2.1.

Inspiration from Condensed Matter Physics

Assuming harmonic fields with angular frequency $\omega$, the electric field can be written as $\overrightarrow{E}(\overrightarrow{r},t)=\overrightarrow{E}(\overrightarrow{r})e^{-i\omega t}$. Then, the propagation of an electromagnetic wave through a linear isotropic dielectric material is governed by the following equation as

where $\epsilon (\overrightarrow{r})$ is the space-dependent dielectric function of the material. In our formulation, we are assuming transverse electromagnetic fields.

Equation (1) is not easy to solve analytically for most realistic geometries of dielectric optical resonators. Numerical techniques to solve it are well developed, but cannot be used to find analytic expressions for mode splittings. Considering the geometry of a ring resonator, we can simplify Eq. (1) significantly. As long as the resonator radius, $R$, is much larger than the local wavelength of light $[R\gg {\lambda }_0/\sqrt{\epsilon (\overrightarrow{r})}]$, the resonator is equivalent to an infinitely long, periodic, two-dimensional waveguide. In this effective waveguide, we further assume the propagation to be along a fixed direction, which we call $\widehat{x}$, so that the wavevector of the electric field is $\overrightarrow{k}(\overrightarrow{r})=k(x)\widehat{x}$. In addition, we will also consider a linearly polarized field, with the polarization axis being defined as the $y$ axis so that $E(\overrightarrow{r})=E(x)\widehat{y}$. Under these assumptions, which should be a good approximation to the fields in rings made with single-mode integrated waveguides, the problem reduces to the well-known Helmholtz equation in 1D as

In the above equation, the dielectric profile $\epsilon (x)$ may depend on the position due to purposeful design (such as by modulating the waveguide width, as will be done later), by the unavoidable introduction of imperfections during the fabrication, or by a combination of both. In all cases, the dielectric profile will be a periodic function of position due to the closed-loop nature of the propagation of light. We can make a formal equivalence between the 1D Helmholtz equation and the time-independent Schrödinger equation in 1D, $\widehat{H}\psi (x)=E\psi (x)$, with $\epsilon (x{)}^{-1}{\partial }^2/\partial x^2$ identified as the position-dependent Hamiltonian operator $\widehat{H}$, $-{\omega }^2/c^2$ as the energy $E$, and $E(x)$ as the wavefunction $\psi (x)$. Then, we can draw from experience with condensed matter systems and consider the formation of frequency bands in the dispersion relation of a ring resonator due to the periodicity of the dielectric function.

2.2.

Central Equation for a Ring Resonator

The periodicity of the resonator is encoded in the periodicity of its dielectric function,

where $P=2\pi R$ is the resonator’s perimeter and the equivalent lattice constant of the problem. This periodicity lets us apply the Bloch-Floquet theorem to Eq. (2) and show that the electric field eigenfunctions must be of the form $\overrightarrow{E}(x)=e^{ikx}E(x)\widehat{y}$, with $E(x)$ possessing the same periodicity:¹³

Due to the periodicity of $\epsilon (x)$, we can expand its inverse in a Fourier series,

where $m\in \mathbb{Z}$ and $\{{\kappa }_m\}$ are the corresponding Fourier coefficients. We can do the same to the electric field $E(x)$, where $l$ is also an integer, and {$E_l$} are the field’s Fourier coefficients.

Introducing Eqs. (5) and (6) into the Helmholtz Eq. (2), identifying all terms with the same exponents, and simplifying, we obtain

Equation (7) shows the main result of this approach, which we can call the central equation for the ring resonator, inspired by solid-state theory.¹⁴ This equation can be written in matrix notation, where it takes the form of an infinite-dimension eigenvalue problem. If we define a matrix $\overline{\overline{M}}$ and a vector $\overrightarrow{E}$ with components

we can compactly represent the central equation as

3.1.

Unperturbed Ring

An ideal, unperturbed ring is a very interesting case to illustrate the use of this framework. In this case, we have $\epsilon (x)={\epsilon }_0$ and a single Fourier coefficient is non-zero, so, using a Kronecker delta, we can write

With this simple form of the inverse dielectric Fourier coefficient, the central Eq. (7) becomes the same equation for all the orders $l$:

We can see that the Fourier components (which are traveling waves) are the eigenfunctions of the system, with dispersion relation

This dispersion relation corresponds to a straight band with band index $l$ and a slope $c/\sqrt{{\epsilon }_0}=c/n$, where $n=\sqrt{{\epsilon }_0}$ is the (constant) refractive index. The bands with $l>0$ take the positive sign and correspond to light traveling in one direction. The bands with $l<0$ take the negative sign and correspond to light traveling in the other direction. We can include this information explicitly, making $l$ a positive integer:

These bands are graphically represented in Fig. 1(a) for the case of no material dispersion. Ring resonances appear when the bands intersect the $k=0$ line, which corresponds to field patterns with the right phase matching after one round trip. Explicitly, we see that

the well-known result for ideal ring resonators, where the resonant modes are equally spaced in frequency, and the free spectral range (FSR) is inversely proportional to the resonator’s optical path $nP$.

This method is also applicable in case the medium has loss. In this particular case, we do it by replacing the refractive index $n$ by a phenomenological complex refractive index $n^{*}=n+i\kappa$, where $\kappa$ is the corresponding extinction coefficient (the extinction coefficient can include different loss mechanisms, such as material absorption, scattering, and diffraction losses). The resonance frequencies are now

which, after some algebra, becomes where ${\mathrm{\Gamma }}_l$ can be identified with the decay rate of the mode and is proportional to the mode linewidth in the resonator spectrum. We can also calculate the corresponding $Q$ factor, inversely related to the losses and the linewidth, as

Finally, the framework can be expanded to include dispersion by replacing the constant refractive index $n$ by a frequency-dependent one, $n(\omega$). This will cause the photonic bands to bend and then affect the resonance frequencies of the modes, as can be seen in the band diagram in Fig. 1(b).

3.2.

Single-Frequency Perturbation

The unperturbed case works well, so we can move on to the simplest periodic perturbation over an average dielectric constant ${\epsilon }_0$, one which has a single spatial frequency $2q/P$ (where $q$ is a natural number). In this case, only three Fourier coefficients in Eq. (5) are non-zero, and we can write the coefficients as

where ${\kappa }_0={\epsilon }_0^{-1}$ and ${\kappa }_{-2q}$ and ${\kappa }_{2q}$ are the complex amplitudes of the perturbation. It is noteworthy that if the perturbation is real (i.e., it does not change the losses in the system) then ${\kappa }_{-2q}=({\kappa }_{2q}{)}^{*}$, where the star symbolizes complex conjugation.

For this perturbation, the central equation, Eq. (7), becomes

This is not a single equation, but rather an infinite family of coupled equations, one for each possible value of $l$. Let us consider only two members of this family, the equations for $l=\pm q$:

The two equations above are linked to an infinite number of other equations due to the presence of the terms proportional to $E_{\pm 3q}$. For an unperturbed ring, as seen in Eq. (14), we can see that the frequency of the mode corresponding to $E_{\pm 3q}$ is three times that of the modes for $E_{\pm q}$. If we consider a situation where only modes with frequencies near ${\omega }_q^{\mathrm{res}}$ have a significant field amplitude (like for an excitation with a bandwidth much smaller than three times the frequency, which is a common occurrence at optical wavelengths), we can safely assume that the excitation of the $\pm 3q$ modes will be negligible, and then we can set $E_{\pm 3q}\approx 0$. Under this approximation, Eq. (20) becomes

or, in matrix form,

We can find the new $q$’th photonic bands by solving for the roots of the above matrix’s characteristic polynomial, keeping only positive frequencies and finding that

