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Measuring spatial coherence by using a mask with multiple apertures
Optics Communications 273 (2007) 428–434 www.elsevier.com/locate/optcom
Measuring spatial coherence by using a mask with multiple apertures Yobani Mejı´a *, Aura Ine´s Gonza´lez ´ ptica Aplicada, Departamento de Fı´sica, Universidad Nacional de Colombia Ciudad Universitaria, Bogota´ DC, Colombia Grupo de O Received 25 October 2006; received in revised form 7 December 2006; accepted 11 January 2007
Abstract A simple method to measure the complex degree of spatial coherence of a partially coherent quasi-monochromatic light field is presented. The Fourier spectrum of the far-field interferogram generated by a mask with multiple apertures (small circular holes) is analyzed in terms of classes of aperture pairs. A class of aperture pairs is defined as the set of aperture pairs with the same separation vector. The height of the peaks in the magnitude spectrum determines the modulus of the complex degree of spatial coherence and the corresponding value in the phase spectrum determines the phase of the complex degree of spatial coherence. The method is illustrated with experimental results. 2007 Elsevier B.V. All rights reserved. Keywords: Spatial coherence; Shift invariant and variant optical fields; Multiple apertures
1. Introduction An important parameter to characterize a light field is the complex degree of spatial coherence since it affects directly the interference and diffraction phenomena. This parameter is a normalized version of the correlation function of the light field at two points [1]. Zernike (1938) showed that the degree of coherence could be determined directly from the visibility of the interference fringes formed in a Young interferometer, supposing that both intensities in the apertures are made equal [2]. To deal with more complex situations, a further generalization was made by Wolf (1954). He showed that the degree of coherence is given by the visibility of the interference fringes multiplied by a factor concerns the intensities of each aperture [2]. Since then several methods to measure the degree of coherence based on the Young interferometer have been used [3–7]. These methods require several masks of two apertures with different mutual distance. Recently, Santarsiero and Borghi have proposed a method that uses a mask with two apertures, but it has to be moved laterally *
Corresponding author. Tel.: +57 1 3165130; fax: +57 1 3165135. E-mail address: [email protected] (Y. Mejı´a).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.01.009
to sample different points of the light field [8]. In any case, to determine the degree of coherence, it is necessary to register several interferograms corresponding to different points of the light field. The modulus and phase is found from the visibility and position, respectively, of the interference fringes. An alternative method proposed by Castan˜eda et al. consists of analyzing the Fourier spectrum of the interferogram produced by a mask with multiple apertures spaced evenly (redundant array) [9,10]. However, they only present simulations for a very particular light field, a Schellmodel beam with uniform illumination. Experimental results are not presented. The Fourier analysis of the interferogram produced by a mask with multiple apertures (non-redundant array) placed in the pupil plane of the imaging system has been used in incoherent imaging to improve the high-frequency value of the modulation transfer function [11], and for aperture synthesis in radio-astronomy in order to remove the effects of the aberrations due to the atmospheric turbulence [12–15]. In this paper, we present an experimental method to measure the complex degree of spatial coherence of a partially coherent quasi-monochromatic light field. In particular,
Y. Mejı´a, A.I. Gonza´lez / Optics Communications 273 (2007) 428–434
we used a Gaussianly correlated light field, but the method could be applied to any other type of illumination. The method analyzes the Fourier spectrum of the interferogram generated by a mask with multiple apertures in terms of classes of aperture pairs [16]. We show that with an appropriate distribution of the apertures the height of the peaks in the amplitude spectrum determines the modulus of the complex degree of spatial coherence and the corresponding value in the phase spectrum determines the phase of the complex degree of spatial coherence. With this type of mask we need only one interferogram to characterize the most significant part of the coherence area. 2. Theory To measure the complex degree of spatial coherence of a partially coherent quasi-monochromatic light field of wavelength k, let us consider a mask with N circular apertures of radius a for sampling the light field. If the radius of the apertures is much smaller than the dimensions of the coherence area and the variations of the light field within each aperture are worthless, the light field sampled by this mask can be expressed as U ðqÞ ¼ hðqÞ
N X
V ðqn Þdðq qn Þ;
ð1Þ
n¼1
where hðqÞ is the function that describes the geometry of each aperture, i.e., hðqÞ ¼ circðq=aÞ; the Dirac’s delta function dðq qn Þ places the center of the nth aperture; V ðqn Þ describes the light field within the nth aperture and q ¼ ðn; gÞ is the position vector in the mask plane. The symbol represents the convolution operation. The far-field interferogram generated by the mask of Eq. (1) can be obtained by placing a convergent lens of focal distance f just behind the mask. If the interferogram is at the focal plane (plane image with Cartesian coordinates ðx; yÞ), the mutual intensity for two points with position vectors r1 and r2 at the focal plane is given as Z Z Z Z expðipðr21 r22 Þ=kf Þ hU ðq1 ÞU ðq2 Þi J ðr1 ; r2 Þ ¼ k2 f 2 2p ð2Þ exp i ðq1 r1 q2 r2 Þ d2 q1 d2 q2 ; kf where angular brackets denote the spatial correlation function. Eq. (2) is given by a couple of two-dimensional Fourier transforms whose spatial frequencies are mk ¼ rk =kf with k ¼ 1; 2. From Eq. (1) and the convolution theorem [17], Eq. (2) becomes J ðm1 kf ;m 2 kf Þ ¼
N X N X expðipðr21 r22 Þ=kf Þ H ðm ÞH ðm Þ 1 2 k2 f 2 n¼1 m¼1
hV ðqn Þ; V ðqm Þi expði2pðqn m1 qm m 2 ÞÞ; ð3Þ where H ðmÞ is the Fourier transform of hðqÞ, then H ðmÞ ¼ pa2 jincð2pamÞ;
ð4Þ
429
where jincð2pamÞ ¼ 2J 1 ð2pamÞ=2pam, with J1( ) the Bessel function of the first kind and order 1. The intensity in a given point at the focal plane is found from Eq. (3) for r1 ¼ r2 ¼ r, then it reduces to Iðmkf Þ ¼
2 N X N jH ðmÞj X J ðqn ; qm Þ expði2pm ðqn qm ÞÞ; k2 f 2 n¼1 m¼1
ð5Þ where J ðqn ; qm Þ is the mutual intensity at the points of the light field sampled by the mask with the apertures n and m. The normalized version of the mutual intensity J ðqn ; qm Þ is known as the complex degree of spatial coherence lnm ¼ J ðqn ; qm Þ=½J ðqn ; qn ÞJ ðqm ; qm Þ1=2 , and the mutual intensities J ðqn ; qn Þ and J ðqm ; qm Þ are the intensities In and Im in the apertures n and m, respectively. Keeping in mind the above-mentioned, Eq. (5) becomes Iðmkf Þ ¼
2 N X N pffiffiffiffiffiffiffiffiffi jH ðmÞj X lnm I n I m expði2pm ðqn qm ÞÞ: 2 2 kf n¼1 m¼1
