17-Phys-A7 Optics · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
98-Phys-A7, Optics — National Exams, December 2016. 3 hours; closed book (formula sheet supplied on pages 6–8). 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. (matrix methods, Ch. 18; fibre waveguides, Ch. 24; thin films, Ch. 15).
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 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 or through apertures comparable to $\lambda$, or polarization — all are wave phenomena that only Maxwell's equations reproduce; it also sets no fundamental resolution limit, unlike the diffraction-based Rayleigh criterion used in Q2(d) and Q5(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) Spherical aberration. A monochromatic aberration in which rays striking a spherical surface or lens far from the axis are focused at a different point than paraxial (near-axis) rays, so a finite-aperture spherical optic has no single sharp focus — the image blurs into a disc even for an on-axis point source.
k) Chromatic aberration. Because $n=n(\lambda)$ (dispersion), a lens has a different focal length for each wavelength; a white-light image therefore shows colour fringing, since different colours come to focus at different planes (longitudinal) or different heights (lateral).
l) Astigmatism. An off-axis (oblique-incidence) aberration in which rays lying in two orthogonal planes through the lens (tangential and sagittal) come to focus at two different axial distances, so the image of an off-axis point is not a single point but two separated, mutually perpendicular line foci — no single sharp image plane exists off-axis.
m) Index of refraction. $n=c/v$, the ratio of the speed of light in vacuum to its phase speed in the medium (equivalently $n=\sqrt{\varepsilon_r\mu_r}$ for a linear medium).
n) Group index of refraction. $n_g=n-\lambda\dfrac{dn}{d\lambda}=c/v_g$, the index that governs the speed of a wave packet's envelope (group velocity $v_g=d\omega/dk$); it equals the ordinary (phase) index $n$ only in a non-dispersive medium ($dn/d\lambda=0$).
o) f-number. $f/\#=f/D$, the ratio of an optical system's focal length to the diameter of its entrance pupil (clear aperture); it sets the light-gathering power and depth of focus, and relates to NA by $f/\#\approx1/(2\,\mathrm{NA})$ for a system imaging from air at small angles.
