## Abstract

We recently reported experimental evidence for double enhancement of spontaneous emission due to increased photon density of states and small group velocity at photonic band-edge frequencies by observing angle-resolved emission and excitation spectra of photoluminescence [
K. Kuroda *et al.*, Opt. Express **17**, 13168 (2009)
]. The specimen we used was a one-dimensional photonic crystal composed of periodic multilayers of ${\mathrm{Ta}}_{2}{\mathrm{O}}_{5}$ and $\mathrm{Si}{\mathrm{O}}_{2}$ with oxygen vacancies as light emitters. In the present study, we report on the lack of any excitation intensity dependence of the emission peak height and width, which excludes possibilities of nonlinear effects, the polarized emission spectra, and their comparison with theoretical calculations to further confirm our finding.

© 2009 Optical Society of America

## 1. INTRODUCTION

Photonic structures that have periodic variation of refractive index, which are called photonic crystals (PCs), on the order of a few hundred nanometers have been attracting much attention [1, 2, 3]. The periodic variation gives rise to a photonic bandgap (PBG) in the wavelength range from visible to near infrared (IR), where photon density of states (DOS) is vanishing. In contrast, light emission in the vicinity of the PBG is expected to be enhanced due to an increased DOS. Many studies have been conducted on modifying and designing the electromagnetic field by PCs so as to control and improve the optical properties of light emitters. The interplay between resonant electronic excitation and electromagnetic field in periodic structures, known as optical lattices, has also been studied [4, 5].

As far as photoluminescence in regular PCs is concerned, there are two enhancement mechanisms for spontaneous emission. One is increased photon DOS at the emission frequency, which accelerates the optical transition according to Fermi’s golden rule, and the other is small-photon velocity at the photoexcitation frequency, which results in a large amplitude of the local electric field due to conservation of Poynting’s vector [6]. As described in [6], the former mechanism is valid only for one-dimensional PCs or for photon emission in particular directions in two- and three-dimensional cases. So, although a change in the emission lifetime of CdSe quantum dots in inverse opal was observed in [7], it is not relevant to the divergent band-edge DOS. Reference [8] reported peculiar spontaneous emission spectra of GaAs quantum wells sandwiched by distributed Bragg reflector (DBR) 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 [9, 10].

As for the second mechanism, the enhanced local electric field increases the efficiency of photoexcitation of fluorescent species. This technique is called the band-edge excitation and was used for lasing [11] and control of the refractive index [12].

Although these two enhancement mechanisms are expected in the case of spontaneous emission of photons in regular PCs, they had not been observed because of the difficulty of fabricating sufficiently regular and thick periodic structures to create distinct peaks in the photon DOS and that of attaining a uniform distribution of light emitters in the PCs. However, we recently reported a clear observation of the double enhancement of spontaneous emission for a one-dimensional PC composed of periodic multilayers of ${\mathrm{Ta}}_{2}{\mathrm{O}}_{5}$ and $\mathrm{Si}{\mathrm{O}}_{2}$ with oxygen vacancies as light emitters [6]. We presented angle-resolved excitation and emission spectra of photoluminescence to show the double enhancement due to the increased photon DOS at the emission wavelength and small group velocity at the excitation wavelength. The light emitters were distributed uniformly and the regularity of the specimen was good enough to observe the band-edge enhancement. We also presented a result of calculations of the enhanced luminescence spectra that supported the experimental observations.

In this paper, we report on the excitation intensity dependence of the emission peak height and width, the polarized emission spectra, and their comparison with theoretical calculations to further confirm our finding. In Section 2, the photonic band structure for two independent polarizations is calculated analytically in the weak modulation approximation. Then the enhancement factor of spontaneous emission in certain directions determined by the experimental conditions is derived, and it is shown that the enhancement factor is divergent at the band-edge frequencies. The experimental setup is briefly described in Section 3. In Section 4, experimental results on the excitation intensity dependence of emission peak height and width are presented, and it is concluded that what we observed was nothing but enhanced spontaneous emission. The polarization dependence of the emission spectra is also presented and compared with the theoretical calculation. Finally, a summary is given in Section 5.

