August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
4927
Using shadows to measure spatial coherence James K. Wood,1 Katelynn A. Sharma,1 Seongkeun Cho,2 Thomas G. Brown,1 and Miguel A. Alonso1,3,* 1 2 3
The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
Mechatronics R&D Center, Samsung Electronics Co., Ltd., Hwaseong-si, Gyeonggi-do, South Korea
Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA *Corresponding author: [email protected] Received June 23, 2014; revised July 16, 2014; accepted July 16, 2014; posted July 17, 2014 (Doc. ID 214526); published August 15, 2014
We present a very simple method for measuring the spatial coherence of quasi-monochromatic fields through the comparison of two measurements of the radiant intensity with and without a small obscuration at the test plane. From these measurements one can measure simultaneously the field’s coherence at all pairs of points whose centroid is the centroid of the obstacle. This method can be implemented without the need of any refractive or diffractive focusing elements. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (050.1940) Diffraction; (050.5080) Phase shift; (070.2580) Paraxial wave optics. http://dx.doi.org/10.1364/OL.39.004927
Insights gained from new optical measurement techniques have often paved the way for broader applications in physics. This is especially true of coherence phenomena where fields must be characterized in four spatial dimensions even for quasi-mononchromatic fields. Fast and accurate ways for measuring spatial coherence are also of great practical importance, e.g., for source characterization in medical imaging or photolithography, as well as in the understanding of novel imaging systems. The canonical experiment for measuring spatial coherence is Young’s two-pinhole setup [1,2]. By placing the pinholes at two test points and observing the position and visibility of the resulting fringes, the complex degree of coherence at these two points is estimated. Variants of this approach use two separate copies of the wavefront to remove the dependence of the fringe period on the pinhole separation [3,4]. However, the full characterization of a field would require scanning each pinhole independently over a transverse plane, something that takes a significant amount of time and effort. Several approaches have been proposed to alleviate this problem by giving simultaneous access to many pairs of points. For example, arrays of pinholes with nonredundant separations allow simultaneous measurements for point pairs corresponding to any two such pinholes [5]. More elaborate interferometric techniques are based on the superposition of two copies of the wavefront that are mutually displaced, reversed, or rotated [6–11]. When projected onto a CCD detector, their interference combined with knowledge of the intensity distribution of each copy gives access to the coherence at a dense sample of pairs of points that either have fixed separation (for wavefront shearing) or are symmetrically distributed around a pivot point or line (for wavefront rotating or reversing). By mutually displacing the wavefronts over two degrees of freedom, a sufficiently dense sample of the full four-dimensional (4D) coherence function is measured. That is, rather than scanning over four dimensions, one only scans over two, as the two-dimensional signal at the CCD provides information over a range of values of the other two. While successful in many cases, these interferometric methods are sensitive to misalignments and vibrations, and are difficult or impossible to implement in some spectral regions. 0146-9592/14/164927-04$15.00/0
A different approach, phase-space tomography [12,13], is based on measuring the intensity distribution at a range of propagation distances. However, this approach only works if the field has a known symmetry (translational or rotational) or separability, so that the coherence function depends only on three or less degrees of freedom. Otherwise, the three-dimensional intensity information is not sufficient to recover the 4D coherence function. A fourth parameter can be introduced in the form of a combination of focal lengths for anamorphic focusing elements (cylindrical lenses or spatial light modulators). In fact, the CCD’s position can be fixed, and the two scanning parameters can be the focal lengths of the anamorphic elements [14,15], but this again means that the field must be manipulated through refraction or diffraction, which