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One way of illustrating the light emitted by real sources is to picture it as sinusoidalwave trains of finite length with randomly distributed phase differences between theindividual tra

Interference

3.1 INTRODUCTION

The superposition principle for electromagnetic waves implies that, for example, two

overlapping fields u1 and u2 add to give u1+ u2 This is the basis for interference.Because of the slow response of practical detectors, interference phenomena are also amatter of averaging over time and space Therefore the concept of coherence is intimatelyrelated to interference In this chapter we will investigate both topics A high degree ofcoherence is obtained from lasers, which therefore have been widely used as light sources

in interferometry In recent years, lack of coherence has been taken to advantage in atechnique called low-coherence or white-light interferometry, which we will investigate

at the end of the chapter

3.2 GENERAL DESCRIPTION

Interference can occur when two or more waves overlap each other in space

Assume that two waves described by

Optical Metrology Kjell J G˚asvik

Copyright  2002 John Wiley & Sons, Ltd.

ISBN: 0-470-84300-4

As can be seen, the resulting intensity does not become merely the sum of the

inten-sities ( = I1+ I2) of the two partial waves One says that the two waves interfere and

2√

I1I2cos φ is called the interference term We also see that when

φ = (2n + 1)π, for n = 0, 1, 2, cos φ = −1 and I reaches its minima The two waves are in antiphase which means

that they interfere destructively

When

φ = 2nπ, for n = 0, 1, 2, cos φ= 1 and the intensity reaches its maxima The two waves are in phase whichmeans that they interfere constructively

For two waves of equal intensity, i.e I1= I2= I0, Equation (3.3) becomes

3.3 COHERENCE

Detection of light (i.e intensity measurement) is an averaging process in space and time

In developing Equation (3.3) we did no averaging because we tacitly assumed the phase

difference φ to be constant in time That means that we assumed u1 and u2 to havethe same single frequency Ideally, a light wave with a single frequency must have aninfinite length Mathematically, even a pure sinusoidal wave of finite length will have afrequency spread according to the Fourier theorem (see Appendix B) Therefore, sourcesemitting light of a single frequency do not exist

One way of illustrating the light emitted by real sources is to picture it as sinusoidalwave trains of finite length with randomly distributed phase differences between theindividual trains

Assume that we apply such a source in an interference experiment, e.g the Michelsoninterferometer described in Section 3.6.2 Here the light is divided into two partial waves

of equal amplitudes by a beamsplitter whereafter the two waves are recombined to interfereafter having travelled different paths

In Figure 3.1 we have sketched two successive wave trains of the partial waves The

two wave trains have equal amplitude and length L c, with an abrupt, arbitrary phase ference Figure 3.1(a) shows the situation when the two partial waves have travelled equalpath lengths We see that although the phase of the original wave fluctuates randomly, thephase difference between the partial waves 1 and 2 remains constant in time The result-ing intensity is therefore given by Equation (3.3) Figure 3.1(c) shows the situation when

dif-partial wave 2 has travelled a path length L clonger than partial wave 1 The head of thewave trains in partial wave 2 then coincide with the tail of the corresponding wave trains

in partial wave 1 The resulting instantaneous intensity is still given by Equation (3.3), butnow the phase difference fluctuates randomly as the successive wave trains pass by As

a result, cos φ varies randomly between+1 and −1 When averaged over many wave

trains, cos φ therefore becomes zero and the resulting, observable intensity will be

Figure 3.1(b) shows an intermediate case where partial wave 2 has travelled a path

length l longer than partial wave 1, where 0 < l < L c Averaged over many wave trains,

the phase difference now varies randomly in a time period proportional to τ = l/c and remains constant in a time period proportional to τ c − τ where τ c = L c /c The result isthat we still can observe an interference pattern according to Equation (3.3), but with areduced contrast To account for this loss of contrast, Equation (3.3) can be written as

I = I1+ I2+ 2
I1I2|γ (τ)| cos φ ( 3.7)

where|γ (τ)| means the absolute value of γ (τ).

