COLOURS OF THIN FILMS ( INTERFERENCE )

 We observe multiple colours on the surface of soap bubble and also on surface of water when oil spreads on the water. 

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DETERMINING RADIUS OF nth DARK RING (NEWTON'S RINGS)

The radius of nth dark ring is directly proportional to

1. the square root of order of the ring,

2.the square root of radius of curvature of the lens and

3.the square root of wave length of the source.

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NEWTON RINGS - FORMATION

Newton rings are formed due to interference of  reflected light through an air film formed by a Plano convex lens.

                                       

newton rings are concentric Circular fringes with a dark fringe at the center and surrounded by bright and dark fringes alliteratively.

               

Formation of newton's rings.

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COSINE RULE--- INTERFERENCE DUE TO REFLECTED LIGHT FROM PARALLEL SURFACES

cosine rule is 2μt cosr =  nλ
it is the condition for dark fringe in interference due to reflected light from parallel surfaces.
μ= refractive index of the surface 
r = angle of reflection , n = order of the fringe 
and t = thickness of parallel film
λ = wavelength of monochromatic light.

The interference pattern consists a dark fringe at the center and surrounded by alternatively bright and dark fringes.

consider a parallel film of thickness t and refractive index μ.
Monochromatic light incidents on upper surface of film at an angle i.
A part of light is reflected along the path AB(light 1 ) from upper surface and the remaining light is refracted along the path AC through the film.
angle of refraction = r
the light AC is again reflected upwards from lower surface of the film in the direction CD and then emerges out from upper surface in the direction 2 ( light 2 ).
There is a path difference between the two lights 1 and 2.
BD is the normal line drawn on  AB.
the light 1 travels an extra path in air which is equal to AB *1
refractive index of air =1
The second light 2 travels an extra path inside the film 
equal to (AC + CD) * μ
hence the total path difference between the two lights is 
Δ =  ( AC + CD ) * μ  - AB
it is clear that AC= CD and AC = EC / cosr 
AC= t / cosr  eq.1
(AC + CD )μ  = 2t *μ / cosr 
also  AB = sin i * AD
where AD = AE + ED =  t * tanr + t* tanr
 AD = 2t (tanr)  * sini
 therefore path difference Δ =  ( AC + CD ) * μ  - AB
 Δ = 2t *μ / cosr  - 2t (tanr)  * sini
applying snell's rule μ = sini/ sinr
 sini  =μ sinr
hence  Δ = 2t *μ / cosr  - 2t (tanr)  * μsinr

Δ = 2t *μ / cosr  - 2t (sinr/cosr)  * μsinr
Δ = 2t *μ / cosr [ 1 - sin^2 r ]
Δ = 2t *μ / cosr [cos^2 r ]

Δ = 2μt cosr

Therefore the condition for dark fringe is 

Δ = 2μt cosr = nλ

INTERFERENCE PATTERN IN DOUBLE SLIT EXPERIMENT

In double slit experiment the monochromatic light from the source S passes through two narrow slits S1 and  S2 , divided into two wave fronts .

The two wave fronts interfere each other and produce interference pattern on the screen .

Nature of interference fringes

In double slit experiment

 (division of wave front)

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ASTIGMATISM ABERRATION

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COMA (COMATIC ABERRATION)

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ACHROMATIC DOUBLET

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MAGNITUDE OF CHROMATIC ABERRATION

Calculating chromatic aberration:

when a lens is suffering with chromatic aberration the focal point is not a single but a multiple focal points are seen from violet to red colour on the principle axis of the lens

We use lens maker’s formula to measure longitudinal chromatic aberration.

  1/f = [ 𝝁 - 1][ 1/R1  - 1/R2  ] 

where f = focal length of given lens , R1 and R2  are radii of curvature of the given lens,

if the given lens suffers with chromatic aberration then the focal points are different for different colours. also the refractive indices are different.

let  the focal length of violet colour = fv

Focal length of red colour = fr 

and focal length of yellow colour = fy

1/fv = [ 𝝁v - 1][ 1/R1  - 1/R2  ] 

1/fr = [ 𝝁r - 1][ 1/R1  - 1/R2  ]  and 

also it is known that fv * fr = fy^2

1/fv - 1/fr = [ 𝝁v - 1][ 1/R1  - 1/R2  ] -  [ 𝝁r - 1][ 1/R1  - 1/R2  ] 

 

fr - fv /fr *fv  = [ 1/R1  - 1/R2  ] [𝝁v -𝝁r]

fr - fv / fy^2 = [ 1/R1  - 1/R2  ] [𝝁v -𝝁r]

fr - fv  = fy^2 * [ 1/R1  - 1/R2  ] [𝝁v -𝝁r]

multiply and divide by [𝝁y-1] ,where 𝝁y = refractive index of  yellow colour.

 fr - fv  = fy^2 * [ 1/R1  - 1/R2  ][𝝁v -𝝁r] [ 𝝁y-1] / [𝝁y-1]

we know that (𝝁v -𝝁r) / (𝝁y-1) = dispersive power of the lens,⍵.

hence

  fr - fv  = fy^2 * [ 1/R1  - 1/R2  ] [ 𝝁y-1] *⍵

 and 

[ 1/R1  - 1/R2  ] [𝝁y-1] = 1/ fy

therefore     fr - fv  = fy^2 *⍵ / fy

So, the magnitude of chromatic aberration is fr- fv = fy *⍵ 

CHROMATIC ABBERATION

Chromatic aberrations

Definition:

Chromatic aberration is defined as the defect produced in the system of a given lens in which the image of a white object is colored.
It is due to the reason that when light form white object refracts through the thick and large lens the angle of refraction varies with colour. 
Hence the focus of different colours if different. Therefore multiple coloured images are formed instead of a single white image.

Types of chromatic aberration:

There are two types of chromatic aberrations. 
They are 1. Longitudinal chromatic aberration and 
2. Lateral chromatic aberration.

1.Longitudinal chromatic aberration :


In this defect the white light from distant object is separated into seven colours after refracting through the lens and different colours converge at different focal points. 
The focal point of violet is nearer to the center of the lens as it refracts at less angle whereas the red colour is focused far away from the center of the lens.

The single focal point of lens without aberration

When the lens is thin:

The multiple focal points with longitudinal chromatic aberration When the lens is thick :

If a point white object is placed on the principle axis in front of the lens at a distance between f and 2f , the image of a white point object is not a single but colored point images from violet to red in the order formed on the principle the axis of the lens.

Magnitude of longitudinal chromatic aberration = fr – fv ,

since fr > fv

2.Lateral chromatic aberration

When an object of certain size is placed on the axis in front of a lens then violet coloured image is formed near to the centre of the lens and red image is formed at far distance on the principle axis. In between them other coloured images are seen.

The image of red colour is larger than that of violet and other colours.