K. Kinnstaetter, Adolf W. Lohmann, Johannes Schwider, and Norbert Streibl
The accuracy of phase shifting interferometers is impaired by mechanical drifts and vibrations, intensityvariations, nonlinearities of the photoelectric detection device, and, most seriously, by inaccuracies of thereference phase shifter. The phase shifting procedure enables the detection of most of the errors listed aboveby a special Lissajous display technique described here. Furthermore, it is possible to correct phase shifterinaccuracies by using an iterative process relying solely on the interference pattern itself and the Fourier sumsused in phase shifting interferometry.
1. Introduction
Phase shifting interferometry (PSI) is a highly effi-cient and accurate phase measuring method, especiallyif 2-D phase distributions are to be measured.1 -3
Typical applications are the measurement of surfacedefects in the macroscopic 4 as well as in the microscop-ic range.5 The accuracy of the detection method islimited by several effects. The accuracy of the refer-ence phase shifter poses one of the most serious limita-tions, and several control and calibration methods forthe reference phase shifter have been published.6 -9
Furthermore, compensation techniques have also beendescribed which enhance the accuracy.8
The effective reference phase shift is not only deter-mined by the movement of a PZT-driven mirror orother phase shifting devices but also by any change ofthe optical path difference. Therefore, error sensingand compensation relying solely on the interferencepattern itself seem to be a worthwhile endeavor. Thispaper presents a new error detection and compensa-tion method relying on the interference pattern itselfand the well-known PSI procedures. The PSI tech-nique uses an R-step phase shift procedure per periodof the interference pattern.1'6 The procedure to begiven is equally applicable to ramp-oriented or inte-grating mode PSI techniques.2
As an introduction to the following methods thebasic equations for PSI are briefly derived. The two-
The authors are with University of Erlangen-Nuremberg, PhysicsDepartment, Erwin-Rommel-str. 1, D-8520 Erlangen, Federal Re-public of Germany.
Received 24 June 1988.0003-6935/88/245082-08$02.00/0.
beam interference pattern is a typical cosine-type ex-pression 9:
I(xy) = IO(x,y)f1 + V(x,y) cos[4(xy) - a (1)
where Io is the mean intensity, V is the visibility afterMichelson, 4' is the phase to be measured, (, is anarbitrary reference phase value, and (x,y) are the coor-dinates in the exit pupil of the interferometric setup,where the fringe pattern is usually detected.
The PSI method uses the arbitrariness of the refer-ence phase to generate a set of R-intensity values Ir byvarying *°r in an R-step procedure over at least oneperiod of the interference pattern:
p, = (r-1)27r/R, with r = 1,2,. _R. (2)
Considering only one point (xo,yo) in the exit pupil, Eq.(1) can be written as
Ir= 1 + IOV cos() - Pr)
= IO + IOV COSb Cor + IOV sinl sinPr. (3)
The set of measured Ir values is processed for PSI inthe following way: first we multiply each equation bycostr and sinsor. Then we sum with respect to r and usethe orthogonality of the trigonometric functions:
This is the basic equation of the PSI technique, and itdescribes most of the proposed methods. But othermethods, such as that given by Carre, 8 are mathemati-cally equivalent and thus have similar problems sincethey also rely on the accuracy of the reference phasevalues.
Since the different reference phase values are ad-justed sequentially, Eq. (5) can be interpreted as asynchronous detection technique,' where the basic fre-quency Q is the inverse of the time that elapses be-tween first and last steps of the PSI procedure.Therefore, all possible mechanical frequencies may beexpressed as multiples of the measuring time T. An-other characteristic time of the process is the integra-tion time r of the detecting array, since these devicescommonly work in the integrating mode for energyreasons.
II. Error Sensing by a Lissajous Display Technique
So far the reference phase values fPr were assumed tobe exact, but in practice they deviate by small values Er
from the theoretical values resulting in physical valuesStr:
'Pr = Pr + er (6)
The deviations can have different origins as there canbe
wrong calibration of the phase shifter in the sensethat the step width A(, is too large or too small;
nonlinearities of the phase shifter, e.g., in the case ofa PZT-driven mirror the relationship between ser andthe applied voltage is monotonic but nonlinear;
any phase shift during the data gathering process,e.g., mechanical or thermal drifts, and vibrations.
