- Although holography requires coherent light, it was invented by Gabor back in 1948, more than a decade before the invention of the laser. - Because of the above-mentioned properties, we shall see that holographic interferometry has many advantages compared to standard interferometry.. - One of the partial waves is directed onto the object by a mirror and spread to illuminate the whole object by means of a microscope objective.. - The object scatters the light in all directions, and some of it impinges onto the hologram plate. - This wave is called the object wave. - This is not essential, but it is important that the reference wave constitutes a uniform illumination of the hologram plate. - Let the object and reference waves in the plane of the hologram be described by the field amplitudes u o and u respectively. - BS = beamsplitter, M = mirrors, MO = micro- scope objectives and (b) Reconstruction of the hologram. - We block the object wave and illuminate the hologram with the reference wave which is now termed the reconstruction wave (see Figure 6.1(b. - just behind the hologram then becomes equal to the field amplitude of the reconstruction wave multiplied by the amplitude transmittance of the hologram, i.e.. - The consequence is that, by looking through the hologram in the direction of the object, we will observe the object in its three-dimensional nature even though the physical object has been removed. - the conjugate of the object wave would have been reconstructed. - u ∗ o therefore represents a wave propagating from the hologram back to the object forming an image of the object.. - By placing a screen in the real wave, we can observe the image of the object on the screen.. - An alternative and more physical description of the holographic process has already been touched on in Section 4.3.1. - This argument can be repeated for all points on the object and give us the virtual reconstructed object wave. - We now consider in more detail the locations of the virtual and real images for the most general recording and reconstructing geometries. - Using quadratic (Fresnel) approximations to the spherical waves, the object and reference fields of wavelength λ 1 incident on the xy-plane may be written. - (6.5) The transmittance of the resulting hologram we write as. - What we have done is to find the coordinates (x i , y i , z i ) of the image point expressed by the coordinates of the object point, the source point of the reference and the reconstruction waves. - Assume the object- and reference waves to be described by. - u r | 2 = 1 4 U 2 t b 2 V 2 (6.21) The diffraction efficiency η of such a hologram we define as the ratio of the intensities of the reconstructed wave and the reconstruction wave, i.e.. - η = I r /I = 1 4 t b 2 V 2 (6.22) From this expression we see that the diffraction efficiency is proportional to the square of the visibility. - when I o = I , which means that the diffraction efficiency is highest when the object and reference waves are of equal intensity.. - Here M is the amplitude of the phase delay.. - The amplitude of the first-order reconstructed object wave is found by multiplying Equation (6.23) by the reconstruction wave u for n = 1, i.e.. - To obtain maximum intensity of the reflected, reconstructed wave, the path length difference between light reflected from successive planes must be equal to λ. - the angles of incidence of the reconstruction and reference waves must be equal. - It can be shown that for a thick hologram, the intensity of the reconstructed wave will decrease rapidly as ψ deviates from θ /2. - A special type of volume hologram, called a reflection hologram, is obtained by send- ing the object and reference waves from opposite sides of the emulsion, as shown in Figure 6.5(a). - Then θ = 180 ◦ and the stratified layers of metallic silver of the developed hologram run nearly parallel to the surface of the emulsion with a spacing equal to λ/2 (see Equation (6.27. - Owing to the Bragg condition, the reconstruction wave must be a duplication of the reference wave with the same wavelength, i.e. - Therefore a reflection hologram can be reconstructed in white light giving a reconstructed wave of the same wavelength as in the recording (see Figure 6.5(b. - In practice the wavelength of the reflected light is shorter than that of the exposing light, the reason being that the emulsion shrinks during the development process and the silver layers become more closely spaced.. - In the description of the holographic recording process we assumed the spatial phases of both the object- and reference waves to be time independent during exposure. - If, for instance, a mirror makes vibrations of amplitude greater than λ/4 during the exposure time, adjacent dark and bright interference fringes interchange their positions randomly, which can lead to a uniform blackening of the hologram film and therefore ruin the experiment. - The exposure time using a 5 mW H-Ne laser is typically of the order of seconds. - This poses stringent requirements on the stability of the set-up. - For instance, we can let it interfere with the wave scattered from the object in a deformed state. - In this method, two exposures of the object are made on the same hologram. - By reconstructing the hologram, the two waves scattered from the object in its two states will be reconstructed simultane- ously and interfere. - This double-exposed hologram can be stored and later reconstructed for analysis of the registrated deformation at the time appropriate for the investigator. - If a lot of different states of the object (e.g. - In this method, a single recording of the object in its reference state is made. - A disadvantage of the method is that the hologram must be replaced in its original position with very high accuracy. - Also the contrast of the interference fringes is not as good as in the double-exposure method.. - Consider Figure 6.6 where a point O on the object is moved along the displacement vector d to the point O due to a deformation of the object. - Here d cos γ is the component of the displacement vector onto the line bisecting the angle between the illumination and observation directions. - Each time the surface of the deformed object intersects one of these planes we get a bright (or dark) fringe. - In this method the two wavefronts corresponding to the object in its two states have different frequencies ν 1 and ν 2 . - Moreover the measurement is indepen- dent of U which affects only the amplitude of the signal. - For the first of these, one of the detectors is tracked across the fringe pattern whilst the other is held static and hence generates the fixed reference frequency. - When the