PHYSICS 415/416 SUPPLEMENTAL PROBLEMS
Part 9
9A.(Fresnel Zones)
You are going to estimate the number of Fresnel zones within a circular aperture as seen by a point P on the
axis of the aperture. You will then examine some ramifications of this result in Problem 9B. Refer to the figure
on the next page to assist you in your derivation. Note that  in the figure should be replace by ro .
(a) Show that the area dS of the very thin ring in the Nth Fresnel zone (the zone is labeled as the lth zone in the
figure) can be expressed as
dS  2ro2 sin  d .
(1)
r 2  ro 2  (ro  ro ) 2  2ro (ro  ro ) cos  .
(2)
(b) Show that
(c) The distance ro is constant and we can approximate ro as being constant over the thin ring. What varies
over the thin ring is  and r. With this in mind, show that
d 
rdr
.
ro (ro  ro ) sin 
(3)
(d) Substitute (3) into (1) to get an expression for dS involving r. Now integrate the expression to get the area S
of the Nth zone. This is an integration over r from one edge of the zone to the other. The value of r varies from
rN 1  ro  ( N  1) / 2 to rN  ro  N / 2 over the zone. Show that the integral can be simplified to yield Eq.
(13-15) in your text:
S
ro ro2
[( / ro )  (2 N  1)( / 2ro ) 2 ]
ro  ro
(4)
*Notice that (4) demonstrates that the zones do get larger in area as N increases.
(e) Now we are assuming in our treatment of Fresnel diffraction that  / ro <<1, i.e. while the screen can be
close to the aperture, it can’t be within several wavelengths of the aperture. Show that this allows (4) to be
written as
S
ro ro

ro  ro
(5)
*Notice that (5) demonstrates that each Fresnel zone has approximately the same area, i.e. there is no
dependence on N. Examining both (4) and (5), we see that the zones do increase in size but that the increase in
area is not dramatic and they can be approximated to have the same area, especially for small values of N.
Interesting!
(f) To estimate the number of zones within a circular aperture of radius R, we simply divide the area of such an
aperture by the area of each zone as given in (5). Do this and show that the number of zones seen is
# zones 
(ro  ro ) R 2
ro ro 
(6)
2
Figure for Problem 9A. Again, our textbook uses ro’ in place of .
Side View of Above Figure (ro’ in place of ).
r + dr
d
ro’

S
ro’
r
ro’ sin
O
ro
P
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9B. (Fresnel Zones)
Let’s investigate the result from Problem 9A. Assume that you are using a He-Ne laser. You focus the light
with a microscope objective and place a pinhole at the position of the focused light so that the light spreading
out from the pinhole is, to a very good approximation, a spherical wave. You place a circular aperture with a
variable diameter 250 mm from the pinhole. A screen is placed 750 mm from the aperture. The figure below
shows the set-up.
aperture
screen
laser
P
250 mm
750 mm
(a) Plot the number of zones that contribute to the intensity at the center of the screen (point P) versus the
radius of the aperture for radii ranging from 0.1 mm to 3 mm. (You are plotting Eq. (6) from Problem 9A).
Describe the dependence of the number of zones on R.
(b) What aperture radius will allow only the first zone to be completely passed?
Calculate this exactly using Eq. (6). Describe the intensity at point P.
(c) What is the smallest aperture radius that will give essentially no intensity at P?
(d) You now increase the aperture radius to 0.5 mm. How many zones contribute to the intensity at P?
(e) Let’s keep the radius at 0.5 mm and now begin to move the screen closer to the aperture. This decreases ro.
Plot the number of zones versus ro from 750 mm down to 50 mm.
(f) What happens to the intensity at P as the screen approaches the aperture? In particular, how many times do
you see essentially no light at P as ro decreases from 750 mm to 50 mm. Indicate these points on your plot.
(g) If the screen is moved the other way, increasing ro, you should see that the number of zones decreases but
that the number approaches a nonzero value. Based on your plot, what does this value look like?
(h) We can calculate this limiting number of zones for large ro by using Eq. (6). If ro becomes much larger than
ro , then show that the number of zones is R 2 ro  . Note that this expression is now independent of ro, proving
that the number of zones remains constant as the screen moves farther and farther away. Calculate this limiting
number of zones for our geometry.
Note 1: Even when the screen is far away, the number of contributing zones is always around 1.6. Fraunhofer
diffraction theory is not valid here! In order to use Fraunhofer results, the screen must be far away and, in
addition, only a small fraction of the first zone must contribute to the intensity. How can we get this to occur?
By moving the source farther away from the aperture (increasing ro ). Physically, this makes the wavefronts at
the aperture flatter, more closely approximating plane waves.
Note 2: The dependence of the number of zones on source-to-aperture distance ( ro ) is exactly the same as the
dependence on aperture-to-screen distance (ro). You can see this in Eq. (6). Swapping ro and ro in the equation
give you back the exact same equation. Thus, changing ro while keeping ro fixed produces the same results as
you examined above where ro was fixed and ro varied.
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9C. (Fresnel Zones for Plane Waves and Far-Field Diffraction)
If a spherical wave travels far enough, then the wave fronts become approximately planar. Thus, we can even
talk about Fresnel zones on planar wave fronts. We can either use our analysis of the zone plate to assist us in
analyzing the zones or extend our spherical wave treatment to plane waves.
(a) Show that the areas of the Fresnel zones on a planar wavefront are approximately constant and equal to ro
for an observation point a distance ro from the wavefront. Use the analysis of the zone plate to derive this.
(b) In Problem 9A, you showed that the Fresnel zones for a spherical wave also had approximately equal areas
given by the expression in Eq. (5) where ro' and ro are the distances from the point source to the aperture and
from the aperture to the observation point, respectively. How can we make this spherical wave in Problem
9A turn into a plane wave without altering the source? Do this and show that Eq. (5) will in fact give the
area you found above in (a) for the zones of a plane wave.
(c) Suppose that a collimated beam of light from a Na lamp (589 nm) hits an opaque screen with a hole in it
such that the first two Fesnel zones are completely transmitted to a point on the center axis 1 meter away
from the hole. What is the diameter of the hole? What is the approximate intensity at this point?
(d) Recall that Fraunhofer (far-field) diffraction analysis would predict a maximum intensity always for this
center point. So this one meter in (c) must not be far enough away. How far is far enough? Answer this not
quantitatively but rather qualitatively in terms of the number of Fresnel zones seen at the center point.