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:  

Young’s Double Slit Experiment.

Optics Series Lecture, Lecture – IX.
“Young’s Double Slit Experiment. Coherent Sources and Conditions of Interference”

This lecture was delivered on 14th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students. At a later date this was delivered as a lecture to honors students as well. The web-version differs slightly from class delivered lectures, in that: any particular idea is explained without reference to what level it must cater to.

That means in class lecture will modulate depending on the actual level of student body and their response. An honors student body who would find a particular discourse difficult will be supplied with further simplified versions of the concepts, verbatim. An elective students body which is well prepared would have no problems grasping the fundamentals at a purported level. Its a happy scenario if that is indeed the case.

The concurrent lecture is particularly divided into two parts. The first part pertains to what are coherent sources and what are the sustainable conditions for interference, for such to be observed. The second part leads us to describe in requisite detail the phenomenon of Young’s double slit interference. Note that we have already discussed the phenomenon of interference in our lecture-VII.

We will only passively mention that there are two kinds of interference the so called wave-front-splitting and the amplitude splitting interference. Later on we will discuss any required details of both kinds. Before we do so we will have several interference phenomenon lectures from both types. Young’s double slit interference is an example of the wave-front splitting interference.

What happens here is there are two primary or secondary coherent sources and two separate waves interfere at a given observation vantage. Another example of wave-front splitting interference is Fresnel’s bi-prism set-up which we will study soon, in an imminent lecture.

For amplitude splitting interference only one wave produces the interference patterns, because the wave amplitude is partially reflected and partially transmitted — or refracted, and both channels meet up somewhere. Just to mention it for the time being, Newton’s Ring Interference patterns are example of amplitude splitting interference. Later we will study the details of all sorts of interference phenomena such as the ones just mentioned.

Coherent sources and conditions for interference:
Let us now discuss the first part of our lecture. Let us for the time being define coherence as the attribute of a light source such that there is no arbitrary and unexpected changes in the phases of different light waves such that when these waves at an observation point meet, we can apply the results of our interference analysis that we discussed in lecture-VII.

Harmonic Plane Waves

Optics Series Lecture, Lecture – VIII.
“Harmonic Plane Waves”
This lecture was delivered on 13th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics honors students.

In our last lecture, lecture-VII we began by discussing what are electromagnetic waves. We also discussed in good detail what are harmonic waves. Harmonic waves are those waves whose wave-profile is either sine, cosine or in general both sine and cosine combined with each other. Shortly (after within a few lectures) we will discuss what is wave profile and how to transform a wave profile into a traveling wave.

A wave profile, wave form or wave shape is simply a time instant view of a more general moving wave. We also discussed what is a plane wave. We applied our harmonic plane waves to the interesting phenomena of interference between two plane waves that are in addition monochromatic that is have same wavelength. Such waves traveling in a homogeneous media do so at a  fixed frequency and as long as they are in free-space their speed remains unaltered at the sped of light value c = 3 × 108 m/s.

A plane wave is one traveling wave where the wave fronts are planar points with equal phases all over the plane. In that order a spherical wave front is the locus of uniform phase over spherical configuration and a cylindrical wave front would be a traveling wave where the locus of uniform phase is nothing but a cylindrical surface.

In one of the future lecture, shortly, we will discuss in much detail what are spherical waves. Cylindrical waves are left to the advanced and willing students to work out by themselves. If time permits sometime in the future we can fall back and make a case for cylindrical wave fronts as well. But no promise at this point.

Note that waves are simply motion of phase points as a function of space or location and time instant. A phase is nothing but the angular argument of the wave described in terms of harmonic functions.

Plane Waves.
Let us begin studying Plane waves in detail. Here are some of its features.
1. A plane wave is the simplest example of a 3-dimensional wave.
2. These are so called, because plane wave wave fronts are planar in shape.
A wave-front is a locus of points on which the phase of the wave is same. Its a surface of wave-disturbances which move together, at the same speed.
3. Optical devices are often tuned to produce plane waves.
This necessitates the study of plane waves as base examples, where more complicated features can be assigned when they become pertinent.
4. Wave fronts are always perpendicular to the direction of wave propagation.

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: 

Application of matrix method to thick lens.

Optics Series Lecture, Lecture – VI.
“Application of Matrix Method to Thick Lenses.”

This lecture was delivered on 7th February in a lecture session of 1 and 1/2 hours. This lecture is slightly bulged out so it might take a little more time than intended. Most of the lectures are intended for honors courses but once in a while the optics series contains lectures suitable for elective courses.

Our previous studies of optical systems were based on two premises.
We assumed a paraxial system.
This means we employed a first order optical theory. Check the article just linked for a good overview of whats paraxial optics and whats first order optical theory. Such assumptions are fraught with various types of aberrations which we studied in detail in lecture-I and lecture-II.
We assumed that our lenses are thin.
This we did for simplicity. In Physics when we assume a simple situation we are not evading the actual complexity of the situation, we are just postponing this to the happy hour, howsoever you define it. Some people go by the Friday happy hour rule. It gives a good substratum on which a disposition can be carried out. Later one develops the nuances and fits it into the substratum and if things are carried out with caution and skill one gets a very effective overview of the pedagogy.
Let us now delve into the complexity of the optical system as a next step from its simple substratum of a thin lens. Our analysis needs to be modified for applying optical principles to optical systems when we consider thick lenses. In our last lecture — lecture – 5 we studied the method of matrices in understanding optical ray tracing. Let us now apply this method to the case of thick lens and see what power it unleashes.

A thick lens can be considered to be an equivalent of multiple thin lenses. Let us draw two suitable thick lens diagrams.

Matrix formulation in Geometrical Optics.

Optics Series Lecture, Lecture – V.

“Matrix formulation in Geometrical Optics.”

This lecture was delivered on 3rd February in a lecture session of 1 and 1/2 hours.

In this lecture, we will discuss about one of the most interesting and powerful methods in Geometrical Optics. As we have discussed, geometrical optics is that segment of optics in which we are limited to a situation when the wavelength of light is negligible eg λ is insignificant compared to the size of the objects light interacts with.

As a consequence light can be considered as rays or geometrical straight lines and the nuances of light as wave undulations can be postponed to a happy hour.

Ray Tracing.

Any general optical system has a ray which can be traced through two basic types of traversal of the ray: Translation and Refraction. The law of refraction is thus the central tool for ray-tracing. A ray can be described in an optical system by its coordinates which we will define soon. 

Our goal is to find the matrix which governs the displacement of the ray from one coordinate to another coordinate of the ray as the ray travels from one geometric point to another. This will enable us to study simple as well as much more complicated systems in the most effective and powerful way as we will see.

Lets discuss the basic matrices available for ray tracing when it travels from one coordinate to another in two cases. I. Translation Matrix for simple straight line motion in a homogeneous medium. II. Refraction Matrix for refraction at the interfaces of two different media.

In general therefore the total traversals of the ray can constitute of any number of translations or refraction. A reflection would merely be two translations and a general refraction might be construed from refraction as well as translations.

A most generalized optical system can therefore have n numbers of translations and s numbers of refraction matrices, so to say, in their respective orders. This total matrix would be called a system matrix.

I. Translation Matrix.

Lets us consider a ray that traverses an optical medium in a straight line, at any angle. This is called as a translation, as, the only change the ray undergoes is a translation or straight-line shifting of coordinates. Needless to say this enforces us to realize, the ray is limited to a single homogeneous medium.

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