average photon lifetime: round-trips

resonator finesse:

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.

**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.

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.

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:

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.

**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].

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}.

[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]

A frequency- (or phase-)modulated carrier signal may be written as

\begin{equation}

E(t) = \frac{1}{2}\Bigl(A_{0}\exp\Bigl[i\omega_0 t + i \varphi(t) \Bigr] + \mathrm{c.c.}\Bigr) \tag{1}

\end{equation}

where $\omega_0$ is the frequency of the modulated carrier and $A_0$ is the carrier amplitude which can be complex to account for an arbitrary initial carrier phase. Then $\varphi(t)$ describes the modulation of the phase, and its derivative $d_t \varphi(t)$ describes the frequency modulation (since frequency is just the derivative of phase). The “c.c.” stands for the complex conjugate of the preceding term and ensures that $E(t)$ is always real-valued.

A carrier frequency other than zero will simply shift the modulated signal within the spectrum; therefore, we can assume $\omega_0 = 0$ without losing much in terms of generality (we will discuss a related point about detection later). Also, we set $A_0 = 1$ for simplicity and disregard the “c.c.” term for now (remembering that it is there, though). Then we are left with the complex phasor

\begin{equation}

\underline{E}(t) = \exp\Bigl[i \varphi(t) \Bigr] = \exp\Bigl[i \intop_0^t \omega(\tau) d\tau \Bigr] \tag{2}

\label{basic-pm}\end{equation}

which describes the (same) signal in terms of a phase and (angular) frequency modulation, respectively. It seems reasonable to first look at a simple harmonic modulation signal such as

\begin{equation}

\varphi(t) = \Delta\varphi \cos \bigl(\Omega t + \Phi\bigr) \tag{3}

\label{phase-function}\end{equation}

with amplitude $\Delta\varphi$, repetition frequency $\Omega$ and initial phase $\Phi$. The corresponding (angular) frequency modulation is

\begin{equation}

d_t\varphi(t) = \omega(t) = - \Omega \Delta\varphi \sin \bigl(\Omega t + \Phi\bigr) = - \Delta\omega \sin \bigl(\Omega t + \Phi\bigr) \tag{4}

\label{frequency-function}\end{equation}

with amplitude $\Delta\omega$. Fortunately, the exponential of a cosine can be expanded into the following series

\begin{equation}

\exp \bigl( i z \cos \theta \bigr) = \sum_{n=-\infty}^{\infty} i^n\, J_n(z)\, \exp \bigl(i n \theta \bigr) \tag{5}

\label{jacobi-anger}\end{equation}

It is called the Jacobi-Anger expansion [1] and the $J_n$ are the $n$th Bessel functions of the first kind.

Combining (2), (3) and (5) we get

\begin{align}

\underline{E}(t) &= \exp\Bigl[i \Delta\varphi \cos \bigl(\Omega t + \Phi\bigr) \Bigr]\\

&= \exp\Bigl[-i \Delta\omega \intop_0^t \sin \bigl(\Omega \tau + \Phi\bigr) d\tau \Bigr]\\

&= \sum_{n=-\infty}^{\infty} J_n(\Delta\varphi)\, \exp \bigl(i n \pi + \Phi \bigr) \, \exp \bigl(i \Omega t \bigr) \tag{6}

\end{align}

with $i^n = \exp(i n \pi)$. The Bessel coefficient in the sum gives the magnitude of each line in the spectrum, the second factor gives its phase, and the third factor determines at which frequency that line appears in the spectrum. Hence the lines are neatly arranged at multiples of the object oscillation frequency.

It appears that the PM and FM spectrum theoretically comprises an infinite number of spectral lines, though the Bessel function becomes very small around $n \approx \Delta\varphi$. In fact, Carson’s rule states that around 98% of the power is contained within a (two-sided) bandwidth of [2]

\begin{equation}

BW = 2 \bigl(\Delta\omega_\text{max} + \Omega_\text{max}\bigr) \tag{7}

\label{carsons-rule}\end{equation}

in which the two terms are the maximum frequency deviation and maximum oscillation frequency of the modulation signal. For modulation with a single cosine this reduces to

\begin{equation}

BW = 2 \Omega\bigl(\Delta\varphi + 1\bigr) \tag{8}

\end{equation}

and basically says to account only for the first $\Delta\varphi + 2$ lines (including the line at zero frequency) which coincides well with the above observation of Bessel coefficient magnitudes. However, Carson’s rule really applies only to modulation signals with a continuous spectrum, or at least one that consists of a multitude of lines. It is fortuitous that it also works for the cosine, and we’ll see below that it doesn’t work always.

