- Because of the slow response of practical detectors, interference phenomena are also a matter of averaging over time and space. - 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.. - 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. - In Figure 3.1 we have sketched two successive wave trains of the partial waves. - 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 head of the wave trains in partial wave 2 then coincide with the tail of the corresponding wave trains in partial wave 1. - To see clearly that this quantity is related to the contrast of the pattern, we introduce the definition of contrast or visibility. - where I max and I min are two neighbouring maxima and minima of the interference pattern described by Equation (3.7). - γ (τ ) is termed the complex degree of coherence and is a measure of the ability of the two wave fields to interfere. - Ordinary discharge lamps have spectral widths corresponding to coherence lengths of the order of 1 µ m while the spectral lines emitted by low-pressure isotope lamps have coherence lengths of several millimetres.. - 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 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).. - The complex amplitude of the two plane waves then becomes (see Equation (1.9a)). - Figure 3.2 Interference between two plane waves. - The interference term is therefore of the form cos 2π. - Here we have put λ = 0.6328 µ m, the wavelength of the He–Ne laser.. - The second equality of this expression is found by trigonometric manipulation of the angles (see Figure 3.2(b. - 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. - Figure 3.6 Intensity distribution in the xy-plane from interference between two plane waves. - V is equal to the amplitude of the distribution divided by the mean value and varies between 0 and 1. - Here we give an alternative description of the method. - Figure 3.7 Laser Doppler velocimetry. - 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. - 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. - 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. - Figure 3.8 Interference between two spherical waves emitted from P 1 and P 2. - Figures 3.8(a) and 3.8(b) we see how the density of the fringes increases as the dis- tance between P 1 and P 2 increases. - Figure 3.9 Interference between a plane wave and a spherical wave. - Figure 3.11. - By observing the interference between the reflected wave and a plane wave, one should in principle be able to determine the topography of the surface. - 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.. - If these surfaces are identical, it should be possible to observe interference between the waves scattered from A 1 and A 2 regardless of the complexity of the scattered wavefronts. - In the case of a plastic deformation of, for example, surface A 2 , it should be possible to do interferometric measurement of the resulting surface height difference between A 1 and A 2 . - the microstructure of the two surfaces must be identical. - This has to do with the mutual spatial coherence of the two scattered waves. - Most interferometers therefore consist of the following elements, shown schematically in Figure 3.12.. - Figure 3.12. - detector for observation of the interference.. - The geometric path length difference s of the light reaching an arbitrary point x on S 2. - from P 1 and P 2 is found from Figure 3.13(b). - Figure 3.13 Young’s interferometer. - The contrast of the interference fringes on S 2 is a measure of the degree of coherence. - As will be shown in Section 3.7 there is a Fourier transform relationship between the degree of coherence and the intensity distribution of the light source. - By assuming the source to be an incoherent circular disc of uniform intensity, one can find the diameter of the source by increasing the separation between P 1 and P 2 until the contrast of the central interference fringe on S 2 vanishes. - Here the amplitude of the incident light field is divided by the beamsplitter BS which is partly reflecting. - The path-length difference between the two partial waves can be varied by moving one of the mirrors, e.g. - x Figure 3.15 Michelson’s interferometer. - By counting the numbers of maxima per unit time, one can find the speed of the object.. - Figure 3.17 The Hewlett-Packard interferometer. - These mirrors are cube-corner reflec- tors in which the reflected beam is parallel to the incoming beam independent of the angle of incidence. - If the movable mirror moves with a velocity v 1 , the frequency of the reflected light will be Doppler-shifted by an amount f = 2v/λ 1 . - For two orthogonally polarized waves to interfere, a polarizer has to be placed in front of the detector. - A fraction of the laser light of frequencies f 1 and f 2 is sent to the detector 6, producing a reference signal of frequency f 2 − f 1 . - If we neglect the spatial term (which is constant at the detector), the electric fields of the optical signal ψ S and of the local oscillator ψ LO are given as. - see Figure 3.18:. - By low-pass filtering of the signals, one is left with. - In the treatment of the Young’s interferometer in Section 3.6.1 we assumed the light to be incident from a point source. - In the. - Then because of the small path-length difference, the intensity at P when only P 2 is open also will be I (α)/2.. - We see that γ 12 is equal to the normalized Fourier transform of the intensity distribution of the source, with the frequency coordinate equal to D/λ.. - With λ = 570 nm, α rad, and from its known distance determined from parallax measurements, the star’s diameter turned out to be about 380 million km or roughly 280 times that of the Sun.. - In the description of the Michelson interferometer in Section 3.6.2 we assumed the light source to be monochromatic and found the output intensity (assuming a 50/50 beamsplit- ter) to be:. - (3.59) where I is the total intensity of the incoming beam, d is the path-length difference and τ = d/c is the time difference between the two paths. - is known as the normalized spectral distribution function of the source. - e iϕ (3.67) is recognized as the complex temporal degree of coherence and is equal to the Fourier transform of the spectral distribution function of the source with the origin of the frequency axis moved to ν 0 . - At this point it should be appropriate to introduce a precise definition of the coherence time τ c (see Section 3.3). - This intensity distribution is illustrated in Figure 3.20. - From Equation (3.73) we see that the envelope of the sinusoidal fringes becomes narrower with increasing spectral width. - From the definition of the coherence time (Equation (3.69)) we get. - For a source covering the whole visible spectrum, ν Hz, we get a coherence length L c = 664 nm, which means that only the zeroth-order fringe has full visibility, while the visibility of the first-order fringe has dropped to almost zero.. - In the development of the spatial degree of coherence, we assumed a quasimonochro- matic source, and in the development of the temporal degree of coherence we assumed a plane wave. - γ (τ )γ 12 (3.75) This equation is known as the reduction property of the complex degree of coherence. - Figure 3.21. - Assuming the intensities of the waves from each mirror to both be equal to I , the intensity at the detector drops from 4I (full coherence) to 2I (no coherence). - By a scanned motion of the reference mirror M 1 , intensity maxima at the detector are found and thereby the depth of such boundaries can be measured. - Figure 3.22 Measurement of the anterior chamber depth using optical low-coherence interferom- etry. - The graph displays the magnitude of the reflected intensity as a function of distance. - Figure 3.23 Grey scale Optical Coherence Tomography image of the anterior chamber of a human eye obtained in vivo. - The image is displayed using a logarithmic mapping of the measured optical signal to brightness. - Figure 3.22 shows an example of axial range measurements performed in the anterior chamber of the eye.. - The graph shows the intensity at the detector as a function of the position of the reference mirror. - The intensity is a measure of the discontinuity of the optical properties of the tissue. - To determine the actual depth of the various boundaries, the distance between the echoes has to be multiplied by the index of refraction of the tissue.. - Figure 3.23 shows an example of a tomographic image of the anterior chamber of the eye displayed in grey scale. - Figure 3.24 shows a schematic representation of a fibre optic version of the interferometer with a superluminiscent LED (see Section 5.4.4) as the light source.. - The terms ω c + ω m constitute what is called the upper sideband while all of the ω c − ω m terms form the lower sideband. - 3.3 Calculate the coherence length of the following sources:. - Calculate the (perpendicular) height of the source slit from the mirror.. - Let the light have a wavelength λ 0 = 500 nm, and the index of refraction of the glass be n = 1.5. - What is the prism angle if the separation of the fringes is 0.5 mm?. - 3.8 As mentioned in the text, the spatial coherence function γ 12 is equal to the Fourier transform of the intensity distribution of the source.
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