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In recent years singular optics has gained considerable attention in science and technology. Up to now optical vortices (phase point dislocations) have been of main interest. This paper presents the first general analysis of formation of phase edge singularities by coplanar three-beam interference. They can be generated, for example, by three-slit interference or self-imaging in the Fresnel diffraction field of a sinusoidal grating. We derive a general condition for the ratio of amplitudes of interfering beams resulting in phase edge dislocations, lateral separation of dislocations depends on this ratio as well. Analytically derived properties are corroborated by numerical and experimental studies. We develop a simple, robust, common path optical self-imaging configuration aided by a coherent tilted reference wave and spatial filtering. Finally, we propose an automatic fringe pattern analysis technique for detecting phase edge dislocations, based on the continuous wavelet transform. Presented studies open new possibilities for developing grating based sensing techniques for precision metrology of very small phase differences.
© 2017 Optical Society of America
1. Introduction
Three-beam interference has a long history. Its first important application, as initiated by Zernike in 1950 [1], concerned precise measurements of very small phase differences [2–6]. Next studies dealt with wave front aberration testing [7–13]. Most recent investigations concern multidirectional shearing interferometry [13,14], and not coplanar three-beam interference for generation of microlens arrays [15] and optical vortex lattices [16–21] including novel metrological applications [22]. Vortices (or screw dislocations) occur when an abrupt undefined phase change is accompanied by zero amplitude at a point. On the other hand relatively few studies have dealt with edge dislocations (or edge singularities) with an abrupt phase change along a line perpendicular to the light propagation direction [23–30]. Their generation with dense wavelength-scale-period gratings was reported [31–33]. As is well known coplanar interference of three lowest diffraction orders in the grating Fresnel diffraction field leads to the so-called self-imaging or Talbot effect [34–36]. In [31–33] authors numerically and experimentally showed the occurrence of phase singularities in self-image planes. In spite of the phenomenon pioneering presentation the cited papers [31–33] lack a general analysis of three-beam interference. The case of the 0th and +/−1st orders of equal intensity levels [33] was only treated.
We present a comprehensive analysis of three-beam interference to show, for the first time, a broad range of the ratio of the amplitude A of side beams and the amplitude C of the central beam for which phase dislocations can be obtained. A general condition A/C>0.5 for the occurrence of phase edge dislocations is derived. Their lateral separation distance depends on the ratio A/C. Although it is well known that the topological charge of an edge dislocation is zero [24], to distinguish the properties of two adjacent singularities we follow the nomenclature introduced in [31,32]. The phase distribution turns around the phase dislocation as the wave propagates and the direction of rotation in two adjacent singularities is mutually opposite. Correspondingly, the opposite sense of rotation indicates the opposite topological charge of the two singularities. The properties encountered in the boundary case of A/C = 0.5 are discussed as well. In that case phase singularity conditions are met in the self-image planes once per a single field lateral period, but the amplitude condition is not accompanied by an abrupt phase change. The derived properties of edge singularities for selected A/C ratio values are corroborated by simulation works. A novel, simple and insensitive to environmental influence common-path experimental method for generation and detection of three-beam interference phase edge dislocations is described. Recorded and subsequently analyzed single and two-shot interference patterns are obtained by adding the fourth coherent beam in the plane perpendicular to the incidence plane of three coplanar beams. An automatic fringe pattern phase processing solution based on the continuous wavelet transform is used for the edge dislocation detection. Potential investigations and applications are envisioned.
2. Analysis of three-beam interference with grating lowest diffraction orders
2.1 Derivation of the edge singularity condition
The analytical studies are conducted on the example of a simple binary amplitude grating. Our goal is to derive the aspect ratio values, defined as the ratio of the transparent slit width and the grating period d (one grating period encompasses a single transparent and single opaque slit) for which three lowest diffraction orders, i.e., 0 and +/−1, generate phase edge singularities in the interference field.
