Page 215 - Modern Optical Engineering The Design of Optical Systems

198 Chapter Nine

Beam truncation
The effect of beam truncation, i.e., stopping down or cutting off the
outer regions of the beam, is discussed by Campbell and DeShazer.
They show that if the diameter of the beam is not reduced below 2(2w),
2
where w is the beam semidiameter at the 1/e points, then the beam
intensity distribution remains within a few percent of a true gaussian
distribution. If the clear aperture is reduced below this value, it will
introduce structure (i.e., rings) into the irradiance patterns, and the
pattern gradually approaches Eq. 9.14 as the aperture is reduced.
A lens aperture large enough to pass a beam with a diameter of 4w
is obviously very inefficient from a radiation transfer standpoint. For
this reason, most systems truncate the beam, very often to the 1/e 2
diameter, and the diffraction pattern is altered accordingly. If the beam
is truncated down to 61 percent of the 1/e diameter, it is difficult to see
2
the difference from a uniform beam, with the ring disk and rings.

Size and location of a new waist formed by
a perfect optical system
When a gaussian beam passes through an optical system, a new waist is
formed. Its size and location are determined by diffraction (and not
by the paraxial equations of Chap. 2). The “waist” and “focus” are at
different locations; in a weakly convergent beam, the separation may
be large. The following equations allow calculation of the new waist
size and location:

2
2 xf
xr 5 (9.28)
2 2
x 1 a w 1 b
2

2 2 xr
2
2
w 2 5 f w 1 5 w 1 a b (9.29)
2 2 2x
x 1 a w 1 b
2

where w 1 the radius (to the 1/e points) of the original waist
2
w 2 the radius of the new waist formed by the optical system
f the focal length of the lens
x the distance from the first focal point of the lens to the
plane of w 1
x′ the distance from the second focal point of the lens to the
plane of w 2
Note that x and x′ are usually negative and positive, respectively. Note
also the similarity to the newtonian paraxial equation (Eq. 2.3).