- Unequal polarization of the interfering waves results in a bias intensity which reduces the contrast of the fringes and, in the limit of opposite polarization, we get no fringes at all. - In analysing finer diffraction effects, the vector property of the light must be taken into account, for example in the reconstruction of a hologram (G˚asvik 1976). - By including the phase constants δ x and δ y we have taken care of the fact that the two components may not have the same phase. - The factor e ikz we omit, since it merely gives the orientation of the z-axis. - u | 2 = U x 2 + U y 2 (9.7) It is evident and easy to prove that the intensity always becomes equal to the sum of the squares of the field components in a Cartesian coordinate system independent of the orientation of the coordinate axes.. - When such a polarized wave passes a fictional plane perpendicular to the z-axis, the tip of the U -vector will in the general case describe an ellipse in that plane. - This has to do with the direction of rotation of the tip of the field vector.. - To alter and to analyse the state of polarization of the light, one places in the beam different types of polarization filters. - A linear or plane polarizer (often called simply a polarizer) has the property of transmitting light which field vector is parallel to the transmission direction of the polarizer only. - The field u t transmitted through the polarizer therefore becomes equal to the component of the incident field onto this direction. - If the incident wave U is plane-polarized at an angle α to the transmission direction of the polarizer, we therefore have. - I dφ = (1/2)(T 1 2 + T 2 2 ) cos 2 α + T 1 T 2 sin 2 α (9.15) where we have substituted the intensity transmittances T 1 = t 1 2 and T 2 = t 2 2 for the cor- responding amplitude transmittances of the polarizer. - Normally, T 1 and T 2 are wavelength dependent and the ratio T 2 /T 1 is a measure of the quality of a linear polarizer. - When the field of the incident wave oscillates parallel to the threads, it will induce currents in the threads. - The energy of the field is therefore converted into electric current, which is converted into heat and the incident wave is absorbed. - Because of the non-conducting spacings of the grid, currents cannot flow perpendicularly to the threads.. - It can be regarded as the chemical version of the metal thread grating. - These long molecules are oriented almost completely parallel to each other and because of the conductivity of the iodine atoms, the electric field oscillating parallel to the molecules will be strongly absorbed.. - By proper cutting and cementing of such crystals, usually calcite, one of the components is isolated and the other is transmitted, thereby giving a linear polarizer. - iU e i(δ 1 +δ 2 )/2 (cos αe x − sin αe y ) (9.19) We see that the outcoming wave is linearly polarized and that the field vector is the mirror image of the incoming field vector about the x-axis, i.e. - A halfwave plate therefore offers a convenient means for rotating the polarization angle of a linearly polarized light wave by turning the axes of the halfwave plate by the desired amount.. - In contrast to the construction of linear polarizers, where one of the doubly refracting components is isolated, both components are transmitted collinearly by proper cutting and orientation of the crystal. - Here two retarders of the same crystal of thicknesses t 1 and t 2 with their axes inclined at 90 ◦ are mounted together. - The total retardance of the unit therefore becomes proportional to the thickness difference t 1 − t 2 . - The result is a retarder with variable retardance which is uniform over the whole field of the compensator.. - This aspect is strongly related to the coherence properties of the light (see Section 3.3). - It should, however, be easy to realize that unpolarized light will (1) be unaffected with regard to intensity when transmitted through a retarder, and (2) become linearly polarized by transmitting a linear polarizer, but with an intensity independent of the transmission direction of the polarizer.. - When light is incident at an interface between two media of different refractive indices, both the reflected and the transmitted light will in general change its state of polarization relative to the state of polarization of the incident light.. - (The plane of incidence is defined as the plane spanned by the surface normal at the point of incidence and the incident light ray.) The corresponding quantities of the reflected light are denoted u rp and u rn and their amplitudes U are related by. - When the transmission axis of the polarization sunglasses is properly oriented, this specularly polarized reflected component will be blocked out.. - Precautions should be taken when calculating the transmittance of the interface. - Trans- mittance is the ratio of the transmitted over the incident flux and is given by. - (9.32) where the projected areas of the incident beams are taken into account since they are unequal. - U is called a Jones vector or state vector, representing the state of polarization of the wave. - The physical interpretation of the scalar product A | B is therefore given as the probability that an A-filter will be transmitted by a B -state and the transmitted intensity is given as the absolute square of the probability, i.e.. - where | E i and | E u are the state vectors of the incoming and outcoming light respectively, and M is the matrix representing the polarization change. - M n , the total polarizing effect is described by the product of the matrices of the individual filters, i.e.. - M 2 M 1 | E i (9.52) Note the order of matrices of the product.. - The standard methods rely on the technique of making a model which is a copy (often on a reduced scale) of the specimen under investigation.. - Thus, a two-dimensional photoelastic model exerted by forces in its own plane will behave as a general retarder, the retardance and the direction of the retarder axes being continuous variables across the model plane. - where σ 1 and σ 2 are the principal stresses, C is the stress-optic coefficient characteristic of the model material and t is the thickness of the model. - The retarder axes h 1 , h 2 coincide with the axes of the principal stresses σ 1 , σ 2. - Let us for the time being assume the model to be a uniform retarder with its prin- cipal axis h 1 making an angle α to the x-axis and try to find the intensity of the light behind plane 3.. - ‘equal inclinations’, and are loci of points where the principal stress axes coincide with the axes of the polarizers. - By synchronous rotation of the two polarizers, we vary α and the isoclinics move, thereby determining the direction of the principal stresses over the whole model. - Figure 9.10 shows the same set-up as in Figure 9.9 except that two quarterwave plates with their axes oriented 45 ◦ to the transmission directions of the polarizers are placed between the polarizers and the retarder. - We thus have a circular polarizer on each side of the retarder (the model). - 1 2 − 1 2 cos δ (9.71) For circular