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.

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.

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:  

Interference of two plane harmonic waves.

Optics Series Lecture, Lecture – VII.

“Conditions of interference, Interference of two plane harmonic waves.”

This lecture was delivered on 7th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students but intended as a lecture towards Honors students at a later date.

Electromagnetic Waves.
Light is an electromagnetic wave. In-fact its a transverse electromagnetic wave which means the oscillation of E and B fields produces light which propagates in a direction that is perpendicular to the plane that contains the E and B fields. In other words E, B and k the vector that denotes the direction of light propagation, are mutually perpendicular vectors.

We will study these details in a later intended lecture. EM waves are not only transverse waves but also vector waves, that is; E and B are vector fields whose undulation is summarized as light.

Light is a general name for all EM waves but visible light is that particular part of EM waves which has frequency of wave such that the wavelength varies from approximately 400 – 700 nm. In vacuum — only in vacuum, light always moves at a fixed speed: namely c = 3×108 m/s. Therefore light whose wavelength lies between 400 – 700 nm is called as visible light: we can write in vacuum c = νλ.

Light as a transverse wave phenomenon of vector fields is comprehensively described by four equations known as Maxwell’s Equations. The Maxwell’s Equations are a summary of important and fundamental laws of electricity and magnetism — together called as electromagnetism, such as Gauss Law and Ampere’s Law. These equations produce the wave equation of motion, a linear, homogeneous, 2nd order differential equation that we will study a few lectures afterwards.

If you are quite serious and technically well equipped though, you can have a glimpse of it all — and may be work out to your satisfaction, by following the link to my slide-share presentations. There are many other important Physics concepts that are worked out in great detail, in those slide-share presentations by me. eg check: Electromagnetic Waves.

Let us therefore write the wave equation of motion, where the 3 components of E field — such as Ex, Ey or Ez or the 3 components of B field such as Bx, By, Bz, are denoted as ψ chosen anyone at one time. eg we can chose Ex = ψ. In general we have: