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Figure 4.1 isa schematic, and Figure 4.2 is a photograph, of a gauge block wrung to a platen.The definition of the length of a gauge block enables the possibility ofrelating the length to optical wavelengths by interferometry. Also, there is noadditional uncertainty due to the wringing as the auxiliary platen could bereplaced by another gauge block, where the wringing would have the sameeffect as the wringing to the platen, which is included in the length definition.Gauge blocks are classified into accuracy classes. The less accurate classesare intended to be used in the workshop. Using mechanical comparators,these gauge blocks can be compared to reference gauge blocks that are relatedto wavelengths using gauge block interferometers.Table 4.1 gives the tolerances for gauge block classes K, 0, 1 and 2according to ISO 3650 [5].

For those to be calibrated by interferometry (classK) the absolute length is not so critical as this length is explicitly measured.However, the demands on parallelism needed for good wringing, and anaccurate length definition, are highest. ISO 3650 gives the basis of demands,tolerances and definitions related to gauge blocks.The method of gauge block calibration by interferometry is a basicexample of how the bridge between the metre definition by wavelength anda material reference artefact can be made.

It will be the main subject of therest of this chapter.FIGURE 4.1 Definition of the length of a gauge block.5758C H A P T ER 4 : Length traceability using interferometryFIGURE 4.2 A typical gauge block wrung to a platen.Table 4.1ClassK012Gauge block classes according to ISO 3650 [5]Tolerance on length, L0.200.120.200.45mmmmmmmmþþþþ4248610106106106LLLLTolerance on parallelism for length, L0.050.100.160.30mmmmmmmmþþþþ2357107107107107LLLL4.3 Introduction to interferometry4.3.1 Light as a waveThis chapter will introduce the aspects of optics that are required to understand interferometry.

For a more thorough treatment of optics the reader isreferred to [7].Introduction to interferometryFor the treatment of light we will restrict ourselves to electromagneticwaves of optical frequencies, usually called ‘visible light’. From Maxwell’sequations it follows that the electric field of a plane wave, with speed, c,frequency, f, and wavelength, l, travelling in the z-direction, is given by ExeiðkzutÞEðz; tÞ ¼(4.1)Eywhere u ¼ 2pf ¼ 2pc/l is the circular frequency and k is the circular wavenumber, k ¼ 2p/l.Here we use the convention that a measurable quantity, for example theamplitude, Ex, can be obtained by taking the real part of equation (4.1) and weassume that Ey ¼ 0, i.e.

the light is linearly polarized in the x direction. At thelocation z ¼ 0, the electric field E ¼ Excosut. This means that the momentaryelectric field oscillates with a frequency f. For visible light, for example greenlight (l ¼ 500 nm), this gives, with the speed of light defined as c ¼ 299 792458 m$s1, a frequency of f ¼ 6 1014 Hz. No electric circuit can directlyfollow such a high frequency; therefore light properties are generallymeasured by averaging the cosine function over time. The intensity is givenby the square of the amplitude, thusIðzÞ ¼ hE,Ei ¼ ðE2x Þhcos2 uti:(4.2)A distortion at t ¼ 0, z ¼ 0, for example of the amplitude Ex in equation(4.1), will be the same as at time, t, at location z ¼ ut/k ¼ ct, so the propagation velocity is c indeed.In equation (4.1), the amplitudes Ex and Ey can both be complex.

In thatgeneral case we speak of elliptical polarization; the E-vector describes anellipse in space. 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.

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