## Introduction to optical resonators An optical resonator is a device - **Collect**: which efficiently collects (or stores) free-space light; - **Concentrate**: has the capability to concentrate the intercepted light down a sub-wavelength volume. ### Fabry-Perot cavity This is the most commonly mentioned optical resonator. It has two mirrors and lights are trapped between these mirrors. Light waves are reflected back and forth by the mirrors. ![[Drawing 2024-03-21 20.12.09.excalidraw.svg]] The cavity resonates when $L=\frac{m\lambda}{2}$ $m$ is integer. This defines the modes. For a given length $L$, we have $f_{m} = \frac{mv}{2L}$ here $v=\frac{c}{n}$, is velocity. These frequencies correspond to those (infinitly) allowed modes. And we have the free spectral range (FSR) of the cavity being $f_{m+1} - f_{m} =\Delta f =\frac{v}{2L}$ Sometimes we write it as $\nu_{F}$. ![[Drawing 2024-08-10 17.11.04.excalidraw.svg]] >[!Note] >This also the minimum frequency that the cavity would work with. The term *free* is likely to describe the fact that the cavity does not "amplify" or "diminish" the wave, and the optical wave can pass through like *free*. > >For medium other than air and vacuum, the FSR becomes >$\nu_{F}=\frac{\nu}{2nL}$ >This means if the medium is not non-dispersive, this FSR could be frequency dependent for the dependence of $n(\omega)$. Now consider the wave inside the cavity, we have the incoming field $E_{i}$, and reflected and transmitted wave $E_{r}$ and $E_{t}$. The mirror has reflection and transmission coefficients being $r$ and $t$, so the field inside the cavity is $E_{cav} = t E_{i} + \alpha E_{cav}$ with $\alpha = r^{2} e^{i\phi}$, $\phi = \frac{2\pi}{\lambda} 2nL$. This $\phi$ is a phase factor, the wave goes total length of $2nL$, namely n round trip, $n$ being an integer. This gives the intensity: $I_{cav} \propto |E_{cav}|^{2} = \frac{I_{i}}{|1-r^{2}e^{-i\phi}|^{2}} = \frac{I_{i}}{(1-r^{2})+4r^{2} \sin^{2}(\phi/2)}$ It is intuitive that the intensity would be maximized at some phase and minimized at others, due to constructive and destructive interference. The same relations holds if converts to $\lambda$. For $\phi = m\pi$, we have destructive interference, light diminished; and $\phi = 2m \pi$, we have constructive interference, light enhanced. Also if the reflectance $r^{2}$ is larger, we would have a sharper peak. >[!Notice]- >Here the transmission is non zero so the cavity is lossy. >[!Note] >People also make ring resonators. A typical one is called Whispering Gallery Mode (WGM) microresonator, the light cycles along the circumference of a ring/disk. We have a similar expression for $E$. >$\frac{E_{3}}{E_1} = \frac{tae^{i \phi}}{1-rae^{i\phi}}$ >and phase $\phi = 2\pi kR$. >![[Drawing 2024-03-21 21.17.18.excalidraw.svg]] >The size of such device is $\sim$ 60 microns. ### Resonate mode #### Mathematical description of mode The mathematical way to determine the mode of a resonator is the following, solve the eigen-problem below: $\begin{equation} \left[ \begin{array}{cc} 0 & \frac{i}{\varepsilon(r, \tilde{\omega})} \nabla \times \\ -\frac{i}{\mu_0} \nabla \times & 0 \end{array} \right] \left[ \begin{array}{c} \tilde{\mathbf{E}}(r) \\ \tilde{\mathbf{H}}(r) \end{array} \right] = \tilde{\omega} \left[ \begin{array}{c} \tilde{\mathbf{E}}(r) \\ \tilde{\mathbf{H}}(r) \end{array} \right] \end{equation} $ The left side is the operator for [[Maxwell equation recap|Maxwell equation]], $\tilde{\mathbf{E}}$ and $\tilde{\mathbf{H}}$ are eigenfunctions, and $\tilde{\omega}$ is eigenvalue. $\tilde{\mathbf{E}}$, $\tilde{\mathbf{H}}$ and $\tilde{\omega}$ are intrinsic to the system, and independent to external field. >[!Notice]- >Some notes on above expression: >- The equation is directly from Maxwell's expression, just did a Fourier transform in time/frequency domain. It's like: >$\nabla \times \mathbf{H} = \frac{\partial\mathbf{D}}{\partial t}$ > Doing Fourier transform