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Natural Science Forum / Physics / General Physics / July 2008decoupling the electric field at the plane of incidence
Hello, I'm writing a software and need some help. I'm trying to decouple the electric field along a ray (using it to simulate behaviour of a wave) into horizontal and vertical components at the plane of incidence. I'm using this article : http://oldeee.see.ed.ac.uk/~dil/thesis_mosaic/subsection2_9_7_1.html#SECTION0097 100000000000000 I have understood how they went about the decoupling thing with one component being perpendicular to the plane of incidence i.e. h and the other component which is parallel to the plane of incidence i.e. vi. Now I have question : what if the ray is normally incident to the interface. If we assume n (normal to surface), ui (direction of ray) are unit vectors then we have: h = (ui X n) -------- (1) This component is perpendicular to the plane of incidence. vi = (h X ui) ---------- (2) This component is parallel to the plane of incidence. The usual procedure is to multiply the maginitude of incident electric field with h and vi and then sum them up to find the total incident electric field vector. But if the ray is normally incident, I observed in my program that h and vi becomes zero resulting in zero incident electric field vector. For eg. ui = (0, 0 , 1) and n = (0, 0 , -1) then h is zero and it follows that vi will also be zero. What is going wrong here ? I doubt this is consistent according to laws of physics(law of conservation of energy etc). | Hello, I'm writing a software and need some help. I'm trying to | decouple the electric field along a ray (using it to simulate | behaviour of a wave) into horizontal and vertical components at the | plane of incidence. I'm using this article : http://oldeee.see.ed.ac.uk/~dil/thesis_mosaic/subsection2_9_7_1.html#SECTION0097 100000000000000 | I have understood how they went about the decoupling thing with one | component being perpendicular to the plane of incidence i.e. h and the [quoted text clipped - 18 lines] | What is going wrong here ? I doubt this is consistent according to | laws of physics(law of conservation of energy etc). The equations are not being displayed on the page you quoted. However, I can suggest that you look carefully and ask yourself what you EXPECT to happen. For one thing, the ray has to reverse direction so there will be a sign change to ui. Now you have h = (-ui X n). > The equations are not being displayed on the page you quoted. > However, I can suggest that you look carefully and ask yourself > what you EXPECT to happen. For one thing, the ray has to reverse > direction so there will be a sign change to ui. > Now you have h = (-ui X n). Ok I will post the equations in that case : The direction of the horizontally polarized component of the incident field is given by : h = ui X N where ui is direction of ray, N is normal to the surface (both are unit vectors). Likewise, the direction of the vertically polarized component of the incident field is given by vi = ui X N From these, the components of the incident electric field are computed as Eh = h dot E Ev = vi dot E where Eh is the component of the electric field, , that is horizontally polarized with respect to the surface, and Ev is the vertically polarized component. E is the incident electric field at the point of reflection. The reflected field can then be expressed as Er = Eh * rh * h + Ev * rv * vr rh and rv are reflection coefficients given as : rh = -1 (This causes the sign change that you were talking about) rv = 1 where vr is in the direction of the vertically polarized component of the reflected field, given by : vr = (N dot vi) n - ( (N X h) dot vi) (N X h) | > The equations are not being displayed on the page you quoted. | > However, I can suggest that you look carefully and ask yourself [quoted text clipped - 16 lines] | | vi = ui X N Stop there for a moment. h = ui X N vi = ui X N ergo h = vi That has to be a confusion, right? | From these, the components of the incident electric field are computed | as [quoted text clipped - 17 lines] | | vr = (N dot vi) n - ( (N X h) dot vi) (N X h) Ok, I think I see what the trouble is, now I have to find the right words to describe it. Terms such as "horizontal" and "vertical" have no meaning except when accompanied by a diagram... Let's do it this way. In this diagram: http://www.androcles01.pwp.blueyonder.co.uk/AC/Photon.gif , the E field is shown horizontal (green +ve, yellow -ve) and the magnetic field (red N, blue S) is vertical. The incident direction is also horizontal, so we have two horizontals at right angles to each other, only one of which reverses direction when the ray reflects. So constant energy (the rotating vector) travels along the spiral, changing its form from magnetic to electric and back again, strikes a plane surface (not shown) and returns along the original path... and there is NO rh = -1, that's the wrong horizontal. Or to put it another way, their cross product reverses, rv and rh do not. http://en.wikipedia.org/wiki/Cross_product "The cross product is not defined except in three-dimensions (and the algebra defined by the cross product is not associative). Like the dot product, it depends on the metric of Euclidean space. Unlike the dot product, it also depends on the choice of orientation or "handedness". " bXa = -aXb http://en.wikipedia.org/wiki/Image:Crossproduct.png > Hello, I'm writing a software and need some help. I'm trying to > decouple the electric field along a ray (using it to simulate [quoted text clipped - 15 lines] > vi = (h X ui) ---------- (2) This component is parallel to the plane > of incidence. Are