Corcovilos AAPT Su2021

Transcript

hello my name is ted corcovalis i'm from duquesne university in pittsburgh pennsylvania and i'm going to discuss geometric algebra as a tool for calculations in geometric optics so geometric algebra is a type of real value clifford algebra and the nice thing about it is it becomes a superset of many common math tools uh you can include vector algebra things like lead algebras tensor analysis uh grassman algebra qualternions lots of things like that and so in this talk what i'm going to do is motivate why we want to use geometric algebra for optics and in particular i'll show you one new result that can be used without prior knowledge of ga but ga is used in the derivation for more detail on how to use ga please see the references at the end of this talk and for an extended version of this talk you can look at my website here to give you an idea where we're going at the end of the talk i'll derive a point transfer matrix that takes a point in the object space of an optical system and transforms it into the corresponding point in the image space so the outline of my talk i'll talk briefly about what geometric algebra and projective geometric algebra are the latter is the flavor of ga that we're using in our calculations and to do that we need to define homogeneous coordinates just to give you the the context for the tools we're using then i'll work through an example of ray tracing by hand in other words what we do in freshman physics with a ruler and and pencil but i'll put equations to it then i'll describe these ray transfer matrices and point transfer matrices that i alluded to on the previous slide and i'll conclude with some comments about how these things generalize to the three-dimensional case so geometric algebra is a real valued clifford algebra that's defined by a geometric product that is associative over vectors and a symmetric bi-linear form in other words a metric so that's a lot of math speak but one of the key results is that the product itself is not closed over vectors so in other words you can multiply two vectors together and get something else you may get a vector but you may also get a scalar piece or a bivector piece etc and so in other words we relax our usual rule that you can't add scalars to vectors and we allow these more composite objects to be valid pieces of our math so why would we want to use this in optics well some nice things happen so raise points areas and volumes they have distinct representations in geometric algebra but they obey unified transformation laws in other words reflections rotations and translations are represent represented the same way for each of these geometric objects it has some nice numerical advantages so these transformations can often be written with faster code and less round off error than more conventional methods the geometric constructions you know typical things that we're going to use in optics have simple algebraic representations and these for example the two examples i give here have an intersection of two lines or a line connecting two points are actually just represented by products in ga so one nice thing here is that often we don't need to explicitly calculate distances and angles they can just kind of be absorbed into our algebra so that saves some of the headaches that students often face of you know remembering trig identities and those sorts of things in particular here we're using the flavor of geometric algebra called projective geometric algebra and this is basically ga using homogeneous coordinates and i'll define what that means in a little bit the nice thing here is that linear transformations in pga are isomorphic to what we call colony collineations of projective geometry and operations like projections reflections rotations and translations can all be done with simple products so there's many different products in geometric algebra i won't go through these in detail but just to realize that all these things have nice geometric interpretations and so we can do things like join two points to produce a line or find them where two lines meet just with a simple product the mathematical language we use in pga is homogeneous coordinates so for example we can represent a line by a vector abc and if we multiply that vector by a scalar then we get an identical geometric element so that leads to this idea of the normal weight of the line which we define to be the square root of a squared plus b squared and if we'd like we can normalize the line by dividing by the the norm and then the value of c represents the distance from the origin to the line and the values of a and b are the signs of the angles between the line and the corresponding axis similarly we can represent points as bivectors these have three components wxy and here we define the norm of the bivector to be the value of w and then if we divide by w to normalize it then we get the usual representation for x and y just to give you an example of what some of these pga calculations look like i've worked through a simple example here so this is what i call ray tracing by hand which is what we teach in freshman physics for drawing the rays to an optical system using just a ruler and so here we have a thin lens and an object point and then we put out three rays one that begins perpendicular to the lens r1 r2 goes to the center of the lens and r3 goes through the front focal point of the lens and then we trace how those behave on the on the image side of the system and so what i have here on the right are the equations to do all those calculations in pga so for example r1 what this equation says is that we we take the plane of the lens walk out perpendicularly to the point o and that's what gives us r1 and then we can do things like intersect uh the rays with the surface of the lens uh projected rays uh between two points etcetera and then at the end we take the intersections of two of those outgoing rays and that gives us the final image point and we can you know insert numbers here so you can see how that works out the key point here is at the end of the day we get a bi vector for the image and then if we normalize that by vector we can read off the location of the image and its height for more complicated optical systems we can represent the ray transfer matrices using geometric algebra as well so here's our kind of standard thing this is that can be found in typical optics textbooks like pedrotty or hect i should note i'm using the convention in padradi which is a slightly different than that in hect and so again this relates the in-going rays and outgoing rays by a matrix multiplication so the key result that i mentioned up at the top of the talk was that we can also derive point transfer matrices using the rules of geometric algebra and so that gives us a new matrix which is used to connect object points to image points if those points are represented in homogeneous coordinates so the way this works so first of all we want to write our rays as vectors in pga and so this is simply just rearranging and identifying those elements abc that i discussed when i defined our lines earlier and then i can just uh basically append an extra row and column to our ray transfer matrices to see the correspondence with the the notation we have above now that we know how our rays transform and homogeneous coordinates we can construct the point transfer matrices and we do this through the outermorphism property what this means is that if we want to find the linear operator that acts on a point we can treat that point as the intersection of two lines transform each of those lines individually and find where the new lines intersect and if we do that we can do that most simply by looking at each of the unit vectors for the the bivector representation and breaking that down into its component vector products and running those through the matrix and that will give us a new column in the bi-vector representation if we put all those three unit vectors together we can get the final representation once we calculate how the three unit bi vectors transform we can stitch those back together and make our matrix that operates on any arbitrary point and that's what we have here and so again if we take the representation of the point and homogeneous coordinates from the object side multiply it by our new matrix we get the resulting point in homogeneous coordinates on the image side of the system or here's a more complex example i pulled this example out of hect so here we're given the ray transfer matrix for a particular photographic lens and uh you're asked to find the image from a particular object and so um you know in heck this takes about a page and a half uh to get the answer but we can do it here in two lines basically just by forming the point transfer matrix multiplying that into the object point normalizing the result and then we can read off the position in x and y coordinates of the the final image so we can look forward to eat 3d this works in 3d but now vectors equal planes uh bi vectors equal lines and skew line pairs which is a new thing and then tri vectors represent points so these ideas of the ray transfer matrix and points transform matrix also work in 3d we can also represent gaussian laser beams using skew line pairs after a paper by arno from the 1980s and then we can follow the same transformation rules as geometric rays and then lastly we can do more complicated things like keeping track of aberrations so there's a lot of fertile ground to look at here so i just want to conclude with a list of references particularly this web page by vectorbind.net that's put together by stephen decanink and he has some great tutorial videos on using geometric algebra and a few other references here i'll stop with that and thank you for your time