ð6Þ The double sum of Eq. (6) can be written as two terms: the first one concerns the sum of the intensities in each aperture, and the second one concerns the superposition of the interference patterns due to contributions from the aperture pairs (m 6¼ nÞ. Therefore, we obtain " 2 N N N 1 pffiffiffiffiffiffiffiffiffi X X jH ðmÞj X Iðmkf Þ ¼ 2 I nI m In þ k f 2 n¼1 n¼mþ1 m¼1 flnm expði2pm ðqn qm ÞÞ # þlnm expði2pm ðqn qm ÞÞg :
ð7Þ
The Fourier spectrum of the interferogram given by Eq. (7) will be: " N N N 1 pffiffiffiffiffiffiffiffiffi X X X eI ðqÞ ¼ KðqÞ I nI m I n dðqÞ þ n¼1
n¼mþ1 m¼1
flnm dðq ðqn qm ÞÞ þ
lnm dðq
# þ ðqn qm ÞÞg ; ð8Þ
where KðqÞ is the Fourier transform of the square of modulus of Eq. (4), i.e. the autocorrelation function of hðqÞ. According to Eq. (8) the Fourier spectrum of the interferogram generated by a mask with multiple apertures is a distribution of peaks whose shapes are determined by KðqÞ. The height of these peaks depends on both the multiplication of the intensities in the apertures and the modulus of the complex degree of spatial coherence. The phase of the peaks determines the phase of the complex degree of spatial coherence. The position of the peaks (except the central peak) is given by the separation vectors of the nth and mth apertures. The peaks of the Fourier spectrum of the interferogram can be interpreted in terms of classes of aperture pairs [16].
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A class of aperture pairs is defined as the set of aperture pairs with the same separation vector dj ¼ constant, i.e. qn qm ¼ dj
ð9Þ
with j ¼ 1; 2; . . . ; M, and M is the total number of classes of aperture pairs of the mask. We find this concept useful to measure the complex degree of spatial coherence, as we will show later on. 2.1. Complex degree of spatial coherence: case shift invariant If the light field under test has a complex degree of spatial coherence which is shift invariant (Schell-model beam), then lnm will be identical for all the aperture pairs that belong to the jth class, therefore it is possible to represent lnm of the jth class by means of lj. With this change, Eq. (8) becomes " # M X eI ðqÞ ¼ KðqÞ S 0 dðqÞ þ S j ðlj dðq dj Þ þ l dðq þ dj ÞÞ ; j
j¼1
PN
ð10Þ PN
wherepwe have defined S 0 ¼ n¼1 I n and S j ¼ n¼mþ1 P N 1 ffiffiffiffiffiffiffiffiffi I I n m provided that it fulfils Eq. (9). m¼1 The Fourier spectrum of the interferogram described by Eq. (10) is a symmetrical and conjugate distribution of peaks with respect to the origin. Therefore, for measuring the complex degree of spatial coherence we omit the peaks of the spectrum given by lj dðq þ dj Þ. Let cj ¼ jcj j expfi/j g be the maximum value of the jth peak of the Fourier spectrum (right side) and lj ¼ jlj j expfiaj g. Then from Eq. (10) we have that cj ¼ Kð0ÞS j lj . Since Sj is of real value, in consequence, the modulus and the phase of the complex degree of spatial coherence are given by jcj j S 0 jlj j ¼ ; jc0 j S j aj ¼ /j ;
and
ð11aÞ ð11bÞ
respectively; jc0 j is the modulus of the central peak. Eq. (11a) is the generalization of the Wolf’s definition for the degree of coherence. Thus the experimental measurement of the complex degree of spatial coherence (Eqs. (11a) and (11b)) from an interferogram generated by a mask with multiple apertures requires to measure: the intensity in each aperture and the height of the peaks in the Fourier spectrum of the interferogram. The corresponding phase at the maximum value of the peaks is the phase of the complex degree of spatial coherence. 2.2. Complex degree of spatial coherence: case shift variant If the light field under test has a complex degree of spatial coherence which is shift variant, in general, lnm will be different for each aperture pair that belongs to the jth class. In consequence, if at least there is one class of aperture pairs with more than a one aperture pair (redundant
array), in this case, the Fourier spectrum of the interferogram can not be represented by means of Eq. (10), and we can not use Eqs. (11a) and (11b). Neither the modulus of the complex degree of spatial coherence will not be proportional to the maximum value of the peaks of the Fourier spectrum nor the phase of the complex degree of spatial coherence coincides with the phase of the maximum value of these peaks. We can state that the jth peak results from the overlapping of peaks with different complex degree of coherence corresponding to each aperture pair that belongs to the jth class. This means that there is destructive interference among the light fields coming from aperture pairs that belong to the same class. Note that it doesn’t happen if the complex degree of spatial coherence is shift invariant, since the light fields coming from the aperture pairs belong to the same class will be in phase (Eq. (7)). Hence the peaks of the Fourier spectrum for the shift variant case can be smaller than the corresponding peaks for the shift invariant case when the mask of apertures is illuminated with a light field whose intensity distribution is equal in both cases. According to the above-mentioned we conclude that for measuring the complex degree of spatial coherence, the mask with multiple apertures has to be designed in such a way that all classes of aperture pairs of the mask be composed by only one pair (non-redundant array). So the Fourier spectrum can be represented again by means of Eq. (10), consequently Eqs. (11a) and (11b) can be used. Then with this type of mask we can test the complex degree of spatial coherence of shift variant light fields as well as shift invariant light fields. 3. Experiment To verify the results obtained in the previous section, we first generated a partially coherent light field following the rotating ground glass method. This method is discussed in detail in Gori [18]. The experimental set-up is sketched in Fig. 1. The He-Ne laser beam (k ¼ 632:8 nm) in the TM00 mode is focused on a rotating ground glass (RGG) by means of a convergent lens L1 of focal distance f1 ¼ 60 mm. In this situation the light field just behind the RGG is an equivalent incoherent source characterized by a Gaussian intensity profile. The light field that diverges from the RGG and arrives to a mask with multiple apertures (MMA) is now partially coherent with a Gaussian intensity profile, according to the Van Cittert-Zernike theorem. The laser intensity is attenuated by a neutral density filter (NDF). The MMA is placed at the distance 450 ± 1 mm from the RGG. Behind the MMA a couple of convergent lenses L2 (spaced 30 ± 1 mm) are placed at 50 ± 1 mm. Each lens is 100 mm of diameter and 450 mm of focal distance (the couple of lenses are used just to shorter the working distance). The far-field interferogram is observed at a plane located at the distance 400 ± 1 mm from L2. In order to avoid the moire´ effect that can be formed between the interference fringes and the pixel array
Y. Mejı´a, A.I. Gonza´lez / Optics Communications 273 (2007) 428–434
RGG
431
MMA
LASER
CCD NDF
L1
L3
L2 Fig. 1. Experimental set-up to measure the complex degree of spatial coherence. With a rotating ground glass (RGG) a partially coherent light field is generated. This field is sampled by means of a mask with multiple apertures (MMA) and from the far-field interferogram on the CCD plane the complex degree of spatial coherence along the coherence area is measured. Only one interferogram is required for this measurement.
