Doug Kerr
Well-known member
In this note I discuss a curious arrangement found in many incident light photographic exposure meters.
Illuminance
The illuminance upon a surface from illumination is defined as the amount of luminous flux that lands on a unit area of the surface.
It turns out that if the illumination arrives as a "beam" from a particular direction, with a luminous flux density of e, then the illuminance E, created on the surface is given by:
The factor cos A does not come from any mysterious physical phenomenon, but merely from geometry. Consider a portion of the beam with a cross section of 1 mm², within which there is traveling a certain amount of luminous flux. If that beam arrives at an angle to the perpendicular, then the landing area of that portion on the surface will be 1/(cos A) mm². Thus the density of flux on the surface (the illuminance) will be less than the flux density in the beam by the factor cos A.
Illuminance measurement
If we wish to measure the illuminance on a surface, it would seem that we could use a meter with a flat photoreceptor (of known area), where the meter would respond to the total luminous flux landing on the receptor (regardless of the alignment of the "beam" that brought the flux). Because of the same logic we saw above, that means that we could say that the "sensitivity" of the meter to a beam of luminous flux with a certain flux density, arriving at a certain angle, was proportional to the cosine of that angle. We can describe that receptor as having a "cosine directivity response" (often just "cosine response" for short).
In fact, for various physical reasons, a simple flat photoreceptor will not usually have a cosine response, especially as we get to greater angles of incidence. We can give it nearly a cosine response by fitting it with a transparent plate populated with tiny lenses or prisms of a certain design. And in fact traditional illuminance meters (such as those used in the field of office lighting design) often have just such a design.
Incident light exposure metering
In the basic concept of incident light exposure metering, we measure the illuminance of the incident illumination on the subject with an incident light exposure meter - just an illuminance meter. From that determination, together with knowledge of the sensitivity of the film or digital sensor, the meter presents a recommended photographic exposure (combination of shutter speed and aperture). This leads to a certain presumably-desirable relationship between the reflectance of each scene element and the photometric exposure it receives on the focal plane.
Note that, precisely, such a meter determines the illuminance as it would be on a plane of a certain orientation (the plane in which the meter receptor lies).
But if the illumination is not "omnidirectional" (comprising in effect beams of equal luminous flux density arriving from every direction) then the illuminance will not be the same on every plane, and thus not on every element of the subject, but will differ for elements facing in different directions.
Thus the concept of incident light exposure measurement cannot be precisely fulfilled. (No kind of metering will overcome that reality.)
A useful accommodation
A (hopefully) useful accommodation of this imperfection might be to determine the average illuminance over all subject surfaces - well, over all those surfaces the camera could see. (Clearly, the illuminance on any part of the "far side" of a subject's head should not enter into this.)
A clever technique for approximating such a determination is to use a meter with a hemispherical receptor (a technique originally introduced by famed cinematographer Don Norwood). This receptor is in effect a proxy for the head of a subject - well, the part whose surface can be seen by the camera.
The total luminous flux landing on this spherical receptor is proportional to the average illuminance created on the receptor (averaged by surface area). The meter reading hopefully will be proportional to that total flux onto the receptor. This will be an indication of the average illuminance on the receptor, and thus hopefully an indicator of the average luminance on the (camera-facing) surfaces of the head of our subject.
It turns out that such a hemispherical receptor gives the meter a "directivity response" that follows not the cosine of the angle of incidence but rather corresponds to a mathematically-defined curve called a cardioid.
Norton, who developed an exposure meter that exploited this principle, soon recognized that the substantial cost of manufacturing a hemispherical photoreceptor could be alleviated by using a conventional flat detector over which was placed a translucent hemispherical dome. Such a dome is a prominent feature of many serious incident light exposure meters today.
But sometimes ...
But there are several tasks for which we want to measure the illuminance on a specific plane (for example, when photographing a truly flat object such as a painting or document). This requires that the meter have a cosine response.
A traditional way to provide for this was to arrange the meter so the hemispherical dome could be removed, and probably replaced with a "cosine response corrector plate". This of course meant that there were loose parts to get lost. (Sometimes a third "front end" was provided to adapt the same receptor to measure reflected light, for use in the reflected light exposure metering scheme, a third doo-dad to lose.)
