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3 Interference 3.1 INTRODUCTION The superposition principle for electromagnetic waves implies that, for example, two overlapping fields u 1 and u 2 addtogiveu 1 + u 2 . This is the basis for interference. Because of the slow response of practical detectors, interference phenomena are also a matter of averaging over time and space. Therefore the concept of coherence is intimately related to interference. In this chapter we will investigate both topics. A high degree of coherence 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 a technique 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 u 1 = U 1 e iφ 1 (3.1a) and u 2 = U 2 e iφ 2 (3.1b) overlap. The electromagnetic wave theory tells us that the resulting field simply becomes the sum, viz. u = u 1 + u 2 (3.2) The observable quantity is, however, the intensity, which becomes I =|u| 2 =|u 1 + u 2 | 2 = U 2 1 + U 2 2 + 2U 1 U 2 cos(φ 1 − φ 2 ) = I 1 + I 2 + 2  I 1 I 2 cos φ (3.3) where φ = φ 1 − φ 2 (3.4) Optical Metrology. Kjell J. G ˚ asvik Copyright  2002 John Wiley & Sons, Ltd. ISBN: 0-470-84300-4 38 INTERFERENCE As can be seen, the resulting intensity does not become merely the sum of the inten- sities (= I 1 + I 2 ) of the two partial waves. One says that the two waves interfere and 2 √ I 1 I 2 cos φ is called the interference term. We also see that when φ = (2n + 1)π, for n = 0, 1, 2, . cos φ =−1andI 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 which means that they interfere constructively. For two waves of equal intensity, i.e. I 1 = I 2 = I 0 , Equation (3.3) becomes I = 2I 0 [1 + cos φ] = 4I 0 cos 2  φ 2  (3.5) where the intensity varies between 0 and 4I 0 . 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 u 1 and u 2 to have the same single frequency. Ideally, a light wave with a single frequency must have an infinite length. Mathematically, even a pure sinusoidal wave of finite length will have a frequency spread according to the Fourier theorem (see Appendix B). Therefore, sources emitting light of a single frequency do not exist. One way of illustrating the light emitted by real sources is to picture it as sinusoidal wave trains of finite length with randomly distributed phase differences between the individual trains. Assume that we apply such a source in an interference experiment, e.g. the Michelson interferometer 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 interfere after 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 dif- ference. Figure 3.1(a) shows the situation when the two partial waves have travelled equal path lengths. We see that although the phase of the original wave fluctuates randomly, the phase 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 partial wave 2 has travelled a path length L c longer than partial wave 1. The head of the wave 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), but now the phase difference fluctuates randomly as the successive wave trains pass by. As COHERENCE 39 (a) (b) (c) L c 1 2 1 2 1 2 L c L c c t L c L c L c Figure 3.1 a result, cos φ varies randomly between +1and−1. When averaged over many wave trains, cos φ therefore becomes zero and the resulting, observable intensity will be I = I 1 + I 2 (3.6) 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 is that we still can observe an interference pattern according to Equation (3.3), but with a reduced contrast. To account for this loss of contrast, Equation (3.3) can be written as I = I 1 + I 2 + 2  I 1 I 2 |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 introduce the definition of contrast or visibility V = I max − I min I max + I min (3.8) where I max and I min are two neighbouring maxima and minima of the interference pattern described by Equation (3.7). Since cos φ varies between +1and−1wehave I max = I 1 + I 2 + 2  I 1 I 2 |γ(τ)| (3.9a) I min = I 1 + I 2 − 2  I 1 I 2 |γ(τ)| (3.9b) 40 INTERFERENCE which, put into Equation (3.8), gives V = 2 √ I 1 I 2 |γ(τ)| I 1 + I 2 (3.10) For two waves of equal intensity, I 1 = I 2 , and Equation (3.10) becomes V =|γ(τ)| (3.11) 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 to interfere. From the previous discussions we must have |γ(0)|=1 (3.12a) |γ(τ c )|=0 (3.12b) 0 ≤|γ(τ)|≤1 (3.12c) where Equations (3.12a) and (3.12b) represent the two limiting cases of complete coher- ence and incoherence respectively, while inequality (3.12c) represents partial coherence. 