We refer to all these as e-type deviations.The PSI evaluation procedure additionally assumes
the constancy of Io and even IoV. Furthermore, theintensities Ir are measured photoelectrically. Due tothe photoelectric detection the photoelectric voltageUr has to be substituted into Eq. (5) instead of Ir. Thismay cause errors if the detector response is nonlinear.The latter deviations are called I-type deviations. Inthis section a procedure for sensing a- and I-type devi-ations is given.
It is assumed that a fringe pattern is adjusted in theinterferometer (about 1-3 fringes per diameter of theexit pupil). Then two detector elements out of thearray detector being approximately in phase quadra-ture are selected. The selection can be done, e.g., byinteractive software. Their photovoltages are dis-played in the following manner: the photovoltage U1of detector 1 is displayed as the abscissa, and thephotovoltage U2 of detector 2 is displayed as the ordi-nate of a Cartesian coordinate system.
Displaying means that the tip of the vector (U1 ,U2)is indicated by a bright dot on an electronic screen.The bright dots form a Lissajous figure which degener-ates in the simplest case to a circle if the phase differ-ence of the detectors is 7r/2 and mean intensity, visibili-ty, and sensitivity of the detecting elements at the twopositions are equal.
Let the intensities at the two locations be Ilr and I2r
where 'Pr is the actual reference phase value differingfrom the mathematically given ideal reference phasevalues sor by Er.
The phase difference between 4', and 4'2 is approxi-mately r/2. The r are the real phase variations em-bracing the movement of the PZT-driven mirror aswell as mechanical relaxations, vibrations, and ther-mal drifts.
The Lissajous display is a convenient diagnostic toolfor detecting the following features:
If the phase steps are not equidistant (e.g., due to thenonlinearity of the phase shifter-for PZT the hyster-esis and nonlinear characteristic curve), the distancesof the dots on the rim of the Lissajous figure are notequidistant.
If the calibration of the phase shifter is incorrect thebright dots on the rim of the Lissajous figure leave anopen gap or overlap in a part of the figure.
If the mean intensity of the interference patternvaries during the data accumulation time the Lissajousfigure shows higher harmonics and indicates, e.g., laserintensity variations.
If the interference pattern is disturbed by mechani-cal vibrations the dots (Uir,U2r) are either for lowfrequency disturbances displaced on the rim or theydeviate in the direction of the center of gravity of theLissajous figure for high frequency disturbancescaused by averaging.
If the detector elements have a nonlinear character-istic, deviations from simple Lissajous geometries (cir-cle or ellipse) occur.
The display of U1 r,U2r can be supplemented by alsodisplaying the averages:
R R
1IR E Uir, 1IR E U2r, (8)r1l r1l
making the asymmetries of the Lissajous figure moreobvious. This point in the U1, U2 plane is the center ofgravity of the Lissajous ellipse since with good approx-imation the averages are proportional to the meanintensities I10,I20.
Here only small vibration amplitudes are consid-ered, where small means amplitudes that are smallcompared to the wavelength of light.
Low frequency vibrations are indicated by systemat-ic or statistical variations of the dot positions on therim of the Lissajous figure. The characteristic cutofffrequency is Po = 1/r, where r is the integration time ofthe detector. In the case of high frequency vibrations,i.e., v >> vo, the integration time leads to an averagingeffect for the modulation of the interference term lead-ing to indentations of the dot positions in the directionof the averages over one period, or, what is the same, inthe direction of the dc term of the interference pattern.
In mathematical terms vibrations and similar effectscan be taken into account by adding a time-dependentphase term to the argument of the cos term:
This phase term can be decomposed into a Fourierseries:
v(t) = 2r/X >7 an sin(n2t + an),n=O
(10)
with = 27r/T, where T is the time needed for onemeasuring cycle. This frequency is the lowest possibledue to the measuring procedure and is well below =2fr/T.