The amplitudes C and A of the 0th and +/−1st diffraction orders, respectively, follow from the grating aspect ratio [37]. Assuming a symmetrical profile of grating transmittance (for simplicity, the origin of the coordinate system is taken to coincide with the center of one of the transparent slits) we have equal amplitudes of side orders, i.e., A+1 = A-1 = A. For the plane wave front beam illumination of wavelength λ along the grating normal the complex amplitude of the three-beam Fresnel diffraction field at a distance z from the grating, with its lines perpendicular to the x axis, is equal to [34–36]
The intensity distribution is calculated asFor the self-imaging planes we put z = md2/λ to obtainWhen m = even integer the intensity distribution matches the three-beam intensity distribution in the object plane, z = 0. For m = odd integer the intensity pattern is laterally shifted by half a grating period.To face the phase singularity the following conditions have to be met [24]:
It follows from Eqs. (1)-(3) that the conditions are met in self-image planes only. For example, for z = 2d2/λ we have We search the values of x for which Re{E} = 0. Correspondinglyand it follows thatThe solutions arewhere kϵZ. The domain of the function f(x) = arc cos(x) is D = [-1; + 1]. To facilitate determination of above equations the following conditions must be metAssuming A and C > 0 we getand the final result isThe Ronchi grating, most common amplitude binary grating, has the aspect ratio equal to 0.5 and the amplitudes C = 0.5, and A = 0.318 [37]. In result A/C = 0.636, this value meets our condition (12). In general, the condition A/C>0.5 can be realized by three lowest diffraction orders of binary amplitude gratings with aspect ratio smaller than 0.6 and binary phase Ronchi gratings with various square groove depths. The boundary case of A/C = 0.5 is implemented using binary amplitude grating with the aspect ratio equal to 0.6033 [37,38].The formation of edge singularities with three-beam coplanar interference in self-image planes can be also explained as the result of in-phase overlap of two-beam interference patterns generated by orders (0, + 1) and (0,-1) [39]. The condition A/C>0.5 is to be met.
Using Eq. (9) we can evaluate the distance Δx between two edge singularities, i.e.,
Normalizing it by the intensity distribution spatial period we have
Figure 1 shows the diagram representation of Eq. (14). The diagram starts with Δx/d = 0 encountered for A/C = 0.5.
Fig. 1 Graphical representation of the separation distance Δx between two edge singularities within a single interference field lateral period as a function of the ratio A/C. Vertical axis units express the values of Δx/d.
2.2 Intensity, amplitude and phase distributions in the self-image planes and their vicinity
First we will present important self-image plane distributions: intensity I(x,z), real Re{E(x,z)} and imaginary Im{E(x,z)} parts of the field complex amplitude, amplitude distribution given by the square root of the sum {Re(E)}2 + {Im(E)}2, and the phase distribution calculated as arctan{Im(E)/Re(E)}, for two exemplifying values of the ratio A/C equal to 0.5 and 1.
Figure 2 shows the functions I(x,z), Re{E(x,z)}, Im{E(x,z)} (left) and the amplitude and phase functions (right) for the ratio A/C = 1 (and, in general, for A/C > 0.5) in the self-image plane at z = 2d2/λ. Two null intensity locations, I = 0, are encountered over a single self-image intensity distribution period. The plots of Re{E(x,z)} and Im{E(x,z)} cross at these points. What is most important, an abrupt phase jump [ + π,-π] occurs between the two points, their lateral location depends on the ratio A/C, see Eq. (14). In the discussed case Im{E(x,z)} = 0 for all x coordinates.
Fig. 2 The case A/C = 1. Left: plots of I(x,z) (black line), Re{E(x,z)} (blue line), and Im{E(x,z)} (red line) of the three-beam interference field for the propagation distance z = 2d2/λ. Right: plots of the amplitude (green line) and phase (black line) functions. Plots are presented for a single lateral period of the three-beam interference field.
The second case of A/C = 0.5 corresponds to the boundary value of Eq. (13). The binary amplitude grating with the aspect ratio equal to 0.6033 implements this case. Figure 3 shows the above mentioned distributions over a single lateral period of the three-beam interference field for the same propagation distance from the grating, z = 2d2/λ, as in Fig. 2.
