Lambert's Photometria

Ernst Mach devotes 7 pages to Lambert in The Principles of Physical Optics. Lambert's basic assumptions were supported by experiments involving the visual comparison of illuminated surfaces. He wrote in Latin near the end of the Enlightenment which makes him difficult for modern readers to understand.

Here is some of what he actually said in his Photometria:

Basic assumptions

"Diximus vero, quod & vulgo notissimum.
1°. Duas pluresue candelas plus illuminare quam unica.
2°. Objectum lumini propius admotum clarius fieri.
3°. Lumen oblique incidens in superficiem, eam minus illuminare."

"We have said, however, what is commonly well known.
1°. Two or more candles illuminate more than one.
2°. The object is made brighter as the light moves nearer.
3°. Light obliquely incident on a surface, illuminates it less."

Sin of inclination factor

"Vis illuminans simulque ipsa illuminatio decrescat in ratione sinus anguli emanations."

"The illuminating power diminishes as the illumination itself in proportion to the sine of the angle of the emanations."

Inverse square law

"Si corpus luminosum, fuerit sphaericum, illuminatio absoluta in A se habebit ad quamlibet aliam normalem, in E, ut se havet quadratum distantiae CE ad quadratum semidiametri corporis CA, adeoque reciproce in ratione duplicata distantiae objecti E a centro corporis C."

"If a luminous body is spherical, it will have absolute illumination at A to that of any normal at E, as it has the square of the distance CE to the square of the radius of the body CA, and so in proportion to the inverse square of the distance E from the center of the body C."

Calculation of the emission into a hemisphere

Note the use of calculus.

The translations are mine with the help of Google translate.

Light and Flux Tubes

The idea of a flux tube comes from the flow of a fluid between stream lines. Here are some links which trace the concept back through time.

1879 Hydrodynamics - Lamb

1873 A Treatise on Electricity and Magnetism - Maxwell

1855 On Faraday's Lines of Force - Maxwell

1852 On the Physical Character of Lines of Magnetic Force - Faraday

Faraday even makes the analogy of lines of force with rays of light! An earlier attempt to arrive at the concept might have been,

1771-81 The Electrical Researches of Henry Cavendish (Maxwell 1879)

Cavendish's canals are remarkably similar to flux tubes. Cavendish studied the electric force in fluids and came close to arriving at the field concept. The idea of an electrical "fluid" is difficult to differentiate from that of electricity. It may have been an early form of an aether theory.

Why Is The Sun Yellow?

The images of the Sun below were exposed at ISO 100, f/8 and for 1/100th, 1/200th, 1/400th and 1/800th of a second respectively moving from left to right with the welding filter. The hue and saturation of the Sun appears to change as the exposure decreases.


The two images on the left are yellowish while those on the right are more greenish. Why does the hue change as the exposure is decreased? For the first two images the green portion of the image saturates near the center of the Sun so all the pixel have values of 255. For the first image we have RGB values of (241, 255, 170). Since the red and green values are nearly equal we perceive the color as yellow. The presence of a blue value shifts the color to make it slightly whitish yellow.

One has to be careful to avoid saturation when measuring relative intensities. Saturation can alter the tristimulus values for pixels in an image and alias the color somewhat. Does the same thing happens with the eye? Human perception appears to be rather subjective at times. What we "see" tends to be projected onto the object.

When colors of the spectrum are rendered as RGB colors the wavelength for the peak intensity of sunlight (555 nm) is seen to be green. But the filter may also be altering the color some.

Observing the Sun through a Welding Lens

One way of observing the Sun is by looking at it through a welding filter in order to avoid injury due to IR and UV radiation. Retinal burns are also possible with prolonged exposure to direct sunlight. The image below was captured with the filter between the Sun and my Kodak Z981 camera's zoom lens and a manual exposure setting (ISO Speed: 100, F-Stop: 8.0 and Shutter Speed: 1/800 sec). I also used the manual focus setting which helped to stablize the camera. To keep the filter in place in front of the camera I taped it to a tube of paper that slipped over the zoom lens. The Eye LCD viewer made it easy to change the settings in daylight since the screen at the back of the camera was totally unusable.


