## Modification of spontaneous emission in Bragg onion resonators

Optics Express, Vol. 14, Issue 16, pp. 7398-7419 (2006)

http://dx.doi.org/10.1364/OE.14.007398

Acrobat PDF (1507 KB)

### Abstract

We formulated an analytical model and analyzed the modification of spontaneous emission in Bragg onion resonators. We consider both the case of a single light emitter and a uniformly distributed ensemble of light emitters within the resonator. We obtain an expression for the average radiation rate of the light emitters ensemble and discuss the modification of the average radiation rate as a function of cavity parameters such as the core radius, the number of Bragg cladding layers, the index contrast of the Bragg cladding, and the refractive index of surrounding medium. We also consider the possibility of non-exponential decay of the light emitter ensemble due to the strong dependence of spontaneous emission on the location and polarization of individual light emitter. We conclude that Bragg onion resonators can both enhance and inhibit spontaneous emission by several orders of magnitude. This property can have significant impact in the field of cavity quantum electrodynamics (QED).

© 2006 Optical Society of America

## 1. Introduction

*Q*) factor and a small modal volume

*V*, have received much attention in recent years [1–6

1. J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. ThierryMieg, and T. Rivera, “Quantum boxes as active probes for photonic microstructures: The pillar microcavity case,” Appl. Phys. Lett. **69**, 449 (1996). [CrossRef]

7. Y. Yamamoto, S. Machida, and G. Bjork, “Microcavity Semiconductor-Laser with Enhanced Spontaneous Emission,” Phys. Rev. A **44**, 657 (1991). [CrossRef] [PubMed]

11. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science **290**, 2282 (2000). [CrossRef] [PubMed]

^{9}. However, to satisfy the condition of total internal reflection, the sizes of such cavities are typically limited to tens of microns or greater. A large cavity size of such order of magnitude can create two significant drawbacks: it reduces the coupling strength between the light emitter and the high Q optical mode, and makes it more difficult to achieve a truly single mode operation (due to the small frequency spacing between adjacent high Q modes). On the other hand, optical microcavities based on Bragg reflections can have much smaller sizes of the order of λ/n, which is ideal for applications demanding strong interactions between the light emitter and the vacuum field. An example of Bragg-confined optical cavities is a semiconductor micropillar [1

1. J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. ThierryMieg, and T. Rivera, “Quantum boxes as active probes for photonic microstructures: The pillar microcavity case,” Appl. Phys. Lett. **69**, 449 (1996). [CrossRef]

5. K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. **83**, 1915 (2003). [CrossRef]

_{bg}. The emission rate into the desired high Q mode can be labeled as Γ

_{mode}. For efficient single photon source, we need to design a cavity structure to achieve Γ

_{mode}≫ Γ

_{bg}.

_{bg}, which corresponds to the radiation into the free space traveling mode. On the other hand, with the presence of the omnidirectional cladding layers, the fully spherically symmetric Bragg onion resonators can almost completely suppress the coupling into the free space radiation mode. Consequently, we can achieve both spontaneous emission enhancement and spontaneous emission inhibition of up to several orders of magnitude. In other words, we can use onion resonators to obtain a significantly increased Γ

_{mode}(the radiation rate into the desired high Q optical mode), and at the same time dramatically reduce the background radiation rate Γ

_{bg}. This unique property should make the Bragg onion resonator a near ideal candidate for single photon devices.

_{free}(∆Ω/4

*π*), where Γ

_{free}is the free space spontaneous emission rate, and ∆Ω is the solid angle spanned by the onion stem. From this estimate, it is also clear that we can significantly reduce this additional background emission rate to a very low level by decreasing the onion stem diameter to the level of 1 μm [14

14. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “High-quality-factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett. **28**, 2144 (2003). [CrossRef] [PubMed]

16. K. G. Sullivan and D. G. Hall, “Radiation in Spherically Symmetrical Structures .2. Enhancement and Inhibition of Dipole Radiation in a Spherical Bragg Cavity,” Phys. Rev. A. **50**, 2708 (1994). [CrossRef] [PubMed]

16. K. G. Sullivan and D. G. Hall, “Radiation in Spherically Symmetrical Structures .2. Enhancement and Inhibition of Dipole Radiation in a Spherical Bragg Cavity,” Phys. Rev. A. **50**, 2708 (1994). [CrossRef] [PubMed]

17. R. R. Chance, Prock R., and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. **37**, 1 (1978). [CrossRef]

14. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “High-quality-factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett. **28**, 2144 (2003). [CrossRef] [PubMed]

## 2. Spontaneous emission of a single dipole in a Bragg onion resonator

16. K. G. Sullivan and D. G. Hall, “Radiation in Spherically Symmetrical Structures .2. Enhancement and Inhibition of Dipole Radiation in a Spherical Bragg Cavity,” Phys. Rev. A. **50**, 2708 (1994). [CrossRef] [PubMed]

**50**, 2708 (1994). [CrossRef] [PubMed]

18. H. Chew, “Radiation and Lifetimes of Atoms inside Dielectric Particles,” Phys. Rev. A. **38**, 3410 (1988). [CrossRef] [PubMed]

19. W. Lukosz and R. E. Kunz, “Light-Emission by Magnetic and Electric Dipoles Close to a Plane Interface .1. Total Radiated Power,” J. Opt. Soc. Am. **67**, 1607 (1977). [CrossRef]

*p*is the electric dipole moment of the atomic transition, ω

_{0}is the intrinsic dipole oscillation frequency in the absence of all damping, and

*b*

_{0}is the spontaneous emission rate of a light emitter in the bulk material. The extra term (

*q*

^{2}/

*m*)

*E*

_{R}(

*t*) in Eq. (1) accounts for the modification of spontaneous emission rate within the microcavity. More precisely,

*E*

_{R}(

*t*) is the component of the reflected field (due to the microcavity) that is located at the position of the dipole source and is parallel to the dipole moment. The two parameters

*q*and

*m*respectively describe the effective charge and the effective mass of the dipole oscillator. The exact values of

*q*and

*m*are not significant, since they only appear in the final expression of the modified spontaneous emission rate in the form of:

_{0}=(ε/ε

_{0})

^{1/2}is the refractive index of the bulk core material, ε

_{0}and

*c*

_{0}are respectively the permittivity and the speed of light in free space. The relation Eq. (2) is derived using the classical radiation dipole model. For a more detailed explanation, the reader can consult Ref. [20].

*p*and the reflected field component

*E*

_{R}(

*t*) oscillate at the same modified complex frequency:

*b*corresponds to the modified spontaneous emission rate, ω is the modified emission frequency,

*p*

_{0}and

*E*

_{0}are respectively the amplitudes of the dipole moment and the reflected field component. By substituting Eqs. (2) and (3) into Eq. (1), we find that the normalized spontaneous emission rate and frequency shift in the presence of the micro-cavity are given by:

*b*and ∆

*ω*=

*ω*-

*ω*

_{0}are much smaller than

*ω*

_{0}. The two terms

*E*

_{s}, and

*E*

_{0}in Eq. (4) are respectively given by:

*ω*

_{0}/

*c*

_{0}is the wave vector,

*ε*and

*μ*

_{0}are respectively the permittivity and permeability of the bulk material. In the case of Bragg onion resonators, since the light emitters are confined within the onion core,

*ε*and

*μ*

_{0}also represent the permittivity and permeability of the core material. In Eq. (5b),

*E*

_{0}is defined after Eq. (3b). The two terms

*l*, and

*m*. The detailed forms of

*E*

_{0}:

*jl*(

*kr*) and

*hl*(

*kr*) are respectively the

*l*th order spherical Bessel and Hankel function, whereas

*X⇀*

_{lm}=

*L̂*∙

*Y*

_{lm}(

*θ,φ*)/√

*l*(

*l*+ 1) is the spherical vector function. The only undefined quantities in Eq. (6) are the two parameters

*l*th order TE and TM multipole modes at the interface between the onion core and the innermost cladding layer [16

**50**, 2708 (1994). [CrossRef] [PubMed]

14. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “High-quality-factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett. **28**, 2144 (2003). [CrossRef] [PubMed]

15. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “Modal analysis of Bragg onion resonators,” Opt. Lett. **29**, 424 (2004). [CrossRef] [PubMed]

*l,m*) multipole orders. Due to the spherical symmetry of the structure, individual multipole fields with different

*l*and

*m*are independent of each other. For a given pair of angular quantum number

*l*and

*m*, the TE or TM components within the

*n*th dielectric layer is:

*Z*

_{n}=(μ

_{0}/ε

_{n})

^{1/2}is the material impedance, and

*k*

_{n}=(ε

_{n}/ε

_{0})

^{1/2}ω

_{0}/

*c*

_{0}is the wave vector within the

*n*th layer. The four linear coefficients

*A*

_{n}

*, B*

_{n}

*C*

_{n}and

*D*

_{n}, are constant within the

*n*th layer. Since the spherical Hankel functions

*kr*

_{co}) and

*kr*

_{co}) represent, respectively, the outgoing and the incoming wave, the amplitude reflection coefficient

*ρ*

_{l}, at the core-cladding interface in Eq. (7) is can be determined from Eq. (8) as [16

**50**, 2708 (1994). [CrossRef] [PubMed]

*E*

_{θ}

*,E*

_{ϕ}

*,H*

_{θ}

*,H*

_{ϕ}at the interface between two adjacent layers and the orthogonality of the spherical harmonics, we can relate the linear coefficients in the onion core (

*A*

_{co},

*B*

_{co},

*C*

_{co},

*D*

_{co}) to those outside the onion resonator (

*A*

_{out}

*, B*

_{out}

*, C*

_{out}

*, D*

_{out}) through two by two matrices

**28**, 2144 (2003). [CrossRef] [PubMed]

*B*

_{out}=0 and

*D*

_{out}= 0, we can express

*B*

_{co}/

*A*

_{co}and

*C*

_{co}/

*D*

_{co}as a function of the individual elements of the two-by-two matrices

*TM,TE*implies summation over both TM and TE modes.

