## Abstract

Double enhancement of spontaneous emission due to increased photon density of states at the emission frequency and the small group velocity of light at the excitation frequency was clearly demonstrated by angle-resolved photoluminescence experiments for dielectric multilayers composed of Ta2O_{5} and SiO_{2} with oxygen vacancies as light emitters. Theoretical emission profiles given by the weak modulation approximation agreed well with the experimental observations.

© 2009 Optical Society of America

## 1. Introduction

Light-matter interaction can be modified by the design of the radiation field — this is one of the key concepts in modern optics. Because the dispersion relation and the photon density of states (DOS) can be designed by photonic crystals (PCs) [1, 2, 3, 4, 5], many experimental studies have focused on this issue. Whereas the acceleration of spontaneous emission of photons by PC microcavities (Purcell effect) was successfully demonstrated [6, 7, 8, 9], that by the increased photon DOS at photonic band edges has not been clearly verified except the case of PC waveguides [10, 11, 12], although it is another crucial test of quantum electrodynamics. This is partly because we need a large number of structured unit cells that are regularly arrayed in the submicron scale to realize a sharp peak in the spectrum of the photon DOS, but it has still been difficult to fabricate satisfactory specimens.

There are two key factors for enhanced spontaneous emission in PCs. One is increased photon DOS at the emission frequency, which brings about an accelerated optical transition according to Fermi’s golden rule. The other is small photon velocity at the photo-excitation frequency, which results in a large amplitude of the local electric field due to conservation of Poynting’s vector.

As for the first factor, this mechanism works only for one-dimensional PCs or for photon emission in particular directions in two- and three-dimensional cases. When we assume a parabolic dispersion relation in the vicinity of a photonic band edge frequency, *ω*
_{0}, such that

where **k** is the photon wave vector, **k**
_{0} is that of the photonic band edge, and *m** is the photon effective mass (see Fig. 1(a)), the photon DOS, *𝓓*(*ω*), is proportional to $\sqrt{\omega -{\omega}_{0}}$ for three dimensions, step function *θ* (*ω-ω*
_{0}) for two dimensions, and $1\u2044\sqrt{\omega -{\omega}_{0}}$ for one dimension. So, the singular enhancement of the optical transition rate is expected only for the one-dimensional case. Because all actual specimens are three-dimensional, we cannot expect a dramatically shortened radiative lifetime at the photonic band edge; rather, we can expect an enhanced emission intensity in certain directions. So, although the change in the emission lifetime of CdSe quantum dots in inverse opal was observed in Ref. [13], it is not relevant to the divergent band-edge DOS. Reference [14] reported peculiar spontaneous emission spectra of GaAs quantum wells sandwiched by DBR (distributed Bragg reflector) structures. However, in this experimental study, the light emitter was located in the vicinity but outside of the one-dimensional PC. As for theoretical studies, Dowling et al. and Scalora et al. presented detailed calculations on finite one-dimensional PCs [15, 16].

As for the second factor, let us describe only briefly the mechanism of the local field enhancement with Fig. 1(b), since this is well known [18, 19]. We assume that a plane wave is propagating into a PC at a normal incidence. We also assume for simplicity that the transmittance of the entrance surface is close to 100% and neglect the reflected wave, which can actually be realized by providing an anti-reflection coating for example. Then at the surface, Poynting’s vector, which is given by the product of the energy density (*U*) and the velocity (*v*) of the electromagnetic wave, should be conserved. Denoting quantities in air and the PC by 0 and pc, we have *U _{0}v_{0}*=

*U*. On the other hand, there is a general rule that the energy velocity is equal to the group velocity [4, 17]. Thus we have

