Re: Graphical Ray tracing on Hangar Floor

On Dec 11, 10:01 pm, "Grandpa" <j.ray...@xxxxxxxx> wrote:
Jim Klein wrote:
Probably with a set of 7 place trig tables and Snell's law and a
mechanical desk calculator. It sounds pretty terrible now but as late
as the 1950's people used 7 place trig tables and a mechanical
calculator like a Marchant or a Frieden. Square roots we performed
iteratively with plus, minus, multiply and divide.

Quite a bit later than that in some places . . .

One Friday afternoon in Summer 1976 I visited Jim Burch (of speckle
interferometry fame, and a great guy) at NPL (National Physical
Laboratory), Teddington, England. At the time, Burch was designing an
ultra-precision alignment system employing a laser beam for positioning
mechanical parts along a line on a big optical table, and was worried
about whether the CG of a coherent light beam really traveled in an
absolutely straight line.

As a conceptual example he considered a hypothetical light beam having a
flat wavefront and a half-triangular (i.e., half-sawtooth) amplitude
profile centered about x=0 at z=0. The CG of this hypothetical beam
profile was therefore offset from x=0 at the input plane z=0 and
presumably for some distance moving outward from it. Since the input
phase front was flat and the input field distribution purely real,
however, the far field angular spread (Fourier transform) should be
exactly symmetric about x=0 and therefore the far-field CG should be
exactly centered at x=0, which worried him.

He had a summer intern from Cambridge Univ working with him on this
problem. Burch had managed to work out an analytical expression for the
complex amplitude profile of this triangular input beam in the
near-field (Fresnel) region, and had the intern numerically calculating
the CG of the near-field profiles for him, to see if or when the CG of
the outward-propagating beam would shift over to be on axis at x=0. To
do the calculations, he'd supplied the hapless intern with an immense,
dusty volume of 7-place tables of complex Fresnel integrals (God only
knows where that came from!), and had him hand-calculating the intensity
profile integrals to find the CG of these near-field beam profiles,
using one of these huge ancient 10 x 10 key Marchant or Frieden
mechanical calculators. On seeing this, I couldn't forebear from
saying to him, "Jim, there are such things as computers!"; and can still
recall him saying, in his usual bluff/gruff fashion, "Aw, you can't
teach an old dog new tricks".

I went home to the house exchange in Virginia Water, near Windsor Castle
and Windsor Great Park, that my family and I were living in for 6 weeks
that summer, and the next day in half an hour or so convinced myself
from simple Fourier transform theory that the CG of *any* propagating
optical beam had to travel in an *exactly* straight line -- something
that quantum theorists had known about for quantum wave packets for many
decades (Parseval's Theorem, is that it?) Intrigued, I then looked at
the second moment of an arbitrary beam about this CG, and with a little
effort convinced myself that it would follow an exactly quadratic
variation with z, just like a gaussian beam, but true for *any* coherent
(or in fact incoherent) beam profile (and, again, just like the second
moment of a quantum wave packet in free space).

When I got back home later that summer I inquired of Joe Goodman and (as
I recall) Amnon Yariv if they were aware of either of these properties.
They weren't, but I decided this result seemed a little too minor to
publish, and so I just stuck my notes on this, pencilled on yellow A4
paper, into a file folder.

Fifteen-odd years later, around 1991, when Mike Sasnett of Coherent who
was working with Tom Johnston on the design of the prototype Coherent
"ModeMaster" beam quality meter stopped by to see me to ask if I had any
thoughts on how one should best characterize the useful width of an
arbitrary optical beam profile ("10%-90% clip levels", "86% power in the
bucket", etc), I was able to pull out these almost forgotten notes --
and that's the story, boys and girls, of how the current ISO Standard
for measuring the M-squared parameter of a laser beam came to be based
on the second-moment definition of beam widths, and not any of the other
possible (but less suitable) approaches.

(It also turned out, when I began digging in the electronic databases
that were available by the 1990s, that the theoretical moment
propagation properties of optical beam profiles had already been
discussed in a number of relatively obscure publications in the optics
literature dating back into the early 1970s and even a bit earlier. The
weakness in Burch's triangular beam profile is left for the NG to work
out.)
.

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