1. Interference of Light Dr. J P SINGH Associate Professor in Physics
Post Graduate Govt. College Sector-11
Chandigarh

2. Interference
The term interference refers to phenomenon involving superposition of waves leading to a modification of their intensity in the region of superposition.
At certain points, the intensity may become more than the sum of the intensities of the individual waves (constructive interference) and at certain other points it may tend to zero (destructive interference).

3. Constructive and Destructive Interference
Two waves (top and middle) arrive at the same point in space.
The total wave amplitude is the sum of the two waves.
The waves can add constructively or destructively
destructive interference
constructive interference

4. When waves from two or more sources arrive at a points in phase, they reinforce each other. The amplitude of resultant wave is the sum of the amplitudes of the individual waves. This is called constructive interference shown in fig below. If the distance of any point P from S1 and S2 is r1 and r2.
For constructive interference the path different must be an integral multiple of wave length i.e.
(path different)
Points a and b in figure below satisfy this equation for n=0 and +2 respectively.

5. Point a and b
Satisfy equation (1) for n=0 and 2 respectively

6. The condition of destructive interference is given by
(b) Condition of constructive interference
Path difference is integral multiple of wave length.
(c) Condition of desstructive interference
Path difference is half-integral multiple of wave length.
The condition of destructive interference is given by
(path different)
Path difference in point is for n=-3

7. Conditions of interference of light
1. The sources must be coherent. They must maintain a constant phase with respect to each other.
2.The waves must have identical wavelengths.
3. The two sources must be narrow, because a broad source of light is equivalent to a large number of narrow sources.
4.Sources should be monochromatic.
5.Distane between two sources should be very small.
6. Wavelength of two sources should be same.
7. Amplitude of waves should be same.
8. Waves should be in the same state of polarization.

8. Temporal Coherence (or longitudinal coherence)
The term coherence refers to a degree of co-relation between the phases at different points and different times in a beam of light or radiation.
Temporal Coherence (or longitudinal coherence)
A beam of light is said to be temporal coherence, if the phase difference of the waves crossing the two points laying along the direction of propagation of the beam is independent. For this reason, the temporal coherence is also called longitudinal coherence.

10. In order to understand the concept of temporal coherence let us consider Michelson interferometer arrangement shown below. Source is extended and near monochromatic .
Light form source S falls upon partially coated glass plate A is divided into two parts on towards M1 and other toward M2 are equidistance from A i.e. when two waves traversing the two different paths take the same time, the interference pattern formed is very good. If the mirror M2 is moved very slowly

12. away from A, then it is seen that for ordinary extended sources of light, the contrast in the fringes goes on decreasing and finally disappear when the path difference AM1-AM2 is about a few mm.
The source S is emitting small wave trains of average duration τc and there is no relationship between the wave trains. When the difference in time taken by the wave trains to travel the paths A to M1 and back, and A to M2 and back, is much less than the average duration τc, the interference is observed between the two wave trains, is one is being derived from the same wave train. Hence even though different wave trains originating from the same source S have no definite phase relationship, but since one is superposing two wave trains derived from the same wave train, therefore, fringes of good contrast are seen. On the other hand, if the difference in time taken to cover the path to M1 and back and to M2 and back is greater than τc, then one is

13. superposing two wave trains which are derived from two different wave trains, interference fringes are not observed since there is no definite phase relationship between the two wave trains emanating from the source. Hence if the mirror M2 is moved, the contrast in the fringes goes on decreasing and for large separations no fringes are observed.
Let d is the distance through which the mirror was moved when no fringes would be observed. The beam which is reflected from the mirror M2 travels an additional distance 2d.
Thus the beam reflected from M1(originates 2d/c second earlier) interfere with the beam reflected from M2.
Then no will exists between two beams therefore no fringes is observed.

14. Spatial Coherence (or transverse coherence)
A beam of light is said to be spatial Coherence, if the phase difference of the waves crossing the two points laying on a plane perpendicular to the direction of propagation of the beam is time independent.

15. l
Two pin holes S1 and S2 on screen 1 are such that their distance from the point source S and the distance from any point O on screen 2 are equal.

