Phase-space approach to lensless measurements of optical field
correlations

Katelynn A. Sharma; Thomas G. Brown; Miguel A. Alonso

doi:10.1364/OE.24.016099

1. Introduction

Phase-space representations such as the Wigner function [1] were defined to provide classical-like descriptions of quantum systems. These representations have been used widely in the characterization of the spatial and temporal statistics of optical fields, in both the classical and quantum regimes. Phase-space tomography, which was proposed by Bertrand and Bertrand [2], was applied by Raymer and collaborators in three contexts: to measure the photon density matrix for the field quadratures of a squeezed state [3] and to measure the field’s temporal [4] and spatial [5] degrees of freedom. This approach was later cast in terms of another phase space representation, the ambiguity function [6], which corresponds to the quasi-characteristic function to Wigner’s quasi-probability distribution. Versions of phase-space tomography using rotating astigmatic lenses [7] or spatial light modulators (SLM) [8] at fixed distances have been implemented to measure spatial coherence over two transverse dimensions. Bartelt et al. [9], Waller et al. [10], and Stoklasa et al. [11] performed measurements based on the Husimi distribution (or spectrogram), which is a convolution in phase space of the optical field’s Wigner function with the Wigner function for the measurement device. Lundeen and collaborators [12,13] used measurements associated with the Dirac (also known as the Kirkwood or Rihaczek) phase-space distribution. When applied to the spatial characterization of the optical field, the main advantage of phase-space-based approaches is their simplicity and robustness: they do not require sensitive interferometric setups in which the beam is split and recombined.

In this work we propose even simpler techniques that involve only free propagation by a fixed distance and simple binary masks that are scanned over the test plane. That is, no lenses or other focusing devices are needed, so these methods could be implemented in spectral regions such as x-rays where focusing elements or beamsplitters are unreliable or even unavailable. Moreover, the analysis presented here, based on the ambiguity function phase-space representation [14], sheds light on the validity of not only the new approach but also of several of the prior techniques just mentioned. While our description and experiments are for measurements of the spatial correlations of quasimonochromatic light beams, mathematically analogous techniques can be used, say, for characterizations in the temporal domain.

2. Theoretical description

Consider a quasimonochromatic optical beam propagating in the positive z direction. Its second order statistics at a test plane z = 0 can be characterized by the mutual intensity function J(x₁; x₂) ∝ 〈x₁|ρ|x₂〉, where x₁ =(x₁, y₁) and x₂ =(x₂, y₂) are two points over the test plane, and ρ is the photon density matrix. The goal is to retrieve this correlation through measurements in which masks are placed at the test plane, and the transmitted field’s intensity, I_A(x), is measured following free propagation by a distance z. This intensity is calculated through a double Fresnel diffraction integral:

I_{A} (x_{0}; x) = \frac{1}{λ^{2} z^{2}} \iint J (x_{1}; x_{2}) A^{*} (x_{1} - x_{0}) A (x_{2} - x_{0}) exp [i k \frac{{(x_{2} - x)}^{2} - {(x_{1} - x)}^{2}}{2 z}] d^{2} x_{1} d^{2} x_{2},

where A(x) is the mask’s transmission function and x₀ is the position at which this mask is centered. For example, this mask could contain two pinholes, leading to the standard Young experiment for which I_A presents a fringe pattern that reveals the coherence between the two pinhole locations only [15]. To gain information for a larger number of pairs of points from a single measurement, plates with an array of pinholes at well-chosen positions have also been considered [16], as well as two non-parallel slits [17]. In the case when propagation is described by a Fourier relation (either in the far-field or through the insertion of a lens in a 2f configuration), several alternative masks have been employed. For example, Bartelt et al. [9] and Waller et al. [10] used opaque masks with narrow apertures that were scanned over the field. The approach by Stoklasa et al. [11] based on a Shack-Hartmann sensor, is also of this type, where x₀ is sampled at discrete points (the centers of the sensor’s lenslets) separated by at least the lenslet width. Cho et al. [18] used a transparent mask with a phase discontinuity, and Wood et al. [19] employed a small obstacle. Here, we explore the use of masks in the Fresnel region, so that measurements can be performed without the need of lenses.

