Scuola di Ingegneria
Dipartimento di Ingegneria Civile e Industriale
Corso di laurea magistrale in Ingegneria Aerospaziale
Tesi di Laurea Magistrale
Design of a Rotation Mechanism
for Interferometric Surface
Topography Measurements of a
Sphere
Supervisors
Prof. Giovanni Mengali
M.Eng. Harald Koegel
Candidate
Francesco Mostallino
Abstract
One of the Airbus DS proposals for the different payloads concepts for LISA mission theo-rized the use of a completely full drag free Gravitational Reference System. This solution uses a spherical proof mass which is isolated from any external spurious accelerations but gravity. An optical read out system tracks the motion of the centre of mass and supplies data to a Disturbance Reduction System. The Disturbance Reduction System compels the spacecraft structure to move around the proof mass which is following a geodesy motion. This innovative system requires an accurate, on ground, topography map of the sphere surface with an accuracy of picometers.
This thesis studies and proposes improvements for an existent tool, developed by Air-bus DS in Friedrichshafen, capable of measuring single circumference topography, at nanometres accuracy, of a spherical proof mass via two high symmetric heterodyne inter-ferometers.
The principal efforts of this work are addressed to the Opto-mechanical design of elements and mechanisms in order to supply the necessary hardware which will enable the complete 2D topography of the sphere.
The designed mechanism rotates the sphere, via friction forces, in order to supplies sev-eral circumference topographies spaced by 17 mrad each. The system is automated using stepper motors and optical incremental encoders controlled with LabView interfaces. The proposed idea is then tested, in order to demonstrate the feasibility of this solution which is planned to supply a sphere rotation accuracy in the order of milli − radiant. An important feature introduced by the new machine set-up is the use of a rotating ring mirror, as reference, for the interferometric measurements. This idea will introduce in the interferometer outcome a double topography, at sub − nanometer level accuracy, from which the final proof mass topography will be extracted.
Contents
1 eLISA mission and different payload concepts 3
1.1 Gravitational Waves . . . 3
1.2 eLISA mission architecture overview . . . 4
1.2.1 eLISA orbit . . . 5
1.2.2 eLISA science . . . 6
1.2.3 eLISA sensitivity curve and noise acceleration model . . . 8
1.2.4 ASTRIUM In-Field Point System . . . 9
1.3 Gravitational Reference System . . . 10
1.3.1 GRS proposals . . . 11
1.4 Out of roundness and CoM errors . . . 12
2 1D Surface topography 13 2.1 Interferometer principles . . . 13
2.2 High symmetric Heterodyne interferometer . . . 15
2.2.1 System optical set-up . . . 17
2.2.2 System mechanical set-up . . . 18
2.3 Data analysis and results . . . 20
2.3.1 Mathematical model . . . 22
2.3.2 First results . . . 25
3 Modification for 2D surface topography 27 3.1 Enhancement required . . . 27
3.2 2nd DOF mechanism . . . 28
3.3 Working principles of the mechanism . . . 30
3.4 Reference mirror configuration . . . 38 iii
4 Design phase 41
4.1 Design guidelines . . . 41
4.2 Design process . . . 42
4.2.1 Fixed dimension elements . . . 43
4.2.2 Mechanism size selection . . . 44
4.3 Test mass support designs . . . 47
4.3.1 Monolithic design . . . 47
4.3.2 Mechanism mounting and alignment . . . 50
4.3.3 Reference Ring Mirror support and fixing system . . . 53
4.3.4 Support-Rotational Stage connection system . . . 54
4.4 Real assembly and key-feature sum-up . . . 56
5 Optical integration 59 5.1 New lasers path . . . 59
5.2 Periscope . . . 60
5.3 Adjustable mirror position and mounting . . . 62
5.4 Cylindrical lens and lenses mounting system . . . 63
6 Mechanism test and software integration 67 6.1 Software implementation . . . 67
6.1.1 Components . . . 68
6.1.2 Software logic description . . . 70
6.1.3 Encoder calibration . . . 73
6.1.4 Errors sources and expected behaviour . . . 74
6.2 Slipping evaluation . . . 76
6.2.1 Encoder missing steps correction . . . 78
6.2.2 Principles error sources . . . 80
7 Conclusion and next step 85
A Data-sheet 87
List of Figures
1.1 Strain sensitivity GW experiments. . . 4
1.2 eLISA mission orbit feature. . . 5
1.3 eLISA spacecraft ”beathing” motion. . . 5
1.4 eLISA components science function. . . 6
1.5 eLISA single arm representation. . . 7
1.6 eLISA strain sensitivity. . . 9
2.1 Michelson interferometer schematic. . . 14
2.2 Surface mapping optical set-up schematic. . . 16
2.3 Surface mapping machine structure. . . 18
2.4 Test mass support structure. . . 19
2.5 Reference flat mirrors mounted. . . 20
2.6 Surface mapping machine noise evaluation. . . 21
2.7 Interferometer signal error schematic. . . 23
2.8 Radius topography result. . . 25
2.9 Diameter evaluation result. . . 26
3.1 New measurement logic flow. . . 29
3.2 Friction wheel schematic. . . 31
3.3 Mechanism idea representation. . . 33
3.4 Schematic mechanism components dimensions. . . 34
3.5 Schematic mechanism components forces. . . 35
4.1 Simplified mechanism components. . . 45
4.2 CAD rendering of the complete assembly design. . . 47
4.3 Partial assembly. . . 48
4.4 CAD design alternative modular solution . . . 49 v
4.5 Exploded view mechanism mounting phase. . . 51
4.6 Partial assembly top view. . . 52
4.7 Exploded view RRM mounting phase. . . 53
4.8 Exploded view of the test mass support rotation stage mounting phase. . . . 55
4.9 Picture mounted structure Encoder side. . . 56
4.10 Picture mounted structure. . . 56
5.1 Schematic of the new optic set-up . . . 60
5.2 Working principle of the periscope used. . . 61
5.3 Schematic of the angle described by the laser path in the adjustable mirror reflections. . . 62
5.4 Picture of the designed structure for the adjustable mirrors housing. . . 63
5.5 Complete assembly final hardware for the 2D surface topography machine. . 65
6.1 Encoder software implementation. . . 71
6.2 Example of encoder signal compared with the theoretical one. . . 75
6.3 Slipping evaluation graphs. . . 77
6.4 Slipping evaluation graphs with encoder corrections. . . 79
6.5 Multiple rotations test. . . 80
6.6 Multiple rotation test component phase evaluations. . . 81
6.7 Mean error curve. . . 82
List of Tables
2.1 Dummy sphere characteristics. . . 21
4.1 Real parameters and nominal parameters differences. . . 57
6.1 Micro-stepper controller settings. . . 69
6.2 SPMs characteristics. . . 76
Introduction and outline of the
thesis
The Gravitational Wave detection represents one of the great challenge of nowadays sci-ence. The detection of such phenomena enables the possibility to observe several events and objects around the universe, giving birth to a new powerful branch of the astronomy observation.
The impact of this new field of research has been a unique possibility for the entire science technology community to develop instruments capable to overcome the challenges related to this purpose.
One of the most ambitious experiment is the eLISA mission (born as LISA mission), which proposed to use two Michelson interferometers, of one million Kilometre arm each, orbiting around the Sun. The interferometers will be able to measure the distances be-tween free falling objects, that are following undisturbed geodetic orbits around the Sun, reaching a strain sensitivity of 10−201/√Hz at frequencies between 10−5Hz and 10−1Hz. In order to isolate the free falling objects from any external disturbances an ultra accurate noise compensation system is needed. Part of this system is in charge of tracking the motion of the centre of mass of the objects inside the housing.