16. Since point source S illuminates the pin holes S1 and S2 with spherical waves, the interference fringes formed near O would be in good contrast. Let us now consider, an another source S’ placed near to S and assume that there is no phase relationship between the waves from S and S’. In such a case, the interference pattern observed on the screen-2 would be a superposition of the intensity distributions of the interference pattern formed due to two point sources S and S’ is in good contrast. If S’ is moved away slowly from S, the contrast of the observed fringes becomes poorer, because the interference pattern formed by S’ is slightly shifted w.r.t. that produced by S. For a particular distance, the interference maximum produced by S falls on interference minimum produced by S’ and vice-versa. For such a situation, the interference fringe pattern is washed away and the uniform illumination is observed on screen-2.

18. and similarly
Thus for disappearance of fringes, the path difference,

19. or

20. For an extended source (of width l) made up of independent point sources, one may, therefore, say that the good interference fringes will be observed as long as
Equivalently, we can also say that interference fringes of good contrast can be formed from a given source of width l by the interference of light from two points S1 and S2 separated by varying distance d as long as

21. Since l/D is an angle, say θ, subtended by the source at the slits, this upper equation can be written as

22. Thus, in practice, for a point (say S1) on the wave front, there is some finite area around it, any point (such as S2), within which will have good phase correlation with S1. By holding one pin hole, say S1 fixed and moving other pin hole S2 about it, one can look for any reduction in fringe contrast (or fringe visibility). The area over which pin hole can be moved and the interference fringes still be seen is called “COHERENT AREA” of the light wave. The distanced “d” between the pin holes for which the fringes just disappear is known as “TRANSVERSE (or LATERAL) COHERENCE LENGTH and the corresponding spatial coherence is termed as TRANSVERSE COHERENCE since it characterizes the spatial variation in coherence across the wave front in the direction transverse to the direction of propagation .

23. Conditions of constructive and destructive interference( Superposition of waves)
Consider two coherent sources of light having displacements as
and
By using principal of superposition at point P, we get

24. Putting
This is simple harmonic wave and Amplitude of resultant wave can be find out by squaring and adding (4) and (5)

25. (ii) Condition for constructive interference
Intensity I is directly proportional to the square of the amplitude of the resultant wave
(ii) Condition for constructive interference

26. (iii) Condition for destructive interference

28. (iv) Energy conservation in interference
Also intensity from two sources at every point is
Which is same as eq. (12)

31. Young’s double slit experiment

32. From triangle S2BP
From triangle S1BP

33. Using eq. (13)
Since both slits are very close

34. Constructive interference (for bright band)

35. Destructive interference (for dark band)

36. White Light Fringes
The fringe width as given above depends upon wave length, it is more for red and less for violet colour. In fact, the fringe width for red light is about double of that for the red light. If white light is used in place of monochromatic light in Young’s double slit experiment, a separate fringe pattern of any p th fringe is obtained for each wavelength. However, the position of central fringe will be same for all colours.

37. Fresnel’s Biprism InterferenceA C D

38. Fresnel’s used a biprism to obtain coherent sources by refraction to produce interference pattern. Fresnel’s biprism ABC consists of two acute angles at B and C are of 30 minutes each and angle A of 179 degrees.
Biprism produces two virtual images S1 and S2 (separated by distance d) of a fine slit S illuminated by a monochromatic source of light by refraction by upper and lower halves of the prism as shown above.
D is the distance between source and screen, then the fringe width as determined in Young’s double slit experiment is given by eq. (15) as
Fringe pattern is formed in the region between b and c as shown in the figure.

39. Interference and diffraction fringes produced in the Fresnel biprism experiment.

40. d is measured by displacement method
d is measured by displacement method. For this a convex lens is placed between eye piece and biprism at L1 as shown in figure and obtain the distance d1 between the sharp image of S1and S2 seen in the eye piece.

