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Abstract
A novel, single-shot, low-cost, multidirectional lateral shear interferometer for extended range wave front phase gradient sensing has been developed. It exploits the Fresnel diffraction field, which is formed by the five lowest diffraction orders of a simple binary amplitude checker grating. The Fresnel intensity pattern encodes information on four directional partial derivatives of the wave front under test. It has been theoretically, numerically, and experimentally shown that for larger gradient phase objects or shear amounts only the diagonal derivative information is easily accessible. The horizontal and vertical direction gradient maps are strongly amplitude modulated. Therefore, their demodulation becomes a challenging task. The same feature has been found in widely used quadriwave interferometer, which was developed at ONERA, France. The results of analytical and numerical studies and experimental works, including fringe pattern processing and phase demodulation, are presented.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Attractivity in shearing interferometry follows from the system compactness, implementation simplicity and increased immunity from environmental instabilities. The configurations providing two orthogonal lateral shear interferograms or multi-directional pairs are of particular interest. They allow retrieving phase information from a single-shot recording and, therefore, enable studies of dynamic phenomena. Such interferometers use diffraction gratings as beam splitters; they facilitate wide-spectral-range applications. Simple crossed binary amplitude Ronchi grating [1] or advanced quadriwave diffraction structure composed of appropriately frequency matched crossed binary amplitude grating and phase checker grating [2] are frequently applied.
The quadriwave interferometer [2–6] is an attractive solution but the hybrid amplitude-phase grating is technologically demanding. Similar remark concerns recently proposed randomly encoded hybrid grating [7]. In this paper we analytically describe the principle and implementation of a novel, single-shot, low cost, multidirectional 5-beam lateral shear interferometry using a simple amplitude checker grating. Spatial filtering of the grating lowest diffraction orders provides 5 beams which generate Fresnel diffraction field with encoded information on four directional partial derivatives of the wave front under test. A pair of horizontal and vertical difference wave fronts along with a pair of diagonal difference wave fronts are available, both having dissimilar shear ratios. With these beams the measurement data is analogous with those encoded in the quadriwave fringe pattern. However, additional interference due to the presence of the axial beam (grating zero diffraction order) results in extra diagonal harmonics. They correspond to the fundamental harmonics of the 5-beam fringe pattern. They are of no interest since their contrast depends on the propagation distance. On the other hand the first diagonal harmonics do not hinder the demodulation of the other harmonics.
It is worthy to compare our technique with methods reported in the literature [8]. One of them, corresponding to a classical Hartmann-Shack test (no spatial filtering) comprises self-imaging. Depending on the longitudinal position of the observation plane (for sensitivity adjustment), the arrays of spots with intensity distribution varying with defocus are detected. The second method reported in [8], named as the three-wave lateral shearing (TWLS) interferometer, uses a bidirectional hexagon grating or microlens array with spatial filtering (blocking the zero order and containing three elliptical holes). Another solution is the six-wave lateral shearing interferometer using an annular mask as the spatial filter to select the first six side orders. It offers simpler filtration at the expense of more reach harmonic content in the recorded multiple beam interference pattern. Angularly separated two-beam lateral shear interferograms are obtained, but because of blocking the zero order, the intensity pattern does not change with the defocus. The basic difference between the three- and six-wave solutions and the 5-beam configuration reported in this paper is the application of the most simple filter in the form of circular or square opening. Additionally, we provide theoretical description of the system which includes the quadriwave configuration as well.
In this paper we focus on developing, for the first time up to our best knowledge, the comprehensive theoretical description of the 5-beam system and its performance interpretation and understanding. Because of its complexity we limit our considerations to monochromatic radiation. The issue of system modifications to possibly obtain achromatic operation is left for separate future publication.
Quadriwave interferometry applications reported in the literature deal with very slowly varying phase specimens [2,3,5] (for achromatic objects the method is essentially achromatic) and stepwise phase objects [4]. These two groups of objects differ considerably. In the latter case quadriwave interferometry is strongly chromatic and this property is successfully exploited for measuring phase steps without ambiguity related to multiples of wavelength. For the reasons mentioned above we do not study chromatic properties of our 5-beam system in this paper. Nevertheless preliminary experiments have shown chromatic effects; they can be attributed to the presence of the zero order beam which is absent in quadriwave interferometry. Let us return, however, to the main scope of our interest, i.e., the 5-beam checker grating interferometer performance characterization under monochromatic illumination. In the presence of large phase gradients (but not so “brutally” large as in the case of stepwise objects) and/or shear amount values, however, the fringe pattern contrast deteriorations are encountered which result in fringe contrast reversals. In consequence, retrieving the horizontal and vertical direction derivatives becomes a challenge. This statement is not relevant to the diagonal directions where the modulation distribution is constant. In consequence, the extended phase (aberration) sensing range can be achieved with two diagonal channels only. One can adopt widely known fringe pattern processing techniques, e.g., the Fourier transform method, to demodulate the diagonal derivatives.