of the CCD camera we insert an imaging lens L3 of focal distance 25 mm. So the interferogram is magnified and imaged by L3 onto the input plane of a CCD camera (1/ 2 in. format, pixels 768 · 494 (H V ), monochrome). Although the far-field interferogram is not formed at the focal plane of L2, as it is supposed in Eq. (2), the Fourier spectrum analysis is completely equivalent. The complex degree of spatial coherence of the light field at the mask plane, according to the theorem of Van CittertZernike [19], is given by ! 2 p ðq q Þ n m lnm ¼ exp i ðq2n q2m Þ exp ; ð12Þ kz w2 where w ¼ kz=pw0 is the size of the coherence area and w0 is the spot size at the beam waist onto the RGG; z is the distance from RGG to MMA. Eq. (12) means that although the modulus of lnm is shift invariant, its phase is not. Therefore, lnm is shift variant.
η
1.5
1 2
6.0
3.0
4.5
3
4
ξ
Since the light field at the mask plane has axial symmetry, it is enough to measure the complex degree of spatial coherence along one direction. The mask is an opaque circular screen with four circular holes along the horizontal direction (Fig. 2), each one has a radius of a ¼ 0:4 mm. The apertures are labeled as {1, 2, 3, 4} and their positions with respect to the intersection of the optical axis with the plane mask are n ¼ f7:5; 6:0; þ3:0; þ7:5g mm, respectively. This distribution of the apertures yields classes of aperture pairs all composed by only one aperture pair. The aperture pairs generated by this screen are shown in Table 1. The far-field interferogram generated by this mask and its profile along the horizontal direction (x) crossing the optical axis are shown in Figs. 3a and 3b, respectively. Since the measure of lnm has been reduced to a problem in one dimension, we analyze the Fourier spectrum of the interferogram in the horizontal direction ðnÞ. Fig. 4 shows the right side of the amplitude spectrum normalized to the height of the central peak jc0 j. In this figure we are able to identify the first 4 classes of aperture pairs. The pair corresponding to class 4 is at the border of the coherence area, therefore this pair contributes very small to the interference process. The pairs of classes 5 and 6 are outside of the coherence area, in consequence neither the light fields from the apertures of class 5 nor the light fields from the apertures of class 6 interfere mutually. On the other hand, to measure lnm we need to determine the intensity terms given by S0 and Sj. The intensity distribution at the mask plane is obtained by using other mask with 27 apertures spaced 2 mm evenly along the horizontal direction (each hole has a radius of b ¼ 0:5 mm). By registering the diffraction pattern Iðx; yÞ generated by R each aperture, and with the equation I p ¼ ð1=pb2 Þ Airy Iðx; yÞ dx Table 1 Classes of aperture pairs yielded by the mask of Fig. 2
Fig. 2. Mask with multiple apertures (non-redundant array). The spatial distribution of the apertures yields classes of aperture pairs all composed by only one aperture pair. Each aperture has a radius of 0.4 mm.
Pair {n, m}
jth class
Separation, dj (mm)
{1, {3, {2, {1, {2, {1,
1 2 3 4 5 6
1.5 4.5 9.0 10.5 13.5 15.0
2} 4} 3} 3} 4} 4}
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432
image planes. The intensity distribution at the mask plane measured in this way is q2 ; ð13Þ IðqÞ ¼ I 0 exp 48:42
Fig. 3a. Far-field interferogram generated by the mask of Fig. 2.
180
Intensity (gray levels 0 - 256)
160 140 120
where I0 = 245 gray levels (in the scale 0–255). The standard error of the fitting is 7 gray levels. From Eq. (13) we found that the width of the Gaussian intensity curve is 96.8 mm. According to Eq. (13) and Fig. 4, Table 2 shows the modulus of the complex degree of spatial coherence corresponding to classes of aperture pairs (Eq. (11a)). The values of Sj are given in gray levels (in the scale 0–255), and the sum of the intensities is S0 = 964. Fig. 5 shows the fitting of jlj j jlðnn ; nm Þj. The dashed curve is the fitting of the experimental data of Table 2. Then, with the method of the mask with multiple apertures the modulus of the complex degree of spatial coherence for the light field under test is ! 2 ðnn nm Þ jlðnn ; nm Þj ¼ exp ; ð14Þ 5:952 where nn and nm are now the coordinates of any point pairs at the mask plane. The standard error of this fitting is 0.03.
100 80 60
Table 2 Modulus of lj
40 20 0 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
x (mm)
Fig. 3b. Profile of the interferogram of Fig. 3a along the horizontal direction (xÞ crossing the optical axis.
jth class
jcj j=jc0 j
Sj 7
jlj j
1 2 3 4 5 6
0.226 0.142 0.028 0.006 0.000 0.000
240 242 243 242 240 239
0.91 ± 0.04 0.57 ± 0.03 0.11 ± 0.01 0.023 ± 0.002 0.00 0.00
dy; ðp ¼ 1; 2; . . . ; 27), we determined the intensity of the light field in the sampled points. This equation takes into account the energy conservation between the mask and
1.2
1.0
0.8
0.8
μ(ξn,ξm)
~ Amplitude spectrum, | I (ξ) |
|c0|
1.0
0.6 0.226
0.4
0.4
0.142 0.028
0.006 0.2
0.2
0
0
0.6
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
Separation of aperture pairs, d j (mm) Fig. 4. Right side of the amplitude of the Fourier spectrum normalized to the height of the central peak jc0 j. From the 6 classes of aperture pairs generated by the mask of Fig. 2 only the first 4 classes are within the coherence area.