41. For the first position of the lens
Similarly move the biprism to a dotted position, so that the images of S1and S2 are again visible sharply in the eye piece and again measure the distance d2.
For the first position of the lens
Also for the second position of the lens at dotted position
Multiplying these two

42. Now the wave length of light can be measured using equation (15)

43. Lloyd’s Mirror Experimentf d M D

44. An arrangement for producing an interference pattern with a single light source
Wave reach screen from source S₁ either by a direct path or by reflection
The reflected ray can be treated as a ray from the source S₂ behind the mirror
The interference pattern is observed in the overlapping region bc
The point f being equal distance from S₁ and S₂, gives position of central fringe, is not visible as no reflected light reaches f.
Nature of fringes and expression are same as in Young’s double slit.
The fringe shift is gives by

45. The distance d can be measured by displacement method as in Fresnel’s biprism experiment and D directly as measured
Nature of central Fringe The central fringe will be dark and it can be seen only when screen touches the edge of mirror. This is because there is a 180° phase change produced by the reflection ray.

46. Achromatic fringes
In the Lloyd’s method of interference, coloured fringes are formed, because the fringe width as given above depends upon wave length, it is more for red and less for violet colour. Therefore overlapping is not complete and as result of which around central maximum, few colours are formed.
However, if one can arrange = constant, the fringes becomes independent of wavelength.
The fringe width become constant for all colours and central maximum will appears white. They are called achromatic fringes with white light.

47. L M
Achromatic fringes with white light are possible because of lateral inversion of image in Lloyd’s mirror. If a spectrum R V is formed, its reflected image V’ R’ is produced.

48. Application of interference
(i) Lateral displacement of fringes and Retardation of optical Path (determination of refractive index of transparent material of thickness t)

49. When a transparent material of refractive index µ and of thickness t is placed in the path of one of the beam, then the path of that very beam become longer by
(µ-1)t.
Becomes as

50. For bright fringe
The distance where the central fringe is formed when the plate is placed in one of the path.

51. The fringe shift Δy in the nth fringe when the plate is placed in one of the paths.
The fringe shift can be calculated by knowing other parameters

52. (ii)Determination of refractive index of a gas at any pressure.
Rayleigh’s Refractometer is determination of refractive index of a gas at any pressure.

53. In the double slit experiment, two beams are made to pass separately through the tube T₁ and T₂ of around one meter length.
The gas whose refractive is to be measured is filled in two tubes T₁ and T₂ are filled at same pressure. The position of fringe pattern is noted using eye piece.
Then gas in one of the tubes is filled at different pressure.
Because of change in pressure in one of the tubes, there is a lateral shift in the fringes.
∆µ = change in the refractive index with the change in pressure of gas.
n = number of fringes displayed.
L = length of the tube.

54. By using this equation Δμ can be calculated
By using this equation Δμ can be calculated. Thus the refractive of any gas can be calculated as (µ - 1) proportional to pressure P i.e.
Where K is constant. Hence

55. (iii) Measurement of small thickness
n fringes t P X’ X

56. Methods Of Obtaining Interference Pattern
Two methods are:-
Division of Wave front - The two interfering light waves were produced by division of wave front. For example, in Young's double slit experiment, Lloyd’s mirror, Fresnel’s Biprism.
Division of Amplitude - In this method, the same beam of light is partially reflected from two surfaces. The two reflected beams so produced interfere as the phase difference is constant , which is an essential condition for coherence. For example, Colour of thin films, Newton’s Ring Experiment, Michelson Morley Interferometer.

57. Interference due to thin filmsA i L E i D B r r r r E i C i F

58. Reflected system
Optical path difference between rays r1 and r2

59. path difference
The ray suffers reflection from denser medium , therefore and additional phase difference of π radian or a path difference of is introduced.

60. For constructive interference
i.e. for bright fringes
For destructive interference
i.e. for dark fringes

61. Transmitted system
Optical path difference between rays r1 and r2

62. Since reflection takes place from rare medium therefore, no additional phase diff. is introduced.
For constructive interference
i.e. for bright fringes
For destructive interference
i.e. for dark fringes
By compression of the interference in both systems we can see that the reflected and transmitted systems are complementary to each other.

