Phase Masks and Holography
Consider a plane wave of light normally incident on a glass sheet as shown in figure 1. A point P
lies a distance z from the glass. Light
that travels along the normal CP
reaches P with a change in phase of
kz. Light that reaches P after traveling
a distance r has undergone a phase
change of kr. These will be in phase if
kr - kz =2 m. What is the condition
on y that will ensure this? Assume
that z >> y. Then
kr − kz = k
(
z2 + y2 − z
)

y 2  k y 2 πy 2
≈ k  z + − z =
=
2z 
2z
λz

The condition for constructive
interference of the two rays is therefore y2 = m (2 z). Suppose we coat the glass with an opaque
paint and then scratch out a transparent disk near y = 0 of radius /10 or less. All the light
reaching P through the hole will be very nearly in phase. If we also scratch out a thin ring of
radius (2 z)½ and width /10 or less, the light that reaches P from the ring will be in phase with
the light from the hole. If we also scratch out rings of radii 2 (2 z)½, 3 (2 z)½, 4 (2 z)½, ...
there will be a large amount of constructive interference at P, so the light intensity will be large
there. The optical device, one of a large category of devices called phase masks, performs
something like a lens, focusing light at P.
To construct a phase mask, we could (in principle) paint a glass plate and then scratch out a small
disk surrounded by concentric thin rings of radii 2 a, 3 a, 4 a, ... where a is some
convenient length. The focal length of our phase mask for light o wavelength will be f = a2/2 .
This phase mask has other important optical properties. Let F be the point on the axis of the mask
located a distance f = a2/2 behind the glass, on the same side as the incident light, as shown in
figure 2. Let 6 be a notional surface, an expanding sphere centered on F whose radius increases
at the speed of light. Light reaching a point A on 6 has come from point C on the glass, while
light reaching point B on 6 must come from point Q on the glass, lying a distance y from C. If, at
the instant shown, the sphere has radius r, then
(
CA − QB = (r − f ) − r −
f +y
2
2
)
 
