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Phase-shift Fizeau interferometer in presence of vibration

Radu Doloca, Hagen Broistedt, Rainer TutschInstitut fr Produktionsmesstechnik, Technical University of Braunschweig,

Schleinitzstr. 20, 38106 Braunschweig, Germany

ABSTRACT

In the context of this article we demonstrate a novel Fizeau interferometric system that copes with the presence of

vibrations. Besides the conventional high spatial, but low temporal resolution detector system (the CCD camera) used inphase shifting interferometry, an additional high temporal, but low spatial resolution detector system was integrated, in

order to measure the random phase shifts that are induced under the influence of the vibrations. The additional sensor

consists of three photodiodes. The acquired analog signals enable the measurement of the occurring phase shifts at threenon-collinear locations on the test surface. The resulting phase shifts at the three individual locations enable the

determination of the random phase shifts over the entire image aperture. To avoid the smear phenomenon at very short

exposure time, a beam shutter was integrated. Another alternative is to integrate a pulsed laser diode, for this purpose the

concept of a wavelength meter is proposed. While the random oscillations of the test object are continuously measured,

the CCD camera acquires several interferograms. In consequence, a phase shifting algorithm for random phase shifts wasapplied. In order to proof the validity of the new interferometer, a test surface of known topography was measured. The

results of the measurements in presence of vibrations show very good concordance with the surface data given by the

supplier. The analysis of the root mean square (RMS) over ten different measurement show a measurement repeatability

of about 0.004 waves (approximately 2.5 nm for 632.8 nm laser wavelength).

Keywords: phase shifting interferometry, Fizeau interferometer, vibrations

1. INTRODUCTIONThe interferometric measurement techniques provide the most accurate non-contact 3D measurement tests for reflective

surfaces and most particularly for high-quality optical components such as lenses, microscope objectives, camera

objectives, prisms, optical flats or even large telescope mirrors. The high accuracy of the measurements is related to the

wavelength of the light, more exactly to the wavelength of the laser beam, which is used as the measuring reference in allinterferometric measurements.

In most cases phase shifting interferometry (PSI) techniques are applied. The relation between the light intensity in the

interference field and the profile of the test surface is applied at each pixel of the CCD sensor array. In this way a

pixelated analysis of the test wavefront results. These methods require a sequence of interference images. By

sequentially shifting the reference plate with well defined steps by means of a piezo-actuator, several interference imagesat well known and predefined phase shifts are sequentially acquired. The sequentially recorded grey values are used to

calculate the 3D information of the test wavefront at each pixel. Because the phase shifted interferograms are

sequentially recorded, and very precise phase shifts are required, the PSI methods prove to be very sensitive to vibration.This is why every PSI interferometer must be mounted and utilised on an optical table with vibration isolation, and even

so, the measurements have to take place under special laboratory conditions far from the manufacturing process. The

measurement accuracy of the PSI methods shows an uncertainty of about /1000 [9].

Nowadays there is a general tendency for integration of the quality testing in the manufacturing process. Asconsequence, there is an increasing demand of new interferometric concepts that make possible the utilisation of theinterferometers concomitantly and closely to the manufacturing process. In this context, the purpose of this work is to

develop and to verify a novel interferometric system that copes with the presence of vibrations in order to use actually no

vibration isolation. As it will be described, the influence of the vibrations is in fact deliberately and specifically used inthe measurement principle.

Interferometry XIV: Applications, edited by Erik L. Novak, Wolfgang Osten, Christophe Gorecki,Proc. of SPIE Vol. 7064, 706403, (2008) 0277-786X/08/$18 doi: 10.1117/12.794031

Proc. of SPIE Vol. 7064 706403-1008 SPIE Digital Library -- Subscriber Archive Copy

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H i g h t e m p o r a l r e s o l u t i o nd e t e c t o r s y s t e m

L 2

C W H e N e - L a s e r? = 6 3 2 . 8 n m

i r i g g e rs i g n a lV i d e os i g n a l

A p e u r e 1

S F

2. VIBRATION TOLERANT METHODSThe data acquisition time in sequential PSI takes several frame times of the electronic camera, adding up to about 100ms.

The frequency spectrum of mechanical vibrations in buildings typically is dominated by the region between 20 Hz and200 Hz [1]. Conventional PSI therefore is very sensitive to floor vibration, making expensive vibration isolation

equipment necessary. In the last few years new PSI techniques have been developed that can be applied in the presence

of vibrations. The basic idea of all these new techniques is to record the sequence of interferograms in a very short timeinterval [2] or even simultaneously [4], [5], [8] resulting in the effect of shifting the sensitivity of the system to higher

frequencies.

