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An analytical expression was derived for light amplification by stimulated emission in arbitrary photonic crystals, which showed an enhancement due to small group velocity. This enhancement was evaluated quantitatively for a two-dimensional crystal with a finite thickness, and an extremely large enhancement due to group-velocity anomaly peculiar to two- and three-dimensional crystals was found even for quite a thin crystal.
©1999 Optical Society of America
1. Introduction
Because of the periodic spatial variation of their dielectric constants, the electromagnetic eigenmodes in photonic crystals form bands and band gaps of eigenfrequencies just as electronic Bloch waves in ordinary crystals do [1–4]. Then the photonic band gaps, especially those in the optical (visible) region, are expected to bring about quite peculiar physical phenomena such as the inhibition of spontaneous emission [5] and energy transfer [6], localized donor and acceptor modes [7–12], stable solitary waves [13,14], nonexponential decay of spontaneous emission [15,16], etc. Here, we have to note that a considerably large three-dimensional variation of the dielectric constant is necessary in order to obtain a complete band gap spanning over whole Brillouin zone for all polarizations of radiation field.
On the other hand, the small group velocity of the eigenmodes that also brings about peculiar optical phenomena, such as the enhancement of sum-frequency generation and other nonlinear optical processes [17,18], can be easily attained even with a low contrast of the dielectric constant. This is particularly true for two- and three-dimensional crystals. Here, let us explain this point with an example in some detail. Figure 1 shows the photonic band structure, or the dispersion relation, of a two-dimensional crystal composed of a regular square array of circular air-rods formed in a dielectric material with a dielectric constant of 2.1, where the essential features of the two-dimensional photonic bands appear. The photonic band structure was calculated for E polarization with the electric field parallel to the rod axis by means of the plane-wave expansion method [19] with 271 plane waves. The computational error was estimated as less than 1 % by changing the number of the basis plane waves. In Fig. 1, the ordinate represents the normalized frequency where ω, a, and c denote the angular frequency of the radiation field, the lattice constant of the two-dimensional photonic crystal, and the light velocity in vacuum, respectively. The abscissa represents the wave vector in the two-dimensional Brillouin zone where Γ, X, and M denote highly symmetric points (0, 0), (π/a, 0), and (π/a, π/a), respectively. On the other hand, S and A denote symmetric and antisymmetric modes [20–23], respectively. Because of the mismatching of the spatial symmetry, the A modes cannot be excited by external plane waves.
It is evident in Fig. 1, as it should be, that the group velocity of the eigenmodes, which is given by the derivative of ω with respect to the wave vector, is equal to zero at the highly symmetric points in most cases and at those points where two modes of the same symmetry repel each other. In addition, a special attention should be paid to the small group velocity of the third lowest S mode and the lowest A mode in the Γ-X direction over their whole frequency ranges. This feature originates from the following fact: For these modes, the wave vector of the unperturbed wave function in the extended zone scheme, i.e., the wave vector when the spatial modulation of the dielectric constant is absent is not parallel to that of the exact eigenfunction in the reduced zone scheme. In addition, the angle between these two vectors is very large. Therefore, the change in the wave vector along the Γ-X direction in the reduced zone scheme results in only a small variation of the length of the wave vector in the extended zone scheme, and hence, results in a small variation of the angular frequency as well. We should note that this feature is peculiar to two- and three-dimensional crystals and is absent for one-dimensional ones, since all wave vectors are parallel to each other in one-dimensional crystals. Because essentially it is not relevant to the amount of the spatial variation of the dielectric constant, this feature is recognizable even for crystals with a small contrast of the dielectric constant as this example. We will call this feature peculiar to two- and three-dimensional photonic crystals group-velocity anomaly hereafter.
Figure 1. The dispersion relation for E polarization in a two-dimensional photonic crystal composed of a regular square array of circular air-rods formed in a dielectric material with a dielectric constant of 2.1. The ordinate is the normalized frequency where ω, a, and c denote the angular frequency of the radiation field, the lattice constant of the two-dimensional crystal, and the light velocity in vacuum, respectively. The abscissa represents the wave vector in the two-dimensional Brillouin zone where Γ, X, and M stand for (0, 0), (π/a, 0), and (π/a, π/a). The radius of the air-rods was assumed to be 0.28 times the lattice constant. Symmetric and antisymmetric modes are denoted by S and A, respectively. Note that the third lowest S mode and the lowest A mode in the Γ-X direction has a small group velocity over their whole frequency ranges.