An arbitrary (periodic) phase modulation can be Fourier-decomposed into a series of cosines and sines (or cosines with some initial phase), so that

\begin{align}

\underline{E}(t) &= \exp\Bigl[i \sum_{n=0}^\infty \Delta\varphi_n \cos \bigl(n \Omega t + \Phi_n) \Bigr] \\

&= \prod_{n=0}^\infty \exp\Bigl[i \Delta\varphi_n \cos \bigl(n \Omega t + \Phi_n) \Bigr]\tag{9}

\end{align}

The Jacobi-Anger expansion can then be applied to each factor in the product separately. The spectrum of the phase-modulated signal is then obtained by convolution of the spectra of all the factors. Thus, the compound spectrum may be quite densely populated over a large bandwidth when there are many such factors. Here is an example:

The values for the interactive spectrum plot at the top of this post are given in interferometric units, for we assume that the phase (or frequency) modulation is generated by a moving mirror in a simple interferometer, e.g. a Michelson interferometer. Our derivation of the spectrum, however, is done in very general terms of the modulated phase $\varphi(t)$ or the modulated frequency $\omega(t)$. Both types of units are related as

\begin{align}

\varphi(t) &= 2 k z(t) = \frac{4 \pi}{\lambda} z(t) \\

\omega(t) &= d_t \varphi(t) = \frac{4 \pi}{\lambda} d_t z(t) = \frac{4 \pi}{\lambda} v(t)\tag{10}

\end{align}

where $z(t)$ is the (relative) position of the mirror, $v(t)$ is its velocity, $k$ is the wavenumber and $\lambda$ is the wavelength of light used in the interferometer. The additional factor of 2 in the equations results from the change in optical path length being twice the change in mirror position, as the light beam within the interferometer must travel the distance $z(t)$ once towards the mirror and then again coming back after being reflected.

At the beginning we assumed the carrier frequency $\omega_0 = 0$ to simplify the analysis. And looking at the symmetric spectra in Figures 1–3 we might think the PM/FM signals $\underline{E}(t)$ are real-valued and could be detected with e.g. a simple photodiode. Looking at (6) we can see that this is not the case. While the magnitudes of the spectrum of negative frequencies are indeed equal to those of the respective positive frequencies because of

\begin{equation}

J_{-n}(x) = (-1)^n J_n(x)

\end{equation}

for integer $n$, the phases do not correspond to complex conjugate pairs. Hence, to properly detect such a complex signal requires additional hardware [3]. Alternatively, to properly detect a PM/FM signal with just a photodiode, we could use a carrier with $\omega_0 \gt 0$. However, the frequency shift $\omega_0$ would have to be large enough so that the whole two-sided modulation spectrum of $\underline{E}(t)$ would no longer generate any significant spectral components at negative frequencies. If $\omega_0$ were too small, the spectra of the two terms in (1) would overlap, as shown below, causing distortions in the detected position / velocity.

While PM or FM is somewhat resilient to interference due to its spread-spectrum nature, it is not immune.

[1] Jacobi-Anger expansion [Wikipedia]

[2] Carson’s rule [Wikipedia]

[3] Homodyne versus Heterodyne

bokeh (disk) size: 0mm

relative bokeh size: 0% (as fraction of sensor height)

`Focal length (FF)`

is the focal length of a full frame camera with the same field of view.

The disks in Figure 1 are essentially the images of point sources located a long distance away and are used here to illustrate the bokeh effect. In a “regular” photograph, each point in the background of the scene is convolved with the shape of the bokeh disk and superposed with the disks from all other background points, just like the overlapping disks in Figure 1. The result is a more or less blurred background which nicely isolates an in-focus foreground. The larger the bokeh disk, the more pronounced the blur effect and foreground isolation will be. For lots of great examples, check out flickr’s *Bokeh - Smooth & Silky* picture pool.