Instantaneous phase-shifting techniques use polarisation components [7] or holographic elements [3], [6], splitting the

beams in multiple paths and phase-shifted interferograms are simultaneously recorded. In conclusion, the already

available phase-shifting interferometric configurations, that handle the presence of vibrations, are very complex andexpensive systems. Nowadays, the trend in testing of optical components shows an increasing demand for low-budget

interferometers that work in presence of vibration.

3. THE RANDOM PHASE SHIFT INTERFEROMETERThe interferometer presented in this paper was designed to work without vibration isolation and to use the floor random

vibration as phase-shifter. In case when the mechanical vibrations are not sufficient, the interferometer (more preciselythe test plate) has to be deliberately perturbed to achieve random vibrations. The challenging goal of this new approach isto determine the random phase shifts that are generated in this manner.

3.1 General description of the interferometric systemThe experimental arrangement of the Fizeau configuration is shown in the figure 1. A continuous He-Ne laser beam of632.8 nm wavelength is directed to the spatial filter SF and collimated with the collimating lens L 1 to the test and

reference round plates T and R. The reflected wavefronts from the test and reference surfaces are deviated by the beam-

splitter BS-1 and trace through a pinhole aperture. Further using the beam-splitter BS-2 the interference field is dividedand directed at two different detector systems.

Fig. 1. The Fizeau interferometric system

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I

The reflected part is projected onto a CCD sensor array of a camera, which actually means a high spatial but low

temporal resolution detector. The transmitted part is collimated by the collimating lens L2 and hits a high temporal butlow spatial resolution detector system consisting of three photodiodes.

The functional principle is basically as follows: the CCD camera records a few interferogram images with random phase-

shifts due to the influence of the vibrations. The camera runs at 25 fps with an exposure time of only 11 s, short enough

to freeze mechanical vibrations. While the camera grabs several interferograms, the detector system consisting of three

photodiodes continuously records the light intensities that occur at three locations in the interference field aperture,

making possible as will be later described, the determination of the phase-shift for the entire test surface at each timeduring the entire measurement.

The reference plate is made of quartz, is 50 mm in diameter and has a wedge angle of about 0.5 to prevent interference

between waves reflected at front and back side. The front-side surface has /20 flatness, and the back-side surface, whichis used as reference surface, has /27 flatness, as measured by the supplier. For the investigation of the new technique we

used a test plate made of BK7 glass with a diameter of 50 mm and a wedge angle of 7. The surface topographies of both

surfaces are measured and specified by the supplier as reference. From the peak-to-valley values flatness deviations of

/8 and /11 result, respectively.

3.2 The oscillating test plateThe Fizeau configuration presents common paths for reference and test waves. This feature reduces the demands for the

adjustment of the optical components against the internal vibration sensitivity and the thermal effects. As result, in caseof a Fizeau arrangement, it can be assumed in good approximation that only the relative movement of the reference and

test plates generates the phase shifts.

The first distinguished element of the interferometer is the postholder of the test plate. In the figure 2 a cross-section of

the mounting system of the test plate is depicted. The postholder is fixed on the rail system of the interferometer. Thegeometric form of the postholder decides the character of the oscillations of the test plate under the influence of the

mechanical vibrations.

Fig. 2: The rigid postholder of the reference surface and the flexible postholder of the test plate

In-plane oscillations of the plate do not have an influence on the interferogram (at least as long as they are small - in the

order of a few m). In case of a postholder in form of a parallelepiped like in the figure above, the width must be much

larger than the thickness in order to suppress the in-plane oscillations. This form allows mostly only out-of-planemovements in form of tilt oscillations of the entire system formed by the postholder, the mounting system and the test

plate, about the X-axis at the bases of the holder. We assume the vibration-induced movements of the plate as rigid-body

shifts and tilts, and we consider the postholder as being the only elastic component of the system.

