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Princeton University
Physics 312
Spring 2011
Holography
1
Introduction and Background
Holograms are ubiquitous and the word “holography” part of our common lexicon. In physics
the concept of holography has even found a place in string theory and quantum gravity. There
is a “holographic principle,” still speculative, that purports that properties of a volume of
space are encoded on the boundary of that space. The goals of this lab are to build a simple
interferometer, to make a transmission hologram, and to make a reflection or “white-light”
hologram.
The aesthetic appeal and commercial usefulness of holography are both related to the
ability of a hologram to store a three-dimensional image. Unlike ordinary photographs,
holograms record both phase and amplitude information. Because phase is a relative property, construction of a hologram requires a “reference beam” in addition to the light reflected
from an object’s surface. Additional requirements include a powerful, coherent, monochromatic light source (e.g., a laser, whose workings you should understand); a vibration-free flat
surface; some common optical devices; and photographic plates, chemicals, and a darkroom.
A hologram can be made by exposing a photographic plate to the interference pattern made
by the reference and object beams. The plate is then developed and dried. When illuminated
by a reference beam similar to the original one, it recreates a three dimensional picture of
the object. There is nothing mystical about this recreation, and you can understand how it
works with the help of some Fourier transform mathematics.
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Mathematical Overview
Consider the light waves reflected from an object (see Figure 1). The details of this “object
beam” obviously depend upon the shape and nature of the particular object. We will take
a general approach and say that the object beam is a superposition of plane waves,
ψ0 (r, t) = A0
ZZ
F0 (kx , ky )ei(ωt−kx x−ky y−kz z) dkx dky ,
1
where (kx2 + ky2 + kz2 )1/2 = k = 2π/λ. Let’s assume that we have permitted ψ0 to interfere
with a reference beam given by
ψ1 (r, t) = A1 ei(ωt−kz) ,
and recorded the resulting intensity pattern on a photographic plate in the plane z = 0. The
recorded intensity will be
I(x, y) = |A1 + A0
ZZ
F0 (kx , ky )e−i(kx x−ky y) dkx dky |2 .
When the right-hand side of this equation is expanded, it will have terms proportional to
|ψ0 |2 , |ψ1 |2 , and |ψ0 ψ1∗ |.
We will assume that the reference beam is much more intense than the object beam, so
that |ψ1 | >> |ψ0 |. Then the term in |ψ0 |2 can be ignored, and the intensity can be written
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I(x, y) ≈ |a1 |
+A0 A∗1
ZZ
−i(kx x+ky y)
F0 (kx , ky )e
dkx dky +A1 A∗0
ZZ
F0 (kx , ky )e+i(kx x+ky y) dkx dky .
The developed plate will have a transmission function T (x, y) = 1 − γI(x, y), where γ is
a function of the exposure and developing processes. The hologram will therefore have a
transmission function
T (x, y) = C0 −γA0 A∗1
ZZ
F0 (kx , ky )e−i(kx x+ky y) dkx dky −γA1 A∗0
ZZ
F0 (kx , ky )e+i(kx x+ky y) dkx dky ,
where C0 is a constant. Put in words, the hologram contains scaled versions of the twodimensional Fourier transform of the object beam, and the inverse two-dimensional Fourier
transform of the object beam, superimposed on each other. If you illuminate this hologram
with a plane wave like the original reference beam, another Fourier transformation takes
place, and what emerges is
ψ(r, t) =
ZZ
0
0
0
F (kx0 , ky0 )ei(ωt−kx x−ky y−kz z) dkx0 dky0
where, as is usual in Fourier optics,
F (kx0 , ky0 ) =
1
2π
2 Z Z
0
0
T (x, y)ei(kx x+ky y) dxdy .
The three terms constituting T (x, y) obligingly integrate as delta functions (the term we
threw out above would not have been so obliging), and we find
F (kx0 , ky0 ) = −γA∗1 A0 F0 (kx0 , ky0 ) − γA1 A∗0 F0∗ (−kx0 , −ky0 ) + C0 δ(kx0 )δ(ky0 ) .
2
Therefore the wave that emerges from the suitably illuminated hologram (see Figure 2)
consists of three parts: (left) a term proportional to the original object beam, reconstructing
the object in all details of amplitude and phase; (center) a so-called “twin reconstruction”
that can be shown to be the original object beam reflected in the plane of the photographic
plate and run backwards in time; and (right) an undeflected piece of the reference beam
traveling along the z axis.
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Transmission and Reflection Holograms
The first holograms were examples of transmission holography. Transmission holograms
require a monochromatic reconstructive beam. In other words, the image is only visible
when the hologram is illuminated by either a laser or other single frequency source. This
limitation arises from the way a hologram stores information. The interference patterns
produced from an object and reference beam, whose formalism has been derived above, are
actually three dimensional hyperbolae of revolution (Beesley, Chap. 12 [1]). The position
of the photographic plate relative to the object determines how this interference pattern is
recorded onto the photographic emulsion.
