In Search of Luminance

Understanding What We See

Ian Ashdown, P. Eng., FIES

Chief Scientist, Lighting Analysts Inc.

[ Please send comments to allthingslighting@gmail.com. ]

The IES Lighting Handbook, Tenth Edition (IES 2010), describes luminance as “perhaps the most important quantity in lighting design and illuminating engineering.” This is an accurate but curious description, as the editors neglected to include an entry for Section 5.7.3, Luminance, in the handbook’s index.

The section itself is a mere five paragraphs long, informing the curious reader that luminance is the “local surface density of light emitting power in a particular direction,” defined mathematically as:

In Search of Luminance - EQN 1

which for most readers will be completely and absolutely … opaque.

This is unfortunate, as luminance is undeniably the most important quantity, and indeed the most fundamental concept, in lighting design and illuminating engineering. More than a mathematical definition, professional lighting designers need to understand what it is that we see.

Luminance Understood

To understand luminance, we begin with a parallel beam of light. Ignore any thoughts of surfaces or light sources; just imagine a beam of light traveling through empty space in a given direction. Imagine also that this beam has a finite width; say, a rectangular beam one meter on a side.

If we take a cross-section of this beam at any point along its length, we can measure so many lumens of light (i.e., photons per second) per unit area. In photometric terms, this is the luminous flux Φ per unit area, or luminous flux density, of the beam. Being parallel, the beam does not diverge or converge, and so the luminous flux density remains constant along the length of the beam.

Now, what happens if the beam illuminates a real or imaginary surface at an angle? We have this:

In Search of Luminance - FIG 1

FIG. 1 – Illuminance of a surface A

The luminous flux per unit area received by the surface A is determined by the cosine of the angle of incidence θ from the surface normal n. Conceptually, as the angle of incidence becomes greater (i.e., more oblique), the illuminance E (lumens per unit area) of the surface decreases. The expression A cos θ represents the projected area of the illuminated surface, and is equal to the cross-sectional area of the beam.

This is nothing more than Lambert’s Cosine Law (Lambert 1760):

In Search of Luminance - EQN 2

If we imagine the area A as being infinitesimally small, we can designate it as dA (for “differential area”). Similarly, the amount of luminous flux Φ within the infinitesimally narrow beam approaches zero, and so we designate it as dΦ. This gives us:

In Search of Luminance - EQN 3

This is basic high school algebra! Ignore the symbols and concentrate on the underlying physical concept.

We can further imagine the beam not as a parallel beam that is infinitesimally narrow, but as an elemental cone whose infinitesimal solid angle we designate as . (See the previous article Solid Angles for an explanation of this concept.)

In Search of Luminance - FIG 2

FIG. 2 – Luminance of a differential surface dA

With this, we have the conceptual framework to understand the formal definition of luminance:

In Search of Luminance - EQN 4

where the factor d2Φ does not mean that the symbol d is being squared. Rather, it simply means that the luminous flux dΦ is being divided by the solid angle of the elemental cone dω and the area dA. Further, the parameter ψ indicates that the luminance may also vary when the beam is rotated horizontally by angle ψ around the surface normal n.

What this equation is saying is that the luminance L of the surface dA is equal to the amount of luminous flux Φ (lumens) leaving dA in the direction θ and contained within the elemental cone (i.e., parallel beam) dω. This is equivalent to the IES Lighting Handbook description of “local surface density of light emitting power in a particular direction.”

There is an important but underappreciated corollary to this definition of luminance. Recalling that the surface can be real or imaginary, we can imagine placing an imaginary surface that is perpendicular to the beam direction (i.e., θ is equal to zero) anywhere along its length. What this means is that the luminance of a parallel beam of light is constant along its length. In other words, luminance is not an intrinsic property of the surface, but of the beam itself. (As an example, the sky has a measurable luminance when viewed from the ground, but it has no real surface.)