In the figure 2 it can be observed that the mounting system is the most massive component, so the masses of the holderand of the plate can be neglected. It can be assumed that the eigenfrequencies of the oscillations are mainly determined

by the post holder form and dimensions and the mass of the mounting system. The characteristics of the oscillations do

not depend on the tested object. Another distinctive feature is the variation of the phase shift across the test surface. Due


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P h o t o d i o d e 3T e s t s u r f a c e

P h o t o d i o d e 2 P h o t o d i o d e I \

to the tilt oscillations, there is a linear increase of the phase shift with the heights of the surface region. The local

intensity of the interference field as a function of time becomes:

][ ),,(),(cos),(),(),,( tyxyxyxIyxItyxI ++= (1)

The main goal still remains to determine the random phase-shifts at every pixel (x,y,t) for each recorded interferogram

with the CCD camera.

3.3 The low spatial resolution detector systemUnder the assumption of rigid-body shifts and tilts of the test plate, the profile of the surface under test does not sufferany changes under the influence of the vibrations. Knowing this, any non-collinear combination of three measurement

points on the test surface defines an oscillating plane that describes exactly the oscillation of the entire surface. In other

words, it is sufficient to measure with high temporal resolution the vibrations at three locations in order to find the

occurring phase-shifts at every point across the test surface. This is the reason why we use in this work a detector system

consisting of three photodiodes.

Fig. 3: The detector system consisting of three photodiodes (left). The three corresponding non-collinear measurementpoints P1, P2 and P3 on the test surface which define the oscillating plane (right).

We have integrated three photodiodes with internal operational amplifier which ensures high signal to noise ratio even

for very low light intensities. The typical response times, (the rise time and the decay time), of about 8 s provide acontinuously high temporal resolution sampling. The sensitive area of each photodiode is 1 mm2. The analog signals of

the photodiodes are connected to a computer and show the dependence in time of the intensity of the He-Ne fringes at

three different sampling points.

In the case of a classic phase shifting method, when the translation of the test or reference plate takes place with constant

velocity in the direction of Z-axis and introduces a linear phase shift in time, the signal of a photodiode has a sinusoidalform. Now having a look at equation 1, where the phase shifts (x,y,t) randomly varies due to the vibrations, we expect

frequency modulated signals of the photodiodes.

In this representation, see figure 4, the signals have an arbitrary intensity bias. According to equation 1, the frequencymodulation character of the signals can be observed. We can also identify on the photodiode signals the particular

extreme values that are related to the extreme positions of the test plate, and we call them main extreme points. Between

two main extreme values, the signals present a series of maxima and minima, concluding that phase shifts up to several

wavelengths are introduced.


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c --(

- ----

Fig. 4: The simultaneous photodiode signals.

3.4 Frames acquisition of the unstable interference fringesAs consequence of the continuous oscillation of the test object, the resulting interference fringe pattern changes andvaries very rapidly in time. A very short exposure time is necessary to freeze the fringe patterns on the CCD sensor.

During the recording of the frames the light intensity must remain constant, otherwise the interference contrast decreases.

The integrating bucket data method is not applicable here, since no linear variation in time of the phase shifts exists.

The high spatial frequency detector system is a standard machine vision camera equipped with a progressive scanningCCD image sensor. We use the camera in trigger mode at the maximum frame rate of 25 fps in order to achieve a shortermeasurement time, which involves the recording of several consecutive interferograms. The acquisition of the images is

governed by the electronic shutter located inside the camera. For the given power of 10 mW of the He-Ne Laser, an

exposure time of 1/90000 sec provides sufficiently bright interference images, up to 255 grey values, on the usual 8 bitdata imaging recording scale.

However, any CCD sensor array is characterised at very short exposure times by the so called vertical smear effect. Thisphenomenon is related to the nature of the interline transfer system employed in the CCD. Smear is a perturbing signal

which appears in form of vertical stripe, above and below of a bright part of the image. Using a continuous light source,

during the shifting of the charges, scattered photons or noise electrons tunnel the storage area, causing deterioration to

the original image.

The modality to overcome this problem is to shut out the light source before and after the exposure times. There are two

alternative solutions to cope with this problem. First, by using an additional mechanical shutter placed before the CCDcamera. The opening of the shutter is synchronized with the electronic shutter on the camera. The other modality, an

additional pulsed laser, which is synchronised with the electronic shutter of the camera, can be used to produce theinterference images on the CCD sensor.

