1. Introduction

Interferometry is commonly used for wavefront determination in visible optics [1], cold atom optics [2], electron optics [3], and X-ray microscopy [4]. A two-beam interferogram is formed by interference between a wave that has passed through a specimen and a reference wave that does not interact with the specimen. Fourier-transform methods are commonly used to reconstruct the intensity and phase of the wave [5]. Although these methods are simple to implement, the spatial resolution of the reconstruction is limited by how isolated the spatial-frequency spectrum is from the central maximum. Transform methods also involve inverting a subsection of the Fourier transform of the interferogram; since a specific region of the desired real-space phase map cannot be localized in Fourier space, the phase is reconstructed non-locally. For example, poor fringe visibility in a subsection of the interferogram can introduce systematic errors in the recovered wave over the entire field of view. Further, a Gibbs-type phenomenon, namely ‘ringing’, can occur when processing subsections of interferograms that exhibit strong scattering or absorption, leading to poor contrast resolution.

The method developed in the present paper uses three-wave interference to generate a uniform lattice of optical vortices, which are used to recover the phase of the input wave-field. Perturbations to the input wave-field due to an object distort the vortex lattice. The resulting transverse displacements of the vortices are then used to determine the phase at the exit surface of the object [6]. The exit surface is the plane where the exit wave resides once it has propagated through the object. We show that the phase is proportional to the positions of each vortex in the lattice, thereby permitting the phase to be retrieved locally without introducing ‘ringing’ artefacts. Further, the method is also applicable to absorbing objects.

The utility of vortex lattices has been investigated in other contexts, such as wavefront tilt [7], small-angle rotations of wave-vectors [8], and to determine polarization parameters of birefringent media [9,10]. Reference [11] reports a method of tracking vortices in a vortex lattice, which plots the relative phase between each vortex in the lattice as a function of their transverse Cartesian coordinates$(x,y)$. A plane is then fitted to this function and the distance between the positions of the vortices relative to the fitted plane is taken as the phase. However, in this method [11] the plane is fitted over the entire field of view and the phase solution at any point depends on all parts of the interferogram; consequently it is also non-local. In contrast, the method reported here reconstructs the phase locally, as it is based on an algebraic method that does not require integration.

The outline of the paper is as follows. In section 2 we detail the method and show how the position of vortices generated by three-beam interference can be used for phase determination. The experimental setup is described in section 3, with results presented in section 4. Section 5 provides a brief discussion of the technique and its potential applications.

2. Theoretical background

Consider the superposition of three coherent plane scalar waves of unit intensity. The first wave, ψ_obj, is directed along the z-axis and transmitted through a phase-amplitude object.The complex transmission function is $A(x,y)\exp [i\varphi (x,y)]$, where $A(x,y)$is a real function of $(x,y)$ and $\varphi (x,y)$is the phase shift imparted to the wave by the object in the projection approximation [12]. Note that the trivial harmonic time dependence $\exp (-i\omega t)$, where ω is angular frequency and t is time, is suppressed throughout. Now consider a second and third plane wave, labeled ${\psi }^A(x,y)=\exp [i2\pi (k_x^{A}x+k_y^{A}y)]$ and ${\psi }^B(x,y)=\exp [i2\pi (k_x^{B}x+k_y^{B}y)]$, respectively, which are both tilted with respect to the $z$-axis. The superposition of the three wave-fields, $\Psi (x,y)$, at $z=0$ is given by

Each term in Eq. (1) represents a phasor in the complex plane. The sum of the three plane waves will only vanish when each phasor in the braces in Eq. (1) forms a closed triangle as shown in Fig. 1(a) (cf [12, 14].). To demonstrate how the phase is related to the position of each vortex, we consider two cases; first a completely transparent object (i.e. where A is unity) and secondly that of an absorbing object.

 

Fig. 1 (a) Phasor geometry for three waves of unit intensity at a vortex point. (b) Phasor geometry for a vortex when the exit wave is transmitted through an absorbing object.

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2.1. Non-absorbing object

For a non-absorbing object, each vortex occurs when the three phasors form an equilateral triangle as shown in Fig. 1(a), where $(\tilde{x},\tilde{y})$ denotes the vortex location. Note that there are two possible phasor arrangements that can form an equilateral triangle. The second triangle is formed by complex conjugation of all three phasors, corresponding to reflection about the $x$-axis in Fig. 1(a). This dual arrangement corresponds to alternate signs of the topological charge of each vortex.

