question on geometrical theory of diffraction.

Thread view:

Enable EMail Alerts

Start New Thread

Thread rating:

pereges - 29 Jul 2008 17:29 GMT

Hello, I am computer science student and I have a question wrt GTD as
I am writing a computer program based on it.GTD allows us to use ray
tracing. I have been given a triangular mesh(object made up of
traingular facets) and I am supposed to apply ray tracing tehcniques
using GTD. The end result is to calculate total incident electric
field on the object(we associate some electric field with the source)
and and the electric field scattered by the object isotropically(in
far field, calculated around a sphere around object). We are assuming
that a plane wave(which I simulated with a array of many parallel
rays) is fired towards the mesh. Now each and every ray has to be
traced of the object. The ray can undergo many multiple reflections,
diffractions or reflection-diffraction etc. When when the rays from
the source(parallel rays) interact with the facet, they will undergo
reflection. When they interact with a sharp edge or a corner
diffraction occurs. I am supposed to consider only specular
reflection. I was reading this source :

http://www.maasdesign.co.uk/WhyGTD/TutorialSection2.html

And I encountered this statement :

"The geometrical Theory of Diffraction (GTD) is based on Geometric
Optics (GO) and Diffraction Theory [ 2-6 ] . It assumes that all waves
are `well-formed' and are locally plane waves. This enables ray
tracing to be used. "

What does the author mean here by saying all waves are "locally plane
waves" ?

I mean what if a plane wave undergoes diffraction at an edge of an
object and results in conical waves(cylindrical incase the ray is
normally incident to the edge).

http://www.maasdesign.co.uk/WhyGTD/TutorialSection2-4.gif

Further, If these conical waves(or lets say a single ray from the
entire bunch of rays representing the conical wave) undergo a
reflection on some other surface of object(talking about multiple
interaction/bouncing of the rays), then should the wave incident on
the surface be considered as plane wave in vicinity of the surface ?
Is this what he/she means by "locally plane". I looked up many books
which deal with GTD ray tracing and all of them provide solutions
only for plane wave incident on edge, corner, surface. Also, if we
are going about tracing rays, then where does wave come into picture.
This confuses me. There are some formulas listed in a thesis that I
have for rays which are used to calculate electric field along a ray:

For a ray from the source(associated with the plane wave) :

E(s) = E(0) exp(-jks) ---- (1)

where E(s) is electric field at distance s from the source and E(0) is
electric field at some rference point. exp(-jks) represents the phase
variation and k is the wavenumber.

For a reflected ray:

E(s) = E(0) exp(-jks)/s ------ (2)

s is distance form the origin of the spherical wavefront that will be
generated because of reflection. E(s) is field at distance s.
Although I fail to see how you can apply this formula in case a
ray(associated with a plane wave) is reflected of a plane triangular
facet . I mean this formula cannot be applied because plane wave
reflection results in another plane wave. Now if we had a triangular
mesh for a sphere, the reflection of all the parallel rays will lead
to a spherical wavefront overall in the far field but if we consider
reflection of individual source rays(associated with plane wave) the
resultant is also rays associated with plane wave. Then we would be
applying formula 1 for calculating field along every reflected ray.

For diffracted ray :

E(s) = D * E(0) exp(-jks)/sqrt(s) ------ (3)

D is diffraction coefficient.

Can some one please help me ?

Reply to this Message

Phil Hobbs - 29 Jul 2008 18:14 GMT

> Hello, I am computer science student and I have a question wrt GTD as
> I am writing a computer program based on it.GTD allows us to use ray
[quoted text clipped - 24 lines]
> What does the author mean here by saying all waves are "locally plane
> waves" ?

He means that the curvature of both the surface and wavefront is assumed
to be small enough on the scale of a wavelength that ray optics is
applicable locally--i.e. that diffraction is not important in most places.

