Q1:(a) Define with the aid of a sketch, electric and magnetic field vectors perpendicular and parallel to the plane of incidence (POI) for a light ray that is undergoing reflection and transmission at a polished air-glass interface.
(b) Starting from Maxwell’s equations for ∮?̅ .??̅ and ∮?̅ .??̅ applied at an interface, and using the relationship between peak magnetic and electric field amplitudes, ?0 = ?0⁄? where v is the phase velocity of the light, derive the following relationship for the amplitude reflection coefficient, ?⊥ of light with electric field perpendicular to the POI:
?⊥ = ?????–???????/ ????? + ???????
where ?? is the glass refractive index, θi is the incident ray angle to the normal, θt is the transmitted ray angle to the normal, and taking the incident refractive index of air to be unity.
(c) Sketch a graph of the intensity reflection coefficients for light with electric field polarised perpendicular and parallel to the POI as a function of θi , and label with values for reflectivity for both polarisations at normal incidence and at Brewster’s angle, for ng = 1.5.
(d) A polariser consists of 8 glass plates arranged at Brewster’s angle for a collimated visible laser beam. Calculate the rejection ratio, ????⁄???? for the polariser where ???? and ???? are the maximum and minimum transmitted intensities as the polariser is rotated about the optical axis of the laser beam
Q2: (a) Derive the system ray transfer matrix for: (i) a single-lens system and; (ii) a two-lens telescope system where lenses are separated by the sum of their focal lengths. Take all lenses to be of the same focal length, f and take the initial and final planes to be separated by s and s’ respectively from the nearest lens.
(b) Use the two system matrices from part (a), to derive how s and s’ are related for each system in order that the initial and final planes are object and image planes respectively, so creating imaging systems for which s = u and s’ = v.
(c) Continuing from part (b) with the system matrices revised in order to now image, derive the magnification for each system.
(d) A transform-limited laser beam is collimated at an object plane with an amplitude function described by: U(r) =exp()
where r is the radius and s is the waist size. Determine the waist size, s’ for the laser beam at the image plane for the telescope imaging system of part (a), where s = 3.0 mm, u = 2.0 m and the focal length for both lenses is, f = 1.0 m.
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