you trying to get unit vectors perpendicular and parallel to the plane of incidence here? If so, you will have to normalize these. > The usual procedure is to multiply the maginitude of incident electric > field with h and vi and then sum them up to find the total incident [quoted text clipped - 5 lines] > What is going wrong here ? I doubt this is consistent according to > laws of physics(law of conservation of energy etc). Your calculations should break down in the case where the ray is normal to the surface, because there is no longer a unique plane of incidence. You will have to figure out what's best for your application, but one possible approach would be to choose a "plane of incidence" arbitrarily in the event the above calculation breaks down. If your code is working right, no matter what you choose as the "plane of incidence," you should get the correct result. The thing I'd be worried about is making sure that your components are perpendicular to each other and to the ray. If the cross product (ui X n) is very small and you then normalize it, you're likely to get a garbage result. I would think carefully about how to handle or avoid this. Look at the larger computation -- you might be able to restructure it so that this division of two near-zero numbers never shows up.  SignatureJim E. Black (domain in headers) How to filter out stupid arguments in 40tude Dialog: !markread,ignore From "Name" +"<email address>" [X] Watch/Ignore works on subthreads > Are you trying to get unit vectors perpendicular and parallel to the plane > of incidence here? If so, you will have to normalize these. Yes but u, n are already normalized. I'm trying to find out the total incident electric field vector on the body. > Your calculations should break down in the case where the ray is normal to > the surface, because there is no longer a unique plane of incidence. You [quoted text clipped - 10 lines] > at the larger computation -- you might be able to restructure it so that > this division of two near-zero numbers never shows up. I'm using double precision data so do you think it will still cause underflow problems ? The object in consideration is a triangular mesh. If the ray is perpendicular to surface, then I can probably find a vector from the point of intersection to one of the triangle vertices and then normalize it and multiply it by the incident electric field magnitude to find the electric field vector at the point. The reflected electric field vector can be found by simple negation. What is bothering me even ore is that this might be wrong way to do it. The elctric field vector can be in any direction perpendicular to the ray dierction. Whereas with this method, you are just finding a fix direction. Summing all the incident electric field vectors to find the total incident electric field vector at the end may produce a wrong result. >> Are you trying to get unit vectors perpendicular and parallel to the plane >> of incidence here? If so, you will have to normalize these. > > Yes but u, n are already normalized. That doesn't make their cross product normalized. > I'm trying to find out the total > incident electric field vector on the body. For that, I don't understand why you need the calculations you were doing in the previous post, but don't take what I have to say too seriously yet, because I still don't understand what you're doing. In particular, this from your previous post >>> The usual procedure is to multiply the maginitude of incident electric >>> field with h and vi and then sum them up to find the total incident >>> electric field vector. is not right; you can't figure out the direction of the electric field given only its magnitude. The website you cited http://oldeee.see.ed.ac.uk/~dil/thesis_mosaic/subsection2_9_7_1.html#SECTION0097 100000000000000 is about starting with the electric field vector, breaking it up into vertical and horizontal components, and then using these components to calculate the electric field vector for the reflected light. I thought this was what you were trying to do, but now I'm not so sure. >> Your calculations should break down in the case where the ray is normal to >> the surface, because there is no longer a unique plane of incidence. You [quoted text clipped - 13 lines] > I'm using double precision data so do you think it will still cause > underflow problems ? If there's a place where you might divide zero by zero, it doesn't matter how good your precision is. > The object in consideration is a triangular mesh. If the ray is > perpendicular to surface, then I can probably find a vector from the [quoted text clipped - 9 lines] > the total incident electric field vector at the end may produce a > wrong result. I'm not sure what you're trying to do. What are you summing over? Is the electric field vector at that point the end result, or just an intermediate step? What is the input to the problem? SignatureJim E. Black (domain in headers) How to filter out stupid arguments in 40tude Dialog: !markread,ignore From "Name" +"<email address>" [X] Watch/Ignore works on subthreads > That doesn't make their cross product normalized. > For that, I don't understand why you need the calculations you were doing > in the previous post, but don't take what I have to say too seriously yet, > because I still don't understand what you're doing. Well I am trying to find out the radar cross section of the object using ray tracing. rays are fired on the object and each ray is followed. The total incident electric field is calculated by taking into consideration the interaction of all rays with the object. The rays undergo reflection and when they leave the object, I calculate the scattered