The metrology system measures the position of the probe relative to the product in the six critical directions in the plane of motion of the probe (the measurement plane). By focussing a vertical and horizontal interferometer onto the Ψ-axis rotor, the displacement of the probe is measured relative to the reference mirrors on the upper metrology frame. Due to the reduced sensitivity in tangential direction at the probe tip, the Abbe criterion is still satisfied. Silicon Carbide is the material of choice for the upper metrology frame, due to its excellent thermal and mechanical properties. Mechanical and thermal analysis of this frame shows nanometer-level stabilities under the expected thermal loads. The multi-probe method allows for in-process separation of the spindle reference edge profile and the error motion. Tests demonstrate the stability of the metrology system.
In this section, the metrology loop concept will be explained. After a short discussion of the expected error sources, several methods of determining the relative position between probe and product are discussed.
Error sources
The probe and product position will deviate from the nominal position. Errors are generally separated in systematic and random errors. Systematic errors are repeatable
and can thus be calibrated. They include guidance straightness, mirror flatness and roundness, position dependent gravity deflection etc. By planning the calibration during the design (section 7.3), it is assured that each of these errors can be calibrated and hence be compensated for in the data-processing.
Random errors do not repeat, and can thus not be compensated for. The term random is disputable, since every error has a cause that can be physically explained. In this deterministic approach ‘non-repeatability primarily depends on time, money and skill (culture) of the user’ (Bryan, 1984). This group of errors includes system dynamics due to for instance to acceleration forces of the probe objective, unbalance of the spindle and floor vibrations. Acoustics may also be a source of dynamic excitation. Further, variations in the supply pressure to the air bearings will cause variations in gap height and thus in variation of the position of the stages. Thermal disturbances from inside the machine (e.g. actuators, read heads, etc) as well as from the environment, cause departures from the nominal position. According to (Bryan, 1990), thermal disturbances can be split in the effects of uniform temperatures other than 20°C and the effects of non-uniform temperatures, acting on a three part system made up of product, machine frame and the reference. By measuring as directly to the point of interest as possible (i.e. probe-tip and product), the above ‘random’ errors are measured and thus cancel from the measurement result.
Measuring the probe tip position
To satisfy the Abbe principle (Abbe, 1890), the measurement systems should be aligned to the point of interest: the probe tip. Intuitively, one aligns the measurement systems with the horizontal and vertical direction. For concave optics, this would require them to be placed outside the measurement volume, far away from the probe. Although the Abbe principle may be satisfied now, many elements have to be introduced that increase the length of the metrology loop. Tilting the measurement systems over 45° allows for measuring to the probe tip into concave optics. A further complication is the rotation of the probe. This requires the measurement systems to move separately from the R- and Z-stage, to maintain alignment with the probe tip in both cases.
One method that is applied to achieve direct measurement in orthogonal CMMs is the addition of laser trackers (Hughes et al., 2000). By measuring the distance and orientation to a target, or by using multi-lateration with several targets, the position and orientation of an object in space can be calculated. Usually the target is a retroreflector or cat’s eye. Pointing of the beam is done by auxiliary pointing mechanisms. These mechanisms tend to be complex and have an accuracy in the order of a micrometer. They are mainly applied in very large Coordinate Measuring tasks, such as used for airplane construction.
4.1 Concept
Measuring the Ψ-axis centre position
As explained in the error budget calculations, the setup is less-sensitive to tangential errors of the probe tip relative to the product. This also includes the ψ-rotation of the probe. This basically means that the Abbe point can be translated along the probe centre line (c-axis). The Abbe criterion can now be satisfied with measurement systems aligned with the Ψ-axis centre, and thus moving along with the R- and Zstage. A further advantage is that for a practical probe length of 100 mm, the Ψ-axis centre never drops below the edge of concave optics. The position measurement system thus continuously have a clear ‘line of sight’ to the Abbe point.
Two measurement systems can provide the required range, resolution and accuracy: optical linear scales and laser interferometers. A good comparison of the two can be found in (Kunzmann et al., 1993). Optical linear scales provide robust measurement at a relatively low cost, but can only measure the displacement between two bodies translating in one direction relative to each other. To measure two orthogonal displacements in Abbe, intermediate bodies can be introduced (Vermeulen, M., 1999; Van Seggelen, 2007), or a capacitive sensor or vacuum preloaded air bearing can be added as suggested in (Hemschoote et al., 2004). A 2D encoder grid may also be applied, but these presently do not have the accuracy of a 1D scale.
Laser interferometers are directly traceable to the SI unit of length, but are more susceptible to environmental disturbances, since the wavelength of light depends on the refractive index of air. Therefore, to obtain high accuracy, the measurement has to be compensated for temperature, pressure and humidity (Estler, 1985). It can also be done in a vacuum environment or have a continuous conditioned air flow. The crucial advantage, however, is that laser interferometers are able to measure the displacement of an object in one direction, while allowing for displacement in other directions, for instance using a plane mirror interferometry setup (Zanoni, 1989). The mirror flatness and orientation can be calibrated. Further, focusing the laser beams onto the Ψ-axis allows for direct position measurement of the Ψ-axis rotor rotating over large angles.
Ideally, the Ψ-axis (1 in Figure 4.1) position is measured directly relative to the product (2) or spindle (3) with a differential beam layout, but this proved to be not practical. It was therefore chosen to only measure the Ψ-axis position interferometrically (4 and 5), relative to reference mirrors (6 and 7) on a metrology frame (8). The product position will be measured using capacitive probes (9), as will be explained later.
In the design, a split is made between the upper part of the metrology frame (8) that holds the mirrors, and the lower part of the metrology frame (10) that holds the capacitive probes. As shown later, the capacitive probes are calibrated in-process and do thus not required a frame with extreme long term stability. In contrast, the upper part should remain straight and square at nanometer level at all times. The base is supposed to be the stable reference here to which the spindle and both metrology frame parts are mounted. A patent has been granted on this metrology system concept (Henselmans and Rosielle, 2004), the main novelty being the application of the focused interferometers to enable the high accuracy position measurement of an optical probe, which has to be rotated over large angles to be kept perpendicular to inclined surfaces.
Figure 4.1: Metrology loop concept with interferometers measuring the probe displacement and capacitive probes measuring the spindle error motion
The components within the dashed area can be translated in r and z-direction, causing the laser beams to move over the reference mirror while maintaining orthogonal alignment to the Ψ-axis centre. The Ψ-rotor can be rotated relative to the dashed area. This way, the displacement of the rotor of the Ψ-axis, and thus of the probe, is measured relative to the reference mirrors on the metrology frame, directly and without Abbe offset. This cancels the non-repeatable r- and z-errors of the probe from the metrology loop.
Measuring the product position
Ideally, the position of the product is measured on the product itself. Since this can be any shape, this is not practical. Next best would be measuring to the intermediate body it may be mounted on. Especially mirrors will have their mounting points incorporated in their structure and thus be mounted directly on the mounting table. Measuring to the edge of the mounting table therefore is the most practical option. Care must be taken to ensure that the product is fixed rigidly to the table, since relative motion between product and table is not measured by the metrology system. This should be done with minimal distortion of the product, for instance with pitch or wax. The fixation of the product is not shown in Figure 4.1.
4.1 Concept
Although the acquired spindle is specified to have an error motion around 25 nm and 0.1 μrad, the position of the rotor needs to be determined with higher accuracy. To determine the position of the spindle rotor relative to the metrology frame, capacitive probes are added. As shown in Figure 4.1, the sensitive directions are r, z and ψ, which are measured by three capacitive probes (9); two vertical and one radial. In principle, this should be enough to measure three degrees of freedom. To separate the edge shape from the error motion, the multi-probe method will be applied, using 7 probes.
Design overview
With this metrology system concept, all six sensitive directions of Figure 2.11 are measured. When measuring a freeform surface, only the probe focusing mechanism is moving dynamically while measuring a circular track. The dynamical displacements that occur during this measurement are recorded by the metrology system and can be corrected for afterwards (off-line).
Figure 4.2 shows the final design of the metrology system. The interferometry system, consisting of the laser and the R and Z-stage optics assemblies will be explained in detail in section 4.2. Next, the upper metrology frame, which holds the reference mirrors, will be described in section 4.3. The lower metrology frame, holding the capacitive probes for the spindle error motion measurement is explained in section 4.4. This chapter will close with the final assembly (section 4.5) and verification of the performance (section 4.6).
Figure 4.2: Metrology system overview
The interferometry system measures the displacement of the Ψ-axis centre relative to the reference mirrors in on the metrology frame. The setup measuring the r and zdisplacement of the Ψ-axis will be explained first. The third axis of the interferometry system measures the displacement of the probe focusing objective. The beam supply to the probe will be explained in section 4.2.2. Next, the influence of environmental disturbances is discussed. After explaining the beam layout and alignment tolerances, the design and realization will be shown. The main components here are the laser mount, the R-stage optics assembly and the Z-stage optics assembly. Test results are included in the metrology system testing (section 4.6). Since interferometers are incremental, they have to be nulled before a measurement. Nulling of the R and Z interferometer and the probe interferometer will be discussed in sections 7.3 and 5.7, respectively.
The Ψ-axis translates 400 mm in r, and 150 mm in z-direction, and rotates from -45° to +120° in ψ-direction. Similar to wafer stage metrology (Zanoni, 1989), one may mount a flat mirror or corner cube on the Ψ-axis bearing housing to measure its displacement. Due to various error sources, such as probe short stroke reaction forces, air pressure fluctuations and thermal effects, the Ψ-axis rotor position will not be equal to the Ψ-axis bearing housing position. Therefore focussing the beams onto the Ψ-axis rotor is applied. This allows for direct measurement of the displacement of the Ψ-axis rotor, and thus of the probe, in the sensitive r and z-direction.
Heterodyne and homodyne interferometry
Two types of laser interferometry systems are commercially available: homodyne and heterodyne. In homodyne interferometry a single frequency laser is applied, which is split in intensity. After recombination, multiple photodetectors are applied to obtain quadrature detection. This makes the system sensitive to intensity variations, for instance from reflectivity changes while translating laterally over the mirrors, from beam intensity profile changes during displacement (e.g. from ambient light) and from beam overlap variation. Since a single frequency laser is used, the beam can be supplied via a fibre. After interference, polarization must be maintained such that the receiving fibre should be kept free of stresses and thermal changes, making fibre reception difficult. It is therefore best to have the detectors at the recombination point. In heterodyne interferometry, a dual wavelength laser is applied that emits two orthogonally linearly polarized frequencies that differ by a few MHz. The recombined
beam passes through a polarizer and is measured with a single photodetector. The beat signal of the two frequencies is compared to a reference beat signal from the laser, to directly determine the relative phase of the reference and measurement beam. The Doppler shift caused by moving one of the mirrors causes a phase change that is converted into displacement. Measuring phase makes this system insensitive to reflectivity variations of the mirrors and overlap changes. Further, quadrature detection is not necessary, so the recombined light can be transported to the detectors using fibres. Since the polarization of the supply beam must be maintained, this cannot be transported to the system by a fibre, and should be done with mirrors if the fibre is subject to varying stress (Knarren, 2003). Generally, a heterodyne system is less difficult to align and is less sensitive to noise (air turbulence, electrical noise and light noise).
The acquired Agilent heterodyne system is the only system that is commercially available in a 3-axis PCI PC-card version, which is convenient to integrate with the other system electronics (see Chapter 6). It consists of a stabilized HeNe Laser (type 5517C-003) with a 3 mm beam diameter output, high-performance fibre coupled remote receivers (E1709A) and the PCI processing electronics board (N1231B). The update frequency of the 36 bits output is 20 MHz. In dual pass, the resolution is 0.15 nm (4096x interpolation) and the maximum target speed is 0.35 m/s. Custom designed optics and alignment mechanisms can be applied, which allows for design and beam layout flexibility.
Focus on Ψ-axis
In the setup of Figure 4.3, the interferometer beam comes in from the top. This beam consists of 2 orthogonally polarized components with a slightly different frequency. At the polarizing beamsplitter (PBS), the out-of-plane polarization component passes straight into the pickup (PU) and the in-plane component is reflected towards the reference mirror (RM). After passing through quarter wave plate 1 (QWP 1), oriented at 45°, the beam becomes circularly polarized. After reflection at the reference mirror, the beam passes through the quarter wave plate again to become out-of-plane polarized. This beam now passes through the polarizing beamsplitter and quarter wave plate 2, and becomes circularly polarized again. The beam is now focused onto the Ψ-axis centre by a cylindrical lens. The Ψ-axis surface has been provided with a mirror by diamond turning, on which the beam reflects. After passing through quarter wave plate 2 again, the beam becomes in-plane polarized and is reflected towards the pickup by the polarizing beamsplitter, to join the first out-of-plane polarized beam. In short, one part of the beam travels straight through into the pickup, and the other part travels an extra distance between the reference mirror and the Ψ-axis. The resulting optical path difference (OPD) is detected by the interferometer electronics and converted into displacement. A similar setup is applied in vertical (z) direction, as already schematically depicted in Figure 4.1.