17. R. R. Chance, Prock R., and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. **37**, 1 (1978). [CrossRef]

19. W. Lukosz and R. E. Kunz, “Light-Emission by Magnetic and Electric Dipoles Close to a Plane Interface .1. Total Radiated Power,” J. Opt. Soc. Am. **67**, 1607 (1977). [CrossRef]

21. H. Chew, “Transition Rates of Atoms near Spherical Surfaces,” J. Chem. Phys. **87**, 1355 (1987). [CrossRef]

*P*

_{cav}, divided by the dipole radiation power in the bulk material,

*P*

_{bulk}. Using this approach, we demonstrate in Appendix B that the normalized modified spontaneous emission rate can also be expressed as

## 3. Numerical results and analysis

15. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “Modal analysis of Bragg onion resonators,” Opt. Lett. **29**, 424 (2004). [CrossRef] [PubMed]

_{2}, the other one with cladding layer composed of SiO

_{2}and Si

_{3}N

_{4}. For the onion resonators considered in this paper, the silicon cladding layers have a refractive index of 3.5 and a thickness of 0.111 μm, whereas the refractive index and the thickness of the SiO

_{2}layers are respectively 1.5 and 0.258 μm. For the Si

_{3}N

_{4}layers, the refractive index is 2.1 and the thickness is 0.185 μm. The parameters of the Bragg cladding pairs are chosen such that the bandgap center is located at 1.55 μm.

### 3.1 A single dipole emitter located at the center of the onion resonator

_{1}mode, since this is the only multipole component that provides a non-zero contribution to the summation in Eq. (13) [16

**50**, 2708 (1994). [CrossRef] [PubMed]

^{4}times in the on-resonance case and spontaneous emission inhibition of the order of 10

^{-4}in the off-resonance case.

### 3.2 Dependence of modified spontaneous emission rate on the dipole position

*ẑ*axis and its corresponding spherical coordinate is

_{0}= (

*r*, 0, 0). In this case, the spontaneous emission rate depends on both the displacement from the center of the onion resonator,

*r*, and the orientation of the dipole oscillator

*b*

^{⊥}(radial) and

*b*

^{//}(transverse). Without a loss of generality, we can also assume that

*x*axis in the case of transverse polarization. Substituting

_{0}= (

*r*, 0, 0) into Eq. (13a) we find:

*b*

^{⊥}/

*b*

_{0}) and the transverse (

*b*

^{//}/

*b*

_{0}) polarization as a function of the dipole position. We first consider three different modes (TE

_{1}, TM

_{2}and TE

_{24}modes) and calculate the spontaneous emission rate at the three corresponding modal wavelengths (

*λ*=1.556445μm, 1.559715μm and 1.541255μm). The radial dependence of the electric field of the TE

_{1}, TM

_{2}and TE

_{24}mode are also plotted In Fig. 3(a). As can be seen from Fig. 3(a)–(d), at the resonant wavelength of the onion cavity mode, the spatial dependence of the spontaneous emission rate of a single dipole emitter follows the electric field distribution of the corresponding resonant mode. Such behavior can be explained by the fact that in the “on-resonance” case, the spontaneous emission process is dominated by the radiation into the resonant high Q mode. The other interesting point is that at the resonance of a TE mode (see Fig. 3(b) and Fig. 3(d), the spontaneous emission of a dipole oscillating along the radial direction (

*b*

^{⊥}/

*b*

_{0}) is strongly inhibited, whereas that of a dipole oscillating along the tangential direction is significantly enhanced. This is due to the fact that the radial component of the electric field of TE modes is zero. Consequently the dipole polarized along the radial direction can only couple to the TM modes, which is off resonance at the given wavelength.

