Metrology and Sensing Lecture 5: Interferometry I 07--6 Herbert Gross Winter term 07 www.iap.uni-jena.de
Preliminary Schedule No Date Subject Detailed Content 9.0. Introduction Introduction, optical measurements, shape measurements, errors, definition of the meter, sampling theorem 6.0. Wave optics Basics, polarization, wave aberrations, PSF, OTF 3 0.. Sensors Introduction, basic properties, CCDs, filtering, noise 4 09.. Fringe projection Moire principle, illumination coding, fringe projection, deflectometry 5 6.. Interferometry I Introduction, interference, types of interferometers, miscellaneous 6 3.. Interferometry II Examples, interferogram interpretation, fringe evaluation methods 7 30.. Wavefront sensors Hartmann-Shack WFS, Hartmann method, miscellaneous methods 8 07.. Geometrical methods Tactile measurement, photogrammetry, triangulation, time of flight, Scheimpflug setup 9 4.. Speckle methods Spatial and temporal coherence, speckle, properties, speckle metrology 0.. Holography Introduction, holographic interferometry, applications, miscellaneous.0. Measurement of basic system properties Bssic properties, knife edge, slit scan, MTF measurement 8.0. Phase retrieval Introduction, algorithms, practical aspects, accuracy 3 5.0. Metrology of aspheres and freeforms Aspheres, null lens tests, CGH method, freeforms, metrology of freeforms 4 0.0. OCT Principle of OCT, tissue optics, Fourier domain OCT, miscellaneous 5 08.0. Confocal sensors Principle, resolution and PSF, microscopy, chromatical confocal method
3 Content Introduction Interference Types of interferometers
4 Interferometry Basic idea: - separation of a wave into two beams (test and reference arm) - every beam surpasses different paths - superposition and interference of both beams - analysis of the pattern Different setups for: - the beam splitting - the superposition - the referencing Different path lengths Difference equivalent of one fringe nt nt N tw t w n collimated laser beam beam splitter detector reference mirror surface under test Measurement of plates: Haidinger fringes of equal inclination Newton fringes of equal thickness Ref: W. Osten
5 Classification of Interferometers Division of amplitude: - Michelson interferometer - Mach-Zehnder interferometer - Sagnac interferometer - Nomarski interferometer - Talbot interferometer - Point diffraction interferometer Division of wavefront: - Young interferometer - Rayleigh interferometer Division of source: - Lloyds mirror - Fresnel biprism Ref: R. Kowarschik
6 Classification of Interferometers Two-beam interferometers: - Michelson - Twyman Green - Sagnac - Young - Mach-Zehnder - Rayleigh - Fizeau - Shearing - Mireau - Linnik Multi-beam interferometers: - Fabry-Perot - Lummer-Gehrke Ref: R. Kowarschik
7 Localization of Fringes Interference volume for a plate incident light front side reflected back side reflected volume of interference fringes Interference volume for a wedge front side reflected incident light back side reflected volume of interference fringes Ref: R. Kowarschik
8 Interference of Two Waves Superposition of two plane waves:. Intensity. Phase difference Spacing of fringes Interference of two spherical waves More complicated geometry ),, ( cos ² ² ),, ( z y x A A A A z y x I r k k z y x z y x z y x ) ( ),, ( ),, ( ),, ( Ref.: B. Dörband sin n s
Interference of a Double Pinhole Interferenz eines kohärent beleuchteten Doppel- Pinhole-Setups Die Struktur hängt von Wellenlänge und Abstand ab decreasing wavelength increasing separation D
0 Two Beam Interference Interference of two point source spherical waves. both wave are radiating outside. one incoming and one outgoing wave
Two Beam Interference Interference of two point source spherical waves with perturbations
Two Beam Interference Interference of two plane waves under different directions Fringe distance s s k k n e e
3 Two Beam Interference Interference of two plane waves with finite spectral width w 0 ))),,, ( )cos( ( ) ( ) ²( ) ²( ( ),, ( 0 d z y x A A A A z y x I
4 Two Beam Interference Interference of two spherical waves with finite bandwidth in x/z Delay rotated cone of maximum contrast bandwidth 0 nm bandwidth 60 nm bandwidth 00 nm no delay delay 5 ms
5 Michelson Interferometer Visibility of fringes S S S S M M M M S S B M B M Ref: R. Kowarschik Haidinger Fringes Fizeau Fringes
