Doug Kerr
Well-known member
In some recent writings on the photometry of photographic lighting, I have mentioned a light source that was "Lambertian", a reference to Lambert's law (which really doesn't exactly apply to this case, I'm afraid). I though I would discuss Lambert's law a little.
Lambert's law - the ideal diffuse reflecting surface
Lambert's law describes the behavior of what is called an ideal diffuse reflecting surface. Elaborated a bit, it tells us that when such a surface is illuminated:
a. It will exhibit the same luminance ("brightness") from any angle of observation (not from behind it, please),
b. That luminance is proportional to the illuminance upon the surface (from the front of course), and
c. Property "a" will be true regardless of the angle from which the surface is illuminated.
By way of example, suppose we have a horizontal ideal diffuse reflecting surface. We illuminate it not from directly overhead but along a line that "leans 45° to the North".
We observe the surface from directly overhead (that is, along a line perpendicular to the surface) and note a certain luminance. We then observe it along a line at an angle of 30°with the perpendicular direction, to the East . We will observe the same illuminance. Then we observe the surface from 45° to the South. We observe the same illuminance.
Property "a" can be looked at in another way. Imagine a tiny patch on the illuminated surface, which we can consider a point source of light, and can thus speak of its luminous intensity in any direction. We find that its luminous intensity is greatest in the direction perpendicular to the surface, and along any other direction, is less by the cosine of the angle that direction makes with the perpendicular.
At first, that does not seem as though it would make the surface exhibit the same luminance from any direction of observation.
But luminance is the total amount of luminous intensity per unit area, where that area is not reckoned on the emitting surface but rather is area in a theoretical plane perpendicular to our line of observation (said to be the "projected area" to the observer).
If we observe a surface at an angle of 30° to the perpendicular, and consider an area of 1 square mm in the theoretical plane, it actually embraces an region of the surface whose area is about 1.155 square mm (that is 1/cos 30°).
Thus, the fact that a certain span of our view embraces more than the corresponding area on the surface, inversely as the cosine of the angle of observation, precisely cancels out the decline in overall luminous intensity per unit area on the surface. And thus we see the same luminance from any angle of observation, for an ideal diffuse reflecting surface.
Note by the way that, with regard to property "b", the angle of incidence of the illumination does figure in; the illuminance is the luminous flux density of the illuminating "beam" times the cosine of its angle of incidence (the angle is line of arrival makes with a line perpendicular to the surface). But, whatever the resulting illuminance, property "a" holds regardless of the angle of incidence that affected the illuminance.
Transmissive surfaces
This concept can be extended to an ideal diffuse transmissive surface ( a "translucent" surface). Here, if we illuminate such a surface "from the rear", then:
a. It will exhibit the same luminance ("brightness") from any angle of observation (not from behind it, please),
b. That luminance is proportional to the illuminance upon the surface (from the rear of course), and
c. Property "a" will be true regardless of the angle from which the surface is illuminated.
The correspondence with the properties of an ideal diffuse reflecting surface means that, for example, if we are using a "diffuser" on our camera lens to equip the camera to "capture" the incident illumination on a scene for purposes of determining its chromaticity, for use in conducting white balance color correction, then for the light collected to have the same effect it would have on a subject if the subject were Lambertian, the diffuser must be Lambertian.
Note that, for a Lambertian diffuser, the fact that the luminance presented on its front is proportional to the illuminance on its rear (regardless of the angle of arrival of the light) can be thought of as a specific description of the "acceptance pattern" of the rear, much as we would plot the "directivity pattern" of a receiving radio antenna. This "acceptance pattern", for a Lambertian diffuser, is a "cosine" pattern. That's because the impact on illuminance, for an arriving beam of any given luminous flux density (the measure of the "potency" of an arriving beam) is proportional to the cosine of its angle of arrival.
When I speak of a light source (such as a softbox-like studio light) being "Lambertian", I am actually applying the term beyond its intended realm (which is really only reflective surfaces). But by that I mean that, for such a source:
• a. It will exhibit the same luminance ("brightness") from any angle of observation (not from behind it, please).
This assumption, along with the assumption that this luminance is constant over the "face" of the light unit, is required to be able to, in any reasonable way, predict mathematically the effect of distance from the light on the illuminance afforded on a point on the subject.
Best regards,
Doug
Lambert's law - the ideal diffuse reflecting surface
Lambert's law describes the behavior of what is called an ideal diffuse reflecting surface. Elaborated a bit, it tells us that when such a surface is illuminated:
a. It will exhibit the same luminance ("brightness") from any angle of observation (not from behind it, please),
b. That luminance is proportional to the illuminance upon the surface (from the front of course), and
c. Property "a" will be true regardless of the angle from which the surface is illuminated.
