# How is the theory of partial coherent light related to quantum-mechanics?

## Background

Let me start this question by a long introduction, because I assume that only few readers will be familiar with the theory of partial coherent light and concepts like a mutual coherence function or a mutual intensity. The coherency matrix and Stokes parameters descriptions of partially polarized light are related concepts which are more widely known.

Correct treatment of partial coherent light is important for an appropriate modeling of optical pattern transfer in computer simulations of proximity and projection lithography as currently used by the semiconductor manufacturing industry. When I came to this industry, my previous optics "training" was insufficient in this area. I found chapter X "Interference and diffraction with partially coherent light" in Principles of Optics by Max Born and Emil Wolf most helpful for filling my gaps in this area. Later, I also "browsed" through "Statistical Optics" by Joseph W. Goodman, which has a nice paragraph in the introduction explaining why insufficient familiarity with statistical optics is so common:

Surely the preferred way to solve a problem must be the deterministic way, with statistics entering only as a sign of our own weakness or limitations. Partially as a consequence of this viewpoint, the subject of statistical optics is usually left for the more advanced students, particularly those with a mathematical flair.

The interesting thing is that Hermitian matrices and eigenvalue decompositions like the Karhunen-Loève expansion are used quite routinely in this field, and they somehow feel quite similar to modeling of coherence and decoherence in quantum-mechanics. I know that there are important obvious (physical) difference between the two fields, but my actual question is what they have in common.

## Question

Some elementary experiments like the double slit experiment are often used to illustrate the particle wave duality of light. However, the theory of partially coherent light is completely sufficient to describe and predict the outcome of these experiments. There are no particles at all in the theory of partially coherent light, only waves, statistics and uncertainty. The global phase is an unobservable parameter in both theories, but the amplitude of a wave function is only important for the theory of partial coherent light and is commonly normalized away in quantum-mechanics. This leads to a crucial difference with respect to the possible transformations treated by the respective theories. But is this really a fundamental difference, or just a difference in the common practices of the respective theories? How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?

One reason for this question is to find out how much familiarity with partial coherence can be assumed when asking questions here. Therefore it explains why this familiarity cannot be taken for granted, and is written in a style to allow quite general answers. However, it also contains specific questions, indicated by question marks:

• How is the theory of partial coherent light related to quantum-mechanics?
• ... the amplitude of a wave function ... But is this really a fundamental difference, or just a difference in the common practices of the respective theories?
• How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?

Don't be distracted by my remark about the double slit experiment. Using it to illustrate the particle wave duality of light seemed kind of cheating to me long before I had to cope with partial coherence. I could effortlessly predict the outcome of all these supposedly counter-intuitive experiments without even being familiar with the formalism of quantum-mechanics. Still, the outcome of these experiments is predicted correctly by quantum-mechanics, and independently by the theory of partial coherent light. So these two theories do share some common parts.

An interesting aspect of the theory of partial coherent light is that things like the mutual intensity or the Stokes parameters can in principle be observed. A simple analogy to the density matrix in quantum-mechanics is the coherency matrix description of is partial polarization. It can be computed in terms of the Stokes parameters $$J=\begin{bmatrix} E(u_{x}u_{x}^{\ast})&E(u_{x}u_{y}^{\ast})\\ E(u_{y}u_{x}^{\ast})&E(u_{y}u_{y}^{\ast}) \end{bmatrix}=\frac12\begin{bmatrix} S_0+S_1&S_2+iS_3\\ S_2-iS_3&S_0-S_1 \end{bmatrix}$$ and hence can in principle be observed. But can the density matrix in quantum-mechanics in principle be observed? Well, the measurement process of the Stokes parameters can be described by the following Hermitian matrices $\hat{S}_0=\begin{bmatrix}1&0\\0&1\end{bmatrix}$, $\hat{S}_1=\begin{bmatrix}1&0\\0&-1\end{bmatrix}$, $\hat{S}_2=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and $\hat{S}_3=\begin{bmatrix}0&i\\-i&0\end{bmatrix}$. Only $\hat{S}_0$ commutes with all other Hermitian matrices, which somehow means that each individual part of the density matrix can be observed in isolation, but the entire density matrix itself is not observable. But we don't measure all Stokes parameters simultaneous either, or at least that's not what we mean when we say that the Stokes parameters can be measured in principle. Also note the relation of the fact that $\hat{S}_0$ commutes with all other Hermitian matrices and the fact that the amplitude of a wave function is commonly normalized away in quantum-mechanics. But the related question is really a serious question for me, because the Mueller calculus for Stokes parameters allows (slightly unintuitive) transformations which seem to be ruled out for quantum-mechanics.

To start with the double slit experiment gives interference even when the beam is composed by one photon at a time. The spot on the screen a photon/particle the statistical accumulation the interference seen as expected classically too.

The joint comes because the photon as a quantum mechanical entity has a wavefunction that is the solutions of Maxwell's equation, treated as operators operating on the wave function. The E=h*nu identity the photon carries is the same as the frequency of the classical solution of Maxwell's equation and that , together with the phase attached to the wavefunction allow the continuity of classical down to quantum single photon level For a formal treatment how classical emerges from an ensemble of photons have a look at this blog entry.

There exists a fundamental difference between classical and quantum equations and their solutions, but also a continuity at the interface.

Edit after question edit

How is the theory of partial coherent light related to quantum-mechanics?

This needs somebody familiar with the formalism of both, but I believe the connection should follow the method in the link, how classical electromagnetic beams emerge from an ensemble of photons.

]>... the amplitude of a wave function ... But is this really a fundamental difference, or just a difference in the common practices of the respective theories?

the square of the wavefunction is the connection with predictions and experiments in quantum mechanics, it is the probabilistic nature that makes the difference with the classical framework, as far as I know.

How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?

Phenomenon is " an observable" , observing something implies a measurement process, measurement implies interaction, picking up a point that will contribute to the quantum mechanical probability distribution (or building up the distribution itself by continuous observations) so there is an inherent contradiction in this part of the question.

• I have nearly finished reading the links now. They are nice links indeed. I was really impressed by the blog entry, both for the nice coverage of the subject, but also by the aggressive tone of its author. Yes, I was also initially misguided with respect to quantum-mechanics, because of the way the subject was treated in the textbooks I read at the time. However, the author somehow seems to imply that I should feel ashamed for having read these textbooks. Or maybe not, but why does he writes so aggressively? – Thomas Klimpel Jul 15 '14 at 9:13
• This is a nice and informative answer, but it doesn't even mention partial coherence. I therefore expanded the text in the question a bit, to explain what I actually would like to learn. Don't know whether you can help there... – Thomas Klimpel Jul 15 '14 at 9:14
• The spectral lines of atoms and molecules are a strange phenomena explained by quantum-mechanics. We might try to explain it by computing the eigenstates and eigenvalues of an appropriate Hermitian operator. At this point there isn't necessarily a quantum-mechanical measurement involved, but the indistinguishablilty of quantum-particles certainly enters into this explanation (hence there is a reference to particles). Roger Penrose tried to explain the strange phenomena of conciousness by reference to quantum-measurement processes (what he calls "objective reduction"). Just an example... – Thomas Klimpel Jul 15 '14 at 13:21
• Spectral lines are measurement in my book, a measurement is a numerical value attached to an observation. Quantum mechanics has nothing to do with the definition of measurement. I see you consider a measurement "strange" because it is not explainable by solutions of classical equations. I suppose that by "measurement processes" you are wrapping around the probabilistic nature of each measurement? My answer to "how much can be explained" would be : none; otherwise quantum mechanics would not have been necessary. The crux is the probability postulate, the Born rule. – anna v Jul 15 '14 at 13:52