Infra-red polarization and Physics - MSCV/VIBOT

Improving software for computer generatedholography for 3D object display

Daniel Martınezdani89mc@gmail.com

Sergi Valverdesergivalv@gmail.com

Bernat Pelachbernatps@gmail.com

January 14, 2012

1 Introduction

Displaying three dimension objects out of acomputer screen is not a trivial task. Thereis a lot of research regarding this topic nowa-days. Holography is an optics technique that isemployed to display objects or scenes in threedimensions (3D). Those 3D images, or holo-grams, can be seen with the unassisted eyeand are very similar to how humans see theactual environment surrounding them. Theholograms were introduced in 1948 by the Hun-garian physicist Dennis Gabor [1] as an unex-pected result from his work improving the elec-tron microscope. Roughly speaking, an holo-gram can be seen as a reconstruction of a pre-viously recorded light field when this is notpresent anymore.

An hologram is recorded into a supportingmedium via the intersection and interferenceof two light distributions. Basically, as it canbe shown in Figure 1 given a light field illu-minating an object for which we want to com-pute the hologram, it is split in two identicallight sources using a beamsplitter. The reflec-tion of the illumination beam from the objectplus the direct incidence of the second lightsource beam form a pattern of intersected andinterfered beams which is recorded in the pho-tographic plate. The interference pattern can

Figure 1: Optical arrangement to record an holo-gram. The light beam is split in two identical lightsources and then the hologram is encoded as theintersection and interference of the sources into thephotographic plate

be said to be an encoded version of the scenewhich requires the original light source in orderto view its contents. On the other hand, Fig-ure 2 depicts the process of retrieving the vir-tual object: when the original reference beamilluminates the hologram, it is diffracted bythe recorded hologram to produce a light fieldwhich is identical to the light field which wasoriginally reflected by the object into the holo-gram. The Gerchberg-Saxton (GS) iterativeprocedure estimates the phase of a pair of lightdistributions via a computation of the Fouriertransform, if their intensities at their respec-

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Figure 2: Optical arrangement to retrieve the vir-tual object. If the original light beam is applied tothe encoded hologram, it is diffracted by it produc-ing a light field which is identical to reflected oneby the object.

tive optical planes are known. In the practice,this procedure becomes an efficient methodol-ogy to estimate the hologram of a light distri-bution or an object. Moreover, the algorithmhas been shown to be efficient in the recon-struction of 2D [3] and 3D [4] dimensions ob-jects holograms. This report implements in thecomputer language Matlab the 3D GS algo-rithm based on the work done in [4] and [2].

This report is structured as follows: section2 introduces in detail the Gerchberg-Saxton al-gorithm in both 2D and 3D while section 3 dis-cusses the results obtained both by computersimulation and in the optics lab. Finally, weend up the report with a conclusions section.The implemented code in Matlab is attachedwith this document.

2 Algorithm

In the introduction section we have shown howto obtain an hologram via optical devices. This

means that the phase image which refers tothe hologram is obtained directly from the realworld. In here, we want to go a little bit fur-ther introducing an algorithm that allows toestimate this phase image from a given Tar-get without any kind of optical devices. Thismethod is known as the Gerchberg-Saxton al-gorithm.

2.1 2D case

Given a 2D image such as the one from Fig-ure 3, the Gerchberg-Saxton algorithm (GS)is able to estimate its hologram by estimatingits phase image. In fact, the GS algorithm isdefined as an iterative algorithm for retrievingthe phase of a pair of light distributions re-lated via a propagating function (such as theFourier transform) if their intensities at theirrespective optical planes are known.

Figure 4 shows the block diagram of the al-gorithm. There are two main elements: theTarget image A which is the input 2D intensityimage and the Hologram B which is estimatedthrough the iterations.At the beginning we initialize the hologramwith random values. Then, we combine thisrandom phase (hologram) with the constantmagnitude from C (in our case fixed to 1). Af-ter that, the Fourier Transform of this com-bined information is computed in E and thephase is keep for the following step. At thatpoint, the obtained phase is combined with themagnitude of the original Target image A andthe Inverse Fourier Transform is carried out inF. The phase of this result is the updated ver-sion of the hologram.By repeating the process, the hologram is im-proved through the iterations and the finalphase image that represents the hologram isachieved at the end of the process. Only themeaning of the image D is missing which isthe resultant magnitude after computing theFourier Transform. This is nothing else than

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the approximation of the Target intensity. Inother words, is the approximation of how thehologram will look like after being projected inthe real world.

Figure 3: Examples of 2D images

Figure 4: Gerchberg-Saxton algorithm block dia-gram

3 3D case

When extending the GS algorithm into the 3Dcase, the main problem is that to project thehologram as it was done in 2D case it is neededa 2D phase. However, the 3D Fourier trans-form gives the results in a 3D space so it neededto reduces the dimensionality.

