The spatial frequency response and resolution limitations of pixelated mask spatial carrier based phase shifting interferometry

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1 The spatial requency response and resolution limitations o pixelated mask spatial carrier based phase shiting intererometry Brad Kimbrough, James Millerd 4D Technology Corporation, 80 E. Hemisphere Loop, Suite 46, Tucson, AZ The spatial requency response o the pixelated phase mask sensor has been investigated both theoretically and experimentally. Using the small phase step approximation, it is shon that the instrument transer unction can be approximated as the product o the system optical transer unction and the spatial carrier processing ilter transer unction. To achieve optimum perormance it is important that the bandidth o the optical imaging system is adequate so that the limiting actor is the detector pixel idth. Actual measurements on a commercial Fizeau intererometer agree very ell ith the theory, and demonstrate detector limited perormance. The spatial resolution o the calculated phase map is algorithm dependent; hoever, both the x and x convolution algorithms result in a requency response that is signiicantly more than hat ould be obtained by a simple parsing o the image. Thereore, a k x k sensor has a spatial requency response that is approximately equal to the detector limited resolution o a 700 x 700 array ith its requency response extending to the ull Nyquist limit o the k x k array. Keyords: Intererometry, optical testing, spatial carrier. INTRODUCTION This paper seeks to explore the phase resolution o the pixelated mask spatial carrier method.,, We start by providing a general overvie o the spatial carrier phase shiting method hich ill allo us to make useul comparisons beteen the pixelated mask spatial carrier and the more typical linear spatial carrier methods. Next e discuss the concept o the instrument transer unction, ITF, and derive an expression or the ITF o the pixelated mask spatial carrier system under limiting conditions. The expected perormance o the pixelated mask spatial carrier system is veriied in three dierent ays. First, a phase calculation is conducted on a simulated surace ith a lat poer spectrum and compared ith the expected theoretical results. Next, modeling results are provided or an ITF calculation based on a small step height measurement and then compared ith actual measured values. Finally, a smooth glass surace is measured and the poer spectral density measurements are compared ith theoretical results.. SPATIAL PHASE MEASUREMENT Spatial phase measurement utilizes a single intererogram to extract phase inormation. In this technique, a spatially dependent carrier phase is added to the averont phase. A commonly used carrier phase is linear and produced by introducing a large tilt beteen the test and reerence beams. The intererence pattern produced may be represented as ollos: I( x, y) = I ( x, y)( + vcos[ θ ( x, y) + φ ( x, y)) () avg c here ν is the ringe visibility, θ( x, y) is the averont phase to be determined, and φ c ( x, y) is the carrier phase. A pixelated mask is another method o producing a carrier phase. Fundamentally the only dierence beteen pixelated mask spatial carrier and other spatial carrier methods is the orm o the carrier averont. For a null averont, the intererograms associated ith both a linear and a pixelated carrier is shon in igure. The linear carrier phase is oriented at +45 deg. ith a phase shit o 90 deg. beteen pixels in the x and y direction.

2 Linear Carrier Intererogram Pixelated Carrier Intererogram Linear Phase Discrete Phase Detector Pixel Figure. Null averont intererograms or a linear carrier and a pixelated carrier. This particular linear carrier as chosen or comparison ith the pixelated mask carrier since its perormance most closely matches that o the pixelated mask. The linear carrier can be expressed analytically as shon in equation. π φ c ( x, y ) = ( x + y ) () For the pixelated mask system, the carrier phase may be represented as to dimensional rectangular unctions convolved ith a to dimensional array o delta unctions eighted ith the necessary phase shit. This representation is shon in equation. φ ( x, y) = c x y x ( y x y x [, [ y + +, ) π π rect[ x, y 0 comb[, comb[, comb πcomb () here rect[ x, y is a to dimensional rectangle unction representing the discrete size o each phase shit area, represents the convolution operation, and the comb[ x, y unction represent a to dimensional grid o delta unctions. Note that the comb[ x, y unction aligned ith the zero phase shit regions has been explicitly included or clariication. Using the linear carrier phase o equation, equation may be expanded and ritten as ollos: I( x, y) = I ( x, y) avg + I ( x, y) vcos[ θ ( x, y) Cos[ π ( x + y) + I ( x, y) vsin[ θ ( x, y) Sin[ π ( x + y) avg avg (4) The intererogram intensity or the pixelated mask system is ound by substituting the pixelated carrier phase o equation 5 into equation, and retaining only the undamental carrier requency terms. In this analysis, the inite extent o each phase shit region is ignored. With the exception o a slight change to ringe contrast, this simpliication does not undamentally alter the analysis since the undamental carrier term is at the Nyquist requency ith all higher order terms aliasing back on top o it. The simpliied result is given in equation 5. I( x, y) = I ( x, y) avg avg θ π avg θ + I ( x, y) vcos[ ( x, y) Cos[ x + I ( x, y) vsin[ ( x, y) Cos[ π y (5) here θ ( x, y) = θ ( x, y) + π. 4

