Page 214 - Modern Optical Engineering The Design of Optical Systems

Stops, Apertures, Pupils and Diffraction 197

The power passed through a centered circular aperture of radius a is
given by
2
P (a) P (1 e 2a /w 2 ) (9.24)
tot
The power passed by a centered slit of width 2s is given by
s 2
P (s) P erf (9.25)
tot w

u
where erf (u) e t 2 dt the error
0
function, which is tabulated in mathematical handbooks.

Diffraction spreading of a gaussian beam
A gaussian beam has a narrowest width at some point, which is called
the “waist.” This point may be near where the beam is focused or near
where it emerges from the laser. As the beam progresses away from
the waist, it spreads out according to the following equation:

2
2
w w 2 z (9.26)
1
z 0 2
w
0
where w z the semidiameter of the beam (i.e., to the 1/e points) at a
2
longitudinal distance z from the beam waist.
w 0 the semidiameter of the beam (to the 1/e points) at the
2
beam waist.
the wavelength
z the distance along the beam axis from the waist to the
plane of w z
At large distances it is convenient to know the angular beam spread.
Dividing both sides of Eq. 9.26 by z , then, as z approaches infinity,
2
we get

a w z 2 4 1.27
5 5 or a 5 5 (9.27)
2 z s2w d s2w d diameter
0
0
zS`
where is the angular beam spread in radians between the 1/e points.
2
For many applications, the gaussian diffraction blur at the image
plane can be found by simply multiplying from Eq. 9.27 by the image
conjugate distance (s′ from Chap. 2).
The Rayleigh Range equals 4 / where is the convergence/
2
deviance angle of the beam. There the beam is 41 percent larger than
at the waist.