_{pc}v_{pc}*v*=

_{pc}*dω/dk*=

*h*̄

^{2}(

*k−k*)/

_{0}*m**. Because the electric field,

*E*, is proportional to the square root of

*U*, we have ${E}_{\mathrm{pc}}\propto {E}_{0}\sqrt{{v}_{0}\u2044{v}_{\mathrm{pc}}}$, which diverges at the photonic band edge and results in an increased efficiency of photo-excitation of fluorescent species. This technique is called the band-edge excitation and was used for lasing [18] and control of the refractive index [19]. Although the decrease in the lasing threshold by the band-edge excitation was confirmed for the liquid crystal laser in Ref. [18], its experiment was apparently performed in the nonlinear region and was not relevant to enhanced spontaneous emission. We should note that the lasing threshold depends on not only the DOS at the emission frequency but also various quantities such as reflectance of the laser cavity and the amount of optical losses of various origins. So, the relation between the lasing threshold to the DOS at the emission frequency is not straightforward at all. Astic et al. observed the refracitve index change due to the enhanced local electric field by the band-edge excitation [19]. But they did not observe the enhanced spontaneous emission.

In the case of one-dimensional PCs, these two factors are closely related to each other as is well known. In fact, the one-dimensional photon DOS, *𝓓*
_{1}, is given by *𝓓*
_{1}=1/2*πv*
_{pc}. So, we can expect a double enhancement for the photo-excitation and successive spontaneous emission process, that is, the enhancement of photo-excitation of fluorescent species due to the small *v*
_{pc} and that of their emission process due to the large *𝓓*
_{1}, if the Stokes shift of the fluorescent species is negligibly small. In practice, the Stokes shift is often larger than the characteristic width of the peak in *𝓓*
_{1}(*ω*). This problem may be overcome by using the upper edge of the photonic bandgap for excitation and the lower edge for emission. Instead, this problem is resolved in this paper by using *the two-dimensional dispersion o f a one-dimensional PC* and we clearly demonstrate the double enhancement of spontaneous emission.

To realize the double enhancement, four crucial conditions have to be satisfied, which all previous experimental studies on this issue failed to fulfill: (1) the regularity of the PC structure should be high enough, (2) the number of unit cells should be large enough, (3) the fluorescent species should be dispersed uniformly in the PC, and (4) the imperfection caused by the specimen’s surface should be relatively small compared with the bulk emission. The first and second conditions are necessary to create a sharp peak in *𝓓*
_{1}(*ω*) at the photonic band edge. The third condition is necessary to average out the spatial variation of the electric field in the PC and clearly observe the pure DOS effect. The fourth condition is necessary to diminish the surface effect that causes a deviation from ideal optical properties of one-dimensional PCs. The fourth condition is, however, automatically fulfilled when the first and second conditions are satisfied.

In this paper, we present the angle-resolved photoluminescence spectra of a one-dimensional PC that clearly show the double enhancement of spontaneous emission caused by the increased photon DOS and small vpc. The light emitters were distributed uniformly and the regularity of the specimen was good enough to observe the gap-edge enhancement. We also present a calculation of the enhanced luminescence spectra that supports the experimental observations.

## 2. Experimental

The specimen used in this study was a commercial optical notch filter made of dielectric multi-layers designed to eliminate the 633 nm line of the He-Ne laser (Semrock, NF01-633U), which satisfies all four conditions mentioned above. This notch filter gives a stop band from 620 to 645 nm with an optical density larger than 6 at the center wavelength. SEM (scanning electron microscope) and EPMA (electron probe microanalyzer) analyses showed that the notch filter consisted of Ta2O_{5} and SiO_{2} multilayers on a SiO_{2} substrate. Both Ta2O_{5} and SiO_{2} have broad luminescence bands around 640 nm due to oxygen vacancies [20, 21], so we can use these bands as the natural photon source embedded uniformly in the one-dimensional PC.

