A few among you already know that I am using set.a.light 3D (let’s call it SAL3D) quite a lot on a regular basis for planning my shoots, creating my educational material and exploring new ideas the effective and inexpensive way. All from my very own desk; no costs involved.

Explore Lighting Via Software: Experimenting & Avoiding Costs.

I really love set.a.light 3D (SAL3D). Period.

OK, I’m a bit nerdy but if you are already using SAL3D, you know what I am talking about and how it feels (to be able to experiment and manifest your ideas without wasting any money).

For the rest of you (the unknowing): For exemplary reasons, I illustrated my bestseller eBook ESSENTIALS. Studio Lighting For Nude Photography and countless articles using SAL3D.

If you are keen to learn about how you can plan and try out real-world lighting designs, quality of light and moods, just hover over set.a.light 3D (PS: You get 20% OFF SAL3D when using code SWP20).

OK, as this write-up is not meant to be a promotional piece for the software itself, let’s now move on to the actual exploration…

 Inverse-Square Law & Light Ratios - Light Quality, Mood & Message

Lighting + Math = Boring?

At first sight, this article’s topic looks a bit dry, yes even kind of boring. Or at least nothing you feel like you should essentially know in order to be able to create alluring photos, right?

Well, you’re miles out!

Johannes Dauner, CEO & Founder of elixxier software (which invented SAL3D) wrote a concise, in-depth, (almost) easy to understand discourse about light-source power, light distribution, light placement (= distance), and impact/relation of light quality, f-stops (and/or) shutter speed.

The following excursion is by far the most accurately written piece about the Inverse Square-Law that I’ve seen for a long time. And it is of course illustrated and proven with simulations done in SAL3D.

I took the liberty to throw in a few real-world examples myself, material that’s been taken from my ESSENTIALS Lighting… ebook.

How The Inverse-Square Law Improves Your Lighting Design

In general, the inverse-square law explains the disproportionate light fall-off with increasing distance of the subject to the light source. This knowledge helps us to better understand how to correlate light and lighting with the distance to the subject and its brightness.

Let’s start by going back to the roots and explaining the aperture (f-stop) first…

Aperture Stops (= f-Stops)

  • Switching from the maximum aperture to the next smaller one reduces the incident amount of light that enters your lens.
  • The aperture diameter is lowered by a factor of 1/√2 each time which also cuts the lens surface and therefore the amount of light in half.
  • Thanks to this gradation, we simply adjust the shutter speed and aperture to the existing lighting: Each f-number (f-stop) is a result of the previous one, multiplied by √2=1,414…
  • At the same time, we round up the result so that, for example, aperture 4 – as per the calculation of 4 x 1.4 – turns into the next higher aperture value of 5.6.
  • Here is an excerpt of this well-known sequence which reads as follows:
    f/1 > f/1.4 > f/2 > f/2.8 > f/4 > f/5.6 > f/8 > f/11 > f/16 > f/22 > f/32

 Inverse-Square Law  f-Number-Sequence

Existing f-Numbers Chart

In rare cases there are also macro lenses with a maximum f-number of 45. Since macro lenses are often times placed very close to the subject, the huge f-number allows you to reach a good depth of field (despite the closeness). Too small f-numbers would oftentimes lead to a too low focus range.

Examining “The Inverse-Square Law Of Light”

The inverse-square law works as follows:

  • If you double the distance between subject and light source, it illuminates a surface area four times greater than the one before. In general, we therefore multiply the distance with itself in order to calculate the enlargement of that surface area.
  • However, a larger surface area leads to a light intensity that is inversely proportional to the square of the distance – since the same amount of light has to be distributed onto a larger surface area respectively. Therefore, we see light fall-off, meaning a decrease of light intensity.

In technical terms the inverse-square law reads as follows: The energy (in our case: light intensity) at location A (subject area) decreases inversely proportional to the square of A’s distance to the energy source (for example, our flash head).