To see clearly that this quantity is related to the contrast of the pattern, we introducethe definition of contrast or visibility

V = Imax− Imin

Imax+ Imin

( 3.8)

where Imaxand Iminare two neighbouring maxima and minima of the interference pattern

described by Equation (3.7) Since cos φ varies between+1 and −1 we have

Imax= I1+ I2+ 2
I1I2|γ (τ)| (3.9a)

Imin= I1+ I2− 2
I1I2|γ (τ)| (3.9b)

which, put into Equation (3.8), gives

which shows that in this case|γ (τ)| is exactly equal to the visibility γ (τ) is termed the

complex degree of coherence and is a measure of the ability of the two wave fields tointerfere From the previous discussions we must have

coher-Of more interest is to know the value of τ c, i.e at which path length difference

|γ (τ)| = 0 In Section 5.4.9 we find that in the case of a two-frequency laser this

coherence length and τ c the coherence time

We see that Equation (3.13) is in accordance with our previous discussion where weargued that sources of finite spectral width will emit wave trains of finite length This isverified by the relation

ν= λ c

which can be derived from Equation (1.2)

As given in Section 1.2, the visible spectrum ranges from 4.3 to 7.5× 1014 Hz which

gives a spectral width roughly equal to ν= 3 × 1014 Hz From Equation (3.13), thecoherence time of white light is therefore about 3× 10−15 s, which corresponds to a

coherence length of about 1µm In white-light interferometry it is therefore difficult toobserve more than two or three interference fringes This condition can be improved byapplying colour filters at the cost of decreasing the intensity

Ordinary discharge lamps have spectral widths corresponding to coherence lengths ofthe order of 1µm while the spectral lines emitted by low-pressure isotope lamps havecoherence lengths of several millimetres

By far the most coherent light source is the laser A single-frequency laser can havecoherence lengths of several hundred metres This will be analysed in more detail inSection 5.4.9

INTERFERENCE BETWEEN TWO PLANE WAVES 41

So far we have been discussing the coherence between two wave fields at one point inspace This phenomenon is termed temporal or longitudinal coherence It is also possible

to measure the coherence of a wave field at two points in space This phenomenon iscalled spatial or transverse coherence and can be analysed by the classical Young’s doubleslit (or pinhole) experiment (see Section 3.6.1) Here the wave field at two points P1 and

P2 is analysed by passing the light through two small holes in a screen S1 at P1 and P2and observing the resulting interference pattern on a screen S2(see Figure 3.13(a)) In the

same way as the temporal degree of coherence γ (τ ) is a measure of the fringe contrast

as a function of time difference τ , the spatial degree of coherence γ12 is a measure ofthe fringe contrast of the pattern on screen S2 as a function of the spatial difference D

between P1 and P2 Note that since γ12 is the spatial degree of coherence for τ= 0, it isthe contrast of the central fringe on S2 that has to be measured

To measure the spatial coherence of the source itself, screen S1 has to be placed

in contact with the source It is immediately clear that for an extended thermal lightsource,|γ12| = 0 unless P1= P2, which gives |γ11| = 1 On the other hand, if we move

S1 away from this source, we observe that |γ12| might be different from zero, whichshows that a wave field increases its spatial coherence by mere propagation We alsoobserve that|γ12| increases by stopping down the source by, for example, an aperture until

|γ12| = 1 for a pinhole aperture The distance D cbetween P1 and P2 for which|γ12| = 0

is called the spatial coherence length It can be shown that D c is inversely proportional

to the diameter of the aperture in analogy with the temporal coherence length, which isinversely proportional to the spectral width Moreover, it can be shown that |γ12| is theFourier transform of the intensity distribution of the source and that|γ (τ)| is the Fourier

transform of the spectral distribution of the source (see Section 3.7)

An experimentalist using techniques like holography, moir´e, speckle and photoelasticityneed not worry very much about the details of coherence theory Both in theory andexperiments one usually assumes that the degree of coherence is either one or zero.However, one should be familiar with fundamental facts such as:

(1) Light from two separate sources does not interfere

(2) The spatial and temporal coherence of light from an extended thermal source isincreased by stopping it down and by using a colour filter respectively