### 3.3 Radiation from a dipole ensemble within the onion resonator core

18. H. Chew, “Radiation and Lifetimes of Atoms inside Dielectric Particles,” Phys. Rev. A. **38**, 3410 (1988). [CrossRef] [PubMed]

*b*(

*b*

_{0}⟨

_{dir}is given by:

*(l, m)*, the contribution to the average spontaneous emission rate involves the integration of

*b*

^{⊥}/

*b*

_{0}and

*b*

^{//}/

*b*

_{0}given in Eq. (15a) and Eq. (15b). Next we need to average ⟨

*b*(

*r⇀*)/

*b*

_{0}⟩

_{dir}over the spatial distribution of the dipole sources to have the ensemble-averaged spontaneous emission decaying rate ⟨

*b*/

*b*

_{0}⟩

_{vol}, which is

*r*

_{co}is the radius of the onion resonator core. Substituting Eq. (15a)–(15b) and Eq. (18) into Eq. (19) and using the integral identity of the Bessel functions [18

18. H. Chew, “Radiation and Lifetimes of Atoms inside Dielectric Particles,” Phys. Rev. A. **38**, 3410 (1988). [CrossRef] [PubMed]

*P*

_{l}and

*Q*

_{l}are the integrals of the Bessel functions and take the form of

*l*into two groups

^{1}: the core modes (which concentrate within the cavity core and are confined by the Bragg reflection), and the cladding modes (which are mainly confined in the cladding layers through total internal reflection (TIR)). The core modes typically have smaller angular quantum number

*l*that satisfies

*l*≤

*n*

_{co}2

*π*∙

*r*

_{co}/

*λ*, where

*n*

_{co}is the refractive index of the onion resonator core. Examples of core modes are shown in Fig. 3(a). The cladding modes generally have angular quantum number

*l*greater than

*n*

_{co}2

*π*∙

*r*

_{co}/

*λ*. Examples of cladding modes are shown in Fig. 4.

*l*less than 10 does not contribute significantly to the cavity modified spontaneous emission rate, which is due to the fact that the light emitter frequency does not coincide with any of the core modes. It is also clear that among the cladding modes, those with a relatively smaller angular quantum number

*l*have a much stronger coupling to the light emitter within the onion core. This can be explained by the observation that the cladding modes with smaller

*l*penetrate deeper into the onion core, as shown in Fig. 4.

**28**, 2144 (2003). [CrossRef] [PubMed]

15. Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “Modal analysis of Bragg onion resonators,” Opt. Lett. **29**, 424 (2004). [CrossRef] [PubMed]

22. M. P. vanExter, G. Nienhuis, and J. P. Woerdman, “Two simple expressions for the spontaneous emission factor beta,” Phys. Rev. A. **54**, 3553 (1996). [CrossRef]

*λ*is the optical wavelength,

*Q*

_{cav}is the quality factor of the cavity mode. Since the quality factor of the core modes increases exponentially as a function of the cladding layer number [14

**28**, 2144 (2003). [CrossRef] [PubMed]

*N*

_{clad}. As an example we calculate the peak enhancement ratio ⟨

*b*/

*b*

_{0}⟩ at the resonant wavelength of the TE

_{1}, TM

_{2}and TE

_{24}modes as a function of the cladding pair number

*N*

_{Bragg}. The results, which are given in Fig. 7, clearly demonstrate an excellent exponential dependence.

### 3.4 Spontaneous emission inhibition in Bragg onion resonators

*N*

_{Bragg}, cladding index contrast, core radius, and the refractive index of the core material. We focus primarily on cases where we may no longer approximate the onion cladding layers as an omnidirectional mirror any more. We demonstrate that even for such “non-ideal” onion resonators, we can still achieve spontaneous emission suppression for at least two orders of magnitude.

*N*

_{Bragg}, assuming the core radius to be 7 μm. It is instructive to first consider the results in Fig. 5. Comparing the two cases (one with

*N*

_{Bragg}= 5 and the other with

*N*

_{Bragg}= 7), we find that the additional cladding layers significantly reduce the spontaneous emission rate into multipole components with small angular quantum number (

*l*<30). The spontaneous emission rate into larger multipole components (38<

*l*<44), however, remains approximately the same even as

*N*

_{Bragg}increases. This is to be expected since the larger multipole components account for contributions from the cladding modes. In Fig. 8, we show the spectra of total average spontaneous emission rate in onion resonators with two different core radii. The results clearly show that we can achieve spontaneous emission inhibition up to two orders of magnitude with seven pairs of cladding layers, and the radius of the onion core does not have a significant impact on the degrees of spontaneous emission inhibition.

**28**, 2144 (2003). [CrossRef] [PubMed]

_{2}/Si

_{3}N

_{4}onion resonators with 15 cladding pairs with SiO2/Si onion resonators of 6 cladding pairs. The quality factors of the TE

_{1}mode in these two onion resonators are respectively 1.9613×10

^{5}and 1.8759×10

^{5}. In Fig. 9(a), we show the averaged spontaneous emission rate into various multipole order

*l*in SiO

_{2}/Si

_{3}N

_{4}and SiO

_{2}/Si onion resonators. We notice that for the SiO

_{2}/Si

_{3}N

_{4}and the SiO

_{2}/Si onion resonators, the spontaneous emission rate into a given multipole order

*l*are very similar in magnitude if the multipole order

*l*is relatively small. On the other hand, for larger multipole orders, the partial spontaneous emission rate of the SiO