 Inverse-Square Law  Light-Fall-Off-to-the-Square

Distance: Light fall-off to the square

Light Falls Off To The Square: Details

It only requires some basic math knowledge to write down the inverse-square law (its formula). However, the physics behind it is generally very complex. Because of this, we are only going to approach this law in an illustrative way and from the viewpoint of photography. For this reason, we are referring to the exposure of the image sensor or film footage – and respectively to the lighting of the subject. When using flash and spotlight, the inverse-square law comes in especially handy.

Inverse-Square Law Overview-Light-Fall-Off-to-the-Square

Light fall-off to the square in %

The light intensity, for example, quadruples (4) upon halving (1/2) the distance to the light source and subject. Respectively, the light intensity decreases to a quarter if we double the distance. According to this, these exemplary pairs of digits are valid (distance: 3-fold; intensity: 1/9) and (4; 1/16) if we multiply the distance respectively.

 Inverse-Square Law-Light-Fall-Off-to-the-Square

The farther the distance, the less rapidly we loose light power.

Practical Application For Real World Shoots

Believe it or not but we all create an ideal lighting situation based the inverse-square law. Even if we do so unintentionally…

Because of the inverse-square relationship of the described law, the light intensity drops rather heavily when the subject is first moved further away from the light source. After that, it continuously decreases on a weaker level.

Inverse-Square Law with Melisa Mendini - in set.a.light 3D - Dan Hostettler

Simulated real-world example: different distance between lighting and group.

For example:

  • If we increase the distance between light source and subject from 1 meter to 2 meters, 75 percent of light intensity are lost on the subject.
  • But when we increase the distance from 4 to 10 meters, we only lose 5 percent.

Which results in real world that:

  • Light intensity close to the light source has especially high values.
  • But in the distance, this intensity (and differences thereof) only reaches a tiny value.

Here’s how we create the appropriate lighting:

  • At constant shutter speed the f-value increases, the closer the subject gets to the light source – the smaller the aperture, the less light enters into the camera.
  • Vice versa, the f-value decreases as the distance of the subject to the light source increases. In both cases the respective shots almost look the same: Simply because the same amount of light enters through the lens.

This is how we theoretically create the correct f-value for each combination of distance, light intensity and shutter speed.

Case A) Only Illuminate One Subject
For static subjects, one fixed f-value is sufficient. Moving subjects, however, require flexible f-values, especially when they are very close to the light source: Due to the inverse-square law, a small change of the distance to the light source leads to an extreme change of lighting. In turn, one fixed f-value is sufficient for a long distance subject – even if it moves around on a larger scale.

Case B) Illumination Of A Few Subjects
Sometimes all subjects are located rather close to the light source in form of a spotlight or the like. In such cases, the front subject may be overexposed whereas the one in the back remains underexposed.

For example, three subjects distributed behind one another cross the f-values within the range of f/22 and f/11. There is an easy solution to this problem that allows you to evenly illuminate all subjects: Simply place all subjects further away from the light source. This way, their relative sizes to one another remain the same and all of them only need one and the same f-value for a good exposure such as f/4, for example.

Inverse-Square Law Light-Fall-Off-between-models

Simulation of light’s power loss due to distance, correlated with f-stops and light quality (soft to hard).

You can see the differences very well in above example ↑:

  • If the lighting is placed at a distance of 8 meters to the first model, the light fall-off to the 4th model is only about 2/3 stops.
  • However, if you were to place the light source only 2 meters away, the difference in brightness between the 1th and 4th model would be a total of 2 1/3 stops (e.g., f-stop 8 -> 3.5)!

Case C) Challenges Of A Correctly Illuminate Backgrounds (f/Stop Differences)
In our images, we often times of course also wish to see contrasts instead of uniform brightness: We have a desire for lighter and darker parts of the image.

