Theory of free space coupling to high-Q whispering gallery modes

Chang-Ling Zou; Fang-Jie Shu; Fang-Wen Sun; Zhao-Jun Gong; Zheng-Fu Han; Guang-Can Guo

doi:10.1364/OE.21.009982

1. Introduction

Whispering gallery modes (WGMs) were first explained by Lord Rayleigh [1] in the case of sound propagation in St Paul’s Cathedral circa in 1878. The same phenomenon is also found in optical domain, where light are almost perfectly guided round by optical total internal reflection in low loss resonators [2]. High quality (Q) factor in excess of 10¹⁰ has been achieved [3]. Combining the ultra-small mode volume with the high-Q factor, the light-matter interaction can be greatly enhanced in WGM microcavities. Therefore, optical WGMs are gaining growing attentions in a wide range of fields including ultra-sensitive sensors [4], low threshold lasers [5], frequency combs [6], cavity quantum electrodynamics (QED) [7], and quantum optomechanics [8–10]. The WGMs also affect the scattering on spherical particles [11], which was found to play an important role in the glory phenomenon [12]. In the last twenty years, the high-Q optical WGMs have been reported in various microcavities, such as liquid droplet [13], microsphere [14], microdisk [15], microcylinder [16] and microtoroid [17].

The excitation and collection of the WGMs are essential issues in practical applications. It is taken for granted that free space coupling to WGMs is very inefficient since the radiation loss of circular shaped microcavity is isotropic and low, which brings difficulties in the energy transference between outside and WGMs. Near field couplers, such as prism [18], fiber taper [19, 20] and perpendicular waveguide [21], enable high efficient excitations and collections to microresonators. Therefore, these near field couplers are widely adapted in experiments. The underlying mechanisms of coupling process between WGMs and couplers have been well understood [22–24] and many phenomena in the spectrum have been reported, such as the analogue of electromagnetically induced transparent (EIT) [25–27], asymmetric Fano line shape [21, 28–30] and ringing phenomena [31]. However, there are some limitations in the near field couplers, such as large footprint, stabilization requirement, and power limit (high excitation energy would cause strong nonlinear effects in fiber tapers).

On the other hand, the asymmetric resonant cavities (ARCs) have been demonstrated to be able to support high-Q WGMs and give highly directional emission [32–40]. The underlying principle and interesting chaotic ray dynamics in such an open billiard have been studied extensively [41–45]. Especially, the unidirectional emission cavities have been successfully designed [46–53], which enable high efficient collection of high-Q WGMs through free space. As the reversal of emission, the focused beam in free space can excite the WGMs, which have been studied in experiments [8, 9, 54–57]. However, the basic features and potentials of this free space beam coupling strategy are not fully explored.

In this paper, we present theoretical studies on free space coupling to high-Q WGMs in both circular and deformed microcavities. First of all, the cylindrical wave incident to a circular cavity is solved analytically and asymptotically in the cylindrical coordinates, and the analytical equations are well consistent with the standard input-output formulation. Then, we consider the excitation of the circular cavity by Gaussian beam for practical applications, the maximal efficiency of about 20% can be achieved under phase matching condition. After that, the excitation of deformed cavity by Gaussian beam is studied by an abstract model of multimode interaction. The result shows that the ARCs not only give good matching parameters, but also can balance radiation and non-radiation losses, which can be well adapted in experiments for high efficient free space coupling. One interesting result is that the asymmetric spectra is universal in the free space coupling as a result of multi-wave interference, and the energy transferring cannot be deduced from the dip depth in the transmission spectrum.

2. Circular cavity

2.1. Cylindrical waves

We start with a two-dimensional circular shaped microcavity, where electromagnetic field (ψ) can be solved in the cylindrical coordinates (r, ϕ) analytically. The basic solutions to the Helmholtz in the cylindrical coordinate are Hankel functions $H_{m}^{(1 (2))} (k r)$ with integral m ∈ (−∞, ∞), and the superscript 1(2) denotes the first (second) kind Hankel functions corresponding to outward (inward) traveling cylindrical waves. Therefore, any electromagnetic field can be decomposed in the basis of $H_{m}^{(1 (2))} (k r)$ . Since the Hankel functions is singular at origin (r = 0), the field with finite intensity at origin should be represented by the Bessel functions $J_{m} (k r) = H_{m}^{(1)} (k r) + H_{m}^{(2)} (k r)$ . So, for the cylinder microcavity with radius is r_c, the field inside the cavity is represented by ∑a_mJ_m(nkr)e^imϕ with r < r_c, and the field outside the cavity is represented by $\sum [b_{m} H_{m}^{(1)} (k r) + c_{m} H_{m}^{(2)} (k r)] e^{i m ϕ}$ with r > r_c.

