Принципы нанометрологии (1027623), страница 20

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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. The interferometer can also be madewith spherical mirrors. In this case the equation for the finesse changessomewhat. This and other details of the Fabry-Pérot interferometer areextensively treated in [13].Fabry-Pérot interferometers have many applications in spectroscopy.However, in engineering nanometrology they are used as the cavity in lasersand they can be used to generate very small, very well defined displacements,either as part of a laser (the so-called ‘measuring laser’) or as an externalcavity.

This is treated in more detail in section 5.7.1.2.7172C H A P T ER 4 : Length traceability using interferometry4.5 Gauge block interferometry4.5.1 Gauge blocks and interferometryAs discussed in section 4.2 the length of a gauge block wrung to a platen canbe measured using interferometry. The ISO definition of a gauge block lengthhas a two-fold purpose: (1) to ensure that the length can be measured byinterferometry, and (2) to ensure that there is no additional length due towringing. An issue that is not obvious from the definition is whether the twosided length of a gauge block after calibration by interferometry coincideswith the mechanical length, for example as measured by mechanical probescoming from two sides.

Up to now no discrepancies have been found thatexceed the measurement uncertainty, which is in the 10 nm to 20 nm range.Figure 4.15 shows a possible definition for a mechanical gauge blocklength. A gauge block with length L is probed from both sides with a perfectlyround probe of diameter d, being typically a few millimetres in diameter. Themechanical gauge block length, L, is the probe displacement, D, in the limitof zero force, minus the probe diameter, or L ¼ D d.4.5.2 Gauge block interferometryIn order to measure gauge blocks in an interferometer, a first requirement forthe light source is to have a coherence length that exceeds the gauge blocklength. Gauge block interferometers can be designed as a Twyman-Green ora Fizeau configuration, where the former is more common. For the majorityof the issues discussed in this section either configuration can be considered.Figure 4.16 is a schema of a gauge block interferometer containing a gaugeblock.

The observer sees the fringe pattern that comes from the platen asshown in Figure 4.10. If the platen has a small tilt this will be a set of straight,FIGURE 4.15 Possible definition of a mechanical gauge block length.Gauge block interferometryFIGURE 4.16 Schema of a gauge block interferometer containing a gauge block.parallel interference fringes. However, at the location of the gauge block,a parallel plate can also be observed, but the fringe pattern may be displaced(see Figure 4.17).If the fringes are not distorted, then an integer number of half wavelengths will fit in the length of the gauge block.

In general this will not be thecase, and the shift of fringes gives the fractional length of the gauge block.The length of the gauge block is given by½4block ðtopÞ 4ref ðtop areaÞ ½4platen ðbaseÞ 4ref ðbase areaÞlnL¼Nþ2p2nðlÞlnðN þ fÞ¼2nðlÞ(4.17)where N is the number of half wavelengths between the gauge block top andthe position on the platen for wavelength l, n is the air refractive index and fis the fraction f ¼ a/b in Figure 4.17.

fblock (top) is the phase on top of thegauge block, fref (top area) is the phase at the reference plate at the location ofthe top area, fplaten (base) is the phase on the platen next tothe gauge block and fref (base area) is the phase at the reference plate at the7374C H A P T ER 4 : Length traceability using interferometryFIGURE 4.17 Theoretical interference pattern of a gauge block on a platen.location next to the image of the gauge block. For a flat reference surface, thephase for the areas corresponding to the base and top of the gauge block arethe same (fref (top area) ¼ fref (base area)) and equation (4.17) simplifiesaccordingly. Equation (4.17) is the basic equation that links an electromagnetic wavelength, l, to a physical, mechanical length, L.Some practical issues that are met when applying equation (4.17) aretreated in the next sections.4.5.3 Operation of a gauge block interferometer4.5.3.1 Fringe fraction measurement – phase steppingAs indicated in Figure 4.17, the fringe fraction can be estimated visually.

Forthis purpose, as a visual aid some fiducial dots or lines can be applied to thereference mirror. Experienced observers can obtain an accuracy of 5%, corresponding to approximately 15 nm.However, more objective and accurate methods for determining the fringefraction are possible by phase shifting; this means that the optical distance ofeither the reference mirror or the platen–gauge block combination is changedin a controlled way [14]. Established methods for phase shifting include:-displacing the reference mirror or the gauge block with platen usingpiezoelectric displacement actuators;-positioning an optical parallel in the beam.

Giving the optical parallela small rotation generates a small controllable phase shift.Having the possibility of shifting the phase, the fraction can be derived ina semi-manual way. For example, the fringes on the platen can be adjusted toa reference line then on the gauge block, and then the next fringe on theplaten can be adjusted to this reference line.

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