Dispensing with the mathematics, we can therefore say:

Luminance is the amount of luminous flux per unit area as measured in a parallel beam of light in a given direction.

Photometry is traditionally taught using the concept that luminance is a property of real or imaginary surfaces. The problem with this approach is that you cannot easily explain why participating media such as the atmosphere, smoke, fog, colloidal suspensions in water, and so forth have measurable luminance. Thinking of luminance as a property of a beam of light rather than of surfaces eliminates this difficulty.

Luminance Perceived

How do we perceive luminance? Imagine that you are looking at a blank sheet of matte white paper. Being an approximately ideal diffuser (except at very oblique angles), this paper will scatter incident light equally in all directions.

Now, imagine that each point of the paper’s surface is a point source of light. In accordance with the inverse square law, the luminous flux density of this light will decrease with the square of the distance from the point source. That is:

In Search of Luminance - EQN 5

where I is the intensity of the point source, d is the distance from the source, and E is the illuminance of a surface (such as the cornea of your eye) at that distance … so why do we see and measure the luminance of the paper as being constant with distance?

To answer this, we need to look at the eye itself, which basically consists of a lens that focuses images onto the cones and rods of the retina. Each cone and rod has a finite width, and so it receives light from a finite area of the surface of the paper.

In Search of Luminance - FIG 3

FIG. 3 – Eye focusing a parallel beam onto the retina

But wait! This area of the paper is dependent on the distance of the paper from the eye. Moreover, it is proportional to the square of the distance … which exactly cancels out the inverse square law for a single point source. Therefore, we perceive the luminance of a finite area surface as being constant regardless of its distance from the eye.

There is a counterexample that emphasizes this point: the night sky. Even though the actual diameter of a star may be a million miles or so, it is so far away that we perceive its light as a parallel beam that is focused onto a single rod or cone of our retina. The luminance of this beam is constant, and so we see the star as having a specific perceived brightness (or visual magnitude). The inverse square law still applies to the star’s emitted light, however – it is after all a point source – and so its magnitude depends on its distance from the Earth. All other things being equal, more distant stars are inherently fainter.

How the eye sees a parallel beam of light, however, is the key point: wherever we look, we see luminance. We do not see luminous intensity or illuminance; we see the luminance of beams of light. Luminance really is the fundamental concept of lighting design.

Conclusion

A famous 20th-century physicist (whose name I regrettably cannot recall, even with Google’s assistance) once observed that until you can visualize a problem, you cannot truly understand the mathematics that describe it. He was likely referring to quantum mechanics, which nobody yet fully understands, but the observation still applies. In particular, knowing the mathematical definition of luminance is not enough; we must understand the concept of luminance. With this understanding, we can better understand its importance to lighting design and illumination engineering.

References

IES. 2010. IES Lighting Handbook, Tenth Edition. New York, NY: Illuminating Engineering Society of North America.

Lambert, J. H. 1760. Photometria (in Latin). English translation by D. L. DiLaura, 2001. New York, NY: Illuminating Engineering Society of North America.

Smith, W. 2008. Modern Optical Engineering, Fourth Edition. New York, NY: McGraw-Hill.

Solid Angles

Truly Understanding Luminous Intensity

Ian Ashdown, P. Eng., FIES

Chief Scientist, Lighting Analysts Inc.

[ Please send comments to allthingslighting@gmail.com. ]

Do you suffer from math anxiety? A surprising number of us do (e.g., Wigfield 1988). I would tell you the exact numbers, but you would need to understand statistical analysis …

Fortunately, we can mostly muddle through our lives without having to deal with statistics, vector calculus, differential geometry, algebraic topology and all that. As an electrical engineer in the 1980s for example, I never needed anything more than a four-function calculator to do my work designing billion-dollar transportation systems.

Our fear (note the implicit “we”) can, however, disadvantage us in subtle ways. In studiously ignoring the mathematics of a topic, we all too often overlook the underlying concepts that help us better understand what we are interested in.

