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If thewavelengths are not the same, or the phase relationship is not constant, theeffect is called beating, which means that the intensity may very witha certain frequency.A fixed phase relationship can be achieved by splitting light, coming fromone source, into two beams and recombining the light again. An instrumentthat accomplishes this is called an interferometer. An example of an interferometer is shown in Figure 4.3.5960C H A P T ER 4 : Length traceability using interferometryFIGURE 4.3 Amplitude division in a Michelson/Twyman-Green interferometer whereS is the source, A and B are lenses to collinate and focus the light respectively, C isa beam-splitter, D is a detector and M1 and M2 are plane mirrors.Consider the fields E1(t) and E2(t) in the interferometer in Figure 4.3which travel paths to and from M1 and M2 respectively and combine at thedetector, D.
According to the principle of superposition we can writeEðtÞ ¼ E1 ðtÞ þ E2 ðtÞ:(4.3)Combining equations (4.1), (4.2) and (4.3), with some additionalassumptions, gives finally,pffiffiffiffiffiffiffiffi4pDLI ¼ I1 þ I2 þ 2 I1 I2 cos(4.4)lwhere DL is the path difference between the two beams and I are intensities,i.e. the squares of the amplitudes.Equation (4.4) is the essential equation of interference.
Depending on theterm 4pDL/l, the resultant intensity on a detector can have a minimum ora maximum, and it depends with a (co)sine function on the path difference orthe wavelength.From equation (4.4) it is evident that the intensity has maxima for4pDL/l ¼ 2pp, with p ¼ 0, 1, 2, ., so that DL ¼ pl/2 and minima forDL ¼ (p þ 0.5)l/2.Introduction to interferometry4.3.2 Beat measurement when u1 s u2If either E1 or E2 are shifted in frequency, or if E1 and E2 originate fromsources with a different frequency, we can write analogous to equation (4.4)pffiffiffiffiffiffiffiffi4pLI ¼ I1 þ I2 þ 2 I1 I2 cosþ ðu2 u1 Þt :(4.5)l2We obtain an interference signal that oscillates with the differencefrequency, which can readily be measured by a photodetector if u1 and u2 arenot significantly different.4.3.3 Visibility and contrastIf the intensities I1 and I2 are equal, equation (4.4) reduces to4pDL2pDL¼ 4I1 cos:I ¼ 2I1 1 þ cosll(4.6)This means that the minimum intensity is zero and the maximumintensity is 4I1.
Also it is clear that if I1 or I2 are zero, the interference term inequation (4.4) vanishes and a constant intensity remains. The relative visibility, V, of the interference can be defined aspffiffiffiffiffiffiffiffiImax Imin2 I1 I2V ¼¼:(4.7)Imax þ IminI1 þ I2The effect of visibility is illustrated in Figure 4.4, for the cases I1 ¼ I2 ¼ 0.5(V ¼ 1); I1 ¼ 0.95, I2 ¼ 0.05 (V ¼ 0.44) and I1 ¼ 0.995, I2 ¼ 0.005 (V ¼ 0.07).Figure 4.4 illustrates that, even with very different intensities of the twobeams, still the fringes can be easily distinguished.
Also note that increasinga single intensity whilst leaving the other constant diminishes the contrastbut increases the absolute modulation depth.FIGURE 4.4 Intensity as a function of phase for different visibility.6162C H A P T ER 4 : Length traceability using interferometryFIGURE 4.5 Intensity distribution for a real light source.4.3.4 White light interference and coherence lengthEquation (4.4) suggests that the interference term will continue to oscillateup to infinite DL.
However, there is no light source that emits a singlewavelength l; in fact every light source has a finite bandwidth, Dl. Figure 4.5shows the general case; if Dl/l < 0.01 we can speak of a monochromatic lightsource. However, for interferometry over a macroscopic distance, lightsources with a very small bandwidth are needed.From equation (4.4) it is evident that an interference maximum appearsfor DL ¼ 0, independent of the wavelength, l.
This phenomenon is calledwhite light interference. If the light source emits a range of wavelengths, infact for each wavelength a different interference pattern is formed and wherethe photodetector measures the sum of all of these patterns, the visibility, V,may deteriorate with increasing path difference, DL.In Figure 4.6 the effect of a limited coherence length is illustrated fora number of different light sources:1. A white light source with the wavelength uniformly distributed overthe visible spectrum, i.e.
between l ¼ 350 nm and l ¼ 700 nm;2. A green light source with the bandwidth uniformly distributedbetween l ¼ 500 nm and l ¼ 550 nm;3. A monochromatic light source with l ¼ 525 nm.Note that for each wavelength (colour) a different pattern is formed. Inpractical white light interferometry these colours can be visibly distinguishedover a few wavelengths. White light interference is only possible in interferometers where the path difference can be made approximately zero.Introduction to interferometryFIGURE 4.6 Illustration of the effect of a limited coherence length for different sources.The path length, DL, over which the interference remains visible, i.e. thevisibility decreases by less than 50 %, is called the coherence length and isgiven byDL ¼l0l0 ¼ Ql0Dl(4.8)where l0 is the wavelength of the light source and Q is the quality factorwhich determines over how many wavelengths interference is easily visible.Table 4.2 gives a few characteristics of known light sources.In the early twentieth century, the cadmium spectral lamp was used forinterference over macroscopic distances.
