17-Phys-A7 Optics · May 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
98-Phys-A7, Optics — National Exams, May 2016. 3 hours; closed book (formula sheet supplied). Notes 1–7 on the cover page require Q1 and Q2 (mandatory) plus any two of Q3–Q6 for a complete paper; every question is solved below as a full study resource, including all four of Q3–Q6.
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).
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) Light. Electromagnetic radiation, generated by accelerating charges, that stimulates the human retina (roughly 380–750 nm in vacuum) or, more broadly, the same band of the EM spectrum treated by optical instruments; each photon carries energy $E=h\nu$.
b) Optics. The branch of physics that studies the generation, propagation, detection and manipulation of light, and its interaction with matter and with instruments (lenses, mirrors, fibres, gratings) built to control it.
c) Geometrical optics. The high-frequency ($\lambda\to0$) limit of wave optics, in which light is treated as rays travelling in straight lines that bend only at reflecting/refracting surfaces (Fermat's principle); valid whenever every aperture and feature is large compared with $\lambda$.
d) Physical optics. The treatment of light as an electromagnetic wave obeying Maxwell's equations; required whenever interference, diffraction, polarization or coherence effects matter.
e) Why physical optics was needed. Geometrical (ray) optics cannot predict interference fringes, diffraction around edges/through apertures comparable to $\lambda$, or polarization — all are wave phenomena that only Maxwell's equations reproduce; it also cannot set a fundamental resolution limit (Sec. 2(c)/(d) below).
f) Plane of incidence. The plane containing the incident ray, the surface normal at the point of incidence, and the reflected/refracted ray.
g) Law of reflection. The incident ray, the normal, and the reflected ray are coplanar (all three lie in the plane of incidence), and the angle of reflection equals the angle of incidence, both measured from the normal: $\theta_r=\theta_i$.
h) Law of refraction (Snell's law). The incident ray, the normal, and the refracted ray are coplanar (again, the plane of incidence), and $n_1\sin\theta_1=n_2\sin\theta_2$.
i) Wavelength/frequency table (using $c=f\lambda$, $c\approx3.00\times10^{8}$ m/s):
| Band | Wavelength range | Frequency range |
|---|---|---|
| UV light | 100–400 nm | $7.50\times10^{14}$–$3.00\times10^{15}$ Hz |
| Red light | 620–750 nm | $4.00\times10^{14}$–$4.84\times10^{14}$ Hz |
| Blue light | 450–495 nm | $6.06\times10^{14}$–$6.67\times10^{14}$ Hz |
| IR light | 750–1000 nm | $3.00\times10^{14}$–$4.00\times10^{14}$ Hz |
j) Transverse plane wave. A plane-wave solution of the source-free Maxwell equations in which $\mathbf E$ and $\mathbf H$ both oscillate perpendicular to the propagation direction $\hat{\mathbf k}$ (and to each other), with constant-phase surfaces that are infinite planes normal to $\hat{\mathbf k}$: $\mathbf E(\mathbf r,t)=\mathbf E_0\cos(\mathbf k\cdot\mathbf r-\omega t)$, $\mathbf E_0\cdot\mathbf k=0$. The condition $\rho=0,\ \mathbf J=0$ (no free charge/current) is exactly what forces $\nabla\cdot\mathbf E=0$, which in turn forces $\mathbf E_0\perp\mathbf k$ — i.e. it is this source-free condition that makes the wave transverse.
k) Are monochromatic plane waves restrictive? No. Any real (broadband, non-planar) field can be built as a superposition of monochromatic plane waves via a Fourier transform in space and time; monochromatic-plane-wave analysis is therefore completely general as a basis set, even though no real source emits a single infinite plane wave of one exact frequency.
l) Speed of propagation of $f(x,t)=A\,e^{\alpha x}\sin\!\big(2\pi(\alpha t+\beta x)\big)$. The phase is $\Phi=2\pi(\alpha t+\beta x)$; a point of constant phase satisfies $\alpha\,dt+\beta\,dx=0$, so $v_{\text{phase}}=dx/dt=-\alpha/\beta$ — propagation is in $-x$ for $\alpha,\beta$ of the same sign, with speed $|\alpha/\beta|$. The multiplicative envelope $e^{\alpha x}$ (amplitude growing or decaying along $x$, e.g. a wave in a gain or lossy medium) plays no role in the phase velocity: only the coefficients inside the sinusoid's argument set how fast the wavefronts move.
m) Dispersion. The dependence of a material's refractive index on wavelength/frequency, $n=n(\lambda)$ — the physical origin of a prism splitting white light into colours, since each $\lambda$ refracts by a different amount.
n) Numerical aperture. $\mathrm{NA}=n\sin\theta_{\max}$, where $\theta_{\max}$ is the half-angle of the widest ray cone the system accepts (or emits) and $n$ is the refractive index of the surrounding medium; it sets the light-gathering power and the diffraction-limited resolution.
o) f-number. $f/\#=f/D$, the ratio of an optical system's focal length to the diameter of its entrance pupil (effective aperture); it sets the light-gathering power and depth of focus, and is directly related to NA by $f/\#\approx1/(2\,\mathrm{NA})$ for a system imaging from air at small angles.