The arrangement of the phasors shown in Fig. 1(a) occurs when the arguments of the second and third phasor take the following values:

2.2. Absorbing object

Here we generalize the analysis of section 2.1, to the case of an absorbing object. The phasor geometry for an absorbing object is shown in Fig. 1(b), where the magnitude of the phasor of the exit wave can vary due to absorption. This means that Eqs. (3a) and (3b) will no longer be valid for every vortex point, since the relative angles between each phasor can vary. The angles that reference waves A and B make with the x-axis are denoted by α and β, respectively. Due to the symmetry of the phasor arrangement in Fig. 1(b) we note that $\beta =2\pi -\alpha$ for any $\left|{\psi }_{obj}\right|$. Therefore we may write

Equation (7) is applicable for strongly absorbing objects, with the exception of the extreme case of total absorption by the object. The magnitude of the phasor of the object beam decreases with increasing absorption; unless the object is completely absorbing, the phasors will still sum to zero and a vortex will be formed. However, strongly absorbing objects will decrease the contrast resolution of the interference pattern. Moreover, low contrast can make it difficult to accurately localize a vortex, which requires the detection of minima in the intensity pattern. However, there are phase shifting methods that use an extra reference arm to form a fringe pattern, which can be exploited to increase the intensity gradient near a vortex core. Phase shifting the reference arm to generate a fringe pattern causes shifts in the intensity maxima, whilst the intensity minima due to vortices remain fixed at the same locations. By taking a series of interferograms, each with the fourth interferometer arm phase shifted, the absolute value of the difference between successive interferograms is calculated. These differences are summed, which increases the local intensity gradient near the vortex core; this compensates for the lack of contrast in the three-beam interferogram of a highly absorbing object [21]. Such an approach could be used to increase the precision with which vortices can be localized in those cases where the absorption is too large to accurately locate each vortex.

The phasor geometry used to derive Eq. (5) considered a single sign of topological charge. Therefore only the coordinates of one particular type of vortex are used to determine the phase using Eq. (5). This is not the case for Eq. (7), where there are two arrangements of the phasors that sum to zero; however, both result in the same equation. This is due to the symmetry in the coefficients of $\tilde{x}$ and $\tilde{y}$ in Eq. (7), which is not present in Eq. (5). Thus Eq. (7) allows us to utilize either vortex sign in the interference pattern to determine the phase. The presence of the third term in Eq. (7) shows that the phase is retrieved modulo π rather than modulo 2π. This is only a disadvantage for objects that have large phase gradients, in which case aliasing may occur and phase unwrapping is difficult.

The proposed method of phase determination proceeds as follows. First we localize all vortices in the lattice, resulting from a three-beam interference pattern, and determine their $(\tilde{x},\tilde{y})$ coordinates by locating intensity zeros. Using either Eq. (5) or (7), we algebraically calculate ϕ for each vortex point $(\tilde{x},\tilde{y})$.The use of either equation is dependent on the object being imaged, i.e. for absorbing objects Eq. (7) is used. If Eq. (5) is utilized, an additional step is required in which we separate vortices by the sign of their topological charge. The wrapped phase is then recovered modulo 2π, if Eq. (5) is used, or modulo π in the case of Eq. (7). Once the phase is computed, interpolation between each vortex is performed to recover the phase over all $(x,y)$ points in the image array.

3. Experiment

The experimental setup is shown in Fig. 2 . The beam from a linearly-polarized Helium-Neon laser (Thorlabs 5 mW) is spatially filtered by focusing it through a 4.51 mm focal length aspheric lens and then passing it through a 20 μm pinhole. The filtered beam is re-collimated using a 100 mm focal length plano-convex lens and an iris, which is adjusted so that the beam is truncated at its first minimum. A neutral density filter located before the first focusing lens is used to attenuate the beam. The filtered beam is then passed through a polarizing beamsplitter cube, oriented to transmit the majority of the beam power; this ensures a pure polarization state for the resulting beam. The beam is then passed into a three-beam interferometer constructed from a pair of Mach-Zehnder interferometers that share a common arm. The path lengths through each arm of the interferometer are matched to maximize the coherence of the interference; 50:50 beamsplitter cubes are used to split and combine beams in each interferometer.