Figure 4.3: Interferometer setup for direct displacement measurement of Ψ-axis rotor
Focus on Ψ-axis surface or centre
In Figure 4.3, the beam is focussed onto the Ψ-axis centre. Focussing the beam onto the Ψ-axis surface is also possible. The main advantage here is that the alignment requirements for the cylinder lens are less critical, especially for rotation around the optical axis. According to Gaussian beam theory (O’Shea, 1985), the wavefront in the waist is flat. For cylindrical optics, the spot width is equal to the waist diameter D0 of spherical optics, and is given by (4.1).
D0 = 4λ(4.1) πθ
With a wavelength λ of 633 nm and a divergence of 0.15 rad (d = 3 mm and f = 20 mm), this gives a spot width of 5.4 m. The small spot size makes it very vulnerable to scratches and dust on the mirror surface, which may well be equal to or larger than the spot. This would result in a total loss of modulation depth. The flat wavefront further reflects on the curved mirror surface, which will introduce curvature to the reflected wavefront that again reduces the modulation depth.
When the beam is focussed onto the Ψ-axis centre, alignment of the cylinder lens relative to the Ψ-axis rotor is critical (μm and μrad level). By fixing the lenses in the Ψ-axis bearing housing, this alignment will only have to be done once and will be sufficiently stable over time. The spot size on the mirror surface is now approximately 2 mm, making it robust to surface imperfections. Further, the cylindrical wavefront matches the mirror, making each ray perpendicular incident on the surface. Apart from aberrations of the lens, no curvature is thus
introduced in the reflected wavefront. Although alignment is to be done with care, focussing the beam onto the Ψ-axis centre is more robust and is thus preferred.
Dual pass concept
When the cylinder lens is properly aligned to the Ψ-axis, this assembly basically acts as a flat mirror. The single pass layout as shown in Figure 4.3 is very susceptible to alignment errors of these two mirrors, as illustrated in Figure 4.4, left. Generally, the alignment between reference and measurement wavefront must be better than λ/8, or 50 rad, to have sufficient modulation depth (Agilent, 2002). Changing this layout to a dual pass configuration resolves this sensitivity, as shown in Figure 4.4, right. Only an overlap error (walk-off) remains, of which a minimum of 10% is required. The doubled measurement path length doubles the resolution, but also doubles the sensitivity to environmental disturbances. Nevertheless, the reduced alignment sensitivity is considered to weigh well up to this.
Single pass Dual pass
Figure 4.4: Sensitivity to mirror tilt of single and dual pass plane mirror interferometry (exaggerated for illustrational purposes)
Besides the R and Z-interferometer, a third interferometer is applied for measuring the displacement of the probe short stroke mechanism. As explained, a heterodyne beam cannot be supplied by a fibre and should thus be supplied by mirrors. It is therefore directed into a hole in the backside of the Ψ-axis rotor (Figure 4.7), aligned with its centre line. The polarizing beamsplitter inside the Ψ-axis rotates, but the polarization direction of the supply beam is constant; one horizontal and one vertical. The polarization direction of the supply beam must hence be rotated to match the orientation of the beamsplitter.
To achieve this, a quarter wave plate on the Z-stage gives the beam circular polarization. A second quarter wave plate on the rotated probe gives the beam a linear polarization again, but now aligned to the also rotated polarizing beam splitter. In
Figure 4.5, this schematically depicted for a general case, with the initial polarization
directions Er and Ez , and the orientation between the components ψ1, ψ2 and ψ3.
Figure 4.5: Polarization rotation with two quarter wave plates
To prove this, the supply beam can be described by the electric vector in Jones form (Hecht, 1998), where z is along the optical axis:
E = EExy ((z tz t,, )) = EE e00,,xye jjσxy with σ ω= kz − t (4.2)
σ
For plane polarized light, σx = σy± mπ with m = 0, 1, 2, … . When σx = σy± m½π with m = 1, 3, 5, … , and E0,x = E0,y = E0, the light is circularly polarized. For right polarized light (the y-component leads the x-component by 90° or π/2), the Jones representation is:
Eℜ = E e0E e0j(σ πxjσ−x /2) = E e0 E ejσ0 xejσ−xjπ/2 = E e0 jσx −1j /2 = E e0 jσx −1j (4.3)
e π
When an incident polarized beam represented by its Jones vector Ei passes through
an optical element, it is transformed to the transmitted wave Et . This transformation can be described by the Jones matrix A:
(4.4)
Et = AEi
A quarter wave plate with its fast axis in vertical direction can be described by:
1 0
AQWP = 0 − j (4.5)
To decompose the incident beam to the orientation of the quarter wave plate, it is multiplied by rotation matrix Ri:
Ri = −cos(sin(ψψii)) cos(sin(ψψii)) with i =1,2 or 3 (4.6)
The beam emerging after the second quarter wave plate, relative to the orientation of the polarizing beamsplitter, is now according to (4.7). In the final step, cos(ψ1) and sin(ψ3) have been replaced by c1 and s3 etc, for shortness.
Et = R A3 QWP R A2 QWP R E1 i =
=−cos(sin(ψ ψψ ψ33)) cos(sin( 33))10 −0j−cos(sin(ψ ψψ ψ22)) cos(sin( 22))10 −0j−cos(sin(ψ ψψ ψ11)) cos(sin( 11))EExy (4.7)
Ex ((c c c1 2 3 + s c s1 2 3)+ j s s c( 1 2 3 + c s s1 2 3))+ Ey ((s c c1 2 3 −c c s1 2 3)+ j s s s( 1 2 3 −c s c1 2 3))
=
Ex ((s c c1 2 3 −c c s1 2 3)+ j(c s c1 2 3− s s s1 2 3))− Ey ((c c c1 2 3 + s c s1 2 3)− j s s c( 1 2 3 +c s s1 2 3))
When the first quarter wave plate is oriented at 45° relative to the supply beam, and the second quarter wave plate at 45° relative to the polarizing beamsplitter, ψ1 = π/4 and ψ3 = -π/4. Now, c1 = c3 = ½√2 and s1 = s3 = -½√2 and (4.7) simplifies to:
Ey (cos(ψ ψ2 )− jsin( 2 )) Ey (cos(− +ψ2 ) jsin(−ψ2 )) E ey − jψ2 Et =Ex (cos(ψ ψ2 )+ jsin( 2 )) = Ex (cos(ψ ψ2 )+ jsin( 2 )) = E ex jψ2 (4.8)
The angle ψ2 is equal to the global angle ψ of the Ψ-axis. Further, if the two plane polarized components f1 and f2are assumed to be in phase and have equal amplitude E, their incident and transmitted Jones vectors are:
E
Ei f, 1 = 0
0
Ei f, 2 = E
|
| Et f, 1 = Ee0jψ
Ee− jψ Et f, = | (4.9) |
2 0
From this result it can be seen that the two quarter wave plates act together as a half
wave plate since both Ei f, 1 and Ei f, 2 are rotated 90°, respectively counter clockwise and clockwise. Further, no mixing occurs and both components are exactly aligned to the splitting directions of the polarizing beam splitter. The horizontal and vertical components get a phase change that is linearly dependent of ψ, resulting in a total phase difference of 2ψ. When the probe objective mirror is fixed, and the Ψ-axis is rotated, an apparent displacement is thus measured equal to 2ψ times the laser wavelength of 633 nm. To compensate for this to an error below 1 nm, the angle of the Ψ-axis needs to be known with an accuracy smaller than 2π/633 = 10 mrad, which is no problem.
A different embodiment of this principle, utilizing a dual pass layout, a corner cube and a fixed detector can be found in (Bockman, 1996). Due to the dual pass layout, the laser beam moves over the optical surfaces when the assembly is rotated.
This method was tested during assembly of the interferometry system. The probe interferometer supply beam is collinear with the Ψ-axis rotor centre line, and with no probe mounted, exits the rotor through a hole in the front into a pickup. A fixed quarter wave plate is on the Z-stage optics assembly as will be explained in section 4.2.6. A temporary quarter wave plate was installed inside the Ψ-axis rotor. With no geometrical path difference between the measurement and reference beam, rotation of the Ψ-axis should result in only the apparent displacement. Figure 4.6 shows the interferometer reading, and the result when compensated for the rotation according to the result of (4.9). The curve in the error is probably due to the coarse alignment of the temporary quarter wave plate and the pickup.
Figure 4.6: Polarization rotation test results
One of the main disadvantages of interferometry compared to optical scales is the sensitivity to environmental disturbances. The optical path length is equal to the geometrical path length times the wavelength of the media in that path. Thermal expansion influences the geometrical path length of the optics, and the refractive index is a function of temperature and for gasses also of pressure and composition.
Thermal expansion
In the proposed dual pass layout, the measurement beam travels four times the combined thickness of the polarizing beamsplitter, two quarter wave plates and the lens more through glass than the reference beam. This is equal to 101.6 mm of BK7, 8 mm crystal quartz and 20 mm Fused Silica, respectively. With αBK7 = 7.1⋅10-6 m/m/K, αquartz = 7.6⋅10-6 m/m/K and αFS = 0.56⋅10-6 m/m/K, the geometrical path length changes 720 nm, 60 nm and 11 nm per Kelvin, respectively. With nBK7 = 1.517, nquarz = 1.458 and nFS = 1.458, the total optical path length change is 1196 nm per Kelvin. The expected temperature change of the (covered) optics during a measurement is expected to be around 10 mK, giving 12 nm OPD error. In the dual pass layout this is equal to 3 nm displacement error.
Refractive index of air
Variation of the refractive index of air is generally the main source of uncertainty with path lengths larger than a few millimetres. The refractive index of air is mainly influenced by temperature, pressure and humidity. Other factors, such as CO2 content, have less influence. The wavelength in air can be calculated with Edlén’s formula (Edlén, 1966), or by later adaptations as summarized in Appendix G of (Cosijns, 2004). The Birch and Downs equation (Birch and Downs, 1993) is most commonly used for precision measurements around 20°C. Here, the pressure (P) in Pa , the temperature (T) is in K and the partial water vapour pressure (H) in Pa:
nPTH = 2.7653 10⋅ −4 ⋅ P ⋅1+10−8 ⋅(0.601− 0.00972 (⋅ T − 273.15))⋅P
(4.10)
As can be calculated from (4.10), the sensitivity to pressure variations is 2.68⋅10-9/Pa, the sensitivity to temperature variations is -9.30⋅10-7/K and the sensitivity to humidity variations is -3.7⋅10-10/Pa. (Cosijns, 2004) also gives the typical daily variation for these parameters: 20 hPa for P, 0.1 °C for T and 1 hPa for H, giving an effect of 5.36⋅10-6, -9.3⋅10-8 and -3.7⋅10-8, respectively. This thus gives large errors, especially pressure. Refractive index compensation is therefore applied by measuring the environmental conditions (Estler, 1985).
The expanded uncertainty of the Vaisala PTU300 sensor that is used for measurement of room pressure, temperature and humidity is 30 Pa, 0.2 K and 1% or 24 Pa, respectively. The dead path error for the interferometer path design is 0.5 m for the Rinterferometer and 0.2 m for the Z-interferometer. Assuming that pressure, temperature and humidity are uncorrelated, gives a worst case uncertainty (2σ) of 102 nm in R and 40 nm in Z.
This does not yet include other uncertainties such as the accuracy of the Birch and Downs model and the vacuum laser wavelength. It must be noted that this is the absolute uncertainty. During a measurement, the trend in the environmental fluctuations is expected to be measured with higher resolution, be it with an offset compared to the absolute values. This will appear as a scaling error in the form measurement, which is relatively acceptable for optics at this level.
To improve this absolute uncertainty, the interferometers can be operated in vacuum or in a gas of which the refractive index is better known. This adds greatly to the complexity of the machine. The uncertainty is therefore accepted for now.
Turbulence
With the above described procedure, global environmental condition variations can be compensated for. Local variations of the refractive index due to turbulence for instance, will cause higher frequency disturbances of the signal. Two main sources of turbulence are the air-conditioning and the releasing boundary layer of the rotating spindle.