is challenging for some frequencies. Other methods [16,17] based on free propagation require some prior knowledge of the coherence properties. Here we present a very simple technique that can, in principle, be implemented without refractive or diffractive focusing elements, and where coherence is measured by scanning a small obstacle over the test plane and then measuring the resulting radiant intensity. For simplicity we introduce the basic ideas for fields that depend only on one transverse coordinate, x, and then generalize them to fields depending on two, x, y. Consider a stationary, quasi-monochromatic, partially coherent scalar field Ux, whose second-order coherence properties at the test plane z 0 are described by the mutual intensity, Jx1 ; x2 hU x1 Ux2 i, where the angle brackets represent a correlation. In the paraxial approximation, the radiant intensity (proportional to the intensity at the far field) is given by ZZ k Jx1 ; x2 exp−ikx2 − x1 pdx1 dx2 2π ZZ k ¯ x; ¯ x0 exp−ikx0 pdx0 dx; ¯ J 2π
Ip
(1)
where k is the central wavenumber, p is a directional variable (the sine of the angle of observation with respect to the z axis), and in the second step we changed to cent¯ x; ¯ x0 Jx¯ − x0 ∕ roid and difference coordinates with J © 2014 Optical Society of America
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2; x¯ x0 ∕2. Ip can be measured either in the far field or by means of a Fourier-transforming lens (as in our experimental implementation). Let a mask now be inserted at z 0, whose amplitude function is given by Ax − x0 , where x0 is a controllable lateral displacement. The resulting radiant intensity is given by
ZZ
x0 x0 A τ τ− 2 2
k A 2π ¯ 0 τ; x0 exp−ikx0 pdx0 dτ; × Jx
I A p; x0
(2)
with τ x¯ − x0 . As we now discuss, different choices for masks A lead to different measurement methods. 1. Standard methods using apertures. Some coherence measurement methods [18,19] use a localized window for Ax (sharp or apodized) to isolate the coherence properties in different locations (measuring the so-called spectrogram or Husimi function). Shack– Hartmann sensors and plenoptic cameras can be considered as special cases of this approach, where the window is the aperture of each lenslet [20]. However, recovering J¯ from I A requires a deconvolution process that is problematic for point separations beyond the window’s extent. To see this, let us expand J¯ in a Taylor series around x0 (the window’s centroid): ¯ 0 τ; x0 Jx
∞ n n¯ X τ ∂ J n0
n! ∂x¯ n
x0 ; x0 :
(3)
The substitution of this relation into Eq. (2) gives IA
∞ Z X
k 2π n0
∂n J¯ x ; x0 An x0 exp−ikx0 pdx0 ; ∂x¯ n 0
(4)
An x
Z
τn x0 x0 A τ− A τ dτ: n! 2 2
(5)
If A is a real symmetric function then An x0 0 for odd n. If the series in Eq. (4) can be well approximated by its first term (n 0), then an estimate of J¯ can be found through inverse Fourier transformation as ¯ 0 ; x0 A0 x0 ≈ Jx
Z
I A p; x0 expikx0 pdp:
−4
(b)
−2
2
(6)
Notice that A0 is the autocorrelation of the aperture function A. If A is chosen as an aperture function with limited extent, then A0 is (significantly) different from zero only over a region slightly larger than the width of A, as shown in Fig. 1(a) for rectangular and Gaussian windows. Therefore, Eq. (6) shows that one cannot recover the coherence for pairs of points whose separation jx0 j is larger than the width of A. This is a fundamental limitation of methods using apertures. 2. Method using binary phase masks. The approach in [21] overcomes this problem by using instead the difference of two measurements, with and without the mask, i.e., by considering the difference of Eqs. (1) and (2):
4 x/w −4
(c) −2
2
x/w 4 −4
−2
2
x/w 4
Fig. 1. (a) Autocorrelation A0 for window functions with rectangular shape Ax rectx∕w (black) and Gaussian shape ¯ 0 x0 for (b) a Ax exp−πx2 ∕w2 (gray). (b), (c) Function A binary phase mask Ax sgnx and (c) a transparent mask with a localized obscuration of rectangular shape ax rectx∕w (black) or Gaussian shape ax exp−πx2 ∕w2 (gray).
Δp; x0 Ip − I A p; x0 ∞ Z n¯ k X ∂ J ¯ n x0 exp−ikx0 pdx0 ; x ; x0 A 2π n0 ∂x¯ n 0 (7) where ¯ n x0 A
Z
τn x0 x0 1−A τ− A τ dτ: n! 2 2
(8)
The inverse Fourier transform of Eq. (7) gives ∞ n¯ X ∂ J
0 ¯ 0 n x0 ; x An x ¯ ∂ x n0
Z
Δp; x0 expikx0 pdp:
(9)
Instead of an aperture, the mask in [21] is a transparent binary phase mask with a single discontinuity, which in the optimal case is given by Ax sgnx, for which ¯ n x0 1 A n!
where 0
(a)
Z
jx0 j∕2 −jx0 j∕2
τn dτ 1 −1n
jx0 jn1 : n 1!