It follows from Fig. 3 that the condition I(x) = Re{E(x)} = Im{E(x)} = 0 is fulfilled once within the field single lateral period. Note that the plots of Re{E(x)} and Im{E(x)} are just tangent at that location (they do not cross), it corresponds to phase singularity because of the zero complex amplitude value, i.e., 0 + i0 (the phase is indeterminable). The phase dislocation (abrupt phase change [ + π,-π]) is not encountered, however. This particular feature is characteristic to the case A/C = 0.5 only.
A similar situation is encountered in the case of the optical vortex generation by interference of three non-coplanar plane wave front beams [17,18,40]. For two beams of equal amplitude and the third beam of the amplitude two times higher (in general, equal to the sum of two others) the two vortices in pair overlap and annihilate.
The maximum modulation amplitude and uniform phase distributions are found in the self-image plane, see right part of Fig. 3. The uniform null phase distribution is a consequence of uniform null value distribution of the imaginary part Im{E(x)}. The parts of Im{E(x,z)} on both sides the self-image plane are sign reversed, the same conclusion concerns phase distributions in those planes, see the discussion below.
Figures 4 and 5 show detailed information about changes of the intensity, amplitude and phase in the close vicinity of the self-image plane, z = 2d2/λ, for the cases A/C = 1 and A/C = 0.5, respectively. In the first case A/C = 1 (in general, A/C>0.5) the intensity and amplitude distribution plots change symmetrically with respect to the self-image plane as a function of defocus. For example, the curves for z = 1.01(2d2/λ) and z = 0.99(2d2/λ) coincide in the diagram (the plots in Figs. 4 and 5 are shown for an increment of +/−0.01 of the propagation distance z normalized by 2d2/λ). This is why we have only six curves in the plots of intensity and amplitude functions for analyzed 11 axial distances (for five shorter and five longer distances than 2d2/λ, and the distance equal to 2d2/λ). As mentioned above, in the self-image planes intensity and amplitude plots show two zeros over a single field lateral period and an abrupt phase change between them. The upper curves with the phase values >0 relate to the propagation distances z>2d2/λ, and the curves below the zero level line display negative phase function values for z<2d2/λ. Concluding, in the case A/C>0.5 phase discontinuities occur in self-image planes.
Fig. 4 The case A/C = 1. Intensity, amplitude and phase distribution cross- sectional plots for the most important part of a single lateral period of the three-beam interference field in close vicinity of the self-image plane at z = 2d2/λ. The curves are drawn with the increment 0.01 of the normalized propagation distance, i.e., z/(2d2/λ).
Fig. 5 The case A/C = 0.5. Intensity, amplitude and phase distribution cross-sectional plots for the most important part of a single lateral period of the three-beam interference field in close vicinity of the self-image plane at z = 2d2/λ. The curves are drawn with the increment 0.01 of the normalized propagation distance, i.e., z/(2d2/λ).
The situation in self-image planes deserves a short comment. Figure 4 could suggest that between phase dislocation locations (where the phase value is indeterminable) the phase assumes, in a random manner, the values of + π or –π. In practice, however, it is the same phase angle and randomness is caused by computing accuracy. Namely, real and imaginary parts are calculated from earlier computed complex amplitude. In effect, the imaginary part is not exactly equal to zero. Once it is slightly larger or smaller than zero. Because in the region under consideration the real part is negative, then depending on the sign of the finite imaginary part value the phase angle becomes equal to + π or –π. As mentioned above we talk about the same angle and we can assume that the phase assumes, simultaneously, the values of + π and –π, and two solid horizontal lines could be drawn in Fig. 4.
The intensity and amplitude distribution plots in Fig. 5, A/C = 0.5, change symmetrically with respect to the self-image plane as in the preceding case of A/C>0.5, e.g., the curves for z = 1.01(2d2/λ) and z = 0.99(2d2/λ) coincide in the diagram. The similarities end here, however. Intensity and amplitude distribution cross-sections show very different character as compared with the ones shown in Fig. 4. As already explained, see the discussion of Fig. 3, the condition I(x) = Re{E(x)} = Im{E(x)} = 0 is fulfilled once within the field single lateral period with the plots of Re{E} and Im{E} just tangent at the null intensity location. The phase dislocation, i.e., abrupt phase change [ + π,-π], is not encountered. This particular feature is characteristic to the case A/C = 0.5 only. In the case of phase distribution, see bottom drawing, the horizontal straight line corresponds to z = 2d2/λ. The upper curves above that line show the phase distribution cross-sections for z>2d2/λ (with the normalized propagation distance increment of 0.01), and the curves below that line display the phase function cross-sections for z<2d2/λ.
