The modulus of the degree of coherence can be derived from interference patterns either by using fringes and next neighbour operations or by using several interferograms produced through phase shifting. Here the latter approach will be followed by using a lateral shearing interferometer exploiting a diffractive grating wedge providing a linearly progressive shear. Phase shifting methods offer pixel-oriented evaluations but suffer from instabilities and drifts which is the reason for the derivation of an error immune algorithm. This algorithm will use five π/2-steps of the reference phase also for the calculation of the modulus of the coherence function.
© 2016 Optical Society of America
Phase shifting interferometry allows for the precise determination of the phase in interference patterns. One of the common methods uses several phase shifted intensity distributions of the same interference pattern . However, the accuracy of the measured phase is hampered by unstable measuring conditions. Under the assumption of stable laser illumination one of the main contributors to phase dependent errors are reference phase mis-calibrations or mechanical instabilities. It has been shown  that a special averaging technique allows the elimination of linear mis-calibrations, and in a more general sense, also the elimination of the linear contribution to the error curve for the reference phase shifter. In the following years up to now a plethora of proposals have been published with an increasing number of phase steps and different algorithms [3–11]. Most of them are dealing with the improvement of the phase accuracy. But, here we want to exploit a set of measured intensities on the time axis for the determination of the modulus of the coherence function in order to avoid spatial neighbourhood operations which means that we are able to measure the visibility pixelwise.
Moreover, phase shifting interferometry allows also for the determination of the mean intensity and the visibility from the detected intensity patterns which are in general also prone to phase dependent errors caused by deviations of the reference phase from the mathematically prescribed values during the data gathering process for a phase shifting evaluation. In this paper, it will be discussed how the measured visibility can be freed from phase dependent variations across the fringe field down to errors of second order and we will show how the a posteriori correction of periodic artefacts enables high quality modulus measurements.
2. Measurement principle
The discussion will be restricted to the 4 and 5 phase step algorithm since the phase-dependent variations of measured visibility and mean intensity values have so far not been discussed even for these simple cases. It shall be assumed that the phase shifter in the interferometer is impaired by only small deviations εr from the ideal π/2-phase steps. Commonly, the predominant error is a linear mis-calibration of the reference phase shifter or drifts of the interferometric setup which allows the derivation of simple equations showing the error impact.
The intensity at the exit of a two-beam interferometer with coherent and monochromatic illumination takes under the assumption of a linear mis-calibration the form:1]. In case of π/2-steps the phase can be retrieved from intensity differences taken out of the set of intensities enabling the derivation of sin- and cos-proportional terms in the following denoted by N and D. The phase Φ and visibility or modulation term follows from: Eqs. (4) and (5) are valid under ideal circumstances, i.e., the absence of errors in the intensity measurement and in the necessary reference phase values.
Phase shifting interferometry requires at least 3 intensity values derived from phase shifted interferograms equally covering the reference phase interval of 2π. It has been shown  that already with 4 interferograms shifted in steps of π/2 especially symmetric equations can be established for the phase which are immune against quadratic nonlinearities of the photodetector array being used in the measuring device. In fact, photo-voltages enter the equations for the determination of the mentioned relevant quantities and not the intensity values directly. To exclude nonlinearities up to second order we will restrict the discussion to the symmetric 4 step and the 5 step algorithm since the latter is resting on two runs with 4 phase steps.
So the discussion will focus on calibration errors of the reference phase stepper using the intensity values directly. The impact of a linear calibration error will be considered in some detail for a 4-phase step and a 5-phase step algorithm since these are simple algorithms which were also the basis for extensions [8, 11] to 6 and even 8 steps. We will deal with slope errors of the reference phase which can either occur through mis-calibrations of the phase shifter or through drifts of the interferometer where the linear portion of the reference phase error will in many cases be the dominating part of the error budget.
3. Four-phase steps versus five-phase steps
For the error discussion, a linearly increasing reference phase error of the following form is assumed:
Under these assumptions the measured intensities Eq. (3) of a measuring run comprising 5 intensities differing in the reference phase by π/2 will result in the following set of equations:
The phase error ΔΦ is then derived using the following relation :
In the physical experiments photovoltages instead of intensity values enter into Eqs. 7 which requires the subtraction of the dark current contribution from the measured photovoltages I1 through I5, if intensity sums are calculated. Difference values are automatically free from dark current contributions.
