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We have developed a differential form of singularimetry, which utilizes phase vortices or intensity gradient singularities as topological fiducial markers in a structured illumination context. This approach analytically measures phase gradients imparted by refracting specimens, yielding quantitative information that is both local and deterministic. We have quantified our phase gradient experiments to demonstrate that lattices of wave field singularities can be used to detect subtle phase gradients imparted by a spherical specimen and fiber optic cylinders.
© 2016 Optical Society of America
1. Introduction
Singular optics is a mature field of research involving phase vortices, nodal lines, diffraction catastrophes, anomalous π phase jumps along ray trajectories, the topology of wavefronts, and singularities in the fine polarization structure embedded within electromagnetic waves [1]. Amidst the discovery of many other generic phenomena [2] foundational research continues, such as the recent formation of isolated vortex knots [3], quasi-crystal optical lattices [4] and the construction of coherence vortices [5], to name a few. Experimental applications have arisen in concert with these developments, such as optical spanners [6], super-resolution [7] and contrast enhancement [8]. Amongst the rich variety of such experiments, we draw attention to the use of optical singularities as wave front sensors. This concept of using optical vortex singularities as high precision fiducial markers was recently coined ‘singularimetry’ [9]. One such technique utilizes multi-wave interference [10–12] to generate light lattices, the disturbances of which can be used to measure wavefront characteristics [13–16], or retrieve optical path length changes created by a specimen in a holography context [17]. In this latter approach, a specimen perturbs one wave in a 3-beam vortex lattice, which causes transverse singularity displacements that are directly proportional to the incurred local phase and attenuation shifts. Whilst this singularimetry method for local holography is readily implemented in light optics, the required use of multiple beam splitters in x-ray or electron microscopy poses significant engineering challenges. Hence we are motivated to consider simpler approaches where all beams pass through an object of interest.
Structured illumination is an established method for detecting subtle phase variations via the specimen-induced movement of features in an incident optical probe. One example is Shack-Hartmann wavefront sensing [18], which incorporates a lenslet array to create illumination structure and is of great utility in optometry. Using simple stripes on a background wall, Massig [19] showed that subtle air refraction variations due to local heating from a human hand can be measured by inferring ray deflections using standard Takeda-type Fourier processing [20] of the background illumination structure. For differential measurement of phase gradients in this work, we are motivated by Massig [19,21,22] and related studies [13–16,23] to use fields of singularities as structured probes. Indeed, structured illumination with three interfering circularly polarized beams was recently employed for volumetric imaging, based upon deconvolution of the optical transfer function for fluorescence microscopy super-resolution applications [24].
Quantitative differential phase contrast has been achieved using a variety of illumination structures for x-rays [25,26], which can be understood using the transport of intensity equation (TIE) [27] framework to describe propagation of the coherent probe structure. In this work, we consider generalizations of the TIE to include phase vortices and thereby use the same ideas for detecting through-focus probe structure variations. Based upon the theory of Rozas et al. [28], using a dedicated paraxial formalism [29], we consider non-diffracting illumination, which does not change upon Fresnel propagation in the absence of a specimen. In the same context, we also analyze structured illumination created from far-field diffraction, although we only experimentally test the former. Recent experimental work has demonstrated that fields of electron singularities can be synthesized using severe lens aberrations [30] or as periodic lattices with multiple bi-prisms [31,32], and so we expect the methodology in this work to be applicable in such settings.
After describing the theory in section 2, the experiments are outlined in section 3. To quantitatively test the main results of the theory, here we use 3-beam interference to create uniform vortex lattices from a He-Ne laser source. A spherical plano-convex lens is selected as an ideal benchmark specimen. Our differential singularimetry method is also applied to a single-mode step index fiber and a graded-index optical fiber, chosen to impart a non-trivial refraction pattern. Optical fibers provide useful tests for comparison with other phase retrieval methods and are commercially important [33–35]. Each singularimetry experiment is compared to TIE reconstructions from identical through-focus measurements. Section 4 analyses the experimental findings and the merits of this work are summarized in section 5.
2. Theory
2.1 Non-diffracting structured illumination
Consider a non-diffracting [36,37] paraxial and monochromatic wavefield of the scalar form
where g(k) is a known complex distribution of the coherent source and r is the transverse position in the imaging plane, downstream from the source. The Dirac delta function confines the transverse wavevector to an infinitesimally narrow annulus of radius |k| = kA; hence the k in the right hand side equality of Eq. (1) is solely a function of angle θ, as are the contributions from the source amplitude and phase. Suppose that ψnon illuminates a specimen, which imparts a phase ϕ(r) and amplitude A(r) upon the wavefield in the imaging plane at z. The free-space evolution of the perturbed wavefield ψ is given by the paraxial Helmholtz Eq. (2)ik∂zψ + ∇2ψ = 0, which we will evaluate for an infinitesimal propagation distance Δz, where k is the wavenumber, ∂z is the longitudinal gradient and ∇2 is the transverse Laplacian. A specimen-induced perturbation ψ = ψnonψs substituted into the paraxial Helmholtz equation describes an infinitesimal propagation along the optic axis by distance Δz, such that ψ(z + Δz) = ψ(z) + Δz∂Zψ(z). We define the complex factor ψs to be the wave that would exit the specimen if a z-directed plane wave had replaced the illumination, such that ψnon = exp(2πikz). The specimen imparts a complex phase shift of the form η(r) = ϕ(r) – iln[A(r)], where ln is the natural logarithm, such that ψs = exp(iη(r)). With this notation, the stated substitution yields,where ε ∝ kA and α = |∇η|. The infinitesimal variation of detectable structure in the propagated illumination is given by the squared modulus of Eq. (2), as the intensity I(r, z + Δz), i.e.,where Iobj is the intensity measured in the imaging plane at z. If we restrict our attention to real η(r) with ∂yη(r) = 0 in Eq. (2) and ignore the Inon(r) pre-factors in Eq. (3), the transverse displacements are then analogous to the deflection of rays along the x-direction through small angles θ = Δx/Δz, due to a specimen phase gradient, which gives a connection to a standard geometric-optics interpretation of differential phase contrast. In the absence of attenuation and for illumination containing vortices, where I(r, z) is strictly zero at each vortex core, Eq. (3) provides a concise interpretation for the longitudinal displacement of the vortices, such that ∂zrvort = −k∇ϕ(r). Hence the focal variation of transverse vortex positions directly measures the specimen-induced phase gradient at localized points, centered at each of the vortex cores. Similarly, if the intensity I(r, z + Δz) is corrected for the Laplacian contributions from the specimen, ∇2ϕ, then other singularities, such as intensity maxima, are also sensitive to the phase, since ∂zrmax = −k∇ϕ(r). For uniformly attenuating or transparent specimens that have constant Iobj(r), the phase Laplacian can be directly measured from a through-focus derivative arising from a z-directed plane wave. For such samples, the Laplacian attenuation can be exactly inverted to improve the detection of maxima in an experimental setting; an approximate correction is also possible for non-uniform attenuation.Isolated singularities, such as vortex zeros, maxima and saddle points, provide ideal fiducial markers of the illumination structure, which can be tracked throughout a focal series. Note that the vortices are somewhat special in this regard since, whilst such intensity minima may change shape, the intensity zeros are nonetheless maintained in the presence of the Laplacian contributions from the specimen-induced phase. Since the phase singularities are zeros of the wavefield ψnon, they are topologically protected from such perturbations. The intensity saddles and maxima are not necessarily topological aspects of ψnon and are not expected to share the same structural stability as the nodal lines along which the phase vortices wind.
For singularimetry using vortices, there is a remarkable interplay between super-oscillations and weak measurements. Specifically, our technique measures phase gradients that would be acquired by a z-directed plane wave, which are hence proportional to weak measurements of the local transverse momentum of that exit-wave [38]. These gradients can be detected from the local movement of the phase vortices, with cores near which the probing wavefield is super-oscillatory, where the local momentum diverges [39].