  • Dark Background: For example, we are in need of a rather dark background against which our attractive model will be illuminated in a correct and bright way – close to the light source. The quadratic light fall-off to the further away background then leads to the underexposed and therefore dark background. This is how strong contrast works to our advantage thanks to the inverse-square law.
  • Well-lit Background: In turn, we also use this “inverse-square law” when we desire an evenly and well illuminated model and background: We then place the light source in a significant distance to both the model and background – this way we achieve a very even illumination.

As you can see in the example below, it is already sufficient to place the light source in a distance of 4 meters to the model in order to evenly illuminate both the model and background. The difference between model and background is only 1 stop now.

Inverse-Square Law W_Light-Fall-Off-between-model-and-wall

Background illumination and shadow cast (light quality soft<>hard) is connected to light source’s distance.

By the way, if the light source is placed too close to the model (figure 1 ↑: 1 meter in illustration above), the light fall-off on the model’s body is already so huge that – in case of a full body shot – the legs would be underexposed. Therefore, such a positioning is only worth to be considered for a portrait shot.

Here again the impact of light fall-off related to the light source distance when doing full-length shots:

 Inverse-Square Law with Melisa Mendini - in set.a.light 3D - Dan Hostettler
Inverse-Square Law with Melisa Mendini - in set.a.light 3D - Dan Hostettler

Light fall-off on model and background related to the light source’s distance

Correlation between Flash Head Output (= Power) <> ISO <> Aperture

All’s connected…

In order to see the dependencies even better, we have set as many values as possible in reference to one another as depicted in the graphic illustration below.

Inverse-Square Law  Reference-Control-Range-Flash-Head-Aperture-ISO-Output

Reference Chart: Correlations

The graphic explained:

  • We are presuming a 1000 Ws flash head with a gigantic control range of 1-10.
  • The aperture and ISO value have been chosen exemplary in order to depict the dependency between these figures.
  • The ISO or aperture value (meaning just one of these two values) needs to be set according to the chart in order to always achieve the same illumination (brightness) of the subject while the flash head’s output controller is being adjusted.

It immediately catches the eye that the flash head’s output power (watt/seconds) has to be doubled per f-stop. This way, the output in the upper range between 9 and 10 is increased by a total of 500 watts whereas the output in the lower range between 1 and 2 only changes by approx. 0.2 watts.

Now by taking a look at these rather extreme output differences you will understand what great development work it took the flash head manufacturers in order to make all his possible with such outstandingly high precision.

Most Important Points To Remember

What should you absolutely keep in mind?

  • Regarding the Aperture
    The difference from one f-stop to the next always either leads to a doubling or halving of brightness.
    For the flash head this also always means a doubling or halving of output (Ws). For example, if the flash head’s output is changed from 5 to 6, then this is exactly equal to one f-stop.
  • Regarding the Distance /1
    If an object is placed close to the light source, the light fall-off on said object is very high in comparison to the background.
    Light fall-off to the square! When doubling the distance to the object, 4-times the light energy is required in order to balance the difference in brightness.
  • Regarding the Distance /2
    Doubling/Halving of distance is always equal to 2 f-stops regarding the difference of brightness!
    A longer distance between light source and subject leads to a more uniform illumination of both the subject and background because the light fall-off keeps decreasing as the distance increases. Therefore, this effect has a huge influence on lighting design.

Here is the chart for a quick overview once again:

Inverse-Square Law Overview-Light-Fall-Off-to-the-Square

Light fall-off to the square in %

Start Experimenting Yourself

Experiment with distances, the aperture and light output at your next shoot – or in set.a.light 3D.

Moods (dark, bright, split,…), shadow falls (short, long, …), light quality (soft, hard, …) – It’s worth it!

Once you have fully understood the correlations between distance, aperture and light fall-off, you can easily use those effects in order to improve your lighting design and selectively control it.

Thanks for reading!
Your Johannes Dauner, CEO & Founder of elixxier software

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