1

Interference

When two light waves pass through a certain position in space at the same time, superposition happens.

The principle of superposition states: When two or more waves move simultaneously

through a region of space, each wave proceeds independently as if the other were not present.

The resulting wave "displacement" at any point and time is the vector sum of the "displacements" of the individual waves.

In other word, the resultant displacement is the sum of the separate displacements of the constituent waves: = y y1 + y2

Superposition principle

2

Interference

Two traveling waves, y1 and y2, whose phases differ only by the constant φ, have amplitude A, angular wave number k and angular frequency ω.

They move in the same direction and have the same frequency and velocity.

To derive an equation for the combined wave, we use the equations for the two waves, the principle of superposition, and a trigonometric identity.

3

Interference To simplify the analysis, the first wave is assumed to

have a phase constant of zero, so the phase difference φ is the phase constant of the second wave.

The principle of superposition says we can add these equations to determine the equation for the combined wave, yc. We use the trigonometric identity

to derive the equation for the combined wave shown in Eq. 1.

2

sin2

cos2sinsinbaba

ba

4

Interference

Eq. 1

y = transverse displacement A = amplitudek = wave numberω = angular frequencyφ = phase constant

5

Interference

The amplitude is the constant factor 2A cos(φ/2). The sine function describes a wave with the same

angular wave number k and angular frequency ω as the two waves that combined to create it.

If there is no phase difference between the initial two waves, φ/2 = 0 and cos(φ/2) = 1, its maximum value. In this case, the combined wave has double the amplitude of the initial waves. In other words, there is fully constructive interference.

In addition, any integer multiple of 2π radians (360°) as the phase difference of the initial waves will cause perfect constructive interference, because the argument of the cosine will be a multiple of π rad (180°), giving the cosine a value of ±1.

6

Interference Conversely, if the phase difference φ is π rad, then φ/2 = π/2

and cos(φ/2) = 0. The function describing the combined “wave” is constantly zero, and there is complete destructive interference.

This is to be expected: Two waves that have a phase difference of π rad cancel out. Any odd integer multiple of π rad means the cosine is zero and the result is fully destructive interference.

We have been focusing on the two extremes, fully constructive and fully destructive interference.

Intermediate interference describes the situation for any phase difference in between.

The resulting traveling wave is sinusoidal, with the same wavelength and frequency as the initial waves, but with an amplitude between zero and double the amplitude of the contributing waves.

7

Interference

Interference is the interaction of two or more waves passing the same point.

Constructive interference occurs when the waves add in phase, producing a larger peak than any wave alone, whereas destructive interference occurs when waves add out of phase, producing smaller peaks than one of the waves alone

When enhancement (constructive interference) and diminution (destructive interference) conditions alternate in spatial display, the interference is said to produce pattern of fringes (interference pattern).

The two sources do not need to be in phase with each other.

If there is some constant initial phase difference between the two sources, the resulting interference pattern will be identical to the original pattern, although it will be shifted in terms of the location of the minima and maxima.

8

Condition for Interference If two beams are to interfere to produce a stable

pattern, they must have very nearly the same frequency

A significant frequency difference would result in a rapidly varying, time dependent phase difference, which in turn would cause the interference intensity to average to zero

Clearest patterns exist when the interfering waves have equal or nearly equal amplitudes.

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Young’s double slit interference

Interference

Young’s observation when two slits were uniformly illuminated.

Consistent with interference between wavefronts emerging from each slit.

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Interference arises due to the path difference between light emerging from each slit varying as d sin.

Young’s double slit interference

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Overall intensity envelope is due to diffraction from each single slit.

,...,,m,msind

,...,,m,msind

210

210

21

ceinterferen eDestructiv

ceinterferen veConstructi

Young’s double slit interference

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Line spacing for double-slit interference

A screen is 1.2m from two slits 0.100mm apart. light of wavelength 500nm is incident upon the slit. What is the separation between bright fringes on the screen?

L

Given: d= 0.100mm = 1x10-4 m, =500x10-9 m, L= 1.20m

mmd

mLx

mmmLx

d

m

mmd

00.122

20.1

00.61000.520.1

1000.51000.1

10500sin

1,sin

22

311

13

4

9

1

1

13

Wavelengths from double-slit interference

White light passes though two slits 0.5mm apart allows the measurement of wavelengths on a screen 2.5m away. The estimate of the violet and red wavelengths may be obtained.

nm

nm

L

x

m

d

m

d

mmd

violet

red

4005.2

100.2

1

100.5

7005.2

105.3

1

100.5

sin,1,sin

34

34

111

mmm

mmm

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Coherence Interference is only possible if there is a fixed phase

relationship between the radiation emerging from each of the two slits.

As the light source for Young’s fringes was a distant source, the sun, and as both slits “sampled” the wavefront at the same time, there was interference between the emerging wavefronts and these slits are then said to be coherent sources.

Most light sources are incoherent. An incandescent light bulb will emit light along the length of the filament. Light emitted at each end of the filament bears no phase relationship to the light emitted at the other. Most lasers are sources of very coherent light. They may have a

phase relationship which extends both across the beam and along the beam.