(3) The visibility function of a multimode laser exhibits maxima at an integral multiple

of twice the cavity length (see Section 5.4.9)

3.4 INTERFERENCE BETWEEN TWO PLANE WAVES

Figure 3.2(a) shows two plane waves u1, u2 with propagation directions n1, n2 that lie

in the xz-plane making the angles θ1 and θ2 to the z-axis We introduce the following quantities (see Figure 3.2(b)): α= the angle between n1 and n2, θ = the angle between

the line bisecting α and the z-axis The complex amplitude of the two plane waves then

becomes (see Equation (1.9a))

n1 z x

Figure 3.2 Interference between two plane waves

where

φ1= kxsin



θ−α2

INTERFERENCE BETWEEN TWO PLANE WAVES 43

p/2 − q q

This is also clearly seen from Figure 3.2 From Equation (3.21) we see that the

dis-tance between the interference fringes (the wavelength d) is dependent only on the angle

between n1 and n2 By comparing Figures 3.2 and 3.4 we see how d decreases as α increases The diagram in Figure 3.5 shows the relation between d and α and f = 1/d and α according to Equation (3.21) Here we have put λ = 0.6328 µm, the wavelength

of the He–Ne laser

The intensity distribution across the xy-plane is found by inserting z= 0 intoEquation (3.19):

From the maxima (or minima) of this equation, we find the inter-fringe distance measured

along the x-axis to be

d

10

50 100

500 1000 5000

0 0.1

0.5 1

3 10 50

For completeness, we also quote the definition of the instantaneous frequency of a

sinu-soidal grating with phase φ(x) at a point x0

INTERFERENCE BETWEEN TWO PLANE WAVES 45

Figure 3.6 Intensity distribution in the xy-plane from interference between two plane waves

When Equations (3.26) and (3.27) are put into the expression for the visibility or contrastdefined in Section 3.3, Equation (3.8), they give

3.4.1 Laser Doppler Velocimetry (LDV)

As the name (also termed laser Doppler anemometry (Durst et al 1991), LDA)

indi-cates, this is a method for measuring the velocity of, for example, moving objects orparticles It is based on the Doppler effect, which explains the fact that light changes itsfrequency (wavelength) when detected by a stationary observer after being scattered from

a moving object

This is in analogy with the classical example for acoustical waves when the whistlefrom a train changes from a high to a low tone as the train passes by

Here we give an alternative description of the method Consider Figure 3.7 where a

particle is moving in a test volume where two plane waves are interfering at an angle α.

In Section 3.4 it was found that these two waves will form interference planes which are

parallel to the bisector of α and separated by a distance equal to (cf Equation (3.21))

As the particle moves through the test volume, it will scatter light when it is passing abright interference fringe and scatter no light when it is passing a dark interference fringe.The resulting light pulses can be recorded by a detector placed as in Figure 3.7

a

Detector

Figure 3.7 Laser Doppler velocimetry

For a particle moving in the direction normal to the interference planes with a

veloc-ity v, the time lapse between successive light pulses becomes

This method does not distinguish between particles moving in opposite directions

If the direction of movement is unknown, one can modulate the phase of one of theplane waves (by means of, for example, an acousto-optic modulator) thereby making theinterference planes move parallel to themselves with a known velocity This velocity willthen be subtracted when the particles are moving in the same direction and added whenmoving in the opposite direction

In Figure 3.7, the particles pass between the light source and the detector If theparticles scatter enough light, the detector can also be placed on the same side of the testvolume as the light source (the laser) Many other configurations of the light source and thedetector are described in the literature For example, one of the two waves can be directlyincident on the detector, or it is possible to have one single wave and many detectors.Laser Doppler velocimetry can be applied for measurement of the velocity of movingsurfaces, turbulence in liquids and gases, etc In the latter cases, the liquid or gas must

be seeded with particles Examples are measurements of stream velocities around shippropellers, velocity distributions of oil drops in combustion and diesel engines, etc

