The inverse square law

[Doug Kerr]

Especially in discussions of lighting technique, we often hear about the inverse square law.

I'll state it here in a quite precise form; I will used numbered notes to state important qualifications so as not to disturb the rhythm of the basic definition.

If we consider emission of light from a point source [1] in a particular direction [2], the luminous flux density of the "beam" varies as the inverse of the square of the distance from the source.

[1] A source whose dimensions are very small compared to the distance at which we consider its influence.

[2] A point source does not by definition necessarily exhibit the same luminous intensity in every direction.​

The variation in luminous flux density is also the variation in the illuminance deposited on a surface at that distance (for any given angle of arrival). In fact, it is this with which we are normally ultimately interested in matters of photographic exposure.

If we are thinking about a shoe-mounted flash unit, and subjects at substantial distances, the "point source" situation is fairly well followed.

But what happens if we do not have a point source - if we have what is called an "extended" source, one whose dimensions are not inconsequential, then the inverse square law does not pertain. This would typically be so for the usual studio or field lighting system.

The relationship of luminous flux density (and thus of illuminance) with distance is very complex in the general case. We can treat it handily if we assume a very ideal light source. Its properties are:

• Its face is circular
• Its luminance is uniform across its face.
• Its emission is Lambertian. One thing that means is that its luminance is the same regardless of from what angle the source is viewed.

Such a source is shown in this figure.

The source is circular, with radius R. Its luminance is L at every place on its face. Its emission is Lambertian.

We consider a subject surface at a distance of S from the source (measured along a line from the center of the source and perpendicular to it). The subject plane is parallel to the source (perpendicular to the aforementioned line). Our interest is in the illuminance, E, created by the source on a point, P, on the subject plane, on the aforementioned line.

The figure shows the mathematical expression for this illuminance. You will see that it does not correspond to the inverse square law (the relationship is more complicated than that).

But, if we envision a situation in which the distance is substantially greater than the radius of the source (that is, the size of the source is "negligible" compared to the distance to the subject - we can consider it be essentially a point source), the expression degenerates to a simpler one - exactly the inverse square law.

What happens if:

• The source face isn't circular?
• The luminance is not uniform across the source face?
• The emission from the source isn't Lambertian?
• The target point of interest is not on the perpendicular line from the center of the source?

For almost any actual situation of photographic interest, one or more of these would be so.

In such a case, the relationship would not be the simple one shown in the figure. Exactly what it would be would of course depend on the specific details of all the behavioral factors mentioned above.

Best regards,

Doug

 

[Asher Kelman]

Especially in discussions of lighting technique, we often hear about the inverse square law.

I'll state it here in a quite precise form; I will used numbered notes to state important qualifications so as not to disturb the rhythm of the basic definition.

If we consider emission of light from a point source [1] in a particular direction [2], the luminous flux density of the "beam" varies as the inverse of the square of the distance from the source.

[1] A source whose dimensions are very small compared to the distance at which we consider its influence.

[2] A point source does not by definition necessarily exhibit the same luminous intensity in every direction.​

The variation in luminous flux density is also the variation in the illuminance deposited on a surface at that distance (for any given angle of arrival). In fact, it is this with which we are normally ultimately interested in matters of photographic exposure.

If we are thinking about a shoe-mounted flash unit, and subjects at substantial distances, the "point source" situation is fairly well followed.

But what happens if we do not have a point source - if we have what is called an "extended" source, one whose dimensions are not inconsequential, then the inverse square law does not pertain. This would typically be so for the usual studio or field lighting system.

The relationship of luminous flux density (and thus of illuminance) with distance is very complex in the general case. We can treat it handily if we assume a very ideal light source. Its properties are:

• Its face is circular
• Its luminance is uniform across its face.
• Its emission is Lambertian. One thing that means is that its luminance is the same regardless of from what angle the source is viewed.

Such a source is shown in this figure.

The source is circular, with radius R. Its luminance is L at every place on its face. Its emission is Lambertian.

We consider a subject surface at a distance of S from the source (measured along a line from the center of the source and perpendicular to it). The subject plane is parallel to the source (perpendicular to the aforementioned line). Our interest is in the illuminance, E, created by the source on a point, P, on the subject plane, on the aforementioned line.

The figure shows the mathematical expression for this illuminance. You will see that it does not correspond to the inverse square law (the relationship is more complicated than that).

But, if we envision a situation in which the distance is substantially greater than the radius of the source (that is, the size of the source is "negligible" compared to the distance to the subject - we can consider it be essentially a point source), the expression degenerates to a simpler one - exactly the inverse square law.]

Doug,

You are indeed a treasure. I do like the way you translate and extend our discussions back and forth and then come our with a clear succinct explanation which can be understood without even understanding the math equations that get simplified. Everyone should review their knowledge and assumptions by rereading and rereading your essays.

What happens if:

• The source face isn't circular?
• The luminance is not uniform across the source face?
• The emission from the source isn't Lambertian?
• The target point of interest is not on the perpendicular line from the center of the source?

For almost any actual situation of photographic interest, one or more of these would be so.

let me add

• the source is in a shaped reflector
• the source is focused with a reflector or lens
• the light is collimated and so does not diverge as expected

Although, each of these can be readily deduced by the exceptions you have just given, these three are the practical implications in the studio or on location with added lights!

In such a case, the relationship would not be the simple one shown in the figure. Exactly what it would be would of course depend on the specific details of all the behavioral factors mentioned above.

Here you have defined well the use of the inverse square law for light sources that are relatively tiny compared to the distances to the subject. Can we define better what that ratio needs to be?

Your short essay here helps to layout the conditions in which the inverse square law might be useful.

Thanks for spending the time to clarify things!

Asher

 

[Doug Kerr]

Hi, Asher.

Here you have defined well the use of the inverse square law for light sources that are relatively tiny compared to the distances to the subject. Can we define better what that ratio needs to be?

Well, as with all these things, there is no clear point at which the simplified relationship is "close enough".

But, to give some points of reference: assuming the various "ideal" properties of my example, if the distance is the times the radius of the source, the departure of the illuminance from the "inverse square law" result is about 10%; if the distance is 20 times the radius, the departure is about 5%. When the distance equals the radius, the illuminance is half that that would be predicted by the inverse square law (assuming we take our comparison level at a substantial distance).

On another front, if we constructed a (fully) collimated beam of finite cross section, then the luminous flux density would be completely independent of distance. But of course the "mouth" of the collimator would have to be as large as the entire subject, not easy to do in the context of normal photography.

Best regards,

Doug