Inverse Square Law and window-light.

[Richard Mc]

A point source of light (eg. the sun) puts out a constant stream of photons in all directions. The intensity is light at any point in space looking back at the source is related to how closely the light rays are packed together. Picture a sphere (the sun) with 1000 evenly spaced arrows emanating out radially from the center. Close to the sun, the light rays are very closely spaced (ie. the arrows seem very closely packed here) and further out, less so. It is this exact phenomenon that gives us the maths for the inverse square rule as the surface area of a sphere is given by 4∏r*r....ie. proportional to the square of the radius. As photons get further out from the source they must spread out over an area that is proportional to the square of the distance from the source, hence the intensity (photons passing through unit area) is inversely proportional to the square of the distance from the source.

There is a very good explanation of this with diagrams at :

http://en.wikipedia.org/wiki/Inverse-square_law

Now, strictly speaking, this equation only applies for a point source radiating equally in all directions in a vacuum. Using this equation for non-point sources is only ever a rough approximation but it is helpful at times in pointing us in the right direction. Light modifiers (eg. reflectors etc.) and atmospheric conditions among other things cause the strictness of the equation to fall down to some degree.

Ignoring any atmospheric effects , the intensity of light from the sun at any two points on earth should be pretty similar, as the distances of these two points to the sun will be pretty similar, as the distance to the sun is large compared to the distances between any two points on earth.

Now once the photons hit something on earth (the side of the building that you can see out of the window or maybe the window shade) then their evenly spaced radially radiating relationship is no longer valid. They scatter in all directions (but not equally), so they become something like a new point source (or thousands of new little point sources) but, as they are not emanating from a single point and equally in every direction, the inverse square law (using the distance from the "new" light source to the subject) will only ever be a poor approximation.... but it will be more accurate than trying to apply the inverse square law using the distance to the sun as this no longer applies.

The light coming in through the window will probably obey the inverse square law to about the same degree as out photo lights and reflectors etc..... none of these strictly obey the inverse square law but it is a useful approximation to nudge us in the right direction. You'll just have to make some determination as to where the light in question is coming from and figure that into your approximation. If its direct sunlight, then the intensity won't change much if the subject is 1 foot from the window or 10 feet from the window (as long as they are still in direct sunlight). If it is reflected/scattered light from the sky then there will be slightly greater effect on moving the subject, and if the light is reflected/scattered from a nearby object (building, window shade), then the light will drop off much more with distance from the window.

The main use to me for the inverse square law is just to say that light drops off A LOT with distance. In the end, a bit of trial and error and a few meter readings usually gives us our correct settings.

[jim kitchen]

Ladies and Gentlemen,

I mention ladies if they decide to contribute or have contributed, but from what I remember about my practical physics in university and any subsequent light wave propagation, diffraction propagation or reflected light studies that I participated in while studying architecture, happens to be that I was taught that the Inverse Square Law did not have a corollary, unless a new science surfaced in the last few years that now indicates how the projected surface area measured from any light source type does not change with distance from the light source. I then believe that your hypotheses, your personal modifications to the ISL, or your interim interpretations of the ISL, whether it is a window, a door and, or a reflected source that allows a light source to strike the subject differently than what ISL emphatically states are absolutely incorrect.

To my knowledge, ISL does not have a corollary, no matter what the light source happens to be, whether it is diffused, diffracted, reflected, a single point source and, or whether the light source happens to be fifty feet away, or equal to the distance from our friendly neighbourhood sun.

I may be wrong after all these years complete with an aging memory, and although you may be able to apply ISL to celestial bodies where a mass in motion might have a Newtonian corollary, I would happily read your ISL dissertation or another physicist's paper, regarding the modification of ISL physics with interest...


jim k

[Jack Dahlgren]

Ladies and Gentlemen,

I mention ladies if they decide to contribute or have contributed, but from what I remember about my practical physics in university and any subsequent light wave propagation, diffraction propagation or reflected light studies that I participated in while studying architecture, happens to be that I was taught that the Inverse Square Law did not have a corollary, unless a new science surfaced in the last few years that now indicates how the projected surface area measured from any light source type does not change with distance from the light source. I then believe that your hypotheses, your personal modifications to the ISL, or your interim interpretations of the ISL, whether it is a window, a door and, or a reflected source that allows a light source to strike the subject differently than what ISL emphatically states are absolutely incorrect.

To my knowledge, ISL does not have a corollary, no matter what the light source happens to be, whether it is diffused, diffracted, reflected, a single point source and, or whether the light source happens to be fifty feet away, or equal to the distance from our friendly neighbourhood sun.

I may be wrong after all these years complete with an aging memory, and although you may be able to apply ISL to celestial bodies where a mass in motion might have a Newtonian corollary, I would happily read your ISL dissertation or another physicist's paper, regarding the modification of ISL physics with interest...


jim k

Not sure what you are going on about. As an example. the intensity of light at a point 4 inches behind a 100mm lens in bright sunlight is enough to kill an ant, a creature impervious to ordinary sunlight.

The window, if acting as a diffuser through presence of dirt, curtain, etc. will do the opposite, diffusing the light to a varying extent.