63. Newton’s rings
Another method for viewing interference is to place a planoconvex lens on top of a flat glass surface. The air film between the glass surfaces varies in thickness from zero at the point of contact to some thickness d. A pattern of light and dark rings is observed.
These rings are called Newton’s rings.
Newton’s Rings can be used to test optical lenses

64. Ray 1 undergoes a phase change of 180 degree on reflection, whereas ray 2 does not undergoes any phase change.
R= radius of curvature of lens
r = radius of Newton’s ring

65. t

66. For air . as the incident ray is normal, hence r = 0,
The path difference between rays reflected from thickness t is given by
For air as the incident ray
is normal, hence r = 0,
Determination of t
For bright fringes
For dark fringes
Thus for dark fringes

67. So, radius of nth dark ring,
Radius of the dark rings is proportional to the square root of natural integral number
Similarly for bright fringe
Radius of the bright rings is proportional to the square root of odd integral number

68. Michelson’s Interferometer

69. In Michelson’s Interferometer a beam of monochromatic light from a extended source is splitted into two beams so that one beam strikes a fixed mirror and the other a movable mirror. When the reflected beams from these two mirrors brought back together, an interference pattern results.
Compensated plate P2 is introduced just to make the optical path equal for both reflected and transmitted beams. Because the reflected beam passes twice from the glass plate P1
The observation in the telescope (detector) due to these two beams is considered to be obtained due to reflection from mirror M1 and the image M2’ of mirror M2 in mirror M1.

70. For dark fringes For bright fringes
Thickness between position of two mirrors M1 and M2’ = t
For dark fringes
Path diff. between coherent beams
As µ = 1 for air therefore path diff.
For bright fringes
Similarly for dark fringes, path diff.
In both the cases for given vales of t, n and λ, the angle r will be constant is constant. Therefore the fringes are circular like Newton’s Rings but they are formed at infinity and hence can be seen using naked eye or with telescope.

71. In Michelson’s experiment fringes of various types can be formed i. e
In Michelson’s experiment fringes of various types can be formed i.e. circular, straight, parabolic, elliptical or even hyperbolic. This depends upon, the separation and the inclination between the mirror M1 and virtual image of M2’ of mirror M2.

72. Circular fringes

73. As in figure above for circular fringes is clear that an additional path difference of λ/2 is introduce in the beam reflected at mirror M2’
Therefore for dark fringes
Therefore for bright fringes
At normal incidence (r = 0), for dark fringes the path diff becomes

75. If are the angles for first, second, third…
If are the angles for first, second, third….rings and are their respective radii of the rings.
Where x is the distance of M2 from source. Therefore, it is clear that
Special case if t = 0 then M1 and M2 coincides i.e. path difference between the rays will be λ/2, therefore, the central fringe is bright.

76. Applications of Michelson’s Interferometer
(i) Measurement of wavelength.
For nth fringe
At center r = 0, therefore cos r = 1
Now decrease and the number of fringes disappearing at the center is N then

77. (ii) Determination of difference of two close wavelengths and width of spectral lines
Let the two closely spaced lines have wavelengths λ₁ and λ₂ . When interferometer is adjusted for circular fringes, each wavelengths produce their own rings. The mirror is moved so that best contrast circular fringes are obtained. It will happen when path difference is such that maximum due to λ₁ coincides with the maximum due to λ₂. Under these conditions, say n₁ order of λ₁ coincides with n₂ order of λ₂.
As the mirror separation in increased gradually, the contrast decreases and becomes worst and then again increases and becomes best again.

78. Let the mirror M₂ is displaced by d, while moving from one contrast to next contrast. The condition occur when (n₁+m) order due to λ₁ coincide with (n₂+m+1) due to λ₂. The new position implies

79. Putting the value of m from this equation into eqn. (3)
λ₁ and λ₂ are very close therefore
Now eqn. (4) becomes
Difference in wave lengths can be determined by knowing the average value of wavelength.

80. (iii) Determination of thickness of transparent sheet.
Using white light fringes are obtained
A transparent sheet of thickness t is introduced in the path of one of the rays.
Net path difference introduced is x = 2(µ - 1)t.
Now the mirror M2 is moved by a distance d such that central white fringe reoccupies its original position.
The thickness of transparent sheet d is given by

81. (iv) Determination of refractive index of transparent film.
First the white light fringes are obtained and then a transparent film of thickness t is introduced in one of the interfering beams.
Then the increase in the path will be and central fringe shifts. Here the factor appears as the ray has to pass the sheet twice.
Let n is the number if fringes crossing the field of view, then.
Thus wave length can be determined.

