Introduction

The word photometry can be broken down into photo- (light) and -metry (measurement), so it should come as no surprise then to hear that this chapter will be focused on discussing my cats’ favourite toys…

Obviously I’m joking; this chapter will explain how we can measure light in the real world and will link this to clinical considerations where appropriate.

Angles and lights

In the introduction, we discussed that the term photometry means ‘measurement of light’, but more specifically than that, it refers to measurement of visible light – the part of the electromagnetic spectrum that is detectable by the human eye. This could include natural light (e.g. from the sun), but more often we use photometry to discuss artificial light sources (e.g. from a lamp). For example, if you’ve ever bought a lightbulb and spent a while wondering what the difference between the watts and the lumens are (and what they mean), then this chapter is for you!

To start with then, let’s consider a light bulb (a bog-standard light bulb), as shown in Fig. 12.1 . Here you can clearly see that the bulb itself is emitting light in all directions (and although this is a two-dimensional (2D) image, please imagine it exists in three-dimensional (3D) space), meaning that we can say it’s emitting light spherically from its centre. However, this means that if we want to measure the angle of light that’s being emitted by the light source, we need to be able to determine the relevant 3D angle.

Diagram of a light bulb to show that it emits light in all directions.

But how do we measure 3D angles? Well, let’s start by reviewing some prinicples of 2D angles. In a 2D circle, like the one in Fig. 12.2 , any planar (flat) angle (θ) measured from the centre can be represented in degrees (°) or radians (rad or r), which is typically expressed relative to pi (π). In the example in Fig. 12.2 , the angle is 90°, which Table 12.1 shows us would convert to 0.25 turns around the circumference (outside edge) of the circle, or π/2 radians.

Circle showing planar angle from the centre (θ) which corresponds to 90°. The distance from the apex of the angle and the circumference of the circle is equal to the radius of the circle (R). The distance along the circumference required to create the angle is called the arc (c).

However, what if we don’t know the angle? To calculate the angle we can utilise the relationship between the radius (R) and the length of the arc (c) through Equation 12.1.1 (radians) and Equation 12.1.2 (same but in degrees). Remember that because these equations use SI units, all measure of distance must be in metres (m).

DEMO QUESTION 12.1

Calculate the planar angle shown as ‘?’ in the image below:

Step 1: Determine what we need to calculate

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  planar angle, θ

Step 2: Define variables

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  c = 0.42 m (we need to convert to metres)

  
  
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  R = 0.14 m (we need to convert to metres)

Step 3: Determine necessary equation

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  θ = c / R ( Equation 12.1.1 )

Step 4: Calculate

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  θ = c / R

  
  
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  θ = 0.42 / 0.14

  
  
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  θ = 3 radians

  
  
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  (now we can divide by π to understand how this number relates to π radians (π is 3.14 so it’s probably going to be close to 1 π radians…))

  
  
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  θ = 3 / π

  
  
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  θ = 0.95 π rad

  
  
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  (or we could also convert to degrees…)

  
  
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  θ = c / R * (180 / π)

  
  
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  θ = 3 * (180 / π)

  
  
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  θ = 171.89°

  
  
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  (in either case, don’t forget the units!)

Practice questions 12.1:

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  12.1.1 Calculate the planar angle shown as ‘?’ in the image below:

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  12.1.2 Calculate the planar angle shown as ‘?’ in the image below:

OK, so now (even if we don’t like them very much), hopefully we can appreciate the link between degrees and radians for 2D angles, but we know at some point we’re going to need to discuss 3D angles.

Three-dimensional angles are tricky. To make them a little less tricky, let’s start by considering a 2D circle; however, the 2D circle is so unbelievably tiny that it looks like a dot ( Fig. 12.3 A). If we put another larger circle in front of the tiny little circle, closer to us ( Fig. 12.3 B), we could join the two together by their entire circumferences to make a cone ( Fig. 12.3 C). We can see from this thought experiment that the shape of the apex (pointy bit) of the cone would be directly related to (1) the distance between the point and the circle (larger distance with same size circle = smaller apex) and (2) the size of the larger circle (larger circle at same distance = larger apex) ( Fig. 12.3 D). This, in essence, is how we define 3D angles, which are termed solid angles .

Diagram showing concept of solid (3D) angles. We start with a tiny circle far away (A) and we superimpose a larger circle closer to us (B). If we viewed this from the side (C), we could join the two circles from circumference to circumference to make a cone shape. This cone shape defines our solid angle. As we can see, the area of the larger circle (D, pink) and the distance away of the larger circle (D, blue) will have an impact on the solid angle.

To this end, the solid angle (ω) can be considered the amount of the field of view (m ² ) from a particular point (termed the apex ). Crucially, the solid angle of the ‘cone’ (field of view) is measured in steradians (sr) which can be thought of as 3D radians. In a real-world example of this, we can imagine the apex as corresponding to an observer’s eye, and the amount of space taken up by the page of this book (or the screen of an e-book) as the field of view ( Fig. 12.4 ).

Diagram showing real-life example of link between solid angle, field of view and distance of object from the observer.

Now, if we turn our example of a solid angle and relate it to a sphere, this can be likened to the 3D version of the 2D planar angles we discussed in Fig. 12.2 . For planar angles within a circle, the angle is related to the distance from the centre of the circle to the outside edge (radius) and the amount of the circle that is covered by the angle (the arc). With solid angles, this principle remains the same, but this time we need to consider the relationship between the square of the distance from the centre of the sphere to the outside edge ( radius, R ) and the amount of surface of the sphere that is covered by the angle ( area, A ) ( Fig. 12.5 ). To see this in action, please review Equation 12.2 (steradians) and remember that because these equations use SI units, all measures of distance must be in metres (m).

Diagram showing a solid angle (ω) in relation to a sphere (3D shape).

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