The Inverse Square Law

If you’ve ever read or been taught about lighting, you’ve probably heard of the Inverse Square Law. It states that light fades in proportion to the square of the distance from the source. But lately I started to wonder if this really applies in all situations. Join me as I attempt to get to the bottom of this…

 

Knowing the law

The seed of this post was sown almost a year ago, when I read Herbert McKay’s 1947 book The Tricks of Light and Colour, which described the Inverse Square Law in terms of light spreading out. (Check out my post about The Tricks of Light and Colour here.)

But before we go into that, let’s get the Law straight in our minds. What, precisely, does it say? Another excellent book, Gerald Millerson’s Lighting for Television and Film, defines it thusly:

With increased distance, the light emitted from a given point source will fall rapidly, as it spreads over a progressively larger area. This fall-off in light level is inversely proportional to the distance square, i.e. 1/d². Thus, doubling the lamp distance would reduce the light to ¼.

The operative word, for our purposes, is “spreads”.

If you’d asked me a couple of years ago what causes the Inverse Square Law, I probably would have mumbled something about light naturally losing energy as it travels. But that is hogwash of the highest order. Assuming the light doesn’t strike any objects to absorb it, there is nothing to reduce its energy. (Air does scatter – and presumably absorb – a very small amount of light, hence atmospheric haze, but this amount will never be significant on the scale a cinematographer deals with.)

In fact, as the Millerson quote above makes clear, the Inverse Square Law is a result of how light spreads out from its source. It’s purely geometry. In this diagram you can see how fewer and fewer rays strike the ‘A’ square as it gets further and further away from the source ‘S’:

Illustration by Borb, CC BY-SA 3.0

Each light ray (dodgy term, I know, but sufficient for our purposes) retains the same level of energy, and there are the same number of them overall, it’s just that there are fewer of them passing through any given area.

So far, so good.

 

Taking the Law into my own hands

During season two of my YouTube series Lighting I Like, I discussed Dedo’s Panibeam 70 HMI. This fixture produces collimated light, light of which all the waves are travelling in parallel. It occurred to me that this must prevent them spreading out, and therefore render the Inverse Square Law void.

This in turn got me thinking about more common fixtures – par cans, for example.

 

Par lamps are so named for the Parabolic Aluminised Reflectors they contain. These collect the light radiated from the rear and sides of the filament and reflect it as parallel rays. So to my mind, although light radiated from the very front of the filament must still spread and obey the Inverse Square Law, that which bounces off the reflector should theoretically never diminish. You can imagine that the ‘A’ square in our first diagram would have the same number of light rays passing through it every time if they are travelling in parallel.

Similarly, fresnel lenses are designed to divert the spreading light waves into a parallel pattern:

Even simple open-face fixtures have a reflector which can be moved back and forth using the flood/spot control, affecting both the spread and the intensity of the light. Hopefully by now you can see why these two things are related. More spread = more divergence of light rays = more fall-off. Less spread = less divergence of light rays = more throw.

So, I wondered, am I right? Do these focused sources disobey the Inverse Square Law?

 

Breaking the law

To find the answer, I waded through a number of fora.

Firstly, and crucially, everyone agrees that the Law describes light radiated from a point source, so any source which isn’t infinitely small will technically not be governed by the Law. In practice, says the general consensus, the results predicted by the Law hold true for most sources, unless they are quite large or very close to the subject.

If you are using a softbox, a Kinoflo or a trace frame at short range though, the Inverse Square Law will not apply.

The above photometric data for a Filmgear LED Flo-box indeed shows a slower fall-off than the Law predicts. (Based on the 1m intensity, the Law predicts the 2m and 3m intensities as 970÷2²=243 lux and 970÷3²=108 lux respectively.)

A Flickr forum contributor called Severin Sadjina puts it like this:

In general, the light will fall off as 1/d² if the size of the light source is negligible compared to the distance d to the light source. If, on the other hand, the light source is significantly larger than the distance d to the light source, the light will fall off as 1/d – in other words: slower than the Inverse Square Law predicts.

Another contributor, Ftir, claims that a large source will start to follow the Law above distances equal to about five times the largest side of the source, so a 4ft Kinoflo would obey the Law very closely after about 20ft. This claim is confirmed by Wikipedia, citing A. Ryer’s The Light Measurement Handbook.

But what about those pesky parallel light beams from the pars and fresnels?

Every forum had a lot of disagreement on this. Most people agree that parallel light rays don’t really exist in real life. They will always diverge or converge, slightly, and therefore the Law applies. However, many claim that it doesn’t apply in quite the same way.

Diagram from a tutorial PDF on light-measurement.com showing a virtual point source behind the bulb of a torch.

A fresnel, according to John E. Clark on Cinematography.com, can still be treated as a point source, but that point source is actually located somewhere behind the lamp-head! It’s a virtual point source. (Light radiating from a distant point source has approximately parallel rays with consequently negligible fall-off, e.g. sunlight.) So if this virtual source is 10m behind the fixture, then moving the lamp from 1m from the subject to 2m is not doubling the distance (and therefore not quartering the intensity). In fact it is multiplying the distance by 1.09 (12÷11=1.09), so the light would only drop to 84% of its former intensity (1÷1.09²=0.84).

