INTERFERENCE IN THIN FILMS�����Dr. U. N. Patel
Newton’s rings are an example of fringes of equal thickness. Newton’s rings are formed when a plano-convex lens P of a large radius of curvature placed on a sheet of plane glass AB is illuminated from the top with monochromatic light (Fig. 1)
Fig. 1
The combination forms a thin circular air film of variable thickness in all directions around the point of contact of the lens and the glass plate. The locus of all points corresponding to specific thickness of air film falls on a circle whose centre is at O. Consequently, interference fringes are observed in the form of a series of concentric rings with their centre at O. Newton originally observed these concentric circular fringes and hence they are called Newton’s rings.
The experimental arrangement for observing Newton’s rings is shown in Fig. 1.
Monochromatic light from an extended source S is rendered parallel by a lens L. It is incident on a glass plate inclined at 45° to the horizontal, and is reflected normally down onto a plano-convex lens placed on a flat glass plate. Part of the light incident on the system is reflected from the glass-to-air boundary, say from point D (Fig. 2). The remainder of
Fig.2
the light is transmitted through the air film. It is again reflected from the air-to-glass boundary, say from point J.
The two rays reflected from the top and bottom of the air film are derived through division of amplitude from the same incident ray CD and are therefore coherent. The rays 1 and 2 are close to each other and interfere to produce darkness or brightness. The condition of brightness or darkness depends on the path difference between the two reflected light rays, which in turn depends on the thickness of the air film at the point of incidence.
1.1 CONDITION FOR BRIGHT AND DARK RINGS :
The optical path difference between the rays is given by Δ = 2μtcosr - λ/2. since μ = 1 for air and cos r = 1 for normal incidence of light.
Δ = 2t - λ/2 -----(1)
Intensity maxima occur when the optical path difference Δ = mλ. If the difference in the optical path between the two rays is equal to an integral number of full waves, then the rays meet each other in phase. The crests of one wave falls on the crests of the other and the waves interfere constructively. Thus if 2t - λ/2 = mλ
Bright fringe is obtained.
Intensity minima occur when the optical path difference is Δ = (2m + 1) λ/2. If the difference in the optical path between the two rays is equal to an odd integral number of half waves, then the rays meet each other in opposite phase. The crests of one wave fall on the troughs of the other and the waves interfere destructively.
Hence, if 2t - λ/2 = (2m + 1) λ/2
2t = mλ ---------(3)
Dark fringe is produced.
1.2 CIRCULAR FRINGES
In Newton’s ring arrangement, a thin air film is enclosed between a plano-convex lens and a glass plate. The Thickness of the air film at the point of contact is zero and gradually increases as we move outward. The locus of points where the air film has the same thickness then fall on a circle whose centre is the point of contact. Thus, the thickness of air film is constant at points on any circle having the point of lens-glass plate contact as the centre. The fringes are therefore circular.
Fig. 3
1.3 RADII OF DARK FRINGES
FIG. 4
Let R be the radius of curvature of the lens (Fig. 4).
Let a dark fringe be located at Q. Let the thickness of the air film at Q be PQ = t. Let
the radius of the circular fringe at Q be OQ = rm. By the Pythagorus theorem.
It means that the radii of the dark rings are proportional to under root of the natural numbers. The above relation also implies that Thus, the radius of the mth dark ring is proportional to under root of wavelength.
Ring Diameter :
Diameter of mth dark ring
1.4. SPACING BETWEEN FRINGES
It is seen that the diameter of dark rings is given by
Where m = 1,2,3, …………….
The diameters of dark rings are proportional to the square root of the natural numbers.
Therefore, the diameter of the ring does not increase in the same proportion as the order of the ring, for example, if m increases as 1,2,3,4, ………… the diameters are
and so on.
Therefore, the rings get closer and closer, as m increases. This is why the rings are not evenly spaced.
