## Interference by wave-front and amplitude splitting.

Optics Series Lecture, Lecture – XIV, XV, XVI.

“Color of thin films, Newton’s rings, Lloyd’s mirror and Phase changes during reflection”

These lectures were delivered on 16th February, 21st February and on 17th March. The lecture sessions were of 1 and 1/2 hours. The lectures were delivered to both Physics honors as well as Physics elective students on different days.

We have previously discussed what is interference and what is wave-front splitting and amplitude splitting interference. We have also discussed in much details two wave-front splitting interference viz. Young’s double slit interference (Lecture – IX) and Fresnel’s bi-prism (Lecture – XI). Today we will discuss one more wave-front splitting interference namely Lloyd’s mirror interference before moving onto the amplitude splitting interference of the Newton’s Rings. Also we will discuss two interesting and related concepts; i. Phase change on reflection and ii. Color of thin films.

Interference in Lloyd’s Mirror.

The Lloyd’s mirror is a set-up for wave-front splitting interference. Here two rays one of which undergoes reflection from a reflecting surface like a mirror meet up at the point of observation and subsequently interfere. The interference pattern that is produced in a Lloyd’s mirror interferometer is complimentary to the pattern produced in a Young’s interferometer. That is because of an additional phase change of ± π during reflection.

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## Waves.

Optics Series Lecture, Lecture – XII and – XIII.

“Traveling waves, Differential wave equations, Particle and wave velocities.”

These lectures were delivered on 17th and 20th February 2017, in two lecture sessions of 1 and 1/2 hours each. The web version has been named “Waves.” and the lectures were delivered to Physics honors students.

In one of our earlier optics session lecture I had hinted at having waves defined by their pulse shape called as wave profile — or alternatively wave shape or wave form, and transcribing them into forms that represent actual wave motion. The later are then called as traveling or progressive waves. The former, the so called wave shape or wave profile are then time-snapshots of the full fledged time varying waves that we just called traveling waves.

Remember that stationary or standing waves are not wave profiles or any snapshots of a single traveling wave, they are rather the superposition of an advanced and a retarded wave — that is one traveling wave moving forward and another exactly shaped traveling wave moving in the reverse direction. We studied advanced and retarded waves, here.

We have also already dealt with traveling waves in much detail, eg, here and here. This lecture will justify what we have been espousing all along. Also in complex waves that are found in quantum mechanical theories, we have what are called as stationary states, these are like the time-snapshots of the quantum mechanical waves, represented through the energy of the system. Since the full energy or wave cycle is not necessarily contained in a given amount of time called as a time window, we have a corresponding uncertainty relation called an energy-time uncertainty relation.

But talking about an instant of time, a stationary state which represents the energy of the wave in that instant, are well defined states of energy and called as eigen-states. But what would happen if one takes a picture of a dynamic system? The fuzzed out region or so called “motion blur” might show up, because these time instants are not well defined eigen states rather superposition of random number of any of them, may be.

One dimensional Traveling Waves.

A traveling wave is a self sustaining oscillation of particles of a medium or oscillations of any physical quantity at different space-time points so that energy is transported across the medium when the oscillation propagates in the medium. There is no motion of the relevant medium in the ideal description of the wave. The oscillating particles move periodically about their equilibrium locations and in the case of physical quantities they take values around their equilibrium values. Examples of waves are mechanical waves:

## Fresnel’s Bi-prism: measurement of wavelength of light.

Optics Series Lecture, Lecture – XI.
“Fresnel’s Bi-prism: measurement of wavelength of light by it.” This lecture was delivered on 16th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students and later to honors students. This web-version does not strictly pertain to 1 and 1/2 hours of regular lecturing session that we have mostly been employing.

That’s because it was created with another part which can be optionally appended to other related subject matter. In this web-version that’s what we will do. Our guiding principle is more in line with the honors course, where the subject matter is quite extensive and deep which brings more flexibility and choices into the lecture compositions.

Today we will discuss another interesting interference set-up, now that we have discussed the Young’s double slit experiment, in lecture – IX. A few words about the general mechanism behind interference. There are two kinds of interference basically that we will be discussing in our lectures. We discussed the Young’s DS interference pattern based on our understandings of intensity or irradiance patterns that we studied here: lecture – VII.

Interference is sustained and visible if the corresponding sources of light are coherent among themselves, that is, if the sources have phase differences that are not arbitrarily or abruptly changing, as a consequence we can safely assume the phase differences are constant and therefore predictable. Incoherent light makes this impossible.

Incoherent light is that light source whose production itself is arbitrary and abrupt and unpredictable, hence nothing can be definitively said on its phase, as a result the coherence is only short lived. If two light sources are so generated that their respective coherence time (or coherence length) are well within each others span, they are said to be coherent light.

Recall the idea of temporal and spatial coherence here that we discussed: here, when we discussed Young’s DS interference. We observe one basic thing about interference here. The two coherent sources S1 and S2 that we considered give rise to two different wave-fronts that meet up after traversing their respective optical paths. When they meet they produce interference. For this reason such type of interference are called as wave-front splitting interference.

Young’s DS experiment is an example of wave-front splitting interference. The Fresnel bi-prism that we will discuss is also an example of wave-front splitting interference. But there is yet another type of interference mechanism. Its called amplitude-splitting interference, examples of which are colors of thin films and Newton’s ring phenomena which we will study soon enough in future lectures.

