide with the numerical results if we make allowance for an error of a factor of three that may be caused by the rough assumption introduced in Sec. 2. We should note that the fact that the lasing threshold is smaller for the third symmetric band by two orders of magnitude than for the first and the second bands is well reproduced by the analytical estimation, which implies that the reduction of the lasing threshold is really brought about by both the enhancement of stimulated emission and the increase of the amplitude reflection coefficient caused by the small group velocity.

### 3.2 H polarization

*H*polarization with magnetic field parallel to the cylinder axis. Figure 5 shows the dispersion relation (left-hand side) and the lasing threshold (right-hand side), where the overall features are common with Fig. 1. We should note that the third lowest symmetric band shows the group-velocity anomaly as

*E*polarization. The effective refractive indices at

*ωa*/2

*πc*= 0.190 (1st band) and 0.565 (2nd band) are 1.34 and 1.37. Then, the lasing thresholds predicted by Eq. (7) are

*∊*″

_{th}= -2.9 and -9.3 × 10

^{-1}, which agree with the numerical results qualitatively as

*E*polarization. On the other hand, the slope of the third band varies considerably with the wave vector and most of its frequency range overlaps that of the fourth symmetric band. Therefore, the comparison with the analytical estimation is not easy for this band. However, it is clearly observed that the lasing thresholds with various magnitude coexist in this frequency range. On the analogy of the numerical results for

*E*polarization, we may regard two longitudinal modes with

*∊*″

_{th}≈ -1.7 × 10

^{-1}at

*ωa*/2

*πc*= 0.763 and 0.797 as originating from the fourth symmetric band, whereas the other four longitudinal modes with |

*∊*″

_{th}| ≤ 1.3 × 10

^{-2}can be attributed to the third symmetric band. In addition, we should note that the longitudinal mode at

*ωa*/2

*πc*= 0.774 with an extremely small threshold of

*∊*″

_{th}≈ -3.0 × 10

^{-4}does not necessarily correspond to a singular point of the dispersion curves. In the previous paper [17

17. K. Sakoda, “Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice,” J. Opt. Soc. Am. B **14**, 1961–1966 (1997). [CrossRef]

*R*for this mode must be especially high, and we may attribute the longitudinal mode at

*ωa*/2

*πc*= 0.774 to this effect.

## 4. Conclusion

## Acknowledgments

## References

1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

2. | C. M. Soukoulis, ed., |

3. | C. M. Soukoulis, ed., |

4. | K. Sakoda, “Photonic crystals,” in |

5. | K. Sakoda, “Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal,” Opt. Express |

6. | K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell’s equations and their applications,” Phys. Rev. B |

7. | K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B |

8. | J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. |

9. | S. Nojima, “Enhancement of optical gain in two-dimensional photonic crystals with active lattice points,” Jpn. J. Appl. Phys. 2, Lett. |

10. | K. Sakoda, “Enhanced stimulated emission in a two-dimensional photonic crystal,” Proc. 1998 Int. Conf. Appl. Phot. Tech., |

11. | M. Sasada, A. Yamanaka, K. Sakoda, K. Inoue, and J. W. Haus, ”Laser oscillation from dye molecules in a 2D photonic crystals,” Technical Digest of the Pacific Rim Conference on Lasers and Electro-Optics, 42–43 (1997). |

12. | K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “Laser action characteristic of a two-dimensional photonic lattice,” 1998OSA Technical Digest Series |

13. | K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, “A two-dimensional photonic crystal laser,” Jpn. J. Appl. Phys. |

14. | M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, “Light-emitting devices with one- and two-dimensional air/semiconductor gratings embedded by wafer fusion technique,” Conference Digest of IEEE International Semiconductor Laser Conference, 211–212 (1998). |

15. | M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, “Surface-emitting laser with two-dimensional photonic band structure embedded by wafer fusion technique,” 1999 OSA Technical Digest Series, in press. |

16. | P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. |

17. | K. Sakoda, “Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice,” J. Opt. Soc. Am. B |

18. | A. Yariv, |

19. | K. Sakoda, “Optical transmittance of a two-dimensional triangular photonic lattice,” Phys. Rev. B |

20. | K. Sakoda, “Transmittance and Bragg reflectivity of two-dimensional photonic lattices,” Phys. Rev. B |

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

22. | W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. |

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

24. | K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B |

25. | K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. III. Group-theoretical treatment,” J. Phys. Soc. Jpn. |