It is assumed in the following discussion that only small mis-calibrations ε̄ << π/2 are present which will allow the following approximations:
3.1. The use of four π/2-phase steps
We start with the equations of the phase Φ4 and visibility V4 for the 4-step method, because this demonstrates the impact of mis-calibrations very convincingly:Eqs. (7) up to I4: Eqs. (12) we arrive at:
3.2. The use of five π/2-phase steps
As has been shown by us  the averaging of the phases of two runs having a phase-offset of π/2 eliminates linear phase errors and can also be applied to two four-step runs resulting in a simple equations:7] derived the equation of the phase Φ5 in the modified form by using an exponential representation for the first Fourier transform for the set of measured intensities. This equation has been generalized for an arbitrary set of intensities covering one period of the interference-function. In this derivation the mean of the first and last intensity provides the elimination of linear phase errors. Exploiting the interpretation by Surrel  we use for the calculation of the mean intensity the sum of I05 = (I1 + I5)/2 + I3 instead of simply the sum of two intensity values being in phase opposition as I1 + I3.
Using the Eq. (7), we arrive at values for N5 and D5 and the mean intensity I05:Eq. (9): Eq. (17) into the phase error function of Eq. (8) one obtains: Eqs. (17) into the expressions for the visibility Eq. (15) results in:
The error phase ΔΦ5 is periodic with 2Φ with a weight of approximately ε̄2/4 [3, 11] which is a considerable improvement over the evaluation using only 4-phase steps where also a 2Φ-periodic error occurs but with a weight ε̄/2.
The visibility in case of five π/2-phase steps shows also a second order dependence on the slope error but it has a stronger Φ-periodic component over a 2Φ-periodic variation. This means that the visibility depends only in second order on the slope error of the reference phase, which should allow for the measurement of the modulation degree with periodic errors below a few percent.
4. Measurement of the modulus of the degree of spatial coherence
Very recently a new interest has been shown in the literature concerning the determination of the spatial degree of coherence using interference fringes produced in a Young double slit or double hole interferometer set up [13,14]. This fact motivated us to resume and to complement work done in our labs before on the subject of measuring and using partially coherent light [15–21]. Here, we will repeat the experiments, done before in a more qualitative way , but now with lithographically made diffractive gratings and by using the phase shifting technique in a quantitative manner making use of the above discussed error tolerant algorithm. For the verification of the quality of the measurement the light field is prepared using binary amplitude masks with a suitable set of transparent slits being illuminated with He-Ne-light which is made spatially incoherent with the help of a moving scatterer . In the past such mastered light fields have been used in some experiments [16–21] enabling lateral shearing interferograms with reduced coherent noise due to dust diffraction and similar defects in the light path.
Our experimental setup (Fig. 1) can be split into two parts: (1) a device for preparing a certain state of partial coherence by using a laser source in combination with a moving scatterer and an absorbing transparency mask and (2) a grating wedge interferometer for the measurement of the modulus of the spatial degree of coherence.
The light of a He-Ne-laser with wavelength λ = 632.8nm illuminates a spot of a few mm’s in diameter on a rotating scatterer producing a time-varying speckle pattern. This speckle pattern is imaged on a screen with transparent regions which function as an extended monochromatic incoherent light source in the time average. The light emerging from the screen is collimated by a collimator lens (500/80). The collimated wave field impinges onto the grating wedge (G1, G2) producing sheared waves with a progressive shear counted from the edge of the grating wedge. The diffracted first order light is then imaged through an imaging telescope with the achromatic doublets (500/80) and (80/20) onto an array detector in such a manner that the plane of the first grating G1 is sharp on the array. The scaling factor of the imaging telescope is therefore 1 : 6.25.
As a photoelectric detector array serves a 1312x1082 pixel CMOS-camera MV1-D1312-80-G2-12 having a S/N-ratio better than 300 provided by Photonfocus AG Lachen . The CMOS-chip is uncovered by a glass window ensuring freedom from spurious reflections and interference patterns.
For the evaluation of the interference pattern a set of 5 interferograms shall be used as has been discussed in connection with Eq. (18) and (19). For the tuning of the reference phase a lateral shift of grating G1 by one grating period perpendicularly to the rulings is made. The on-line PC controls the PZT phase shifter and processes the stored intensity data due to Eqs. (13), (14) and (18), (19).
The quality of the measurement of the modulus of the degree of spatial coherence will become visible if the degree of coherence covers a broad range of values form 0 to 1. Especially visibility values close to one would be distorted through systematic errors caused by drifts and slope errors of the reference phase shifter.