0.0 -25
-20
-15
-10
-5
0
5
10
15
20
25
Coherence area, ξn − ξm (mm) Fig. 5. Modulus of the complex degree of spatial coherence. The dashed curve is the fitting of the experimental data obtained using the mask with multiple apertures (Table 2). The continuous curve is the fitting of the experimental data obtained using several masks with two apertures (Fig. 6).
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Fig. 6. Far-field interferograms obtained by means of the traditional method based on the Young interferometer. The separations of the two apertures in the masks are: (a) 2 mm, (b) 4 mm, (c) 6 mm, (d) 8 mm, (e) 10 mm and (f) 12 mm. Each aperture has a radius of 0.5 mm.
From Eq. (14) we deduce that the coherence area has a diameter equal to 11.9 mm (width of the Gaussian curve). The continuous curve in Fig. 5 corresponds to the fitting of the experimental data obtained by means of the traditional method based on the Young interferometer (several masks with two apertures). To end this, we used 6 masks with two apertures (each one has a radius of 0.5 mm). The separations of the two apertures in the masks are 2, 4, 6, 8, 10 and 12 mm, respectively. The interferograms obtained in each case are shown in Fig. 6. We determine the modulus of the complex degree of spatial coherence for each interferogram from their respective Fourier spectrum. In this case, the magnitude spectrum consists of a central peak and two symmetrical peaks. Because the mask with two apertures (labeled as {1, 2}) has just one class of aperture pairs (j ¼ 1Þ, Eq. (11a) becomes jl12 j ¼
jc1 j ðI 1 þ I 2 Þ pffiffiffiffiffiffiffiffi ; jc0 j I 1I 2
ð15Þ
where jc0 j is the height of the central peak and jc1 j is the height of the peak in the right side of the magnitude spectrum; I1 and I2 are the intensities within the two apertures. Note that Eq. (15) is equal to the Wolf’s equation for the degree of coherence. The fitting for the experimental data obtained with Eq. (15) is ! 2 ðnn nm Þ jlðnn ; nm Þj ¼ exp ; ð16Þ 6:302 with a standard error of 0.03. In this case, the diameter of the coherence area is 12.6 mm. Thus the error in the measurement of the coherence area with the method of a mask with multiple apertures and with the method based on the Young interferometer is approximately 6%. It shows that
the method of a mask with multiple apertures is reliable to measure the complex degree of spatial coherence. The phase of lnm can be obtained from Eq. (11b). This phase, for the partially coherent beam generated with the RGG, corresponds to the difference of the optical paths for two optical waves from the origin of coordinates on the RGG plane (intersection of the optical axis with the RGG plane) to the nth and mth apertures on the mask plane [19]. According to our experimental set-up (Fig. 1), in the phase term of Eq. (12), kz ffi 0:31 mm2 , while the numerator varies from zero up to some hundreds of mm2. Therefore the phase of lnm changes quickly along the coherence area; only for very small regions within the coherence area the phase changes slightly, i.e. where the light field is highly coherent. As the phase of lnm , aj ¼ /j , in practice, is obtained from the inverse tangent function of imaginary part divided by the real part of the corresponding points in the Fourier spectrum (phase spectrum), this phase is wrapped. Table 3 shows the phase of lnm of the light field sampled by the mask. As the phase of lnm is shift variant, then the phase for a coherence area located in a particular place, in general, will be different for a coherence area located in other place.
Table 3 Phase of lj of the light field sampled by the mask of Fig. 2 Pair {n, m}
jth class
aj (radians)
{1, {3, {2, {1, {2, {1,
1 2 3 4 5 6
0.49 0.87 1.16 0.43 – –
2} 4} 3} 3} 4} 4}
434
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Consequently, if we move laterally the mask with multiple apertures, the far-field interferogram changes its shape. Therefore, we can check quickly by shifting the mask if the light field is shift invariant. If the modulus of lnm is shift variant the Fourier analysis of a single interferogram produced with a one-dimensional aperture array is not longer sufficient to characterize the spatial coherence of the light field. In this case, we would register several interferograms or it would be possible to use a mask with a two-dimensional aperture array. This will be considered in a future work. Finally, in the above discussion we do not have specified how the aperture distribution of the mask is obtained. There was not a special reason for its choice, in fact, the distribution that we used is not the optimum to sample the light field, but it allows us to show how the method works. The ideal distribution of apertures in the mask would produce uniformly spaced and non-redundant spectral peaks (excluding the zero-frequency peak). Such distributions exist only in a few cases. For instance, an array with 4 apertures placed at distances 0, d, 4d and 6d generates 6 classes of aperture pairs spaced fd; 2d; 3d; 4d; 5d; 6dg, respectively. So this array with 4 apertures produces uniformly spaced and non-redundant spectral peaks. But there is not an array with 5 apertures which can produce uniformly spaced and non-redundant spectral peaks. In this case, the best way to order 5 apertures is at 0, 2d, 7d, 8d, 11d. It generates 10 classes of aperture pairs spaced fd; 2d; 3d; 4d; 5d; 6d; 7d; 8d; 9d; 11dg, respectively; the distance 10d is missing. Such non-redundant aperture arrays that sample the light field (one-dimensional case) in such a way as to produce almost uniformly spaced spectral peaks can be obtained according to Russell et al. [11].
If the phase and modulus of the complex degree of spatial coherence are shift variant, a single interferogram is no longer sufficient to characterize the light field. To end this, the mask with a non-redundant aperture array should be placed at different locations of the light field. Other method could be to use a two-dimensional aperture array. This will be discussed in a future work. The method proposed in this paper is very simple and from the results obtained experimentally we state that this is a reliable method. From the Fourier spectrum of the farfield interferogram we have found a generalization of the Wolf’s equation for measuring the degree of coherence. The height and phase of the peaks in the Fourier spectrum determines the modulus and phase of the complex degree of spatial coherence, respectively. With our method we overcome the drawback of the techniques based on the Young interferometer where it is necessary to take sequentially several interferograms to measure the whole coherence area. Acknowledgement The authors thank Roma´n Castan˜eda (Universidad Nacional de Colombia – Sede Medellı´n) for helpful discussions. References [1] [2] [3] [4] [5] [6] [7]
4. Conclusions In this paper we have shown a method to measure the complex degree of spatial coherence from a single interferogram generated by a mask with multiple apertures in the following cases: (a) If the complex degree of spatial coherence of the light field under test is shift invariant (Schell model-beam). We can use a mask with a redundant or non-redundant aperture array. (b) If the modulus of the complex degree of spatial coherence is shift invariant but its phase is shift variant. The distribution of the apertures in the mask has to be in such a way that all classes of aperture pairs be composed of just one pair (non-redundant array).
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[19]
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Measuring spatial coherence by using a mask with multiple apertures Yobani Mejı´a *, Aura Ine´s Gonza´lez ´ ptica Aplicada, Departamento de Fı´sica, Universidad Nacional de Colombia Ciudad Universitaria, Bogota´ DC, Colombia Grupo de O Received 25 October 2006; received in revised form 7 December 2006; accepted 11 January 2007
Abstract A simple method to measure the complex degree of spatial coherence of a partially coherent quasi-monochromatic light field is presented. The Fourier spectrum of the far-field interferogram generated by a mask with multiple apertures (small circular holes) is analyzed in terms of classes of aperture pairs. A class of aperture pairs is defined as the set of aperture pairs with the same separation vector. The height of the peaks in the magnitude spectrum determines the modulus of the complex degree of spatial coherence and the corresponding value in the phase spectrum determines the phase of the complex degree of spatial coherence. The method is illustrated with experimental results. 2007 Elsevier B.V. All rights reserved. Keywords: Spatial coherence; Shift invariant and variant optical fields; Multiple apertures
1. Introduction An important parameter to characterize a light field is the complex degree of spatial coherence since it affects directly the interference and diffraction phenomena. This parameter is a normalized version of the correlation function of the light field at two points [1]. Zernike (1938) showed that the degree of coherence could be determined directly from the visibility of the interference fringes formed in a Young interferometer, supposing that both intensities in the apertures are made equal [2]. To deal with more complex situations, a further generalization was made by Wolf (1954). He showed that the degree of coherence is given by the visibility of the interference fringes multiplied by a factor concerns the intensities of each aperture [2]. Since then several methods to measure the degree of coherence based on the Young interferometer have been used [3–7]. These methods require several masks of two apertures with different mutual distance. Recently, Santarsiero and Borghi have proposed a method that uses a mask with two apertures, but it has to be moved laterally *
Corresponding author. Tel.: +57 1 3165130; fax: +57 1 3165135. E-mail address: [email protected] (Y. Mejı´a).
0030-4018/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.01.009
to sample different points of the light field [8]. In any case, to determine the degree of coherence, it is necessary to register several interferograms corresponding to different points of the light field. The modulus and phase is found from the visibility and position, respectively, of the interference fringes. An alternative method proposed by Castan˜eda et al. consists of analyzing the Fourier spectrum of the interferogram produced by a mask with multiple apertures spaced evenly (redundant array) [9,10]. However, they only present simulations for a very particular light field, a Schellmodel beam with uniform illumination. Experimental results are not presented. The Fourier analysis of the interferogram produced by a mask with multiple apertures (non-redundant array) placed in the pupil plane of the imaging system has been used in incoherent imaging to improve the high-frequency value of the modulation transfer function [11], and for aperture synthesis in radio-astronomy in order to remove the effects of the aberrations due to the atmospheric turbulence [12–15]. In this paper, we present an experimental method to measure the complex degree of spatial coherence of a partially coherent quasi-monochromatic light field. In particular,
Y. Mejı´a, A.I. Gonza´lez / Optics Communications 273 (2007) 428–434
we used a Gaussianly correlated light field, but the method could be applied to any other type of illumination. The method analyzes the Fourier spectrum of the interferogram generated by a mask with multiple apertures in terms of classes of aperture pairs [16]. We show that with an appropriate distribution of the apertures the height of the peaks in the amplitude spectrum determines the modulus of the complex degree of spatial coherence and the corresponding value in the phase spectrum determines the phase of the complex degree of spatial coherence. With this type of mask we need only one interferogram to characterize the most significant part of the coherence area. 2. Theory To measure the complex degree of spatial coherence of a partially coherent quasi-monochromatic light field of wavelength k, let us consider a mask with N circular apertures of radius a for sampling the light field. If the radius of the apertures is much smaller than the dimensions of the coherence area and the variations of the light field within each aperture are worthless, the light field sampled by this mask can be expressed as U ðqÞ ¼ hðqÞ
N X
V ðqn Þdðq qn Þ;
ð1Þ
n¼1
where hðqÞ is the function that describes the geometry of each aperture, i.e., hðqÞ ¼ circðq=aÞ; the Dirac’s delta function dðq qn Þ places the center of the nth aperture; V ðqn Þ describes the light field within the nth aperture and q ¼ ðn; gÞ is the position vector in the mask plane. The symbol represents the convolution operation. The far-field interferogram generated by the mask of Eq. (1) can be obtained by placing a convergent lens of focal distance f just behind the mask. If the interferogram is at the focal plane (plane image with Cartesian coordinates ðx; yÞ), the mutual intensity for two points with position vectors r1 and r2 at the focal plane is given as Z Z Z Z expðipðr21 r22 Þ=kf Þ hU ðq1 ÞU ðq2 Þi J ðr1 ; r2 Þ ¼ k2 f 2 2p ð2Þ exp i ðq1 r1 q2 r2 Þ d2 q1 d2 q2 ; kf where angular brackets denote the spatial correlation function. Eq. (2) is given by a couple of two-dimensional Fourier transforms whose spatial frequencies are mk ¼ rk =kf with k ¼ 1; 2. From Eq. (1) and the convolution theorem [17], Eq. (2) becomes J ðm1 kf ;m 2 kf Þ ¼
N X N X expðipðr21 r22 Þ=kf Þ H ðm ÞH ðm Þ 1 2 k2 f 2 n¼1 m¼1
hV ðqn Þ; V ðqm Þi expði2pðqn m1 qm m 2 ÞÞ; ð3Þ where H ðmÞ is the Fourier transform of hðqÞ, then H ðmÞ ¼ pa2 jincð2pamÞ;
ð4Þ
429
where jincð2pamÞ ¼ 2J 1 ð2pamÞ=2pam, with J1( ) the Bessel function of the first kind and order 1. The intensity in a given point at the focal plane is found from Eq. (3) for r1 ¼ r2 ¼ r, then it reduces to Iðmkf Þ ¼
2 N X N jH ðmÞj X J ðqn ; qm Þ expði2pm ðqn qm ÞÞ; k2 f 2 n¼1 m¼1
ð5Þ where J ðqn ; qm Þ is the mutual intensity at the points of the light field sampled by the mask with the apertures n and m. The normalized version of the mutual intensity J ðqn ; qm Þ is known as the complex degree of spatial coherence lnm ¼ J ðqn ; qm Þ=½J ðqn ; qn ÞJ ðqm ; qm Þ1=2 , and the mutual intensities J ðqn ; qn Þ and J ðqm ; qm Þ are the intensities In and Im in the apertures n and m, respectively. Keeping in mind the above-mentioned, Eq. (5) becomes Iðmkf Þ ¼
2 N X N pffiffiffiffiffiffiffiffiffi jH ðmÞj X lnm I n I m expði2pm ðqn qm ÞÞ: 2 2 kf n¼1 m¼1
ð6Þ The double sum of Eq. (6) can be written as two terms: the first one concerns the sum of the intensities in each aperture, and the second one concerns the superposition of the interference patterns due to contributions from the aperture pairs (m 6¼ nÞ. Therefore, we obtain " 2 N N N 1 pffiffiffiffiffiffiffiffiffi X X jH ðmÞj X Iðmkf Þ ¼ 2 I nI m In þ k f 2 n¼1 n¼mþ1 m¼1 flnm expði2pm ðqn qm ÞÞ # þlnm expði2pm ðqn qm ÞÞg :
ð7Þ
The Fourier spectrum of the interferogram given by Eq. (7) will be: " N N N 1 pffiffiffiffiffiffiffiffiffi X X X eI ðqÞ ¼ KðqÞ I nI m I n dðqÞ þ n¼1
n¼mþ1 m¼1
flnm dðq ðqn qm ÞÞ þ
lnm dðq
# þ ðqn qm ÞÞg ; ð8Þ
where KðqÞ is the Fourier transform of the square of modulus of Eq. (4), i.e. the autocorrelation function of hðqÞ. According to Eq. (8) the Fourier spectrum of the interferogram generated by a mask with multiple apertures is a distribution of peaks whose shapes are determined by KðqÞ. The height of these peaks depends on both the multiplication of the intensities in the apertures and the modulus of the complex degree of spatial coherence. The phase of the peaks determines the phase of the complex degree of spatial coherence. The position of the peaks (except the central peak) is given by the separation vectors of the nth and mth apertures. The peaks of the Fourier spectrum of the interferogram can be interpreted in terms of classes of aperture pairs [16].
