Reflection from a curved surface

[Anton Shepelev]

Hello, all

Is there a symbolic method to calculate the intensi-
ty distribution on a flat screen from a flat wave-
front after it reflected from a curved mirror whose
surface is described by an equation?

The normal of the wavefront, and the relative posi-
tion of screen and mirror are known.

I know of two ways to solve it:

1. numerically, by tracing the rays reflected
from tiny sections of the mirror, and

2. approximately, via the first N moments of the
distribution, which can be found by integra-
tion over the mirror surface.

Is there a method that will give an exact symbolic
equation?

P.S.: I have crossposted the question to sci.math
because it is essentially a mathematical prob-
lem with a thin physical coating.

[Phil Hobbs]

Post by Anton Shepelev
Hello, all
Is there a symbolic method to calculate the intensi-
ty distribution on a flat screen from a flat wave-
front after it reflected from a curved mirror whose
surface is described by an equation?

I don't think so, in general. Even in the scalar field approximation,
you have to deal with multiple reflections to have full generality.

Post by Anton Shepelev
The normal of the wavefront, and the relative posi-
tion of screen and mirror are known.
1. numerically, by tracing the rays reflected
from tiny sections of the mirror, and
2. approximately, via the first N moments of the
distribution, which can be found by integra-
tion over the mirror surface.

Well, the usual approach in physical optics is to use the scalar
approximation and apply either the Huyghens propagator for paraxial
fields or the Kirchhoff or Rayleigh-Sommerfeld propagators for more
general situations. All of these fail to model the boundary
contribution to the diffracted light, but they're okay for normal use.

A more accurate method in general is PTD (physical theory of
diffraction). See the papers and books of Pyotr Y Ufimtsev and Joseph
B. Keller. It handles multiple scattering and models the boundary waves
much better, but it's far from exact.

Cheers

Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

[Phil Hobbs]

Post by Phil Hobbs

Post by Anton Shepelev
Hello, all
Is there a symbolic method to calculate the intensi-
ty distribution on a flat screen from a flat wave-
front after it reflected from a curved mirror whose
surface is described by an equation?

I don't think so, in general. Even in the scalar field approximation,
you have to deal with multiple reflections to have full generality.

Post by Anton Shepelev
The normal of the wavefront, and the relative posi-
tion of screen and mirror are known.
1. numerically, by tracing the rays reflected
from tiny sections of the mirror, and
2. approximately, via the first N moments of the
distribution, which can be found by integra-
tion over the mirror surface.

Well, the usual approach in physical optics is to use the scalar
approximation and apply either the Huyghens propagator for paraxial
fields or the Kirchhoff or Rayleigh-Sommerfeld propagators for more
general situations. All of these fail to model the boundary
contribution to the diffracted light, but they're okay for normal use.
A more accurate method in general is PTD (physical theory of
diffraction). See the papers and books of Pyotr Y Ufimtsev and Joseph
B. Keller. It handles multiple scattering and models the boundary waves
much better, but it's far from exact.

I should add that discussions of diffraction in the physical optics
literature are mostly erroneous. The propagators come from the Green's
function for the wave equation in a half space with a planar boundary,
and folks blithely apply them to curved surfaces with no attention even
to the Jacobian (analogous to the area of a ray bundle).

Aberration theory (where the moments idea comes from) is an asymptotic
theory of the propagation of optical phase in the limit of large Fresnel
number. It completely ignores amplitude and polarization besides.

An amplitude-only Fresnel zone plate focuses light reasonably well, but
aberration theory predicts that the light propagates undeviated.

Cheers

Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

[Yuri Kretin]

Post by Anton Shepelev
Hello, all
Is there a symbolic method to calculate the intensi-
ty distribution on a flat screen from a flat wave-
front after it reflected from a curved mirror whose
surface is described by an equation?
The normal of the wavefront, and the relative posi-
tion of screen and mirror are known.
1. numerically, by tracing the rays reflected
from tiny sections of the mirror, and
2. approximately, via the first N moments of the
distribution, which can be found by integra-
tion over the mirror surface.
Is there a method that will give an exact symbolic
equation?
P.S.: I have crossposted the question to sci.math
because it is essentially a mathematical prob-
lem with a thin physical coating.

interesting problem, I have worked with non imaging optics using
reflective surfaces, and the exact solution can become very difficult
for a simple curve. One problem was reflecting from a sectional
paraboloid with axis tilted, truncated, rotated in 3D. I ended up using
your #1 above. There were a few exact soultions that hold at certian
points only.

[Anton Shepelev]

Is there a symbolic method to calculate the inten-
sity distribution on a flat screen from a flat
wavefront after it reflected from a curved mirror
whose surface is described by an equation?
The normal of the wavefront, and the relative po-
sition of screen and mirror are known.

Thank you for replies, Phil and Yuri.

I am currently working on two analytical solutions
under the assumptions of geomtrical optics (indepen-
dent linear rays) of the simple case when the mirror
deformations are so small that not only secondary
reflexions are impossible but also no concave region
may focus parallel rays before they hit the screen.

