An experimental apparatus for white light imaging by means of a spherical obstacle Andrzej Kolodziejczyka) Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland Zbigniew Jaroszewiczb) Institute of Applied Optics, Kamionkowska 18, 03-805 Warsaw, Poland Rodrigo Henao and Orlando Quintero Facultad de Ciencias, Universidad de Antioquia, A.A. 1226 Medellı´n, Colombia ͑Received 23 April 2001; accepted 20 September 2001͒ We discuss the use of a spherical obstacle as an imaging device for white light illumination. A brief description of the phenomenon is given and then a simple, compact experimental apparatus is described in detail. The experimental results underline the importance of the spatial incoherence of light used in the imaging arrangement. The superiority of incoherent over coherent illumination is demonstrated and some interesting features of imaging by a spherical obstacle are discussed. © 2002 American Association of Physics Teachers. ͓DOI: 10.1119/1.1419099͔ I. INTRODUCTION tained before the invention of the laser, we assume that in both cases a light source with a low spatial coherence was When a plane or spherical wave illuminates a circular or applied. spherical obstacle, one can observe in the shadow’s center a Walker attempted to repeat the experiments cited by Som- 9 diffractive light pattern consisting of a central small spot merfeld using visible laser light. However, in spite of many surrounded by rings. This spot is known as the Arago or different experimental arrangements, the use of coherent il- Poisson spot.1,2 When Fresnel presented his work on the lumination did not lead to satisfying results. Walker’s setup wave theory of light to the French Academy in 1818, Poisson had lengths up to 80 m, similar to the one used by von — an adherent of the ray theory — noted that according to Angerer. Recently, a detailed description of Arago spot for- the new theory, a bright spot should appear in the middle of mation and its action as a point spread function was given by Sommargren and Weaver in their two papers, which consti- the shadow of the circular object, thus arguing for the incor- 10,11 rectness of Fresnel’s theory. However, the wave theory tute the main and most detailed reference to the problem. gained strong experimental support when the spot was ob- They presented an imaging arrangement with a steel ball for served soon afterwards by Arago. which their theoretical approach was verified. The input ob- This effect is discussed in many textbooks on optics as a ject was illuminated by spatially incoherent monochromatic phenomenon that confirms the wave theory of light. It also light obtained by means of a laser beam and probably a ro- constitutes one of the favorite examples of experimentum tating ground glass. The length of the setup was about 4 m, crucis, that is, an experiment that allows one to choose be- still quite long. Although the work of Sommargren and tween two competing theories. Because of the important his- Weaver is theoretically important, the authors did not pay torical and didactical value of the Arago spot, there have much attention to the importance of the spatial coherence of been several articles devoted to this effect in this journal.3–7 light to the quality of the output images formed by a steel These articles described simple optical arrangements that al- ball. In our article we concentrate on the experimental aspects low the phenomenon to be demonstrated to college and uni- of the problem. The theoretical details can be found in Refs. versity students. 10 and 11. We compare imaging by a spherical obstacle in In this article we discuss an interesting and less known incoherent white light and a coherent laser beam and show a application of the Arago spot, namely, the appearance of im- clear hitherto underestimated advantage of the first source. ages within the shadow zone, where the Arago spot acts as a After the general introduction, we describe the compact op- point spread function of the imaging process. To our knowl- tical setup, which makes it possible to obtain images of two- edge, only a few workers have reported imaging with a cir- dimensional input objects illuminated by white light on an cular disk or spherical obstacle. The first person who noted ordinary laboratory table. Because of its simplicity, the this possibility was probably R. W. Pohl, who presented an present arrangement can be used for educational purposes in image of a simple monogram created by a steel ball. Som- the context of diffraction. Some interesting features of imag- merfeld cited Pohl’s experiment in his book on optics,8 where the experiment done by E. von