Michelson’s determination of thecadmium lamp wavelength related to the metre standard was a breakthroughto a metre definition based on physical constants. The orange-red line of the86Kr spectral lamp was used as the metre definition from 1963 until 1983.This definition was possible as, with some effort, interference over a metrelength difference was possible and a length up to one metre could bemeasured using interferometry.Table 4.2The quality factor and coherence length of some light sourcesLight sourceBulbHg lampCd lamp86Kr lampHe-Ne laser (multiple mode)He-Ne laser (single mode)Q1.818003.1 1051.4 1068 104108DL/m60.8 101 1030.20.80.0560l0/nmColour525546644565633633whitegreenredorange-redredred6364C H A P T ER 4 : Length traceability using interferometry4.4 Interferometer designsFor precision measurements, many interferometer types are used.
It isimportant that for almost all types of interferometer the principles outlinedin section 4.3 are valid.4.4.1 The Michelson and Twyman-Green interferometerWhere Michelson was a major pioneer in interferometry and carried outexperiments that achieved major breakthroughs in physics, one often refersto a Michelson interferometer where in fact a Twyman-Green interferometeris intended. The original Michelson interferometer does not operate withcollimated light, but with a point source, S, as shown in Figure 4.7.A beam-splitter, A, with a 50 % coating splits the input beam. Theinterference fringes are detected from B.
The compensator, C, is a glass platewith the same thickness as A which makes the optical path length throughglass equal for both beams. This ensures that chromatic effects in glass plate,A, are compensated and white light interferometry is possible.Optically, the system as viewed from B consists of two sources, M1 and M2,behind each other. If the two image planes, M1 and M2, are parallel, this isequivalent to sources in line behind each other and one detects circular fringes.If M1 and M2 intersect, the crossover is the position of zero path differenceand, as this region is a straight line of intersection, white light fringes willappear on the straight line of the intersection.
The fringes appear to belocalised at the front mirror, M1, i.e. the detector must be focused on thisFIGURE 4.7 Schema of the original Michelson interferometer.Interferometer designssurface in order to obtain the sharpest fringes. With increasing displacementthe fringes become spherical because of the divergent light source.4.4.1.1 The Twyman-Green modificationIn the Twyman-Green modification to the Michelson interferometer, thesource is replaced by a point source, S, at the focus of a well-corrected concavelens (see Figure 4.8).
The lens B collects the emerging light and the detectorobserves the interference pattern at the focal plane, D.Consider the case where the mirror and its image are parallel. Now thecollimated point source leads to a field of uniform intensity. Variations of thisinterferometer are the Köster gauge block interferometer [8], displacementmeasuring interferometers (see section 5.2) and the Linnik- and Mirau-typeinterference microscopes (see section 6.7.3.2).An important characteristic of the Twyman-Green interferometer is thatthe paths in both beams can be made equal so that white light interferenceoccurs.
A disadvantage is that both beams have a macroscopic path lengthand can be sensitive to turbulence and vibration. The reflectivity of bothmirrors can be up to 100 %. If the reflectivity of the mirrors is different, thevisibility decreases, as is illustrated in Figure 4.4. In the interferogram, thedifference between the two mirrors is observed. For example, if both mirrorsare slightly convex, and one mirror is slightly tilted, the interferogram willconsist of straight lines (the same as with perfectly flat mirrors).FIGURE 4.8 Schema of a Twyman-Green interferometer.6566C H A P T ER 4 : Length traceability using interferometry4.4.2 The Fizeau interferometerIn Fizeau interferometry, the reference surface and the surface to bemeasured are brought close together.
Compared to Figure 4.3, mirror M1 istransparent and partially reflecting, and the partial reflecting side is positioned close and almost parallel to mirror M2. This gives a configuration asshown in Figure 4.9.For a wedge angle, a, and perfectly flat mirrors, the intensity of theinterference pattern between the mirrors is given bypffiffiffiffiffiffiffiffi(4.9)IðxÞ ¼ I1 þ I2 þ 2 I1 I2 cos 2kðDL þ xaÞwhere x is the position of the interference pattern from the left edge of themirrors. In two dimensions, with circular mirrors, this gives a characteristicinterference pattern consisting of straight lines (see Figure 4.10).The Fizeau interferometer gives a direct way of observing geometricalfeatures in an interferogram.
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