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Here is lab handout with complete explanation of physics experiment. Keywords are: Holographic Interferomentry, Introduction, Hologram Formation, Double Exposure Interferometry, Beam Deflection, Optical Setup, Cantilever Beam, Making Holograms, Furture Reading
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HOLOGRAPHIC INTERFEROMETRY
February 2003
Optical interferometry is a powerful tool for measuring displacements of the order of a wavelength of light. One significant limitation of common interferometric methods is that they require specular reflectors, so that one is limited to measuring the displacement of a mirror. This limitation can be removed by utilizing holography, allowing very small motions of arbitrary, diffusely reflecting, objects to be detected. Holography, or wavefront reconstruction, is a process by which the amplitude and phase variation across a wavefront may be recorded and subsequently reproduced. Recall that if monochromatic, coherent light enters the eye or any other optical instrument, the only information furnished is the amplitude and phase variation across the input aperture. It follows, therefore, that the holographically reconstructed wavefront is completely equivalent to the original optical wave. Any measurement that could be made on the original optical signal can also be made with the reconstruction. In double-exposure holographic interferometry this principle is applied to obtain an interference pattern between two reconstructed images of the same object. A holographic recording is first made with the object in some reference state. The object is then moved or deformed as desired, and a second hologram is made on the same photographic film. During reconstruction two waves are created and interfere in the space beyond the hologram. The resulting interference fringes record the local displacement of the surface of the object. The experiment to be done here consists of making a double-exposure hologram of a cantilever beam and using the fringe pattern to deduce the deflection under load. The measured deflection can be compared with a simple theory. If desired, the deformation of more interesting objects can be examined as well.
ER = E 0 R cos[ ωt +φ (^) R (x,y)] (1)
The amplitude is constant, while the phase varies across the film because the wave does not arrive at normal incidence. The object wave can similarly be expressed as
EO (x, y) = E 0 O (x,y)cos[ ωt +φO (x, y)] (2)
but now both the amplitude and phase are complicated functions of position because the wavefront is quire irregular. The photographic film records the time-average intensity of the combined beams
I(x, y) = ( ER + EO)^2 = E^0 R
2 2
2 2
as a pattern of light and dark fringes. To reconstruct the image we illuminate the developed film with a replica of the reference beam,
Er = E 0 r cos[ ωt +φ (^) R (x,y)] (4)
The transmitted wave is proportional to the product I(x,y)Er(x,y), which can be written out as
E(x, y) = 1 2
E 0 r ( E 02 R^ + E 02 O)cos[ ωt +φR (x, y)]
1 2 E 0 r E 0 R E 0 O cos[ ωt + 2 φR −φO ]
1 2 E 0 r E 0 R E 0 O cos[ ωt +φO ]
The first term is the undeflected reference beam, while the other two contain information about the object beam. The last is, in fact, the desired replica of the object beam, except for the constant E0RE 0 r/2. The middle term is similar, except that the phase factor has the wrong sign.
Although it is not obvious from the equations, an analogy with the simple grating discussed above suggests that the beams carrying object information are deflected to either side of the reconstructing beam, and this is in fact the case. It is also true that the last term describes a virtual image of the original object, located at the object position. The phase-reversed term
describes a real image which is inverted in a peculiar way. In practice, we use the virtual image, which can be examined by looking through (not at) the hologram as though it were a window. Making and viewing a plane transmission hologram, as described above, demands excellent stability of all the components and fairly careful placement of the reconstruction beam. By going to the geometry shown in Fig. 2a, both these problems are eased. The reference and object beams now mix in such a way that interference fringes are formed in the volume of the photographic emulsion, more or less parallel to the surfaces. The hologram is therefore a three-dimensional grating, more like a crystal than the familiar flat pattern. Since there are fewer components and they can be closer together, alignment and vibration during hologram exposure are less critical. During reconstruction, shown in Fig. 2b, the hologram is viewed by reflected light, which need not be coherent or even monochromatic. Like a crystal, the three-dimensional grating will only reflect wavelengths that meet the Bragg condition, so it acts as a monochromator at the same time as it forms the image. Finally, we note that the photographic process can be further manipulated to enhance the hologram. Exposure to light sensitizes some of the silver halide grains in the emulsion in such a way that they can be chemically reduced to metallic silver. This process is called "development", and yields the familiar black photographic image. (Small grains of silver appear black, not shiny.) For a hologram the blackened regions absorb light, and serve as the barriers of a grating, but it would obviously be more efficient to use all the incident light. This can be accomplished by converting the metallic silver to a different silver halide compound. The new compound formed in the exposed and developed regions has a different index of refraction than the original silver halide left in the unexposed areas, so there is a differential phase shift for light which traverses different parts of the emulsion. Interference still occurs as it does in the absorptive grating, but with greater intensity because no light is actually lost.
film
r
a b
virtual image
Fig. 2 Arrangement for (a) making and (b) viewing a reflection hologram.