_{2}/Si

_{3}N

_{4}onion resonator is generally larger than that of the SiO

_{2}/Si onion resonators. Such behavior can be attributed to the fact that the SiO

_{2}/Si cladding layers can provide optical confinement equally well for both larger and smaller multipole orders (due to their large index contrast), whereas the SiO

_{2}/Si

_{3}N

_{4}cladding layers are less effective in providing confinement for radiation fields with larger multipole orders. In Fig. 9(b) and Fig. 9(c), we show the total average spontaneous emission rate in a SiO

_{2}/Si

_{3}N

_{4}onion resonator and a corresponding SiO

_{2}/Si onion resonator. The results demonstrate that for two Bragg onion resonators with similar quality factors, the one with SiO

_{2}/Si cladding layers can achieve better spontaneous emission inhibition as compared to the resonator with SiO

_{2}/Si

_{3}N

_{4}cladding layers. However, it should be noted that the degree of spontaneous emission inhibition in a SiO

_{2}/Si

_{3}N

_{4}onion resonator is only marginally less than that of a comparable SiO

_{2}/Si onion resonator, which is approximately a factor of two.

*n*

_{co}= 1.33), the wave vector in a water-filled onion resonator is smaller compared to that in an air core onion resonator, which means an onion resonator with a water-filled core can support higher order multipole modes. Therefore, the light emitter within the solution filled onion cavity can couple to more mutlipole components, which is clearly shown in Fig. 10(a). As a result, we expect that the inhibition of spontaneous emission should be less pronounced in the water core onion resonator as compared to the air core onion resonator. In Fig. 10(b) and (10c), we show the total average spontaneous emission rate in a water core onion resonator and the corresponding air core resonator. As expected, the off resonance spontaneous emission rate in a water core onion resonator is similar or slightly larger than that in an air core resonator. However, our calculations also demonstrate that it is possible to achieve a two orders of magnitude reduction in the spontaneous emission rate in a solution-filled onion resonator.

### 3.5 Non-exponential averaged decaying

*t*= 0. Subsequently at a later time

*t*, the total number of light emitters at the excited state is

*n*(

*θ,φ*) . To further simplify calculations, we assume the initial distribution function

*n*(

*n*

_{0}and consider the local spontaneous emission rate to be independent of the dipole polarization. In reality, the spontaneous emission rate of an individual dipole source may have a strong dependence on the dipole polarization (as shown in Fig. 3). However, the assumption of a polarization independent local spontaneous emission rate greatly simplifies the integration in Eq. (24), and also allows a qualitatively analysis of the non-exponential spontaneous decay in the onion resonator. With these considerations, Eq. 24 is simplified as:

*b*(

*b*

_{0}⟩ is the local spontaneous emission rate average over all the possible dipole orientation and is given by Eq. (18). The optical radiation power due to the dipole ensemble in the onion core can be obtained from Eq. (25):

*ħυ*is the energy of a single photon. From Eq. (26), we can define a time dependent average damping rate as

_{2}/Si cladding pairs, we analyze the time dependent spontaneous emission at λ = 1.55972μm, which correspond to the resonant wavelength of the TM

_{2}mode, and λ= 1.548 μm, which is off cavity resonance. In Fig. 11, we plot the temporal variation of the spontaneous radiation power and the time dependent spontaneous emission rate. From Fig. 11, we notice that initially, the total radiation power calculated from Eq. (26) decays faster than the results obtained assuming exponential decay. On the other hand, at later time the rate of non-exponential decay becomes smaller than the average spontaneous emission rate obtained assuming exponential decay. Since the spontaneous emission from the light emitter ensemble contains both fast-decaying and slow-decaying dipoles, it is reasonable to expect that the total radiation from the dipole ensemble begins with contributions from fast-decaying light emitters followed by slow-decaying light emitters, as shown in Fig. 11. From the results shown in Fig. 11, we can also conclude that the average spontaneous emission rate as defined in Eq. (22) provides a reasonably good description of the decay of the dipole ensemble within the onion resonator, even though the time dependent spontaneous emission rate changes after the initial excitation.

## 4. Discussion

**28**, 2144 (2003). [CrossRef] [PubMed]

12. J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. **81**, 1110 (1998). [CrossRef]

12. J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. **81**, 1110 (1998). [CrossRef]

25. M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and nhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. **86**, 3168 (2001). [CrossRef] [PubMed]

**28**, 2144 (2003). [CrossRef] [PubMed]

*b*

_{stem}/

*b*

_{0}= (

*r*

_{stem}/2

*r*

_{co})

^{2}. The practical value for the stem radius and the core radius are 1 μm and 7μm, thus we obtain

*b*

_{stem}/

*b*

_{0}~ 0.5%. In our analysis (e.g., Fig. 9), a suppressed of ~1% can be achieved with

*N*

_{Bragg}= 7. Thus we expect that the suppression ratio can be limited by the presence of the stem only for very large

*N*

_{Bragg}> 8.