However, comparing the both alternatives to handle the smear phenomenon, the integration of the beam shutter proves to

be advantageous and more convenient. First of all, only the He-Ne laser source is necessary. The coherence length of the

He-Ne laser enables the applicability of the interferometer also for spherical surfaces tests, where the test surface isplaced at a distance depending on the curvature radius of the surface. Due to the relative low power of the He-Ne beam,

the exposure time of the camera must be at least 10 s, a circumstance when the stroboscopic fringes recording still

remains effective. As consequence, the alternative using the beam shutter was chosen and implemented.

3.5 Description of the methodThe time dependency of the intensity measured by the photodiodes can be described as:

[ ] 3,2,1,)(cos)( =++= itIItI iiiii (2)

where i is the photodiodes index, I'i is the intensity bias, I''i is half the peak-to valley intensity, i is the time varyingphase shift introduced to the test beam corresponding to the three sampling points. The equation enables the mod

determination of the phase-shift time dependence at the three points on the test surface. Without loss of generality,

considering i = 0 we obtain:


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2 N o r m a l i s e d f i t t e d p o l y n o m i a l s P N 1 ( t )

A A I A I I I A A A A M A A A A K A A A A iV \ f l J V \ J- 2

4 a c o s [ P N j j ( t ) ]2

T w o c o n s e c u t i v e z e r oc r o s s i n g p o i n t s

o P+ - + - + + - + - + H +D e c o n v o l u t i o n p h a s e r e s u l t ( r a d )

J 1 2 5 1 5 0 I 7 5 T i m e ( m s )0 1 0- 2 0 2 0 0

i

iii

I

ItIat

=)(

cos)( (3)

The successive deconvolution of the mod phase from the equation 3 provides the time variation of the phase over the

entire interval of the measurement. Our approach is to fit, and further to normalize the signals. By properly dividing thesignals in intervals, the fitting can be processed successively on each interval, see figure 5. First we subtract the average

value I'i, and then we define the intervals between two consecutive zero crossing points. We apply a polynomial fit, anddue to the fact that the signals present one or three maxima on each interval, the fitting process with polynomials Pij(t) of6th grade is accurate enough.

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

The absolute value of the fitting polynomials Pij(t) at the extreme points, gives the value of the amplitudes I'' ij for each jinterval. Dividing now each Pij(t) polynomial by I''ij, we get all normalized polynomials P

Nij(t). After this we can apply

the equation 3 under the form:

)(cos)( tPat Nijij = (4)

obtaining the modulo variation of the phase-shift, for the related sampling point on the test surface, during the entiremeasurement time. By deconvolution of the mod phase, results in the real variation in time of the phase-shift at the

sampling point. In the figure the intervals in which the phase result is added or subtracted, can be distinguished.

Knowing the movement of three surface points in time and assuming the test plate to move as a rigid body thecalculation of the local phase-shift (x,y,t) as a function of time is straightforward.


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i t 3 i c0 i t I \ f \ Iu 1 x , y ) u 2 x , y ) o 3 x , y ) o 4 x , y4P P P )

3.6 The equation of the plane described by the three sampling pointsLet us consider the image coordinates (x1, y1), (x2, y2), (x3, y3) of the three sampling points in pixels on the CCD sensor

array. Having now the phase-shifts 1(t), 2(t), 3(t), the plane equation can be applied for existing phase-shift across theimage plane on the CCD sensor. The random oscillating phase-shift plane across the interferogram images recorded with

the CCD camera can be described as:

DCzByAx =++ (5)The resulting determinants are:

)(1

)(1

)(1

33

22

11

ty

ty

ty

A

= ,

)(1

)(1

)(1

33

22

11

tx

tx

tx

B

= ,

1

1

1

33

22

11

yx

yx

yx

C= ,

)(

)(

)(

333

222

111

tyx

tyx

tyx

D

= (6)

The resulting phase-shifts at each pixel become:

C

ByAxDtyx

=),,( (7)

In consequence, according to the equation 1 this enables the application of a PSI algorithm for random phase-shifts(x,y,t).

3.7 Four-step algorithm with random phase-shiftsSince the basic equation of phase-shifting interferometry has three unknowns, a system of minimum three equations is

necessary. However, an algorithm with more than three equations will reduce the sensitivity to errors in the phase-shifts.

In our case, the phase-shifts are not only random, but they are also different at each measured point on the test surface,see figure 6.

