Coherent Sources. Wavefront splitting Interferometer.

Coherent Sources

Wavefront splitting Interferometer

Young’s Double Slit Experiment

Young’s double slit

© SPK

Path difference: PSSP

2222 22 dxDdxD

dxD ,

1 2 1 22 2

2 2

2 21 1

For 1

1 1n

x d x dD D

D D

y

y nx

222 2

2

2 2

x d x d

D

x d d xd D

For a bright fringe,

For a dark fringe,

SP S P m

2 1 2SP S P m

m: any integer

For two beams of equal irradiance (I0)

204 cos

xdI I

D

Visibility of the fringes (V)

max min

max min

I IV

I I

Maximum and adjacent minimum of the fringe system

Photograph of real fringe pattern for Young’s double slit

The two waves travel the same distance– Therefore, they arrive in phase

S

S'

• The upper wave travels one wavelength farther–Therefore, the waves arrive in phase

S

S'

• The upper wave travels one-half of a wavelength farther than the lower wave.

This is destructive interference

S

S'

• Young’s Double Slit Experiment provides a

method for measuring wavelength of the light• This experiment gave the wave model of light a

great deal of credibility.

Uses for Young’s Double Slit Experiment

Wavefront splitting interferometers

•Young’s double slit

•Fresnel double mirror

•Fresnel double prism

•Lloyd’s mirror

Confocal hyperboloids of revolution in 3D

S

S

Path difference

20, 1, 2, 3

SP S P m

m

- confocal hyperbolae with S and S as common foci

2 2

2 2 2

11 1

( )4 4

x y

d

D=ml

Transverse section –Straight fringes

S

S

d

P

D

O

x

The distance of mth bright fringe from central maxima

Fringe separation/ Fringe width

Dx

d

d

mDDx mm

sin

Longitudinal section –Circular fringes

P

O

rn

S

Sd

D

N

q

Path difference = d

For central bright fringe

0

dm

2 00

cos

2( ) 2 ( )

m

m

SP SP SN d m

m m nn m m

d d

Radius of nth bright ring

For small qm

22 2 2 2n m

D nr D

d

Wavefront splitting interferometers

•Young’s double slit

•Fresnel double mirror

•Fresnel double prism

•Lloyd’s mirror

Interference fringes

Real Virtual Localized Non-localized

Localized fringe

Observed over particular surface

Result of extended source

Non-localized fringe

Exists everywhere

Result of point/line source

Concordance

Discordance

= (q+1/2)

Division of Amplitude

Phase Changes Due To Reflection• An electromagnetic wave undergoes a phase change of 180°

upon reflection from a medium of higher index of refraction than the one in which it was traveling– Analogous to a reflected pulse on a string

21 μ1

μ2

Phase shift 0k

D

nfn1 n2

B

d

A

C

ti

t

t

A

B

C

D

Optical path difference for the first two reflected beams

f 1

t

f

1

n [AB BC] n (AD)

AB BC d /cos

nAD ACsin 2d tan sin

ni t t

f2n dcos t

Condition for maxima

cos (2 1) 0, 1, 2,...4f

f tdn m m

Condition for minima

cos 2 0, 1, 2,...4f

f tdn m m

Fringes of equal thickness

Constant height contour of a topographial map

Wedge between two plates1 2

glassglass

air

Dt

x

Path difference = 2tPhase difference = 2kt - (phase change for 2, but not for 1)

Maxima 2t = (m + ½) o/n

Minima 2t = mo/n

Newton’s Ring• Ray 1 undergoes a phase change of 180 on

reflection, whereas ray 2 undergoes no phase change

R= radius of curvature of lens

r=radius of Newton’s ring

R

r

R

rRR

rRRd

2

2

22

2

1

...2

11

2

12 ( )

2

1 12 ( )

2 2

1 1( ) ( ) , 0,1,2...

2 2

For bright ring

bright

d n

rn

R

r n R n R n

...2,1,0,

n2d

ringdark For

nnRrdark

Reflected Newton’s Ring

Newton’s Ring

1. Optics Author: Eugene Hecht Class no. 535 HEC/O Central library IIT KGP