1. INTRODUCTION

Optical whispering gallery mode resonators (WGMRs) [1] are optical structures that confine light due to total internal reflection. Their properties have attracted significant attention in the last decade [2,3]. Different WGMRs allow the application of variable experimental conditions to change their internal properties [4,5]. Most studied WGMRs 3D structures are balls [6], toroids [7], and 2D structures, i.e., rings [8]. The light is introduced in these structures most often by a prism [9] or tapered fiber coupling [10]. These resonators are outstanding because they form optical resonances with high $ Q $-factors up to ${10^7} - {10^{10}}$ [11,12]. When the external environment, e.g., temperature [13], humidity [14], or refractive index [15], changes, the resonances shift; therefore, WGMRs are usable as sensors to monitor these changes [16]. In biosensing, WGMRs are used as units where tested molecules stick to their surface [17,18]. Another significant aspect of WGMRs is a high density of light confined in these structures, thus forming conditions for studies of nonlinear optical effects [19]. For example, whispering gallery mode frequency combs [20] are formed through such effects.

While most of the studies in this field are concentrated on the particular peculiarities and application issues of WGMR, there is a lack of a detailed description of the theoretical aspects of these resonators. In this paper, we provide the advanced classical analytical description of optical resonances with mentioning only some results obtained via Maxwell’s equations (ME) [21,22]. This gives space to accent physical processes operated in resonators, e.g., interference of waves, and not to dive into ME formalism, which gives more precise results of resonances but is ambiguous and practically usable only in some general resonator geometries, e.g., balls and cylinders. Formulas provided here give a basic level of deep understanding of resonances of optical resonators, including WGMR and Fabry–Perot resonators. Finite element simulations of WGMRs and light propagation in them can be made, for example, in COMSOL Multiphysics software and can be used as a supplementary material to analytical theory [23–26].

This paper contains the description of the main parameters of optical resonances such as resonance wavelengths and frequencies, free spectral range, $ Q $-factor, summation principle of $ Q $-factors of various processes, finesse, etc. The intensity distribution of resonance spectra is derived from the interference of an infinite number of waves of smaller amplitudes and equal phase differences. Resonances of Fabry–Perot resonators with different and equal reflection coefficients of their mirrors are described. Their similarity to resonances of whispering gallery mode resonators with single and two waveguides coupling is presented. Provided formulas with underlying proofs form a concept system for an in-depth understanding of the formation of optical resonances.

2. BASIC ANALYTICAL FORMULAS FOR RESONATOR DESCRIPTION

Optical resonances observed in WGMR can be described by wavelength positions ${\lambda _i}$ and widths $\Delta {\lambda _i}$ or frequency positions ${\nu _i}$ and widths $\Delta {\nu _i}$. The whole spectra can be expressed as $I(\lambda)$ or $I(\nu)$.

A. Wavelength in Media

Resonance frequency ${\nu _i}$ is the same across various media. However, a resonance wavelength inside the resonator ${\lambda _{\rm mat}}$ differs from the wavelength in vacuum ${\lambda _{\rm vac}}$ as

B. Resonance Positions

A resonance condition appears when light interferes positively. If two waves have phases ${\phi _1}$ and ${\phi _2}$, respectively, the resonance condition is

The corresponding resonance frequency for index $m$ is

For the first radial TE mode, ${m_r} = 1$ and $\alpha (1) = 2.33811$. When $m \gg 1$ and the resonator is surrounded by air ${n_2} = 1$, the resonance position in Eq. (10) turns into [1]

In the case when refractive index $n$ is nonhomogenous in a media, the resonance condition in Eq. (4) turns into

C. Free Spectral Range

The distance between the two closest resonances is called the “free spectra range” (FSR).

