More on the Ray Matrix Formalism Tuesay, 10/17/2006 Physics 158 Peter Beyersorf Document info 12. 1
Class Outline Paraxial Ray Matrices General Imaging Systems 12. 2
Compoun Optical Systems Compoun optical systems can be analyze using ray tracing an the thin an thick lens equations A compoun optical system can be escribe in terms of the parameters of a thick lens Is there an easier way? Yes using paraxial ray matrices 12. 3
ABCD matrices Consier the input an output rays of an optical system. They are lines an so they can be escribe by two quantities r in Input plane r in optical system r out r out Output plane Position (r) Angle (r ) Any optical element must transform an input ray to an output ray, an therefor be escribable as a 2x2 matrix [ r out r out = [ A B C D [ A B C D note this is the convention use by Hecht. Most other texts have [r r [ r in r in 12. 4
Free Space Matrix Consier an optical system compose of only a length L. What is the ABCD matrix r out =r in r in r in r out =r in +Lr in Input plane optical system Output plane [ 1 0 L 1 12. 5
Refraction Matrix Consier an optical system compose of only an interface from a material of inex n 1 to one of inex n 2 What is the ABCD matrix r out (n 1 /n 2 )r in r in Input plane r in n 1 n 2 optical system r out =r in Output plane [ n1 /n 2 0 0 1 12. 6
Slab matrix What is the ABCD matrix for propagation through a slab of inex n an thickness L? r in r in n 1 n 2 optical system [ n2 /n 1 0 0 1 r out r out [ 1 0 L 1 [ 1 0 [ n1 /n 2 0 n 1 n 2 L 1 0 1 12. 7
Curve surface Consier a ray isplace from the optical axis by an amount r. The normal to the surface at that point is at an angle r/r. The ray s angle with respect to the optical axis if r, so its angle with respect to the normal of the surface is θ=r +r/r θ r R n 1 r n2 optical system [ n1 /n 2 (n 1 /n 2 1)/R 0 1 12. 8
Thick Lens A thick lens is two spherical surfaces separate by a slab of thickness r r n 1 n 2 optical system [ n2 /n 1 (n 2 /n 1 1)/R 2 0 1 [ 1 0 L 1 [ n1 /n 2 (n 1 /n 2 1)/R 1 0 1 12. 9
Thin Lens For a thin lens we can use the matrix for a thick lens an set 0 Alternatively we can make geometrical arguments r in =r/s o r in r out =r inr out =-r/s i =-r(1/f-1/s o )= r in -r/f [ 1 1/f 0 1 12. 10
Curve Mirrors For a thin lens we can use the matrix for a thin lens an set f -R/2 Alternatively we can make geometrical arguments r in =r/s o r in r out =r in θ in =r in +r/r θ out =r out +r/r=-θ in [ 1 2/R When you reflect off a mirror, the irection from which you measure r gets flippe. Also note in this example R<0 0 1 12. 11
Curve Mirror Correction For rays emanating from a point off the optical axis the curvature of the mirror that is seen by the rays is istorte For tangential rays R Rcosθ For sagittal rays R R/cosθ [ 1 2/R 0 1 s 4 12. 12
Break 12. 13
Example Fin the back focal length of the following compoun system f f 12. 14
Example f f Step 1, fin the ABCD matrix for the system [ 1 1/f [ 1 0 [ 1 1/f 0 1 [ 1 1/f 1 0 1 [ 1 1/f 0 1 1 /f [ 1 /f 2/f + /f 2 [ 2r/f + r/f 2 1 /f [ 1 /f 2/f + /f 2 [ 0 r r r/f = 1 /f 12. 15
Example [ r out 0 Step 2. Require input rays parallel to the optical axis pass through the optical axis after the lens system an an aitional path length equal to the back focal length = [ 1 0 b.f.l. 1 [ 1 /f 2/f + /f 2 1 /f [ 0 r in Step 3. solve for b.f.l. 0 = ( b.f.l. ( 2/f + /f 2) + 1 /f ) r in b.f.l. = 1 /f 2/f + /f 2 = f 2 f 2f 12. 16
Example Fin the effective focal length of the following compoun system f f 12. 17
Example f f Step 1, fin the ABCD matrix for the system [ 1 1/f 0 1 1 0 1 [ [ 1 1/f 1 1/f [ 1 0 [ 1 1/f 0 1 1 /f [ 1 /f 2/f + /f 2 1 /f 12. 18
Example Step 2, Consier what happens to an input ray that is parallel to the optical axis at the principle plane at h 2 f f [ r out r in = [ A B C D [ 0 r in Step 3, At what point oes, a istance f eff away oes that ray cross the optical axis? r M h 2 θ=-2r/f-r/f 2 r-r/f feff 1 r in f eff = r out r out = Br in f eff = 2/f + /f 2 12. 19
Example After how many roun trips will the beam be re-image onto itself? (this is calle a stable resonator ) R R 12. 20
Example Step 1. Fin the ABCD matrix for 1 roun trip R R ([ 1 2/R [ 1 0 ) 2 M rt = 0 1 1 [ 1 2/R 2/R 2 M rt = 1 Step 2. Require that after N roun trips the ray returns to its original state M N rt = 1 12. 21
Example Relate the eigenvalues of M N to M recalling that so thus λ 2N λ v = M v ow= one way ow = λ N rt = λ N = 1 λ ow = e ±iθ from where R M N rt = 1 2Nθ = 2πm R M ow Iλ ow = [ 1 2/R λow 2/R 1 λ ow = 0 12. 22
Example R R Explicitly computing λ ow gives et(m ow Iλ ow ) = 1 2/R λ ow 2/R 1 λ ow = 0 (1 2/R λ ow )(1 λ ow ) 2/R = 0 λ ow = 1 R ± ( 1 R ) 1 = 1 R ± i 1 ( 1 ) 2 R λ ow = e ±iθ with cos θ = 1 R 12. 23
Example R R Relating the expressions for λ ow 2Nθ = 2πm an cos θ = 1 R N = πm cos 1 ( 1 R ) The so-calle g-factor for a resonator is g=cosθ=1-/r. Note the beam can only be re-image onto itself if <2R. Another way of saying that is 0 g 2 1 the stability criterion for a resonator. 12. 24
Summary Thick lenses require more parameters to escribe than thin lenses If the parameters of a thick lens are properly escribe, its imaging behavior can be etermine using the thin lens equation ABCD matrices are convenient ways to eal with the propagation of rays through an arbitrary optical system 12. 25