By way of example, suppose we have a horizontal ideal diffuse reflecting surface. We illuminate it not from directly overhead but along a line that "leans 45° to the North".
We observe the surface from directly overhead (that is, along a line perpendicular to the surface) and note a certain luminance. We then observe it along a line at an angle of 30°with the perpendicular direction, to the East . We will observe the same illuminance. Then we observe the surface from 45° to the South. We observe the same illuminance.
Property "a" can be looked at in another way. Imagine a tiny patch on the illuminated surface, which we can consider a point source of light, and can thus speak of its luminous intensity in any direction. We find that its luminous intensity is greatest in the direction perpendicular to the surface, and along any other direction, is less by the cosine of the angle that direction makes with the perpendicular.
At first, that does not seem as though it would make the surface exhibit the same luminance from any direction of observation.
But luminance is the total amount of luminous intensity per unit area, where that area is not reckoned on the emitting surface but rather is area in a theoretical plane perpendicular to our line of observation (said to be the "projected area" to the observer).
If we observe a surface at an angle of 30° to the perpendicular, and consider an area of 1 square mm in the theoretical plane, it actually embraces an region of the surface whose area is about 1.155 square mm (that is 1/cos 30°).
Thus, the fact that a certain span of our view embraces more than the corresponding area on the surface, inversely as the cosine of the angle of observation, precisely cancels out the decline in overall luminous intensity per unit area on the surface. And thus we see the same luminance from any angle of observation, for an ideal diffuse reflecting surface.
Note by the way that, with regard to property "b", the angle of incidence of the illumination does figure in; the illuminance is the luminous flux density of the illuminating "beam" times the cosine of its angle of incidence (the angle is line of arrival makes with a line perpendicular to the surface). But, whatever the resulting illuminance, property "a" holds regardless of the angle of incidence that affected the illuminance.
Transmissive surfaces
This concept can be extended to an ideal diffuse transmissive surface ( a "translucent" surface). Here, if we illuminate such a surface "from the rear", then:
a. It will exhibit the same luminance ("brightness") from any angle of observation (not from behind it, please),
b. That luminance is proportional to the illuminance upon the surface (from the rear of course), and
c. Property "a" will be true regardless of the angle from which the surface is illuminated.
The correspondence with the properties of an ideal diffuse reflecting surface means that, for example, if we are using a "diffuser" on our camera lens to equip the camera to "capture" the incident illumination on a scene for purposes of determining its chromaticity, for use in conducting white balance color correction, then for the light collected to have the same effect it would have on a subject if the subject were Lambertian, the diffuser must be Lambertian.
Note that, for a Lambertian diffuser, the fact that the luminance presented on its front is proportional to the illuminance on its rear (regardless of the angle of arrival of the light) can be thought of as a specific description of the "acceptance pattern" of the rear, much as we would plot the "directivity pattern" of a receiving radio antenna. This "acceptance pattern", for a Lambertian diffuser, is a "cosine" pattern. That's because the impact on illuminance, for an arriving beam of any given luminous flux density (the measure of the "potency" of an arriving beam) is proportional to the cosine of its angle of arrival.
One manufacturer of white balance measurement diffusers speaks proudly of the fact his diffusers have a "narrow" acceptance pattern (the words used to present this do not make it immediately clear that is what is being said) - a pattern "narrower" than the cosine pattern generally thought desirable for this type of measurement. He asserts that this makes them especially suitable for one particular variation on the standard measurement technique.
Does he take special steps in design to attain this? No. In fact, any simple diffuser unavoidably has an "acceptance" pattern much narrower than the "cosine" pattern of a Lambertian diffuser. In fact, most professional measurement diffusers have special features (such as a pattern of micro-lenses) to attain the Lambertian cosine pattern. In this manufacturer's diffuser, there are just no such special features employed.
A "Lambertian" light sourceDoes he take special steps in design to attain this? No. In fact, any simple diffuser unavoidably has an "acceptance" pattern much narrower than the "cosine" pattern of a Lambertian diffuser. In fact, most professional measurement diffusers have special features (such as a pattern of micro-lenses) to attain the Lambertian cosine pattern. In this manufacturer's diffuser, there are just no such special features employed.
When I speak of a light source (such as a softbox-like studio light) being "Lambertian", I am actually applying the term beyond its intended realm (which is really only reflective surfaces). But by that I mean that, for such a source:
• a. It will exhibit the same luminance ("brightness") from any angle of observation (not from behind it, please).
This assumption, along with the assumption that this luminance is constant over the "face" of the light unit, is required to be able to, in any reasonable way, predict mathematically the effect of distance from the light on the illuminance afforded on a point on the subject.
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Best regards,
Doug