y2
≈ (r − f ) − r −  f +
2f
 
18.1
  y 2 λy 2
=
=
a
 2 f
fig. 2. Construction showing a virtual
light source produced by a phase mask
Light can only be emitted from a point Q on the mask that lies on one of the transparent rings, so
that y2 = ma2, i.e. CA - QB = m . This means that 6 is a surface of constant phase; it is a
wavefront, from which it follows that some of the light coming from the phase mask appears to
be coming from F. The point F is a virtual point source.
A construction similar to the above shows that there is also a transmitted plane wave, identical to
the incident wave (except that it is diminished in intensity). Summarizing, our phase mask has
the following properties when illuminated with normally incident plane waves: it produces
1. A real focus with focal length f = a2/2
2. An unaltered plane wave.
3. A virtual focus, also with focal length f.
Problem 1: Find the types of waves produced by the phase mask when the plane wave is not
normally incident, but rather is incident at an angle .
Making a phase mask by scratching out paint is not very practical. An easier method is to use
photography. Suppose that the light from the expanding spherical wave of a point source, located
a distance f from the photographic plate, falls on that plate as well as the light from a normally
incident plane wave, as shown in figure 3. Both waves have the same wavelength . In the
simplest case, when we suppose that the light from the point source is in phase with the plane
wave at C, the intensity will be large at C. It will also be large at point on the plate whose
distance y from C satisfies
f 2 + y 2 − f = mλ . These are just the rings of the phase mask
considered above. The developed plate will be most exposed where we want a phase mask to be
transparent and it will also be exposed (but less so) where we would want the mask to be opaque.
18.2
This plate (or its negative image) is a useful phase mask.
Fig. 3.
Using photography to
produce a phase mask.
We have just described the essential elements of holography. A hologram is basicallly just a
phase mask, made as above, but instead of a single luminous point, all the points on the surface
or surfaces of an entire three dimensional object or scene act as sources of light, usually because
they are reflecting the light of a laser beam. A photographic plate is illuminated directly by the
laser beam and also by this reflected light. The direct light is called the reference beam. When
developed, the photograph looks nothing like the object. However, when it is exposed to laser
light of the same wavelength as the light that produced the photograph, the resulting virtual
image of the object is a three dimensional image, often much more realistic than any standard
photograph.
18.3
Step-by-Step Instructions for Reflection Holography
(This section was written by Clint Densham)
SETUP
Lighting:
The room must sufficiently dark so that you cannot read with the light in the room. You
can use a common green incandescent "Party Bulb" in a desk lamp for light since the plates are
red-sensitive. Set the desk lamp near a wall and point the lamp towards the wall (leaving a 1"-2"
gap works well). This will diminish the overall brightness of the bulb and reduce much of the
remaining red light emitted by the bulb (It’s not ALL green light).
Diode Laser:
Remove the lens from the laser if it is attached. Place the laser in a clothespin and set in a
cup of sand, salt or sugar. This gives maximum maneuverability and mechanical stability.
Object:
The choice and preparation of the object is crucial: (1) it should be made of a solid
material (no furry or cloth objects); (2) it must appear bright when illuminated with the red laser
light; and (3) it must not move or deform. If there is any doubt, it should be hot-glued to the table
or floor where the hologram will be made.
Note: Shiny objects may not be suitable for holograms. Shiny metal is especially not
conducive to holograms; it reflects only light at a certain angle for every perspective. This
will result in points or lines of light, and irregular coverage.
To eliminate the need for vibration damping you can lean the plate directly against the
object. You can use a plate that has already been developed to get an idea of what your setup
will be like during the actual process. If the object is stable enough to support the plate, that’s
great, but some objects might need to be supported in order to hold the weight of the plate.
When leaning the plate against the object, try to get the plate to lean as vertically as
possible, while still resting against the object for support. The more vertical the plate is, the less
light the edges will reflect and refract. This light can interfere tremendously in your final
hologram.
If you wish to separate the object from the plate you need to have a way
to suppress vibrations in your plate and object. This can be accomplished by
setting all of the components for the hologram in sand. To make sure not to get
sand on the plate, you should use a stand to elevate the plate. A few clothes pins
clipped to the bottom of the plate work well. See figure at right.
Final Preparation Thoughts:
• Make sure that the room is dark.
18.4
•
•
x
Are there any objects, or walls near your setup that will reflect the laser light and interfere
with the hologram? Make sure any of these objects that are around are removed, or
covered to minimize harmful interference.
Is there a lot of vibrations in the room? If possible, wait for them to diminish before
making your hologram.
Always wear protective gloves!
The chemicals used in the processing can be harmful. Ask your professor for more
information about the dangers of using the chemicals.
Making the Hologram:
Prepare the chemical processing solutions and layout the processing trays as directed by
1.
the instructions that accompany the kits. Do this before you start the actual making of the
hologram.
Adjust the laser in the holder so that the beam spreads out horizontally.
2.
Place the object at a distance 35 to 40 cm from the laser.
3.
Place a white card behind the object and adjust the laser until the object is optimally
4.
illuminated by the laser light. This is an important step. If your object is not evenly
illuminated, you will only see the illuminated parts in your final hologram. You may
remove the white card after finishing this step, it is not used during the making of the
hologram.
Place your shutter between the laser and the object. A piece of cardboard works well.
5.
Remove a plate from it’s container in the darkest part of the room to minimize exposure
6.
of the other plates. Close the container.
Note: We need to know which side of the holoplate has the emulsion on it. The
side with the emulsion is the ‘sticky’ side. The emulsion faces the same way on all
of the plates in the container and should face towards you if the label is upright
when you hold the container in front of you. (For Slavich PFG-01M plates)
Lean the holoplate against the object, making sure it will not slip or move. The sticky
7.
side should touch the object.
Allow ten seconds for settling, and have everyone in the room stop moving.
8.
Lift the shutter lightly off the table, but still blocking the laser light and wait two seconds
9.
for the vibration to subside.
Lift the shutter all the way up to expose the plate and object for 5 seconds. Then block
10.
18.5
the light again.
Note: Exposure times can vary. These short exposure times are sufficient, but
brighter holograms have been produced using longer exposure times (20 to 30
seconds). Holograms made with the prolonged exposure times have signs of
overexposure including unwanted interference patterns on the object in the image
and also on the plate. Coloring also seems affected by changing the exposure
times.
Process the exposed holoplate (JD-2 Kit):
11.
a.
b.
Mix equal parts of Solutions A and B before making the hologram. The mixed
solution will remain usable for about 8 hours.
Develop the plate in the working solution for 2 minutes.
c. Wash in running water for 3 minutes.
d. Bleach until hologram is transparent, less than 2 minutes. (1:30 to 1:45 is sufficient)
e. Wash in running water for 3 minutes.
f. Soak in photoflo solution for three minutes.
g. Hang up vertically to dry, using clean and dry clothespins.
Processing Notes (JD-2 Kit):
• The processing solutions work best when made with distilled water.
Some steps are not necessary:
•
o
The photoflo solution is a cosmetic step that reduces water spots on the
final hologram and seems to help the film dry a bit faster.
o
The water soaking step between the developing solution and the bleach
solution is also optional. This step is used to preserve the bleaching solution
for more use. If you are only making one hologram, this step is not needed.
Using the thin latex gloves makes handling the plates a lot easier than with thick
•
chemical gloves. However, these gloves are not very durable. Be careful when
handling the plates not to cut the gloves or your hands on the sharp edges of the
plates.
If you are not comfortable with handling the plates, or with sticking your gloved
•
hands into the solutions, tongs can be used to retrieve the plates from the
solutions.
o
Start at the edge of the tray and slide one tong across the bottom until it
touches the plate, then set the other tong down on the other side of the plate
and firmly grab the plate. Tongs tend to not be stable when handling the
plates. Be sure to transport the plate in your hand, or hold your hand under
the plate when moving between steps.
18.6
o
Be sure to handle the plate out of the trays carefully as they are fragile and
the emulsion scratches easily.
18.7