1.5 FRINGES OF EQUAL THICKNESS
Newton’s rings are formed as result of interference between light waves reflected from the top and bottom surfaces of a thin air
film enclosed between a plano-convex lens and a plane glass plate. The occurrence of alternate bright and dark rings depends on the optical path difference arising between the reflected rays. If the light falls normally on the air film the optical path difference between the waves reflected from the two surfaces of the film is
Δ = 2t - λ/2
It is seen that the path difference between the reflected rays arises due to the variation in the thickness ‘t’ of the air film. Reflected light will be of minimum intensity for those thickness for which the path difference is mλ and maximum intensity for those thickness for which the path difference is (2m+1)λ/2. Thus, each maxima and minima is a locus of constant film thickness. Therefore, the fringes are known as fringes of equal thickness.
1.6. DARK CENTRAL SPOT
The central spot is dark as seen by reflection. Newton’s rings are produced due to superposition of light rays reflected from the top and bottom surfaces of a thin air film enclosed between a plano-convex lens and a plane glass plate.
Fig. 5
The occurrence of brightness or darkness depends on the optical path difference arising between the reflected rays. The optical path difference is given by Δ = 2t - λ/2.
At the point of contact ‘O’ of the lens and glass plate (Fig. 5),
the thickness of air film is negligibly small compared to a wavelength of light.
The wave reflected from the lower surface of the air film suffers a phase change of π while the wave reflected from the upper surface of the film does not suffer such change.
Thus, the superposing waves are out of step by λ/2 which is equivalent to a phase difference of 180° (or π rad). Thus the two interfering waves at the centre are opposite in phase and produce a dark spot.
1.7 DETERMINATION OF WAVELENGTH OF LIGHT
A plano-convex lens of large radius of curvature (about 100 cm) and a flat glass plate are cleaned. The lens is kept with its convex face on
The glass plate and they are held in position with the help of a metal ring arrangement. The system is held under a low power travelling microscope kept before a sodium vapour lamp. It is arranged that the yellow light coming from the sodium lamp falls on a glass plate held at 45° light beam. The light is turned through 90° and is incident normally on the lens-plate system. The microscope is adjusted till the circular rings came into focus. The centre of the cross-wire is made to come into focus on the centre of the dark spot, which is at the centre of the circular ring system. Now, turning the screw the microscope is moved on the carriage slowly towards one side, say right side.
AS the cross-wire move in the field of view, dark rings are counted. The movement is stopped when the 22nd dark ring is reached. Then the microscope is moved in the opposite direction and stopped at the 20th or 19th dark ring. The vertical cross-wire is made tangential to the 19th ring and the reading is noted with the help of the scale graduated on the carriage. Thus, starting from the 19th ring, the tangential positions of the 18th,17th, 16th, ………,5th dark rings are noted down. Now, the microscope is moved quickly to the left side of the ring system and it is stopped at the 5th dark ring. The cross-wire is again made tangential to the 5th dark ring and its position is noted.
The difference between the readings on right and left sides of the 5th dark ring gives its diameter value. The procedure is repeated till 19th ring is reached and its reading is noted. From the value of the diameters the square of the diameters are calculated. A graph is plotted between and the ring number ‘m’. A straight line would be obtained as shown in Fig. 6.
Fig. 6
We have
For the (m+p)th ring,
The slope of the straight line (Fig. 6) gives the value of 4λR. Thus,
The radius of curvature R of the lens may be determine using a spherometer and λ is computed with the help of the above equation.
1.8 REFRACTIVE INDEX OF A LIQUID
The liquid, whose refractive index is to be determined, is filled in the gap between the lens and plane glass plate. Now the liquid substitutes the air film. The condition for interference may then be written as
Where μ is the refractive index of the liquid. For normal incidence the equation becomes
Following the above relation, the diameter of mth dark ring may be expressed as
1.9 NEWTON’S RINGS IN TRANSMITTED LIGHT
Newton’s rings in transmitted light may be observed with the arrangement made as in Fig. 7.