Holography
Emulsion
P2
Reference
Point Source
Object Point
Source
P1
Figure 1: Figure
A two 1:dimensional
projection
of the
photographic p
A two dimensional
projection
of interference
the interferencepatterns
patterns produced.
produced. A A
photoat P1 would
result
inata P1
transmission
hologram,
a plate
at P2a in
a reflection
hologram.
graphic
plate
would result in
a transmission
hologram,
plate
at P2 in a reflection
hologram.
object beam, superimposed on each other. If you illuminate this hologram with a plane wave
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the original reference beam, another Fourier transformation takes place, and what emerges is
ψ(r, t) =
ZZ
0
0
0
F (kx0 , ky0 )ei(ωt−kx x−ky y−kz z) dkx0 dky0
where, as is usual in Fourier optics,
F (kx0 , ky0 )
1
=
2π
2 Z Z
0
0
T (x, y)ei(kx x+ky y) dxdy .
Holography
3
Recording
plate
Recording
plate
Plane
reference
beam
Object
source
Recording
Plane
reference
beam
Object
source
Recording
Primary
image
Hologram
Conjugate
Image
Hologram
Plane
reconstructing
beam
Conjugate
Image
Plane
reconstructing
beam
Primary
image
Transmission Reconstruction
(a)
Reflection Reconstruction
(b)
Figure
recording
andand
reconstruction
setups
for (a)
and (b)
holograms.
Figure2: 2:The
The
recording
reconstruction
setups
for transmission
(a) transmission
andreflection
(b) reflection
The
transmission
figure
was
taken
from
[3].
holograms. The transmission figure was taken from Beesley [1].
can Figure
pass through
If this
hologram
werephotographic
viewed in white
light,relative
every frequency
1 showsthe
twoplate.
possible
placements
of the
emulsion
to the ob- would
create
image,one
the (P1)
combined
simply beingofa wash-out.
This hologram.
is why transmission
ject. an
Position
showseffect
the development
a transmission
Note how holograms
the
are
not visiblepattern
in whiteis light.
interference
recorded perpendicular to the plane of the plate. Consequently, any
However of
if light
we place
the photographic
plate at
interference
patterniniswhite
recorded
frequency
can pass
through the plate.
If P2,
this the
hologram
were viewed
light,parallel
toevery
the plate
and along
thickness
of the
This thickness,
irrelevant
to transmission
frequency
would the
create
an image,
the emulsion.
combined effect
simply being
a wash-out.
This
holograms
is essential holograms
for reflection
Reflection
holograms get their name from the fact
is why transmission
are holograms.
not visible in
white light.
that instead of creating an image by transforming the reconstructive beam into a replica of the
However
if we
place through
the photographic
at P2,a the
interferenceofpattern
is recorded
object
beam as
it passes
the plate, plate
they reflect
reconstruction
the object
beam. Figure
parallel
to
the
plate
and
along
the
thickness
of
the
emulsion.
This
thickness,
irrelevant
to
2 illustrates the difference between the two types of holograms.
transmission
is essential
for reflection
holograms.
holograms
their light.
The uniqueholograms
characteristic
of reflection
holograms
is that Reflection
they can be
viewed get
in white
name
from
the
fact
that
instead
of
creating
an
image
by
transforming
the
reconstructive
beam
Therefore reflection holograms are commonly known as white light holograms. Exactly how does
into a replica of the object beam as it passes through the plate, they reflect a reconstruction
this happen? As we mentioned earlier, the thickness of the emulsion is crucial. In reflection
of the object beam. Figure 2 illustrates the difference between the two types of holograms.
holography, the interference pattern is recorded along the thickness of the emulsion, creating what
are known
as Bragg
diffraction
These
planes isact
similarly
to be
half-silvered
mirrors:
some of
The unique
characteristic
of planes.
reflection
holograms
that
they can
viewed in white
light
the
light
used to
illuminate
object
each embedded
fringe.
As shown in
even
though
they
are madethe
with
a redreflects
laser. off
Therefore
reflectioninterference
holograms are
commonly
figure 3, there is a path difference between light rays reflecting off different Bragg diffraction planes.
These reflected rays then interfere with each other, and if they are not in phase, the interference
4 rays reflected off a Bragg diffraction grating have
will be destructive. Consequently, the only light
path length differences equal to an integral number of wavelengths.
The geometry shown in figure 3 allows us to derive a quantitative description of Bragg diffraction
[5]. The beam passing through the top grating and reflecting off the lower layer travels a distance
AB + BC longer than the beam reflected off the top layer. The angle of incidence θ is equal to
the angle of reflection, so AB = BC = d sin θ. The path length difference must be equal to an
integral multiple of wavelengths for constructive interference to occur. Therefore, the condition for
lography
q
q
A
q
C
d
B
ure 3: Diagram
Bragg diffraction.