3 Resolution limitations o the spatial carrier technique are most easily understood by examining the phase modulation and calculation process in the Fourier domain. The spectrum o the linear carrier intererogram is ound by taking the Fourier transorm o equation 4 ith the results given in equation 6. Likeise the spectrum o the pixelated mask intererogram is given by the Fourier transorm o equation 5 ith the results given in equation 7. I { I( x, y) } = I{ Iavg ( x, y) } +I{ Iavg ( x, y) vcos[ θ ( x, y) } ( δ[ x, [, 4 y + δ 4 x + 4 y + 4 ) i +I{ Iavg ( x, y) vsin[ θ ( x, y) } ( δ[ x, y δ[ x +, y + ) I { I( x, y) } = I{ Iavg ( x, y) } + I { Iavg ( x, y) vcos[ θ ( x, y) } ( δ[ x + δ[ x ) + I{ Iavg ( x, y) vsin[ θ ( x, y) } ( δ[ y + δ[ y ) (6) (7) here δ[ is the Dirac delta unction, ( x, y ) are the spatial requency coordinates given in cycles/pixel and represents the convolution operation. The spectrums given in equations 6 and 7 are shon graphically in igure. Linear carrier spectrum Pixelated carrier spectrum 5. Average intensity term 4 4. Waveront positive exponential term. Waveront negative exponential term 4. Waveront cosine term 5. Waveront sine term 5 Figure. Linear and pixelated carrier intererogram spectral magnitudes. In each case the average intensity term has been broadened or clarity and the maximum averont slope is 0.05 aves/pixel. As is demonstrated in igure, adding the spatial carrier phase to the averont phase has separated the averont terms rom the average intensity term in the Fourier domain. There are several dierent techniques or determining the averont phase rom the spatial carrier intererogram. In the most general sense they involve suppression o the average intensity term, component, and iltering out the to averont terms, components and in the spectrums o igure. These calculation techniques may be carried out in the spatial domain, the Fourier domain or both. 4 One technique or determining the averont phase rom the spatial carrier intererogram is the spatial lo pass iltering method. In this technique the spatial carrier intererogram intensity values are irst multiplied by the sine and cosine o the carrier phase orming spatial heterodyne averonts. These to products are then lo-pass iltered by convolution ith a ilter kernel in the spatial domain to remove the DC and double requency terms. The demodulated phase is given by the arctangent o the ratio o the iltered sine and cosine products. Figure provides an outline o the spatial lo-pass iltering process.

4 Spatial Lo Pass Filtering I ( x, y ) Sin [ φ c Cos [ φ c I ( x, y ) Sin [ φ c k k I ( x, y ) Cos [ φ c I ( x, y ) vsin [ θ avg I ( x, y ) vcos [ θ avg Sin [ θ Cos [ θ ArcTan 4 θ Figure. Spatial lo pass iltering process steps.. Form to heterodyne signals by multiplying the spatial carrier intererogram by the sine and the cosine o the carrier phase.. Filter out the DC and double requency terms in the heterodyne signals by convolving ith a ilter kernel.. Form the ratio o the iltered heterodyne terms. 4. Take the arctangent o the terms ratio to determine the phase. Although this technique is generally carried out strictly in the spatial domain, e ill continue to analyze the steps in the Fourier domain or clarity. The spectral magnitude o the spatial heterodyne signals or the linear and pixelated carriers is shon in Figure. In each case the heterodyne signal as ormed by multiplying each intererogram ith the cosine o the carrier phase. Linear spatial heterodyne spectrum Pixelated spatial heterodyne spectrum. Average intensity term at carrier requency. Waveront cosine term. Waveront cosine term at double carrier requency Figure 4. Linear and pixelated carrier spatial heterodyne spectral magnitudes. The heterodyne signal is ormed by multiplying the spatial carrier intererogram by the cosine o its carrier phase. The critical step ith regard to the phase resolution o our phase calculation technique involves the lo pass iltering o the central averont term. Containing the averont phase term o interest, the central term, items in igure, must be iltered out ithout signiicant attenuation. The average intensity terms and the double requency averont terms, items and in igure, must be signiicantly attenuated. I iltered too much, then the phase resolution o the measurement ill be degraded. I not iltered enough, then the average intensity terms and the double requency terms ill introduce high requency noise into the result. 5 The accuracy o the calculated averont phase associated ith the detected intererogram is ultimately set by the sharpness o our ilter. As averont slopes increase, the central averont terms ill expand and the double requency terms ill alias in toard the center, making the iltering process more diicult. The spectrum o the linear and pixelated heterodyne signals or a averont containing large slopes is shon in igure 5.