The development of a simple amplitude checker grating based interferometer and the comparison of its performance with the quadriwave system represent scientific and technical innovations. The results of analytical calculations and laboratory experiments are presented.
Former studies [9,10] on the amplitude checker grating Talbot shearing interferometry dealt with the use of all grating diffraction orders within the aperture of the system. Smaller wave front deformations were investigated and no spatial filtering was applied. Two mutually orthogonal, diagonal phase gradient maps were obtained. Although they suffice to retrieve phase information a higher number of directional pairs of interferograms with orthogonal shears provides more accurate phase reconstruction [11–13].
2. Principle of the 5-beam amplitude checker grating shearing interferometer – analytical description and comparison with quadriwave interferometry
As in the proposed interferometer system, see Section 4, the interference of aberrated quasi-plane wave front beams is encountered, the theoretical analysis below is conducted for such beams [14–16]. The transmittance of a binary amplitude checker grating can be expressed by the following equation [17]
where d denotes grating spatial period (understood as the sum of widths of single transparent and opaque squares along x and y directions) and m, n denote the number of diffraction orders along x and y directions, respectively. Grating binary structure generates many diffraction orders (m,n) but we are interested in the five lowest ones, i.e., (0,0), ( + 1, + 1), ( + 1,-1), (−1, + 1) and (−1,-1). In our experimental works they are obtained by spatial filtering. Figure 1 shows, for presentation clarity, propagation directions of the five orders of interest to aid analytical calculations of the 5-beam Fresnel diffraction field of the checker grating illuminated by a deformed, quasi-plane wave front under test. In the case of quadriwave interferometer configuration the (0,0) axial beam is not present, the propagation directions of side diffraction orders in both 4- and 5-beam systems are the same. Our analytical description presented below allows for common presentation of the two systems and their performance comparison when testing larger gradient phase functions.
Fig. 1 Schematic representation of the principle of multidirectional 5-beam lateral shear interferometry using an amplitude checker grating G1. φ(x,y) is the phase distribution of the impinging aberrated quasi-plane wave front under test. OP indicates one of the observation planes (coinciding with a CCD/CMOS matrix).
The complex amplitude distribution at a distance z from the grating is given by the sum of 5 beams [14–16], i.e.,
where φ(x,y) designates the phase distribution over the wave front under test, |Δ| is the lateral displacement of side diffraction orders with respect to the zero order along x and y axis directions (the lateral shear amount is equal to (λ/d)z in the configuration of Fig. 1), λ is the light wavelength, z is the diffraction field propagation distance from the grating G1 to observation plane OP, and a0 and a+1 = a-1 = a1 are the amplitudes of the central and side diffraction orders. The term (2πλz/d2) denotes the phase difference between side and zero order.The intensity distribution given by the modulus square of Eq. (2) is calculated as
Note that for diagonal directions (x + y) and (x – y) the change of the coordinate system x, y to x + y, x-y is encountered. Correspondingly, the period of fringe families along x + y and x-y directions is reduced byandfor the first and second harmonics of the diagonal fringe intensity distribution as compared with the fringe families along x and y axes, respectively. Accordingly, the shear amount is larger along diagonals.A Taylor series expansion cut on the second term, i.e.,
has been used when deriving Eq. (3) to allow for the analysis of wider range of wave front deformations as opposed to the case of slower phase changes for which the Taylor series expansion is cut on the first partial derivative term. Obviously, Eq. (4) is valid for the y direction as well.Let us group the intensity terms of Eq. (3) to enable clear understanding of contributions of the 4-beam (quadriwave) and 5-beam interference. The first group of terms includes:
- 1) the background term equal to 16a14;
- 2) the sum of two cosine terms describing two lateral shear interferograms with carrier fringes of spatial period d/2, perpendicular to x axis, encoding spatial partial derivatives ∂φ(x,y-Δ)/∂x and ∂φ(x,y + Δ)/∂x of tested phase distribution φ(x,y). These two interferograms are formed by the diffraction order pairs (−1, +1), ( + 1, +1) and (−1,-1), ( + 1,-1), respectively. Note that the two patterns do not coincide in space and are mutually displaced along the y direction by 2Δ. Calculating the sum we obtain:
- (5)
In the obtained product of two cosine functions the first cosine term with mixed derivative ∂2φ(x,y)/∂x∂y in its argument expresses contrast modulation of the second cosine with carrier fringes of spatial period d/2. They map the gradient ∂φ(x,y)/∂x of interest. Contrast modulations follow from the lack of coincidence in space of the two component lateral shear interferograms. The modulations do not allow for easy retrieval of the gradient map ∂φ(x,y)/∂x. An incoherent addition of two identical functions (in our case two dense fringe patterns added in Eq. (5), which are slightly mutually displaced in the y direction, results in so-called mechanical differentiation in the y direction, see for example [18,19]. Additive superimposition moiré fringes appear as the contrast modulation bands. Up to our best knowledge this feature of quadriwave interferometry is pointed out for the first time in the literature.