In addition to the enhancement of nonlinear optical processes mentioned previously, the small group velocity of electromagnetic eigenmodes brings about the enhancement of stimulated emission if optical centers with population inversion are dispersed in the photonic crystals [24–26]. This phenomenon originates from the long interaction time between radiation field and matter caused by the small group velocity as was pointed out by Dowling et al. [24]. They calculated the temporal evolution of a Gaussian wave packet incident on a one-dimensional photonic crystal and showed the enhancement of an optical gain at a photonic band edge. This enhancement is similar to that of distributed feedback (DFB) lasers caused by the formation of a standing wave [27]. On the other hand, Nojima calculated the dispersion relation of a two-dimensional photonic crystal assuming a typical gain function for semiconductors [25]. He obtained complex wave vectors as functions of the real angular frequency, ω. Then he could evaluate the spatial light-amplification characteristics and showed the enhancement at the band edges.
In this paper, we will report that the group-velocity anomaly brings about the enhancement of light amplification more efficiently than the small group velocity at the band edges. Especially, we will report that a large enhancement is expected even for a photonic crystal with a very small number of lattice layers. We will show this fact by calculating the light amplification spectra of the two-dimensional photonic crystal dealt with in Fig. 1. In Sec. 2, an analytical expression for light amplification factor will be derived. The light amplification spectra will be evaluated numerically and compared with the dispersion relation for both E and H polarizations in Sec. 3. A brief summary will be given in Sec. 4.
2. Analytical Expression for Light Amplification
Before we present the numerical results that show quite a large enhancement of the stimulated emission due to the group-velocity anomaly in the two-dimensional photonic crystal, let us derive an analytical expression for the light amplification factor in order to clarify the origin of the enhancement. The following derivation is valid when the density of optical centers is small. We assume in what follows that impurity atoms (or molecules) are doped in the photonic crystal and their population inversion is attained by some means such as optical pumping. Let us denote the polarizability and the density of the impurity atoms by α(ω) and n(r), where r stands for the position vector. For simplicity, we assume that n(r) has the same periodicity as the host crystal. When an eigenmode with a unit amplitude whose electric field is denoted by E kμ(r), where k is a wave vector in the first Brillouin zone and μ is the band index, is propagated in the photonic crystal, it induces an extrinsic polarization field P ex due to the impurity atoms. Because n(r) is small as we assumed, we can neglect the local field correction and P ex is given by
Note that α(ω) has a negative imaginary part because of the population inversion. We will set ω equal to the eigen-angular frequency ω kμ at the end of the calculation in order to avoid an unphysical divergence.
In a previous paper [17], the author and his coworker derived a general expression for the electric field E ex(r,t) induced by the extrinsic polarization P ex(r,t):
where ∈(r) and V denote the position-dependent dielectric constant and the volume of the photonic crystal. In Eqs. (1) and (2), E kμ(r) is normalized such that
Note that E kμ(r) is dimensionless by this definition. The second term in the right-hand side of Eq. (2) describes the propagating part of the induced electric field. Next, we convert the summation with respect to k′ into an integral in frequency region. After following the same procedure as in Sec. 3 of Ref. [17], we finally obtain
where l is the length of the optical path and vg (k μ) is the component of the group velocity of the eigenmode parallel to k. F(k μ) is given by
where V 0 denotes the volume of the unit cell. Then, combined with the incident wave, the total electric field is given by
Therefore, the amplitude amplification factor is proportional to and a large enhancement of light amplification is expected when vg is small. Since the energy velocity is equal to the group velocity in the photonic crystals [28], we can regard this fact as a consequence of the long interaction time for the impurity atoms and the radiation field as Dowling et al. pointed out [24].
3. Numerical Results and Discussion
3.1 E polarization
Because we are interested in the influence of the photonic band structure on the stimulated emission and the nature of the impurity atoms is irrelevant to the following discussion, we assume that the polarizability of the impurity atoms is independent of ω. In addition, we assume for simplicity that the impurity atoms are uniformly distributed in the dielectric material. Then this situation can be modeled by simply assuming a complex dielectric constant with a negative imaginary part for the dielectric material. Then the transmission and reflection spectra were calculated for the two-dimensional photonic crystal with 2 to 16 layers of the air-rods in the dielectric material by means of the plane-wave expansion method formulated previously [29–31].
