Richard Leach - Fundamental prinsiples of engineering nanometrology (778895), страница 19
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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. If the distance DL is increased, the fringes willmove from left to right (or right to left).
If the tilt angle is changed, thedistance between the fringes changes. If either of the mirrors is not flat, this isobserved as distortions in the straightness of the fringes.If the interference term in equation (4.9) can be changed in somecontrolled manner, the phase f ¼ 2kDL can be determined by makingFIGURE 4.9 The Fizeau interferometer.Interferometer designsFIGURE 4.10 Typical interference pattern of a flat surface in a Fizeau interferometer.intensity measurements in one location (x, y).
The phase can be changed bya small displacement, DL, or by a wavelength change. If DL is changed in foursteps of l /8 each, and the intensities are labelled as IA, IB, IC and ID, then itcan be shown thatIB ID4ðx; yÞ ¼ arctan:(4.10)IA ICThis is an example of deriving the phase, and DL, by phase stepping. Thiscan only give an estimate of DL within an unknown integer number, N, ofhalf wavelengths.Considered over the surface, the distance between the surfaces S1 and S2can be expressed as!4S2 ðx; yÞ 4S1 ðx; yÞlDLðx; yÞ ¼:(4.11)Nþ2p2If the upper surface deviations of both S1 and S2 are to be consideredpositive in the glass–air interface direction then, apart from a constant termand a constant tilt, the deviations can be expressed asS2 ðx; yÞ ¼ 42 ðx; yÞl=4p and S1 ðx; yÞ ¼ 41 ðx; yÞl=4p(4.12)In S1 the coordinates can be (x, y) or (x, y), depending on the definitionand the (flipping) orientation of the (optical) surface.
However, in a Michelson interferometer for S1 the equivalent of S2 holds.If S1 is perfectly flat, or has a known flatness deviation, the form of the othersurface can be derived, either by visually observing the interference pattern or byanalysing the phase using equation (4.11). This method of surface interferometry is a research field of its own and is covered in several textbooks (see [9,10]).6768C H A P T ER 4 : Length traceability using interferometryBecause the lateral resolution is usually limited, this is form rather than surfacetexture measurement.
Uncertainties can be in the nanometre region in thedirection perpendicular to the surface. Limitations are in the roughness and themaximum angle that can be measured. For engineered surfaces this method isapplicable for polished, and precision turned, lapped and ground surfaces. Forsuch surfaces, Fizeau interferometry is a very powerful tool to obtain very rapidlythe complete geometry of the surface.Some characteristics of Fizeau interferometers should be mentioned, alsoin comparison to Michelson set-ups:-white light interference is not possible; one always needs a light sourcewith a coherence length of a few millimetres or more;-the reference mirror must be partially transmitting, and the back side ofthis reference mirror should not interfere with its front side.
This can beachieved by, for example, an anti-reflection coating or by a wedge;-if mirror S2 has a reflectivity of around 100 %, it is difficult to achievegood visibility, as the reference mirror must be transmitting;-the ambiguity of N can be a problem if it varies over the surface ina complicated way (i.e. the fringe pattern is complex and/or noisy). Thedetermination of the proper variation in N over the surface can becomplicated; this process is called phase unwrapping;-as mirror S1 is held upside-down, the interferometer measures the sumof the surface deviations of both surfaces. This enables an absoluteflatness calibration when a third flat is used. However, because of thecoordinate flipping the measurement in all three combinations must becombined with additional rotations of one of the flats [11].
Ina Michelson set-up an absolute calibration is not possible;-instead of flats, spheres can be measured and, with somemodifications, even parabolas can be measured. This is outside thescope of this book (but see [12]).4.4.3 The Jamin and Mach-Zehnder interferometersThe Jamin interferometer is depicted in Figure 4.11. The beams are split in Aand recombine at D.
A first important application of the Jamin interferometer was the measurement of the refractive index of gases (T1 and T2represent gas cells in Figure 4.11). The Jamin arrangement can also be usedto make an image interfere with itself, but slightly displaced, for example bytilting one mirror relative to the other. This is called shearing interferometry.Interferometer designsFIGURE 4.11 Schema of a Jamin interferometer.A modification of the Jamin arrangement is known as the Mach-Zehnderinterferometer and is depicted in Figure 4.12.
As in the Michelson interferometer, white light interference is possible and there is no limitation to thereflectance at, for example, points C and F.The Mach-Zehnder interferometer can be used for refractometry, i.e. formeasurement of the refractive index of a medium in either arm.
It can also bemodified in order to enable displacement measurement.FIGURE 4.12 Schema of a Mach-Zehnder interferometer.6970C H A P T ER 4 : Length traceability using interferometry4.4.4 The Fabry-Pérot interferometerIf in the Fizeau interferometer in Figure 4.9 both mirrors are placed almostparallel and the reflectance of both mirrors is increased, a particular type ofinterferometer is obtained, called the Fabry-Pérot interferometer (seeFigure 4.13).Light enters from the left, and B and B’ are the reflecting faces betweenwhich the interference occurs. P and P’ are spacers to put flats B and B’ asparallel as possible.
Between B and B’ multiple reflections occur. Equation(4.4) no longer holds if the reflectance, R, of both plates becomes significantlylarge, for example, R > 0.1. Summation of all reflected and transmittedcomponents leads to an infinite series, which can be expressed asF sin2T ¼Ll1 þ F sin2L(4.13)lwhere F is defined asF ¼4Rð1 RÞ2:(4.12)The reflectance of the whole system is given by R ¼ 1 T, where T is givenby equation (4.13) and where it is assumed that no absorption takes place.The transmittance as a function of the distance, L, between the plates, fora wavelength l ¼ 600 nm, is shown in Figure 4.14.Figure 4.14 shows (co)sine-like behaviour similar to that described inequation (4.4) for low reflectances, but for high reflectance of the mirrorsthere are sharp transmittance peaks.
This has the disadvantage that inbetween the peaks the position is hard to estimate, but it has the advantageFIGURE 4.13 Schematic of the Fabry-Pérot interferometer.Interferometer designsFIGURE 4.14 Transmittance as a function of distance, L, for various reflectances.that once a peak reflectance is achieved, one is very sure that a displacementof exactly an integer number of half wavelengths has taken place.The reciprocal of the full width of a fringe at half of the maximum intensityexpressed as a fraction of the distance between two maxima is given bypffiffiffiffip RppffiffiffiNR ¼¼F:(4.15)1R2The term NR is called the finesse of the interferometer.
For example, forR ¼ 0.9, NR ¼ 30. This means that 1/30th of a half wavelength can readily beresolved by this interferometer; compare this to half of a half wavelengthusing the same criterion for the cosine function in equation (4.4).At a fixed distance, L, the possible frequencies that fit in the cavity can becalculated as followslccc/ fm ¼ m/Df ¼ fmþ1 fm ¼:L ¼ m ¼ m22nf2nL2nL(4.16)Here m ¼ 0, 1, 2, . and n is the air refractive index, which is approximately 1. The frequency difference between two successive possiblefrequencies is called the free spectral range. For example, for a cavity lengthL ¼ 100 mm, Df ¼ 1.5 GHz. Clearly, in a Fabry-Pérot interferometer whitelight interferometry is not possible.
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