HolographyLaboratory & Computational Physics 2

Last compiled August 8, 2017

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Contents

1 Introduction 31.1 Prelab questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background theory 42.1 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The mathematics of holograms . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Intensity of a wave . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.2 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.3 Wave interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.4 The transmission function and phase . . . . . . . . . . . . . . . . . 6

2.3 Hologram setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Light from the object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Hologram types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Storing the hologram in the film . . . . . . . . . . . . . . . . . . . . . . . 8

3 Holography - Procedure 93.1 Note: Qualitative results . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 SAFETY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Setting up and taking laser power measurements . . . . . . . . . . . . . . . 103.4 Investigating the inverse square law and exposure times . . . . . . . . . . . 113.5 Improving the clarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Measuring the power at the film holder . . . . . . . . . . . . . . . . . . . . 123.7 Creating your hologram! . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.8 Re-entering the room . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.9 Viewing the hologram outside of the dark room . . . . . . . . . . . . . . . 133.10 Viewing with a different wavelength laser . . . . . . . . . . . . . . . . . . 143.11 Creating a reflection hologram . . . . . . . . . . . . . . . . . . . . . . . . 14

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1 Introduction

The word ‘holography’ likely means a variety of things to various people. Holograms oncredit cards, holographic wrapping paper, robotic characters projecting three-dimensionalimages... Ultimately, the defining feature of any hologram is the ability to display (or recre-ate) phase information. Phase information allows us to observe beyond simple two dimen-sional intensity photographs and have our view of a flat surface change depending on theangle at which we look at it.

This experiment is rare among the labs here in the school, both in that your observations andwrite up will be largely qualitative, but also that you get to take your results home! There areone or two quantitative sections but these aren’t the main focus of the lab.

1.1 Prelab questions

1. At what laser power and classification can a laser cause damage to the eyes? Does itmatter if the laser is visible or invisible, e.g.: infra red?

2. We are using a coherent, monochromatic light source in this experiment. What does‘coherent and monochromatic’ mean? Does sunlight fit this criteria?

3. Make a table listing at least three differences between a photograph and a hologram.

4. If you cut a hologram into smaller pieces, what would be seen in each piece? Do youlose any information?

5. Use the spherical wave intensity (equation 6) with the hologram function (equation 10)to find the spherical wave transmission function. Does this transmission function dif-fer much from the plane wave transmission function? Write down relevant equationsbefore you start.

6. Re-draw figure 1 to show how the equipment should be set up to create a reflectionhologram.

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2 Background theory

Unlike a regular photograph, a hologram can store both the amplitude and phase informa-tion of light from a given object.

The phase is measured and recorded through the interference patterns of two monochromaticand coherent light beams. We are able to then store these interference patterns in holographicfilms. If the film is then viewed using appropriate light, the interference patterns in the filmdiffract the light and reproduce the light initially reflected from the object. This produces athree-dimensional image of the object that can be viewed from a variety of angles.

2.1 Interference

As second year physics students, the idea of interference of light waves shouldn’t be foreignto you. As you will have learnt, light has some frequency, f and corresponding wavelength,λ.

You may also be familiar with the idea that a light source is considered coherent if all ofthe photons over some relevant scale are of the same frequency and their wavelengths are inphase. If coherent sources diffract from or through an object, meaningful diffraction patternscan be observed that relate to the structure of the diffracting object.

In Young’s two slit experiment, a diffraction pattern can be observed on a two dimensionalscreen. Here, using holographic plates, we are able to store the complete diffraction patternfor later reproduction.

2.2 The mathematics of holograms

2.2.1 Intensity of a wave

Before we can discuss concepts like interference or phase information, we need to mathe-matically define electromagnetic waves.

Electromagnetic waves are constructed from transverse electric and magnetic fields; bothare transverse to the direction of propagation and perpendicular to each other. It is veryimportant to note that what we observe as ‘light’ is the intensity I of the wave, not theindividual magnetic or electric components.

The intensity of a wave is defined as the square of the amplitude of the electric field. Startingwith a plane wave of the form

E = E0 cos (k · r− ωt) (1)

The instantaneous intensity at time t is then given by

Iinst = ε0c∣∣E2

∣∣ = ε0cE20 cos

2 (k · r− ωt), (2)

where E0 is the amplitude of the electric field, k is the wave number and ω is the frequency.Normal photography is an example where the time-averaged intensity is recorded. This

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intensity is given by

I = ε0c⟨|E|2

⟩= ε0c

⟨E2

0 cos2 (k · r− ωt)

⟩=ε0cE

20

2(3)

Note that the frequency and wavenumber have disappeared and we are left with a simplescalar, so we are only recording a single aspect of the wave at any given point on the film.