For wavelength scale, the FSR is

In frequency scale, the FSR is

D. Quality Factor

Quality factor of the resonator system is defined as [29,30]

Let us analyze the case when the amplitude $U(t)$ of the optical signal oscillates in time $t$ with an angular frequency ${\omega _o}$. It is related to its intensity as $I(t) \sim U{(t)^2}$. When the intensity $I(t)$ of the optical signal decays according to Eq. (16), we obtain $U(t) = {U_0}{e^{- \frac{t}{{2\tau}}}}{e^{i{\omega _0}t}}$, where ${U_0}$ is a coefficient. By taking the Fourier transform, we obtain

E. Finesse

Finesse ${\cal F}$ describes the resonator and is defined as the FSR divided by the full width of resonance at the half maximum (FWHM):

Taking into account Eqs. (14), (19), and (23), we obtain

For the resonance condition in Eq. (4), Eq. (32) turns into

F. Intensity Distribution of a Resonance Spectra

Let us examine the interference of an infinite number of waves of progressively smaller amplitudes ${U_i}$ and equal phase differences [29], where $i$ is the index of the wave changing from 1 to infinity. The first wave has the intensity ${I_0}$ and an amplitude ${U_1} = \sqrt {{I_0}}$. The next wave is smaller by the factor of $h = |h|{e^{{i\phi}}}$, $|h| \lt 1$, compared with the previous wave, and incorporates the decay of the amplitude and a phase shift $\phi$. Thus, a series of waves is formed:

If we assume $|h|$ to be fixed and $\nu$ to be variable, then the maximal value of intensity is achieved when $\sin (\frac{{\pi nL}}{c}\nu) = 0$, thus giving the resonance condition

Full width at half maximum $\Delta \nu$ of the intensity in Eq. (48) is obtained from equation

Thus, $Q$-factor is obtained as

$|h|$ can be expressed as

For slow decay, Eqs. (61) and (62) turn into

By combining Eqs. (30), (14), and (63), we obtain

Close to resonance described by ${\phi _{\rm res}}$ [see Eq. (46)] or ${\nu _{\rm res}}$, the intensity distribution in Eq. (69) becomes Lorentzian:

G. Resonance Shift

The advantage of optical resonators is ease of use for sensing applications [16]. The most used mechanism that realizes the sensing process is the shift of resonance positions when the external environment, e.g., temperature, changes. This is realized by the expansion of the resonator and the change of its refractive index, as resonance positions depend on the light path length and refractive index [see Eq. (4)].

Thermal expansion of materials is described by the coefficient of thermal expansion ${\alpha _0}$. For material with length $L$, the expansion $dL$ for the temperature change $dT$ is described as

3. ADVANCED ANALYTICAL FORMULAS FOR RESONANCE DESCRIPTION

A. Fabry–Perot Resonator

The Fabry–Perot resonator is an optical system with two parallel semitransparent mirrors placed at a distance $d = L/2$. Laser light is irradiated on one of the mirrors, and transmitted light of the whole system is measured [33,34]. There are two main types of Fabry–Perot resonators, i.e., bulk glass with parallel surfaces that are covered with reflection coatings [FP Type—1, Fig. 1(a)] and air-spaced plain parallel surfaces, which are covered with reflecting coatings on inner surfaces and with antireflecting coatings on outer surfaces [FP Type—2, Fig. 1(b)].

 

Fig. 1. Types of Fabry–Perot resonators: (a) solid etalon (Type 1); (b) air-spaced plain parallel surfaces (Type 2).

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Let us derive a significant property of transmitted and reflected light that falls on the boundary of two optical medias. We suppose that incident light has amplitude $a$ and is transmitted from media with refractive index ${n_1}$ to media with refractive index ${n_2}$ (Fig. 2). The amplitude of the ray in the second media becomes $at$, where $t$ is the transmittance coefficient. The amplitude of the reflected ray is $ar$, where $r$ is the reflectance coefficient. The time-reversal principle can be used, i.e., when the direction of light propagation changes to the opposite, the amplitudes of the field have to remain the same. Let us use $r^\prime $ as the reflection coefficient when the ray comes from media ${n_2}$ and reflects from media with ${n_1}$, and $t^\prime $ is the corresponding transmittance coefficient. Then, the time-reversal gives ray $at$ to reflect as $atr^\prime $ and to be transmitted as $att^\prime $ and ray $ar$ to be transmitted as $art$ and reflected as $a{r^2}$ [see Fig. 2(b)]. Thus, we have

