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INTERFERENCE OF LIGHT Light is an electromagnetic radiation we can see with our eyes light has the properties of a wave. One of these properties is interference, caused by the addition of waves

Introduction: If two or more waves cross one another in the same medium, each wave produces its own effect totally independent of the effects due to the other. Principle of superposition tells that at any instance the resultant displacement is equal to the vector sum of the individual displacements produced by each wave. This is based on the theory of interference of light discovered by Thomas Young in 1801.

With single source of light the energy distribution in the surrounding medium is uniform. But when there are two adjacent sources of light giving out light waves of the same wavelength, amplitude and having zeroed or constant phase difference, the distribution of energy is no longer uniform. At some points where the crest of one wave falls upon the crest of the other or the trough of one wave falls on the trough of the other, the resultant amplitude is large and hence the intensity become maximum. At other points where the crest of one wave falls on the trough of the other wave and vice versa, the resultant amplitude is reduced to zero and intensity is minimum. The modification in the distribution of light energy due o superposition of two or more waves of light is called interference of light. It should be noted that in this phenomenon there is only transference of energy from one region to another. The energy missing at one region reappears at the other region. There is no creation destruction of energy in any region as a result of the superposition of the waves of light.

Because of interference there are alternate bright and dark regions observed and they are called interference bands or interference fringes. The color pattern observed on soap bubbles and oil on the wet roads are due to interference of light. If a convex lens of large radius of curvature is placed on a glass plate and illuminated with monochromatic light source, alternately bright and dark concentric rings (called as Newton’s rings) are observed because of interference. Similarly if a wedge shaped air film between two plane glass plates (air wedge) is illuminated with monochromatic light, alternate bright and dark bands are formed because of interference, the bands being parallel to the edge of the wedge.

Concepts:

Light travels in waves. Sometimes, two or more waves join together. Interference is the addition, or coming together, of several waves. Constructive Interference happens when two or more waves come together to form a larger and stronger wave, matching their crests and troughs. Destructive Interference is when two or more waves come together and cancel each other out to make a weaker wave.

Principles:

Since light has wave properties, it will experience interference (the addition of waves). This interference is like that seen with water waves. Whether you get constructive or destructive interference depends on the wavelength (color) of the light. White light is made up of many colors. These different colors have different wavelengths. We can see the many colors of light on bubbles. The separation of white light into many colors on a bubble happens because of interference.

Thomas Young’s double slit Experiment: One of the most important experiments of wave theory is that of Young's double slits. It is a clear example of the diffraction of light conducted with essentially basic scientific equipment. Thomas Young was a not only a physicist but also a physician and Egyptologist, who was responsible for deciphering the Rosetta stone. He devised an experiment in the early 1800's that proved that light is a wave. The experiment has been used subsequently to show that wave behaviors exists in many other areas of nature and therefore it is worth spending a little time going into the experiment in detail. When two light beams interact they create interference which can be constructive or destructive as we have discussed earlier. The places where constructive and destructive interference occur are subject to constant change, since electromagnetic waves emitted are capable of varying phase. Using one light source and splitting it into two beams you can create two coherent sources, meaning they are of identical frequencies and have a constant phase difference (the distance between a peak of wave 1 and wave 2 is always the same) It is also important to use monochromatic light for this experiment as the location of interference occurs is wavelength dependent.

Interference pattern on completely flat surface

The apparatus consists of a source matching the above requirements, a screen with two very thin identical slits or the order of a wavelength in width, and a screen to view the interference on. In principle, a light bulb, or lamp can be used, but the light must be reduced to a monochromatic source with filters. It can also be done by splitting the light into its various frequencies using a prism, a frequency can be selected by channeling the light with a further screen with only a pin prick in it to allow the light out. These days’ lasers are used to provide the light source. When the light is switched on, it travels up to the first screen and is split into two beams by the slits, we have seen that when this happens waves are diffracted and bulge outwards causing two curved wave fronts to propagate the other side of the slits, at many places between the slits and the viewing screen there are areas of constructive interference and by moving the viewing screen it is possible to get a picture of where they are occurring

Conditions for Sustained Interference: Light waves from two sources can form a sustained (well defined and observable) interference pattern where they meet, only if they satisfy the following conditions: Two light waves superposing at a point must have the same wavelength or frequency The amplitude of superimposing light waves should be equal or almost equal. Two light waves superposing at a point should either have the same phase or constant phase difference. The two sources emitting light waves should be very narrow. The sources emitting light waves should be very close to each other.

Coherent Sources: Any two sources of light continuously emitting light waves having zeroed or constant phase difference are called coherent sources. Thus coherent sources have a definite phase relationship between them and emit light waves of equal frequency or wavelength. Interference can occur only with coherent sources. But two independent sources cannot be coherent because even though they may emit light waves of equal wavelength and equal amplitude, they may not have constant phase difference. Therefore, to obtain two interfering beams, a single source is used and its beam is split into two by following ways:

Division of wave front: In this case the wave front is divided into two parts by reflection, refraction or diffraction so hat those two parts reunite at a small angle and produce interference. (Example: Fresnel’s biprism, Lloyd’s mirror)

Division of Amplitude: In this case the wave front is split up into two parts by partial refraction and reflection at a surface and these two parts are later made to reunite to produce interference. (Example: Air wedge, Newton’s Rings)

Air Wedge:

Newton’s Rings:

Newton’s Rings Explanation:

Given a hemisphere of radius R resting on a plane, monochromatic light falling normally on the top of the hemisphere reflects off both the plane and the hemisphere. Near the center of the hemisphere, refraction can be neglected, and the rays create rings of interference. On the curved surface, a point a distance r from the center will be a height d above the plane (1)

(2)

Reflection from hemisphere undergoes a phase change of

. Reflection from the

plane gives a phase change of . The phase difference between rays reflected off the plane and hemisphere is therefore

