1. INTRODUCTION

The concept of optical coherence was formulated precisely in terms of the second-order correlation function by Zernike [1] and subsequently refined by Wolf [2,3]. The first experimental verifications of their theoretical predictions were made by Thompson and Wolf [4], who demonstrated elegantly the effect of partial coherence on two-beam interference best known as Young’s experiment. Since then, much progress, largely due to the efforts of Emil Wolf, has been made toward establishing optical coherence theory [5–7]. It is now widely known that optical coherence, especially the state of spatial coherence, is an indispensable element in a wide range of applications relating to optical sciences and technologies [8]. Thus, characterizing the spatial coherence of light has been of great importance to understand and control these implementations.

Many methods of measuring the spatial coherence of light have so far been proposed. Most of them are based on two-beam interference, following the celebrated Thompson and Wolf experiment [4] and its modern electronic counterpart [9]. The representative examples include the method with a (modified) wavefront-folding interferometer [10–13], a grating interferometer [14], a pair of nanoparticles [15,16], a non-parallel double slit [17], and a digital micromirror device (DMD) [18–20]. Optical configurations beyond simple two-beam interference have also been put forward, such as the spatial coherence measurement with a discontinuous phase mask [21,22], a kind of intensity interferometry similar to the Hanbury Brown–Twiss experiment [23,24], and self-referencing holography [25]. These methods generally need somewhat complicated procedures in data acquisition and analysis, depending on their principles. It should be noted, however, that multiple-beam interference using either a uniformly redundant array of apertures [26–29] or a nonredundant array of apertures [30–34] gives rise to rather beneficial features in spatial coherence measurement. Specifically, the underlying principle is basically the same as the well-established Young’s two-beam experiment, but the performance is greatly improved in the sense that the number of experiments to fully characterize the spatial coherence may be significantly reduced as compared to the conventional Young’s method.

In this study, we propose to utilize a temporally modulated nonredundant array of slits in the spatial coherence measurement, in accordance with the previous study [20], where a double slit is modulated temporally with a DMD. A nonredundant slit array consists of multiple slits arranged such that separations between any pair of slits are different from each other, i.e., each slit separation occurs only once. Accordingly, it is found that the interference fringe pattern produced by a nonredundant slit array is a superposition of the interference fringe patterns resulting from multiple Young’s experiments with different slit separations. Each interference fringe pattern associated with a single Young’s experiment can, thus, be restored on the basis of the uniqueness of slit separations. The temporal modulation applied to the slit array is expected to yield immunity to background light in the experiment. These features enable one to achieve fast and reliable spatial coherence measurement without any cumbersome experiment and analysis.

This paper is organized as follows. In Section 2, we describe the general theory of the spatial coherence measurement with a (static) nonredundant array of apertures including slits in some detail and then discuss the effect of temporal modulation applied to the aperture array. Section 3 deals with the setup for the proof-of-principle experiment and specifies all the steps necessary to the experiment and the subsequent analysis. In Section 4, we show a number of experimental results to validate the proposed method of measuring the spatial coherence of light. Finally, we briefly summarize the present study in Section 5.

2. PRINCIPLE

A. Spatial Coherence Measurement with a Nonredundant Array of Apertures

Let us first consider a statistically stationary optical field of any state of coherence, propagating along the $z$ axis, as illustrated in Fig. 1. The incident field at a point $\boldsymbol{\rho} ^\prime = (x^\prime ,y^\prime)$ in a plane ${\cal A}$ may be represented by an ensemble of space–frequency realizations ${U_I}(\boldsymbol{\rho} ^\prime,\omega)$ with $\omega$ being the angular frequency ([6], Section 4.7; [7], Chap. 4). The second-order coherence properties of the field are characterized by the cross-spectral density defined by the following expression:

 

Fig. 1. Notation relating to spatial coherence measurement with a nonredundant slit array.