This is a standard move in asymptotic theories. For instance, when you
derive the Stirling series for the gamma (factorial) function, you start
with an integral relationship for gamma(x), do a Taylor expansion for x'
< epsilon (epsilon assumed small), integrate term by term, and then let
epsilon go to infinity so that the upper limit of the integral goes
away. At different times you assume that epsilon is very small, and
then that it's infinity. This works because the size of the
contribution from large epsilon is actually negligible--smaller than any
term in the asymptotic series, as x -> infinity. It's fun once you get
used to it.

> I mean what if a plane wave undergoes diffraction at an edge of an
> object and results in conical waves(cylindrical incase the ray is
> normally incident to the edge).

GTD and PTD put that case in by hand, by adding "diffracted rays".

> http://www.maasdesign.co.uk/WhyGTD/TutorialSection2-4.gif
>
[quoted text clipped - 3 lines]
> interaction/bouncing of the rays), then should the wave incident on
> the surface be considered as plane wave in vicinity of the surface ?

Yep. Once you have a diffracted ray, you treat it like any other ray.
Pure ray optics is an asymptotic theory for lambda->0. Roughly
speaking, GTD takes it one more perturbation order, and PTD one more
order after that.

Cheers,

Phil Hobbs

Reply to this Message

pereges - 29 Jul 2008 20:33 GMT

On Jul 29, 10:14 pm, Phil Hobbs <pcdhSpamMeSensel...@pergamos.net>
wrote:

> > "The geometrical Theory of Diffraction (GTD) is based on Geometric
> > Optics (GO) and Diffraction Theory [ 2-6 ] . It assumes that all waves
[quoted text clipped - 18 lines]
> term in the asymptotic series, as x -> infinity. It's fun once you get
> used to it.

thanks, for the reply. Ok please consider this following diagram which
shows multple interactions of rays with the body:

http://i34.tinypic.com/29up1l0.jpg

Let's say the ray going from 1 to 2 is associated with plane wave. It
strike s sharp edge at point 2, undergoes diffraction and one of the
ray 2 to 3 hits a triangular facet and undergoes reflection at point
3. Let's say I want to calculate field at 4.
If field at 1 is suppose E1. Then since ray from 1 to 2 is plane
wave :

E(2) = E(1) * exp(-jks12) . This is by formula 1 of first post.

where s12 is distance between 1 and 2 and E(2) is field incident at
point 2.

When plane wave undergoes diffraction at the edge, it results in a set
of conical wave. Let's say one of the ray is 2 to 3.

field incident at 2 = E(2), let D= Diffraction coefficient, s23 =
Distance betwee. So field incident at 3

E(3) = D * E(2) * exp(-jks23) / sqrt(s23). This is by formula 3 of
first post.

or do you think it should be

E(3) = E(2) * exp(-jks23) as we consider all waves locally plane ??
Then I don't understand the purpose behind the thesis mentioning the
other formulas. Is it for far field ? But anyway let us assume the
first formula for E(3) is correct and proceed.

The ray 2-3 hits a facet and undergoes reflection. Now here I get
stuck up because if the incident wave is cylindrical wave, then I do
not know what would be the nature of reflected ray. IF it is a
cylindrical wave again, I need to change formula 1 in first post to
accomodate for 1/sqrt(s) attenuation instead of the 1/s attenuation
factor I had written (which was for spherical wave). This is what
leads to too many questions like what to do when cylindrical
wave/"ray" had hit another edge etc like the one I asked in first
post. However none of this is a problem if you assume all such waves
are plane wave. In that case:

Field at 2 is E(2) = E(1)exp(-jks12). Field at 3 is E(3) = E(2) *
exp(-jks23) and field at 4 is E(4) = E(3) * exp(-jks34).