 

Fig. 2. (a) Scheme of directions of incident, reflected, and transmitted field rays when light is irradiated on the boundary between medias with different refractive indexes ${n_1}$ and ${n_2}$. (b) Scheme for comparison of incident, reflected, and transmitted fields with rays in time-reversal situation. The light propagates along one horizontal axis. Only for visualization purposes, rays have angles to the boundary that separates medias.

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The reflection coefficient of the mirror (or semitransparent mirror) is assumed to be $R$, and it describes the proportion of intensity that is reflected. For the amplitude, this becomes $r = \sqrt R$ and $r^\prime = - \sqrt R$ with $r$ used when the ray reflects from media with a larger refractive index and $r^\prime $ used when the ray reflects from media with a smaller refractive index. In the second case, it can be described as a reflection with a coefficient $r$ and a phase shift $\pi$. The transmission coefficient of light intensity is defined as $T = 1 - R$. For amplitude transmission, it becomes $t = t^\prime = \sqrt T$.

 

Fig. 3. Schematics of light field propagation in a Fabry–Perot resonator. The light propagates along one horizontal axis. Only for visualization purposes, rays are separated vertically.

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Let us explore the Fabry–Perot resonator of Type 1 [Fig. 1(a)]. The resonator is filled with media with refractive coefficient $n$. The resonance condition appears when the distance between both mirrors is equal to positive natural number $m$ of half wavelength:

The left reflective layer (first mirror) of the Fabry–Perot resonator (see Fig. 3) is characterized by the intensity reflection coefficient ${R_1}$, transmittance coefficient ${T_1}$, corresponding amplitude reflection coefficient ${r_1} = \sqrt {{R_1}}$, and transmittance coefficient ${t_1} = \sqrt {{T_1}}$. The right reflective layer (second mirror) is characterized similarly by coefficients ${R_2}$, ${T_2}$, ${r_2}$, and ${t_2}$. The incident light with intensity ${I_0}$ and field amplitude ${U_0} = \sqrt {{I_0}}$ travels from left to right and hits the left side of the resonator. This field is transmitted through the first mirror as ${U_{01}} = {t_1}{U_0}$. When it reaches the second mirror, its phase is shifted by $\phi /2 = (2\pi n/\lambda)d$, thus obtaining ${U_{02}} = {e^{i\phi /2}}{U_{01}}$. Part of it is transmitted through the second mirror ${U_{T0}} = {t_2}{U_{02}} = {e^{i\phi /2}}{t_1}{t_2}{U_0}$. The reflected part obtains the phase shift by $\pi$, thus giving ${U_{03}} = - {r_2}{U_{02}}$. Further, we find that ${U_{04}} = {e^{i\phi /2}}{U_{03}}$, ${U_{11}} = - {r_1}{U_{04}}$, ${U_{12}} = {e^{i\phi /2}}{U_{11}}$, and ${U_{T1}} = {t_2}{U_{12}} = {e^{i3\phi /2}}{r_1}{r_2}{t_1}{t_2}{U_0}$. Additional steps show that ${U_{T2}} = {t_2}{U_{22}} = {e^{i5\phi /2}}r_1^2r_2^2t_1^2t_2^2{U_0}$. Thus, the summary field amplitude transmitted through the system becomes