2.2.2 Spherical waves

As we are dealing with spherical waves in this experiment, we should quickly make thenecessary adjustments to the plane wave equations. The initial wave E becomes

E =E0

rcos (kr − ωt). (4)

with instantaneous intensity

Iinst = ε0c∣∣E2

∣∣ = ε0cE2

0

r2cos2 (kr − ωt), (5)

and time averaged intensity

I = ε0cE2

0

2r2. (6)

Essentially, unlike a plane wave, a spherical wave expands radially uniformly, but decreasesin intensity as the square of the distance to the source.

2.2.3 Wave interference

At any point in space or time any two waves E1 and E2 will add vectorially. The resultingintensity pattern will depend on how these vectors add. For E = E1 +E2, the instantaneousintensity is given by

Iinst = ε0c∣∣E2

∣∣ = ε0c∣∣(E1 + E2)

2∣∣

= ε0c(E2

1 + E22 + 2E1 · E2

)=ε0c

r2(E2

1 cos2 (k1r − ω1t) + E2

2 cos2 (k2r − ω2t)

+ 2E1E2 cos2 (k1r − ω1t) cos

2 (k2r − ω2t) cos θ12)

(7)

for the case of two planes. The angle θ12 is the relative phase between the two waves.

It is this, the relative phase, that we are interested in storing for later reproduction.

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2.2.4 The transmission function and phase

We can define the transmission function of our holographic film as

t(x, y) = t0 + βI (8)

= t0 +βε0c

r2∣∣E2

1 + E22 + 2E1 · E2

∣∣ , (9)

where the terms t0 and β are somewhat equivalent to the post-development light amplitudeand the recorded phase, respectively. These terms also depend on the properties of the un-exposed film and the developing process, as a poorly developed film will hold no phaseinformation.

We can write, from the transmission function, the hologram function:

Eholo = t(x, y)Eref (10)= (t0 + βI)Eref

Where Eref is the beam used to re-create the hologram after exposure. While the phaseinformation of the hologram is provided by the I term, the amplitude term t0 only dependson the reference (sometimes called viewing or imaging) beam.

By substituting in equation 3 for the intensity I , we arrive at:

Eholo ={t0 + βε0c

(E2

1 + E22 + 2E1 · E2

)}E1

= t0E1 + βε0c{E2

1 cos2 (k1x+ ωt) + E2

2 cos2 (k2x+ ωt)

+ 2E1E2 cos (k1x+ ωt) cos (k2x+ ωt) cos θ12}E1 cos (k1x+ ωt) . (11)

This is a rather complicated equation1 but it illustrates how the relative phase is recovered.

1Refer to the prelab questions, you will substitute in the spherical intensity yourself.

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2.3 Hologram setup

LASER

Planemirror

Planemirror

Sphericalmirror

Sphericalmirror

Beamsplitter

Film

Object

Reference beam

Object beam

Figure 1: Experimental set-up for the transmission hologram.

The figure above shows one setup possible to make a transmission hologram. Note that thelight from the reference beam is passing through the film to the observer. This is shown tohelp understand how light is incident on the object and how it reflects.

2.4 Light from the object

Film

Object

Incidentlight

Reflectedlight

P

X

(a) Light ray diagram

Film

Object

IncidentlightReflected

light

P

(b) Close up with more accurate diffracted light

Figure 2: Diagrams showing light reflecting from object, with b) showing what actually hap-pens at each point on the object.

It’s important to make several points about the light coming from the object.

Firstly, it should be clear that only sections of the object that are visible from the locationof the film will be imaged. In figure 2a), no light from point X will reach the film, becauseof the point relative to the angle of the film. However, for all the other points on the object

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facing the film, figure 2a) is a simplification. Figure 2b) shows what is happening at eachpoint, where the light from every point on the object diffracts to every point on the film2

This means every (film-facing) point of the object is stored from every (film-facing) angle.But if we just recorded the intensity of every point on the object, we’d end up with a feature-less blur. We need then to know how the intensity is modulated depending on the viewingangle. This is where the phase information is important. We first need to discuss the twotypes of holograms you can make today.

2.5 Hologram types

The two main types of hologram are transmission and reflection holograms. You can choosewhich type you’d like to make depending on your setup and results. Both types of holo-grams require a viewing beam that is coherent and monochromatic, with the viewing beamat roughly the same angle to the initial reference beam.

For transmission holograms, light is shone on the rear of a plate and the observer sees thethree-dimensional image on the opposite (transmitted) side as a virtual image. Reflectionholograms use light reflected back from the interference pattern, so the viewing beam is onthe same side as the observer.

2.6 Storing the hologram in the film

Okay, so we have some idea from the maths how to describe the transmission function of thehologram, and we know that light diffracts from the object over the entire film, but what isactually being stored in the film? How can a two-dimensional film store a three-dimensionalimage?