## 5. Conclusion

^{2}~ 10

^{3}upon resonance and a suppression ratio of ~10

^{-2}off resonance can be achieved with 7 Bragg cladding layers. Finally, the assumption that the averaged radiation decays exponentially is examined. The analysis presented in this paper should provide a quantitative foundation for future experimental investigation of onion resoantors.

## Appendix

## A. Derivation of the reflected electric field

**50**, 2708 (1994). [CrossRef] [PubMed]

_{0}in the Bragg onion core and polarized along direction

*iω*

_{0}

*t*) harmonic time dependence)

*ω*

_{0}is the angular frequency,

*μ*

_{0}and

*ε*are the constant permeability and permittivity of the core material,

*iω*

_{0}

*p*

_{0}

*δ*(

_{0})

*p*

_{0}is the amplitude of the dipole moment. Employing the dyadic Green’s function method [26

26. P. Das and H. Metiu, “Enhancement of Molecular Fluorescence and Photochemistry by Small Metal Particles,” J. Phys. Chem. **89**, 4680 (1985). [CrossRef]

*J*

_{0}=-

*iω*

_{0}

*p*

_{0}is the current amplitude,

*g*(

_{0}) = exp(

*ik*|

_{0}|) /

_{0}| is the three-dimensional scalar Green’s function. The subscript “

*0*” in the gradient operator

_{0}means the operator is acting on the source variable

_{0}∙

*g*(

_{0}) can be expanded in terms of spherical waves such that

*j*(

*kr*

_{<}) and

*h*

_{l}

_{1}(

*kr*

_{>}) are respectively the spherical Bessel function and the spherical Hankel function of the first kind,

*Y*

_{lm}(

*θ, φ*) is the spherical harmonics function,

*r*

_{>}(

*r*

_{<}) is the greater (lesser) of

*r*and

*r*

_{0}. The reflected field is coming from the source field reflected by the cladding and should have similar forms to the source field. We can thus express the reflected fields according to Eq. (A2a)–(A3) as

*A*

_{lm})

^{TE,TM}are constant amplitudes decided by boundary condition,

*L̂*=

*i*is the angular momentum operator,

_{0}) and

_{0}) are operators acting on the source variable, which can be expressed as

*j*

_{l}(

*kr*), because the reflected field must be nonsingular at the center

*r*= 0. The total field in the core is the sum of the source field and the reflected field and must satisfy the boundary conditions: at the core-cladding interface, the sum of the inward-traveling radial waves is equal to the sum of the product of the outward-traveling radial waves and the field amplitude reflection coefficient. Employing the boundary condition we obtain the constant amplitudes

*ρ*

_{l}is the field amplitude reflection coefficient of the

*l*th order multipole mode at the core-cladding interface,

*r*

_{co}is the radius of the core. Substituting Eq. (A6) into Eq. (A4a)–(A4b), and applying the result to Eq. (5b), the reflected electric field component along the dipole moment orientation at the source position normalized by parameter

*E*

_{s}(Eq. (5a)) is

## B. Direct evaluation of the modified decaying rate by calculating the radiation power

17. R. R. Chance, Prock R., and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. **37**, 1 (1978). [CrossRef]

19. W. Lukosz and R. E. Kunz, “Light-Emission by Magnetic and Electric Dipoles Close to a Plane Interface .1. Total Radiated Power,” J. Opt. Soc. Am. **67**, 1607 (1977). [CrossRef]

21. H. Chew, “Transition Rates of Atoms near Spherical Surfaces,” J. Chem. Phys. **87**, 1355 (1987). [CrossRef]

26. P. Das and H. Metiu, “Enhancement of Molecular Fluorescence and Photochemistry by Small Metal Particles,” J. Phys. Chem. **89**, 4680 (1985). [CrossRef]

*P*

_{cav}, divided by the dipole radiation power in the bulk material,

*P*

_{bulk}. For an oscillating dipole located at

_{0}, the radiative electric field at

*r*>

*r*

_{0}in the core can be expanded in terms of the multipole modes [28]

*Z*

_{co}= √

*μ*

_{0}/

*ε*

_{co}is the material impedance and

*k*

_{co}= √

*ε*

_{co}/

*ε*

_{0}ω

_{0}/

*c*

_{0}is the wave vector. Here

*ε*

_{co}is the constant permittivity of the core material. The source current is

*iω*

_{0}

*p*

_{0}

*δ*(

_{0})

*ρ*= -

*p*

_{0}

*δ*(

_{0})

*J*

_{0}= -

*iω*

_{0}

*p*

_{0}is again the dipole current amplitude. The total field inside the cavity core is composed of the direct radiation field of the dipole (Eq. (B1)) plus the field reflected from the boundary, i.e.