Considering one single Fourier component and de-noting n = , an = a, and an = a, for the timeaverage of the photovoltages one obtains
(Uir)/Ui 0 = 1 + V[cos(Qj - 'P,)Jo(ak)
- 2 sin(4) - r)Ji(ak) J sin(wt - a)dt]
(14)
(U2 ,)/U2 0 = 1 + V2 [cos(42 - r)Jo(ak)
- 2 sin(s 2 - 'r)Ji(ak) J sin(wt - a)dt],
where Jo and J1 are Bessel functions of the first kindand an uninteresting common factor has been omitted.
Equation (14) can be written as a single trigonomet-ric term with an amplitude factor and a phase sum-mand using the addition theorems of the trigonometricfunctions:
(U) = (U011 + VJ' + 4J1 sin 2(t - a) COS(I - 'P + 01),
(15)tanf: = 2J1 (ak)(sin(wt - a))/J0 (ak).
This form of Eq. (15) suits our discussion rather wellsince the positions of the dots on the Lissajous curveare also determined by an amplitude factor and aphase term.
The square root factor can be approximated by
- a)dt,
(12)
For high frequency vibrations this expression becomesvery simple because all time averages containing trigo-nometric functions cancel. The frequency must begreater than the inverse of the integrating time of thedetector; with the typical TV norm the frequencieshave to be greater than -20 Hz. The physical effect ofsuch vibrations is a loss in visibility since the photovol-tage becomes
(U,) = Uo[1 + VJ0(ka) cos(I -'pr)]. (13)
This type of vibration does not seriously influence theattainable accuracy. For small vibration amplitudesJo can be approximated by [ - (ak)2 /4].
For frequencies v < vo the situation is more compli-cated and general statements are difficult to make.Keeping this restriction in mind one can consider smallvibration amplitudes and use the fact that the Besselfunctions form a null series in the neighborhood ofzero, i.e., the higher harmonics have forefactors whichform a null series. In addition the time average overthe higher harmonics also tends to zero, since the har-monic frequencies eventually reach the frequency vo,resulting in a cancellation of their contribution to thephase error. Therefore, we can restrict the discussionto the basic frequency and we get the following resultfor the photovoltages:
+ .. J(ak) + 2J1(ak)(sin2(t -a))/JO(ak). (16)
Due to our assumption, frequencies v < v must now bediscussed. The amplitude factor varies between Joand Jo + 2Jl2/Jo because of the variation of (sin 2(Cot -a)) between 0 and 1. As the mean expectation valuefor the amplitude factor we assume the time averageover (sin 2(ct - a)) = 1/2. With this assumption therelative variation o- of the amplitude factor becomes
= 2J2/(J2 + J2). (17)
In a first approximation we set Jo 1 - (x/2)2 and J (x/2) [1 - (x/2)2]. Substituting these approximationsinto Eq. (17) o- becomes
In a similar manner phase fl can be estimated. Themaximum value of this phase error follows from Eq.(15) by assuming that (sin2 (Wt - a)) = 1:
Itan/lmax = 2IJ11/J0 ak. (19)
For values ak << 1 the amplitude factor varies onlyslightly while the phase variation caused by low fre-quency vibrations has to be taken into account. Sincemechanical disturbances show statistical behavior,their effect on the accuracy of a phase measurementcan only be diminished by proper mechanical design ofthe interferometer and by taking averages over manyindependent phase measurements.
Ill. Iterative Algorithm to Determine the ReferencePhase Values
The aim of the following algorithm is to improve theset of reference phase values 'Pr with the help of timeaverages and averages in the spatial domain. Theaverages are derived from the intensity values of theactual interference pattern. Averaging in the timedomain means that the summation is carried out overan ensemble of values covering one period of the step-ping process; averaging in the spatial domain means asummation over values spread out in the x,y exit planeof an interferometer. As a first step the time domainaveraging process is now explained.
