Focus has bugged me from a lot of years. Light having a dual property is very difficult to understand. I started experimenting to find out more about focus. Here are the results along with the explanation. If you have a different interpretation, please share it with me.

 

Introduction to light: I really do not have a high-fi knowledge about the behavior or the theory of light; I will only try to explain the behavior as I found it during my experiments. There might be other interpretations to it, if so please share it with me. My experimental images are also presented here, so go ahead and verify. I will be assuming the ray theory of light here. This means that each and every point is visible to us due to the light reflected by it. The reflected light is spherically oriented in space (proved in SECTION 3). When it encounters an optical medium (henceforth will be assumed to be a generalized double convex lens), it is only a part of the sphere that comes in contact with it. This section wrt the point source forms a cone of light. Depending on the focal length of the lens the light rays are converged. If the image sensor is at the right place the point source forms a point image, else a circular patch of lesser intensity is formed, the center of the circle being the point where the focused image would have been formed. The radius and intensity of the circular patch depends on the distance between the point of focus and the sensor plane. Greater the distance, greater is the radius and lesser is the intensity.

If,

D = distance between the focus point and sensor.

R = radius of the circular patch formed.

I = intensity of the circular patch formed.

D = k1*R;

                                      D = K2*I;

    The figure shows a sphere, at the center of which is a point source. The sphere shows the divergence of light only to a certain distance, it is later continued by the portion the intercepts the lens. This is the cone that emerges out of the sphere. The ellipsoid is the biconvex lens. The lens will converge the light to a point, as shown by the other cone to the immediate left of the lens. The rectangular plate is the sensor. If placed at the right distance (here it the point at which the central cone converges), a focused image is formed. If the sensor plate is moved either to the front or back, it intersects the cone between the base and the vertex. The image formed will now be an unfocused image. The circular patch is the area of the circle that the plane intersects. The area multiplied by its intensity value is a constant, so as the radius of the circle increases the intensity of the circle decreases. Beyond the point of focus the image gets laterally inverted (will be proved in SECTION 3). This will only be observed if the light suffers an occlusion. This theory will be proved in a series of experiments that will now follow. The theory is that “Light spreads spherically in 3D from a point && it criss-crosses after the focus point”.

 

SECTION 1: Here even though there is a match stick occluding the central slant section of the light source present at some distance from the match-stick, the light source is completely recovered when it is focused.

Fig (2): The focus point is the match-stick. The light source has still not reached its focus point, so forms a circular patch. From many days I used to wonder, why does not the light, from the still not focused light source disturb the image of the focused match-stick when it spreads? Obviously the threat is from the points of the light source that lie at the edge of the match-stick. In fig(1) let the point source be at the edge, such that in the zoomed view the object in front of it cuts it exactly into half in the vertical direction (since the divergence of light is spherically symmetric any direction can be taken). Following from this, the sphere also gets cut into half and the cone also gets cut into half. Now since the object in front of it is focused, the cone is cut before its vertex, which is now a half circle and the circle exists away from the occluding object. The object in front of it stays intact.

Fig (3): The focus point is somewhere in between the light source and the stick. The stick is a solid object, as each and every point in it spreads there will be overlap on the central axis and hence the central line persists for a while.