For an absorbing object, Eqs. (2) and (3) require thoughtful interpretation, since η is then a complex function. We can obtain the required insight by studying the movement of vortices in the structured illumination. Near a first-order vortex core, a useful local model of a non-diffracting wavefield is proportional to x + iy or x − iy, depending upon the sign of the unit topological charge, where x and y are defined in a suitable coordinate system centered on the core of the vortex. Suppressing the action of the phase, in the presence of a specimen-induced amplitude modulation, ln[A], our analysis indicates that the vortex is shifted upon propagation by Δz according to: x ± iy → x ∓ Δzk−1∂yln[A] + i(y ± Δzk−1∂xln[A]), which agrees with the findings of Rozas et al. [28]. Hence, for the sole amplitude contribution, the vortices are displaced in an orthogonal direction to the amplitude gradient, with an explicit sign difference between the x and y displacements. The simpler specimen-induced phase shifts act in the same direction as ∇ϕ, which linearly add to the amplitude-induced vortex displacement.
Before ending this section, we emphasize that a subtle but important aspect of the singularity motion is the non-diffracting condition for the structured illumination. It is this property which prevents ε in Eq. (2) from varying with respect to the angle θ. From an experimental point of view, non-diffracting structured illumination is conceptually desirable; for then the only quantity that can cause a through-focus movement of fiducial markers is the specimen. In the next section, we describe how scalar fields arising from far-field diffraction can mimic this useful behavior.
2.2 Far-field structured illumination
Far-field diffracted waves behave in a similar manner to non-diffracting wavefields, in the sense that the form of the light field is almost invariant with respect to further propagation; the main difference being transverse scaling due to a constituent spherical wavefront. Consider a structure that is illuminated with a z-directed plane wave, imparting a complex distribution g(R), which diffracts waves to the far field at distance Z from the structure to an imaging plane, where a specimen is located. Including a complex specimen phase shift of η(r), the propagated wave ψF is given in the far-field by [29],
where R specifies transverse distances in the plane of the structure, λ is the wavelength and the intrinsic expanding spherical wavefront is denoted as S = exp[ikZ−1(x2 + y2)/2].The similarities between Eq. (1) and Eq. (4) suggest that similar analytical results may be obtained to describe the propagation-induced transverse displacement of singularities. To this end, consider ψF imparted onto a specimen at Z, such that ψF → ψF exp(iη). Upon absorbing the phase of the spherical factor S into η and denoting this process as η →η′, substitute Eq. (4) into the paraxial Helmholtz equation. The transverse Laplacian then acts only on the R•r term in Eq. (4) and η′ to give
whereand N(R) = |R|2/(λZ). The maximum value of N(R) is the Fresnel number NF, which obeys NF<<1 in the far field, so the πΔzN(R)/Z contribution to Eq. (6) can be ignored for infinitesimal Δz. We then have:where, for compactness of notation, we have suppressed the r dependence of η′. Like Eq. (2), we see that the structured wave is transversely displaced in accordance with the specimen-induced phase gradient, albeit with an additional offset from the far-field spherical wavefront S. One could consider collimation at the specimen plane to remove this component of ψF, such that η′ → η in Eq. (6). With such a correction of the far-field scaling, the square modulus of Eq. (7) would then be identical to Eq. (3), so the interpretation of the defocus-induced transverse singularity displacement is exactly the same (if the far-field wave contains vortices and intensity maxima).2.3 The effect of noise
By tracking an isolated singularity, we can analyze the accuracy of local phase gradient measurements in the presence of statistical fluctuations and systematic errors. For this purpose we use existing insights from in-line holography [40] based upon the TIE. Suppose repeated measurements of a vortex position rvort are statistically distributed with a standard deviation σ. Now consider measuring the focal variation of rvort using a pair of images; one over-focused by Δz, the other under-focused by Δz. Using a central-difference approximation to estimate ∂zrvort, the truncation error added in quadrature with the statistical uncertainty is given by (Δz)2∂3rvort/3! + σ/(Δz√2), where we have assumed that σ is the same for each image. The condition for accurate phase gradient estimation is thus: |∇ϕ| >> k(Δz)2∂3rvort/3! + kσ/(Δz√2). So we see that a smaller defocus Δz amplifies the effect of noise, yet reduces contributions from the 3rd order focal gradient. These terms must therefore be balanced in the same manner as for in-line holography with the TIE, with one significant difference. We have not specified the form of σ, which depends upon the method used to detect the singularity position as well as fluctuations in the intensities. If intensity extrema are widely spaced apart in a vortex lattice, then the individual position of extremum can be measured using a large number of pixels to drastically reduce σ. Hence spatial sampling can be traded for phase gradient precision in this context.