(10)
¯ n 0 for odd n. Again, if the series in Eq. (9) is That is, A approximated by its leading term, we get ¯ 0 x0 ≈ ¯ 0 ; x0 A Jx
Z
Δp; x0 expikx0 pdp:
(11)
Note that unlike the case in Eq. (6), there is no upper bound for the point separation jx0 j for which we can re¯ 0 x0 2jx0 j [shown in cover the coherence since A Fig. 1(b)] grows linearly. There are however problems with recovering J¯ for small jx0 j, which in [21] were addressed through interpolation. Also problematic is the extension of this approach to two dimensions [22]. 3. Method using small obstacles. Such problems are completely bypassed if instead of an aperture or a phase object, the mask is transparent and uniform except for a localized obscuration, i.e., Ax 1 − ax where 0 ≤ ax ≤ 1 is a real distribution localized to a region near x 0. In particular, we can get from Eq. (11) with x0 ¯ denoted as J¯ 0 , given by x¯ a basic estimate of J, ¯ x; ¯ x0 ≈ J¯ 0 x; ¯ x0 J
R
¯ expikx0 pdp Δp; x ; ¯ 0 x0 A
(12)
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
where from Eq. (8) for n 0 and Ax 1 − ax, ¯ 0 x0 2 A
Z
Z axdx −
x0 x0 a τ dτ; a τ− 2 2
J¯ 0 ¯x; x0
N X ∂n J¯ N−1 n1
¯n
∂x
¯ x0 A ¯ x0 n 0 : x; ¯ 0 x A
(18)
and further corrections can be calculated as
(14)
Note that if ax a−x, then all terms with odd n ¯ n 0. In particular, disappear since A ¯ x0 ∂2 J¯ A ¯ x0 J¯ 0 x; ¯ x0 − 20 x; ¯ x0 2 0 : J¯ 2 x; ¯ 0 x ∂x¯ A
Δp; x¯ expikx0 · pdpdq ; ¯ 0;0 x0 A
(13)
namely a constant minus the obstacle’s autocorrelation. That is, the difference of the measurements with and without the obstacle leads to an estimate of the coherence between all pairs of points symmetrically distributed around the obstacle’s centroid. The main advantage of this approach over the previous two is that the function we must ¯ 0 , never vanishes. In fact, it goes to a constant divide by, A for jx0 j larger than the obstacle’s width, and it is never smaller than half this constant, as shown in Fig. 1(c) for rectangular and Gaussian obscurations. To measure coherence over many pairs of points with many centroids, one must scan the obstacle across the field (i.e., vary x0 ). Further, if the sampling in x0 is sufficiently fine, one can find higher-order estimates J¯ N by using iteratively truncated versions of Eq. (9): ¯ x0 J¯ 0 x; ¯ x0 − J¯ N x;
RR
4929
(15)
J¯ N ¯x; x0 J¯ 0 ¯x; x0 −
N X n 0 ¯ 0 X 0 ∂n J¯ N−1 0 An ;n−n x : ¯ x ; x 0 0 ¯ 0;0 x0 ¯ n ¯ n−n A n1 n0 0 ∂x ∂y
(19) As in the one-dimensional case, the leading correction is negligible if both the width of a and the local coherence width are much smaller than the local length scale of variation of J¯ with respect to the centroid coordinates. Our experimental implementation is shown in Fig. 2(a). First, the partially coherent field to be measured is prepared by focusing a green laser (λ 532 nm) onto a rotating ground-glass diffuser followed by two lenses that collimate it. The measurement system consists of a transmissive liquid-crystal spatial light modulator (SLM) placed at the test plane and surrounded by suitably oriented polarizers, a converging lens (f 340 mm) and a CCD detector. The lens is used in Fourier-transforming configuration, with the SLM and CCD at its focal planes, so that the intensity measured at the CCD is proportional to the radiant intensity at the test plane. The obscurations are written onto the SLM so that their size, shape, and location can be controlled at will. In Fig. 2 we show the basic estimate J¯ 0 that results from using obscurations of 2(b)
Suppose the local characteristic length scale of variation ¯ x; ¯ x; ¯ x; ¯ x0 in x¯ is R, so that ∂2 J ¯ x0 ∕∂x¯ 2 ∼ J ¯ x0 ∕R2 . of J 0 0 2 02 ¯ ¯ Then, since A2 x ∕A0 x w 3x ∕24 for jx0 j ≥ w (and something not too different for jx0 j < w), the second-derivative correction in Eq. (15) is insignificant if R is significantly greater than both w and the local coherence width. That is, the obstacle’s size is not constrained by the local coherence width but by how spatially inhomogeneous the field is. This approach is easily generalized to two transverse dimensions. The difference of two radiant intensity measurements, with and without an obscuration at z 0 given by ax, where x x; y can be written as Δp; x0 Ip − I A p; x0 ∞ ZZ k2 X ∂nm J¯ ¯ n;m x0 2 x ; x0 A ∂x¯ n ∂y¯ m 0 4π n;m0 × exp−ikx0 · pdx0 dy0 ;
(16)
where p p; q are the directional/Fourier-conjugate variables and τn ηm x0 x0 a τ− a τ n!m! 2 2 0 0 x x a τ dτdη; −a τ − 2 2
¯ n;m x0 A
Z
with τ τ; η. The basic estimate is
(17)
Fig. 2. (a) Diagram of the optical set-up. (b)–(g) Real parts of J¯ 0 0; x0 ∕J¯ 0 0; 0 obtained by using square obscurations of side (b) 1.855 mm and (d), (f), and (g) 3.81 mm, and circular obscurations of diameter (d) 1.855 mm and (e) 3.81 mm. In (b)–(e) the obscuration was at the center of the SLM, while in (f)/(g) it was shifted to the left/right by 2.93 mm. The insets in (b)–(g) show the measured intensities without and with the obscuration.
