Experiment 17. Group 728. Advanced Laboratory in Physics. (Holography) Author: Kay Jahnke Mail: Signature:...

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1 Experiment 17 (Holography) Group 728 Advanced Laboratory in Physics Author: Kay Jahnke Mail: Signature:... Author: Matthias Hocker Mail: Signature:... Adviser: Fengzhen Zhang Date of the experiment: Date of the disposal:

2 Contents 1 Theoretical Basics Maxwell Equations The wave character of the light Plane waves spherical waves Bessel waves Interference and Intensity Coherence Time Coherence Spacial Coherence Laser Holography Double-Exposure Procedure Conguration Digital In-line-Holography O-axis-Holography Results Digital In-line-Holography Hair TEM nets AFM-Cantilever O-axis holography Deformation of a sheet metal Rotation of a metal plate Appraisal of results In-line holography O-axis holography References 16 2

3 1 Theoretical Basics 1.1 Maxwell Equations In this experiment we produce Holograms using coherent light. But rst we have to make clear, what light is: Light has both a particle as well as a wave character. In our experiment we use light to do some interference eects so only the waves are important for us. Thats the reason, why we discuss only waves in this report. The light is an electromagnetic wave that is determined by the Maxwell Equations. This equations look the following way: div D = ρ (1) div B = 0 (2) rote = B t roth = j + D t With this four equations you can describe the behavior of every magnetic or electric eld. The rst equation says, that an electric eld is produced by charges. The second equation makes the statement there are no sources and drains in a magnetic eld, so all magnetic eld lines are closed. The statement of the third equation is that an changing magnetic eld produces a magnetic invertebrate eld. In the last equation you can see that a oating current or a change in the electric eld causes an electric invertebrate eld. To have a context between the Maxwell Equations you need the two material equations: (3) (4) D = ɛ 0 ɛr E (5) B = µ 0 µr H (6) ɛ r and µ r are tensors whose appearance depends on the material the magnetic and electric elds are in. In pure vacuum this tensors are the unity matrix. To make it easier to solve the equations the material equations are often projected in one direction, so all the components become the character of a scalar. 1.2 The wave character of the light Now we have some equations that determine the behavior of magnetic and electric elds in general, but how can we evaluate the propagation of the light waves from that? You can get a meaningful dierential equation if you use the rot operation once more on equation (4) and say that you are in vacuum with no current at all: rot(rote) = rot B t = t rot B = 1 D µ 0 t t = 1 2 E µ 0 ɛ 0 t 2 (7) 3

4 The rot(rot E) can be written as E because rot rot = grad div and dive = 0 in pure vacuum. In the End you get the equation: E = 1 2 E µ 0 ɛ 0 t 2 = 1 c 2 2 E t 2 (8) This is the dierential equation which has many dierent possible solutions. Here we will present the three most important kinds of solutions: Plane waves The easiest kind of solutions are the plane wave which can be described in the following way: E( r, t) = E 0 exp(i( k r ± ωt)) k r = const (9) This waves are not localized in space and propagate into the direction k with the velocity c = ω k. The wave fronts of the wave are planes if you look at them at a constant time. That's why they are called plane waves. The waves have a minus in the exponent if the wave is propagating in the direction of the wave vector k and they have a plus if the are propagating against the direction of k spherical waves Apart from Cartesian coordinates their are other coordinates systems. An important one are the spherical coordinates, because they t perfectly into the Huygens principle. This principle says that you can construct every wave by starting new spherical waves on the whole wave front. The transformation from Cartesian coordinates to spherical coordinates is: x = r sin θ cos φ (10) y = r sin θ sin φ (11) z = r cos θ (12) The spherical waves have a spherical symetrie, so the propagation in every direction is the same and you only need to evaluate the part of the equation which depends on the distance from the source of the wave. When you put the transformation on equation (8) you get: r 2 (r E(r, t)) = 1 c 2 (r E(r, t)) (13) t2 The solution of that is: E(r, t) = E 0 exp(i(k r ωt)) (14) r with the relation c = k/ω. As you can see the amplitude of this wave is E 0 /r. This means that the wave gets weaker as the distance from the source becomes bigger. 4