We are not concerned at all with nature of the wave.
What I find confusing here is that this actually means that the field
has not changed from 1 to 4 if we consider all waves locally plane.
While this might be true if these distances 1-2 2-3 3-4 are pretty
small but it may not be true if object is pretty huge like a
helicopter or something. eg: A ray diffracted from the wings may
travel a few meters before it hits some other surface of helicopter.
Also I the question I asked previously i.e why the thesis mentions
other formulas at all. I have one book on GTD by graeme L james and he
calculates a divergence factor only in case of reflection(the formulas
for diffraction and plane wave remain same). His formula is :

E(s) = Einc * R * DF * exp(-jks)

R is reflection coefficient
DF is divergence factor
E(0) is electric field incident at the point of reflection
s is distance from the point of reflection along the reflected ray at
which we are interested in calculating E(s) field.
k is wavenumber

DF in case of reflecting wavefront being spherical is 1/s, in case
for cylindrical its 1/sqrt(s) and I guess for plane wave it is 1. In
his book he used some complicated derivation for divergence factor for
curved surfaces etc but I don't know how to do it with plane facets.
Any way the formula in his book is :

DF = p1p2 / (p1 + s) (p2 + s)

where s is distance from point fo reflection, p1 and p2 are principal
radii of curvature.

I can figure out the DF in case of incident spherical wave though.
This is because the outgoing wavefront or the reflected wavefront will
also be spherical and you can find the origin for the reflected
spherical wavefront by extending the reflected ray by same distace
behind the facet that the origin of the incident ray is in front of
the facet.(same thing we do with the plane mirrors, here we assume
that the facet act as a mirror) :

Now we know the field which was incident on the facet say at some
point 1 because of some incident ray. We know the distance between 1
and O, so the field at O is obviously field at 1 multiplied by field
at 1 and distance between 1 and O. (electric field is attenuated by 1/
distance along spherical wave). Using field at O we can determine
field at any point along the reflected ray based on distance between O
and that point.

> GTD and PTD put that case in by hand, by adding "diffracted rays".

Well, what I usually do is trace the rays (and their children rays) as
they come and add up their contributions to electric fields.

> Yep. Once you have a diffracted ray, you treat it like any other ray.
> Pure ray optics is an asymptotic theory for lambda->0. Roughly
[quoted text clipped - 4 lines]
>
> Phil Hobbs

Reply to this Message

Phil Hobbs - 29 Jul 2008 20:50 GMT

> thanks, for the reply. Ok please consider this following diagram which
> shows multple interactions of rays with the body:
[quoted text clipped - 18 lines]
> field incident at 2 = E(2), let D= Diffraction coefficient, s23 =
> Distance betwee. So field incident at 3

I think I see the difficulty. You're mixing up a field calculation with
a ray optics calculation. In ray optics, you just have to live with the
combinatoric explosion of multiple bounces and multiple diffractions.
You add up the field contributions at the end, as a function of
observation position.

If you were doing a full EM calculation, you'd be worrying about the
field values at each point, but you aren't, you're doing geometric
optics. Rays can cross, and it's each ray that corresponds locally to a
plane wave. You can't add them all up and then try to pick some single
compromise plane wave for them all. You've got light going in all
different directions, so it's intrinsically more complicated.

The plus side of all this is that there's more information in a ray
optics plot than in a wave optics one...you can have several source
contributions at once and still keep them straight.

As a suggestion, you might want to code this up with just the geometric
optics part, then put the diffraction part in afterwards. The rhythm of
the two are similar, but the bookkeeping is outrageously more difficult
in GTD, especially if you're doing concave things with lots of multiple
bounces.

Cheers,

Phil Hobbs

Reply to this Message

pereges - 29 Jul 2008 22:25 GMT

On Jul 30, 12:51 am, Phil Hobbs <pcdhSpamMeSensel...@pergamos.net>
wrote:

> I think I see the difficulty. You're mixing up a field calculation with
> a ray optics calculation. In ray optics, you just have to live with the
[quoted text clipped - 22 lines]
>
> Phil Hobbs

Thing is I need the total incident electric field which is the only
reason I keep finding the incident field at each point. I keep adding
the field contribution of every ray hitting the object to the total
field value. However, if I do consider every bouncing ray as a plane
wave itself, then multiple bouncing calculations become extremely
easy. However, I don't think this is applicable when finding the
scattered electric field i.e. when the ray has exited the object
finally (after all those multiple bounces) and I want to find out the
field on a spherical surface around the object.

