Back radiation from the target

Far-field approximation of the radiant flux from a horizontal disk. The region closer than one disk radius from the center of the disk has been blacked out. If the disk is at the temperature of the sun, the remotest black fringe is at 0 suns of flux, the next at 1000 suns, etc.

In a perfect solar concentrator, the target reaches thermal equilibrium (radiative balance) upon reaching the temperature of the sun's surface. Radiant flux is then net zero throughout the system due to the presence of thermal radiation radiated back from the target. In the forward direction, sunlight uniformly illuminates the target; in the reverse direction, back radiation from the target is collimated and beamed toward the sun. When a perfect optical system is in thermodynamic equilibrium, these two fluxes cancel out everywhere.

At any point in empty space, the divergence pattern of the beam of sunlight will be exactly mimicked by the back radiation traveling in the opposite direction. At an optical surface—where light might be redirected and its divergence pattern reshaped—the fluxes on the two sides of the surface must still sum to zero.

Flux on a surface is proportional to cosθ, the cosine of the angle the incident rays make with the surface normal. This is Lambert's Law. The divergence pattern of light at a particular point, P, on a surface can be described as the interior of region drawn on a unit sphere. In geometrical terms, Lambert's Law says that the flux at the surface is not the area of this region on the unit sphere, but rather it is the area of this region after it has been projected orthogonally onto the surface at point P. The orthogonal projection supplies the factor of cosθ.   Our requirement for net zero flux through the optical surface is then that the projected areas for both sides of the optical surface are equal.

There is one part of the optical design we have no control over: the pattern of thermal radiation that will be emitted from the target. In the case of a solar glass-melting furnace, the target is the opening of the furnace, the oculus, which is effectively a horizontal disk. Calculating the flux of back radiation from this disk is merely a problem in visual perspective because it is a perspective view of the disk that determines the shape and area of the region on the unit sphere. According to Lambert's Law, we should follow upon that exercise in visual perspective with an orthogonal projection onto some surface (whose orientation would be somewhat unclear at this point,) but, if we assume the angles of incidence to be the same on both sides of the optical surface (a desirable condition for both efficiency and simplicity,) that step is unnecessary. We will make that assumption.

The problem in visual perspective gets a bit complicated close to the disk, but in a far-field approximation the horizontal disk appears as an ellipse, one angular dimension being proportional to 1/r, and the other to cosθ/r.

More precisely, taking the radius of the oculus = 1, and r = sqrt(x² + y² + z²),

the angular extent of the ellipse in the horizontal (non-foreshortened) dimension = 2 * arctan(1/r), or approximately 2/r,

and the angular extent of the ellipse in the vertical (foreshortened) dimension, is approximately cosθ * 2/r.

Therefore, our perspective view of the disk translates approximately into an elliptical region on the unit sphere having an area, measured in steradians, of cosθ * 4/r² * π/4 = πcosθ/r².

The angular diameter of the sun is .01 radians, so a flux of 1000 suns corresponds to an image of the sun covering 1000 * π * (.005)²  =  .079 steradians.

The graphic above shows the far-field approximation of the thermal flux from the oculus. When the disk is at the sun's blackbody temperature, each fringe represents 1000 suns, that is, the remotest black fringe is at 0 suns, the next black fringe is at 1000 suns, etc. To give a sense of scale, points closer to the center of the disk than one disk radius have been blacked out, but the far-field approximation is not actually good enough to go in that close.