## the Fabry-Pérot resonator

Calculate the transmitted or reflected intensity and phase for a Fabry-Pérot cavity. Both reflectivities and resonator losses are accounted for.

reflectivity 1: %

reflectivity 2: %

cavity round-trip loss (power): %

transmissionreflection

Fig. 1: normalized intensity vs. round-trip phase [rad]

resonator finesse:

#### Formulas

Note: These formulas are for the optical Fabry-Pérot resonator with normal incidence. The round-trip phase then depends only on the length $L$ of the resonator and the wavenumber $k$ of the light inside it, so that $\delta = 2 k L$. For oblique incidence it is usually sufficient to alter the round-trip phase to account for the additional path length due to the angle.

Fig. 3: Illustration of the components constituting the reflected and transmitted fields in a Fabry-Pérot resonator. As incidence is assumed normal to the material interfaces, the vertical axis denotes passage of time.

Reflection
Assuming an incident external field with (complex) Amplitude $E_i$ as in Fig. 3, the reflected field is composed of infinitely many components of decreasing magnitude which can be obtained by simply tracing the trajectories in the figure and accounting for reflections, transmissions, etc.

$\label{basic-reflected} E_r = E_{r0} + E_{r1} + E_{r2} + E_{r3} + \ldots$

with

\begin{equation*}
\begin{aligned}
E_{r0} &= E_i \Bigl[ r_{1}^+ \Bigr]\\
E_{r1} &= E_i \Bigl[ t_{1}^+ r_{2}^+ t_{1}^- a \exp\bigl[ i\delta \bigr] \Bigr]\\
E_{r2} &= E_i \Bigl[ t_{1}^+ r_{2}^+ t_{1}^- a \exp\bigl[ i\delta \bigr] \Bigl( r_{1}^- r_{2}^+ a \exp\bigl[ i\delta \bigr] \Bigr) \Bigr]\\
E_{r3} &= E_i \Bigl[ t_{1}^+ r_{2}^+ t_{1}^- a \exp\bigl[ i\delta \bigr] \Bigl( r_{1}^- r_{2}^+ a \exp\bigl[ i\delta \bigr] \Bigr)^2 \Bigr]\\
\end{aligned}
\end{equation*}

which can be written

$\label{reflection-sum} E_r = E_i \Bigl[ r_{1}^+ + t_{1}^+ r_{2}^+ t_{1}^- a \exp\bigl[ i\delta \bigr] \sum_{m=0}^\infty \Bigl( r_{1}^- r_{2}^+ a \exp\bigl[ i\delta \bigr] \Bigr)^m \Bigr]$

where the term in (round) parentheses is a complex scaling factor for each additional round-trip the light makes within the resonator.
Except for the first term, the reflected field components form a geometric series [Wikipedia] whose sum converges when the magnitude of the parenthesis term is less than 1:

$\label{reflection-asymptotical} \frac{E_r}{E_i} = r_1^+ + \frac{t_{1}^+ t_{1}^- r_{2}^+ a \exp\bigl[ i\delta \bigr]}{1 – r_{1}^- r_{2}^+ a \exp\bigl[ i\delta \bigr]}$

In the preceding, real-valued $a$ is a per-round-trip amplitude factor due to absorption or scattering losses with $a=1$ meaning no losses at all and $a=0$ meaning total absorption within one round-trip. The angle $\delta$ is the round-trip propagation phase as discussed previously, $r_i$ is the (complex) reflection coefficient and $t_i$ the (complex) transmission coefficient at interface $i$. For both, the superscripts “+” and “−” indicate the direction from which the beam is incident on the interface, e.g. “+” meaning left-to-right in Fig. 3, as illustrated.

As shown in the Appendix, for nonabsorbing (reciprocal) interfaces with real-valued $n$ for all involved media we can find the following relations:

\begin{align}
n_+ t^- &= \vphantom{\Bigl(\Bigr)} n_- t^+ \label{Stokes-1} \\
\sqrt{R^+} &= \vphantom{\Bigl(\Bigr)} \sqrt{R^-\vphantom{R^+}} = \sqrt{R\vphantom{R^+}} \label{Stokes-2}\\
\sqrt{T^+ T^-} &= \vphantom{\Bigl(\Bigr)} 1 – R \label{Stokes-3} \\
n_- T^+ &= n_+ \Bigl( 1 – R \Bigr) \label{Stokes-4}\\
2\theta – \rho^+ – \rho^- &= \vphantom{\Bigl(\Bigr)} \pm \pi \label{Stokes-5}
\end{align}