We assume that the photoelectric voltage Ur of thedetector array is proportional to the intensity, i.e.,
Ur = aIr. (20)
In a first approach it is assumed that a set of approxi-mately correct reference phase values already exists.The photovoltages of the two detector elements beingin approximate phase quadrature are
UIr = U01[1 + V1 CoS(4)1 - r)],(21)
U2r = U02[1 + V2 COS('12 - 'r)],
where 'r are the actual reference phase shifts and 4)1A?2
the object phase at positions P1 ,P2.The phase values 4b can be estimated by
R R \
arctan Ur sin'Or/ Ur cos'Pr = 4)(modvr) (22)
rlr=1 or 4) mod27r, if the sign information of the arguments isused in addition. The sor are the ideal mathematicalvalues. Furthermore, one can define the following
Phase jumps occurring in the evaluation of Eq. (27) aretreated by checking the phase difference of adjacent rvalues and correcting them accordingly. The actualphase shift r differs from the wanted shift (Pr by Er:
'Pr - , = Er, (28)
where the e, values are small corrections. With thehelp of these small deviations a new set of values isgenerated: (1) = O() - a/q, where q > 1 to ensureconvergence of the algorithm. We chose
q = 1.2, i.e., P(l) = P(o) - c/1.2. (29)
This procedure can be repeated several times until thecorrections become negligible.
Since the r phase shifts are generated, e.g., by themovement of a PZT-driven mirror, the correctionsmust be provided as voltages for the piezotransducer.The first phase value is arbitrary and can be set to acertain value. For r-values >1 the voltages are cor-rected in a similar manner as given by Eq. (29):
Vj1) = VO) -V'(). (30)
The starting set of PZT voltages v) can be found bytechniques described elsewhere.6 A great help forfinding the first set can be the Lissajous figure becausebig calibration errors show clearly as a gap or as anoverlap in the Lissajous figure and can be easily cor-rected. If the corrections are displayed as a curve thedegree of error compensation can be estimated.
In a previous publication6 it was shown that incor-rect reference phases 'Pr cause b-dependent errors:
A = arctan er, - C cos21 - S sin2) /(R- C sin21 + S cos2c>),
quantities: A = (U1 r - U0 l)/Uol, and Br = (U2U02)/U 02 with the average values
R R
U01 = hIR U1r, and U0 2 = IR E U2r-r=l1 ~
By using Eq. (21) we have
Ar = V1 cOs(l - 'Pr); Br = V2 COS(42 - 'r)-
The values 4)1,4)2 are chosen in such a manner that i41) + 7r/2 + x, (where x << 7r/2). With this substituiEqs. (23) can be written as
Ar = V 1 cOS(41 - 'r); Br =-V 2 sin(4QI + X - 'Pr).
First estimates can also be derived for V1,V 2 fromaverages of a first run with provisional referephases:
(R Ur COSr).
r=1Ur) (, Ur sinor) +
Dividing Br by A, one gets
withR R
C = , er cos2'Pr and S = Eer sin2(Pr.r=1 r=1
This type of error is also present in the correction(23) process discussed above. Therefore, due to the 4)-
dependency of the error an averaging process over a2= whole number of periods in the spatial domain resultsLtion in a considerable improvement. This is an averaging
process in the spatial region in contrast to the use of(24) temporal averages to determine the other unknowns
V, Uo, etc.the The averaging can be carried out as follows: Sincence the fringe pattern is adjusted for the Lissajous display
anyway, corrections are estimated for a set of pointpairs being in phase quadrature along the scan line of
(25) the array detector. Within the approximations madethe following pixels of the detector will also be in phasequadrature and this will also be true for the followingpixels of the scan line in the case of weak wave aberra-
tions as is common in optical testing. The point pair isshifted as an ensemble across the fringe pattern. Foreach pair the corrections are calculated and then theiraverage value (of the, e.g., M values) is taken:
MC = 1/M grm-
m=l(32)
The resulting averages are stable against drifts of theinterferometer and independent of the mean phase inthe interferometer as required for a stable optimiza-tion algorithm.
IV. Influence of a Nonlinear Characteristic of thePhotoelectric Detector
In the following we assume that the photoelectricvoltage can be expanded into a power series:
U=aI+bI2 +cP +d14 +.... (33)Stetson and Brohinsky10 discussed the influence ofsuch nonlinearities for small numbers of phase stepsper period and published a table containing the degreeof nonlinearity causing phase errors for different num-bers of phase steps per period. They stated correctly,e.g., that for four-phase steps/period second-ordernonlinearities do not contribute to phase errors. Theycame to a similar conclusion for all even order nonlin-earities; this we cannot confirm as will be shown in aproof for fourth-order nonlinearities.