Fig. 7
The condition for maxima or bright rings is
As μ=1 for air and r = 0 for normal observation, the above expression may be simplified to
1.10 NEWTON’S RINGS FORMED BY TWO CURVED SURFACES
Case 1 : Lower surface concave :
Let us consider two curved surfaces of radii of curvature R1 and R2 in contact at the point O.
A thin air film is enclosed between the two surfaces. The dark and bright rings are formed and can be viewed with a travelling microscope.
Suppose the radius of the mth dark ring is r. The thickness of the air film at P is
PQ = PT – QT
From geometry,
But PQ = t. The condition for dark rings in reflected light is given by
As μ = 1 and cos r = 1 for normal incidence, the above condition reduces to 2 t = mλ.
For bright fringes the condition is
Case 2 : Lower surface Convex :
Let us consider two curved surfaces of radii of curvature R1 and R2 in contact at the point O.
A thin air film is enclosed between the two surfaces. The dark and bright rings are formed and can be viewed with a travelling microscope. Suppose the radius of the mth dark ring is r. The thickness of the air film at P is
PQ = PT + QT
From geometry
But PQ = t. The condition for dark rings in reflected light is given by
For bright fringes the condition is
2. MULTIPLE BEAM INTERFERENCE
We assumed that the high order reflections occurring at interfaces of thin film are negligible. However, if for any reason the reflectance of the interfaces is not negligible, then the higher order reflections are to be taken into account. When the reflected or transmitted beams meet,
multiple beam interference takes place. We are specifically interested in the fringes associated with an air space between two reflecting surfaces. Usually, these surfaces consist of metal films deposited on glass plates.
Let us consider the reflected rays 1,2,3, etc as
Fig. 1
shown in Fig. 1. The amplitude of the incident ray is a. Let ρ be the reflection coefficient, τ the transmission coefficient.
The amplitude coefficient of reflection is
If the film does not absorb light, the amplitude of the reflected and transmitted waves are aρ and a (1-ρ) respectively.
2.1 INTENSITY DISTRIBUTION
Let a be the amplitude of the light incident on the first surface. A certain fraction of this light, aρ, is reflected and another fraction, aτ is transmitted (Fig. 1). The factors ρ and τ are known as the amplitude reflection coefficient and amplitude transmission coefficient respectively. Again, at the second surface, part of the light is reflected with amplitude aρ2 and part is transmitted with amplitude aτ2.
The next ray is transmitted with an aρ2τ2, the next one with after that with aρ4τ2 and so on. If T and R be the fractions of the incident light intensity which are respectively transmitted and reflected at each silvered surface, then τ2=T and ρ2=R. Therefore, the amplitude of the successive rays transmitted through the pair of plates will be
a T, a T R, a T R2, ……..
In complex notation, the incident amplitude is given by E = a ei ω t.
Then the waves reaching a point on the screen will be
By the principle of superposition, the resultant amplitude is given by
Using the expression for sum of the terms of geometrical progression, we get
When the number of terms in the above expression approaches infinity, the term RN e –I N δ tends to zero, and the transmitted amplitude reduces to
The complex conjugate of A is given by
The intensity will be maximum when , i.e. δ= 2 m π, where m= 0,1,2,3,…… Thus
The intensity will be a minimum, when
where m = 0,1,2,3, ….. Thus,
We can now rewrite the equ. (4) as
Similarly, the interference intensity from the reflected light beams can be shown to be
2.2 COEFFICIENT OF FINESSE
we now introduce a quantity F, which is called the coefficient of finesse. It is defined as
Then the relative interference intensity distribution can be expressed as
2.3 VISIBILITY OF FRINGES
The visibility of fringes is given by
Substituting the values of Imax and Imin, we get
Equ. (12) shows that the visibility of fringes is a function of reflectivity only. The visibility of fringes increases with increase in the value of R. V reaches the value 0.8 when R = 0.5 and approaches unity as R approaches 1. Thus the higher the reflectivity, the greater is the contrast of the fringes.