2.4 Dynamical diffraction
The singularimetry we have described is best suited to smoothly varying specimens in the projection approximation, such that attenuation and phase gradients imparted upon the non-diffracting illumination can be sensitively detected and readily interpreted. However, the theory can be applied to more general dynamical diffraction. The key assumption we have made is that the wave exiting the specimen has the form ψ = ψnonψs. This assumption will hold if the range of angles in the structured illumination is sufficiently small, as determined by kA in Eq. (1) or the Fresnel number for Eq. (4).
2.5 Algorithm to detect the optical flow of singularities
Equations (3) and (7) indicate that phase gradients can be measured from as few as two images, to determine the local displacements of singularity structures. However, improved accuracy of the required numerical differentiation can be obtained using many images to fit a Taylor series for the focal variation of the singularity positions. To this end, we employed a singular value decomposition (SVD) routine adapted from Numerical Recipes [41] to fit smooth quadratic functions for each r(z) and estimate variances for the focal (z) variation of the transverse displacement of each singularity.
Singularities were detected using a simplified detection method: pre-smoothing by convolution with a Gaussian of width matching the extrema size was used to reduce noise and select the appropriate scale-space [42], which was largely fixed by the spatially uniform vortex lattice. Fourier transforms were then used to compute the Laplacian of each image, to highlight minima and maxima. Vortices and maxima were separated according to the sign of the curvature and the local extrema were detected to sub-pixel accuracy by fitting each image Laplacian with two mutually orthogonal Gaussian functions, using five points along the x- and y-directions centered about each intensity extremum [43].
Singularity types (minima or maxima) were sorted and then tracked using a simple search between pairs of images of incrementally expanding radius, centered about each singularity in each sequential image. This scheme was employed to avoid false detections arising from transversely adjacent lattice points, which would otherwise overly limit the densities of the lattices that could be utilized. After neighboring singularities had been identified for an image pair, local displacement vectors were computed. Upon repeating these steps through all successive pairs, the singularity tracks were longitudinally integrated to yield all of the sparse rvort(z) and rmax(z) tracks, which were aligned to the central in-focus lattice image in preparation for through-focus polynomial fitting.
3. Experiment
To design a suitable non-diffracting wavefield, we interfered collimated Gaussian beams to approximate δ(kA-k)g(k) in Eq. (1) as three plane waves with discrete transverse k vectors forming an equilateral triangle. Mirror angles were adjusted to alter the size and orientation of the resulting vortex lattice, along with micrometers to center the triangle on the optic axis.
3.1 Apparatus
Three-beam lattices of phase vortices and intensity gradient singularities were created using a He-Ne laser (Thorlabs, HRP 050-1, 5 mW, λ = 633 nm) and a 3-arm Mach-Zehnder interferometer, as depicted in Fig. 1. For the benchmark lens specimen, an achromatic doublet lens of focal length f = 127 mm was used, corresponding to imaging system 1 in the figure. Both of the fiber optic specimens were analyzed with imaging system 2, replacing the stage lens with a 40 × objective having a NA of 0.65 (Olympus P040). For both systems, the imaging lens and CCD (Prosilica GE 1650) were translated with micrometers to acquire focal series. The condensing lenses included a 10 × microscope objective to scale the vortex lattice onto a smaller field of view.
