1 INTRODUCTION

One of the goals of quantum nanophotonics is to fabricate efficient single-photon sources, where quantum dots serve as dipole emitters. The key method for the enhancement of radiation and its most efficient directional extraction from emitters is the use of microcavity structures. Resonant photonic crystals [1, 2] and vertical circular microcavities [3] are among the most well-known and promising microcavities for single-photon sources. In particular, circular microcavities with a radius of 1–4 μm with distributed AlGaAs/GaAs Bragg reflectors above and below the GaAs active region with InGaAs/GaAs quantum dots [4] are used to implement single-photon sources in the spectral range of ~920 nm. A cavity is designed so that an optical transition occurs near the energy of the HE11 waveguide optical mode degenerate in polarization and is located at the maximum amplitude of the implemented optical mode. This makes it possible to accelerate the emission of quantum dots owing to the Purcell effect and to efficiently extract radiation ou-tside.

However, a number of problems described in review [5] can worsen the optical characteristics of such sources, e.g., the degree of photon indistinguishability. One of these problems is the pumping of a quantum dot high into the conduction band with its further scattering with an unpredictable emission time. This problem can be solved by using resonant or quasiresonant pumping, which allows one to avoid loss in the degree of indistinguishability owing to a higher definiteness of the lifetime of the emitting state. Simultaneously, detection in experiments should be separated from scattered laser pump radiation: pump radiation and detection in the case of resonant photon waveguides are separated in space [2], whereas one of the methods in the case of circular microcavities is spectral filtering in polarization [6]. The latter method reduces the efficiency of a source by at least 50%, which fundamentally restricts the scaling of such sources.

One of the ways to overcome a decrease in the efficiency of single-photon sources is the reduction of the symmetry of the system so that the previously doubly degenerate fundamental optical mode of the cavity is polarization split [7]. This can be ensured by modifying the geometry of the circular cavity, changing the circular cross section to an elliptical one [8, 9]. This splitting in the far field region results in the linear polarization of radiation from the cavity. Varying the eccentricity, one can achieve the situation where two modes of such a cavity are almost degenerate and polarized in mutually perpendicular directions in the far field region. In this cavity, where two optical modes with the V and H polarizations are allowed and the fr-equency spacing between them \(\Delta \omega \) is small, e.g., the V-polarized mode can be excited by an external laser and the H-polarized mode is also excited; as a result, a single-photon source is excited. Since exciting radiation and emitted photons are polarized in mutually perpendicular directions, a high efficiency of radiation extraction up to 93%, as estimated by the authors of [7], can be obtained.

Molecular beam epitaxy is primarily used to obtain circular and elliptical semiconductor nanopillars with an insert of quantum emitters. This method allows the atomically accurate growth of planar Bragg mirrors and layers of quantum dots with the minimum number of defects for ensuring a quantum efficiency of quantum dots of almost unity. These cavities can provide a high Q factor of about 104. The circular or elliptical nanopillars are obtained by electron lithography and subsequent ion etching [10]. Thus, the current technology makes it possible to fabricate cavities with almost any shape and size.

A preliminary numerical simulation is needed to control the parameters of a cavity. This simulation usually involves the solution of Maxwell’s equations in the time or frequency domain. Unfortunately, a large number of layers in a structure (e.g., in Bragg mirrors) make problems computationally difficult; they require large memory and high computational capacity. Furthermore, the variation of a selected parameter in order to choose an ideal design significantly increases the time required for a given algorithm.

In this work, a method is presented to rapidly select the necessary parameters of the elliptical cavity increasing the efficiency of single-photon sources. We propose a method to determine the energies of optical modes, calculate their spatial distribution in a two-dimensional metallic cavity, and estimate errors in the transition to a dielectric zero-dimensional cavity with three-dimensional quantization owing to Bragg mirrors. In addition, a method is described to excite almost degenerate states to obtain an efficient source of single and entangled photons.