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A class of aperture pairs is defined as the set of aperture pairs with the same separation vector dj ¼ constant, i.e. qn qm ¼ dj
ð9Þ
with j ¼ 1; 2; . . . ; M, and M is the total number of classes of aperture pairs of the mask. We find this concept useful to measure the complex degree of spatial coherence, as we will show later on. 2.1. Complex degree of spatial coherence: case shift invariant If the light field under test has a complex degree of spatial coherence which is shift invariant (Schell-model beam), then lnm will be identical for all the aperture pairs that belong to the jth class, therefore it is possible to represent lnm of the jth class by means of lj. With this change, Eq. (8) becomes " # M X eI ðqÞ ¼ KðqÞ S 0 dðqÞ þ S j ðlj dðq dj Þ þ l dðq þ dj ÞÞ ; j
j¼1
PN
ð10Þ PN
wherepwe have defined S 0 ¼ n¼1 I n and S j ¼ n¼mþ1 P N 1 ffiffiffiffiffiffiffiffiffi I I n m provided that it fulfils Eq. (9). m¼1 The Fourier spectrum of the interferogram described by Eq. (10) is a symmetrical and conjugate distribution of peaks with respect to the origin. Therefore, for measuring the complex degree of spatial coherence we omit the peaks of the spectrum given by lj dðq þ dj Þ. Let cj ¼ jcj j expfi/j g be the maximum value of the jth peak of the Fourier spectrum (right side) and lj ¼ jlj j expfiaj g. Then from Eq. (10) we have that cj ¼ Kð0ÞS j lj . Since Sj is of real value, in consequence, the modulus and the phase of the complex degree of spatial coherence are given by jcj j S 0 jlj j ¼ ; jc0 j S j aj ¼ /j ;
and
ð11aÞ ð11bÞ
respectively; jc0 j is the modulus of the central peak. Eq. (11a) is the generalization of the Wolf’s definition for the degree of coherence. Thus the experimental measurement of the complex degree of spatial coherence (Eqs. (11a) and (11b)) from an interferogram generated by a mask with multiple apertures requires to measure: the intensity in each aperture and the height of the peaks in the Fourier spectrum of the interferogram. The corresponding phase at the maximum value of the peaks is the phase of the complex degree of spatial coherence. 2.2. Complex degree of spatial coherence: case shift variant If the light field under test has a complex degree of spatial coherence which is shift variant, in general, lnm will be different for each aperture pair that belongs to the jth class. In consequence, if at least there is one class of aperture pairs with more than a one aperture pair (redundant
array), in this case, the Fourier spectrum of the interferogram can not be represented by means of Eq. (10), and we can not use Eqs. (11a) and (11b). Neither the modulus of the complex degree of spatial coherence will not be proportional to the maximum value of the peaks of the Fourier spectrum nor the phase of the complex degree of spatial coherence coincides with the phase of the maximum value of these peaks. We can state that the jth peak results from the overlapping of peaks with different complex degree of coherence corresponding to each aperture pair that belongs to the jth class. This means that there is destructive interference among the light fields coming from aperture pairs that belong to the same class. Note that it doesn’t happen if the complex degree of spatial coherence is shift invariant, since the light fields coming from the aperture pairs belong to the same class will be in phase (Eq. (7)). Hence the peaks of the Fourier spectrum for the shift variant case can be smaller than the corresponding peaks for the shift invariant case when the mask of apertures is illuminated with a light field whose intensity distribution is equal in both cases. According to the above-mentioned we conclude that for measuring the complex degree of spatial coherence, the mask with multiple apertures has to be designed in such a way that all classes of aperture pairs of the mask be composed by only one pair (non-redundant array). So the Fourier spectrum can be represented again by means of Eq. (10), consequently Eqs. (11a) and (11b) can be used. Then with this type of mask we can test the complex degree of spatial coherence of shift variant light fields as well as shift invariant light fields. 3. Experiment To verify the results obtained in the previous section, we first generated a partially coherent light field following the rotating ground glass method. This method is discussed in detail in Gori [18]. The experimental set-up is sketched in Fig. 1. The He-Ne laser beam (k ¼ 632:8 nm) in the TM00 mode is focused on a rotating ground glass (RGG) by means of a convergent lens L1 of focal distance f1 ¼ 60 mm. In this situation the light field just behind the RGG is an equivalent incoherent source characterized by a Gaussian intensity profile. The light field that diverges from the RGG and arrives to a mask with multiple apertures (MMA) is now partially coherent with a Gaussian intensity profile, according to the Van Cittert-Zernike theorem. The laser intensity is attenuated by a neutral density filter (NDF). The MMA is placed at the distance 450 ± 1 mm from the RGG. Behind the MMA a couple of convergent lenses L2 (spaced 30 ± 1 mm) are placed at 50 ± 1 mm. Each lens is 100 mm of diameter and 450 mm of focal distance (the couple of lenses are used just to shorter the working distance). The far-field interferogram is observed at a plane located at the distance 400 ± 1 mm from L2. In order to avoid the moire´ effect that can be formed between the interference fringes and the pixel array
Y. Mejı´a, A.I. Gonza´lez / Optics Communications 273 (2007) 428–434
RGG
431
MMA
LASER
CCD NDF
L1
L3
L2 Fig. 1. Experimental set-up to measure the complex degree of spatial coherence. With a rotating ground glass (RGG) a partially coherent light field is generated. This field is sampled by means of a mask with multiple apertures (MMA) and from the far-field interferogram on the CCD plane the complex degree of spatial coherence along the coherence area is measured. Only one interferogram is required for this measurement.
