Jim Worthey, Lighting and Color Research
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Lighting Quality and Light Source Size
This article was published in 1990:
James A. Worthey, "Lighting quality and light source size," Journal of the IES 19(2):142-148 (Summer 1990).
In the version below, I have copied the original with a minimum of editing. The 3 photographs are in color below, although they were black-white in the journal. The text has been retyped, so there could be typographical errors. Any inserted or updated material will be in [square brackets] or otherwise indicated.

Lighting Quality and Light Source Size
James A. Worthey, PE, PhD

When we see an object, the information that the eye receives results from the optical interaction of the object's surface with the light striking the surface. Potentially, every detail of the object's optical environment has an effect on the way that its surface properties are revealed. For example, the first two photos show a bank of windows on the Teamster's Union headquarters in Washington, D.C., from two different vantage points. When the dome of the US Capitol is imaged in the glass, you see three things: that the windows are mirrored; that the Capitol Dome is imaged in the mirrors; and that the mirrors must not be perfectly flat, because of the distortion of the reflections.

Teamsters Bldg reflecting the Capitol
Teamsters Bldg reflecting only Sky

When the windows' mirror coating images only the sky, you cannot be sure that they are mirrored, and there is no clue that they are not flat.

Are we to conclude that the "illuminant" for the mirrored windows is the Capitol dome, and that the mind compares its remembered image of the dome with the distorted mirror image, and makes an inference about the non-flatness of the glass? The answer is obviously yes. The eye is a sophisticated detector connected to a sophisticated brain, and we "see" the distortion of the glass instantly, but only if the information becomes available because of contrasts in the glass's environment.

The mirror is a curious example of an illuminated object, because its contrasts are those of the rest of the environment, however great or small those may be. The flat or almost-flat mirror is an extreme example, but not a trivial one, because many objects are mirror-like to a greater or lesser degree. Metallic objects, such as Christmas-tree ornaments, may be as shiny as a mirror, but are not flat. A wine glass has a shiny surface but is also transparent, so that it acts as both a distorting mirror and a distorting lens. Thus, the simple mirror illustrates an important principle: contrasts beget more contrast.

The example of the distorted mirror plus the US Capitol shows another principle, as already implied. If the object displays visual cues---almost any kind of contrast---the human visual system probably can interpret the cues in some useful (and correct) way. A whole article could be devoted to examples supporting this principle, but the important thing here is that it might be a useful and non-trivial axiom for engineering analysis. At present, for instance, robot vision systems are highly specialized and limited, and could be counted on to make no sense of a distorted image of the Capitol, or of most other scenes. With humans, an engineer who just tries to provide the contrast is on the right track. She can safely assume that the eye will make use of all contrasts. A series of interesting articles could also be devoted to the question of how one would "count up" the contrasts in a complex scene, but let's move on.

Another unusual example
The third photograph again shows that everything about a light source matters. The photo is of the mottled shadows seen beneath a deciduous tree on a sunny day.  The usually circular bright spots are peculiar in this picture however. Each bright spot is like a cookie with a bite taken out of it.
Eclipse Shadows 1984 May 30
The picture was taken in Cleveland, NC, during the annular solar eclipse of May 30, 1984, perhaps a half-hour after the moment of annularity. We see that the usual circular light spots do not just happen: they are in effect, pinhole-camera images of the sun. To generalize, we may say that where cast shadows are concerned, the exact size and shape of the light source matter: Even the sun, which is one of the smallest familiar sources, cannot be treated as a mere point.

Source size
Now if  contrast begets contrast, and contrast is usually useful, it makes sense to ask how contrast can be introduced into a scene. This question has many answers involving such topics as interior design and color rendering.1 The remainder of this paper focuses on just one parameter, the size of the light source. Familiar lights vary greatly in area, as seen in Table 1.2 This table shows that lights vary in the solid angle they present---their "visual area"---by a factor of at least a million.