Fig. 1 Experimental configuration. Imaging system (1) was used for the benchmark specimen. The fiber optic experiments used system (2) with a condensing pair of lenses, to shrink the probing vortex lattice, in conjunction with a higher numerical aperture microscope objective lens. Similarly, the initial polarizing beam splitter was reproducibly interchanged with a mirror to flip between 3-beam laser measurements and LED imaging, for which two of the three beams were blocked. The vortex lattice is an unprocessed 3-beam laser image from the experiment, in the absence of the specimen, showing a 142 μm field of view.
As an example, the vortex lattice shown in Fig. 1 is an unprocessed experimental intensity distribution (1024 × 1024 pixels) which was used in the fiber-optic specimen measurements (the spacing between adjacent maxima is approximately 5 μm).
A light emitting diode (LED) was incorporated to perform in-line holography, based upon the TIE, in conjunction with the through-focus singularity measurements. The nominal wavelength of the LED (Thorlabs M625F1) was 625 nm, with a nominal FWHM in the wavelength of 17 nm for the spectrum and output power of 2.53 mW. The LED was coupled to an SMA terminated multi-mode 200 μm core fiber optic patch cable with a NA of 0.39. A kinematic base with either a mirror or a polarizing beam splitter cube was used to reproducibly switch between the two sources, maintaining the same field of view for a given specimen, keeping the three lattice beams and single LED beam accurately centered about the optic axis. In principle, an on-axis laser beam would suffice for the TIE measurements. However, in practice, the high laser coherence was found to be detrimental in this set-up, as weak fringes and high spatial frequency ripples caused by reflections from optical surfaces invalidated the assumption of a small defocus limit, which is essential for accurate inversion of the TIE. A thick achromatic lens was found to produce the most uniform collimation of the LED illumination from the end of the fiber. In every through-focus experiment, the standard deviation of background intensity variations over the field of view was less than 2% of the mean intensity.
3.2 Specimens and measurements
The simplest test of Eq. (3) is a non-absorbing specimen that produces a known monotonically increasing phase gradient across a field of view and for which the Laplacian modulation of the intensity can be neglected. For a field of view much smaller than the radius of curvature, a lens centered on the optic axis fulfils all of these criteria, as the imparted phase shift is a simple function of the focal length and refractive index, whilst the transverse Laplacian is almost constant. For these reasons an anti-reflection coated plano-convex lens (Thorlabs LA1433-A) was chosen as a benchmark. Two other specimens of interest were studied and characterized; namely, a single-mode step-index fiber (Thorlabs SM600, NA 0.10-0.14) and a graded-index multi-mode fiber (Newport FMLD, NA 0.29).
For each specimen, two sets of focal series were collected from each light source. For every measurement, a centered window of size 1024 × 1024 pixels was cropped from each 1200 × 1600 pixel CCD image and saved for image processing. Further details of the through focus experiments and specimens are listed in Table 1.
Table 1. Experimental details.
Given the field of view in Table 1 and the nominal radius of curvature R ≈77 mm, the lens specimen expectedly imparts approximate phase gradients of the form ∂xϕ ∝ x/R, ∂yϕ ∝ y/R and |∇ϕ| ∝ (x2 + y2)1/2/R, with a series expansion truncation error orders of magnitude smaller than estimated experimental uncertainties. Similarly, the phase curvature obeys ∇2ϕ ∝ 1/R to an excellent approximation. Hence for the LED measurements using the TIE, we avoided solving a differential equation for the phase retrieval by employing the following relation: ∂zln[I] ∝ 1/R, where |∇I| = 0. Ignoring subtle dispersion for the refractive index, it should be noted that the lens-specimen focal length measured from this relation is independent of λ and the transverse scale of the images; it depends only upon the though-focus z variation of ln[I]. As with the vortex lattice optical flow measurements, the TIE through-focus gradients were computed using quadratic SVD fits.