(3)

for

, and the interference pattern goes as

Constructive interference occurs when

. with m = 0, 1, 2, ..., i.e.,

(4)

so (5)

(6)

(7)

(8)

Solving for

gives (9)

which, for

, gives

(10)

Theory of Interference (Mathematical Model): Consider two light waves of same frequency, wavelength traveling in same directions. Let a1 and a2 be the amplitudes of the two waves. The displacements of any particle in the medium due to these waves at any instance of time t are y1 = a1 sin ωt and y2 = a2 sin (ωt + δ) Where δ is phase difference between the waves and ω = 2πf is the angular frequency of the waves. Thus the resultant displacement of the particle is the sum of both equations = y1 + y2 = a1 sin ωt + a2 sin (ωt + δ) = a1 sin ωt + a2 (sin ωt cos δ + cos ωt sin δ) = a1 sin ωt + a2 sin ωt cos δ + a2 cos ωt sin δ = (a1 + a2 cos δ) sin ωt + a2 cos ωt sin δ -------------- (1) Let Rcosθ = (a1 + a2 cos δ) -------------------------------- (2) And Rsinθ = a2 sin δ ----------------------------------------- (3) Thus y = R [sin ωt cos θ + cos ωt sin θ] y = R sin (ωt + θ) --------------------------------------------- (4) y

Equation (4) represents the a simple harmonic vibration of amplitude R and angular frequency ω. Thus the resultant wave is also a simple harmonic wave with same frequency as those individual waves. Squaring and adding equation (2) and (3) R2 cos2θ + R2 sin2θ = (a1 + a2 cosδ) 2 + (a2 sinδ) 2 R2 (cos2θ + sin2θ) = a12 + a22 cos2 δ + 2a1 a2 cos δ + a 22 sin2 δ i.e. R2 = a12 + a22 + 2a1 a2 cos δ --------------------------------------- (5) The phase angle θ is given by tanθ = ( Rsinθ / Rcosθ ) = ( a 2 sinδ /{a1 + a2 cosδ})------------(6)

Condition for constructive interference: When there is constructive interference the amplitude is maximum and hence the intensity is also maximum. From the equation (5) the resultant amplitude R is maximum when cosδ = +1 i.e, δ = 2nπ where n = 0, 1, 2, 3…….. Thus R will be maximum when the phase difference between the two waves is δ = 0, 2π, 4π, 6π …………., that is even multiple of π If Δ is the path difference between the waves corresponding to the phase difference δ, δ = (2π/λ) Δ or Δ = (λ/2π)δ = (λ/2π) 2nπ = nλ = 2n λ/2

For constructive interference the phase difference between the two light waves must be equal even multiple of π or the path difference between the waves must be equal to even multiple of λ/2 where λ is wave length of light. Rmax = (a12 + a22 + 2a1a2)1/2 = a1 + a2. The maximum intensity is I max α (a1 + a2)2 If a1 = a2 = a then I max α 4a2

Condition of destructive interference:

When there is destructive interference the amplitude is minimum and hence the intensity is also minimum. From equation (5) the resultant amplitude is R is minimum when cos δ = -1. δ = (2n + 1)π where n= 0, 1, 2, 3……… Thus R will be minimum when the phase difference between the two waves is δ = π, 3π, 5π, 7π………………. That is the odd multiple of π If Δ is the path difference between the waves corresponding to the phase difference δ, Δ = (λ/2 π) δ = (λ/2 π)(2n+1) π = (2n+1) λ/2 For destructive interference the phase difference between the two waves must be equal to odd multiple of π or the path difference between the waves must be equal to odd multiple of λ/2 where λ is wavelength of light. The minimum amplitude R min = (a12 + a22 - 2a1a2)1/2 = a1 + a2. The minimum intensity is I max α (a1 - a2)2 If a1 = a2 = a then I max = 0 and R min = 0

Width of Interference Fringes:

P

A

x

Q

O B

R D

In figure A and B represents the two coherent sources (slits in young’s double slit experiment) of light separated by a distance d. let a screen be placed at a distance D from light source. Then the point O on the screen is equidistant from A and B so that the path difference between the two light waves from A and B reaching O is zero. Thus the point O has the maximum intensity. Consider a point P at a distance x from O. The path difference between the light waves from A and B reaching the point P is Δ = BP – AP From the figure BP2 = BR2 + PR2 = D2 + [x + (d/2)] 2 AP2 = AQ2 + PQ 2 = D2 + [x - (d/2)] 2 BP2 - AP2 = {D2 + [x + (d/2)] 2} – {D2 + [x - (d/2)] 2} = [x + (d/2)] 2 - [x - (d/2)] 2 = 2xd Or (BP-AP) (BP+AP) = 2xd (BP-AP) = {(BP+AP)/2xd} Since P is very close to O, (BP+AP) = 2D Therefore path difference Δ = (BP-AP) = 2xd/2D = xd/D …………. (1) Thus the equation (1) represents the path difference between light waves from A and B interfering at the point P.

For Bright Fringes: For bright fringes or maximum intensity at P, the path difference must be an even multiple of λ/2 where λ is the wavelength of light used. xd/D = 2nλ/2 or x = nλD/d ……………………… (2) Equation (2) gives the position of bright fringes from the point O For n = 0, x0 = 0 For n = 1, x1 = λD/d For n = 2, x2 = λD/d For n = 3, x3 = λD/d and so on. The distance between the centers of any two consecutive bright fringes is called fringe width of bright fringes. It is given by β = x1 – x0 = x2 – x1 = x3 – x2 = …………. = λD/d Therefore β = λD/d ……………….. (3) Thus the widths of all the bright fringes are same.