5 Linear spatial heterodyne spectrum Pixelated spatial heterodyne spectrum. Average intensity term at carrier requency. Waveront cosine term. Waveront cosine term at double carrier requency Aliasing limit Figure 5. Linear and pixelated carrier spatial heterodyne spectral magnitudes or a averont ith large slopes. In each case the average intensity term has been broadened or clarity and the maximum averont slope is 0. aves/pixel. The dashed line indicates the sharp ilter limit or a averont ith only tilt and a lat average intensity. Reerring to igure 5, in both the linear and pixelated cases e are limited to hal the Nyquist requency in directions along the diagonals. Note that this is equal to times the Nyquist limit along the horizontal or vertical directions. This is because this is the point at hich separation o the central term rom the average intensity term and the double requency term in the linear case, or rom the double requency term in the pixelated case, ould be impossible. It should be noted, that since the pixelated heterodyne signal places the average intensity term out at the Nyquist limit it is more tolerant than the linear carrier technique to larger variations in the average intensity. I a symmetric ilter ith a step edge cuto ere possible, the dashed line in each spectrum indicates the maximum spatial requency to prevent aliasing or a averont ith only tilt and a lat background intensity. Lo pass iltering o the pixelated heterodyne spectrum is done in the spatial domain through convolution ith one o to ilter kernels, a x ilter kernel or a x ilter kernel. Both kernels and their associated ilter unctions are shon in igure 6. A cross sectional plot o the ilter unctions is also shon. Filter Kernel Function x Cos [ π x Cos [ π y x 4 Cos [ π x Cos [ π y Filter Amplitude Filter Function Cross Sections X Hor. or Vert. Diagonal X Figure 6. Pixelated mask ilter kernels and their associated ilter unctions. Line pairs / pixel x In igure 7 the pixelated heterodyne spectrum is shon overlaid ith contour plots o the to dierent ilter unctions.

6 x ilter contour x ilter contour Pixelated spatial heterodyne spectrum. Average intensity term. Waveront cosine term. Waveront cosine term at double carrier requency Filter Contour Figure 7. Pixelated spatial heterodyne spectrum ith contour overlays o the processing ilter unctions. It is obvious rom looking at the above ilter unction plots that the x kernel has better high requency perormance due to a steeper roll-o. Hoever, note that the x ilter has a very sharp slope at the Nyquist limit, hereas the x ilter has a broader area o attenuation. The ilter must remove the DC term rom the intererogram at this point. I the illumination is non uniorm, then the average intensity term can have a signiicant spectral idth. In this case the x ilter ould be a better choice or attenuating this term, preventing it rom introducing high requency noise into the phase calculation.. INSTRUMENT TRANFER FUNCTION The primary actors aecting the ability o an intererometer to accurately determine the phase proile o the object under test are the illumination source spatial and spectral characteristics, the imaging system, the detector sampling, and the phase calculation technique. The basic process o an intererometric measurement or a coherent spatial carrier system is outlined belo in igure 8. The process has been separated into three sections; imaging, detection, and computation. Taken by themselves, each o the sub processes is characterized by a linear transer unction. Hoever, due to the non-linearity o the intererometric phase calculation process, the net system transer unction relating the calculated phase spectrum to the object phase spectrum can only be associated ith the product o the individual transer unctions under limited conditions. Goodman 7 provides a rigorous treatment o linear system theory as applied to optics. Additionally, an excellent general discussion concerning an intererometric instrument transer unction is provided by de Groot. 8 Ε obj Ε img Detector θ obj I img I det Algorithm θ cal θ img Transer Function Imaging Detection Computation H H H NA pix alg Figure 8. Intererometric measurement process. For an intererometric system measuring surace height, the instrument transer unction (ITF) is the ratio beteen the calculated surace height and the actual object surace height as a unction o spatial requency. In general, the instrument transer unction is not independent o the object being measured due to the non-