In this axial symmetric geometry, the scattering of cylindrical wave should conserve the angular-momentum. For example, for m-th cylindrical wave $H_{m}^{(2)} (k r) e^{i m ϕ}$ incident to the cavity, only m-th components of cavity field and reflected outgoing wave are nonzero. Therefore, for a stationary electromagnetic field represented by coefficients a_m, b_m and c_m, we can obtain the relationships between these coefficients

a_{m} = \frac{H_{m}^{(1)} (z) H_{m}^{{(2)}^{'}} (z) - H_{m}^{{(1)}^{'}} (z) H_{m}^{(2)} z}{n^{p} J_{m}^{'} (n z) H_{m}^{(1)} (z) - J_{m} (n z) H_{m}^{{(1)}^{'}} (z)} c_{m},

b_{m} = - \frac{n^{p} J_{m}^{'} (n z) H_{m}^{(2)} (z) - J_{m} (n z) H_{m}^{{(2)}^{'}} (z)}{n^{p} J_{m}^{'} (n z) H_{m}^{(1)} (z) - J_{m} (n z) H_{m}^{{(1)}^{'}} (z)} c_{m},

by applying boundary conditions. Here, z = kr_c, and p = −1(1) for transverse electric (TE) field (transverse magnetic (TM) field).

In above formulas, there are singularities at z = z₀ where

n^{p} J_{m}^{'} (n z_{0}) H_{m}^{(1)} (z_{0}) - J_{m} (n z_{0}) H_{m}^{{(1)}^{'}} (z_{0}) = 0 .

This corresponds to the quasi-bound eigenmodes (WGMs) with dimensionless eigenfrequency (kr_c = z₀), which is a complex number. When the material refractive index n is real, i.e. there no material absorption, we can write

z_{0} = z_{0}^{r} - i κ_{0}

with

z_{0}^{r}

and κ₀ are real numbers. Here, κ₀ is the pure radiation loss of WGM, which corresponds to the width of the resonance in spectrum as

κ_{0} = z_{0}^{r} / 2 Q_{0}

with Q₀ is the radiation quality factor.

For nearly resonant frequency z = z₀ + Δ (|Δ| ≪ |z₀|), by expanding Eq. (3) to the first order of Δ, we can approximately obtain

n^{p} J_{m}^{'} (n z) H_{m}^{(1)} (k z) - J_{m} (n z) H_{m}^{{(1)}^{'}} (z) \approx F (z_{0}) Δ,

with

F (z_{0}) = (n^{p - 1} - 1) [(n J_{m}^{'} (n z_{0}) H_{m}^{{(1)}^{'}} (z_{0}) + \frac{m (m - 1)}{z_{0}^{2}} J_{m} (n z_{0}) H_{m}^{(1)} (z_{0})] - (n^{p + 1} - 1) J_{m} (n z_{0}) H_{m}^{(1)} (z_{0})

.

For the real materials, the non-radiation linear loss should be taken into account. For example, the material absorption can be considered in the characterize equation by adding a small imaginary number (n_i ≪ n) to the refractive index as ñ = n + in_i. For a monochromatic light with $z = z_{0}^{r} - δ$ , we have $\tilde{n} z = (n + i n_{i}) (z_{0}^{r} - δ) \approx n (z_{0} - δ - i κ_{0} - i κ_{1})$ with κ₁ = z₀n_i/n denoting the non-radiation loss. Substitute Δ = i(iδ + κ₀ + κ₁) into Eq. (4), Eq. (1), and Eq. (2). For high-Q mode ( $κ_{0} ≪ z_{0}^{r}$ ) and near resonance ( $δ ≪ z_{0}^{r}$ ), and with the approximation that most of WGM energy is confined inside the cavity [58], we can get

a_{m} (δ) = \frac{1}{i δ + κ_{0} + κ_{1}} \frac{- 4}{π z_{0}^{r} F (z_{0}^{r})} c_{m},

b_{m} (δ) = - \frac{i δ - κ_{0} + κ_{1}}{i δ + κ_{0} + κ_{1}} \frac{F^{*} (z_{0}^{r})}{F (z_{0}^{r})} c_{m} .

