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terms are best described using a diagram as shown in figure 3.2. The incident ray (red arrow) is defined in terms of the incident angle,α, which is the angle between the incident ray and the vector normal to the crystal interface. The plane of incidence is outlined in red.

The coordinate system is chosen so that (x, y) span the crystal interface, andz points in the direction normal to the interface. The vector c points in the direction of the optical axis and is determined by the two angles: δ which subtends the optical axis and the plane of incidence, andϑwhich subtends the optical axis and the crystal interface. The thickness of the plate in the direction of transmission is given by L.

Figure 3.2: Geometry of a uniaxial crystal as defined by Veiras in his analytical model [79]. The

incident light (red) is defined by the angleαon the plane of incidence. The optical axis is c. The

anglesδandϑdescribe the position of the optical axis with respect to the plane of incidence and

the crystal interface, respectively. The thickness of the crystal is given by L.

The Veiras formula has been implemented in this work to calculate the phase map for various optical components. This aids in the discussion regarding the operation of these components and allows numerical simulation of the expected interferogram for individ- ual plates as well as combined optical systems. The accuracy of the formula has been confirmed by comparing the interferogram produced by real uniaxial birefringent crystals (with known orientation) with the interferogram produced by taking the cosine of the phase calculated using Veiras formula. This is shown as part of the discussion in the following two sections which describe the main optical components required in coherence imaging.

3.2

The waveplate

The waveplate or retarder plate is a uniaxial birefringent crystal, cut such that the prin- cipal (optical) axis lies parallel to the entrant and exit surfaces of the crystal [80]. The components of the impinging light which have perpendicular and parallel polarisation to the optical axis direction will experience an offset in phase due to the anisotropy of the refractive indices. The function of the waveplate is shown in figure 3.3 for a ray directed normal to the entrant plane. The extraordinary components, polarised parallel to the

38 Coherence imaging using polarisation interferometry

optical axis, are depicted by the green arrows and the ordinary components, polarised orthogonal to the optical axis, are vectors oriented normal to the page and depicted as the red dots. As the ray passes through the optic, the components separate in phase in the direction of the ray propagation. The resulting phase offset is determined by the crystal refractive indices, wavelength of the incident light and the plate thickness.

Figure 3.3: Transmission of light at normal incidence through a waveplate showing the delay in path length between the polarization components perpendicular (red) and parallel (green) to the optical axis.

Waveplates can be cut so that they change the phase by a large number of waves (referred to here as delay plates or retarders), or, they can shift the phase by only a portion of a single wave. The latter types of plates are designed to delay the phase by either π/2 or

π and are known as quarter and half-waveplates respectively [80]. Half-wave plates, delay the phase by a half-wave and hence result in the polarisation of the wave being flipped about the axes. A quarter waveplate shifts the phase by a quarter-wave and can change the polarisation from linear to circular. These particular plates can be useful in techniques such as field widening which will be discussed in section 3.4.

Substituting ϑ= 0 into Veiras formula, the phase of the waveplate is given by,

φ0(α, δ) = 2πL λ0 (n2_O−sin2α)12 − 1 n2 O n2_En2_O−(n2_E−n2_O) sin2δsin2α 1 2 (3.4)

Considering only the on-axis ray (α = 0), the on-axis phase shift between the E and O ray simplifies to,

φ0= 2πLB

λ0

(3.5)

which is well documented in literature [41, 42, 46, 48].

The interferogram for a 20 mm α-barium borate plate was imaged at 488 nm using a 75 mm focal length lens attached to a 1392×1040 pixel CCD (square pixel width of 6.45µm). The plate was placed between crossed polarisers and rotated around thez axis so that the optical axis was 45◦ to the polariser axes. The interferogram measured is shown in figure 3.4 a).

§3.2 The waveplate 39

For comparison, the interferogram was also calculated using equation 3.4. The focal length and pixel offset values in the model were adjusted manually to obtain the best fit with the measured pattern. A difference of 1.5 mm in focal length, a 0.5◦ rotation of the viewing angle and an offset of 80 and 25 pixels were required in the i and j pixel directions, respectively, to obtain the best fit between measurement and model (see figure 3.4 b).

a) b) c) d) i. d) ii. R esidual Fringe Dept h R esidual Fringe Dept h

Figure 3.4: Interferometric fringe pattern from a 20 mm α-barium borate waveplate obtained from (a) a computer model using Verias formula and (b) a waveplate placed between crossed polarisers for the same uniaxial plate orientation. Image (c) shows the residual difference between

(a) and (b). Plots (d) i. and (d) ii. are cross sections of (c) atj= 1000 andj= 100, respectively.

The difference image shown in figure (c), reveals a good fit across the image except in the upper left quadrant. Minor deviations between the model and the measurement are most likely due misalignment between the optic and lens and/or to imperfections in the crystal such slight changes in the thickness and orientation of the optic. The localised ‘bad fit’ in the upper left quadrant is due to a decrease in the fringe contrast on the periphery of the measurement image. On inspection it is clear that the model still accurately predicts the position of the fringes in this region.

The 2D phase profile for a waveplate is a saddle surface. The phase surfaces are shown for two plate thicknesses (20 mm and 80 mm crystal thicknesses of α-barium borate at a wavelength of 488 nm) in figure 3.5 a i) and b i). These two examples were chosen as 20

40 Coherence imaging using polarisation interferometry

a i)

a ii)

b i)

b ii)

Figure 3.5: (i) The phase surface and (ii) the interferometric fringe pattern is shown for aα- barium borate waveplate of length (a) a 20 mm and (b) an 80 mm.

mm plates are readily available in the laboratory and are the standard delay plates used to construct the snapshot imaging system for this work. An 80 mm thickness of delay plate is considered as this is the reference length of delay used for instruments in this work.

The crystals modeled in these figures have been rotated around thezaxis by 45◦(compared with the waveplate shown in figure 3.4) and the DC offset has been subtracted for each of these saddles so that the magnitude of the surface variation can be compared. The DC offset is indicated as the value forφDC. The saddle surface is dependent on the polar angle,

δ and is therefore governed by the sin2δ term in equation 3.4. This term scales asL/nO and become more pronounced at larger interferometric delays (large crystal thickness L). This results in more densely packed hyperbolic fringes for thicker crystals as can be seen by comparing figure 3.5 a ii) and b ii).