For further illustration we calculated equiphase lines for three-beam interference field in close vicinity of a self-image plane. Detailed discussion of spiral singularities (optical vortices) can be found, for example, in [41]. It can be extended to edge singularities. Figure 6 presents the propagating field equiphase lines for the case A/C = 1 in the vicinity of the plane z = 2d2/λ (the ordinate axis description expresses the propagation distance z normalized by 2d2/λ). In this case, two adjacent edge singularities are of opposite sign [31,32]. Equiphase lines are displayed within single lateral period of the field (along the abscissa axis). Besides presenting selected phase value contour lines the color representation is added.
Fig. 6 Equiphase lines in the case A/C = 1 in the vicinity of the three-beam interference self-image plane, z = 2d2/λ (the ordinate axis shows the propagation distance z normalized by 2d2/λ). Phase distribution is presented within single lateral period of the field.
Figure 7 presents the equiphase line calculation results for A/C = 0.5.
Fig. 7 Equiphase lines in the case A/C = 0.5 in close vicinity of the three-beam interference self-image plane, z = 2d2/λ (the ordinate axis shows the propagation distance z normalized by 2d2/λ). Single lateral period of optical field is displayed along the abscissa axis direction.
3. Experimental phase edge dislocation detection using two-dimensional continuous wavelet transform
In the experimental part we focus on detecting phase edge singularities without the necessity to determine their sign. To avoid employing complex Twyman-Green or Mach-Zehnder interferometer configurations to unambiguously distinguish the phase singularities from local intensity minima [42], we have developed very simple common path experimental setup using two crossed linear diffraction gratings with spatial filtering. It allows generation of phase edge dislocations and their subsequent detection by adding the fourth beam in the plane perpendicular to the one containing three coplanar beams. In result the reference (secondary) fringes run perpendicularly to three-beam interference fringes, in particular to zero intensity lines encountered in self-image planes. The secondary fringes should be denser than the spacing between the zero intensity (phase edge dislocation) lines.
Laboratory experiments were conducted in a robust optical system resistant to environmental conditions shown schematically in Fig. 8. To relax the conditions for sampling the interference image by CCD matrix pixels the configuration with spherical wave front illumination was chosen. Now the lateral period of three-beam interference pattern under test can be easily magnified in comparison with the one obtained with plane wave front illumination. The distance z0 is selected to comfortably perform spatial filtration, see Fig. 9(b), of diffraction orders of the grating G and the plane OP is set at an appropriate distance z’ from the grating. The three-beam interference pattern period is now equal to [(z’ –z0)/z0]d.
Fig. 8 Schematic geometry of the experimental setup with coplanar three-beam interference of spherical wave fronts aided by the fourth beam in the plane perpendicular to the drawing to generate reference fringes. OL – focusing objective; G – crossed diffraction structure (see Fig. 9 below and the text for detailed explanation); SF – spatial filter; OP – observation (CCD matrix) plane.
Fig. 9 Schematic drawing of a magnified part of the diffraction structure designed for experiments (a). Low spatial frequency amplitude Ronchi grating with horizontal lines (its lateral extent of 2.5 period is shown) is used to generate three coplanar interfering beams. Denser Ronchi grating with vertical lines provides the fourth beam to produce reference (secondary) fringes. The situation in the spatial frequency plane and the position of T-shaped filter are schematically shown in (b). The filter passes three lowest orders of horizontal line grating and + 1 order of denser vertical line grating (indicated as + 1f in the drawing).
The heart of the experimental setup is the beam splitting element composed of crossed linear binary amplitude gratings. The first one was the Ronchi amplitude grating of aspect ratio equal to 0.5 and spatial frequency of 5 lines/mm. The aspect ratio 0.5 results in the amplitudes of the zero and +/−1 orders equal to C = 0.5 and A = 0.318, respectively. Therefore A/C = 0.636. These three lowest orders implement coplanar three-beam interference to generate phase edge dislocations in the self-image planes. The second grating was the binary amplitude Ronchi grating of frequency of 50 lines/mm and aspect ratio equal to 0.5. One of its first diffraction orders served as the fourth beam to generate fork fringes. Figure 9(a) shows a magnified small area of our crossed grating beam splitter. The mask was set in a convergent beam behind the lens OL to enable filtering of diffraction orders by the spatial filter SF.