Several additional measurements were conducted in order to detect systematic errors and estimate statistical uncertainties, for the benchmark lens specimen. Namely, dark backgrounds were estimated for both the LED and laser acquisitions and were only found to be significant for the LED images, which were thus corrected. To test the uncertainty in the detection of minima and maxima, 100 in-focus 3-beam lattice images were acquired without the specimen over a period of time shorter than average duration between focal increments in the singularimetry experiments (approximately by an order of magnitude). Measured standard deviations of the maxima and minima displacements averaged to 0.17 ± 0.02 pixels over the field of view, with the vertical average double that of the horizontal displacements. The minima and maxima standard deviations were not found to be significantly different, despite the differing transverse sizes and shapes of these illumination singularities. The separate vertical and horizontal standard deviations were used as input for the through-focus polynomial fitting of each singularity to estimate the uncertainty of the fitted coefficients for each transverse phase gradient. With the specimen removed, a 3-beam lattice focal series of 21 images was acquired with focal steps of 200 μm. Standard deviations of the vortex displacements were 0.28 ± 0.04 and 0.53 ± 0.06 pixels in the respective x and y directions. The phase gradient measured in the y-direction was uniform over the field of view with an average of −0.012 ± 0.002 μm−1. Given that the vortex shifts in both directions were of the same order as that of 100 image test, the specimen-free focal series indicated sufficient alignment of the vortex lattice with the optical axis. Lastly, a focal series for the LED data matching that in Table 1 (including 100 images for each defocus) was acquired in the absence of the specimen. The SVD-fitted through-focus derivative was more than two orders of magnitude smaller than that with the specimen inserted, indicating sufficient collimation of the flat field illumination and the absence of other background artefacts. Therefore no background corrections of this sort were made to the LED lens-specimen data.
The coating of each fiber was removed using a dry micro-stripping tool. Each optical fiber was index-matched in glycerol at room temperature (refractive index ngly = 1.471 [45]) by wedging each cleaved fiber between two coverslips of nominal thickness 130-160 μm, which were separated by two pairs of narrow double-spaced strips of cleaved coverslip glass on either side of the fiber. Note that nclad<nglyc<ncore for the graded index fiber in Table 1. The coverslip assembly was clamped on one side and mounted with the centered fiber parallel to the optical bench. To remove sizeable wedge-artefacts arising from coverslip strain and asymmetric clamping, data matching that in Table 1 was also collected from the glycerol/glass-only regions adjacent to the fiber for each lattice experiment. Significant phase gradients arising from the glycerol assembly were measured to be consistent with an air-glass wedge angle less than 1°. Backgrounds from each wedge were vertically extrapolated into the fiber region, using a smooth quadratic fit to the horizontally-averaged measurements of the wedge gradients ∂xϕ and ∂yϕ. All data (TIE and vortex lattice) were collected within a period of less than 10 minutes, to counteract detrimental hydroscopic effects of the glycerol interacting with laboratory air humidity, which would otherwise measurably reduce the refractive index.
For the benchmark lens specimen, |∇ϕ| was combined from both the sparse maxima and vortex phase gradient vectors, then tessellated and cubic-spline interpolated to produce a continuous map, using native MATLAB routines (MATLAB R2014a, The Math Works Inc.). An 875 × 875 pixel subsection of the tessellated |∇ϕ| provided an estimate of the center of curvature, calculated with a dedicated center-finding algorithm, which iteratively tests for rotational isometry [46]. The radially symmetric pattern was then rotationally averaged into 647 radial bins, with the variance estimated for each radius, which was largely independent of the phase gradient magnitude, implying that the uncertainties were dominated by background artefacts as opposed to noise. Indeed, systematic focus-dependent intensity fluctuations were observed in single beam laser and LED images, which were interpreted as arising from imperfections in the various optical components and extraneous reflections. Using SVD, the radial variation of |∇ϕ| was least squares fitted with a quadratic function to compute the transverse specimen curvature and thus focal length, along with the estimated standard deviation.