An alternative way [35] to obtain Eq. (93) is as follows. We use the summary amplitudes of fields propagating outside and inside the resonator: incident field amplitude ${U_0}$, reflected field amplitude ${U_R}$, transmitted field amplitude ${U_T}$, field amplitude in the resonator close to left mirror propagating in the right direction ${U_1}$, and the field amplitude in the resonator close to left mirror propagating in the left direction ${U_4}$. They have relations ${U_1} = {t_1}{U_0} - {r_1}{U_4}$, ${U_R} = {t_1}{U_4} + {r_1}{U_0}$, ${U_4} = - {U_1}{r_2}{e^{{i\phi}}}$, and ${U_T} = {U_1}{t_2}{e^{i\phi /2}}$ from which ${U_T}$ can be derived. This alternative provides a fast way to obtain the final equation but lacks the clarity of its relation to the interference phenomena that is highlighted in this paper.

Equation (93) can be transformed using operations similar to those used for Eqs. (36) and (37); then, we obtain the transmitted field intensity of the Fabry–Perot resonator [36]:

Minimal value ${I_{T\min}}$ of the transmitted intensity in Eq. (94) is obtained when $\sin (\phi /2) = 1$:

To obtain the sharp lines of the Fabry–Perot resonator, reflection coefficient $R$ has to be close to 1. In this case,

As in Eq. (120), phase dependence comes from the denominator, which is equivalent to the intensity of transmitted light in Eq. (94); the finesse of reflected light ${{\cal F}_R}$ is the same as the finesse of transmitted light in Eq. (97):

B. Circular Resonator Coupled to One Waveguide

Let us explore the situation when a circular whispering gallery mode resonator with radius $a$ is coupled to a waveguide (Fig. 4). The field in this resonator can be modeled as reflected field ${U_R}$ of the Fabry–Perot resonator when the second mirror is fully reflective, ${R_2} = 1$ and ${R_1} = R$. In this case, the absorption and dissipation of the field were not taken into account. Thus, intensity distribution is obtained from Eq. (120) and becomes $I = {I_0}$, which means that all fields are transmitted through the system.

 

Fig. 4. Light propagation in a circular resonator coupled to a waveguide.

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We will describe a model of a waveguide coupled to a circular resonator, taking into account field decay in the system. This system is called an “optical all-pass filter.” Coupling of the waveguide and the resonator will be described by the reflection coefficient $r = \sqrt R$, transmission coefficient $t = \sqrt {1 - {r^2}}$, one loop light path length in the resonator is $L$, giving the phase shift per loop $\phi = 2\pi nL/\lambda$, and the field decay rate ${e^{- \beta}}$ with

Let us obtain the summary transmitted light amplitude ${U_{P1}}$ of the all-pass filter in Port 1 (Fig. 4). The light with amplitude ${U_0} = \sqrt {{I_0}}$ enters the waveguide from the left side. Part of this amplitude ${U_{P10}} = r{U_0}$ is passing through the waveguide without entering the resonator. Another part ${U_{11}} = t{U_0}$ enters the resonator. After travelling one loop in the resonator, the amplitude of the wave becomes ${U_{12}} = - {e^{{i\phi}}}{e^{- \beta}}{U_{11}}$. This field reflects back into the resonator as ${U_{21}} = - r{U_{12}}$. Another part is transmitted to the waveguide as amplitude ${U_{P11}} = t{U_{12}} = - {e^{{i\phi}}}{e^{- \beta}}{t^2}{U_0}$. The field amplitude ${U_{21}}$ after travelling the next loop in the resonator turns into ${U_{22}} = - {e^{{i\phi}}}{e^{- \beta}}{U_{21}}$. This field is transmitted to the waveguide as ${U_{P12}} = t{U_{22}} = - {e^{i2\phi}}{e^{- 2\beta}}r{t^2}{U_0} = {U_{P11}} \cdot ({e^{{i\phi}}}{e^{- \beta}}r)$ and is reflected into the resonator as ${U_{31}} = - r{U_{22}}$. Further, ${U_{31}}$ after one loop in the resonator turns into ${U_{32}} = - {e^{{i\phi}}}{e^{- \beta}}{U_{31}}$. It is transmitted to the waveguide as ${U_{P13}} = t{U_{32}} = {e^{{i\phi}}}{e^{- \beta}} {{rtU}_{22}} = {e^{{i\phi}}}{e^{- \beta}}r{U_{P12}} = {U_{P11}} \cdot {({e^{{i\phi}}}{e^{- \beta}}r)^2}$. In a similar manner, the further series of reflected and transmitted signals can be found. Finally, the summary transmitted light amplitude ${U_{P1}}$ of the all-pass filter in Port 1 is obtained as a sum of series:

The transmitted field intensity in Port 1 is

The maximal value ${I_{P1\max}}$ of the intensity ${I_{P1}}$ in Eq. (146) is obtained when $\mathop {\sin}\nolimits^2 (\phi /2) = 1$:

Now the $ Q $-factor of the optical all-pass filter can be described as

 

Fig. 5. Light propagation in circular resonator coupled to two waveguides.

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C. Circular Resonator Coupled to Two Waveguides

A circular resonator with two waveguides can be analyzed (see Fig. 5). Such a resonator is called an “add-drop filter.” It can be modeled as a Fabry–Perot resonator with various reflection coefficients of mirrors. The decay of signal in the system can be described in similarity with the description of the all-pass filter. By comparing Eq. (118) when ${r_2} = 1$ and ${r_1} = r$ with Eq. (143), we see that decay could be introduced by substituting ${e^{{i\phi}}}$ with ${e^{{i\phi}}}{e^{- \beta}}$ in Eq. (118). This is logical, as a phase shift $\phi$ was obtained by light travelling one loop in the resonator; in this path, the decay ${e^{- \beta}}$ was obtained. Now transmitted light field amplitude ${U_{P1}}$ through Port 1 can be expressed from Eq. (118) by substituting ${e^{{i\phi}}}$ with ${e^{{i\phi}}}{e^{- \beta}}$:

Table 1. Summary of Main Equations That Describe Parameters of Resonances

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The dissipated intensity ${I_D}$ of the add-drop filter can be obtained as

When the coupling of the resonator to both waveguides is equal, which means ${r_1} = {r_2} = r$, then the characteristic parameters of the field in Port 1 become

4. CONCLUSION

Main derived formulas as a result of this paper are summarized in Table 1. They are ordered in a way that asserts the similarity of Fabry–Perot and whispering gallery mode resonances with those of an interference of an infinite number of waves of progressively smaller amplitudes and equal phase differences.

We presented the classical analytical description of resonances in Fabry–Perot and whispering gallery mode resonators. Basic terms such as wavelength in media, resonance condition for wavelength and frequency, including an integral form of resonance condition in case of nonhomogenous media, free spectral range, $ Q $-factor, summation principle of $ Q $-factors of various processes, and finesse were introduced. Interference of an infinite number of waves of progressively smaller amplitudes and equal phase differences were described, its intensity distribution, maximal intensity, minimal intensity, resonance depth, resonance condition, resonance width, $ Q $-factor, and finesse were derived. The case of a small decay was analyzed.

Fabry–Perot resonators with nonequal and equal reflection coefficients of their mirrors were described. The amplitudes of fields in a resonator, summary amplitude of transmitted and reflected fields, intensity distribution, maximal and minimal intensities, resonance depth, resonance width, finesse, $ Q $-factor, and corresponding values for slow decay were analyzed.

Circular resonators coupled to one and two waveguides were described. Field decay in the resonator was introduced. Characteristics of resonances were derived and presented in the form that allows them to compare with Fabry–Perot resonances and general case of the interference of an infinite number of waves of progressively smaller amplitudes and equal phase differences.

A description of the resonances provided in this paper is a useful tool for reference when forming an in-depth understanding of optical resonances, analyzing experimental data, and searching for ways to optimize resonator systems.

Summarizing our paper represents a detailed description of the theoretical approach describing features of the resonators for general introduction in the topics of optical resonances.