To simplify the treatment, we change variables of integration according to x₁ = x₀ + τ − x′/2 and x₂ = x₀ + τ + x′/2. We also write J̄(x̄, x′) = J(x̄ − x′/2; x̄ + x′/2). This makes Eq. (1) take the form

I_{A} (x_{0}; x) = \frac{1}{λ^{2} z^{2}} \iint \bar{J} (x_{0} + τ; x^{'}) A^{*} (τ - \frac{x^{'}}{2}) A (τ + \frac{x^{'}}{2}) exp [\frac{i k (x_{0} + τ - x) \cdot x^{'}}{z}] d^{2} τ d^{2} x^{'} .

In what follows, we consider linear combinations

Δ = \sum_{n} c_{n} I_{A_{n}}

(where c_n are constant coefficients) of consecutive measurements resulting from using different masks A_n. These linear combinations have the form

Δ (x_{0}; x) = \frac{1}{λ^{2} z^{2}} \iint \bar{J} (x_{0} + τ; x^{'}) 𝒜 (τ, x^{'}) exp [\frac{i k (x_{0} + τ - x) \cdot x^{'}}{z}] d^{2} τ d^{2} x^{'},

where

𝒜 (τ, x^{'}) = \sum_{n} c_{n} A_{n}^{*} (τ - x^{'} / 2) A_{n} (τ + x^{'} / 2)

. The goal is to recover J̄ by scanning the masks (varying x₀).

It is easy to show that Eq. (3) can be solved for the mutual intensity by using the ambiguity function [14] representation of the quantities in Eq. (3). Let us express all three functions in this representation via Fourier transformation:

\tilde{\tilde{Δ}} (p; x^{'}) = \frac{1}{λ^{4}} \iint Δ (x_{0}; x) exp [i k (\frac{x \cdot x^{'}}{z} - x_{0} \cdot p)] d^{2} x d^{2} x_{0},

\tilde{\bar{J}} (p; x^{'}) = \frac{1}{λ^{2}} \int \bar{J} (x; x^{'}) exp (- i k x \cdot p) d^{2} x,

\tilde{𝒜} (p; x^{'}) = \frac{1}{λ^{2}} \int 𝒜 (τ; x^{'}) exp (- i k τ \cdot p) d^{2} τ,

where p is a dimensionless variable. With these substitutions, Eq. (3) leads to the simple relation

\tilde{\tilde{Δ}} (p; x^{'}) = \tilde{\bar{J}} (p - \frac{x^{'}}{z}; x^{'}) \tilde{𝒜} (- p; x^{'}),

from which we can solve for

\tilde{\bar{J}}

and then find J̄ through inverse Fourier transformation.

3. Mask choices

The main hurdle in the recovery of J̄ from Eq. (7) is that 𝒜̃(−p; x′) can have localized zeros or be zero over extended regions of the (p; x′) phase space. Therefore, a requirement for a good measurement is to choose the mask combination (and if possible the propagation distance z) so that these zeros are largely absent from regions where $\tilde{\bar{J}} (p - x^{'} / z, x^{'})$ is significant. We now illustrate this with three different mask combinations, each leading to a different function 𝒜̃:

