(2011) Random-Phase-shift Fizeau Interferometer

download (2011) Random-Phase-shift Fizeau Interferometer

of 12

Transcript of (2011) Random-Phase-shift Fizeau Interferometer

  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    1/12

    Random-phase-shift Fizeau interferometer

    Hagen Broistedt,1,* Nicolae Radu Doloca,2 Sebastian Strube,2 and Rainer Tutsch1

    1Institut fr Produktionsmesstechnik, Technische Universitt Braunschweig,

    Schleinitzstrae 20, 38106 Braunschweig, Germany

    2Physikalisch-Technische Bundesanstalt, Bundesallee 100D-38116 Braunschweig, Germany

    *Corresponding author: [email protected]

    Received 6 June 2011; revised 28 September 2011; accepted 28 September 2011;

    posted 4 October 2011 (Doc. ID 148803); published 13 December 2011

    A new and potentially cost efficient kind of vibration-tolerant surface measurement interferometer basedon the Fizeau-principle is demonstrated. The crucial novelty of this approach is the combination of twooptoelectronic sensors: an image sensor with high spatial resolution and an arrangement of photodiodeswith high temporal resolution. The photodiodes continuously measure the random-phase-shifts causedby environmental vibrations in three noncollinear points of the test surface. The high spatial resolutionsensor takes several frozen images of thetest surface by using short exposure times. Under theassump-tion of rigid body movement the continuously measured phase shifts of the three surface points enablethe calculation of a virtual plane that is representative for the position and orientation of the whole testsurface. For this purpose a new random-phase-shift algorithm had to be developed. The whole systemwas tested on an optical table without vibration isolation under the influence of random vibrations. Theanalysis of the root-mean-square (RMS) over ten different measurements shows a measurement repeat-ability of about 0.004 wave (approximately 2:5nm for 632:8nm laser wavelength). 2011 OpticalSociety of America

    OCIS codes: 120.3180, 120.3940, 120.6650, 110.3175.

    1. Introduction

    The interferometric measurement techniques pro-vide the most accurate noncontact 3D measurementtests for reflective surfaces and most particularlyfor high-quality optical components such as lenses,microscope objectives, camera lenses, prisms, opticalflats, or even large telescope mirrors. The high accu-racy of the measurements is related to the wave-length of the light, which is used as the measuringreference in all interferometric measurements. In

    many applications frequency stabilized lasers areused.

    The basic principle of the interferometric 3D formtests of reflective surfaces is that the 3D shape of thesurface is encoded in the reflected wavefront (knownas the test wavefront). The test wavefront is com-pared to a reference wavefront, which is generated byreflection from a reference surface. The interference

    between the two coherent wavefronts generates theso called interference fringe pattern, which can bedigitized using an image sensor array. In case ofa flatness test, the deviations from planarity of thesurface under test deform the ideal straight-lineequidistant interference fringes.

    A single fringe pattern can be numerically anal-yzed using either data interpolation methods orFourier analysis methods. These techniques arecapable of a resolution of about =50, where is the

    laser wavelength [1]. In most cases, however, phaseshifting interferometry (PSI) techniques are applied.These methods require a sequence of interferenceimages. By sequentially shifting the reference platewith well-defined steps by means of a piezo-actuator,several interference images at well-known and pre-defined phase-shifts are sequentially acquired. Thesequentially recorded gray values are used to calcu-late the 3D information of the test wavefront at eachpixel. In this way a pixilated analysis of the testwavefront results.

    0003-6935/11/366564-12$15.00/0 2011 Optical Society of America

    6564 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

    http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    2/12

    Because the phase-shifted interferograms are se-quentially recorded and very precise phase-shiftsare required, the PSI methods prove to be very sen-sitive to vibration. This is why every PSI interferom-eter must be mounted and utilized on an optical tablewith vibration isolation and, even so, the measure-ments have to take place under special laboratoryconditions far from the manufacturing process.The measurement accuracy of the PSI methods is

    anyway better, compared to the fringe pattern anal-ysis methods. Commercial PSI interferometers typi-cally show a resolution of about =1000.

    Nowadays there is a general tendency for integra-tion of the quality testing in the manufacturingprocess. As a consequence, there is an increasing de-mand for new interferometric concepts that allow theutilization of the interferometers concomitant withand close to the manufacturing process. In this con-text, the purpose of this work is to develop and to ver-ify an innovative interferometric system that copeswith the presence of vibrations in order to omit vibra-tion isolation equipment. As it will be described, thisnew measurement principle actually makes use of

    the vibrations.

    2. Vibration-Tolerant Interferometers

    The data acquisition time in sequential PSI takesseveral frame times of the electronic camera, addingup to about 100ms. The frequency spectrum ofmechanical vibrations in buildings typically is domi-nated by the region between 20Hz and 200Hz [2].Conventional PSI therefore is very sensitive tofloor vibration, making expensive vibration isola-tion equipment necessary. In the last few yearsnew PSI techniques have been developed that canbe applied in the presence of vibrations. The basic

    idea of all these new techniques is to record thesequence of interferograms in a very short time in-terval or even simultaneously, resulting in the effectof shifting the sensitivity of the system to higherfrequencies.

    In Wizinowich [3], a 2 1 algorithm is describedthat uses two interferograms with =2 phase-shifttaken very rapidly by using a Pockels cell to switchbetween two phase-shifted, orthogonally polarizedwaves synchronously to the switch between the twohalf-frames of an interline-transfer CCD image sen-sor. A third image gives the average intensity acrossthe aperture, which is obtained by superposition

    and averaging of two interferograms phase-shiftedby . Several other approaches use the simultaneousphase-shifting interferometry.

    In Koliopoulos [4] and Millerd, et al. [5], a methodis described where polarizing beam splitters and re-tardation plates are used to create four phase-shiftedinterferograms on four image sensors. In a similarway in Kihm [6] and Smythe and Moore [7] fourphase-shifted interferograms are generated andimaged on one image sensor. In Millerd, et al. [8] animproved point diffraction interferometer that ap-plies a polarizing point diffraction plate is presented.

    With a further implementation of a holographicelement in combination with a birefringent maskof four elements in front of the image sensor, also fourphase-shifted interferograms are created on oneCCD-chip.

    An alternative approach to the others is the use ofa pixilated phase mask, which uses an array of mi-cropolarizers very precisely matched to a CCD sensorarray [9]. The array contains different types of

    elements with the transmission axis at0

    ,45

    ,90, 135, which are arranged in groups of four andform the superpixel. Different types of phase-shift-interferometers that make use of this method canbe seen in Millerd [10] and Kimbrough [11].

    Several vibration-tolerant phase-shifting solutionshave been developed and demonstrated. Recordingthe interferograms at higher frame rates has theeffect of moving the sensitivity to higher vibrationfrequencies. Instantaneous phase-shifting techni-ques use polarization components or holographic ele-ments, splitting the beams in multiple paths, andphase-shifted interferograms are simultaneously re-corded. In conclusion, the already available phase-shifting interferometric configurations that handlethe presence of vibrations are quite complex andexpensive systems. There is still a demand for low-budget interferometers that work in the presenceof vibration.

    3. Random-Phase-Shift Interferometer

    The interferometer presented in this paper wasdesigned to work without vibration isolation and touse the random floor vibration as phase-shifter.In the case when the mechanical vibrations are notsufficient, the interferometer (more precisely the testplate) has to be deliberately perturbed to achieve ran-dom vibrations. This could be easily done by e.g.slightly striking the mechanical holder of the testplate with a finger or by mounting a small electro-motor to the mechanical holder that has an imbal-ance. The challenging goal of this new approach isto determine the random-phase-shifts that are gener-ated in this manner.

    We will begin by describing the experimental setup(Section 3.A) and randomly oscillating test plate(Section 3.B). We will then discuss the low spatial re-solution detector system (Section 3.C), how it is usedto determine the phase-shift (Section 3.D and 3.E)between interferograms, and how it is registered tothe interferograms (Section 3.F). We complete thissection by discussing a four-step algorithm appropri-ate to these random-phase-shifts.

    A. Experimental Setup

    The experimental arrangement of the Fizeau config-uration is shown in Fig. 1. A He-Ne laser beam of632:8nm wavelength is directed to the spatial filterSF, using the mirrors M1 and M2 and is collimatedwith the collimating lens L1 to the test and referenceround plates T and R.

    20 December 2011 / Vol. 50, No. 36 / APPLIED OPTICS 6565

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    3/12

    The reflected wavefronts from the test and refer-ence surfaces are deviated by the beam splitterBS-1 and traced through a pinhole aperture. Furtherusing the beam splitter BS-2 the interference fieldis divided and directed at two different detectorsystems. The reflected part is projected onto aCCD sensor array of a camera, which actually meansa high spatial but low temporal resolution detector.The transmitted part is collimated by the collimatinglens L2 and hits a high temporal but low spatialresolution detector system consisting of threephotodiodes.