For Dark Fringes: For dark fringe or minimum intensity at P, the path difference must be an odd multiple of λ/2. xd/D = (2n + 1) λ/2 or x = (2n + 1)λD/2d ………………………(4) Equation (2) gives the position of bright fringes from the point O For n = 0, x0 = λD/2d For n = 1, x1 = 3 λD/2d For n = 2, x2 = 5 λD/2d For n = 3, x3 = 7 λD/2d and so on. The distance between the centers of any two consecutive dark fringes is the fringe width of dark fringes. It is given by β = x1 – x0 = x2 – x1 = x3 – x2 = …………. = 2λD/2d = λD/d Therefore β = λD/d ……………….. (5) Thus the widths of all the dark fringes are same. From the equation (3) and (5), it is found that the fringe widths for bright fringes and the dark fringes are equal. The width β of interference fringes increases when Wavelength of light λ is increased. The distance D between the coherent sources and the screen is increased. The distance d between the two coherent sources or slits is decreased.

Interferometry: Interferometry is the science, and the art, of making measurements with coherent light. Light moves in waves, and light from a single source can travel different paths and then recombine in ways we call interference. When we measure that interference, we learn about the different paths the light has taken; and the yard-stick, that is, the lengthscale of the measurement, is the wavelength of light itself.

Primary Applications of Interferometry: Interferometry has got a wide range of applications following are some of the important applications of interferometry. 1. Optical Testing: Generally applications of interferometry used in testing are Measurement of surface quality. a. Flat surfaces b. Spherical surfaces c. Surface roughness d. Aspherical surfaces 2. Inspection: Interferometry has following applications in the field of inspection. a. Inspection of slip gauges b. Inspection of measurement standards 3. Direct phase measurements a. Multiple wavelengths b. Phase stepping and phase shifting 4. Aligning very high quality lenses, like those in cameras, telescopes, and photolithography tools called steppers, which are used in the fabrication of the intricate circuit patterns. 5. Sagnac interferometers are used as an alternative to gyroscopes to measure changes with respect to an inertial reference frame in fighter planes.

6. Space applications a. Radio Astronomy. b. Measuring light intensity c. Used in retrieving images from telescope. 7. To measure angular sizes as small as 0.0005 arc seconds from distant stars.

Interferometer: Interferometer is an instrument that measures the wavelengths of light and distances. It transmits two radio signals or beams of light and uses the interference principle (how they reinforce or neutralize each other) to determine the measurement.

Types of Interferometers:

Michelson Interferometer: Fabry-Perot Interferometers Spherical Interferometers Fringe Counting Interferometers Goge Block Interferometers Correlation Interferometer Intensity Interferometer Stellar Interferometer Sagnac Interferometer Pohl Interferometer Martin-Puplett Interferometer Mach-Zehnder Interferometer Single Frequency DC interferometer Twyman-Green Interferometer N.P.L. Flatness Interferometer

Types of light sources used for Interferometry:

Mercury Mercury 198 Cadmium Krypton Krypton 86 Thallium Sodium Helium Neon Gas lasers

Michelson Interferometer: This is the oldest type of all interferometers, which has subsequently been modified in several resects and lot of sophistication introduced. However Michelson using this interferometer, established exact relationship between meter and red wavelengths of Cadmium lamps. The Michelson interferometer used in the MichelsonMorley experiment of 1887 to determine the velocity of light had a half-silvered mirror to split an incident beam of light into two parts at right angles to one another. The two halves of the beam were then reflected off mirrors and rejoined. Any difference in the speed of light along the paths could be detected by the interference pattern. The failure of the experiment to detect any such difference threw doubt on the existence of the ether and thus paved the way for the special theory of relativity. Another type of interferometer devised by Michelson has been applied in measuring the diameters of certain stars.

Fabry-Perot Interferometer:

An interferometer in which the mirrors are movable. A coefficient describing the reflectivity of the mirrors in a Fabry-Perot interferometer and equal to (1)

where r is a Fresnel reflection coefficient from the Fresnel equations, so that the reflection and transmission ratios may be simply expressed as

(2)

(3)

where (4) with d the thickness of the glass the light is transmitted through, n is its index of refraction, and is the wavelength of the light. Equation (3) defines the Airy function

Interferometry Applied in Flatness Testing: The essential equipment for measurement by light wave interference is a monochromatic light source and a set of optical flats. An optical flat is a circular piece of optical glass or fused quartz having its two planes faces flat and parallel and the surfaces are finished to an optical degree of flatness. Optical flats very from 25mm diameter to 300 mm diameter. Quartz has minimum expansion due to heat and the thickness used insures distortion from freedom in use. The importance optical flats are apparent in light wave measurement because all inspection applications are performed with reference to the master surfaces of the optical flats. If an optical flat placed upon another flat reflecting surface (without pressure) it will not for an intimate contact, but it will lie on some angle θ making an inclined plane. If the optical flat be now illuminated by monochromatic source of light, the eye if placed in proper position will observe a number of bands. These are produced by the interference of the light rays reflected fro the lower surface of the top flat and the top surface of the lower flat through a very thin layer of air between the flats.

EYE

OPTICAL FLAT

FLAT SURFACE

Flatness testing By Interferometry In the figure the angle θ the two flats shown exaggerated too much. S is the source of monochromatic light. At point A, the wave of incident beam from S is partially reflected along AB and is partially transmitted across the air gap along AC. At C, again the ray is reflected along CD and passes out towards the eye along CDE. Thus two reflected components, reflected at A and C are collected and recombined by the eye, having traveled paths whose lengths differ by an amount ACD.