in time domain >$\nabla \times \tilde{\mathbf{H}} = -i \tilde{\omega} \tilde{\mathbf{D}}$ >$\nabla \times \tilde{\mathbf{H}} = -i \tilde{\omega} \varepsilon\tilde{\mathbf{E}}$ >$\frac{i}{\varepsilon} \nabla \times \tilde{\mathbf{H}} = \tilde{\omega} \tilde{\mathbf{E}}$ > The $\tilde{\omega}$ is just for indicating this frequency has a complex value. > >- One may see that the left side, in the $\varepsilon$ term, we also have a frequency term. This means the procedure for solving this question would be hard since it's a self-consistency problem. FEM or [FDTD](https://en.wikipedia.org/wiki/Finite-difference_time-domain_method) may be applied to solve it. #### Quasi-normal mode It is normal we have not a lossless material, so the system gonna be dissipative (non-Hermitian). Quasi-normal modes are introduced to represent this case. And the lossy behavior, we have a broadening in the resonance intensity peak and it has a finite bandwidth (FWHM) $\Gamma$. Such peak broadening can be caused by - Absorption of the constitutive material - Scattered to free space (or say transmission) And mathematically, such lossy behavior can be described by a complex frequency $\tilde{\omega}$, and $\tilde{\omega} = \omega - i\Gamma /2$ The $\Gamma$ term accounts for the losses. ![[Drawing 2024-03-21 23.44.07.excalidraw.svg]] (This plot should be Lorentzian but its does not seem like a Lorentzian) So the temporal part of the phase term gonna be $e^{-i \tilde{\omega} t} = e^{-i (\omega - i\Gamma /2) t} = e^{-i {\omega} t} e^{-(\Gamma /2) t}$ $e^{-(\Gamma /2) t}$ is called the Attenuation term. The peak boarding is contributed by this term. >[!Note] >Look at the expression of $\Gamma$, it should have the unit of $s^{-1}$, and can be interpreted as the parameter of the field amplitude decay over the initial amplitude (consider exponential decay). Now consider a generalized case, an optical resonator supporting a set of quasi-normal modes: $\left[ \tilde{E}_1(\mathbf{r}), \tilde{E}_2(\mathbf{r}), \tilde{E}_3(\mathbf{r}), \ldots \right] $ Then the response field $\mathbf{E}_{s} (\mathbf{r},t)$ to an incident field $\mathbf{E}_{0} (\mathbf{r},t)$ would be $E_{s}(\mathbf{r}, t) = \mathfrak{R} \left[ a_1(t)\tilde{E}_1(\mathbf{r}) + a_2(t)\tilde{E}_2(\mathbf{r}) + a_3(t)\tilde{E}_3(\mathbf{r}) + \ldots \right]$![[Drawing 2024-03-29 15.51.38.excalidraw.svg]] Here $a_{m}$, $m=1,2,...$ are the **time-dependent modal excitation coefficient**, i.e., how much each quasi-normal mode is coupled to by the incident field $\mathbf{E}_{0} (\mathbf{r},t)$. And this can be interpreted as, a given incident field may cause the excitation of several quasi-normal modes. (one may be the dominant) >[!Note]- >$\mathfrak{R}$ typically refers to the "real part", and $\tilde{a}$ indicates the Fourier transform to frequency domain, or in later the $\tilde{\omega}$, an indication of complex form. >More about tilde, see https://en.wikipedia.org/wiki/Tilde#Mathematics and https://en.wikipedia.org/wiki/Tilde#Physics. >In above mode, the response field is linked to the resonator behavior by $\tilde{E}_1(\mathbf{r})$, which is a space description (this is reasonable cause the resonator itself should have no time dependence), and to the incident field by $a_{m}$, which is a time description. This way we decoupled the space and time in $\mathbf{E}(\mathbf{r}, t)$. #### Solving $a_{m}$ with phenomenological model Use a phenomenological model to see the behavior of the resonator, namely the expression of $a_{m}$. Consider $\frac{d a_{m}}{\mathrm{d}t}$ for a specific mode $\tilde{E}_m(\mathbf{r})$, $\frac{\mathrm{d} a_m}{\mathrm{d} t} = - i \tilde{\omega}_m a_m - i \rho_m u$ $\tilde{\omega}_m$ is the frequency of the specific mode, $u$ is a generic excitation ($u=u(t)$, only time dependency). Here we assumption that $u(t)$ has the same nature as $a_{m}(t)$. $\rho_{m}$ is like a coupling factor between $a_{m}$ and $u$. It can also be considered as a