7 linear nature o the intererometric process. Since the surace height values are directly proportional to the calculated phase, the ITF may be expressed in terms o averont phase as ollos: { θcal} ITF { θobj} I I (8) The irst step in the intererometric measurement process is the imaging o the object electric ield at the detector plane. Since the system being discussed is strictly coherent, the relationship beteen the object and image electric ields is given by equation. { img } H NA { obj} I Ε = I Ε (9) here I{ } designates the Fourier transorm and H NA is the coherent traner unction. At the detector plane the image test and reerence ield interere and are sampled by the detector array. The relationship beteen the image intererogram and the detected intererogram is shon in equation 0. here H pix calculation process. { Idet} H pix { Iimg } I = I (0) is the detector transer unction associated ith the inite pixel size. The inal step is the phase As discussed in section, the lo pass iltering method o spatial carrier intererometry requires iltering the intererogram heterodyne signals to separate out the averont component. This process may be represented as ollos: i c I HET Idete φ () { I } H alg { I HET } I = I () θ = modulus[ I () calc here Ĩ HET is the complex heterodyne signal, H is the phase processing algorithm ilter unction, Ĩ alg is is the complex iltered heterodyne signal and ill be reered to as the algorithm intensity, and θcalc is the calculated phase. Combining equations,, and gives: alg i c { Ialg } H ( H pix { Iimg } { e φ }) I = I I (4) here represents the convolution operation. The image intensity spectrum is given by the autocorrelation o the electric ield spectrum given in equation 9. 7 This operation is shon belo in equation 5. { I } ( H { }) ( H { }) img NA obj NA obj I = I Ε I Ε (5) here represents the auto correlation operation. Clearly, the relationship beteen the image and the object intensity spetrums is non-linear. Restricting the test object to surace heights << λ 8 4, equation 5 may be simpliied to: Substituting equation 6 into equation 4 gives: I { Iimg } H NA I{ Iobj } (6)

8 ( ) i c { Ialg } H H pix H NA { Iobj} { e φ } I = I I (7) Substituting the object intensity o equation into equation 7 gives: iθ iφc iθ + φc { Ialg} H pix H NAH ( { Iavgve } { Iavge } { Iavgve }) I = I + I + I (8) Assuming that the iltering is suicient to remove both the single and double requency carrier terms, and noting that H NAH pix is simply the system optical transer unction, equation 8 may be reduced to: Converting equation 9 to the spatial domain gives: i { Ialg } OTF H { Iavgve θ } I = I (9) i I alg = kotf k Iavgv e θ (0) ere kotf and k are the ilter kernels associated ith the OTF and H respectively, and represents the convolution operation. Assuming the variance o the average intensity term is lo, it is taken outside the convolution operation. i I = I v ( k k e θ ) () alg avg OTF Using the previous assumption o small averont deviation the exponential term o equation is expanded through the irst order. iθ i( kotf k θ ) OTF OTF ( + θ ) = + ( OTF θ ) () k k e k k i i k k e here it as assumed that the ilter kernels are normalized. Using this result, equation may be ritten as: i ( kotf k ) I θ alg = Iavgv e () Finally, substituting equation into equation, the calculated phase is given in both the spatial and Fourier domain as: θ k k θ I calc OTF { θ } OTF H I{ θ } calc A comparison o equations 8 and 4 gives the inal result or the instrument traner unction o a spatial carrier intererometer or averonts ith small phase deviations: here once again, (4) ITF OTF H (5) H is the spatial carrier ilter transer unction given in igures 6 and 7. In summary, ith the assumption o small surace deviations, << λ 4, the instrument transer unction, ITF, o a spatial carrier intererometer may be approximated by the product o the system OTF and the spatial carrier processing ilter, H. Additionally, just as the detector pixel spacing deines a spatial requency limit, the Nyquist requency, necessary to prevent aliasing o high requency inormation; the carrier

