1. Introduction

Generation of optical pulses consisting of exactly one photon is a hot topic of modern quantum optics. Application of such pulses to quantum cryptography substantially improves robustness of quantum communication channels against enemy attacks [1]. I has been suggested that single-photon states can also be used for quantum computation [2]. Ordinary weak laser pulses have Poisson probability distribution function for the photon numbers. Accordingly, the probability to have more than one photon per pulse for such pulses is 1-(1+n̄)exp(-n̄), where n̄ is the average photon number, and is quite high unless n̄≪1. The presence of multi-photon pulses can substantially compromise security of the quantum channel and actually limits the maximum possible length of the optical link between the sender and the recipient [3]. It is also important to maintain a small probability for pulses without any photons (that is to have n̄≈1) because such pulses make communication inefficient. All these shortcomings of weak laser pulses are eliminated if a perfect single-photon source is used.

A large number of experimentally demonstrated or proposed single-photon sources are based on the fundamental property of a single quantum system (SQS) to emit one photon when making transition from excited to the ground state [4–13].

In most cases the single-photon emission is a spontaneous process following excitation of the quantum system with a short pumping pulse. The downside of a single-photon source based on single-photon emission by an SQS is that, as a rule, such a photon can be emitted in any direction with almost equal probability making efficient preparation of a collimated single-photon beam very difficult. This problem can be efficiently resolved by placing an SQS in a microcavity [14]. If only one mode of the microcavity is in resonance with the optical transition of the quantum system, photons will be most effectively emitted in that particular mode. The important parameter of the microcavity is its Purcell factor F which reads [15, 16]

where τ _⊥ is the decay time of the electric field amplitude, n is the refraction index, c is the speed of light in vacuum, ν is the field frequency, and V _m is the mode volume which is defined by the equation

The Purcell factor should be much larger than one to achieve efficient generation of single-photons into a single-mode. The probability P_m of spontaneous photon emission in the cavity mode is given by

The validity of Eq. (3) implies irreversible energy exchange between the cavity and the SQS. Therefore the cavity and the SQS should be weakly coupled [15] that is the cavity bandwidth and the homogeneous linewidth of the SQS should be larger than the SQS-cavity coupling constant g ₀/π≡[3c ³Γ₀/(8π ³ ν²$n_p^{\mathit{2}}$V_m ]^1/2, where Γ₀ is the free-space spontaneous emission rate of the SQS (note that strong coupling is required for quantum networking [17]). A relatively fast field decay-rate also simplifies requirements for the stability of the cavity length and hence of the cavity resonance frequency (which should be kept in resonance with the SQS).

Several cavity designs have been proposed and realized [14]. A bulk Fabry-Perot cavity made of two spherical mirrors [17] has a relatively large (V_m > 2000 ×λ ³) mode volume and therefore a very long mode decay-time is required. Practically, such cavity can operate only in vacuum and requires expensive mirrors with losses below 10^-6. A micropost type of cavity [6–8] exploits microchip production technology and is hardly usable for organic molecules and diamond nanocrystals. High quality factors of microsphere resonances [18] can also be exploited but this type of microcavity has a mode volume similar to a bulk Fabry-Perot cavity.

2. Cavity design and numerical simulations

We propose a cavity which can be built using moderate quality, relatively cheap and easily available mirrors with losses of about 10^-3. The active region of the cavity is filled with a polymer. This polymer can be used to host a large variety of single-photon emitters including organic molecules [4, 5, 9], diamond nanocrystals [10], and semiconductor quantum dots [6, 19].

The design of the cavity is shown in Fig. 1. It is made of two flat highly reflecting dielectric mirrors with a small cylindrically shaped spacer made of a polymer. The spacer is placed and squeezed between the two mirrors. If squeezing is very gentle, the spacer stays cylindrical. If a substantial force if applied, the side surface of the spacer is curved. Such a cavity also can be fabricated, for example, by spin coating on one cavity mirror a very dilute solution of polymer microspheres doped with organic dye molecules or other single-photon emitters. After the spheres are distributed, they are squeeze with the second mirror. Spin coating will ensure that the micro cavities are spread over the mirror surface without touching each other. It is assumed that the concentration of the SQSs is sufficiently low so that only one SQS per cavity is in resonance with a cavity mode.