**OCIS Codes**

(250.4480) Optoelectronics : Optical amplifiers

(260.2110) Physical optics : Electromagnetic optics

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 26, 1999

Published: June 7, 1999

**Citation**

Kazuaki Sakoda, Kazuo Ohtaka, and Tsuyoshi Ueta, "Low-threshold laser oscillation due to
group-velocity anomaly peculiar to two- and three-dimensional photonic crystals," Opt. Express **4**, 481-489 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-12-481

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### References

- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
- C. M. Soukoulis, ed., Photonic Band Gaps and Localization (Plenum, New York, 1993).
- C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer, Dordrecht, 1996). [CrossRef]
- K. Sakoda, "Photonic crystals," in Optical Properties of Low-Dimensional Materials, Vol. 2, T. Ogawa and Y. Kanemitsu, ed. (World Scientific, Singapore, 1998). [CrossRef]
- K. Sakoda, "Enhanced light amplification due to group-velocity anomaly peculiar to two- and three-dimensional photonic crystal," Opt. Express 4, 167-176 (1999). http://www.opticsexpress.org/oearchive/source/8698.htm [CrossRef] [PubMed]
- K. Sakoda and K. Ohtaka, "Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwell's equations and their applications," Phys. Rev. B 54, 5732-5741 (1996). [CrossRef]
- K. Sakoda and K. Ohtaka, "Sum-frequency generation in a two-dimensional photonic lattice," Phys. Rev. B 54, 5742-5749 (1996). [CrossRef]
- J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, "The photonic band edge laser: a new approach to gain enhancement," J. Appl. Phys. 75, 1896-1899 (1994). [CrossRef]
- S. Nojima, "Enhancement of optical gain in two-dimensional photonic crystals with active lattice points," Jpn. J. Appl. Phys. 2, Lett. 37 L565-L567 (1998). [CrossRef]
- K. Sakoda, "Enhanced stimulated emission in a two-dimensional photonic crystal," Proc. 1998 Int. Conf. Appl. Phot. Tech., Applications of Photonic Technology 3, Vol. SPIE 3491, edited by G. A. Lampropoulos and R. A. Lessard (SPIE, Washington, D.C., 1998) 248-253.
- M. Sasada, A. Yamanaka, K. Sakoda, K. Inoue, and J. W. Haus, "Laser oscillation from dye molecules in a 2D photonic crystals," Technical Digest of the Pacific Rim Conference on Lasers and Electro-Optics, 42-43 (1997).
- K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, "Laser action characteristic of a two-dimensional photonic lattice," 1998 OSA Technical Digest Series 7, 47-48 (1998).
- K. Inoue, M. Sasada, J. Kawamata, K. Sakoda, and J. W. Haus, "A two-dimensional photonic crystal laser," Jpn. J. Appl. Phys. 38, L157-L159 (1999). [CrossRef]
- M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, "Light-emitting devices with one- and two- dimensional air/semiconductor gratings embedded by wafer fusion technique," Conference Digest of IEEE International Semiconductor Laser Conference, 211-212 (1998).
- M. Imada, S. Noda, A. Chutinan, and Y. Ikenaga, "Surface-emitting laser with two-dimensional photonic band structure embedded by wafer fusion technique," 1999 OSA Technical Digest Series, in press.
- P. Yeh, "Electromagnetic propagation in birefringent layered media," J. Opt. Soc. Am. 69, 742-756 (1979). [CrossRef]
- K. Sakoda, "Numerical analysis of the interference patterns in the optical transmission spectra of a square photonic lattice," J. Opt. Soc. Am. B 14, 1961-1966 (1997). [CrossRef]
- A. Yariv, Quantum Electronics (Wily, New York, 1967) Sec. 19.6.
- K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995). [CrossRef]
- K. Sakoda, "Transmittance and Bragg reflectivity of two-dimensional photonic lattices," Phys. Rev. B 52, 8992-9002 (1995). [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]
- W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, "Measurement of photonic band structure in a two-dimensional periodic dielectric array," Phys. Rev. Lett. 68, 2023-2026 (1992). [CrossRef] [PubMed]
- K. Sakoda, "Symmetry, degeneracy, and uncoupled modes in two-dimensional photonic lattices," Phys. Rev. B 52, 7982-7986 (1995). [CrossRef]
- K. Sakoda, "Group-theoretical classification of eigenmodes in three-dimensional photonic lattices," Phys. Rev. B 55, 15345-15348 (1997). [CrossRef]
- K. Ohtaka and Y. Tanabe, "Photonic bands using vector spherical waves. III. Group-theoretical treatment," J. Phys. Soc. Jpn. 65, 2670-2684 (1996). [CrossRef]

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