∆y = n^ λ 2(cos α + cos β)
where n counts both dark and light fringes from some undeflected reference point.
C. Beam deflection Because of engineering interest, the deflection of various beams and plates has been calculated for many different load conditions. The results can be found in the references. Our system is a uniform rectangular bar of length l , width w and thickness t, clamped at one end. The total weight of the bar is W' and there is a small additional weight W at the end farthest from the clamp. The deflection y(x) is given by
EIy = −W 6 (l − x)^3 − W^ ′(^ l^ −^ x)
4 24 l − Wl
(^2) x 2 − W^ ′l
(^2) x 6
3 6
3 24
where x is the distance measured from the clamped edge. E is Young's modulus for the material, and I is the moment of area, wt^3 /12 for a rectangular bar deflected normal to the thickness. In our experiment we are actually interested in the additional deflection produced by the added weight W, so we need only consider ∆y,
EI∆y = −W 6 (l − x)^3 − Wl
(^2) x 2
3 6
This is the formula we need to compare with the measured deflections deduced from Eq. 9.
Several steps are needed to construct the holographic interferogram and measure the beam deflection. A laser beam must be expanded and directed to uniformly illuminate the object. The photographic film must be positioned and exposed to record the holograms of the bar with and without load. The latent image in the film must then be developed and viewed under appropriate conditions to visualize the interference fringes. The fringe positions can then be measured, the deflection as a function of position deduced, and a comparison made with the expected variation. We consider each step in turn.
A. Optical set up The beam emerging from the laser is non-uniform because of inhomogeneities in the lasing medium, and is too narrow to illuminate the object. These problems are cured with a spatial filter. The expanded beam can then be directed onto the subject with mirrors. A typical arrangement is shown in Fig. 4. The spatial filter consists of a microscope objective, which is effectively a short focal length lens, and a pin hole in a piece of metal. The objective focuses the plane-wave portion of the laser output to a very small diameter beam which can pass through the pin hole. The remainder of the light from the laser is intercepted by the metal, so that the output of the spatial filter assembly is a spherically diverging wave. The spatial filter is usually left in position. If not, you will need to align it. Put a white screen in the laser beam, and note where the beam strikes it. Remove the pin-hole from the filter assembly, and position the filter in the beam so that the expanding light beam is centered on the previously noted position. Use the focusing screw to move the objective away from the pin hole position, and install the pin hole. Roughly align the aperture with the focused spot by looking at
laser
spatial filter
large rectangular mirror
round mirror
film holder
isolation table shutter
beam stop
Fig. 4 Typical layout of the optical table for making a reflection hologram.
4x5 plate Fig. 5 When positioned as shown, the emulsion (sensitive) side is facing you.
background. A slide projector makes a convenient source of bright, collimated light. Arrange the projector and a mirror so that the reconstruction beam is at the same angle as the laser beam was during exposure. It is also necessary to orient the hologram correctly, that is emulsion side down and with the reconstruction beam coming from the proper direction.
D. Analysis To analyze the fringe pattern, set up the hologram for viewing as explained above. Lay a ruler on top of the plate, and estimate the center of each dark fringe, relative to the edge of the clamp. Note that there will be some parallax because the ruler is not in the plane of the fringes. Alternatively, you can support the plate above the viewing surface and hold the ruler in the plane of the cantilever. In either case, it is probably most convenient to look straight down on the image so that β = 0. Convert your fringe count to a displacement using Eq. 9, and compare with that expected on the basis of Eq. 11. A detailed error analysis is probably not justified, but you should quantitatively estimate major uncertainties for both the position and deflection measurements. Discuss any deviations you do (or don't) see between measurements and calculation.
