Ray & Wave Optics

Ray Optics

IMAGES :

1. Light rays forms two types of images

a) real image

b) virtual image

2. A convergent beam of light rays from an optical device forms a real image.

3. A divergent beam and parallel beam of light rays emitted out of an optical device forms a virtual image.



REFLECTION OF LIGHT THROUGH PLANE MIRROR :

The phenomenon of returning of incident light in the same medium is known as reflection.


Laws of reflection

(a) The incident ray, the reflected ray and the normal all lie in the same plane.

(b) In regular reflection the angle of incidence is equal to the angle of reflection i.e

When two mirrors are at an angle , the number of images of an object kept between the mirrors formed by the multiple reflection is

( if is odd)

(if is even)



REFLECTION AT SPHERICAL SURFACES:

A mirror which is a part of a sphere is called spherical mirror which are of two types

a) Convex mirror

b) Concave mirror

The distance of the principal focus from the pole of the mirror is called focal length (f) and it is related to radius of curvature (R) of the spherical surface as R=2f


CONCAVE MIRROR:

It forms both real and virtual images. Real images are magnified or diminished while all virtual images are magnified. Real images are inverted while virtual images

are erect.

If u and v are object and image distance from the pole of the mirror and R is radius of curvature, then


CONVEX MIRROR:

It always forms virtual image.

As the object moves towards the convex mirror the virtual image moves towards the mirror.

For real objects and images distances are taken positive while for virtual objects and images distances are taken negative.

A real magnification is

For a mirror of focal length 'f' & magnification 'm',

a) The real image distance from mirror v = f (m + 1)

b) The virtual image distance from mirror v = f (m - 1)

c) The object distance for real image ,

d) The object distance for virtual image


REFRACTION OF LIGHT

When a ray of light enters from one medium to another then the boundary of separation it deviates from its initial path. This phenomenon is known as refraction.

Laws of refraction

(a) The incident ray, the refracted ray and the normal all three lie in the same plane.

(b) For given two media and for a given wavelength, the ratio of the sine of angle of incidence to the sine of angle of refraction always remain constant i.e

This is also known as Snell's law.

For a given colour of light and for a given pair of media (const). is called refractive index of the second medium w.r.t. the first medium or relative

refractive index.


Critical angle and total internal reflection:

1. The angle of incidence in the denser medium whose corresponding angle of refraction in the vacuum is 900is called absolute critical angle of the denser medium

For a medium of refractive index ,





Note : The angle of incidence in the denser medium whose corresponding angle of refraction in the rarer medium other than vacuum is 900is called critical angle of

denser medium with respective to rarer medium.


Refraction of light through a prism:

Angle between the two refracting faces at the refracting edge of the prism is called its apex angle or refracting angle or angle of the prism.

The angle between the direction of emergent ray and the direction of incident ray is called the angle of deviation.

A = r1+ r2and d = i1+ i2- A

The phenomenon of splitting of a ray of light into different colours by a prism is defined as dispersion

If vand r are refractive indexes for blue light and red light, the average refractive index is .

For a small angled prism the deviation produced for red light and blue light is and

The angular dispersion


Refraction of light through thin lenses:

A lens is a transparent medium bounded by either two spherical surfaces or a spherical surface and a plane surface.

The spherical surfaces need not have the same radii of curvatures.


Lens maker's formula:



R1is positive if a light ray is incident on a convex surface. R1 is negative if a light ray is incident on a concave surface. R2 is positive if light ray inside the lens is

incident on the second surface which is convex and negative if it is concave with respect to the first surface. R is infinite for a plane surface.


Focal length of the combination of lenses:
When two lenses of focal lengths f1and f2are put in contact, their equivalent focal length is given by (proper sign must be given to f1and f2)

When two lenses are separated by a distance 'd', then the effective focal length of the combination is given by

(proper sign must be given to f1and f2)


SILVERED LENSES

When one of the surfaces of a lens is silvered, it behaves like a spherical mirror.

If one the surfaces of a convex lens is silvered, its effective focal length is given by

where fl= focal length of the lens.

fm= focal length of the silvered surface (R/2).

If the plane surface of a plano convex lens is silvered, then it behaves like a concave mirror.

Fm= infinite or .

If its curved surface is silvered, then it behaves like a concave mirror of .

If one of the surfaces of a plano concave lens is silvered; it behaves like a convex mirror.


Chromatic Aberration:

A lens fails to form a white point image of a white point object in white light. This defect is called chromatic aberration.

For achromatism, when two lenses are separated by a distance d, then or when w1=w2



SIMPLE MICROSCOPE:

A simple microscope or magnifying glass consists of a single convex lens of small focal length.

It is used for observing magnified images of tiny objects.

The image may be made to form at least distance of distinct vision (25cm for normal human eye).

In this case the magnification is

The image may be made to form at infinite distance with magnification

Note:- In simple microscope m > 20 is not possible



COMPOUND MICROSCOPE:

Compound microscope consists of two convex lens, one is called objective (towards the object) and the other is called eye piece (towards the eye). Focal length

of eyepiece is greater than focal length of objective.