On the other hand, the standard input-output formulation for a cavity mode is [59]

\frac{d}{d t} E_{m} = (- i δ - κ_{0} - κ_{1}) E_{m} + \sqrt{2 κ_{0}} E_{m}^{in},

where E_m is the cavity mode field,

E_{m}^{in}

is the incoming field in the form of radial inward cylindrical waves

H_{m}^{(2)} (z_{0})

, and the output field is

E_{m}^{out} = - E_{m}^{in} + \sqrt{2 κ_{0}} E_{m}

. When the system is in the steady state,

\frac{d}{d t} E_{m} = 0

should be satisfied, thus

E_{m} = \frac{\sqrt{2 κ_{0}}}{i δ + κ_{0} + κ_{1}} E_{m}^{in},

and the output field is

E_{m}^{out} = - \frac{i δ - κ_{0} + κ_{1}}{i δ + κ_{0} + κ_{1}} E_{m}^{in} .

Comparing those equations with Eq. (5) and Eq. (6), the cavity and outgoing fields deduced by input-output formulation are consistent with the results derived by boundary conditions, only different in constants. The conversion relationships between two sets of formulas are

a_{m} = \frac{1}{\sqrt{2 κ_{0}}} \frac{- 4}{π z_{0}^{r} F^{*} (z_{0}^{r})} E_{m}

,

b_{m} = E_{m}^{out}

, and

c_{m} = E_{m}^{in} \frac{F (z_{0}^{r})}{F^{*} (z_{0}^{r})}

.

For the case of the near field coupler, the cavity field reads [24]

E_{m} = \frac{\sqrt{2 κ_{ext}}}{i δ + κ_{int} + κ_{ext}} E_{m}^{in},

where κ_int is intrinsic loss including radiation and absorption losses, and κ_ext is external loss induced by coupler. Comparing with Eq. (8), the free space and near field coupling manners are similar, with κ₀ and κ₁ replaced by κ_int and κ_ext.

From Eq. (8), the extremum of cavity field emerges when ∂E/∂κ₀ = 0, i.e. κ₀ = κ with δ = 0. Similar to the case of a near field waveguide coupling to WGMs, there are three coupling regimes: under coupling regime κ₀ < κ₁, critical coupling regime κ₀ = κ₁, and over coupling regime κ₀ > κ₁. The electric field distribution in Fig. 1(a) clearly shows the excitation of WGM at critical coupling by an inward cylindrical wave $H_{30}^{(2)} (k r) e^{i 30 ϕ}$ with the spiral-like propagation. No reflection wave is excited in this case, where all energy is perfectly absorbed by the cavity, as shown by the reflectivity in Fig. 1(b). Figures 1(c) and 1(d) are the intensity and the phase of outgoing wave against the detuning in three regimes. In the under and over coupling regime, the energy transfer efficiency without detuning is low. Moreover, the phase of the outgoing wave is strongly modified by the WGM in the over coupling regime with small detuning value, while there is only small perturbation for under coupling regime.

Fig. 1 (a) Electric field distribution for a cylinder cavity critical coupling with m = 30 cylinder wave, and detuning δ = 0. (b) The normalized reflectivity and the total Q factor against the angular number m, with the refractive index n = 1.45+0.000117i. (c) and (d) are the intensity and phase of outgoing field for different coupling conditions, with κ₁/κ₀ = 0.2 (Red Dotted lines), 1.0 (Blue Dashed lines) and 5.0 (Black Solid lines).

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In the case of near field coupler, the extra coupling loss κ_e can be controlled by simply adjusting the gap between coupler and cavity, while κ_i is an intrinsic parameter that does not change. Differently, in the free space coupling case, κ₁ is an intrinsic parameter determined by material and fabrication which is almost a constant for WGMs. It is only possible to change κ₀ by change the cavity size or boundary shape. As shown in Fig. 1(b), the Q-factor [Q = kr_c/2(κ₀ +κ₁)] and the outgoing intensity change against m with n = 1.45+0.000117i. The Q-factor increases with the increasing m and shows a saturation value of about 6×10³. The critical coupling (κ₀ ≈ κ₁) happens when m = 30, corresponding to a specific cavity size (r_c ≈ m/nk). For larger or smaller cavity size, the coupling efficiency reduces.

2.2. Gaussian beam

Gaussian beam is most widely applied in practical experiments, thus we use this beam in following studies for realistic consideration. Supposing a Gaussian beam centered at (x₀, y₀) propagation along x-axis, with the waist width w and the maximum field intensity is normalized to unity. Then, the electric field at the waist is

G (x_{0}, y) = e^{- \frac{{(y - y_{0})}^{2}}{w^{2}}}

Correspondingly, the angular spectrum is

\begin{array}{l} \tilde{G} (θ) & = & \frac{1}{2 π} \int_{- \infty}^{\infty} e^{- i y k sin θ} e^{- \frac{y^{2}}{w^{2}} d y} \\ = & \frac{w}{2 \sqrt{π}} e^{- \frac{k^{2} w^{2}}{4} {sin}^{2} θ}, \end{array}

with θ ∈ (

- \frac{π}{2}

,

\frac{π}{2}

). So, in the cylindrical coordinate in Fig. 2(a), the field distribution can be expressed as

E (r, ϕ) = \int_{- k}^{k} \frac{w}{2 \sqrt{π}} e^{- \frac{k^{2} w^{2}}{4} {sin}^{2} θ} e^{i k (sin θ y + cos θ x)} d (k sin θ),

where

x = r cos (ϕ + θ_{0}) - r_{0} cos (ϕ_{0} + θ_{0}),

y = r sin (ϕ + θ_{0}) - r_{0} sin (ϕ_{0} + θ_{0}) .

Substituting them to the Eq. (13) and employing the Jacobi-Anger expansion,

e^{i k r cos (ϕ + θ_{0} - θ)} = \sum_{m = - \infty}^{\infty} J_{m} (k r) e^{i m [π / 2 - (ϕ + θ_{0} - θ)]},

we can derive the coefficient of inward cylindrical waves of the Gaussian beam as

c_{m} = \frac{e^{i m (π / 2 - θ_{0}) - i k q - \frac{{(k r_{0} d - m)}^{2}}{k^{2} w^{2} + i 2 k q}}}{2 \sqrt{1 + i \frac{2 q}{k w^{2}}}},

in the limit of kw ≫ 1 and d = r₀sin(ϕ₀ + θ₀) and q = r₀cos(ϕ₀ + θ₀). Here, only the part of sinθ ≪ 1 contributes in the integral, and the approximation

cos θ = \sqrt{1 - {sin}^{2} θ} \approx 1 - \frac{1}{2} {sin}^{2} θ

is applied. Substituting the coefficients to Eq. (1) and (2), the intracavity field and outgoing field can be solved analytically for a cylinder cavity. For example, Fig. 2(b) shows the near field distribution when a Gaussian beam is incident to the cavity with the on-resonance frequency of WGM with m = 40. Obviously, most energy are directly transmitted. Only a small portion of energy can be coupled into the cavity field. The ripples in the incident beam are due to the interference of incident beam and reflected light.

Fig. 2 (a) Schematic illustration of a Gaussian beam incident to a circular microcavity. The direction of beam is defined by θ₀, and the location of center is (r₀, ϕ₀) in the cylindrical coordinate. (b) The electric field intensity distribution when the incident Gaussian beam is resonant with the WGM, with kr_c = 30.10, r₀ = 1.1r_c, ϕ₀ = π/2 and θ₀ = 0.

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By integrating the Poynting vector of the electromagnetic field, we can calculate the power (energy flux) contained in the Gaussian beam

P_{G} = \sqrt{\frac{π}{2}} \frac{k w}{2 ω μ},

where ω is angular frequency, and μ is relative permeability. Power of normalized cylindrical wave

H_{m}^{(1 (2))} (k r) e^{i m ϕ}

is

P_{m} = \frac{2}{ω μ} .

Therefore, the ratio of power contained in the m-th cylindrical waves to that in the Gaussian beam is

η_{m} = {| c_{m} |}^{2} \frac{P_{m}}{P_{G}} .

From Eq. (17), η_m is maximal when q = 0 and d = m/k. The first condition guarantees the smallest expansion of cylindrical waves in momentum space (m). The second condition corresponds to the phase matching condition that the momentum of the pump beam (kd) should be equal to the momentum of the WGM (m). It is worth noting that the phase matching condition, also known as “excitation localization principle”, have already been demonstrated experimentally by focused Gaussian beam coupling to high-Q WGMs in spherical microcavities [60, 61].

In practical applications, we can manipulate the beam position d to satisfy the conditions, then

η_{\max} = \sqrt{\frac{2}{π}} \frac{1}{k w} .

Therefore, the rate of power transfer from a Gaussian beam to the cavity WGM is limited to η_max. Under the paraxial approximation, the waist w of a Gaussian beam should be larger than 4/k, i.e. kw ≥ 4, corresponding to the energy transferring rate η_m ≤ 0.2 for a cylinder cavity.

3. Deformed cylinder

3.1. General model

When the cavity boundary is slightly deformed [32–40, 47–52] or small perturbations are introduced to the cavity [46, 53], highly directional emission can be obtained. Because the boundaries of ARCs are not regular, different momentum (m) components can couple to each other. In addition, WGMs always exist in pairs, clockwise (A_c) and anti-clockwise (A_a) traveling modes, in a circular-like shape microcavity, then the general Hamiltonian of ARC reads

ℋ = ℋ_{modes} + ℋ_{int} + ℋ_{pump} .