3.1 Ewald’s sphere

In 2003, Shabtay [2] pointed out that Ewald’ssphere could be done to reduce the dimen-

Figure 5: 3D GS algorithm block diagram

sion of the hologram. Ewald’s sphere can beuse to express the differential and wave equa-tion. In our case, it was shown that for a 3Dobject measured with a beam light of wave-length λ, its 3D Fourier transform should bezero everywhere except the sphere that satis-fies k2

0 = k2x + k2

y + k2z , where k0 = 2π

λ . Thus,by projecting this surface in a 2D plane we canget a 2D hologram.

It is important to say that Ewald’s surfaceonly will keep the information seen for the laserbeam, which is the one in the z direction. Sothat we only need to keep the hemisphere withpositive z of the Ewald’s sphere. Due to thisphysical constraint in this case the hologramwill be in the Fourier space (in 2D case thehologram is done in the inverse Fourier space).

3.2 Algorithm

Thus the GS algorithm for the 3D case willstart be taking the 3D space informationud(x, y, z) and applying the Fourier transformto get ud(νx, νy, νz). Then the all the informa-tion that is not in the Ewald’s sphere, S, willbe set to 0:

uc(νx, νy, νz) =

{uc(νx, νy, νz), (νx, νy, νz) ∈ S0, (νx, νy, νz) /∈ S

Then the inverse of the Fourier transformwill be applied to uc(νx, νy, νz). Only the phaseof the results will be kept. The magnitude

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Figure 6: 3D GS algorithm block diagram withEwald’s surface

obtained from the acquisition and the phaseobtain in this iteration will be combined andused as an input for the next iteration. For thefirst iteration we have used a random phase.

At each iteration we can see how the itera-tion evolves by projecting the sphere in a plane.In Figure 6, it is shown the algorithm.

4 Results

4.1 Lab Setup

Once the implementation of the algorithmis done, it is needed to test if the resultsthat we achieve are the expected ones or not.Figure 7 shows the SETUP that we used. Itis composed by mainly three components: acomputer, a Spatial light modulator (SLM)and a laser beam. The idea in here is to loadthe phase hologram obtained after running theimplemented code and send this phase imageto the SLM. This device is a digital devicethat models the light from the projected laserbeam encoding some information on it. TheSLM is a digital device that is lighted by aplane wave (homogeneous intensity on all thebeam) and it modulates the wave encodingsome information on it. In fact each pixel actas a retarder element. The SLM is betweentwo polarizers that tune the final configurationof the SLM in order to have either amplitude

(a) Components (b) Laser beamprojection

Figure 7: Lab setup

or phase modulations only. After the SLMencodes the phase information, the desiredhologram becomes in the resultant projectionon the final white plane illustrated in Figure 7(b).

4.2 2D case

The results from Figure 8 and 9 proves thewell-functioning of the implemented algorithm.From the two inputs images (a) we estimatetheir respective phase images (b) that will pro-duce the hologram as shown in (d). Note thatthe projected hologram is similar to the ap-proximation done in (c).

4.3 3D case

For the 3D case experiments, we could not dothe acquisition with a laser beam so we used avirtual image. To find the radius of the Ewald’ssphere we did it experimentally because as themodeled object was a virtual object we couldnot use the wavelength λ. We used a thicknessof 0.1 for the Ewald sphere.

The virtual object was a 3D representationof the face used in the 2D case along the z di-rection. Along the x direction we put an otherimage but as it was discussed in section 3.1,only the one in the z direction can be seen in

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Figure 10: Evolution of the hologram during 15 iterations

(a) Target image (b) Hologram

(c) Approximation toTarget intensity

(d) Projected hologram

Figure 8: Happy-face result for the 2D case

(a) Target image (b) Hologram

(c) Approximation toTarget intensity

(d) Projected hologram

Figure 9: Vibot logo result for the 2D case

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Figure 11: Final simulated 3D Hologram approx-imation

the hologram. In order to see how the holo-gram evolves during the iteration, at each stepwe projected the Ewald’s surface and we calcu-lated the approximation of the target intensityin the same way we did for the 2D case. Theresults are shown in Figure 10 and the the ap-proximation of the final hologram is shown inFigure 11. The final result could not be test inthe lab due to time constrain.

5 Conclusions

During this project we have seen that holo-gram have a big potential in many applicationsand it would probably became more importantin few years.

We have seen that for the 2D case theGerchberg-Saxton algorithm achieves verygood results only using the an virtual imageso it can be very useful.

However, for the 3D case the results ob-tained are not as good. In order to see thereal potential of the GS algorithm in 3D willbe needed to test it acquiring the real imagewith a laser beam.

References

[1] D. Gabor. Microscopy by ReconstructedWave-Fronts. Royal Society of London Pro-ceedings Series A, 197:454–487, July 1949.

[2] Gal and Shabtay. Three-dimensional beamforming and ewald’s surfaces. Optics Com-munications, 226(16):33 – 37, 2003.

[3] R. W. Gerchberg and W. O. Saxton. Apractical algorithm for the determinationof phase from image and diffraction planepictures. Optik, 35:237–246, 1972.

[4] G Whyte, Johannes Courtial, andG Whyte. Experimental demonstra-tion of holographic three-dimensional lightshaping using a gerchberg-saxton algo-rithm. NEW JOURNAL OF PHYSICS, 7,2004.

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