*P*

_{bulk}= √

*ε*

_{co}/

*ε*

_{0}∙

*πε*

_{0}

*p*

_{0}|

^{2}

^{3}. Substituting Eq. (B3a)–(B3b) and Eq. (B8) into Eq. (B9), the normalized decaying rate is

*TM,TE*means summation over both TM and TE modes.

*l,m*) described by Eq. (8a)–(8b), by integrating the poynting vector over all solid angle, we obtain the radiation power in the

*n*th layer [28]

*B*

_{out}= 0 and

*D*

_{out}= 0 , we can prove

*c*

_{E}=

*a*

_{E}and

*c*

_{M}=

*a*

_{M}, then Eq (B10) leads to

## References and links

1. | J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. ThierryMieg, and T. Rivera, “Quantum boxes as active probes for photonic microstructures: The pillar microcavity case,” Appl. Phys. Lett. |

2. | J. M. Gerard and B. Gayral, “Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities,” J. Lightwave Technol. |

3. | B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. |

4. | D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature |

5. | K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho, and C. Gmachl, “Experimental demonstration of a high quality factor photonic crystal microcavity,” Appl. Phys. Lett. |

6. | K. J. Vahala, “Optical microcavities,” Nature |

7. | Y. Yamamoto, S. Machida, and G. Bjork, “Microcavity Semiconductor-Laser with Enhanced Spontaneous Emission,” Phys. Rev. A |

8. | M. H. Macdougal, P. D. Dapkus, V. Pudikov, H. M. Zhao, and G. M. Yang, “Ultralow Threshold Current Vertical-Cavity Surface-Emitting Lasers with Alas Oxide-Gaas Distributed Bragg Reflectors,” IEEE Photon. Technol. Lett. |

9. | V. Sandoghdar, F. Treussart, J. Hare, V. LefevreSeguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A. |

10. | D. L. Huffaker, L. A. Graham, H. Deng, and D. G. Deppe, “Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors,” IEEE Photon. Technol. Lett. |

11. | P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science |

12. | J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. |

13. | M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. |

14. | Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “High-quality-factor Bragg onion resonators with omnidirectional reflector cladding,” Opt. Lett. |

15. | Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S. Y. Lin, “Modal analysis of Bragg onion resonators,” Opt. Lett. |

16. | K. G. Sullivan and D. G. Hall, “Radiation in Spherically Symmetrical Structures .2. Enhancement and Inhibition of Dipole Radiation in a Spherical Bragg Cavity,” Phys. Rev. A. |

17. | R. R. Chance, Prock R., and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. |

18. | H. Chew, “Radiation and Lifetimes of Atoms inside Dielectric Particles,” Phys. Rev. A. |

19. | W. Lukosz and R. E. Kunz, “Light-Emission by Magnetic and Electric Dipoles Close to a Plane Interface .1. Total Radiated Power,” J. Opt. Soc. Am. |

20. | W. K. H. Panofsky and M. Phillips, Classical |

21. | H. Chew, “Transition Rates of Atoms near Spherical Surfaces,” J. Chem. Phys. |

22. | M. P. vanExter, G. Nienhuis, and J. P. Woerdman, “Two simple expressions for the spontaneous emission factor beta,” Phys. Rev. A. |

23. | E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. |

24. | J. Vuckovic, M. Pelton, A. Scherer, and Y. Yamamoto, “Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics,” Phys. Rev. A. |

25. | M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald, and A. Forchel, “Inhibition and nhancement of the spontaneous emission of quantum dots in structured microresonators,” Phys. Rev. Lett. |

26. | P. Das and H. Metiu, “Enhancement of Molecular Fluorescence and Photochemistry by Small Metal Particles,” J. Phys. Chem. |

27. | W. C. Chew, Waves and Fields in inhomogeneous Media (Van Nostrand Reinhold, New York, 1990). |

28. | Jackson, |

**OCIS Codes**

(230.1480) Optical devices : Bragg reflectors

(230.5750) Optical devices : Resonators

(270.5580) Quantum optics : Quantum electrodynamics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: May 25, 2006

Revised Manuscript: July 19, 2006

Manuscript Accepted: July 20, 2006

Published: August 7, 2006

**Citation**

Wei Liang, Yanyi Huang, Amnon Yariv, Yong Xu, and Shawn-Yu Lin, "Modification of spontaneous emission in Bragg onion resonators," Opt. Express **14**, 7398-7419 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-7398