9 requency also deines a maximum spatial requency limit necessary to prevent the aliasing o high requency heterodyne components into the measured phase. 4. SIMULATED SURFACE WITH A FLAT POWER SPECTRUM An easy ay to determine the aects o the pixelated mask sensor on the ITF is to simulate a measurement on a surace ith a lat poer spectrum. A 000 x 000 array o uniormly distributed random numbers as generated or the surace. The distribution had a mean o zero and an RMS o 0. aves. For this simulation the OTF as assumed to be equal to or all requencies out to Nyquist. The surace phase as then added to the pixelated mask phase and a simulated intererogram as generated. Next, phase calculations ere conducted using the x and the x algorithms. Finally, the PSD or the calculated phase as determined. In this case, since the initial PSD as lat, the ITF is simply the square root o the calculated PSD ith the necessary normalization applied. In order to reduce noise, the inal ITF curves are the average o 00 simulations. Both measured, dashed lines, and theoretical, solid lines, are shon in igure 9. The theoretical curves are the ilter unctions given in igure 6. With the exception o the noise due the random surace generation, the simulated and theoretical curves are in agreement. Vertical or Horizontal ITF Diagonal ITF x x ITF ITF x x Line pairs / aperture Line pairs / aperture Theoretical Calculated Fig. 9. Measured ITF o a simulated surace ith a lat poer spectrum using the x, and x pixelated algorithms. The dashed lines indicate the calculated ITF hile the solid curves are the theoretical values. 5. MODELING A STEP RESPONSE In order to determine the eect o the pixelated mask sensor and associated algorithms on the ITF, a simulated step height measurement as perormed. The simulation assumed a step height o 0. aves oriented along the y-axis. The imaging as strictly coherent ith the system numerical aperture set to produce a cuto requency o ½ ave per pixel at the camera. The system as assumed to be diraction limited. The camera sensor as simulated by an array o 000 x 000 pixels ith 00% ill actor (i.e. there are no gaps beteen the pixels). The ITF curves shon in Figure 0 ere determined by the olloing procedure:. Generate an electric ield associated ith a 0. ave step height, and a lat reerence surace. Oversample ield by a actor o 0 compared ith the CCD pixel sampling.. Propagate the electric ield rom the object and reerence to the pixelated mask.. Add pixelated mask phase oset to object and reerence electric ields. Skip this step or calculating the temporal phase shit ITF curve. 4. Calculate the intensity at the CCD by summing the electric ields and then squaring. 5. Convolve the intensity ith a square pixel and then sample the result at the pixel spacing.

10 6. Calculate the phase using the x parsed algorithm, the x convolution algorithm, the x convolution algorithm and a simple 4 step temporal algorithm. 7. Determine the poer spectral density (PSD) or each phase calculation 8. Normalize each PSD by the PSD o a perect step. 9. Take the square root o the normalized PSDs to obtain the ITFs. Fig. 0. Instrument transer unction calculated or the pixelated mask sensor and associated algorithms. The calculation is based on a simulated 0. ave step measurement. The camera sensor is a 000 x 000 array o square pixels. The ITF o a temporal measurement, hich is shon or comparison, is the limit due to the pixel idth. The 500 x 500 trace represents the temporal ITF that ould be obtained by an array ith pixels tice as ide (i.e ¼ the number o pixels). The X and X traces sho the response o the convolution method or the respective kernel sizes. The Linear Carrier trace shos the eective resolution o a Fourier Transorm method utilizing 00 ringes o tilt as the carrier and a Gaussian ilter. These calculations sho that using the convolution technique ith the pixelated sensor preserves a signiicant portion o the spatial requency spectrum. In this example or a 000 x 000 array, the curve labeled temporal is the maximum resolution that can be achieved, assuming a diraction limited optical system and pixels ith 00% ill actor. The curve labeled 500x500 represents the requency response that ould be achieved or an array ith ¼ the number o pixels that are tice as ide (equal to 4 pixels o the 000x000 sensor). For the pixelated phase sensor, the 500x500 curve represents the resolution that ould be achieved i the array as sub-divided into unique groups o 4 neighboring pixels, and a single phase value as calculated or each sub-group. Utilizing the convolution algorithm extends the requency response out to the limit o the ull sensor resolution, signiicantly beyond the 500 x 500 limit. Also shon or comparison in Figure 9 is the response o a phase measurement utilizing the spatial carrier method. 00 ringes o linear tilt are used to generate the carrier requency on a 000x000 sensor. In this case, the ITF is signiicantly limited by the requency response o the spatial carrier processing ilter. 6. FIZCAM 000 STEP HEIGHT MEASUREMENT ITF CALCULATION In order to veriy the results o the simulation in section three, the ITF o a FizCam000 as determined by measuring a 0. ave step height standard. 9 The imaging in the FizCam000 is strictly coherent. The step standard is ormed on a inch optical lat and has uniorm relectivity on each side o the step. As in the simulation above, the step as oriented along the y-axis. The calculated ITF along ith the corresponding simulated ITF curve is shon belo in igure. These measurements conirm that the instrument ITF ollos the theoretical response.