6 Haidinger Fringes Fringes of equal inclination: Haidinger Every inclination creates an individual delay in the plate
Two Beam Interference Two beam interference of two waves: - propagation in the same direction - same polarization - phase difference smaller than axial length of coherence Coherent superposition of waves I I E E I I I cos Difference of phase / path difference Number of fringes location of same phase Conrtast s s N K I I max I min max I min I I I I
Interference Fringes at a Plane Plate Two beam interference at a plane plate - Fresnel fringes of equal thickness - Haidinger fringes of equal inclination Path difference s nd cos d n sin detector n : source transparent plane plate d
9 Interference at a Plane-Parallel Plate Multiple reflection superposition Airy formulas T: tranmittance R: Reflectance I ( r) ( R) 4Rsin 4Rsin I ( i) I ( t) ( R) T 4Rsin I ( i) Plane monochr. wave n r, t Reflection, Transmission Coeff. n n r, t Reflection, Transmission Coeff. n n n h n Ref: R. Kowarschik
Interference at a Plane Plate Multi beam interference Intensity of pattern I T ( R) R R cos Finesse determines the contrast F / R R n d I( ) R = 0. R = 0.6 m R = 0.9 (m+) (m+)
Interferometers Accuracy of interferometers Ref: F. Hoeller
Test by Newton Fringes Reference surface and test surface with nearly the same radii Interference in the air gap Reference flat or curved possible Corresponds to Fizeau setup with contact to detector Broad application in simple optical shop test Radii of fringes beamsplitter r m mr illumination test surface path difference reference surface here: flat Ref: W. Osten
3 Newton Fringes Movement of fringes Determination of the OPD sign Ref: B. Doerband
Example Interferograms spherical aberration coma tilt astigmatism
5 White Light Interferograms Typical interferograms monochromatic / white light Additional information by different wavelengths Ref: B. Dörband
Autocollimation Principle Spherical test surface: - incoming and outgoing wavefront spherical - concentric waves around center of curvature: autocollimation auxiliary lens spherical test surface center of curvature wavefronts spherical Aspherical test surface auxiliary lens outcoming wavefront aspherical aspherical test surface paraxial center of curvature incoming wavefront spherical
7 Michelson Interferometer Types of fringes:. equal thickness. equal inclination Fizeau Haidinger S S S S M M S S M M B M B M Ref: R. Kowarschik
Fizeau Interferometer Fizeau surface as part of the system work as reference Fizeau surface near to test surface: - large common path, insensitiv setup - small cavity length The test surface is imaged onto the detector collimator light source beam splitter stop detector Fizeau surface plane test surface
Fizeau Interferometer Long common path, quite insensitive setup Autocollimating Fizeau surface quite near to test surface, short cavity length Imaging of test surface on detector Straylight stop to bloc unwanted light Curved test surface: auxiliary objective lens (aplanatic, double path) Highest accuracy collimator auxiliary lens convex surface under test beam splitter light source stop detector Fizeau surface
Mach-Zehnder Interferometer no common path setup, sensitive long distances, measurement of samples with small effects mirror sample beam combiner test arm detector source reference arm beam splitter mirror
Michelson Interferometer Test and reference arm separated: setup sensitive Both arms aligned: fringes of equal inclination Tilt in reference arm: fringes of equal thickness Setup corresponds to Twyman-Greeninterferometer reference beam reference mirror compensator plate laser source test beam beam splitter surface under test screen
Testing with Twyman-Green Interferometer Short common path, sensible setup reference mirror Two different operation modes for reflection or transmission collimated laser beam Always factor of between detected wave and component under test beam splitter objective lens stop. mode: lens tested in transmission auxiliary mirror for autocollimation. mode: surface tested in reflection auxiliary lens to generate convergent beam detector