I got the welding lens several years ago at a local hardware store. It appears to be a polycarbonate filter and with a "10" stamped on it (Shade 10?). The filter blocked practically all of the blue portion of the image above and reduced the intensity of the Sun's light considerably.


The Sun is not a perfect Lambert radiator but exhibits limb darkening. The plot below was obtained from the image of the Sun by computing rounded values of the pixel distance, r, from the center of the Sun and then averaging over those pixels with the same distance. The plot was rescaled so that the peak value is 1. Note also that the vertical scale is nonlinear due the picture's encoding format.


It is difficult to accurately determine the angular size of the Sun because of the Sun's corona which is just noticable in the plot. There also appears to be evidence of a solar atmosphere. Some of the extraneous light might be due to scattering off the Earth's atmosphere or in the filter. With maximum zoom one gets an image of the Sun that is about 800 wide for 14 megapixel images. The resolution of the image is limited by the aperature of the camera's lens. It appears that this method will work for simple images of the eclipse a few months from now.

Supplemental (Feb 25): GONG Sun images

The Irradiance For Two Lambert Sources

One can compute the theoretical irradiance for Lambert sources by integrating over all points on its surface visible from the illuminated area. The following curve is for a sphere of radius R and a small area normal to and a distance z along the ray from the center.


The curve is just the inverse square law. Replacing the sphere with a disk of the same radius one gets a curve which is fairly constant near the disk but begins to follow the inverse square law as one moves farther away from it.


I evaluated the curve for the sphere numerically and it is a good fit to the formula below. The formula for the disk was evaluated analytically.


In both cases the irradiance just at the surface of the source is πL.

Lambert Radiators

We derived some of the radiometric laws by assuming a uniformly radiating point source and considering a flux tube linking it to a illuminated surface. To handle an extended radiating surface we can divide the surface into differential elements which will approximate a point source. A complication is that we have to use Lambert's cosine law in order to determine the projected areas at both ends of the flux tube since the surface areas of the source and the illuminated object are not necessarily normal to the connecting "ray." To find the total flux emitted by the surface element we have to sum or integrate over the hemisphere exterior to the surface. We can divide the hemisphere that receives the flux into elements of solid angle

dΩ = sin(θ)dθdφ.
The sides of the element of solid angle are dθ and sin(θ)dφ. For the flux in the direction of angle θ relative to the surface normal we have,

dΦ = L cos(θ) dAdΩ
where L is the Luminance. The total flux for the hemisphere is,


The assumptions we have made are those of geometrical optics which just considers the rays involved in the illumination process. The more general case which treats light as waves and takes interference into account is physical optics. It is used by astronomical optical interferometry to study the emission from stellar sources.

Radiometric Quantities and Flux Tubes

As usually presented the definitions for the radiometric quantities can be difficult to follow. Let's start with a point source emitting light energy at a rate of Φ_T watts uniformly in all directions. For a ray in any given direction we need to construct a flux tube with a solid angle ΔΩ about the ray in order to define the quantities. The flux traveling through the tube will be ΔΦ = Φ_T · ΔΩ/4π since the solid angle for all directions is 4π and ΔΩ/4π is the fraction of Φ_T that enters the tube. Calculations are simplified by defining the Intensity, I = Φ_T/4π, which is the same in all directions and at all points along a ray from the source. The flux through the tube is ΔΦ = I·ΔΩ and this flux will pass through all cross-sections of the tube.


The normal cross-sectional area for a flux tube is ΔS = R² ΔΩ where R is the distance of the surface from the point source. The Exitance of a finite source is defined as the amount of flux traveling through a unit of surface area or,

   M = ΔΦ/ΔS = I·ΔΩ/R²·ΔΩ = I/R².