Small aperture. Consider using a single mask corresponding to an opaque plaque with a localized aperture, as done in [9,11] for the case when a Fourier transforming lens is used (which is mathematically equivalent to letting z → ∞ in our treatment). Let us describe this aperture by A_a(x) = a(x) where, henceforth, a(x) represents a function that is zero outside a region of width w centered at the origin. This could describe a hard aperture that is circular, square, of some other shape, or an apodized aperture. We define 𝒜⁽¹⁾(τ; x′) = a^*(τ − x′/2)a(τ + x′/2). Then, 𝒜̃⁽¹⁾ is simply the ambiguity function of the aperture, given by ${\tilde{𝒜}}^{(1)} (p; x^{'}) = \frac{1}{λ^{2}} \int a^{*} (τ - \frac{x^{'}}{2}) a (τ + \frac{x^{'}}{2}) exp (- i k τ \cdot p) d^{2} τ .$
Difference of measurements without and with a small obstacle. Consider instead using the complementary mask, A_o(x) = 1 − a(x), which describes a transparent mask with a localized obstacle of size w, and then subtracting the corresponding measurement from one where no mask is used (A = 1), as done in [19] for the Fourier case (z → ∞). One can show that this results in ${\tilde{𝒜}}^{(2)} (p; x^{'}) = \tilde{a} (p) exp (\frac{i k x^{'} \cdot p}{2}) + {\tilde{a}}^{*} (- p) exp (- \frac{i k x^{'} \cdot p}{2}) - {\tilde{𝒜}}^{(1)} (p; x^{'}),$ where ã(p) = 1/λ² ∫ a(x) exp(−ikx · p) d²x is the Fourier transform of a, and 𝒜̃⁽¹⁾ is the function in Eq. (8). For amplitude masks (where a is real), this expression simplifies to ${\tilde{𝒜}}^{(2)} (p; x^{'}) = 2 \tilde{a} (p) cos (\frac{i k x^{'} \cdot p}{2}) - {\tilde{𝒜}}^{(1)} (p; x^{'}) .$
Sum of the previous two options. Given that Eqs. (9) and (10) include a term equal to minus that in Eq. (8), we also consider a combination of three measurements: no mask minus obstacle plus aperture. This gives 𝒜⁽³⁾(τ; x′) = 𝒜⁽¹⁾(τ; x′) + 𝒜⁽²⁾(τ; x′), and therefore ${\tilde{𝒜}}^{(3)} (p; x^{'}) = \tilde{a} (p) exp (\frac{i k x^{'} \cdot p}{2}) + {\tilde{a}}^{*} (- p) exp (- \frac{i k x^{'} \cdot p}{2}) = 2 \tilde{a} (p) cos (\frac{i k x^{'} \cdot p}{2}),$ where the last expression is valid for amplitude-only masks (real a).

4. Limitations due to zeros

As mentioned earlier, the main limitations to the measurements are due to the zeros of the corresponding functions 𝒜̃. To visualize these limitations, consider first the 1D case where only one transverse coordinate, x, is taken into account. Let us assume the simplest possible mask: a(x) is unity for |x| ≤ w/2 and zero otherwise. By using the 1D definition of the Fourier transform, one finds

{\tilde{𝒜}}^{(1)} (p; x^{'}) = \frac{sin [k (w - | x^{'} |) p / 2]}{π p} Θ (w - | x^{'} |),

{\tilde{𝒜}}^{(3)} (p; x^{'}) = 2 \frac{sin (k w p / 2)}{π p} cos (k x^{'} p / 2),

{\tilde{𝒜}}^{(2)} (p; x^{'}) = {\tilde{𝒜}}^{(3)} (p; x^{'}) - {\tilde{𝒜}}^{(1)} (p; x^{'}),

where Θ(t) = (1 + t/|t|)/2 is the Heaviside unit step distribution. These three functions are illustrated in Figs. 1(a)–1(c), respectively. They are all real since a(x) is real and even, but they all include regions where they are negative and, more critically, zero. Notice that 𝒜̃⁽¹⁾, shown in Fig. 1(a), vanishes for |x′| > w, so the first type of measurement cannot be used to recover the coherence for pairs of points whose separation is larger than the aperture. Additionally, 𝒜̃⁽¹⁾ presents zero curves of hyperbolic shape which cross the p axis at non-zero integer multiples of λ/w. On the other hand, 𝒜̃⁽²⁾ and 𝒜̃⁽³⁾, shown in Figs. 1(b) and 1(c) respectively, do not vanish over extended areas but only along certain lines (segments of straight lines and hyperbolas). The regions (curves and areas) where these three functions vanish are shown in Fig. 2.

