summary
Summary
The performance of high-precision optical systems using spherical optics is limited by aberrations. By applying aspherical and freeform optics, the geometrical aberrations can be reduced or eliminated while at the same time also reducing the required number of components, the size and the weight of the system. New manufacturing techniques enable creation of high-precision freeform surfaces. Suitable metrology (high accuracy, universal, non-contact, large measurement volume and short measurement time) is key in the manufacturing and application of these surfaces, but not yet available. In this thesis, the design, realization and testing of a new metrology instrument is described. This measurement machine is capable of universal, noncontact and fast measurement of freeform optics up to ∅500 mm, with an uncertainty of 30 nm (2σ).
A cylindrical scanning setup with an optical distance probe has been designed. This concept is non-contact, universal and fast. With a probe with 5 mm range, circular tracks on freeform surfaces can be measured rapidly with minimal dynamics. By applying a metrology frame relative to which the position of the probe and the product are measured, most stage errors are eliminated from the metrology loop. Because the probe is oriented perpendicular to the aspherical best-fit of the surface, the sensitivity to tangential errors is reduced. This allows for the metrology system to be 2D. The machine design can be split into three parts: the motion system, the metrology system and: the non-contact probe.
The motion system positions the probe relative to the product in 4 degrees of freedom. The product is mounted on an air bearing spindle (), and the probe is positioned over it in radial (r), vertical (z) and inclination (ψ) direction by the R-stage, Z-stage and Ψaxis, respectively. The motion system provides a sub-micrometer repeatable plane of motion to the probe. The Z-stage is hereto aligned to a vertical plane of the granite base using three air bearings, to obtain a parallel bearing stage configuration. To minimize distortions and hysteresis, the stages have separate position and preload frames. Direct drive motors and high resolution optical scales and encoders are used for positioning. Mechanical brakes are applied while measuring a track, to minimize power dissipation and to exclude encoder, amplifier and EMC noise. The motors, brakes and weight compensation are aligned to the centres of gravity of the R and Zstage. Stabilizing controllers have been designed based on frequency response measurements.
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 Summary
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. Simulations of the multi-probe method show capabilities of in process separation of the spindle reference edge profile and the spindle error motion with sub-nanometer uncertainty.
The non-contact probe measures the distance between the Ψ-axis rotor and the surface under test. A dual stage design is applied, which has 5 mm range, nanometer resolution and 5° unidirectional acceptance angle. This enables the R and Z-stage and Ψ-axis to be stationary during the measurement of a circular track on a freeform surface. The design consists of a compact integration of the differential confocal method with an interferometer. The focussing objective is positioned by a flexure guidance with a voice coil actuator. A motion controller finds the surface and keeps the objective focused onto it with some tens of nanometers servo error.
The electronics and software are designed to safely operate the 5 axes of the machine and to acquire the signals of all measurement channels. The electronics cabinet contains a real-time processor with many in and outputs, control units for all 5 axes, a safety control unit, a probe laser unit and an interferometry interface. The software consists of three main elements: the trajectory planning, the machine control and the data processing. Emphasis has been on the machine control, in order to safely validate the machine performance and perform basic data-processing.
The performance of the machine assembly has been tested by stability, single track and full surface measurements. The measurements focus on repeatability, since this is a key condition before achieving low measurement uncertainty by calibration. The measurements are performed on a ∅100 mm optical flat, which was calibrated by NMi VSL to be flat within 7 nm rms. At standstill, the noise level of the metrology loop is 0.9 nm rms over 0.1 s. When measuring a single track at 1 rev/s, 10 revolutions overlap within 10 nm PV. The repeatability of three measurements of the flat, tilted by 13 μm, is 2 nm rms. The flatness measured by the uncalibrated machine matches the NMi data well. Ten measurements of the flat tilted by 1.6 mm repeat to 3.4 nm rms.