frequency or rate. This model is like a driven oscillator (recall [[Classical Lorentz model]]), $- i \tilde{\omega}_m a_m$ part is like the intrinsic oscillation (with loss), and the later term is the external driving (in this way $\rho$ can be understand as a driving coupling rate easily). With $\tilde{\omega}_m = \omega_m - i \frac{\Gamma_m}{2}$, we can write a more clear term: $\frac{\mathrm{d}a_m}{d t} = - i\omega_m a_m - \frac{\Gamma_m}{2} a_m - i \rho_m u$ The three terms in RHS are oscillatory term, losses, and excitation. If we assume that the excitation is harmonic, $u(t) = u_0 e^{-i\omega t}$, since $a_{m}$ and $u$ has the same behavior, $a_m(t) = \tilde{a}_m e^{-i\omega t}$, $i\tilde{a}_m [(\omega - \omega_m) + i \frac{\Gamma_m}{2}] = \rho_m u_0 $ $\rightarrow \left|\tilde{a}_m(\omega)\right|^2 = \frac{|\rho_m|^2 u_0^2}{(\omega - \omega_m)^2 + \left(\frac{\Gamma_m}{2}\right)^2} $ No surprise, we get the typical Lorentzian response, centered at $\omega_{m}$ and with FWHM $\Gamma_{m}$. ![[Drawing 2024-03-29 16.57.31.excalidraw.svg]] >[!Note] >This coupling rate $\rho_{m}$ has the proportionality of $\rho_{m} \propto t$, this t is transmission (transmission loss of mirror). #### Critical coupling Consider a practical problem: we have a photodetector made of quantum wall, placed inside a FP cavity (i.e., 2 mirrors), and we wanna maximize the absorption of the QW to have a better response, or say signal. ![[Drawing 2024-03-29 17.27.50.excalidraw.svg]] Consider the phenomenological model, we should have the losses terms modified to $\gamma_{A}$ and $\gamma_{M}$ (for mirror its transmission loss), so we have $\frac{\mathrm{d}a_m}{d t} = - i\omega_m a_m - \gamma_A a_m - \gamma_M a_m - i \rho_m u$ Assume incident wave is monochromatic ($u$ has the form of $u(t) = u_0 e^{-i\omega t}$), $\frac{\mathrm{d}a_m}{d t} = - i\omega \tilde{a}_m$ so we have $\tilde{a}_m(\omega) = \frac{\rho_m u_0}{(\omega - \omega_m) + i(\gamma_A + \gamma_M)}$ $ |\tilde{a}_m|^2 = \frac{\rho_m^2 u_0^2}{(\gamma_A + \gamma_M)^2} \propto \frac{\gamma_M}{(\gamma_A + \gamma_M)^2} u_0^2 $ This means the modal excitation coefficient (proportional to intensity), is maximized when $\gamma_{A}=\gamma_{M}$, at $\omega = \omega_{m}$. >[!Question] >Here it stated that $\gamma_{A}=\gamma_{M}$ is the condition for the critical coupling. But the equal of two losses terms are kind of strange. (Interpreting the loss of the mirror as transmission loss is somehow acceptable but still strange) >The definition of critical coupling is that no energy get released from the resonator and all gets absorbed. See the link https://en.wikipedia.org/wiki/Optical_ring_resonators#Optical_coupling for critical coupling inside the ring resonator. >Or maybe for this problem, the definition of critical coupling is different from the one we use at ring resonators. ### Quality factor and mode volume Quality factor $Q$ is defined as the **temporal confinement of the resonant mode**, and the mode volume $\tilde{V}$ is defined as the **spatial confinement**. These parameters are used to quantify the capability of cavity modes to enhance light-matter interaction. #### Finesse For a typical intensity-frequency profile for a cavity, we may have the ratio of free spectral range (FSR) and FWHM to define finesse to see the selective of wavelength. As shown before, the frequency difference between resonance peaks is FSR, describing how far away are the resonance frequencies from each other. It also indicates the minimum allowed frequency inside the resonator. ![[Drawing 2024-08-10 17.11.04.excalidraw.svg]] Define finesse $F$ as $F=\frac{\nu_{F}}{\delta \nu} = \frac{\nu_{F}}{\text{FWHM}}$ If write FWHM explicitly (with respect to intensity and refractivity), we have $F = \frac{\pi r}{\sqrt{1 - r^2}} \implies \frac{I_{\text{inc}}}{1 + \left(\frac{2F}{\pi}\right)^2 \sin^2(k \cdot L)}$ #### Mode volume The mode volume of a given mode supported by an optical resonator