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Report: Prof. Hari Dass makes light coherence coherent

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Report: Prof. Hari Dass makes light coherence coherent

Prof. N. D. Hari Dass from TIFR-TCIS, in a very brief seminar, explains the heart of the meaning of light coherence, and its consequences on the HBT effect.

Last Friday, Prof. N. D. Hari Dass from the Tata Institute of Fundamental Research-TICS, Hyderabad, presented a very stimulating seminar on “Light Coherence and Quantum Mechanics” here at the Department of Physics, University of Mysore.

Right at the heart of coherence

The professor began the talk disclosing that his motivation for the choice of the seminar’s topic was the ongoing observance of 2015 as the International Year of Light. He then straightaway mentioned what he called the crux of the entire seminar, as the significance of the difference between “squaring followed by averaging and averaging followed by squaring” – more precisely, calculate the average of given set of data and square it; the number you get, will, in general, be different from what you’d get if you were to square the set elements first and then average. Mathematically,

(1)   \begin{equation*} <X^2> \neq <X>^2 \end{equation*}

The difference,

(2)   \begin{equation*} <X^2> - <X>^2= <\Delta X>^2 \end{equation*}

is technically called the variance.

Considering the situation with two random variables, he continued, we are dealing essentially with the difference between calculating the average of the product of the variables and the product of their averages,

(3)   \begin{equation*} <XY>-<X><Y> \end{equation*}

Such correlations between variables, measure the degree of coherence of light fields, he added.

How did Young do it?

Then, picking up the simplest situation in which light coherence manifests – the Young’s double slit experiment – Prof. Dass exclaimed that this is one of the deepest experiments in all of physics. He praised the simpleness of the set up and added that often deep experiments in Physics do not require billions of dollars of investment! He also pointed out that unlike most bygone experiments which lose their credibility especially after the advent of Quantum Mechanics, the Young’s experiment remains undiminished and is in fact greatly reassured by the Quantum theory.

Young's experiment

Set up of Young’s double slit experiment. How far must S1 be from S2, that’s the question! Image Courtesy: Wikimedia Commons

The professor pointed out a very interesting fact here – while conducting the experiment, Thomas Young was uncertain about the nature of light and had in fact set up the experiment in order to understand it. Had he not obtained the interference pattern on the screen, he wouldn’t have any basis to assert the wave nature of light, and the reality is that since the source he used for the experiment was a discharge tube, the light emanating from it is largely incoherent – light is emitted in a random fashion from excited atoms, causing random fluctuations in the phase of the emitted light; and these fluctuations are so rapid that they’d essentially wash out any possibly observable pattern on the screen. How did Young then obtain a pattern at all?

Solving the riddle, the professor pointed at the importance of the distance at which the source ought to be placed from the slits. He called this detail, the heart of the experiment. The closer the source is to the slits, the less definite is the phase relation between the two secondary sources at the slits. Therefore, to obtain an as coherent a light from the source as possible, place the source correspondingly farther from the slits. Young of course had the source placed some distance from the slits, resulting in a partially coherent light, and a partial yet visible interference pattern, which ultimately turned our understanding of light, on its head.

Prof. Dass then continued explaining the analysis with some mathematics. The electric field operator of an electromagnetic wave is generally written as,

(4)   \begin{equation*} E(t)=2E_0 cos (\omega t) \end{equation*}

If however we separate it into its positive and negative frequency parts, as was first suggested by the Hungarian physicist, Dennis Gabor,

(5)   \begin{equation*} E(t)=E_0 e^{-i\omega t} + E_0 e^{i\omega t}=E^+ (t) +E^- (t) \end{equation*}

Now a photo-detector placed at any specific point r on the screen measures only the modulus squared of the field amplitude (called the intensity) at any instant t,

(6)   \begin{equation*} I=|E^+ (\textbf{r},t)|^2=E^- (\textbf{r},t)E^+ (\textbf{r},t) \end{equation*}

Here Prof. Dass stressed that as far as Classical Physics is concerned, the order of multiplication is insignificant: E^- (\textbf{r},t)E^+ (\textbf{r},t) or E^+ (\textbf{r},t)E^- (\textbf{r},t), it’s all the same.
We may write the field amplitudes measured on the screen as a superposition of the fields present at the slits at an earlier time (t minus the transit time between the slits and the screen). That is,

(7)   \begin{equation*} E^+(\textbf{r},t)=\lambda_1 E^+(\textbf{r}_1,t_1)+\lambda_2 E^+(\textbf{r}_2,t_2) \end{equation*}

Invoking the expression for E^+ inspired from equation (5), we may write,

(8)   \begin{equation*} E^+(\textbf{r},t)=\alpha e^{i \textbf{k}.\textbf{r}_1 -\omega t_1}+\beta e^{i \textbf{k}.\textbf{r}_2 -\omega t_2} \end{equation*}

Or, pulling the phase information to the first term,

(9)   \begin{equation*} E^+(r,t)=\alpha e^{i \textbf{k}.\textbf{r}_1 -\phi}+\beta e^{i \textbf{k}.\textbf{r}_2} \end{equation*}

Considering the analogous equation for E^-(r,t), and putting them into equation (6), yields,

(10)   \begin{equation*} I=|\alpha|^2+|\beta|^2+2|\alpha||\beta|\cos(\Delta \phi +\frac{\Delta d}{\lambda}) \end{equation*}