Table 1---Light source sizes
Light Source
Area, (m2)
Solid Angle at 2 meters distance, microsteradians
Unfrosted 60 W incandescent bulb
The Sun (distance = 93 million mi)
Ordinary frosted 60-W incandescent bulb
Soft White 60-W incandescent bulb
F40T12 fluorescent tube
Luminous ceiling, extending to ∞ (2π steradians)
(2π million)

Much can be learned about the effects of light source size from the simplified apparatus of Figure 1.  The test object shows the essence of the veiling reflections: The black glass acts as a partially reflecting mirror, forming clear images, but with only 4 percent the luminance of the thing imaged. 

Idealized apparatus
Figure 1---Idealized apparatus for thought experiments concerning source size. A test object consists of a flat piece of black glass mounted next to a diffusely-reflecting (Lambertian) white surface. The test object is positioned under a circular luminaire of uniform luminance.
The white area next to it reflects all the light that strikes it, but diffusely. If a light-source image is visible in the black glass from a particular vantage point, that's a veiling reflection. A simple measure of veiling reflection amplitude is the gray level of the black glass, that is the luminance of the source image, divided by the luminance of the white area.

We assume that only a single light source llluminates the white surface, a circular uniformly bright area which subtends an angle 2ρ and is centered over the test object, as shown at the bottom of Figure 1. We assume that everything is centered vertically, including the viewer's eye, but we ignore any effect of the viewer's head blocking the light.

Basics of veiling reflections
Let us now review some results related to veiling reflections. Details are available in recent publications.2,3 For the apparatus of Figure 1, the gray level of the black glass is
g = 0.04 / sin2ρ                   (1)
where g is the gray level. For a light source that covers the hemisphere above the test object, ρ = 90 degrees. The gray level is then 4 percent, which is not negligible because it may be high, relative contrasts that are being veiled, and because of the non-linearity in the way the eye sees lightness. The constant 0.04 in Equation 1 is a typical value based on a convenient value for the refractive index of the black glass.3

Veiling reflections and color
A look at veiling reflections in colored objects reinforces the view that such reflections are never negligible. Again, recent work on this topic2 will be summarized. The eye sees colored objects in comparison to a white object, even if the white object is not there. Furthermore, the eye responds nonlinearly to color variation, so that a small physical change in a color stimulus has a bigger visual effect when the starting point is a dark gray or a deep color. All these facts and more can be summarized by a color solid, which is a three-dimensional representation of the range of colors that people can see.

veiling reflections on surface colors
Figure 2---Reduction in the range of lightness and saturation of surface colors due to veiling reflections. Data are plotted in the cylindrical-polar version of the CIELAB uniform color space, at four selected hue angles (h*). Radial coordinate c* represents saturation, while axial coordinate L* is a measure of lightness. The solid lines represent the limits attainable with real pigments. Chains of arrows show successive shifts as veiling reflection is increased to 4, 8, 12, and 16 percent of white. At all hue angles, including those not shown, a similar systematic loss of saturation and of lightness occurs.2

Several variations on the basic idea of a color solid are in use by visual scientists and practical color users such as designers, all meant to represent the same facts. The so-called CIELAB uniform color space provides a convenient basis for drawing the color solid when calculated results are important, since it is defined by formulas. Figure 2 shows four radial slices through the color solid in CIELAB space. The vertical axis represents the variation from black, at the bottom, to white, at the top. Each graph represents a particular hue, and the abscissa is the radial coordinate, corresponding to the saturation of colors. The solid line in each graph is the limit of practical colors in that plane, according to data of Pointer.4 The chains of arrows show the effect of veiling reflections in successive increments of 4 percent. The net effect of veiling reflections is to reduce the volume of the color solid. For instance, the initial 4 percent veiling reflection, corresponding to spherical lighting, shrinks the range of colors seen by 37 percent.2 Again, this shows that veiling reflections are never negligible. Having reviewed two recent papers at some length, we now move on to new material.