A new non-contact measurement machine prototype for freeform optics has been developed. The characteristics desired for a high-end, single piece, freeform optics production environment (high accuracy, universal, non-contact, large measurement volume and short measurement time) have been incorporated into one instrument. The validation measurement results exceed the expectations, especially since they are basically raw data. Future calibrations and development of control and dataprocessing software will certainly further improve these results.
Nomenclature
Symbol Description Unit
A Amplitude [m]
A, AS, Ap Area, Surface area, Porous plug surface area [m2]
A Jones matrix - B Beam width [m]
B Magnetic flux density [T]
D Diameter [m] D0, Dph, Dz Beam waist diameter, Pinhole diameter, e-2 diameter [m]
E Young’s modulus [N/m2]
Ex ,Ey Electric field vector [-]
Ec Contact modulus [N/m2]
F Force [N]
Fc, Fg, Fa Spring force, gravity force, acceleration force [N]
F1,2 View factor [-]
G Shear modulus [N/m2]
Gc Contact shear modulus [N/m2]
H Beam height [m]
H Partial water vapour pressure [Pa]
H Complex transfer coefficient column [-]
I Current [A]
I Irradiance [W/m2]
I Planar moment of inertia against bending [m4]
J, Ji,j Moment of inertia (of part i in direction j) [kgm2]
L Length [m]
LP Probe length [m]
M Moment [Nm]
M Mass flow [kg/s]
P Pressure [Pa]
PS, PA, PR Bearing pressure (supply, atmospheric, restriction) [Pa]
P Line load [N/m]
P Power [W]
Pax, Prad Profile (axial and radial) [m] Q Normalized volume flow rate [l/min]
R Radius [m]
R Radial position [m]
R Flow resistance [Pa.s/m3]
R Gas constant [J/(kmol⋅K)]
R Rotation matrix [-] Nomenclature
Ra Raleigh number [-]
Ra Roughness [m]
Rc Radius of curvature [m] Rc Contact radius [m]
Sax, Srad Sum signal (axial, radial) [m]
T Temperature [K] or [°C]
T Tangential force [N]
V Volume [m3]
Z Height, Vertical position [m]
a,b,c,α,β,γLocal probe coordinate system [m]
a,b,c,d,e,f,g Sum factors [-] a Major contact radius [m] b Minor contact radius [m] b Half contact width [m] b Beam wall thickness [m] b Gap width [m]
c, ci,j Stiffness (of part i in direction j) [N/m]
cp Specific heat [J/kg/K] e Relative radial bearing eccentricity [-] eax , erad Phase vector [-]
f, fe Frequency, eigenfrequency [Hz] f Focal length [m]
g Gravitational acceleration [m/s2] g Geometry factor [m-3] h Hinge thickness [m] h Bearing gap height [m]
h Convection coefficient [W/m2/K]
hax, hrad Complex transfer coefficient [-] j Complex number [-]
k, ki,j Rotation stiffness (of part i in direction j) [Nm/rad] k Thermal conductivity [W/m/K] k, klimit Harmonic number, harmonic limit [upr] kF Motor force constant [N/A] km Motor constant [N/√W]
kp Permeability [m2]
kT Motor torque constant [Nm/A] l Length [m] m Mass [kg]
mi Measurement signal of probe i [m]
n Refractive index [-] n Number of coil windings [-] n Normal vector [-]
|
n |
Noise level column |
[m] |
|
q |
Heat transfer rate |
[W] |
|
q” |
Heat flux |
[W/m2] |
|
r |
Radius |
[m] |
|
r,y,z,ϕ,ψ,θ |
Modified Cartesian coordinate system as used in this thesis |
[m] |
|
s |
Bearing porous plug thickness |
[m] |
|
t |
Time |
[s] |
|
t |
Thickness |
[m] |
|
t t |
Hinge width Tangential vector |
[m] [-] |
|
u |
Position from focus along optical axis |
[m] |
|
w |
Relative beam displacement (walk-off) |
[m] |
|
w |
Width |
[m] |
|
z |
Distance from focus along optical axis |
[m] |
|
zr |
Rayleigh range |
[m] |
|
zg |
Floor vibration amplitude |
[m] |
|
x,y,z
|
Cartesian coordinate system |
[m] |
|
Greek |
Description |
Unit |
|
|
|
|
|
Δn, Δt, Δs, |
Resulting error (normal, tangential, slope) |
[m] |
|
Φ |
Contact tangential displacement correction factor |
[-] |
|
|
|
|
|
α |
Thermal expansion coefficient |
[m/m/K] |
|
α,β,γ |
Rotations in a,b,c coordinate system |
[rad] |
|
δ |
Deflection |
[m] |
|
δi,j |
Position error of part i in direction j |
[m] |
|
δn, δt |
Position error at probe tip (normal, tangential) |
[m] |
|
δ |
Vibration amplitude at probe |
[m] |
|
ε |
Emissivity |
[-] |
|
ε |
Wavefront alignment difference |
[rad] |
|
ζ |
Rotation in polar coordinate system |
[rad] |
|
η |
Dynamic viscosity |
[Pa.s] |
|
η |
Local surface slope |
[°] |
|
θ |
Divergence |
[rad] |
|
θi |
Angular position of probe i |
[°] |
|
κ |
Wavefront curvature difference |
[m] |
|
λ |
Wavelength |
[m] |
|
|
Friction coefficient |
[-] |
|
ν |
Poisson’s ration |
[-] |
Nomenclature
|
ρ |
Density |
[kg/m3] |
|
ρc |
Resistivity |
[Ωm] |
|
σ |
Stress |
[N/m2] |
|
σ |
Boltzmann constant |
[W/m2/K4] |
|
σ |
Standard deviation ( = rms) |
[m] or [rad] |
|
σHz |
Hertzian contact stress |
[N/m2] |
|
τ |
Shear stress |
[N/m2] |
|
φ, φt, φκ |
Optical phase error from wavefront tilt or curvature |
[rad] |
|
ϕ,ψ,θ |
Rotations in r,y,z coordinate system |
[rad] |
|
ω
|
Angular frequency |
[rad/s] |
|
Subscript |
Description |
|
|
|
|
|
|
CG |
Centre of gravity |
|
|
P |
Probe |
|
|
R |
R-stage |
|
|
S |
Spindle |
|
|
SUT |
Surface under test |
|
|
Z |
Z-stage |
|
|
Ψ
|
Ψ-axis |
|
|
a,b,c |
In direction of local coordinate system |
|
|
ax |
Axial |
|
|
dl |
Dimensionless |
|
|
e |
Erroneous component |
|
|
enc |
Encircled |
|
|
eq |
Equivalent |
|
|
f |
Frame |
|
|
g |
Floor |
|
|
i |
Index |
|
|
i,j |
Property of part i in direction j |
|
