## 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.

©1999 Optical Society of America

## 1. Introduction

Because of the periodic spatial modulation of their dielectric constants, the dispersion relations of electromagnetic eigenmodes in photonic crystals are quite different from those in uniform materials [1–4]. In addition to the formation of bands and band gaps of eigenfrequencies, extremely small group velocities can easily be realized at the photonic band edges and by group-velocity anomalies [5]. The group-velocity anomalies, which are peculiar to two- and three-dimensional photonic crystals and are absent for one-dimensional ones, are the phenomena that the group velocities of certain bands of electromagnetic eigenmodes become small over their entire spectral ranges. The origin of the group-velocity anomalies was described in detail in Ref. [5]. The small group velocity of the eigenmodes brings about the enhancement of various optical processes. For example, we reported the enhancement of sum-frequency generation [6,7]. The enhancement of stimulated emission was predicted as well [5, 8–10]. Dowling et al. estimated the enhancement factor for stimulated emission at the band edges of a one-dimensional photonic crystal by analyzing the temporal evolution of a wave packet propagated in the crystal [8]. Later, Nojima calculated the dispersion relation of a two-dimensional photonic crystal assuming a typical gain function for semiconductors [9]. He obtained complex wave vectors as functions of the real angular frequency. Then he could evaluate the spatial light-amplification characteristics and showed the enhancement at the band edges. In addition to the theoretical investigations, the laser oscillation that was attributed to the peculiar dispersion relations in two-dimensional photonic crystals has already been observed experimentally [11–15].

On the other hand, one of the present authors derived an analytical expression for the enhancement factor and reported that the enhanced stimulated emission takes place more efficiently for eigenmodes characterized by their group-velocity anomaly than for those at the band edges [5,10]. Because quite a large enhancement of stimulated emission is expected even for specimens with very small thickness, the experimental observation of the enhancement does not seem difficult. In addition, we will show in this paper that the threshold of laser oscillation is quite low for those eigenmodes with the group-velocity anomaly in comparison with ordinary eigenmodes. In Sec. 2, an analytical expression for the lasing threshold will be derived. The threshold will be evaluated numerically for a two-dimensional photonic crystal and compared with the analytical results in Sec. 3. A brief summary will be given in Sec. 4.

## 2. Theory

Before we present the numerical results that show quite a low threshold of lasing due to the group-velocity anomaly in a two-dimensional photonic crystal, let us treat the problem analytically. In a recent paper [5], one of the present authors derived an expression for the amplitude amplification factor due to stimulated emission, which showed clearly the enhancement caused by the small group velocity of the eigenmode. Let us first summarize the result. In what follows, we assume that impurity atoms or molecules are doped in the photonic crystal and their population inversion is attained by some means such as optical pumping. We denote the polarizability and the density of the impurity atoms by *α*(*ω*) and *n*(**r**), where *ω* and **r** stand for the angular frequency and the position vector. For simplicity, we assume that *n*(**r**) has the same periodicity as the host crystal. When we denote the wave vector and the band index of the eigenmode by **k** and *μ*, we can prove that the amplitude amplification factor in a unit length, *β*(**k**
*μ*), is given by

where *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}:

where *V*
_{0} denotes the volume of the unit cell. **E**
_{kμ} is normalized as follows.

where *∊*(**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], we can regard this enhancement as a consequence of the long interaction time for the impurity atoms and the radiation field as Dowling et al. pointed out [8].

Next, in order to estimate the lasing threshold, let us assume a photonic crystal with a thickness of *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*:

where *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.

Now, it was shown in a previous paper [17] that the interference patterns observed in the transmission spectra of the photonic crystals can be described quite well by an effective refractive index defined by *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}
:

Substituting Eq. (5) into Eq. (4), we obtain

where *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.

In this equation, *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 ${\upsilon}_{g}^{2}$, and hence, its large reduction is expected. Here, we should note that the reduction of the lasing threshold is brought about by both the enhancement of stimulated emission and the increase of the amplitude reflection coefficient. The imaginary part of Eq. (6) determines longitudinal modes, or the wavelength of lasing, which we will not discuss in detail in this paper.