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### References

- J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L. Manin, E. Costard, V. Thierry Mieg and T. Rivera, "Quantum boxes as active probes for photonic microstructures: The pillar microcavity case," Appl. Phys. Lett. 69, 449 (1996). [CrossRef]
- J. M. Gerard, and B. Gayral, "Strong Purcell effect for InAs quantum boxes in three-dimensional solid-state microcavities," J. Lightwave Technol. 17, 2089 (1999). [CrossRef]
- B. Gayral, J. M. Gerard, A. Lemaitre, C. Dupuis, L. Manin and J. L. Pelouard, "High-Q wet-etched GaAs microdisks containing InAs quantum boxes," Appl. Phys. Lett. 75, 1908 (1999). [CrossRef]
- D. K. Armani, T. J. Kippenberg, S. M. Spillane and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925 (2003). [CrossRef] [PubMed]
- K. Srinivasan, P. E. Barclay, O. Painter, J. X. Chen, A. Y. Cho and C. Gmachl, "Experimental demonstration of a high quality factor photonic crystal microcavity," Appl. Phys. Lett. 83, 1915 (2003). [CrossRef]
- K. J. Vahala, "Optical microcavities," Nature 424, 839 (2003). [CrossRef] [PubMed]
- Y. Yamamoto, S. Machida and G. Bjork, "Microcavity semiconductor-laser with enhanced spontaneous emission," Phys. Rev. A 44, 657 (1991). [CrossRef] [PubMed]
- M. H. Macdougal, P. D. Dapkus,V. Pudikov, H. M. Zhao and G. M. Yang, "Ultralow threshold current vertical-cavity surface-emitting lasers with alas Oxide-Gaas distributed Bragg reflectors," IEEE Photon. Technol. Lett. 7, 229 (1995). [CrossRef]
- V. Sandoghdar, F. Treussart, J. Hare, V. Lefevre-Seguin, J. M. Raimond and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A. 54, R1777 (1996). [CrossRef] [PubMed]
- D. L. Huffaker, L. A. Graham, H. Deng and D. G. Deppe, "Sub-40 mu A continuous-wave lasing in an oxidized vertical-cavity surface-emitting laser with dielectric mirrors," IEEE Photon. Technol. Lett. 8, 974 (1996). [CrossRef]
- P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. D. Zhang, E. Hu and A. Imamoglu, "A quantum dot single-photon turnstile device," Science 290, 2282 (2000). [CrossRef] [PubMed]
- J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard and V. Thierry-Mieg, "Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity," Phys. Rev. Lett. 81, 1110 (1998). [CrossRef]
- M. Pelton, C. Santori, J. Vuckovic, B. Y. Zhang, G. S. Solomon, J. Plant and Y. Yamamoto, "Efficient source of single photons: A single quantum dot in a micropost microcavity," Phys. Rev. Lett. 89299602 (2002). [CrossRef] [PubMed]
- Y. Xu, W. Liang,A. Yariv,J. G. Fleming and S. Y. Lin, "High-quality-factor Bragg onion resonators with omnidirectional reflector cladding," Opt. Lett. 28, 2144 (2003). [CrossRef] [PubMed]
- Y. Xu, W. Liang, A. Yariv, J. G. Fleming and S. Y. Lin, "Modal analysis of Bragg onion resonators," Opt. Lett. 29, 424 (2004). [CrossRef] [PubMed]
- K. G. Sullivan and D. G. Hall, "Radiation in spherically symmetrical structures.2. Enhancement and inhibition of Dipole Radiation in a Spherical Bragg Cavity," Phys. Rev. A. 50, 2708 (1994). [CrossRef] [PubMed]
- R. R. Chance, A. Prock and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1 (1978). [CrossRef]
- H. Chew, "Radiation and lifetimes of Atoms inside dielectric particles," Phys. Rev. A. 38, 3410 (1988). [CrossRef] [PubMed]
- W. Lukosz and R. E. Kunz, "Light-emission by magnetic and electric dipoles close to a plane interface.1. Total radiated power," J. Opt. Soc. Am. 67, 1607 (1977). [CrossRef]
- W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, (Addison-Weskley, MA, 1956).
- H. Chew, "Transition rates of Atoms near Spherical Surfaces," J. Chem. Phys. 87, 1355 (1987). [CrossRef]
- M. P. van Exter, G. Nienhuis and J. P. Woerdman, "Two simple expressions for the spontaneous emission factor beta," Phys. Rev. A. 54, 3553 (1996). [CrossRef]
- E. Yablonovitch, "Inhibited spontaneous emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58, 2059 (1987). [CrossRef] [PubMed]
- J. Vuckovic, M. Pelton, A. Scherer and Y. Yamamoto, "Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics," Phys. Rev. A. 66023808 (2002). [CrossRef]
- M. Bayer, T. L. Reinecke, F. Weidner, A. Larionov, A. McDonald and A. Forchel, "Inhibition and enhancement of the spontaneous emission of quantum dots in structured microresonators," Phys. Rev. Lett. 86, 3168 (2001). [CrossRef] [PubMed]
- P. Das, and H. Metiu, "Enhancement of molecular fluorescence and photochemistry by Small Metal Particles," J. Phys. Chem. 89, 4680 (1985). [CrossRef]
- W. C. Chew, Waves and Fields in inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
- J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New York, 1999).

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