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.
a) Optics, in layman terms. The branch of physics/engineering concerned with how light is produced, travels, and interacts with lenses, mirrors, and other matter — in short, the science of light and vision, and of the instruments (cameras, telescopes, fibres, lasers) built to control it.
b) Brewster condition. At the Brewster angle $\theta_B=\arctan(n_2/n_1)$, the reflected and refracted rays are exactly perpendicular to one another ($\theta_r+\theta_t=90^\circ$), and the reflected ray contains no electric-field component in the plane of incidence — only the perpendicular (s) polarization reflects, so light reflected at Brewster's angle is completely linearly polarized.
c) Wave train sketch. A monochromatic wave train is a sinusoidal disturbance $E(x)=A\cos(kx)$ travelling along $x$; the amplitude $A$ is the peak displacement from zero, and the wavelength $\lambda$ is the spatial period (distance between two successive points of identical phase). Two points differing in phase by $4\pi$ are exactly $2\lambda$ apart, since $\Delta\phi=k\,\Delta x=\dfrac{2\pi}{\lambda}\Delta x=4\pi\Rightarrow\Delta x=2\lambda$.
d) Maxwell's equations (differential form, general media):
$$\nabla\cdot\mathbf D=\rho_f,\qquad \nabla\cdot\mathbf B=0,\qquad \nabla\times\mathbf E=-\dfrac{\partial\mathbf B}{\partial t},\qquad \nabla\times\mathbf H=\mathbf J_f+\dfrac{\partial\mathbf D}{\partial t}$$with $\mathbf D=\varepsilon\mathbf E$ and $\mathbf B=\mu\mathbf H$. Equivalently, in integral form:
$$\begin{aligned} \oint_S\mathbf D\cdot d\mathbf a&=Q_{f,\text{enc}}\\[2pt] \oint_S\mathbf B\cdot d\mathbf a&=0\\[2pt] \oint_C\mathbf E\cdot d\boldsymbol\ell&=-\dfrac{d}{dt}\int_S\mathbf B\cdot d\mathbf a\\[2pt] \oint_C\mathbf H\cdot d\boldsymbol\ell&=I_{f,\text{enc}}+\dfrac{d}{dt}\int_S\mathbf D\cdot d\mathbf a \end{aligned}$$These are, in order: Gauss's law (electric), Gauss's law (magnetic, no monopoles), Faraday's law, and the Ampère–Maxwell law. Combining the curl equations in a source-free, linear, homogeneous medium yields the wave equation and $v=1/\sqrt{\mu\varepsilon}$ — the basis of electromagnetic optics.
e) Circular polarization. The state in which the tip of the electric-field vector, viewed head-on along the propagation direction, traces a circle at constant angular rate as the wave advances — equal-amplitude $x$- and $y$-components exactly $90^\circ$ out of phase, $\mathbf E=E_0(\hat{\mathbf x}\cos(kz-\omega t)\pm\hat{\mathbf y}\sin(kz-\omega t))$, so $|\mathbf E|$ is constant in time while its direction rotates (right- or left-circular depending on the sign).
f) Three of the five primary (Seidel) aberrations.
Spherical aberration. Rays striking a spherical surface far from the axis focus closer to the lens than paraxial rays, so there is no single sharp focus on-axis — a point object images as a blur circle.
Coma. An off-axis point source images as an asymmetric comet-shaped flare (a series of overlapping circles of different size and lateral position, one per annular lens zone) because the magnification produced by outer zones differs from that of the central zone.
Astigmatism. For an off-axis point, rays in the tangential (meridional) plane and the sagittal plane focus at two different axial positions, so no single image plane gives a sharp point image — the image is a line (tangential focus), then a line rotated $90^\circ$ (sagittal focus), with a circle of least confusion in between.
g) Amplitude, irradiance and power for a plane-wave beam. For a plane wave of electric-field amplitude $E_0$ in a medium of refractive index $n$, the (time-averaged) irradiance is $I=\dfrac{1}{2}n\varepsilon_0 c E_0^2$ — irradiance scales as the square of the amplitude, not linearly. The power carried through a beam cross-section of area $A$ is $P=I\,A$, so doubling the amplitude quadruples both irradiance and (for fixed beam area) power.