 

Fig. 2 Experimental setup of the three-beam vortex interferometer. A He-Ne laser (λ = 633nm) is spatially filtered by the pinhole and a single polarization direction is selected as it passes through the polarizing beam splitter. Two beamsplitters are used to create the three arms of the interferometer. The central beam is transmitted through an object after which two lenses are used to focus the exit surface of the object into the camera. Each reference arm contains λ/2 and λ/4 wave plates; which are used for phase stepping. Neutral density filters (ND) ensure that the two reference waves have the same intensity as the object wave.

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The common central arm of the combined interferometer (object beam) is used as the source of illumination for the object. The object is located close to the first beamsplitter cube in the arm and is imaged onto a CCD camera via a system of two plano-convex lenses arranged to image the object plane onto the camera with a magnification of two. The lenses are arranged so that the first lens is located one focal length away from the object $(f=75\text{mm})$ with the second lens $(f=150\text{mm})$ spaced 225 mm away from the first lens and 150 mm away from the camera. The monochrome CCD camera (Prosilica GE1650) has $1600\times 1200$ pixels and 12 bits per pixel.

The pixel size is 7.4 μm × 7.4 μm. The two remaining interferometer arms (reference beams) each contain a neutral density filter to match the intensity of the beam through each arm to that of the object beam. A λ/2 and λ/4 wave plate allows the phase of the beam exiting each arm to be varied in discrete steps. The phase is stepped by either λ/2 or λ/4 (depending on which wave plate is used) by changing the alignment of the wave plate so that the fast optical axis is either parallel or perpendicular to the polarization axis of the beam.

To form a vortex lattice, two reference beams are adjusted in angle relative to the object beam, such that the wave-vectors of the three beams are non-coplanar. The angles of the beams of the reference arm can be adjusted by tilting the final beamsplitter in the top arm of the interferometer and the final mirror in the bottom arm. Different geometries of the lattice are generated depending on the mutual angle of the three beams, with a hexagonal lattice being produced when the angle between any two beams is ${120}^{\circ }$. The period of the three-beam interferogram maxima was adjusted to be approximately 20 pixels. A phase shift of either reference beam causes a shift in the vortex in the direction of that beam’s transverse wave-vector.

Our technique was experimentally tested on two objects: a Thorlabs spherical lens (part number LA1464) made from N-BK7 optical glass with a diameter of 25.4 mm, and the wing of a common house fly (Musca Domestica). For the spherical lens, in addition to the three-beam pattern, two-beam interference patterns of ${\psi }_{obj}+{\psi }^A$ and ${\psi }_{obj}+{\psi }^B$were individually acquired by blocking each of the reference arms in turn. Both reference arms were then blocked to record an image of the Gaussian illumination. Wave-vectors of each arm were measured by locating the peak maximum in the power spectrum of the three-beam interference pattern. The interferogram was smoothed, to reduce noise, by convolution with a Gaussian filter of approximately 60 μm full width half maximum (FWHM) and then flat-field corrected using a specimen-free illumination image. This was performed to increase the signal in the outer edges of the interferograms.

To determine the accuracy with which vortices can be localized we computed the normalized autocorrelation of a non-deformed section of our vortex lattice. Since the lattice is uniform, perfect localization gives complete autocorrelation between vortex points (i.e. the autocorrelation value is unity). However, small errors in localization decrease the autocorrelation for the vortex lattice, giving some width to the central maximum of the autocorrelation function. Using the half width of the maximum as a measure of the precision of vortex localization, it was found that the technique has a precision of 1.5 pixels. Based on Eq. (5) this translates to a measured phase accuracy of ± 0.2 radians.

4. Results

A section of the three-beam interferogram of the lens along with a simulated interferogram is reproduced in Fig. 3(a) and 3(b) respectively; these show excellent agreement is found between theory and experiment. To locate the vortices in the image, it is only necessary to locate the intensity zeros in the interference pattern. A zero in the intensity is a necessary but not sufficient condition for a vortex; a three-beam interference pattern of planar waves will always have a vortex at each intensity zero [12,14]. Note, in this context, that non-vortical zeros are not stable with respect to perturbation, whereas vortical zeros are stable with respect to perturbation. The vortices were separated depending on the sign of their topological charge using the methods outlined in Ref [22]. The phase of the lens was calculated using Eq. (5) for vortices of positive topological charge. Two-dimensional linear interpolation was then performed between each vortex point to map the phase over all Cartesian pixel coordinates within the field of view. This was performed by taking the real and imaginary parts of the wavefunction. The real and imaginary parts were then linearly interpolated, which allowed for the complete wrapped phase $\varphi (x,y)$ to be recovered using

 

Fig. 3 Comparison of the experimental three-beam interference pattern with a numerical simulation. (a) Experimental three-beam interference pattern. (b) Simulated three-beam interference pattern. The plus signs and crosses in the top right insert indicate the locations of vortices of positive and negative topological charge, respectively.