4.1. Core interferometer of the shearing setup
The core of the grating wedge interferometer is depicted in Fig. 2 and consists of two gratings G1 and G2 with identical grating periods p = 2μm. Grating G2 is rotated by an angle identical to the diffraction angle of the first diffraction order α = arcsin(λ/p) forming the wedge. From the scheme of Fig. 2 two relations can be derived - first, for the lateral shear σ in relation to the impinging height x of a ray onto the first grating and, second, for the optical path difference (OPD) of interfering rays which gives also an estimate for the longitudinal shift of selected points in the depth of the wave field.
For the lateral shear σ follows:
For the optical path difference OPD the following holds:
To point out the order of magnitude of the possible shear σ and the maximum OPD to be expected in our experiment we give two figures for the diffraction angle of α = 18.4° (sin(α) = 0.316) and the shearing region we use, a rectangular region of 70mmx40mm of the collimator aperture. The maximum shear is σmax ≈ 7mm and the maximum optical path difference between the interfering rays OPDmax < 1.2mm.
The last figure indicates that for the experiment the laser is a monochromatic source even if several longitudinal modes are oscillating. In contrast to the usual Young-double slit experiment, in our experiment the lateral shear is connected with a small longitudinal shift of the interfering wave amplitudes. The present case can be regarded as a lateral shearing experiment as the modulus of the degree of coherence varies by one order of magnitude slower in the longitudinal direction [24–27] compared with its lateral variation. A similar argument holds also for periodic light source distributions since the Talbot distance can be taken as a guide line. Since we consider only rather coarse structures with periods of the order of 1000λ the longitudinal maximum depth deviation amounts only to 0.1% of the Talbot distance so the impact of these deviation can be neglected.
4.2. Selected Screens
Since the production of general absorbing masks is difficult, here only binary masks shall be discussed. The van-Cittert-Zernike theorem  furnishes the theoretical background for the design of suitable binary screens producing a great variety of suitable coherence functions. The following types of intensity masks have been realized: slits, double slits and gratings. For the diffraction pattern we may adopt the mathematical equations from a textbook chapter  covering diffraction of monochromatic waves at binary structures. The intensity distributions of the far-field diffraction pattern for diffraction at a slit with the slit width s, at a double slit with the slit distance d, and at a finite grating with period d are sinc2-functions depending on the slit-width s multiplied either with a two beam interference cos-term for a double slit or with a multiple beam interference term known from grating diffraction at a grating with period d.
Furthermore, also sinusoidal intensity distributions have been realized via binary transparency masks by using the degree of freedom perpendicular to the shear direction for encoding the sinusoidal modulation (Fig. 3). In this case only one coherence region beside the zero shear region exists due to the fact that the Fourier-transform of the sine-function delivers at the carrier frequency a 0.5 − δ -function.
5. Experiments, Results, and Discussion
A qualitative overview concerning the spatial degree of coherence can be obtained from shearing interferograms with a set of moderate carrier fringes which can be adjusted by varying the wedge angle or by a small rotation of one of the gratings around the grating normal. This enables a first estimate of the contrast distribution in dependence of the lateral shear.
As a demonstration a shearing interferogram for a periodic light source is shown in Fig. 4. In the following we will discuss the influence of reference phase deviations in the phase shifting procedure on the stability of the modulation degree and show how the impact of reference phase slope defects can be greatly reduced.
5.1. Mis-calibration of the reference phase shifter
As it has been depicted in the theoretical part only very simple algorithms namely a 4-phase step algorithm and a compensating 5-phase step algorithm with π/2-steps will be investigated. In case of a linear calibration error the measured modulation shows periodic errors being dependent on the adjusted phase in the interferometer in form of Φ and 2Φ periodic contributions (see Eq. (13), (14) and (18), (19)).
By using the PZT calibration algorithm it is possible to provoke a linear slope error of the reference phase shifter. Using the 4-step algorithm for V4 in Eq. (10) one obtains a strong periodic error in the case of the linear calibration error (see Fig. 5 left). The proposed 5-step algorithm V5 in Eq. (15) reduces the amplitude of the periodic artefacts by a factor of 5 which is a direct result of the method for the calculation of the mean intensity. Since the two summands I13 = I1 + I3 and I35 = I3 + I5 are in phase opposition (Fig. 5 right) their addition will cancel the main part of the periodic artefacts. Although the compensating 5-step algorithm reduces the remaining artefacts a certain amount of periodic errors will in most cases prevail since stochastic phase errors are not systematic as linear mis-calibrations. But, the impact of drifts of the interferometric setup and other linear phase contributions will experience a reduction of periodic artefacts from the proposed 5-step algorithm to some extent.