The same is true for any cross-section of the flux tube and its Irradiance is,

   E = I/R².

The quantity of flux passing through an element of surface, ΔS, normal to the ray at a distance R from it will be,

   ΔΦ = E ΔS = I·ΔS/R²

which is the inverse square law.

The sides of a flux tube are determined by rays which are normal to a cross-sectional surface and so the inverse square law applies only to spherical sources. If one had a large plane which was the source of the light then the rays coming from it would not diverge and the cross-sectional areas would all be the same size. The irradiance would have approximately the same value as one moved away from the plane. The inverse square law will work for plane surfaces unless the distance from the source is very large compared to its dimensions.

For multiple sources one needs to consider the flux tubes from all of them and the angles that they make with the irradiated surface.

Books on Photography circa 1900

1886 An investigation in stellar photography - Pickering

1892 The first principles of photography - Leaper

1898 Photography - Blake

1906 Photography for students of physics and chemistry - Derr

1906 The complete photographer - Bayley

1911 Photography, its principles and applications - Watkins

Radiometry and Photometry

Anyone interested in observing an eclipse should be familiar with radiometry and photometry. These topics are important for an understanding of illumination and exposure in photography. Photometry is also useful in astronomy to measure the magnitude of stars.

The history of the subject can be traced back to ancient times. Hero of Alexandria and Ptolemy both wrote books on optics. These works were essentially geometric in nature but Ptolemy touches on the idea of the cone of vision. Although ancient philosophy considered vision to be projective in nature, i.e., ideas projected onto objects, with visual rays "emanating" from the eye they recognized that light line of sight were essential for the visual process.

Here are some selected references which show the subject's developement over time,

1866 Emissive and Absorptive Power - Kirchhoff

1888 An elementary treatise on heat - Stewart

1914 Theory of Heat Radiation - Planck

1914 Standardization rules of the AIEE

1918 Radiation, light and illumination - Steinmetz

1920 Journal of the Optical Society of America

1950 Radiative transfer - Chandrasekhar

1998 Introduction to radiometry - Wolfe

2012 Applied Photometry, Radiometry and Measurements of Optical Losses - Bukshtab

Port Said and Nika

Is history repeating itself with the Port Said clashes earlier this week? The event involved rival factions at a football game assaulting each other while the authorities stood idlely by. There was a similar incident during the reign of Justinian when rival factions at the hippodrome confronted their ruler and demanded concessions. This and the subsequent events are recorded in history as the Nika Riots. It was the result of the excesses by a weak ruler.

The rabble rousers on both sides Port Said incident appear to be trying put the blame on the Egyptian military for not doing anything. But those involved should be made aware that democracy is not mob rule but requires individual responsibility on the part of the people. Perhaps the Middle East is conditioned against expressions of democratic impulses because of bad experiences in its past. I doubt that this is merely a new tactic to frustrate those seeking democratic reforms or an opportunity to make Obama look bad by those seeking power. One should be on guard against oppressors lurking in the background and playing factions against itself. There are people who are too quick to lay blame.

A history of the events of the Nika Riots can be found in,

Gibbon, The History of the Decline and Fall of the Roman Empire, p. 222 & p. 533

A Simple Imager

One can create a simple model for the eye by treating it as a camera with a variable focal length. The image distance is fixed at 2 cm and the focal length required ranges from 2 cm for an object at ∞ and 1.8 cm for an object distance of 18 cm from the lens.


We can solve the thin lens formula for the focal length in terms of o and i.


Assuming an image distance of 2 cm one can plot the required focal length as a function of the object distance.


If one looks at an object 1 cm in diameter at various distances from the imager its image will be 20 μ, 200 μ and 2 mm for 10 m, 1 m and 10 cm respectively. The symbol μ represents a micron or 10^-6 m. The results show how small images in the human eye can be. The acuity of the model compares well with that of the fovea of the eye with a maximum of about 50 cones per 100 microns.