Angerer was also re- ported. von Angerer successfully repeated Pohl’s demonstra- tion and obtained an image of a picture of a female face. A circular disk of diameter 2aϭ50 mm was used, and the ex- perimental arrangement was 70 m long ͑the object distance d and the image distance z were both equal to 35 m͒. Unfortu- nately, Pohl and Sommerfeld did not describe the sources of Fig. 1. The Arago spot at a distance z behind the spherical obstacle illumi- light used in the experiments. Because the images were ob- nated by a spherical source placed at the distance d. 169 Am. J. Phys. 70 ͑2͒, February 2002 http://ojps.aip.org/ajp/ © 2002 American Association of Physics Teachers 169 ing by a spherical obstacle are discussed in the conclusions. the spherical obstacle, that is, there does not exist any well- We have chosen a spherical ball rather than a circular disk, defined image plane. Consequently, we do not observe a su- because the ball possesses a circular cross section indepen- perposition of one sharp image for a specific wavelength dent of the direction of illumination, thus avoiding an aber- with defocused images for other wavelengths, but rather a ration that occurs when the circular disk is applied for wide superposition of almost identical images for different wave- objects. In other words, all object points are axial for the lengths. spherical ball. Second, incoherent illumination should be much better for the image formation process. In the coherent case the point II. GENERAL REMARKS spread function has strong subsidiary ring maximums, for example, the first amplitude maximum is 40% of the central When a spherical ball of a diameter 2a is illuminated by a peak value and therefore strong interference effects should spherical divergent wave with radius of curvature d,wecan be observed. As a result almost complete information is lost observe a diffraction pattern in the form of the Arago spot in about small details of the object. In the Walker experiment a plane located at an arbitrary distance z behind the obstacle this loss was probably the main reason for his lack of success ͑see Fig. 1͒. The amplitude of the output field U is described in repeating the imaging experiment.9 On the other hand, by the complicated Lommel functions, but inside the ball although the incoherent illumination produces a glare effect, shadow near the optical axis, the amplitude can be approxi- the image is still easily recognizable, because in this case the mated by the Bessel function of the first kind and zero intensities add and the subsidiary minima are weaker. order:10,11 ͒ϭ ͒ ͑ ͒ U͑r AJ0͑kar/z , 1 where Aϭexp͓ik͑zϩd͔͒exp͓ika2͑zϩd͒/2zd͔/͑zϩd͒, ͑2͒ III. EXPERIMENT ϭ ␲ ␭ ␭ and k 2 / and is the wavelength of the spherical wave. It is crucial for the successful performance of the experi- The resultant diffractive pattern forms a focal segment and ment to focus the highest possible intensity on the object. In the Arago spot can be included into the family of our setup we have used a 300-W white light projector, and 12–14 ͑ axicons. An axicon is a figure of revolution that images we have focused its light in the object plane over the circle a point source on its axis to a range of points along its with an approximate diameter of 3 mm. Only then was the 12͒ ͑ ͒ axis. From Eq. 1 and the fact that the function J0(x) has energy in the image plane sufficient to be recorded by a its first zero at xϭ2.403, the radius of the central Arago spot charge coupled device ͑CCD͒ camera. ͑All our experiments is equal to were done using an ordinary CCD camera Sony SSC-M370.͒ ␳ ϭ ␭ ͑ ͒ 0 0.38 z/a. 3 Earlier we had used a projector with 60-W power and al- ␳ though very dim images were observed, their intensity was For visible light illumination 0 is usually very small. For example, for ␭ϭ0.63 ␮m, zϭ54 cm, aϭ3 mm, ␳ ϭ43 ␮m not sufficient to be detected by the CCD camera. In our 0 arrangement the object distance d was equal to 74 cm, and is nearly 70 times smaller than the radius of the ball. whereas the image distance z was 54 cm. We used a steel ball The radius of the Arago spot within this approximation does not depend on the curvature of the illuminating wave bearing of diameter 2a equal to 6 mm. It was glued to an front.10 Because of the radial symmetry of the ball and small ordinary microscope glass substrate and then the glass was size of the central diffractive spot, one can expect that the placed perpendicular to the optical axis of the setup. A sketch spherical obstacle can be used for imaging purposes and the of the experimental arrangement is shown in Fig. 3. The results of the experiments are given in Figs.