11 Fig.. Measured instrument transer unction (ITF) or the Fizcam 000 using the x algorithm. The calculation is based on a 0. step height measurement. The system aperture size is 00mm. The theoretical ITF o the x is shon as the dotted line. 7. SMOOTH SURFACE MEASUREMENT An additional check on the traner unction characteristic o the pixelated sensor as conducted by measuring a relatively smooth glass surace, < 0.5 nm RMS. The let graph in igure shos the horizontal or vertical PSD curves. The right graph provides the corresponding curves or the diagonal, 45 deg., PSD. In each case the theoretical curves or the x and x algorithms are given as dashed gray lines. The theoretical curves er generated by itting the linear portion o the measured PSD curves and then modiying the curve to take into account the sample spacing and the ilter transer unction. For the diagonal PSD curves, signiicant departure rom the theoretical curves occurs at higher spatial requencies due to aliasing ith the double requency heterodyne terms as discussed in section 4. Additional aliasing occurs in the sub sampled curve due to the don sampling aect. Vertical or Horizontal PSD Diagonal PSD Sub Sampled Sub Sampled PSD x PSD x x x Line pairs / aperture Line pairs / aperture Fig.. Measured PSD o a smooth glass surace using the x, x, and sub sampled x algorithms. The dashed gray lines indicate the theoretical PSD curves or the x, and x algorithms.

12 8. CONCLUSIONS The spatial requency response o the pixelated phase mask sensor has been investigated both theoretically and experimentally. It as shon that ith the assumption o small surace deviations, << λ 4, the instrument transer unction, ITF, o a spatial carrier intererometer may be approximated by the product o the system optical transer unction, OTF, and the spatial carrier processing ilter, H. Additionally, just as the detector pixel spacing deines a spatial requency limit, the Nyquist requency, necessary to prevent aliasing o high requency inormation; the carrier requency also deines a maximum spatial requency limit necessary to prevent the aliasing o high requency heterodyne components into the measured phase. The pixelated mask spatial carrier operates at the maximum carrier requency possible or a given detector. This allos non-aliased measurements on averonts ith spatial requency components up to / times the Nyquist sampling limit at its most restrictive. Actual measurements on a production Fizeau intererometer agree very ell ith the theory, and demonstrate detector limited perormance. The spatial resolution o the calculated phase map is algorithm dependent; hoever, both the x and x convolution algorithms result in a requency response that is signiicantly more than hat ould be obtained by a simple parsing o the image into 4 pixel cells and, ith some attenuation, extends to the ull Nyquist limit o the sensor along the x and y axis. REFERENCES. J. E. Millerd, N. J. Brock, J. B. Hayes, M. B. North-Morris, M. Novak, and J. C. Wyant, "Pixelated phase-mask dynamic intererometer," Proc. SPIE 55, 04-4, (004).. J. E. Millerd in Fringe 005, edited by W. Osten, (Springer, Ne York, 005), pg B. Kimbrough, J. Millerd, J. Wyant, J. Hayes, Lo Coherence Vibration Insensitive Fizeau Intererometer, Proc. SPIE 69, (006). 4. D. Malacara, M. Servin, and Z. Malacara, Intererogram analysis or optical testing, (Marcel Dekker, Ne York, 998). 5. Womack, K. H. "Intererometric phase measurement using spatial synchronous detection." Optical Engineering, (4): 9-95, Kimbrough, B. "Pixelated mask spatial carrier phase shiting intererometry-algorithms and associated errors." Appl. Opt. 45, , Goodman, Introduction to Fourier Optics, McGra-Hill, Peter de Groot, Xavier Colonna de Lega, Interpreting Intererometric height measurements using the instrument transer unction, in Fringe 005, edited by W. Osten, (Springer, Ne York, 005) 9. E. Novak et al., Optical resolution o phase measurements o laser Fizeau intererometers, Proc. SPIE 4, 4- (997)