5.2. A posteriori-elimination of Φ-periodic modulations of the modulus of the complex degree of coherence
From our error estimates made for the normal 4-step Bruning-algorithm , and, the compensating algorithm using five π/2-steps  under the assumption of systematic calibration errors of the phase shifter, functionals follow which comprise periodic Φ and 2Φ members. Instigated by a proposal for the elimination of such artefacts  we have used a least squares algorithm to extract periodic artefacts from the measured modulus values in accordance with the Eqs. (14), (13) and (19), (18) for the two algorithms. Although stochastic reference phase deviations will not lead to such closed expressions similar functional dependencies can be expected. While the dominant artefacts for the 4-step algorithm are of the first order with the phase errors and follow a combination of sin(Φ) and sin(2Φ) members the 5-step algorithm delivers second order effects in a combination of cos(Φ) and cos(2Φ) members.
Figure 6 demonstrates the reduction of periodic artefacts. The sine-functionals for V4 have been used in the lsq-fit which allows already a considerable improvement of the measured data. However, about 6% periodic artefacts remain due to neglecting of second order effects. If a similar procedure is applied to the V5-values resulting from our compensating algorithm the artefact suppression works even better.
To demonstrate the quality of the modulus measurement using the 5-step algorithm in combination with the a posteriori error elimination we show in Fig. 7 a 3D-graph of the modulus of the degree of coherence for an incoherently radiating double-slit.
5.3. Experimental realisations of different coherence functions
In the following we will show that the modulus of the coherence functions can satisfactorily be measured with the help of the proposed shearing interferometer which allows a direct access to the coherence function due to the progressive shear across the measuring aperture. For the realization of a set of coherence functions we have chosen one-dimensional binary intensity screens in front of a monochromatic incoherent light source providing mathematically unique distributions.
Slit sources will produce sinc-distributions of the coherence function where the slit width determines the region with modulation. This is demonstrated for two slit widths of wf = 100λ and wg = 250λ, respectively (Figs. 8 and 9) with the first minima occurring at a shear of σf ≈ 5mm and σg ≈ 2mm.
In the following Figs. the coherence functions for double slit sources are given. Figure 10 shows the case of a symmetric double slit with slits of 100λ at a distance of 1000λ which produces the typical pattern known from the diffraction at a double slit aperture.
By breaking the symmetry of the double slit source (Fig. 11) it is to be expected that zeros of the modulus will be absent in the high contrast region near the zero-shear position in contradistinction to the case shown in Fig. 10.
From Fourier-theory it is known that a periodic function in position space will produce a periodic function in frequency space which is here on display in the exit plane of the linearly progressive shearing interferometer (Figs. 12 and 13). Depending on the slit width the envelope changes.
A sinusoidal intensity distribution of the spatially incoherent light source produces in the linearly progressive shearing interferometer only one region with contrast beside the zero-shear position (Fig. 14). Theory tells us that the modulus is only 0.5 in this region.
It has been shown that the measuring error for the visibility or the modulus of the degree of coherence through linear calibration errors of the reference phase shifter or drifts of the interferometer in general can be reduced by using a 5-step algorithm in a similar manner as known from the phase error reduction . This can be especially related to the modification of the calculation concerning the mean intensity by using two intensity sums which are in phase opposition. On the one hand the exploitation of the time axis can introduce systematic phase dependent errors but offers on the other hand a pixel-true normalized modulus value in contrast to methods deriving the modulation through neighbourhood operations in multi-fringe patterns. A comparison between 4 and 5 step algorithms in case of small mis-calibrations of the phase shifter showed an error reduction by a factor of 5–6. Remaining periodic variations of the measured modulus values have been eliminated through least squares fitting of suitable sine- and cosine-functionals. In the case of the 5-step algorithm this procedure did reduce the artefacts below the 2%-level.
The modulus of the degree of coherence was measured experimentally by a grating wedge interferometer with a linearly progressive shear. As a partially spatial coherent light source an extended laser spot on a moving scatterer was used illuminating binary intensity masks. Since only binary intensity masks had been used the resulting coherence functions had been derived from a simple Fourier-transform with a high degree of accuracy. In this way theory did provide a high degree of soundness for benchmarking of the experimental data. The measured modulus values have been within a few % in agreement with theory.
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