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. 4–7. images should be observed in the shadow zone. A schematic ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ of the optical arrangement is presented in Fig. 2. A two- Figures 4 a ,5a ,6a , and 7 a present the input objects dimensional uniformly illuminated object of height h is in- with the indicated dimensions. The images obtained in white light illumination are shown in Figs. 4͑b͒,5͑b͒,6͑b͒, and serted in the input plane at a distance d before the ball. In ͑ ͒ this case each point of the object forms its own Poisson 7 b . The images were seen by the eye as white, which con- diffractive pattern, which plays the role of the impulse re- firms the achromatic character of imaging. We then repeated sponse of the imaging setup.11 The conjugate points lie on the imaging experiment in spatially coherent illumination. In spheres, but in the paraxial approximation (dӷa, dӷh), the this case the objects were illuminated in the same setup by a 20-mW He–Ne laser beam. The images obtained are pre- sphere fragments can be replaced by the proper fragments of sented in Figs. 4͑c͒,5͑c͒,6͑c͒, and 7͑c͒. The photographs the planes. Therefore, we expect that the object image is ͑4b͒, ͑5b͒, ͑6b͒, ͑7b͒ and ͑4c͒, ͑5c͒, ͑6c͒, ͑7c͒ are presented formed in the output plane at any distance z behind the ob- in the same scale. Observe the big difference between the stacle. According to geometrical optics, the image is real, images produced in the coherent and the incoherent cases. In inverted, and has the height the first case the images are completely unrecognizable, Hϭhz/d. ͑4͒ whereas in the second case the image can be easily seen, although a glare decreases the contrast. Finally, we placed a A detailed analysis of the imaging with a spherical ob- pinhole with a diameter 25 ␮m instead of the object and stacle can be found elsewhere.10,11 Here we pay attention to illuminated it by a laser beam. In this way we obtained in the two important and not fully appreciated features. First, the output imaging plane the Arago spot ͑Fig. 8͒ which was the image formation process is almost achromatic in spite of its coherent impulse response in our imaging setup. The quality apparently diffractive character. Usually diffractive imaging of the spot is not high. One can see deformation of the outer elements illuminated by polychromatic light form output im- rings probably caused by the slight spherical asymmetry of ages in different planes for different wavelengths. In the case the ball and phase errors introduced by the microscopic sub- of the Arago spot the image appears at any position behind strate glass into the Fresnel zones surrounding the ball.

170 Am. J. Phys., Vol. 70, No. 2, February 2002 Kolodziejczyk et al. 170 Fig. 3. A schematic of the experimental setup. P is the white light 300-W projector; T is the telescopic system that focuses light onto the object; O is the two-dimensional object; B is the 6-mm-diam steel ball that is glued to the microscope glass substrate; and CCD is the CCD camera that registers the output images.

Fig. 2. The image formation in the full shadow zone of an object of width h.

Fig. 4. The input object and its images obtained in the experimental arrangement shown in Fig. 3: ͑a͒ the input object with indicated dimensions; ͑b͒ the object image formed in white light; ͑c͒ the corresponding image ob- tained when the object was illuminated by the laser beam.

Fig. 5. The input object and its images obtained in the experimental arrangement shown in Fig. 3: ͑a͒ the input object with indicated dimensions; ͑b͒ the object image formed in white light; ͑c͒ the corresponding image ob- tained when the object was illuminated by the laser beam.

Fig. 6. The input object and its images obtained in the experimental arrangement shown in Fig. 3: ͑a͒ the input object with indicated dimensions; ͑b͒ the object image formed in white light; ͑c͒ the corresponding image ob- tained when the object was illuminated by the laser beam.

Fig. 7. The input object and its images obtained in the experimental arrangement shown in Fig. 3: ͑a͒ the input object with indicated dimensions; ͑b͒ the object image formed in white light; ͑c͒ the corresponding image ob- tained when the object was illuminated by the laser beam.

171 Am. J. Phys., Vol. 70, No. 2, February 2002 Kolodziejczyk et al. 171 Nevertheless, for practical purposes the distance between a ball and an imaging plane should be reasonable. Images in distant planes have large dimensions, and therefore their in- tensities are very low.