Airbus DS analysed different configurations for the payload concepts, one of these concerns the possibility to use a total drag free GRS with whole optical read out. This solution uses spherical proof masses and avoid any active forces.
This possibility is enabled by the use of a 2D topography map of the sphere surface at picometre accuracy level. Airbus Friedrichshafen started developing a machine capable of recording 1D circumferences topographies of a sphere similar to the one designed for
LISA-like GRS.
The work in this thesis is addressed to enhance the capabilities of the machine and prepare the hardware to record a full 2D surface topography.
A short introduction about the eLISA mission and the different payload proposals with special emphasis on the GRS different solutions and errors is given in chapter 1.
Chapter 2, starting from the necessity of a surface topography for the total drag free GRS, shows the working principles and the set up of the machine for the surface characterization developed at Airbus. The final part of the chapter illustrates the first results concerning a single circumference topography.
Chapter 3 illustrates the solutions proposed for the 2D topography, exploring the impact of these choices on the instrument working principles.
The proper requirements and tasks described in Chapter 3 are the fundamentals for the design phase largely explained chapter 4.
The optical integration of the new designed elements is presented in Chapter 5. The end of this chapter shows a rendering of the definitive hardware of the machine for the 2D topography.
Chapter 6 explores the feature of the mechanism automation studying the feasibility of the system designed with data regarding the real SPM motion.
Chapter 1
eLISA mission and different
payload concepts
The first part of this chapter presents the mission target, the gravitational waves detection and the mission concept proposed in the past years.
The second part will show the alternative payload concepts, focusing in particular on the alternative ideas for the Gravitational Reference System, in order to introduce the reader to the thesis scope.
1.1
Gravitational Waves
The Gravitational Waves are ripples in the spacetime curvature that propagate, at the speed of light, as waves, with certain amplitudes and certain frequencies [1]. The sources of this waves are the asymmetrical accelerations of masses that cause changes in the geometric structure of the spacetime. The spacetime, hit by the gravitational waves, ex-perienced a stress-strain on its structure. The amplitudes of this waves are very small and difficult to be appreciable with state of art technology, in fact in order to be detected they need huge masses involved.
Taking into account the difficulties related to GW detection, these phenomena made possible to study extraordinary events and objects around the universe, giving birth to a new and powerful branch of the astronomy observation, which aims to use GW to collect data about astronomical events such those visible in fig. 1.1.
Figure 1.1: The picture relates the strain sensitivity of various experiments to which events are possible to observe. SOURCE [3]
The GR150914 is the merging of two black holes and is the first gravitational wave ever detected [2]. This result has been achieved by the twin LIGO stations (on-ground experiment), on 14 September 2015.
The most recent and practical space mission proposal for GW detection is the eLISA (evolved Laser Interferometer Space Antenna) mission, carried out by the ESA (European Space Agency).
The original project, LISA, was a join mission between ESA and NASA, but after NASA departure, in 8 April 2011, ESA scaled down the project in order to adapt the costs for European budget.
The eLISA mission proposes a gravitational wave observatory which has been selected for the L3 ESA Cosmic Vision Program, with a planned launch date in 2034.
1.2
eLISA mission architecture overview
The fundamental idea of the experiment is to measure the distance changes among three free falling objects via two long baseline Michelson interferometers.
1.2. ELISA MISSION ARCHITECTURE OVERVIEW 5
1.2.1 eLISA orbit
The constellation is composed by a Mother spacecraft, that serves as the central hub and defines the apex of a V. Two other, simpler spacecraft Daughter are situated at the ends of the V-shaped constellation.
The swarm of satellites orbits around the Sun forming an equilateral triangle, spaced by 1 million Kilometre. The constellation barycentre is shifted from the Earth centre about 10◦ - 30◦. Every component rotates around the orbit path with a period of one year. In fig. 1.2 a picture of the orbit.
Figure 1.2: eLISA mission heliocentric orbit. SOURCE [4]
The constellation performs a ”breathing” movement of about 1◦ in the angle formed by two sides of the space triangle, fig. 1.3.
Figure 1.3: The graph shows the variation of the inner angles between S/C due to orbital mechanics over the mission time. SOURCE [5]
In order to follow the breathing motion of the Daughters the central spacecraft, the Mother, has a moveable structure which permits to address the laser lights.
1.2.2 eLISA science
The constellation of the three spacecraft constitutes the science instrument. The Mother houses two send/receive laser while the Daughters one each. In fig. 1.4 the constellation is shown, highlighting the spacecraft science components.
Figure 1.4: The constellation of the three spacecraft constitutes the science instrument. The mother S/C has double optical instrument. The laser in the end spacecraft is phase-locked and sent back. SOURCE [1]
The direct reflection of laser light, such as in a normal Michelson interferometer, is not feasible due to the large distance between the spacecraft. Diffraction widens the laser beam so that for each Watt of laser power sent, about 250pW are received. Therefore, lasers at each end of each arm operate in a transponder mode.
A laser beam is sent out from the Mother to the Daughters. The laser in the Daughter is then phase-locked to the incoming beam thus returning a high-power phase replica. The returned beam is received from the Mother and its phase is in turn compared with the phase of the local laser.
1.2. ELISA MISSION ARCHITECTURE OVERVIEW 7 The change in the arm length causes a Doppler shift of the laser phase which can be associated to gravitational wave signals.
For practical reasons, this measurement is divided into three distinct parts: the mea-surement between the spacecraft optical benches and the meamea-surement between each of the test masses and its respective optical bench. By combining the three measurements, the distance between the test masses is reconstructed and kept insensitive to the noise in the position of the spacecraft with respect to the test masses. Fig. 1.5 shows a single arm measurement.
Figure 1.5: The green objects are the lasers sources. The orange boxes are the test masses in free fall. The blue dots indicate the phase detectors and the red lines the lasers path. SOURCE [1]
The measurement between the free falling test masses and their local optical benches provide, also, the GRS (Gravitational Reference System).
The eLISA GRS (mother S/C) has two cubical test masses, of 46mm length, made out of a gold platinum alloy, shielded by external disturbances (magnetic fields, solar wind, interaction S/C-test mass). One of these masses is used as inertial reference and the other one is carried by the first. The single cube is not totally in a drag free mode. Its attitude, around the axes perpendicular to the measurement one, is controlled in order to avoid wrong laser reflections. The test mass is, also, electrostatic suspended. The movements on the measurement axes and the rotations around it remain in free falling mode.
The read out is a mix of capacitive sensors for the non-measurements direction (nanometer accuracy) and optical sensors for the science measurement (picometer accuracy).
Whereas the interferometric measurement system allows to measure the distance be-tween the test masses to picometer accuracy, the DRS (Disturbance Reduction System) is responsible to render these measurements meaningful, as it ensures that the test masses follow gravitational orbits as much as possible.
The system consists in several µN thrusters which, combined with the GRS data, moves the S/C around the test mass in order to reducing the spurious accelerations [6].
1.2.3 eLISA sensitivity curve and noise acceleration model
The sensitivity is, conceptually, the inverse of the transfer function of the instrument that it represents. The sensitivity tells how small the signal might be, in order to be still ap-preciable from the detectors [1].