5 1.2.3 Bessel waves The third form of waves we want to discuss here are Bessel waves. This waves have a certain characteristic: they can propagate without diraction. That means that intensity distribution in the direction of their propagation is independent from the intensity distribution in the plane upright to the direction of the propagation. The easiest solution of these waves have the form: E(r, t) = E 0 J 0 (αρ) exp(i(β z ωt)) (15) with the parameters: β 2 = k 2 α, β 0, 0 < α k = 2π/λ, ρ 2 = x 2 + y 2, r = (x, y, z) J 0 is the Bessel equation to the index 0. In general Bessel equations are dened in the following way: ( 1) r ( ) x 2r+n 2 J n (x) = (16) Γ(n + r + 1)r! while the Bessel equation to the index 0 is relatively easy dened as: r=0 J 0 (x) = 1 π π 0 e i x sin ξ dξ (17) The Bessel wave transforms itself into a plane wave, if you set the parameter α = Interference and Intensity Until now we only had a look on the electric eld of the light wave but our eye and the common detectors can not detect this fast oscillating eld. What we see is the intensity of the light. In general the intensity of any wave is dened as energy per area per time. For the light we dene the intensity as: I( r, t) = E( r, t) E( r, t) = E( r, t) 2 (18) This intensity is time dependent because the electric eld is oscillating. But this oscillation is so fast that our eyes and the most of the detector can not resolve it. We can only see the time average of the intensity which can be written as: I( r) = 1 T T/2 T/2 E( r, t 2 dt (19) T is the duration the detector or the eye needs to register the intensity. When the waves of two dierent rays of light are in the same time in the same space there are two situations, at which the rst is only a special case of the second one: 1.) If between the waves of the two rays there is no constant phase respectively the intensities of the two rays are simple add. This is for example the case, if you focus the light of two light bulbs in on spot. 5

6 2.)If between the waves of the two rays there is a constant phase respectively you have to add the two electric elds. I(t) = E 1 (t) + E 2 (t) 2 = E 1 (t) 2 + E 2 (t) 2 + E 1 (t)e 2 (t) cos Θ (20) Θ represents the angle between the two rays. As you can see in equation (20) there are two terms where you simply add the intensity but there is also the third term which causes the interference. When the phase dierence φ between the waves is φ = 0 the resulting intensity is high but when the dierence is φ = π the waves annihilate each other and you can only see darkness. 1.4 Coherence When the waves of two rays of light have a constant phase respectively they are call coherent. The phase respectively can only be constant if the two waves are from the same source out of the same emission tact. For the practical use you dene two dierent kinds of coherence you can measure directly: Time Coherence The time of one emission tact is nite. The Time Coherence is dened as the time τ in which the phase dierence between two waves is smaller then ϕ = 2π. In the sunlight or the light of a light bulb this Time Coherence is very small, for the sun it is in the area of τ = 1ns. That means for the experimenter that it is complicated to do inference eect with such an incoherent light source Spacial Coherence Spacial Coherence means that the dierence of the ways two rays go must be so small, that the phase dierence is smaller than ϕ = 2π. The length in which a laser beam is coherent is called coherence length l c. For sunlight the coherence length is in the area of few micrometers but for lasers the coherence length may be up to a couple of kilometers. The best way to measure the Spacial Coherence is to use a Michelson Interferometer. The light from the source is split into two rays at the Beam Splitter. One ray is reected by the stationary mirror and reected back to the Beam Splitter from where it runs to the detector. The second ray is reected by the moving mirror back to the Beam Splitter and also runs into the detector. Now you can variegate the location of the moving mirror to change the way dierence between the two rays. If the way dierence is smaller than the coherence length you will get interference in the detector. But if it is larger than the coherence length the intensities will simply be added and you only get one bright dot. 6

7 Figure 1: The design of the Michelson Interferometer [2] 1.5 Laser To have a large coherence length, we use a laser as light source for our experiments. The word Laser is articial and only a simplication for the expression "`Light Amplication by Stimulated Emission of Radiation"'. The most simple concept of a laser contains only three dierent parts: ˆ The gain medium is a transparent material that emits the the photons which build up the laser beam. ˆ It has to be supplied with power by a pump light. ˆ befor the light leaves the laser it is reected inside through the gain medium by an optical cavity. 7

8 Figure 2: Energy levels of a laser [3] The pump light lifts the electrons from the basic energy level E 0 to the higher level E 3. this level has a very short lifetime and falls back into the level E 2 very fast. The upper laser level E 2 has a relatively long lifetime whilst the lower laser level E 1 has a short lifetime additionaly the electrons fall very fast from E 1 to E 0. With this conditions we have a population inversion between E 2 and E 1. This is one of the conditions for stimulated emission. The second condition is, that there is an other photon in the same space as the excited electron with the exact energy of hν = E = E 2 E 1. This condition can be fullled by photons that are reected by the optical cavity and run through the gain medium several times. If it is possible for the experimenter to get the stimulated emission to dominate over the spontaneous emission the light in the medium is amplied and you get a strong coherent beam, because the photon emitted by the stimulated emission has the exact same phase as the original photon. 8