electric field based on that. Using the total scattered electric field and total incident electric field one calculate the RCS. > In particular, this from your previous post > is not right; you can't figure out the direction of the electric field [quoted text clipped - 6 lines] > calculate the electric field vector for the reflected light. I thought > this was what you were trying to do, but now I'm not so sure. If that's the case then I think he is trying to find the components (or magnitude) of the electric vector E along the directions h and vi.Is that right ? > If there's a place where you might divide zero by zero, it doesn't matter > how good your precision is. Ok, now I understand. I will take a look into some floating point exception libraries. > I'm not sure what you're trying to do. What are you summing over? Is the > electric field vector at that point the end result, or just an intermediate > step? What is the input to the problem? I want to find out the total incident electric field vector. Find the incident electric field vector for a single ray is an intermediate step, yes. You have to do it for all rays. And there will also be situations when the ray would be reflected of the surface and it may undergo second reflection against the same object. The input to the problem is a triangular mesh of an object i.e. an object converted into triangles. >> That doesn't make their cross product normalized. > [quoted text clipped - 10 lines] > Using the total scattered electric field and total incident electric > field one calculate the RCS. Make sure you consider what the wavelength of the radar beam is, and the size of the smallest features on the object. Only if the wavelength is much smaller should you be thinking about "rays" at all. >> In particular, this from your previous post >> is not right; you can't figure out the direction of the electric field [quoted text clipped - 10 lines] > (or magnitude) of the electric vector E along the directions h and > vi.Is that right ? Yes. The reason he is doing this is because not all of the light is reflected; some is transmitted into the surface. The two components of polarized light he's computing have different coefficients for reflection and transmission. >> If there's a place where you might divide zero by zero, it doesn't matter >> how good your precision is. > > Ok, now I understand. I will take a look into some floating point > exception libraries. The point was that it is not enough to handle the 0/0 case; you will also get tiny/tiny. If this happens, it's probably a good idea to restructure the calculation somehow. But at this point, I doubt it was relevant at all. SignatureJim E. Black (domain in headers) How to filter out stupid arguments in 40tude Dialog: !markread,ignore From "Name" +"<email address>" [X] Watch/Ignore works on subthreads > Make sure you consider what the wavelength of the radar beam is, and the > size of the smallest features on the object. Only if the wavelength is > much smaller should you be thinking about "rays" at all. well, we are considering only high radio frequencies hence the ray tracing method. > Yes. The reason he is doing this is because not all of the light is > reflected; some is transmitted into the surface. The two components of > polarized light he's computing have different coefficients for reflection > and transmission. > The point was that it is not enough to handle the 0/0 case; you will also > get tiny/tiny. If this happens, it's probably a good idea to restructure > the calculation somehow. But at this point, I doubt it was relevant at > all. I only need to consider the reflection bit not transmission/ refraction. I'm also referring a book (Geometrical Theory Of Diffraction - Graeme . L. James) and it says that if the second medium is highly conducting (i.e. the object), only reflection occurs, and provided that the conducting is sufficiently large : Rh = -1 , Rv = 1 i.e. reflection coefficients along the horizontal direction, Rh (perpendicular to plane of incidence) and the vertical direction, Rv (parallel to plane of incidence) > I only need to consider the reflection bit not transmission/ > refraction. I'm also referring a book (Geometrical Theory Of [quoted text clipped - 7 lines] > (perpendicular to plane of incidence) and the vertical direction, Rv > (parallel to plane of incidence) I drew some pictures to make this clearer: http://img135.imageshack.us/img135/9239/horizontallx0.png http://img135.imageshack.us/img135/1175/verticalvz1.png You can figure out the direction of polarization from the requirement that the total electric field (incident + reflected) is perpendicular to the surface. The reason for this is that we're modeling the surface as a perfect conductor, so if there was an electric field perpendicular to the surface, the electrons in the surface would instantly rush in the opposite direction, eliminating the field. From what I can see, all you need to do is find the components parallel and perpendicular to the surface, and reverse the parallel component. If n is a unit vector normal to the surface, and E the incident field, (E dot n) n is the perpendicular component, and thus E - (E dot n) n is the parallel one. So the result would be (E dot n) n - (E - (E dot n) n) = 2 (E dot n) n - E. SignatureJim E. Black (domain in headers) How to filter out stupid arguments in 40tude Dialog: !markread,ignore From "Name" +"<email address>" [X] Watch/Ignore works on subthreads > I drew some pictures to make this clearer: > [quoted text clipped - 22 lines] > > = 2 (E dot n) n - E. Shouldn't (E dot n)n be the parallel component(parallel to plane of incidence) and E - (E dot n)n the perpendicular