Light sources with spikes, an interesting but not yet finished story

Version 0.1, Mon, 17 Nov 2014 00:09:20 +0100
not yet finished, see below therefor

Abstract

When taking a photo or recording a video, it occasionally happens that light sources appear as stars. The prevalent opinion is that the observed spikes have to be seen as results of Fraunhofer diffraction. Fraunhofer diffraction is well studied and an important result of that theory is that, under certain conditions, the optical disturbance observed from an aperture is nothing than the Fourier transform of the appropriate aperture function. On the other hand, the actually observed experimental material prompts questions. So the mentioned camera phenomena show sharp and long spikes without the typical diversity of fringes and color and texture patterns as they normally occur in diffraction figures. One consequence can therefore be to ask whether the Fraunhofer approximations actually apply to the Kirchhoff integral theorem in the present configuration. Another consequence is that there is plenty room for alternative approaches as the reflection one, though the latter approach has serious problems to explain, for example with stray light, the high symmetry properties of the patterns. After rigorously deriving a diffraction theory based on scalar spherical waves and applying that theory to known camera situations, it seems to be possible to validate that diffraction actually yields the experimentally observed material. Nevertheless, the author is still not ready to sweep the reflection approach completely away, whereat an acceptable quantitative ansatz is, at the present time, not available.

Photographs

Photograph 1: Camera Sony DSC-R1, sensor detail 3.3 x 2.2 mm²,
sensor detail center offset 1.5 mm horizontal and 3.7 mm vertical,
ISO 400, f = 23 mm, D = f / 8, t = 1/320 s, postprocessed

Photograph 2: Camera Sony DSC-R1, sensor detail 3.3 x 2.2 mm²,
sensor detail center offset 1.5 mm horizontal and 3.7 mm vertical,
ISO 400, f = 23 mm, D = f / 8, t = 1/20 s, postprocessed

Photograph 3: Camera Sony DSC-R1, sensor detail 3.3 x 2.2 mm²,
sensor detail center offset 1.5 mm horizontal and 3.7 mm vertical,
ISO 400, f = 23 mm, D = f / 8, t = 1.6 s, postprocessed

Photograph 4: Camera Sony DSC-R1, sensor detail 3.3 x 2.2 mm²,
sensor detail center offset 1.5 mm horizontal and 3.7 mm vertical,
ISO 400, f = 23 mm, D = f / 3.5, t = 1.3 s, postprocessed

Facts

 (1) Photographs 1 through 4 above were taken with the same camera and all of them show the same motif, except of course for some window that was closed at any time between taking photograph 3 and photograph 4. (2) The light sources are high-pressure sodium vapor streetlights. So the light is yellow but not monochromatic. (3) Exposure data belonging to photograph 1 was chosen in such a manner that the luminous objects appear brightly, near the saturation limit, but the image sensor was, according to the information the camera gave, still not saturated. (4) Assuming that the image sensor saturation limit would have just reached with photograph 1 exposure data, photograph 2 exposure data would mean a 16-fold image sensor saturation, photograph 3 exposure data a 512-fold saturation, and photograph 4 data a 2173-fold one. (5) The extension of the color fringes seen in photographs 1 and 2 roughly equals the one of 2 image pixels or 10 μm. That is about 20 times the average wavelength of light. (6) The color fringe patterns in photographs 1 and 2 correlate with certain high-contrast areas on the streetlight heads. (7) The color fringe patterns in photographs 1 and 2 are not visibly repeated. (8) Details of the tilt-and-turn windows seen in photograph 4 allow to guess the resolution of the optical system in all, being of the order of the extension of 1 image pixel or 5 μm. That is about 10 times the average wavelength of light. (9) The camera has a 7-blade iris diaphragm. The blades are dark with a matte finish. The blades can be seen and can be counted. (10) Luminous sources in photographs 2 and 3 appear with 14 spikes of high symmetry. (11) Experiences with spacing templates, these are tools made from steel, show that the mechanical stability of sheet metal significantly falls if the sheet thickness goes essentially under 50 μm. That is about 100 times the average wavelength of light. (12) The 14 spikes originating in the luminous sources in photographs 2 and 3 do not show color fringes, they appear constantly yellow but with radially decreasing intensity. (13) The visible length of the 14 spikes originating in the luminous sources in photographs 2 and 3 is about 1000 times the average wavelength of light. (14) Photograph 4 was taken with the iris diaphragm completely open. (15) Photograph 4 does not show spikes originating in the luminous sources but it shows large circular halos. (16) The width of the spikes seems to correlate with the extension of the luminous source while the diameter of the circular halo does not.