82. (v) Standardization of meter.
Michelson calibrated standard meter in
1 meter is found to be equal to red cadmium wavelengths and the accuracy was of the order of
Wave length of red line of cadmium

83. Stokes’ Law
This shows that there is reversal of phase when the ray of light is reflected by a denser medium and there is no absorption of light.

84. Multiple Beam Interference

86. The path difference between (1) and (2) is
The ray (1) undergoes a phase change of π at reflection, while ray (2) does not, since it is internally reflected. Then the condition
becomes a condition of destructive interference as far as ray (1) and (2) are concerned. The path difference between (2) and (3) is given by
But here only internal reflections are involved, the ray (3) is in phase with (2). The same holds for successive pairs, so under these conditions (1) and (2) will be out of phase, but the rays (2), (3), (4)….will be in phase with each other. On the other hand if the condition are such that
Minima
Maxima

87. Ray (2) will be in phase with (1), but (3), (5), (7)…
Ray (2) will be in phase with (1), but (3), (5), (7)…. Are out of phase with (2), (4), (6),….since (2) is more intense than (3), (4) is more intense than (5), etc., these pair cannot cancel each other, and since the stronger series combines with (1), the strongest of all, there will be a maximum of intensity.
For the minimum intensity, ray (2) is out of phase with ray (1), but (1) considerable greater amplitude than (2), so that these two will not completely annul each other. The addition of (3), (4), (5), …, which all in phase with (2), will give a net amplitude just sufficient to make up the difference and to produced complete darkness at the minimum. Using a for the amplitude of the incident wave, r for the fraction of this reflection, t and t’ for the fraction transmitted in going from rare to dense or dense to rare

89. From the above figure, the resultant amplitude is obtained as
Since r is necessarily less than 1, therefore the geometrical in bracket has finite sum given as
From Stokes’ law

90. This is just equal to the amplitude of the first reflected ray, so there will be complete interference.

91. Complex Amplitude
The simple harmonic wave in exponential form is written as
Where kx = δ

92. the time-varying factor exp(iωt) is of no importance in combining waves of same frequency, since the amplitudes and relative phases are independent of time. The factor exp( -iδ), is called complex amplitude. The negative sign merely indicate that the phase is behind the standard phase. In general vector a is given by
The advantage of using complex amplitudes lies in the fact that the algebraic addition of two or more is equivalent two vector addition of the real amplitudes.

94. Fabry Perot Interferometer

95. Principle It is based on the principle of interference by multiple reflections.
Construction it consists of two quartz plates, placed parallel to each other, so that a film of air is enclosed between them as in the Fig. shown above. Air film form the medium in which the multiple reflection occur. The inner plates are coated with a thin layer of silver to make the surface partially reflecting. The outer surfaces of the plates are made slightly inclined w.r.t. the inner surfaces, to avoid the spurious interference due to plates themselves acting as the medium multiple reflections.
Working The transmitted beams of light are brought focus on the screen with the help of a lens L. the separation between the plates is ‘d’ , then the point P₂ at which the rays comes to focus will be maximum or minimum depending upon the path difference.

96. Derivation of intensity in Fabry Perot Interferometer

97. For the fringe system formed by transmitted light, the sum of complex amplitudes is
The infinite geometric series in the second parentheses has the common ration r²exp(iδ), and has a finite sum because r²<1. summing the series, one obtains
Intensity is at point P₂ proportional to the square of the amplitude as

98. And maximum intensity =

100. For minimum intensity

101. Sharpness of fringes The fringes are said to be sharp if the intensity falls rapidly on either side of maxim. sharpness of the fringes is given by
The sharpness if the fringes is studies in terms of half fringe width i.e., width of fringe corresponding to half of the maximum intensity.
For half fringe width
Intensity from eq. (1)

102. For δ is small
For r =1, then δ = 0
i. e. maximum visibility, for fringes to be sharp δ should be as small as possible. The value of δ can be controlled by varying r only.

103. The visibility is maximum

104. For Michelson’s interferometer the intensity is given by
At half width
eq. (X) and (Y) shows that the fringes obtained with Fabry Perot interferometer are comparatively much sharper than those obtained with the Michelson’s interferometer.