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M E A S U R I N G   L I G H T (continued)

6. Using the Inverse Distance Squared Law (continued)

6.1 Point Source and Distance
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Key words:
INVERSE DISTANCE SQUARED-LAW | BEAM WAIST | ANISOTROPIC SOURCE | SOURCE SIZE LIMIT

Deriving the inverse distance squared law:

POINT SOURCEs have already been used in this essay: see
fig. 4.2.3-a in para. 4.2.3,
fig. 5.2.2   in para. 5.2.2,
fig. 5.3.1-g in para. 5.3.1.

The latter has even some connection to the real world: a "100 W bulb" is treated as a point source!

Surely you're curious about what a point source really is. But right now I'll just admit that it's a small source. More details follow at the end of this paragraph.

First we will look one more time at a figure which you already saw named "fig. 5.2.2": All the luminous flux  (Phi)  that is emitted from the source hits the inner surface of a (thought-of) sphere. If the light is emitted isotropically, then it is evenly distributed across surface area  A  of the sphere with radius  r  and hence, producing illuminance  E :
E = (Phi) / A
= (Phi) / [4 * (pi) * r^2]

Since we need the influence of distance  r  on illuminance  E , we'd better write:
E(r) = (1/r^2) * (Phi) / [4 * (pi)]
...and this is why I call it the "INVERSE DISTANCE SQUARED"-LAW.

Usually people add the cosine factor (1) for the possibly tilted position of the light receiving plane:
 E(r) = (1/r^2) * (Phi) * cos[(epsilon)2] / [4 * (pi)]

where "(epsilon)2" is defined by fig. 4.3.3-b in para. 4.3.3 .

Now we take the first two steps into the real world direction:

Coping with ANISOTROPIC SOURCEs:

These sources are frequent. Remember LEDs, fluorescent tubes, all sorts of reflector lamps, ... How can we use our law here, where we got no even distribution of light all over the inner surface of the surrounding sphere?

Just remember that light rays (within the limits of our context) are -- and will always stay -- straight lines.
So, if the detector aperture is filled homogeneously at the distance  r1 , it will also be filled homogeneously at the distance  r2 > r1 , no matter how big r2 might become ... of course staying in the same angular position to the source.
Detector aperture represents one part of the sphere surface that we considered above. And, no matter what the rest of the sphere does: this part is stretched out with the square of the radius like the entire sphere is.
But beware of BEAM WAISTs: Focussed beams of light often have a point of least diameter. The above mentioned distances  r2  and  r1  always have to be measured beginning at this waist!

Consequence for the cookbook:

 Keep detector aperture small enough so as to have it homogeneously (over-)filled; cling to the same angle with respect to the source If there is a BEAM WAIST, take this waist as starting point for distance measurements now you can increase distance  r  from the ANISOTROPIC point source   (or, from the BEAM WAIST) and inverse distance squared law will stay valid

SOURCE SIZE LIMIT for point sources:

The limit depends on the accuracy you need.
In most cases, your instrument's accuracy will set the limit.
Usual instruments have errors no better than +/- 2%; very often just +/- 10% or even worse.
Let us call this error percentage  "+- e" .

Now look at fig. 6.1-b with different distances from source rim to detector and from source center to detector. We take the distance  rc  from source center to detector and calculate sphere surface  Ac :
Ac = 4 * (pi) * rc^2

Then we take the distance  rr  from source rim to detector and calculate sphere surface  Ar :
Ar = 4 * (pi) * rr^2

And here you get a percentage of sphere area error  +-EA :
+-EA = 100% * (Ar - Ac)   /   (Ar + Ac)
= 100% * (rr^2 - rc^2) / (rr^2 + rc^2)

And now you see that as long as
|+-EA| << |+-e|
you can use the inverse distance squared law confidently.

Using this law means dealing with quite a range of distances  r . For staying on the safe side you'd better use the smallest distance out of this range for our accuracy test.

Too much ado for this little test? - Instead, you may use two rules of thumb from (1):

 * If your source is a disk of diameter  D  and if it is a Lambertian emitter (fig. 4.3.3-c in para. 4.3.3), then the error will be less than 5% if distance  r  is at least two times  D. * If your source is a system like search lights or projectors, then better keep distance  r  equal to or greater than tenfold source diameter  D .

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 Subject used in source "fotometrisches Entfernungsgesetz" text ref. (1) R.Baer, "BELEUCHTUNGSTECHNIK GRUNDLAGEN", Verlag Technik Berlin, 1996, p. 30

Continued: 6.2 Line Source and Distance

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