Data about the instruments sensitivity permits to set the acceleration noise levels, essential data to write the requirements for the GRS and for the DRS.
From [6] it is possible to write the strain noise spectral density:
δh
h = TT DI q
(δLmeas)2+ ((2 π f )δa 2)2
L (1.1)
The term TT DI represents the science payload transfer function which is the conversion of
single-link position uncertainties into detector strain response, including geometry factors in order to eliminate higher noise values in the measures and TDI technique (Time Delay Interferometry) developed by JPL [7].
The terms δLmeas and δa represent, respectively, the fluctuations in arm measurement
length and spurious accelerations of the test mass inside the housing due to disturbances. The term L is the nominal arm length.
The sensitivity curve designed for the experiments, results in the slope visible in 1.6.
The sensitivity curve has a quasi-flat bottom level of 1.3×10−20√Hz between 3×10−3 Hz and 3 × 10−2 Hz. For frequencies below 3 × 10−3 the curve rises as a f−2 function.
1.2. ELISA MISSION ARCHITECTURE OVERVIEW 9
Figure 1.6: eLISA strain sensitivity curve. The colours indicate the difference between the requirement (red) and the goal (blue). SOURCE [1]
1.2.4 ASTRIUM In-Field Point System
As a part of the LISA formulation study, under ESA contract, EADS Astrium (now Air-bus DS) proposed a different payloads concept for the LISA mission (technology under validation study for eLISA).
The so-called ”In-Field Pointing” architecture [8] is based on different concepts of payload that allow simpler and reliable solutions. Here the key-features:
• No steering structure.
Moveable mirrors within the optical bench-telescope laser path instead of moveable telescope structure. The benefits of this idea are a simpler method for the ”breath-ing” movements of the telescope which permits to have a simplified structure and a full on ground testability. The disadvantages are related to the topography charac-terization of the rotating mirror surface.
• One proof mass.
It could be possible to keep just one test mass, placing it in the SC centre. In this configuration the test mass reflects the two incoming laser beams, using the same proof mass for the GRS. The advantages are a potential redundancy, placing a second system in case of failure of the designed one.
The weakness is the most demanding necessity to ”control the attitude” of the test mass in order to properly reflect the incoming laser beams.
• Better drag free system accuracy.
According to the geometry selected for the test mass, a completely full drag free-mode, without electrostatic alignment, is enabled, allowing a more accurate GRS.
1.3
Gravitational Reference System
The first major studies about drag free inertial gravitational systems are attributed to Lange in 1964 [9] and then DeBra in 1974 [10] with the DISCOS system mounted on the TRIAD spacecraft, which applied the drag free concepts to realize a successful experiment. Mission such as GP-B [11] or GRACE used similar devices concepts.
The basic concept is to provide a device in which a test mass is shielded from all the external disturbances but gravity, allowing a perfect geodetic motion. Any movements between the test mass and its housing represent input for the SC attitude control. Basically, from an inertial point of view, in a perfect drag free system the movements of the test mass inside the housing are movements of the S/C toward the test mass.
Generally the GRS configurations are classified in pure drag free and accelerations modes.
The acceleration-modes, unlike the pure drag free mode, use the test mass forced to a pre-set position by active forces and control its attitude by an inner control loop. The efforts spent to stabilize the position of the test mass represent the input for the outer S/C control loop [13].
This kind of configurations might have one or more drag free degrees of freedom. An example is the ongoing GRS belonging the eLISA mission that uses controlled degrees of freedom for rotations, which axis are perpendicular to the measurement one.
1.3. GRAVITATIONAL REFERENCE SYSTEM 11 The ongoing mission LISA pathfinder (technology test for the LISA-like GRS) shown that the inertial sensors fulfil the high-level requirements in differential acceleration noise, specified to be 3 × 1014 ms−2/√Hz, around 10−3 Hz [27].
1.3.1 GRS proposals
Four different configurations for LISA-like GRS have been proposed in [13].
Configuration 1 is similar to the ongoing eLISA (mother S/C) concept with 2 cubic masses, one acting as GRS and the second one carried by the other. Optical read out for science measurement and capacitive read out for the other degrees of freedom. This configuration necessities, also, of an electrostatic suspension and a moveable telescope as-sembly.
Configuration 2 is already contained within the ”In Field Pointing architecture” [8], using one cubic mass, total optical read-out, moveable mirrors inside the telescope assem-bly and forced electrostatic orientation in order to do not spread the reflection of the laser beams.
Configuration 3 uses a spherical test mass that permits to eliminate completely the active electrostatic suspension because its shape avoid reflection lasers issues.
The test mass is manufactured with an hollow shape in order to have I1= I2≈ 0.9 I3 and
it is spun up at 10 Hz, above science frequency band, in order to spectrally shift eventual out-of-roundness errors.
Configuration 4 is similar to configuration 3. For this option there is no active spin up and the sphere does not present a difference in inertia moments. In this case the out of roundness problems are overcome by an highly precision on-ground map that permits to let completely free the test mass inside the housing, enabling a complete pure drag free gravitational system.
Configurations 3 and 4, the ones with spherical test masses, have also the possibility to decouple the attitude control, since the attitude of the test mass does not influence any more the output signals.
1.4
Out of roundness and CoM errors
In the target of GRS the motion of the CoM (Centre of Mass) is the one to be followed. Material density inhomogeneities introduce position uncertainties and the GC (Geometric Centre) differs from the CoM. These results in false displacements signal output.
Taking into consideration the materials proposed for the SPM in [13], alloys of Au-Pt or eventually Cu-Be, the actual manufacturing processes limit the CoM position uncer-tainty to ≈ 1 µm. Advanced manufacturing processes and accurate measures might lower the CoM uncertainty value to ≈ 100 nm. These CoM measures are explained in [12] where is illustrated also the manufacturing techniques for the test mass of configuration 3.
The solution adopted for Out-of-roundness problems is, for configuration 3, a frequency spectral shift of surface errors by spinning up the SPM at a frequency that permits to over-come the real shape.
Instead, for configuration 4 an highly accurate topography surface [20] (picometer ac-curacy) of the SPM, should permits to distinguish real displacements from surface profile. This will be used for analysis and correction of the measurement data during the science mission.
Chapter 2
1D Surface topography
This chapter explores the achievements reached in the surface topography of the spherical proof mass.
In the first part of this text a brief introduction to interferometer basics is given in order to understand the working principles of the machine developed at Airbus for the Surface characterization.
In the second part the machine and its key-features are illustrated, ending showing the measurements done. This part of the chapter is a synthesis of the work done in [20].
2.1
Interferometer principles
Interferometry is a family of techniques in which electromagnetic waves, are superimposed in order to extract informations, for example, about distances between objects. The same technique is used for eLISA gravitational waves detection.
Interferometry systems are generally classified as either homodyne (single-frequency) or heterodyne (two-frequency) systems.
In the figure 2.1 a basic homodyne Michelson interferometer scheme is illustrated. The laser source, on the left, sends a laser to the beam splitter, in the centre of the picture. The laser in then split into two identical beams. Beam B is reflected by the movable mirror, while beam A by the reference mirror. The passage, after reflection, in the BS (Beam Splitter) divides each laser in two identical beams directed to the photo detector. The two electric fields constructively (or destructively) interfere when they overlap, de-pending on the relative path lengths of the two arms. Changing the relative path lengths of the arms causes a relative phase change, which is detected by the measured radiance. The photodiode converts incident optical power into an electrical current. Photodiodes
behave in a manner similar to a low-pass filter, based on the photodiode area [15].