An enclosure around the metrology system, telescopic tubes or bellows around the beams, and a box around the beams are discussed in Appendix F. Besides these atmospheric solutions, the air inside the telescopic tubes may also be replaced with Helium. This reduces the effect of turbulence by a factor of 10 (Bryan, 1979; Bryan and Carter, 1979). An even more rigorous approach would be to apply a (low) vacuum in these tubes, as done in (Donaldson and Patterson, 1983) and (Thompson, 1989). A pressure of only 10 mbar would improve the uncertainty by a factor of 100. The required enclosure and vacuum forces would have a large impact on the design. An ‘air shower’ of turbulent, well-mixed and well-conditioned air may also be blown through the beams, such that the average index of refraction is constant over the path length (Bobroff, 1987).
Since little quantitative data is available on the effect of turbulence, tests were performed on the machine to determine what level of shielding was required. Appendix F shows typical results of these tests. The rms noise level clearly decreases when the volume around the metrology system is enclosed by a tent. Fluctuations with a typical wavelength of 10-30 s are further reduced by telescopic tubes or a box, but fluctuations with a wavelength of several minutes are amplified now. The best results were eventually obtained with the easiest solution: the enclosure around the metrology system.
In Figure 4.7 the beam layout is shown that supplies the interferometer laser to the 3 interferometer axes, around the structure of the R- and Z-stage and the Ψ-axis. Note that the assembly is viewed from the backside for clarity. The laser (1) is aligned to the R-stage, on which also the R-stage optics assembly (2) is mounted. When the Rstage is repositioned (and thus also the Z-stage and Ψ-axis), the R-stage optics assembly moves away from or towards the laser at a constant distance under the horizontal reference mirror (3). Since an air bearing guides are used, the motion perpendicular to the laser will be in the order of micrometers.
At the R-stage optics, the beam is split by a 66% non-polarizing beamsplitter (NPBS1), where 33% passes through towards the Z-interferometer and 66% is reflected in y-direction toward a fold mirror (FM1). The Z-interferometer consists of a polarizing beam splitter (PBSZ), two quarter wave plates (QWPZ), a corner cube (CCZ) and a pickup (PUZ).
Figure 4.7: Interferometry system beam layout (backside view)
Fold mirror 1 and 2 (FM1 and FM2) together form a periscope between the R- and the Z-stage optics assembly (4). This allows the Z-stage optics assembly to move vertically relative to the R-stage optics assembly. After reflecting on fold mirror 3 (FM3), the beam arrives at a 50% non-polarizing beamsplitter (NPBS2). Here, 50% is reflected into the Ψ-axis rotor, aligned with its centre line. It passes through a quarter wave plate (QWPP), towards the probe interferometer.
The remaining 50% of the beam is directed towards the R-interferometer by fold mirror 4 (FM4). Just as the Z-interferometer, the R-interferometer consists of a polarizing beamsplitter (PBSR), two quarter wave plates (QWPR), a cube corner (CCR) and a pickup (PUR). Also shown here are the two cylinder lenses (CLR and CLZ), which are mounted into the Ψ-axis bearing housing. These lenses focus the interferometer beams onto the cylindrical mirror (CM) on the Ψ-axis.
Misalignment of the optical elements causes measurement errors, originating in wavefront tilt and curvature, cosine error and polarization mixing. A tolerance analysis was performed to investigate alignment requirements for the optical elements.
Wavefront tilt and curvature
Ideally, the measurement and reference wavefront are flat and perfectly overlapping. Due to alignment differences between the two wavefronts, they will be tilted and / or curved. When this is combined with relative displacement between the two, the average phase of the overlap area changes, which causes a measurement error. When w is the relative beam displacement (walk-off) over the stroke, ε is the total alignment difference, is curvature difference and is the wavelength, the phase error from tilt (φt) and curvature (φκ) is given by (Johnstone et al., 2004):
φt =πεw and φκ =π κ2wλ2 (4.11) λ
With the phase written as an error part δ of the wavelength λ:
φ=2π (4.12)
The resulting errors from tilt δt and curvature δκ are:
δt = wε and δκ = w2κ (4.13)
2 4
For a given error budget, one is free to choose the ratio between beam tilt and relative displacement, using equation (4.13). In the system described in this thesis, the relative beam displacement is expected to be small. In lithography systems for instance, the rotation of the stage is measured using two interferometers. A dual pass setup ensures the alignment of the wavefronts, but the relative beam displacement may amount up to several millimeters. The stages only translate in this application, giving minimal walk-off.
The alignment error between the measurement and reference wavefront should not be chosen larger than λ/8 at the edge of the spot, to prevent fringes over the aperture (Agilent, 2002). Fringes will reduce the intensity of the spot, which decreases the modulation depth. The described system uses a beam diameter of 3 mm, which allows for a maximum misalignment of 50 μrad. With a maximum remaining relative beam displacement of 0.05 mm, this would result in an error of 1.25 nm. Similarly, a curvature difference of 2 m also results in 1.25 nm. Generally, these large curvature differences do not occur, making wavefront tilt the main alignment requirement.
Cosine error
When the measurement beams are not aligned with the global axes of the coordinate system, a cosine error will occur. This error is linear with the stroke and repeatable, and thus in principle calibratable. The R-stroke is 400 mm, but the geometrical path length travelled by the beam is 2 m. The maximum walk-off over this stroke is 0.05 mm, giving a measurement error of 0.6 nm. With the Z-path length of 0.7 m and similar walk-off values, the expected cosine error here is 1.8 nm.
Polarization mixing
The polarizing components (polarizing beamsplitters, quarter wave plates and polarisers) are not ideal. Their limited extinct ratio causes one polarization state to leak into the other (polarization mixing). Misalignment of these components relative to the chief polarization directions of the laser increases this mixing (Cosijns, 2004). Polarization mixing causes a periodic (non-cumulative) error, generally in the order of a few nanometers with proper alignment.
Alignment tolerances
The interferometer setup has been modelled in ZEMAX, to calculate the sensitivity to alignment and position errors of the individual components (Figure 4.8). This has been calculated for focussing onto the surface as well as the centre of the Ψ-axis, and for different path lengths. As already discussed before, focussing onto the centre of the Ψ-axis is more sensitive to alignment but more robust in practice. The alignment sensitivity is largest when the beam path is longest, which is 400 mm for the Rinterferometer in the left-most position. These (worst-case) values have been used as alignment tolerances for the system.
Figure 4.8: ZEMAX model of Z-interferometer (focus on Ψ-axis surface case)
With this model, the sensitivity of each individual direction of each component has been determined. Next, the alignment budget has been divided over the components such that the system can be assembled and aligned piece by piece, and that the alignment directions are decoupled as much as possible. The most critical alignments are the laser parallelism to the R- and Z-stage and the 3 beamsplitters, for which alignment resolutions between 5 and 30 rad are to be achieved. Hereto alignment mechanisms are designed as further explained in the next section.
The model also gives the nominal wavefront error due to aberrations. With a slightly different lens from the one eventually used, the rms error is 0.0016 waves (1.0 nm) and the peak-to-valley error is 0.015 waves (9.5 nm).
Alignment procedure
The interferometry system will be assembled in the same sequence at which the beam arrives at the components. To minimize the relative beam displacement, the beams will be aligned to the motion direction of the stages. This minimizes walk-off errors as described by equation (4.13), but may give rise to cosine errors in case of misalignment of the stages. These will have to be calibrated later.
Beams in r and z-direction will be aimed at a Position Sensitive Detector (PSD), with which the spot position can be measured to a few micrometer (after averaging). While moving the stages through their range, the components are manipulated until the spot displacement is minimal. This gives sufficient resolution to achieve the above mentioned alignment requirements.
A pentaprism will be inserted in beams in y-direction. This deflects the beam at an inherent square angle, allowing for alignment in one direction. By executing the procedure in horizontal and vertical direction, the beam can be aligned in ϕ and direction.
The cylinder lenses in combination with the cylindrical Ψ-axis mirror act as a flat mirror. Alignment can therefore be done using a Fizeau interferometer, to minimize the fringes over the aperture. This is described in section 4.2.7.
The interferometry system consists of an adjustable laser mount and the R and Zstage optics assemblies. This section describes the design of these assemblies in general and some of the components in detail.
Laser mount
The laser is to be aligned parallel to the R-stage in ψ and -direction within 5 μrad. In Figure 4.9, the laser mount assembly and exploded view are shown. The laser (1) is bolted to a rectangular tube (2) with three spacers (3) in between. The coarse alignment of the laser is done by sliding the laser while tightening this connection and adapting the spacers if necessary. The tube is mounted to the two plates (4 and 5), which are connected by a triangular plate (6), and bolted to inserts in the granite base (7). An aluminium shield (8) is also added to prevent heat from the laser reaching the base block or metrology frame by direct radiation or convection.
Figure 4.9: Laser mount design
The rectangular tube has two pieces glued to either end. The right one (9) has a plate spring machined into it by wire-EDM. A strut (10) is glued in to form a cross-spring that leaves the ψ and -rotations free. The fixed end of this cross spring is mounted to the fixed right plate (4).
At the left end-piece of the tube (11), two orthogonal micrometer screws (12) are positioned under 45°, in contact with two 45° faces that are machined in the left endpiece. This allows for independent actuation of both rotations, while the micrometer screws are reachable and the laser weight provides the preload on the contacts. Once the laser is aligned, the position is locked by an A-shaped locking plate (13), that constrains the ψ and -rotations to the fixed left plate (5). The micrometer screws can then be screwed back, leaving the laser constrained statically determined, with four degrees of freedom constrained at the front and two at the backside. The laser mount assembly is shown in Figure 4.10.
Figure 4.10: The laser mount
Adjustable optics mounts
Several of the previously discussed optical components have to be aligned to within several tens of microradians, for which dedicated alignment mechanisms have been designed. An alignment mechanism generally consists of a guidance, an actuator and a means of locking.
Due to the absence of friction, flexure hinges allow for infinite positioning resolution, and can well be machined into the mount by wire-EDM. The stroke limits are best machined into the mount as well, to prevent failure of the hinge. Several degrees of freedom can be incorporated into one mechanism this way, resulting in a compact and stable mechanism. These are therefore common practice in opto-mechanical instruments (Van der Lee et al., 2003).
To achieve microradian resolution for lever lengths in the order of 50 mm, micrometer resolution actuation is required. Micrometer screws or differential screws can be applied to achieve this, in combination with a separate locking mechanism to exclude the instability of these screws from the structural loop. These screws are relatively large compared to the optics to be adjusted, and it is difficult to prevent a shift during locking. The micrometer screws may be removed after locking to save space.
Two small screws, in push-pull setup, can also be applied. An M2 with a pitch of 0.4 mm allows for a coarse adjustment of approximately 0.01 mm, and the second screw is used for fine adjustment by deformation of the thread and contact points. Small screws are best used due to the lower stiffness of the threads and smaller contact surface. The adjustment is somewhat cumbersome, but the main advantage is that a separate locking mechanism is not required as long as the two screws preload each other. Further, the required space is much less compared to the micrometer or differential screws. Since tightening the two screws against each other introduces a moment, the two screws should be aligned perpendicular to the hinge to prevent parasitic rotations. Push-pull screws are applied throughout the alignment mechanisms of the interferometry system.
The optics are bonded to the mechanism. To minimize deformation, they are mounted onto three raised areas. In the centre of these a 0.1 mm deep recesses are machined in which a drop of adhesive is deposited. Shrinking of the adhesive during UV curing preloads the contacts, providing a stable connection.
The mounts of NPBS1 and FM1 must be adjustable for one rotation, and the mounts FM3, NPBS2 and FM4 for two rotations. These mounts are monolithically machined from aluminium and black anodised against stray light reflections. The M2 push-pull screws are aligned perpendicular to and at the centre of the hinge.
FM1 Mount NPBS2 Mount
Figure 4.11: Alignment mechanisms for 1 (left) and 2 (right) rotations
The polarizing beamsplitters are anti-reflection coated, just as both sides of the quarter wave plates. The quarter wave plates were therefore bonded to the beamsplitters with spacers in between to create an air-gap. The PBSR and PBSZ assemblies (Figure 4.12) are to be aligned in two rotations with higher resolution than the previously discussed mounts. Space unfortunately was limited, since the corner cube and pickup are close by and two beams pass through the beam-splitter on four sides.
