Holography (A13) Christopher Bronner, Frank Essenberger Freie Universität Berlin Tutor: Dr. Fidder July 1, 2007 Experiment on July 2, 2007 1 Preparation 1.1 Normal camera If we take a picture with a camera, a photo film is exposed to the scattered light of the object. The optical density ρ of the film is proportional to the intensity I of the incoming light (during the exposure time t e ) and the intensity is defined by ρ(x 0, y) te 0 dt I(x 0, y, t) I = E(x o, y, t) 2 = 1 2 E o e +ikxxo+ikyy + c.c. 2 = E 2 0 Therefore, all information about the phase Φ(x 0, y) = iωt + ik x x 0 + ik y y is lost. This is not the case for the procedure of wave-front reconstruction (or holography) developed by Gabor 1947. 1.2 Holography 1.2.1 Recording the hologram Figure 1: Picture of a normal camera We have now a special film (see Sect. 1.3) which is penetrable for the incoming light. Its optical transmissivity τ (like ρ) is proportional to the incoming light. τ(x 0, y) te 0 dt I(x 0, y, t) I(x 0, y). 1
We asume, for the last proportionality, that 1 ω t e. Also, the direction of the beam is changed as seen in figure 2. It is also importent to have a coherent light source. The polarisition should be perpendicular to the optical plane (table). Figure 2: Recording the hologram The beam, which is directed to the film by a mirror, is called reference beam E r (x, y, t) and the other one is refered to as object beam E 0 (x, y, t). Thus the total intensity is I = e iωt 2 E r e i k r r + E 0 e i k 0 r 2 = E 2 r + E 2 0 + E r E 0 ( e i( k r k 0) r + e i( k 0 k r) r ) 1.2.2 Reconstruction of the hologram If the taken hologram is illuminated with a single beam U 0 similar to the reference beam we get a transmitted beam U T. For this beam it is U T τ U 0 I U 0. If we take U 0 (x, y, t) = E r (x, y, t) we find: U T = (Er 2 + E0 2 + E r E ( )) 0 e i( k r k 0) r + e i( k 0 k r) r Er e i k r r Where the terms are: = E r (Er 2 + E0)e 2 i k r r + E }{{} r E 0 e i(2 k r k 0) r + E }{{} r E 0 e i k 0 r. }{{} =T erm 1 =T erm 2 =T erm 3 Term 1: These two parts are similar to the reconstruction beam with a changed amplitude. Term 2: This part disturbes the recreation of the hologram and creates the real image. Term 3: This part is the recreated scattered wave from the object. image. This creates the virtual The holography acts like a a diffraction grating. It produces a direkt beam an two first order diffracted beams. 2
Figure 3: reading out the holography 1.2.3 Amplitude and phase holograms There are two different types of holograms. In general, the transmissivity of the holographic film is a function of both amplitude and phase. τ(x, y) = A(x, y)e iφ(x,y) In case of a phase hologram, the amplitude A is considered constant, for an amplitude hologram, φ = const. Since both amplitude and phase are approximable with a linear function of the intensity I, one obtains the transmissivity as τ p A + Aiφ τ a (a bit e ) e iφ for phase and amplitude holograms, respectively. In any case, the transmissivity is proportional to the intensity. When a holographic film is developed, the areas which were exposed less than the rest are darker and one has an amplitude hologram. By bleaching the film one can transform the amplitude hologram to a phase hologram which is advantageous in case of low intensity. 1.3 Material of the film Normal photographic layers can be used. They consist of a silver halide, which is embedded in gelatine. The silver halide crystals are very small, so the resolution of the film is very high. If a photon hits such a small crystal, the crystal is split into its components and the silver atom remains in the film as a black point. For developing the picture a fixer is needed, which dissolves the unreacted crystals, and only the stable exposed elementary argent particals remain in the film. 1.4 Transmission and reflection holography Basically, there are two different ways of recording a hologram. The first is to illuminate the photographic plate with reference and object beam from the same side as shown in fig. 4. 3
Figure 4: Recording a transmission hologram [1] The second possibility is to have the object beam hit the plate from behind (which means, from the opposite side of the reference beam). The simplest way is shown in figure 5. The reference beam illuminates the object to be recorded, which is placed behind the plate, and the scattering light acts as the object beam. This method allows one to fix the object to the plate and therefore eliminate vibrational disturbances. Figure 5: Simple way of recording a reflection hologram [1] An alternative method for creating a reflection hologram is shown in figure 6. Here, the object is illuminated by a split-off part of the reference beam. This is useful, if the photographic plate has bad transmission properties or if the object is not reflective enough. 4
Figure 6: Alternative for recording a reflection hologram [1] 1.5 Michelson interferometer In a michelson interferometer, the beam of light emitted in the source is split up by a beamsplitter in the center of the apparatus. the two perpendicularly directed beams propagate on and are reflected by a mirror. the beamsplitter reunites (parts of) them again and the superposition of both can be detected. If the lengths of the two arms of the interferometer are exactly equal, constructive interference happens. This is also the case if the length difference is a multiple of the light s wavelength. On the other hand, if the difference is half a wavelength or an odd multiple of λ 2, destructive interference will occur and no light is seen on the detector. Since only coherent light is capable of interference, the coherence length can be determined by this type of interferometer by increasing the arm lengths until no more interference effect are obeserved. Figure 7: Michelson interferometer 5