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The purpose of this study is to determine the modulus $|{\mu _I}({\boldsymbol{\rho} ^\prime_1},{\boldsymbol{\rho} ^\prime_2},\omega)|$ of the spectral degree of coherence experimentally. To this end, we set a nonredundant array of slits in the plane ${\cal A}$ and observe the resultant interference fringe pattern in a plane ${\cal B}$ placed some distance from the plane ${\cal A}$. The nonredundant array of slits is constructed in such a way that equivalent slits are spaced according to a so-called Golomb ruler [35]. Consequently, separations between all possible pairs of slits are necessarily distinct, so that each slit separation is unique. Although we employ a one-dimensional array of slits in the present method, the following analysis is based on a two-dimensional formalism to cover a two-dimensional array of (circular) apertures for generality. In a way similar to the one-dimensional case, the two-dimensional nonredundant array of apertures may be generated with the help of a so-called Costas array [36]. However, to date only a limited number of studies of spatial coherence measurement have been performed using two-dimensional arrays of apertures [31,32], whether nonredundant or uniformly redundant.

Let $T(\boldsymbol{\rho} ^\prime)$ represent the transmission function of the array of circular apertures located in the plane ${\cal A}$. It is written in the following form:

In the previous studies dealing with the spatial coherence measurement with a nonredundant array of apertures including slits [30–34], the resultant interference fringe pattern is observed in the far field of the aperture array. However, this condition may not be easily met in some practical situations. Accordingly, in the treatment to follow, we slightly relax the condition and permit the placement of the observation plane ${\cal B}$ closer to the plane ${\cal A}$. The spectral density describing the interference fringe pattern at a point ${\boldsymbol \rho} = (x,y)$ in the plane ${\cal B}$ is then expressible, using the paraxial propagation equation for the cross-spectral density ([6], Section 5.6.3), in the following form:

To proceed, we suppose that each aperture is small enough to ensure that the spectral density and the spectral degree of coherence of the field across it are uniform, as in all the related studies performed previously [30–34]. This approximation is commonly employed in the formulation of Young’s experiment, except for a few studies [37–39]. The cross-spectral density ${W_I}({{\boldsymbol \rho ^{\prime \prime}_1}} + {{\boldsymbol q}_l},{{\boldsymbol \rho ^{\prime \prime}_2}} + {{\boldsymbol q}_m},\omega)$ in the integrand of Eq. (8) may reasonably be replaced by ${W_I}({{\boldsymbol q}_l},{{\boldsymbol q}_m},\omega)$ in this case. We further suppose that, due to the small aperture size, the observation plane ${\cal B}$ is located in the far field of a single aperture (rather than the whole array of apertures). Then, the quadratic phase term with respect to ${{\boldsymbol \rho ^{\prime \prime}_1}}$ and ${{\boldsymbol \rho ^{\prime \prime}_2}}$ in the integrand of Eq. (8) may be removed to a good approximation. As a consequence, after interchanging the order of the integral and the summation, Eq. (8) reduces to the following expression:

Let us now consider a nonredundant array of apertures, where the separations between all possible pairs of apertures are distinct. The interference fringe pattern produced by such an array is a superposition of different interference fringe patterns, each of which has a unique spatial frequency determined by the aperture separation. These elemental interference fringe patterns have different information about the spatial coherence of light, depending on the aperture separation. Therefore, we perform a Fourier analysis to retrieve this information about the spatial coherence from the resultant interference fringe pattern given by Eq. (11).

For the purpose of the subsequent analysis, we number each pair of apertures so as to arrange the separations in ascending order. Specifically, we define the difference for the $j$th pair of apertures as ${{\boldsymbol d}_j} = {{\boldsymbol q}_{{m_j}}} - {{\boldsymbol q}_{{l_j}}}$ under the condition of ${l_j} \lt {m_j}$ and assume that $|{{\boldsymbol d}_1}| \lt |{{\boldsymbol d}_2}| \lt \cdots \lt |{{\boldsymbol d}_M}|$, where $M = N(N - 1)/2$ is the maximum number of $j$ giving a unique separation. On taking the Fourier transform of Eq. (11) using the following formula (see Fig. 1):

According to Eqs. (15) and (16), and recalling that the aperture size is assumed to be small enough, one may readily expect that $C(\textbf{0})$ is almost the same as ${C_j}(\textbf{0})$ and, thus, the approximation $C(\textbf{0}) \simeq {C_j}(\textbf{0})$ holds. This relation becomes exact when the plane ${\cal B}$ is located in the far field of the whole array of apertures for the reason mentioned above. If we suppose that each aperture in the array is illuminated evenly, the spectral densities at all the apertures are the same, i.e., ${S_I}({{\boldsymbol q}_{{l_j}}},\omega) = {S_I}({{\boldsymbol q}_{{m_j}}},\omega)$. On combining Eqs. (17) and (18) under this condition, we find

