iprom Optical Interferometry Prof. Dr. -Ing. Rainer Tutsch Institut für Produktionsmesstechnik IPROM Technische Universität Braunschweig

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1 Optical Interferometry Prof. Dr. -Ing. Rainer Tutsch Institut für Produktionsmesstechnik IPROM Technische Universität Braunschweig Frontiers of Metrology April 1, 01 I P NSTITUT FÜR RODUKTIONSMESSTECHNIK TECHNISCHE UNIVERSITÄT BRAUNSCHWEIG

2 Basics of interferometry Length measurement Michelson interferometer Homodyne/heterodyne evaluation Discussion of error sources Straightness, orthogonality, angle Absolute interferometry Optical testing Twyman-Green interferometer Fizeau interferometer Transmission spheres Phase evaluation techniques Phase unwrapping Testing of aspheric surfaces Microinterferometry Michelson-, Mirau-, Linnick-setup White light interferometry Structure

3 Basics of interferometry Interferometry: Deviation from the additivity of light intensity Electromagnetic wave: Intensity: Superposition of two waves: Uncorrelated wave-components (incoherent light) -> The term within the brackets averages to 0 ( ) 0 0 ω +φ = r k t i e E E r r r r t E I > < r t t t t t E E I I E E E E E E I > < + + = > < + > + < > =< > + < r r r r r r r r

4 E(t) a) E E ges t E(t) b) E ges E E 1 t E 1 Destructive interference Constructive interference The resulting intensity depends on the relative phase shift. Constructive and destructive interference

5 Coherence length: l c λ = Δλ Ideally: monochromatic light Natural light: Unstabilized laser: Stabilized laser: l c < μm l c 1m l c > 1km Coherence length

6 Condition for interference: a sinϑ << λ Ideally: a -> 0 (point source) ϑ -> 0 (collimated light) Spatial coherence

7 Ideally: a -> 0 (point source) -> 0 (collimated light) Δ ϑ λ -> 0 (monochromatic light) Interference contrast: I I max min γ = 0 γ 1 max I + I min Interference contrast

8 Length measurement

9 Reference reflector Δx Laser Detectors Beam splitter Measurement reflector Corner-cube prisms: tolerant against tilting beam-shift prevents reflected beam from entering the laser cavity Michelson interferometer

10 Signal A Different movements: a) From x 1 to the right side b) From x to the left side I 1 x 1 x x Ambiguity is an inherent problem of incremental measurement techniques. Common remedy: Generation of A quad B signals Ambiguity of the sinusoidal output

11 Solving the ambiguity problem using A quad B signals

12 Superposition of phase-shifted orthogonally polarized waves

13 Michelson interferometer with homodyne evaluation

14 Heterodyne interferometer

15 Abbe-error (1st order) if laser-beam is not coincident with path to be measured Deviation of nd order if laser-beam is not parallel to path to be measured Stability of laser wavelength Stabilization of the laser frequency -> uncertainty caused by refraction index n n of air is dependent of temperature, pressure, humidity,... a) Measurement of atmospheric parameters -> Edlén formula b) Direct measurement of n (refractometer) Nonlinearity caused by interpolation Errors caused by optical elements Temperature coefficient of workpiece... Error sources

16 Modification for straightness measurement

17 Penta prism Testing of orthogonality

18 Modification of laser interferometer for angle measurement

19 Two-wavelength interferometry: λ synth = λ1 λ λ λ 1 Unambiguous measurement range can be extended Typical limitation to a factor of < by signal/noise-ratio Further extension by multiple-wavelength interferometer With 3 laser wavelengths an absolute range of several meters is possible Absolute interferometry

20 Optical Testing

21 Twyman-Green interferometer

22 Fizeau interferometer

23 The aplanatic lens is often refered to as Transmission Sphere Measurement of spherical surfaces

24 Phase shifting interferometry PSI

25 Common Path Phase shifting interferometer Point diffraction interferometer Pinhole Object wavefront Object wavefront Spherical reference wavefront Vibration resistant interferometers

26 Simultaneous Phase shifting interferometer Point diffraction element Surrounding region Spherical reference wavefront and object wavefront Holographic optical element CCD Polarizing phase-mask Orthogonal linear polarizers Phase offsets of: 0, π/, π und 3π/ Four phase-shifted interferograms are recorded simultaneously on one image sensor Reduction of spatial resolution 1 Vibration resistant interferometers

27 Pixelated Phase-Mask Polarizing phase mask on CCD Sensor Unit cell Simultaneous recording of four phase-shifted interferograms on a single CCD image sensor The four interferograms share the same image sensor -> Reduction of spatial resolution High-end technology, therefore quite expensive 1 Vibration resistant interferometers

28 Most phase unwrapping algorithms work under the condition, that the phase-difference between adjacent pixels is < 180 -> Limitation of local surface slope! In fact this is an example of Shannon s sampling theorem, applied to the sampling of the quasi-periodic interference pattern by the camera. This can be overcome by adding information, e.g. by using a multiple-wavelength evaluation. Phase unwrapping

29 Large deviation between the regular test wavefront (plane or spherical) and the aspheric surface under test -> high fringe density (apart from other problems) Different approaches: a) Modelling an aspheric test wavefront that fits the nominal shape of the aspheric surface under test either by specially designed reflective or refractive optical elements or by a calculated computer generated hologram (CGH) b) Extension of the measurement range by multiple-wavelength interferometry c) Extension of the measurement range by using IR wavelength (e.g. 10,6µm) if reduced resolution is acceptable Aspheric optical testing

30 Microinterferometry

31 Typical range of magnification: Michelson interf. : Mirau interf. : Linnick interf. : 1x - 15x 0x - 40x 50x - 100x Different types of microinterferometers

32 White light interferometer

33 Thank you for your attention! End