Again, we find the modified resonances for the $q$’th modes evaluating the perturbed photonic bands at $k=0$. The result is then

This result is exact, regardless of the amplitude of the perturbation. However, if the perturbation is small enough so that $|{\kappa }_{-2q}{\kappa }_{2q}|\ll |{\kappa }_0{|}^2$ is satisfied, then we can expand the outer square root to the first order and obtain a simpler expression for the resonances,

where ${\omega }_q^0=2\pi (c{n}_0/P)q$ is the $q$’th resonance frequency for the unperturbed ring ($n_0=1/\sqrt{{\kappa }_0}$ is the unperturbed index of refraction).

For many relevant applications, a particularly important quantity is the gap between these two resonances, which is the perturbation-induced mode splitting. We can easily compute it from Eq. (25):

When the perturbation modifies the dielectric profile, but does not add any gain or loss to the system, then it will be modeled by a real function, and its Fourier coefficients will satisfy ${\kappa }_{-2q}={\kappa }_{2q}^{*}$. Then, the magnitude of the induced mode splitting becomes

consistent with what was found in Ref. 5. If the perturbation affects the gain and/or loss in the system, we can still use Eq. (26), with no general relationship between ${\kappa }_{-2q}$ and ${\kappa }_{2q}$.

Furthermore, as long as resonant modes are excited with narrow-band light (with a bandwidth smaller than the FSR of the resonator, to avoid exciting more than one mode), we can extend the result from Eq. (26) to perturbations involving multiple Fourier coefficients, so that the splitting observed at a resonance with mode number $q$ is proportional to the product of the $2q$’th and $-2q$’th Fourier coefficients of the perturbation.

For illustration purposes, let us consider an optical resonator constructed of a waveguide. As explored in Ref. 5, we can introduce a perturbation by modulating the waveguide width, leading to a position-dependent effective refractive index. Let us consider a harmonic modulation of the waveguide width, leading to an effective index

where $n_0$ is the effective refractive index of the unperturbed waveguide, $a$ depends on the strength of the width modulation, and the spatial frequency of the modulation is $2q/P$ with $q$ an integer.

From the effective refractive index, we can define an effective dielectric constant $\epsilon$,

When the modulation is weak ($|a|\ll n_0$), we can then approximate the inverse of the effective dielectric function as

which has three non-zero Fourier coefficients:

Using the result from Eq. (25), then we can see that in a ring with such a perturbation, the mode with number $q$ (that is, with half the spatial frequency as the modulation), will then be split into two resonances with frequencies

with the splitting proportional to the modulation strength $a$.

If the modulation also introduces losses with the same spatial periodicity, the split-mode linewidths will also be affected. We can model this effect in a straightforward fashion by making both the mode frequency and the modulation amplitude complex. If ${\tilde{\omega }}_q^0={\omega }_q^0-i2\pi {\mathrm{\Gamma }}_q^0$ and $\tilde{a}=a^{\prime }+i{a}^{\prime \prime }$, then

In a typical microfabricated ring, in the optical telecommunications wavelength range, we will have ${\omega }_q^0\gg 2\pi {\mathrm{\Gamma }}_q^0$, and (as long as no purposeful loss/gain modulation is introduced) we can expect that $a^{\prime }$ and $a^{\prime \prime }$ will have a similar order of magnitude. In this case, Eq. (33) becomes somewhat simpler:

and one of the split modes will have a larger decay rate than the original one (larger imaginary part of the frequency), while the other will show a smaller decay rate. Thus, we not only expect a splitting in frequency, but also an asymmetry on the split-mode linewidth (and, correspondingly, on the Q factor).

As previously mentioned, if we were to apply a modulation with more than one non-zero Fourier coefficient and use a narrowband light source so that only one resonant mode is excited, we can consider that each mode (with mode number $q$) will only be affected by the 2$q$’th Fourier component modulation.