is the **spatial extension (or confinement)** of this mode, and it's independent on the magnitude of field itself. $ \tilde{V} = \int \frac{\varepsilon(r) |E(r)|^2}{\varepsilon(r_{\text{max}}) |E(r_{\text{max}})|^2} \, \mathrm{d}r $ The $\varepsilon$ is the permittivity at $r$, and $E$ is the field (inside (in some case, if not confined, outside) the resonator, for certain mode) at $r$, and this quantity is normalized by the maximum value of this product inside the cavity. The problem of this mode volume is that optical resonators are in many cases "open resonators", and the energy will leak to space. This means one cannot set the integral to the entire space for practical calculations, and people set the limits to a reasonable value, like ~hundreds of the radius of the cavity. This makes the $\tilde{V}$ becomes an approximation. #### Quality factor The quality factor can be defined as the energy stored inside the cavity divided by the energy loss, namely, $Q=\omega \frac{E_{stored}}{E_{lost}}$ It can also be related to finesse, by $Q = \frac{\nu_0}{\nu_F} \cdot F = \frac{\nu_0}{\delta \nu}$ Here $\nu_{0}$ is the working frequency (the frequency we care about). But an noticeable problem is this definition is not well defined for open resonators, cause the energy stored is hard to differentiate the energy "inside" and "outside" the resonator. And such difficulties are universal for all resonators with big loss. Therefore, another definition is applied for such systems with high energy dissipation. $ Q = -\frac{1}{2} \frac{\Re(\tilde{\omega})}{\Im(\tilde{\omega})} \rightarrow Q = \frac{\omega}{\Gamma} $ This is unambiguously defined for any cavity. >[!Note] > It is worth to say that above expression can be rewritten to the energy expression: > $Q = \frac{\Re(\tilde{\omega}) \frac{1}{4}\int_V\left[ \Re(\varepsilon(\tilde{\omega})) |\mathbf{E}|^2 + \mu_0 |\mathbf{H}|^2 \right]\, dV }{\frac{1}{2} \int_{\Sigma} \Re(\mathbf{E} \times \mathbf{H}^*) \cdot dS + \frac{\Re(\tilde{\omega})}{2} \int_V \Im(\varepsilon(\tilde{\omega})) |\mathbf{E}|^2 \, dV}$ > the upper part $\int_V\left[ \Re(\varepsilon(\tilde{\omega})) |\mathbf{E}|^2 + \mu_0 |\mathbf{H}|^2 \right]\, dV$ is time-average EM energy stored in the volume V, $\frac{1}{2} \int_{\Sigma} \Re(\mathbf{E} \times \mathbf{H}^*) \cdot dS$ is the power flowing out from surface $\Sigma$, and $\frac{\Re(\tilde{\omega})}{2} \int_V \Im(\varepsilon(\tilde{\omega})) |\mathbf{E}|^2 \, dV$ is the power dissipated by absorption enclosed in volume V. > This is valid for any surface $\Sigma$ enclosing a volume $V$. This makes it become applicable for any resonator even has high dissipation. > See GPT reply at https://chat.openai.com/share/ad1c7d05-a136-4fa3-8899-7682609dcdd8 ### Good cavity, bad cavity We use finesse and quality factor to describe how "good" a cavity is. In general, a good cavity of high finesse $F$ and/or high quality factor $Q$ can be used as - Spectrum analyzer. Small changes in the frequency around a resonance could cause huge intensity change. - Precision measurement. Small change in mirror distances could dramatically affect the resonance frequency. - Or other applications requires small loss, like laser. However, a "bad" cavity is also useful. Antennas are essentially bad cavities that couple to the environment. They have short ringdown and have no "echo" from signal (which is nice as an antennas). The applications could be: - Receiver. Radiation from the environment couples to the internal mode, and measure the internal mode to get received signals. - Sender. High driving field couples to the outside mode, and by engineering the cavity the desired mode could be obtained. >[!Info] >See more at >https://en.wikipedia.org/wiki/Optical_cavity >https://en.wikipedia.org/wiki/Q_factor >https://en.wikipedia.org/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer >https://en.wikipedia.org/wiki/Polarization_(waves)#s_and_p_designations