Figure 2.1: Simple Michelson interferometer scheme.
The two electric fields of lasers A and B can be expressed as:
EA= E0ei(f t+ ~K~xA+φA) (2.1)
EB= E0ei(f t+ ~K~xB+φB) (2.2)
Where E0 is the amplitude of the wave, f is the wave frequency, ~K is the propagation
vector, ~x is the position vector and φ is the initial phase.
Considering now the current on the photo detector proportional to the average value of the two electric fields:
I ∝ |EA+ EB|2 (2.3)
Supposing same frequency f and same initial phase φ the current I is proportional to:
2.2. HIGH SYMMETRIC HETERODYNE INTERFEROMETER 15 The last equation shows the relation between the detected signal I and the different mirrors positions ∆r = rB− rA.
This concepts applies in many fields, for example, dilatometry or surface topography. In [16] an heterodyne interferometer works either for dilatometry measures either for sur-face characterization.
The experiment for the new definition of the Kilogram used a Fizeou interferometer for the diameter estimation of a Silicon sphere reaching the sub-nanometer accuracy level [18]. The purpose of the experiment was the determination of the number of atoms contained in the sphere in order to define a new Avogadro constant.
The GRS system of Gravitational Probe B, GP-B, used spherical proof masses polished and characterized by a tactile instrument reaching the nanometre accuracy level [19].
2.2
High symmetric Heterodyne interferometer
In heterodyne interferometry, a signal of interest at frequency f1is mixed with a reference
laser beam at different frequency f2. The desired outcome is the frequency differences,
which carries the information of the original higher frequency signal, but is oscillating at a lower, more easily processed, carrier frequency.
The tool developed at Airbus DS for spherical surface characterization is a symmetric double heterodyne interferometer [20].
This application used an enhanced version of the interferometer set-up cited before, con-cerning the dilatometry [17].
In the fig. 2.2 there is a schematic of the symmetric double interferometer.
The SPM is placed in the centre of the set-up and rotates due to a RS (Rotational Stage) A, while the reference mirrors are fixed and are decoupled from the SPM mo-tion. During the rotation the SPM topography induces changes in the measurement beam length, which, compared to the fixed value of the reference path gives the topography signal.
The resultant current I, with λ as lasers wavelength, is proportional to:
I ∝ cos(2π
λ∆r) (2.5)
Finally, by referring to the centre of the set-up, ∆r = rref − rmeas is the distance
between the local radius of the SPM and the reference mirror hit point.
Figure 2.2: Schematic of the optical set-up. After the Koestner prism the laser at 78,01 MHz is divided in reference (blue) and measurement (red) beam.
The topography obtained is not the absolute radius value of the SPM, but instead, is a differential distance. The knowledge of an absolute measure is enabled only with the knowledge of the position of the reference mirror with the same accuracy used for the differential distance.
2.2. HIGH SYMMETRIC HETERODYNE INTERFEROMETER 17
2.2.1 System optical set-up
Each interferometer in fig. 2.2 utilizes a Nd:YAG Non Planar Ring Oscillator laser (In-nolight Mephisto NE500) with a laser wavelength of 1064 nm. The Heterodyne frequency of 10KHz is generated by AOM (Acoustic-Opto Modulators) 1, which shift the original laser frequency to 78, 00M Hz and 78, 01M Hz. These frequencies belong to the local laser (pale blue) and principle lasers (purple).
The use of KP (Koestner Prism) enable the sub-division of each lasers in two identical beams.
The local lasers ahead toward a BS, passing first through a couple of AM (Adjustable Mir-rors) which permits to, manually, correct the laser direction tilting the reflective surface of each AM thanks to a mechanism inside their supports. Finally the lasers arrive to the QPDs (Quadrant Photo Detectors)2, where they will be mixed with the other lasers. The paths of the reference (blue) and measurement (red) beams point toward, first a PBS (Polarized Beam Splitter) which addresses the lasers outside the interferometer to a couple of AM. The lasers are then directed to the SPM and the reference mirror.
A spherical lens, with focus length of ≈ 100mm, is used for the measurement beam in order to focus the laser onto the SPM. The measurement beam diameter at the SPM is Dmeas= 175µm.
After back-reflections the beams repeat the same path back to the PBS, which this time addresses those to the QPDs.
In each QPD the reference and measurement beams are mixed with one of the local laser beams to enable the interference and then each signal is multiplied and processed in order to extrapolates the desired outcome.
1
Acousto-optic modulators are devices which allow the frequency, intensity and direction of a laser beam to be modulated. Within these devices incoming light Bragg diffracts off acoustic wavefronts which propagate through a crystal. Modulation of this incoming light can be achieved by varying the amplitude and frequency of the acoustic waves travelling through the crystal, [21].
2
The sensible surface is divided in four quadrant in order to be able to identify the angle direction of the laser due to different power signals felt in each quadrant. In this set-up, however, the angle identification has not been used.
2.2.2 System mechanical set-up
In fig. 2.3 the machine structure is visible. A main aluminium support houses the ZERO-DUR 3 cylinders that connect the Interferometers chassis. The optics elements are fixed inside aluminium chassis, IFO1 and IFO2.
The AM supports are mounted directly on the interferometer chassis with bolt connec-tions.
The spherical lenses, L1 and L2, are mounted on xyz linear stages able to shift on their plate support.
Figure 2.3: Machine structure with dummy sphere at centre of the set-up.
All the set-up is placed in a vacuum chamber closed by a thick glass cover. Outside the vacuum chamber, at ambient laboratory condition, are placed the Heterodyne frequency generator and the data analysis system.
3
Zerodur, is an inorganic non-porous lithium aluminium silicon oxide glass ceramic characterized by a Coefficient of Thermal Expansion (CTE) of 2 × 10−8K1.
2.2. HIGH SYMMETRIC HETERODYNE INTERFEROMETER 19 Test Mass Support
The test mass support, in fig. 2.4 is fixed on the RS and rotates with this. The material used for the test mass support structure is stainless steel.
A key-feature of this test mass support is the possibility of, manually, shifting the initial position angle of the sphere, φSP M, by a rotative mechanism inside the support.
The positions used are φSP M = 0, π/3, 2π/3, π, 4π/3.
This, added, degree of freedom is used to perform various set of measures with different initial angle values in order to recognize and suppress periodical errors in the measures.
Figure 2.4: Test mass support with dummy sphere positioned on top. The little spheres in the base plate are part of the mounting system.
Reference mirrors
The reference mirrors are flat surfaces of BK7 mounted on a ZERODUR support, fig. 2.5. The plate support and the reference mirrors have the possibility of being oriented using spring connections and flexure hinges. The hole in the centre of the ZERODUR plate is the space for the SPM. This reference mirror support is fixed on the main aluminium support and is independent from the RS and SPM motion.
Figure 2.5: Fixed flat mirrors with their ZEROUR support plate.
2.3
Data analysis and results
The dummy sphere chosen to get experience for the eventual eLISA spherical test mass is a off-the-shelf sphere made of Tungsten-Carbide used for large ball bearing, quality Class G20 according to DIN5401. The sphere characteristic are illustrated in the table below 2.1.
