theory

Tex/ffz.bib

@@ -1497,6 +1497,19 @@ @Book{van2001
   lccn      = {2001055348},
 }
 
+@Article{pollnau2020,
+  author   = {Markus Pollnau and Marc Eichhorn},
+  journal  = {Progress in Quantum Electronics},
+  title    = {Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation},
+  year     = {2020},
+  issn     = {0079-6727},
+  pages    = {100255},
+  volume   = {72},
+  abstract = {The degree of spectral coherence characterizes the spectral purity of light. It can be equivalently expressed in the time domain by the decay time τ or the quality factor Q of the light-emitting oscillator, the coherence time τ coh or length ℓcoh of emitted light or, via Fourier transformation to the frequency domain, the linewidth Δν of emitted light. We quantify these parameters for the reference situation of a passive Fabry-Pérot resonator. We investigate its spectral line shapes, mode profiles, and Airy distributions and verify that the sum of all mode profiles generates the corresponding Airy distribution. The Fabry-Pérot resonator is described, as an oscillator, by its Lorentzian linewidth and finesse and, as a scanning spectrometer, by its Airy linewidth and finesse. Furthermore, stimulated and spontaneous emission are analyzed semi-classically by employing Maxwell′s equations and the law of energy conservation. Investigation of emission by atoms inside a Fabry-Pérot resonator, the Lorentz oscillator model, the Kramers-Kronig relations, the amplitude-phase diagram, and the summation of quantized electric fields consistently suggests that stimulated and spontaneous emission of light occur with a phase 90° in lead of the incident field. These findings question the quantum-optical picture, which proposed, firstly, that stimulated emission occurred in phase, whereas spontaneous emission occurred at an arbitrary phase angle with respect to the incident field and, secondly, that the laser linewidth were due to amplitude and phase fluctuations induced by spontaneous emission. We emphasize that the first derivation of the Schawlow-Townes laser linewidth was entirely semi-classical but included the four approximations that (i) it is a truly continuous-wave (cw) laser, (ii) it is an ideal four-level laser, (iii) its resonator exhibits no intrinsic losses, and (iv) one photon is coupled spontaneously into the lasing mode per photon-decay time τc of the resonator, independent of the pump rate. After discussing the inconsistencies of existing semi-classical and quantum-optical descriptions of the laser linewidth, we introduce the spectral-coherence factor, which quantifies spectral coherence in an active compared to its underlying passive mode, and derive semi-classically, based on the principle that the gain elongates the photon-decay time and narrows the linewidth, the fundamental linewidth of a single lasing mode. This linewidth is valid for lasers with an arbitrary energy-level system, operating below, at, or above threshold and in a cw or a transient lasing regime, with the gain being smaller, equal, or larger compared to the losses. By applying approximations (i)-(iv) we reproduce the original Schawlow-Townes equation. It provides the hitherto missing link between the description of the laser as an amplifier of spontaneous emission and the Schawlow-Townes equation. Spontaneous emission entails that in a cw lasing mode the gain is smaller than the losses. We verify that also in the quantum-optical approaches to the laser linewidth, based on the density-operator master equation, the gain is smaller than the losses. We conclude this work by presenting the derivation of the laser linewidth in a nut shell.},
+  doi      = {10.1016/j.pquantelec.2020.100255},
+  keywords = {Spectral coherence, Optical resonance, Fabry-Pérot resonator, Resonator linewidth, Finesse, Lasers, Laser theory, Laser resonators, Laser linewidth, Schawlow-Townes equation},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}
 
 @Comment{jabref-meta: grouping:

Tex/theory.tex

@@ -1,23 +1,11 @@
 \chapter{Theory}
-
 \label{chap:theory}
-
-%\cite{goodman2000,goodman2007,agarwal2013,classen2017,cowley1995,born1980,trigg2005,attwood1999,griffiths2005,agarwal2013,classen2017,loudon2000,mandel1995,hanbury1956,glauber2006,baym1997,zernike1938,rosen1996,yabashi2002,singer2013,santra2009,krause1979,trost2020,inoue2019,sorum1987,lajunen04,mpccd,tono2013}
-
-
-
+The main principle of IDI is to use intensity correlations of X-Ray fluorescence. To have a basic understanding of the underlying definitions necessary for both simulations and the experimental application, a (semi-) classical approach to coherence and the experiment of Hanbury-Brown Twiss suffices. 
 
 \paragraph{Stationarity and Ergodicity}
-A process $f(t)$ is wide-sense stationary, if the expectation value $E[f(t)]$ is independent of the time $t$ and $E[f(t_1)f(t_2)]$ depends only on the time difference $\tau=f_1-f_2$. A process, for which the time average and the ensemble average are equal is called ergodic. Stationarity is a necessity for ergodicity.
-
-
+We will first assume all considered fields to be both stationary and egodic. A process $f(t)$ is wide-sense stationary, if the expectation value $E[f(t)]$ is independent of the time $t$ and $E[f(t_1)f(t_2)]$ depends only on the time difference $\tau=f_1-f_2$. A process, for which the time average and the ensemble average are equal is called ergodic. Stationarity is a necessity for ergodicity. Deviations from this assumption will be considered in \fref{sec:specklecontrast}.
 