If the path lengths of the two components differ by an odd number of half wavelengths, then the condition for complete interference is achieved. If the surface is completely flat then condition of complete interference is satisfied in a straight line across the flat surface as the surface at right angles to the plane of the paper is parallel to the optical flat. Therefore a straight dark line will be seen passing through point C. Consider another ray passing along path SFH. Again this ray is also spitted in two components. It is obvious that the path difference of two component rays will keep on increasing along the surface due to angle θ. Thus if path difference FHI be 3λ/2 or the next odd number of half wavelengths, then interference will occur and similar fringe will be seen. Next when the path difference is 5λ/2, again there will be another dark fringe. At the intermediate point between the points C and H, the path difference will be an even number of half wavelengths and the two components will be in phase producing a light band. Thus, in case of perfectly flat surface, we will have pattern of alternate dark and bright straight lines on the surface, any deviation from this pattern will be a measure of error in the flatness of surface being inspected.

Checking of flatness by above method: Interference pattern on completely flat surface Referring the same previous figure if θ is very small then AC = DC = λ/4 , and FH=HI=3 λ/4 Therefore, the change in separation between the optical flat and the surface between the optical flat and the surface between the two similar adjacent fringes is difference between AC and FH = 3 λ/4 - λ/4 = λ/2 Thus it is obvious that each adjacent fringe represents a change in elevation of the work surface relative to the optical flat of λ/2 and total change in elevation from point of contact to the outermost fringe will be n * λ/2, if it contains n number of adjacent fringes. Now there are four possible cases when the contact between the optical flat and the work surface occurs at one point only. If surfaces are perfectly wrung together, there no gap exists and no fringe pattern will be observable. If angle θ is increased, then point C and H will be closer together and fringes are brought closer together. If angle θ is reduced then fringes spacing increase as the points C and H will occur at greater distances. If angle θ is too large, then fringes will be closely spaced as to be indistinguishable and no observable pattern will be visible.

In practice, on the surface to be tested, we have large number of hills and valleys and in these circumstances following figure will show the behavior of optical flat if it is made to rest on the hills. If optical flat is resting on a hill then it will just behave as if it were placed on some spherically convex surface. In such case the contact is made at the central high point and in centre a bright circle will be visible. Around it there will be concentric dark and bright circular fringes. Optical Flat Convex Surface

Convex Surface Specimen

Optical Flat

As the distance from the centre increases the separation between the optical flat and the surface keeps on increasing and the fringes become narrow and more closely spaced. Optical Flat New Contact Position

Specimen

Optical Flat

From the above illustrations we can use this application of optical flats and Interferometry for studying various surface contours, optical flat is kept at certain inclination over the surface to be tested.

Study of surface contours: In the study of surface contours it is important to know as to where the optical flat is in contact with the surface being tested. The point or line of contact is usually where the pressure is applied causing the air film to be squeezed thin at that place. In monochromatic light the bands are sharper near the point of contact; and in day-light the point of contact can be seen as the color of the surface being tested. Let XX be the line of contact of specimen and the optical flat. Then following types of fringes will be seen of particular object.

Concave surface will show such fringes

Fringes produced by the convex surface gauge.

These type of fringes are produced when a surface is having a scratch in it. If D is the distance between the two fringes and d is the distance due to scratch then depth of the scratch can be calculated as Depth = dλ / 2D

By applying pressure at points the surfaces with high and low spots will be observed.

Such type of fringes are produced when the gauge surface is warned out at edges.

Testing the parallelism of any surface with reference to standard flat surface by using optical flat: There are two blocks, one whose surface is made optically flat and other whose surface is to be checked are wrung on a flat reference plane surface. An optical flat is then placed on the top of both the blocks. Here A is the standard reference gauge and B is the gauge under the test. The distance L between the two gauges is measured very accurately. In case A and B perfectly flat but different in height then there will be equal wedge shaped air layers between the gauges A and B. in case the surfaces are not parallel, the angles formed by the surfaces of the two gauges with the optical flat will be different. Let the angle be θ and θ’ respectively. If width of the gauge block l is same for both, the number of fringes per unit width will be different. These possibilities have been shown in the figure

Once it is assured that surface of the two gauges are flat and parallel to each other, then it is possible to check the height of the two gauges also. Let the number of fringes on the reference gauge block be N in the width of l units. The difference in the height of blocks h = L/l * N * λ/2 = λLN / 2l This is however a rough method and meant to explain principle only. For this purpose the use of interferometer is best.

N.P.L. Flatness interferometer:

This instrument mainly used for checking the flatness of flat surfaces. This interferometer is designed by National Physical Laboratory. The flatness of any surface is judged by comparing with an optical flat surface which is generally the base plate of the instrument. This instrument essentially consists of a mercury vapor lamp. As we are interested in having single monochromatic source of light, the radiations of the mercury lamp are passed through a green filter. This radiation is then brought to focus on pinhole in order to obtain an intense point source of light. A mirror is used in order to deflect the light beam through 900. The pinhole is placed in the focal plane of collimating lens, thus the radiation out of the lens will be parallel beam of light. This beam is directed on the gauge to be tested via an optical flat. The fringes formed are viewed directly above by means of a thick glass plate semi-reflector set at 45o to the optical axis. The gauge to be tested is wrung on the base plate whose surface is finished to a degree comparable to that of the highest quality gauge face. As the optical flat is placed above it in a little tilted position, interference fringes are formed; one between rays

reflected from the under surface of the optical flat and those reflected from the surface of the gauge, and the other between rays reflected from the undersurface of the optical flat and those reflected from the base plate.

Illustrations by NPL interferometer Following are some inferences which can be drawn from the fringe pattern obtained when a gauge is tested under N.P.L. Interferometer. (1)

(2)

Gauge Face is flat and parallel to the base plate. Optical flat being equally inclined on both surfaces (4)

Gauge surface is of concave or convex in shape. Fringes on the gauge are curved lines.

(3)

Gauge is flat but not parallel to the base plate. Straight and parallel fringes but of different pitches produced

Gauge is flat and parallel to the base plate but slightly rounding of at the corners of the gauge.