PBSZ Mount PBSR Mount
Figure 4.12: Beamsplitter mechanisms for 2 rotations
Both mechanisms are assembled from two parts. A lever is glued to the moving part of hinge one (Figure 4.12), with the other end interlocking with the monolith to form stroke limit 1. Due to the space constraints, these levers actuate the hinges on the side instead of in the centre, which will cause parasitic rotation. This can be compensated with the second available rotation, since two of the three rotations of a beam splitter have the same first order effect.
R-stage optics assembly
The R-stage optics assembly consists of a central aluminium beam (1 in Figure 4.13) to which the mounts of NPBS1 (2), PBSZ (3), CCZ (4), FM1 (5) and the pickup (6) are mounted. This beam is bolted to the read-head arm (7) that is connected to the Rstage position frame (8), as discussed in section 3.5.4. At the other end, the beam is bolted to a plate (9) which is also connected to the R-stage position frame (8). The Rstage position frame provides a stable base for the optics, as explained in Chapter 3. A dowel pin (10) in the read-head arm and shoulder bolts (11) in the plate connection provide some repeatability in case the R-stage optics assembly needs to be removed. Since the construction is statically determined, no tight position tolerances are required, the position only needs to be repeatable. Some realignment can probably not be avoided.
Figure 4.13: R-stage optics assembly
Figure 4.14 shows the R-stage optics components (left) and the assembly (right). The assembly is black anodized against stray light reflections. This assembly will be protected by a cover.
Figure 4.14: R-stage optics assembly
Z-stage optics assembly
The Z-stage optics assembly consists of two beams (1 and 2), which are mounted left and right of the Ψ-axis mount (3). A shoulder bolt (4) constrains three degrees of freedom at the backside. A plate (5) is connected to the Z-stage mounting interface (6), to which the two beams are connected via two shoulder bolts (7) that constrain the remaining three degrees of freedom. In between the two beams, a bridge (8) is connected, again with two shoulder bolts (9), to which the NPBS2 (10) and QWPP (11) mount are attached. This bridge has a flexure to relieve the distance between the beams which was already constrained by the shoulder bolts and the Ψ-axis mount. No tight position tolerances are required for the shoulder bolts since the structure is statically determined. They only provide some repeatability in case of disassembly. The mounts for FM2 (12) and FM3 (13) are connected to the left beam (1), and the mounts for FM4 (14), PBSR (15), CCR (16) and the pickup (17) are connected to the right beam (2).
Figure 4.15: Z-stage optics assembly
Figure 4.16 shows the optics mounts (left) and the Z-stage optics assembly (right) as it is assembled apart from the Ψ-axis mount. After alignment, this assembly is also protected by covers.
Figure 4.16: Realized Z-stage optics mounts and assembly (upside down)
Figure 4.17 shows the interferometry system assembled to the machine. A general purpose mounting plate is attached to the Ψ-axis mount for testing. The laser was aligned by attaching a PSD to the R-stage. Over the 400 mm stroke, the displacement measured at the PSD was 5 m in y-direction, and 1 m in z-direction, which is equal to an alignment of 12.5 and 2.5 rad. The periscope beam between the R and Z-stage was aligned parallel to the Z-stage to within 2.5 m over 130 mm stroke. Together with the alignment of the interferometers, this has resulted in 10 m over 130 mm for the Z-interferometer and 10 m over 300 mm for the R-interferometer. This is within the cosine error and walk-off budget.
Figure 4.17: Interferometry system as assembled on the machine
To aid aligning the probe interferometer collinear to the Ψ-axis, a central through hole is present in the rotor through which the beam exits at the front. To align the beam parallel to the Ψ-axis centre line, a pentaprism is attached to the rotor that points the beam down in z-direction onto a fixed PSD. Moving the Z-stage up and down shows the misalignment. Rotating the Ψ-axis 90° allows for the same with the R-stage. Next the pentaprism is replaced with a PSD, such that rotating the rotor shows the eccentricity. The alignment and position were iteratively adjusted this way, until the beam was parallel to the Ψ-axis centre line within 100 rad in ϕ and 12.5 rad in θdirection, and centred to 30 m in r and 50 m in z-direction.
Cylinder lens alignment
The cylinder lenses in the Ψ-axis housing are bonded into two cups, which are in turn bonded to the Ψ-axis housing. Shims are applied to adjust focus, and a removable
2DOF manipulator is used to adjust alignment while the adhesive is curing (Figure
4.18, left).
Figure 4.18: Cylinder lens alignment manipulator and measured interferogram
During alignment, a Phase-Shifting Interferometer is used to measure the reflected wavefront quality. The two final interferograms are shown in Figure 4.18, right. In Rdirection some tilt is still present due to shifting during curing of the adhesive. The reflected wavefront error is 56 nm PV in R and 79 nm PV in Z over the ∅3 mm pupils, with tilt removed. This wavefront error mainly consists of the spherical aberration of the cylinder lens.
The reference mirrors are to be supported by the upper metrology frame (Figure 4.1), either as separate mirrors or as an integral part. The flatness of the mirrors should be highly stable at all times, as well as their alignment relative to the lower metrology frame.
The main parameter in the design of the upper metrology frame is the choice of material, where the optimum trade-off between mechanical and thermal behaviour must be found. Availability and manufacturability play a role here since large pieces of (sometimes exotic) materials are involved. One has to take this into account in the design.
Disturbances
Thermal disturbances are the main error source, and originate in environmental temperature fluctuations, internal heat sources (actuators, read heads etc.) and operator body heat radiating onto the machine. Mechanical disturbances are vibrations and deflections of the granite machine base and acoustical excitation. The sensitivity of the metrology frame to these disturbances is minimized by optimizing the thermo-mechanical behaviour, the eigenfrequency, by statically determined mounting to the base and by enclosing it with a shield.
Material selection
The beams with the reference mirrors should stay as straight as possible under thermal gradients. The parameters that describe the transient thermal behaviour of a material are the expansion coefficient α, the thermal conductivity k and the specific heat cp. The curvature of a beam due to a non-uniform heat flux (Figure 4.19) is proportional to the thermal sensitivity α/k and to the ratio of the wall thickness b and the beam width B, according to (4.14) (Breyer and Pressel, 1991).
Figure 4.19: Deflection due to a temperature gradient from a single sided heat flux
δα B (4.14) k 2b
For a beam to stay straight under the influence of a heat flux, either temporary (operator) or continuous (read heads), the ratio of expansion coefficient and conductivity should thus be small, and the conductive area between top and bottom should be large.
The time dependent behaviour of a body is described by (4.15). For natural convection, the convection coefficient h is generally constant (5 W/m2/K). The volume V and surface area As are determined by the geometry, and the density ρ and specific heat cp are material properties.
dT hAs T −T∞ = e−ρhAc Vps (t t− c ) (4.15)
dt = − p (T −T∞) 1 −T∞ ρc V T
In this formula, the temperature T of the body is supposed to be uniform. To also include the temperature distribution, the volumetric thermal diffusivity is defined as k/(ρcp), with conductivity k. This parameter describes how fast a body adapts to a new environmental temperature, and how long it takes for the distribution to be uniform again.
If the expansion coefficient is also included, the volumetric thermal stability is defined as k/(αρcp). This also takes the amount of deformation into account that is caused by the non-uniform temperature distribution, which the thermal diffusivity does not.
One can go two ways now. Either one goes for a large thermal time constant, such that the frame is continuously but slowly changing temperature and curvature, but does not drift too much during a short measurement, or one goes for a small time constant that swiftly follows room temperature at a uniform distribution and thus continuously stays straight. The dimensions of the part determine whether one should go for a high or low thermal diffusivity.
From a mechanical point of view, high stiffness (Young’s modulus E) is required against distortion due to internal and external loads. For dynamic loads a high first resonance and/or large material damping is required to minimize amplitude. Since material damping for most high stiffness materials is very small, the specific stiffness E/ρ is the parameter used. Formula (4.16) gives the relation between geometry, material and eigenfrequency. It can be seen that a high structural frequency is obtained by high E/ρ, and a large percentage of material on the outer perimeter of the beam cross-section (i.e. a large thin walled structure).
I ⋅ E (4.16) fe ~A ρ
In Table 4.1, the above described properties are listed for some possible upper metrology frame candidate materials. Aluminium and Invar were for instance compared in (Ruijl, 2001), Zerodur is used in (Becker and Heynacher, 1987; Jäger et al., 2001) and granite was applied in (Takeuchi et al., 2004). Further, it has also been indicated whether a material can be polished or diamond turned, such that the reference mirrors can be applied directly onto the frame, perhaps with an extra reflective coating.
| Property [1] | Unit | |||||||
| E | x109 N/m[2] | 69 | 1412 | 145 | 90 | 410 | 303 | 65-85 |
| ρ | x10[3] kg/m3 | 2.7 | 8.1 | 8.1 | 2.5 | 3.15 | 3.7 | 2.9 |
| E/ρ | x106 Nm/kg | 26 | 17.4 | 17.9 | 36 | 131 | 81 | 22-29 |
| α | x10-6 m/m/K | 24 | 1.3 | 0.6 | 0.053 | 2.1 | 6.1 | 5-6 |
| k | W/m/K | 180 | 10.5 | 10.5 | 1.46 | 175 | 24.7 | 3.2 |
| cp | J/kg/K | 880 | 515 | 515 | 800 | 658 | 880 | 800 |
| α/k | x106 m/W | 0.13 | 0.12 | 0.06 | 0.03 | 0.012 | 0.25 | 1.7 |
| k/(ρcp) | x106 m2/s | 76 | 2.5 | 2.5 | 0.73 | 84.4 | 7.6 | 1.4 |
| k/(αρcp) | m2K/s | 3.2 | 1.9 | 4.2 | 14.6 | 40 | 1.24 | 0.25 |
| Machinability | ++ | + | +/- | +/- | - | +/- | +/- | |
| Mirror (polish / SPDT) | Yes | No | No | Yes | Yes | Yes | No | |
Table 4.1: Material properties (optima marked in gray)
Many types of Silicon Carbide exist (Van Veggel, 2007), but Sintered Silicon Carbide (SSIC) has the best properties. As can be seen from the table, it has twice the stiffness of steel and only slightly higher density than Aluminium. This material has thus a superior E/ρ. The expansion coefficient of Zerodur is by far the lowest. The thermal conductivity is also very low, especially compared to SSiC and Aluminium. Invar also has very low conductivity. Remarkably, the thermal sensitivity is better for SSiC than for Zerodur. The volumetric thermal diffusivity, describing the uniformity and magnitude of the temperature distribution, is slightly higher for SSiC than Aluminium. For Zerodur this is two orders of magnitude smaller. Zerodur will thus constantly have large temperature gradients with little deformation as a consequence due to the low α. This is also reflected by the volumetric thermal stability. SSiC is even better. SSiC and Zerodur basically operate in two different ways to have good thermal stability: SSiC has a very uniform temperature with a small but present thermal expansion, while Zerodur always has thermal gradients but these have no effect due to the extremely low α. All other materials perform much worse compared to SSiC and Zerodur.
Based on the thermal and mechanical properties, Silicon Carbide is the material of choice for the metrology frame. Care must be taken in the design, since it is very hard and brittle, and thus difficult to machine. It can only be ground with diamond and even then it is very time consuming. It is only recently becoming available in highprecision custom shapes. The fact that the material is polishable, is also a great advantage. Machining the mirrors directly onto the side of the beams avoids the need for assembly and alignment, and thus greatly improves mirror flatness stability. Polishing and coating beams of this size requires special tools and expertise. Manufacturer CoorsTek (www7) agreed on fabricating these parts, and was closely involved in the further design process of the upper metrology frame.
Layout
The length of the horizontal beam is approximately 950 mm and the cross-section is 100 mm wide and 120 mm high. The vertical beam is approximately 330 mm high and has the same cross-section as the horizontal beam. The base design allows for supporting the frame in vertical direction to the lower metrology frame, and backwards directly to the granite base. Providing support points on the granite base directly behind the frame has greatly affected the choice of motion system concept, which has already been commented in section 3.1.3.
To keep the horizontal mirror beam stationary relative to the lower metrology frame, it will be supported by two Super Invar struts, which are connected to the mirror surface of the beam on one side, and to the steel lower metrology frame on the other side (Figure 4.20). Here, a material with the smallest expansion coefficient is desired, which is Zerodur. For machinability and robustness, however, Super Invar was chosen. Since the spindle is made of steel that sits on the granite surface, the steel lower metrology frame has the same height as the mounting surface of the spindle.