The phase edge dislocations can be detected using single and two-shot approaches. The latter one employs two images with phase shifted secondary fringes. For this purpose the crossed diffraction structure is laterally shifted between exposures in the direction perpendicular to denser grating lines. The optimum phase shift value is equal to π, but the method works for other shift values as well [43]. Two frames are subsequently subtracted for background and noise correction. Figure 10(a) shows the image recorded in one of the self-image planes. The visibility of reference fringes in Fig. 10(a) is not high because of low intensity level in the regions with edge dislocations (in accordance with the theory and numerical simulations, see Section 2B). The result of subtracting two self-images recorded with grating lateral displacement by half its period is shown in Fig. 10(b). Information on phase dislocations can be readily apprehended after the subtraction.
Fig. 10 Image recorded in one of the self-image planes (a) and the result of subtraction of two self-images recorded with grating lateral displacement by half its period (b).
Single and two-shot images can be processed using automatic fringe pattern processing techniques, e.g., the Two-Dimensional Continuous Wavelet Transform (2D CWT) [44], and the Hilbert-Huang Transform (HHT) [43] both being continuously developed in our group. They represent important tools in the field of fringe pattern phase and modulation analysis. They are known for their excellent abilities to filter out high frequency noise and low frequency background signal, very significant features for our investigations. In this paper a novel 2D CWT method for finding phase edge dislocations is presented.
Calculating the 2D CWT of an image results in a 4D matrix of complex wavelet coefficients which enable evaluation of phase and amplitude distributions. To find phase edge dislocations our novel algorithm progresses as follows.
1. The modulus of the phase difference in adjacent pixels along the vertical direction is evaluated from the wrapped phase map calculated by the 2D CWT. The obtained intensity image contains information about phase jumps with values close to 2π in the places where the edge of a phase fringe deviates from the vertical direction. In the case of presence of phase dislocation the modulus of the phase jump value is approximately equal to π. On the other hand, in the regions without phase dislocation lines, the change of phase in neighboring pixels is much smaller than π.
2. Basing on these observations the algorithm searches for pixels with phase jump values in the range [π-1, π + 1]. In this way a binary image containing thin bright lines which indicate the localization of phase edge dislocations is obtained.
The described method concerns the detection of horizontal phase edge dislocations, i.e., with a priori known orientation. Nevertheless the proposed algorithm can be generalized for an arbitrary orientation of dislocation lines. The detection accuracy of phase edge dislocations by our technique depends on the accuracy of phase evaluation by the 2D CWT algorithm. The novelty of above presented 2D CWT algorithm is attributed to the fact that it bases on phase detection in contrast to the intensity processing method [44] formerly developed to detect optical (screw) vortices.
Figures 11(a), 11(b), 11(c) and 11(d) show, respectively: a magnified part of the filtered four-beam interference pattern with reference fringes recorded in one of the three-beam interference self-image planes, the wrapped phase distribution, the normalized fringe pattern and thin bright lines indicating the presence of phase edge dislocations. Images 11(b), 11(c) and 11(d) were calculated from the difference of two frames recorded with mutual lateral shift of the grating G by half its period. Two frame subtraction operation showed better pre-filtration results than background and noise removal by the wavelet transform itself. The reason for presenting a filtered image 11(a) instead of a row one, Fig. 10(a), is explained below (clearer presentation of a short movie Visualization 1).
Fig. 11 Magnified part of the CWT filtered four-beam interference image recorded in the three-beam interference self-image plane (a) (Visualization 1, top left), wrapped phase distribution (b) (Visualization 1, top right), normalized fringe pattern (c) (Visualization 1, bottom left), and the processing result obtained by our novel phase detection based 2D CWT method (d) (Visualization 1, bottom right).