 \section{Basic Conecepts of Coherence}
-
-
-
-
-
 Spherical waves as solutions wave equation in spherical coordinates,
 \begin{equation}
 	\vec{E}(\vec{r},t)=\vec{E}_0(k,t) \frac{e^{i\vec{r}\vec{k}-iwt}}{R}
@@ -63,21 +51,29 @@ \section{Basic Conecepts of Coherence}
 \end{equation}
 and (again for equal intensities) the visibility is $V(\tau)=\left|\gamma(\tau)\right|$ \cite{zernike1938,loudon2000}.
 
-The \textit{coherence time }shall describe at which delay the visibility falls to approximately zero. One possible definition by Mandel based on the complex degree of coherence is
+The \textit{coherence time }shall describe at which delay the visibility falls to approximately zero. One possible definition by Mandel based on the complex degree of coherence,
 \begin{equation}
 \tau_c = \int_{-\infty}^{\infty} \left| \gamma(\tau)\right|^2 \diff \tau 
-\end{equation} and will be used in the following. Other definitions commonly used are equivalent up to a constant factor and lead generally to coherence times of the same order of magnitude \cite{mandel1959,goodman2000}.
+\end{equation} will be used in the following. Other definitions commonly used are equivalent up to a constant factor and lead generally to coherence times of the same order of magnitude \cite{mandel1959,goodman2000}. 
 
-For an exponentially decaying electric field $E(t)=\Theta(t)e^{-t/\tau}$, the spectrum is Lorentian with an angular frequency FWHM of $\frac{2}{\tau}$ as 
-\begin{equation*}
-\left|\int_{0}^{\infty}  e^{-t/\tau} e^{-iwt} \dif t \right|^2 \propto  \frac{1}{1/\tau^2+w^2} .
-\end{equation*}
-Therefore, an Lorentian spectrum with a FWHM of $\Delta E$ corresponds to an lifetime of $\frac{2\hbar}{\Delta E}$ and vice versa.
-The Wiener Khinchin theorem connects the spectrum $F(w)$ with $\gamma$
+Furthermore, the Wiener Khinchin theorem connects the power spectrum $F(w)$ with $\gamma$
 \begin{equation}
 	F(\omega)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \gamma(\tau) \exp (\mathrm{i} \omega \tau)  \dif \tau
-\end{equation}, leading to
-$<\gamma(\tau)>=e^{-|\tau/| \tau_c}$ and showing that the the coherence time  $\tau_c$ with the definition above is the time, after which $\gamma$ has decreased to $\sfrac{1}{e} $.
+\end{equation}.
+
+As a relevant example, for an exponentially decaying electric field envelope\footnote{Corresponding to an intensity/photon number decay with time constant $\sfrac{\tau}{2}$} $E(t)=\Theta(t)e^{-t/\tau_E}$, the power spectrum is Lorentzian with an angular frequency FWHM of $\Delta \omega=\frac{2}{\tau_E}$,
+\begin{equation*}
+\left|\int_{0}^{\infty}  e^{-t/\tau_E} e^{-iwt} \dif t \right|^2 \propto  \frac{1}{1/\tau_E^2+w^2} .
+\end{equation*}
+and
+\begin{align*}
+\left\langle E(t) E^{*}(t+\tau)\right\rangle
+&=\int_{0}^{\infty} E(t) E^{*}(t+\tau)\diff t
+=\frac{\tau_E}{2}   e^{-\frac{\left| \tau \right| }{\tau_E }}\\
+\left\langle E(t) E^{*}(t)\right\rangle
+&=\frac{\tau_E}{2} 
+\end{align*}
+Hence, $\left<\gamma(\tau)\right>=e^{-|\tau/| \tau_E}$,   and the coherence time as defined is above is $\tau_c=\int_{-\infty}^{\infty} \left| \gamma(\tau)\right|^2 \diff \tau=\tau_E=\sfrac{2\hbar}{\Delta E}$ \cite{goodman2000,pollnau2020}.
 
 
 
@@ -169,11 +165,9 @@ \section{Basic Conecepts of Coherence}
 
 
 \section{Hanbury Brown Twiss}
-Hanbury Brown and Twiss 
-
-The interpretation based on (semi-)classical coherence theory for thermal light sources \cite{baym1997,goodman2000}
+Hanbury-Brown Twiss first used the correlation between the intensities captures by photomultiplier tubes to measure the angular size of stars  \cite{hanbury1956}. The underlying effect can be explained based on (semi-)classical coherence theory for chaotic light sources \cite{baym1997,goodman2000}:
 
