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Studies done on soft light & inverse square law


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Does anyone know of any studies done on soft light's relation to the inverse square law? Books, websites, forum members?

I'm very curious to know how deep people have gotten into the subject. My web searches get littered with basic lighting tutorials and the like.

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No two rays are perfectly parallel. So all light follows the inverse square, even lasers. The rate of fall is just scaled. Soft sources exhibit compounded math that describes the readings we get. I'm really looking for any available studies on the subject.

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I've never come across one. All light sources have a size; what matters is whether the size of the source is any significant proportion of the distance between the source and the subject, which anything that we'd call a "soft light" would be, by definition. A better mathematician would probably be able to trivially work out how to integrate it.

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3 hours ago, Stephen Sanchez said:

I'm really looking for any available studies on the subject.

https://en.wikipedia.org/wiki/Etendue to get one started. 

1 hour ago, Phil Rhodes said:

I've never come across one. All light sources have a size; what matters is whether the size of the source is any significant proportion of the distance between the source and the subject, which anything that we'd call a "soft light" would be, by definition. A better mathematician would probably be able to trivially work out how to integrate it.

As a quick rough guess, to get illuminance E(L) at distance L you'd have to integrate Intensity(r)*cos(angle with the normal)*dx/(r+L)^2 over r, which's a radius of a large light source. Then you can make some series expansions that will show E(L) is close to inverse square law and approaches it at big enough L/r. 

It's simple like this if every point of the large source emits a wide enough (and uniform over angle) cone of light that its rays reach the observer at any distance. The ideal case is a lambertian source - a frame of thick diffusion is pretty close to it. If you move far enough from a spotlight (there's a so called beam forming distance - not sure what's the correct English term), you become illuminated by rays coming from all parts of its lens' aperture and then the illuminance starts to decrease with the square of distance (no wonder as the lit area increases with the square of it - the whole inverse square thing comes from any surface being proportional to squared linear dimension - and conservation law, of course). Close to the (quasi-)collimated light source (less than around D/tan(beam spread/2)) you won't observe an inverse-square relationship as the 'apparent lens aperture' will be widening.

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Wow. What a response!

On 10/31/2020 at 4:50 PM, Michael Rodin said:

https://en.wikipedia.org/wiki/Etendue to get one started. 

This is great! It's a little over my head, though. It seems to refer to lensing, which is not my focus. I'll have to study it more.

On 10/31/2020 at 4:50 PM, Michael Rodin said:

As a quick rough guess, to get illuminance E(L) at distance L you'd have to integrate Intensity(r)*cos(angle with the normal)*dx/(r+L)^2 over r, which's a radius of a large light source. Then you can make some series expansions that will show E(L) is close to inverse square law and approaches it at big enough L/r. 

It's simple like this if every point of the large source emits a wide enough (and uniform over angle) cone of light that its rays reach the observer at any distance. The ideal case is a lambertian source - a frame of thick diffusion is pretty close to it. If you move far enough from a spotlight (there's a so called beam forming distance - not sure what's the correct English term), you become illuminated by rays coming from all parts of its lens' aperture and then the illuminance starts to decrease with the square of distance (no wonder as the lit area increases with the square of it - the whole inverse square thing comes from any surface being proportional to squared linear dimension - and conservation law, of course). Close to the (quasi-)collimated light source (less than around D/tan(beam spread/2)) you won't observe an inverse-square relationship as the 'apparent lens aperture' will be widening.

As I understand it, a Lambertian reflection is kin to a matte surface bounce? I understood half the math you supplied. I have to correctly identify and understand some of those terms.

Visually, is your description similar to the drawing below?

LIGHT.thumb.jpg.6e54f998638e00beca606156deb0498a.jpg

Once I am educated on some of these concepts, do you mind chatting sometime, Michael?

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Etendue is a pretty fundamental thing, certainly not limited to imaging or focused light. You can think of it as 'spread' or 'divergence' of light. Or you could think of brightness as a 'conversion multiplier' to get luminous flux of a beam with a given etendue. It's ddФ = L*ddG for any point emitting or receiving light. If we integrate it over the directions of rays (to get dФ, full flux from a single point) and then over the surface of light source/receiver, we'll get flux Ф, which's basically power in either watts or lumens (for visible spectrum). Since it's obviously dW/dt, W for energy, in a closed system it's conserved. As long as no diffuse reflection or scattering is involved, we can assume a beam of light is kind of an 'isolated volume' (that's very imprecise language, so don't take it as a legitimate definition) where flux is 'confined'. And a pretty remarkable fact of geometric (or maybe actually Hamiltonian) optics is that etendue is conserved too. This means, even brightness L is conserved and is thus the same for an object and image point! 

For practical lighting calculations, the general ddФ = L*cos(normal angle)*d(area)*d(spread) formula plus the conservation laws give us a multitude of useful physical quantities (that are derivatives of Ф) and formulae such as Lambert's.

E.g. irradiance of a point dE = ddФ/dS = L*cos(norm.ang.)*d(spread) - and illuminance is the same, but for the receiving side. For a Lambertian source, L=const - it's equally bright from any angle. And yes, a diffuse bounce is close to Lambertian. But cosine is still there and it means that from a sharp angle the apparent area of a source is 1/cos(norm.ang) times smaller. That's why rays coming from the edges on your sketch contribute less to the illumination than the center rays. No need for a concept of collimation here, it'll solely confuse. Actually, a perfectly collimated beam (which's impossible) has no falloff since its area is constant.

Feel free to have a chat. It's not easy to find the right read when you've just started out with the topic. So don't wait till you consider yourself educated - it'll sure be a long trip down the rabbit hole 🙂

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On 11/3/2020 at 8:16 PM, Michael Rodin said:

And yes, a diffuse bounce is close to Lambertian. But cosine is still there and it means that from a sharp angle the apparent area of a source is 1/cos(norm.ang) times smaller. That's why rays coming from the edges on your sketch contribute less to the illumination than the center rays. No need for a concept of collimation here, it'll solely confuse. Actually, a perfectly collimated beam (which's impossible) has no falloff since its area is constant.

Feel free to have a chat. It's not easy to find the right read when you've just started out with the topic. So don't wait till you consider yourself educated - it'll sure be a long trip down the rabbit hole 🙂

Yes, and as you pull away from the sharp angle, the more the edges contribute to the illumination. That's my main suspicion on why falloff is so slight or plateaued up close. Up until a certain angle of view.

I've related my observations to perspective. But you've put formulas and definitions to what I can only describe in concepts.  I'll contact you at some point when my schedule loosens up.

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On 11/4/2020 at 11:16 AM, Michael Rodin said:

For practical lighting calculations, the general ddФ = L*cos(normal angle)*d(area)*d(spread) formula plus the conservation laws give us a multitude of useful physical quantities (that are derivatives of Ф) and formulae such as Lambert's.

Sir,

Is it possible to see an example of this formula used in a practical location? This interests me greatly however cannot find useful education resources on the topic. Maybe I'm not looking in the right place.

 

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