The magnification of the microscope is





(approximately) where L is Length of microscope



ASTRONOMICAL TELESCOPE:

It consists of an objective lens (convex) of larger focal length of large aperture and a convex lens eyepiece of very short focal length (fe) of small aperture.

The optical length of tube of telescope is under normal adjustment position.

The magnifying power is given by

If final image is formed at least distance of distinct vision, then

If final image is formed at infinity, then


Wave Optics

Interference:

Interference of light is the phenomenon in which two or more coherent light waves superpose giving rise to modification of light intensity.

Because of interference of light, the resultant intensity is not uniform. At some places it is maximum and at some other places it is minimum.

In Young's double slit experiment, the two slits which are equidistant from a narrow slit work as coherent sources and produce a stable interference pattern.

If the displacements of the interfering waves are y1= a1sint and y2= a2sin(t + ) the resultant displacement is given by y = y1+ y2.

y = a1sint + a2sin(t + )

y = A sin(t + )

where A =

If a1= a2= a; A = 2a cos /2 and = /2

Resultant intensity at any point I A2.

Considering the proportionality constant as unity,

I = A2= 4a2cos2(/2)

The difference in the distances traveled by the two interfering beams as they reach the point of observations is called path difference.

Phase difference = and x is path difference.

Condition for maxima or bright bands path difference = n (n = 0, 1, 2, 3, .)

n = 0 central maximum.

Phase difference = 2n

Condition for minima or dark bands

path difference = (2n - 1)/2 (n = 1, 2, 3, )

n = 1 1stdark band

n = 2 2nddark band

Phase difference = (2n-1)

Resultant intensity I = I1+ I2+ 2 =

; =

Imin= I1+ I2- 2 =

Width of the bright band () is the distance between two successive maxima = width of the dark band () is the distance between two successive minima.

= D/d. Where l is the wave length, D is the distance between the sources and the screen and 'd' is the distance between the sources. Thus,

l, D, 1/d

Angular Fringe width (). It is the ratio of the fringe width to the distance of the screen from the sources. =



DIFFRACTION

Bending of light rays around the edges of an object is called the diffraction. Due to diffraction light encroaches into the geometrical shadow of an obstacle. Bands

of maximum intensity and minimum intensity are formed due to diffraction near the edge of the geometrical shadow of an obstacle.

To determine the diffraction effect the wave front is divided into different zones. These zones are called Fresnel zones or Half period zones.


Fresnel's diffraction

a) The source and the screen are at finite distances from the obstacle.

b) No lenses are required to observe the pattern.

c) The wave fronts are spherical or cylindrical.


Fraunhoffer diffraction

a) The source and screen are at infinite distances.

b) A convex lens is needed to study this diffraction pattern.

c) The wave fronts are plane.

Fresnel's diffraction due to a small circular obstacle (disc) or a small coin or a small lead shot is (i) is circular geometrical shadow with a bright spot at the center

(ii) the shadow is surrounded by alternate bright and dark rings. (the bright spot at the center is called Poisson's spot)


Fraunhoffer diffraction due to a single slit.

a) Condition for minimum intensity is a sin = n

Where a is the width of the slit, is the angle of diffraction.

is the wavelength of the light and n = 1, 2, 3 ... etc.,

b) Condition for (secondary) maximum intensity a sin = (2n + 1)/2

Where n = 1, 2, 3 ... etc.



POLARIZATION

Polarized wave is that wave in which vibrations are confined along one specific direction which is normal to the direction of propagation of the wave. Polarization

is exhibited by transverse waves. Longitudinal waves do not exhibit Polarization.

By convention, we define the 'direction of polarization' of an EM wave to be the direction along which the electric field (vector) of EM wave vibrates

(oscillates) with a definite frequency.

"Plane of polarization" the plane determined by the (vector) and the direction of propagation of the EM wave or the plane perpendicular to the direction of

polarization.


Unpolarized Wave :

a) In the case of an unpolarised EM wave {light wave as an example} the directions of are symmetrically oriented in all directions confined to the plane normal to the propagational direction of the wave.

b) For an unpolarized wave, there are many directions of polarization {i.e., vibrates (oscillates) along many directions (axis)}

c) In ordinary sources (sun light, incandescent light bulb) the atoms behave independently rather than cooperatively.



Polarization By Reflection

a) When an unpolarized beam of light is incident on a glass plate {light traveling from air to glass} at an angle of incidence p{Brewester's angle (or) angle of

polarization } the reflected beam is completely plane polarized (with weak intensity) with (vectors) vibrating normal to the plane of incidence and the transmitted

beam {in glass} is partially polarized {with greater intensity} as it contains vectors vibrating both "Normal" to plane of incidence as well as "parallel" to the plane of incidence.

b) The component with polarization parallel to the plane of incidence is completely refracted while the perpendicular component is partially reflected and partially

refracted.



If is the critical angle of the medium

.



MALUS LAW

Im E2; I E2cos2;

I = Imcos2; (Malus's law, polarized light passing through an analyzer)

Same proportional constant.

If "Im" is the intensity of the plane polarized light incident on the analyzer and "" is the angle between the transmission axes of polarizer and analyzer then the

intensity of the transmitted light from the analyzer is 'I'.