In an amplitude splitting interference what happens is there is only one wave (or its wave-fronts) which splits into different components such as reflected or transmitted (refracted) parts according to the respective coefficients for these processes. So the amplitude has a fraction which is reflected and another which is refracted.

Naturally the question of coherence does not deter the production of interference effects. There always is inherent coherence in the amplitude splitting processes. When these different components meet up later, they produce interference.

## Electromagnetic Nature of Light — A brief history of light

Optics Series Lecture,

“Electromagnetic Nature of Light — A brief history of light”

This lecture was delivered on 16th March, yesterday, in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics honors as well as Physics elective students.

As I promised in the last lecture, lecture-X we have our one of the interesting historical and technical perspective about light that is also one of my favorite, as I discovered yesterday, or shortly before that, the night before, when I was composing the lecture from scratch. We will name this lecture with its proper number, only after its clear to us what chronological number must be associated with it. Its like an advanced wave, it reached us before in time, before it was intended to be taken up for its web-version.

Let us begin this lecture which has roughly two parts, 1. the history of light and its understanding through the centuries and 2. the electromagnetic nature of light. The second part is intended as the course material for honors as well as elective students but you will be in amusement if you also cover the first part.

A brief history of light.
Various optical devices and optical phenomena have been known since close to 4000 years. The optical devices of ancient time includes mirrors, burning glasses, lenses and other magnifying devices.

Accordingly various properties and laws of light were understood and developed since these times. Eg light was understood to propagate rectilinearly, light was understood to reflect and refract. There were various laws that were known since these times which catered to the need for explanation of these phenomena. eg Reflection was understood to be a phenomena explained by the principle of shortest path — follow link to know this and other related ideas and their history: Hero of Alexandria. Laws of refraction were understood either partially or completely as the centuries or even millennia passed.

Apart from rectilinear propagation of light it was understood that light moves at infinitely large speed. Advanced optical devices such as telescopes were developed based on partial and faulty understanding of light which was gradually refined to accommodate better credits of advancement. Eyes as optical devices were understood and eye defects could be corrected by using suitable optical devices such as eye-glasses. Albeit all this light was never understood properly before the 17th CE.

The last 4 centuries saw tremendous leaps of understanding and applications of light. In the 17th CE great progress were made to understand various phenomena exhibited by light such as reflection, refraction and total internal reflection etc. Descartes proposed light as a longitudinal pressure vibration in elastic medium. Human beings understand by way of imitation and this was the reason light as waves were considered exactly in the image of sound as waves. The mantra lies in keeping the mind open for successive refinements through acceptance of truth as ordained eg by experimental facts.

Thus light was considered to be a wave. During 17th CE, discoveries were made that depicted the diffraction of light. This way light was considered as a rapid vibratory motion of a medium propagating at great speed. In these similar times Newton had opposing ideas regarding nature of light. According to him light was vibrations of corpuscles or particles with certain emission properties. Despite of this light was most successfully understood to be a phenomena of wave.

During the same 17th CE Romer performed astronomical experiments on Jupiter’s moon Io and the work of Newton and Huygens helped ascertain the speed of light to be c = 2.4 × 108 m/s and c = 2.3 × 108 m/s respectively.

In the 19th CE wave theory of light received many supporting evidences. The phenomena of interference and polarization were discovered or understood. Colors of thin films were understood and wavelength of light were determined. The wave theory successfully explained rectilinear propagation of light. In-fact it was this difficulty about wave theory which kept Newton a staunch supporter of the corpuscular theory of his rather than the wave theory. But one by one all hurdles of wave theory of light disappeared at the master strokes of many genius scientists. Similarly the need for explanation for polarization led to the correct view of light as a transverse wave rather than a longitudinal one.

Terrestrial determinations as opposed to astronomically cosmic determinations as evinced by the work of Romer became order of the day for speed of light. Fizeu by his toothed wheel method carried out an experiment that established the value of speed of light to a respectable c = 3.15 × 108 m/s. Speed of light in water was found to be reduced in comparison to speed of light value in air. This was in conflict with the corpuscular theory of light held in esteem by Newton. Not many supporters of this view remained any more in the annals of Physics, due also to the demise of the giant that Newton was, to be disproved easily or amicably.

## Harmonic Spherical Waves

Optics Series Lecture, Lecture – X.

“Harmonic Spherical Waves”

This lecture was delivered on 16th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics honors students.

In our lecture ( lecture-VIII ) we worked out the form of plane harmonic traveling waves. Note that soon we will barge into the concept of wave profile and how to convert a wave profile into its corresponding time-dependent or traveling form. But before we do that here is yet another general form of a traveling wave which we often meet in the Physicists Den. The traveling spherical wave fronts. Let us work out its details.

Spherical Waves.
When a stone is dropped in water it sends out circular waves. Similarly a sphere or a glob of matter that oscillates inside of a water body would send out 3-dimensional waves or ripples. Sources of light wave, which we will study in great detail, in this course, to fulfill our insatiable hunger for understanding the nature of optical phenomena, similarly, send out oscillations which propagate radially and uniformly in all directions. These are the spherical waves and the points or region that move out with equal phase are the wave fronts in this case, spherical in shape, called as spherical wave fronts.

We evidently need to describe the spherical wave fronts in spherical polar coordinate system, owing to the spherical symmetry in problems of 3-dimensional propagation of light waves.

Let us recall that the Laplacian in spherical polar coordinate system is given as: