Review calculation of Fresnel zones
For off-center points on screen, Fresnel zones on
aperture are displaced …harder to “integrate” mentally.
When white and black areas are equal, light at P’ is _____
a) dimmest b) brightest
P’
Fresnel zone plates as “lenses”
f is the
distance
used to
calculate the
zones. Will
change with l
Spatial resolution of
image is  r ,width of
narrowest zone 1.22r
 min 
D
Fresnel zone plates as “lenses”
Thinnest zone included:
r=25 nm. Used to image cell
structures with soft xrays
Fresnel zones are widely used in antenna-receiver tech
L+l
L+l/2
L
No aperture: The zones here have to do with the phase of
reflections from other objects
How high to raise an antenna from the ground?
If the ground reflects zone 2, you may do better lowering
the antenna
How tightly can we focus light?
Ray picture
Wave picture
Diffraction from the focus of a gaussian beam
Every finite-width beam is constantly diffracting. Must
be always either diverging or converging!
Self-divergence is largest for narrowest beams or
tightest focus
Fresnel method works well
E  x, y, z  
i
ikz
e e
i

k 2 2
x y
2z
lz




 dxdy  Eoe
 x 2  y 2
2
wo  w2  z 
 E  , z   Eo
e
e
w z

wo2
e

i

k
x 2  y 2
2z

e
i
k
xx  yy  

z
k2
z
ikz  i
i tan1
2 R z 
zo
kwo2 zo: Rayleigh range. When z > zo, we are far enough
zo 
2 from focus to go back to rays (Fraunhofer regime).
w  z   wo 1  z 2 zo2 beam width in terms of beam waist
Wavefronts on gaussian beams
Wavefronts are flat only at z=0, and have radius R for z > zo
R  z   z  zo2 z
w  z   wo 1  z 2 zo2 (radius)
kwo2
zo 
2
D
Given the ray geometry of
focusing, find wo in terms
of f,D
f
2fl
 wo 
D
Laser cavities: rays vs waves
Laser cavity : besides finding the waist size,
need also to find where focus (origin) is.
R  z   z  zo2 z
It’s simple algebra: R must match mirrors,
measured from the unknown focus.
Fourier optical manipulation
The “transform plane” is f after the lens.
Suppose the aperture were removed. How would the plane
wave look in the transform plane? With a different angle?
In this FT of the letter E, sketch what you would
see in the image plane if you blocked most of the
dots arranged in the horizontal direction.
a) I got it mostly right
b) I got it mostly wrong.
Where do you block the high spatial frequencies?.
K-space Undersampling
Jorge Jimenez
Fully sampled k-space
Low frequencies
High Frequencies
Vertical Image resolution
Horizontal Image resolution