Here,

ℋ_{modes} = \sum_{j} \bar{h} δ_{j} (A_{j, c}^{†} A_{j, c} + A_{j, a}^{†} A_{j, a})

represents the energy of cavity modes, with δ_j is the frequency difference between cavity mode and excitation laser, and

A_{j, c}^{†} (A_{j, c})

is the creation (annihilation) operator of cavity mode. The coupling between different modes is

\begin{array}{l} ℋ_{int} & = & \sum_{j} \sum_{k > j} (g_{j, k, c} A_{j, c}^{†} A_{k, c} + g_{j, k, a} A_{j, a}^{†} A_{k, a} + h . c .) \\ + \sum_{j, k} (β_{j, k} A_{j, c}^{†} A_{k, a} + h . c .), \end{array}

where β_j,k is the coupling coefficient between counter propagation WGMs and g_j,k,a(c) is the coupling coefficient between co-propagation modes. The external pumping on modes is given by

ℋ_{pump} = \sum_{j, k} (i h_{j, m, c} A_{j, c}^{†} u_{m}^{in} + i h_{j, m, a} A_{j, a}^{†} u_{- m}^{in} + h . c .) .

Here, h_j,m,c is the coupling strength of the cavity mode to the outside cylindrical waves, with m = 1···∞, where the external incoming field and outgoing field are represented by cylindrical waves as

\sum u_{m}^{in} e^{i φ_{m}} H_{m}^{(2)} (k r) e^{- i m ϕ}

and

\sum u_{m}^{out} e^{- i φ_{m}} H_{m}^{(1)} (k r) e^{- im ϕ}

with

e^{- i 2 φ_{m}} = - H_{m}^{(2)} (k r_{c}) / H_{m}^{(1)} (k r_{c})

. Actually, A_c and A_a are time reversal to each other, so the coupling coefficients satisfy

g_{j, k, c} = g_{j, k, a}^{*} = g_{j, k}

and

h_{j, m, c} = h_{j, m, a}^{*} = h_{j, m}

.

Here, we only concern about the ARCs with smooth boundary and small deformation which can support high-Q WGMs. Then we can omit the coupling between counter propagation WGMs (β_i,j = 0) and the direct scattering induced transition between incoming and outgoing cylindrical waves. Thus the Heisenberg equation of cavity field are written as

\begin{array}{l} \frac{d}{d t} A_{j, c} & = & - i \sum_{k > j} g_{j, k} A_{k, c} - i \sum_{k < j} g_{k, j}^{*} A_{k, c} \\ - χ_{j} A_{j, c} + \sum_{m} h_{j, m} u_{m}^{in}, \end{array}

where χ_j = iδ_j + κ_j,0 + κ_j,1, κ_j,0 = ∑_m |h_j,m|²/2 and κ_j,1 are intrinsic radiation loss and non-radiation loss, respectively. Supposing there is only one high-Q WGM with j = 1 near resonance of the excitation, and other modes are low-Q or largely detuned, we can adiabatic eliminate the fast varying terms by

\frac{d}{d t} A_{j, c} = 0

for j ≥ 2, i.e.

A_{j, c} |_{j \geq 2} = - i \frac{g_{1, j}^{*}}{χ_{j}} A_{1, c} + \sum_{m = 1}^{\infty} \frac{h_{j, m}}{χ_{j}} u_{m}^{in} .

Therefore,

\frac{d}{d t} A_{1, c} = - {\tilde{χ}}_{1} A_{1, c} + \sum_{m = 1}^{\infty} {\tilde{h}}_{1, m} u_{m}^{in} .

Here, χ̃₁ = χ₁ + κ_e + iδ_e with

κ_{e} = \sum_{j \geq 2} \frac{κ_{j, 0} + κ_{j, 1}}{{| χ_{j} |}^{2}} {| g_{1, j} |}^{2}

and

δ_{e} = - \sum_{j \geq 2} \frac{δ_{j}}{{| χ_{j} |}^{2}} {| g_{1, j} |}^{2}

, corresponding to the extra loss and frequency shift induced by low-Q modes, and

{\tilde{h}}_{1, m} = h_{1, m} - i \sum_{j \geq 2} g_{1, j} \frac{h_{j, m}}{χ_{j}}

are effective coupling coefficients to outside for high-Q WGM. Denoting the coupling efficiencies and the incoming field by vectors h⃗ = {h̃_1,m} and