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 hap- pens when τ = τ c = L c c = 1 ν (3.13) where ν is the difference between the two frequencies. It can be shown that this relation applies to any light source with a frequency distribution of width ν. L c is termed the coherence length and τ c the coherence time. We see that Equation (3.13) is in accordance with our previous discussion where we argued that sources of finite spectral width will emit wave trains of finite length. This is verified by the relation ν = λ c λ 2 (3.14) which can be derived from Equation (1.2). As given in Section 1.2, the visible spectrum ranges from 4.3 to 7.5 × 10 14 Hz which gives a spectral width roughly equal to ν = 3 × 10 14 Hz. From Equation (3.13), the coherence time of white light is therefore about 3 × 10 −15 s, which corresponds to a coherence length of about 1 µm. In white-light interferometry it is therefore difficult to observe more than two or three interference fringes. This condition can be improved by applying colour filters at the cost of decreasing the intensity. Ordinary discharge lamps have spectral widths corresponding to coherence lengths of the order of 1 µm while the spectral lines emitted by low-pressure isotope lamps have coherence lengths of several millimetres. By far the most coherent light source is the laser. A single-frequency laser can have coherence lengths of several hundred metres. This will be analysed in more detail in Section 5.4.9. INTERFERENCE BETWEEN TWO PLANE WAVES 41 So far we have been discussing the coherence between two wave fields at one point in space. 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 is called spatial or transverse coherence and can be analysed by the classical Young’s double slit (or pinhole) experiment (see Section 3.6.1). Here the wave field at two points P 1 and P 2 is analysed by passing the light through two small holes in a screen S 1 at P 1 and P 2 and observing the resulting interference pattern on a screen S 2 (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 γ 12 is a measure of the fringe contrast of the pattern on screen S 2 as a function of the spatial difference D between P 1 and P 2 . Note that since γ 12 is the spatial degree of coherence for τ = 0, it is the contrast of the central fringe on S 2 that has to be measured. To measure the spatial coherence of the source itself, screen S 1 has to be placed in contact with the source. It is immediately clear that for an extended thermal light source, |γ 12 |=0 unless P 1 = P 2 , which gives |γ 11 |=1. On the other hand, if we move S 1 away from this source, we observe that |γ 12 | might be different from zero, which shows that a wave field increases its spatial coherence by mere propagation. We also observe that |γ 12 | increases by stopping down the source by, for example, an aperture until |γ 12 |=1 for a pinhole aperture. The distance D c between P 1 and P 2 for which |γ 12 |=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 is inversely proportional to the spectral width. Moreover, it can be shown that |γ 12 | is the Fourier 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 photoelasticity need not worry very much about the details of coherence theory. Both in theory and experiments 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 is increased 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 u 1 , u 2 with propagation directions n 1 , n 2 that lie in the xz-plane making the angles θ 1 and θ 2 to the z-axis. We introduce the following quantities (see Figure 3.2(b)): α = the angle between n 1 and n 2 , 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)) u 1 = U 1 e iφ 1 (3.15) u 2 = U 2 e iφ 2 (3.16) 42 INTERFERENCE n 2 n 1 z x (a) (b) d x n 2 n 1 q 1 q 2 q a Figure 3.2 Interference between two plane waves where φ 1 = k  x sin  θ − α 2  + z cos  θ − α 2  (3.17) φ 2 = k  x sin  θ + α 2  + z cos  θ + α 2  (3.18) The intensity is given by the general expression in Equation (3.3) by inserting φ = φ 1 − φ 2 = k  x  sin  θ − α 2  − sin  θ + α 2  + z  cos  θ − α 2  − cos  θ + α 2  = 2k sin α 2 {−x cos θ + z sin θ} (3.19) The interference term is therefore of the form cos 2π d (z sin θ − x cos θ) (3.20) By comparing this expression with the real part of Equation (1.9a), we see that Equation (3.20) can be regarded as representing a plane wave with its propagation direction lying in the xz-plane making an angle θ with the x-axis as depicted in Figure 3.3, INTERFERENCE BETWEEN TWO PLANE WAVES 43 p/2 − q q z x Figure 3.3 Figure 3.4 and with a wavelength equal to d = λ 2sin(α/2) (3.21) 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 n 1 and n 2 . 