Sketchy conclusions

 |1| (5), (6) and (8) together suggest that the chromatic artifacts cannot be explained by chromatic aberrations of the optics. |2| (7) and (8) together suggest that the chromatic artifacts should not be explained by diffraction. |3| (3), (5), (6) and (7) make it probable that the chromatic artifacts seen in photographs 1 and 2 come from interpolation problems at edges with sharp intensity transitions, when transforming image sensor pixel data into image pixel data. Taking the image contents itself into account, it seems that, in HSV representation, Value and Saturation are good whereas Hue is not. In the present case the mentioned interpolation was performed by the camera, not by any external raw-format converter. |4| (9), (10), (14) and (15) suggest that the spikes originating in the luminous sources come from interactions between the incoming light and the iris diaphragm. |5| (8), (11), (12) and (13) might us persuade to believe that the spikes originating in the luminous sources should not be explained by diffraction effects. The resolution of the optical device is better than 5 μm, the effect under discussion here is a 1000-wavelength effect, and no part of the optical device has dimensions that come into the order of the wavelength. So some calculation is required. |6| In a properly focused and aligned imaging optics, there are two sets of conjugate planes that occur along the optical pathway through the system. In a camera as the Sony DSC-R1, one set consists of the object plane and the sensor plane and is referred to as the field or image-forming conjugate set, while the other set consists of only one member, the aperture stop or iris diaphragm plane, and is referred to as the pupil conjugate set, or as the illumination conjugate set in case of microscopes or electron beam microlithography machines, for example. Each plane within a set is said to be conjugate with the others in that set because the imaging condition between them is pairwise fulfilled. Contrariwise, each plane within a set is said here to be disjoint with the planes in the other set because some anti-imaging condition between them is pairwise fulfilled. The optical train of certain systems exhibits a segment where one finds two disjoint planes beside each other without an optically active device between them. In such a case, the relationship between the ray coordinates in the disjoint planes is quite obvious. Unfortunately, cameras do not belong to those systems. The point with respect to disjoint planes is that the ray coordinates exchange their roles when passing the way from a plane of the pupil conjugate set, the iris diaphragm plane, for example, to a plane of the field conjugate set, the image sensor plane, for example. So, in the iris diaphragm plane, the whole image information the light bundle carries is fairly completely encoded as a ray slope distribution, whereas the intersection points, the points where the rays pass that plane, are, with respect to the image contents, of nearly no importance. Therefore, iris diaphragms as a special implementation of aperture stops are normally mounted in pupil planes to control the image brightness without changing the field of view. If now the ray slope distribution, containing the essential image information, is somehow affected by certain perturbations in a pupil plane, then an image contents modification has to be expected in a subsequent field plane, i.e. in the sensor plane. Such modifications can be seen here in photographs 2, 3 and 4. We have three cases. Firstly, an important part of the rays passes the iris diaphragm without interaction, otherwise, there would not be a nice image on the sensor. Secondly, another important part of the rays is stopped down, either heating up the diaphragm or being reflected back, otherwise, the image on the sensor would probably be too bright. Finally, a small part of the rays hits the edges of the diaphragm, which are always rounded off, recall (11), and which are always somewhat reflecting, like a mirror, like a Lambertian, or like a mixture of both. Since the latter reflections take place in or near a pupil plane, the image on the sensor must suffer a certain modification. Specular reflections, the mirror-type ones, will cause quite sharp image contents changes that will correlate with the diaphragm shape. Diffuse reflections, which obey Lambert's law, will cause halos that will or will not correlate with the diaphragm shape. (3) and (9) might explain why photograph 1 does not show spikes, the percentage of by reflections additionally tilted rays is small. (4) might explain why photographs 2, 3 and 4 do show spikes and/or halos, the percentage of additionally tilted rays is small but the intensity is immense. With (9), we furthermore understand why photographs 2 and 3 show not only spikes but also halos, even matte surfaces can reflect anything, probably more like a Lambertian. (14) and (15) explain that there are no spikes but halos in photograph 4. There should be any circular limitation in or near the diaphragm plane. (10) suggests that there must occur a second reflection a little after the iris diaphragm to come from 7 blades to 14 spikes. We are in agreement with (12), single or double reflections do not change colors here. Summarizing, the spikes could also be explained by specular reflections at the iris diaphragm edges, with the option of other reflections somewhat after.