    The functional principle is basically as follows: theCCD camera records a few interferogram imageswith random-phase-shifts due to the influence ofthe vibrations. The camera runs at 25 fps with anexposure time of only 11s, short enough to freezemechanical vibrations. While the camera grabs sev-eral interferograms, the detector system consisting ofthree photodiodes continuously records the lightintensities that occur at three locations in the inter-ference field aperture, making possible as will be la-ter described, the determination of the phase-shiftfor the entire test surface at each moment during theentire measurement.

    The reference plate is made of quartz, is 50mm indiameter, and has a wedge angle of about 0:5 toprevent interference between waves reflected atfront- and back-side. The front-side surface has=20 flatness, and the back-side surface, which isused as reference surface, has =27 flatness, as mea-sured by the supplier. For the investigation of thenew technique we used a test plate made of BK7glass with a diameter of50mm and a wedge angleof7. The surface topographies of both surfaces aremeasured and specified by the supplier as reference.

    From the peak-to-valley values flatness deviations of=8 and =11 result, respectively.

    B. Randomly Oscillating Test Plate

    A characteristic advantage of the Fizeau configura-tion against the TwymanGreen interferometer isthe common paths for reference and test waves. Thisreduces the sensitivity of the optical componentsagainst internal vibrations and thermal effectsconsiderably. In good approximation, only the rela-tive movement of the reference and test platesgenerates the phase-shifts. The first distinguishedelement of the interferometer is the flexible post-

    holder of the test plate. In Fig. 2 a cross-sectionof the mounting system of the test and referenceplate is depicted. The postholders are fixed on therail system of the interferometer. The geometric

    Fig. 1. (Color online) The experimental setup.

    Fig. 2. The mounting systems for reference and test plate. Bothconsist of a mechanical holder, but with a rigid postholder for thereference surface and a flexible postholder for the test plate.

    6566 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

    http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    4/12

    form of the flexible postholder decides the characterof the oscillations of the test plate under the influ-ence of the mechanical vibrations. In-plane oscilla-tions of the plate do not have an influence on theinterferogram (at least as long as they are smallin the order of a few m).

    In case of a flexible postholder in form of a paral-lelepiped like in the figure above, the width must bemuch larger than the thickness in order to suppress

    the in-plane oscillations. This form allows mostlyonly out-of-plane movements in form of tilt oscilla-tions of the entire system formed by the flexible post-holder, the mounting system and the test plate, aboutthe X-axis at the base of the holder. We assume the

    vibration-induced movements of the plate as rigid-body shifts and tilts and we consider the flexible post-holder as being the dominant elastic component ofthe system.

    In Fig. 2 it can be seen that the mechanical holderis the most massive component, so the mass of theplate can be neglected. It can be assumed that theeigenfrequencies of the oscillations are mainly deter-mined by the flexible postholder form and dimen-

    sions and the mass of the mechanical holder. Thecharacteristics of the oscillations depend onlyslightly on the tested object because its mass is con-siderably smaller than that of the mechanical holder.Several tests were made with postholders manufac-tured from aluminum with different profile thick-ness. As a result, different elastic constants andimplicit different eigenfrequencies of the oscillatingsystem were observed.

    The local intensity I of the interference field as afunction of time t becomes

    Ix;y; t I0x;y; t I00x;y; t cosx;y x;y; t;

    1

    where I0i is the intensity bias and I00i the half peak-

    to-valley intensity. The x- and y-directions are per-pendicular to the z-direction, which runs parallelto the optical axis. The main goal still remains to de-termine the random phase shifts x;y; t at everypixel for each interferogram recorded with theCCD camera. Preliminary tests were performed ona vibration isolation optical table, so to produce oscil-lations we used a piezoelectric transducer that acteddirectly on the flexible postholder. Starting with sim-ple oscillating functions, like a sinusoidal signal, the

    system was gradually complicated with random os-cillations, to simulate the influence of the floor vibra-tions. In this way, the oscillations of the test plateproduced a continuous phase-shift between the testand reference plate.

    C. Low Spatial Resolution Detector System

    Under the assumption of rigid-body shifts and tilts ofthe test plate, the profile of the surface under testdoes not suffer any changes under the influence ofthe vibrations. Knowing this, any noncollinear com-bination of three measurement points on the test

    surface defines an oscillating plane that describesexactly the oscillation of the entire surface [Fig. 3(b)].In other words, it is sufficient to measure the vibra-

    tions at three locations with high temporal resolutionin order to find the occurring phase-shifts at everypoint across the test surface. This is the reasonwhy we use in this work a detector system consistingof three photodiodes, shown in Fig. 3(a).

    We have integrated three photodiodes with inter-nal operational amplifier that ensures high signal-to-noise ratio even for low light intensities. The typicalresponse times (the rise time and the decay time) ofabout 8s provide a continuously high temporal re-solution sampling. The sensitive area of each photo-diode is 1mm2. The analog signals of the photodiodesare connected through a National Instruments data

    acquisition card to a computer and show the depen-dence in time of the intensity of the He-Ne fringes atthree different sampling points.

    In classic phase-shifting interferometry the refer-ence plate is shifted with constant velocity in thedirection of the Z-axis and introduces a linear phase-shift in time. In this case the signal of a photodiodehas a sinusoidal form. Now having a look at Eq. (1),where the phase-shift x;y; t randomly varies dueto the vibrations, we expect frequency modulatedsignals of the photodiodes. In order to make a corre-lation analysis between the photodiode signals andthe test plate oscillations, a laser vibrometer from

    the company Polytech was used as a reference in theearly phase of the project. The laser vibrometer sig-nal is also connected through the acquisition card tothe computer. Directing the laser spot onto the me-chanical holder of the test plate, we obtain a quanti-tative evaluation of the test plate oscillation in thedirection of the laser vibrometer beam. In Fig. 4the analog signals from the photodiodes are showntogether with the vibrometer signal on the same timeaxis. In this special case a sinusoidal oscillation with100Hz frequency was generated using the piezoelec-tric actuator. The signals have an arbitrary intensity

    Fig. 3. (Color online) (a) Detector system consisting of threephotodiodes. (b) The three corresponding noncollinear measure-ment points P1, P2, and P3 on the test surface which define theoscillating plane .

    20 December 2011 / Vol. 50, No. 36 / APPLIED OPTICS 6567

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    5/12

    bias. According to Eq. (1), the frequency modulation

    character of the signals can be observed. We can alsoidentify on the photodiode signals the particularextreme values that are related to the extreme posi-tions of the test plate, and we call them main extremepoints.

    Between two main extreme values, the signalspresent a series of maxima and minima, concludingthat phase-shifts up to several wavelengths are in-troduced. The value of the introduced phase-shiftis a realistic assumption for the movement of the testplate under the influence of room vibrations. Thiswas proved by experiments with random oscillationsof the test plate caused by vibrations. A feature that

    has to be mentioned is the synchronous occurrence ofthe main extreme points at the three signals, a factthat proves the rigid-body oscillation of the plate andthe dominance of a single bending mode of the postholder. The intervals defined by consecutive extremepoints of the signals are related to =4 shifting ofthe three corresponding sampling points on the testsurface. The signals present unequal numbers ofextrema during a period of oscillation of the plate.This is caused by the different heights of the sam-pling points and proves the tilt oscillation of theplate about a rotation axis that is located outsideof the plate.

    D. Description of the MethodThe time dependency of the intensity measured bythe photodiodes can be described as

    Iit I0i I

    00i cosi it; i 1;2;3;

    it 4

    zit; 2

    where i is the photodiodes index, it is the timevarying phase-shift introduced to the test beam cor-responding to the three sampling points, andzit are

    the related shifts. The Eq. (2) enables the modulo2determination of the phase-shift time dependenceat the three points on the test surface. Without loss ofgenerality, consideringi 0 we obtain

    ijt cos1Iit I

    0i

    I00i; t ftjg; 3

    wherejassigns the time intervals tjdefined by con-secutive extreme values of the signals. The succes-sive deconvolution of the modulo phase from theEq. (3) provides the time variation of the phase overthe entire interval of the measurement. We can useforI0i the average value of the signal. But this processcorrectly functions only when the sameI00i would existalong the entire interval. The continuous variation ofthe fringes spatial frequency has influence on thefringe contrast and a slight amplitude modulation ofthe photodiode signals will always exist. But if we tryto find an average value for I00i during the entire mea-surement time, there is an imminent risk to get the

    ratio jIit I0=I00i j > 1 for some regions of the inter- vals. The noise of the signal can also lead to thisunwanted effect. In these cases the Eq. (3) cannotbe applied because the cos1 function would not bedefined.

    E. Fitting Process of the Photodiode Signal

    As a consequence of the amplitude modulation and ofthe noise, our approach is to fit, and further to nor-malize the signals. In case of pure sinusoidal orattenuated sinusoidal vibration oscillation, the fit-ting process may work on the entire measurementinterval. But we must take into account that in rea-

    lity we have random oscillations, and finding a prop-er fitting function becomes impossible. By properlydividing the signals in intervals, the fitting can beprocessed successively on each interval, see Fig. 5.First we subtract the average value I0i and then wedefine the intervals between two consecutive zerocrossing points.

    One may first tend to perform sinusoidal fits onthe intervals, but due to the frequency modulatedsignals, this would not work correctly. We apply apolynomial fit and, due to the fact that the signalspresent one or three maxima on each interval, thefitting process with polynomials Pijt of sixth-ordershould be accurate enough. With the increase of thefringe spatial frequency, the modulation in ampli-tude becomes observable, but at the same time thenumber of the extreme points increases accordinglyand the intervals get considerably narrower. Underthese circumstances, it can be considered with goodapproximation that during each interval Pijt thesignal contrast does not change.

    The absolute value of the fitting polynomials Pijtat the extreme points gives the value of the ampli-tudes I00ij for each j interval. Dividing now each Pijt

    polynomial by I00ij, we get all normalized polynomials

    Fig. 4. (Color online) The simultaneous photodiode signalsand the vibrometer signal indicating correlation between the mainextreme values and the extreme positions of the oscillating testplate.

    6568 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    6/12

    PNijt. After this we can apply the Eq. (3) in the

    form

    ijt cos1PNijt; 4

    obtaining the modulo variation of the phase shift,for the related sampling point on the test surface,during the entire measurement time. By deconvolu-

    tion of the modulo output the real variation in timeof the phase-shift at the sampling point is obtained.

    In Fig. 5 the intervals in which the phase result isadded or subtracted, can be distinguished.

    Knowing the movement of three surface points intime and assuming the test plate to move as a rigidbody the calculation of the local phase shift x;y; tas a function of time is straightforward.

    Knowing the position of three points of the shiftedplane they can be inserted into the plane equation:

    A XB Y C Z D; 5

    Fig. 5. Deconvolution of the modulo phase, resulting in the phase variation at the sampling point on the test plate.

    20 December 2011 / Vol. 50, No. 36 / APPLIED OPTICS 6569

    http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    7/12

    A, B, C define a vector normal to the plane, D theperpendicular distance between the plane and theorigin. X, Y, Z are the coordinates of a point lyingin the plane. With the three points X1;Y1;Z1,X2;Y2;Z2, X3;Y3;Z3 of a plane, the equationcan be solved by calculation ofA, B, C, and D usingthe determinants

    A

    1 Y1 Z1t

    1 Y2 Z2t

    1 Y3 Z3t

    ;

    B

    X1 1 Z1t

    X2 1 Z2t

    X3 1 Z3t

    ;

    C

    X1 Y1 1

    X2 Y2 1

    X3 Y3 1

    ;

    D

    X1 Y1 Z1t

    X2 Y2 Z2t

    X3 Y3 Z3t

    : 6

    Having only small amounts of tilt (about 5 105) ofthe test plate, the in-plane movement of the surfaceis neglected and only the rotational out-of-planemovement is considered to create phase-shifts.Hence the ZX;Y; t coordinates of the plane repre-sent the phase-shifts due to vibration at every pointof the testplate,

    ZX;Y; t D AXBY

    C: 7

    In the same manner the phase-shift of the inter-

    ference images for every pixel on the camera canbe evaluated. The whole plane movement of the testsurface can be calculated due to the measurement ofthe plane movement or phase-shift in three pointsby the photodiodes. Let x1;y1, x2;y2, and x3;y3be the pixel coordinates of the sampling points onthe CCD sensor and the Z1, Z2, Z3 coordinatesare replaced by the phase-shifts 1t, 2t, 3tin Z-direction. Applying Eq. (5) with the mea-sured phase-shifts 1t, 2t, 3t, the determinantsare:

    M

    1 y1 1t

    1 y2 2t

    1 y3 3t

    ;

    N

    x1 1 1t

    x2 1 2t

    x3 1 3t

    ;

    O

    x1 y1 1

    x2 y2 1

    x3 y3 1

    ;

    P

    x1 y1 1t

    x2 y2 2t

    x3 y3 3t

    : 8

    The resultant phase-shift at each x;y pixelbecomes:

    x;y; t P Mx Ny

    O: 9

    Fig. 6. Measured photodiode signal dependent on the apertureposition of the calibration mask in front of the sensitive photo-diode area. (a) Shows the signal with partial overlapping betweenaperture and sensitive photodiode area, (b) with maximumoverlapping.