The numerical simulation shows (see details below) that the exact form of the spacer side shape is of little importance even in an extreme case when the radius of the curvature is equal to the spacer thickness s. (see Table 1, superscript a) and therefore most simulations are done for a cavity with cylindrical spacer (Table 1, no superscript or superscripts b, c or d).

This cavity resembles a Fabry-Perot microcavity used for single photon generation by De Martini et al [20] but has a substantial difference. The space between the two mirrors forming the Fabry-Perot resonator is not filled with a homogeneous dielectric. This inhomogeneity leads to strong localization of the electromagnetic field if the field oscillation frequency is close to the cavity resonances.

The cavity has been analyzed using 3-dimentional finite-difference time-domain (FDTD) simulation technique with perfectly matching boundary conditions at the boundary of the simulation region (a commercially available package Lumerical FDTD Solution has been exploited). Distributions of the electric field amplitude in the cavity modes were obtained by simulating propagation of a short electromagnetic pulse (initial polarization is parallel to the x-axis) inside the resonator. Simulations were repeated with different space and time resolution to verify their stability and reproducibility. An example of such distribution taken at a frequency close to the lowest EM11 cavity resonance is shown in Fig.1. The numerical values of the refraction indexes, the thicknesses of the dielectric layers, and the geometry of the polymer spacer are given in the figure caption. The cavity characteristics were systematically investigated as a function of the diameter D of the polymer spacer. The results are given in Table 1. Note that the best performance of the cavity can be achieved if the SQS is placed at the highest electric field intensity with SQS transition dipole moment parallel to the electric field vector. This can be achieved approximately by selecting a best performing micro cavity from a number of cavities with randomly distributed SQSs.

3. Discussion

If the mirrors were perfectly reflecting plains, the solution of the problem would be a single period of a standing wave in an infinitely long cylindrical fiber. It is expected that for large D the volume of the lowest axial mode is proportional to the volume of the polymer spacer or to D ², if the spacer length stays constant. As follows from the Table 1, this dependence holds quite well already when D>2λ. When D gets smaller, the electromagnetic field extends substantially outside the spacer and the proportionality to D ² violates. Note, that for the cavity with D=0.6 µm the mode volume is just 0.83·(λ/n_p )³ and is about 2 times smaller than a minimum mode volume of a micropost cavity reported in [15].

If the mirrors were perfectly reflecting plains, there would be no losses except for the losses due to absorption and/or scattering in the material of the fiber (polymer spacer). Because the mirrors are distributed Bragg reflectors (DBR) the electromagnetic field can penetrate into the mirrors and can propagate sidewise inside the layers. To get a simple physical picture for the losses due to this sidewise propagation we first consider a ray of light incident on the mirror at a small angle θ with the z-axis (see Fig. 1). The sidewise displacement δ of this ray during a unit time will be proportional to sin(θ). The energy transmitted by all rays propagating at the angle θ irrespective of their polar angle φ in the x,y-plane is proportional to sin(θ)dθ, where we take into consideration that the problem is cylindrically symmetrical. Therefore the sidewise energy flow transmitted by these rays is proportional to δ·sin(θ)dθ∝θ²dθ, where we have substituted θ for sin(θ). The total energy flow out of the cavity is proportional to ${{\int }_0^{{\theta }_{max}}{\theta }^{2}d\theta \propto {\theta }_{max}^3\propto (\lambda ⁄D)}^3$, where we take into account that for a confined electromagnetic field θ _max∝λ/D due to diffraction. Thus we expect the decay time of the cavity mode to be proportional to D ³. Such scaling together with V_m ∝D ² means that F/D≈ constant. This holds surprisingly well (see Tab. 1) despite of quite a rough model.

 

Fig.1. istribution the electric field (|E_x |) oscillating at the resonance frequency of the microcavity is shown as a false color image. The color bar on the right shows the color code for the relative amplitude of the electric field. The relative values of $E_x^{\mathit{2}}$n² are shown for z=0 (upper-right panel) and r=0 (lower-right panel). Thirty pairs of the dielectric layers form two mirrors (with 15 pairs for each mirror). The alternating refraction indexes of the layers are 2.4 (TiO₂) and 1.46 (SiO₂). Their thicknesses are t ₁=0.06 µm and t ₂=0.1 µm (for the layers with the smaller refraction index). The polymer spacer has a refraction index of n_p =1.45 and a thickness of s=0.24 µm. Two possible shapes of the polymer spacer (cylindrical and elliptical) are shown. The influence of the spacer shape on the cavity characteristics was marginal. The cavity characteristics have been studied as a function of the diameter of the polymer spacer (see Table 1). For the simulation results presented in this figure D=0.8 µm.