Calculating the expected size of bokeh disks – and thus the strength of the bokeh effect – is not at all that hard. The only formula from optics that we need is the *thin lens formula*

$$\frac{1}{f} = \frac{1}{a} + \frac{1}{b} \tag{1}$$

where $f$ is the focal length of the lens, $a$ is the object distance, and $b$ is the image distance. The rest is just basic trigonometry. We will assume a thin lens throughout to make our life easier. However, the results then will only hold for $a \gg f$, since otherwise the geometry of the lens becomes relevant. Therefore, macro photography is not covered by this post.

The above quantities and a few more are shown in Figure 2. Here, $\Delta b$ is the difference in image distance between an object at distance $a$ and one at infinity, as can be determined using (1) – for the object at infinity, $b = f$. Furthermore, $A$ is the (absolute) aperture size of the lens, as related to its f-number $N_f$ by $A = f / N_f$, $w$ is the size of the sensor, and $d$ is the diameter of the bokeh disk on the sensor.

Now from Figure 2 we have (red ray)

$$d = \Delta b \frac{A}{f} = \frac{\Delta b}{N_f}\tag{2}$$

with

$$\Delta b = \frac{f^2}{a-f}\tag{3}$$

which can be obtained from (1) by setting $b = f + \Delta b$. Remember that $a \gg f$, so that actually $\Delta b \ll f$, unlike shown in the figure. Combining (2) and (3) we get

$$d = A \frac{f}{a-f} \tag{4}$$

which for $a \gg f$ can be simplified to

$$d \approx A \frac{f}{a} = \frac{f^2}{a N_f}\tag{5}$$

So, a large aperture $A$ (small f-number $N_f$) will produce large bokeh disks, as well as does a large focal length. At a constant aperture, the bokeh disk is proportional in size to the focal length; at a constant f-number, the bokeh disk sizes increases with the square of focal length. Also, the closer the object is to the lens the more blurry the bokeh will be.

What happens when the aperture becomes smaller because the lens is “stopped down?” Figure 3 shows.

Comparing with Figure 2, we can easily see how the bokeh disk becomes smaller because the angle of the outermost (red) rays becomes less acute as they pass through the focal point.

For great bokeh, your aperture cannot be large enough, i.e. your lens f-number cannot be small enough.

If you’ve ever compared a photo shot with a full-frame camera to one shot with a phone camera, you already know the result of this section. But we’ll calculate it anyway.

To image the same field of view (same object size for a given distance), the focal length of a camera with a smaller sensor must also be smaller, as shown in Figure 4.

The field of view $\theta$ is given by

$$ \tan \frac{\theta}{2} = \frac{w}{2 f} \tag{6}$$

and for an equal field of view $\theta$ we have $w \propto f$. Hence, according to (5), $d \propto w^2$ for an equal f-number. The corresponding ray trajectories are shown in Figure 5.

However, since the sensor is smaller, the image of the bokeh disk takes up a larger part of the whole image, and to be fair we should normalize the disk size $d$ by the sensor size $w$ in order to give the size of the bokeh disk as fraction of the whole image (which is what you will eventually see when looking at the picture). Then,

$$\frac{d}{w} \propto \frac{w}{a N_f} \tag{7}$$

Thus, the bokeh size as fraction of the whole image for a full-frame sensor will be about eight times as large as that for a 1/3.2″ sensor, as can be found in e.g. the iPhone 5, at the same f-number and field of view. More generally, the bokeh fraction scales inversely with the “crop factor” for the various sensor sizes, with APS-C having about 2/3 that of full-frame, Four-Thirds having about 3/4 that of APS-C, and so on.

If you simply use a lens with some focal length $f$ and f-number $N_f$ on a camera with a smaller sensor, you will get bokeh disks which are even larger than those on the large sensor (multiplied by the crop factor), but your field of view $\theta’$ becomes smaller, as also shown in Figure 4, and you might not image the object completely.

For great bokeh, your sensor cannot be large enough.

When the object is to be imaged at a certain size on the sensor, we have to increase the object distance $a$ when we increase the focal length $f$. How does this affect the bokeh?