9 1.6 Holography Holography was invented in 1947 by the physicist Dennis Gabor. He developed a method to improve the resolution in the microscopy. Because of the early time, laser had not been invented jet, so he used a mercury arc lamp as light source. The light had a pretty small coherence length, so his results were not of a good quality, but he could show that the concept would work. To produce his hologram he use the In-Line-Holography, as we call it nowadays: Figure 3: Design of the apparatus to produce an In-Line-Hologram [4] In this kind of holography you position the laser, the object and the photo plate on which you want to capture the hologram in one line. The waves of the coherent beam are scattered on the object. This means with the Huygens principle there are starting new elementary wave fronts from the surface of the object. This new waves have other phases then the original reference wave. So when the reference wave and the elementary waves meet on the photo plate there might be an interference. The condition for interference is that the depth of the object is maximal the size of the coherence length. To reconstruct the hologram, you rst have to develop the photo plate. Then you can hold the plate in the pure reference beam and look at it from the back of the photo plate. Now you can see the hologram as a 3D image of the original object. If you change your perspective the picture changes too. This is because in the hologram all the informations about the absolute intensity and the wavelength as well as the informations about the phase of the object are saved. In contrast in a conventional picture only the information about the intensity and the 9

10 wavelength is saved while you loose every information about the phase. Because you use interference eects to reconstruct your picture there is an other new eect in hologram: you can cut the hologram into pieces and every piece contains the information about the whole object. But by cutting the hologram in pieces you loose intensity as well as resolution when reconstructing the object. After the discovery of the laser in the year 1960 the quality of the holograms became much better. And in 1963 E. Leith and Upatnjeks invented a new, better way to produce holograms: the O-Axis-Procedure. Figure 4: Design of the apparatus to produce an O-Axis-Hologram [5] In the O-Axis-Holography You use a Beam Splitter to separate the reference beam from the object beam. It is advantageous to adjust the Beam Splitter in a way that the object beam has a higher intensity then the reference beam, because the object beam looses a lot of his intensity when it is scattered on the object. With this design you can take holograms of much bigger objects then with the In-Line- Holography. An other advantage is, that you have only one, the real picture, and not an overlay of the real and the virtual picture, you would have with the In-Line-Holography. Due to this advantages the O-Axis-Holography is used for very accurate measurements. In the following we will discuss the method we used to measure small deformations of a bar and the rotation of a metal object. 10

11 1.6.1 Double-Exposure Procedure In the Double-Exposure Procedure you rst take a hologram of the object with the O- Axis-Holography. then then you change the object a bit and take an other hologram on the same photo plate as the rst. You have to be careful and do only very small changes in the area of the wavelength of the light you use, because otherwise you can not measure the interference eects. In this experiment we rst will deform a bar out of metal. Now we want to derive a formula which tells us how big was the deformation when we know the interference stripes. Let E 1 and E 2 be the amplitudes of the electric elds of the two rays (reference wave and object wave): E 1 = Ee iϕ 1 E 2 = Ee iϕ 2 (21) The intensity I is proportional the square of the the sum of the amplitudes of the electric elds. So you get: I = 2E 2 (1 + cos(ϕ 2 ϕ 1 )) (22) The dierence ϕ 2 ϕ 1 is the phase dierence between the two waves and named ϕ. To express the phase dierence in terms of the deformation we say that the movement is upright to the surface and Θ 1 and Θ 2 are the angle of incidence and the emergent angle measured from normal of the surface. You get for the way dierence of the two waves: d = D(cos Θ 1 + cos Θ 2 ) (23) While D is the movement of the surface. For the phase dierence you get from this: φ = 2π λ D(cos Θ 1 + cos Θ 2 ) (24) When you put this in the formula for the intensity, you get: ( )) 2π I = 2E (1 ( 2 + cos λ D(cos Θ 1 + cos Θ 2 ) = 4E 2 cos 2 π ) λ D(cos Θ 1 + cos Θ 2 ) (25) So, the intensity is proportional to cos 2 (x), while x is proportional to the movement. So we can conclude that the movement for two interference stripes is: D = When you count N stripes on the bar, the deformation is: For the rotation of the object you get the relation: λ cos Θ 1 + cos Θ 2 (26) λ D = N (27) cos Θ 1 + cos Θ 2 α = N r λ cos Θ 1 + cos Θ 2 [rad] (28) 11