One version involves the caluclation of two orthogo-
mal curvature radii, of which one is in the plane of
the ray, and the other of only the second derivative
of the mirror surface. In the two-dimensional case,
when the screen is placed orthgonally to the re-
flected rays (for under the said assumptions their
directions will not differ much), the second solu-
tion gives the following intensity on the one-dimen-
sional screen:

I(x) = I0 [ 1 - 2d Sec(alpha) S''(x Sec(alpha)) ]^-1

where I0 is the intensity of the incoming light, d
the distance to the screen, alpha the angle of inci-
dence, and y = S(x) the one-dimensional mirror sur-
face.

The concave parable, for example, will focus light
uniformly, i.e. increase intensity in all points.
Does the resemble the truth?

[Kuri Yretin]

Post by Anton Shepelev

Is there a symbolic method to calculate the inten-
sity distribution on a flat screen from a flat
wavefront after it reflected from a curved mirror
whose surface is described by an equation?
The normal of the wavefront, and the relative po-
sition of screen and mirror are known.

Thank you for replies, Phil and Yuri.
I am currently working on two analytical solutions
under the assumptions of geomtrical optics (indepen-
dent linear rays) of the simple case when the mirror
deformations are so small that not only secondary
reflexions are impossible but also no concave region
may focus parallel rays before they hit the screen.
One version involves the caluclation of two orthogo-
mal curvature radii, of which one is in the plane of
the ray, and the other of only the second derivative
of the mirror surface. In the two-dimensional case,
when the screen is placed orthgonally to the re-
flected rays (for under the said assumptions their
directions will not differ much), the second solu-
tion gives the following intensity on the one-dimen-
I(x) = I0 [ 1 - 2d Sec(alpha) S''(x Sec(alpha)) ]^-1
where I0 is the intensity of the incoming light, d
the distance to the screen, alpha the angle of inci-
dence, and y = S(x) the one-dimensional mirror sur-
face.
The concave parable, for example, will focus light
uniformly, i.e. increase intensity in all points.
Does the resemble the truth?

yes that is true.

I was designing essentially an unfocused parabala that would guide light
toward a centrial shperical surface, with a cuttoff of +_45 degrees, and
the light was from any angle. Goal was to have highest light captured
overall. (Non-imaging optics).

What I did one time was a 2 dimentional analysis, where I could specify
all the angles, angle of arival, surface angle, angle of reflection,
and I had a surface of intercept (photomultipler tube surface). so I
came up with an equation relating incoming ray angle and offset, to
intercept on photomultiplier tube surface.

With that I could come up with equations that would determine if the ray
(from x offset at angle k) would hit the photomultiplier surface.
The variable was the curvature of the surface, from squared to quartic.
I would change that run to run to see what type of surface curve works
best, turned out the 4th order worked better. (photomultiplier has light
bulb shaped surface)

There was a boundry set of acceptance of +- 45 degrees, so the mirror
would bounce the ray out if more than 45 degrees and not hit the
photomultiplier.

the math for such was stright forward, some complexity. I did not sum up
the intensity at points though, I guess that is another step where one
would sum up the # of rays at a location within an acceptance angle at
the surface.

that was all 2 dimentional. I did not go for exact for 3 dimentional, as
rotating the 2 dimentional solution was a first order approximation, and
the additional cases it missed did not seem to be that large.

I did double check this using a strip of reflective surface cut one of
those skylight light tunnels on top of a paper with the shapes of the
PMT, and cutoffs using a flashlight. Cheap and easy.

[Phil Hobbs]

Post by Anton Shepelev

Is there a symbolic method to calculate the inten-
sity distribution on a flat screen from a flat
wavefront after it reflected from a curved mirror
whose surface is described by an equation?
The normal of the wavefront, and the relative po-
sition of screen and mirror are known.

Thank you for replies, Phil and Yuri.
I am currently working on two analytical solutions
under the assumptions of geomtrical optics (indepen-
dent linear rays) of the simple case when the mirror
deformations are so small that not only secondary
reflexions are impossible but also no concave region
may focus parallel rays before they hit the screen.
One version involves the caluclation of two orthogo-
mal curvature radii, of which one is in the plane of
the ray, and the other of only the second derivative
of the mirror surface. In the two-dimensional case,
when the screen is placed orthgonally to the re-
flected rays (for under the said assumptions their
directions will not differ much), the second solu-
tion gives the following intensity on the one-dimen-
I(x) = I0 [ 1 - 2d Sec(alpha) S''(x Sec(alpha)) ]^-1
where I0 is the intensity of the incoming light, d
the distance to the screen, alpha the angle of inci-
dence, and y = S(x) the one-dimensional mirror sur-
face.
The concave parable, for example, will focus light
uniformly, i.e. increase intensity in all points.
Does the resemble the truth?

At low numerical aperture, yes. At high numerical aperture, no. This
is because at higher NA the edge rays get squashed closer together due
to obliquity, so in planes far from focus the beam is brighter at the edges.

Cheers

Phil Hobbs
--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

[Boxman]

Post by Anton Shepelev
Hello, all
Is there a symbolic method to calculate the intensi-
ty distribution on a flat screen from a flat wave-
front after it reflected from a curved mirror whose
surface is described by an equation?

You may want to check out this article and the references listed in the
article. If anyone has a mathematical formulation for the problem,
Vladimir Oliker (author of the article) has probably found it.

http://spie.org/newsroom/5600-controlling-light-with-freeform-optics?ArticleID=x110544