To deal with a more general case of uneven illumination, we next consider individual apertures in the array. When only a single aperture is located at ${{\boldsymbol q}_l}$ in the plane ${\cal A}$, the resultant spectral density produced in the plane ${\cal B}$ is given, according to Eq. (11), by the following expression:

B. Temporal Modulation of the Aperture Array

In the spatial coherence measurement described in the previous section, one needs to obtain experimentally the spectral densities produced by the array of apertures and each aperture, which are given theoretically by Eqs. (11) and (20), respectively. To do this, we consider that the fields across the apertures in the plane ${\cal A}$ are modulated temporally. It has been shown recently that this can be conveniently achieved using a DMD [20]. DMDs are reflective pixelated spatial light modulators that perform binary amplitude modulation. Thus, each aperture displayed on the DMD acts as a modulator that modulates the reflected field temporally. Although DMDs were originally developed for projector applications, many new applications have recently been proposed [40,41].

Let $m(t)$ be the modulation function. We assume that the modulation function is given by a square wave of unit amplitude to model the behavior of a DMD. We also assume that the fields across all the apertures are modulated at the same frequency ${f_m}$. The frame rate of current DMDs is at most 20 kHz, which is much smaller than the spectral width of typical light sources. Under this condition, the modulation due to the DMD is expected to have no appreciable effect on the spectrum of light. Consequently, the cross-spectral density of the field across the apertures after modulation may be written, to a good approximation, in the following form [20]:

We now consider a practical experimental condition that background light is not necessarily avoidable. The spectral density in this case may be written, instead of Eq. (30), in a more general form,

Most methods of spatial coherence measurement including the method described in the previous section are sensitive to background light; thus, all the experiments relating to light detection must be performed in a well-conditioned dark room. Otherwise, subtraction of the background light is indispensable in data processing. However, the present method of detection is immune to background light; thus, its suppression or subtraction is not necessary. To conclude, it is expected that, combining the spatial coherence measurement detailed in the previous section with the modulation technique for light detection outlined in the present section, one can achieve fast and reliable spatial coherence measurement without any difficulties.

 

Fig. 2. (a) Setup for a proof-of-principle experiment. (b) Schematic diagram of micromirror arrangement forming, e.g., a double slit on the DMD. (c) Nonredundant array of six slits on the DMD employed in the present experiment. DMD, digital micromirror device; MM fiber, multimode optical fiber; CL, cylindrical lens; LS CCD, line scan CCD camera.

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3. SETUP AND METHOD FOR A PROOF-OF-PRINCIPLE EXPERIMENT

Figure 2(a) shows a setup for a proof-of-principle experiment. Light to be characterized originates from an LED (Thorlabs, M625F2), which has a Gaussian-like spectrum centered at ${\lambda _0} = 625\;{\rm nm} $ with FWHM bandwidth $\Delta \lambda = 15\;{\rm nm} $. The light of this kind is effectively quasi-monochromatic. Under this and other general assumptions, the spectral degree of coherence at center frequency ${\omega _0}$ is shown to be equivalent to the complex degree of coherence,

The LED is connected to a collimator (Thorlabs, F810FC-635) via a step-index multimode optical fiber. The collimator consists of an air-spaced doublet with effective focal length $f = 35.4\;{\rm mm} $. Since the end of a step-index multimode fiber is considered as a circular incoherent source with a uniform intensity distribution [43], the modulus of the complex degree of coherence of the light from the collimator is given, according to the van Cittert–Zernike theorem (e.g., [7], Section 3.2), by the following expression:

Light from the collimator is normally incident on the DMD (ViALUX, V-Module V-7001). The DMD is a two-dimensional array of tiltable square micromirrors with one side 13.68 µm and diagonal 19.35 µm. Each micromirror has two states, called on- and off-state, depending on the tilt angle, where the rotation axis for tilting is along its diagonal. The present setup is arranged so that the incident light reflected by on-state micromirrors is directed to a line scan CCD camera, while the light reflected by off-state ones escapes out of the camera. A schematic diagram of the micromirror arrangement forming, e.g., a double slit on the DMD is depicted in Fig. 2(b). The slit width is roughly equivalent to the diagonal of each micromirror, i.e., 19.35 µm. A cylindrical lens of focal length 50 mm is placed before the camera with a view to enhance the intensity of the interference fringe pattern to be captured by the camera. It should be stressed that a static double slit displayed on a similar DMD has so far been proven to be effective in various spatial coherence measurements [14,18,19,44,45].