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the measurements can be performed in the far field, so no lens is needed. Further, the method involves very little numerical processing that can be performed in real time during the data collection. Here, a Taylor expansion of J¯ was used, leading to a simple estimate and an iterative formula for corrections. However, a rigorous deconvolution in four (phase space) dimensions is also possible, as will be discussed elsewhere. ¯ x0 ; y0 ∕J¯ 0 0; y; ¯ 0; 0j, for (a) x0 0 and Fig. 3. Plots of jJ¯ 0 0; y; 0 (b) y 0. The curves in the insets show three slices corresponding to y¯ −3.71 (black), 0 (gray), and 3.71 mm (pale gray), respectively.
and 2(d) square and 2(c) and 2(e) circular shapes, whose sides/diameters are 2(b) and 2(d) 1.855 mm and 2(d), and 2 (e) 3.81 mm, all at the SLM’s center. In all cases, the measured mutual intensity as a function of point separation is approximately a rotationally symmetric real Gaussian (the imaginary part is about two orders of magnitude smaller) with half-width-half-maximum of about 30 μm. The differences between these measurements are within a few percent of their peak (normalized to unity), even though different shapes and sizes were used. It is clear from Figs 2(b) and 2(c) that noise affects the results more for smaller obscurations, as the measured distributions (shown in the insets) being subtracted are more similar. Parts 2(f) and 2(g) of Fig. 2 correspond to square obstacles of side 3.81 mm displaced 2.93 mm to the left 2(f) and right 2(g) of the center of the SLM. These figures show that in ¯ with centroid insignificant, the this case the variation of J, corrections in Eq. (19) are negligible and the basic estimate J¯ 0 is sufficient. The previous example, an approximately translationally invariant field with Gaussian statistics, shows the method’s robustness and repeatability. However, the power of approaches like this one is in the ability to characterize translationally variant fields. To show this we disrupt the translation invariance and isotropy of the field’s coherence by introducing a horizontal knife edge a few millimeters along the axis away from the intermediate image of the rotating diffuser. The resulting measure¯ using as an obstacle a square of side 3.8 mm ments for J, in the SLM, are shown in Fig. 3 for y¯ varying from −4.1 to 4.1 mm in steps of 0.195 mm. Vignetting from the knife ¯ edge makes the coherence vary with y. To summarize, we proposed a simple method to measure simultaneously the spatial coherence between a large number of point pairs by studying an obstacle’s shadow/penumbra. In implementations, say, with x rays,
We acknowledge support from the National Science Foundation (PHY-1068325). References 1. F. Zernike, Physica 5, 785 (1938). 2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Chap. 10. 3. M. Santarsiero and R. Borghi, Opt. Lett. 31, 861 (2006). 4. D. P. Brown and T. G. Brown, Opt. Express 16, 20418 (2008). 5. A. I. González and Y. Mejía, J. Opt. Soc. Am. A 28, 1107 (2011). 6. H. P. Jauch and K. M. Baltes, Opt. Lett. 7, 127 (1982). 7. C. Iaconis and I. A. Walmsley, Opt. Lett. 21, 1783 (1996). 8. D. Mendlovic, G. Shabtay, A. W. Lohmann, and N. Konforti, Opt. Lett. 23, 1084 (1998). 9. D. L. Marks, R. A. Stack, and D. J. Brady, Appl. Opt. 38, 1332 (1999). 10. C. C. Cheng, M. G. Raymer, and H. Heier, J. Mod. Opt. 47, 1237 (2000). 11. G. A. Swartzlander, Jr. and R. I. Hernandez-Aranda, Phys. Rev. Lett. 99, 163901 (2007). 12. K. A. Nugent, Phys. Rev. Lett. 68, 2261 (1992). 13. M. G. Raymer, M. Beck, and D. F. McAlister, Phys. Rev. Lett. 72, 1137 (1994). 14. A. Cámara, J. A. Rodrigo, and T. Alieva, Opt. Express 21, 13169 (2013). 15. D. L. Marks, R. A. Stack, and D. J. Brady, Appl. Opt. 25, 1726 (2000). 16. C. Rydberg and J. Bengtsson, Opt. Express 15, 13613 (2007). 17. J. C. Petruccelli, L. Tian, and G. Barbastathis, Opt. Express 21, 14430 (2013). 18. H. O. Bartelt, K.-H. Brenner, and A. W. Lohmann, Opt. Commun 32, 32 (1980). 19. L. Waller, G. Situ, and J. W. Fleischer, Nat. Photonics 6, 474 (2012). 20. B. Stoklasa, L. Motka, J. Rehacek, Z. Hradil, and L. L. Sánchez-Soto, Nat. Commun. 5, 3275 (2014). 21. S. Cho, M. A. Alonso, and T. G. Brown, Opt. Lett. 37, 2724 (2012). 22. S. Cho, “Studies on partially coherent fields and coherence measurement methods,” Doctorate thesis (University of Rochester, 2013).