ACKNOWLEDGMENTS This work was supported by Departamento de Fisica, Uni- versidad de Antioquia-CODI, Medellı´n, Colombia. The main part of this work was performed while ͑A.K.͒ was a visiting Fig. 8. The Arago spot playing the role of the coherent impulse response in professor at the Facultad de Ciencias, Universidad de Antio- the imaging setup shown in Fig. 3. quia, A.A. 1226 Medellı´n, Colombia, and ͑Z.J.͒ was a pro- fessor at Facultad de Ciencias, Universidad Nacional sede Medellı´n, A.A. 3840 Medellı´n, Colombia. Both authors would like to express their sincere gratitude for the hospital- IV. CONCLUSIONS ity that they received during their stay in Colombia.

We have shown a simple and compact optical setup with a a͒ steel ball acting as an imaging device, and have compared Electronic mail: [email protected] b͒Electronic mail: [email protected] the quality of the images obtained in spatially incoherent and 1J. Z. Buchwald, The Rise of the Wave Theory of Light ͑The University of coherent light. The Arago spot plays the role of the point Chicago Press, Chicago, 1990͒. spread function. Some interesting features of this uncommon 2Ch. F. Meyer, The Diffraction of Light, X-rays, and Material Particles imaging device include the following. ͑Edwards, Ann Arbor, MI, 1949͒. ͑1͒ A striking advantage of the incoherent illumination in 3P. M. Rinard, ‘‘Large-scale diffraction patterns from circular objects,’’Am. J. Phys. 44, 70–76 ͑1976͒. comparison with the coherent one. The images obtained in 4 the spatially coherent illumination are completely spoiled by J. E. Harvey and J. L. Forgham, ‘‘The spot of Arago: New relevance for an old problem,’’ Am. J. Phys. 52, 243–247 ͑1984͒. the interference between the outer rings surrounding the 5E. A. Hovenac, ‘‘Fresnel diffraction by spherical obstacles,’’ Am. J. Phys. Arago spots which correspond to images of the object points. 57,79–84͑1989͒. Therefore, a steel ball can form images of extended objects 6G. R. Wein, ‘‘A video technique for the quantitative analysis of the Pois- only when light of low spatial coherence is used. son spot and other diffraction patterns,’’ Am. J. Phys. 67, 236–240 ͑1999͒. 7 ͑2͒ A precise adjustment of the imaging arrangement is not M. E. Harrison, C. T. Marek, and J. D. White, ‘‘Rediscovering Poisson’s ͑ ͒ necessary. Phase errors introduced by the circular asymmetry spot,’’ Phys. Teach. 35,18–19 1997 ; Letters to the Editor, ibid. 35, 197–198 ͑1997͒. of the setup do not disturb the output images in incoherent 8A. Sommerfeld, Optik ͑Dieterich’sche Verlagsbuchhandlung, Wiesbaden, light. The arrangement is simple, compact, and includes eas- 1950͒; English translation: Optics ͑Academic, New York, 1954͒. ily available optical components. The setup does not need 9J. Walker, ‘‘A ball bearing aids in the study of light and also serves as a any special stabilization so a demonstration can be per- lens,’’ Sci. Am. 251 ͑5͒, 154–162 ͑1984͒. formed on an ordinary table. The only crucial thing is a 10G. E. Sommargren and H. J. Weaver, ‘‘Diffraction of light by an opaque sufficiently high powered white light source. Hence, the ex- sphere. I: Description and properties of the diffraction pattern,’’ Appl. Opt. 29, 4646–4657 ͑1990͒. periment can be conducted in many schools as a beautiful 11G. E. Sommargren and H. J. Weaver, ‘‘Diffraction of light by an opaque illustration of the wave theory of light. The Arago spot is a sphere. II: Image formation and resolution consideration,’’ Appl. Opt. 31, direct manifestation of the existence of a boundary wave. 1385–1398 ͑1992͒. ͑3͒ Because of the practically achromatic character of im- 12J. H. McLeod, ‘‘The axicon: A new type of optical element,’’ J. Opt. Soc. aging, we can observe images of objects in their natural col- Am. 44, 592–597 ͑1954͒. 13L. M. Soroko, ‘‘Axicons and meso-optical imaging devices,’’ in Progress ors by means of a steel ball illuminated by white light. ͑ ͒ ͑4͒ A very long depth of focus. The Arago spot belongs to in Optics, edited by E. Wolf Elsevier, New York, 1989 , Vol. 27, pp. 109–160. the family of axicons and forms a long focal segment behind 14Z. Jaroszewicz, ‘‘Axicons. Design and propagation properties,’’ in Re- the ball. Therefore the inverted, real images in white light search and Development Treatises, edited by M. Pluta ͑Polish Chapter of can be practically observed in any plane behind the ball. SPIE, Warsaw, 1997͒, Vol. 5.

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