To validate the method of spatial coherence measurement proposed in this study, we employ a nonredundant array of six slits displayed on the DMD, as illustrated in Fig. 2(c). These slits are spaced in a manner proportional to one of the six-mark Golomb rulers characterized by the set (0, 1, 8, 12, 14, 17) [35] so that separations between the slit numbers 1–2, 2–3, 3–4, 4–5, and 5–6 are given by 38.7 µm (2 pixels), 270.9 µm (14 pixels), 154.8 µm (8 pixels), 77.4 µm (4 pixels), and 116.1 µm (6 pixels), respectively. The nonredundant slit array of this kind was successfully employed to measure the spatial coherence of synchrotron radiation in the previous study [34], where another set of the ruler, i.e., (0, 1, 4, 10, 15, 17), was adopted. Referring to the discussion in Section 2.A, we define the difference for the $j$th pair of slits as ${{\boldsymbol d}_j} = {{\boldsymbol q}_{{m_j}}} - {{\boldsymbol q}_{{l_j}}}$ under the condition of ${l_j} \lt {m_j}$. Since $N = 6$, the number $M = N(N - 1)/2$ of distinct slit separations $|{{\boldsymbol d}_j}|$ is 15. The index $j$, the slit numbers ${l_j}$ and ${m_j}$, and the corresponding separation $|{{\boldsymbol d}_j}| = |{{\boldsymbol q}_{{m_j}}} - {{\boldsymbol q}_{{l_j}}}\!|$ are listed in Table 1.

Table 1. Index $j$, the Slit Numbers ${l_j}$ and ${m_j}$, and the Corresponding Separation $|{{\boldsymbol d}_j}| = |{{\boldsymbol q}_{{m_j}}} - {{\boldsymbol q}_{{l_j}}}|$ under the Condition of ${l_j} \lt {m_j}$ for All 15 Distinct Separations of the Nonredundant Slit Array Employed in the Present Experiment

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The nonredundant array of slits displayed on the DMD is modulated temporally by changing the on/off state of micromirrors alternatingly at ${f_m} = 1\;{\rm kHz} $. This can easily be achieved by using a control software provided by the manufacturer of the DMD module. The resultant interference fringe patterns are captured by a line scan CCD camera (e2v, AVIIVA EM2) in synchronization with the modulation, with the help of a commercial frame grabber and the accompanying software (EPIX, PIXCI-E4, and XCAP-Ltd). The number of frames captured in a single series is 2048, where each frame has a one-dimensional 2048 pixel image.

To retrieve the original interference fringe pattern from the modulated signal, we perform a fast Fourier transform (FFT) on the series of the one-dimensional images with respect to the frame number for each image position ${\boldsymbol \rho}$ [20]. On selecting the resultant Fourier spectrum at the modulation frequency ${f_m}$, we obtain the original interference fringe pattern without background light, even though the background light is captured unexpectedly by the camera. Similarly, we can also obtain the spectral densities produced by each slit if the array of slits is replaced by one of the slits in the array. By incorporating these experimental data obtained in this way into the principle of the spatial coherence measurement described in Section 2.A [specifically, Eq. (19) or (28), depending on the experimental condition], one can determine the modulus of the spectral degree of coherence (or, equivalently, the complex degree of coherence under the present quasi-monochromatic condition) for 15 different separations simultaneously.