-If the sources $A$ and $B$ produce each spherical electromagnetic waves with random but constant phases $\phi_{A,B}$, $E_{A,B}=c_{A,B} e^{i k\left|\vec{r}-\vec{r}_{1,2}\right|+i \phi_{A,B}} /\left|\vec{r}-\vec{r}_{A,B}\right|$ and ignoring polarization, the total intensity at detector 1 $I_{1}$ (and similar $I_{2}$ at detector 2) is
+If the sources $A$ and $B$ (in the original experiment different parts of the observed star) produce each spherical electromagnetic waves with random but constant phases $\phi_{A,B}$, $E_{A,B}=c_{A,B} e^{i k\left|\vec{r}-\vec{r}_{1,2}\right|+i \phi_{A,B}} /\left|\vec{r}-\vec{r}_{A,B}\right|$ and ignoring polarization, the total intensity at detector 1 $I_{1}$ (and similar $I_{2}$ at detector 2) is
 \begin{equation}
 	I_{1} =
 	\frac{1}{L^{2}}\left(
@@ -184,7 +178,7 @@ \section{Hanbury Brown Twiss}
 	\right)
 \end{equation}
 with $r_{1 a}$ the distance from source $A$ to detector 1 etc. 
-Averaging over many independent realisations of the random phases leads to the average intensities in the two detectors,
+Averaging over many independent realizations of the random phases leads to the average intensities in the two detectors,
 \begin{equation}
 	\left\langle I_{1}\right\rangle=\left\langle I_{2}\right\rangle=\frac{1}{L^{2}}\left(\left\langle|c_A|^{2}\right\rangle+\left\langle|c_B|^{2}\right\rangle\right)
 \end{equation}, 
@@ -200,13 +194,13 @@ \section{Hanbury Brown Twiss}
 	=1+2 \frac{\left\langle|\alpha|^{2}\right\rangle\left\langle|\beta|^{2}\right\rangle}{\left(\left\langle|\alpha|^{2}\right\rangle+\left\langle|\beta|^{2}\right\rangle\right)^{2}} \cos \left(k\left(r_{1 a}-r_{2 a}-r_{1 b}+r_{2 b}\right)\right)
 \end{equation}
 
-For large separation between the sources and detectors $(L \gg R)$, the small angle approximation can be made and
+For a large enough separation between the sources and detectors $(L \gg R)$, the small angle approximation can be made and
 \begin{equation}
 	g^2\left(\vec{k_1}-\vec{k_2}\right)-1\propto \cos{\left(\vec{R} \cdot\left(\vec{k}_{2}-\vec{k}_{1}\right)\right)}
 \end{equation} 
-where $\vec{k}_{1,2}=k \vec{r}_{1,2}$ is the wavevector of the light seen in detector $1$/detector $2$. As the modulation frequency of the the intensity correlation is depended $R$, the separation of the two sources can be recovered.
+where $\vec{k}_{1,2}=k \vec{r}_{1,2}$ is the wavevector of the light seen in detector $1$/detector $2$. As the modulation frequency of the the intensity correlation is depended $R$, the separation of the two sources (the size of the star) can be recovered.
 
-A detailed quantum mechanical treatment  considers the mixtures of the probability amplitudes of emission and detection at the two detectors (as the individual paths are indistinguishable), leading what was shown as an equivalent description of the HBT effect for thermal light sources \cite{fano1961,sudarshan1963,glauber2006}, but allows the treatment of non classical light sources such as single photon emitters \cite{mandel1995,classen2017}. This introduces a correction term in $g^2$ of the order of $1/N$ with $N$ the number of emitters, which for the considered cases is negligible.
+A more detailed quantum mechanical treatment considers the mixtures of the probability amplitudes of emission and detection at the two detectors (as the individual paths are indistinguishable), leading what was shown as an equivalent description of the HBT effect for thermal light sources \cite{fano1961,sudarshan1963,glauber2006}, but allows the treatment of non classical light sources such as single photon emitters \cite{mandel1995,classen2017}. This introduces a correction term in $g^2$ of the order of $1/N$ with $N$ the number of emitters, which for the considered cases is negligible.
 
 
 
@@ -236,7 +230,7 @@ \section{Hanbury Brown Twiss}
 \end{figure}
 
 \section{Photon Statistics}
-Considering a complex sum of phasors of constant amplitude $A$ and independent uniformly in $(-\pi,\pi)$distributed phases $\phi_k$,
+Considering a complex sum of many phasors (e.g. the superposition of many electrical fields or probability amplitudes) of constant amplitude $A$ and independent uniformly in $(-\pi,\pi)$distributed phases $\phi_k$,
 \begin{align}
 	c=\sum^N_k A e^{i\phi_k}
 \end{align}
@@ -291,6 +285,7 @@ \section{Signal to Noise Considerations}
 The signal strength is determined by $\left|S(q)\right|^2$, which is dependent and the sample, and the contrast of the recorded speckle image. The noise in the measurement will consist of three parts: First, noise inherent to IDI caused by the random distribution of phases, second, the poissonian noise caused by quantized photons, and third, experimental noise \cite{trost2020, goodman2007}. 
 
 \section{Speckle Contrast}
+\label{sec:specklecontrast}
 The speckle contrast $\beta =\tfrac{1}{M}$ is governed by the number of independent modes $M$ overlaid in the measurement \cite{goodman2000}. 
 If the measurement is performed over a finite amount of time, the number of temporal modes is
 \begin{equation}