For an object of extent $h$ from the optical axis, we need a observation distance $a$ that depends on the field of view $\theta$ as

$$\tan \frac{\theta}{2} = \frac{h}{a} = \frac{w}{2f}$$

and thus

$$a = \frac{2f h}{w}$$

where we also used (6). Then with (5),

$$d\bigr|_{h = \mathrm{const}} = \frac{f w}{2h N_f} = A \frac{w}{2h} \tag{8}$$

If you have a zoom lens that has a constant aperture $A$ throughout the focal length range, the bokeh disk size will not change if you adjust the object distance so as to have a constant object size in the image. At a fixed f-number, however, the bokeh disk size increases with focal length, though only linearly instead of quadratically as in (5).

There you have it. For a given field-of-view to be imaged, nothing beats a camera with a large sensor and a lens with a small f-number, both usually not cheap. Whether the bokeh quality is “good” or “bad” is a whole different story, though.

In principle, similar calculations can be used to determine the depth-of-field which is basically those distances $\Delta a_+$ and $\Delta a_-$ (instead of the infinite distance used here) at which the size of the bokeh disk has some defined value such that an imaged point is just perceived as sharp. The bokeh disk in this context is called circle of confusion. See also the Wikipedia article on depth of field or Paul van Walree’s article on the depth of field [toothwalker.org].

]]>Since no project ever goes according to plan, it’s sensible to plan ahead by inserting buffers which can eat up some of the delay. Buffers shouldn’t be added to every task, but at the end of project phases or before milestones. A buffer of 15% of the phase duration was recommended in [1]. A buffer can be added by

- inserting a dedicated buffer task between project phases, or
- using positive
`Lag`

between tasks (phases) that are dependent on another (see below), or - creating a
*manually scheduled*summary task for the project phase to which the buffer is added. In this case MS Project separately keeps track of the subtasks’ duration whose progress bar turns red when it exceeds the summary task’s set duration.

Don’t calculate with 100% productive-towards-the-project resources (people). Nobody works without distractions. This can be accounted for by

- setting
`Max Units`

of the resources to less than 100% (`Resource`

sheet,`Max Units`

cell), or - using shorter workdays in the project (
`Project Options`

→`Schedule`

)

The first option is more flexible and more visible.

If you also schedule equipment resources, you can probably leave `Max Units`

at 100%. Don’t use the second option in this case.

If you have documents describing a task / work package in detail, link the tasks in Project to these documents by choosing `Hyperlink`

in the context menu of relevant tasks.

When you make changes to a task’s duration, resource units or work, Project will calculate the other two quantities automatically according to these rules:

Task Type | User Changes | Program Adjusts |
---|---|---|

Fixed Units | Duration | Work, based on Units |

Units | Duration, based on Work | |

Work | Duration, based on Units | |

Fixed Duration | Duration | Work, based on Units |

Units | Work, based on Duration | |

Work | Units, based on Duration | |

Fixed Work | Duration | Units, based on Work |

Units | Duration, based on Work | |

Work | Duration, based on Units |

It’s a good idea to change the task type only while you are making changes to a task, and afterwards resetting it to the standard `Fixed Units`

type, in order to avoid unexpected behavior and confusion in the future.

This may sound counter-intuitive, but material resources are consumables that are planned by quantity, not by availability. If you have two pieces of some particular equipment, set `Max Units`

to 200%, and analogous for more. Equipment resources likely don’t need slack adjustment to their `Max Units`

(see above).

When you use `Automatic Scheduling`

(and you should), make use of the different task dependency options. These are

- Finish-to-Start (FS):
`Successor`

task starts when`Predecessor`

task has finished. Most common option. Positive`Lag`

creates a delay between tasks, negative`Lag`

leads to overlap. - Start-to-Start (SS):
`Successor`

task starts when`Predecessor`

task starts. Positive`Lag`

delays the start time of the`Successor`

, negative`Lag`

advances it. - Finish-to-Finish (FF):
`Successor`

task finishes when`Predecessor`

has finished. Positive`Lag`

delays the finish time of the`Successor`

, negative`Lag`

advances it. - Start-to-Finish (SF):
`Successor`

task finishes when`Predecessor`

task starts. Positive`Lag`

leads to overlap, negative`Lag`

creates a delay between tasks.

