Richard Leach - Fundamental prinsiples of engineering nanometrology (778895), страница 18
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If Ex and Ey are both real, the light is called linearly polarized.Another special case is when Ey ¼ iEx, in which case the vector describesa circle in space; for that reason this case is called circular polarization.When light beams from different sources, or from the same source but viadifferent paths, act on the same location, their electric fields can be added.This is called the principle of superposition, and causes interference. Visible,stable interference can appear when the wavelengths are the same and thereis a determined phase relationship between the superimposed waves.
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.
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