In order to confirm the accuracy of our numerical method, a transmission spectrum with a real dielectric constant was calculated and is compared with the dispersion relation in Fig. 2. The same parameters as Fig. 1 were used for numerical calculation. For the calculation of the transmission spectrum, it was assumed that the number of the lattice layers was 16 and the incident light was propagated in the Γ-X direction. We also assumed that 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-rod was half a lattice constant. As can be clearly observed, there is a band gap at ωa/2πc = 0.702 to 0.728 and it corresponds to the spectral range with low transmittance very well. The periods of interference patterns at ωa/2πc = 0.732 to 0.782 and below 0.683 also correspond very well to the effective refractive indices evaluated from the group velocities of the relevant bands. The coincidence between the two independent calculations give clear evidence for the accuracy of our numerical method.
Figure 2. The comparison between the dispersion relation (the left-hand side) and the transmission spectrum (the right-hand side) for E polarization in the Γ-X direction. The same parameters as Fig. 1 were assumed for numerical calculation. The number of the lattice layers was assumed to be 16 for the calculation of the transmission spectrum. The dashed line in the left-hand side represents an A (antisymmetric) mode.
Now let us proceed to the quantitative evaluation of the enhancement of stimulated emission. In the following calculation, the dielectric constant of the host material was assumed to be 2.1 - 0.01i. Figure 3 shows the sum of the transmittance and the reflectance for an incident wave with E polarization propagated in the Γ-X direction where each solid line with filled circles denotes that of the two-dimensional crystal and each dashed line denotes that of a uniform plate of the same thickness with a spatially averaged dielectric constant. The latter was calculated for comparison. The calculation was done for crystals with (a) 16, (b) 8, (c) 4, and (d) 2 lattice layers. Note that the sum can be greater than unity because of the stimulated emission that takes place in the crystals and the uniform plates. Let us first examine Fig. 3(a) in detail. The solid line clearly shows a large enhancement of the stimulated emission at ωa/2πc = 0.788, which exactly coincides with the upper edge of the third lowest S mode where vg = 0. It also shows a peak at ωa/2πc = 0.701 and periodic peaks at ωa/2πc = 0.733 to 0.784. The former exactly coincides with the upper edge of the second lowest S mode whereas the latter coincides with the frequency range of the third lowest S mode. Therefore it is evident that the enhancement of the stimulated emission originates from the low group velocity of the eigenmodes. The enhancement factor, which was calculated as the ratio of the stimulated emission in the photonic crystal and that in the uniform plate was as large as 45 at ωa/2πc = 0.788.
When the number of the lattice layers is decreased, the spectrum changes considerably. Especially, the peaks at the band edges become small. In Fig. 3(c), the peaks that were observed at the upper edges of the second and the third lowest S modes in Fig. 3(a) are absent. But two peaks in the frequency range of the third lowest S mode, where the group velocity is quite small, can be clearly observed. The enhancement factor is surprisingly large, that is, 130 at ωa/2πc = 0.755 and 68 at ωa/2πc = 0.776 even for such a thin geometry. Because the assumed geometry is far from the infinite crystal, it is quite reasonable that the overall correspondence between Fig. 3(c) and the photonic band structure in Fig. 1 or Fig. 2 is obscure. Especially, the enhancement just at the band edges where vg = 0 may be difficult to attain with a thin geometry, since the wave vector parallel to the propagation direction is not well defined and the correspondence with the band structure is not assured. However, when the frequency range with a small group velocity is wide enough as was realized in the third lowest S mode of the present example, the thin geometry has traces of the perfect photonic band structure and we can still expect a large enhancement of the light amplification. A small enhancement is observed even for two layers as shown in Fig. 3(d). Because we do not need a large contrast of the dielectric constant and a very thin crystal is enough to observe the enhancement, we believe that the experimental confirmation of this effect is not difficult.