`Successor`

and `Predecessor`

merely describe the dependency (which task controls the other), not the temporal order of the tasks, which can certainly cause confusion in the SF case.

`Lag`

can also be entered as a percentage of the duration of `Predecessor`

, which can be useful for creating buffers (see above).

Summary tasks are a helpful visual aid when the project is divided into different phases, each with its own set of tasks. With a single click all subtasks can be hidden from view when a particular phase is no longer relevant or when the structure of subtasks is of no current interest, e.g. individual acquisition tasks before the start of assembly.

Also, changing a summary task to `Manually Scheduled`

in Project adds a duration indicator for the contained subtasks as part of the task bar. Should the duration of the subtasks exceed the duration of the summary task, e.g. because of unforeseen delays, this duration indicator turns red, warning about the problem. Manual scheduling also allows for inserting some buffer time as described above by making following tasks depend on the finish of the summary task instead of a particular subtask.

If you find yourself using similar views over and over again, save your setting as custom view(s) and make it available to your other projects.

To show e.g. a Gantt / Entry Table with custom columns and bar styles, and with the Task Details Form in a second pane below the chart, follow these steps:

- In the
`View`

ribbon, select the`Gantt Chart`

view. - In the
`View`

ribbon, select`Data`

→`Tables`

→`Entry`

(or your favorite table) and use the`Format`

ribbon’s`Columns`

section to format the table to your liking. Often, less (columns) is more and you can also get e.g.`Start`

and`Finish`

data by hovering the mouse over the task bars in the Gantt chart or in the`Task Details Form`

that we’ll setup later. - In the
`Format`

ribbon use the`Bar Styles`

section to format the task bars to your liking. - Go to
`File`

→`Info`

→`Organizer`

, select the`Tables`

tab, select the appropriate table on the right side and click`<< Copy`

. In the pop-up warning box, click`Rename...`

and give the table a new name in the global template (do not first rename and then copy). Do the same with the view in the`Views`

tab of the`Organizer`

(as the table to use is saved in the custom view).

- In any of the drop-downs in the
`View`

ribbon,`Task Views`

section, select`More Views...`

- In the dialog box, click on
`New...`

and select`Combination View`

.

- Give the view a new name and select your custom view for the
`Primary View`

and e.g. the`Task Details Form`

for the`Details Pane`

.

- Go back to the
`Organizer`

and copy your combination view to the global template to make it available in your other projects.

[1] Bonnie Biafore, *Microsoft Project 2010: The Missing Manual*, O’Reilly Media, 2010.

Enter a number into any field to convert it to the other bases.

[1] Humidity [Wikipedia]

]]>The previous post on the Gaussian beam noted that this beam is a solution to the paraxial wave equation and thus only accurate for small divergence angles. The limit given in [1] was $w_0 \gt 2\lambda/\pi$ or $\theta \lt 0.5\mathrm{rad}$ or $\approx 29^\circ$. The Gaussian beam model is of course not exact up to that limit angle and then suddenly becomes inaccurate. There actually is always a small error which grows as the divergence angle increases.

To get a notion of how accurate the approximation is for various beam waists, the Fourier diffraction propagation method as explained in this post can help, unlike the Fresnel or Fraunhofer propagation methods which were shown in this post and which also make the paraxial approximation are thus not of great use here.

Using Fourier propagation on a Gaussian field distribution with $w_0 = \lambda$ in the source plane, we can compare the exact field and the Gaussian beam model at any propagation distance. This is the result at $10\lambda$ from the waist position:

The Gaussian and exact fields differ significantly away from the optical axis. The difference between the exact and Gaussian fields is plotted below for various beam waist to wavelength ratios. Refer to this post to calculate any other beam parameters for these cases.

The amplitude error is normalized to the exact on-axis field. This way the errors at different distances remain comparable without giving undue significance to the weak fields far away from the axis. The phase difference is not normalized. All plots show a transverse cross section of the beam up to a radius of $2 w(z)$ as computed from the Gaussian beam model.