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The reconstruction of the lens in Fig. 4(a) shows a series of concentric circles where the phase has been wrapped by 2π, as would be expected for a spherical lens since its projection is rotationally symmetric. To demonstrate our technique quantitatively, the wrapped lens phase was unwrapped using the methods outlined in Ref [23]. In this paper, Volkov and Zhu showed that the image may be represented in terms of the Fourier transforms of its x and y derivatives. The gradient of the wrapped phase suffers abrupt changes of 2π per pixel at each point where the phase undergoes wrapping. Thresholding the x and y phase gradients eliminates these discontinuities. The unwrapped phase is then recovered using Fourier transforms to reconstruct the phase from the thresholded x and y derivatives. Once the lens data had been unwrapped, the projected thickness was recovered by dividing the unwrapped phase by the factor $\delta k$, where δ is the difference of the refractive index of the lens from unity and k is the wavenumber. A circular arc was fitted to the profile centered on the recovered projected thickness in order to measure the radius of curvature of the lens. The profile of the lens and the fitted curve are shown in Fig. 4(b). The fitted curve returned a value of 519±1 mm, in good agreement with the quoted radius of curvature of the lens of 515.5 mm. This shows that our method is able to quantitatively reconstruct the object function.

 

Fig. 4 Experimental results for phase reconstruction of a spherical lens using the three-beam vortex interferometer. (a) The unwrapped reconstructed phase of the lens. The greyscale in this image ranges from 0 (black) to 2π (white). (b) Unwrapped phase profile of the lens. The solid curve corresponds to the experimental data whilst the dashed curve is the fitted curve. The radius of curvature of the fitted profile is 519 ± 1 mm.

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The method was also applied to a three-beam interferogram of the wing of a common house fly. As this sample contained higher spatial frequency information than the lens, the sampling of the phase, and consequently, the number of vortices needed to be increased. This was achieved by using the λ/2 and λ/4 wave plates described in section 3 to perform phase shifting in each reference arm of the interferometer. Three-beam interferograms were recorded for the fly’s wing, with 11 phase shifts of π/4 between images used to reconstruct the phase. Each interferogram was smoothed by convolution with a Gaussian filter of approximately 60 μm FWHM and the transverse wave-vectors and vortex points were found using the same method as for the spherical lens data. Vortices were not separated by sign for this object as Eq. (7) is applicable to vortices of either topological charge. For each of the 11 phase-stepped interferograms, the phase was recovered using Eq. (7). Before the phase of each image could be combined for interpolation, the relative offset between the phase of each image needed to be determined. Two-dimensional cross-correlation of a small portion of the vortex lattice was used to determine the offset between each phase step. The coordinates of the maximum in the cross-correlation data were substituted into Eq. (7), which gave the value of the phase difference between each vortex lattice. This phase difference was added to each reconstruction and linear interpolation was applied to every vortex point in all 11 phase stepped lattices. This results in a single phase image which incorporates information from all11 recorded vortex lattices. The recovered wrapped and unwrapped phase reconstruction of the fly’s wing is shown in Figs. 5(a) and 5(b), respectively. The phase determined by the Takeda method [5] using a mask of 25 $\times$ 25 pixels applied to a two-beam interference pattern is given in Fig. 5(c) for comparison. The two-beam interference pattern was obtained by blocking one of the reference arms of the three-beam interferometer. This was also applied to a subsection of the three-beam interference pattern spatial-frequency spectrum, producing a consistent result to the phase obtained for the two-beam case. This is expected as a three-beam interference pattern implicitly contains all the information of a two-beam interference pattern. The phase recovered from the vortex lattice method was unwrapped using the same method [23] as for the lens data and is shown in Fig. 5(d). As Eq. (7) has been used to recover the phase of the fly wing, the phase is wrapped from $-\pi /2$ to $\pi /2$. The phase recovered from the Takeda method [5] has also been wrapped between these values, which facilitates comparison of the two results. Both images in Figs. 5 (a) and 5(b) show good quantitative agreement with each other. Minor differences are observed around the frame of the wing, where the absorption of the light is strongest. However, the vortex lattice phase determination technique is mostly unaffected by the absence of information in these regions. The local nature of our technique allows the phase reconstruction to simply ignore these strongly absorbing regions, which are then interpolated. For Fourier-based methods these regions result in larger phase gradients. This can be observed at the bottom edge of the wing where the phase rapidly increases, which is not observed in Fig. 5(b). Ringing artefacts in Fig. 5(c) are also observed, corresponding to spatial frequencies not present in the fly’s wing. Our vortex lattice method is not susceptible to this type of artifact as the technique uses a local algebraic solution to recover the object’s phase.