An example from lighting design: luminous intensity. We measure the luminous intensity of a light source in candela, which is defined as “one lumen per steradian” (IES 2010). A lumen is easy enough to understand, but what the blazes is a “steradian”?

The all-knowing Wikipedia has an answer: it is the measure of a “solid angle.” Going to the Wikipedia definition of this phrase, we see:

Solid Angle Equation

Anxiety? What anxiety?

But now for a trade secret: most mathematicians do not think in terms of equations like these double integrals. Instead, they visualize. Just as lighting designers can look at architectural drawings and imagine lighting designs, mathematicians can look at a set of equations – which are really nothing more than an arcane written language – and visualize new mathematical concepts and proofs.

I learned this from a professor of mine whose specialty was hyperspace geometry – he could “easily imagine” four- and five-dimensional objects by mentally projecting them into three-dimensional shapes and imagining how their shadows changed as he rotated the objects in his mind. Some people …

So, we start by visualizing a circle (FIG. 1):

FIG. 1 - Circle

FIG. 1 – Circle with radius r

If you remember anything at all from mathematics in school, it is that the circumference C of a circle with radius r is equal to two times pi times its radius, or:

C = 2 * pi * r

where pi is approximately 3.14159. (Remember that 1980s-era four-function calculator – it is all you will need for this.)

What this means is that if we take a piece of string with length r, we will need to stretch it by a factor of two pi (6.28328 …) to wrap around the circumference of the circle.

But suppose we wrap the string with length r part way around the circle (FIG. 2). The resultant angle is precisely one radian, which is abbreviated rad.

FIG. 2 - One radian

FIG. 2 – One radian

Most of us are used to thinking of angles in terms of degrees – there are 360 degrees in a circle. (The reason for the magic number 360 is lost in history, according to Wikipedia.) This means that one radian is equal to 360 / (2 * pi) = 180 / pi degrees, which is approximately 57.3 degrees. Radians are more useful simply because they are related to the geometry of the circle rather than some magic number – they are easier to visualize and so understand.

Now, imagine a sphere with radius r, and with a cone-shaped section whose base has a surface area of r * r, or r2 (FIG. 3):

FIG. 3 - Solid angle

FIG. 3 – Solid angle

This cone has a solid angle of precisely one steradian (or one “solid radian”), which is abbreviated sr.

No mathematics required – easy.

(To be precise, a solid angle does not need to be a circular cone-shaped section as shown in FIG. 3. The top of the cone can be any shape; all that matters is the ratio of the surface area of the base to the radius r.)

How many “square degrees” in a steradian? That’s also easy: if one radian is equal to 180 / pi degrees, then one steradian is equal to (180 / pi) * (180 / pi), or approximately 3282.8, square degrees.

To be honest, I also suffer from math anxiety when first reading a set of equations. I do not really understand them until I can visualize what they mean. Mathematical equations are just the formal written language we use to express what we have visualized.

… now if only I could understand batting averages in baseball and cricket …

References

IES. 2010. IES Lighting Handbook, Tenth Edition. New York, NY: Illuminating Engineering Society of North America.

Wigfield, A., and J. L. Meece. “Math Anxiety in Elementary and Secondary School Students,” Journal of Educational Psychology 80(2):210-216.

Mesopic Photometry and Statistics

Ian Ashdown, P. Eng., FIES

Chief Scientist, Lighting Analysts Inc.