Figure 4.20: Metrology frame layout options (horizontal support discussed later)
The vertical mirror beam can now be mounted upwards to the horizontal beam (Figure 4.20, left), or downwards directly onto the lower metrology frame (Figure 4.20, right). In both cases there is a large thermal length between the spindle centre line and the vertical mirror face. In the second case, less mass is to be supported, but small deflections in the granite base are amplified in mirror displacements at the top of the vertical beam. Since keeping the mirror alignment constant is critical, the first option is preferred. The horizontal constraint (dashed), and the out of plane constraints will be further discussed later.
Silicon Carbide manufacturing
Extensive information on properties, manufacturing processes and design guidelines of different types of SiC can be found in (Van Veggel, 2007), on which this section is based. Sintered Silicon Carbide (SSiC) is manufactured by cold isostatic pressing αSiC powder combined with organic binders and sintering aids, at about 2000 bar. The compacted powder exhibits chalk-like behaviour and is called the ‘green’ state. In the green state the substrate can be machined by conventional milling to make holes, pockets etc. The minimum wall thickness is generally about 2 mm, and all sharp edges must be rounded or chamfered. Further, shape limitations are similar to standard CNC machining. The green shape is pressureless sintered next, at approximately 2100°C in a non-oxidizing environment. Shrinkage during sintering is about 17%, which can be predicted to approximately 0.4%.
After sintering, the material becomes extremely hard but also brittle. This means that all machining (grinding or polishing) must be done with diamond abrasives. Machining is very time consuming and tool wear is high. The attainable roughness after grinding is Ra ≈ 0.3 μm, and polishing can improve this to 1 nm. SSiC contains an open porosity of about 3%, so micro holes (excluded from the roughness figure) must be expected to cause stray light.
[1] AD-96 (96% purity)
[2] Annealed
[3] Expansion class 1 (class 0 = 0.02⋅10-6m/m/K and class 2 = 0.1⋅10-6m/m/K)
Frame structure
Several frame structures have been evaluated from thermal and mechanical point of view (Figure 4.21). For straightness of the beams, the solid version is best since it is least susceptible to thermal gradients. Mechanically however, this is not optimal since the inner core material contributes little to the bending stiffness (4.16), resulting in a first eigenfrequency of 494 Hz. Since the outer dimensions are fixed, removing the inner core to obtain a square tube significantly increases the eigenfrequency, typically to 617 Hz for 15 mm wall thickness.
Figure 4.21: Possible frame structures and indicative 1st eigenfrequencies
From a manufacturing point of view, pressing a solid and then machining it in the ‘green state’ from the outside is easier. The I-beam and I-beam with webbing are made this way and typically result in 338 Hz and 385 Hz eigenfrequency, respectively. To make a more torsionally stiff structure this way, one may machine two open back halves and bond or braze them together, obtaining an internally webbed tube. This would typically result in 552 Hz eigenfrequency, but production is relatively difficult.
Curvature due to thermal gradients in steady state conditions for instance occurs when a continuous heat flux is applied on the bottom of the beam, such as coming from the linear scale read heads. When this flux is transported upwards to the top of the beam, some of it is extracted from the beam by natural convection. The amount of convection is linearly dependent on the surface area in contact with outside air. This area is minimal for the solid beam. Next comes the I-beam, then the square tube and next the I-beam with webbing. The amount of convection on the inside of the closed box structure depends on the pocket size.
In close cooperation with the beam supplier, it was chosen to apply the square tube option since it is the best compromise between thermal and mechanical behaviour and manufacturability. The 900 mm long beam is made out of one piece, and the mirror is polished onto one side. No CVD coating is applied, since sufficiently low roughness is achievable without it.
Reducing thermal disturbances
Besides minimizing the sensitivity to thermal disturbances of the upper metrology frame, the magnitude of these heat loads can also be minimized. In (Ruijl, 2001) the metrology frame is enclosed in an aluminium box, which acts as a low-pass filter for dynamic thermal disturbances such as operators and environmental fluctuations. The high conductivity of aluminium creates a uniform temperature distribution over the shield, thereby also creating a uniform heat load on the metrology frame inside and thus reducing thermal gradients in the frame. Simulations in the next section will show that this simple solution is sufficient, and no active thermal stabilization is required.
Conclusion
Silicon Carbide is the material of choice for the upper metrology frame, since it is least susceptible to thermal gradients under steady state as well as dynamic heat loads. The frame is built out of 2 rectangular tubes forming a square that is supported by Super Invar struts. The mirrors are polished directly onto the sides of the tubes. Passive shielding is applied by an aluminium box. Thermal and mechanical simulations will be performed in the next sections to simulate the performance of this concept, and to optimize the dimensions.
To estimate the stability of the square tube Silicon Carbide metrology frame and to optimize the wall thickness, a thermal analysis has been conducted. After identifying the heat loads and heat conduction mechanisms that are present, the response is simulated.
Heat loads
Three main disturbances have been identified for which the response of the upper metrology frame is calculated.
σ(T14 −T24 )
q1 2− = 1−ε1 + 1 + 1−ε2 (4.17)
ε1A1 A F1 12 ε2 A2
Required thermal stability
The position as well as the flatness of the mirrors should be constant over time, which is determined by rigid body motion of the frame and thermally induced deformation of the frame.
The temperature variation in the SuperInvar struts during a 15 minute measurement is estimated at 0.01K and their length is 380 mm. This results in small z-displacement of 2.3 nm. If this occurs in only one of the struts, the mirror tilts with a negligible 2.5 nrad. The three struts in y-direction may cause small drift in y, ϕ and θ, to which the interferometry system is insensitive. A horizontal Super Invar strut couples the vertical mirror plane to the horizontal capacitive sensor, as will be explained in later. This eliminates the sensitivity to rigid body motion in r-direction.
A vertical thermal gradient (dT/dz) causes curvature of the horizontal mirror (Figure 4.22), and out of squareness between the two mirrors. A horizontal gradient (dT/dr) causes bending of the vertical mirror. A gradient in y-direction (dT/dy) causes out of plane deflection of the frame, which has no first order effect on the mirrors. The elongation of the beams has negligible effect on the measurement due to the high flatness of the mirrors.
Figure 4.22: Metrology frame deflection due to thermal gradients
Slope ψ and deflection δ of a beam due to a temperature difference dT between top and bottom as shown in Figure 4.23, is given by (4.18) (Appendix G):
Figure 4.23: Beam deformation due to thermal gradient
dL =α⋅L0 ⋅dT
ψ= = L0 ⋅α⋅dT (4.18)
dT
8H 8H
For dT/dz, the metrology frame is most sensitive to out of squareness of the vertical beam. To obtain 10 nm stability of the vertical mirror, the length difference between top and bottom of the horizontal beam must be smaller than 8 nm, which is equal to 3.4 mK. For dT/dr, the length difference for the vertical sides must be smaller than 16 nm, or 10.7 mK. Rocking of the vertical beam due to curvature of the horizontal beam is thus the most sensitive to thermal gradients, where the maximum vertical gradient should be in the order of a few mK over the height of the beam.
Convection
Natural convection will occur along the vertical walls. For the convection from the outside walls to the environment, a convection coefficient hair of 5 W/m2K will be used. To assess whether conduction or convection is the main heat transfer means in the air layer between the shield and the frame, their contributions are compared using (Bejan, 1993). For still air with a conduction coefficient kair = 0.025 W/mK, an air layer thickness tair = 5 mm and an estimated temperature difference dT = 0.01 K, the heat flux through conduction can be calculated with (4.19) to be 0.05 W/m2.
q'' = kair dT (4.19)
tair
With a gap height H of 120 mm, the Rayleigh number can be calculated with (4.20) to be 4.4 ⋅104. The flow is thus laminar.
RaH = gβ⋅ q H" 4 with gβ=107cm K−3 −1 (4.20) αυ k αυ
The thickness of the air layer tair is initially set to 5 mm. This gives a H/tair ratio of 24. From (Bejan, 1993) it can be seen that the flow is in the tall enclosure regime, where the heat transfer rate is practically equal to the pure conduction estimate. This is the case down to H/tair = 14, so the air pocket may be slightly enlarged (to 8.6 mm) to obtain a higher thermal resistance of the air layer before convection has to be taken into account.
For a vertical wall, the heat loss to the environment on one side, and the air pocket on the other, can be estimated with:
q'' = hair (T −T∞) + kair (T −T∞) (4.21)
tair
In this case hair is 5 W/m2K and kair/tair is also 5 W/m2K. The heat loss to the environment and the air-pocket are thus equal.
Radiation
The radiation between for instance the shield wall and the frame can be calculated with (4.17). The emissivity is in the order of 0.1 for aluminium and 0.8 for SiC (estimation), the temperature difference is about 0.01 K, the area is 0.09 m2 and the view factor is 1. This gives a radiation heat transfer of 0.5 mW, which is an order of magnitude smaller than the conductive heat transfer. Radiation will therefore not be taken into account in the further calculations.
Finite Element Model
To calculate the time-dependent and steady state behaviour of the horizontal beam, a finite element model was made in Matlab Simulink (Figure 4.24). The model consists of wall elements that are composed of a shield, air and SiC part, of which the equations used are explained in Appendix G. Since the model is symmetrical, only half has to be modelled. The shield is modelled with the opening at the bottom for the mirror, such that part of the heat input is into the shield and part of it is directly into the metrology frame beam.
Figure 4.24: Thermal FEM model of the horizontal metrology frame beam
It should further be noted, that the vertical beam should be connected to the horizontal beam with high thermal resistance, otherwise it will start to function as a cooling fin for the bottom of the beam. This would increase the temperature difference between the top and bottom at the location of the connection. As long as this is done, the model is valid.
Simulations
The thermal behaviour of the system was simulated for shield thickness (ts) of 1 to 10 mm and frame thickness (tf) of 5 to 25 mm. These values are related to the available space in the design. With this geometry, the response to periodic fluctuations and temporary heat loads (e.g. operators) was simulated.
The typical response of a SiC frame with 15 mm wall thickness and 2 mm shield to periodic fluctuations of the environment temperature is shown in Figure 4.25. The low-pass behaviour of the shield is clearly visible. At 10-2 cycles per hour, or about 4 days, both shield and frame follow the environment almost one to one. At 2 cycles per hour, the shield response is 0.1 and the frame response is about 0.02, or a factor 50 reduction in amplitude compared to the environmental fluctuations.
0 0
10-2 10-1 100 101 102 10-2 10-1 100 101 102 Frequency [/hour] Frequency [/hour]
Figure 4.25: Frequency dependent transfer coefficient of temperature (left) and temperature difference (right)
Due to the opening in the shield, a phase and/or amplitude difference between top and bottom occurs. This gives rise to a temperature difference and thus to curvature of the beams. In Figure 4.25 (right), the maximum transfer coefficient for the temperature difference is shown to be 2⋅10 -3 at approximately 4 periods per hour (15 min.). In SiC the (small) temperature difference is mainly caused by an amplitude difference due to heat lost to convection in the side walls.Simulating the frequency dependent transfer coefficient for variable frame and shield thickness for SiC and Zerodur, results in Figure 4.26. Since the thermal mass (ρVcp) is almost equal, the difference in response can only be caused by the large difference in conductivity. Due to this, the peak temperature difference of Zerodur is about 23 times higher and the peak frequency is about 10 times lower. For materials with high volumetric diffusivity, shield thickness is of less influence to the response than for materials with low volumetric diffusivity.
Figure 4.26: Transfer coefficient for temperature gradient for SiC and Zerodur
To translate the transfer coefficient to deflection, it is multiplied by the temperature amplitude, the expansion coefficient and the frame dimensions, according to (4.18). A frame thickness of 15 mm in combination with a shield thickness of 2 mm was chosen as a compromise between transfer coefficient, frame weight and shield robustness. The transfer coefficient and deflection of the vertical beam are shown in Figure 4.27 for several materials for comparison. The temperature amplitude is chosen 0.2°C and constant over the frequency spectrum. Due to the almost equal volumetric thermal diffusivity, the response of Invar and SuperInvar, and of Aluminium and SiC are equal.
Figure 4.27: Transfer coefficient for temperature difference and deflection for various frame materials
The thermal deflection of the vertical beam is lowest for Zerodur over the entire spectrum, at about 1.6 nm at a cycle of about 6.5 hours. Up to this frequency, the response of SiC is approximately equal. Above this frequency, the deflection rises to 2.4 nm at a cycle of 17 minutes (which unfortunately is equal to the expected duration of a measurement). The low conductivity of Zerodur is starting to make the beam too slow to follow the input. The maximum response of SuperInvar, Invar and Aluminium is 5.4, 11.8 and 27.5 nm, respectively.