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Table 1. Characteristics of the Micro Resonator as Calculated by 3D-FDTD Method

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The decay time or the quality factor Q≡π _τ ν of proposed cavity (see Table 1) is comparable to the quality factor of micropost [15] and microdisk [21] cavities (at equal mode volumes). The Purcell factor of this cavity is from 2 to 16 times (depending on the spacer diameter) larger than a theoretically estimated value reported in [8]. The comparison with very high-Q cavities [14] is more difficult because these cavities have much larger mode volumes than presented in Table 1. Theoretically, by increasing D one can increase Q (according to the scaling derived above one gets Q≈10⁶ when V_m ≈10³ µm³) and in this way even a strong coupling regime [14] can be achieved but such a high Q will be very sensitive to any imperfections and losses in the cavity (see below about losses in the polymer spacer and DBR penetration). This may compromise the main idea of this cavity – its low cost and simplicity.

It follows from Eq. (3) and the Purcell factor obtained in the simulations, that the probability of spontaneous emission in the fundamental cavity mode is about 96% for D=0.6 µm and even higher for larger D. The Purcell factor marginally depends on the shape of the spacer (compare the results labeled with a superscript a to those without a subscript).

The applicability of the theory outlined in the introduction is limited. As an example we consider a dye molecule with a typical Γ₀=50 MHz [22]. At room temperatures, a characteristic homogeneous linewidth of such a molecule is around 15 THz [23] but this number drops below 0.3 THz at 80 K [24] and can be as small as 50 MHz at 2K [22]. The cavity bandwidth γ_cav ≡1(πτ _⊥) ≈2.2 THz (at D=0.6 µm) and g ₀/π≈23 GHz (at Γ₀=50 MHz and D=0.6 µm) are practically temperature independent. Therefore the validity of Eq. (3) depends on the temperature and the cavity diameter. For example, using scaling g ₀∝D ^-1 and γ_cav ∝D ^-3 one can prove that the strong coupling conditions are fulfilled at liquid He temperatures if D>8 µm.

To estimate the DBR penetration effect on the decay rate of the cavity field, we performed simulations with just 8 pairs of dielectric layers on each side of the spacer (see simulation results labeled in Table 1 with a superscript b). The reflectivity R of such mirrors is only 0.998, but if D<0.8 µm additional losses introduced by mirrors are negligible. Therefore relatively cheap mirrors can be used to achieve a Purcell factor up to 30. ${\tau }_m^{-1}$≈ν(1-R). For even smaller reflectivity (0.996 and 0.989, see Table 1), the effect of DBR penetration becomes significant but not totally destructive (the corresponding Purcell factors are still larger than 10).

The field decay times τ _⊥ were calculated assuming no absorption or scattering in the polymer. The effect of the losses in the polymer can be estimated as follows. A round trip inside the resonator tikes time equal to one period of oscillations of the electromagnetic field of the lowest longitudinal cavity mode. If the energy loss per one passage through the polymer insert (including both absorption and scattering) is L, the related field decay rate ${\tau }_p^{-1}$ can be estimated as ${\tau }_p^{-1}$≈νL. Optical loses in PMMA fibers are less than α=1 dB/m [25] but as a conservative estimate we take 100 dB/m and obtain L=1-10^-sα/10≈6×10^-6 for a polymer length s=0.24 µm. The total decay rate of the cavity field is given by ${\tau }_{\perp }^{-1}$+${\tau }_p^{-1}$. The probability that a photon is absorbed or scattered by the polymer is given by the ratio of absorption and/or scattering rate to the total rate ${\tau }_p^{-1}$/(${\tau }_{\perp }^{-1}$+${\tau }_p^{-1}$). This is smaller than 1% if τ _⊥<5 ps. Therefore losses in the polymer can be neglected unless D>2.5 µm (see Table 1).

4. Conclusions

The proposed micro cavity can be used for substantial improvement of the fidelity of single-photon sources based on spontaneous emission of a single quantum system. The achievable theoretical efficiency of this cavity is comparable to that of already known cavities but the design of this cavity is suitable for a larger variety of quantum emitters (including organic molecules, diamond nanocrystals, and others) which can be used for efficient single-photon generation.