This high degree of coherence is required to improve the contrast of interference fringes and in holography.

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Intensity in the double-slit interference pattern The electric field is given by the sum of the fields from each slit.

These vary with angle in terms of a phase difference .

The intensity in the double slit interference pattern has periodic maxima corresponding to d sin =ml or, in terms of the phase difference :

The intensity in between these maxima may be determined as a function of angle .

)cos(cos

00 21

21

tEtE

EEE

sin

2d

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Intensity in the double-slit interference pattern

A phasor diagram illustrates the phase summation of the two equal fields.

2sin2cos2

2sin)()(

cos22

cos2)(

)2

cos2(2)(

)cos1(2)(

cos22)(

cos2)(

2 ,

0

0

000

220

20

20

20

20

20

20

2122

220

021

00001

00

tE

tEE

EEE

EE

EE

EEE

EEEEE

EEE

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Observe intensity rather than E field

Assuming I1 = I2 = Io . The cosinus term is the characteristics of the interference pattern while sin2/2 is the term for the diffraction.

where

Intensity in the double-slit interference pattern

)sin

2)(cos1(cos2)(2

22

2121 oAIIIII

4Io

2cos4cos2)( 2

2121

oIIIIII

2cos2cos1 2

18

Multiple beam interference

The difference in optical path length between adjacent raysinterference between multiple reflection beams.

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Interference in thin films

Suppose that a very thin film of air is trapped between two pieces of glass, as shown in the diagram below.

If monochromatic light is incident almost normally to the film then some of the light is reflected from the interface between the bottom of the upper plate and the air, and some is reflected from the interface between the air and the top of the upper plate.

The eye focuses these two parallel light beams at one spot on the retina. The two beams produce either destructive or constructive interference, depending on whether their path difference is equal to an odd or an even number of half-wave-lengths, respectively.

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Interference in thin films

Let t be the thickness of the air film.

The difference in path-lengths between the two light rays shown in the diagram is clearly D = 2 t.

Naively, we expect that

constructive interference (brightness)

D=m

destructive interference (darkness)

D = (m+1/2).

where m is an integer

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Interference in thin films Phase difference is introduced between the two rays on reflection.

The first ray is reflected at an interface between an optically dense medium (glass), through which the ray travels, and a less dense medium (air). There is a 180o phase change on reflection from such an interface.

The second ray is reflected at an interface between an optically less dense medium (air), through which the ray travels, and a dense medium (glass). There is no phase change on reflection from such an interface.

Thus, an additional 180o phase change is introduced between the two rays, which is equivalent to an additional path difference of l/2.

When this additional phase change is taken into account, the condition for constructive interference becomes

2 t = (m + 1/2) l , Similarly, the condition for destructive interference becomes

2 t = ml

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Interference in thin films

If the thin film consists of water, oil, or some other transparent material of refractive index n then the results are basically the same as those for an air film, except that the wavelength of the light in the film is reduced from l (the vacuum wavelength) to l/n.

It follows that the modified criteria for constructive and destructive interference are

2 n t = (m + 1/2) l

and

2 n t = ml

respectively.

For white light, the above criteria yield constructive interference for some wavelengths, and destructive interference for others.

Thus, the light reflected back from the film exhibits those colours for which the constructive interference occurs.

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Interference in thin films

If the angle of transmittance qt is taken into account, the path difference is expressed by

D=2ntcos qt

Then the condition for

constructive interference: D +Dr = mland

destructive interference: D +Dr = (m + 1/2) lwhere m=integer and Dr is the path difference arising from path change on reflection.

Fringes formed by a dielectric film with an extended source are referred to as Haidinger fringes, or fringes of equal inclination (circular fringes). The fringes are formed by parallel beams from an extended source.

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Michelson interferometer Extremely precise measurements of

distance are possible using an optical interferometer.

Varying the position of M1 by as little as 100nm would alter the fringes of 400nm light from dark to bright.

By sweeping M1 over a large known distance from the beam splitter, it is possible to perform a Fourier analysis on the allowed frequencies and obtain very high resolution spectra.

Constructive interference: 2tcos =(m+ ½)

25

Anti-reflection coating

The deposition of a “quarter-wave” thickness of a mechanically hard material possessing an appropriate refractive index allows the surface reflection from glasses, camera lens and optical surfaces to be minimised.

Usually MgF2 is used as it’s refractive index is almost halfway between air and glass.

Destructive interference occurs at 2nt =m

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Wedges and optical testing

Fringes may easily be generated by placing a thin wire or sheet of paper along one edge of a piece of glass resting on an optically flat surface and illuminating with monochromatic light.

Bright bands occur when 2nt =(m+ ½) .

Extremely useful in determining the flatness of optical surfaces.

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Interference in thin films An interference effect is seen in the reflection of light

from water with a layer of oil on the surface. Rainbow coloured fringes are visible.

Light passing from material of lower to higher refractive index undergoes a 180° phase change upon reflection. (Higher to lower has no associated change.)

The additional optical path from A to B to C allows constructive interference when equal to a integer multiple of /noil. (It is assumed that nair > noil > nwater.)

The film thickness for normal incidence corresponding to a bright fringe is (/noil)/2.