Fig. 1 Plots of (a) 𝒜̃⁽¹⁾, (b) 𝒜̃⁽²⁾, and (c) 𝒜̃⁽³⁾.

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Fig. 2 Regions in ambiguity phase space that are zeros of 𝒜̃⁽¹⁾ (blue areas and lines), 𝒜̃⁽²⁾ (yellow lines), and 𝒜̃⁽³⁾ (dotted red lines). Note that for |x′| > w the zero lines of 𝒜̃⁽²⁾ and 𝒜̃⁽³⁾ coincide. The background green-level distribution represents $\tilde{\bar{J}} (p - x^{'} / z; x^{'})$ . The inset shows how the inclination of this distribution depends on z.

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It is worth stressing that the fact that 𝒜̃⁽¹⁾ = 0 for |x′| > w poses the only hard constraint, separating areas of zero and nonzero values. The transition could be made smooth by using non-binary apodized masks in which a is continuous. However, in practice, the values of 𝒜̃⁽¹⁾ would still drop to small levels that are effectively zero at separation points |x′| > w. Note that such apodization could also remove at least some of the vertical zero lines of 𝒜̃⁽²⁾, and 𝒜̃⁽³⁾, but at the cost of making these functions decay faster in p. As shown later, it is possible to obtain estimates for the coherence by regularization or fitting when zero regions of lower dimensionality, such as those for 𝒜̃⁽²⁾ and 𝒜̃⁽³⁾, overlap with the data.

For the sake of illustration, Fig. 2 shows also the ambiguity function for a Gaussian-Schell-model beam (where both the intensity profile and the correlation are Gaussian) as a green distribution in the background. Regardless of whether the actual measured field is of this type or not, its ambiguity function’s (horizontal) width in the p direction is proportional to λ divided by the scale of spatial variation of the field, so provided that this spatial variation is over widths larger than the chosen w, the significant parts of the distribution should not interact with zeros of any of the three 𝒜̃ along the p axis. The (vertical) width in the x′ direction, on the other hand, is proportional to the typical coherence width of the beam. Clearly, when this coherence width is larger than the chosen w, the ambiguity function of the field overlaps with the zero regions of 𝒜̃⁽¹⁾. Finally, the tilt (or shear) of the distribution away from the vertical axis is proportional to 1/z (as indicated in the inset) so in the far-field limit the distribution is vertical and can in principle avoid all the zeros of 𝒜̃⁽²⁾ and 𝒜̃⁽³⁾. For finite z, on the other hand, the tilt can cause the field’s ambiguity function to overlap with the zero lines of these mask functions. Importantly, the diagram in Fig. 2 clarifies that there are two regimes for the measurement parameters w and z that are independent of the measured field: i) when 2w²/λz < 1 (meaning that the aperture/obstacle fits in a single Fresnel zone when observed from the detector plane) the critical zeros are those of 𝒜̃⁽¹⁾, so it is best to use measurements involving obstacles; ii) when 2w²/λz > 1, on the other hand, the critical zeros are those of 𝒜̃⁽²⁾ followed by those of 𝒜̃⁽³⁾. In this second case, the aperture-only approach works best within |x′| < w, and outside of this region one can use obstacles.

Analogous results hold for 2D masks. For example, if the aperture/obstacle were a square with side w, the expressions would be

{\tilde{𝒜}}^{(1)} (p; x^{'}) = \frac{sin [k (w - | x^{'} | p_{x} / 2)]}{π p_{x}} \frac{sin [k (w - | y^{'} | p_{y} / 2)]}{π p_{y}} Θ (w - | x^{'} |) Θ (w - | y^{'} |),

{\tilde{𝒜}}^{(3)} (p; x^{'}) = 2 \frac{sin (k w p_{x} / 2)}{π p_{x}} \frac{sin (k w p_{y} / 2)}{π p_{y}} cos (k x^{'} \cdot p / 2),

{\tilde{𝒜}}^{(2)} (p; x^{'}) = {\tilde{𝒜}}^{(3)} (p; x^{'}) - {\tilde{𝒜}}^{(1)} (p; x^{'}) .