I tried to confirm this using the Arri Photometrics App, but the data it gives for Arri’s fresnel fixtures conforms perfectly with an ordinary point source under the Law, leaving me somewhat confused. However, I did find some data for LED fresnels that broke the Law, for example the Lumi Studio 300:

As you can see, at full flood (bottom graphic) the Law is obeyed as expected; the 8m intensity of 2,500 lux is a quarter of the 4m intensity of 10,000 lux. But when spotted (top graphic) it falls off more rapidly. Again, very confusing, as I was expecting it to fall off less rapidly if the rays are diverging but close to parallel.

A more rapid fall-off suggests a virtual point source somewhere in front of the lamp-head. This was mentioned in several places on the fora as well. The light is converging, so the intensity increases as you move further from the fixture, reaching a maximum at the focal point, then diverging again from that point as per the Inverse Square Law. In fact, reverse-engineering the above data using the Law tells me – if my maths is correct – that the focal point is 1.93m in front of the fixture. Or, to put it another way, spotting this fixture is equivalent to moving it almost 2m closer to the subject. However, this doesn’t seem to tally with the beam spread data in the above graphics. More confusion!

I decided to look up ETC’s Source Four photometrics, since these units contain an ellipsoidal reflector which should focus the light (and therefore create a virtual point source) in front of themselves. However, the data shows no deviation from the Law and no evidence of a virtual point source displaced from the actual source.

 

I fought the law and the law won

I fear this investigation has left me more confused than when I started! Clearly there are factors at work here beyond what I’ve considered.

However, I’ve learnt that the Inverse Square Law is a useful means of estimating light fall-off for most lighting fixtures – even those that really seem like they should act differently! If you double the distance from lamp to subject, you’re usually going to quarter the intensity, or near as damn it. And that rule of thumb is all we cinematographers need 99% of the time. If in doubt, refer to photometrics data like that linked above.

And if anyone out there can shed any light (haha) on the confusion, I’d be very happy to hear from you!

SaveSave

SaveSave

SaveSave

SaveSave

The Inverse Square Law

9 Fun Photic Facts from a 70-year-old Book

Shortly before Christmas, while browsing the secondhand books in the corner of an obscure Herefordshire garden centre, I came across a small blue hardback called The Tricks of Light and Colour by Herbert McKay. Published in 1947, the book covered almost every aspect of light you could think of, from the inverse square law to camouflage and optical illusions. What self-respecting bibliophile cinematographer could pass that up?

Here are some quite-interesting things about light which the book describes…

  

1. SPHERES ARE THE KEY to understandING the inverse square law.

Any cinematographer worth their salt will know that doubling a subject’s distance from a lamp will quarter their brightness; tripling their distance will cut their brightness to a ninth; and so on.  This, of course, is the inverse square law. If you struggle to visualise this law and why it works the way it does, The Tricks of Light and Colour offers a good explanation.

[Think] of light being radiated from… a mere point. Light and heat are radiated in straight lines and in all directions [from this point]. At a distance of one foot from the glowing centre the whole quantity of light and heat is spread out over the surface of a sphere with a radius of one foot. At a distance of two feet from the centre it is spread over the surface of a sphere of radius two feet. Now to find an area we multiply two lengths; in the case of a sphere both lengths are the radius of the sphere. As both lengths are doubled the area is four times as great… We have the same amounts of light and heat spread over a sphere four times as great, and so the illumination and heating effect are reduced to a quarter as great.

 

2. MIRAGES ARE DUE TO Total internal reflection.

This is one of the things I dimly remember being taught in school, which this book has considerably refreshed me on. When light travels from one transparent substance to another, less dense, transparent substance, it bends towards the surface. This is called refraction, and it’s the reason that, for example, streams look shallower than they really are, when viewed from the bank. If the first substance is very dense, or the light ray is approaching the surface at a glancing angle, the ray might not escape at all, instead bouncing back down. This is called total internal reflection, and it’s the science behind mirages.

The heated sand heats the air above it, and so we get an inversion of the density gradient: low density along the heated surface, higher density in the cooler air above. Light rays are turned down, and then up, so that the scorched and weary traveller sees an image of the sky, and the images looks like a pool of cool water on the face of the desert.

 

3. Pinhole images pop up in unexpected places.

Most of us have made a pinhole camera at some point in our childhood, creating an upside-down image on a tissue paper screen by admitting light rays through a tiny opening. Make the opening bigger and the image becomes a blur, unless you have a lens to focus the light, as in a “proper” camera or indeed our eyes. But the pinhole imaging effect can occur naturally too. I’ve sometimes lain in bed in the morning, watching images of passing traffic or flapping laundry on a line projected onto my bedroom ceiling through the little gap where the curtains meet at the top. McKay describes another example:

One of the prettiest examples of the effect may be seen under trees when the sun shines brightly. The ground beneath a tree may be dappled with circles of light, some of them quite bright… When we look up through the leaves towards the sun we may see the origin of the circles of light. We can see points of light where the sun shines through small gaps between the leaves. Each of these gaps acts in the same way as a pinhole: it lets through rays from the sun which produce an image of the sun on the ground below.