\vec{u} = {u_{m}^{in}}

with m = 1···∞, the expected cavity field at steady state becomes

< A_{1, c} > = ξ | \vec{h} | | \vec{u} | / {\tilde{χ}}_{1},

where ξ = h⃗· u⃗/|h⃗| |u⃗| is the beam matching parameter. The output for arbitrary incoming field reads

u_{m}^{out} = - u_{m}^{in} + f_{m} < A_{1, c} > + \sum_{j \geq 2} \sum_{m^{'}} \frac{h_{j, m}^{*} h_{j, m^{'}}}{χ_{j}} u_{m^{'}}^{in},

where

f_{m} = h_{1, m}^{*} - i \sum_{j \geq 2} \frac{g_{1, j}^{*}}{χ_{j}} h_{j, m}^{*}

corresponds to the radiation of A_1,c.

From Cauchy-Schwartz inequality, |ξ| ≤ 1, the maximum of matching can be achieved only when

\vec{u} \propto {\vec{h}}^{*} / | \vec{h} | .

This corresponds to the optimal beam to excite the WGM

E_{inc} (r, ϕ) \propto \sum_{m} e_{m}^{i φ_{m}} {\tilde{h}}_{1, m}^{*} H_{m}^{(2)} (k r) e^{- i m ϕ}

, which is the reverse of the radiation of anti-clockwise WGM

E_{a}^{rad} (r, ϕ) \propto \sum_{m} {\tilde{h}}_{1, m} e^{- i φ_{m}} H_{m}^{(1)} (k r) e^{i m ϕ}

, meaning that the optimal excitation of clockwise WGM can be achieved by the reverse of the radiation of anti-clockwise WGM.

When on resonance, the cavity field reads A_1,c = ξ|h⃗| |u⃗|/(κ_1,0 + κ_1,1 + κ_e), and the energy transferring depends on both ξ and |h⃗| / (κ_1,0 +κ_1,1 +κ_e). All the parameters mentioned above can not be solved analytically because of the irregular boundary shape. Realistic details about the free space coupling should be solved by numerical simulation. In the following, we analytically solve the free space coupling to ARCs by an abstract model without the specific boundary shape. In this simplified model, we assume directional emission is in the form of Gaussian beam ignoring the specific boundary of a ARC. The general underlying mechanism and basic properties are revealed with several reasonable approximations of a near circular boundary shape.

3.2. Coupling through barrier tunneling

In a very slightly deformed ARC, the coupling between high-Q and low-Q modes are weak, and the directional emission is mainly due to the direct tunneling [62]. The center of the emission beam is departed from the boundary, as a result of barrier tunneling [63]. We assume that the parameters of emission Gaussian beam are d_t = 1.2r_c, w_t = 0.1r_c and direction along x-axis, as shown in Figs. 3(a) and 3(b). Thus, the coupling coefficients can be solved from the mode emission

h_{1, m} = 𝔥_{0} e^{- i m π / 2 - \frac{{(k d_{t} - m)}^{2}}{k^{2} w_{t}^{2}} + i φ_{m}},

where

𝔥_{0} = \sqrt{κ_{0} / \sqrt{\frac{π}{2}} \frac{k w}{2}}

, corresponding to the normalization ∑ |h_1,m|² = 2κ₀.

Fig. 3 (a) Field distribution of directional emission of anti-clockwise WGM in a near circular cavity (The field is shown in Logarithm scale, and the intracavity field is not shown since the exact boundary shape is not known in our abstract model). The emission is in the form of Gaussian beam with d_t = 1.2r_c, w_t = 0.1r_c and direction is shown by arrows. (b) Field distribution when an on-resonance Gaussian beam coupling to the cavity in the over coupling regime with κ₁ = 0.1κ₀, d = 1.2, w = 0.2r_c and direction is shown by arrows. (c) Beam matching parameters against the d for different w. (d) The far field intensity of outgoing wave when a Gaussian beam incident to a cylinder, for excitation frequency off-resonance and on-resonance at difference coupling conditions. (e), (f) and (g) are spectra detected at different angle (ϕ) in far field.

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We discussed the coupling in detail using a Gaussian beam from left with θ₀ = 0 and on-resonance to a high-Q mode of kr_c = 31.1. Firstly, we calculate the beam matching parameter ξ versus the beam position d for different beam widths w, as shown in Fig. 3(c). The result shows that good match with ξ > 0.8 can be easily realized with good fault tolerant to the imperfect focus or align of beam in experiments. In the near integrated system, we can neglect the coupling to other low-Q modes ( $κ_{0}^{1, c} + κ_{1}^{1, c} ≫ κ_{e}$ ), then $A_{c} \approx \sqrt{2 κ_{1, 0}} / (κ_{1, 0} + κ_{1, 1})$ . Similar to the cylinder case, the critical coupling with κ_1,0 = κ_1,1 guarantees the maximum energy transfer.