(5)

Gauge is flat but its surface is inclined to the base plate at some other angle

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Introduction: If two or more waves cross one another in the same medium, each wave produces its own effect totally independent of the effects due to the other. Principle of superposition tells that at any instance the resultant displacement is equal to the vector sum of the individual displacements produced by each wave. This is based on the theory of interference of light discovered by Thomas Young in 1801.

With single source of light the energy distribution in the surrounding medium is uniform. But when there are two adjacent sources of light giving out light waves of the same wavelength, amplitude and having zeroed or constant phase difference, the distribution of energy is no longer uniform. At some points where the crest of one wave falls upon the crest of the other or the trough of one wave falls on the trough of the other, the resultant amplitude is large and hence the intensity become maximum. At other points where the crest of one wave falls on the trough of the other wave and vice versa, the resultant amplitude is reduced to zero and intensity is minimum. The modification in the distribution of light energy due o superposition of two or more waves of light is called interference of light. It should be noted that in this phenomenon there is only transference of energy from one region to another. The energy missing at one region reappears at the other region. There is no creation destruction of energy in any region as a result of the superposition of the waves of light.

Because of interference there are alternate bright and dark regions observed and they are called interference bands or interference fringes. The color pattern observed on soap bubbles and oil on the wet roads are due to interference of light. If a convex lens of large radius of curvature is placed on a glass plate and illuminated with monochromatic light source, alternately bright and dark concentric rings (called as Newton’s rings) are observed because of interference. Similarly if a wedge shaped air film between two plane glass plates (air wedge) is illuminated with monochromatic light, alternate bright and dark bands are formed because of interference, the bands being parallel to the edge of the wedge.

Concepts:

Light travels in waves. Sometimes, two or more waves join together. Interference is the addition, or coming together, of several waves. Constructive Interference happens when two or more waves come together to form a larger and stronger wave, matching their crests and troughs. Destructive Interference is when two or more waves come together and cancel each other out to make a weaker wave.

Principles:

Since light has wave properties, it will experience interference (the addition of waves). This interference is like that seen with water waves. Whether you get constructive or destructive interference depends on the wavelength (color) of the light. White light is made up of many colors. These different colors have different wavelengths. We can see the many colors of light on bubbles. The separation of white light into many colors on a bubble happens because of interference.

Thomas Young’s double slit Experiment: One of the most important experiments of wave theory is that of Young's double slits. It is a clear example of the diffraction of light conducted with essentially basic scientific equipment. Thomas Young was a not only a physicist but also a physician and Egyptologist, who was responsible for deciphering the Rosetta stone. He devised an experiment in the early 1800's that proved that light is a wave. The experiment has been used subsequently to show that wave behaviors exists in many other areas of nature and therefore it is worth spending a little time going into the experiment in detail. When two light beams interact they create interference which can be constructive or destructive as we have discussed earlier. The places where constructive and destructive interference occur are subject to constant change, since electromagnetic waves emitted are capable of varying phase. Using one light source and splitting it into two beams you can create two coherent sources, meaning they are of identical frequencies and have a constant phase difference (the distance between a peak of wave 1 and wave 2 is always the same) It is also important to use monochromatic light for this experiment as the location of interference occurs is wavelength dependent.

Interference pattern on completely flat surface

The apparatus consists of a source matching the above requirements, a screen with two very thin identical slits or the order of a wavelength in width, and a screen to view the interference on. In principle, a light bulb, or lamp can be used, but the light must be reduced to a monochromatic source with filters. It can also be done by splitting the light into its various frequencies using a prism, a frequency can be selected by channeling the light with a further screen with only a pin prick in it to allow the light out. These days’ lasers are used to provide the light source. When the light is switched on, it travels up to the first screen and is split into two beams by the slits, we have seen that when this happens waves are diffracted and bulge outwards causing two curved wave fronts to propagate the other side of the slits, at many places between the slits and the viewing screen there are areas of constructive interference and by moving the viewing screen it is possible to get a picture of where they are occurring

Conditions for Sustained Interference: Light waves from two sources can form a sustained (well defined and observable) interference pattern where they meet, only if they satisfy the following conditions: Two light waves superposing at a point must have the same wavelength or frequency The amplitude of superimposing light waves should be equal or almost equal. Two light waves superposing at a point should either have the same phase or constant phase difference. The two sources emitting light waves should be very narrow. The sources emitting light waves should be very close to each other.

Coherent Sources: Any two sources of light continuously emitting light waves having zeroed or constant phase difference are called coherent sources. Thus coherent sources have a definite phase relationship between them and emit light waves of equal frequency or wavelength. Interference can occur only with coherent sources. But two independent sources cannot be coherent because even though they may emit light waves of equal wavelength and equal amplitude, they may not have constant phase difference. Therefore, to obtain two interfering beams, a single source is used and its beam is split into two by following ways:

Division of wave front: In this case the wave front is divided into two parts by reflection, refraction or diffraction so hat those two parts reunite at a small angle and produce interference. (Example: Fresnel’s biprism, Lloyd’s mirror)

Division of Amplitude: In this case the wave front is split up into two parts by partial refraction and reflection at a surface and these two parts are later made to reunite to produce interference. (Example: Air wedge, Newton’s Rings)

Air Wedge:

Newton’s Rings:

Newton’s Rings Explanation:

Given a hemisphere of radius R resting on a plane, monochromatic light falling normally on the top of the hemisphere reflects off both the plane and the hemisphere. Near the center of the hemisphere, refraction can be neglected, and the rays create rings of interference. On the curved surface, a point a distance r from the center will be a height d above the plane (1)

(2)

Reflection from hemisphere undergoes a phase change of

. Reflection from the

plane gives a phase change of . The phase difference between rays reflected off the plane and hemisphere is therefore