of the CCD camera we insert an imaging lens L3 of focal distance 25 mm. So the interferogram is magnified and imaged by L3 onto the input plane of a CCD camera (1/ 2 in. format, pixels 768 · 494 (H V ), monochrome). Although the far-field interferogram is not formed at the focal plane of L2, as it is supposed in Eq. (2), the Fourier spectrum analysis is completely equivalent. The complex degree of spatial coherence of the light field at the mask plane, according to the theorem of Van CittertZernike [19], is given by ! 2 p ðq q Þ n m lnm ¼ exp i ðq2n q2m Þ exp ; ð12Þ kz w2 where w ¼ kz=pw0 is the size of the coherence area and w0 is the spot size at the beam waist onto the RGG; z is the distance from RGG to MMA. Eq. (12) means that although the modulus of lnm is shift invariant, its phase is not. Therefore, lnm is shift variant.
η
1.5
1 2
6.0
3.0
4.5
3
4
ξ
Since the light field at the mask plane has axial symmetry, it is enough to measure the complex degree of spatial coherence along one direction. The mask is an opaque circular screen with four circular holes along the horizontal direction (Fig. 2), each one has a radius of a ¼ 0:4 mm. The apertures are labeled as {1, 2, 3, 4} and their positions with respect to the intersection of the optical axis with the plane mask are n ¼ f7:5; 6:0; þ3:0; þ7:5g mm, respectively. This distribution of the apertures yields classes of aperture pairs all composed by only one aperture pair. The aperture pairs generated by this screen are shown in Table 1. The far-field interferogram generated by this mask and its profile along the horizontal direction (x) crossing the optical axis are shown in Figs. 3a and 3b, respectively. Since the measure of lnm has been reduced to a problem in one dimension, we analyze the Fourier spectrum of the interferogram in the horizontal direction ðnÞ. Fig. 4 shows the right side of the amplitude spectrum normalized to the height of the central peak jc0 j. In this figure we are able to identify the first 4 classes of aperture pairs. The pair corresponding to class 4 is at the border of the coherence area, therefore this pair contributes very small to the interference process. The pairs of classes 5 and 6 are outside of the coherence area, in consequence neither the light fields from the apertures of class 5 nor the light fields from the apertures of class 6 interfere mutually. On the other hand, to measure lnm we need to determine the intensity terms given by S0 and Sj. The intensity distribution at the mask plane is obtained by using other mask with 27 apertures spaced 2 mm evenly along the horizontal direction (each hole has a radius of b ¼ 0:5 mm). By registering the diffraction pattern Iðx; yÞ generated by R each aperture, and with the equation I p ¼ ð1=pb2 Þ Airy Iðx; yÞ dx Table 1 Classes of aperture pairs yielded by the mask of Fig. 2
Fig. 2. Mask with multiple apertures (non-redundant array). The spatial distribution of the apertures yields classes of aperture pairs all composed by only one aperture pair. Each aperture has a radius of 0.4 mm.
Pair {n, m}
jth class
Separation, dj (mm)
{1, {3, {2, {1, {2, {1,
1 2 3 4 5 6
1.5 4.5 9.0 10.5 13.5 15.0
2} 4} 3} 3} 4} 4}
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432
image planes. The intensity distribution at the mask plane measured in this way is q2 ; ð13Þ IðqÞ ¼ I 0 exp 48:42
Fig. 3a. Far-field interferogram generated by the mask of Fig. 2.
180
Intensity (gray levels 0 - 256)
160 140 120
where I0 = 245 gray levels (in the scale 0–255). The standard error of the fitting is 7 gray levels. From Eq. (13) we found that the width of the Gaussian intensity curve is 96.8 mm. According to Eq. (13) and Fig. 4, Table 2 shows the modulus of the complex degree of spatial coherence corresponding to classes of aperture pairs (Eq. (11a)). The values of Sj are given in gray levels (in the scale 0–255), and the sum of the intensities is S0 = 964. Fig. 5 shows the fitting of jlj j jlðnn ; nm Þj. The dashed curve is the fitting of the experimental data of Table 2. Then, with the method of the mask with multiple apertures the modulus of the complex degree of spatial coherence for the light field under test is ! 2 ðnn nm Þ jlðnn ; nm Þj ¼ exp ; ð14Þ 5:952 where nn and nm are now the coordinates of any point pairs at the mask plane. The standard error of this fitting is 0.03.
100 80 60
Table 2 Modulus of lj
40 20 0 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
x (mm)
Fig. 3b. Profile of the interferogram of Fig. 3a along the horizontal direction (xÞ crossing the optical axis.
jth class
jcj j=jc0 j
Sj 7
jlj j
1 2 3 4 5 6
0.226 0.142 0.028 0.006 0.000 0.000
240 242 243 242 240 239
0.91 ± 0.04 0.57 ± 0.03 0.11 ± 0.01 0.023 ± 0.002 0.00 0.00
dy; ðp ¼ 1; 2; . . . ; 27), we determined the intensity of the light field in the sampled points. This equation takes into account the energy conservation between the mask and
1.2
1.0
0.8
0.8
μ(ξn,ξm)
~ Amplitude spectrum, | I (ξ) |
|c0|
1.0
0.6 0.226
0.4
0.4
0.142 0.028
0.006 0.2
0.2
0
0
0.6
1.5
3.0
4.5
6.0
7.5
9.0
10.5
12.0
13.5
Separation of aperture pairs, d j (mm) Fig. 4. Right side of the amplitude of the Fourier spectrum normalized to the height of the central peak jc0 j. From the 6 classes of aperture pairs generated by the mask of Fig. 2 only the first 4 classes are within the coherence area.
