Reversing the Inverse Square Law

Admittedly, one of the most boring subjects in photography is the inverse square law. But before you begin scratching your head and yelling out “the inverse-what?” Just hold on a sec.

First off, one does not need to memorize the laws of light to go and start taking pictures, or even to become a professional. But anyone with an SLR that truly wants to master all the variables in an exposure, you should at least know about it, and have a good sense of how it works. And of course the main purpose to know about the inverse square law (if you haven’t figured it out already) is when making the dive into flash photography.
Therefore, if you’re one of those who claim “I don’t shoot with flash, I only use available light,” then have fun on your little pedestal making up excuses on why you pretend not to be interested in flash photography. Sure your brand new Canon 5Dmkii or Nikon D3 with a 50mm f/1.4 lens is going to destroy the darkness with stunning images at ISO 3200, but we are talking about professional studio portraiture / magazine quality photography that utilizes flash to create amazingly sharp, colorful, and beautifully lit photos. Not to mention, your flashes are “always available” — use them to your advantage.
Alright, now that we have that out of the way, let’s look at some basics that we might already know.

Shutter Speed, Aperture, and ISO sensitivity

Shutter Speed

By now everyone knows that the longer the shutter speed, the more light you let in. Inversely, the faster the shutter, you let in less light. And of course, stops of light (in terms of shutter speed) work in factors of two. That means if you double a shutter speed of 1/100, it becomes 1/50. That is ‘one stop of light’ brighter. Inversely, if you cut it in half from 1/100 → 1/200, you have made your exposure ‘one stop darker’.
Great, let’s move on.


SB-900 @ 1/2 power – 3ft away

We should all have our f stops memorized. If not, it’s a simple scale where we can double our numbers, starting at f/1, and each time we double, we will increase by two stops of light. Therefore:
1 → 2 → 4 → 8 → 16 → 32
To calculate our increments in-between, we simply multiply each stop by our magic number 1.4 (which is, coincidentally, the rounded number of the square root of 2 (which is ~ 1.414)). This makes our first stop easy. 1 x 1.4 = 1.4! We can now fill in our gaps accordingly!
1.0 → 1.4 → 2.0 → 2.8 → 4 → 5.6 → 8 → 11 → 16 → 22 → 32
We now have an aperture scale displaying full one stop increments. Just for kicks, let’s have a brief look at what a classic third aperture scale looks like (since many of you will be working with these numbers on your SLR):
… → 1.4 → 1.6 → 1.8 → 2.0 → 2.2 → 2.5 → 2.8 → …

Ok, I get the aperture numbers, remind me of the exposure relationship?

Let’s say we had a correct exposure of 1/125 at f/8. Your model sitting their patiently is waiting for their beautiful portrait to be taken. You (the photographer) have made a design decision to go for a more shallow depth of field. So you drop your aperture to f/4. That’s two stops of light brighter. f/8 → f/5.6 → f/4. So to compensate, you speed up your shutter by two stops. 1/125 → 1/250 → 1/500. Easy peezy, makes sense, been there done that.

ISO Speed

SB-900 @ 1/32 power – 3ft away

We all know that our lowest ISO produces our cleanest files. Our light stops will look fairly similar and our scale usually looks something like this:
100 → 200 → 400 → 800 → 1600
Some of the latest cameras like the Nikon D700/D3, or the Canon 5Dmkii go up to crazy ISO’s like 25,600, which is four stops brighter than ISO 1600!
Jumping quickly back to our example, if we were originally at ISO 400 and needed to drop two stops of light, we could have simply went from ISO 400 → 200 → 100

Ok, now what about this flash thing?

Flashes have stops of light too! These increments are measured in half increments, just like shutter speed. This is what we call the “flash power,” or rather, how much light it spits out. Our scale looks like this:
1/1 → 1/2 → 1/4 → 1/8 → 1/16
From left to right, we say “full power,” “half power,” “quarter power,” etc. Some flashes can let out light as dim as 1/128 like the SB-800/900 or the Canon 580EXii.
One of the main things you should consider when buying a flash, is to look at how powerful they are. Meaning, “how bright will this be at full power compared to this other flash at full power?” We can tell how powerful they are by looking at the flash Guide Number. This number is extremely important to know when buying a flash. Almost nearly as important as knowing how many millimeters your lens is. For example, you wouldn’t go out and buy a 200mm lens without knowing it’s, well, 200mm! Right?