    6570 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

    http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    8/12

    In consequence, according to the Eq. (1) this en-ables the application of a PSI algorithm for randomphase-shifts x;y; t.

    F. Calibration of the Interferometer

    Before the measurement, the interferometer was ca-librated by determining the x, y position of the threephotodiodes which measure the phase 1t, 2t,3t. Therefore a calibration mask is applied betweenreference and test plate crossing the beam path. Thismask has an aperture with a diameter, which is equalto the width and height of the photodiode sensitivearea. The light is reflected only through the apertureof the mask from the test plate. Thus interference canbe recognized only at the location of the mask aper-ture on the CCD sensor. The photodiode can recognizeinterference in a case of the aperture overlapping[Fig. 6(a)]. If the mask aperture is accurately adjustedin x- and y-directions, the photodiode shows maxi-mum intensity for the interference [Fig. 6(b)]. Afteradjustment, the aperture position, respectively the

    photodiode position, has to be found. For this purposeone image from a sequence is chosen, on which abright interference fringe overlaps the aperture.Hence a bright spot with aperture size at the photo-diode position appears on the CCD sensor. An algo-rithm for ellipse finding gives the x- and y-positionsx1;y1, x2;y2, x3;y3 in pixels of the center pointon the CCD sensor with subpixel accuracy.

    G. Four-Step Algorithm with Random Phase-Shifts

    Since the basic Eq. (1) of phase-shifting interferome-try has three unknowns, a system of at least threeequations is necessary. However, an algorithm with

    more than three equations will reduce the sensitivityto errors in the phase-shifts. We propose in our casean algorithm with four equations. Let us considerfour interference frames recorded at the momentstk, with k 1, 2, 3, 4 and the random phase-shiftsat each pixel kx;y. The following system results[12]:

    I1x;y I0x;y I00x;y cosx;y 1x;y;

    I2x;y I0x;y I00x;y cosx;y 2x;y;

    I3x;y I0x;y I00x;y cosx;y 3x;y;

    I4x;y I0x;y I00x;y cosx;y 4x;y; 10

    with three unknowns: I0x;y, I00x;y, x;y for eachpixel. Since the phase-shifts are random, the systemdoes not simplify as in conventional PSI. The analy-tical solution can be given as

    x;y tan1Rx;yc34x;y c12x;y

    s12x;y Rx;ys34x;y

    ; 11

    with

    Rx;y I1x;y I2x;y

    I3x;y I4x;y;

    c12 cos 1x;y cos 2x;y;

    s12 sin 1x;y sin 2x;y;

    c34 cos 3x;y cos 4x;y;

    s34 sin 3x;y sin 4x;y:

    12

    The mod 2ambiguities are then removed by apply-ing an unwrapping algorithm as explained in detailin Greivenkamp [1]. The measured wavefront x;yis related to the height profile hx;y of the testedsurface

    hx;y

    4x;y: 13

    4. Noise Sensitivity

    This section deals with specific noise and its origin,

    which can appear during the analysis of the realsurface topography from the taken interferograms,that leads to invalid data in some points. It is furtherdescribed how to eliminate this noise by replacementof the invalid data points by valid data points.

    According to the phase-shift algorithm describedabove, the first measurements were performed by re-cording four interference images with random phase-shifts, shown in Fig. 7.

    For the topography analysis with the phase-shiftalgorithm, only the relative phase-shifts betweenthe interference images are relevant. To obtain therelative phase-shift for every pixel it is possible to

    subtract one of the phase planes from all the otherphase planes. The result is a quantitative analysisof the wavefront. Subtracting e.g. the first phaseplane x;y; t1, where tn depicts the acquisition timefor the respective interference image, the phasesbecome

    x;y; t11 x;y; t1 x;y; t1 0;

    x;y; t21 x;y; t2 x;y; t1;

    x;y; t31 x;y; t3 x;y; t1;

    x;y; t41 x;y; t4 x;y; t1: 14

    The result for the modulo 2 phase map is the firstinterference image. Because the first phase plane issubtracted we call it map 1234. The solution of the

    Fig. 7. Measurement set with four sequentially recorded interfer-ence images with random phase shifts.

    20 December 2011 / Vol. 50, No. 36 / APPLIED OPTICS 6571

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    9/12

    equation system (10) provides the raw map of themeasured wavefront shown in Fig. 8(a), with discon-tinuities at =2 to =2.