The film itself stores the information as planes of interference fringes. Figure 3 shows howthe interference planes are stored in the two different types of holograms. The differentrefractive indices of the fringes shift the reference (viewing) beam across the different layersand reproduce an image of the original object.

The interference fringes are formed in the film as the laser provides energy to set the holo-graphic film. During development it’s important to keep the setup as motionless as possible.Any movement will result in new sets of interference fringes developing, resulting in a red-dish blur in the film.

2Note that the light actually diffracts beyond the film, too, to all angles,, but the figure is relevant for ourpurposes.

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θ1

θ1

FilmObjectbeam

Referencebeam

Diffractedreference beam

Interferencefringes

Observer

(a) Transmission hologram.

θ2 θ2

Film

Objectbeam

Referencebeam

Diffractedreference beam

Interferencefringes

Observer

(b) Reflection hologram.

Figure 3: Hologram interference fringes and viewing angles.

3 Holography - Procedure

3.1 Note: Qualitative results

Unlike other experiments in this subject, this majority of this experiment requires you tobe largely qualitative. This means you aren’t searching for a specific number or physicalconstant, but that you are looking to describe the qualities of your holograms. Your reportshould largely be about the conditions you chose to make the hologram, and how this helpedyou understand more about holograms, light, and optics.

Some hints for good directions for this report:

• Before beginning, look over the setup and consider what experimental parameters youare able to change. What factors do you think are important?

• You will be given a maximum of 4 (four!) holographic plates. Carefully consider eachand every plate you use.

• Make sure to vary only ONE condition at a time. If you vary more than one parameterat a time you won’t be able to make decisive observations about the effects.

• You should at some point have a table with a list of all the parameters you used foreach hologram. You want to try to find the combination of parameters that producesthe clearest, sharpest (or most interesting!) hologram. Include your own DRAWINGSof the holograms to better describe what exactly you see.

• Ask yourself: What have I always wanted to know about holograms? What parametersare important?

3.2 SAFETY

Throughout this experiment you will be using a (relatively) high-powered laser beam. It’sextremely important you take several steps to ensure the safety of your eyes. Your demon-strator will show you how to engage the interlock and laser, but there are other points toalways keep in mind:

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• First and foremost, NEVER EVER lower your head to the height of the laser path.There are always reflections and unpredictable light behaviours!

• Wear the green-tinted safety goggles when the laser is on and when before you re-enterthe room.

• When aligning the spherical mirror, look AWAY from the mirror. Follow where thelaser ends up on the wall.

• Make sure the beam path is clear before turning the laser on, and remove any reflectivejewellery.

• Under no circumstances try to look at the laser box or behind the shield in frontof it.

• Under absolutely no circumstances disable the interlock.

• Be careful in low-light conditions. Your pupils are more open and will take in morelight, and you might bump in to things!

3.3 Setting up and taking laser power measurements

Note that you shouldn’t take any holographic films out of their boxes until later. For nowyou’ll be investigating the system.

1. Identify all of the necessary elements, including the various interlocks and switches.Safety first!

2. IMPORTANT: Wear safety glasses!

3. Check the optical table is clear of obstructions and turn the laser on as follows:

a) Close the door!

b) Turn on both the laser power and interlock at the wall.

c) Switch the interlock on FIRST and only then turn the laser on.

d) On the interlock, press the ‘RESET’ button.

e) Now, if you flick the shutter control (light switch connected to wires), the laserwill come on.

f) If the laser doesn’t appear, consult your demonstrator.

g) When developing, make sure to press the button at the door to give you 10 sec-onds to open the door and leave the room.

Question 1 Find a sheet of blank white paper and use it to follow the beam path. Whathappens to the beam at various points along the path? Are the beams identical after thebeamsplitter? What does the beam look like after hitting one of the spherical mirrors?

4. Position the object and angle the mirrors and film appropriately, given that:

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(a) The spherical mirror and filmstand

(b) The beamsplitter

Figure 4: The spherical mirror and beamsplitter.

• The beam should be centered on the object and appear ‘clean’ (see figure 4b).

• Any movement of the setup during the reading will ruin the hologram. Is every-thing fixed in place?

5. Without the holographic plate in place, turn on the laser and note any sources of vibra-tion or objects that could ruin your exposure. How you could minimise motion?

6. Take a few measurements (carefully!) of the beam power along the beam path.

Question 2 Compare these power values to those you researched in the prelabs. Where isthe laser most dangerous?

3.4 Investigating the inverse square law and exposure times

Question 3 How will you define R, the distance from the spherical mirror to the object, toensure your measurements are consistent? Take a series of measurements of the laser poweralong the path of the cone of laser light.

Plot the laser power versus distance from the centre of the spherical mirror.

Question 4 Does the relationship between laser power and distance follow the inverse squarelaw? If not, explain what factors could be influencing this.