The four-phase step method is especially suitablefor evaluations with simple microcomputers, since ithas a simple form which enables the handling of bigdata sets gathered by, e.g., a TV-type CCD array.
To simplify the understanding, the formula forphase 4) is derived from first principles.
The intensity distribution for two-beam interfer-ence has been given as Eq. (1). For the four-phase stepalgorithm the reference phase values are steppedthrough the values 0, r/2, 7r, 37r/2. The corre-sponding intensities are
I1 =p + q cos-, I3 = p-q cost,(34)
I2= p + q sin4, I4 = p-q sinb.
Substituting Eqs. (34) and (33) into Eq. (5) yields thefollowing expression for 4), the phase to be measured,which differs slightly from the correct phase 4):tan~ =aI+-bI+cg+ dI2+...)-(aI4+bI2+cI +dI4+ ..
(al +b,+ c+ d4,+...) - (aI3+ bI+ cI+ dI43+ ..
By substituting Eq. (34) into the last expression andneglecting contributions higher than fourth order onecan derive the following expression:
tani = (kl sin4 - k3 sin34)/(kl cos + k 3 cos3f), (36)
with
k = a2q + b4pq + c(6p2q + 32q3) + d(8p3q + 6pq3),
k3 = cq3 /2 + d2pq3 .
This expression contains contributions from the thirdand fourth orders in the intensity as a forefactor of
SAMPLE J LREFERENCESURFACE SURFACE
Fig. 1. Schematic of a planeness Fizeau interferometer used in theexperiments.
sin34) and cos34) in the numerator and denominator,respectively. If only second-order nonlinearities arepresent the phase measurement is free from distor-tions. We find, however, that fourth-order terms doinfluence the phase to be measured, which is in con-trast to Ref. 10. The contribution of the fourth orderexists but is proportional to the third harmonic of thephase term. From Eq. (36) it can be deduced that forsmall contrasts of the interference pattern (q << 1) allnonlinear effects can be neglected, as should be expect-ed.
Coarse nonlinearities can be detected visually asdeformations of the Lissajous figure. Numerical val-ues of the constants a,b,c,d, ... can be derived from aleast-squares algorithm relying on the Fourier sums:
R R R
U E Ur sin'Pr, E U, cosPr,r=l r=l r=l
R R
E Ur sin2',.r, 2 Ur cos2',0r, etc.r=l I
The quantities Io, V, and can be derived from thefirst three sums with sufficient accuracy in the samemanner as given in Eqs. (22) and (25).
V. Experiments
In our experiments we used a Fizeau interferometerfor planeness testing (Fig. 1). One of the plane mirrorsis driven by a PZT ceramic. The intensity is detectedby a CCD camera with 512 X 512 pixels. The CCDcamera is connected to a 1-M byte memory.
(35)
For experimental verification of the theoretical pre-dictions several disturbances can be introduced. Tofind the optimum phase values )1,4)2 the followingprocedure has been chosen. A fringe pattern with onlya few fringes is adjusted and one scan line of the CCDarray is read into the computer memory. Figure 2shows a typical scan line. By convolving this scan lineintensity with a rect-wave function one period longhaving equal positive and negative portions, the firstderivative of the intensity distribution is generated."
Fig. 5. Intensity variations due to mode shifting of an unstabilizedHe-Ne laser during the warmup time (R = 40).
2504U2
200 +
150+
_ ' o o
0- .. 1 i ! 11 - - -F. I I I i I50 100 150 200- 250
Fig. 3. Lissajous figure for an almost ideal adjustment of all param-
eters of the interferometer and the PSI process. The number of
phase steps is R = 40. The central spot indicates the averages due toEq. (8).
2 0 U 2 OO 0 00 000
o
8o
oo
o
o
00
00 00~~~~
0 0 0 0 0~ ~ ~0.0
0 - . .. . . .
U
0 50 100 150 200 250
Fig. 4. Deformations of the Lissajous figure due to nonlinearities of
the photoelectric detector (R = 64).
Its zeros correspond to the extrema of the interferencepattern. From this procedure a suitable set of pointsbeing in phase quadrature is obtained.