1.6 Building a lattice With the apparatus for making a hologram, we are also able to produce an optical grating. We must only remove the object out of the beam (see figure 8). So the film is getting black only where we have interference maxima. Figure 8: Creation of a lattice We immediately get the grid constant of the produced grid as 2 Experiment and analysis 2.1 Michelson interferometer d = λ sin(θ). Figure 9: Picture of the Michelson interferometer At the beginning of the experiment, we set up a Michelson interferometer as shown in fig. 11. All the experiments were conducted on an optical table and inside a wooden case to avoid light disturbing. The laser beam of a HeNe laser (633 nm) was guided through a gate which was connected to a remote control. The gate could be opened continuously or only for a specified time. An aperture 6
widened up the laser beam which was then collimated by a lens with focus on the aperture. Diffraction rings were visible on a screen behind the aperture and we used an iris to filter out all but the first order. From the lens, the collimated beam was directed onto the interferometer s beamsplitter. After quite some adjustment we were able to obtain an interference pattern on a screen positioned in the outgoing beam of the interferometer. The lengths of the interferometer s arms in this situation were a = 17 cm and b = 15cm, respectively. An error of 0.5 cm is assumed for these measurements since the beamsplitter was held in a relatively wide fixture. 2.2 Stability of the setup Figure 10: Interference pattern Figure 11: Setup for the stability test with a Michelson interferometer 7
In order to investigate the stability of the setup we replaced the screen by a mirror which reflected the outgoing beam to a photo diode. Its signal was recorded versus time on a chart recorder. We conducted a short time (6 min) and a long time measurement (36 min). The respective charts are shown below. Figure 12: Short time measurement Figure 13: Long time measurement During the short time measurement we tested the inluence of some actions on the interference pattern, e.g. touching the desk of the setup, hitting it (slightly), jumping and screaming around. We noticed that perturbations to the desk had a much larger effect than anything else. We expected this result since the desk is supported in a dampening way. For the long time measurement we left the room but obtained a small peak for every time the door was opened. We obtained a rather stable result except for the continuous increase in the beginning. 8
As we were told, an oscillation with a period of several minutes should have been obtained. Altough we haven t seen it, we will address this issue in the discussion. 2.3 Coherence length Now, we replaced the miror and the diode by a screen again. Changing the length of the b arm, we determined the coherence length of the laser. We increased the length in steps of a few centimeters until no more interference pattern was obtained. We observed that the pattern did not vanish suddenly but its contrast decreased continuously over a distance of a few centimeters. The length at which we couldn t see any more interference, was measured to be b = (33.0 ± 0.5) cm. The coherence length is now calculated from the difference of the two possible ways length which is just the difference of b and a, multiplied by two since the way was gone twice by the beam. The coherence length is therefore 2.4 Recording the holograms l c = (32 ± 1) cm. We recorded four holograms using the simple reflection method as shown in fig. 5. During the whole recording procedure, every light source (except for the laser) was turned off. The shutter gate of the laser was closed and opened only for the recording via the remote control for the exposure times of 2 s and 4 s, respectively. The object to be recorded was placed behind the holographic film which was mounted in a double glass fixture under an angle of about 45. The exact angle is not relevant but the film shouldn t be perpendicular to the beam plane because then both the real and virtual image would overlap. Before recording the holograms we waited 5 minutes every time to have the table stable and not oscillating. Figure 14: Object in the fixture illuminated with the laser beam 2.5 Developing the holograms The holographic films were developed in complete darkness by putting them in a bath of a development solution for 2 minutes first, then secondly in water for a moment and finally in sulfur acid. After this 9
procedure the holograms are hung up in an oven for drying. In the development solution, the hologram is fixed by dissolving the remaining silver halides. The sulfur acid is used for bleaching of the initial amplitude hologram in order to convert it into a phase hologram. Unfortunately, during the development in the dark, we grabbed two films at once and thus they were not developed in the spot where the gripper touched them. 3 Discussion The holograms we made are quite satisfying as the objects can clearly be seen in all four cases. With the shorter exposure time the holograms are sharper but slightly darker, as expected. The oscillation in the long time measurement is due to the dampened table. These tables are constructed in a way so that their resonance frequency is very low. Therefore by the usual desturbances they cannot be perturbed too much. We presume that an oscillation with a period of a few minutes corresponds to that low frequency of the table. As for the measurement of the coherence length we have calculated an error of 1 cm. This error is only due to the measuring with our tape and does not consider the fact that for every length we had to find the interference pattern again by adjusting the mirrors. Since these adjustments were quite difficult it might be that we simply haven t found an existing interference pattern for larger lengths. References [1] P. Hariharan: Optical holography. Principles, techniques and applications. Cambridge University Press, Cambridge 1991. 10