The above described effect of lateral displacement of two shearing interferograms in the direction perpendicular to shear direction is also observed in the spatial frequency spectrum of the interference pattern. The spectrum and its interpretation will be presented in the experimental part of the paper.
Note that the contrast modulation cosine term with mixed derivative ∂2φ(x,y)/ ∂x∂y in its argument does not include the distance z. In fact this statement is only ostensible since Δ depends on z and with an increase of Δ the number of modulations bands grows. Nevertheless once we hopefully solve the problem of demodulating the fringes with contrast reversals the two additional informative x and y channels will become available.
- 3) the sum of two terms describing two lateral shear interferograms with carrier fringes of spatial period d/2, perpendicular to y axis, encoding partial spatial derivatives ∂φ(x-Δ,y)/∂y and ∂φ(x + Δ,y)/∂y of the phase distribution φ(x,y) under test. These two interferograms are formed by diffraction order pairs (−1, +1), (−1,-1) and ( +1, + 1), ( + 1,-1), respectively. They do not coincide in space and are mutually displaced along the x direction by 2Δ. Their further discussion is analogous to the one presented above for the orthogonal fringe family.
- 4) two single-cosine terms representing two-beam interference fringes carrying information on the partial derivatives ∂φ(x,y)/∂(x + y) and ∂φ(x,y)/∂(x-y) along the diagonals of the coordinate system. These diagonal second harmonic fringe patterns are free of contrast modulations.
All above described terms are formed by the diffraction orders ( + 1, + 1), ( + 1,-1), (−1, + 1) and (−1,-1). They correspond, therefore, to the interference pattern generated in the quadriwave interferometer [2–6] applied now to sensing extended range phase deformations. Note that only diagonal derivatives can be readily separated; the phase maps of ∂φ(x,y)/∂x and ∂φ(x,y)/∂y cannot be easily demodulated because of the described contrast modulations. In places where the modulation fringe pattern assumes zero value, discontinuous phase jumps by π in carrier fringes are encountered; they lead to contrast reversals of carrier fringes.
The second group of terms in Eq. (3) comprises:
- 5) additional background term equal to 4a04;
- 6) two terms given by the product of two cosine functions. In this product one cosine function describes the carrier fringes of spatial period d perpendicular to the diagonal direction (in a changed coordinate system with axes parallel to diagonals). This cosine term encodes partial spatial derivatives ∂φ(x,y)/∂(x +y) or ∂φ(x,y)/∂(x-y), respectively, of the tested phase distribution φ(x,y). These two wave front gradient fringe patterns are contrast modulated by cosine functions with second order derivatives ∂2φ(x,y)/∂(x + y)2 and ∂2φ(x,y)/∂(x-y)2, respectively, in their argument. The contrast modulation bands change their spatial localization with the distance z (in our case of spherical aberration the lengths of axes of elliptical modulation bands change). Discussed terms represent first harmonics of interferograms along diagonals. Note that the fringe pattern contrast depends on the propagation distance z. The Talbot effect precludes the demodulation of spatial derivatives from these two fringe pattern terms.
