November 1962 LETTERS TO THE EDITOR 1305 Reflecting Surfaces as Magnifiers* JURGEN R. MEYER-ARENDT, MANFRED E. BAYER,† AND J. KENT NELSON Physics Department, Utah State University, Logan, Utah (Received April 20, 1962) HEN a bundle of light falls on a highly polished convex mirror, dust particles lying on it produce greatly magnified shadow images. 1 Such a system, because of the vanishing numerical aperture, has a rather limited resolving power and low intensity. Objects of finite thickness, furthermore, give shadows with doubled contours. On the other hand, objects of zero thickness can easily be represented by dull areas in the reflecting surface. This case is shown in Fig. 1 where the light is incident on a mirror S of radius R. The object extends from A to B. Since the "object distance" is virtually zero, it is obvious that the conventional mirror equation cannot be applied. Instead, it follows from construction that AB produces a shadow (neglecting diffraction) which extends laterally as I = b+p—a. We also see that b=(i+CD) tan2β. Since BD = R sinβ and AC = R sinα, p = BD — AC = R (sinβ—sinα). The image size then becomes To eliminate CD which is equal to CO —DO, we equate cosα = (CO)/R and cosβ=(DO)/R so that CD = R(cosα-cosβ). Setting β=α+γ we have where γ describes the size of the object in radians. If the object is w FIG. 1. Ray diagram for reflection by a convex spherical mirror and "zero object distance. FIG. 2. Theoretical (solid line) and experimental data (O) showing distortion of the shadow image produced by a convex mirror (i′ = 10 cm; R—5 cm; object size =2.4 mm; γ =2° 40'). small, term R[sin(α+γ)-sino:] becomes negligible and may be omitted. For 7 = 0.01 rad, for instance, it would be 0.3%-0.6%, depending on the angle of incidence. We have determined, for a variety of experimental conditions and for mirror radii from 3 mm to 29 cm, the distortions produced by such systems. Fig. 2 shows some of the results. It can be seen that distortion becomes a significant factor only when the angle of incidence exceeds a value of about α = 20°. Besides this, the plot shows that the experimental data perfectly fit the theoretical curve. * Supported by a research grant from the National Institutes of Health. † Present address: Institute for Cancer Research, Fox Chase, Phila- delphia, Pennsylvania. 1 M. E. Bayer, Naturwissenschaften 49, 177 (1962).

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Page 1: Reflecting Surfaces as Magnifiers

November 1962 L E T T E R S T O T H E E D I T O R 1305

Reflecting Surfaces as Magnifiers* JURGEN R. MEYER-ARENDT, MANFRED E. BAYER,† AND J. KENT NELSON

Physics Department, Utah State University, Logan, Utah (Received April 20, 1962)

HEN a bundle of light falls on a highly polished convex mirror, dust particles lying on it produce greatly magnified

shadow images.1 Such a system, because of the vanishing numerical aperture, has a rather limited resolving power and low intensity. Objects of finite thickness, furthermore, give shadows with doubled contours. On the other hand, objects of zero thickness can easily be represented by dull areas in the reflecting surface.

This case is shown in Fig. 1 where the light is incident on a mirror S of radius R. The object extends from A to B. Since the "object distance" is virtually zero, it is obvious that the conventional mirror equation cannot be applied. Instead, it follows from construction that AB produces a shadow (neglecting diffraction) which extends laterally as I = b+p—a. We also see that b=(i+CD) tan2β. Since BD = R sinβ and AC = R sinα, p = BD — AC = R (sinβ—sinα). The image size then becomes

To eliminate CD which is equal to CO —DO, we equate cosα = (CO)/R and cosβ=(DO)/R so that CD = R(cosα-cosβ). Setting β = α + γ we have

where γ describes the size of the object in radians. If the object is

FIG. 1. Ray diagram for reflection by a convex spherical mirror and "zero object distance.

FIG. 2. Theoretical (solid line) and experimental data (O) showing distortion of the shadow image produced by a convex mirror (i′ = 1 0 cm; R—5 cm; object size =2.4 mm; γ =2° 40').

small, term R[sin(α+γ)-s ino:] becomes negligible and may be omitted. For 7 = 0.01 rad, for instance, it would be 0.3%-0.6%, depending on the angle of incidence.

We have determined, for a variety of experimental conditions and for mirror radii from 3 mm to 29 cm, the distortions produced by such systems. Fig. 2 shows some of the results. It can be seen that distortion becomes a significant factor only when the angle of incidence exceeds a value of about α = 20°. Besides this, the plot shows that the experimental data perfectly fit the theoretical curve.

* Supported by a research grant from the National Institutes of Health. † Present address: Institute for Cancer Research, Fox Chase, Phila

delphia, Pennsylvania. 1 M. E. Bayer, Naturwissenschaften 49, 177 (1962).