Atelier Bonryu(E)
pinhole photography
Laboratory: Pinhole Photography
Taking Pinhole Photographs - Remark
Remark#4: Optimum Pinhole Diameter
Huygens’ Principle: In order to understand why there is an optimum value for a pinhole diameter, we consider the mechanism of the pinhole phenomenon by using a simple model. We analyze the problem on the basis of the famous Huygens’ principle (1690, Holland, Christiaan Huygens, 1629 -1695). It should be noted that, as by the original Huygens’ principle the diffraction phenomenon of a light wave cannot be explained properly, the principle of Huygens and Fresnel is used for the analysis, where the effect of the interference is taken into account.
According to the Huygens’ principle, secondary spherical waves are emitted from points on a surface wavefront of an incident light wave and the envelop of the secondary waves form a new surface wavefront (Fig.1). Therefore, a light wave with a flat surface (a plane wave) remains a plane wave without change in every point.
Fig.1: Huygens’s principle
A new surface wavefront is formed by an envelope (a red line) of secondary waves each of which is emitted from a point on a previous surface wavefront as a source.
Fig.2: Small pinhole case
When a diameter of a pinhole is extremely small a light of a plane wave passing through the pinhole becomes a spherical wave.
Fig.3: Large pinhole case
When a diameter of a pinhole is very large a light of a plane wave passing through the pinhole remains a plane wave.
Behavior of a Diffracted Light Wave: It may be instructive to display the dependence of the behavior of the light beam passing through the pinhole on the pinhole size. We consider the behavior of a light wave with a wavelength of 
emitted from a point source at infinity. For simplicity we assume that the light wave passes through a slit instead of a pinhole and calculate the wave propagation on the basis of the Fresnel diffraction. Figure 4 shows contours of intensity distribution (white: high intensity, black: low intensity) of the light for cases of different slit widths, d=0.05, 0.1, 0.15, and 0.3 mm. Pink bands denote paths of the light wave when the diffraction is not taken into account. Though both the units of the vertical and the horizontal axes are millimeters, it should be noted that the horizontal axis is extremely expanded. Because the effect of the diffraction is more noticeable in the distance from the pinhole plate and in the case of small pinhole, an image projected on a distant screen becomes more sharper with increasing width of the slit.
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(b)
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Fig.4: Dependence of diffracted light intensity on pinhole size (slit width)
A slit is located on the horizontal axis and the incident light comes from beneath. The pink bands denote the light beam without the diffraction phenomenon. Sub-figures (1), (b), (c), and (d) show the cases with the slit width d=0.05, 0.1, 0.15, and 0.3 mm, respectively.
From Fig.4 it is not difficult to understand that the diffracted light diverges rapidly in the case of a narrow slit as d=0.05 mm but by using a wider slit as d=0.15 mm the width of the light beam is confined within the width of the slit at the distance of 100 mm from the pinhole plane. We made calculation of the distribution of the light intensity at the focal plane for the case of the slit width of d=0.15mm (Fig.5). It is also clearly seen that the minimum beam width is attained for d=0.15 mm.
Fig.5 Distribution of the light intensity on the focal plane (f=100 mm) for different slit widths, d=0.05, 0.1, 0.15, and 0.3 mm.
However, when the plane wave encounters a wall with a very small pinhole, a new secondary spherical wave is emitted from the pinhole as a point source (Fig.2). The new wavefront is that of the secondary wave itself because the pinhole is very small and the source is considered as one point without an area. By applying the Huygens’ principle to this spherical wave, it is understandable that the light wave after passing through the pinhole remains still a spherical wave in every point. In this case the ”pinhole phenomenon“ cannot be observed because the light wave does not go straight to the traveling direction of the incident plane wave.
Optimum diameter of a pinhole: A candidate of the optimum pinhole diameter is the diameter of the central circle of a zone plate,
. But it is not easy to justify this equation as a formula to determine the optimum pinhole diameter. Next, we consider the Fraunhofer diffraction. Exactly speaking, it is questionable whether the equation of Fraunhofer diffraction can be applied or not to this problem. However, the equation is not so bad anyway and may give a good approximate result in some measure. On the basis of the Fraunhofer diffraction one can derive an equation describing the intensity distribution of the diffracted light on the focal plane. The result shows that in the central circle the light is most intense and concentric circler zones surround the central circle, where the light intensity is very weak. The radius of the central circle is derived as 
. For the pinhole diameter optimization we impose a condition that the radius of the central circle of the image is equal to the radius of the pinhole (r=a). Then the pinhole radius is given as follows.
This equation is the same as that in the main text. There are more other formulas with different coefficient but differences are not essential by considering the difference due to the wave length of the light adopted for the calculation.
Fig.6 Determination of the optimum pinhole diameter
Determine the pinhole diameter from the diameter of the central circle of the distribution of the light diffracted by a circular pinhole.