point source and the point of observation are placed at finite distances from the object, the illumination and observation directions (n 1 and n 2 ) will vary across the object surface. - It is therefore also impossible to measure deformations if the microstructure of the object changes drastically, as for example in plastic deformations.. - It is an annoying fact that the fringes in holographic interferometry only in special cases are localized on the object surface.. - Loss of fringes in the plane of the object or other planes therefore means that their contrast is too low to be detected in that plane.. - In the analysis below, we shall show that this problem might be overcome by stopping down the aperture of the imaging system, thereby increasing the depth of focus to obtain simultaneously an image of sufficient quality of both the fringe pattern and the object surface. - Consider Figure 6.10 where the holographic interferogram due to displacement of the object surface (drawn as a single surface in the figure) is imaged onto a viewing screen by a lens through a rectangular aperture. - The central ray passing through Q emanates from a point P(x 0 , y 0 ) on the object surface. - The intensity at Q is the integral of the intensity of all the ray pairs within the ray cone defined by the aperture. - kd(cos θ 1 + cos θ is the phase difference due to the displacement of each point on the object between the two exposures (see Equation (6.33. - Therefore, by decreasing the aperture, relatively distinct fringes can be observed over an extended region and fringes can be seen in the plane of the object even though it is at some distance from the region of localization.. - The determination of the regions of fringe localization must therefore be calculated in each separate case by maximizing Equation (6.44). - (2) For a rigid body rotation about an axis inside the object’s surface and normal to the illumination and observation directions the fringes localize on the object.. - Assume that the object point in Figure 6.6 executes harmonic vibrations given by. - where D(x) is the amplitude, x represents the spatial coordinates of the object point and ω is the vibration frequency. - Let u o of Equation (6.46) be the object wave in a hologram recording and u the refer- ence wave. - For exposure times much longer than the vibrating period of the object (Løkberg 1979) this time averaging is equivalent to averaging over one vibration period T . - e iη cos ξ dξ = J o (η) (6.50) The observable intensity distribution of the reconstructed wave becomes. - In Figure 6.11(b) we have drawn the Bessel function squared which represents the intensity distribution that will be observed in the reconstruction of a holo- gram recording of the vibrating bar. - Figure 6.11 (a) Vibration of a bar about an axis and (b) Intensity distribution of the resulting time- average holographic recording. - For g = 2 (illumination and observation directions parallel to the displacement) and λ = 632.8 nm (wavelength of the He–Ne laser) this gives:. - A very detailed map of the amplitude distribution is therefore obtained. - Here one observes fringes due to the displacement of the object, when at rest and when illuminated by the light pulse. - In that way, by slowly varying the phase between the light pulse and the object vibration, one can observe the vibration of the object in slow motion. - Thus it is also possible to observe the phase of the displacements on the different parts of the object surface.. - This can be done by placing a vibration mirror in the light path of the reference wave. - The argument of the Bessel function of Equation (6.51) then changes from (4π/λ)D(x) (for g = 2) to. - the vibrating amplitude of the object,. - R = the vibrating amplitude of the reference mirror,. - ψ r = the phase difference between the vibration of the object and reference mirror.. - The result of this reference wave modulation is that the centre of the Bessel function is moved from the nodal points of the object to points that vibrate at the same amplitude and phase as the modulating mirror. - By varying the phase of the reference mirror it is also possible to trace out the areas of the object vibrating in the same phase as the reference mirror, thereby mapping the phase distribution of the object.. - Reference mirror modulation can also be used to measure very small vibration ampli- tudes by moving the steepest part of the central maximum of the Bessel function to coincide with zero object vibration amplitude (Metherell et al. - The quantity actually measured by this method is the change in refractive index due to some change in the object volume. - The change in refractive index due to air compression can be found by counting the number of fringes starting from the tip of the cone.. - Density, denoted by ρ, is related to the refractive index of the gas by the Gladstone-Dale equation:. - where K, the Gladstone-Dale constant, is a property of the gas. - K is nearly inde- pendent of the wavelength of light and of temperature and pressure under moderate physical conditions.. - Where P is the pressure, M is the molecular weight of the gas, T is the absolute temper- ature and R is the universal gas constant. - In plasma diagnostics, interferometry is used to determine the spatial distribution of densities of the plasma which is a collection of atoms, ions and electrons created by very high temperatures. - The refractive index of a plasma is the sum of the refractive indices of these particles weighted by their number densities. - 6.3 In Section 6.5 we stated that maximum diffraction efficiency for a thin amplitude hologram is obtained when the visibility (or contrast) of the intensity distribution is unity, which means that the ratio R = I o /I r between the object and reference intensities is equal to 1.. - Calculate R in terms of E 1 and E 2 when the whole linear portion of the t − E curve is utilized.. - of the reconstructed wave and the ray angle magnification M α sin θ s / sin θ o . - What should preferably be the angle between the object- and reference waves when using this film?. - θ 2 = 20 ◦ and an He–Ne laser is used, what is the maximum deflection of the plate?. - (b) Assuming g = 2 and λ = 632.8 nm, what is the maximum vibration amplitude on the left and right side of the turbine blade in B of Figure 6.9(d) (Consult a table of Bessel functions).. - Assume that we record a time-average hologram with a modulated reference wave with a vibrating amplitude of the reference mirror equal to R and the phase ψ r = π . - Sketch (qualitatively) the intensity distribution of the resulting interferogram.
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