Fig 6: Classic phase-shifting method with /2 at each step (left). Random phase-shifting method with variation of the phaseshift across the test surface (right)

We propose in our case an algorithm with four equations. Let us consider four interference frames recorded at the

moments tk, with k = 1, 2, 3, 4, and the random phase-shifts at each pixel k(x,y). The following system results:

[ ]

[ ][ ]

[ ]),(),(cos),(),(),(

),(),(cos),(),(),(),(),(cos),(),(),(

),(),(cos),(),(),(

44

33

22

11

yxyxyxIyxIyxI

yxyxyxIyxIyxIyxyxyxIyxIyxI

yxyxyxIyxIyxI

++=

++=++=

++=

(8)

with three unknowns: I'(x,y), I''(x,y),(x,y) for each pixel. The analytical solution can be given as:


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- 0 , 1 5 10 , 1 0 1O , O 5 i H e i g h t ( w a v e )C C p pp o s i t i o n ( p i x e l ) - .C 0 1 C 0 13 0 0 4 0 0

1 0 0 1

I I

poson(pixl)

o

o

0

0

4 o o j0 0

= ),(),(),(

),(),(),(tan),(

3412

12341

yxsyxRyxs

yxcyxcyxRyx (9)

with

),(),(

),(),(),(

43

21

yxIyxI

yxIyxIyxR

= and

),(sin),(sin),(

),(cos),(cos),(

),(sin),(sin),(

),(cos),(cos),(

4334

4334

2112

2112

yxyxyxs

yxyxyxc

yxyxyxs

yxyxyxc

+=

=

+=

=

(10)

The mod 2 ambiguities are then removed by applying an unwrapping algorithm as explained in detail in [10]. The

measured wavefront (x,y) is related to the height profile h(x,y) of the tested surface: ),(4

),( yxyxh

=

4. FIRST MEASUREMENTS RESULTSFigure 7 shows the surface topography map of our test plate. It is in good agreement to the data supplied by the

manufacturer of the test plate. Even more important to us was the repeatability of the results, because this is governed by

the random phase shift evaluation technique and not primarily by the quality of the optical components.

Fig. 7: Surface topography of the test plate, measured by random phase shift interferometry

We repeated the measurement ten times within two working days to get an estimation of the long-term stability. The rootmean square (RMS) over the ten different measurements was calculated. According to the definition, the RMS error

function was applied to each data point (x,y) as follows:

[ ]2

1

),(),(1

1),(

=

=N

i

yxhyxhN

yxRMS (11)


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0 , 0 1 00 , 0 0 8! H O H

where h(x,y) is the surface height, )y,x(h is the height average value at each pixel, and N=10 is the number of

measurements. The RMS map is shown in the figure 8. Analysing the map, an RMS 0,004 can be evaluated,corresponding to approximately 2.5 nm for the 632.8 nm laser wavelength. The few spots of higher RMS values can be

assigned to dust particles or multiple reflections.

Fig. 8: Distribution of the local RMS error, demonstrating the repeatability of random phase shift interferometry over two

days

5. INTEGRATION OF A PULSED LASER DIODE. WAVELENGTH DETERMINATIONOF THE LASER PULSES

The second modality to avoid the undesirable smear phenomenon is to use a pulsed laser diode for the acquisition of theinterference images. The system becomes a two wavelength Fizeau interferometer whereas the laser pulses must be

synchronised with the shutter of the CCD camera. The fringes from the pulsed laser are projected onto the sensor of the

CCD camera, and the He-Ne fringes hit the detector system consisting of three photodiodes. While the oscillations of thetest plate are continuously measured, the CCD camera records a number of interferograms of the laser diode wavefronts.

The trigger signal of the pulse laser is connected in parallel with the photodiodes. The same analysis of the photodiode

signals, which was previously described, can be applied. The only difference appears in the form of the surface heightresult which is now a function of the laser diode wavelength [11].

A positive feature of using a laser diode is the higher power of the beam, which ensures sufficient light intensity foracquiring good interference images even for very short exposure times. The stroboscopic images become more effective,

ensuring no changing of the fringe pattern during the exposure times. However, the laser diode beams are characterised

by short coherence length. Two times the distance between the test and reference plates must be shorter than the

coherence length of the laser diode, otherwise poor contrast and unstable fringe patterns on the CCD chip appear. The

test plate must be accordingly placed very close to the reference. Under these circumstances, the system is limited tointerferometric tests of optical flats only. One more disadvantageous feature of the laser diodes is the wavelength

variation with the injection current and the ambient temperature. A very stable current source and a temperaturecontroller are necessary. And even so, the laser diode presents sometimes non repeatable properties for the same current

level and temperature. The wavelength proves to be sensitive and instable. In consequence, the integration of a

wavelength meter is necessary. The proposed extension of the experimental setup is shown in the figure 9:


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A

f l P u l s e g e n e r a t o rL - i- J LC C D P u l s - L a s e r d i o d eX 2 7 8 0 n m> >

X 1 = 6 3 2 , 8 n m ML 4

C W H e N e - L a s e rM 4/ M

i f i f B S - 1 t o i n t e r f e r o m e t e rS F I IHS h e a r - P I a t t e L 3

D i v e r g i n g b e a m C o l l i m a t e d b e a m C o n v e r g i n g b e a m

S h e a r - P l a t e P l a n e w a v e f r o n t

Fig. 9: The representation of the wavelength meter proposed for the measurement of the laser diode pulses

The beams of the He-Ne laser and the laser diode are orthogonally polarised and are coupled through the beam splitterBS-3. The transmitted beams through the beam splitter BS-1 go into the Fizeau interferometer. The reflected parts enter

the wavelength meter, whereas after the collimation by the lens L3, the beams are directed to the shear-plate, the central

piece of the system, a wedged, uncoated high quality optical flat, which itself represents a simple interferometer. Usually

the shear-plates are used as collimation testers. Previously we used it for the collimation of the Fizeau interferometer.The wavefront of the beam to be tested is reflected under the incident angle of 45. Both the front and rear surfaces

produce reflections of almost equal intensity. Due to the thickness of the plate, the reflected wavefronts are laterally

sheared, and interference occurs in the overlapping region. In the figure 10 the interference of the reflected wavefronts issimulated.

Fig 10: Simulation of a shear-plate collimation tester. The test wavefront is collimated when the fringes are horizontally

oriented.

The fringes that occur by the interference between the reflected wavefronts from the surfaces are recorded with a CCD

camera. The equidistant fringes are separated by the distance:

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2 5 0

2 0 0 1" I i 'o i V V V V V V V V- 1 X D 0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0

x p o s i t i o n ( p i x e l )

8 5 , 9 98 5 , 9 48 5 , 8 9I8 5 , 6 98 5 , 6 48 5 , 5 98 5 , 5 4

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0T i m e ( s )

S

nx

sp

i ==)2sin(

(12)

The equation 12 shows that the periodicity of the fringes depends linearly on the wavelength. The parameter S depends

on the shearing plate properties only: the refractive index of the material nsp, and the wedge angle. We use the He-Ne

wavelength as reference. Several fringes across the aperture of the plate are sufficient to fit the modulated signal with asinus function. We start with He-Ne fringes of known wavelength in order to make the calibration of the setup. The

Gauss intensity profile of the laser beam must be also considered. A simple sinus function may not be sufficient to fit the

signal. The product of a Gauss function and a sinus function, as in the equation 13, describes with sufficient accuracy themodulated signal.

+

++=

2

8

7654

3

212

1exp

2sin)(

p

pxppp

pppxf

(13)

The parameter p3 defines the period of the sinus function, and it is directly related to the separation distance of thefringes, see equation 12.

Fig 11: The interference fringes of the reflected He-Ne wavefronts from the shear-plate (left). Analysis of the fringe width

over a y-line: grey values of the fringes fitted by the function f(x) from the equation 13 (right).The figure 11 shows the He-Ne interference image after the reflections on the shear-plate. The grey values across a y-lineare fitted with the function f(x) from the equation 13. Measurements over a period of one minute were performed in

order to determine the statistical error of the calibration. As can be seen on the figure 12, the measurement values

oscillate in interval of about 0.115 pixel, corresponding to approximately 0.85 nm.

Fig 12: Variation of the measured width of the He-Ne fringes over one minute.

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6. CONCLUSIONSRandom phase shift interferometry has been demonstrated to be a low-cost alternative to sophisticated concepts of

simultaneous phase shifting interferometers. Not only the vibration isolated table can be omitted but also the precisiondevice for introducing well-defined phase-shifts. Implemented on a quite simple experimental Fizeau setup our first

results show excellent repeatability. In case a pulsed laser diode is used in order to avoid the smear phenomenon, the

concept of a wavelength meter was developed. This encourages us to continue the work. Our current focus is ondeveloping a calibration procedure to transfer repeatability to measurement uncertainty.

The authors gratefully acknowledge project funding by Deutsche Forschungsgemeinschaft DFG

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