2.3. DATA ANALYSIS AND RESULTS 21
Nominal Diameter 40,0mm
Diameter Fluctuation 0,5 µm
Roughness 0,032 µm
Mass 0,4963 Kg
Table 2.1: Dummy sphere characteristics.
The experiment is performed in vacuum at P ≈ 0, 1 P a, in order to avoid air fluctua-tions which modify the diffraction environment constant.
A first noise evaluation in fig. 2.6, done running the machine without RS motion, shows that the accuracy level of this set-up is extremely high, reaching picometer level at high frequency (0.4 × 100Hz < f < 100Hz). The noise level, for frequency > 10−3Hz remains in the sub-nanometre length order. This graph demonstrates the high symmetry of the two interferometers.
Figure 2.6: Surface mapping machine noise evaluation. This noise is mainly attributed to thermal fluctuations and set-up’s electronic components. ADAPTED [20]
Single interferometers have higher accuracy, reaching a sensitivity of ≈ 1 pm/√Hz [23]. The noise level in the graphs is higher due to the all set-up structure.
2.3.1 Mathematical model
A mathematical model is developed in order to overcome accuracy limiting factors:
• Eccentricity of the SPM rotation axis.
• Rotational stage noise.
The correct position of the SPM centre, CSP M, in the machine centre, CIF O, (where
the measurements beams are supposed to encounter) and the correct alignment of the SPM axis with the RS axis is a fundamental process. Eccentricities result in false dis-placements.
The two principal eccentricities are introduced by the static offset ~c, between RS and CIF O, and by ~d, between CSP M and RS axis.
The alignment reduces the offsets between SPM and interferometer centre to < 20µm.
A secondary noise source derives from the RS bearing. The contacts of the rolling ele-ments of the bearing introduce offsets during the revolution, probably due to a non perfect shape of the elements. Since the reference mirrors are independently fixed from RS and SPM, these offsets results in relative motion and consequently in false displacements. The vector ~s is the error movements of the rotation axis caused by the ball bearing, with ~s = ~sse+ ~sae, where ~sse represents the synchronous error movements, which figure in a
constant shape every revolution, and ~sae represents the asynchronous error movements,
which are basically random. This offset, unlike ~d and ~c, is dynamic and compares when the RS start moving.
The sum of all this offsets is ~m and represents the offset between CSP M and CIF O.
2.3. DATA ANALYSIS AND RESULTS 23
Figure 2.7: Interferometer signal error schematic. ADAPTED [20]
Considering φ the RS rotation, for a perfect sphere of radius R the interferometer signals are related to the error ~m with:
R2 = [±x1,2(φ)~ex− ~m]2 (2.6)
Introducing the topography ρ(φ, θ) as the spherical integral of a sphere surface:b
R2 = 1 4 π Z π 0 Z 2π 0 b ρ2(φ, θ) sin(θ) dφ dθ (2.7)
Considering θ = π, for this measures, the interferometer signals finally are: ± x1,2(φ) ≈ (ρ( π 2 − φ + ( π 2 + cy R) + φSP M) + ~m)~ex (2.8)
In the dependences of ρ appears cy
R which derives from the misalignment between RA
and CIF O and also φSP M which corresponds to the initial angle selected by the mechanism
in the test mass support.
The RS errors can be considered small s ≈ 1 − 2µm, while d and c are in the order of ≈ 10 − 20µm.
Performing a Taylor expansion of eq.(2.8) and considering, as a first step, the dynamic RS errors s ≈ 0 it is possible to define 1,2 as a periodical error function representing the
error due to ~d and ~c. 1,2(φ) = D eiφd× ±x 1,2(φ) E = ±d 2e i(−φd± cy R)+ dO(cy R) 2 (2.9)
Where φd is the angular error due to the eccentricity ~d.
The cx errors are eliminated considering the symmetry of the interferometers signals. Is
possible to obtain this value via linear combination of the two signal x1,2.
Taking eq.(2.9) the offsets expressions are: cy ≈
R
2[arg(1) − arg(2)] (2.10)
d ≈ |1| − |2| (2.11)
φd≈ −arg(1− 2) (2.12)
The equations of the dominant errors lead to a correct interferometer signals xc1,2:
xc1,2(φ, φSP M) = x1,2(φ, φSP M) ± d cos(φ − φd) +
[cy+ d sin(φ − φd)]
2R (2.13)
The errors of the RS ~s are evaluated comparing the interferometer signals with different φSP M (initial angle shift) value:
4sx(φ) ≈ xc1(φ, 0) − xc2(φ, 0) + xc1(φ, π) − xc2(φ, π) (2.14)
While the sy and sz are evaluated considering the function:
σ(φ) = sy(φ)cy
R +
sz(φ)cy
R (2.15)
These errors are acting during the rotation modifying the offset of the RS to the in-terferometer centre, for this reason they are evaluated in the cy
R component.
Unfortunately the shape of the σ function, after operation, remains ambiguous which leads to a solution without high frequency contents.
The final topography expression, on the basis of the recorded signal x1,2, is:
ρ1,2(φ) =ρb1,2(φ, φSP M) − R ≈ x
c
2.3. DATA ANALYSIS AND RESULTS 25
2.3.2 First results
The first milestone reached is the topography of a circumference of the dummy sphere at nanometre accuracy level. The experiment used an SPM rotation velocity of ≈ 1, 2◦/sec and a sampling frequency of 20Hz, resulting in a resolution of ≈ 0, 06◦.
In fig. 2.8 the final ρ1,2(φ) function is shown. The ρ plotted is a mean value among the five
set of multiple measure. Each set of measure starts from the same initial angle positions φSP M, set by the mechanism inside the test mass support. Each set of measure is averaged
resulting in a standard deviation < 15nm.
Figure 2.8: Radius mean value. This average is evaluated over four set of measures each of that with different initial angle. Every set of measure has ≈ 80 single measures. The standard deviation of each single measure is generally < 15nm.
The double interferometers signals can be used in order to minimize the errors and reduce the standard deviation of every measure as seen in fig. 2.9.
Figure 2.9: Diameter evaluation, obtained summing the two mean radius values obtained separately from each interferometer.
The standard deviation of each set of measure, using the opposite interferometer signal is lowered to < 7, 9nm, albeit losing the topography. In this data the two signals from the interferometers are indistinguishable because the references are loose. The measure represent a mean diameter of the sphere, while the purpose of the machine should be the measure of the distance between a fixed element and the local radius of the sphere. However this measure is not useless for the GRS, since it can be used for calibration of the set-up in space, as detailed described in [14].
Chapter 3
Modification for 2D surface
topography
This chapter, starting from the results shown in chapter 2, explores the requirements for a 2D topography and formulates solutions.
The solutions proposed are conceptually studied in order to understand their impact on the machine working principles and the practical requirements arising for the design phase.
3.1
Enhancement required
In order to achieve the final result of a complete 2D surface topography a set-up modifi-cation is necessary. The principal requirements are listed below:
• Two independent degrees of freedom for the SPM. • Faster and automatic experiment session.
• Reference mirrors different configuration.
The two independent degrees of freedom enable a complete visibility of the surface by the laser spot. This is a primary task for the new set-up.
A second request is the necessity of faster and automatic experiment session.
The topography results in fig. 2.8 and 2.9 are averages of five sets of measures with dif-ferent initial angles. Each set is an average of ≈ 80 single topographies. Also, the initial angle shift is a manual process, which leads to open the vacuum chamber four times during
the whole topography session.