Again, note that 𝒜̃⁽¹⁾ only differs from zero for separations x′ that fit within the aperture. Beyond these separations, 𝒜̃⁽²⁾ and 𝒜̃⁽³⁾ are indistinguishable and their zeros come from the zeros of ã(p) (which correspond to the straight lines in Fig. 2), but also to the zeros of the mask-shape-independent factor cos(kx′ · p/2). That is, even if the mask were apodized to replace a series of discrete zero manifolds with a smooth decay, the equivalent of the hyperbolic zero lines in Fig. 2 would still be present. As in the 1D case, there are two regimes corresponding to whether or not the aperture/obstacle occupies less than a Fresnel zone. In the experimental results presented below we explore both regimes.

5. Slowly-varying approximation

For all the measurement combinations described earlier, the mutual intensity can be found as

\bar{J} (\bar{x}, x^{'}) = \int \tilde{\bar{J}} (p, x^{'}) exp (i k \bar{x} \cdot p) d^{2} p = \int \frac{\tilde{\tilde{Δ}} (p + x^{'} / z; x^{'})}{\tilde{𝒜} (- p - x^{'} / z; x^{'})} exp (i k \bar{x} \cdot p) d^{2} p .

Of course, to evaluate this expression, a large number of measurements must be performed for x₀ sampling thoroughly the test plane. Further, given the integral nature of this expression, determining the coherence for any pair of points requires all the measurements. We now discuss a limiting situation in which, for a single set of measurements with the masks centered at a given location x₀, one can estimate the coherence at all pairs of points whose centroid is this location. The required assumption is that w must be much smaller than the scale of spatial variation for the field, so that the width in p of

\tilde{\bar{J}}

is much smaller than that of 𝒜̃. We can then approximate 1/𝒜̃(−x′/z − p; x′) by its Taylor expansion in p around −x′/z. For simplicity, we only keep the leading two terms of this expansion. This results in the expression

\bar{J} (x_{0}, x^{'}) = [1 + f_{1} \cdot \nabla_{x_{0}} + \dots] \frac{\tilde{Δ} (x_{0}, x^{'})}{\tilde{𝒜} (- x^{'} / z; x^{'})},

where

f_{1} (x^{'}, z) = \frac{\nabla_{p} \tilde{𝒜} (- x^{'} / z; x^{'})}{i k \tilde{𝒜} (- x^{'} / z; x^{'})},

\tilde{Δ} (x_{0}, x^{'}) = \frac{1}{λ^{2}} \int Δ (x_{0} + τ, x) exp [i k \frac{τ \cdot x^{'}}{z}] d^{2} x .

If the second term in brackets in Eq. (19), which is a correction proportional to the gradient in x₀, can be neglected, then the estimate requires only measurements at the corresponding x₀. If the correction term is needed, on the other hand, then measurements at at least two nearby points to x₀ are also required to estimate this gradient.

For measurements involving obstacles, choosing small aperture/obstacle sizes w is advantageous not only because it makes the approximation above valid, but also because it extends the validity of the methods by moving some of their zeros away from the region occupied by the field. This is illustrated in Fig. 3 and Visualization 1, which show how the position of the zeros change with z and w. These are the zeros of the functions 𝒜̃(−x′/z − p; x′) over the plane x′. When only apertures are used, then using larger w would allow extending the region over which correlations can be measured, but with the constraint that w still needs to be small compared to the scale of change of the coherence function. From a practical point of view, however, letting the value of w be too small makes the measurements susceptible to the effects of noise, regardless of the approach used. Similar arguments can be made about the effect of changing the propagation distance z. Theoretically, it is better to let z be as large as possible, as this removes the effects of the zeros for measurements involving obstacles. However, larger z implies lower intensities at the detector plane as well as the need for larger detectors.

