17-Phys-A7 Optics · December 2018
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
17-Phys-A7, Optics — National Exams, December 2018. 3 hours; closed book (approved Sharp/Casio calculator only). Total 78 marks. Questions 1–6 are mandatory; the paper then offers a choice of Question 7 or 8, and a choice of Question 9 or 10. Every question is solved in full below as a complete study resource, including both members of each either/or pair.
Reference texts. Hecht, Optics, 5th ed.; Pedrotti, Pedrotti & Pedrotti, Introduction to Optics, 3rd ed.; Griffiths, Introduction to Electrodynamics, 4th ed. (Ch. 9, EM waves in matter, for Maxwell's equations).
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Part (a) — irradiance through crossed polarizers (Malus's law).
Given. Unpolarized incident beam, $I_0=1$ unit; ideal (lossless) linear polarizers; input polarizer transmission axis along $y$; exit polarizer (analyzer) at angle $\theta$ to $y$.
Find. $I_{\text{out}}(\theta)$.
Approach. An ideal polarizer transmits exactly half the irradiance of an unpolarized beam (it passes only the component along its axis, and an unpolarized beam has equal power in every direction on average); the light leaving the input polarizer is then fully linearly polarized along $y$, so the analyzer applies Malus's law.
Part (b) — photoelastic fringes in the clamped fork. Between crossed polarizers, an isotropic (unstressed) transparent plastic transmits no light at all — the analyzer blocks whatever the polarizer passes. The stressed fork instead shows bright/dark fringes because clamping induces mechanical stress that makes the plastic birefringent (the stress-optic effect): the local stress field creates two different refractive indices along the principal stress directions, so light passing through splits into two orthogonal components travelling at different speeds. On exit, these components have accumulated a stress-dependent relative phase (retardation) proportional to the local stress difference and the path length; the analyzer recombines them, and wherever the retardation is an integer number of wavelengths the recombined light is again linearly polarized along the original (blocked) axis, giving a dark fringe, while intermediate retardations partially leak through as bright regions. The resulting fringe pattern is a map of stress concentration — dense fringes crowd around the highly stressed tines and the clamp point, exactly the classic photoelasticity technique used to visualize stress in transparent models.
Part (c) — linear to circular polarization. Yes. Passing a linearly polarized beam through a quarter-wave plate oriented with its fast and slow axes at $45^\circ$ to the incident polarization direction splits the beam into two equal-amplitude orthogonal components (along the plate's fast/slow axes) and introduces exactly a $90^\circ$ ($\lambda/4$) relative phase retardation between them — precisely the equal-amplitude, quadrature-phase condition that defines circular polarization (part (e) of Q1). Any other retardation, or unequal component amplitudes (axes not at $45^\circ$), gives elliptical rather than circular polarization.