The process described lasts several hours and is obviously inapplicable for a 2D surface measurement, which counts a number of points to be measured largely higher than the 1D case.
A large test session time may introduce noise in the measures. A faster option for the single measurement is required. A solution which does not demand such high rotations number for the average may decreases, consistently, the time for each single topography. Also an automatic process, without the need for opening the chamber decreases drastically the execution time. When the vacuum chamber is opened, the experiment environment changes and the vacuum needs to be re-established. This operation is time expensive and an homogeneous environment for every experiment session is recommended.
The third request, for this new set-up, is a different configuration for the reference mirrors. During the measures taken for the 1D topography, it was noted a relative mo-tion between reference mirrors and SPM, caused by noise introduced by the RS. Part of these errors are suppressed by the mathematical model in post processing, anyway different arrangements may lead to higher accuracy levels without the need for correction functions.
3.2
2nd DOF mechanism
A second rotation, around an axis perpendicular to the RS one, provides an independent motion variable for the SPM. The two rotations, provided by the RS and the mechanism of the new set-up, allow different data keeping modes depending on the duty cycle assessed for the set-up. This two rotations could act ”one per time” or ”simultaneously”, allowing with both of them the possibility to cover all the sphere surface.
The requirement about ”points density” derives from the request of covering the whole sur-face of the sphere with the laser spot. The laser diameter on the SPM is dmeas≈ 0.175mm
and the dummy sphere has a radius of 20mm, thus the angle distance between two radius vectors of the sphere should be less than ≈ 0.017rad.
The selected mode is the ”one per time rotation”. With this configuration the surface topography becomes a set of single great circle topographies. The primary rotation φ is acting during the interferometric measurements while the θ is the rotation of the sphere in between and is not necessary to be registered by the interferometers.
3.2. 2ND DOF MECHANISM 29 The rotational stage is selected as principal DOF, that means that is the one which escort the SPM during the single topography capture. The 2nd DOF mechanism is responsible for rotating the SPM after every single topography session.
In fig. 3.1 the logic flow illustrates the loop process and an example of measures collection.
Figure 3.1: New measurement logic process and whole surface topography example. The angle between the single topographies is not the designed one for clarity of the picture.
The alternative mode, the ”simultaneous” rotations, would have used two contempo-rary rotations. This would have drawn a tight spiral representing the surface topography. This alternative choice seems to be faster than the designed idea, however the 2nd rotation would have had similar values of accuracy and repeatability given by the RS. This would have highly complicated the mechanism design process, requiring performances which can be avoid.
The designed topography duty cycle permits more relaxed requirements for the second motion accuracy and stability.
A drawback of the ”one per time rotation” method is the different points density near the poles (φ = ±90◦; θ = 0◦, 180◦). More points are registered in those areas and the density decreases with the latitudes.
The data collected may also, be represented in a iso-density region. A Cartesian frame with φ, θ and ∆r (the differential radius found) is an iso-density point representation of the surface topography, since for every couple φ and θ there is a unique value of ∆r.
3.3
Working principles of the mechanism
The task of the mechanism is to enable a second rotation of the SPM, but primary is the knowledge of the SPM position with the highest accuracy possible. A deep study of the system working principles is essential in order to understand the accuracy limits.
The kind of motion required would have perfectly matched with a system of spur gears, or shaped rack, directly between SPM and mechanism. Unfortunately, since the SPM can not be shaped for optical and functionality reasons, the direct gear solutions are inopera-ble.
The solution proposed is similar to the so called ”frictions wheels”. Friction wheels found place especially in past applications of power transmission between rotating elements. Basically a rotating wheel, in contact with another wheel with parallel rotation axis such those in 3.2, has a kinematic constraint on the rotation increments.
The friction forces acting on the contact point drag the driver wheel but, in the meanwhile, the opposite force on the carried wheel make it follow the rotation. The contact point seen
3.3. WORKING PRINCIPLES OF THE MECHANISM 31 from the two elements has the same speed, which can be related to the angular rotation using the rigid body equations.
Figure 3.2: Friction wheels working method and forces exchanged.
The friction between two surface is composed by two major contributions: macroscop-ically by surface contacts, which create shear stresses, and microscopmacroscop-ically by electrostatic and bonding between molecules.
When two elements, at rest, are in contact the molecules between the surfaces form elec-trostatic bond connections. One element is able to move with respect to the other if it is subjected to a force higher than the whole friction resistance. During the motion the electrostatic bondings are remarkably less because these phenomenons are too weak to form.
This phenomena set the well known difference between static friction and dynamic friction: µstatic> µdynamic (3.1)
The friction is then a phenomena in which two surfaces share stresses till the relative force applied on one of them is higher then all the resistance forces acting on the surfaces in
contact.
The application limits are set principally by the slipping phenomena in which the con-straint about rotations are not respected and the knowledge of angular positions becomes difficult to identify.
These events can be predicted, macroscopically, using a simple mathematical model. The design of the mechanism is largely influenced by the slipping evaluations which limits the values of torque that might be released in the system.
The tangential forces exchanged at the contact point ought to be less or equal than the friction static forces. When the transmitted force Ft is higher than the friction force the
wheels start slipping. This phenomena does not permits to use the kinematic constraints, which means that all the data about second wheel position, velocity and acceleration are not available any more.
This model considers a constant friction contribution over the whole surfaces. It does not take into account possible local slipping phenomena due to local surface status.
In power transmission application, generally, the wheels are pressed each other in or-der to increase the normal forces and consequently higher the friction force. Another used option is to increase the friction value, using high friction material for surface cover.
The solution proposed uses the same principles of the friction wheels. The mechanism concept in Fig. 3.3 represents the SPM and two wheels, the cylindrical red one is called Driver and the symmetric conical blue one is called Carried.
The conical shape is chosen in order to accommodate the spherical geometry, providing three contact points which guarantee a stable position, ensuring a geometric symmetry. From a kinematic point of view the ”carried” contact points would have also belonged to a static structure and the SPM would have, theoretically, slide on them. However a rotating element helps the rotation avoiding the possibility of damages of the SPM surface and could also be used as an additional reference for the SPM angle position, due to the same kinematic principles illustrated before.
In fact the SPM rotation can not be directly measured, the application needs indirect proofs. Since the motor stage drives the Driver, the commands sent to the motor are a possible indirect measure for the SPM position. Another useful data may be the carried
3.3. WORKING PRINCIPLES OF THE MECHANISM 33
Figure 3.3: Mechanism representation with test mass (green), Driver (red) and Carried (blue).
wheel angle position. The conical wheel is directly carried by the SPM. This supplies a powerful measure about the sphere rotation because is the proof that the SPM is rotating.
The motor transmits a torque τ , forcing the Driver to rotate. It represent the only system input. The outputs are the direct measures which is possible to register. Those data are related to the SPM rotation, which is the goal of this application.
A brief mathematical analysis is developed with the purpose of geometrical and physi-cal characterize the system and finds relevant geometric variables to be taken into account during the mechanism design phase.
Geometrical characterization
The geometrical characterization intends to study the radius ratio values between each mechanism component. These ratios are related with the contact points among the el-ements. While the cylinder-sphere ratio finds a quite simple solution, the contact cons-sphere needs to be derived.
this suggests that the real contact points between sphere and conical wheels belong to planes shifted from the contact circumference Driver-sphere.