2 THEORETICAL DESCRIPTION OF OPTICAL MODES

To describe the mode structure of the elliptical cavity, we consider the exactly solvable problem of the elliptical cavity with ideal metallic walls. This problem is partially considered in [11]. Cavity modes are determined by the following equation for the z component of the electric (TM mode) or magnetic (ТЕ mode) field (other components are expressed in terms of this component):

$$\left( { - \frac{{{{n}^{2}}{{\omega }^{2}}}}{{{{c}^{2}}}} - \Delta } \right)A = 0,$$
(1)

where A = \({{E}_{z}}\) (\({{H}_{z}}\)) for the TM (ТE) mode with the frequency ω, n is the refractive index, and c is the speed of light in vacuum. The variables in Eq. (1) for the metallic cavity are separated, and the operator Δ can be represented in the form \(\Delta = {{\Delta }_{2}} + \frac{{{{\partial }^{2}}}}{{\partial {{z}^{2}}}},\) where \({{\Delta }_{2}}\) is the two-dimensional Laplace operator. Assuming \(A = B(x,y)Z(z)\), we obtain

$$\frac{{{{d}^{2}}Z}}{{d{{z}^{2}}}} = - k_{z}^{2}Z,$$
(2)
$${{\Delta }_{2}}B = - k_{\parallel }^{2}B,$$
(3)

where \({{k}_{{||}}}\) is the wave vector of the mode in the plane of the layer and \({{k}_{z}}\) is the transverse wave vector, which satisfy the relation \(k_{\parallel }^{2} + k_{z}^{2} = \frac{{{{n}^{2}}{{\omega }^{2}}}}{{{{c}^{2}}}}\).

We introduce elliptical coordinates in the plane of the cavity. Let the foci of the ellipse be located on the x axis at the points x = a and x = –a. Then, the x and y coordinates are represented in the form

$$x = a{\text{cosh}}\mu \,{\text{cos}}\nu ,$$
$$y = a{\text{sinh}}\mu \,{\text{sin}}\nu ,$$

where \(\;\mu \geqslant 0\) and \(\nu \in [0,2\pi ]\). After this change of variables, the \(\mu = {\text{const}}\) contours are ellipses, one of which should coincide with the edge of the cavity. The parameters of the ellipse in the canonical ellipse eq-uation

$$\frac{{{{x}^{2}}}}{{b_{1}^{2}}} + \frac{{{{y}^{2}}}}{{b_{2}^{2}}} = 1$$
(4)

are \(b_{1}^{2} = {{a}^{2}}{\kern 1pt} {\text{cosh}}{{{\kern 1pt} }^{2}}\mu \) and \(b_{2}^{2} = {{a}^{2}}{{\sinh }^{2}}\mu .\) The parameter μ = arctanh(b2/b1) in this case characterizes the eccentricity of the ellipse; in particular, μ = ∞ in the circular case, and μ → 0 with an increase in the eccentricity. After this change of variables, Eq. (3) is transformed to

$$\frac{1}{{{{a}^{2}}({\text{sin}}{{{\text{h}}}^{2}}\mu + {{{\sin }}^{2}}\nu )}}\left( {\frac{{{{\partial }^{2}}B}}{{\partial {{\mu }^{2}}}} + \frac{{{{\partial }^{2}}B}}{{\partial {{\nu }^{2}}}}} \right) + k_{\parallel }^{2}B = 0.$$
(5)

Introducing \(B = M(\mu )N(\nu )\) and using the formula \(2{{\sin }^{2}}x = 1 - \cos 2x\) and its hyperbolic analog, after the separation of variables, we obtain the pair of equations

$$N{\kern 1pt} ''\; + \left( {\alpha - \frac{{k_{\parallel }^{2}{{a}^{2}}}}{2}\cos 2\nu } \right)N = 0,$$
(6)
$$M{\kern 1pt} ''\; - \left( {\alpha - \frac{{k_{\parallel }^{2}{{a}^{2}}}}{2}\cosh 2\mu } \right)M = 0,$$
(7)

where α is an additional separation constant, the function N should be \(2\pi \)-periodic, and the function M at the edge of the metallic cavity should satisfy one of the boundary conditions

$$N(\nu + 2\pi ) = N(\nu ),$$
(8)
$$\left\{ {\begin{array}{*{20}{c}} {M({{\mu }_{0}}) = 0\;\quad {\text{for TM modes}}} \\ {M{\kern 1pt} '({{\mu }_{0}}) = 0\quad \,{\text{for TE modes}},} \end{array}} \right.$$
(9)

where \({{\mu }_{0}}\) is the μ value at which the μ contour coincides with the edge of the cavity. Equation (6) coincides with the Mathieu equation

$$y{\kern 1pt} ''(z) + (\alpha - 2q\cos 2z)y(z) = 0,$$
(10)

and Eq. (7) coincides with the modified Mathieu equation obtained from Eq. (10) by the substitution z = iu:

$$y{\kern 1pt} ''(u) - (\alpha - 2q\cosh 2u)y(u) = 0,$$

where

$$q = \frac{{k_{\parallel }^{2}{{a}^{2}}}}{4}.$$
(11)

