1 Matrix methods in paraxial optics Wednesday September 25, 2002

2 Matrices in paraxial Optics Translation (in homogeneous medium) 0000 L yoyoyoyo y 

3 Matrix methods in paraxial optics Refraction at a spherical interface  y ’’’’φ  ’’’’ nn’

4 Matrix methods in paraxial optics Refraction at a spherical interface  y ’’’’φ  ’’’’ nn’

5 Matrix methods in paraxial optics Lens matrix n nLnLnLnLn’ For the complete system Note order – matrices do not, in general, commute.

6 Matrix methods in paraxial optics

7 Matrix properties

8 Matrices: General Properties For system in air, n=n’=1

9 System matrix

10 System matrix: Special Cases (a) D = 0   f = Cy o (independent of  o ) yoyoyoyo ffff Input plane is the first focal plane

11 System matrix: Special Cases (b) A = 0  y f = B  o (independent of y o ) oooo yfyfyfyf Output plane is the second focal plane

12 System matrix: Special Cases (c) B = 0  y f = Ay o yfyfyfyf Input and output plane are conjugate – A = magnification yoyoyoyo

13 System matrix: Special Cases (d) C = 0   f = D  o (independent of y o ) Telescopic system – parallel rays in : parallel rays out oooo ffff

14 Examples: Thin lens Recall that for a thick lens For a thin lens, d=0 

15 Examples: Thin lens Recall that for a thick lens For a thin lens, d=0  In air, n=n’=1

16 Imaging with thin lens in air oooo ’’’’ ss’ yoyoyoyoy’ Inputplane Output plane

17 Imaging with thin lens in air For thin lens: A=1 B=0 D=1 C=-1/f y’ = A’y o + B’  o

18 Imaging with thin lens in air y’ = A’y o + B’  o For imaging, y’ must be independent of  o  B’ = 0 B’ = As + B + Css’ + Ds’ = 0 s (-1/f)ss’ + s’ = 0 For thin lens: A=1 B=0 D=1 C=-1/f

19 Examples: Thick Lens n nfnfnfnfn’ yoyoyoyo y’ H’ h’ x’ f’ ’’’’ h’ = - ( f’ - x’ )

20 Cardinal points of a thick lens

21 Cardinal points of a thick lens

22 Cardinal points of a thick lens Recall that for a thick lens As we have found before h can be recovered in a similar manner, along with other cardinal points