[jim kitchen]

Not sure what you are going on about. As an example. the intensity of light at a point 4 inches behind a 100mm lens in bright sunlight is enough to kill an ant, a creature impervious to ordinary sunlight.

The window, if acting as a diffuser through presence of dirt, curtain, etc. will do the opposite, diffusing the light to a varying extent.

Dear Jack,

I can only agree with your examples, but what I seem to be going on about, as you politely put it, happens to be about the known physical characteristic of the ISL, where the light source's intensity is inversely proportional to the distance from the source. I believe that no corollaries exist that modify the original law.

Your examples talk to a light source's modified intensity over a modified surface area. Your first example points to and introduces a physical change and reconcentration to the original light source intensity through an optical device, where the light source's intensity is obviously dispersed within a smaller concentrated surface area as it strikes the poor ant. Your first example will surely follow the ISL as the refocused and reconcentrated light source intensity leaves the lens's rear element.

The intensity of the light source within the ISL, as you state in your second example, will surely change too as the light source becomes diffused, dispersed and, or diffracted over a larger surface area. Every light source and modified light source complete with its inherent intensity will always follow the ISL, whether the light source passes through a window, a door opening, or happens to be reflected off a surface material and, or diffused by any translucent material.

I thought my ISL comments were clear about its known characteristic, but then again, I apologize if I was not clear enough...


jim k

[cjbroadbent]

We are talking about a window, a large rectangular light source facing the north sky. The conclusion, with regard to the Inverse Square Law, seems to be that the origin of the source is at the window frame, regardless of whether the window is covered with a diffuser or left wide open. Is that it?

[cowanw]

Yes. (in the northern hemisphere)
Regards
Bill

[Ben Syverson]

The soft light from a window is the collected reflections of the outside world, sky, etc. The sun and sky obviously don't suffer from ISL (at least on Earth). So what gives?

Well, imagine the window from the perspective of someone standing directly in front of it. From their perspective, the window is huge. Their face gets light from many different directions (top of the window, bottom, left, right), so the quality of the light is soft and bright. If they look out the window, they can see the landscape and clouds.

Now imagine the window from the perspective of someone standing 100 feet away (we'll assume it's the only window in a great hall). From their perspective, the window is tiny. Their face is only getting light from the very specific direction of the window, so the quality of light is hard, like a bare bulb. The light is also dimmer, which is NOT because the window light itself is suffering from ISL.

The light is dimmer because the percentage of reflected light in the world that can reach the subject has plummeted. The person standing at the window can see buildings, landscape and clouds, which all contribute reflected light to their face. The person at the end of the great hall perhaps only sees the mountaintop or church steeple through the window. The only light that's reaching them is the meager amount of light reflected off the steeple.

So it has nothing to do with ISL!

[jim kitchen]

We are talking about a window, a large rectangular light source facing the north sky. The conclusion, with regard to the Inverse Square Law, seems to be that the origin of the source is at the window frame, regardless of whether the window is covered with a diffuser or left wide open. Is that it?

Dear Christopher,

That is correct, and to add to your comment, a new and totally unique ISL is created from every reflective surface, if one could completely isolate, identify, and measure the reflected light source...

jim k

[Jim Michael]

The window scenario is open to a lot of hand waving, so I'll just assume a broad source where the window is providing the north light illumination. If you have a different interpretation then the following won't work.

Broad sources do follow ISL but it's complicated by the fact that you have to account for more than one point source, hence my reference to integrating over the source. For a source orthogonal to a line extending from the source to the subject, let d0 be the distance from the center of the source to the subject and d1 be the distance from the source at a point x units from the center, then d1 = sqrt(d0*d0 + a*a) and the change in intensity of the light originating from d0 to d1 follows the inverse square law. The intensity at the subject is the sum (integral) of those intensities.

And, ISL is nothing more than saying that as the light leaves a point source the area of the sphere onto which the light falls at a distance gets bigger by a predictable amount (look up area of sphere and calculate change in area between two radii of f-stop feet like 8 and 11). You just have to decide from where the light originates.

I looked up what I wrote before re the sources with a stated beam spread, e.g. fresnel:

Assume a source with a beam spread angle of phi and intensity I. At distance d, the intensity is I/area or I/pi(d tan phi)^2. The ratio of intensity at distances d' and d is i'/i = ((d' tan(phi))^2)/((d1 tan(phi))^2) so i' = i((d' tan(phi))^2)/((d1 tan(phi))^2) or i'=i*d'^2/d^2. Picking a couple of f-stop feet, 8 and 16, i' = 4.

And re the virtual point source:

Assume a beam from a source of radius r and beam divergence phi. We can think of this as a point source originating at a distance d0 behind the source where d0 = r/tan phi. Then when comparing illumination at a distances d' and d'', use the distance to the virtual point source in place of the physical source. In the case of a perfectly collimated beam the virtual point source is at infinity, so intensity i' = i''.

Check my math.

[cjbroadbent]

Jim K, thanks, you always have a balanced and well-informed reply to our primitive questions.
Jim M, Wow! I'ts going to take me awhile to work that out. I'll print it and hang it on the wall. The great thing about LF users is their wide knowlege base from other fields.