To determine the constants α and \({{k}_{\parallel }}\) and, as a result, the spectrum of the cavity, boundary conditions (8) and (9) should be used. To this end, it is necessary to select \(2\pi \)-periodic solutions of the Mathieu equation and solutions of the modified Mathieu equation that have a root at a given \({{\mu }_{0}}\) value. It is known that the Mathieu equation with a pair (α, q ≠ 0) can have no more than one π- or \(2\pi \)-periodic solution [12]. At a fixed q value, periodic solutions exist at a discrete set of values αm, where m is the number of roots of the function in the period. An important difference from the case of the conventional cylinder is the nondegeneracy of these αm.

For the cylinder, q = 0 and two periodic solutions \(\sin \nu \) and \(\cos \nu \) with the same frequency correspond to each αm > 0. These solutions for the ellipse are split in frequency. An even solution cem and odd solution sem generally having different frequencies appear as analogs of cosine and sine, respectively. For the modified Mathieu functions, the corresponding functions of the imaginary argument are introduced \(C{{e}_{m}}(x) = c{{e}_{m}}(ix)\) and \(S{{e}_{m}} = - is{{e}_{m}}(ix),\) where the subscript m corresponds to the even or odd solution of the Mathieu equation.

According to [11], the solution of Eq. (5) has the form

$${{B}_{{{{z}_{m}}}}} = {{C}_{m}}ce(\nu ,{{q}_{{mr}}})Ce(\mu ,{{q}_{{mr}}})\;{\text{for TM mode}},$$
(12)
$${{B}_{{{{z}_{m}}}}} = {{C}_{m}}se(\nu ,{{q}_{{mr}}})Se(\mu ,{{q}_{{mr}}})\;{\text{for TE mode}},$$
(13)

where \({{B}_{z}}\) = \({{E}_{z}}\) (\({{H}_{z}}\)) for the ТМ (TE) mode. The other combinations \(c{{e}_{m}}(\nu ,{{q}_{{mr}}})S{{e}_{m}}(\mu ,{{q}_{{mr}}})\) are forbidden by the condition of continuity in the [–a, a] segment. The subscript r specifies the roots of the function \(C{{e}_{m}}\) or \(S{{e}_{m}}\). Two generally nondegenerate frequencies \(q_{{mr}}^{{(i)}}\) appearing in a given geometry of the ellipse are determined from the conditions

$$0 = \left\{ {\begin{array}{*{20}{c}} {Ce({{\mu }_{0}},q_{{mr}}^{{(1)}})} \\ {Se({{\mu }_{0}},q_{{mr}}^{{(2)}})} \end{array}} \right.\quad {\text{for TM mode}}{\text{,}}$$
(14)
$$0 = \left\{ {\begin{array}{*{20}{c}} {{{\partial }_{\mu }}Ce({{\mu }_{0}},q_{{mr}}^{{(1)}})} \\ {{{\partial }_{\mu }}Se({{\mu }_{0}},q_{{mr}}^{{(2)}})} \end{array}} \right.\quad {\text{for TE mode}}.$$
(15)

2.1 Examples of Optical Modes

We compare the circular and elliptical cavities in the case of the ТМ mode. As known, the solution for \({{E}_{z}}\) in the circular cavity is doubly degenerate [13] and has the form

$${{E}_{z}} = C{{J}_{m}}(qR)\left\{ {\begin{array}{*{20}{c}} {\cos (m\varphi )} \\ {\sin (m\varphi )} \end{array}} \right.,$$

where \({{q}^{2}} = \frac{{{{n}^{2}}{{\omega }^{2}}}}{{{{c}^{2}}}} - k_{z}^{2}\). The frequency of the mode is determined from the condition

$${{J}_{m}}(qR) = 0,$$
(16)

where R is the radius of the cylinder. In the elliptical case for the TM polarization (12), two eigenvalues \(q_{{mr}}^{{(1)}}\) and \(q_{{mr}}^{{(2)}}\) are determined from Eq. (14) instead of the doubly degenerate optical mode (16). The frequency is determined from the condition

$$k_{{\parallel ,mr}}^{{(i)}} = \sqrt {\frac{{4q_{{mr}}^{{(i)}}}}{{{{a}^{2}}}}} .$$
(17)