With these approximations, we can calculate the far field distributions by substituting parameters into Eq. (30). The far field distributions at different coupling regime are shown in Fig. 3(d) with w = 0.2r_c. At critical coupling, the outgoing energy is great reduced but larger than 0 as a result of imperfect matching. In under coupling, there is a small reduction of energy in all directions. However, in over coupling regime, the peak at ϕ = 0 is split into two peaks, which is from the interference between the transmitted beam and the emission of the cavity mode, as a result of the strong phase shift at the over coupling regime.

The spectra are also calculated (Figs. 3(e)–3(g)) in different far field angles. It is revealed that the line shapes strongly depend on the positions of detectors. At ϕ = 0°, the resonances show regular Lorentzian-shape dips similar to traditional near field coupler. In contrast with Fig. 1(c), similar dip depths are given in the over and critical coupling condition regimes, which indicates that the dip depth can not tell the energy transfer rate in the free space coupling. Asymmetric line shapes (Fano-like and EIT-like) are observed at ϕ = 15° and 30°, as a result of multiple cylindrical waves interference formalized in Eq. (30).

3.3. Coupling through refraction

When the degree of deformation is large, the directional emission is mainly due to the radiation from low-Q modes. This process is also known as the dynamical tunneling in the study of ARC, where rays in high-Q mode tunnel into chaotic sea and refracted out at some specific region in phase space [36, 37]. The center of Gaussian beam should be adjusted on the cavity as a result of refraction [64]. For the refraction dominated emission, the extra loss induced by low-Q mode is much larger than the intrinsic radiation loss (i.e. κ_e ≫ κ₀), thus we can neglect the direct tunneling loss. Assuming the emission is Gaussian beam with the parameters d_r = 0.9r_c, w_r = 0.1r_c and direction along x-axis (Fig. 4(a)), we have

\sum_{j \geq 2} \frac{g_{1, j}}{χ_{j}} h_{j, m} e^{- i φ_{m}} = 𝔥_{e} e^{- i m π / 2 - \frac{{(k d_{t} - m)}^{2}}{k^{2} - w_{t}^{2}}} .

Here,

𝔥_{e} = \sqrt{κ_{e} / \sqrt{\frac{π}{2}} \frac{k w}{2}}

with

κ_{e} = \frac{1}{2} \sum_{m \geq 2} \frac{{| g_{1, m} |}^{2} {| h_{m, m} |}^{2}}{{| χ_{m} |}^{2}}

. In addition, the interactions between low-Q modes are not taken into consideration. We also assume the deformation just slightly influences the low-Q modes for simplicity, thus h_j,m = 0 if j ≠ m. Therefore

\frac{g_{1, m}}{χ_{m}} h_{m, m} = 𝔥_{e} e^{- i m π / 2 - \frac{{(k d_{t} - m)}^{2}}{k^{2} w_{t}^{2}} + i φ_{m}} .

Similar to the procedure in previous section, we solved the outgoing field when a Gaussian beam is incident from left with θ₀ = 0, w = 0.2r_c, d = 0.9r_c and on-resonance to a high-Q mode with kr_c = 31.1. As shown in Figs. 4(b) and 4(c), the maximal far field intensity is located away from ϕ = 0 as a result of the refraction of incident beam. The far field distribution is much more complex than the case of tunneling coupling in Fig. 3(d), as there are more peaks for the off resonance excitation due to the refraction by the cavity. The far field amplitudes at different collection positions (ϕ) for the on-resonance case can be larger, equal or smaller than the off-resonance case, as a result of multiple cylindrical wave interference, which lead to the EIT, asymmetric Fano or dip lineshapes in the spectra.

Fig. 4 (a) Field distribution of directional emission of anti-clockwise WGM in a near circular cavity (The field is shown in normal scale, and the intracavity field is not shown since the exact boundary shape is not known in our abstract model). The emission is in the form of Gaussian beam with d_r = 0.9r_c, w_r = 0.1r_c and direction is shown by arrows. (b) The field distribution when an on-resonance Gaussian beam coupling to the cavity in the over coupling regime with κ₁ = 0.1κ₀, d = 0.9, w = 0.2r_c and direction is shown by arrows. (c) The far field intensity of outgoing wave when a Gaussian beam is incident to the ARC with Gaussian beam emission, for off-resonance and on-resonance under different coupling conditions.