The second group terms are formed by interference of the zero order beam with all four side diffraction orders. The contrast modulations can be explained by the influence of the self-imaging phenomenon (Talbot effect) [14–16] and mechanical differentiation approach [18,19]. For example, for the diagonal direction (x + y) the addition of two interferograms generated by the diffraction orders pairs (0,0), (−1,-1) and (0,0), ( + 1, + 1) is encountered. These two interferograms are laterally separated along the diagonal direction, which is perpendicular to carrier fringes.
The second group terms are treated as parasitic ones since we focus on retrieving the phase gradient maps in two directions from the first group terms. Note, therefore, that in case of testing extended phase deformations the quadriwave and 5-beam systems are equivalent with a common challenge to demodulate horizontal and vertical phase gradient maps. Remarkable simplicity of the amplitude checker grating make our interferometer solution attractive and worth of presentation.
3. Simulations
In this Section we present numerical simulations of the key issues discussed in the previous Section. As the phase function under test we have assumed the spherical aberration with defocus with the optical path difference OPD described by OPD = A(x2 + y2)2 + D(x2 + y2) and the parameters equal to A = 20 and D = −20. We use the aberration notation found in [20].
Figure 2 shows the intensity distribution pattern generated by all terms of the first group of Eq. (3), i.e., its first 5 rows, in the case of spherical aberration present, Fig. 2(a), and without the aberration, Fig. 2(b). The patterns result from the interference of diffraction orders ( + 1, + 1), ( + 1,-1), (−1, + 1) and (−1,-1). Such a situation would be encountered, therefore, in the quadriwave interferometer. Note the contrast modulation bands in Fig. 2(a) characteristic to the presence of spherical aberration, see the analysis in Section 2.
Fig. 2 Simulated intensity patterns generated by interfering four side orders of the checker grating [see first 5 rows of Eq. (3)) in the case of (a) the beam under test with spherical aberration and (b) without it].
Another visualization of these contrast reversal bands is obtained by calculating the intensity pattern described by first three rows of Eq. (3). It is presented in Fig. 3, its detailed discussion was given in points 2 and 3 of the previous Section. Contrast reversal bands can be interpreted as additive type superimposition moiré fringes formed by the geometric moiré differentiation process [18,19] due to the presence of two pairs of lateral shear interferograms along x and y directions, respectively, slightly separated in space in the direction perpendicular to the shear direction.
Fig. 3 Simulated intensity pattern generated by interference of four side orders of the checker grating taking into consideration only the first two rows of Eq. (3); the beam under test with spherical aberration.
The last but the very important part of the first group of terms of Eq. (3) is described by rows number 4 and 5. They represent the second harmonic terms of the whole interferogram intensity distribution described by Eq. (3). Figure 4 shows simulated intensity pattern corresponding to the diagonal direction x + y, see the fourth row of Eq. (3). According to the theoretical description the two beam interference pattern without contrast modulation bands is encountered. It allows further comfortable interferogram phase demodulation.
Fig. 4 Simulated intensity distribution of two-beam interference providing one of the two second harmonic terms of intensity distribution given by Eq. (3). Here the pattern along the x + y diagonal direction, see the fourth row of Eq. (3), is presented.
After simulating the first group of terms of Eq. (3) which effectively characterizes the performance of the 4-beam (quadriwave) interferometer in case of testing extended gradient phase disturbances (in our case large spherical aberration), it is time to present simulations of 5-beam interference. Taking into consideration all terms of Eq. (3) the calculated intensity pattern is shown in Fig. 5.
Fig. 5 Simulated intensity patterns generated by interference of five lowest diffraction orders of the checker grating and described Eq. (3) in the case of the presence of spherical aberration, Fig. 5(a), and its absence, Fig. 5(b), respectively.
Note characteristic changes in both images of Fig. 5 as compared with the images of Fig. 2. The difference is due to the terms appearing in last three rows of Eq. (3). Two last rows describe the first harmonic terms along diagonal directions x + y and x – y. They are characterized by strong contrast modulations, see the discussion in point 6 of Section 2. Because of that they are of no use for further interferogram phase demodulation. To illustrate the issue in more detail we present in Fig. 6 the simulated intensity pattern described by the one before last term of Eq. (3) (i.e., the first harmonic of the intensity distribution along the diagonal direction x + y).
Fig. 6 Simulated intensity distribution according to the one before last term of Eq. (3). It displays the first harmonic of intensity distribution along the diagonal direction x + y. The pattern is effectively generated by three-beam interference which leads to characteristic contrast modulations due to the Talbot effect (self-imaging phenomenon).