## 2. THEORY

#### 2A. Photonic Band Calculation

In this section, we calculate the dispersion relation of the electromagnetic eigenmodes and the photon DOS. Figure 1 schematically shows the experimental configuration. The specimen used in this study was a commercial optical notch filter made of dielectric multilayers designed to eliminate the $633\text{\hspace{0.17em}}\mathrm{nm}$ line of the He–Ne laser (Semrock, NF01-633U). This notch filter gives a stop band from $620\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}645\text{\hspace{0.17em}}\mathrm{nm}$. SEM (scanning electron microscope) and EPMA (electron probe microanalyzer) analyses showed that the notch filter consisted of ${\mathrm{Ta}}_{2}{\mathrm{O}}_{5}$ and $\mathrm{Si}{\mathrm{O}}_{2}$ multilayers on an $\mathrm{Si}{\mathrm{O}}_{2}$ substrate. In Fig. 1, *a* is the lattice constant of the one-dimensional PC (the dielectric multilayers), *θ* is the observation angle, and *ω* and *k* are the angular frequency and wave number of emitted photons.

We have to consider two independent polarizations of the electromagnetic field. One is the *p* polarization for which the electric field is pointing in the $x\u2013y$ plane. The other is the *s* polarization for which the electric field is pointing in the *z* direction, which is perpendicular to the $x\u2013y$ plane. It is convenient to calculate the *z* component of the magnetic field, ${H}_{z}$, for the *p* polarization and that of the electric field, ${E}_{z}$, for the *s* polarization. From the Maxwell equations, we obtain the following eigen equations:

*c*is the velocity of light in free space and $\u03f5\left(x\right)$ is the spatially periodic dielectric constant. Here we assumed that the permeability of the specimen is equal to unity, since it does not contain magnetic materials.

The ratio of the width of the first PBG to its center wavelength is $25\text{\hspace{0.17em}}\mathrm{nm}\u2215633\text{\hspace{0.17em}}\mathrm{nm}\approx 4\%$, which is small enough to justify the weak modulation approximation. That is, it is a good approximation to truncate the expansion of $\u03f5\left(x\right)$ up to the first-order Fourier components provided we calculate the dispersion relation in the vicinity of the first PBG:

where $G=2\pi \u2215a$ is the elementary reciprocal lattice constant. Because the structure is uniform in the*y*direction, spatial variation of the eigenfunctions in the

*y*direction is simply described by a phase factor ${e}^{i{k}_{y}y}$ with the

*y*component of the wave vector, ${k}_{y}$. On the other hand, spatial variation in the

*x*direction is described by the Bloch theorem because of the periodicity of the structure. Thus the eigenfunctions of the

*p*and

*s*polarizations should have the following forms, respectively: whereis the reciprocal lattice constant and ${k}_{x}$ is the

*x*component of the wave vector.

Let us consider the *p* polarization first. By substituting Eqs. (3, 4) into Eq. (1) and comparing the Fourier component of the same order, we obtain

From the secular equation, we finally obtain

For the *s* polarization, we substitute Eqs. (3, 5) into Eq. (2) and obtain

For ${k}_{y}=0$, we obtain the following dispersion relation common to the *p* and *s* polarizations:

*a*, ${\eta}_{0}$, ${\eta}_{1}$) were determined from the refractive indices of the constituent materials (2.08 for ${\mathrm{Ta}}_{2}{\mathrm{O}}_{5}$ and 1.46 for $\mathrm{Si}{\mathrm{O}}_{2}$), and the center frequency and width of the first bandgap found in the transmission spectrum. Thus we obtained $a=155\text{\hspace{0.17em}}\mathrm{nm}$, ${\eta}_{0}=0.240$, and ${\eta}_{1}=0.00877$. ${\eta}_{1}\u2215{\eta}_{0}$ is sufficiently small as we expected.

#### 2B. Enhancement Factor

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 light emission spectrum is given by the genuine spectrum in free space times the enhancement factor of the photon DOS for the designated emission angle.

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 $x\u2013y$ plane, and its direction is specified by the observation angle *θ*. Thus the two dimensions are reduced by these two experimental conditions, and it is sufficient to take into consideration the one-dimensional volume $ds=\sqrt{d{k}_{x}^{2}+d{k}_{y}^{2}}$ in the PC and $dk=d\omega \u2215c$ in free space. Since $d{k}_{y}=dk\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta $ and ${k}_{x}$ is implicitly determined by Eq. (10) or Eq. (13),

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 Eq. (18) and is plotted in Fig. 2 , where thick lines denote the *s* polarization and thin lines denote the *p* polarization. Taking into account the lifetime of the eigenmodes in the actual specimen of a finite thickness $(\approx 30\text{\hspace{0.17em}}\mu \mathrm{m})$, a small constant imaginary part was added to $\partial \omega \u2215\partial {k}_{x}$. In Fig. 2, we can clearly observe the following features. First, there is a frequency range in each spectrum where the emission intensity is equal to zero, which originates from the PBG and shifts to the shorter wavelength with increasing observation angle. Second, on each side of the PBG, there is one distinct emission peak due to a divergent photon DOS. Thirdly, the center of the PBG is nearly the same for the *s* and *p* polarizations, whereas its width is wider for the *s* polarization. In Section 4, we show that these features are actually observed in the experiment.

