Angles Of View
When we speak about screens and projectors we are always interested in gauging their brightness. But when we declare that some particular display is "bright," how do we know exactly how much light we're looking at? To find out we need to survey the ways in which light is quantified and to define its
Units of MeasureSurely the most effective light meter ever invented is the human eye. When it comes to registering brightness, this instrument will respond to light over a range of intensities which extends from 1 to 500 million. The dark adapted eye can detect the smallest packet of electromagnetic energy there is: a single photon. Yet that same instrument functions perfectly for us in the middle of an equatorial desert at high noon where the brightness level is exponentially larger.
Since all of us possess this extraordinary photosensitivity, how come we can't see if the edge of a projected image is not nearly as bright as its center? The answer is that the eye-brain interface doesn't permit any one portion of the visual field to outshine the remainder. Moreover it tends to ignore continuous portions of its field in favor of changes, discontinuities and motion. We don't perceive the projected center-to-edge fall off because it is smoothly continuous. Gradual brightness differentials of as much as 50% are thus rendered hardly noticeable.
To be sure, there are light levels so low that we can't really "see" anymore, when it really is "pitch dark." And of course there are circumstances when there is so much light flooding our retinas that the visual system goes into overload and we experience what we call glare. (It is interesting to note that common to the onset of both extremes is the impairment of our ability to distinguish detail...)
But in between there is this enormous range of light levels to which our eyes are able to adjust automatically. How, then, are we to decide what is and isn't bright?
When people first set out to quantify visible light they chose as their standard a source that was familiar and common to all of them: a candle. Yes, it had to be a specifically sized candle, made of a specific material and molded in a specific way, but an ordinary candle nonetheless. The amount of light emitted from such a candle became our first and most fundamental unit of brightness. It is called 1 candlepower.
If we visualize such a candle lighted at the center of an otherwise darkened room, we can see from walking around it that its energy is radiating equally in all directions. It is also apparent that the farther we retreat from its flame the less light it appears to be shedding. Although those two facts are straightforward enough, some powerful deductions can be made from them. Generalizing from the observations, we can state that light from a point source (the candle) radiates outward in all directions such that it uniformly illuminates the surface of an ever expanding sphere. As the radius of that sphere gets larger and larger, the surface area grows at an even faster rate and thus the energy from our candle is spread ever thinner.
Since the surface area of a sphere of radius r is given by 4pr2, we can see that a radius of 1 foot will give us a surface area of 12.56ft2 (or 4pft2, since r2 =1). Increase the radius to 2 feet, however, and the surface area jumps to 50.27 ft2. When r=3 feet, the surface area has ballooned to 113.10 ft2; and so on. This is the Inverse Square Law and among the things it explains is how come a 6 x 8-foot screen (48 ft2) isn't half as bright as a 3 x 4-foot display (12 ft2), it is a fourth as bright, even though the throw distance (r) has merely doubled.
Now suppose that in addition to defining a unit of emitted light (the candlepower) we wish to establish some sort of standard for measuring light incident to a surface area. Let's place our candle at the center of a sphere with a convenient radius of 1 foot. And now let's calculate the amount of our candle's energy that is incident to just one square foot of the sphere's surface. And, because it makes perfect sense to, let's call that quantity 1 foot-candle.
Armed with this definition we can establish and communicate all sorts of useful standards by taking all sorts of repeatable measurements. The noonday sun can deliver 10,000 foot-candles onto the roof of your car, for instance; while the full moon will deposit only 0.02. Workspaces in our offices and operating rooms want to have 15 or more foot-candles; an auditorium may need only 5. For comfortably sustained reading it's nice to have 10 and for close machine work we'd better have more than 30. In a darkened theater, the bright parts of the movie will have about 15 foot-candles, the night scenes as little as 2.
The next step we want to take concerns deciding exactly what fraction of the energy in 1 candlepower is expended in the production of 1 foot-candle. We need this new unit because when we come to consider light sources other than candles we recognize that some of them, video projectors, for example, do not radiate spherically but beam all their output in just one specific direction. They illuminate, therefore, only a section of the sphere surrounding them and we want some way of accounting for that.
As the radius of the sphere is still 1, the total surface area is 12.56 ft2 (4p again). Since we are only interested in 1 of those square feet, it follows that our new unit will equal 1 candlepower divided by 12.56. Let's call this unit 1 lumen.
