Принципы нанометрологии (1027623), страница 32

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This effect also accounts for the aperture correction in gauge block interferometry (see section 4.5.4.6), but it has127128C H A P T ER 6 : Surface topography measurement instrumentationFIGURE 6.8 Numerical aperture of a microscope objective lens.a larger effect here; it may cause a step height to be measured up to 15 % short.This correction can usually be determined by measuring a step artefact witha calibrated height value and it can be directly determined using a grating [40].The second limitation is the optical resolution of the objective. Theresolution determines the minimum distance between two lateral features ona surface that can be measured.

The resolution is approximately given byr ¼l2NA(6.3)where l is the wavelength of the incident radiation [41]. For a theoreticallyperfect optical system with a filled objective pupil, the optical resolution isgiven by the Rayleigh criterion, where the ½ in equation (6.3) is replaced by0.61. Yet another measure of the optical resolution is the Sparrow criterion,or the spatial wavelength where the instrument response drops to zero andwhere the ½ in equation (6.3) is replaced by 0.82. Equation (6.3), and theRayleigh and Sparrow criteria, can be used almost indiscriminately, so theuser should always check which expression has been used where opticalresolution is a limiting factor. Also, equation (6.3) sets a minimum value. Ifthe objective is not optically perfect (i.e.

aberration-free) or if a part of thebeam is blocked (for example, in a Mirau interference objective, or whena steep edge is measured) the value becomes higher (worse).Optical instrumentsTable 6.1Minimum distance between features for different objectivesMagnificationNAResolution/mmPixel spacing/mm1020500.30.40.51.000.750.601.750.880.35For some instruments, it may be the distance between the pixels (determined by the image size and the number of pixels in the camera array) in themicroscope camera array that determines the lateral resolution.

Table 6.1gives an example for a commercial microscope – for the 50 objective, it isthe optical resolution that determines the minimum distance betweenfeatures, but with the 10 objective it is the pixel spacing.The optical resolution of the objective is an important characteristic of anoptical instrument, but its usefulness can be misleading. When measuringsurface texture, one must consider the ability to measure the spacing ofpoints in an image along with the ability to accurately determine the heightsof features. We need an optical equivalent of equation (6.1) for stylusinstruments. This is not a simple task and, at the time of writing, the exactdefinitions have not been decided on.

Also, there may not be a commonexpression that can be used for all optical instruments. One such definition isthe lateral (50 %) resolution or the wavelength at 50 % depth modulation.This is defined as one half the spatial period of a sinusoidal profile for whichthe instrument response (measured feature height compared to actual featureheight) falls to 50 %. The instrument response can be found by directmeasurement of the instrument transfer function (see [42] and annex C in[43]). Note that this definition is not without its faults – the value of thelateral (50 %) resolution will vary with the height of the features beingmeasured (as with equation (6.1) for a stylus instrument).Another important factor for optical instruments that magnify thesurface being measured is the optical spot size.

For scanning type instruments the spot size will determine the area of the surface measured as theinstrument scans. To a first approximation, the spot size mimics the actionof the tip radius on a stylus instrument, i.e. it acts as a low-pass filter [44].The optical spot size is given byfld0 ¼(6.4)w0where f is the focal length of the objective lens and w0 is the beam waist (theradius of the 1/e2 irradiance contour at the plane where the wavefront isflat [41]).129130C H A P T ER 6 : Surface topography measurement instrumentationIn a non-scanning areal instrument it will be the field of view thatdetermines the lateral area that is measured. In the example given inTable 6.1 the areas measured are 0.3 mm 0.3 mm and 1.2 mm 1.2 mmfor the 50 and 10 objectives respectively.Many optical instruments, especially those utilizing interference, can beaffected by the surface having areas that are made from different materials[45,46].

For a dielectric surface there is a p phase change on reflection (atnormal incidence), i.e. a p phase difference between the incident andreflected beams. For materials with free electrons at their surfaces (i.e. metalsand semiconductors) there will be a (p d) phase change on reflection, whered is given by2n1 k2d ¼(6.5)1 n22 k22where n and k are the refractive and absorption indexes of the surrounding air(medium 1) and the surface being measured (medium 2) respectively. For theexample of a chrome step on a glass substrate, the difference in phase changeon reflection gives rise to an error in the measured height of approximately 20nm (at a wavelength of approximately 633 nm) when measured using anoptical interferometer. A stylus instrument would not make this error inheight. In the example of a simple step, it may be possible to correct for thephase change on reflection (if one has prior knowledge of the opticalconstants of the two materials) but, when measuring a multi-materialengineered surface, this may not be so easy to achieve.Most optical instruments can experience problems when measuringfeatures with very high slope angles or discontinuities.

Examples includesteep-sided vee-grooves and steps. The NA of the delivery optics will dictatethe slope angle that is detectable and in the case of a microscope objective itwill be the acceptance angle. For variable focus and confocal instruments (seesections 6.7.2.2 and 6.7.3.1) sharp, overshooting spikes are seen at the top ofsteps and often the opposite at the bottom of the step. These are usuallycaused by the instrument not measuring the topography correctly, sometimes due to only a single pixel spanning the discontinuity. For lowcoherence interferometers (see section 6.7.3.4) there can be problems thatare caused by diffraction and interference from the top and bottom surfacewhen a step height is less than the coherence length of the source [47,48].These effects give rise to patterns known as batwings (see Figure 6.9).

Ingeneral, care should be taken when measuring steep slopes with opticalinstruments. Note that some optical instruments can extend the slopelimitation of the objective by making use of diffusely scattered light. This canonly be achieved when the surface of the slope is sufficiently rough to obtainOptical instrumentsFIGURE 6.9 Example of the batwing effect when measuring a step using a coherencescanning interferometer.enough diffuse scatter.

It is also possible to extend the slope limitation withsome surfaces using controlled tilting of the sample and specialist imageprocessing [49].Many optical instruments for measuring surface topography utilizea source that has an extended bandwidth (for example, coherence scanninginterferometers and confocal chromatic microscopy). Such instruments canbe affected by dispersion in the delivery optics or due to thin films at thesample surface.

For example, due to dispersion, coherence scanning interferometers can miscalculate the fringe order, giving rise to what are referredto as 2p discontinuities or ghost steps [50]. Dispersion effects can also befield or surface gradient dependent [51]. Also, all optical instruments will beaffected by aberrations caused by imperfections in the optical componentsand these will affect the measurement accuracy and optical resolution (suchsystems will not be diffraction limited).Finally it is important to note that surface roughness plays a significantrole in measurement quality when using optical instrumentation. Manyresearchers have found that estimates of surface roughness derived fromoptical measurements differ significantly from other measurement techniques [52–55]. The surface roughness is generally over-estimated by opticalinstrumentation (this is not necessarily true when considering areaintegrating instruments) and this can be attributed to multiple scattering.Although it may be argued that the local gradients of rough surfaces exceedthe limit dictated by the NA of the objective and, therefore, would be classified as beyond the capability of optical instrumentation, measured valueswith high signal-to-noise ratio are often returned in practice.

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