0.0 -25
-20
-15
-10
-5
0
5
10
15
20
25
Coherence area, ξn − ξm (mm) Fig. 5. Modulus of the complex degree of spatial coherence. The dashed curve is the fitting of the experimental data obtained using the mask with multiple apertures (Table 2). The continuous curve is the fitting of the experimental data obtained using several masks with two apertures (Fig. 6).
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Fig. 6. Far-field interferograms obtained by means of the traditional method based on the Young interferometer. The separations of the two apertures in the masks are: (a) 2 mm, (b) 4 mm, (c) 6 mm, (d) 8 mm, (e) 10 mm and (f) 12 mm. Each aperture has a radius of 0.5 mm.
From Eq. (14) we deduce that the coherence area has a diameter equal to 11.9 mm (width of the Gaussian curve). The continuous curve in Fig. 5 corresponds to the fitting of the experimental data obtained by means of the traditional method based on the Young interferometer (several masks with two apertures). To end this, we used 6 masks with two apertures (each one has a radius of 0.5 mm). The separations of the two apertures in the masks are 2, 4, 6, 8, 10 and 12 mm, respectively. The interferograms obtained in each case are shown in Fig. 6. We determine the modulus of the complex degree of spatial coherence for each interferogram from their respective Fourier spectrum. In this case, the magnitude spectrum consists of a central peak and two symmetrical peaks. Because the mask with two apertures (labeled as {1, 2}) has just one class of aperture pairs (j ¼ 1Þ, Eq. (11a) becomes jl12 j ¼
jc1 j ðI 1 þ I 2 Þ pffiffiffiffiffiffiffiffi ; jc0 j I 1I 2
ð15Þ
where jc0 j is the height of the central peak and jc1 j is the height of the peak in the right side of the magnitude spectrum; I1 and I2 are the intensities within the two apertures. Note that Eq. (15) is equal to the Wolf’s equation for the degree of coherence. The fitting for the experimental data obtained with Eq. (15) is ! 2 ðnn nm Þ jlðnn ; nm Þj ¼ exp ; ð16Þ 6:302 with a standard error of 0.03. In this case, the diameter of the coherence area is 12.6 mm. Thus the error in the measurement of the coherence area with the method of a mask with multiple apertures and with the method based on the Young interferometer is approximately 6%. It shows that
the method of a mask with multiple apertures is reliable to measure the complex degree of spatial coherence. The phase of lnm can be obtained from Eq. (11b). This phase, for the partially coherent beam generated with the RGG, corresponds to the difference of the optical paths for two optical waves from the origin of coordinates on the RGG plane (intersection of the optical axis with the RGG plane) to the nth and mth apertures on the mask plane [19]. According to our experimental set-up (Fig. 1), in the phase term of Eq. (12), kz ffi 0:31 mm2 , while the numerator varies from zero up to some hundreds of mm2. Therefore the phase of lnm changes quickly along the coherence area; only for very small regions within the coherence area the phase changes slightly, i.e. where the light field is highly coherent. As the phase of lnm , aj ¼ /j , in practice, is obtained from the inverse tangent function of imaginary part divided by the real part of the corresponding points in the Fourier spectrum (phase spectrum), this phase is wrapped. Table 3 shows the phase of lnm of the light field sampled by the mask. As the phase of lnm is shift variant, then the phase for a coherence area located in a particular place, in general, will be different for a coherence area located in other place.
Table 3 Phase of lj of the light field sampled by the mask of Fig. 2 Pair {n, m}
jth class
aj (radians)
{1, {3, {2, {1, {2, {1,
1 2 3 4 5 6
0.49 0.87 1.16 0.43 – –
2} 4} 3} 3} 4} 4}
434
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Consequently, if we move laterally the mask with multiple apertures, the far-field interferogram changes its shape. Therefore, we can check quickly by shifting the mask if the light field is shift invariant. If the modulus of lnm is shift variant the Fourier analysis of a single interferogram produced with a one-dimensional aperture array is not longer sufficient to characterize the spatial coherence of the light field. In this case, we would register several interferograms or it would be possible to use a mask with a two-dimensional aperture array. This will be considered in a future work. Finally, in the above discussion we do not have specified how the aperture distribution of the mask is obtained. There was not a special reason for its choice, in fact, the distribution that we used is not the optimum to sample the light field, but it allows us to show how the method works. The ideal distribution of apertures in the mask would produce uniformly spaced and non-redundant spectral peaks (excluding the zero-frequency peak). Such distributions exist only in a few cases. For instance, an array with 4 apertures placed at distances 0, d, 4d and 6d generates 6 classes of aperture pairs spaced fd; 2d; 3d; 4d; 5d; 6dg, respectively. So this array with 4 apertures produces uniformly spaced and non-redundant spectral peaks. But there is not an array with 5 apertures which can produce uniformly spaced and non-redundant spectral peaks. In this case, the best way to order 5 apertures is at 0, 2d, 7d, 8d, 11d. It generates 10 classes of aperture pairs spaced fd; 2d; 3d; 4d; 5d; 6d; 7d; 8d; 9d; 11dg, respectively; the distance 10d is missing. Such non-redundant aperture arrays that sample the light field (one-dimensional case) in such a way as to produce almost uniformly spaced spectral peaks can be obtained according to Russell et al. [11].
If the phase and modulus of the complex degree of spatial coherence are shift variant, a single interferogram is no longer sufficient to characterize the light field. To end this, the mask with a non-redundant aperture array should be placed at different locations of the light field. Other method could be to use a two-dimensional aperture array. This will be discussed in a future work. The method proposed in this paper is very simple and from the results obtained experimentally we state that this is a reliable method. From the Fourier spectrum of the farfield interferogram we have found a generalization of the Wolf’s equation for measuring the degree of coherence. The height and phase of the peaks in the Fourier spectrum determines the modulus and phase of the complex degree of spatial coherence, respectively. With our method we overcome the drawback of the techniques based on the Young interferometer where it is necessary to take sequentially several interferograms to measure the whole coherence area. Acknowledgement The authors thank Roma´n Castan˜eda (Universidad Nacional de Colombia – Sede Medellı´n) for helpful discussions. References [1] [2] [3] [4] [5] [6] [7]
4. Conclusions In this paper we have shown a method to measure the complex degree of spatial coherence from a single interferogram generated by a mask with multiple apertures in the following cases: (a) If the complex degree of spatial coherence of the light field under test is shift invariant (Schell model-beam). We can use a mask with a redundant or non-redundant aperture array. (b) If the modulus of the complex degree of spatial coherence is shift invariant but its phase is shift variant. The distribution of the apertures in the mask has to be in such a way that all classes of aperture pairs be composed of just one pair (non-redundant array).
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19]
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