A scheme that avoided this problem was to arrange the hemispherical dome so it could be slid to the side, out of play, and perhaps provide a cosine response corrector plate, normally out of the way, that could be slid into place. But this took up a lot of "real estate" on the meter.
The retractable dome scheme
A more handsome arrangement has been provided in several contemporary incident light exposure meters, including models by Sekonic and others. Here, to place the meter into the "cosine response" mode, a lever, or perhaps a knurled ring around the base of the dome, is turned, the visible result being that the dome (and receptor) is retracted into the meter until the apex of the dome is about flush with the surrounding surface.
This having been done, the meter will exhibit a cosine directivity pattern, just as would result from an "ideal" flat receptor (or a cosine-corrected real flat receptor).
Wow! That must involve some really tricky optical theory. Not at all - the principle is almost agricultural in its working.
How does this happen?
Consider the "mouth" of this chimney. This is the orifice through which any light that is to reach the dome, and thus ultimately the receptor beneath, must enter.
In fact, for a "beam" of light from any direction, with a certain flux density, the amount of flux entering the chimney will be proportional not to the actual area of the mouth but to the area as it is "projected" from the direction in which the beam arrives. So for any arrival other than along the axis of the chimney, the effective port is an ellipse. And its area is just the actual area of the mouth times the cosine of the angle from which the beam comes.
Aha! The plot thickens.
Now, the flux that enters the mouth strikes either the dome or the shiny wall. In the later case, it is reelected and eventually strikes the dome (maybe after more than one bounce) (with negligible absorption).
We assumed before (without explicitly saying so) that any flux that strikes the surface of the dome, at whatever point, from whatever angle, is taken into account in the same way by the meter. If we still accept that, then we see that our "dome retracted into a chimney" configuration should exhibit a cosine response.
How about that!
I have yet been unable to find the patent on thus very clever arrangement.
Best regards,
Doug
Illuminance
The illuminance upon a surface from illumination is defined as the amount of luminous flux that lands on a unit area of the surface.
It turns out that if the illumination arrives as a "beam" from a particular direction, with a luminous flux density of e, then the illuminance E, created on the surface is given by:
e = E cos A
where A is the angle of incidence of the beam, measured from a line perpendicular to the surface.The factor cos A does not come from any mysterious physical phenomenon, but merely from geometry. Consider a portion of the beam with a cross section of 1 mm², within which there is traveling a certain amount of luminous flux. If that beam arrives at an angle to the perpendicular, then the landing area of that portion on the surface will be 1/(cos A) mm². Thus the density of flux on the surface (the illuminance) will be less than the flux density in the beam by the factor cos A.
Illuminance measurement
If we wish to measure the illuminance on a surface, it would seem that we could use a meter with a flat photoreceptor (of known area), where the meter would respond to the total luminous flux landing on the receptor (regardless of the alignment of the "beam" that brought the flux). Because of the same logic we saw above, that means that we could say that the "sensitivity" of the meter to a beam of luminous flux with a certain flux density, arriving at a certain angle, was proportional to the cosine of that angle. We can describe that receptor as having a "cosine directivity response" (often just "cosine response" for short).
In fact, for various physical reasons, a simple flat photoreceptor will not usually have a cosine response, especially as we get to greater angles of incidence. We can give it nearly a cosine response by fitting it with a transparent plate populated with tiny lenses or prisms of a certain design. And in fact traditional illuminance meters (such as those used in the field of office lighting design) often have just such a design.
Incident light exposure metering
In the basic concept of incident light exposure metering, we measure the illuminance of the incident illumination on the subject with an incident light exposure meter - just an illuminance meter. From that determination, together with knowledge of the sensitivity of the film or digital sensor, the meter presents a recommended photographic exposure (combination of shutter speed and aperture). This leads to a certain presumably-desirable relationship between the reflectance of each scene element and the photometric exposure it receives on the focal plane.
Note that, precisely, such a meter determines the illuminance as it would be on a plane of a certain orientation (the plane in which the meter receptor lies).