Figure 2 schematically shows the experimental configuration, where *a* is the lattice constant, *ϕ* is the observation angle, and *ω* and *k* are the angular frequency and wave number of the emitted photons. The sample was excited by the 532 nm line of a CW (continuous wave) green laser (Coherent, Verdi V6) and a CW tunable dye laser with a linewidth of less than 0.02 nm (Coherent, 699-21) through a lens with focal length of 100 mm. The estimated beam size on the specimen was 15 *µ*m. Emitted light was collected by another lens with the same focal length. A pinhole was located between the specimen and the collection lens to decrease the uncertainty of the detection angle down to 2°. The emission spectra were measured by a triple-grating polychrometer (Bunkoh-Keiki, M330-T) with a CCD (charge-coupled device) detector (ANDOR, DU401A-BR-DD) at various observation angles by rotating the specimen on a rotation stage.

## 3. Results and Discussion

Figure 3(a) is the emission spectrum at *ϕ*=0°. Two emission peaks at 620 and 645 nm, and a pronounced dip between them are clearly observed. The dip was created by the photonic bandgap, which can be verified by comparing it with the dip of the transmission (T) spectrum shown in Fig. 3(b). The two peaks evidence the enhancement of spontaneous emission due to the increased photon DOS at the edges of the photonic bandgap. While we will discuss this point in more detail later, we note here that the emission spectrum in Fig. 3(a) can not be explained by the filter effect. Because of the presence of the photonic bandgap, the emission intensity in the gap frequency range is reduced according to the optical density of the specimen. This reduction is often called the filter effect. To roughly estimate the spectrum profile caused by the filter effect, we may calculate the product of the transmission spectrum and the genuine emission spectrum of the light emitters in a uniform matrix. As for the latter, we may use the emission spectrum of the present specimen at a large oblique emission angle, which is not essentially influenced by the presence of the photonic bandgap. The product is shown in Fig. 3(c). This figure clearly shows the photonic bandgap around 633 nm on which the broad emission band of the oxygen vacancies is superimposed. Figures 3(a) and 3(c) are clearly different from each other, thus we can exclude the possibility of the mere filter effect.

We can also exclude the possibility of ASE (amplified spontaneous emission) and lasing by examining the excitation intensity dependence of the emission peak intensity, which is shown in Fig. 3(d). In this figure, the solid and open circles stand for the intensity of the lower and upper frequency peaks, respectively, while the two solid lines represent the best fit by the power law dependence. (Emission spectra shown in Fig. 3(a) were measured with an excitation intensity of 10 mW.) The power of the excitation intensity dependence thus obtained was 0.97 for the lower frequency peak and 0.95 for the upper frequency peak, both of which are very close to unity. This fact precludes the possibility of any nonlinear process and is apparent evidence for the spontaneous emission.

Figure 4 shows the emission spectra observed at various detection angles, *ϕ*, from −15° to 45°. In each spectrum, we can find two pronounced peaks and a dip between them. The peaks and the dip shift to the shorter wavelength for increasing |*ϕ*|. This feature is consistent with the theoretical prediction given below. The fairly large difference between the intensities of the spectra of *ϕ*=+15° and −15° cases mainly comes from the uncorrected reflection coefficient for each excitation angle.

In the present specimen, the luminescent species are distributed uniformly in each dielectric layer. Thus the effect of the spatial variation of the electric field intensity on the luminescence spectra is averaged out and can be neglected in the lowest order approximation. So, the angle-resolved photoemission spectrum is given by the genuine spectrum in free space times the enhancement factor of the photon DOS for the designated emission angle. Because of the translational symmetry, the *y* component of the wave vector, *k _{y}*, is conserved in the emission process (see Fig. 2), while its

*x*component in the photonic crystal is determined by the photon dispersion relation such that

*ω*(

*k*)=

_{x},k_{y}*ω*

_{em}, where

*ω*

_{em}is the emission angular frequency. For weakly modulated systems like the present specimen, the dispersion relation in the vicinity of the first photonic bandgap can be calculated accurately by truncating the expansion of the position dependent dielectric constant,

*ε*(

*x*), up to the first-order Fourier components:

where *G*=2*π/a* is the elementary reciprocal lattice constant. The three sample parameters were determined by the refractive indices of the constituent materials (2.08 for Ta2O_{5} and 1.46 for SiO_{2}), and the center frequency and width of the first bandgap found in the transmission spectrum. Thus we obtained *a*=155 nm, *η*
_{0}=0.240, and *η*
_{1}=0.00877.