 

4. The sun isn’t a point source.

“Shadows are exciting,” McKay enthuses as he opens chapter VI. They certainly are to a cinematographer. And this cinematographer was excited to learn something about the sun and its shadow which is really quite obvious, but I had never considered before.

Look at the shadow of a wall. Near the base, where the shadow begins, the edge of the shadow is straight and sharp… Farther out, the edge of the shadow gets more and more fuzzy… The reason lies of course in the great sun itself. The sun is not a mere point of light, but a globe of considerable angular width.

The accompanying illustration shows how you would see all, part or none of the sun if you stood in a slightly different position relative to the hypothetical wall. The area where none of the sun is visible is of course in full shadow (umbra), and the area where the sun is partially visible is the fuzzy penumbra (the “almost shadow”).

  

5. Gravity bends LIGHT.

Einstein hypothesised that gravity could bend light rays, and observations during solar eclipses proved him right. Stars near to the eclipsed sun were seen to be slightly out of place, due to the huge gravitational attraction of the sun.

The effect is very small; it is too small to be observed when the rays pass a comparatively small body like the moon. We need a body like the sun, at whose surface gravity is 160 or 170 times as great as at the surface of the moon, to give an observable deviation…. The amount of shift depends on the apparent nearness of a star to the sun, that is, the closeness with which the rays of light from the star graze the sun. The effect of gravity fades out rapidly, according to the inverse square law, so that it is only near the sun that the effects can be observed.

 

6. Light helped us discover helium.

Sodium street-lamps are not the most pleasant of sources, because hot sodium vapour emits light in only two wave-lengths, rather than a continuous spectrum. Interestingly, cooler sodium vapour absorbs the same two wave-lengths. The same is true of other elements: they  emit certain wave-lengths when very hot, and absorb the same wave-lengths when less hot. This little bit of science led to a major discovery.

The sun is an extremely hot body surrounded by an atmosphere of less highly heated vapours. White light from the sun’s surfaces passes through these heated vapours before it reaches us; many wave-lengths are absorbed by the sun’s atmosphere, and there is a dark line in the spectrum for each wave-length that has been absorbed. The thrilling thing is that these dark lines tell us which elements are present in the sun’s atmosphere. It turned out that the lines in the sun’s spectrum represented elements already known on the earth, except for one small group of lines which were ascribed to a hitherto undetected element. This element was called helium (from helios, the sun).

 

7. Moonlight is slightly too dim for colours.

Our retinas are populated by two different types of photoreceptors: rods and cones. Rods are much more sensitive than cones, and enable us to see in very dim light once they’ve had some time to adjust. But rods cannot see colours. This is why our vision is almost monochrome in dark conditions, even under the light of a full moon… though only just…

The light of the full moon is just about the threshold, as we say, of colour vision; a little lighter and we should see colours.

 

8. MAGIC HOUR can be longer than an hour.

We cinematographers often think of magic “hour” as being much shorter than an hour. When prepping for a dusk-for-night scene on The Little Mermaid, I used my light meter to measure the length of shootable twilight. The result was 20 minutes; after that, the light was too dim for our Alexas at 800 ISO and our Cooke S4 glass at T2. But how long after sunset is it until there is literally no light left from the sun, regardless of how sensitive your camera is? McKay has this to say…

Twilight is partly explained as an effect of diffusion. When the sun is below the horizon it still illuminates particles of dust and moisture in the air. Some of the scattered light is thrown down to the earth’s surface… Twilight ends when the sun is 17° or 18° below the horizon. At the equator [for example] the sun sinks vertically at the equinoxes, 15° per hour; so it sinks 17° in 1 hour 8 minutes.

 

9. Why isn’t Green a primary colour in paint?

And finally, the answer to something that bugged me during my childhood. When I was a small child, daubing crude paintings of stick figures under cheerful suns, I was taught that the primary colours are red, blue and yellow. Later I learnt that the true primary colours, the additive colours of light, are red, blue and green. So why is it that green, a colour that cannot be created by mixing two other colours of light, can be created by mixing blue and yellow paints?

When white light falls on a blue pigment, the pigment absorbs reds and yellows; it reflects blue and also some green. A yellow pigment absorbs blue and violet; it reflects yellow, and also some red and green which are the colours nearest to it in the spectrum. When the two pigments are mixed it may be seen that all the colours are absorbed by one or other of the components except green.

 

If you’re interested in picking up a copy of The Tricks of Light and Colour yourself, there is one on Amazon at the time of writing, but it will set you back £35. Note that Herbert McKay is not to be confused with Herbert C. McKay, an American author who was writing books about stereoscopic photography at around the same time.

9 Fun Photic Facts from a 70-year-old Book