where

\begin{split}
R^\pm = \bigl(r^\pm\bigr)^* r^\pm &\quad T^\pm = \bigl(t^\pm\bigr)^* t^\pm \vphantom{\Bigl(\Bigr)}\\
t^\pm = \sqrt{T^\pm} \exp \bigl[\, i \theta \,\bigr] &\quad r^\pm = \sqrt{R^\pm} \exp \bigl[ i \rho^\pm \bigr] \vphantom{\Bigl(\Bigr)}
\end{split}

and $n_-$ and $n_+$ are the refractive indices of the side on which the respective beam originates (e.g. if $t^+$ refers to a left-to-right beam then $n_+$ is the index of the medium on the left side of the interface and $n_-$ is the index on the right). The interface can be a simple interface between two dielectrics, for which $\theta = 0$ and $r$ is real-valued as described by the Fresnel equations [Wikipedia], or a multilayer (stratified) interface with which the reflectivity can be tailored as needed.

We can now rewrite $\eqref{reflection-asymptotical}$ using

$\psi \equiv \delta + \rho_1^- + \rho_2^+$

as

$\frac{E_r}{E_i} = \sqrt{R_1^+} \exp\Bigl[ i \rho_1^+ \Bigr] + \frac{a \sqrt{T_{1}^+ T_{1}^- R_{2}^+} \exp\Bigl[ i\bigl( 2\theta_1 + \psi – \rho_1^- \bigr) \Bigr]}{1 – a \sqrt{R_{1}^- R_{2}^+} \exp\Bigl[ i\psi \Bigr]}$

Apart from the round-trip phase $\psi$, the phase relations are determined solely by the coefficients of the first interface. We can thus use the various relations stated above, starting with \eqref{Stokes-5} and \eqref{Stokes-2}, then applying \eqref{Stokes-3} and some algebraic manipulation to get

\begin{align}
\frac{E_r}{E_i} &= \biggl( \sqrt{R_1\vphantom{R_1^+}} – \frac{a \sqrt{T_{1}^+ T_{1}^- R_{2}} \exp\Bigl[ i \psi \Bigr]}{1 – a \sqrt{R_{1} R_{2}\vphantom{R_1^+}} \exp\Bigl[ i\psi \Bigr]} \biggr) \exp\Bigl[ i \rho_1^+ \Bigr] \notag\\
&= \frac{\sqrt{R_1\vphantom{R_1^+}} – a \sqrt{R_2\vphantom{R_1^+}} \exp\Bigl[ i \psi \Bigr]}{1 – a \sqrt{R_1 R_2\vphantom{R_1^+}} \exp\Bigl[ i \psi \Bigr]} \exp\Bigl[ i \rho_1^+ \Bigr] \label{reflection-coefficient}
\end{align}

as a general expression for the reflection coefficient of a Fabry-Pérot interferometer with dielectric materials solely in terms of the reflection and scattering coefficients and effective resonator phase.

For metallic reflection coatings equations \eqref{Stokes-1} through \eqref{Stokes-5} no longer hold, as such layers are absorbing and thus not reciprocal. Refer to [1] for a thorough discussion. As a result, the properties of such a Fabry-Pérot resonator can differ significantly from the one shown here, where e.g. the transmission maximum does not coincide with the reflection minimum. Therefore, resonators with interfaces coated in this way will not be considered here.

The intensity reflection coefficient, or reflectance, from the resonator is then given by

$\frac{I_r}{I_i} = \frac{\cfrac{c n_1 \epsilon_0}{2}\Bigl|E_r\Bigr|^2}{\cfrac{c n_1 \epsilon_0}{2}\Bigl|E_i\Bigr|^2} = \biggl|\frac{E_r}{E_i}\biggr|^2$

Transmission
From Fig. 3, the sequence of transmitted field components through the Fabry-Pérot resonator is

