Class Note for OPTI 510L at UA
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Date Created: 02/06/15
OPTI 420 Introductory Optomechanical Engineering 6 Mirror matrices Matrix formalism is used to model reflection from plane mirrors Start with the vector law of re ection 2 31 0 fly k2 re ected ray IE2 1 2l 10 The hats indicate unit vectors k1 meldemray k1 incident ray A k2 re ected ray 739 n surface normal surface normal For a plane mirror with its normal vector n with Xyz components nxnynz Using the standard vector representation with unit vectors nnx1nyjnzk The matrix representation of this vector is nx I l I Zy Hz J H Burge University of Arizona 1 Optical Sciences 420 The vector law of re ection can be written in matrix form as k2Mk1 Where the mirror matrix M is calculated to be le ZnnT M can be expanded as 10 0 x M1 010 72 ny ltnX ny n2 0 01 HZ 0r 1 2 2 2 2 39IlX 39IlX39Ily 39IlX39IlZ 2 Me 72nXny 172ny 72nynz 72nXnZ 72n nZ liZHZ2 3 After calculating this mirror matrix any vector k1 gets changed by re ection from the mirror to a new vector k2 calculated by simple matrix multiplication 1lt2IMk1 lfthe initial vector k1 is the direction the ray incident on the mirror then k2 is the direction ofthe re ected ray J H Burge University of Arizona Optical Sciences 420 A series of reflections is modeled by successive mirror matrix multiplications lf light bounces off mirror 1 then 2 then 3 the net effect ofthese three re ections is which reduces to a single effective mirror matrix M eff M 3M 2M 1 M2 k4 nal re ected raV k1 incident ray k4 M3 M2 M1 k1 Meff k1 M1 M3 80 the effect ofany set of mirrors can be reduced to a single 3x3 matrix J H Burge University of Arizona Optical Sciences 420 The mirror matrix shows the re ected coordinates not just the incident ray Initial coordinates ijk get reflected to a new set i j k For example a mirror with its normal in the z direction would be described by M2 100 MZ010 0071 A set of coordinates would be re ected so that An incident ray traveling in the 2 direction will be re ected to travel in the z direction lmages ofthe X and y axes do not change lmage orientation can be computed by transforming the quotupquot and quotrightquot axes in object space using the mirror matrix M to find the orientation and parity in image space Each direction ofthe coordinate system is transformed by the mirror matrix J H Burge University of Arizona Optical Sciences 420 Parity The parity ofthis one mirror is of course odd 1 The image ofa right handed coordinate system will appear to be left handed in the reflection This means that clockwise rotation about any basis vector will appear counterclockwise in the image In general the determinant of the mirror matrix gives the parity ofthe system 0 An even number of re ections will cause the image to be righthanded or to have parity detM 1 o A system with an odd number ofre ections will cause the image to be left handed or to have parity detM 1 J H Burge University of Arizona Optical Sciences 420 Mirrors with any orientation can be de ned using rotations The matrix method uses well de ned coordinate transformations which use simple matrix multiplications The effect of rotating a mirror M or system of mirrors that has equivalent matrix M is T M R M R where Mr is the new matrix and R is the rotation matrix given below 1 0 0 x rotation RX 0 00501 Sinoi 0 sind cosd 008B 0 sinl5 y rotation Ry 0 1 0 rsini 0 cosB cosy esiny 0 zrotation R21 51110 005W 0 0 0 1 Transpose operation swap rows with columns 1 2 3 T 1 4 7 4 5 6 2 5 8 7 8 9 3 6 9 J H Burge University of Arizona Optical Sciences 420 for the simple case of the mirror starting with its normal in the z direction rotation at about the X axis gives M Rxloc Mz RxocT We can reduce this using the identify A BT BT AT Since MZ CDCDl A OHO M Rxloc Rxlocl MleT For this special case Rxoc M2T Rxoc M2 not generally true Rxloc Rxlocl MA Rxlocl Rxoc MA R4200 MZ Likewise the effect ofy rotation is Mr ma 3 M The effect of z rotation is NOT the same You can use trig identities to show that Vlr RAY Mz RZTW M2 J H Burge University of Arizona Optical Sciences 420 Some common types of mirrors Free space X mirror y mirror 2 mirror 90 X roof 90 y roof 90 2 roof 45 X roof cube corner J H Burge Optical Sciences 420 OOH Ogt O 0 0 1 O O 1 W University of Arizona ED B 4 A insensitive to x rotation 29 for y and z rotations insensitive to y rotation 29 for x and z rotations insensitive to z rotation 29 for x and y rotations insensitive to x rotation 29 for y and z rotations insensitive to y rotation 29 for x and z rotations insensitive to z rotation 29 for x and y rotations 90 deviation insensitive to x rotation 9 for y and z rotations retrore ects insensitive to all rotations In many cases you can write down the mirror matrix by inspection You can trace the x y and 2 unit vectors through the prism by reflecting the vectors one at a time using the bouncing pencil paradigm In fact you only need to trace two axes through and use the parity to get the third Use these coordinates to evaluate how object motion relates to image motion both for translation and rotation Remember to reverse the direction of rotation if the system has 1 parity To evaluate the effect of prism rotation you can 1 Apply the transformation equations for each degree of freedom 2 By inspection determine how object rotation couples into image rotation then shift coordinates to object frame Use symmetry when possible For nearly all cases prism rotation 9 about the xyor z axis does one ofthree things 1 causes image rotation about same axis by an amount 29 2 has no effect on image about any axes 3 causes image to rotate an amount or 9 about the other two axes J H Burge University of Arizona Optical Sciences 420
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