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]

Fig. 2: out­put phase [rad] vs. round-trip phase [rad]

aver­age pho­ton life­time: round-trips
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.

Illustration of the components constituting the reflected and transmitted fields in a Fabry-Pérot resonator.

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

\begin{equation}
\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]
\end{equation}

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.

Pho­ton Life­time

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]

\begin{equation}
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]}
\end{equation}

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].

Notation

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

\[
r_- = -\frac{t_+}{t_+^*} r_+^*
\]

leads to
\[
\sqrt{R_+} = \sqrt{R_-} = \sqrt{R} \tag{5}\\
\]

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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