|
isol |
Isolator |
|
|
max |
Maximum |
|
|
mov |
Moving |
|
|
norm |
Normalized |
|
|
ph |
Pinhole |
|
|
rad |
Radial |
|
|
s |
Shield |
|
|
stat |
Stationary |
|
|
t |
True component |
|
|
t-b |
Top – Bottom |
|
tilt Tip – tilt (no axial rotation) tot Total
Nominal
Bulk
Abbreviation Description
ADC Analog to Digital Converter
C Controller
CC Corner Cube (retro reflector)
CL Cylinder lens
DAC Digital to Analog Converter
DOF Degree of freedom FES Focus Error Signal FJP Fluid Jet Polishing FL Force Linearization
FM Fold mirror
FTP Fractional Transferred Power
IF Interferometer
I/O Digital input / output MZC Minimum Zone Circle NA Numerical Aperture
NPBS Non-polarizing beamsplitter
P Plant
PBS Polarizing beamsplitter
PD Photo Diode
PH Pinhole
PSD Position Sensitive Detector PTH Pressure, Temperature, Humidity
PU Pickup
PV Peak-to-valley
QWP Quarter wave plate
SH Sample & Hold
SPDT Single Point Diamond Turning SSiC Sintered Silicon Carbide SUT Surface under test TG Traject Generator
TIR Total Indicator Reading
rms Root-mean-square upr Undulations per revolution
Table of contents
Summary i
Samenvatting iii
Nomenclature v
Chapter 1 Introduction 1
1.1 Aspherical and freeform optics ........................................................................1
1.1.1 Advantages of aspherical and freeform optics.................................2
1.1.2 Applications and trend.....................................................................3
1.1.3 Surface properties and definitions....................................................6
1.1.4 Current manufacturing methods.......................................................9
1.1.5 Current metrology methods .............................................................9
1.2 The NANOMEFOS project............................................................................12
1.2.1 Objective........................................................................................12
1.2.2 Methods .........................................................................................13
1.3 Thesis outline .................................................................................................15
Chapter 2 Conceptual design 17
2.1 Conceptual considerations..............................................................................17
2.2 Machine concept.............................................................................................22
2.3 Error budget....................................................................................................25
2.3.1 Coordinate system definition.........................................................25 2.3.2 Error sensitivity..............................................................................26 2.3.3 Error budget...................................................................................28
2.4 Machine design overview...............................................................................30
2.4.1 Main design aspects.......................................................................30 2.4.2 Machine design overview ..............................................................33
Chapter 3 Motion system 35
3.1 Concept...........................................................................................................35
3.1.1 Requirements .................................................................................35 3.1.2 Basic components and principles used...........................................36 3.1.3 Motion system concept..................................................................43 3.1.4 Dynamic analysis...........................................................................51 3.1.5 Design overview ............................................................................52
3.2 Base................................................................................................................53
3.2.1 Base block......................................................................................53
3.2.2 Vibration isolation .........................................................................55
Table of contents
3.2.3 Transportation................................................................................59
3.2.4 Base assembly................................................................................60
3.2.5 Vibration measurement..................................................................60
3.3 Spindle ...........................................................................................................61
3.3.1 Air bearing spindle ........................................................................61
3.3.2 Product mounting table and intermediate body.............................63
3.3.3 Spindle brake.................................................................................64
3.3.4 Spindle assembly...........................................................................65
3.3.5 Error motion measurement............................................................66
3.4 Z-stage............................................................................................................69