polarizers of the same handedness. - where n is the order of the isochromatics and where we have put S = λ. - When using the same material of the same thickness and a monochromatic light source of the same wavelength, S is a system constant. - where F is the applied force, D is the diameter of the disc, and n is the measured fringe order.. - From Equation (9.73) we see that for n = 0, δ becomes independent of the wavelength. - Finally, we mention that on a load-free boundary of the model, the principal stress normal to the boundary is zero, and the value of the other principal stress is found directly.. - This can be done by first orienting, for example, a Babinet–Soleil compensator parallel to the direction of the principal stresses, i.e. - We will observe that during this rotation, the nth-order isochromatics would have moved to the initial position of the (n + 1)th-order isochromatics.. - For an intermediate value χ p of the rotation angle, the nth-order isochromatics will intersect our point P (see Figure 9.12(c). - As shown in the previous chapter, the directions and the difference σ 1 − σ 2 of the principal stresses can be determined by using a standard polariscope. - For a complete solution of the stress distribution inside a two-dimensional model, however, the absolute values of σ 1 and σ 2 must be known.. - This phase is unaffected by the birefringence of the model, and is given by. - 2n 0 ]t (9.85) where K is a constant of the model material and n 0 is the refractive index of the unstressed model. - If β could be determined, the principal stress sum σ 1 + σ 2 is given, which together with the values of σ 1 − σ 2 gives a complete solution of the stress-distribution problem.. - The method consists of making two exposures of the model, first in its unloaded and then in its loaded condition. - To get a registration of the unloaded model, it is therefore essential to have a light-field polariscope.. - By reconstruction of this doubly exposed hologram, the reconstructed field will be proportional to the sum of the partial exposures and the intensity becomes. - These characteristics of the combined pattern are shown in Figure 9.14 for the case where the isochromatics and the isopachics cross each other nearly perpendicularly.. - In the case of perpendicular crossing of the two patterns, their analysis becomes fairly easy, but when the two patterns are nearly parallel, the quantitative interpretation can give erroneous results. - By passing the rotator the orthogonal polarizations are reversed and the birefringence effect is cancelled in the second pass of the model resulting in a isopachics pattern only.. - In this method, advantage is taken of the multiphase (diphase) nature of plastics used as model materials to conserve strain and birefringence in the model after the load has been removed. - Consider a slice cut from a three-dimensional model in such a way that the directions of two of the three principal stresses, e.g. - σ 1 and σ 2 , lie in the plane of the slice (see Figure 9.17). - This pre-supposes that the direction of at least one of the principal stresses, e.g. - σ 3 , is known, which is frequently the case, for example on a section of symmetry or a free surface of the model. - To determine the principal stress difference in the other two principal planes, a subslice may be cut from the original slice parallel to one of the principal stresses, say σ 1 (see Figure 9.17). - If this subslice is observed in a polariscope in the direction of σ 2 , the resulting isochromatics will. - When none of the directions of the three principal stresses is a priori known, the situ- ation is more complicated. - For the detailed description of the various methods for solving this problem, we direct the reader to the specialized literature given in the Bibliography.. - If a beam of light passes through a transparent isotropic medium, it will be scattered by small particles in suspension or by the molecules of the medium. - The field amplitude of the light scattered from a certain point, will be proportional to the component of the amplitude of the incident light normal to the scattering direction.. - The field amplitude of the scattered light then becomes equal to the y-component of the field of the primary beam at the scattering point. - The calculation of the scattered intensity now becomes equivalent to the problem of calculating the intensity of the light transmitting a retarder given by Equation (9.61) placed between a polarizer. - From Equation (9.100) we see that the intensity is constant and independent of when either the polarization or observation direction is parallel to one of the principal stresses. - No interference effects will then be observed along the path of the primary beam.. - The directions of the principal stresses can therefore be determined by making the polarization and observation. - directions parallel and then rotating the model about the axis of the primary beam until a uniform minimum intensity is observed. - Expressed in terms of the fringe order, this equation can be written as. - The principal stress difference at any point along the primary beam is therefore propor- tional to the gradient or inversely proportional to the spacing of the fringes in the scattered light. - Alternatively, a change of sign may be indicated by reversal of the colour sequence of the fringes when using white light.. - Another variant of the scattered light method is to apply an unpolarized primary beam propagating in, for example, the z-direction as in Figure 9.19. - To apply the method, the primary beam is moved parallel to itself in small steps along the line of observation from the surface towards the interior of the model. - For each step, the phase difference of the emergent light components is measured. - If the results are plotted against the position of the scattering point, the slope of the resulting curve multiplied by the stress-optic coefficient gives the principal stress difference at that point.. - Since the efficiency of the scattering process is low, a source of high intensity is required. - Provision for rotational and translational movement of the model in the tank is needed.. - A fraction of the light, determined by the transmittance coefficients, will propagate into the second medium.. - When the surface consists of a thin layer, by taking account of the reflection and trans- mittance coefficients and of multiple reflections in the layer, the state of polarization of the reflected light will depend both on the optical constants and the thickness of the layer.. - 9.1 Determine the emergent state in each of the following cases:. - (a) A P-state incident on a quarter-wave plate with the line of the P-state midway between the principal axes of the plate.. - (b) A P-state incident on a half-wave plate with the line of the P-state midway between the principal axes of the plate.. - P y -basis, the matrix T of the same element in the | R. - e − iβ/2 0 0 e iβ/2 (a) What is the matrix N of the same circular retarder in the |R , |L -basis? Hint:
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