 

Fig. 5 Experimental results for the wing of a fly. (a) Experimental three-beam interferogram of the fly's wing. (b) Recovered phase of the fly wing using Eq. (7). (c) Recovered phase using the Takeda method [5]. (d) The unwrapped phase in (b). The greyscale in (b) and (c) is $[-\pi /2,\pi /2]$ from black to white.

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5. Discussion

A surprising aspect of the technique described in this paper is that vortices, which correspond to points at which the phase becomes undefined, are used to determine the phase shifts of an object. In many forms of phase retrieval vortices are often undesirable, since their presence can be detrimental to accurate phase retrieval [24]. The separation of vortex sublattices was performed using Eq. (5), which depends on a particular geometry of the phasors of the three waves; hence it applies to a particular topological sign of the sublattice. The ability to separate vortices based on their sign is a consequence of the vortex sign rule [25]. A vortex occurs at the intersection of lines, where the real and imaginary parts of the wavefunction vanish; the sign rule states that each time a line intersects with another, the sign of the topological charge alternates. If we assign any vortex to a particular charge, the sign rule allows us to determine the sign of every vortex present in the lattice. Hence we may separate the vortices in the lattice based on their topological sign. To reconstruct the phase without a sign ambiguity, using Eq. (5), the sign of $k_x$ and $k_y$ for each reference wave and the topological charge of each vortex must be determined. This is because the sign of the vortex determines the form of the equation used to calculate the phase, and the sign of $k_x$ and $k_y$determines the value of the coefficients of $\tilde{x}$ and $\tilde{y}$ . These steps can be bypassed by using a priori knowledge of the object, e.g., an object with refractive index greater than unity results in a positive sign in the phase shift, as a consequence of the retardation of the wave as it passes through the medium. If it is necessary to determine these parameters, the sign of $k_x$ and $k_y$ for each reference arm and the topological charge of the vortex sublattice can be experimentally measured using the method outlined in Ref [22].

The spatial resolution of the technique is determined by the spacing of the vortices in the lattice, which increase as the angle that each wave-vector makes with the z-axis increases; this allows the spatial resolution to be tuned. However, increasing the vortex density too much results in aliasing, which can lead to inaccurate vortex localization. Changing the relative angles between the transverse wave-vectors can also be used to increase sampling, as this affects the geometry of the lattice, e.g., a hexagonal lattice geometry occurs when the angle between each wave-vector is ${120}^{\circ }$, whilst a rectangular geometry occurs when two wave-vectors are orthogonal. This could also be used as an alternative to phase stepping, obviating the need for wave plates; in this case the relative angles between wave-vectors can be changed for each image. The changing lattice would result in a shift of the vortices and therefore sample more of the object.

An advantage of using vortex interferometry for phase retrieval is its robustness in the presence of noise. Because the method uses a real space calculation, the power spectrum of the noise has little or no effect on the reconstruction. In this case, noise only affects the vortex localization itself, since small variations due to noise may either lead to false detection of a vortex or inaccurate localization of the vortex points. However, the noise will have a significant effect on the interferogram, since shot noise is reduced for low intensity regions, such as areas in the vicinity of the vortex core. These effects can be significant if the vortex lattice spacing is small and the intensity “blobs” in Fig. 3 are under-sampled.

6. Conclusion

This paper presents a new method of phase determination in interferometry. By interfering three plane waves, a uniform vortex lattice is generated. Distortions in this lattice due to phase shifts induced in the object wave have been shown to be related to the position of each vortex in the interference pattern (see Eqs. (5) and (7)). Localizing each vortex in the lattice allows for the phase to be algebraically calculated. The technique has been demonstrated on a spherical lens and a house fly’s wing. Both objects show good agreement with existing Fourier methods. An advantage of the presented technique is its robustness to noise and its capacity to reconstruct the phase locally.