Light Measurement Handbook, by Alex Ryer Measurement  Geometries

Solid  Angles

One of the key concepts to understanding the relationships between measurement geometries is that of the solid angle, or steradian.  A sphere contains 4p steradians.  A steradian is defined as the solid angle which, having its vertex at the center of the sphere, cuts off a spherical surface area equal to the square of the radius of the sphere.  For example, a one steradian section of a one meter radius sphere subtends a spherical surface area of one square meter.
Fig. 7.1  Section view of a steradian removed from a sphere.
The sphere shown in cross section in figure 7.1 illustrates the concept.  A cone with a solid angle of one steradian has been removed from the sphere.  This removed cone is shown in figure 7.2.  The solid angle, W, in steradians, is equal to the spherical surface area, A, divided by the square of the radius, r.

Most radiometric measurements do not require an accurate calculation of the spherical surface area to convert between units.  Flat area estimates can be substituted for spherical area when the solid angle is less than 0.03 steradians, resulting in an error of less than one percent.  This roughly translates to a distance at least 5 times greater than the largest dimension of the detector.  In general, if you follow the ďfive times ruleĒ for approximating a point source, you can safely estimate using planar surface area.

Fig. 7.2  One steradian.

Radiant and Luminous Flux

Fig. 7.3 Total flux output.
Radiant flux is a measure of radiometric power.  Flux, expressed in watts, is a measure of the rate of energy flow, in joules per second.  Since photon energy is inversely proportional to wavelength, ultraviolet photons are more powerful than visible or infrared.

Luminous flux is a measure of the power of visible light.  Photopic flux, expressed in lumens, is weighted to match the responsivity of the human eye, which is most sensitive to yellow-green.  Scotopic flux is weighted to the sensitivity of the human eye in the dark adapted state.

Units Conversion: Power

 
l 
nm
Photopic 
Luminous 
Efficiency 
Photopic 
lm / W 
Conversion 
Scotopic 
Luminous 
Efficiency 
Scotopic 
lm / W 
Conversion
380 
390 
400 
410 
420 
430 
440 
450 
460 
470 
480 
490 
500 
507 
510 
520 
530 
540 
550 
555 
560 
570 
580 
590 
600 
610 
620 
630 
640 
650 
660 
670 
680 
690 
700 
710 
720 
730 
740 
750 
760 
770
0.000039 
.000120 
.000396 
.001210 
.004000 
.011600 
.023000 
.038000 
.060000 
.090980 
.139020 
.208020 
.323000 
.444310 
.503000 
.710000 
.862000 
.954000 
.994950 
1.000000 
.995000 
.952000 
.870000 
.757000 
.631000 
.503000 
.381000 
.265000 
.175000 
.107000 
.061000 
.032000 
.017000 
.008210 
.004102 
.002091 
.001047 
.000520 
.000249 
.000120 
.000060 
.000030
0.027 
0.082 
0.270 
0.826 
2.732 
7.923 
15.709 
25.954 
40.980 
62.139 
94.951 
142.078 
220.609 
303.464 
343.549 
484.930 
588.746 
651.582 
679.551 
683.000 
679.585 
650.216 
594.210 
517.031 
430.973 
343.549 
260.223 
180.995 
119.525 
73.081 
41.663 
21.856 
11.611 
5.607 
2.802 
1.428 
0.715 
0.355 
0.170 
0.082 
0.041 
0.020
0.000589 
.002209 
.009290 
.034840 
.096600 
.199800 
.328100 
.455000 
.567000 
.676000 
.793000 
.904000 
.982000 
1.000000 
.997000 
.935000 
.811000 
.650000 
.481000 
.402000 
.328800 
.207600 
.121200 
.065500 
.033150 
.015930 
.007370 
.003335 
.001497 
.000677 
.000313 
.000148 
.000072 
.000035 
.000018 
.000009 
.000005 
.000003 
.000001 
.000001
1.001 
3.755 
15.793 
59.228 
164.220 
339.660 
557.770 
773.500 
963.900 
1149.200 
1348.100 
1536.800 
1669.400 
1700.000 
1694.900 
1589.500 
1378.700 
1105.000 
817.700 
683.000 
558.960 
352.920 
206.040 
111.350 
56.355 
27.081 
12.529 
5.670 
2.545 
1.151 
0.532 
0.252 
0.122 
.060 
.030 
.016 
.008 
.004 
.002 
.001

 

Spectroradiometry  is the calibrated analysis of light from radiant sources, e.g. the sun, lamps and other light sources.

Photometry  involves measurement of radiation visible to the human eye.

Light source

Accessory

Radiometric unit

Photometric unit

Tungsten halogen lamp

Integrating sphere

Radiant power
[W/nm]

Luminous flux
[lm]

LED

LED adapter

Radiant intensity
[W/sr nm]

Luminous intensity [cd]

Sun

External optical probe

Irradiance
[W/m2 nm]

Illuminance
[lux]

Display

Telescope head

Radiance
[W/cm2 sr nm]

Luminance
[cd/m2]

 
 

  Irradiance   and   Illuminance:

Irradiance  is a measure of radiometric flux per unit area, or flux density.
Irradiance  is typically expressed in W/cm2 (watts per square centimeter) or W/m2 (watts per square meter).