## 3. EXPERIMENTAL

The experimental setup for the angle-resolved photoluminescence measurement was essentially the same as in [6]. In brief, the $532\text{\hspace{0.17em}}\mathrm{nm}$ emission line of a frequency-doubled continuous Nd:YAG laser was used as an excitation light source. The laser beam was weakly focused onto the specimen by a lens with a focal length of $100\text{\hspace{0.17em}}\mathrm{mm}$. The beam diameter on the sample surface was about $15\text{\hspace{0.17em}}\mu \mathrm{m}$. Luminescence light from the specimen was collected by another lens with the same focal length. A pinhole was placed before the collection lens to decrease the uncertainty of the detection angle down to 4° (full width). The dependence of the emission intensity on the detection angle $\left(\theta \right)$ was measured by rotating the sample with a rotation stage. To measure the polarization of the emission, a polarizer was placed behind the collection lens. Emission intensity was measured by a triple-grating spectrometer with a charge-coupled device (CCD) detector.

## 4. RESULTS AND DISCUSSION

#### 4A. Excitation Power Dependence

First, we should confirm that all emission spectra were measured in the linear region and that lasing, or any other nonlinear emission process, was not involved. Figure 3 shows emission spectra in the normal direction $(\theta =0)$ measured with various excitation powers. The scale of the vertical axis of each spectrum is proportional to the excitation power.

It is clearly seen in Fig. 3 that there is one distinct peak on both edges of an opaque range from $620\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}645\text{\hspace{0.17em}}\mathrm{nm}$ caused by the PBG. These peaks originate from enhanced spontaneous emission due to the increased photon DOS discussed in Subsection 2B and coincide well with Fig. 2. The line shapes of the emission spectra with five different excitation powers in Fig. 3 are quite similar to each other. Particularly, the peak intensity is proportional to the excitation intensity and the peak width is independent of the excitation intensity.

To analyze these two features further, we examined the peak height and width more accurately by a curve fitting analysis. As shown in Figs. 4a, 4b , each peak was fitted with a Lorentzian function, and height and width were estimated. As we can see, the line shape on the gap frequency side is reproduced well by the Lorentzian curve, because the emission probability in the gap is mainly determined by the photon escape time through the specimen surface, whereas there is an appreciable deviation in the opposite frequency side where spontaneous emission takes place according to the photon DOS in the specimen, which cannot be reproduced by a Lorentzian function.

The peak intensity and width are plotted as functions of excitation intensity in Figs. 4c, 4d, respectively. In Fig. 4c, open (solid) circles denote the peak intensity of the longer (shorter) wavelength peak and open squares denote the mean value of the intensity between 680 and $700\text{\hspace{0.17em}}\mathrm{nm}$, where the enhanced spontaneous emission by an increased photon DOS is not relevant. Solid lines are the best fit by the power-law dependence. As we can see, the observed emission intensity is well reproduced by the power law, and the exponents of all three data were close to unity $(1.00\pm 0.08)$. So, we can conclude that no nonlinear process such as stimulated emission, amplified spontaneous emission, or lasing was involved in our experiment. This fact is further corroborated by the constant width of the emission peaks given in Fig. 4d. Thus we can conclude that the two emission peaks on both ends of the PBG originate from spontaneous emission.

#### 4B. Polarization Dependence

Figure 5 depicts the polarized emission spectra observed for detection angles *θ* from 0° to 45°, where thick and thin lines denote *s*- and *p*-polarized emissions. The overall features of this figure agree well with Fig. 2. First, the PBG and two emission peaks shift to the shorter wavelength with increasing detection angle. Second, the center wavelength of the PBG is nearly the same for the *s* and *p* polarizations. Third, the distance between the two peaks is larger for the *s* polarization than for the *p* polarization.