To study potential differences between vortex and maxima detection, a separate “discrete” analysis was also employed for the measured ∂xϕ and ∂yϕ benchmark data. Namely, vertically and horizontally aligned rows and columns of singularity points were extracted from the sparse ∇ϕ vector field, producing approximately 40 graphs for the maxima and 60 graphs for the vortices, whilst nearest-neighbor interpolation was used to align stray points within ± 5 pixels of each line. The respective x- and y-variations of ∂xϕ and ∂yϕ were measured using linear least squares fits for every graph. The transverse gradients of ∇ϕ were then averaged for a given singularity type, along with the standard deviation in the means, and used to estimate R−1.
For the fiber optic specimens, the Paganin-Nugent algorithm and Fourier transforms were used to solve the TIE for the phase, in conjunction with quadratic SVD-based least squares fitting in a similar manner to earlier work [47]. Here we have tailored the boundary conditions by horizontally repeating the SVD fitted ∂zI(r, z = 0) and I(r, z = 0) data sets, which were then vertically padded with their respective averages [48] to create synthetic 3 × 3 tilings. These pre-processing steps removed background artefacts, such as excessive low-spatial frequency undulations. Recently it has been shown that the effects of partial spatial coherence can be corrected to quantitatively recover the specimen optical path length [49]. However, such refinements were not included in this work.
4. Results and discussion
Table 2 shows that all measurements of the lens specimen focal lengths are comparable, within stated uncertainties that represent ± one standard deviation. The nominal focal length of the lens is f = 149.5 mm ± 1% at a design wavelength of λ = 587.6 nm (or f = 150.0 mm at λ = 633 nm), which agrees with our measurements. The vortices and maxima performed equally well as suitable fiducial markers.
Table 2. Results for benchmark lens specimen.
The LED result displays substantial uncertainty, despite the fact that the 100 image frame averaging had reduced the standard deviation to below 1% for every pixel for the intensities in all 21 through-focus images. On close inspection of the unprocessed images, it was apparent that the aforementioned focus-dependent backgrounds were responsible for this variability across the field of view. It should be noted that the TIE primarily detects phase curvature, which is strongest near specimen edges or features that have high spatial frequencies. For non-absorbing objects, the TIE cannot detect linear phase gradients and so the constant phase curvature exhibited by the lens specimen represents the lowest detectable spatial frequency to which the TIE is sensitive. It is well established that the TIE exhibits the poorest signal to noise for low spatial frequencies [40], so the significant variance of the focal length in Table 2 is not surprising.
Phase gradients of an analytical spherical surface were computed using the focal length in Table 2, the nominal refractive index of N-BK7, and measured center of curvature. The RMS difference between the analytical result and that of the combined vortex and maxima results were summed over the set of sparse points, giving errors of 3.6% and 3.8% for the horizontal and vertical directions, respectively. Similarly, a commensurately sparse analytic vector field was created. Figure 2(a) shows the vortex lattice measurements of ∇ϕ for the lens specimen, compared to the analytical estimate in Fig. 2(b).
Fig. 2 a) Experimentally measured phase gradient due to the lens specimen (left), compared to the analytical calculation in b) (right). Colors indicate the magnitude of the phase gradient.
The discrepancies between Figs. 2(a) and 2(b) are subtle but perceptible near the center (where the phase gradient is smallest) as slight deviations from radial directions are evident. For a fractional uncertainty less than 30% for the tessellated and rotationally averaged |∇ϕ|, we have estimated a detection limit of 4.3 × 10−3 μm−1 for this vortex lattice benchmark experiment.