I-type errors can be detected as deformations of theLissajous figure. Figure 3 shows the case of a nearlyerror-free Lissajous figure. Nonlinearities in the pho-toelectric detection lead to deformations of the ellipticLissajous figure, i.e., this figure becomes unsymmetri-cal (Fig. 4) and in extreme cases even clipped. Varia-tions of the mean intensity may occur if an unstabi-lized laser is used in the interferometric setup (Fig. 5).
e-type errors can be detected in the case of vibrationsor drifts. Figure 6 shows the decrease of the fringe
100.
5 0
O 1 -- Maa WMW M| eM || 0 50 100 150 200 250
Fig. 6. High frequency mechanical vibrations. Note the loss in
contrast of the interference pattern. The irregularities of the Lissa-jous figure are due to stochastic phase shifts in the interferometer
(R = 64).
contrast due to high-frequency vibrations as a reduc-tion in the size of the Lissajous ellipse. A typicalpicture for an uncalibrated phase shifter is shown inFig. 7(a). The open gap indicates that the single volt-age step selected is too small. As pointed out in Sec.III the PSI process can be used for a self-calibratingprocedure. In an iterative process the reference phasevalues are corrected, which is shown in Fig. 7(b). Inthis way the nonstochastic phase errors can be elimi-nated. This is evident when the e values are displayedin one graph [Fig. 7(c)]. The convergence of the cor-recting algorithm can be demonstrated even with ex-tremely unsatisfactory starting values for the refer-ence phase shifter. The correction algorithm succeedsafter several iteration cycles even if only a part of theLissajous figure is covered by the starting values [Figs.8(a)-(d)]. For a comparison the corrections e are givenin Fig. 8(e) and the corrections for the driving voltagesin Fig. 8(f). The ideal state of adjustment of the sovalues is reached if the corrections e tend to zero for allphase steps r.
VI. Conclusion
It has been shown that the most serious systematicerror sources of the PSI technique, i.e., intensity varia-tions during the data gathering time, reference phase
-0 0.20- 1I I a : I.... ... . ..0 5 10 15 20 25 30
Fig. 7. (a) Lissajous figure for an uncalibrated phase shifter butwith linear slope of the piezovoltage (R = 32). (b) The same as Fig. 6but after one iteration cycle. Note the closing of the gap in theLissajous figure. (c) Display of the E-curves for the runs document-ed in (a) lower curve and (b) upper curve. The ordinate values aregiven in radians and the abscissas values represent the phase step
number.
errors, vibrations, and nonlinearities of the photode-tector can be detected and diagnosed with the help of aLissajous display technique.
While reference phase errors can be eliminated by aself-calibrating algorithm developed on the basis of thePSI algorithm itself, the interferometric setup has tobe safeguarded against other influences. The self-calibration is carried out iteratively. The iterativeprocedure relies only on measured intensity values andtheir Fourier sums. The iterative process convergesagainst the ideal values even with strongly misadjustedstarting parameters.
U2 250-
200-
150-
100.
50-
0
I
o~
(a) °O 00 0 U'
0 50 100 50 200 250
U2
o o
o oo o
0 o
(b) .. . . . .0 .... 50 ... 10 150 200
250 U2
200 +
150-
100-
50
250
oo 0
* o 0000r~ . .I
0 5 0 , . .1 ..5 . .. 25_
0 0 0 0 0
Cd) 0 0 0
0 50 o00 i50o oo
0.
0.
-0.
0 -0.
0 - 0.
-0.
- .
a i-U,250
. (rad)
after 8 ierations0 ,
2 -
a~ I
o. 3 arlius a ues ,
I(e) -2 i
2200 +
VPZT
after 8 iterations
after 5 iterations2000 -.
1800'
1600'
after I iteration
'starting values4(f)
0 tO o 0 io -- 30 40
Fig. 8. Demonstration of the convergence of the phase correcting algorithm underlying Eqs. (28)-(30) (R = 40): (a) starting sequence ofvoltages is rather unsatisfactory, (b) the same after one iteration cycle, (c) the same after five iteration cycles, (d) the same after eight iteration
cycles, (e) comparison of -curves for runs (a)-(d), (f) same as (e), but for the driving voltages of the PZT.
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D. Psaltis of the California Institute of Technology, photographedby W. J. Tomlinson of Bellcore at the Topical Meeting on Photonic