Figure 3. The sum of the transmittance and the reflectance for E polarization as a function of the normalized frequency calculated for a crystal with (a) 16, (b) 8, (c) 4, and (d) 2 layers of air-rods formed in the dielectric material with a dielectric constant of 2.1-0.01i (solid line with filled circles) and a uniform plate of the same thickness with a spatially averaged dielectric constant (dashed line). The negative imaginary part of the dielectric constant stands for the inverted population of the impurity atoms. The incident light was assumed to be propagated in the Γ-X direction. Note that the sum can be greater than unity because of the stimulated emission.
3.2 H polarization
Now let us examine the case of H polarization with magnetic field parallel to the rod axis. Figure 4 shows the dispersion relation for this case. The same parameters as Fig. 1 were used for numerical calculation. The group-velocity anomaly is observed for the third lowest S mode and the lowest A mode in the Γ-X direction as E polarization.
Figure 4. The dispersion relation for H polarization in the two-dimensional photonic crystal composed of a regular square array of circular air-rods formed in a dielectric material. The same parameters as Fig. 1 were assumed for numerical calculation. Symmetric and antisymmetric modes are denoted by S and A as before. Note that the third lowest S mode and the lowest A mode in the Γ-X direction show the group-velocity anomaly as E polarization.
The dispersion relation is compared with the transmission spectrum in Fig. 5. The band gap at ωa/2πc = 0.728 to 0.742 corresponds to a spectral range with low transmittance very well. The period of interference patterns below 0.720 also corresponds very well to the effective refractive index evaluated from the group velocity of the second lowest S mode. Therefore, we may conclude that our numerical method is accurate for H polarization as well. The third lowest S mode is convex downward near the Γ point, and most of its frequency range overlaps that of the fourth S mode. This fact implies that the incident plane wave excites two eigenmodes simultaneously for the most part of the third S branch. This is the origin of quite singular interference patterns observed in that frequency range.
Figure 5. The comparison between the dispersion relation (the left-hand side) and the transmission spectrum (the right-hand side) for H polarization in the Γ-X direction. The same parameters as Fig. 2 were assumed for numerical calculation. The dashed line in the left-hand side represents an A mode.
The sum of the transmittance and the reflectance for H polarization is presented in Fig. 6. The same parameters as Fig. 3 were used for numerical calculation. In Fig. 6(a), which was calculated for a crystal with 16 lattice layers, the enhancement of stimulated emission can be clearly observed near the upper band edge of the second S mode, ωa/2πc = 0.728, and just at the lower band edge of the third S mode, ωa/2πc = 0.742. When the number of the lattice layers is decreased to four, the peak near the upper band edge of the second S mode disappears completely. However, the peak in the frequency range of the third S mode can be clearly observed even for a crystal with only two lattice layers! In the case of H polarization for this example, the main peak of the light amplification spectrum happens to coincide with the lower band edge of the relevant mode where vg = 0. Therefore, it may be difficult to distinguish whether the enhancement is caused by the vanishing group velocity at the band edge or the group-velocity anomaly. However, judging from the disappearance of the enhancement at the upper band edge of the second S mode with decreasing lattice layers, we may conclude that the group-velocity anomaly surely contributes to the enhancement of stimulated emission predicted for the two-dimensional crystal with the small contrast of the dielectric constant and the very small number of lattice layers.
Figure 6. The sum of the transmittance and the reflectance for E polarization as a function of the normalized frequency calculated for a crystal with (a) 16, (b) 8, (c) 4, and (d) 2 layers of air-rods formed in the dielectric material with a dielectric constant of 2.1- 0.01i (solid line with filled circles) and a uniform plate of the same thickness with a spatially averaged dielectric constant (dashed line). The incident light was assumed to be propagated in the Γ-X direction.
4. Conclusion
We have derived an analytical expression for the light amplification in arbitrary photonic crystals that is valid when the density of optical centers with population inversion is small, and shown the enhancement due to the small group velocity of the electromagnetic eigenmodes. In addition, the enhancement was evaluated quantitatively for both E and H polarizations for a two-dimensional crystal composed of a square array of air-rods formed in a dielectric material with a dielectric constant of 2.1. An enhancement factor as large as 130 was found for E polarization for quite a thin geometry with only four lattice layers. In the case of H polarization, the enhancement was clearly observed even for two lattice layers. The origin of the enhancement was attributed to group-velocity anomaly peculiar to two- and three-dimensional photonic crystals.
Acknowledgments
This work was supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports, and Culture of Japan. It was also financially supported by the Sumitomo Foundation and the Inamori Foundation.
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