Young's double slit experiment: interference occurs when there is coherence between the sources and the frequency of light are close enough, also closeness in amplitude leads to clarity of the interference pattern. Young's double slit experiment is a wave front splitting type of interference. Photo Credit:

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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.

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Harmonic wave: Plane wave front of a harmonic wave. Photo Credit:

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Harmonic plane waves

Optics Series Lecture, Lecture – VIII.

“Harmonic Plane Waves”

This lecture ( 1.5 hours ) was delivered on 13th February 2017 to 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 a combination of both sine and cosine. You can learn more about what is wave profile and how to transform a wave profile into a traveling wave in the following lecture.

A wave profile, wave form or wave shape is simply a spatial snapshot view of a more general moving wave, at a suitable time. We have also discussed what is a plane wave. We studied the interesting phenomena of interference between two plane waves in the context of our harmonic plane waves. We assumed that our harmonic plane waves are also monochromatic waves, that is they have the same same wavelength.

These waves traveling in a homogeneous media have a fixed frequency and as long as they are in free – space their speed remains unaltered at the speed of light value c = 3 × 108 m/s.

A plane wave is a traveling wave where the wave fronts are planar points with equal phases anywhere on the plane. In a similar manner a spherical wave front is the locus of uniform phase over a 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 lecture we have discussed in much detail what are spherical waves. Cylindrical waves have been 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 I make no promises at this point.

Note that the waves are simply a motion of the phase points and therefore waves are a function of the space ( or location ) and time instant by which they are described. A phase is nothing but the angular argument of the wave described in terms of harmonic functions. Since harmonic functions ( another name for sinusoidal functions ) are periodic functions and trigonometric functions have a certain periodicity in terms of angular values, a phase represents the angular value ( or argument ) of these functions.

Plane Waves
Let us begin studying the plane waves in detail. Here are some of its features.

1. plane wave: a plane wave is the simplest example of a 3-dimensional wave.

2. why named so: plane waves are so called, because plane wave have wave fronts that 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. simplest example: 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.

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Interference of plane harmonic waves: This diagram explains the interference caused by two plane harmonic waves. Plane harmonic waves are plane waves with sinusoidal variations of the field vectors, E and B. The E field has a linear or plane polarisation, i.e. it vibrates on a fixed plane. There are two such waves with different planes coming from a coherent source which leads to interference of these plane waves. Photo Credit:

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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: 

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Matrix method for thick lens: Matrix transformation of the ray coordinate of a thick lens; the ray coordinates are an important parameter for description of optical systems under the assumption of geometrical or ray optics. How does the coordinate transform under various optical phenomena like reflection and refraction is important in understanding and predicting various optical phenomena. The matrix method gives a powerful and useful way for achieving this. Photo Credit:

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Application of matrix method to thick lens

Optics Series Lecture, Lecture — VI
All Optics series articles: 

Application of Matrix Method to Thick Lenses

Topics covered in this lecture 

A. Cardinal points 

B. Thick lens equation and matrix for thick lens 

C. System matrix for thin lens 

D. Unit and Nodal planes 

E. Matrix for a system of 2 thin lenses

Our previous studies of optical systems were based on two premises.

a. We assumed a paraxial system

b. We assumed that our lenses are thin

a. We assumed a paraxial system, see what this means and how its defined in a previous article, here.

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 lectures, here and here. 

b. We assumed that our lenses are thin

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Matrix formulation in geometrical optics: Translation refers to ray tracing in a single homogeneous medium. Ray coordinates are transformed from one point to another through a powerful matrix formulation of optical systems. Photo Credit:

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Matrix formulation in geometrical optics

Topics covered in this lecture

Ray tracing
a. Translation matrix

b. Refraction matrix

c. System matrix 

In this lecture, we will discuss about one of the most interesting and powerful methods in Geometrical Optics. As we have discussed here (, 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 the  ray 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.

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