2.4 SHARPNESS OF THE FRINGES
If a plot is drawn for I against δ at different values of R, we obtain a set of curves, as shown in Fig. 2.
Fig. 2
It is noted from the graphs that the intensity falls off on both sides of the maximum at higher values of R.
The sharpness of fringe is measured by the half-width of the curve. The half-width is the width of the I - δ curve at the position where
3. FABRY-PEROT INTERFEROMETER AND ETALON
The Fabry-Perot interferometer is a high resolving power instrument, which makes use of the ‘fringes of equal inclination’, produced by the transmitted light after multiple reflections in an air film between two parallel highly reflecting glass plates.
Fig. 1
The interferometer consists of two optically plane glass plates A and B with their accurately parallel to each other. Screws are provided to secure parallelism if disturbed.
This system is difficult to manufacture and is no more in use. Instead an etalon which is much more easily manufactured is used. The etalon consists of two semi-silvered plates rigidly held parallel at a fixed distance apart. The reflectance of the two surfaces can be as high as 90 to 99.9%. Although both reflected and transmitted beams interfere with
each other, the Fabry-Perot interferometer is usually used in the transmissive mode.
S is a broad source of monochromatic light and L1 a convex lens (not shown in Fig. 1) which makes the ray parallel. An incident ray suffers a large number of internal reflections successively at the two silvered surfaces, as shown in Fig. 1. At each reflection a small fraction of the light is transmitted also. Thus, each incident ray produces a group of coherent and parallel, transmitted rays with a constant path difference between any two successive rays. A second convex lens L brings these rays together to a point in its focal plane where they interfere.
Hence the rays from all points of the sources produce an interference pattern on a screen placed in the focal plane of L.
3.1 FORMATION OF FRINGES
Let d be the separation between the two silvered surfaces and θ the inclination of a particular ray with the normal to the plates. The path difference between any two successive transmitted rays corresponding to the incident ray is 2d cos θ. The condition for these rays to produce maximum intensity is given by
2 d cos θ = m λ
Where m is an integer. The locus of points in the source, which give rays of constant inclination,
3.2 DETERMINATION OF WAVELENGTH
When the reflecting surfaces A and B of the interferometer are adjusted exactly parallel, circular fringes are obtained. Let m be the order of the bright fringe at the centre of the fringe system. As at the centre θ = 0, we have
2 t = m λ
If the movable plate is moved a distance λ/2, 2t changes by λ and hence a bright fringe of the next order appears at the centre.
If the movable plate is moved from the position x1 the number of fringes appearing at the centre during this movement is N, then
Measuring x2, x1 and N, we can determine the value of λ.
3.3 MEASUREMENT OF DIFFERENCE IN WAVELENGTH
The light emitted by a source may consist of two or more wavelength, as D1 and D2 lines in case of sodium. Separate fringe patterns corresponding to the two wavelengths are not produced in Michelson Interferometer.
Hence, Michelson interferometer is not suitable to study the fine structure of spectral lines. On the other hand, in Fabry-Perot interferometer, each wavelength produces its own ring pattern and the patterns are separated from each other. Therefore, Fabry-Perot interferometer is suitable to study the fine structure of spectral lines.
Difference in wavelength can be found using coincidence method. Let λ1 and λ2 be two very close wavelengths in the incident light. Let us assume that λ1> λ2 . Initially, the two plates of the interferometer are brought into contact. Then the rings due to λ1 and λ2 coincide partially.
Then the movable plate is slowly moved away such that the ring systems separate and maximum discordance occurs. Then the rings due to λ2 are half way between those due to λ1. Let t1 be the separation between the plates when maximum discordance occurs. At the centre
Using the value of m1 in equ. (2), we get
(since λ1 λ2 = as λ1 - λ2 is very small).
When the separation between the plates is further increased, the ring systems coincide again and then separate out and maximum discordance occurs once again. If t2 is the thickness now
Using the above expression into equ. (6), we obtain