$E_t = E_{t1} + E_{t2} + E_{t3} + \ldots$

with

\begin{align*}
E_{t1} &= E_i \Bigl[ t_1^+ \sqrt{a} \, t_2^+ \exp\bigl[i\delta/2\bigr] \Bigr]\\
E_{t2} &= E_i \Bigl[ t_1^+ \sqrt{a} \, t_2^+ \exp\bigl[i\delta/2\bigr] \Bigl( r_2^+ r_1^- a \exp\bigl[i\delta\bigr] \Bigr) \Bigr]\\
E_{t3} &= E_i \Bigl[ t_1^+ \sqrt{a} \, t_2^+ \exp\bigl[i\delta/2\bigr] \Bigl( r_2^+ r_1^- a \exp\bigl[i\delta\bigr] \Bigr)^2 \Bigr]
\end{align*}

which can be written

\label{transmission-sum}
E_t = E_i \Bigl[\sqrt{a}\, t_1^+ t_2^+ \exp\bigl[i\delta/2\bigr] \sum_{m=0}^\infty \Bigl( r_1^- r_2^+ a \exp\bigl[i\delta\bigr] \Bigr)^m \Bigr]

where again the term in parentheses is the scaling factor for each round-trip. This also is a geometric series whose sum converges to

$\label{transmission-coefficient} \frac{E_t}{E_i} = \frac{\sqrt{a \, T_1^+ T_2^+} \exp\Bigl[i\bigl(\delta/2 + \theta_1 + \theta_2 \bigr)\Bigr]}{1 – a \sqrt{R_1 R_2\vphantom{R_1^+}} \exp\Bigl[ i\psi \Bigr]}$

where \eqref{Stokes-4} and \eqref{Stokes-1} can be used to relate the transmission coefficients to the corresponding reflection coefficients.

The intensity transmission coefficient, or transmittance, from the resonator is then given by

$\frac{I_t}{I_i} = \frac{n_3}{n_1} \biggl|\frac{E_t}{E_i}\biggr|^2$

Equations \eqref{reflection-coefficient} and \eqref{transmission-coefficient} are all that is needed to produce the graphs above.

Dynamic Response

The formulas above and the interactive plots give the stationary behavior of the resonator which is, according to \eqref{reflection-sum} and \eqref{transmission-sum}, obtained after an infinite number of round-trips. While a large number, certainly less than infinity, of round-trips may be sufficient to approach the stationary behavior, it is not obtained immediately after an external source is switched on. Instead, the step responses in terms of round-trips are obtained again from \eqref{reflection-sum} and \eqref{transmission-sum}, when the upper limit for $m$ is increased from 0 to infinity.

They are shown for the resonance case ($\psi = 0$) and assuming $n_1=n_3$ in the following figures.

refresh

Fig. 4: Evolution of the normalized reflected field (red) and intensity (blue) vs. number of round-trips for reflectivity and loss setting from above; phases are relative as in Fig. 2. Press the refresh button after changing values.

refresh

Fig. 5: Evolution of the normalized transmitted field (red) and intensity (blue) vs. number of round-trips for reflectivity and loss setting from above; phases are relative as in Fig. 2. Press the refresh button after changing values.

As is evident from the figures, it can take quite a while for the fields to approach their steady state. Note that the changes in $E_t$ occur at the middle of full round-trips.

The optical intensity in the cavity also increases gradually in resonance and approaches a stationary value at which the injected energy exactly balances the round-trip losses. The energy at any point in time after turning on the external source can be determined from the fields inside the cavity which in turn are obtained analogous to \eqref{reflection-sum} and \eqref{transmission-sum} from Fig. 3. This is done in the Appendix.

refresh

Fig. 6: Evolution of the normalized stored energy vs. number of round-trips for reflectivity and loss setting from above. Press the refresh button after changing values.

The asymptotic value of the normalized energy at resonance is

$\label{asymptotic-energy} \frac{W(t)}{W_i}\biggr|_{t \to \infty} = \frac{1}{2} \frac{\bigl( 1 – R_1 \bigr)\bigl( 1 + a R_2 \bigr)}{1 + a^2 R_1 R_2 – 2\,a \sqrt{R_1 R_2}}$

where $W_i$ is the incident energy per round-trip time. This normalization is also used in the figure above.