Next, the influence of an operator standing in front of the machine for 1 hour and then leaving again is simulated. Most of the radiative heat load will be applied to the front of the shield, causing a gradient in y-direction with negligible effect to the mirrors. At worst case, it is estimated that 10%, or 0.1 W, radiates onto the bottom of the shield and frame. For the frame thickness of 15 mm and shield thickness of 2 mm, the response of the bottom and top temperature is shown in Figure 4.28.
Figure 4.28: Top and bottom temperature of frame from operator heat load
As can be seen, the low conductivity of Zerodur gives a high temperature rise at the bottom, from where the heat load is applied. The top temperature clearly falls behind. For SiC and Aluminium, the bottom and top temperatures are almost equal. The low response of Invar is due to its large ρVcp.
The temperature difference and resulting deflection of the vertical beam are shown in Figure 4.29. SiC and Aluminium reach equilibrium swiftly, while the other materials are still drifting after 1 hour. When the operator leaves again, SiC and Aluminium are stable again after a few minutes, while the other materials require more than one hour. The actual temperature difference and deflection are 0.4 mK and 2,2 nm for SiC and 15 mK and 2.1 nm for Zerodur. Coincidentally, the performance is thus equal for this timescale, heat load and geometry. If steady state conditions were reached, the ratio in deflection would have been equal to the ratio of α/k.
Figure 4.29: Temperature difference and deformation from operator heat load Another source of continuous heat flux is the linear scale read-heads, situated directly underneath the horizontal beam. The read heads produce 1 W each, most of which will flow into the stages by conduction. The part that flows into the frame by radiation will be much less then the worst case flux of the operator calculated above, and be steady state as well. This is therefore estimated to have negligible effect.
Silicon Carbide and Zerodur are the two materials that are best used for the metrology frame. Their stability relies on either a very high, and very low volumetric thermal diffusivity, respectively. For the described geometry and thermal loads, Zerodur is more stable at higher frequencies while Silicon Carbide is better under continuous or slowly varying heat flux conditions. Both materials show stabilities in the order of a few nanometers, which is well below specification.
For the dimensions chosen in the last section, the first three eigenmodes and frequencies are shown in Figure 4.30. The first mode only has second order influence on the mirror position. The second mode at 658 Hz does have first order influence. This frequency is expected to be sufficiently high.
Figure 4.30: First three eigenmodes of the upper metrology frame
The total gravity deflection is 520 nm at the centre of the horizontal beam and 550 nm at the bottom of the vertical mirror, as shown in Figure 4.31. This is a steady state deflection, but is has to be taken into account in the alignment budget of the interferometry system. The slope at the ends of the horizontal mirror is approximately 0.6 rad, and the tilt of the vertical mirror is approximately 1.5 rad. These tilts are small compared to the budget explained in interferometry alignment analysis.
Figure 4.31: Gravity deflection
The two Silicon Carbide beams containing the mirrors are connected with high stability and aligned as square as possible. This assembly will then be mounted to the base and the lower metrology frame in a statically determined manner. The shielding is mounted separately from the metrology frame.
SiC beams with mirrors
The upper metrology frame beams have their mirrors directly polished onto the beams, and are manufactured by CoorsTek (Figure 4.32). Reflectivity is enhanced by a protected Aluminium coating. Polishing has taken much more effort than expected, but finally resulted in a flatness of about 1 m PV over the entire mirror length and a Ra roughness of 5.7 nm, as shown in Figure 4.32 (right).
Figure 4.32: Metrology frame vertical beam with mirror, and confocal microscope measurement of mirror surface.
In the surface finish measurement the porosity of Sintered Silicon Carbide can clearly be distinguished. The remaining pores vary in size from 100 nm to a few of 10 m and are up to 0.5 m deep. The area covered by the pores is approximately 1.2 %. The pores will mainly cause scattering, which is not expected to influence the measurement as long as the area between the pores is flat enough to preserve the reflected wavefront flatness.
Beam connection
The horizontal and vertical beams are to be connected with high stability and aligned as square as possible. Since the top face of the vertical beam is not very square to the mirror, the beam connection should allow for alignment. As explained in the thermal analysis, the beam connection should further have high thermal resistance to prevent the vertical beam from acting as a cooling fin on the bottom of the horizontal beam. It should also be possible to disassemble the frame, in case of mirror damage for instance.
A pull rod through the beams or a bolted connection between two glued-in inserts can easily be disassembled. The forces required to obtain sufficient contact pressure will deform the beam and thus the mirrors. Further, alignment should be done by adjusting shims at the contact points, either by lapping or by elastic deformation.
A dowel pin connection is stressless and can be aligned while the adhesive is curing. Since a high thermal resistance is required, it was chosen to leave a 1 mm gap between the beams. This gap also allows for sawing the pins with a wire saw in case disassembly is required. The pins are further provided with M3 thread through the centre for extracting them after heating the adhesive. The stiffness transfer is through the pin’s outer cylindrical surface.
Figure 4.33: Dowel pin connection of metrology frame beams
Invar 36 was chosen since it is easier to machine and its expansion coefficient better matches that of SiC, compared to Super Invar. The maximum pin diameter in the 15 mm thickness of the beam walls is 8 mm. The pins were made to fit the holes to 0.02 mm after the beams were delivered. With the half pin length of 17 mm, the surface area of the glue layer is 855 mm2. Araldite 2020 adhesive has a Young’s modulus of approximately 2 GPa and a Poisson’s ratio ν of 0.3, giving a shear stiffness of one glue layer of 3.3⋅1010 N/m. For the total 4-pin connection, the axial stiffness is 6.6⋅1010 N/m.
Upper metrology frame mounting
The upper metrology frame is mounted to the base kinematically, to prevent frame deformations due to thermally or mechanically induced deformation of the base. In z and ψ-direction this is done with two Super Invar struts. These struts have been heat treated after machining to regain the original low expansion coefficient. They further have hinges to reduce the sideways stiffness and are bolted to Invar blocks that are glued to the Silicon Carbide. The alignment of the horizontal mirror relative to the spindle is done by adjusting this connection. Since the two supports are almost 900 mm apart, sufficient angular resolution can be obtained here relatively easily.
In r-direction the frame has to be constrained such that the vertical mirror has a stable or known position relative to the capacitive probe measuring the error motion of the spindle in r-direction. The distance between this probe and the mirror surface is 155 mm, due to the space required by the Ψ-axis and Z-stage optics in the outer right position. This distance is bridged by a Super Invar strut.
In the first option of Figure 4.34, the probe is fixed and the Super Invar strut constrains the r-direction of the upper metrology frame. Deflections of the base will not cause relative motion between the upper and lower metrology frame, but the rsupport is far below the centre of gravity of the upper metrology frame. This will lead to the first eigenmodes being bending of the vertical beam that is trying to constrain the r-direction of the (heavier) horizontal beam, typically at 260 Hz.
Figure 4.34: Constraining of horizontal degree of freedom of the metrology frame
In the second option, the y-support is aligned with the centre of gravity of the frame, Since it is connected to the base, the upper metrology frame will now shift sideways with bending of the base due to R-stage motion. By mounting the capacitive probe to an elastic parallelogram that is linked to the mirror face by the Super Invar strut, this probe will shift along with the frame. This will cause the R-interferometer and the capacitive probe to measure an equally larger or smaller distance and thus cancelling this shift from the metrology loop. This option is preferred since its first eigenmodes is typically at 620 Hz and thermally equal compared to the first option.
Invar pads are bonded to the SiC frame to interface with the mounting elements. The above described constraint in r-direction is combined with a constraint in y-direction into a steel A-plate (1), as shown in Figure 4.35. The ϕ and θ-directions are constrained by two steel struts (2 and 3). Steel is used here since no high thermal stability is required in these directions. Both the A-plate and the struts extend backwards to inserts in the granite base (4).
Figure 4.35: Upper metrology frame mounting
The two vertical Super Invar struts (5 and 6), the A-plate and the two steel struts now constrain the upper metrology frame in six degrees of freedom. The dimensions of the struts and hinges is chosen such that the first rigid body mode is around 600 Hz, which is similar to the first eigenfrequency of the SiC frame. The Super Invar strut (7) that connects the mirror face of the short beam to a parallelogram on which a capacitive probe is mounted will be further discussed in the design of the lower metrology frame (section 4.4.3).
Shielding
The thermal shielding consists of a 5 mm Aluminium back plate that is bolted to the base with rubber dampers in between, as shown in Figure 4.36. The covers (3 and 4) for the horizontal and vertical beam and the left support strut (5) are made from single pieces of folded 2 mm thick Aluminium plate, to maintain high conduction between the front and the top and bottom. The covers are bolted to the back plate using M3 screws that can be reached through the small holes in the front.
To reduce acoustically induced vibrations, a 1 mm rubber strip is applied between the back plate and the covers to increase the damping. This virtually eliminates thermal conduction between the cover and back plate, which is different compared to the model. For uniform environmental temperature changes this makes little difference since the shield is uniformly heated with negligible in-plane conduction. Operator induced heat is applied from the front and conducted to the top and bottom, which also makes little difference compared to the model. The shielding is connected to the base via rubber mounts, completely isolated from the upper metrology frame. The lower metrology frame is also enclosed with a cover.
Figure 4.36: Metrology frame shielding
Realization
The two beams are assembled on a flat granite table. First, an autocollimator (MöllerWedel Elcomat) was adjusted parallel to the table by pointing the beam downwards with a pentaprism to an optical flat on the table. This flat was rotated 180° to eliminate the non-parallelism between its front and back surface. Next, the beams were placed on their sides in front the autocollimator to verify the perpendicularity between the sides and the mirror face. The beams were mounted on spacers to match the height of the autocollimator. A calibrated pentaprism was then aligned in front of the autocollimator, to measure the alignment of the two mirrors (Figure 4.37, left).
The Araldite 2020 adhesive was then applied to the dowel pins and the parts are assembled. This adhesive requires 24 hours to cure, giving about 45 minutes to align the two mirrors.
Figure 4.37: Metrology frame mirror alignment and bonding of the interfaces
The NMi VSL pentaprism used has a calibrated deviation of 1.18 arcsec with an uncertainty of 0.27 arcsec. FEM simulations show a gravity deflection of 0.4 arcsec, so the beams are to be aligned to 1.58 arcsec. With a noise level of about 0.1 arcsec, the averaged result was 1.6 arcsec. The alignment was checked after curing for several days, and had remained stable.
Next, the Invar interface blocks and the linear scale were mounted to the frame, again with Araldite 2020 adhesive. After curing, the frame was installed on the machine (Figure 4.38).
Figure 4.38: Upper metrology frame after installation
The metrology frame mirrors must be aligned to the spindle axis of rotation and the measurement plane. To achieve this, an autocollimator is mounted to the granite base and a 45° mirror directs the beam upwards through the hole in the spindle (Figure 4.39A). A mirror is placed upside down on the spindle and adjusted perpendicular to the spindle axis by rotating the spindle (Figure 4.39B). The spindle tilt is specified to be 0.1 rad, which is below the resolution of the autocollimator (0.05 arcsec or 0.24 μrad. The wobble of the mirror can therefore be adjusted to almost zero. Next, the autocollimator was aligned perpendicular to the mirror, after which it was collinear with the spindle axis of rotation to within ±0.05 arcsec, which is equal to the noise level of the autocollimator. The mirror on the spindle was removed, revealing the horizontal metrology frame mirror to the autocollimator. By adjusting the bolted connection of the struts, the horizontal mirror was set perpendicular to the autocollimator in ϕ and ψ-direction.
Figure 4.39: Upper metrology frame alignment
Next, a calibrated pentaprism of NMi VSL was positioned above the hole (Figure 4.39C), pointing the beam backwards towards the vertical granite base plane. This plane determines the orientation of the measurement plane. A 2-sided mirror with front and backside parallel to within 0.09 arcsec (calibrated) was then pressed to the granite face, after which the spindle was rotated to set the pentaprism perpendicular to this mirror.
The pentaprism was rotated exactly 90° by the spindle and its encoder, to point it towards the vertical mirror of the metrology frame (Figure 4.39D). The accuracy of the encoder is specified to be 1 arcsec, and the spindle controller positions the spindle well below this. The vertical mirror was now adjusted in -direction, such that it is square to the measurement plane.