4. EXPERIMENTAL RESULTS

To examine the basic features of the experimental setup, we first connect a He–Ne laser to the collimator via a single-mode optical fiber and illuminate the DMD coherently. By measuring the fringe spacing of the interference fringe pattern produced by a double slit displayed on the DMD, we estimated the distance $z$ between the DMD and the camera to be 197.4 mm. Recalling the far-field condition ${b^2}/(\lambda z) \ll 1$, with $b$ representing the radius of the largest extent of the object [46], we find that the distance $z$ in the present setup fully satisfies the far-field condition with respect to a single slit ($2b = 19.35 \;{\unicode{x00B5}{\rm m}}$) but may possibly violate that condition with respect to the whole slit array [$2b \simeq 678 \;{\unicode{x00B5}{\rm m}}$, see Fig. 2(c)].

Figure 3 shows an interference fringe pattern produced by the slit array displayed on the DMD illuminated coherently and its (spatial) Fourier spectrum computed by means of an FFT. The interference fringe pattern was captured conventionally by the camera without the use of the modulation technique for detection. Each pixel on the line scan CCD camera has a width of 14 µm, so that the actual range of detection in Fig. 3(a) [and also Figs. 4(a) and 5(a)–5(f)] is ${\sim}28.7\;{\rm mm} $. In the Fourier spectrum illustrated in Fig. 3(b), one large peak at the zero frequency and other 15 peaks at different frequencies are clearly isolated. This result obtained by experiment is entirely consistent with the preceding discussion on the behavior of Eq. (14). Therefore, it is assured that the peak magnitudes ${b_0}$ and ${b_j}$ in the Fourier spectrum of the interference fringe pattern at the zero frequency and other frequencies are given fairly accurately by Eqs. (17) and (18), respectively. The spatial frequencies ${\nu _j}$, ($j = 1,2, \cdots ,15$), at which those 15 peaks appear, are determined by the separations between all possible pairs of slits in the array, and these values ${\nu _j}$ are necessary in the subsequent analysis.

 

Fig. 3. (a) Interference fringe pattern produced by the slit array illuminated coherently and (b) modulus of its spatial Fourier spectrum computed by means of a fast Fourier transform (FFT). These curves are normalized by their maximum values.

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Fig. 4. (a) Interference fringe pattern produced by the slit array and (b) modulus of its spatial Fourier spectrum computed by means of an FFT, where the slit array is illuminated by spatially partially coherent light from the collimator. The collimator is coupled with the LED via a step-index multimode fiber with nominal core diameter 105 µm. These curves are normalized by their maximum values.

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Fig. 5. Intensity distributions produced solely by (a) slit #1, (b) slit #2, (c) slit #3, (d) slit #4, (e) slit #5, and (f) slit #6. Other experimental conditions are the same as in Fig. 4. These curves are normalized by the maximum value of the interference fringe pattern produced by the slit array illustrated in Fig. 4(a).

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Let us now turn our attention to the main problem of the spatial coherence measurement on the basis of the setup and the method described in Section 3. We first characterize the spatial coherence of the light from the collimator coupled to the LED via a step-index multimode fiber of nominal core diameter 105 µm. This experiment was performed in a dark room, although we employed the modulation technique for light detection. As some typical examples, the interference fringe pattern produced by the slit array and its Fourier spectrum computed by using an FFT are illustrated in Fig. 4. Furthermore, the intensity distributions produced by each slit in the array are shown in Fig. 5. All the intensity distributions are seen to be almost the same in Fig. 5, but a closer look at these curves reveals that the peak positions are slightly different from each other, depending on the location of the slit. This is evidently due to the fact that the camera is not accurately located in the far field of the whole slit array, as alluded above. The theory formulated in Section 2.A allows the camera to be located closer to the slit array, but we note that the main expressions [Eqs. (19) and (28)] (for determining the modulus of the spectral degree of coherence) derived are valid under the conditions of $C(\textbf{0}) \simeq {C_j}(\textbf{0})$ and ${C^\prime _j}(\textbf{0}) \simeq {C_j}(\textbf{0})$, respectively [see Eqs. (15), (16), and (26)].