Source luminance
The lights in Table 1 were chosen on the basis that each approximates a familiar lighting situation. They would not give equal illuminances. Although this intuitive approach shows that lights do vary greatly in size, there is a deeper reason that familiar sources vary in solid angle. The size variation must occur because of the tremendous variation in luminance.

Refer again to Figure 1. Let E be the illuminance at the object under the circular source, and let L be the lominance of the source. Then, taking a standard formula for illuminance and making semi-subtense the dependent variable tells us that
Equation 2.
A short calculation gives the solid angle Ω subtended by the circular source:

Ω = 2π(1-cosρ)     .     (3)

[Applying a trig identity puts Eq. (3) in an improved form:  Better equation for solid angle. Original article had Eq. (3) as above.]

Equations 2 and 3 express exactly the tradeoff between source luminance and source size. Repeated use of the small-angle approximation lets us combine and simplify to get
Equation 4    .     (4)

This is just what intuition would say, that to increase illuminance on the task (E) one must increase the light's size as seen from the task (Ω), but if you can use a source of higher luminance (L), then you can make it smaller. Taking the small angle approximation in Equation 2 and electing to express ρ in degrees gives
Equation 5    .         (5)
For many purposes, the approximations of Equations 4 and 5 are sufficient, as we shall see.

Table 2 lists the luminances of various light sources and the corresponding size that the circular source of Figure 1 must have in order to give E = 1000 lx. Size is given both as ρ in degrees or minutes of arc, and as Ω in microsteradians. All luminances are taken from Reference 6. Note that the solid angles are not the actual solid angles of the lights as in Table 1, but are the solid angles that would be necessary to give 1000 lx on the task.

Table 2---Actual source luminances and the corresponding size that would be necessary for 1000 lux on the task in Figure 1.
Luminance [cd/m2]
ρ [deg]
Ω [microsteradians]
1.58 min
Carbon arc, maximum
1.94 min
Carbon arc, minimum
5 min
Tungsten, vacuum
2×106 43.4 min
Tungsten, inside frost
1.2×105 2.95 deg
Cool white fluorescent
7×103 12.3 deg

Figure 3 will help to illustrate Equations 2-5 and Table 2. The graph shows how light-source size varies with luminance, for a fixed assumption of 1000 lx on the task. Source luminance is on the x-axis, while the y-axis is left unlabelled because two dissimilar quantities are graphed, the solid angle Ω covered by the circular source and the angular radius ρ, that it subtends. The solid curves, based on Equations 2 and 3 are exact given the assumption of a circular uniform source, symmetrically positioned.
Source size in relation to luminance
Figure 3---Variation of light source size with luminance. As indicated on the face of the graph, the upper solid and dashed curves represent solid angle (Equations 2 and 3), while the lower solid and dashed curves are semi-subtense (Equations 4 and 5). [This version of the figure has a couple additional light sources not in the printed article.]

The dashed lines show the approximations of Equations 4 and 5, based on small-angle assumptions. The fluorescent tube is lower in luminance than the sun by a factor of more than 100,000. The other familiar sources are scattered along the interval between these lights, and throughout this range the "approximate" formulae are seen to give accurate results.

There are clear limits on the range of luminances, which has been indicated by terminating the graph of solid angle with circles. The sun when directly overhead in a clear sky has the highest luminance of any familiar source. At the other extreme, a luminous ceiling which covers the hemisphere above the work and has a luminance of 1000/π [candelas/m2] is the dimmest source that can give an illuminance of 1000 lx. Setting a lower figure for illuminance would lengthen the graph, but the same principle would apply.