The whole procedure was iterated several times. The horizontal mirror finally was aligned to +0.05 arcsec in the most critical ψ-direction, and to 0.1 arcsec in the less critical ϕ-direction. The vertical mirror was aligned to 0.75 arcsec in -direction. It must be noted that the flatness of the mirror is not taken into account here, and the alignment and perpendicularity measurements are done on a single point of the mirror. The actual flatness has not yet been calibrated, but is specified to be 200 nm/200 mm, which is equal to 1 rad or 0.2 arcsec.
The perpendicularity was finally measured in a setup as shown in Figure 4.40, resulting in a perpendicularity error of ψ = -3.53 arcsec (17.1 μrad) after correction for the deviation of the NMi VSL pentaprism. This will only cause a measurement error for products with large height differences, and can of course be compensated for in the data processing.
Figure 4.40: Mirror perpendicularity measurement
The lower metrology frame measures the product position relative to the horizontal and vertical reference mirrors. The horizontal beam of the upper metrology frame is coupled to the lower metrology frame by the two vertical Super Invar struts. In rdirection, the vertical mirror should be referenced to the axis of rotation of the spindle. To minimize the thermal loop length between this mirror and the spindle centre, a coupling is made with a Super Invar strut and a capacitive probe as described in the upper metrology frame design.
The product is rotating on the air bearing spindle, and is supposed to be rigidly attached to the mounting table. Although the acquired Professional Instruments BlockHead 10R spindle is specified to have an error motion smaller than 25 nm, it is too much to meet the design error budget (δS,r < 15 nm, δS,z < 15 nm, δS, < 0.1 μrad (3σ)). The error motion will thus have to be measured to determine the product position relative to the reference mirrors with sufficient accuracy.
Ideally, the product position is directly measured relative to the lower metrology frame. The product can be any shape, so measuring to the intermediate body is next best. This body may for instance be provided with a reference edge. The diameter of this edge depends on the product, and some products such as mirrors may not be mounted on an intermediate body at all. Measuring to the edge of the mounting table is thus the most practical solution, provided that the product does not move relative to this edge during a measurement. The edge of the mounting table will obviously not be perfect, so it will have to be calibrated. Due to product specific mass and mounting positions, the edge shape will also vary between measurements.
First, methods of measuring spindle error motion will be briefly discussed, after which the multi-probe method is explained. Next, the design and realization are discussed and experimental results will be given.
The error motion of the spindle consists of a synchronous and an asynchronous part (ASME, 1985). The synchronous part is generally repeatable, but may also be dependent on product mass. The asynchronous part, caused by for instance air pressure variations, unbalance and other external excitation, is not repeatable and should thus be monitored continuously.
A probe measuring to a rotating object measures the error motion plus the roundness error of the object. In this case, the rotating object is a reference edge on the table, either as an integral part of the table or as a separate ‘metrology frame’ ring mounted underneath. In either case the edge is probably not sufficiently stable due to the
varying loading of the table due to the variable product shape and mass. Preferably, the edge shape is thus determined in-process, before or during each measurement with the product mounted. Several methods exist to separate the object roundness error and the spindle error motion (Marsh et al., 2006).
Compare to a calibrated artefact
Spindle calibration is mostly done by mounting a reference sphere or other artefact with a known form, to which a measurement is performed with (capacitive) sensors. This artefact would have to be mounted over the product before a measurement, to include the deformation caused by the product itself.
Reversal methods
With Donaldson ball reversal (Donaldson, 1972) and Estler face motion reversal (Estler, 1986), a number of revolutions are averaged to obtain the sum of the synchronous error motion and the profile form. This measurement is then repeated with the artefact and probe orientation reversed. Subtracting the two measurements results in an exact solution for the edge form and the synchronous error motion, provided that no other errors are introduced during the reversal.
If this method is to be performed with the product mounted, it would require a separate edge and probe which can be reversed. The form of this edge should remain stable during this operation. Although this is the only method that results in an exact solution, it is not very practical in this case.
Multi-step method
In the multi-step method, the probe remains stationary and the edge is measured under multiple (equidistant) orientations (Chetwynd and Siddall, 1976). Averaging the measurements separates the error motion from the edge form, except for frequencies that are integer multiples of the number of steps. Numerous measurements thus have to be applied to avoid errors from harmonic suppression, especially for objects with a large circumference. Similar to the reversal methods, this method is thus not practical in this case.
Multi-probe method
When multiple probes are placed around the rotating table, the error motion is measured in phase by all the probes, while the form error of the edge is phase shifted with the probe spacing. By Fourier decomposition, the edge profile and error motion can be separated, even for a single revolution (Mitsui, 1982). A minimum of seven capacitive probes are required for this procedure to measure the five degrees of freedom of the spindle and the axial and radial profiles. As with the multi-step method, harmonic suppression occurs at certain frequencies, depending on the chosen probe spacing. Some therefore apply more than seven probes to obtain redundant data to increase the harmonic suppression limit (Zhang et al., 1997).
This method does not require repositioning of the edge and the probes, and can be applied with the product mounted on the table. It may even be applied during measurement of the surface under test. It is thus a true in-process calibration of the edge form of the table. It may, however, suffer from harmonic suppression (Whitehouse, 1976). Each sensor must be individually calibrated, and differences due to mounting must be minimized. This method was further analyzed to assess the applicability and achievable measurement uncertainty.
The principle of the multi-probe method will first be explained, followed by the risk of harmonic suppression and the harmonic content of the reference edge. Next, the optimum probe locations are summarized, based on the analysis in Appendix H. Simulations show the suitability of the method. When preparing for testing the method in practice, the spindle was found to show excellent error motion properties. Other parts of the machine have therefore been completed first. Eventually, time did unfortunately not permit the method to be validated in practice.
Principle
A probe measuring to an object rotating on a spindle as shown in Figure 4.41, measures the sum of the error motion and the axial or radial profile (flatness / roundness error). The axial and radial profiles of the object Pax(θ) and Prad(θ), respectively, are considered stable during a revolution (typically one second in time).
Axial Radial
Figure 4.41: Axial and radial probe measuring a profile
The asynchronous error motion varies per revolution, making the error motion a function of the angle θ and time t. For a single revolution however, the error motion can be considered as being only a function of θ. The axial and radial measurement signals max,i(θ) and mrad,i(θ), of probe located at θi and radius Rax,i or Rrad,i, can now be expressed as:
| max i, ( )θ θθ θ= Pax ( + +i ) zS ( ) +ϕS ( )θ θsin( i )Rax −ψS ( )θ θcos( i )Rax
| (4.22) |
| m ( )θ = P (θθ θ θ θ θ+ ) + r ( )cos( ) + y ( )sin( ) | (4.23) |
rad i, rad i S i S i
Besides the rotation θ, there are 7 degrees of freedom in the system: 3 translations, 2 tilts and 2 edge profiles. Applying seven probes provides enough information to solve all 7 variables. Hereto, four axial probes (i = 1 – 4 in Figure 4.42) and three radial probes (i = 5 - 7) are applied. To prevent thermal expansion of the table being interpreted as radial motion, an 8th probe is added, opposite to probe number 5.
Figure 4.42: Four axial (1-4) and three radial (5-7) probes
The multi-probe error motion reconstruction is shown for the radial profile here. An analogous derivation for the axial direction is shown in Appendix H. The radial measurements m1(θ), m2(θ) and m3(θ) are expressed as:
m5( )θ = Prad (θθ θ θ θ θ+ 5) + rS ( )cos( 5) + yS ( )sin( 5)
m6( )θ = Prad (θθ θ θ θ θ+ 6 ) + rS ( )cos( 6 ) + yS ( )sin( 6 ) (4.24) m7 ( )θ = Prad (θθ θ θ θ θ+ 7 ) + rS ( )cos( 7 ) + yS ( )sin( 7 )
Where Prad is the radial edge profile, rs is the spindle error motion in r-direction and ys is the error motion in y-direction. These measurements are multiplied by sum factors f, g and h and then summed to Srad(θ):
Srad ( )θ = fm5( )θ + gm6( )θ + hm7 ( )θ
= fPrad (θ θ+ 5) + gPrad (θ θ+ 6 ) + hPrad (θ θ+ 7 )
+ rS ( )θ ( f cos(θ5) + gcos(θ6 ) + hcos(θ7 )) (4.25)
+ yS ( )θ ( f sin(θ5) + gsin(θ6 ) + hsin(θ7 ))
The coefficients f, g and h can now be chosen such that the error motions cancel out. This is the case if the following equations are satisfied:
f cos(θ5) + g cos(θ6) + hcos(θ7) = 0
f sin(θ5) + gsin(θ6) + hsin(θ7) = 0 (4.26)
(f 2 + g2 + h2)=1 (unit vector)
The resulting summed measurement now simplifies to (4.27). This is only a function of the radial profile Prad, the probe orientations θ5-7 and the spindle orientation θ. The error motion is cancelled from the equation.
Srad ( )θ = fPrad (θ θ+ 5) + gPrad (θ θ+ 6 ) + hPrad (θ θ+ 7 ) (4.27)
Since the edge profile is closed, only discrete Fourier terms are present. With harmonic number k, the discrete Fourier transform of formula (4.27) is equal to:
| rad rad rad rad
= Prad ( )k ( fe jkθ5 + ge jkθ6 + he jkθ7 )
The Fourier transform of the radial profile can be calculated with:
| (4.28) |
| P ( )k = Srad ( )k | (4.29) |
S ( )k = fP ( )k e jkθ5 + gP ( )k e jkθ6 + hP ( )k e jkθ7 rad fe jkθ5 + ge jkθ6 + he jkθ7
By taking the inverse Fourier transform of this profile, Prad(θ) can be calculated from a single revolution. By phase-shifting and substituting the obtained profiles into the original measurement signals (4.24), the error motion is found.
Harmonic suppression
Three error sources are considered in Appendix H: harmonic suppression, probe noise and probe angular position errors. Only harmonic suppression will be briefly addressed here.
The denominator of equation (4.29) becomes zero for certain combinations of probe positions and harmonics. Physically, this means that this harmonic generates measurement signals of which the mutual phase is such that it cannot be determined whether it is error motion or edge form, as schematically shown in Figure 4.43.
Figure 4.43: Harmonic suppression example
To overcome this problem, one may either choose to leave out these specific harmonics in the reconstruction, or to stop reconstruction before the first suppressed harmonic (klimit) is encountered. All harmonics required to describe the profile and error motion with sufficient accuracy must then be below klimit. Probe locations can be optimized to achieve a high klimit.
Harmonic content of mounting table edge
The diameter of the edge is ∅600 mm and the measurement spot of the probes is ∅1.7 mm. Harmonics up to 1100 upr can thus reliably be measured, above this frequency aliasing may occur.
Axial and radial measurements of the actual edge are shown in Figure 4.44 (left), along with the harmonic content (right). In axial direction the signal is dominated by a second order component, which indicates that the table is warped but level. In radial direction a first order component is visible, which is eccentricity.
Figure 4.44: Typical measurement signals and harmonic content
This edge form was not yet known during the analysis and simulations of the multiprobe method. It was expected that some low order frequencies would dominate the signal, and that a more or less the other content would be inversely proportional to the harmonic number. A 1/k function is also shown in Figure 4.44. Generally, this assumption fits the data, but some sharp high order frequencies are present. These probably originate in the properties of the manufacturing machine, such as the number of rolling elements in the bearings. Judging from the above figure, significant content is present up to 600 Hz (3 mm wavelength), but a higher harmonic limit is desirable.
Probe locations
The analysis shown in Appendix H, shows that a smaller angle between the probes allows for a higher achievable harmonic limit. The probe signals will, however, also become correlated. When measuring a sinusoidal profile with amplitude A, harmonic number k and probe spacing Δθ , the nominal difference between the two probe signals is:
Δm( )θ = A(sin(kθ) − Asin( (k θ θ− Δ )))
with (4.30)
Δm( )θ = 2Acosk θ− Δθsink Δθ
2 2
The weakest profile to be distinguished is assumed to be a second order harmonic with 2.5 nm amplitude. The capacitive probes used have a resolution of 0.4 nm rms, therefore the minimum difference between the measurement signals is set to 1 nm. This results in a minimum probe spacing of approximately 12°. The test setup is provided with adjustable mounts to vary the probe positions around this position.