(3)

for

, and the interference pattern goes as

Constructive interference occurs when

. with m = 0, 1, 2, ..., i.e.,

(4)

so (5)

(6)

(7)

(8)

Solving for

gives (9)

which, for

, gives

(10)

Theory of Interference (Mathematical Model): Consider two light waves of same frequency, wavelength traveling in same directions. Let a1 and a2 be the amplitudes of the two waves. The displacements of any particle in the medium due to these waves at any instance of time t are y1 = a1 sin ωt and y2 = a2 sin (ωt + δ) Where δ is phase difference between the waves and ω = 2πf is the angular frequency of the waves. Thus the resultant displacement of the particle is the sum of both equations = y1 + y2 = a1 sin ωt + a2 sin (ωt + δ) = a1 sin ωt + a2 (sin ωt cos δ + cos ωt sin δ) = a1 sin ωt + a2 sin ωt cos δ + a2 cos ωt sin δ = (a1 + a2 cos δ) sin ωt + a2 cos ωt sin δ -------------- (1) Let Rcosθ = (a1 + a2 cos δ) -------------------------------- (2) And Rsinθ = a2 sin δ ----------------------------------------- (3) Thus y = R [sin ωt cos θ + cos ωt sin θ] y = R sin (ωt + θ) --------------------------------------------- (4) y

Equation (4) represents the a simple harmonic vibration of amplitude R and angular frequency ω. Thus the resultant wave is also a simple harmonic wave with same frequency as those individual waves. Squaring and adding equation (2) and (3) R2 cos2θ + R2 sin2θ = (a1 + a2 cosδ) 2 + (a2 sinδ) 2 R2 (cos2θ + sin2θ) = a12 + a22 cos2 δ + 2a1 a2 cos δ + a 22 sin2 δ i.e. R2 = a12 + a22 + 2a1 a2 cos δ --------------------------------------- (5) The phase angle θ is given by tanθ = ( Rsinθ / Rcosθ ) = ( a 2 sinδ /{a1 + a2 cosδ})------------(6)

Condition for constructive interference: When there is constructive interference the amplitude is maximum and hence the intensity is also maximum. From the equation (5) the resultant amplitude R is maximum when cosδ = +1 i.e, δ = 2nπ where n = 0, 1, 2, 3…….. Thus R will be maximum when the phase difference between the two waves is δ = 0, 2π, 4π, 6π …………., that is even multiple of π If Δ is the path difference between the waves corresponding to the phase difference δ, δ = (2π/λ) Δ or Δ = (λ/2π)δ = (λ/2π) 2nπ = nλ = 2n λ/2

For constructive interference the phase difference between the two light waves must be equal even multiple of π or the path difference between the waves must be equal to even multiple of λ/2 where λ is wave length of light. Rmax = (a12 + a22 + 2a1a2)1/2 = a1 + a2. The maximum intensity is I max α (a1 + a2)2 If a1 = a2 = a then I max α 4a2

Condition of destructive interference:

When there is destructive interference the amplitude is minimum and hence the intensity is also minimum. From equation (5) the resultant amplitude is R is minimum when cos δ = -1. δ = (2n + 1)π where n= 0, 1, 2, 3……… Thus R will be minimum when the phase difference between the two waves is δ = π, 3π, 5π, 7π………………. That is the odd multiple of π If Δ is the path difference between the waves corresponding to the phase difference δ, Δ = (λ/2 π) δ = (λ/2 π)(2n+1) π = (2n+1) λ/2 For destructive interference the phase difference between the two waves must be equal to odd multiple of π or the path difference between the waves must be equal to odd multiple of λ/2 where λ is wavelength of light. The minimum amplitude R min = (a12 + a22 - 2a1a2)1/2 = a1 + a2. The minimum intensity is I max α (a1 - a2)2 If a1 = a2 = a then I max = 0 and R min = 0

Width of Interference Fringes:

P

A

x

Q

O B

R D

In figure A and B represents the two coherent sources (slits in young’s double slit experiment) of light separated by a distance d. let a screen be placed at a distance D from light source. Then the point O on the screen is equidistant from A and B so that the path difference between the two light waves from A and B reaching O is zero. Thus the point O has the maximum intensity. Consider a point P at a distance x from O. The path difference between the light waves from A and B reaching the point P is Δ = BP – AP From the figure BP2 = BR2 + PR2 = D2 + [x + (d/2)] 2 AP2 = AQ2 + PQ 2 = D2 + [x - (d/2)] 2 BP2 - AP2 = {D2 + [x + (d/2)] 2} – {D2 + [x - (d/2)] 2} = [x + (d/2)] 2 - [x - (d/2)] 2 = 2xd Or (BP-AP) (BP+AP) = 2xd (BP-AP) = {(BP+AP)/2xd} Since P is very close to O, (BP+AP) = 2D Therefore path difference Δ = (BP-AP) = 2xd/2D = xd/D …………. (1) Thus the equation (1) represents the path difference between light waves from A and B interfering at the point P.

For Bright Fringes: For bright fringes or maximum intensity at P, the path difference must be an even multiple of λ/2 where λ is the wavelength of light used. xd/D = 2nλ/2 or x = nλD/d ……………………… (2) Equation (2) gives the position of bright fringes from the point O For n = 0, x0 = 0 For n = 1, x1 = λD/d For n = 2, x2 = λD/d For n = 3, x3 = λD/d and so on. The distance between the centers of any two consecutive bright fringes is called fringe width of bright fringes. It is given by β = x1 – x0 = x2 – x1 = x3 – x2 = …………. = λD/d Therefore β = λD/d ……………….. (3) Thus the widths of all the bright fringes are same.