Reply to this Message

Phil Hobbs - 30 Jul 2008 00:11 GMT

> On Jul 30, 12:51 am, Phil Hobbs <pcdhSpamMeSensel...@pergamos.net>
> wrote:
[quoted text clipped - 35 lines]
> finally (after all those multiple bounces) and I want to find out the
> field on a spherical surface around the object.

That's a separate issue. Once you have all the rays, propagate them out
to some boundary far enough away that they can't hit the surface again,
compute the field distribution there (you've carried around the Jacobian
for all the oblique reflections and curves surfaces, I presume), and
then decompose that into a plane wave spectrum.

Cheers,

Phil Hobbs

Reply to this Message

pereges - 30 Jul 2008 12:42 GMT

On Jul 30, 4:11 am, Phil Hobbs
<pcdhSpamMeSensel...@electrooptical.net> wrote:

> That's a separate issue. Once you have all the rays, propagate them out
> to some boundary far enough away that they can't hit the surface again,
[quoted text clipped - 5 lines]
>
> Phil Hobbs

I do not know what jacobian means but I will look up thanks.

I have problem with calculating scattered electric far field. For eg
consider a sphere which is illuminated with a plane wave. Each facet
facing the source is illuminated with plane wave. Then, the individual
rays are reflected off the plane facets. Reflection of a plane wave of
plane facet results in a plane wave. So all rays will be associated
with plane wave ? This doesn't seem right becuase in far field the
reflected rays clearly form a spherical wavefront. There will be a
problem while calculating electric field. If all reflected rays are
associated with plane wave, then the attenuation of the field will be
1 and if they are associated with spherical wave, then attenuation
will be 1/s wher s is distance travelled by the ray. This is the
biggest problem in dealing with plane facets and not actual object and
curves and everything. You have to follow the laws of optics.

Reply to this Message

pereges - 30 Jul 2008 13:22 GMT

On Jul 30, 4:11 am, Phil Hobbs
<pcdhSpamMeSensel...@electrooptical.net> wrote:

> That's a separate issue. Once you have all the rays, propagate them out
> to some boundary far enough away that they can't hit the surface again,
[quoted text clipped - 5 lines]
>
> Phil Hobbs

I do not know what jacobian means but I will look up thanks.

I have problem with calculating scattered electric far field. For eg
consider a sphere which is illuminated with a plane wave. Cosnider the
following diagram:

http://i36.tinypic.com/zlexht.jpg

Each facet facing the source is illuminated with plane wave. Then, the
individual
rays are reflected off the plane facets. Reflection of a plane wave of
plane facet results in a plane wave. So all rays will be associated
with plane wave ? This doesn't seem right becuase in far field the
reflected rays clearly form a spherical wavefront. There will be a
problem while calculating electric field. If all reflected rays are
associated with plane wave, then the attenuation of the field will be
1 and if they are associated with spherical wave, then attenuation
will be 1/s wher s is distance travelled by the ray. This is the
biggest problem in dealing with plane facets and not actual object and
curves and everything. You have to follow the laws of optics.

Reply to this Message

Phil Hobbs - 30 Jul 2008 14:03 GMT

> On Jul 30, 4:11 am, Phil Hobbs
> <pcdhSpamMeSensel...@electrooptical.net> wrote:
[quoted text clipped - 23 lines]
> with plane wave ? This doesn't seem right becuase in far field the
> reflected rays clearly form a spherical wavefront.

> There will be a
> problem while calculating electric field. If all reflected rays are
[quoted text clipped - 3 lines]
> biggest problem in dealing with plane facets and not actual object and
> curves and everything. You have to follow the laws of optics.