But if the illumination is not "omnidirectional" (comprising in effect beams of equal luminous flux density arriving from every direction) then the illuminance will not be the same on every plane, and thus not on every element of the subject, but will differ for elements facing in different directions.
Thus the concept of incident light exposure measurement cannot be precisely fulfilled. (No kind of metering will overcome that reality.)
A useful accommodation
A (hopefully) useful accommodation of this imperfection might be to determine the average illuminance over all subject surfaces - well, over all those surfaces the camera could see. (Clearly, the illuminance on any part of the "far side" of a subject's head should not enter into this.)
A clever technique for approximating such a determination is to use a meter with a hemispherical receptor (a technique originally introduced by famed cinematographer Don Norwood). This receptor is in effect a proxy for the head of a subject - well, the part whose surface can be seen by the camera.
The total luminous flux landing on this spherical receptor is proportional to the average illuminance created on the receptor (averaged by surface area). The meter reading hopefully will be proportional to that total flux onto the receptor. This will be an indication of the average illuminance on the receptor, and thus hopefully an indicator of the average luminance on the (camera-facing) surfaces of the head of our subject.
It turns out that such a hemispherical receptor gives the meter a "directivity response" that follows not the cosine of the angle of incidence but rather corresponds to a mathematically-defined curve called a cardioid.
Norton, who developed an exposure meter that exploited this principle, soon recognized that the substantial cost of manufacturing a hemispherical photoreceptor could be alleviated by using a conventional flat detector over which was placed a translucent hemispherical dome. Such a dome is a prominent feature of many serious incident light exposure meters today.
But sometimes ...
But there are several tasks for which we want to measure the illuminance on a specific plane (for example, when photographing a truly flat object such as a painting or document). This requires that the meter have a cosine response.
A traditional way to provide for this was to arrange the meter so the hemispherical dome could be removed, and probably replaced with a "cosine response corrector plate". This of course meant that there were loose parts to get lost. (Sometimes a third "front end" was provided to adapt the same receptor to measure reflected light, for use in the reflected light exposure metering scheme, a third doo-dad to lose.)
A scheme that avoided this problem was to arrange the hemispherical dome so it could be slid to the side, out of play, and perhaps provide a cosine response corrector plate, normally out of the way, that could be slid into place. But this took up a lot of "real estate" on the meter.
The retractable dome scheme
A more handsome arrangement has been provided in several contemporary incident light exposure meters, including models by Sekonic and others. Here, to place the meter into the "cosine response" mode, a lever, or perhaps a knurled ring around the base of the dome, is turned, the visible result being that the dome (and receptor) is retracted into the meter until the apex of the dome is about flush with the surrounding surface.
This having been done, the meter will exhibit a cosine directivity pattern, just as would result from an "ideal" flat receptor (or a cosine-corrected real flat receptor).
Wow! That must involve some really tricky optical theory. Not at all - the principle is almost agricultural in its working.
How does this happen?
First, my apologies for not having illustrations to support this discussion. I hope to prepare some soon.
The dome actually retracts into a snug-fitting cylindrical "chimney" with a shiny reflective wall.Consider the "mouth" of this chimney. This is the orifice through which any light that is to reach the dome, and thus ultimately the receptor beneath, must enter.
In fact, for a "beam" of light from any direction, with a certain flux density, the amount of flux entering the chimney will be proportional not to the actual area of the mouth but to the area as it is "projected" from the direction in which the beam arrives. So for any arrival other than along the axis of the chimney, the effective port is an ellipse. And its area is just the actual area of the mouth times the cosine of the angle from which the beam comes.
Aha! The plot thickens.
Now, the flux that enters the mouth strikes either the dome or the shiny wall. In the later case, it is reelected and eventually strikes the dome (maybe after more than one bounce) (with negligible absorption).
We assumed before (without explicitly saying so) that any flux that strikes the surface of the dome, at whatever point, from whatever angle, is taken into account in the same way by the meter. If we still accept that, then we see that our "dome retracted into a chimney" configuration should exhibit a cosine response.
How about that!
I have yet been unable to find the patent on thus very clever arrangement.
Best regards,
Doug