component(perpendicular to plane of incidence). In that case, the perpendicular component as indicated by your diagram 1 will be reversed. Is that right ? I was a little confused about E being a vector or scalar because of the formulas given in my thesis that I have been referring. For eg. for a direct ray travelling from the source towards the object, if the electric field at the reference point is E(0) then the field at a distance s from the reference point is given as : E(s) = E(0) exp(-jks) where E(s) is electric field at distance s from the reference point k is wave number E(0) is electric field at the reference point If you take only real part, E(s) = E(0) cos (ks) Now usually one would expect that E(0) is a scalar and not a 3d vector. Isn't that how wave equations are ? I know less about it as I'm a computer science student. Considering it as a 3d vector also brings in the question of direction of the vector and there are infinite possible ways for the electric field vector to be around around the ray and still be perpendicular to the direction. > For eg. > for a direct ray travelling from the source towards the object, if the [quoted text clipped - 11 lines] > Now usually one would expect that E(0) is a scalar and not a 3d > vector. Isn't that how wave equations are ? Not necessarily. E(0) is in fact a 3d vector. SignatureJim E. Black (domain in headers) How to filter out stupid arguments in 40tude Dialog: !markread,ignore From "Name" +"<email address>" [X] Watch/Ignore works on subthreads I was a little confused about E being a vector or scalar because of the formulas given in my thesis that I have been referring. For eg. for a direct ray travelling from the source towards the object, if the electric field at the reference point is E(0) then the field at a distance s from the reference point is given as : E(s) = E(0) exp(-jks) where E(s) is electric field at distance s from the reference point k is wave number E(0) is electric field at the reference point If you take only real part, E(s) = E(0) cos (ks) Now usually one would expect that E(0) is a scalar and not a 3d vector. Isn't that how wave equations are ? I know less about it as I'm a computer science student. Considering it as a 3d vector also brings in the question of direction of the vector and there are infinite possible ways for the electric field vector to be oriented around around the ray and still be perpendicular to the direction of the ray. |I was a little confused about E being a vector or scalar I'm sure you are. Like the gravitational field of the Earth, the electric field is omnidirectional as this image clearly demonstrates. http://www.affordablehousinginstitute.org/blogs/us/hair_standing_on_end_small.jpg Therefore it cannot be a vector, but it is a field. However, in a conductor it is very much a vector. http://www.androcles01.pwp.blueyonder.co.uk/AC/m.gif Reversing the electric vector reverses the rotation of the motor. Vectors have direction. They also have magnitude, which is a scalar. Fields can have magnitude also. If you wish to describe that mathematically then you should first deconfuse what the symbols are referring to, field or vector. >> I drew some pictures to make this clearer: >> [quoted text clipped - 28 lines] > perpendicular component as indicated by your diagram 1 will be > reversed. Is that right ? (E dot n)n is the component of the vector E perpendicular to the surface -- a different thing entirely. It's not even perpendicular to the direction of travel. SignatureJim E. Black (domain in headers) How to filter out stupid arguments in 40tude Dialog: !markread,ignore From "Name" +"<email address>" [X] Watch/Ignore works on subthreads On Jul 16, 1:18 am, Jim Black <fmla...@organization.edu> wrote: > Are you trying to get unit vectors perpendicular and parallel to the plane > of incidence here? If so, you will have to normalize these. Yes I'm trying to find out unit vectors perpendicular and parallel to the plane of incidence. u, n are already normalized. My ultimate goal is to find out the total incident electric field vector on the body. I will do it for every ray. > Your calculations should break down in the case where the ray is normal to > the surface, because there is no longer a unique plane of incidence. You [quoted text clipped - 3 lines] > matter what you choose as the "plane of incidence," you should get the > correct result. > The thing I'd be worried about is making sure that your components are > perpendicular to each other and to the ray. If the cross product (ui X n) > is very small and you then normalize it, you're likely to get a garbage > result. I would think carefully about how to handle or avoid this. Look > at the larger computation -- you might be able to restructure it so that > this division of two near-zero numbers never shows up. I'm using double precision data so do you think it will still cause underflow problems ? The object in consideration is a triangular mesh. If the ray is perpendicular to surface, then I can probably find a vector from the point of intersection to one of the triangle vertices and then normalize it and multiply it by the incident electric field magnitude to find the electric field vector at the point. The reflected electric field vector can be found by simple negation. What is bothering me even ore is that this might be wrong way to do it. The elctric field vector can be in any direction perpendicular to the ray dierction. Whereas with this method, you are just finding a fix direction. Summing all the incident electric field vectors to find the total incident electric field vector at the end may produce a wrong result. |
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