Suppression of Straylight by Polarization Straylight suppression in Twyman- Green interferometer reference mirror Polarization of both arms by /4 plates Analyzer in front of detector: only signal light is passing Optimization of azimuthal orientations of the plates: - reflectivity of test surface - splitting of power in both arms - largest contrast of interferogram collimated laser beam / 4 plate polarization beam splitter surface under test tan A tan i R R / auxiliary plate / 4 lens plate lens analyzer detector
Shearing Interferometer Separation of wavefront: self reference Interferograms are looking completly different Aperture reduced due to shear Splitting and shift of wavefront: - by thin plate - by grating shear distance source
Grating Shearing Interferometer Shearing interferometer with two identical Ronchi gratings with distance d Self referencing system Lateral shear offset d limizes transverse resolution Interference by only the orders + and - Quite different interferogram pictures obtained s d sin d g d orders + - + s (+/+) (+/-) (-/+) (-/-) - g g
Shearing Interferometer Schematic drawing of sheared wavefronts wavefront W x Typical interferogram shear distance
Radial Shearing Interferometer Compact setup wavefront under test beam splitter lens Modified Mach-Zehnder setup with telescope wavefront with radial shear mirror test arm beam splitter detector telescope for change of diameter source beamsplitter reference arm mirror
38 Shearing Interferometer Types of shear
Point Diffraction Interferometer Focussing onto a transparent plagte with pinhole Pinhole creates a reference spherical wave Optimization of contrast: - size of pinhole - numericalaperture - transparency of the plate Very stable setup transparent plate with pinhole wavefront under test reference wavefront
40 Point Diffraction Interferometer Full setup according to Smartt
4 Point Diffraction Interferometer Setup integrated into Mach-Zehnder interferometer
Rayleigh Interferometer Beam splitting by stop / slit Application: measurement of inhomogeneities of refractive index liquids must be in polishes glass cuvette source stop test arm detector reference arm
Nomarski Interferometer Separation of both arms by polarization Shear principle Used in microscopy for differential interference contrast (DIC) phase imaging Point spread function E psfdic Parameter: ( x, y) ( R) e. Shear distance. Phase offset 3. Splitting ratio Re i i E E psf psf ( x x, y) ( x x, y) shear distance x -R R analyzer Wollaston prism adjustment phase objective object condenser splitting ratio Wollaston prism compensator polarizer
44 DIC Phase Imaging Orientation of prisms and shift size determines the anisotropic image formation Ref.: M. Kempe
Differential Interference Contrast Orientation of the shear DIC-Psf object image, x-shear image, y-shear image, x-and y-shear
Fiber Interferometer Interference of the the light from two coherent excited monomode fibers Interferogram: fringes with gaussian envelope I( x) I 0 e x w akx cos R Spacing of fringes depends on fiber distance a z x a s x a z fiber s screen
Sagnac Interferometer Mirror ring setup both arms are defined by opposite round trip direction perfectly common path guarantees stable setup alignment of mirror critical mirror beamsplitter mirror detector mirror
Further Types of Interferometers Jamin Interferometer detector test arm reference arm source Köster Interferometer detector reference arm test arm source
49 Fabry-Perot Interferometer Setup of an etalon Point source Fabry-Perot Etalon n B Applications: - spectral line resolution - laser mode selection h Ref: R. Kowarschik
50 Fabry-Perot Interferometer Intensity Finesse Transmission Contrast Ref: R. Kowarschik ) ( ) ( sin R R R A I I i t R R F max ) ( ) ( R A I I i t p min ) ( ) ( max ) ( ) ( 4 F R R I I I I C i t i t
5 Fabry-Perot Interferometer Intrumental functions Properties W( ) RP F Perfectly planeparallel plate R m0 ( ), m m m d 0 Absorption R T R 4R ( R) sin ( d A d A F A Surface imperfections R, h H( ) f ( h) h H h F H Finite range of incidence R, (cos ) F( ) f ( cos( ) (cos) F (cos ) F F Ref: R. Kowarschik
More complex Setup of an Interferometer Spectral filtering Straylight suppression Diameter adaptation stop B lens L distance L stop B lens L spectral filtering D :.5 mm D : straylight suppression and 3.8 mm diameter adaptation prism group lens L4 distance L stop B3 lens L3 D : 3.8 x 9.49 mm D : 0.0 mm distance D : L3 7.7 mm disrance s beam splitter M lens L5 Linse L6 distance s CCDcamera detection test surface M reference arm
53 Real Interferometers Ref: R. Kowarschik