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 = 0into Equation (3.19): I = I 1 + I 2 + 2  I 1 I 2 cos  2kx sin α 2 cos θ  (3.22) From the maxima (or minima) of this equation, we find the inter-fringe distance measured along the x-axis to be d x = 1 sin θ 2 − sin θ 1 = λ 2sin α 2 cos θ = d/cos θ(3.23) 44 INTERFERENCE f d 10 50 100 500 1000 5000 0 0.1 0.5 1 3 10 50 50 100 150 α (degrees) d (µm) f = 1/ d (lines/mm) l = 0.6328 mm d = l 2 sin (a/2) Figure 3.5 The second equality of this expression is found by trigonometric manipulation of the angles (see Figure 3.2(b)). Accordingly, the spatial frequency becomes f x = 1/d x = 2sin α 2 cos θ λ = cos θ/d (3.24) For completeness, we also quote the definition of the instantaneous frequency of a sinu- soidal grating with phase φ(x) at a point x 0 . f x (x = x 0 ) = dφ(x) dx     x=x 0 (3.25) The intensity distribution given in Equation (3.22) is sketched in Figure 3.6. We see that it varies between I max = I 1 + I 2 + 2  I 1 I 2 (3.26) and I min = I 1 + I 2 − 2  I 1 I 2 (3.27) with a mean value equal to I 0 = I 1 + I 2 (3.28) INTERFERENCE BETWEEN TWO PLANE WAVES 45 l d x x l 1 + l 2 + 2√ l 1 l 2 l 1 + l 2 l 1 + l 2 − 2√ l 1 l 2 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 contrast defined in Section 3.3, Equation (3.8), they give V = I max − I min I max + I min = 2 √ I 1 I 2 I 1 + I 2 (3.29) V is equal to the amplitude of the distribution divided by the mean value and varies between 0 and 1. We see that V = 1forI 1 = I 2 V = 0 for either I 1 or I 2 = 0 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 or particles. It is based on the Doppler effect, which explains the fact that light changes its frequency (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 whistle from 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)) d = λ 2sinα/2 (3.30) As the particle moves through the test volume, it will scatter light when it is passing a bright 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. 46 INTERFERENCE v 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 t D = d v (3.31) and thus the frequency f D = 1/t D = 2v sin α/2 λ (3.32) If there are many particles of different velocities, one will get many different frequencies. They can be recorded on a frequency analyser and the resulting frequency spectrum will tell how the particles are distributed among the different velocities. 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 the plane waves (by means of, for example, an acousto-optic modulator) thereby making the interference planes move parallel to themselves with a known velocity. This velocity will then be subtracted when the particles are moving in the same direction and added when moving in the opposite direction. In Figure 3.7, the particles pass between the light source and the detector. If the particles scatter enough light, the detector can also be placed on the same side of the test volume as the light source (the laser). Many other configurations of the light source and the detector are described in the literature. For example, one of the two waves can be directly incident 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 moving surfaces, 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 ship propellers, 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 1 and P 2 on the z-axis interfere. From [...]... on a screen 1 m away from the source are 1 mm apart Calculate the (perpendicular) height of the source slit from the mirror 3.6 Show that a for the Fresnel biprism of Figure P3. 1 is given by a = 2d(n − 1)α S′ a a S0 n S′ d Figure P3. 1 PROBLEMS 65 3.7 The Fresnel biprism is used to obtain fringes from a point source which is placed 2 m from the screen and the prism is midway between the source and the