displacement measuring interferometery measurement uncertainty Mike Holmes and Chris Evans (contact info:

Transcription

1 displacement measuring interferometery measurement uncertainty Mike Holmes and Chris Evans (contact info: 1

2 agenda heterodyne displacement measuring interferometer (DMI) overview case study: 4-axis 4 DMI system overview uncertainty analysis results of one DMI configuration beam mixing contributions in more detail analytical predictions experimental measurements conclusions acknowledgements 2

3 heterodyne DMI target mirror two-frequency laser mixing polarizers at 45 degrees with respect to polarization axes vertical displacement measuring interferometer measurement signal reference signal horizontal z phase interpolator s analog front end 3

4 DMI signals (simplified) V ref = 0.56k(E1 2 + V = 0.56k(E1 2 meas + E k = R G 2η 0 E E E cos[( ω ω )t] E E cos 1 2 ω ω t n meas l 8π meas - λmeas n ref l ref λ ref The phase interpolator measures changes in phase of the measurement signal with respect to the reference signal and calculates the displacement of the target mirror. 4

5 4-axis DMI system moving DMI U.S. and foreign patents pending 5

6 moving DMI reference mirror/quarter-wave plate polarization beam splitter quarter-wave plate stage constant deviation prism output prism fiber optic coupler target mirror f1 f2 input beams retro-reflector 6

7 things that result in heterodyne DMI measurement uncertainty light source wavelength instability frequency mixing interferometer refractive index instability measurement arm reference arm polarization leakage ghost beams wavefront deformation/beam shear phase interpolator noise 7

8 uncertainty summary uncertainty source laser head: frequency mixing wavelength instability moving DMI: thermal sensitivity beam mixing stray light beam shear phase interpolation electronics vacuum index uncertainty 0.15 nm 0.32 nm total (RSS): 0.73 nm moving DMI measurement uncertainty (k=1) contribution to DMI measurement uncertainty 0.05 nm 0.20 nm 0.05 nm 0.10 nm 0.01 nm 0.60 nm 8

9 some possible beam mixing paths signals sources of uncertainty measurement arm full-cycle leakage half-cycle leakage half-cycle leakage reference arm full-cycle leakage half-cycle leakage half-cycle leakage 9

10 beam mixing model input parameters model input description expectation standard uncertainty AR AR coating efficiency (0.9995) 0.5 (0.0003) 0.5 Tpbs polarization beam splitter transmission efficiency (0.98) 0.5 (0.0115) 0.5 Rpbs Polarization beam splitter reflection efficiency (0.98) 0.5 (0.0115) 0.5 Rpmt plane mirror target reflection efficiency (0.935) 0.5 (0.0375) 0.5 Rretro Retro-reflector reflection efficiency (0.97) 0.5 (0.0173) 0.5 Rleak polarization beam splitter leakage into reference arm (0.001) 0.5 (0.0006) 0.5 Tleak polarization beam splitter leakage into measurement arm (0.001) 0.5 (0.0006) 0.5 Rpmr Plane mirror reference reflection efficiency (0.9975) 0.5 (0.0014) 0.5 qwleak quarter-wave plate induced leakage (0.0055) 0.5 (0.0032) 0.5 qwloss quarter-wave plate efficiency (0.9945) 0.5 (0.0032)

11 machine DMIs 4-sided straight-edge cyclic error characterization X axis SE cal tool DMIs 11

12 cyclic error characterization 12

13 SE calibration tool 13

14 SE cal tool DMI (se12) output during 1 µm/s scan in X raw data X (µm) time (s) slope removed X (nm) time (s) peak mag (nm) spectrum 2.7 Hz frequency (Hz) 14

15 SE cal tool DMIs SE12 and SE14 output data spectrums 2.7 Hz individual spectrums The top graph shows output data spectrums for both DMIs SE12 and SE14. The three largest peaks, as well as most of the smaller ones, stem from machine motion. difference spectrum 6.3 Hz 12.6 Hz The bottom graph shows the spectrum for the difference between DMIs SE12 and SE14. The two significant peaks stem from full-cycle and two-cycle SE cal tool DMI nonlinearities. 15

16 SE cal tool HSPMI uncertainty summary uncertainty source laser head: frequency mixing wavelength instability moving DMI: thermal sensitivity beam mixing stray light beam shear phase interpolation electronics vacuum index uncertainty 0.15 nm 0.01 nm total (RSS): 0.18 nm HSPMI measurement uncertainty (k=1) contribution to DMI measurement uncertainty 0.05 nm 0.01 nm 0.02 nm 0.07 nm 0.02 nm 0.00 nm 16

17 Did we get what we expected for the 6.3 Hz nonlinearity? we expected 0.07 nm (k=1) per HSPMI therefore, we should see something between 0 and 0.14 nm (k=1) in the difference data depending on the phasing of SE12 with respect to SE14 we measured 0.12 nm and therefore did get what was expected 17

18 Did we get what we expected for the 12.6 Hz nonlinearity? we expected 0.02 nm (k=1) per HSPMI therefore, we should see something between 0 and 0.04 nm (k=1) in the difference data depending on the phasing of SE12 with respect to SE14 we measured 0.32 nm and therefore did NOT get what was expected 18

19 a closer look at the stray light primary measurement signal 4-pass source of 2-cycle nonlinearity 3-pass source of 1.5-cycle nonlinearity 19

20 Why 2-cycle and not 1.5-cycle? (diagrams) 2-cycle beating occurs because the wavefronts of interfering beams are parallel 1.5-cycle beating does not occur because the wavefronts are not parallel 20

21 conclusions Uncertainty analysis plays a vital role in capturing/bounding the sources of uncertainty and provides the means by which a successful design can be realized. Here we have presented an overview of our analysis for a particular application and have demonstrated one approach to characterizing beam mixing/stray light influences on the measurement data. 21

22 acknowledgements Dr. Henry Hill Dr. Peter de Groot Lars Selberg Michael Schroeder Mike Metz Dave Newton Andy Stein Matt Van Doren 22

23 What is the source of the 6.3 Hz nonlinearity? beam mixing within the interferometer occurs due to polarization axes misalignment (light to interferometer) elliptical polarization states (light) polarization beamsplitter leakage (interferometer) retardation plate misalignment (interferometer) retro-reflector rotation of polarization states (interferometer) the 6.3 Hz (full-cycle) nonlinearity stems from beam mixing 23

24 What is the source of the 12.6 Hz nonlinearity? a two-cycle nonlinearity goes through 4π of phase for every nm displacement of the target mirror therefore, the source of this nonlinearity must be making 4 passes to the target mirror a source for this type of nonlinearity stems from stray light reflections from the measurement arm quarter-waveplate window 24

25 glossary of symbols and terms E 1 measurement arm beam electric field magnitude (V/m) E 2 reference arm beam electric field magnitude (V/m) ω 1 = 2πf 1 (rad/s) ω 2 = 2πf 2 (rad/s) n meas measurement arm index n ref reference arm index l meas measurement arm path length (m) l ref reference arm path length (m) λ meas measurement arm wavelength (m) λ ref reference arm wavelength (m) R responsivity of silicon (A/W) G transimpedance amplifier gain (V/A) η 0 wave impedance of free space (Ω) V ref reference signal Voltage (V) V meas measurement signal Voltage (V) DMI: Displacement Measuring Interferometer HSPMI: High Stability Plane Mirror Interferometer SE: Straight Edge 25