12 2 Conguration 2.1 Digital In-line-Holography Figure 5 shows the schematic conguration of the digital in-line-holography. The laserbeam projects the object onto the screen. This image of the object is taken by a digital camera. Afterwards, another picture of the screen without the object is taken as reference. For technical reasons, the images are converted to greyscale. Then, the hologram is reconstructed by a computer. Because the distance between the object and the laser is not known exactly, we let the software calculate with a range of 2 millimeters. At the end, the best image is selected from the 50 calculated holograms. The quality of the pictures is evaluated by the sharpness of the edges of the three dierent objects. We have taken the holograms of three dierent objects. An hair, an AFM cantilever and a TEM nets. The width of the screen is known, so we can calculate the scale of the resulting hologram. With this scale, the parameters of the lattice and the dimensions of the candelabra can be measured in the digital hologram. 2.2 O-axis-Holography In gure 6, the schematic conguration for the o-axis holography is shown. The laser beam is splitted into two parts: The major beam goes straight through the splitter and illuminates the object, we want to image. The beam is scattered at the surface of the object. The scattered light exposures a photoplate. The minor part of the laser exposures the photoplate directly without any scattering. This is the reference beam. The interference of the two laser beams creates the hologram on the photoplate. Then the photoplate has to be processed before the hologram can be seen. To display the image, the photoplate is illuminated with an unsplitted laserbeam. 3 Results 3.1 Digital In-line-Holography In the digital hologram, the dimensions of the pictured object can be measured in pixels. To get the scale to calculate the real dimensions in SI-units, we need one xed dimension in the image. The real width of the square is 10.0 cm. The width of the digital image (not the hologram) is 2048 pixels, the height 1536 pixels. With the software ImageJ, we measured the width of the screen in pixels in the image as well. So we can set up the following equation: 10 cm 1950 px = x 2048 px = y 1536 px (29) where x and y are the dimensions in cm of the real image on the screen. To reconstruct the holograms on the computer we need also the distances between the laser and the object, and between the object and the screen. The distance between the laser and the screen constitutes 42.7 cm. For the distance between the laser and the 12

13 object, we could only approximate the value. For all of the three measurements, we approximated a value of 0.1 cm. The program which reconstructs the hologram, oers the possibility to input a range for this distance. We took a range from 0.0 cm to 0.2 cm and let the program calculate the holograms in 50 steps. Afterwards, the best image could be chosen. The last measurements we need are the dimensions of the backwards projection of the screen into the layer of the object. We can set up a simple equation via the ratios: x x = y y = 0.1 cm 42.7 cm (30) where x and y are the dimensions of the projection. The others variables are the same as in (29) above. The 0.1 cm are the distance between object and laser. Latter value can be calculated more exactly afterwards by choosing the best of the 50 reconstructed holograms. Later, we want to measure the dimensions of the pictured objects. The values v are measured in pixels with the software ImageJ again. The image size of the digital hologram is pixels, so the real size S of our object in cm is: Hair S = x v 512 px (31) Figure 7 shows the digital reconstructed hologram of one hair. With ImageJ we can measure the thickness of the hair. The average of 5 measurements is pixel with 2.9 pixels of standart deviation. So we get a real thickness of the hair of (1.134 ± 0.066) 10 4 m. In some cases one can reconstruct the hologram one more time from the rst hologram. But this second iteration of reconstruction was not useful in this case TEM nets Figure 8 shows the hologram of a TEM (transmission electron microscope) nets. In the hologram we measured the distances between the two outer edges and the distance between the two inner edges of one notch of the lattice. The results are: ˆ ˆ Distance of inner edges: (5.56 ± 0.29) 10 5 m Distance of outer edges: (14.47 ± 0.65) 10 5 m AFM-Cantilever In gure 9, the hologram of a AFM (atomic force microscope) cantilever is shown. We measured the height, the width and the angle at the top of the object. The height is only the height of the visible part of the cantilever. The results are: ˆ Width: (5.25 ± 0.36) 10 5 m 13