Fig. 3 Regions where the functions 𝒜̃(−x′/z, x′) vanish, for λ = 532nm, z = 470mm, and for w = (a) .27mm and (b) .76mm. In both figures, blue areas are the zeros of 𝒜̃⁽¹⁾, yellow line are the zeros of 𝒜̃⁽²⁾, and dotted red lines are the zeros of 𝒜̃⁽³⁾. Visualization 1 shows the effect of the variation of w and z on the location of these zeros.

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6. Experimental description

A diagram of the experimental setup is shown in Fig. 4. The spatially partially coherent field to be measured is generated by focusing a diode-pumped doubled Nd:YAG laser (λ = 532nm) onto a spinning diffuser. The diffused field is then collimated, creating approximately an elliptical Gaussian Schell-model field at the test plane [20] (due to ellipticity of the focused spot). Small longitudinal shifts of the diffuser can be used to change the area of the illuminated region, therefore modifying the spatial coherence of the measured field. Note from the diagram that an image of the focus at the diffuser is formed before collimation. As discussed later, this intermediate image allows the introduction of a vignetting object to disrupt the spatial homogeneity of the coherence at the test plane, making the measured field differ significantly from the Gaussian-Schell model.

Fig. 4 Diagram of the experimental setup. Just after a relayed focus, the dotted line indicates the location of an obstruction (shown in the top image) inserted to create an inhomogeneous field. The bottom images indicate three examples of data taken with their respective masks displayed on the SLM.

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The field’s coherence is measured by placing a SLM at the test plane, and then letting the field propagate freely for 470 mm to an electron-multiplying CCD. The SLM was extracted from a liquid crystal display projector system and is used to display the desired aperture and obstacle masks. The obstacle/aperture used for our measurements was a square, and two different sizes were used, 0.27mm and 0.76mm (14 and 40 pixels) across. Each of these sizes falls in one of the two regimes mentioned earlier, since 2w²/λz equals 0.583 and 4.62, respectively. The choice of a square aperture/obstacle shape against, say, a circle, was chosen for simplicity due to the square pixels in the SLM. It was shown in [19] for the case of obstacles in the far field that the shape has a negligible effect on the measurements, particularly for small w. In addition to the open, aperture, and obstacle measurements, a dark mask measurement was also used for background subtraction.

7. Data processing and results

Figure 4 also shows an example of measurements taken when the SLM displays an aperture (with the background subtracted), when it has an obstacle, and when it has no mask. Both the obstacle measurement and the difference of the measurements with no mask and an obstacle are localized within a limited region of the CCD. A super-Gaussian filter was applied around these regions to remove the effects of noise from the rest of the detector. Basic estimates of J̄(x₀, x′) can be obtained by using the leading term of Eq. (19), ignoring the correction proportional to f₁, under the assumption that w is smaller than the scale of significant change of J̄. (This assumption can be verified after making measurements at nearby points and, if needed, new measurements can be performed with smaller w.) The functions 𝒜̃(−x′/z; x′) that appear in the denominator can either be calculated or obtained experimentally from the measurement of a spatially coherent field (removing the rotating diffuser) using the corresponding masks, leading to nearly indistinguishable results. A linear phase factor was applied to the results of Eq. (19) to compensate for residual misalignment of the CCD. Also, problems of division by zero were avoided by replacing division by 𝒜̃(−x′/z; x′) in Eq. (19) with multiplication by $\tilde{A^{*}} (- x^{'} / z; x^{'}) / [{| \tilde{A} (- x^{'} / z; x^{'}) |}^{2} + ε^{2}]$ , where ε was chosen as 10% of the peak value of 𝒜̃(−x′/z; x′)|. Since the numerator in the equation has zeros that in theory coincide with those of the denominator, the results in the figures vanish at the corresponding locations.