Figure 1 illustrates the search for the elliptical of TM-polarized optical modes with the prespecified quantum numbers (m, r) and \({{k}_{z}} = 0\) in the two-dimensional elliptical dielectric microcavity with metallic walls filled with GaAs (\({{n}_{{{\text{GaAs}}}}} = 3.5\)). To this end, we solved transcendental equations (14) with the parameters b1 = 1000 nm and b2 = 700 nm. The split TM(1) and TM(2) modes differ in symmetry with respect to the principal axis of the ellipse; even modes are below in energy than odd modes with the same quantum numbers (m, r), by analogy with the quantized levels in a quantum well. The first intersection of the modes with the quantum numbers (m = 1, r = 1) with the x axis occurs at long wavelengths \({{\lambda }^{{(1)}}} \approx 2{{b}_{1}}{{n}_{{{\text{GaAs}}}}}\) = 7000 nm for the \({\text{TM}}_{{11}}^{{(1)}}\) mode and \({{\lambda }^{{(2)}}} \approx 2{{b}_{2}}{{n}_{{{\text{GaAs}}}}}\) = 4200 nm for the \({\text{TM}}_{{11}}^{{(2)}}\) mode comparable with the principal axis of the ellipse. Optical modes with quantum numbers (m > 1, r = 1) tend to the edge of the ellipse; the wavelength of both optical modes in the limit m → ∞ can be estimated as \(\lambda _{{m \to \infty }}^{{(1,2)}} = {{n}_{{{\text{GaAs}}}}}L{\text{/}}m\), where L is the perimeter of the ellipse; and the frequency spacing between these modes will decrease. This approach makes it possible to recognize split optical modes and to determine their quantum numbers (m, r), whereas the numerical calculation allows one to determine only the energies of optical modes and the distribution of the electromagnetic field.

Fig. 1.
figure 1

(Color online) Illustration of the search for optical modes of a two-dimensional elliptical dielectric cavity with metallic walls in TM polarization with the use of transcendental equation (14) for two arbitrarily chosen angular quantum numbers m = (top panel) 1 and (bottom panel) 5. The intersection of lines with the x axis corresponds to different radial quantum numbers r, beginning with \(r = 1\) at the first intersection at long wavelengths. The two-dimensional spatial distribution of the z component of the electric field of waves is shown for each intersection point.

Full size image

2.2 Search for the Parameter μ

The inverse problem of the search for the necessary geometry (the parameter μ) for an optical mode having a certain frequency is illustrated in Fig. 2. This figure presents the (μ, λ) color map of the absolute value of the modified Mathieu function \(Ce = Ce(\mu ,\lambda )\) for the symmetric optical mode \({\text{TM}}_{{rm}}^{{(1)}}\) with m = 1. We assume that \({{k}_{z}} = 0\) and express the frequency \(\omega \) in Eq. (11) in terms of λ. Optical modes with quantum numbers (m, r) in this cavity correspond to zeros of the function \(Ce\), which are indicated by a thin white line for several first modes. By analogy with Fig. 1, red points mark a particular case of the cavity with b1 = 1000 nm and b2 = 700 nm. It is seen that the energy of each optical mode increases with decreasing μ. As μ increases, the ellipse becomes similar to the cylinder and the dispersion curves of split modes approach each other. Thus, the field in the elliptical cavity can be represented as the product of a function of one variable μ (analog of the radius in the circular case) and a function of the second variable ν (analog of the angle in the circular case).