The contact radius are called rd, rc, dc. They represent the distance between each element
rotation axis and contact point. The angles θd and θc are the position angles of the
ro-tation axis of each element with respects to the sphere centre that is taken as frame centre.
Figure 3.4: Left picture is a profile view of the mechanism with illustrated the geometrical variables characterizing the system.
Defining ψc the aperture angle of the cons and rmin as the minimum radius of the
conical shape, geometric relations among the characteristic lengths of the cons can be found:
tan(ψc) = rtm cos(θcl) cos(ψc)2
tan(ψc) = ∆rl
rc= rmincos(θcos(θcc))+∆r
(3.2)
The value of the contact radius of the Carried is:
3.3. WORKING PRINCIPLES OF THE MECHANISM 35 While the distance between the contact point of the Carried and the sphere rotation axis is:
dc= rtm
p
(cos(θc) cos(ψc)2)2+ (sin(θc))2 (3.4)
The cons geometry is thus characterized by the rmin and ψc values.
Force equilibrium
In Fig. 3.5 is shown a schematic of the forces exchanged.
Figure 3.5: Schematic of mechanism forces system.
Every component exchanges forces equal and opposite on the contact points locations.
Defining a Cartesian frame in the sphere centre and assessing the value m for the sphere mass, the static equilibrium equation is:
m~g + ~Rd+ 2 ~Rc= 0 (3.5)
The force Rc is doubled because it represents the Carried, which has two separately
Developing the system above the normal force modulus are :
|Rc| = mg/(2 cos ψc cos θc tan θd+ 2 sin θc)
|Rd| = mg/(sin θd+cos θcos ψd tan θc c)
(3.6)
Next step is to find relations about forces during the motion phase.
The tangential forces can be interpreted as the friction released in the contact points. Then, since these forces are equal and opposed they can be used in the internal moment balance for each element.
The driver wheel balances its own inertia with the difference between motor torque and friction torque received from the sphere.
The sphere is directly moved by the Driver friction force and hampered by the Carried contacts.
Finally the Carried is forced to move by the two forces exerted by the SPM.
Considering the element inertia moments in the rotation axis as given and the same for the friction coefficients, the moment equations for each rotation axis are:
τ − Ftm/d rd= IDα¨d Driver equilibrium
−Ftm/d rtm+ 2 Fc/tm dc= Itmγ¨tm T est mass equilibrium
2 Ftm/c rc= Icβ¨c Carried equilibrium
(3.7)
The no slip condition sets the kinematic constraints:
rdα˙d= −rtm˙γtm N o slip cond. rcβ˙c= −dc˙γtm N o slip cond. (3.8)
The derivatives of the no slip conditions give relations about the angular accelerations. After substitutions in the system, the angular accelerations of every component are found. In order to have compact equations a new element, representing a sort of whole system inertia, is defined: Is= Itm+ ( rtm rd )2Id+ ( dc rc )2Ic
3.3. WORKING PRINCIPLES OF THE MECHANISM 37 The angular accelerations can be written as function of geometrical parameters and input torque τ : ¨
γtm= −rrtmd I1sτ T est mass ang. acc.
¨
αd= (rrtmd )2 1Isτ Driver ang. acc.
¨
βc= drccrrtmd I1sτ Carried ang. acc.
(3.9)
Since the no-slide condition imposes that the tangential forces between two bodies must be less than the friction static force, a condition on the torque can be extrapolated. For the Driver contact:
Fd/tm< µd |Rd|
and for the Carried contact:
Ftm/c< µc|Rc|
The terms of the tangential forces are directly connected to the angular accelerations of the bodies. From these is possible to found the limits about the input torque τ :
τd< µd |Rd| rd ( Is r2d r2 d Is−r 2 tm Id) τc< µc |Rc| dc (2 Is r 2 c Ic d2c ) (3.10)
Finally, if the τ constraints are respected, the angular position relations are:
γtm= −rrtmd αd γtm= −drccαd βc= rdccrrtmd αd (3.11)
These are the governing equations of the mechanism.
The geometrical parameters rd, rc, θd, θc, ψcare the most influencing variables. They
in the slipping ignition. The parameters such as axial lengths or elements mass properties figure in the inertia moments expressions. However since the rotation axis is the symmetry axis their impacts have secondary roles.
This model sets important design key-points: • Relation between SPM and Driver rotation.
• Relation between SPM and Carried rotation. The Carried angular position can be measured with an encoder, providing a strong data about the SPM rotation. • Limit torque values for having a pure rolling motion, essential for this application.
The motors commercially available, especially for small applications, do not have a large available torque range. Characteristic which introduces a high stiffness in the configuration choice.
• Set of influencing values which permits a faster mechanism size selection.
3.4
Reference mirror configuration
The previous machine set-up used RMs fixed on a static support. The advantages of this configuration are a compact design solution.
The limit of a fixed reference mirrors configuration is the unpredictable motion of the SPM, introduced by an active noise source which belongs only to the test object, the RS.
A solution which can decreases the RS noise impact is an integral design for SPM and reference mirror. Placing both elements on the same support, they are more equally subjected to the same RS noise. This allows to have a higher accuracy without the use of massive post processing corrections.
The major witnesses of such a configuration are: • Mismatching of classical flat mirrors.
• Substantial differences in data interpretation.
The reference mirror reflective surface must always be orthogonal with respect to the laser coming. If the mirror is rotating with the test mass a classical flat geometry is inop-erable. The solution is a ring shape, with its lateral external surface always perpendicular
3.4. REFERENCE MIRROR CONFIGURATION 39 to the incoming beams. Such a solution has been also used in [12].
This set-up change introduces modifications about the lasers paths. The lasers plane must be flipped in order to be correctly reflected by the target elements (SPM and reference mirror). This topic is illustrated in chapter 5.
The other key-feature introduced by the RRM (Reference Ring Mirror) configuration is the interferometer output data interpretation.
Calling the interferometer output data as IFO, placing a polar coordinate frame in the sphere centre and using the symbols for the rotations already mentioned, the equations which represent the, simplified, output data are shown below.
It has to be notice that this derivation is conceptual, in order to understand the key points, and is not taken into account the far more complex mathematical model shown in chapter 2.
For the fixed flat reference mirror:
IF O(φ) = rref(φ) − ρtm(φ) (3.12)
The r and ρ functions are the absolute measures of the objects, corresponding to the position of reference mirror and the local radius of test mass. The equation can not be solved without the knowledge, at the same accuracy of the relative displacement, of one of the two functions (denoting the origin of interferometer differential displacement measures).
However it is possible to eliminate the angular dependence in the function rref(φ), since
it is fixed, of φ reducing its dimension to a scalar instead of a one-dimensional function. This allows to have the topography of the SPM surface, for one circumference since the function IFO is the differential displacement along the variable φ.
Calling the constant representing the reference mirrors with A, the output signal becomes the topography function:
IF O(φ) = A − ρtm(φ) = topography (3.13)
In the RRM configuration, the IFO equation changes. Inserting the 2nd DOF variable and taking into account that now the reference is representing a radius, the equation is:
IF O(φ, θ) = ρref(φ, θ) − ρtm(φ, θ) (3.14)
The last expression illustrates the new dependences of the signals. Since a new degree of freedom is provided the single data collected for one measure will have a collocation in a
two dimensional space, composed by the φ and θ variables.
It is not possible now to eliminate the φ dependence in ρref(φ, θ), because the reference
mirror is fixed on the same test mass support and is rotating with it. Instead, the θ variable might be cancelled because the RRM has no other degrees of freedom.