Because the dispersion relation in the frequency range of interest is not appreciably different between the p and s polarizations, we shall deal with the p polarization in the rest of this paper. The *z* component of the magnetic field, *H _{z}*, satisfies the following equation:

where *c* is the velocity of light in free space. Using the approximation given in Eq. (2), we obtain

$$+\genfrac{}{}{0.1ex}{}{{\mathrm{ach}}^{2}}{2}\genfrac{}{}{0.1ex}{}{{\eta}_{0}\mp \mid {\eta}_{1}\mid \pm \genfrac{}{}{0.1ex}{}{2{\pi}^{2}{\eta}_{0}^{2}}{\mid {\eta}_{1}\mid \left({\pi}^{2}-{a}^{2}{k}_{y}^{2}\right)}}{\sqrt{{\pi}^{2}\left({\eta}_{0}\pm \mid {\eta}_{1}\mid \right)+{a}^{2}{k}_{y}^{2}({\eta}_{0}\mp \mid {\eta}_{1}\mid )}},$$

where *h*=*k _{x}*−

*π/a*is the distance from the Brillouin zone boundary and

*ω*

_{+}(

*ω*

_{−}) gives the dispersion relation of the upper (lower) branch. This dispersion relation is accurate in the vicinity of the first photonic bandgap and for |

*h*|≪

*π*/

*a*.

The enhancement factor of the photon DOS is obtained by calculating the ratio of the volume in the phase space in the PC to that in free space. In the present experiment, the wave vector is located in the *x-y* plane, and its direction is specified by the observation angle *ϕ*. Thus the two dimensions are reduced by these two experimental conditions, and it is sufficient to take into consideration the one-dimensional volume $\mathrm{ds}=\sqrt{d{k}_{x}^{2}+d{k}_{y}^{2}}$ in the PC and *dk*=*dω/c* in free space. Since *dky*=*dk*sin*ϕ* and *k _{x}* is implicitly determined by Eq. (4), we obtain the enhancement factor as

Note that this factor is divergent at the Brillouin zone boundary for any *k _{y}*, since

*∂ω/∂k*=0 there.

_{x}The theoretical emission intensity at each observation angle was obtained as a product of the genuine emission spectrum and the enhancement factor calculated according to Eqs. (4) and (5), and is plotted in Fig. 5. Taking into account the lifetime of the eigenmodes in the actual specimen of a finite thickness (≈30 *µ*m), a small constant imaginary part was added to *∂ω/∂ k _{x}*. The positions of emission peaks at the upper and lower gap edge frequencies are reproduced well for each observation angle. Note that the photon DOS in the gap frequency range is exactly equal to zero in the calculation because we assumed the dispersion of an ideal infinite PC. On the other hand, those emitters located in the vicinity of the sample surface can emit photons even in the gap frequency range because the influence of the PC environment is imperfect. So, we observed emission intensity in the gap range as shown in Fig. 4. Apart from this difference, the overall agreement between the observation and the theoretical calculation is quite good.

Let us consider the difference between the peak height of the lower and upper frequency peaks in Fig. 3(a). In the vicinity of the first photonic bandgap, the electric field is more concentrated in the high index (Ta_{2}O_{5}) layers for the lower branch and in the low index layers (SiO_{2}) for the upper branch [14]. So, Ta_{2}O_{5} gives the main contribution to the lower frequency peak. Because its genuine emission is located around 640 nm [21], the position of the lower frequency peak (645 nm) matches the emission peak of Ta_{2}O_{5}. On the other hand, SiO_{2} should give the main contribution to the upper frequency peak. However, its genuine emission band is located around 690 nm [20], and hence, there is a large mismatch with the peak frequency. These facts may explain the difference in the peak height, although information on the density of photoemitters in each layer is necessary for a more quantitative discussion.