Obviously, the solid angles given for the various sources in Table 2 and Figure 3 are not the solid angles that these sources actually subtend, but the solid angle that they would need to subtend in order to give 1000 lx on the task in Figure 1. Comparison of Table 1 and Table 2 shows that only one 1/100 of the full sun area is needed to give 1000 lx, for instance. Further comparison shows that the solid angle of fluorescent tube required to give 1000 lx is about 100 times that of the real sun. This shows what is known intuitively, that where "sparkle" is concerned, meaning the compactness and luminance of highlights and the sharpness of cast shadows, the fluorescent lamp cannot compete with the sun. On the other hand a fluorescent tube or luminaire showing the luminance of a fluorescent tube need cover only 140,000 microsteradians, or about 1/40 of the hemisphere above the object, in order to provide a respectable illuminance of 1000 lx. If such a light were positioned next to a large dark area, a small tipping of the task could eliminate the veiling reflection.3 As a basis for understanding luminaire design and application, Equations 2-5 can be used in conjunction with the optical principles limiting image luminance. An aerial image of a source has the luminance of the source, reduced by any losses or partial reflections along the path.5 It is quite possible for a luminaire, within a certain range of viewing angles, to show a luminance approaching that of the source. It is not posssible, however, to make an "optical funnel" that brings all the source rays together in a small area in order to present a higher luminance than that of the sources.

Therefore, the solid angle of Equation 3 presents a simple goal for luminaire designers. No luminaire can present a smaller bright area; one that approaches it can be considered compact for the specified source. If the illuminance E used in Equation 3 to evaluate the luminaire is in fact the task illuminance needed in a complete installation, then it follows that only one luminaire need shine on the task. This is desirable from the point of view of veiling reflections, so the real working illuminance can and should be used in evaluating luminaire compactness. Referring to Table 2 and remembering the definition of solid angle, Ω = A/r2, we see, for example, that 1000 lx on the task from a single bright area with the luminance of a fluorescent tube requires that area to be 0.56 m2. A lower illuminance would permit a smaller area, as would a lower ceiling. Using focused luminaires so that each task only sees one or two compact sources implies that the luminaires must have sharp cutoffs in their photometric distributions. Such luminaires are often sought in order to reduce direct illumination of the eye. The present analysis shows other distinct effects of focused luminaires. Technically, Equations 2-5 apply only to the circular source of Figure 1. However, it is more than plausible that Equation 4, which expresses a simple conservation-of-flux principle, would be valid for other shapes, so long as their semi-subtense in the longest dimension did not exceed about 35 degrees, the size at which the approximate formula begins to be inaccurate. This technicality can be examined in the future.

At this point, many readers may note that I am using optical principles that are well known to luminaire designers, and perhaps not even explaining them very well. My goal, of course, is not to re-invent optics itself, but to apply optical principles to questions of lighting quality. In this same spirit, let us now take a look at the optics of highlights.

Highlights are images of a light source as imaged in the surface of a shiny object, whether the object is metallic or dielectric. The term is apt because highlight luminance, even in a dielectric object, can be quite high when the source is small. For instance, applying Equation 1 when the sun is the source shows that highlight luminance is some 2000 times the luminance of a white. While we could consider a luminaire's image in a plane mirror to be a highlight by definition, highlights are of particular visual interest in curved surfaces, even highly curved surfaces. For instance, highlights tend to pile up on the rim of a glass tumbler. If the source is small, this highlight region gives high contrast and serves to localize sharply an important detail of the glass. As for the luminance of such a highlight, it is equal to the source luminance times a reflected fraction that we can take to be 0.04 for dielectric objects3 (as in Equation 1). The highlight luminance does not depend on surface curvature, so long as the surface is shiny.

What does happen as surface curvature increases is that the reflection (image) of the source gets smaller. Let us do a thought-experiment with mirrors. Imagine that it is night, the sky is clear, and the moon is full. We have a flat mirror, tipped so that we can see the image of the moon. The moon's image will be 384,400 km behind the mirror and with a diameter of 3476 km will subtend about 32 min of arc, the same as in direct viewing. Since the moon's luminance is about 2470 cd/m2, the image will have a luminance just a bit lower, say 2200 cd/m2. Now in place of the plane mirror, put a mirrored sphere of 1 m radius, in effect a very large Christmas tree ornament. The ball's diameter is 2 m; its focal length is 0.5 m. The moon's image is no longer so far away; it is at the focal point, or 0.5 m below the surface of the ball. The image size is now 4.7 mm. The image luminance is still 2200 cd/m2.