Summarizing from the analysis in Appendix H, the optimum angular probe position has its suppression limit (klimit) at a harmonic above which the harmonics have negligible contribution to the edge form and error motion. This optimization results in angular limits between which the probes physically have to be positioned to not be blind to certain harmonics below klimit. Further, a position must be chosen that has maximum clearance from zero points to obtain minimum alignment sensitivity. Hence, the limits between which the probes have to be positioned must be chosen as far apart as possible. This also eases the physical alignment of the probes. Finally, the cumulative sum of the transfer coefficient harmonics must be minimal to minimize the influence of measurement noise. Judging from the preliminary results, the probes should be positioned near, but not onto, the θ2 = -θ4 and θ6 = -θ7 lines. This should allow for a harmonic limit of 1100 upr.
Simulated results
To test the reconstruction and to estimate the measurement uncertainty, simulations have been performed. In this simulation, edge profiles are generated with a 1/k amplitude and random phase content up to a specified limit. The same is done for the error motion in five degrees of freedom. The profile and error motion are sampled at the nominal probe locations to generate simulated probe signals. The above described reconstruction algorithm is then executed to calculate the profile and the error motion. The difference between the nominal input and the simulated results is the error of the method.
The randomly generated edge profile is shown in Figure 4.45 (left). These profiles were generated with 10000 points/rev and a harmonic content up to 1100 upr. The error motion is expected to have less high harmonics due to the averaging effect of the air gap. The error motion shown in Figure 4.45 (right) has harmonics up to 50 upr. Note that the error motion is to be detected in a signal that is two orders of magnitude larger.
Figure 4.45: Simulated edge profile and error motion
The above profiles and error motion were measured by probes positioned at 12° and 12.25°. These values are based on the analysis of Appendix H. The reconstruction algorithm was first executed without noise and probe misalignment. For probe orientations without harmonic suppression below1100 upr, the results are exact. Next, noise of 0.4 nm rms was added to each probe signal, and a probe misalignment with an uncertainty of 0.0001° was introduced. This high alignment accuracy is expected to be achievable since the probe alignment can be directly derived from the signals due to the large ratio between the profile and the error motion. The reconstructed error motion is compared with the real input of Figure 4.45. The resulting profile reconstruction error and error motion reconstruction error are shown in Figure 4.46. In this example, the axial and radial profile are measured with an rms error of 0.98 and 0.96 nm. The r, z and ψ-error motion are determined with an rms error of 1.13 nm, 0.89 nm and 2.8 nrad, respectively. Running this simulation with similar parameters shows consistent results.
Figure 4.46: Simulated profile and error motion reconstruction error
The simulations show that the multi-probe method is suitable for in-process determination of the spindle error motion. Finding the optimum probe position is still subject of future work. When preparing for testing this method in practice, it was found that the spindle already showed excellent rotation accuracy (section 3.3.5). It was therefore decided to first focus on the rest of the machine. Unfortunately, time did eventually not permit the method to be validated in practice.
The lower metrology frame consists of two steel base plates on which adjustable mounts are placed. These probe brackets can be adjusted to bring the probes into range, and the angular position can be varied. The rotor is grounded to improve the noise level. The whole assembly is protected by a cover.
Probes
The probes used are the Lion Precision C7-C with CPL190 driver modules, specified to have 0.4 nm rms resolution, 10 μm range and 10 kHz bandwidth. The driver modules have an analog +/- 10 V BNC output that is fed into the data acquisition system. The probes are ∅8 mm and 50 mm long and the measurement spot is ∅1.7 mm. Seven probes are used for the multi-probe method; the 8th probe is employed for various measurement tasks during machine assembly, and will finally be applied as an extra radial probe opposite to probe number 5. This allows for determining thermal expansion of the table during measuring, which would otherwise be interpreted as drift of the spindle in positive r-direction.
Reference edge
The reference edge is machined on the side of the steel mounting table. For this ∅600 mm disc a roundness of 2.3 μm PV and a flatness of 5.1 μm PV was achieved. With the tilt and eccentricity which are introduced by bolting the steel table to the spindle taken into account, the total indicated readout (TIR) is 5.2 μm in axial direction and 4.8 μm in radial direction. The probes are to be positioned in range to about 1 μm.
Base blocks and brackets
To match the expansion coefficient of the spindle and the mounting table (1), the lower metrology frame is made of similar steel. It consists of two blocks (2 and 3) that are mounted to the granite base (Figure 4.47). The fixation points of the blocks have hinges (4) to create a thermal centre on the measurement plane. The blocks have a square groove (5) that is concentric with the spindle axis. The movable probe brackets (6-11) have two pins that slide through this groove when the orientation is adjusted, to maintain perpendicular alignment and to assure that each probe measures the same track on the axial (12) and radial (13) reference edge. The adjustment range of the probes is ±1°. The axial and radial probe located at zero degrees do not need adjustment, therefore these are mounted to a single bracket (14) which has two pins that fit into a hole and a slot.
Figure 4.47: Lower metrology frame design
Probe brackets
Figure 4.48 shows the probe brackets in more detail. The movable brackets are constrained in radial and local θ-direction by the two pins in the groove (1). Once in position, they are locked by tightening a central bolt (2), of which the friction torque under the head is diverted to the base plate by a torsionally stiff coupling (3). During alignment and testing of the multi-probe method, the probe location will be changed frequently. Since the probes have an axial measurement range of only 10 μm, they will have to be adjusted many times. To ease adjustment, especially of the axial probes (4) that are hard to reach underneath the mounting table, the bracket has a hinge (5), an adjustment screw (6) and a stroke limit (7). Turning the screw tilts the top part of the bracket to bring the probe into range. A clamp (8) at the backside of the mechanism clamps a plate (9) that locks the mechanism, after which the screw can be loosened. This avoids the instability of the adjustment screw.
Figure 4.48: Probe brackets
As explained in section 4.3, the upper metrology frame may move in r-direction due to bending of the base. To compensate for this, the radial probe (10) in the fixed bracket is mounted on a parallelogram (11). A Super Invar strut (12) connects it to the vertical mirror face (13) on the upper metrology frame. The vertical mirror and radial probe now experience equal shift in r-direction, cancelling the effect.
Grounding of the rotor
To achieve optimal performance of the probes, the target should be grounded to the driver unit. Since the spindle rotor is floating on air, no electrical conducting contact is present. A plate with a small bump is therefore pressed lightly onto a plug in the bottom hole of the spindle. Since the contact point is located at the centre line, the relative speed is (almost) zero and thus the friction is negligible. Tests showed no detectable difference of the spindle performance with and without this contact. This grounding significantly reduced the rms noise level from 5.6 nm to 1.9 nm. This is measured with the air pressure turned on, so it also includes air gap variations.
Shielding
Similar to the upper metrology frame, the lower metrology frame is enclosed in a box (Figure 4.36). This is mainly to protect the probes and brackets, and to enclose the
reference edge. The covers are connected to the granite base separate from the upper and lower metrology frame.
Realization
The assembled lower metrology frame is shown in Figure 4.49, with and without the shielding, together with a side view of a movable bracket and the parallelogram.
Figure 4.49: Lower metrology frame, probe brackets and shielding
Figure 4.50 shows the metrology system assembly, with all covers installed. The noncontact probe is already installed in this photograph. The pressure-temperaturehumidity (PTH) sensor is situated in the upper right corner of the of the upper metrology frame.
Figure 4.50: Metrology system assembly
In this section, the metrology system stability experiments over 0.1 s and 15 minutes will be shown. The harmonic contents of the measurement error and the displacement is compared. The metrology frame shift compensation mechanism is also tested.
In section 3.8.2 a test setup was described in which three orthogonal capacitive probes mounted on the spindle measured the displacement of an Invar dummy probe (Figure 3.100). With this setup the stability of the metrology system has also been tested before the optical probe was included in the metrology loop. These tests have been performed with the dummy probe oriented at ψ = 0° and 90°. The results shown below are with the probe at ψ = 0°, since this will be the most common orientation.
Figure 4.51 shows the displacement in z-direction as measured by the reference capacitive probe and the metrology loop over 0.1 s, sampled at 10 kHz. An offset of 5
4.6 Experiments
nm has been added to the reference probe signal for clarity. The difference between the signals is the part of the displacement that is not measured by the metrology loop. This error has a PV value of 4 nm and an rms value of 0.6 nm.
Figure 4.51: Comparison of reference probe and metrology loop in z-direction
Figure 4.52 shows a similar measurement over a period of 15 minutes in r, y and zdirection, sampled at 1 kHz. The drift during this measurement as measured by the reference probes is about 50 nm in r, 25 nm in y and 85 nm in r-direction. These values are an indication of the stability of the motion system with all stages locked. Also note the shock in the nearby workshop at around 10 minutes in z-direction.
R-displacement Error in R-direction, rms = 8.2 nm
Figure 4.52: r,y,z-displacement measured by reference probes and metrology system
The metrology system measures the displacement of the Ψ-axis and the spindle in r and z-direction. The difference with the measurements of the reference probes is the error at the probe tip. In these measurements, the refractive index is compensated with a 3rd order polynomial fit. In r-direction the faster, uncorrected turbulence and refractive index variations are the main remaining error. The global drift is removed by the metrology loop. The error in r-direction is 8.2 nm rms. In y-direction the error is equal to the actual displacement, coincidently also 8.2 nm rms including approximately 20 nm of linear drift. In z-direction remaining error is 2.3 nm rms, with virtually no remaining drift. The shock around 10 minutes is also compensated for. The remaining fluctuations are probably also turbulence, but with a smaller amplitude than in the r-direction due to the shorter air-path.
The harmonic content of a 10 s measurement at 10 kHz is shown in Figure 4.53. The actual displacement as measured by the reference probes shows all the eigenmodes of the motion system, similar to section 3.8.2. The harmonic content of the measurement error is also shown here. In r and z-direction it can clearly be seen that most of the vibrations are compensated for by the metrology system, especially in the most important z-direction. The remaining peak at 278 Hz in r-direction is rotation of the Ψ-axis rotor on its brake stiffness, which is too small to be measured by the Ψencoder.
Figure 4.53: Harmonic content of the displacement and the error
4.6 Experiments
As explained in section 3.2.1, the metrology frame may shift in r-direction due to bending of the base when the R-stage is moving, or due to thermal expansion of the base. Radial capacitive probe 5 was hereto mounted on an elastic parallelogram and linked to the upper metrology frame via a Super Invar strut. To test this compensation, a force has been applied to the upper metrology frame at the r-constraint to shift it in r-direction (Figure 4.54). While applying this force, the output of the R-interferometer and the capacitive probe are compared. Little correlation between the error and the displacement can be seen, showing that the compensation works.
Figure 4.54: Force applied to upper metrology frame and resulting error
When the R-stage is moved through its range, the capacitive probe measures 760 nm displacement (Figure 4.55). In section 3.2.1 a shift of 170 nm was predicted from FEM simulations, but apparently the real shift is larger. The R-stage was moved through its range four times, in which some hysteresis can also be seen. As long as this is mutual for the capacitive probe and the interferometer, this does not cause a measurement error.
Figure 4.55: Capacitive probe 5 shift due to moving R-stage
The metrology system measures the position of the probe relative to the product in the six critical directions in the plane of motion of the probe (the measurement plane). The interferometry system measures the Ψ-axis rotor position relative to reference mirrors on the upper metrology frame. A horizontal and vertical interferometer are focussed onto a cylindrical mirror on the face of the Ψ-axis rotor. Due to the reduced sensitivity in tangential direction at the probe tip, the Abbe criterion is still satisfied. Silicon Carbide is the material of choice for the upper metrology frame, due to its excellent thermal and mechanical properties. Mechanical and thermal analysis of this frame shows nanometer-level stabilities under the expected thermal loads. The multiprobe method allows for in-process separation of the spindle reference edge profile and the error motion. Simulations show nanometer level uncertainty, but this has not yet been tested in practice.
Experiments show motion amplitudes in the order of 15 nm PV over 0.1 s, which is compensated to 0.6 nm rms by the metrology system. Over 15 minutes, 80 nm of drift in the most critical in z-direction is compensated to 2.3 nm rms. In the tangential rdirection, the stability is 8.2 nm rms over 15 minutes, just as in the out-of-plane ydirection. Comparing the harmonic content of the error to the displacement clearly shows the eigenfrequencies that are compensated with the metrology system, and those which are not. The upper metrology frame shift in r-direction is measured to be 760 nm, which is cancelled by the compensation mechanism.
At standstill, the metrology system short term stability exceeds the requirements. The long term drift is also small, and can be compensated for in the surface measurement procedure as shown in section 7.2.3. After further experiments with a rotating spindle, moving stages and the optical probe included, a similar conclusion will be drawn in Chapter 7.
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