TheThe
path
length
is AB
− BC and th
Figurerepresenting
3: Diagram representing
Bragg diffraction.
path
length difference
difference is AB
− BC
and the
angle
of reflection
θ.
gle of reflection
is θ.
Figure
takenis from
[5].
known as white light holograms. Exactly how does this happen? As we mentioned earlier,
In reflection holography, the interference pattern is
recorded along the thickness of the emulsion, creating what are known as Bragg diffraction
planes. These planes act similarly tonλ
half-silvered
some of the light used to illumi= 2d sinmirrors:
θ,
(1
nate the object reflects off each embedded interference fringe. As shown in Figure 3, there
ich is knownis as
Bragg’s
Law.
a path
difference
between light rays reflecting off different Bragg diffraction planes. These
reflected
rays
then
with wavelength
each other, andtoif they
are not in at
phase,
the interference
Bragg’s law allows only a interfere
particular
be reflected
a specific
angle. Therefor
will be destructive. Consequently, the only light rays reflected off a Bragg diffraction grating
en viewed inhave
white
light, a reflection hologram will essentially select one particular frequency t
path length differences equal to an integral number of wavelengths.
nstructive interference
is the emulsion is crucial.
the thickness of
onstruct the image, preventing the wash-out effect seen in transmission holograms.
The geometry shown in Figure 3 allows us to derive a quantitative description of Bragg
diffraction. The beam passing through the top grating and reflecting off the lower layer
travels a distance AB + BC longer than the beam reflected off the top layer. The angle
dditional Details
of incidence θ is equal to the angle of reflection, so AB = BC = d sin θ. The path length
Following certain
youmultiple
to ensure
success for
in constructive
making your
holograms.
differenceprocedures
must be equalwill
to anhelp
integral
of wavelengths
interference
to occur. Therefore, the condition for constructive interference is
1. Recall that in the derivation above we ignored the contribution of |ψ0 |2 in comparison wit
|ψ1 |2 ; that is, we assumed that the reference
beam
nλ = 2d sin
θ, was more intense than the
(1)object beam
However, the derivation did not tell us how much more intense the reference beam should b
It has been found empirically that a reference/object intensity ratio between 3 and 10 yield
5
good results. A photocell is provided to measure
the intensities of the reference and objec
beams.
2. Higher intensities (consistent with a suitable reference/object ratio), will need shorter exposur
times, and, will, in general, produce better holograms.
3. A clean, uniform reference beam is needed for a high quality hologram. A pinhole apparatu
can help you achieve this goal. Set it up carefully in order to minimize the diffraction an
which is known as Bragg’s Law.
Bragg’s law allows only a particular wavelength to be reflected at a specific angle. Therefore, when viewed in white light, a reflection hologram will essentially select one particular
frequency to reconstruct the image, preventing the wash-out effect seen in transmission holograms.
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Experimental Procedure
CAUTION: be careful with laser light! Never look directly into the undiverged beam. When
the beam is 5-10 cm across it is safe to look at directly.
4.1
Michelson interferometer
Using the output of one of the beam divergers, set up a Michelson interferometer so that you
can gain some familiarity with the various pieces of equipment. This will give you practical
experience with elements of the optical table and with aligning optical elements. A diagram
of the Michelson configuration can be found in many books. Particularly good discussions
may be found in Hecht’s Optics[3] and Born and Wolf’s Principles of Optics [2]. Never look
down the beam line. Demonstrate interference phenomena to yourselves and investigate
the coherence length of this particular helium-neon laser. The Michelson is just one of many
possible configurations and has its own specific properties. Questions to ponder:
• Why does the interference pattern show up as concentric rings of light and dark zones?
• Why do the zone spacings depend as they do on the path-length difference in the two
arms of the interferometer?
• What happens if you increase the path-length difference well beyond the coherence
length? What does this behavior suggest about the spectral distribution of the laser
light?
• What is the coherence length of the laser? Think carefully about what you are doing
here, and record your measurements and the details of your interferometer setup to
the necessary degree of accuracy for at least a semi-quantitative result.
• What is a speckle pattern? How should one think about it?
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4.2
Transmission hologram
Set up the optics to produce a transmission hologram. From Figures 1 and 2 one can see that
you’ll want to set up a reference beam that illuminates the holder for the film. The other
beam from the laser should illuminate the object and reflect off of it so that it interferes with
the reference beam on the film. The path lengths of the reference and object beams should
differ by less that the coherence length of the laser. Why?
Recall that in the derivation above we ignored the contribution of |ψ0 |2 in comparison with
|ψ1 |2 ; that is, we assumed that the reference beam was more intense than the object beam.