On the other hand, the increase in the spectral width of the emission peaks with increasing *ϕ* is mainly caused by a constraint in the experiment. We used a pinhole to decrease the detection angle uncertainty down to 2°. So, for example, the spectrum for *ϕ*=40° in Fig. 4 represents a weighted average of emission spectra from *ϕ*=38° to 42°. Because the variation of the peak position with the detection angle is larger for larger *ϕ*, the observed linewidth in the averaged spectrum becomes larger with increasing *ϕ*.

To observe the enhancement of the excitation electric field due to small *v*
_{pc}, the excitation laser has to be tuned to a band edge wavelength, which should be different from the observation wavelength because of the non-zero Stokes shift. There are various possibilities for the combination of excitation and observation wavelengths. For the sake of simplicity and clarity of the experiment, we chose excitation around 584 nm at 45° incidence and emission around 646 nm with *ϕ*=0°. From the top-most curve in Fig. 4, we see that the longer wavelength band edge of *the two-dimensional band structure of the one-dimensional PC* is located at 584 nm for 45°, and hence, the above excitation condition produces resonant excitation of the band edge mode with small *v _{pc}*.

Emission spectra measured by 579 to 593 nm excitation at 45° incidence with 1 nm interval are plotted in Fig. 6(a), where the black line shows the excitation wavelength and the blue line is the position of the band edge. We see a pronounced increase in the emission peak intensity when the black line crosses the blue line, that is, when the band edge mode is excited by the incident laser. Excitation-wavelength dependence of the emission peak intensity is plotted in Fig. 6(b) (red curve for *ϕ*=45°), which clearly shows the resonance enhancement. When the excitation wavelength goes away from the resonance condition to the shorter wavelength side, the emission intensity decreases rapidly because the excitation wavelength is in the photonic band gap and the incident laser cannot go into the specimen. We performed the same measurement using a slightly different configuration, for which the incident and detection angles were 40° and 5°, respectively. The results are also plotted in Fig. 6(b), where we see a similar resonance peak at 598 nm, which is the same position as the longer wavelength peak in the case of *ϕ*=40° shown in Fig. 4.

We can roughly estimate the enhancement factors due to increased *𝓓*
_{1} and small vpc based on comparison of the emission intensity between the resonant and non-resonant cases. As for the enhancement by *𝓓*
_{1}, we must compare the emission intensity between the PC and a uniform material for its rigorous evaluation. However, as is found in Fig. 4, the bandgap is located around 590 nm for *ϕ*=40°, which implies that its emission spectrum for wavelengths longer than 610 nm is scarcely influenced by the PC structure. So, we may use the spectrum in this range as a reference. Thus we compared the emission intensity at 645 nm for *ϕ*=0° and *𝓓*=40° to obtain the enhancement factor of 4.5 for the longer wavelength peak of the *𝓓*=0° spectrum. As for the enhancement by *v _{pc}*, we compared the emission intensity of 45° excitation at 584 nm (resonant) and 593 nm (non-resonant). Thus we obtained an enhancement factor of 3. The overall enhancement was 13.5.

## 4. Conclusion

In summary, we carefully observed spontaneous emission from oxygen vacancies uniformly distributed in a one-dimensional PC composed of alternating dielectric multilayers of Ta_{2}O_{5} and SiO_{2}. Thanks to ideal characteristics of the specimen, we clearly demonstrated the double enhancement of spontaneous emission due to increased photon DOS at the emission frequency and reduced photon velocity at the excitation frequency by measuring the emission and excitation spectra as a function of the emission and excitation angles. The enhancement factor was estimated at 4.5 for the former and 3 for the latter to give an overall enhancement of 13.5. The theoretical analysis based on the weak modulation approximation showed good agreement with the observed angle-resolved emission spectra.

## Acknowledgments

The authors appreciate the SEM and EPMA analyses by Toray Research Center. This work was supported by KAKENHI (20340080).

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