For each photon within the cavity there is a certain probability per round-trip that the photon will exit the cavity through the mirror or be absorbed. This probability is derived from the fractional energy loss per round trip, which is

\begin{aligned} \label{success-probability} \mathcal{P}\bigl[\text{photon is lost}\bigr] &= 1 – \Bigl( r_1 r_2 a \exp\bigl[i\delta\bigr] \Bigr)^* \Bigl( r_2 r_1 a \exp\bigl[i\delta\bigr] \Bigr) \\ &= 1 – R_1 R_2 \, a^2 \end{aligned}

where the term in parentheses is again the field scaling factor for each round-trip from \eqref{basic-reflected}. The process of the photon making its round-trips is then analogous to Bernoulli trials [Wikipedia] where success is defined as the loss of the photon from the cavity. The probability distribution of the number of round trips the photon makes inside the cavity before being lost is then the geometric distribution [Wikipedia] with success probability $\mathcal{P}$ from \eqref{success-probability}. On average, the photon is thus lost in round-trip

$\mathcal{E} \bigl[n\bigr] = \frac{1}{\mathcal{P}} = \frac{1}{1 – R_1 R_2 \, a^2}$

The lifetime distribution for the above parameters looks like this:

refresh

Fig. 7: PMF (red) and CDF (blue) of the number of round-trips after which a photon is lost from inside the cavity. Press the refresh button after changing values.

Alternatively, we could model the photon losses as distributed decay by writing the (energy) scaling factor per round-trip as an exponential so that the number of photons left after $n$ round-trips is

$N_n = N_0 \bigl( R_1 R_2 \, a^2 \bigr)^n \equiv N_0 \exp \bigl[-\tau_p \, n\bigr]$

with a lifetime parameter $\tau_p$ in terms of round-trips and an initial number $N_0$ of photons in the cavity assuming no further injection of light. We quickly find that

$\tau = -\ln \bigl( R_1 R_2 \, a^2 \bigr) = -\ln \bigl( 1 – \mathcal{P} \bigr)$

with the average for the resulting exponential distribution [Wikipedia] of

$\mathcal{E}\bigl[n\bigr] = \frac{1}{\tau_p}$

which is usually slightly smaller than that for the geometric distribution due to the latter’s discrete nature. However, for e.g. laser resonators which have low mirror losses and distributed absorption within the cavity the exponential description is more accurate.

Finesse
The finesse of the resonator is defined as [3]

F = \cfrac{\pi}{2 \sin^{-1} \Biggl[ \cfrac{1-\sqrt{R_1 R_2 \, a^2}}{2 \bigl(R_1 R_2 \, a^2\bigr)^{1/4}} \Biggr]}

and gives the ratio of the free spectral range (which corresponds to $2\pi$ in the interactive plots) to the FWHM bandwidth of the resonance.

#### Appendix

Stokes relations
A number of amplitude and phase relation between the reflected and transmitted components for a lossless beam splitter such as a more or less reflecting mirror can be derived from the principles of reciprocity and energy (intensity) conservation. For these relations to hold it does not matter if the mirror is a simple interface between two dielectrics or a multilayer / stratified structure. The approach was first used by G. G. Stokes in 1849; we will follow a slightly modified version as proposed in [2].

Fig. 8: Notation

For a reciprocal object, a beam of light passing through the object will take exactly the same path backwards if properly reversed, i.e. the wavefront and time-dependence are conjugated. Hence, for the lossless, non-magnetic interface between two media of refractive index $n_+$ and $n_-$ in Fig. 8, a beam incoming e.g. from the left is split so that

$\label{appendix-original-beam} E_0 = r_+ E_0 + t_+ E_0$

where the index “+” denotes the left-to-right direction and no assumptions are yet made about $r$ and $t$. If both outgoing beams are (conjugate) reversed and again split at the interface, they must reconstruct the original (conjugate) beam,

\begin{align}
\label{appendix-plus}
r_+ \bigl(r_+ E_0\bigr)^* + t_- \bigl( t_+ E_0 \bigr)^* &= E_0^* \quad \text {or} \notag\\
r_+ r_+^* + t_- t_+^* &= 1
\end{align}

and there must not be an outgoing beam on the other side of the interface since there was no beam incoming on that side at the beginning,

$\label{appendix-minus} t_+ r_+^* + r_- t_+^* = 0$

For normal incidence (and thus equal beam size on both sides), the optical intensities in \eqref{appendix-original-beam} must be equal on both sides of the equation if we are to uphold the conservation of energy,