For Dark Fringes: For dark fringe or minimum intensity at P, the path difference must be an odd multiple of λ/2. xd/D = (2n + 1) λ/2 or x = (2n + 1)λD/2d ………………………(4) Equation (2) gives the position of bright fringes from the point O For n = 0, x0 = λD/2d For n = 1, x1 = 3 λD/2d For n = 2, x2 = 5 λD/2d For n = 3, x3 = 7 λD/2d and so on. The distance between the centers of any two consecutive dark fringes is the fringe width of dark fringes. It is given by β = x1 – x0 = x2 – x1 = x3 – x2 = …………. = 2λD/2d = λD/d Therefore β = λD/d ……………….. (5) Thus the widths of all the dark fringes are same. From the equation (3) and (5), it is found that the fringe widths for bright fringes and the dark fringes are equal. The width β of interference fringes increases when Wavelength of light λ is increased. The distance D between the coherent sources and the screen is increased. The distance d between the two coherent sources or slits is decreased.

Interferometry: Interferometry is the science, and the art, of making measurements with coherent light. Light moves in waves, and light from a single source can travel different paths and then recombine in ways we call interference. When we measure that interference, we learn about the different paths the light has taken; and the yard-stick, that is, the lengthscale of the measurement, is the wavelength of light itself.

Primary Applications of Interferometry: Interferometry has got a wide range of applications following are some of the important applications of interferometry. 1. Optical Testing: Generally applications of interferometry used in testing are Measurement of surface quality. a. Flat surfaces b. Spherical surfaces c. Surface roughness d. Aspherical surfaces 2. Inspection: Interferometry has following applications in the field of inspection. a. Inspection of slip gauges b. Inspection of measurement standards 3. Direct phase measurements a. Multiple wavelengths b. Phase stepping and phase shifting 4. Aligning very high quality lenses, like those in cameras, telescopes, and photolithography tools called steppers, which are used in the fabrication of the intricate circuit patterns. 5. Sagnac interferometers are used as an alternative to gyroscopes to measure changes with respect to an inertial reference frame in fighter planes.

6. Space applications a. Radio Astronomy. b. Measuring light intensity c. Used in retrieving images from telescope. 7. To measure angular sizes as small as 0.0005 arc seconds from distant stars.

Interferometer: Interferometer is an instrument that measures the wavelengths of light and distances. It transmits two radio signals or beams of light and uses the interference principle (how they reinforce or neutralize each other) to determine the measurement.

Types of Interferometers:

Michelson Interferometer: Fabry-Perot Interferometers Spherical Interferometers Fringe Counting Interferometers Goge Block Interferometers Correlation Interferometer Intensity Interferometer Stellar Interferometer Sagnac Interferometer Pohl Interferometer Martin-Puplett Interferometer Mach-Zehnder Interferometer Single Frequency DC interferometer Twyman-Green Interferometer N.P.L. Flatness Interferometer

Types of light sources used for Interferometry:

Mercury Mercury 198 Cadmium Krypton Krypton 86 Thallium Sodium Helium Neon Gas lasers

Michelson Interferometer: This is the oldest type of all interferometers, which has subsequently been modified in several resects and lot of sophistication introduced. However Michelson using this interferometer, established exact relationship between meter and red wavelengths of Cadmium lamps. The Michelson interferometer used in the MichelsonMorley experiment of 1887 to determine the velocity of light had a half-silvered mirror to split an incident beam of light into two parts at right angles to one another. The two halves of the beam were then reflected off mirrors and rejoined. Any difference in the speed of light along the paths could be detected by the interference pattern. The failure of the experiment to detect any such difference threw doubt on the existence of the ether and thus paved the way for the special theory of relativity. Another type of interferometer devised by Michelson has been applied in measuring the diameters of certain stars.

Fabry-Perot Interferometer:

An interferometer in which the mirrors are movable. A coefficient describing the reflectivity of the mirrors in a Fabry-Perot interferometer and equal to (1)

where r is a Fresnel reflection coefficient from the Fresnel equations, so that the reflection and transmission ratios may be simply expressed as

(2)

(3)

where (4) with d the thickness of the glass the light is transmitted through, n is its index of refraction, and is the wavelength of the light. Equation (3) defines the Airy function

Interferometry Applied in Flatness Testing: The essential equipment for measurement by light wave interference is a monochromatic light source and a set of optical flats. An optical flat is a circular piece of optical glass or fused quartz having its two planes faces flat and parallel and the surfaces are finished to an optical degree of flatness. Optical flats very from 25mm diameter to 300 mm diameter. Quartz has minimum expansion due to heat and the thickness used insures distortion from freedom in use. The importance optical flats are apparent in light wave measurement because all inspection applications are performed with reference to the master surfaces of the optical flats. If an optical flat placed upon another flat reflecting surface (without pressure) it will not for an intimate contact, but it will lie on some angle θ making an inclined plane. If the optical flat be now illuminated by monochromatic source of light, the eye if placed in proper position will observe a number of bands. These are produced by the interference of the light rays reflected fro the lower surface of the top flat and the top surface of the lower flat through a very thin layer of air between the flats.

EYE

OPTICAL FLAT

FLAT SURFACE

Flatness testing By Interferometry In the figure the angle θ the two flats shown exaggerated too much. S is the source of monochromatic light. At point A, the wave of incident beam from S is partially reflected along AB and is partially transmitted across the air gap along AC. At C, again the ray is reflected along CD and passes out towards the eye along CDE. Thus two reflected components, reflected at A and C are collected and recombined by the eye, having traveled paths whose lengths differ by an amount ACD.