The equation becomes:
IF O(φ, θ) = ρref(φ, 0) − ρtm(φ, θ) (3.15)
However the equation is still impossible to be solved in a closed form. The signal ρref is
now a one-dimensional function instead of a scalar, while ρtm is two-dimensional.
A solution that can overcame this issue is the knowledge of a first RRM topography, for example removing the test mass and letting the RRM acting as measurement mirror and the end path laser as new reference constant. Basically this configuration should run with-out the SPM.
Defining the new reference as a constant named B, the equation representing the reference RRM topography, called with the IF Oref is:
IF Oref(φ) = B(0) − ρref(φ) (3.16)
The IFO functions is now:
IF O(φ, θ) = B(0) − IF Oref(φ) − ρtm(φ, θ) (3.17)
And finally the test mass topography equation becomes:
ρtm(φ, θ) = B(0, 0) − IF Oref(φ, 0) − IF O(φ, θ) (3.18)
The last equation shows how the single topography might be extrapolated with a cylin-drical shape mirror as reference.
An interesting feature of this new system is the possibility to modify the mathematical model seen in 2.7. The error sx is no longer influencing the topography since an offset of
the RS axis in the x direction moves the SPM as well as the RRM, while the errors sy
and sz are still influencing the topography.
However an sx = 0 permits to avoid the deconvolution process for the derivations of
σ leading to less ambiguous shape of the function and a topography containing high frequency contents.
Chapter 4
Design phase
In order to update the Surface Map machine with: two DOF for the SPM, RRM configu-ration and the automation of the process, the test mass support needs to be re-designed. Chapter 3 illustrated the necessities for the new set-up configuration, this chapter explores the design phase of the test mass support and its side components, illustrating the design process and the mechanical solutions proposed.
4.1
Design guidelines
The guidelines for the design of the new elements derive from the tasks that the elements shall fulfil in order to be useful for the application and from the integration of themselves into the existent machine.
The tasks and requirements for the test mass support are: • 2nd DOF mechanism housing.
• RRM housing.
• High isolation from mechanical and thermal noise.
• Rotational stage mounting interface with calibration motion possibility. • Reference ring independent calibration system.
• Space encumbrance less than a cylinder of φ110mm × 70mm. 41
The first three points are practical consequences of the new set-up requests, illustrated in chapter 3.
Thermal and mechanical stability is a fundamental skill for opto-mechanical applica-tions. The material used for each element composing the 2nd DOF mechanism assembly is the stainless steel.
This material is a trade off between high stiffness (Young modulus = 200 MPa) and low thermal expansion (CTE = 16 × 10−6m×Km ). Only the RRM and its support are made out of copper for optical reasons.
The calibration is an essential operation in order to prepare the set-up for the measure-ments. Axis misalignments, as seen in 2.3.1, introduce eccentricities in the shapes seen in the lasers output. The SPM and the RRM, need to be moved in order to align their axis to the RS axis.
Since the RRM is an optical component, it needs special manufacture and storage features. It is desirable to have it as separated element, not directly integrated with the test mass support. For this reason it is mounted separately and it needs independent alignment solution.
The space encumbrance limits are imposed by the integration of the test mass support in the main structure of the set-up.
4.2
Design process
The design process starts with a first evaluation of the elements dimensions. The principal components, which have to be mounted together, are:
• Mechanism, Driver and Carried wheels. • Motor stage.
• Encoder. • RRM.
These elements are divided in custom design elements (mechanism components, Driver and Carried) and fixed dimension elements (Motor, Encoder and RRM).
4.2. DESIGN PROCESS 43 In the following text a brief description of motor and encoder selected is given. These elements have fixed dimensions because are common commercial parts. Since the space encumbrance is a limiting factor, the dimensions of these elements represent a starting point of the mechanism size selection. While the custom designed elements such as the mechanism might be better adapted to space constraints.
The RRM is custom designed, however since it is an optical element it needs a non trivial manufacture process. Its design has been mostly influenced by the external com-pany in charge to produce this object. For this reason it is classified as external component.
4.2.1 Fixed dimension elements
In this chapter the motor and the encoder are analysed from the dimension point of view while the proper working principles are illustrated in chapter 6.
Stepper Motor
Stepper motors are special types of electric motors that moves in increments, or steps. Inside the device, sets of coils produce magnetic fields that interact with the fields of per-manent magnets. The coils are switched on and off in a specific sequence to cause the motor shaft to turn through the desired angle.
The step size depends by the structure inside the motor and by the controller used to guide the motor. The technique of micro-stepping allows to sub-dived the single increment that derives from the motor structure.
After a brief research study, regarding the parameters of the motors commercially available, it has been noted that generally the size of the motor corresponds to a known torque range, which for small applications, does not have a large set of guaranteed values. The size NEMA8 (28mm × 28mm × 32mm) 1 is small enough to be easily included in the support design. Its nominal torque range varies, generally, from 12 mNm to 16 mNm. Starting from this parameter, every mechanism configuration which does not respect this limit is discarded.
1
NEMA denomination is a common dimension system used for stepper motors. In this work it is chosen to use a device that follows this line in order to be more interchangeable and allows an eventual future substitution with a more accurate element.
Bigger motor sizes are not considered: their dimensions are generally uncomfortable for such small applications. Also the higher torques values implies much current passing through the coils. This may cause problems related to the higher temperatures reached by the motor during the experiment.
However a consideration about the motor torque should be pointed out. The parameter which the stepper motors are referring is the holding torque, that may not match with the classical knowledge of motor torque considered. The holding torque is the torque to be applied to the shaft in order to move it from its position when the coils are energized. The torque considered for the slipping, instead, is referring to the angular acceleration of the shaft during the step phase. This value is less or equal to the holding torque and its knowledge is enabled with an accurate study of the motor dynamics. For this application, anyway, the holding torque is considered as reference parameter for the maximum torque the motor can release.
Encoder
The encoder selected is an Optical Incremental rotary encoder, partial data-sheet in A. It is composed by a glass ring (φext = 36mm × φint = 12mm ) and a readhead
(20mm × 14mm × 8mm).
Reference Ring Mirror
The ring mirror is a copper ring with external reflectivity of 98 %. The dimensions are, for the external diameter φext = 49mm ± 2µm, for the internal diameter φint = 44mm
and axial length of 7mm. The external surface is the one hit by the laser. For this reason the tolerance is specified just for this measure.
4.2.2 Mechanism size selection
In 3.3 a geometrical and physical characterization of the system shows the limit values concerning the torque input and the geometric relations among the elements. From the same study a set of principal measures are chosen in order to trade off between space encumbrance and torque limit values. These values are rd, rmin, θd, θc, ψc.
In order to evaluate the torque limits, the inertial parameters of the elements are neces-sary. Considering a simplified geometry for the mechanism elements, such as the one in
4.2. DESIGN PROCESS 45 fig. 4.1 is possible to estimate these values.
Figure 4.1: Simplified mechanism components. The shaft diameter measure is chosen in accordance with a common ball bearing size 26mm × 10mm × 8mm.
The inertia moments of this simplified geometry can be used in order to rapidly eval-uate large amount of rd, rmin, θd, θc, ψc combinations.
The nominal parameters selected are: rd= 18mm rmin= 9mm θd= 62◦ θc= 75◦ θd= 10◦ (4.1)
