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M E A S U R I N G   L I G H T (continued)
4.2 Radiometry at Light Emitting Plane (continued)

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Key words:
POINT SOURCE | SOLID ANGLE | POWER PER SOLID ANGLE | WATT PER STERADIAN | W/sr | FAR DISTANCE | ISOTROPIC | ANISOTROPIC | OVERFILLED APERTURE | GONIO-RADIOMETER | PIXELIZED CAMERA

Description, definition, unit:

First: what is "intensity"?
Well equipped with knowledge about SOLID ANGLEs, we will look (from a safe distance) with our bare eyes into a light source (which hopefully is not as intense as the sun is). In a dark night, you ask your friend to hold two incandescent lamps out of his or her window:
* one
bulb of 5 Watts and
* one
bulb of 100 Watts.

Then you take a walk of, say, 7 minutes away from his or her house. Now that you have reached a distance of about half a kilometer, you turn around and look back at the two shining lamps. Are you going to see any difference between the two lamps? - Of course you will.

The 5 Watt-bulb will be far less brilliant than its bigger brother ... if visible at all. Though their radiant exitances  M  (see para. 4.2.1) may be of nearly the same amount.
Looking at size and wattage of both lamps, we easily understand the reason. But: In their filaments' very near field both lamps are characterized by the same exitance  M  and from the distance of 0.5 kilometers, both bulbs resemble POINT SOURCEs - how can we quantify the difference?

We simply consider the radiated power per unit SOLID ANGLE. Into the full SOLID ANGLE (entire sphere) of 4*(pi) steradian, the big lamp emits about 20 times as much power as the smaller one does. And within the very small SOLID ANGLE that our eye's pupil subtends with the 500-meter distance, the ratio is the same (since the bulbs emit nearly ISOTROPICally).

Not to bore you, but just for practice, we'll calculate this very small SOLID ANGLE:
Area of pupil is     A2 = (8E-3m/2)^2 * (pi) = 5E-5 m^2
At the radius of 500m, spherical surface and flat area of this small size is nearly the same.
So, we get the SOLID ANGLE
(Omega)  =  area / (radius^2)  =  5E-5 m^2 / ((500 m)^2)  =  20E-11 steradian  = 6E-11*(pi) sr
The SOLID ANGLE of the entire sphere, that is  4*(pi) sr , is  7E10  times  as big!

Into this very narrow SOLID ANGLE, of course the 100 W-bulb radiates 20 times more power, than the 5 W-bulb does.

So we take this for a measure of distant light sources' intensity:
It is called Radiant Intensity  I  and it is calculated as  radiated POWER  (Phi)  PER UNIT SOLID ANGLE  (Omega) :

I  =  (Phi) / (Omega)

The unit of choice is
WATT PER STERADIAN = W/sr .

Intensity  I  is the one-and-only interesting property of light sources that shine from a FAR DISTANCE. And in this respect, a distance is "far", if the source's size is no more discernible ... that is, if the source can be called a POINT SOURCE.

Now we take a look at measurement procedures:

The light to be measured has obviously to illuminate the entrance aperture of the radiant power meter. And, as opposed to para. 4.1.1, the sensor's aperture has to be   O V E R F I L L E D   because we want to measure a flux density.
This is o.k. for light sources that radiate "ISOTROPICally", that is, with equal intensity into all directions. At these ISOTROPIC sources, you compute the SOLID ANGLE  (Omega)  that your detector aperture subtends with the source point and you divide the power meter readig by this SOLID ANGLE.

But this is not o.k. for light sources with "ANISOTROPIC" radiation. Here, you'd better know where in the source's radiation pattern you are measuring. For this point, you can call the sensor's aperture "dA" and its reading "d(Phi)". And then you compute in the same way
I  =  d(Phi) / d(Omega) ,
knowing that this is only a local value of radiant intensity  I .

With these ANISOTROPIC sources, you're going to be interested in the angular distribution of I. And you'll watch out that the detector's aperture is small enough to resolve the angular radiation pattern that your source produces.

So, stop down your detector! ... Stop it down? This will sharply cut the meter's reading!
Never mind. The meter's reading decreases linearly with the SOLID ANGLE that it subtends and that you use in your calculation. So, intensity  I  won't be impaired. Just don't get into difficulties with your meter's power resolution.

Having installed the optimum aperture stop on your sensor, you'll slowly swing the sensor at constant distance around the source and take one power meter reading at every interesting angle step. And then you may call your setup a goniometer or a GONIO-RADIOMETER.

Instrumentation details:

First let us consider sensitivity and lenses.
Looking from the source to the detector, the detector's aperture area subtends a certain solid angle. Look at fig. 4.2.3-a: there, the aperture area has 8 mm diameter.
And the detector will deliver a signal strength (and hence, a signal-to-noise ratio) proportional to the received radiation power. And this in turn is proportional to the solid angle, i.e. to the detector aperture area. Calls for large and expensive detectors.
That's why some intensity meters are equipped with a lens. The lens collects all light that it receives from the distant source onto its total entrance pupil. And focusses the collected light into its focal point. Where a small and inexpensive detector will exhibit good signal-to-noise ratio.
On the expense of an aiming problem: The focal point is only hit if your source is positioned on the optical axis of your lensed detector head.
So, if you intend to buy an intensity meter with lensed detector head: chose one that has some aiming aid, e.g. a mirror view-finder with crosshairs.

You have certainly understood that with lensed detector heads, "lens entrance pupil" has to be taken for "detector area" in the solid angle defining equation. That's why in goniometers with good angular resolution, lensed heads are of questionable use.

Can we use a pixelized (solid state) camera as a detector?
Of course we can.
Concerning "sensitivity and lenses", see the sentences just above.
Your application software would have to allow defining the source area in the image interactively.
And it's necessary for the software to have exact knowledge about dark signal and sensitivity of the image sensor. For exact measurements, individual values of dark signal and sensitivity will be needed for each and every pixel. For even more demanding tasks, you will need the entire light-to-signal transducing characteristics for every pixel.
Spectral sensitivity characteristics of the image sensor of course must be (corrected to be) broad-band flat out to specified wavelength limits.

If there is no intensity meter at hand, what substitute instrumentation can we use?
Well, we look at the definition
I  =  (Phi) / (Omega)
and find that we needn't but find a "fluxmeter", also called "radiant power meter", or "radiometer". With this instrument we measure  (Phi) , but this time with overfilled aperture.
Then we determine the aperture area of this radiometer's detector (in m^2) and the distance to the source (in m).
And now
I  =  (Phi) times distance-squared divided by area
is the intensity we wanted to find.
Be sure that distance is big enough for the plane aperture area to be nearly equal to the spherical area (radius of curvature is the distance to the source).
Especially for photometric measurements, the inexpensive luxmeters might tempt you to try them for measuring intensities; see para. 5.2.2.

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 Subject used in source human eye's pupil size text reference (1) Heinz Pforte: "Feinoptiker Teil II", VEB Verlag Technik Berlin, 1972, page 177