On the other hand, there are two main differences between the calculation and the experiment. One difference is the nonzero emission intensity in the gap frequency range observed in the experiment, which is a consequence of the finite number of the periodic dielectric layers of the specimen. 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. 5. As for the height of the two peaks, it is proportional to the photon DOS in the PC. In an ideal one-dimensional PC with an infinite number of layers, the photon DOS diverges on both ends of the PBG. However, in the actual PC, the number of periodic layers is finite, which leads to a finite lifetime of electromagnetic modes, and hence, to the decrease of the photon DOS. So, generally speaking, we can expect a larger peak height for the larger number of periodic layers.

The second difference is the larger peak width for the larger detection angle. This is caused by a constraint in the experiment. We used a pinhole to decrease the detection angle uncertainty down to 4° (FWHM). So, for example, the spectrum for $\theta =40\xb0$ in Fig. 5 represents a weighted average of emission spectra from $\theta =38\xb0$ to 42°. Because the variation of the peak position with the detection angle is larger for larger *θ*, the observed linewidth in the averaged spectrum becomes larger with increasing *θ*.

To further confirm the agreement between the calculation and observation, we compared the difference of the peak positions between the *s* and *p* polarizations as a function of the detection angle. Figures 6a, 6b show a magnified view of the two peaks of the *s*-(thick curve) and *p*-polarized (thin curve) emissions at $\theta =20\xb0$, from which we obtain the difference of the peak wavelengths. Then, the experimental observation was compared with the calculation in Fig. 6c, where it is clearly seen that the experiment and calculation agree well with each other.

## 5. CONCLUSION

We investigated both theoretically and experimentally the photoluminescence of a one-dimensional PC composed of periodic multilayers of ${\mathrm{Ta}}_{2}{\mathrm{O}}_{5}$ and $\mathrm{Si}{\mathrm{O}}_{2}$ with oxygen vacancies as light emitters. The photonic band structures for *s* and *p* polarizations were calculated analytically in the weak modulation approximation, which was justified by the small PBG width of the specimen. The enhancement factor of spontaneous emission derived from the band structure was shown to be divergent at the band-edge frequencies.

Then we presented the excitation intensity dependence of the emission peak height and width to exclude possibilities of nonlinear processes such as amplified spontaneous emission and lasing. Thus we confirmed that what we observed was nothing but enhanced spontaneous emission due to increased photon DOS. We also compared polarized emission spectra measured at various detection angles with theoretical calculations to confirm their good agreement.

## ACKNOWLEDGMENTS

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

**1. **J. D. Joannopoulos, R. D. Meade, and J. N. Winn, *Photonic Crystals: Molding the Flow of Light* (Princeton Univ. Press, 1995).

**2. **K. Sakoda, *Optical Properties of Photonic Crystals*, 2nd ed. (Springer, 2004).

**3. **K. Sakoda, “Optics of photonic crystals,” Opt. Rev. **6**, 381–392 (1999). [CrossRef]

**4. **M. Hübner, J. P. Prineas, C. Ell, P. Brick, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Optical lattices achieved by excitons in periodic quantum well structures,” Phys. Rev. Lett. **83**, 2841–2844 (1999). [CrossRef]

**5. **E. L. Ivchenko, M. M. Voronov, M. V. Erementchouk, L. I. Deych, and A. A. Lisyansky, “Multiple-quantum-well-based photonic crystals with simple and compound elementary supercells,” Phys. Rev. B **70**, 195106 (2004). [CrossRef]

**6. **K. Kuroda, T. Sawada, T. Kuroda, K. Watanabe, and K. Sakoda, “Doubly enhanced spontaneous emission due to increased photon density of states at photonic band edge frequencies,” Opt. Express **17**, 13168–13177 (2009). [CrossRef] [PubMed]

**7. **P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature **430**, 654–657 (2004). [CrossRef] [PubMed]

**8. **M. D. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A **53**, 2799–2803 (1996). [CrossRef] [PubMed]

**9. **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]

**10. **M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band-gap materials,” Appl. Phys. B **60**, S57–S61 (1995).

**11. **Y. Matsuhisa, Y. Huang, Y. Zhou, S.-T. Wu, Y. Takao, A. Fujii, and M. Ozaki, “Cholesteric liquid crystal laser in a dielectric mirror cavity upon band-edge excitation,” Opt. Express **15**, 616–622 (2007). [CrossRef] [PubMed]

**12. **M. Astic, Ph. Delaye, R. Frey, G. Roosen, R. André, N. Belabas, I. Sagnes, and R. Raj, “Time resolved nonlinear spectroscopy at the band edge of 1D photonic crystals,” J. Phys. D **41**, 224005 (2008). [CrossRef]