Illuminance  is a measure of photometric flux per unit area, or visible flux density.
Illuminance  is typically expressed in lux (lumens per square meter) or foot-candles (lumens per square foot).

Fig. 7.4 Irradiance.
In figure 7.4, above, the lightbulb is producing 1 candela.  The candela is the base unit in light measurement, and is defined as follows:  a 1 candela light source emits 1 lumen per steradian in all directions (isotropically).  A steradian is defined as the solid angle which, having its vertex at the center of the sphere, cuts off an area equal to the square of its radius.  The number of steradians in a beam is equal to the projected area divided by the square of the distance.

So, 1 steradian has a projected area of 1 square meter at a distance of 1 meter.  Therefore, a 1 candela (1 lm/sr) light source will similarly produce 1 lumen per square foot at a distance of 1 foot, and 1 lumen per square meter at 1 meter. Note that as the beam of light projects farther from the source, it expands, becoming less dense. In fig. 7.4, for example, the light expanded from 1 lm/ft2 at 1 foot to 0.0929 lm/ft2 (1 lux) at 3.28 feet (1 m).

Cosine Law

Irradiance measurements should be made facing the source, if possible.  The irradiance will vary with respect to the cosine of the angle between the optical axis and the normal to the detector.

Calculating Source Distance

Lenses will distort the position of a point source.  You can solve for the virtual origin of a source by measuring irradiance at two points and solving for the offset distance, X, using the Inverse Square Law:
E1(d1 + X)2 = E2(d2 + X)2

Figure 7.5 illustrates a typical setup to determine the location of an LEDís virtual point source (which is behind the LED due to the built-in lens).  Two irradiance measurements at known distances from a reference point are all that is needed to calculate the offset to the virtual point source.

Fig. 7.5 Solving for the distance from source to sensor.

Units Conversion: Flux Density

Radiance and Luminance:

Fig. 7.6  Radiance.
Radiance is a measure of the flux density per unit solid viewing angle, expressed in W/cm2/sr.  Radiance is independent of distance for an extended area source, because the sampled area increases with distance, cancelling inverse square losses.

The radiance, L, of a diffuse (Lambertian) surface is related to the radiant exitance (flux density), M, of a surface by the relationship:

L = M / p

Some luminance units (apostilbs, lamberts, and foot-lamberts) already contain p in the denominator, allowing simpler conversion to illuminance units.
 

Irradiance From An Extended Source:

The irradiance, E, at any distance from a uniform extended area source, is related to the radiance, L, of the source by the following relationship, which depends only on the subtended central viewing angle, q, of the radiance detector:
E = p L sin2(q/2)

So, for an extended source with a radiance of 1 W/cm2/sr, and a detector with a viewing angle of 3°, the irradiance at any distance would be 2.15 x 10-3 W/cm2.  This assumes, of course, that the source extends beyond the viewing angle of the detector input optics.

Units Conversion: Radiance & Luminance

Radiant and Luminous Intensity:

Fig. 7.7  Radiant intensity.
Radiant Intensity is a measure of radiometric power per unit solid angle, expressed in watts per steradian.  Similarly, luminous intensity is a measure of visible power per solid angle, expressed in candela (lumens per steradian).  Intensity is related to irradiance by the inverse square law, shown below in an alternate form:
I = E * d2

If you are wondering how the units cancel to get flux/sr from flux/area times distance squared, remember that steradians are a dimensionless quantity.  The solid angle equals the area divided by the square of the radius, so d2=A/W, and substitution yields:

I = E * A / W

The biggest source of confusion regarding intensity measurements involves the difference between Mean Spherical Candela and Beam Candela, both of which use the candela unit (lumens per steradian).  Mean spherical measurements are made in an integrating sphere, and represent the total output in lumens divided by 4p sr in a sphere.  Thus, a one candela isotropic lamp produces one lumen per steradian.

Fig. 7.8  both mean spherical candela and beam candela are expressed in cd.
Beam candela, on the other hand, samples a very narrow angle and is only representative of the lumens per steradian at the peak intensity of the beam.  This measurement is frequently misleading, since the sampling angle need not be defined.

Suppose that two LEDís each emit 0.1 lm total in a narrow beam: One has a  10° solid angle and the other a 5° angle.  The 10° LED has an intensity of 4.2 cd, and the 5° LED an intensity of 16.7 cd.  They both output the same total amount of light, however -- 0.1 lm.

A flashlight with a million candela beam sounds very bright, but if its beam is only as wide as a laser beam, then it wonít be of much use.  Be wary of specifications given in beam candela, because they often misrepresent the total output power of a lamp.
 

Units Conversion: Intensity

Converting Between Geometries

Converting between geometry-based measurement units is difficult, and should only be attempted when it is impossible to measure in the actual desired units.  You must be aware of what each of the measurement geometries implicitly assumes before you can convert.  The example below shows the conversion between lux (lumens per square meter) and lumens.
Fig. 7.9  Geometry conversion.

 

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