The effects of using different aperture/obstacle sizes for fields of varying coherence are shown in Fig. 5. In the nine images on the left half of the figure, the aperture/obstacle size was 0.27mm (14 SLM pixels) so that 2w²/λz = 0.583, while for the nine images on the right it was 0.76mm (40 SLM pixels) so that 2w²/λz = 4.62. In each case, three longitudinal positions of the rotating diffuser were used, corresponding to fields with different coherence widths, going from most coherent (top row) to least coherent (bottom row). The figure compares the three different amplitude mask combinations discussed earlier: 1) an aperture (first and fourth columns), 2) the difference of an open measurement and an obstacle (second and fifth columns), and 3) the sum of the previous two (third and sixth columns). Notice that in (a), (d) and (j), the aperture alone is too small to characterize the fields. On the other hand, approaches using obstacles can retrieve correlations over scales much larger than w, with the exceptions of certain ranges of distances corresponding to the zeros of 𝒜̃⁽²⁾(−x′/z; x′) and 𝒜̃⁽³⁾(−x′/z; x′). When 2w²/λz > 1 (right-hand side), the central zero-free region is largest for the aperture-only approach, but the obstacle-based methods allow finding correlations over separations larger than w. Hence, a combination of the two would be most effective. Finally, note from the plots in the bottom row that, when the regions where J̄ is significant do not overlap with any zeros, all methods give similar results.

Fig. 5 Estimations of J̄ found using (a,d,g,j,m,p) an aperture, (b,e,h,k,n,q) the difference of an open mask and and obstacle, and (c,f,i,l,o,r) an aperture plus an open mask minus and obstacle. The three rows correspond to three test fields with decreasing coherence widths. The plots in (a–i) used w = 0.27 mm (so that 2w²/λz = 0.583) while those in (j–r) used w = 0.76 mm (so that 2w²/λz = 4.62). In all cases, the plot ranges are |x′, y′| ≤ 1.7 mm, and the color schemes for amplitude (normalized to peak value) and phase are as indicated by the palette on the right.

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The gaps left by the zero lines can be filled through a fitting process in which an expansion in, say, a Hermite-Gaussian function basis in x′ is proposed for J̄(x₀, x′):

\bar{J} (x_{0}, x^{'}) \approx \sum_{m, n = 0}^{N} a_{m n} {HG}_{m} (\frac{x^{'}}{W}) {HG}_{n} (\frac{x^{'}}{W}),

where HG_m is the mth order Hermite-Gaussian function, given by the product of a Hermite polynomial of order m, a Gaussian, and a normalization factor. For a given truncation value N, one can optimize for the scaling factor W and the expansion coefficients so that |J̄(x₀, x′) 𝒜̃(−x′/z; x′) − Δ̃(x₀, x′)|² is minimized. (For given W, the expansion coefficients a_mn can be found in closed form.) The fitting works best when the residual linear phase is removed prior to the fitting. Clearly, the results cannot be trusted for measurements that only give access to a very restricted range of correlations. The results of this type of fitting are shown in Fig. 6 for two of the measurements.

Fig. 6 Measured coherence distributions and the results of using Hermite-Gaussian fitting. Parts (a) and (c) correspond to parts (c) and (f) of Fig. 5, while parts (b) and (d) show the corresponding results when the Hermite-Gaussian fitting procedure is used. In all cases, the range of separations in x′ and y′ is from −1.7 mm to 1.7 mm.

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8. Results for spatially inhomogeneous fields

To show that the system can characterize the four-dimensional coherence properties for a spatially inhomogeneous field, a thin cardboard tip was inserted a few millimeters after the intermediate image of the focused field. This obstruction causes a vignetting effect that introduces significant inhomogeneity in the spatial coherence at the test plane, so that the field can no longer be approximated by a Gaussian-Schell model. An image of this tip following the focus is shown in the upper inset in Fig. 4. The resulting variation of J̄(x₀, x′) for several values of x₀ over the test plane is shown in Fig. 7, which corresponds to the estimate that uses three measurements: an aperture, an obstacle (both squares of side 0.76 mm), and a clear mask. Visualization 2 shows an animation of the combination of the three measurements, and the corresponding estimate of J̄(x₀, x′) as the the aperture/obstacle scans the test plane.

