24 Sep 2026

I: Coherence Intro

Coherence

Main differences b/w Classical & Quantum:

1. Determinism
2. Effect of measurement
3. Equation of motion
4. Duality
5. Superposition Principle

Coherence

A real, physical source consists of infinitely many emitters all of which emit in a statistically independent manner. As a result the total field emitted by the source becomes random. In the case of random fields, interference occurs to the extent that the fields at two different space-time points are mutually correlated. The nature and the degree of this correlation is described through what is known as a coherence function, and the equation that dictates the dynamics of a coherence function is known as the Wolf equation.

Two interfering field-amplitudes in the classical theory and the two interfering wave-functions in the quantum theory need to be mutually coherent for the interference to take place. Within quantum theory the concept of coherence can also be cast in terms of indistinguishability arguments which say that if a photon in two separate alternatives remain indistinguishable, that is, if there is no way of figuring out which particular interfering alternative a photon took then the two alternatives are mutually coherent and therefore interference will take place.

Coherence is the order in a random field. Within classical description, randomness is in principle removable if complete information about the system becomes available. But it is not always possible to obtain the complete information about a system, and this is why one studies random fields through correlation functions. On the other hand, in quantum description, the randomness in not removable even in principle. So, the only concrete quantity that can be studied within quantum theory is anyway the correlation function.

For two random variables $x_1$ and $x_2$ the simplest correlation function that can be defined is:

\[\left< x_{1} x_{2} \right>=\int x_{1} x_{2} p ( x_{1} , x_{2} ) d x_{1} d x_{2} .\]

We note that if $x_{1}=x_{2}$ , it becomes the second moment $\langle x_{2}^{2} \rangle$ of random variable $x_{2}$ . Therefore it is clear that $\langle x_{1} x_{2} \rangle$ contains more information than just $\langle x_{1}^{2} \rangle$ or $\langle x_{2}^{2} \rangle$ . In general a whole hierarchy of correlation functions can be defined for two random variables. For example,

\[\langle x_{1}^{m} x_{2}^{n} \rangle=\int x_{1}^{m} x_{2}^{n} p ( x_{1} , x_{2} ) d x_{1} d x_{2} .\]

is the correlation function with $m^{\mathrm{t h}}$ moment for $x_{1}$ and $n^{\mathrm{t h}}$ moment for $x_{2}$ . By adding time arguments the above correlation function definitions can be made even more general. For example, $\langle x_{1} ( t_{1} ) x_{2} ( t_{2} )\rangle$ is the two-time correlation function of two random variable $x_{1} ( t_{1} )$ and $x_{2} ( t_{2} )$

Two complex time correlation function $\langle z_{1}^{*} ( t_{1} ) z_{2} ( t_{2} ) \rangle$ is called the mutual correlation function of random variables $z_{1} ( t_{1} )$ and $z_{2} ( t_{2} )$. In situation in which $z_{1} = z_{2} = z ,$, the correlation function $\langle z^{*} ( t_{1} ) z ( t_{2} ) \rangle$ is referred to as the auto-correlation function.

Complex Analytic Signal Representation

Let $x( t )$ be a real function of a real variable $t$. Let us also assume that the Fourier transform of $x( t )$ exists such that

\[\begin{aligned} x(t) &= \int_{-\infty}^{\infty} \tilde{x}(\omega) e^{-i \omega t} d\omega \\ &= \int_{-\infty}^{0} \tilde{x}(\omega) e^{-i \omega t} d\omega + \int_{0}^{\infty} \tilde{x}(\omega) e^{-i \omega t} d\omega \\ &= z^*(t) + z(t) \end{aligned}\]

Here, $z ( t )$ is the complex analytic signal associated with the real random variable $x ( t )$ .We can write $z ( t )$ as a Fourier transform:

\[z ( t )=\int_{-\infty}^{\infty} \tilde{z} ( \omega) e^{-i \omega t} d \omega\]

where

\[\begin{aligned} \tilde{z}(\omega) &= \tilde{x}(\omega) \quad \text{when } \omega \geq 0 \\ &= 0 \quad\quad \text{when } \omega < 0 \end{aligned}\]

We note that $z ( t )$ is a complex analytic signal and therefore it is single valued and has continuous derivatives. Moreover, $x^{*}(t)$ can be written as

\[\begin{aligned} x^{*}(t) &= \int_{-\infty}^{\infty} \tilde{x}^{*}(\omega) e^{+i \omega t} d\omega \\ &= \int_{\infty}^{-\infty} \tilde{x}^{*}(-\omega) e^{-i \omega t} d(-\omega) \quad \text{(By substituting } \omega \text{ with } -\omega\text{)} \\ &= \int_{-\infty}^{\infty} \tilde{x}^{*}(-\omega) e^{-i \omega t} d\omega \end{aligned}\]

This simply implies that $\tilde{x}^{*}(-\omega) = \tilde{x}(\omega)$, and therefore the positive frequency part of the signal contains as much information as the negative frequency part.

It is known that for a complex analytic signal $z(t)$, the real part is exactly $\frac{1}{2}x(t)$. Let us denote its imaginary part by $\frac{1}{2}y(t)$, so that $z(t) = \frac{1}{2}[x(t) + iy(t)]$. The real signals $x(t)$ and $y(t)$ then form a Hilbert transform pair:

\[\begin{aligned} y(t) &= \frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{x(t')}{t' - t} dt' \\ x(t) &= -\frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{y(t')}{t' - t} dt' \end{aligned}\]

where the Cauchy’s principle value is defined as

\[P \int_{-\infty}^{\infty} \frac{x(t')}{t' - t} dt' = \lim_{\delta \to 0} \left[ \int_{-\infty}^{t-\delta} \frac{x(t')}{t' - t} dt' + \int_{t+\delta}^{\infty} \frac{x(t')}{t' - t} dt' \right]\]

The correlation functions in classical optics

So far, we have seen how a real signal corresponding to a physical process can be represented in terms of a complex analytic signal for the ease of mathematical calculations. We have also studied how, from a given random variable, one can construct correlation functions of different orders so as to extract relevant information up to various levels.

Let us take the electric field to be a real random signal $V^{(\mathrm{r})}(\boldsymbol{r}, t)$ representing the classical electric field due to a chaotic light source. We represent this real signal in terms of the complex analytic signals as follows:

\[\begin{aligned} V^{(\mathrm{r})}(\boldsymbol{r}, t) &= \int_{-\infty}^{\infty} \tilde{v}(\boldsymbol{r}, \omega) e^{-i \omega t} d\omega \\ &= \int_{-\infty}^{0} \tilde{v}(\boldsymbol{r}, \omega) e^{-i \omega t} d\omega + \int_{0}^{\infty} \tilde{v}(\boldsymbol{r}, \omega) e^{-i \omega t} d\omega \\ &= V^{*}(\boldsymbol{r}, t) + V(\boldsymbol{r}, t) \end{aligned}\]

Here, $V(\boldsymbol{r}, t)$ is the complex analytic signal corresponding to the real random variable $V^{(\mathrm{r})}(\boldsymbol{r}, t)$. Also, since $V^{(\mathrm{r})}(\boldsymbol{r}, t)$ is real, we have $\tilde{v}^{*}(\boldsymbol{r}, \omega) = \tilde{v}(\boldsymbol{r}, -\omega)$. Let us look at the first few correlation functions that can be constructed using this analytic signal:

• Intensity: $I(\boldsymbol{r}, t) \equiv \langle V^{*}(\boldsymbol{r}, t) V(\boldsymbol{r}, t) \rangle$

  This expectation value is the simplest second-order correlation function corresponding to the analytic signal. The quantity $V^{*}(\boldsymbol{r}, t) V(\boldsymbol{r}, t)$ represents the instantaneous intensity of the field at $(\boldsymbol{r}, t)$, and its expectation value is the ensemble-averaged intensity of the field at $(\boldsymbol{r}, t)$.

• Cross-correlation function: $\Gamma(\boldsymbol{r}_1, t_1; \boldsymbol{r}_2, t_2) \equiv \langle V^{*}(\boldsymbol{r}_1, t_1) V(\boldsymbol{r}_2, t_2) \rangle$

  The next expectation value we consider is the two-point correlation function, which is commonly referred to as the cross-correlation function.

• Higher-order correlation function: \(\Gamma(\boldsymbol{r}_1, t_1; \boldsymbol{r}_2, t_2; \boldsymbol{r}_3, t_3; \boldsymbol{r}_4, t_4) \equiv \langle V^{*}(\boldsymbol{r}_1, t_1) V^{*}(\boldsymbol{r}_2, t_2) V(\boldsymbol{r}_3, t_3) V(\boldsymbol{r}_4, t_4) \rangle\)

  This is a fourth-order (four-point) correlation function. Here, we will not be studying it extensively within classical theory, but in the context of the quantum theory of coherence, four-point correlation functions will be analyzed in greater detail.