3.4.1 Position frame................................................................................69 3.4.2 Preload frame.................................................................................74
3.4.3 Weight compensation ....................................................................76
3.4.4 Motor, brake and linear scale.........................................................80
3.4.5 Emergency brake...........................................................................82
3.4.6 Z-stage assembly ...........................................................................83
3.5 R-stage ...........................................................................................................84
3.5.1 Position frame................................................................................85 3.5.2 Preload frame.................................................................................88
3.5.3 Z-stage tube bearings.....................................................................91
3.5.4 Motor, brake and linear scale.........................................................92
3.5.5 Cable guidance ..............................................................................94
3.5.6 R-stage assembly...........................................................................95
3.6 Ψ-axis.............................................................................................................96
3.6.1 Air bearing.....................................................................................97
3.6.2 Motor, brake and encoder............................................................104
3.6.3 Probe nulling target .....................................................................105 3.6.4 Ψ-axis mount...............................................................................106
3.6.5 Ψ-axis assembly ..........................................................................111
3.7 Motion system assembly and alignment ......................................................112
3.7.1 Z-stage alignment ........................................................................112 3.7.2 Ψ-axis alignment .........................................................................113 3.7.3 Motion system assembly..............................................................114
3.7.4 Motion system assembly property summary ...............................115
3.8 Experiments and Calibration........................................................................116
3.8.1 Noise level...................................................................................116
3.8.2 Eigenfrequencies .........................................................................117
3.8.3 Stage tilt calibration.....................................................................117
3.9 Motion control..............................................................................................121
3.10 Conclusion ...................................................................................................123
Chapter 4 Metrology system 125
4.1 Concept ........................................................................................................125
4.2 Interferometry system ..................................................................................130
4.2.1 R and Z interferometer concept ...................................................130
4.2.2 Probe interferometer polarization rotation...................................133
4.2.3 Environmental disturbances.........................................................137
4.2.4 Beam layout.................................................................................139
4.2.5 Alignment tolerance analysis.......................................................140
4.2.6 Design and realization..................................................................143
4.2.7 Assembly and alignment..............................................................149
4.3 Upper metrology frame ................................................................................150
4.3.1 Concept........................................................................................151
4.3.2 Thermal analysis..........................................................................157
4.3.3 Mechanical analysis.....................................................................165 4.3.4 Design and realization..................................................................166
4.3.5 Alignment and calibration............................................................172
4.4 Lower metrology frame................................................................................174
4.4.1 Spindle error motion measurement..............................................174
4.4.2 Multi-probe method.....................................................................176
4.4.3 Design and realization..................................................................182
4.5 Metrology system assembly .........................................................................185
4.6 Experiments..................................................................................................186
4.6.1 Stability measurements................................................................186
4.6.2 Metrology frame shift..................................................................189