4927
Using shadows to measure spatial coherence James K. Wood,1 Katelynn A. Sharma,1 Seongkeun Cho,2 Thomas G. Brown,1 and Miguel A. Alonso1,3,* 1 2 3
The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
Mechatronics R&D Center, Samsung Electronics Co., Ltd., Hwaseong-si, Gyeonggi-do, South Korea
Center for Coherence and Quantum Optics, University of Rochester, Rochester, New York 14627, USA *Corresponding author: [email protected] Received June 23, 2014; revised July 16, 2014; accepted July 16, 2014; posted July 17, 2014 (Doc. ID 214526); published August 15, 2014
We present a very simple method for measuring the spatial coherence of quasi-monochromatic fields through the comparison of two measurements of the radiant intensity with and without a small obscuration at the test plane. From these measurements one can measure simultaneously the field’s coherence at all pairs of points whose centroid is the centroid of the obstacle. This method can be implemented without the need of any refractive or diffractive focusing elements. © 2014 Optical Society of America OCIS codes: (030.1640) Coherence; (050.1940) Diffraction; (050.5080) Phase shift; (070.2580) Paraxial wave optics. http://dx.doi.org/10.1364/OL.39.004927
Insights gained from new optical measurement techniques have often paved the way for broader applications in physics. This is especially true of coherence phenomena where fields must be characterized in four spatial dimensions even for quasi-mononchromatic fields. Fast and accurate ways for measuring spatial coherence are also of great practical importance, e.g., for source characterization in medical imaging or photolithography, as well as in the understanding of novel imaging systems. The canonical experiment for measuring spatial coherence is Young’s two-pinhole setup [1,2]. By placing the pinholes at two test points and observing the position and visibility of the resulting fringes, the complex degree of coherence at these two points is estimated. Variants of this approach use two separate copies of the wavefront to remove the dependence of the fringe period on the pinhole separation [3,4]. However, the full characterization of a field would require scanning each pinhole independently over a transverse plane, something that takes a significant amount of time and effort. Several approaches have been proposed to alleviate this problem by giving simultaneous access to many pairs of points. For example, arrays of pinholes with nonredundant separations allow simultaneous measurements for point pairs corresponding to any two such pinholes [5]. More elaborate interferometric techniques are based on the superposition of two copies of the wavefront that are mutually displaced, reversed, or rotated [6–11]. When projected onto a CCD detector, their interference combined with knowledge of the intensity distribution of each copy gives access to the coherence at a dense sample of pairs of points that either have fixed separation (for wavefront shearing) or are symmetrically distributed around a pivot point or line (for wavefront rotating or reversing). By mutually displacing the wavefronts over two degrees of freedom, a sufficiently dense sample of the full four-dimensional (4D) coherence function is measured. That is, rather than scanning over four dimensions, one only scans over two, as the two-dimensional signal at the CCD provides information over a range of values of the other two. While successful in many cases, these interferometric methods are sensitive to misalignments and vibrations, and are difficult or impossible to implement in some spectral regions. 0146-9592/14/164927-04$15.00/0