Numerical simulations corroborate, to the full extent, the physical interpretation of analytically derived representation of the 5-beam interference pattern obtained from the lowest diffraction orders of the binary amplitude checker grating. The 4-beam (quadriwave) interference case can be treated as the special one.
4. Experimental works
The experiments were conducted in the optical system shown schematically in Fig. 7. A binary checker grating of spatial frequency 10 lines/mm was used as the beam splitter. The grating was produced by conventional etching technology of a thin chromium layer on a glass substrate. Next both sides of the plate were antireflection coated. He-Ne laser served as the light source. For optical wave front aberration generation the objective OL1 (f1 = 200 mm) was set opposite to its correction direction to introduce spherical aberration. The aperture mask set in the spatial frequency (beam focus) plane passes 5 lowest diffraction orders of the checker grating. The grating frequency should be properly selected to avoid the overlapping of the diffraction spots (point spread functions, PSFs) in the grating spectrum. For example, in the case of turbulent wave front sensing, the angular size of the PSF is approximately equal to 2λ/r0, where λ is the light wavelength, and r0 denotes the so-called Fried’s parameter [21]. Since the angular separation between adjacent PSFs (along diagonal directions) is λ/dchecker, where dchecker = d√2 is the checker grating period, we obtain dchecker < r0/2. Situation in the spatial frequency plane in the case of square opening filter is shown in Fig. 8.
Fig. 7 Schematic geometry of our experimental amplitude checker grating based 5-beam lateral shear interferometer. Five lowest diffraction orders of the checker grating G1 are passed through the circular opening filter in the spatial frequency (light source image) plane. OL1 and OL2 – objectives with focal lengths f1 and f2, respectively, OP – output plane.
Fig. 8 The situation in the central part of the spatial frequency plane of the amplitude checker grating; the position of the square shaped filter is indicated be red dashed line.
We present the results for testing conditions with the second-order derivative of the spherical aberration (phase function) under test influencing the contrast of the recorded fringe image. Figure 9 shows the interference pattern for the checker grating placed in front of the spatial frequency plane, Fig. 7, at the distance equal to z0 = Md2/λ ≈63 mm (M = 4) [16,22]. The pattern was recorded in the output plane OP located at the back focal plane of the objective OL2. As was shown in [18] the horizontal/vertical shear amount in that plane is equal to Δ = (λ/d)f2. Changing the axial separation distance between the checker grating and the spatial frequency (light source image) plane allows for adjusting the density of carrier fringes. Five quasi-plane wavefront beams form the interferogram shown in Fig. 4 with properties described by equations derived in Section 2. Parasitic, quasi-vertical curved fringes are additionally seen in Fig. 9. They are generated by a glass plate protecting the CCD detector matrix. Since their spatial frequency is much lower as compared with analyzed image (carrier fringe) lines, they can be filtered out, to a large extent, during the fringe pattern processing.
Fig. 9 Multidirectional interference pattern influenced by the aberration of lens OL1 tested in the optical system shown in Fig. 7 for the axial separation distance between the checker grating G1 and the spatial frequency plane equal to Md2/λ ≈63 mm (M = 4). Opening mask in the frequency plane passes 5 lowest diffraction orders of the checker grating to form a fringe pattern.
In Fig. 10 we show the enlarged central part of modulus of the spectrum of the interferogram presented in Fig. 9. Note horizontal and vertical bright bands modulating the spectrum diffraction spots along x and y directions. Equidistant fringe-like bright bands modulate comatic shape diffraction spots characteristic to spherical aberration tested by shearing interferometry (explaining the issue in the interferogram domain, the detected interference fringes for coma tested in the reference beam type, e,g., Twyman-Green configuration, have the same form as the fringe pattern obtained when testing spherical aberration by lateral shear interferometry [20]). The modulating bands are parallel to shear direction for ∂φ(x,y)/∂x and ∂φ(x,y)/∂y gradient maps. On the other hand they are perpendicular to the diagonal derivatives directions ∂φ(x,y)/∂(x + y) and ∂φ(x,y)/∂(x-y), respectively (in the fundamental harmonics along diagonal directions), see Fig. 10. All spectrum modulation bands are directly related to the interference pattern zero contrast bands with sudden phase changes of π in the fringes, Fig. 9.