[ Please send comments to allthingslighting@gmail.com. ]

Related Posts

Understanding Mesopic Photometry

One of the joys of statistics is that you can never be proven wrong …

In a previous All Things Lighting article titled “Understanding Mesopic Photometry” (October 8th, 2013), I wrote:

Some publications on mesopic lighting have indicated that the S/P ratio of a lamp can be estimated from its correlated color temperature (CCT), but this is incorrect …

I continued on with an example that compared the spectral power distributions and scotopic-to-photopic (S/P) ratios of a phosphor-coated white light LED:

FIG 5A

Fig. 1– Phosphor-coated LED module

and a red-green-blue LED:

FIG 5B

Fig. 2 – Red-green-blue LED module

Both lamp modules had the same correlated color temperature (CCT) of 3500K, but their S/P ratios were 1.41 and 2.02 respectively. I concluded that:

Simply put, the only way to accurately determine the S/P ratio of a light source is through calculation using its spectral power distribution.

While this statement is technically correct, it is not particularly useful when you need to know the S/P ratio of a lamp or lamp module for mesopic roadway or area lighting calculations.

Measurements and Equations

One of the publications I chose not to reference was the “City of San Jose Public Streetlight Design Guide” [Anon. 2011]. This report presented a list of eight light sources with their reported S/P ratios, which were derived from [CIE 2010] and [Berman 1992]:

Source S/P Ratio CCT
Low pressure sodium 0.25 1700
High pressure sodium 0.65 2100
Warm white metal halide 1.35 3500
Daylight metal halide 2.45 5500
Warm white fluorescent 1.00 3000
Cool white fluorescent 1.46 3700
Triphosphor fluorescent 1.54 4100
Daylight fluorescent 2.22 7500

Table 1 – S/P Ratio versus CCT [Anon. 2011]

This list is somewhat selective, as Berman reported the S/P ratio versus CCT of sixteen light sources:

FIG. 3

Fig. 3 – S/P Ratio versus CCT [Berman 1992]

The report noted:

Although the S/P ratio is derived from the spectral power distribution of the light source, it approximately corresponds to the correlated color temperature of that source.

However, this was immediately followed by:

To determine the S/P ratio for any given CCT, the following equation can be used:

S/P ratio = -7 * 10-8 (CCT)2 + 0.001 * CCT – 1.3152

While I otherwise agree with the report, I must disagree with this statement. Of the tens of thousands of lamp types that are commercially available, you cannot fit a quadratic curve through a mere eight data points and generalize it to any light source. This is especially true when the light sources include the near-monochromatic spectral power distribution (SPD) of low-pressure sodium (LPS) lamps.

Worse, there is no indication of the expected error with this equation. You may calculate an S/P ratio for a given CCT, but you have no idea whether it is accurate. Is it for example 1.65, 1.6, or somewhere between 1.0 and 2.0?

Statistics

Based on the work of [Berman 1992], it is evident that the S/P ratio of a white light source “approximately corresponds” to its CCT. However, the evidence in support of this conclusion is statistically weak, and further does not consider today’s phosphor-coated white light LEDs.

What is needed is a random sampling of many commercial white light sources. Ideally, the work would be done by an independent photometric testing laboratory so as not to inadvertently skew the results towards the products of a single lamp manufacturer.

Having the results for many different light sources serves two purposes. First, it provides enough data points to have confidence that an equation fitted to the data fairly represents most commercial lamps and LED lamp modules.

Second, it provides the all-important confidence interval for any given S/P ratio. That is, given a calculated S/P ratio for a specific CCT, you can have (say) 95% confidence that the value is accurate to within a given range of values.

This is important because photometric measurements and calculations always include implicit confidence intervals. For example, electric lighting calculations are typically accurate to within ±10 percent when compared to careful in situ measurements of the completed project. It makes no sense therefore to perform (for example) mesopic roadway lighting calculations if your assumed S/P ratio varies by ±0.5.

Recommendations

The good news is that we now have the necessary information. LightLab International Inc. (www.lightlabint.com) recently collated the results of some 90 tests of LED-based roadway and area lighting luminaires that they performed for their customers. In accordance with the requirements of LM-79 testing procedures [IESNA 2008], the test reports included spectral power distribution measurements, and with them (although not required by LM-79) calculated S/P ratios.