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4. Discussion

According to the above analysis, the coupling to high-Q WGMs through free space can be efficient. For the cylinder cavity, the largest coupling efficiency can be achieved under the phase matching condition. And the maximal coupling efficiency depends on the power ratio of the cylindrical wave with certain mode order, which is limited by finite waist width of a focused Gaussian beam. For the deformed cavity, the coupling efficiency can be higher since the pump Gaussian beam can be well matched to the directional emission beam.

The energy transferred to the cavity is also determined by the coupling regime, i.e. the ratio of radiation loss to non-radiation loss. The strongest coupling happens when κ₀ ≈ κ₁. The radiation loss κ₀ decreases exponentially with the increase of cavity size, but the non-radiation loss κ₁ is almost constant. So, in the axis symmetric cavities, the efficiency coupling can only be achieved with certain size. Taking the Fig. 1(b) as an example, the material absorption limits the cavity Q-factor to be about 6000. The high efficient free space coupling to WGMs in such cavity can only be achieved for WGMs with m ≈ 30, which corresponds to r_c/λ ≈ 3.8. While for ARC, the near critical coupling can be achieved for larger m, as the κ₀ of ARC can be tuned by adjusting the deformation, similar to the tuning of gap between near field coupler and microcavity. We should also note that the κ₀ of ARC can only be adjusted to be lower than that for the circular cavity of same size.

In above studies, the surface scattering is not taken into consideration. Similar to the ARCs, the surface scattering can also induce the momentum non-conservation when cylindrical wave incidents to the cavity, and lead to the extra radiation loss of WGMs to multiple cylindrical waves. So, for the scattering loss dominated WGMs, the free space coupling may be more efficient by coupling through low-order cylindrical waves with higher tunneling rate to WGM, then the optimal excitation of WGMs can be achieved when kd < m. This deviation to phase matching condition was firstly observed by Lin et al. [61].

In contrast to near field coupling to microcavity, the asymmetric line shapes are universal in free space coupling to WGMs [52]. In more realistic cases, multiple high-Q modes near to each other in spectrum are involved, and then the interactions between high-Q modes are enhanced by the low-Q modes mediated interaction. From the view of ray dynamics in ARC, the high-Q modes usually correspond to regular periodic or quasi-periodic orbits. Due to dynamical tunneling, the high-Q modes can couple to chaotic modes which correspond to chaotic ray trajectories. Therefore, the chaotic sea mediates the indirect coupling between separated regular orbits in phase space (high-Q modes), which is also known as chaos assisted tunneling [43, 65, 66]. Similar to the case of waveguide coupled microcavity [27, 28], we can expect more profound asymmetric spectra when more high-Q modes are taken into consideration.

5. Conclusion

In summary, we have presented the theoretical study of the free space coupling to high-Q WGMs in both regular and deformed microcavities. The coupling by free space beam depends on the phase matching and beam matching conditions in regular cylinder cavities and ARCs, respectively. Three coupling regimes of free space coupling are discussed. The cavity should work near the critical coupling regime for high efficiency energy transfer. In cylinder cavity, the maximal energy transfer efficient is limited, and critical coupling can only be achieved with specific cavity size. The ARCs not only give good beam matching between focused Gaussian beam and highly directional emission, but also enable the tuning of the cavity degree of deformation to achieve critical coupling. Therefore, the efficiency approaching unity is possible in realistic experiments of free space coupling to unidirectional emission cavity. It is found that the asymmetric spectra or peak like spectra instead of the Lorentz-shape dip is universal in spectra, and the coupling efficiency cannot be estimated from the absolute depth of dip. Our results provide guidelines for free space coupling to high-Q WGMs, which will be valuable for further experiments and applications of WGMs based on free space coupling.

Acknowledgments

We thank Prof. Hailin Wang and Prof. Kyungwon An for discussions and comments. This work was supported by the 973 Programs (No. 2011CB921200), the National Natural Science Foundation of China (NSFC) (No. 11004184), the Knowledge Innovation Project of the Chinese Academy of Sciences (CAS). F.-J. Shu is supported by the Foundation of He’nan Educational Committee (No. 2011A140021) and the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11204169).

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Theory of free space coupling to high-Q whispering gallery modes

Abstract

1. Introduction

2. Circular cavity

2.1. Cylindrical waves

2.2. Gaussian beam

3. Deformed cylinder

3.1. General model

3.2. Coupling through barrier tunneling

3.3. Coupling through refraction

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (34)

Optics Express