Fig. 2.
figure 2

(Color online) (μ, λ) color map of the absolute value of the modified Mathieu function \(Ce = Ce(\mu ,\lambda )\) in transcendental equation (14) for the symmetric optical mode \({\text{TM}}_{{rm}}^{{(1)}}\) with the angular quantum number m = 1 and several radial quantum numbers \(r = 1{\kern 1pt} - {\kern 1pt} 5\), where μ is the parameter of the cavity and λ is the wavelength of the mode. Red points mark the points where the plot of the right-hand side of Eq. (14) for \({{\mu }_{0}} = 0.867\) intersects the x axis, which correspond to b1 = 1000 nm and b2 = 700 nm shown in Fig. 1.

Full size image

Comparison with numerical calculations shows that the search for a desired optical mode with small quantum numbers (m, r) in the dielectric cavity in air requires the correction of the initial frequency by 20‒30% with respect to the frequency in the same cavity with metallic walls or the corresponding correction of the dielectric function. Moreover, the described procedure for the fast determination of optical modes becomes applicable for the dielectric cavity with a large refractive index n because, according to the known theorem of electrodynamics, solutions of Maxwell equations in dielectric media are transferred to solutions with metallic boundary conditions in the limit of infinite difference between refractive indices at interfaces.

2.3 Case of Degenerate Frequencies

To obtain the most efficient single-photon source under resonant pumping, it is appropriate to use optical modes at an almost degenerate frequency. In this case, laser pumping can be carried out through one optical mode, whereas detection is performed through the other optical mode at a shifted frequency because of a small overlapping of studied optical modes. In addition, the use of modes of different natures—TE–TM, Ce–Se, and other asymmetric combinations—for different quantum numbers (m, r) allows the natural polarization filtering of light. For illustration, Fig. 3 presents several dispersion curves of optical modes \({\text{TM}}_{{mr}}^{{(1)}}\) and \({\text{TM}}_{{mr}}^{{(2)}}\) with small quantum numbers (m, r) and different polarizations in different geometries of the elliptical cavity. The energies of optical modes strongly depend on the parameter μ of the cavity. The ellipse with μ ≫ 1 approaches the circle. The difference between the semiaxes of the ellipse is related to the difference cosh x – sinh x and becomes small at x ≫ 1. For this reason, the dispersion curves corresponding to, e.g., \({\text{TM}}_{{41}}^{{(1)}}\) and \({\text{TM}}_{{41}}^{{(2)}}\) smoothly approach each other with increasing μ. Red points in Fig. 3 mark the wavelength regions where orthogonal modes become doubly degenerate. To design single-photon sources, it is necessary to theoretically determine regions of degeneracy and to slightly detune from them if weak overlapping is required for, e.g., optical filtering. It is also noteworthy that almost triply degenerate states (a doubly degenerate state and nearby third state) appear in such structures. In particular, the inset of Fig. 3 demonstrates the region where three pairwise orthogonal \({\text{TE}}_{{51}}^{{(2)}}\), \({\text{TE}}_{{61}}^{{(1)}}\), and \({\text{TM}}_{{11}}^{{(2)}}\) optical modes with close frequencies can exist simultaneously. Such states can be used for optical pumping through the closely located mode, thus exciting a doubly degenerate entangled state.

Fig. 3.
figure 3

(Color online) Wavelengths of selected (for illustration) split modes \({\text{TM}}_{{mr}}^{{(1)}}\) and \({\text{TM}}_{{mr}}^{{(2)}}\) with quantum numbers \(m = 1{\kern 1pt} - {\kern 1pt} 4\) and \(r = 1\) versus the parameter μ of the dielectric cavity with metallic walls; μ = 0.9 corresponds to a flattening of 20%. Red points mark the regions of double degeneracy of optical modes. The inset shows the case of double degeneracy of optical modes and a closely located third state.

Full size image

3 CONCLUSIONS

To summarize, a theory of optical modes in elliptical dielectric microcavities with metallic walls has been developed using Mathieu functions in elliptical coordinates. It has been shown that estimates obtained with this theory can be used as the zeroth approximation to calculate dielectric cavities. A key difference from the classical circular cavity is the splitting of doubly degenerate modes. In the case of a large difference between the refractive indices of the cavity and its environment, the classification of modes in the metallic cavity can be approximately used for dielectric cavities, as done for circular waveguides, where the hybridization of modes can be disregarded in the first approximation. A method has been proposed to choose the parameters of a cavity for problems of quantum photonics such as nearly degenerate modes for single-photon sources and multiple entangled states.