Understanding the relationship between foot-candle and lumen enables us, for instance, to calculate precisely how much light will fall on a screen of any specified size from a projector of some specified lumen output. All we need to know is the requisite throw distance (the r of our formula). Or, given the foot-candles, we are equally adept at solving backwards for the lumens.
So now we know how to express numerically the brightness of a light source and we also know how to quantify the light emanating from that source as it illuminates a distant surface. What happens if that surface is a screen which is, therefore, reflective (or transmissive), how will we quantify the brightness it may re-radiate?
First, let's stick with the square foot concept and make our newest unit a measure of the light coming off 1 ft2 of surface. And what energy unit would be appropriate to choose? Let's use the lumen again and declare that 1 square foot of surface radiating 1 lumen of light is producing 1 foot-Lambert.
To tie all this together neatly we need just a few more terms, the first of which is the word flux. Technical people like to say flux when they are referring to a flow of energy. (When they quantify such a flow onto a surface, incidentally, they'll say flux density.)
Another phrase popular with scientific types is solid angle. An ordinary angle of the kind we draw on a piece of paper has just two dimensions (and is often called a plane angle). The angle formed at the vertex of the cone of a projection beam, however, has three dimensions and is, therefore, deemed to be "solid." (Thought about in this light, we can generalize that a solid angle must always be formed by at least three intersecting planes, the intersection of two walls and a ceiling being a typical example.)
With this small vocabulary in mind we should be ready to decipher a full-blown, scientific definition:
A lumen is equal to the luminous flux through a unit solid angle from a uniform point source of one candle, or to the flux on a unit surface area all points of which are at a unit distance from a uniform point source of one candle.
We can also state with equal rigor (but a tad less pomp) that an intensity of 1 lumen/ft2 equals 1 foot-candle.
And we will concisely define the foot-Lambert as a unit of luminance equal to 1 lumen/ft2.
In fairness we now have to acknowledge that all of the brightness units we've developed so far count on one foot. And although citizens of the United States and a few other enclaves remain comfortable with this 12-inch length, the rest of the planet may find it bewildering.
Can we convert our definitions to the metric system without stumbling? Of course. Notice that our expanded lumen definition has already freed us from reliance on any particular distance unit. Any consistent unit will serve.
Let's start with the same candle. When we light it in the metric system, however, we don't assign any specific number (like 1 meter) to the radius of its surrounding sphere. We pay attention instead to the solid angle (whose vertex is our candle) formed by the radius and a "square" section of the surface whose sides are equal to that radius. We call this solid angle a steradian. We should not be surprised to discover that there are 4p (that number again!) steradians in every sphere.
Metric people use as their basic unit 1 lumen/steradian. They call this unit a candela.
If 1 lumen per square foot equals 1 foot-candle, 1 lumen per square meter equals 1 lux. If the surface area is reduced to 1 cm2, the unit becomes a phot. And 1 candela/m2 is known as a nit.
If you recall that there are 10.76 ft2 in 1 m2, you could extract foot-Lamberts from candelas by dividing the latter by p times the square feet (ftL=cd/đ ft2), although you might not want to.
Now that we are conversant with these various units, something useful that we could do with them is discover why so many manufacturers of LCD projectors choose to express the brightness of their products in lux rather than in lumens.
If, say, the luminance of one of these machines is stated to be "600 lux (@ 40" screen size)" the first thing we do is elucidate the parenthetical phrase and convert a screen diagonal given in inches (a unit wholly inappropriate to start with) into a screen area measured in square meters. That calculation yields a convenient answer: .50m2 exactly.
That established, we are ready to plug our values into the formula
Lumens = (Lux x Area) / (Screen Gain)
Since we may take ScreenGain to equal 1, we are left with 600 lux (a largish number) to multiply by .5 (the area) to get 300 lumens (a smaller number). To complete the analysis, suppose that we consider a screen size a good deal more typical than a 40-inch diagonal; suppose we take a 100-inch diagonal screen. Can you see why the number of lux from a 300 lumen projector will plummet from 600 to 97?
There is nothing wrong of course with wanting to see a product in the best possible light. A clear understanding of the units chosen to express brightness, however, can also be illuminating. Knowing how to interpret the illuminance of a projector or the luminance from a screen requires that we keep a sharp eye on the units used to express them. Brightness specifications are not meant to obfuscate or disguise. Surely they are meant to enlighten.