However, the derivation did not tell us how much more intense the reference beam should
be. It has been found empirically that a reference/object intensity ratio between 3 and 10
yields good results. A photocell is provided to measure the intensities of the reference and
object beams. There are attenuators that can be placed in the beam line to help obtain the
desired intensity. Higher intensities (consistent with a suitable reference/object ratio), will
need shorter exposure times, and, will, in general, produce better holograms.
Check that the reference beam is smooth across the film. In addition, you should make
sure that the optical surfaces are clean and free of dust. A clean, uniform reference beam is
needed for a high quality hologram. We produce one with a commercial beam expander but
could have also used a simple pinhole. A pinhole acts as a “spatial filter” to assure that you
have clean wavefronts, and makes the system relatively forgiving of dirty or imperfect optics
upstream of the pinhole.
Now you can make some holograms! Instructions for the photography aspects of the
experiment can be found in the darkroom. He is a quick guide to augment the complete
instructions.
1. Always wear a pair of disposable gloves.
2. Mix the chemicals if they do not always exist. Make sure there is enough deionized
water for various rinses. Set up five trays in a row on the stainless bench.
3. In the first tray mix equal parts Developer A and B. The combination should be about
0.7 cm deep.
4. In the second tray put about 1 liter of deionized water. This is a rinse.
5. Into the third tray pour the bleach to a depth of about 0.7 cm deep.
6. In the fourth tray put about 1 liter of deionized water. This is the second rinse.
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7. In the fifth tray put the wetting solution.
8. Set up for the hologram and familiarize yourself with the lab.
9. Put a black cloth over the laser. Float the table by opening the small valve at the base
of the regulator. Don’t forget to close the valve before you leave.
10. In the dark, or with the safelight, remove film from the pack, put it in the glass film
holder, and put that in the holder.
11. Remove the black cloth and expose the film. Experiment with the exposure time.
Something around 1 minute should work. There is a timer in the room.
12. Put the black cloth over the laser. Remove the film and quickly place it into the
developer. Slosh it around for 2 minutes. It should turn black.
13. Next put it in the rinse, gently agitating it for 3 minutes.
14. Next put it in the bleach, again gently agitating it for 2 minutes. The film should now
be clear. You may now turn on the light.
15. Next put it in the next rinse, gently agitating it for 3 minutes.
16. Next put it in the wetting solution for 20 seconds, remove it, and hang it up to dry.
17. To view the hologram, put it back in the holder and illuminate it with the reference
beam. Look through the film to see the image.
After you have made a successful hologram following this prescription, vary the parameters.
You can adjust the reference/object intensity ratio, to see how magical are the limits 3 to 10.
And you can play with the exposure times or try unequal path lengths for the object and
reference beams. Use your imagination to discover which factors are physically important,
and discuss your findings in your write-up. Questions to ponder:
• Where is the conjugate image?
• Does the orientation of the film make a difference?
• If you cut the film in half would you still get the full image?
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4.3
Reflection hologram
Using just the reference beam, set up the configuration shown in Figures 1 and 2. Here the
reference beam reflects off the object and interferes with itself on the film. Coins are good
objects to try because they are fairly reflective. Follow the same steps as above. Here are
some questions to ponder.
• What emulsion thickness is required to make white light holograms?
• What does this specifically mean for our setup? Check to make sure that the photographic plates fulfill these requirements.
• Can white light holograms still be viewed with laser light, and what differences are
there between reflection and transmission holograms when illuminated by laser light?
• When viewed in white light, your hologram will appear in different colors when viewed
at different angles. Why is this?
• When doing white light holography, is it more or less important to have a strong
reference beam?
• How should exposure time be different for white light versus transmission holograms?
There is a bright white desk-top light in the room which may be useful for illuminating the
hologram. You might want to try making a hologram with two object beams or recording
different objects at different angles. What factors are important when using two object
beams? How can you avoid contaminating your beam with a more complicated optical
setup? See what other creative setups you can think of.
The paper by Gabor [4] is the author’s Nobel Prize address. Jeong [6] gives a very helpful
discussion of experimental approaches, and the collection of articles from Scientific American
[5] is also good.
References
[1] M. J. Beesley. Lasers and Their Applications. London: Taylor and Francis, 1971.
[2] M. Born and E. Wolf. Principles of Optics. Pergamon Press, 1980.
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[3] Hecht E. Optics. Addison Wesley, 1974.
[4] D. Gabor. Holography, 1948-1971. Science, 177:299, 1972.
[5] A. L. Schawlow. Lasers and Light. San Francisco: W.H. Freeman and Company, 1969.
[6] Jeong T. Physics Demonstrations. New York: Ronald Press, 1970.
Updated L. Page, 2011
Previous versions by J. H. Taylor (1990), A. Erickcek and Z. Kermish (2002)
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