    To generate the real surface topography of the mea-sured test surface by the unwrapping algorithm, firsta conversion of the =2 phase map to a wrappedphase map with the interval 0 to 2had to be imple-mented. The phase jumps of this wrapped phase mapshown in Fig. 8(b) characterize the intensity maxi-

    mum of the interference image. By comparing thewrapped phase map with the first interference image,it is obvious that the position of the intensity maxi-mum fits exactly to the position of the phase jumps.That is a first proof for the validity of the measure-ment results generated by the algorithm.

    For the map 1234 the whole phase map has validdata without any gaps. But most often it appearsthat in some regions of the phase map a significantnoise influence leads to invalid surface information.This case is shown in Fig. 9 where the tilt angle be-tween reference and test plate is related to the fourthinterference image, the phase of which is subtractedfrom all the others and thus results in the map 4123.

    Further investigation showed that the noise iscaused by the argument of the tan1 function thatis used for the solution of the measured wavefrontin Eq. (11). In Fig. 9 the values for the nominatorand denominator over the image pixels are shown.The intensity signals, which form the nominatorand denominator (Eq. (12)), are effected by a highfrequency noise that appeared during the image ac-quisition process. The errors on the phase map occurin case, when both, the values of the nominator anddenominator are close to zero. Thus the noise has asignificant influence on the result of the tan1 func-tion because in a very small interval the phase can e.

    g. change very often the algebraic sign and varybetween large or small values. Taking into account

    Fig. 8. (a) Direct raw phase map solution 1234 of the algorithm

    given by the Eq. (11). (b) Wrapped phase map 1234 with values inthe interval 0; 2.

    Fig. 9. Phase map 4123 with the calculated nominator Nand denominator D over a pixel column xi. The interval at which both Nxi;yand Dxi;y present small absolute values is characterized by nonaccurate data points.

    6572 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    10/12

    the Eqs. (12), it is also obvious that it is possible toget undefined values for two cases. First case istan10=0 if the first image pair has the same inten-sity values at x and y position with an integer num-ber of2phase difference. Second case is tan1=if the second pair of images has the same intensityat x and y position. In conclusion, four interferenceimages are not sufficient to provide accurate results

    across the entire test surface. Each measurementmay present regions of nonaccurate data.

    To overcome this problem it is possible to evaluateseveral phase maps with different error regions andcut out the invalid data points. Followed by aver-

    aging the grey values of the correspondent pixelsover all phase maps, the error regions will be filledwith obtained valid data. This is done by using theapproach applied in the last example. By just chan-ging the order of the interference images the result-ing phase maps show the invalid data points indifferent positions. By taking e.g. six instead of fourimages, fifteen different combinations for four inter-ference images are possible. This is already equiva-lent with fifteen multiple measurements with fourimages. The advantage of this method compared to

    Fig. 10. Several modulo 2 phase maps for different combina-tions of four out of six interference images. The last imageillustrates the final unwrapped result.

    Fig. 11. (Color online) Surface topography of the test plate, measured by random-phase-shift interferometry.

    Fig. 12. (Color online) Surface topography of the test plate,measured by the manufacturer.

    20 December 2011 / Vol. 50, No. 36 / APPLIED OPTICS 6573

    http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    11/12

    the multiple measurements method is that everyphase map is related to the same tilt angle of the testplate for four images. As told before, the subtractionof the tilt involves the subtraction of the fitting planeequation of the phase map. Consequently, the fittingplanes may not exactly describe the tilts, since eachmeasurement presents different domains with lackof data points.

    In Fig. 10 seven modulo 2phase maps with differ-ent combinations of four interference images from

    totally six taken images are shown. The combina-tions are always related to the first interferenceimage.

    The error regions with invalid data points werefirst identified by a threshold value of 0.3 for the ab-solute value of nominator N and denominator D,

    Nx;y jRx;yc34x;y c12x;yj < 0:3;

    Dx;y js12x;y Rx;ys34x;yj < 0:3; 15

    and finally cut out. An image fusion technique isapplied for all raw phase maps in Fig. 9 that allowsobtaining the complete raw phase map. This is done

    by averaging the valid image information pixel bypixel over all raw phase maps. In this manner theerror regions are filled with the average values ofthe remaining valid raw phase image information.The last image depicts the complete unwrappedsurface map with the error regions filled with validdata points.