You need to distinguish between the asymptotic (lambda ->0) theory and
the full electromagnetic theory. You aren't doing a full EM
calculation, so the rules are going to be different. In the asymptotic
theory, each ray behaves locally like a plane wave for the purpose of
figuring out which way it's going when it leaves the surface--a plane
wave is the only EM thing that really has a well-defined direction, just
as a delta function is the only thing that has a well-defined position.
(Plane waves are delta functions in k-space.)

Also each ray really has to occupy some patch of solid angle, or else
you'll be computing the far field pattern of a bunch of delta-functions
arranged on a sphere someplace. The area of that patch will have got
larger or smaller by a factor of the Jacobian J, and its amplitude will
have changed by a factor of 1/sqrt(J).

I've never gone through the far-field calculation with GTD myself, but
it's a standard move in the radar cross-section (RCS) literature. One
could imagine assigning each ray a bit of solid angle to start with,
using ray optics and the Jacobian to turn that into a patch on an
indefinitely large sphere, and summing up all the contributions at each
point on the sphere. The RCS folk must have some way of reducing the
uncertainty in the computed pattern due to not having enough rays to
represent the fields adequately--presumably some adaptive scheme that
traces extra rays (with appropriately diminished amplitudes) to fill in
any blank spots. Putting soft edges on the patches (like Gaussian
beams) would also probably help.

BTW you also have to watch what happens to the polarization for any rays
whose complete path doesn't lie in a plane. (The rotation of linear
polarization due to nonplanar paths has a mysterious-sounding
name--topological phase--but really it's just a consequence of spherical
trigonometry. Equivalently, 2D rotation matrices commute, but 3D ones
don't.

If you think about the far field pattern as lying on an extremely large
sphere (big enough that the angular pattern has stopped changing with
R), a lot of these confusions will go away, I think.

Cheers,

Phil Hobbs

Reply to this Message

pereges - 29 Jul 2008 20:40 GMT

>>He means that the curvature of both the surface and wavefront is assumed
>>to be small enough on the scale of a wavelength that ray optics is
>>applicable locally--i.e. that diffraction is not important in most places.

>>This is a standard move in asymptotic theories. For instance, when you
>>derive the Stirling series for the gamma (factorial) function, you start
[quoted text clipped - 6 lines]
>>term in the asymptotic series, as x -> infinity. It's fun once you get
>>used to it.

thanks, for the reply. Ok please consider this following diagram which
shows multple interactions of rays with the body:

http://i34.tinypic.com/29up1l0.jpg

Let's say the ray going from 1 to 2 is associated with plane wave. It
strike s sharp edge at point 2, undergoes diffraction and one of the
ray 2 to 3 hits a triangular facet and undergoes reflection at point
3. Let's say I want to calculate field at 4.
If field at 1 is suppose E1. Then since ray from 1 to 2 is plane
wave :

E(2) = E(1) * exp(-jks12) . This is by formula 1 of first post.

where s12 is distance between 1 and 2 and E(2) is field incident at
point 2.

When plane wave undergoes diffraction at the edge, it results in a set
of conical wave. Let's say one of the ray is 2 to 3.

field incident at 2 = E(2), let D= Diffraction coefficient, s23 =
Distance betwee. So field incident at 3

E(3) = D * E(2) * exp(-jks23) / sqrt(s23). This is by formula 3 of
first post.

or do you think it should be

E(3) = E(2) * exp(-jks23) as we consider all waves locally plane ??
Then I don't understand the purpose behind the thesis mentioning the
other formulas. Is it for far field ? But anyway let us assume the
first formula for E(3) is correct and proceed.

The ray 2-3 hits a facet and undergoes reflection. Now here I get
stuck up because if the incident wave is cylindrical wave, then I do
not know what would be the nature of reflected ray. IF it is a
cylindrical wave again, I need to change formula 1 in first post to
accomodate for 1/sqrt(s) attenuation instead of the 1/s attenuation
factor I had written (which was for spherical wave). This is what
leads to too many questions like what to do when cylindrical
wave/"ray" had hit another edge etc like the one I asked in first
post. However none of this is a problem if you assume all such waves
are plane wave. In that case:

Field at 2 is E(2) = E(1)exp(-jks12). Field at 3 is E(3) = E(2) *
exp(-jks23) and field at 4 is E(4) = E(3) * exp(-jks34).