The quoted exposure for the holographic film is:

• 20 mJ/cm2 at 635 nm

• 30 mJ/cm2 at 532 nm

• 50 mJ/cm2 at 450 nm

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Question 5 Can you determine roughly the minimum exposure time for these films, basedon the power readings you’ve taken, and considering you will be using both the object andreference beam?

Question 6 What are the advantages of positioning the object and plate further away fromthe laser, despite the longer exposure time?

3.5 Improving the clarity

Sphericalmirror

Film

Object

Laserblocks

Collimator

Figure 5: Isolating the laser beams to the film and object. Compare to relevant earlier setupfigure.

One additional step in creating bright, sharp holograms is to limit the ambient or ‘cross-over’light from both of the laser beams. By limiting this ambient light, we will reduce the ‘noise’in the image and only light involved in creating the interference pattern will be incident onthe holographic film.

There should be equipment available to isolate the laser beams as in the figure above. Thelaser stop to the left of the object isn’t as important, but you might find it frames the back-ground well.

Question 7 Draw a diagram showing how exactly you isolated the laser beams on the objectand film. Does the view from the film stand appear brighter, or perhaps has more contrast?

3.6 Measuring the power at the film holder

The power of both of the beams is quite important. You should adjust your setup until thereference beam is more than 3 times but less than 10 times the power of the object beam, atthe film holder.

Question 8 How will you measure the power of the beams individually? What is your ratio?Will you adjust it at all throughout the experiment?

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3.7 Creating your hologram!

The box of holographic plates should only be opened in low light conditions. They consistof a holographic film stuck on to a piece of glass. Before they are exposed, they should beblue (though you shouldn’t really check them before exposure...) After exposure they shouldbe red.

Take care handling your plates and only handle the plates by the edges. Do not touch thecentre as this will ruin the area for the hologram.

When placing a plate in the stand, write down which way around you have your plate relativeto the beam. You can feel the film if you run your finger on the edge of the glass, it shouldstick out slightly.

When you think you’re ready, consult with your demonstrator before exposing your firsthologram. When ready, turn off the lights and flick the shutter switch. Quietly and care-fully exit the room. Don’t slam the door or nudge any tables as this could cause vibrations.While you wait outside for the hologram to expose (ten minutes at least) you should continuediscussing with your partner and writing up outside.

Question 9 Which objects did you choose? Did you place more than one object at a time?

3.8 Re-entering the room

• Put on the safety glasses, turn the key outside the door and enter with the laser on.

• double check that the laser hasn’t drifted and that your objects are still in place.

• with the laser on, remove the object and you should see your hologram!

Remember: you can only make four holograms. Repeat the process and discuss the differ-ences in your holgrams relative to their exposure conditions, relating it back to what youknow about the physics of holograms. In particular discuss the sharpness and contrast ofyour images.

3.9 Viewing the hologram outside of the dark room

As well as the powerful laser, you can use something that emits at a single wavelength toview your holograms. Luckily, the LED on the bottom of an optical computer mouse servesthis purpose nicely. Remove your hologram from its stand and use a mouse LED to view thehologram. Remember the angle of incidence is important!

Question 10 Discuss the focus and range of the hologram. Does your object appear three-dimensional object from only one angle, or can you move the plate and see more detail?

As your light source is no longer fixed, you should be able to comment on:

• the importance of the laser angle,

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• whether the object moves in or out of focus as you move the light source,

• more detailed comparisons between holograms (as you can easily swap between mul-tiple plates)

• whether any hologram can be viewed with the light source reflecting from (rather thantransmitting through) the film.

Question 11 Do you see any evidence of your plate functioning as a reflection hologram?Is the image any different?

Question 12 What do you think were the most important factors in making a clear and sharphologram? What advice would you give to someone just starting this experiment?

As you now have a method of viewing your holograms without a laser, decide betweenyourselves which you’d like to take home. Feel free to wrap them in the tissues provided forsafe transport.

3.10 Viewing with a different wavelength laser

Additionally, for the film you will use, a different wavelength viewing beam can cause theimage to vary in size. You will be able to investigate this using a green laser.

There should be a laser pointer available that can emit both red and green monochromaticlight.

Question 13 Look at the laser type and power and decide what safety precautions you needto take.

Using equipment available in the dark room, align the green laser so it diffracts from one thespherical mirrors used for the reference beam.

Can you see your hologram? Is it any different in size?

Explain your observations. Do you notice any other differences between the red and greenwavelength holograms?

3.11 Creating a reflection hologram

If you still want to go further, try making a reflection hologram. Follow as much or as little(but including safety!) of the transmission hologram instructions as you like.

Comment on the differences you noticed between the two hologram types, and whether theinterference fringe directions are noticeable in any way.

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