3.5 INTERFERENCE BETWEEN OTHER WAVES

Figure 3.8 shows the geometric configuration of the fringe pattern in the xz-plane when

two spherical waves from two point sources P and P on the z-axis interfere From

INTERFERENCE BETWEEN OTHER WAVES 47

(b)

Figure 3.8 Interference between two spherical waves emitted from P1 and P2

Figures 3.8(a) and 3.8(b) we see how the density of the fringes increases as the tance between P1 and P2increases Note that the figure shows the situation for the actualwavelength The distance between the point sources in Figure 3.8(a) is about seven wave-

dis-lengths, which for light with λ = 0.5 µm would give 3.5 µm Two real point sources

separated by the same distance as in the figure therefore would have resulted in a pattern

of much higher density; but the form of pattern would be the same

Figure 3.9 shows the interference pattern in the xz-plane when a spherical wave from

a point P on the z-axis interferes with a plane wave propagating in the z-direction.

In the same way as in the case of two plane waves, we can observe the intensitydistribution over a plane of arbitrary orientation in space A special distribution can be

observed over the xy-plane in Figure 3.9 This is further illustrated in Figure 3.10 which

shows the case of a spherical wave and a plane wave The intensity distribution in the

xy-plane is given as

I = I1+ I2+ 2
I1I2cos(βr2) ( 3.33) with r2 = x2+ y2, β = constant This is a sinusoidal pattern of linearly increasing fre-quency and is called a circular zone plate pattern

z x

be able to determine the topography of the surface However, for surfaces of roughnessgreater than the wavelength, phenomena such as interference between light scattered fromdifferent points on the surface, multiple scattering and diffraction effects will occur Inthis case, it therefore becomes impossible to derive the surface topography from a giveninterference pattern For smoother surfaces, however, such as optical components (lenses,mirrors, etc.) where tolerances of the order of fractions of a wavelength are to be measured,that kind of interferometry is quite common.

Figure 3.11 shows a Michelson interferometer (see Section 3.6.2, Figure 3.15) wherethe mirrors are exchanged for two non-optical surfaces A1 and A2 If these surfacesare identical, it should be possible to observe interference between the waves scatteredfrom A1 and A2regardless of the complexity of the scattered wavefronts In the case of aplastic deformation of, for example, surface A2, it should be possible to do interferometricmeasurement of the resulting surface height difference between A1 and A2 The problem

is, however, that the phrasing ‘identical surfaces’ in this context must be taken literally,i.e the microstructure of the two surfaces must be identical This has to do with themutual spatial coherence of the two scattered waves This requirement on the two surfacesmakes this interference experiment impracticable However, when we learn later on aboutholography, we shall see that this type of measurement becomes more than an imaginaryexperiment

3.6 INTERFEROMETRY

Interference phenomena can be observed in interferometers As stated in Section 3.3, lightwaves can interfere only if they are emitted by the same source Most interferometerstherefore consist of the following elements, shown schematically in Figure 3.12

source

Wave splitting

Introduction of phase difference

Wave combination

Observation of interference

Figure 3.12

• light source;

• element for splitting the light into two (or more) partial waves;

• different propagation paths where the partial waves undergo different phase tions;

contribu-• element for superposing the partial waves;

• detector for observation of the interference

Depending on how the light is split, interferometers are commonly classified as dividing or amplitude-dividing interferometers; but there are configurations which falloutside this classification

wavefront-3.6.1 Wavefront Division

As an example of a wavefront-dividing interferometer, consider the oldest of all ence experiments due to Thomas Young (1801) (Figure 3.13) The incident wavefront isdivided by passing through two small holes at P1 and P2 in a screen S1 The emergingspherical wavefronts from P1 and P2 will interfere, and the resulting interference pattern

interfer-is observed on the screen S2 This is in analogy with the case of two point sources in

Figure 3.8 with the plane of observation oriented parallel to the yz-plane.

The geometric path length difference s of the light reaching an arbitrary point x on S2

from P1 and P2 is found from Figure 3.13(b) When the distance z between S1 and S2 is

much greater than the distance D between P1 and P2, we have, to a good approximation,