For the single-mode step-index fiber, interlaced maxima and vortex data were combined for the vortex lattice measurement, denoted as “lattice”, which is compared to the “in-line” experimental data obtained using the TIE (see Fig. 3). Both measurements are compared to an analytical profile (with λ = 633 nm), where the glycerol refractive index was reduced by 0.1% to fit the vortex lattice result and account for possible partial atmospheric water absorption. Note the slight vertical offset in the lattice profile, which may have arisen from imperfect coverslip wedge correction. More noticeable are the large deviations at the extremities of the fiber profile, where the strong phase gradients destroyed the vortex lattice structure in the majority of through focal images. Within the fiber interior, the profile of the in-line gradient is notably smaller and can be fitted quite well using nglyc = 1.467. Whilst the room humidity could have slowly reduced nglyc over time, the in-line TIE data was in fact collected before the vortex lattice measurements. One possibility for the underestimation of the phase curvature is that some Fresnel fringes at either end of the focal range fell outside the field of view. More importantly, the TIE profile is missing an uncorrected phase ramp ∇ϕramp, measured in the lattice experiments to an average of |∇ϕramp| ≈0.1 μm−1, which substantially offset and also skewed the lattice profile prior to correction. In this context the insensitivity of the TIE towards linear phase gradients, together with the padded boundary conditions, artificially prevented the retrieval of any such ramps for the in-line profile. Lastly, whilst both the in-line and lattice profiles identify the step-index boundary, the internal gradient is underestimated, which is to be expected given the small core size relative to the large focal range (see Table 1) and finite wavelength. Similarly, the nearest distance between singularities in the lattice was approximately 2 μm, so very few points probed the under-sampled core.
Fig. 3 Horizontally averaged vertical phase gradient for the single-mode fiber measured with vortex lattice singularimetry and deterministic in-line holography. Both measurements are compared to an analytical estimate based upon the projection approximation, which assumes a perfect cylinder containing a cylindrical step-index core.
For the graded-index fiber, the vector field in Fig. 4(a) shows the gradients measured using vortex lattice extrema, while Fig. 4(b) compares these measurements from an identical field of view using in-line holography with the LED source, sparsely sampled at the same locations. Note the reversal of gradient directions in Fig. 4(a), which occur at the core/cladding interface, which is expected for the nominal refractive indices in Table 1. Whilst symmetric and smooth, the in-line result from the TIE measurements does not show the same trend and appears to significantly overestimate the phase curvature by comparison. This may also be explained by slight clipping of Fresnel fringes outside the analyzed field of view in the pre-processing of the data. Similar to the step-index fiber, the measured coverslip wedge gradients were substantial, with an average <∂xϕramp> = 0.1 μm−1 in the horizontal direction. Although the vertical wedge gradients were also corrected, some residual asymmetry is apparent in Fig. 4(a) and it is difficult to determine if this is a real feature of the specimen. The in-line vector field was not corrected, but is nonetheless only composed of vertically-oriented gradients, indicating that the wedge effects were artificially filtered out by the phase retrieval process, much like the results for the step-fiber. One final interesting aspect of Fig. 4(a) is that the combined maxima and vortex derived phase gradients are virtually indistinguishable, suggesting that the Laplacian attenuation in Eq. (3), which was not accounted for in this analysis, has had little effect.
Fig. 4 Measured phase gradients for the multi-mode graded-index fiber, which was immersed in glycerol. The field of view is 142 μm in each direction and contains the horizontal fiber of width 140 μm. The singularimetry measurements are shown in a) and the TIE based in-line holography phase gradients have been sampled at the same points in b).
5. Conclusion
We have proposed a form of singularimetry for direct measurement of phase gradients, which utilizes vortices and intensity gradient singularities in non-diffracting or far-field structured illumination. The propagation-induced transverse optical flow of these singularities was shown to be analytically proportional to phase and amplitude gradients to first order in the focal variation. Using a plano-convex lens as a benchmark specimen, phase gradients were quantified to estimate the focal length to within 1% and the measured vector field was consistent with that from a spherical surface. Compared with quantitative in-line holography experiments, based upon the transport of intensity equation, our vortex lattice method also measured phase gradients from fiber optic specimens and was found to be sensitive to small refraction variations. Our singularimetry approach should be particularly advantageous for slowly varying and subtle specimen-induced gradients, due to the local nature of the measurements, coupled with the detection precision of wavefield extrema which can be increased by trading spatial sampling. One prime application is the detection of nano-scale electromagnetic fields in the bright-field modality of transmission electron microscopy.
Acknowledgments
The authors are grateful for the input of M. Weyland in useful discussions of this research. D. M. Paganin and M. J. Morgan acknowledge the financial support of the Australian Research Council.
References and links
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