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.