Some errors in the displays are noted, e.g., appearance of thin lines is not uniform over the processed image, Fig. 11(d). This feature follows from the fact that etched chromium lines of the Ronchi amplitude grating are on a glass substrate which introduces slow phase errors into the illuminating beam. Additionally, optical elements in the optical system were slightly locally contaminated (e.g., dust particles). In result we detect shearing interference of not perfectly spherical wave fronts. In consequence the self-imaging condition is fulfilled over local areas across the interference field. This feature can be carefully followed in all images in Fig. 11 and related movies.
The experiments were conducted with binary amplitude gratings of different opening ratio values to check the accuracy of our method against the derived relationship expressed by Eq. (14). The ratio Δx/d was determined, each time, from Fig. 11(d) and compared with the value calculated from Eq. (14) for A and C values corresponding to a particular opening ratio of the grating used. Very good coincidence was obtained with small differences in the second number after the decimal point. This discrepancy can be attributed to grating technology, mainly to non-zero transmittance of dark lines of the grating resulting in the values of A and C slightly different from the theoretical ones.
Most characteristic changes in the three-beam interference field which occur along the propagation distance z can be clearly exemplified by short movies, see Fig. 11 (Visualization 1). The choice of presenting intensity changes in a filtered image, Fig. 11(a), instead of intensity changes in the interferogram itself was dictated by low intensity levels in the regions with phase edge dislocations. A movie made of filtered images (top left) clearly shows the evolution of reference fringes (deformed proportionally to the phase distribution in the field) into two half-period shifted interference patterns in the self–image planes where the phase value is indeterminable. Second movie, Visualization 1 top right, illustrates field phase changes with the propagation distance z including the transient phase indetermination when passing through the self-image plane. Third movie (Visualization 1 bottom left) demonstrates the changes of normalized reference fringes including the transient change of their deformation direction, and the fourth movie (Visualization 1 bottom right) exemplifies the axial resolution of the 2D CWT method proposed for the phase edge dislocation detection. At all processing stages the reference fringes serve the purpose very well.
The data for the movies were collected in the optical system shown schematically in Fig. 8. The distance z0 was set equal to 110 mm. Starting from z' = 263 mm twenty two out-of-phase interference image pairs were recorded for z' increasing by 0.5 mm. Stepwise discontinuities in the position of fringes in the movies are caused by manual axial displacements of the CCD camera along the beam propagation direction. Note the above mentioned influence of laterally sheared, non-spherical wave front shape beams on the interference result. The features of the three-beam interference field as a function of propagation distance are truly displayed by the conducted experiment. We believe that our movie presentation approach is quite competitive and advantageous in comparison with another possible tool, i.e., optical carpet [45].
4. Discussion
In summary we presented the first general theoretical analysis of coplanar three-beam interference resulting in periodic formation of phase edge dislocations along the light propagation direction. The condition for the amplitudes of interfering beams was derived and detailed studies of the field properties were given. A simple and robust experiment for phase edge dislocation generation and detection was designed. A novel 2D CWT algorithm, based on the interference field phase detection, was developed to process the experimental fringe patterns. Obtained results fully corroborate conducted theoretical, numerical and laboratory investigations.
We believe that our discovery should open new possibilities for studies of related phenomena and applications of diffraction grating based sensing methods. This remark concerns several scientific disciplines since the self-imaging phenomenon (Talbot effect) gains wider interest in optics, photonics and scientific communities working with electromagnetic radiation, starting from very short X-ray radiation up to far infrared. New, potential investigations are envisioned in high precision measurements of very small phase differences (testing of phase filters, thin film thickness and surface roughness). Instead of various schemes for photoelectric detection of characteristic intensity profiles in the three-beam interference field (e.g., detection of the constant modulation amplitude envelope of interference fringes, see [6] and references therein) we propose to detect the three-beam interference self-image planes by localizing phase edge dislocations. Other potential applications include testing of the beam wave front and collimation, diffraction grating period and radiation wavelength which influence the parameters of the three-beam self-imaging phenomenon.
Funding
Narodowe Centrum Nauki (NCN) (2014/15/B/ST7/04650), Faculty of Mechatronics Warsaw University of Technology statutory funds.
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