The magnitudes of each peak at ${\nu _j}$ in Fig. 4(b) determine the values of ${b_0}$ and ${b_j}$ given by Eqs. (17) and (18). On the other hand, each intensity distribution in Fig. 5 is characterized by Eq. (20) and, hence, determines the values of ${b^\prime _0}$ and ${b^\prime _j}$ given by Eqs. (23) and (27). These values (i.e., ${b_0}$, ${b_j}$, ${b^\prime _0}$, and ${b^\prime _j}$ for $j = 1,2, \cdots ,15$) are all derived from experimental data obtained by seven measurements (one for the interference fringe pattern produced by the slit array and six for the intensity distributions produced by each slit). Using these values of ${b_0}$, ${b_j}$, ${b^\prime _0}$, and ${b^\prime _j}$, we can evaluate at once the modulus of the spectral degree of coherence for 15 different separations in accordance with Eq. (28). Provided that each slit is illuminated evenly, we can characterize the spatial coherence more easily using the values of ${b_0}$ and ${b_j}$ (which are obtainable by a single measurement) in accordance with Eq. (19). These results are plotted in Fig. 6(a). In this figure, the solid curve indicates theoretical values calculated using Eq. (35) under the condition of $a = 110 \;{\unicode{x00B5}{\rm m}}$, which may be the actual value for the core diameter of this fiber [20]. In a similar way, we also perform the spatial coherence measurement using a multimode fiber with nominal core diameter 200 µm, in place of the nominal 105 µm core fiber. The results are summarized in Fig. 6(b), where the solid curve indicates the theoretical results based on Eq. (35) under the condition of $a = 210 \;{\unicode{x00B5}{\rm m}}$, which is the probable value for the actual core diameter. Figure 6 clearly shows that all the experimental results agree well with the theoretical values calculated according to the van Cittert–Zernike theorem. Furthermore, we find that the experimental results derived on the basis of Eqs. (19) and (28) are almost the same. This feature demonstrates that all the six slits in the array are illuminated evenly in the present setup. Accordingly, in the following analysis, we characterize the spatial coherence only on the basis of Eq. (19) for simplicity.

 

Fig. 6. Modulus of the spectral degree of coherence of light from the collimator coupled with the LED via step-index multimode fibers with nominal core diameter (a) 105 µm and (b) 200 µm. The solid curves indicate the theoretical results calculated according to the van Cittert–Zernike theorem, i.e., Eq. (35), under the condition of $a = 110\,\,\unicode{x00B5}{\rm m}$ in (a) and 210 µm in (b). Circles and triangles indicate the experimental results evaluated on the basis of Eqs. (28) and (19), respectively. These experiments are performed in a dark room.

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We next conduct the same experiments under background light. This environment was realized by turning on a halogen lamp suspended from the ceiling. For comparison, we also perform the spatial coherence measurement in a conventional fashion, i.e., by means of the static slit array, rather than the temporally modulated slit array. The results are summarized in Figs. 7 and 8 for the nominal 105 µm and 200 µm core fibers, respectively. To enhance the small variations in each figure, the results plotted on a linear scale in Figs. 7(a) and 8(a) are replotted on a logarithmic scale in Figs. 7(b) and 8(b), respectively. The error bars represent the standard deviations of the 10 experimental values. It is obvious from these figures that the experimental results obtained with the modulation technique for detection are in excellent agreement with the theoretical values even when background light is not negligible, whereas those obtained without the modulation technique are much lower than the theoretical values. Such deviations are, of course, due to the effect of background light.

 

Fig. 7. Modulus of the spectral degree of coherence of light from the collimator coupled with the LED via a step-index multimode fiber with nominal core diameter 105 µm, plotted on (a) linear and (b) logarithmic scales. The solid curves indicate the theoretical results Eq. (35) under the condition of $a = 110\,\,\unicode{x00B5}{\rm m}$. The experimental results obtained by means of the modulation technique for detection and those obtained in a conventional way (i.e., without the modulation technique) are indicated by circles and rectangles, respectively. The error bars in (b) represent the standard deviations of the 10 experimental values. These experiments are performed under background light.

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Fig. 8. Modulus of the spectral degree of coherence of light from the collimator coupled with the LED via a step-index multimode fiber with nominal core diameter 200 µm, plotted on (a) linear and (b) logarithmic scales. The solid curves indicate the theoretical results Eq. (35) under the condition of $a = 210\,\,\unicode{x00B5}{\rm m}$. The meanings of the marks and the error bars, and the experimental conditions, are exactly the same as in Fig. 7.