What is a Guide Number?

In simple mathematical terms, it is aperture multiplied by distance, in which your flash can properly expose a subject at a given distance (aperture and distance). The standard Guide Numbers you should be looking for are measured at ISO 100 (film speed), at the 35mm head position, at full 1/1 power.
Let’s take a basic Nikon flash, for example the SB-600 has a G.N. of 98. To make our math easier, let’s just round it off to 100. What this means is the following. Take a look at our ascii diagram. We have a flash (F), a subject (S). They are twenty feet apart in distance.
To properly expose our subject, we would need an aperture of f/5.
f/5 × 20′ = 100 G.N.
Now if we moved our subject 20′ further…
To compensate or this distance, we would need to let in more light! This means we’ll have to open up our aperture to f/2.5
f/2.5 × 40′ = 100 G.N.
If we kept our aperture at f/5, we would underexpose our subject. Likewise, if we opened up to much, for example, to f/1.4, we would overexpose our subject. Got it?

Ok, I’m ready for that inverse square thingy

Alright, if you insist: The inverse square law states that “the intensity of light radiating from a point source is inversely proportional to the square of the distance from the source.
Therefore an object twice as far away, receives only 1/4 the amount of light. Or if it’s twice as close, then it’s 4x as bright. Each of these is a difference of two stops.

This is confusing, I know. But luckily, and coincidentally, we already know a little about this already with our aperture scale. Just know that light has “depth” in the same way that our focal plane does. By now you know that the closer you get to an object, the shallower your depth of field looks, of which we can conclude that all objects behind your subject “quickly fall out of focus.” This same exact rule applies to light (Thank heavens!). Let’s look at a simple example, and we’ll use the numbers on our aperture scale to make it easy.
In this illustration, we added a background (B):
Let’s assume these settings:
ISO 100
flash is at 1/4 power
It is safe to assume that our background is two stops underexposed. (4′ → 5.6′ → 8′)
Now what if we moved our subject two feet closer toward the flash?
Since we just moved our subject “twice as close” we made it two stops brighter! Therefore to make up for this overexposure, we need to do one of two things:
A) Dial down the flash two stops
1/4 → 1/8 → 1/16
B) Close down the aperture
f/4 → f/5.6 → f/8
Each will keep your subject properly exposed, but it’s an artistic decision for you to make if you want to keep your shallow depth of field (option A), or remove more ambient light and get a slightly sharper image (option B). Also pay close attention that due to the inverse square law, our background is now “four stops” underexposed!
8′ → 5.6′ → 4′ → 2.8′ → 2′

Cool! What flash do I buy? The one with the higher Guide Number, right?

Well, kind of, yes. There are some cases where newer, more expensive flashes will have lower Guide Numbers than say for instance, the 30 year old Vivitar 285HV for $89 (via B&H) that has a G.N. of 120 whereas we’ll see Canon selling their flagship 580EXii for over $400 which has the same G.N. Or even the new Nikon SB-900 has a G.N. of 111. What’s up with that? Well, clearly if you buy a new Nikon or Canon speedlight for your system, it will talk to your camera with all the latest and greatest i/e-TTL technology, the stops of light will range from full 1/1 power to 1/128, whereas the Vivitar only has stops of 1/1 → 1/2 → 1/4 → 1/16 (yes, it skips 1/8). The SB-900 will have a 17-200mm range of light to cover larger or tighter spaces. It has gel detection, three different patterns to throw at your subjects, it even has super quick recycling time.
But hey, don’t knock on the Vivitar; off camera, that’s a helluva bargain for that much power! Hook it up to your camera with a PC chord, or a pair of Pocket Wizards, and you’re golden.
With all that’s been said, go out, get a flash (if you don’t already have one), throw it on manual, and experiment by taking a ton of pictures.
This article has been republished for use at Digital Photography School.
About the Author: Dustin Diaz is an Engineer at Twitter Inc. and a professional photographer with his wife Erin Caton. Together they run Flash Bullet Photography in San Francisco.

A Guest Post by Dustin Diaz from Flash Bullet Photography

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