Click on the buttons to see the results for the respective beam:

Clearly, the validity limit $w_0 = 2\lambda/\pi$ for the Gaussian beam model already has a significant phase error in the far field, which could be relevant for mode coupling calculations. This should be kept in mind before blindly trusting such calculations.

[1] Gaussian beam [Wikipedia]

]]>Below the graphic, various plots will give the beam cross-section (amplitude, intensity and phase) and the evolution of beam width and wavefront curvature near the waist position $z=0$. At the bottom of the page, the various formulas used for the calculations here are given.

Also note that the Gaussian beam is a solution of the *paraxial Helmholtz (wave) equation*, i.e. it is only a good approximation as long as the divergence angle is not too large. This is the case when approximately $w_0 k > 4$ or $w_0 > 2\lambda / \pi$.

Check out the entry on Paraxiality for more information.

The complex electric field amplitude is given by [1]

\begin{equation}

E(\mathbf{r}, z) = E_0 \frac{w_0}{w(z)} \exp \left( -\frac{\mathbf{r} \cdot \mathbf{r}}{w^2(z)} -ikz -ik \frac{\mathbf{r} \cdot \mathbf{r}}{2R(z)} +i \zeta(z) \right)

\end{equation}

with transverse coordinate $\mathbf{r} = (x,y)^T$ and longitudinal coordinate $z$, defined such that the location of minimal beam width is at $z=0$. The quantity $k = 2\pi / \lambda$ is called the wavenumber.

The (time-averaged) intensity of the beam is then given by

\begin{equation}

I(\mathbf{r},z) = \frac{ E(\mathbf{r},z) E^*(\mathbf{r},z)}{2 \eta} = I_0 \left( \frac{w_0}{w(z)} \right)^2 \exp \left( -\frac{2 \mathbf{r} \cdot \mathbf{r}}{w^2(z)} \right)

\end{equation}

where $*$ denotes the complex conjugate, and $\eta$ is the characteristic impedance of the medium. The initial amplitude and intensity are

\begin{equation*}

E_0 = E(x=0,y=0,z=0) \quad \text{and} \quad I_0 = I(x=0,y=0,z=0)

\end{equation*}

respectively. The $1/e$ beam width is given by

\begin{equation}\label{beam-width}

w(z) = w_0 \, \sqrt{ 1+ {\biggl( \frac{z}{z_R} \biggr)}^2 } = w_0 \, \sqrt{ 1+ {\biggl( \frac{2 z}{k w_0^2} \biggr)}^2 }

\end{equation}

where

\begin{equation}\label{rayleigh-range}

z_R = \frac{\pi w_0^2}{\lambda} = \frac{k w_0^2}{2}

\end{equation}

is called the Rayleigh range and $w_0 = w(0)$ is the minimum beam width, or beam waist. At the Rayleigh range, $w(\pm z_R) = \sqrt{2} w_0$.

The wavefront curvature $R(z)$ is given by

\begin{equation}\label{curvature}

R(z) = z \left[{ 1+ {\biggl( \frac{z_R}{z} \biggr)}^2 } \right] = z \left[{ 1+ {\biggl( \frac{k w_0^2}{2 z} \biggr)}^2 } \right]

\end{equation}

At the waist, the wavefronts are plane, $R(0) = \infty$; the curvature radius decreases to a minimum at the Rayleigh range, and far from $z_R$ increases again as $R(z) \approx z$.

The confocal parameter, or depth of focus, $b$ is the distance between the Rayleigh ranges on both sides of the waist,

\begin{equation}

b = 2 z_R = k w_0^2

\end{equation}

The longitudinal phase delay $\zeta(z)$, or Guoy phase, is

\begin{equation}\label{guoy}

\zeta(z) = \mathrm{arctan} \left( \frac{z}{z_R} \right) = \mathrm{arctan} \left( \frac{2 z}{k w_0^2} \right)

\end{equation}

which imposes an additional phase shift of π on the beam as it passes through the focus.

The far-field divergence angle $\theta$ (in radians) of the beam is given by

\begin{equation}\label{divergence-angle}

\theta \simeq \frac{\lambda}{\pi w_0} = \frac{2}{k w_0} = \frac{w_0}{z_R}

\end{equation}

and the numerical aperture is

\begin{equation}

\mathrm{NA} = \sin \left(\theta \right)

\end{equation}

At the waist, the Gaussian beam is a plane wave with a Gaussian apodization, while in the far-field it is a spherical wave, also with Gaussian apodization, making it cone-shaped.

Interestingly, all spatial quantities in the above formulas scale with wavelength, even if it’s not immediately apparent. Hence, we can normalize the equations describing the properties of the Gaussian beam to the wavelength of the radiation to obtain the canonical, or standard, or normalized forms. Starting with the normalized quantities

\begin{equation*}

z’ = 2\pi \frac{z}{\lambda} = k \cdot z \quad \mathbf{r}’ = k \cdot \mathbf{r} \quad w’_0 = k \cdot w_0

\end{equation*}

we obtain from $\eqref{rayleigh-range}$

\begin{equation*}

z’_R = k \cdot z_R = \frac{k^2 w_0^2}{2} = \frac{{w^\prime_0}^2}{2}

\end{equation*}

and from $\eqref{beam-width}$ and $\eqref{curvature}$

\begin{gather*}

w’(z’) = k \cdot w\biggl(z = \frac{z’}{k}\biggr) = w’_0 \, \sqrt{ 1+ {\biggl( \frac{2 z’}{{w’_0}^2} \biggr)}^2 } \\

R’(z’) = k \cdot R\biggl(z = \frac{z’}{k}\biggr) = z’ \left[{ 1+ {\biggl( \frac{{w’_0}^2}{2 z’} \biggr)}^2 } \right]\\

\end{gather*}

Also, from $\eqref{divergence-angle}$ and $\eqref{guoy}$

\begin{gather*}

\theta \simeq \frac{2}{w’_0} = \frac{w’_0}{z’_R}\\

\zeta’(z’) = \zeta\biggl(z = \frac{z’}{k}\biggr) = \mathrm{arctan} \left( \frac{2 z’}{{w’_0}^2} \right)

\end{gather*}

Note that since $\zeta’$ is not a coordinate or length, it does not scale with $k$ – it is simply the Guoy phase written in terms of $z’$.

Then finally, the canonical form of the complex field amplitude is

\begin{equation*}

E(\mathbf{r’}, z’) = E_0 \frac{w’_0}{w’(z’)} \exp \left( -\frac{\mathbf{r’} \cdot \mathbf{r’}}{{w’}^2(z’)} -iz’ -i \frac{\mathbf{r’} \cdot \mathbf{r’}}{2R’(z’)} + i \zeta’(z’) \right)

\end{equation*}

and analogously the beam intensity.

Also check out post #145 which takes a closer look at the difference between the Gaussian beam model and the real beam resulting from diffraction.

[1] Gaussian beam [Wikipedia]

]]>Since the plugin code is quite long, I will just post the few lines that need to be patched / added to Eric’s original plugin. My additions are shown in red. They are based on v1.8.1 of the EditSectionPlugin.

In `config.macros.editSection.createPanel`

add

```
this.showPanel(p,here,ev);
jQuery(f).find("INPUT[value=cancel]").focus();
return this.ok(ev);
```

In `config.macros.editSection.initForm`

add

```
form.content.value=store.getTiddlerText(tid,'');
jQuery(form).keydown( function(eventObject) {
if (eventObject.which == 27) { // ESC
config.macros.editSection.cancel(jQuery(this).find("INPUT[value=cancel]").get(0),eventObject);
}
if (eventObject.which == 13 && eventObject.ctrlKey) { // CTRL-ENTER
config.macros.editSection.saveForm(jQuery(this).find("INPUT[value=save]").get(0),eventObject);
}
});
if (version.extensions.TextAreaPlugin) new window.TextAreaResizer(form.content);
```

In `.initForm`

a keyboard event handler is added to pop-up (form) that fires a function on every keypress, which listens specifically for ESC or CTRL+ENTER. When ESC is pressed, the “Cancel” button is passed to the `.cancel`

function as originating element. Same with the “Save” button for CTRL-ENTER. The new line in `.createpanel`

initially gives focus to the “Cancel” button (any form element would do), so that pressing e.g. ESC works without having a form element first.

The code could have been more concise with some changes to the `EditSectionTemplate`

, but that would require editing two tiddlers.