Now that the moon's image is nearby, we must know the eye's position in order to find the angle that the image subtends. Assume that the eye is 0.5 m from the sphere's surface. Then the image subtense is approximately 16 min. Now let a series of smaller and smaller balls be substituted for the one of 1 m radius, always putting the ball's surface 0.5 m from the eye. The moon's image will get smaller and smaller, while keeping the same luminance. The image will always be 1/2 the radius below the surface, 1/4 of the way through the ball. When the ball is about 6 mm in diameter, the moon's image will subtend only one min of arc, which is about the resolution limit of the eye; 20/20 acuity means that details of 1 min can be resolved. As the balls get smaller after that, the eye no longer perceives the image to shrink, it only sees it getting dimmer.

Familiar objects are most often convex, with a wide range of curvatures. In discussing highlights, it is of interest to know if the source images often fall below the limit of resolution. Suppose that you are designing the lighting for the Acme Shiny & Convex Co., which produces such items as ball-bearings, light bulbs and 1956 Oldsmobiles. You wish to choose between two light sources. If both sources are so small that the common objects in the factory all give highlights below the limit of visual resolution, then highlights will look the same under both lights. Otherwise, the smaller source will give smaller highlights of higher luminance.

Theory of highlights
In order to write some equations describing highlights, let us define r = radius of curvature of the object, d = viewing distance, eye to object, σ = angular subtense of the light source at the object, α = visual angle of the highlight.

Symbol σ is intended to represent the subtense of the light's long dimension. Thus, if the source is the circular one of Figure 1, σ  = 2ρ. As stated before, the highlight will lie below the surface of a convex object at depth r/2. Simple optics7 and repeated use of the small-angle approximation gives
Equation 6     .    (6)
If the object radius is small compared to the viewing distance, the term r/2 can be neglected to give

α = σr/(2d)        .     (7)

Because Equations 6 and 7 involve just two angles and two distances, they can be interpreted in any two convenient units, such as minutes of arc and millimeters. If a concave object's radius of curvature is taken to be a negative number, these equations apply to it also. To use Equation 7, let us assume a fixed viewing distance of 0.5 m, and a source-object distance of 2 m, then compute the critical radius r, such that α = 1 min of arc. This results in Table 3.

Table 3---Object radius for highlight to be seen as a point.
Light source
Subtense, min arc
Object radius for α = 1 min arc, mm
The Sun (distance = 93 million miles)
Unfrosted 60 W incandescent bulb
Soft White 60 W incandescent bulb
F40T12 fluorescent tube
Luminous ceiling, extending to ∞ 10800
no highlights

We see that when the source is the sun, many common objects such as ball bearings fall below the critical radius, while many other objects such as bowling balls fall above it. For other sources, even more objects will be above the critical radius, meaning that the highlights will have a perceptible size and shape.

Dynamic range
The luminous ceiling gives nothing that can be called a highlight, only a veiling reflection of gray level 0.04. With a small source, highlights add to the contrast and dynamic range of a scene. Consider the setup of Figure 1, but assume realistically that there is a slight haze on the surface of the black glass, so that its gray level away from the source image is 0.01, rather than 0.0.

Table 4 shows the variation in highlight luminance and dynamic range of the test object, as the lighting is changed. Again in this table, lights, other than the sun, are 2 m overhead. Highlight luminance is expressed as a multiple of the white area's luminance. Except in the case of the luminous ceiling, the maximum and minimum luminances both occur in the black glass. Under the luminous ceiling, the highest luminance is the white, while the black glass is covered by an unavoidable veiling reflection. Dynamic range means the ratio of highest to lowest luminance. From Table 4, we see that walking in from sunlight to a space with a luminous ceiling decreases the dynamic range by a factor of about 10,000.