14 ˆ Height: (8.52 ± 0.45) 10 5 m ˆ Top angle: ± 3.92 Because the top angular doesn't have to be converted, the error is represented by the standard deviation. 3.2 O-axis holography The rst hologram we created was a simple intake of a little motorbike without changing at the object during the exposure time. The exposure time was about 2 minutes. We expected a good quality of the hologram but the result was not usable. Only some edges can be seen in the hologram. At the image of taken with the same digital camera as in the in-axis apparatus, nothing can be seen, except the diuse red light from the laserbeam. The quality of the other holograms was much better than this one and can be used to decree the deformation and the rotation of the two objects Deformation of a sheet metal A small sheet metal was xed at the bottom. We used the double-exposure procedure to measure the deformation caused by a micrometer calliper. The sheet metal is exposured twice: First one minute in the original state, the second time after the deformation. Figure 10 shows the resulting hologram. By counting the stripes caused by the interference, the deformation D of the sheet metal can be calculated. From the top to the button of the sheet, 28 stripes can be counted. Because of the very weak contrast at the button of the sheet, we dene the individual error of this number as 2 for the calculation of the maximum deviation. The two angles are: Θ 1 = 10 ± 3 and Θ 2 = 30 ± 3. So we get a deformation: D = N λ sin Θ 1 + sin Θ 2 = (9.57 ± 0.87) 10 6 m (32) The wave length λ of the laser is nm, N is the number of interference stripes. The error is calculated by: D = λ N λ N + cos Θ 1 + cos Θ 2 (cos Θ 1 + cos Θ 2 ) 2 ( sin Θ 1 Θ 1 + sin Θ 2 Θ 2 ) Rotation of a metal plate The conguration of the experiment is the same as for the deformation measurement. Instead of the deformation, the plate was turned a very small angle between the two exposures. The illumination of the plate is much more equable than the illumination of the sheet metal. The interference stripes can be counted exactly. Figure 11 shows the digital image of the hologram. So we get an angle (in radian): D = N r (33) λ cos Θ 1 + cos Θ 2 = (3.42 ± 0.12) 10 4 rad (34) 14

15 where N = 60 is the number of stripes again and r = 60 mm the width of the object. The other parameters are the same like in the calculation for the deformation of the sheet metal. The error is: D = N r λ cos Θ 1 + cos Θ 2 4 Appraisal of results 4.1 In-line holography ( sin Θ1 Θ 1 + sin Θ 2 Θ 2 cos Θ 1 + cos Θ 2 + N N + r ) r The normal thickness of a hair is between 0.04 and 0.12 millimeters 1 (full developed hairs). The diameter of the hair we measured is about 0.11 millimeters. So our calculated value is realistic. For the other two object, we haven't got other values to compare with. But compared with the diameter of the hair, these values seem to be in a realistic range. The quality of the in-axis holography is not very good. Some things of the conguration of the experiment can be improved: The screen where the image of the object and the reference image are projected, is not very plain. There are some scratches and blains in it. The fastening of the digital camera is not very tight, so that the camera can tilt over a bit while the releaser is pressed or the conguration of the camera is manipulated. The software we used oers the possibility to recalculate the hologram with the new parameters got from the rst calculated image. But in none of our three holograms the second calculation resulted in an usable new image. 4.2 O-axis holography The quality of the rst hologram of the motorbike was very bad. One can see only some edges from the top of the object. The contrast between the scattered light and the reference beam seems to be very low. Maybe the problem can be solved by moving the motorbike closer to the photoplate where the hologram is taken, so that more scattered light illuminates the photoplate. The other holograms are much more useable. The objects are imaged clearly and the interference stripes can be seen well. Only at the bottom of the sheet metal, the contrast is weak again, so we calculated with a deviation of 2 for the number of stripes. On the plate which was rotated a bit, the stripes can be counted easily because the contrast is very good and the edges are sharp enough to distinguish between bright and dark stripes. The most inexact values are likely the two angles Θ 1 and Θ 2 in the last two measurements. Maybe it would be helpfull to draw a line onto the table under the parts of the assembling so the second angle could be measured more exactly. (35) 1 from: [6] 15

16 5 References References [1] Holographic Nondestructive Testing: Robert K. Erf; Academic Press, Inc.; Orlando, Florida: 1974 [2] [3] [4] [5] [6] 16

17 Figure 5: Schematic conguration for digital in-line-holography Figure 6: Schematic conguration for o-axis-holography 17

18 Figure 7: Hologram of an hair 18

19 Figure 8: Hologram of a TEM nets 19

20 Figure 9: Hologram of AFM cantilever 20

21 Figure 10: Hologram of a deformed sheet metal with double-exposure procedure 21

22 Holography May 21, 2009 Kay Jahnke Matthias Hocker Figure 11: Hologram of rotated plate, recorded with double-exposure procedure 22