As you might expect, the lamp CCTs clustered around the industry-standard nominal values:

FIG. 4

Fig. 4 – Lamp CCTs

Perhaps less expected is that they exhibited a reasonably linear relationship between S/P ratio and CCT:

FIG. 5

Fig. 5 – S/P ratio versus CCT

I will not repeat the curve-fitting equation here, as it has a meaningless precision of 15 decimal points. What is important is this table of recommended values (where the 2700K values were extrapolated from the measured data):

CCT S/P Ratio Range
2700K 1.1 – 1.4
3000K 1.2 – 1.5
3500K 1.3 – 1.6
4000K 1.4 – 1.8
5000K 1.6 – 2.0
6000K 1.9 – 2.2

Table 2 – S/P Ratio versus CCT for LED lamps

Note carefully that this table applies to LED-based white light sources only; it does not apply to fluorescent (including magnetic induction) or HID lamps, and certainly not to LPS lamps. (Metal halide lamps in particular as reported in Table 1 are outside of the range of this table.)

Looking at Figure 1, it is perhaps not surprising that LED lamps exhibit a strong correlation between S/P ratio and CCT. Virtually all of today’s high-flux LEDs for roadway lighting applications rely on a blue pump LED (which accounts for the 450 nm peak in Figure 1) and broadband emission phosphors between 500 and 700 nm. With minor differences due to different phosphor combinations, most white light LED SPDs will look something like Figure 1, with the CCT mostly determined by the ratio of the blue peak to the phosphor emissions.

The counterexample of course is the red-green-blue LED SPD shown in Figure 2, with its anomalous S/P ratio of 2.02 for a CCT of 3500K. It is not coincidental that the SPD somewhat resembles that of a triphosphor fluorescent or metal halide lamp.

Ideally, we would have S/P ratio versus CCT data for thousands of white light sources. It is unlikely that the recommended S/P values above would change by more than 0.1 units, but it would improve our statistical confidence in the results.

On the other hand, these results show that the S/P ratio varies by ±0.2 for any given CCT, or about ±10 percent of the median value. This is commensurate with the expected accuracy of most electric lighting applications.

Granted, it would be preferable to have S/P ratios available for every lighting product. (S/P ratios are also integral to IES TM-24-13, Incorporating Spectral Power Distribution into the IES Illuminance Determination System for Visual Task Categories P through Y [IESNA 2013].) As was explained in “Understanding Mesopic Photometry” however, there are practical reasons why this is unlikely to occur.

In retrospect, this likely does not matter.. Lighting designers can rarely assume the use of particular product when performing photometric calculations. With competitive bidding for commercial and government projects, it is best to simply specify luminaires with a given CCT. Given that most new roadway and area lighting installations will involve LED-based luminaires, Table 2 provides lighting designers with the confidence that they can assume a usefully narrow range of S/P ratios for design and specification purposes.

References

Anonymous. 2011. City of San Jose Public Streetlight Design Guide. Available as www.sanjoseca.gov/DocumentCenter/Home/View/242.

Berman, S. W. 1992. “Energy Efficiency Consequences of Scotopic Sensitivity,” Journal of the IES 21(1):3-14.

CIE. 2010. Recommended System for Mesopic Photometry Based on Visual Performance. CIE Technical Report 191:2010. Vienna, Austria: Commission International de l’Eclairage.

IESNA. 2008. IES LM-79-08. Electrical and Photometric Measurements of Solid-State Lighting Products. New York, NY: Illuminating Engineering Society of North America.

IESNA. 2013. IES TM-24-13. Incorporating Spectral Power Distribution into the IES Illuminance Determination System for Visual Task Categories P through Y. New York, NY: Illuminating Engineering Society of North America.

Acknowledgements

Thanks to Eric Southgate of LightLab International Inc. (www.lightlabint.com) for sharing the S/P versus CCT data on which this article was based.

Thanks also to Dawn DeGrazio of Lighting Analysts Inc. for invaluable editorial assistance.