If the sources were to be uncorrelated, i.e., if we performed the experiment leaving only one of the slits open at a time, the above equation would reduce to,

(11)   \begin{equation*} I=I_1+I_2 \end{equation*}

In terms of the photon-picture, we may say that the intensity at any specific point on the screen is just the total number of photons received there, and so, is additive.
The professor here mentioned that looking at equation (10) we see that if \Delta \phi, the phase difference between the sources (at the slits) were fixed, we would obtain a perfect interference pattern with sharp alternating dark and bright fringes, and the resultant intensity at the screen with both slits open would be (from equation (10)),

(12)   \begin{equation*} I=I_1+I_2+2\sqrt{I_1 I_2} \cos \delta \end{equation*}

Here we have assumed that the intensities at the two slits I_1 and I_2 are not fluctuating – a rather ideal case.

He further added that the distinctness of the interference pattern may be analyzed by defining a quantity called the visibility by,

(13)   \begin{equation*} \nu = \frac{I_{max} - I_{min}}{I_{max} + I_{min}} \end{equation*}

Using equation (12) yields,

(14)   \begin{equation*} \nu = \frac{\sqrt{I_1} \sqrt{I_2}}{2(I_1+I_2)} \end{equation*}

It’s clear from the last equation that if the intensities of the two sources were to be exactly the same, I_1=I_2, \nu=1; and for any other case, \nu < 1
The professor here mentioned an analogy with the contrast adjustments done in photography. If we go on making the dark spots on a photograph appear darker and light ones, lighter, we end up with poor clarity of the picture. In a similar way, a larger difference between the intensities at the slits causes the interference pattern on the screen to be less visible (or less distinct).

So far we have dealt with the ideal situation wherein we consider zero amplitude-fluctuations at the sources, but practically, we cannot have such a source – fluctuations are inevitable. Therefore we have to consider “ensemble averages” as is done in Statistical mechanics. Then the average intensity at a point on the screen is,

(15)   \begin{equation*} <I>=I_1+I_2+2 <\alpha \beta^*> \cos\delta \end{equation*}

Note that here <\alpha \beta^*> isn’t the same as \sqrt{I_1 I_2} in equation(12). We relate the two quantities with,

    \begin{equation} \gamma=\frac{<\alpha \beta^*>}{\sqrt{I_1 I_2}} $$and$$ |\gamma|\leq 1 \end{equation}

The average intensity is then,

(16)   \begin{equation*} <I>=I_1+I_2+2 \gamma \sqrt{I_1 I_2}\cos\delta \end{equation*}

With \gamma defined so, the modified visibility would be

(17)   \begin{equation*} \nu=|\gamma|\nu_{ideal} \end{equation*}

\nu_{ideal} is the visibility in the ideal case we earlier dealt with.

Since |\gamma|\leq 1, the visibility calculated in lab experiments with real sources is always less than that in the ideal case. And the basis for our transition from the ideal to the real case was the consideration of amplitude fluctuations in the source. Therefore, a concern of the fluctuations in the source due to the randomness of the atomic processes costs us the visibility – the formation of a distinct interference pattern on the screen. As the professor explained this point, we realized this was one of the most important and beautiful results up to this point of the seminar. We’d confronted the meaning of coherence eyeball to eyeball.

Hanbury Brown, Twiss, et al…

Prof. Hari Dass then briefly mentioned the Hanbury Brown and Twiss effect which is used to actually measure the angular diameters of radio-source-stars by measuring the correlation between the intensities of a source wave through two detectors (generally called intensity interferometers).

If I_1 and I_2 are intensities of a wave entering the two detectors, with a phase difference \phi, the fluctuation in the intensities may be defined as the deviation from the average intensity,

(18)   \begin{equation*} \Delta I=I-<I> \end{equation*}

Then its correlation would be,

(19)   \begin{equation*} <\Delta I_1 \Delta I_2>=<(I_1-<I_1>)(I_2-<I_2>)> \end{equation*}

(20)   \begin{equation*} <\Delta I_1 \Delta I_2>=<I_1 I_2>-<I_1><I_2> \end{equation*}

Therefore, in the presence of fluctuations,

(21)   \begin{equation*} <I_1 I_2> \neq <I_1><I_2> \end{equation*}

Which underlines the importance of the order of carrying out multiplication and averaging that was stressed in the beginning, which ultimately decides the presence of fluctuations and the degree of coherence of light.

The HBT effect was triumphant for radio sources, but when applied to the visible electromagnetic spectrum, it appeared to fail and was initially criticized harshly by even the bigwigs such as Richard Feynman. But it was later accepted once Hanbury Brown and Twiss explained the apparent discrepancies in a series of papers. Brown and Twiss did “deserve the last laugh” the professor quipped.

All in all, the entire seminar by Prof. Hari Dass, summing up the meaning of light coherence in as concise and elegant way as possible, was quite illuminating and left in us an urge to dig deeper into Quantum Optics and its “coherence”.

Further Reading:
  1. Profile, “Prof. N. D. Hari Dass”, on ResearchGate
  2. Article, “Hanbury Brown and Twiss Intensity Interferometer” for further details on the HBT effect.

Cover image by Josh Myer

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Roshan Sawhil is a Physics postgraduate who rejoices both doing and explaining Physics. He also finds doing Philosophy as a leisure activity quite interesting. You can find and connect with him on Facebook and Twitter.

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