$\label{appendix-intensities} n_+ r_+^* r_+ + n_- t_+^* t_+ = n_+$

Comparison of \eqref{appendix-intensities} and \eqref{appendix-plus} yields

$n_+ t_- = n_- t_+ \tag{4}$

Rewriting \eqref{appendix-minus} as

$r_- = -\frac{t_+}{t_+^*} r_+^*$

$\sqrt{R_+} = \sqrt{R_-} = \sqrt{R} \tag{5}\\$

where again $R_\pm = r_\pm^* r_\pm$ and $T_\pm = t_\pm^* t_\pm$. Since according to \eqref{Stokes-1} $t_+$ and $t_-$ are in phase, we obtain from \eqref{appendix-plus}

$\sqrt{T_+ T_-} = 1 – R \tag{6}$

From \eqref{appendix-intensities} we readily obtain with \eqref{Stokes-2}

$n_- T_+ = n_+ \Bigl( 1 – R \Bigr) \tag{7}\\$

Rewriting as before

$t^\pm = \sqrt{T_\pm} \exp \bigl[\, i \theta \,\bigr] \quad r_\pm = \sqrt{R_\pm} \exp \bigl[ i \rho_\pm \bigr] \notag$

in \eqref{appendix-minus} we obtain with \eqref{Stokes-2}

$\exp\bigl[i \theta – i \rho_+\bigr] = – \exp\bigl[i \rho_- – i \theta\bigr]$

or

$2\theta – \rho_+ – \rho_- = \pm \pi \tag{8}$

Cavity energy

The average energy in the cavity is given by the volume integral

$W = \intop_V \frac{n_2^2 \epsilon_0}{2} \bigl|E\bigr|^2 dV$

where $E$ is the sum of a forward-propagating field $E_+$ and a backward-propagating field $E_-$. From Fig. 3 we obtain analogously to \eqref{reflection-sum} and \eqref{transmission-sum} the fields $E_+$ after $M$ round-trips and $E_-$ after $M+1/2$ round-trips as

\begin{align}
E_+\bigl(M \tau_{rt}\bigr) &= E_i t_{1}^+ \sum_{m=0}^M \Bigl( r_{1}^- r_{2}^+ a \exp\bigl[ i\delta \bigr] \Bigr)^m \\
E_t\Bigl(M \tau_{rt} + \frac{\tau_{rt}}{2}\Bigr) &= E_i \sqrt{a}\, t_1^+ r_2^+\exp\bigl[i\delta/2\bigr] \sum_{m=0}^M \Bigl( r_1^- r_2^+ a \exp\bigl[i\delta\bigr] \Bigr)^m \quad
\end{align}

where the lumped loss factor $\sqrt{a}$ is accounted for after each half round-trip. The equations change slightly when loss is lumped after each full round-trip or distributed over the cavity. In the lumped loss case the fields at arbitrary times can be obtained by simple interpolation, in the distributed case there appears an exponential dependence.

We normalize the energy to the external energy $W_i$ injected into the system per round-trip time $\tau_{rt}$,

$W_i = \tau_{rt} \intop_A I_i dA = \frac{c n_1\epsilon_0 \tau_{rt}}{2} \intop_A \bigl|E_i\bigr|^2 dA = n_1 n_2 \epsilon_0 L \intop_A \bigl|E_i\bigr|^2 dA$

where $A = V/L$ is the cross-section of the cavity of length $L$ and we used

$\frac{c}{n_1}\tau_{rt} = 2 L \frac{n_2}{n_1}$

as the propagation distance of the incident beam within the round-trip time $\tau_{rt}$. Assuming the transversal dependence of the intensity to be equal for all fields, the area integrals cancel and we obtain for the normalized cavity energy

\begin{align}
\frac{W(t)}{W_i} = \frac{n_2}{2 n_1} \frac{\bigl|E_+(t)\bigr|^2 + \bigl|E_-(t)\bigr|^2}{\bigl|E_i\bigr|^2}
\end{align}

For $t \to \infty$ we can use the known relations for the geometric series and use \eqref{Stokes-4} so that all $n$ cancel, then expand the denominator and use $\delta + \rho_2^+ + \rho_1^- = 0$ as the resonance condition to obtain \eqref{asymptotic-energy}.

#### References

[1] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed., Cambridge University Press, 1999.
[2] Masud Mansuripur, Classical Optics and its Applications, 2nd ed., Cambridge University Press, 2009.
[3] Finesse [RP Photonics Encyclopedia]

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