If the path lengths of the two components differ by an odd number of half wavelengths, then the condition for complete interference is achieved. If the surface is completely flat then condition of complete interference is satisfied in a straight line across the flat surface as the surface at right angles to the plane of the paper is parallel to the optical flat. Therefore a straight dark line will be seen passing through point C. Consider another ray passing along path SFH. Again this ray is also spitted in two components. It is obvious that the path difference of two component rays will keep on increasing along the surface due to angle θ. Thus if path difference FHI be 3λ/2 or the next odd number of half wavelengths, then interference will occur and similar fringe will be seen. Next when the path difference is 5λ/2, again there will be another dark fringe. At the intermediate point between the points C and H, the path difference will be an even number of half wavelengths and the two components will be in phase producing a light band. Thus, in case of perfectly flat surface, we will have pattern of alternate dark and bright straight lines on the surface, any deviation from this pattern will be a measure of error in the flatness of surface being inspected.

Checking of flatness by above method: Interference pattern on completely flat surface Referring the same previous figure if θ is very small then AC = DC = λ/4 , and FH=HI=3 λ/4 Therefore, the change in separation between the optical flat and the surface between the optical flat and the surface between the two similar adjacent fringes is difference between AC and FH = 3 λ/4 - λ/4 = λ/2 Thus it is obvious that each adjacent fringe represents a change in elevation of the work surface relative to the optical flat of λ/2 and total change in elevation from point of contact to the outermost fringe will be n * λ/2, if it contains n number of adjacent fringes. Now there are four possible cases when the contact between the optical flat and the work surface occurs at one point only. If surfaces are perfectly wrung together, there no gap exists and no fringe pattern will be observable. If angle θ is increased, then point C and H will be closer together and fringes are brought closer together. If angle θ is reduced then fringes spacing increase as the points C and H will occur at greater distances. If angle θ is too large, then fringes will be closely spaced as to be indistinguishable and no observable pattern will be visible.

In practice, on the surface to be tested, we have large number of hills and valleys and in these circumstances following figure will show the behavior of optical flat if it is made to rest on the hills. If optical flat is resting on a hill then it will just behave as if it were placed on some spherically convex surface. In such case the contact is made at the central high point and in centre a bright circle will be visible. Around it there will be concentric dark and bright circular fringes. Optical Flat Convex Surface

Convex Surface Specimen

Optical Flat

As the distance from the centre increases the separation between the optical flat and the surface keeps on increasing and the fringes become narrow and more closely spaced. Optical Flat New Contact Position

Specimen

Optical Flat

From the above illustrations we can use this application of optical flats and Interferometry for studying various surface contours, optical flat is kept at certain inclination over the surface to be tested.

Study of surface contours: In the study of surface contours it is important to know as to where the optical flat is in contact with the surface being tested. The point or line of contact is usually where the pressure is applied causing the air film to be squeezed thin at that place. In monochromatic light the bands are sharper near the point of contact; and in day-light the point of contact can be seen as the color of the surface being tested. Let XX be the line of contact of specimen and the optical flat. Then following types of fringes will be seen of particular object.

Concave surface will show such fringes

Fringes produced by the convex surface gauge.

These type of fringes are produced when a surface is having a scratch in it. If D is the distance between the two fringes and d is the distance due to scratch then depth of the scratch can be calculated as Depth = dλ / 2D

By applying pressure at points the surfaces with high and low spots will be observed.

Such type of fringes are produced when the gauge surface is warned out at edges.

Testing the parallelism of any surface with reference to standard flat surface by using optical flat: There are two blocks, one whose surface is made optically flat and other whose surface is to be checked are wrung on a flat reference plane surface. An optical flat is then placed on the top of both the blocks. Here A is the standard reference gauge and B is the gauge under the test. The distance L between the two gauges is measured very accurately. In case A and B perfectly flat but different in height then there will be equal wedge shaped air layers between the gauges A and B. in case the surfaces are not parallel, the angles formed by the surfaces of the two gauges with the optical flat will be different. Let the angle be θ and θ’ respectively. If width of the gauge block l is same for both, the number of fringes per unit width will be different. These possibilities have been shown in the figure

Once it is assured that surface of the two gauges are flat and parallel to each other, then it is possible to check the height of the two gauges also. Let the number of fringes on the reference gauge block be N in the width of l units. The difference in the height of blocks h = L/l * N * λ/2 = λLN / 2l This is however a rough method and meant to explain principle only. For this purpose the use of interferometer is best.

N.P.L. Flatness interferometer:

This instrument mainly used for checking the flatness of flat surfaces. This interferometer is designed by National Physical Laboratory. The flatness of any surface is judged by comparing with an optical flat surface which is generally the base plate of the instrument. This instrument essentially consists of a mercury vapor lamp. As we are interested in having single monochromatic source of light, the radiations of the mercury lamp are passed through a green filter. This radiation is then brought to focus on pinhole in order to obtain an intense point source of light. A mirror is used in order to deflect the light beam through 900. The pinhole is placed in the focal plane of collimating lens, thus the radiation out of the lens will be parallel beam of light. This beam is directed on the gauge to be tested via an optical flat. The fringes formed are viewed directly above by means of a thick glass plate semi-reflector set at 45o to the optical axis. The gauge to be tested is wrung on the base plate whose surface is finished to a degree comparable to that of the highest quality gauge face. As the optical flat is placed above it in a little tilted position, interference fringes are formed; one between rays

reflected from the under surface of the optical flat and those reflected from the surface of the gauge, and the other between rays reflected from the undersurface of the optical flat and those reflected from the base plate.

Illustrations by NPL interferometer Following are some inferences which can be drawn from the fringe pattern obtained when a gauge is tested under N.P.L. Interferometer. (1)

(2)

Gauge Face is flat and parallel to the base plate. Optical flat being equally inclined on both surfaces (4)

Gauge surface is of concave or convex in shape. Fringes on the gauge are curved lines.

(3)

Gauge is flat but not parallel to the base plate. Straight and parallel fringes but of different pitches produced

Gauge is flat and parallel to the base plate but slightly rounding of at the corners of the gauge.

(5)

Gauge is flat but its surface is inclined to the base plate at some other angle

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