    5. First Measurement Results

    Figure 11 shows the surface topography map ofour test plate. It is in good agreement to the data

    supplied by the manufacturer of the test plate(Fig. 12). The peak-to-valley distance measured bythe random-phase-shift interferometer is 0.134 waveand 0.125 wave measured by the manufacturer.It has to be mentioned that in our interferometerthe mounting ring of the test plate reduced the visi-ble aperture of the test plate by 4mm diameter,so the outer rim of the test surface in Fig. 12 isnot visible in Fig. 11.

    Our laboratory setup was built from off-the-shelf

    components and is by no means optimized. The focusof our work was to investigate the repeatability of theresults. This is not primarily governed by the qualityof the optic but by the capability of the evaluationtechnique.

    We repeated the measurement ten times withintwo working days to get an estimation of the long-term stability. Between the single tests there werealways time lapses of a few hours. The test plateremained in the mounting system and was adjustedbefore each measurement. The root-mean-square(RMS) over the ten different measurements wascalculated. According to the definition, the RMSerror function was applied to each data point x;yas follows:

    RMSx;y

    ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

    N 1

    XNi1

    hx;y hx;y2;

    vuut 16

    where hx;y is the surface height, x;y is the aver-age height value at each pixel, and N 10 is thenumber of measurements. The RMS map is shownin Fig. 13. Analyzing the map, an RMS 0; 004can be evaluated, corresponding to approximately

    Fig. 13. Distribution of the local RMS error, demonstrating the repeatability of random-phase-shift interferometry over two days.

    6574 APPLIED OPTICS / Vol. 50, No. 36 / 20 December 2011

    http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-
  • 8/2/2019 (2011) Random-Phase-shift Fizeau Interferometer

    12/12

    2:5nm for the 632:8nm laser wavelength. The fewspots of higher RMS values can be assigned to dustparticles or multiple reflections.

    6. Conclusions

    Random-phase-shift interferometry has beendemonstrated that has the potential of offering a low-cost alternative to sophisticated concepts of simulta-neous-phase-shifting interferometers. Compared toconventional PSI systems not only the vibrationisolated table can be omitted but also the precisiondevice for introducing well-defined phase-shifts. Im-plemented on a quite simple experimental Fizeausetup our first results show excellent repeatability.This encourages us to continue the work. Our currentfocus is on developing a calibration procedure totransfer repeatability to measurement uncertainty.

    The authors gratefully acknowledge project fund-ing by Deutsche Forschungsgemeinschaft (DFG)under grant TU 135/15.

    References

    1. J. E. Greivenkamp and J. H. Bruning, Phase shifting inter-

    ferometry, in Optical Shop Testing, D. Malacara, ed. (Wiley,2007), pp. 504505, 514515.

    2. J. Hayes, Dynamic interferometry handles vibrations,in Proceeding of Laser Focus World, March 2002, pp. 109113.

    3. L. Wizinowich, Phase shifting interferometry in the presenceof vibration: a new algorithm and system, Appl. Opt. 29,32713279 (1990).

    4. C. Koliopoulos, Simultaneous phase-shift interferometer,Proc. SPIE 1531, 119127 (1992).

    5. J. E. Millerd, N. J. Brock,and J. B.Hayes, Modern approachesin phase measurement metrology, Proc. SPIE 5856, 1422(2005).

    6. H. Kihm, A point-diffraction interferometer with vibration-desensitizing capability, Proc. SPIE 6293, 62930B (2006).

    7. R. Smythe and R. Moore, Instantaneous phase measuringinterferometry, Opt. Eng. 24, 361364 (1984).

    8. J. E. Millerd, N. J. Brock,J. B.Hayes, and J. C. Wyant, Instan-taneous phase-shift point-diffraction interferometer, Proc.SPIE 5531, 264272 (2004).

    9. N. Brock, J. B. Hayes, B. Kimbrough, J. E. Millerd, M. B.North-Morris, M. Novak, and J. C. Wyant, Dynamic interfero-metry, Proc. SPIE 5875, 58750F (2005).

    10. J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, Pixelated phase-mask dynamic interferometer, Proc. SPIE 5531, 304314(2004).

    11. B.Kimbrough, J. E. Millerd, J. C.Wyant,and J. B.Hayes, Lowcoherence vibration insensitive Fizeau interferometer, Proc.SPIE 6292, 62920F (2006).

    12. N.Doloca and R. Tutsch,Random phase shift interferometer,

    inProceeding of Fringe 2005The 5th International Workshopon Automatic Processing of Fringe Patterns, W. Osten, ed.(Springer Verlag, 2005), pp. 167174.

    20 December 2011 / Vol. 50, No. 36 / APPLIED OPTICS 6575