4.7 Conclusion....................................................................................................190
Chapter 5 Non-contact probe 191
5.1 Concept.........................................................................................................191
5.1.1 Required probe characteristics.....................................................192
5.1.2 Optical absolute distance measurement principles ......................194
5.1.3 Probe concept...............................................................................195
5.1.4 Rough or diffuse surfaces ............................................................199
5.2 Differential confocal analysis and testing ....................................................199
5.2.1 Gaussian beam theory..................................................................200
5.2.2 Normalized dimensionless focus error signal..............................201
5.2.3 Test setup.....................................................................................203
5.2.4 Optimization ................................................................................206
5.3 Design and realization..................................................................................206
5.3.1 Optical design ..............................................................................206
5.3.2 Main optics body..........................................................................209
5.3.3 Objective guidance and actuator..................................................211
5.4 Probe assembly.............................................................................................220 5.5 Experiments..................................................................................................221
5.6 Installation and alignment on machine.........................................................224
5.7 Motion control .............................................................................................225
5.8 Conclusion....................................................................................................230
Table of contents
Chapter 6 Electronics and software 231
6.1 Electronics....................................................................................................231 6.2 Software .......................................................................................................234
Chapter 7 Machine validation 241
7.1 Machine assembly........................................................................................241
7.2 Machine validation.......................................................................................242
7.2.1 Stability measurements................................................................243
7.2.2 Single track measurements..........................................................246
7.2.3 Surface measurements.................................................................247
7.3 Calibration and nulling.................................................................................253
7.4 Achievable uncertainty estimation...............................................................255
7.5 Conclusion ...................................................................................................256
Chapter 8 Conclusion and recommendations 257
8.1 Conclusion ...................................................................................................257
8.2 Recommendations........................................................................................261
|
References |
263 |
|
Appendix A Current fabrication methods |
279 |
|
Appendix B Current metrology methods |
283 |
B.1 Imaging techniques ......................................................................................284 B.2 Scanning techniques.....................................................................................286 B.3 Conclusion ...................................................................................................291
|
Appendix C Brake stiffness calculation |
293 |
|
Appendix D Intermediate body concept |
299 |
|
Appendix E Motion system stiffness and eigenfrequencies |
301 |
E.1 Stiffness measurements................................................................................301
E.2 Eigenfrequency measurements.....................................................................302
Appendix F Interferometer shielding experiments 305
Appendix G Upper metrology frame thermal analysis 307
G.1 Beam deflection ...........................................................................................307
G.2 FEM model ..................................................................................................309
Appendix H Multi-probe method 313
H.1 Axial profile reconstruction .........................................................................313 H.2 Analysis........................................................................................................316
H.3 Optimization.................................................................................................319
|
Appendix I Optical distance measurement principles |
323 |
|
Dankwoord |
327 |
|
Curriculum Vitae |
331 |