A different approach, phase-space tomography [12,13], is based on measuring the intensity distribution at a range of propagation distances. However, this approach only works if the field has a known symmetry (translational or rotational) or separability, so that the coherence function depends only on three or less degrees of freedom. Otherwise, the three-dimensional intensity information is not sufficient to recover the 4D coherence function. A fourth parameter can be introduced in the form of a combination of focal lengths for anamorphic focusing elements (cylindrical lenses or spatial light modulators). In fact, the CCD’s position can be fixed, and the two scanning parameters can be the focal lengths of the anamorphic elements [14,15], but this again means that the field must be manipulated through refraction or diffraction, which is challenging for some frequencies. Other methods [16,17] based on free propagation require some prior knowledge of the coherence properties. Here we present a very simple technique that can, in principle, be implemented without refractive or diffractive focusing elements, and where coherence is measured by scanning a small obstacle over the test plane and then measuring the resulting radiant intensity. For simplicity we introduce the basic ideas for fields that depend only on one transverse coordinate, x, and then generalize them to fields depending on two, x, y. Consider a stationary, quasi-monochromatic, partially coherent scalar field Ux, whose second-order coherence properties at the test plane z 0 are described by the mutual intensity, Jx1 ; x2 hU x1 Ux2 i, where the angle brackets represent a correlation. In the paraxial approximation, the radiant intensity (proportional to the intensity at the far field) is given by ZZ k Jx1 ; x2 exp−ikx2 − x1 pdx1 dx2 2π ZZ k ¯ x; ¯ x0 exp−ikx0 pdx0 dx; ¯ J 2π
Ip
(1)
where k is the central wavenumber, p is a directional variable (the sine of the angle of observation with respect to the z axis), and in the second step we changed to cent¯ x; ¯ x0 Jx¯ − x0 ∕ roid and difference coordinates with J © 2014 Optical Society of America
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OPTICS LETTERS / Vol. 39, No. 16 / August 15, 2014
2; x¯ x0 ∕2. Ip can be measured either in the far field or by means of a Fourier-transforming lens (as in our experimental implementation). Let a mask now be inserted at z 0, whose amplitude function is given by Ax − x0 , where x0 is a controllable lateral displacement. The resulting radiant intensity is given by
ZZ
x0 x0 A τ τ− 2 2
k A 2π ¯ 0 τ; x0 exp−ikx0 pdx0 dτ; × Jx
I A p; x0
(2)
with τ x¯ − x0 . As we now discuss, different choices for masks A lead to different measurement methods. 1. Standard methods using apertures. Some coherence measurement methods [18,19] use a localized window for Ax (sharp or apodized) to isolate the coherence properties in different locations (measuring the so-called spectrogram or Husimi function). Shack– Hartmann sensors and plenoptic cameras can be considered as special cases of this approach, where the window is the aperture of each lenslet [20]. However, recovering J¯ from I A requires a deconvolution process that is problematic for point separations beyond the window’s extent. To see this, let us expand J¯ in a Taylor series around x0 (the window’s centroid): ¯ 0 τ; x0 Jx
∞ n n¯ X τ ∂ J n0
n! ∂x¯ n
x0 ; x0 :
(3)
The substitution of this relation into Eq. (2) gives IA
∞ Z X
k 2π n0
∂n J¯ x ; x0 An x0 exp−ikx0 pdx0 ; ∂x¯ n 0
(4)
An x
Z
τn x0 x0 A τ− A τ dτ: n! 2 2
(5)
If A is a real symmetric function then An x0 0 for odd n. If the series in Eq. (4) can be well approximated by its first term (n 0), then an estimate of J¯ can be found through inverse Fourier transformation as ¯ 0 ; x0 A0 x0 ≈ Jx
Z
I A p; x0 expikx0 pdp:
−4
(b)
−2
2
(6)
Notice that A0 is the autocorrelation of the aperture function A. If A is chosen as an aperture function with limited extent, then A0 is (significantly) different from zero only over a region slightly larger than the width of A, as shown in Fig. 1(a) for rectangular and Gaussian windows. Therefore, Eq. (6) shows that one cannot recover the coherence for pairs of points whose separation jx0 j is larger than the width of A. This is a fundamental limitation of methods using apertures. 2. Method using binary phase masks. The approach in [21] overcomes this problem by using instead the difference of two measurements, with and without the mask, i.e., by considering the difference of Eqs. (1) and (2):
4 x/w −4
(c) −2
2
x/w 4 −4
−2
2
x/w 4
Fig. 1. (a) Autocorrelation A0 for window functions with rectangular shape Ax rectx∕w (black) and Gaussian shape ¯ 0 x0 for (b) a Ax exp−πx2 ∕w2 (gray). (b), (c) Function A binary phase mask Ax sgnx and (c) a transparent mask with a localized obscuration of rectangular shape ax rectx∕w (black) or Gaussian shape ax exp−πx2 ∕w2 (gray).