Fig. 10 Modulus of the Fourier spectrum of the multidirectional fringe pattern shown in Fig. 9. The comatic shape of diffraction spots is characteristic to the wave front spherical aberration tested by lateral shear interferometry [20,23]. See text for detailed explanation of rectilinear bands modulating comatic shape diffraction spots. Two diagonal diffraction spots are encircled with dashed lines to symbolically indicate filtering masks used in the fringe pattern Fourier transform processing described below.
A straightforward interpretation of the above described modulation bands follows from the well-known property of the Fourier transform. Namely, the visually observed energy Fourier transform of the sum of two identical signals slightly displaced in space is given by the product of the signal energy spectrum and the function cos2(pΔ/2), where p is the signal spatial frequency along the displacement direction and Δ is the lateral displacement amount. In the case under discussion this cosine square function expresses rectilinear bands modulating the energy spectrum of the interferogram. Note that for the x and y directions these bands are perpendicular to the direction of displacement between two lateral shear interferograms which add incoherently, see the discussion of the intensity distribution expressed by Eq. (3), points 2 and 3 of Section 2.
In result the spectrum modulating bands cut out some frequencies in the gradient maps, e.g., ∂φ(x,y)/∂x lying along straight lines. In the special case of φ(x,y) = const the discussed two fringe patterns formed, as mentioned above, by the order pairs (−1, + 1), ( + 1, + 1) and (−1,-1), ( + 1,-1) perfectly coincide (the two interferograms are just rectilinear fringes). However, in the presence of even very small object phase function φ(x,y) the Fourier transform will detect the sum of two slightly displaced interferograms, i.e., after performing spectrum logarithmization the modulation fringes will be seen in the spectrum. They will not overlap, however, with the diffraction spots of a weak phase function. This is because the recorded interferogram will encompass only a small fraction of a contrast modulation period.
It follows from the discussion of Section 2, Eq. (3) and Fig. 10, that only the second harmonics of the component directional interferograms encoding diagonal derivatives ∂φ(x,y)/∂(x + y) and ∂φ(x,y)/∂(x-y) provide ready access to phase gradient images; they are free of fringe contrast modulations. These two orthogonal phase gradient maps enable the investigated phase function retrieval.
For diagonal phase gradient maps demodulation well-established Fourier Transform method is used. It can be easily applied due to complete separation of second harmonic diagonal comatic spectral lobes of interest, which is to be seen as one of main advantages of the proposed extended range wavefront gradient sensing. The single-shot interferogram analysis path follows.
- 1) Calculation of the interferogram Fourier transform, its modulus is depicted in Fig. 10;
- 2) Filtering the selected diffraction spot by use of a circular Hanning masks as shown in Fig. 10;
- 3) Calculating the inverse Fourier transform. As a result one obtains complex valued interferogram with an easy access to phase and amplitude (analytic signal);
- 4) Amplitude modulation calculation as the magnitude of complex interferogram;
- 5) The phase angle calculation for each pixel of complex valued interferogram. The resulting angles lie between ± π rad;
- 6) Unwrapping the mod(2π) phase distribution obtained in the previous step to retrieve wavefront phase gradient map;
- 7) Subtracting spatial carrier frequency by a least squares fit of a plane to the continuous unwrapped phase distribution. The resultant phase map is a diagonal difference wave front.
In Figs. 11(a) and 11(b) we present the demodulation results for the two diagonal channels. Please note that the obtained difference wave fronts are characteristic for spherical aberration present in our experimental optical system. In Figs. 11(c) and 11(d) we show the cosine of demodulated phase maps to help assess their quality.
Fig. 11 Demodulated diagonal derivatives of the tested aberrated wave front. (a) and (b) correspond to 45 and 135 degrees directions, respectively, phase maps presented in radians; (c) and (d) cosine functions of demodulated phase maps – 45 and 135 degrees directions, respectively.
The amplitude modulation distribution calculated for the second harmonic along the 45 degrees direction is presented in Fig. 12(a). The same has been obtained for the orthogonal direction. If do not take into account parasitic fringes influencing the modulation its distribution is constant. This is the experimental verification of our earlier theoretical analyses. We have attempted to demodulate phase information encoded in the amplitude modulated harmonic, e.g., the x direction harmonic. In Fig. 12(b) demodulated phase map is presented. To allow clearer observation of characteristic 2π phase jumps predicted theoretically we have computed the cosine function of the demodulated phase, Fig. 12(c). Jump lines correspond to dark amplitude modulation bands shown in Fig. 12(d). In Fig. 13 we showed amplitude modulation maps for y harmonic, Fig. 13(a), and for first 45 degree diagonal harmonic, Fig. 13(b). All amplitude modulation fringes represent the wave front mixed derivatives. Naturally, for the orthogonal directions the amplitude maps are the same but rotated by 90 degrees. As has been mentioned before diagonal first harmonics predominate the 5-beam fringe pattern. This can be clearly seen if we look at the modulations present in the interference pattern from Fig. 9. They are a combination of 45 and 135 degrees direction elliptical fringes depicted in Fig. 13(b).