Fig (4): The focus point here is the light source. The stick has spread more than in the earlier case. It is amazing to see the image of the entire light source, even though it was not seen completely when the stick was in focus. How then did it appear? This kind of an appearance depends on the intensity of the point in focus. It is not seen for image points that reflect small amount of light. Here it is a LIGHT SOURCE. The stick is sufficiently far from the source; hence a part of the light coming out of the source from the occluded space will be collected by the lens. If this is very small it will not be able to register itself, but here since the occlusion is small and source is relatively, sufficiently bright it WORKS! Since the light source is focused, the image of the point should be itself and not from some other neighboring point. The explanation goes back to the half cut cone concept explained earlier, but here it’s more than just a half cone. Earlier the point formed an edge, here it is fully masked! Now the vertex of the cone does not exist at all. So the first cut to the cone has to be made parallel to the base and at an offset from the vertex perpendicular to the base. The cone after the first cut is as shown in the figure below. The second cut will be done at an offset to the central axis passing through the centers of the circles at the left and right and perpendicular to the base plane. Of the two unequal pieces, the smaller one is what we are interested in. If the area of this dilapidated circle * the intensity value is sufficiently large the image gets restored. If this were to be true, why wasn’t there any light at the center when the focus point was in between the stick and the source in fig (3)? The final cone obtained is not shown in any figure, but imagining it once again, we can see that till the point actually gets nearly focused it will not register any light at its focus point. This is because we have cut the cone at an offset from the center. This offset is going to be preserved nearly till the end, hence the GAP!

Fig (6): Nearly the same kind of a situation continues as the focus is taken beyond the light source, but only till an interesting behavior is seen in the next image.

Fig (7): The focus point remains the same as the earlier case. But we observe that the right semicircle has disappeared, but why? This is because as seen TTL the light from the light source entering the lens from its left half is obstructed by placing some other object (not seen in the image since it is fully out of focused). Due to the criss-crossing of light after the focus point is reached, the closing of the left portion of the lens actually masks the left portion of the light source but the image records it as the right one.

Fig (8): Bringing back the focus point to the match-stick and obstructing the light from the left of the source, the left of the image gets masked properly, since now the focus point is not yet reached and so there is no criss-cross.

 

SECTION 2: The images below strictly concentrate to show only the inversion property. The circle is cut on the upper left side when the focus point is in front of the source and lower right side when the focus point moves ahead of it. But there are a few catches here, why did the entire image of the light source itself invert? It should have been just be the edge points, since the other points are seen fully. This contradicts the earlier explanation! Actually NO, in the earlier explanation in order to make the problem look simpler and enhance imagination only the edge points were taken into account. What the edge points suffer is also suffered by the other points of the source, but to a lesser extent! Since the light is diverging spherically and wrt the lens it is a cone, all the cones (from each and every point of the light source) are cut by the match stick, but since their images overlap we do not observe it. This is what is observed after the focus point is taken beyond the source. Now since the image is crossed the cut appears on the other side.

SECTION 3: Light from reflected surfaces: The concept of the cone changes slightly, when light from reflected surfaces is taken into account. It is this light that we generally see/perceive in our surrounding, because objects reflect light and not produce light. This reflection is not the same in all the directions and hence the circular cross section of the cone is not of uniform intensity. When we perceive it as a point it is the sum of all the light rays that meet at that point that we see. These light rays might not be coming from a single point. In all the above examples the circle was uniform in color because it was a source, but one should not expect the same thing from an object (a non-source). A circle formed by a non-source can be of varying intensity. An example is given below. When I take a photograph, light will already be added, so I can’t show you this. What I can show you here, is that over a wide space around the point of observation, light reflected by it shows variations. This is the reason I am moving over a wide space. The arc is the space over which I will move. Black circles are either your eye or in this case the camera lens. The point on the object under consideration is the point from which the lines are originating and moving towards the circles. Even though a lot of rays actually come from the point to each circle, only its average is considered here, which is the line. It is because in this kind of a setup it is difficult to detect individual rays. That is not necessary also, because if it fails for a bunch it fails for the individual ones too.

        If this sounds too complicated, just place a CD near you and try to observe a particular point where colors can be seen. From different view points you will be able to see different colors. This means that the same point on the CD is diffracting different colors. So if the aperture is big enough to accommodate all these colors, the color of the actual point will be the addition of these, but out of focusing this point would reveal all the individual colors. One more example is the mirror. Assume that in the above diagram the rectangular plate is a mirror and you are looking at a point on it from different places around. You would definitely see the images of different points in the surrounding, at the same point on the mirror. So for all you photographers out there, bigger aperture might solve ISO problems, but definitely not color reproduction!

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