Now, let us proceed to the method for the quantitative evaluation of the lasing threshold. Since it is a process of light emission without input signals, which is analogous to the oscillation of electric circuits, the onset of lasing is equivalent to the divergence of the transmittance and/or the reflectance of the assumed specimen. This criterion was described clearly and applied to the analysis of distributed feedback (DFB) lasers in a book of Yariv [18]. We applied the same method to the present problem. In what follows, we assume that optical impurities with population inversion are doped uniformly in the host material. Then the situation can be modeled by a complex dielectric constant with a negative imaginary part. Because we are interested in the effect of the photonic band structure on the laser oscillation characteristics and the nature of the optical impurities is irrelevant to the following analysis, we assume a frequency-independent dielectric constant. We calculated the transmittance and the reflectance of the assumed specimen as functions of the frequency and the imaginary part of the dielectric constant, and attributed their divergence to the laser oscillation. Because this analysis neglects the energy loss caused by spontaneous emission, it only describes the threshold semi-quantitatively. However, it is sufficient, as will be shown below, to clarify the roll of the group-velocity anomaly.

For the numerical calculation of the transmission and reflection spectra, the same geometry as Refs. [5] and [10] was assumed for which quite a large enhancement of stimulated emission due to the group-velocity anomaly was found. That is, the assumed specimen was a two-dimensional photonic crystal composed of a square array of circular air cylinders formed in a host material with a dielectric constant of 2.1, which is a typical value for transparent organic polymers. The radius of the air cylinders was 0.28 times the lattice constant. The number of the lattice layers was eight and the incident light was propagated in the *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 [17,19,20]

## 3. Numerical Results and Discussion

#### 3.1 E polarization

The left-hand side of Fig. 1 shows the dispersion relation for *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] with 271 basis waves from Γ to *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]. Here, we should note that the third lowest band, besides the uncoupled mode, shows a group-velocity anomaly, i.e., over its entire spectral range, it has a small group velocity, which is given by the slope of the dispersion curve.

Now, Fig. 2 shows an example of the divergence mentioned above, where the sum of the transmittance and the reflectance for the third symmetric band is presented in a logarithmic scale as a function of the normalized frequency and the imaginary part of the dielectric constant. The real part of the dielectric constant was assumed to be 2.1 as before. The numerical calculation was performed for 100 × 40 sets of (*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.

The divergent points are plotted on the right-hand side of Fig. 1. As can be clearly seen, the laser oscillation in the frequency range of the third lowest band takes place with - *∊*″_{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.

The high threshold at *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.

Now, let us compare the numerical results with the analytical estimation based on Eq. (7). Because the slope of the dispersion curves varies considerably in the vicinity of the band edges, the estimation of the group velocity, and hence, that of the lasing threshold is difficult at those frequency ranges. Therefore, we compare the numerical and the analytical results in the middle of the frequency range of each band. The effective refractive indices at *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

Now let us examine the case of *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], quite sharp and irregular patterns were found in the transmission spectrum in this frequency range, which were attributed to the interference between the relevant two bands. Judging from the extraordinary sharpness of the interference patterns, we may conclude that standing waves with extremely high quality factors were realized as a consequence of the interference. Therefore, the amplitude reflection coefficient *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

We derived an analytical expression to evaluate the threshold of lasing in the photonic crystals and showed that the threshold is proportional to the square of the group velocity of the relevant electromagnetic eigenmode when it is sufficiently smaller than the light velocity in vacuum. The reduction of the lasing threshold was brought about by the enhancement of stimulated emission due to the long interaction time between the electromagnetic field and the matter, and by the increase of the reflection coefficient at the surface of the crystal. The lasing threshold was also evaluated numerically by analyzing the divergence of the transmittance and the reflectance for a two-dimensional photonic crystal composed of a square array of circular air cylinders formed in a dielectric host material with a dielectric constant of 2.1, for which a large enhancement of stimulated emission due to the group-velocity anomaly had been found. A large reduction of the lasing threshold due to the group-velocity anomaly was shown by numerical calculation for a specimen composed of eight lattice layers. The numerical results agreed with the analytical evaluation qualitatively. We also found that the lasing with mutually different thresholds can coexist when the frequency ranges of more than one bands overlap each other.

## Acknowledgments

This work was supported by the Grant-in-Aid for Scientific Research of Priority Area “Photonic Crystals” from the Ministry of Education, Science, Sports, and Culture of Japan. K. S. was financially supported by the Sumitomo Foundation and the Inamori Foundation. K. O. and T. U. were supported by the Casio Foundation.

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