Fig. 12 (a) amplitude modulation map of the second harmonic along 45 deg direction of the intensity distribution of the aberrated 5-beam fringe pattern, Fig. 9; (b) the x direction harmonic phase distribution, and (c) cosine function of this phase map presenting phase 2π jumps along dark amplitude modulation bands; (d) amplitude modulation map of the x harmonic. Additional high spatial frequency fringes seen in all the amplitude modulation images result from parasitic interferences present in the optical experimental system.
Fig. 13 Amplitude modulation maps of (a) the y direction harmonic and (b) first 45 deg direction harmonic of the intensity distribution of the aberrated 5-beam fringe pattern, Fig. 9. Additional high spatial frequency fringes seen in all the amplitude modulation images result from parasitic interferences present in the optical experimental system.
5. Conclusions
The paper is focused on introducing a new multidirectional shearing interferometer for the wave front aberration testing and adding further refinement to the theory of the 4-beam free-space diffraction interference. We propose to deploy an amplitude binary checker grating for splitting an examined beam into five interfering replicas. The resulting 5-beam fringe pattern is closely related to the quadriwave interference one. The only difference between them is the presence of additional diagonal harmonics of a lower spatial frequency in the 5-beam pattern. These parasitic fringe families do not hamper the demodulation of other intensity harmonics. Our innovative multi-channel shearing interferometer can be treated as a low-cost version of the quadriwave interferometer.
It should be realized, however, that the 5-beam system has two major drawbacks. Firstly it is necessary to utilize the optical coherent processor to spatially filter the five lowest diffraction orders of the checker grating. This makes the optical system more complex. Note, however, that the optics which images extended gradient phase objects onto the conjugate detection plane is indispensable anyhow. In contrast, for slowly varying phase objects the detection plane defocus is intentionally used to introduce continuously varied shear amount, see Talbot and quadriwave interferometry. The lack of possibility to change the shear amount in a continuous manner represents the second demerit; in our system shown in Fig. 7 the shear is fixed and equal to (λ/d)f2.
It is straightforward to note that since spatial filtering is conducted in the spatial frequency plane of the optical imaging system, Fig. 7 (e.g., optical coherent processor), we are able to switch from the 5-beam interferometer system to the 4-beam (quadriwave) one by adding central occultation to block the zero order beam.
We have further advanced the theoretical description of the interference of four quasi-plane wave front beams propagating at symmetrical angles with respect to the z axis, each in a different diagonal direction of the x-y plane. This a geometry of the well-known quadriwave interferometer which has significantly moved the field of wave front sensing research ahead. Our aim has been to understand the nature of the contrast modulation appearance in the 4-beam interference pattern. It turns out that fringe contrast reversals, inherent to large phase gradients or shear ratios, result from summing of two identical, spatially separated aberrated fringe families. This mechanism applies, however, only to the x and y pattern intensity harmonics. It has been analytically and experimentally proved that no matter what aberrations are present in the tested wave front the modulation of diagonal fringe families is constant. This is because the diagonal intensity harmonics result from two-beam interference. The above mentioned characteristics of the 4-beam interference remain valid for our 5-beam configuration proposed. The formation of additional diagonal intensity harmonics of lower spatial frequency by three-beam interference results (similarly to the x and y direction harmonics) from the incoherent addition of two spatially displaced fringe patterns. For this reason the contrast deteriorations are encountered.
Both quadriwave and 5-beam interferometers possess two modulation contrast-free orthogonal channels. They allow reconstruction of a two-dimensional wave front. This finding potentially opens new frontiers in the field of large aberration studies.
Funding
National Science Center Poland (NCN) (2017/25/B/ST7/02049); Statutory Funds Department of Mechatronics Warsaw University of Technology.
Acknowledgments
We acknowledge the support of National Science Center (Poland) grant OPUS 13 2017/25/B/ST7/02049 and Faculty of Mechatronics Warsaw University of Technology statutory funds.
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