We are not concerned at all with nature of the wave.
What I find confusing here is that this actually means that the field
has not changed from 1 to 4 if we consider all waves locally plane.
While this might be true if these distances 1-2 2-3 3-4 are pretty
small but it may not be true if object is pretty huge like a
helicopter or something. eg: A ray diffracted from the wings may
travel a few meters before it hits some other surface of helicopter.
Also I the question I asked previously i.e why the thesis mentions
other formulas at all. I have one book on GTD by graeme L james and he
calculates a divergence factor only in case of reflection(the formulas
for diffraction and plane wave remain same). His formula is :

E(s) = Einc * R * DF * exp(-jks)

R is reflection coefficient
DF is divergence factor
E(0) is electric field incident at the point of reflection
s is distance from the point of reflection along the reflected ray at
which we are interested in calculating E(s) field.
k is wavenumber

DF in case of reflecting wavefront being spherical is 1/s, in case
for cylindrical its 1/sqrt(s) and I guess for plane wave it is 1. In
his book he used some complicated derivation for divergence factor for
curved surfaces etc but I don't know how to do it with plane facets.
Any way the formula in his book is :

DF = p1p2 / (p1 + s) (p2 + s)

where s is distance from point fo reflection, p1 and p2 are principal
radii of curvature.

I can figure out the DF in case of incident spherical wave though.
This is because the outgoing wavefront or the reflected wavefront will
also be spherical and you can find the origin for the reflected
spherical wavefront by extending the reflected ray by same distace
behind the facet that the origin of the incident ray is in front of
the facet.(same thing we do with the plane mirrors, here we assume
that the facet act as a mirror) :

http://i36.tinypic.com/2me4qpt.jpg

Now we know the field which was incident on the facet say at some
point 1 because of some incident ray. We know the distance between 1
and O, so the field at O is obviously field at 1 multiplied by field
at 1 and distance between 1 and O. (electric field is attenuated by 1/
distance along spherical wave). Using field at O we can determine
field at any point along the reflected ray based on distance between O
and that point.

> GTD and PTD put that case in by hand, by adding "diffracted rays".

Well, what I usually do is trace the rays (and their children rays) as
they come and add up their contributions to electric fields.

> Yep. Once you have a diffracted ray, you treat it like any other ray.
> Pure ray optics is an asymptotic theory for lambda->0. Roughly
> speaking, GTD takes it one more perturbation order, and PTD one more
> order after that.

> Cheers,

> Phil Hobbs

Ok what if I want to find out thescatered electric field in the far
field of the object. The far field radiation is spherical and I want
to calculate the scattered field on a sphere of radius R (R is
extremely huge as we are talking about far field radiation), The
emissions coming from the object may be categorised into many groups
like plane wave, spherical wave or diffraction wave. So does that mean
I need to check the type of ray (plane , spherical or diffraction) and
then apply appropirate formulas(1-3) to calculate contribution of this
ray to total far field scattered radiation ?
My ultimate aim is calculate radar cross section of the object. The
formula for RCS being :

RCS = 4 * PI * R^2 * (|Es|)^2 / (|Ei|)^2

where R is the radius of sphere (tending to infinity), Es is total
scattered electric field vector and Ei is the incident electric field
vector.

Reply to this Message

pereges - 29 Jul 2008 20:51 GMT

> >>He means that the curvature of both the surface and wavefront is assumed
> >>to be small enough on the scale of a wavelength that ray optics is
[quoted text clipped - 136 lines]
> scattered electric field vector and Ei is the incident electric field
> vector.

sorry p1 and p2 are the principal radius of curvature of incident
wavefront

Reply to this Message

Natural Science Forum / Physics / Optics / July 2008

question on geometrical theory of diffraction.