Table 4---Highlight luminance & dynamic range
Light Source
Highlight Luminance
Dynamic Range
Unfr. 60 W
Ord. Fr. 60 W
Soft White
1 Fluor. Tube
Lum. Ceiling

These calculations concerning amplitudes of veiling reflections and highlights, and their effect on dynamic range can be reduced to man-in-the-street terms. Suppose that the man in the street is standing in front of a drugstore, diffusely lit by fluorescent lamps, looking in the plate glass windows. Suppose that it's a clear day, but the sun is fairly low in the western sky, so that the mean luminance of the outdoor scene is equal to that inside the drugstore. What the man in the street will see, or what you and I will see is that the scene in the drugstore looks washed out, compared to the scene outdoors.

What we now understand from calculation is that the drugstore does not look washed out for some mysterious reason involving fluorescence of the crystalline lens, or flicker, or some quirk of the visual system. It looks washed out because it is washed out. Highlights are dim and large; blacks and saturated colors are covered by veiling reflections. This is in addition to the loss of color contrast because of the inferior color rendering of fluorescent lights, the loss of black-white contrast because of the lack of shadows in the drugstore, and the enhancement of color contrast outdoors due to the fact that light from the west is reddish while that from the east is bluish.

Summary and conclusions
The examples of the mirror object and the eclipsed sun show that in general everything about a light source's size and shape can matter. One important form of variation among lights is their size. This paper has concentrated primarily on the role of source size, and its interaction with reflections from shiny objects. For a specified illuminance, lights must vary in size because they vary in luminance. Veiling reflections are never negligible; in colored objects, they cause a loss of deep colors. Highlights, reflections of the light source, can be an important source of contrast and information in a scene. With a compact source such as the sun, highlights will appear as points in highly curved surfaces, but will have a size and shape in objects of greater radius of curvature. When a luminous ceiling is the source, highlights do not exist, only a nearly unavoidable veiling reflection. The result is a huge reduction in the dynamic range of a scene.

Many simplifying assumptions have been made in order to create a simple picture of the effects of source size. The quantitative treatment extended only to reflection from perfectly shiny surfaces. The small-angle approximation and the example of a circular uniform light source were used repeatedly. The goal has been to provide a basic description of some effects of source size, while taking account of such facts as the luminance variation among light sources, the optics of image size and image luminance, the eye's limit of angular resolution, and the way in which object colors are seen.

1. Worthey, J. A. 1985. An analytical visual clarity experiment. J. of the IES 15(no. 1):239-251.
2. Worthey, J. A. 1989. Effect of veiling reflections on vision of colored objects. J of the IES 18(no. 2).
3. Worthey, J. A. 1989. Geometry and amplitude of veiling reflections. J. of the IES 19(no. 2)
4. Pointer, M. R. 1980. The gamut of real surface colors. Color Res. Appl. 5:145-155.
5. Martin, L. C. 1932. An Introduction to Applied Optics, Volume II. Pitman: London. The general issue of light concentration by optical systems is discussed in W. T. Welford and R. Winston, The Optics of Nonimaging Concentrators (Academic, New York, 1978). Any reader who knows of yet another textbook or article that derives or explains the theorem about image luminance, is requested to contact the author.
6. Wyszecki, G. and Stiles, W.S. 1967. Color Science: Concepts and Methods, Quantitative Data and Formulas, John Wiley: New York.
7. Jenkins, F.A. and White, H.E. 1957. Fundamentals of Optics, Third Edition. McGraw-Hill: New York.
8. Gilchrist, A.L. and Jacobsen, A. 1983. Lightness constancy through a veiling luminance. J. Exp. Psych: Human Perception and Performance 9:926-944.

[Once again, this article was originally published as: James A. Worthey, "Lighting quality and light source size," Journal of the IES 19(2):142-148 (Summer 1990). Color versions of the photos happened to be available, so those are used above. Figure 3 is slightly updated from the printed version. Figure 2 is a little ugly above, but is essentially the same as in the original article. Otherwise editing has been kept to a minimum. Your comments are invited. Jim Worthey, 2005 January 24.]

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