illuminating a sphere

The back is in shadow, but the law of reflection sends the reflected light off at twice the surface angle, so you only need to illuminate a hemisphere.

It isn't difficult to prove. It's obviously true in azimuth, by symmetry. In polar angle, the obliquity cancels out the increase in circumference with radius, making the whole thing isotropic. I first learned that from my colleague Doug Goodman at IBM--it's pretty neat and occasionally very useful, as in the case John alludes to.

Cheers

Phil Hobbs

Oops, it DOES get up to 4*pi steradians. It's not terribly practical, though, unless you have a perfect source. The 'uniformity' of the scattered light doesn't preclude polarization effects, of course.

I'm familiar with the machining of MgO blocks to make coupled spherical cavities; feed light through a small port in cavity #1, and make a pinhole from cavity #1 to cavity #2, and a second pinhole in cavity #2 has a light meter behind it. The intent is to eliminate from the measurement any dependence on the planarity or other directionality of the incident light source. Collimated controlled-aperture source not required.

Well, obviously, we're only talking about the light reflected off the ball, not the light that misses it 19 miles away.

No, it flings lots of light into the "back half" of the universe. Try it.

John

It's useful if you're interested in calculating how much light bounces off a shiny sphere. Elegant simplification.

How do you machine spherical cavities? Hemispherical cavities that fit together?

John

But you haven't answered that other question - what do you see when the sphere is directly between the flashlight and your eye?

Thanks, Rich

Close to the sphere, obviously nothing. Farther away, slightly off-axis, with a finite sized source, you seem to get a bright arc that almost circles the ball. In the classic case, with a point source, there is a ball-sized shadow which approaches zero angular diameter as you get far-field. In reality, there's diffraction which bends light back into the shadow.

The point is, if you know how many watts of light hit the ball, it's easy to calculate how many watts head off in any direction. Well, I thought it was cool.

John

Does it also have to be polarized? That's what "plane" implies to me.

I guess all phase-coherent light is polarized in practice. Not sure if it's possible to create non-polarized light in phase.

The theorem is actually in ray optics, which ignores both diffraction and polarization, unless you put them in by hand (which is usually fairly straithforward). Nobody actually does wave-optics simulations of optical systems--you do ray optics until you get to the exit pupil, keeping track of the propagation phase, then construct a wavefront there that you propagate to the image using wave optics. Works great, and is easy to follow.

Cheers

Phil Hobbs

I will because I think that as you have stated it the result is only approximately true in the far field. It is certainly true that a ray that hits the sphere making an angle theta with the sphere centre is reflected away from the axis by 2*theta but there is also geometrical spherical aberration even in the classical ray tracing case.

The position that the point source appears to be in the reflection moves depending on the angle of the outgoing ray. Taking a concrete example for paraxial rays they appear to radiate from r/2 but by the time you get out to pi/4 the reflected ray cuts the axis at r/sqrt(2) and for pi/3 it is out to r. A sketch will confirm this.

If my back of the envelope algebra is right I think that with an ideal plane wavefront the ray leaving the vicinity of the sphere centred at (0,0) in a direction 2*theta appears to have come from a point source on axis and positioned at ( r/(2cos(theta)), 0)

It does not behave like a point source at a fixed position so that close to the axis behind the sphere the brightness is lower.

You would have to have a convex parabolic reflector for the point source to stay fixed for any incident ray.

It is approximately true for the far field. Fine for garden ornaments. Don't rely on it for anything quantitative.

Regards, Martin Brown

There are a great many applications where the far field result is very useful quantitatively. The calculation John was alluding to was part of a photon budget for a laser-based detection system for small metal spheres, where it's good to a few percent over all sizes of interest (i.e. 10*lambda < radius < 20*lambda, observation distance > 10k lambda). There are diffraction rings in the forward-scatter direction due to the shadow punching a hole in the wavefront, but at large scattering angles they die away like 1/(ka sin theta)**3, so the angular spectrum of the backscatter is pretty well featureless.

Surface films and minor surface irregularities will be a more serious source of error than the ray optics approximation, but the whole thing ought to be accurate to 1 dB or so, which is better than good enough for a photon budget.

The curved surface of the sphere gives rise to a lot of interesting boundary wave physics, which is the topic of the geometrical theory of diffraction (GTD, of stealth aircraft fame). GTD effects are mostly seen inside shadow boundaries and near caustics, and would be pretty important in the forward direction in this case, but in backscatter it's not a big issue.

Finding the right sleazy approximation for the job is one of the major skills you learn in physics, and an analytical expression contains more information than a year's worth of simulations. A well-stocked bag of tricks is part of my stock-in-trade, so I collect them fairly assiduously.

Cheers

Phil Hobbs

As a physicist, you have to learn sleazy tricks to get approximations to system behavior. As engineers, it comes natural to us.

John

Sleaziness comes naturally, I grant you. ;)

Cheers

Phil Hobbs

Yes; I believe the starting point was a split block of the ceramic, and the halves were clamped to a lathe faceplate, roughed out, and a carefully profiled scraper finished the hemispherical surface. It's a great lab convenience to have an instrument shop in the basement...