Fig. 7 Changes in coherence as a function of point separation x′ within a range of −0.714 mm to 0.714 mm, for nine locations of the pivot point x₀ (the center of the aperture/obstacle) over the test plane with a spacing of 1.425mm. Visualization 2 shows an animation with a finer sampling in x₀ with spacing of 0.45 mm.

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This spatially-variant field also allows us to evaluate the corrections in the second term of Eq. (19). The factor f₁ is computed for the three-measurement approach, and it is multiplied by the estimated gradient in x of the basic estimate of J̄, calculated through finite differences from values of ±0.45mm in x and y around x₀. As seen in Fig. 8, these corrections give a small yet appreciable contribution to the result, particularly to the side lobes. The maximum amplitude of the correction is approximately 16% of the peak amplitude of the basic estimate. That is, the value of w used here is indeed smaller than the range of variation of J̄, but not much smaller, since the leading correction is small but not negligible.

Fig. 8 (a) The basic estimate for J̄ [also shown in Fig. 7(e)] and (b) a corrected estimate that includes the corrections in the second term of Eq. (19).

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9. Conclusion

We presented an approach for measuring the four-dimensional spatial correlations of a quasimonochromatic field at a test plane. Three variations were discussed and implemented experimentally: one using an aperture, one using an obstacle, and one using both. These methods are based exclusively on simple binary amplitude masks and intensity measurements at a fixed distance; no refractive, focusing, or wavefront-splitting optical elements are needed. A phase-space description in terms of the ambiguity function was used to validate the approaches. It was shown that the methods that use an obstacle allow for the measurement of correlations over point separations beyond the size of the aperture-obstacle used.

Our focus was on the use of simple binary masks that modify only the amplitude of the field. This is motivated by two facts. First, we wanted to explore the simplest possible setup for measuring coherence at a large number of pairs of points simultaneously. Second, for some spectral regimes, creating masks that impart phase or that have a smoothly varying transmissivity might be challenging. Nevertheless, the ideas presented here apply also to apodized masks and to masks that impart phase. While apodization would avoid some of the oscillations of the functions 𝒜̃, it would also make these functions decay faster, which is not desirable. On the other hand, some of the constraints imposed by zeros could be relaxed by using phase masks. For example, consider a binary transparent mask with an added small transparent layer of size w which imparts an extra phase θ, described by the transmission function A_t(x) = 1 + [exp(iθ) − 1]a(x). By using a combination of measurements with this mask, an opaque obstacle of the same size, and no mask, according to Δ = (I_{A_t} − I_{A_o}) − exp(−iθ)(I₁ − I_{A_o}), one can remove the z-dependent zeros that are prevalent in the methods described earlier. Depending on the application, this could be an interesting option to follow.

Finally, let us mention that the method discussed here can resolve ambiguities in the coherence intrinsic to other methods that use only free propagation and intensity measurements. Such ambiguities, pointed out by Gori et al. [21], occur for example for fields that include coherent modes with definite vorticity. When placed away from these modes’ vortices, the apertures and obstacles used here would probe sections of their wavefronts, so that the intensity measured at the CCD would depend on the vortices’ handedness. That is, off-centered apertures or obstacles break the rotational symmetry, just as cylindrical lenses do [22].

Acknowledgements

The authors would like to express gratitude to Amber Betzold, Anthony Vella for programming help and advice, as well as QImaging for loaning the EM-CCD used to obtain this data. We acknowledge support from the National Science Foundation (PHY-1507278).

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Name	Description
Visualization 1: MOV (3800 KB)	regions where the A function vanishes for varying values of z and w
Visualization 2: MOV (4421 KB)	change in coherence function as obstacle position changes

Phase-space approach to lensless measurements of optical field correlations

Abstract

1. Introduction

2. Theoretical description

3. Mask choices

4. Limitations due to zeros

5. Slowly-varying approximation

6. Experimental description

7. Data processing and results

8. Results for spatially inhomogeneous fields

9. Conclusion

Acknowledgements

References and links

Supplementary Material (2)

Cited By

Figures (8)

Equations (22)

Optics Express