Δp; x0 Ip − I A p; x0 ∞ Z n¯ k X ∂ J ¯ n x0 exp−ikx0 pdx0 ; x ; x0 A 2π n0 ∂x¯ n 0 (7) where ¯ n x0 A
Z
τn x0 x0 1−A τ− A τ dτ: n! 2 2
(8)
The inverse Fourier transform of Eq. (7) gives ∞ n¯ X ∂ J
0 ¯ 0 n x0 ; x An x ¯ ∂ x n0
Z
Δp; x0 expikx0 pdp:
(9)
Instead of an aperture, the mask in [21] is a transparent binary phase mask with a single discontinuity, which in the optimal case is given by Ax sgnx, for which ¯ n x0 1 A n!
where 0
(a)
Z
jx0 j∕2 −jx0 j∕2
τn dτ 1 −1n
jx0 jn1 : n 1!
(10)
¯ n 0 for odd n. Again, if the series in Eq. (9) is That is, A approximated by its leading term, we get ¯ 0 x0 ≈ ¯ 0 ; x0 A Jx
Z
Δp; x0 expikx0 pdp:
(11)
Note that unlike the case in Eq. (6), there is no upper bound for the point separation jx0 j for which we can re¯ 0 x0 2jx0 j [shown in cover the coherence since A Fig. 1(b)] grows linearly. There are however problems with recovering J¯ for small jx0 j, which in [21] were addressed through interpolation. Also problematic is the extension of this approach to two dimensions [22]. 3. Method using small obstacles. Such problems are completely bypassed if instead of an aperture or a phase object, the mask is transparent and uniform except for a localized obscuration, i.e., Ax 1 − ax where 0 ≤ ax ≤ 1 is a real distribution localized to a region near x 0. In particular, we can get from Eq. (11) with x0 ¯ denoted as J¯ 0 , given by x¯ a basic estimate of J, ¯ x; ¯ x0 ≈ J¯ 0 x; ¯ x0 J
R
¯ expikx0 pdp Δp; x ; ¯ 0 x0 A
(12)
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
where from Eq. (8) for n 0 and Ax 1 − ax, ¯ 0 x0 2 A
Z
Z axdx −
x0 x0 a τ dτ; a τ− 2 2
J¯ 0 ¯x; x0
N X ∂n J¯ N−1 n1
¯n
∂x
¯ x0 A ¯ x0 n 0 : x; ¯ 0 x A
(18)
and further corrections can be calculated as
(14)
Note that if ax a−x, then all terms with odd n ¯ n 0. In particular, disappear since A ¯ x0 ∂2 J¯ A ¯ x0 J¯ 0 x; ¯ x0 − 20 x; ¯ x0 2 0 : J¯ 2 x; ¯ 0 x ∂x¯ A
Δp; x¯ expikx0 · pdpdq ; ¯ 0;0 x0 A
(13)
namely a constant minus the obstacle’s autocorrelation. That is, the difference of the measurements with and without the obstacle leads to an estimate of the coherence between all pairs of points symmetrically distributed around the obstacle’s centroid. The main advantage of this approach over the previous two is that the function we must ¯ 0 , never vanishes. In fact, it goes to a constant divide by, A for jx0 j larger than the obstacle’s width, and it is never smaller than half this constant, as shown in Fig. 1(c) for rectangular and Gaussian obscurations. To measure coherence over many pairs of points with many centroids, one must scan the obstacle across the field (i.e., vary x0 ). Further, if the sampling in x0 is sufficiently fine, one can find higher-order estimates J¯ N by using iteratively truncated versions of Eq. (9): ¯ x0 J¯ 0 x; ¯ x0 − J¯ N x;
RR
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(15)
J¯ N ¯x; x0 J¯ 0 ¯x; x0 −
N X n 0 ¯ 0 X 0 ∂n J¯ N−1 0 An ;n−n x : ¯ x ; x 0 0 ¯ 0;0 x0 ¯ n ¯ n−n A n1 n0 0 ∂x ∂y
(19) As in the one-dimensional case, the leading correction is negligible if both the width of a and the local coherence width are much smaller than the local length scale of variation of J¯ with respect to the centroid coordinates. Our experimental implementation is shown in Fig. 2(a). First, the partially coherent field to be measured is prepared by focusing a green laser (λ 532 nm) onto a rotating ground-glass diffuser followed by two lenses that collimate it. The measurement system consists of a transmissive liquid-crystal spatial light modulator (SLM) placed at the test plane and surrounded by suitably oriented polarizers, a converging lens (f 340 mm) and a CCD detector. The lens is used in Fourier-transforming configuration, with the SLM and CCD at its focal planes, so that the intensity measured at the CCD is proportional to the radiant intensity at the test plane. The obscurations are written onto the SLM so that their size, shape, and location can be controlled at will. In Fig. 2 we show the basic estimate J¯ 0 that results from using obscurations of 2(b)