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In general, not only the interference fringe pattern arising from a nonredundant array of slits but also the intensity distributions produced by each slit are required to characterize the spatial coherence in the proposed method. The experimental procedures to do this are mostly troublesome and time-consuming, if masks (spatial filters) are employed to realize the slit array and the individual slits as in all the related studies in the past because one has to prepare these masks beforehand and change them in turn in the experiment. However, our method utilizes a DMD on which various slit patterns can be formed very easily and, hence, mitigates all these problems. This is one of the advantageous features of the proposed method over the existing methods based on a mask, rather than a DMD.

In closing, some remarks are made on the use of slits, instead of circular apertures, in the proposed method (and also in the conventional Young’s experiment). It is readily inferred that the spatial coherence measurement can be performed accurately using slits for one-dimensional fields that are uniform along the $x$ axis, parallel to the slits (see Fig. 2). However, care must be taken for realistic two-dimensional fields. Whereas circular pinholes can be employed to measure the coherence at any pair of points of a two-dimensional optical field, slits can only be used for coherence measurement in a plane perpendicular to the slits and passing through their centers. On physical grounds, it seems obvious that the method using slits provides reasonably accurate results for the spatial coherence of a two-dimensional field if the field is effectively uniform in each slit. The field generated by a spatially incoherent source in our proof-of-principle experiment meets this condition, as our system characterization tests indicate. In other situations, the validity of utilizing slits in a Young-type spatial coherence measurement must be assessed in terms of the properties of the fields in question.

5. CONCLUSION

In this paper, we proposed a method of measuring the spatial coherence of light by means of a temporally modulated nonredundant array of slits implemented on a DMD. The conventional theory of the spatial coherence measurement with a (static) nonredundant array of apertures including slits was formulated under the condition that the observation plane is located in the far field of the aperture array. However, this condition may be somewhat cumbersome in reality. We have, thus, extended the theory so as to accept the observation plane closer to the aperture array. The main expressions [Eqs. (19) and (28)] derived in the theory established are applicable, in so far as the relations $C(\textbf{0}) \simeq {C_j}(\textbf{0})$ and ${C^\prime _j}(\textbf{0}) \simeq {C_j}(\textbf{0})$ hold, respectively.

It is shown theoretically that the interference fringe pattern produced by a nonredundant slit array is a superposition of Young’s double-slit fringes with different slit separations; thus, information pertaining to a single Young’s experiment can be extracted by a Fourier analysis on the basis of the uniqueness of slit separations. Consequently, the method of measuring the spatial coherence of light with a nonredundant slit array is quite beneficial in practice since the number of experiments to fully characterize the spatial coherence may be greatly reduced in comparison with performing the conventional Young’s experiment repeatedly. The reconfigurable nature of the DMD forming the slit array serves to relieve one from the troublesome preparation for the experiment. The temporal modulation applied to the slit array yields immunity to background light, so that its suppression or subtraction is not absolutely necessary. These features particularly associated with the DMD serve to extend the range of applications of the proposed method. One important example may be an in-line characterization of the spatial coherence of light from optical sources in a factory production line, where quick operation is indispensable and background light can hardly be controlled in most cases.

We have combined the principle of the spatial coherence measurement with the modulation technique for detection using the DMD and confirmed experimentally that all the beneficial features associated with the above principle, technique, and device arise simultaneously. Specifically, in our proof-of-principle experiment, a single measurement for the interference fringe pattern is sufficient to evaluate the spatial coherence for 15 different slit separations. This can readily be performed without any hard prior task for the experiment. The result is always consistent with the theory, even under background light. If the slit array is illuminated unevenly, one needs seven measurements to obtain the degree of coherence on the basis of Eq. (28), rather than Eq. (19). The proposed method is still effective since more than 21 measurements (including background light measurements) are required to obtain the same information when using the conventional Young’s experiment. As a consequence, we conclude that fast and highly reliable spatial coherence measurement is possible with the proposed method without any difficulties in data acquisitions and analyses. The proposed method aims at acquiring only the modulus of the spectral degree of coherence. Note finally that, although it provides significant information in many practical problems, the phase of the spectral degree of coherence is equally important, e.g., in x ray crystallography [47].