Exam 3 Review Chapters 9-11
Equations that will be given on the exam. All of the equations/constants from exam 2, plus the following:



Rays: R(r )  n(r ) sˆ(r )
ABCD Matrices
1 d 
Translation: 

0 1 
Diffraction formulas
Helmholtz equation: 2 E (r)  k 2 E (r)
Fresnel approximation:
E ( x, y, d )  
0 
1
Flat surface refraction: 

0
n

1 n2 
1
0 


Curved surface refr:  1
 n1 n2  1 n1 n2 
R

R = + for convex, – for concave
 1
Spherical mirror/thin lens: 
 1 f
flens   n2 n1  11 R1  1 R2  
0

1

ie e
i

k 2 2
x y
2z
z
E ( x, y,0)e
i

Top hat:
FT  a 2 J1  k  a  k  a  a 2 jinc(k  a )


k
x 2  y  2
2z

e
i
k
 xx yy 
z
Spectrometer:
dxdy
Fraunhofer approximation:
i
k
 x2  y 2 
ieikz e 2 z
E ( x, y, z )  
z


E ( x, y,0)e
i
k
 xx yy 
z
dxdy
Fourier Transforms:
Comb function (N total deltas):


2 sin  Nt0 2  sin t0 2 

Single slit: FT  1
1.22
l
Gaussian Beams
xh ,

 
md
mN
2
aperture
FT  1

 min 
aperture
1
R = + for convex surface; – for
concave surface
R
f mirror  , R = positive for concave
2
p1 = (1–D)/C, p2 = (1–A)/C, f = -1/C
Cavity stability:  1  A  D  1
ikz
Rectangle:
FT  1 2  ab sinc  k x a 2  sinc  k y b 2 
ikz 
w
2
E  x, y, z   E0 0 e w e
w
 z 
ik  2
i tan 1  
2R
 z0 
kw0 2
z2
z2
, R  z  0 , w  w0 1  2
2
z
z0
Aq1  B
q2 
; q  z  iz0
Cq1  D
z0 

2 a sinc  k x a 2 
2
Equations that you won’t need to know by heart. (Or, if you do, I will give them in the problem statement.)
 ABCD matrix for thick lens
 Numerical aperture
 “Fresnel’s diffraction formula”
 “The Fresnel-Kirchhoff diffraction formula”
 Bessel function formulas
 Complicated formula for diffraction through a lens (Eqn 11.14)
Equations/derivations/other stuff that you may need to know by heart. All of the items listed for exam 2, plus the
following (not an exhaustive list):
 OPL   nd












Fermat’s principle of least time
If ABCD matrix includes whole trip, then B = 0 is requirement for focusing, and A = magnification
Thin lens equations: 1/f = 1/p + 1/q; M = -q/p
Aberrations: names & effects
f-number, including how it relates to angle of light cone
Babinet’s principle
IFraunhofer = I0  |2DFT of aperture function|2 with kx = kx/z, ky = ky/z (z = distance to screen, x & y = coordinates
on screen)
Convolution theorems (in context of diffraction)
2 slit diffraction formula (because 2 slit aperture function is just single slit  2 delta functions)
FT transform of function shifted from origin (because it’s just function at origin  shifted delta function)
Array theorem (because an array is just an aperture  a bunch of shifted delta functions)
Intensity from diffraction grating (because it’s a single slit  a comb function)
Exam 3 Review