Suppose the local characteristic length scale of variation ¯ x; ¯ x; ¯ x; ¯ x0 in x¯ is R, so that ∂2 J ¯ x0 ∕∂x¯ 2 ∼ J ¯ x0 ∕R2 . of J 0 0 2 02 ¯ ¯ Then, since A2 x ∕A0 x w 3x ∕24 for jx0 j ≥ w (and something not too different for jx0 j < w), the second-derivative correction in Eq. (15) is insignificant if R is significantly greater than both w and the local coherence width. That is, the obstacle’s size is not constrained by the local coherence width but by how spatially inhomogeneous the field is. This approach is easily generalized to two transverse dimensions. The difference of two radiant intensity measurements, with and without an obscuration at z 0 given by ax, where x x; y can be written as Δp; x0 Ip − I A p; x0 ∞ ZZ k2 X ∂nm J¯ ¯ n;m x0 2 x ; x0 A ∂x¯ n ∂y¯ m 0 4π n;m0 × exp−ikx0 · pdx0 dy0 ;
(16)
where p p; q are the directional/Fourier-conjugate variables and τn ηm x0 x0 a τ− a τ n!m! 2 2 0 0 x x a τ dτdη; −a τ − 2 2
¯ n;m x0 A
Z
with τ τ; η. The basic estimate is
(17)
Fig. 2. (a) Diagram of the optical set-up. (b)–(g) Real parts of J¯ 0 0; x0 ∕J¯ 0 0; 0 obtained by using square obscurations of side (b) 1.855 mm and (d), (f), and (g) 3.81 mm, and circular obscurations of diameter (d) 1.855 mm and (e) 3.81 mm. In (b)–(e) the obscuration was at the center of the SLM, while in (f)/(g) it was shifted to the left/right by 2.93 mm. The insets in (b)–(g) show the measured intensities without and with the obscuration.
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the measurements can be performed in the far field, so no lens is needed. Further, the method involves very little numerical processing that can be performed in real time during the data collection. Here, a Taylor expansion of J¯ was used, leading to a simple estimate and an iterative formula for corrections. However, a rigorous deconvolution in four (phase space) dimensions is also possible, as will be discussed elsewhere. ¯ x0 ; y0 ∕J¯ 0 0; y; ¯ 0; 0j, for (a) x0 0 and Fig. 3. Plots of jJ¯ 0 0; y; 0 (b) y 0. The curves in the insets show three slices corresponding to y¯ −3.71 (black), 0 (gray), and 3.71 mm (pale gray), respectively.
and 2(d) square and 2(c) and 2(e) circular shapes, whose sides/diameters are 2(b) and 2(d) 1.855 mm and 2(d), and 2 (e) 3.81 mm, all at the SLM’s center. In all cases, the measured mutual intensity as a function of point separation is approximately a rotationally symmetric real Gaussian (the imaginary part is about two orders of magnitude smaller) with half-width-half-maximum of about 30 μm. The differences between these measurements are within a few percent of their peak (normalized to unity), even though different shapes and sizes were used. It is clear from Figs 2(b) and 2(c) that noise affects the results more for smaller obscurations, as the measured distributions (shown in the insets) being subtracted are more similar. Parts 2(f) and 2(g) of Fig. 2 correspond to square obstacles of side 3.81 mm displaced 2.93 mm to the left 2(f) and right 2(g) of the center of the SLM. These figures show that in ¯ with centroid insignificant, the this case the variation of J, corrections in Eq. (19) are negligible and the basic estimate J¯ 0 is sufficient. The previous example, an approximately translationally invariant field with Gaussian statistics, shows the method’s robustness and repeatability. However, the power of approaches like this one is in the ability to characterize translationally variant fields. To show this we disrupt the translation invariance and isotropy of the field’s coherence by introducing a horizontal knife edge a few millimeters along the axis away from the intermediate image of the rotating diffuser. The resulting measure¯ using as an obstacle a square of side 3.8 mm ments for J, in the SLM, are shown in Fig. 3 for y¯ varying from −4.1 to 4.1 mm in steps of 0.195 mm. Vignetting from the knife ¯ edge makes the coherence vary with y. To summarize, we proposed a simple method to measure simultaneously the spatial coherence between a large number of point pairs by studying an obstacle’s shadow/penumbra. In implementations, say, with x rays,
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