17-Phys-A7 Optics · May 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. National Examinations, May 2015, 98-Phys-A7 Optics. Three hours, closed book, one approved Casio or Sharp calculator. Question 1 is mandatory; the rubric asks for Question 1 plus any four of Questions 2–6 (only the first four questions appearing in the answer book are marked) — all six are worked here, because the set is a study resource.
Reference texts. E. Hecht, Optics, 5th ed.; F. L. Pedrotti, L. M. Pedrotti and L. S. Pedrotti, Introduction to Optics, 3rd ed.
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 that stimulates the human retina (and, more generally, the band of the EM spectrum — roughly 380–750 nm in vacuum — that a light-sensitive detector responds to); 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 between media, bending only at reflecting/refracting surfaces (Fermat's principle / the eikonal equation); valid when every aperture and feature is large compared with $\lambda$.
d) Physical optics. The treatment of light as an electromagnetic wave (Maxwell's equations), which is needed whenever interference, diffraction, polarization or coherence effects matter — phenomena geometrical optics cannot predict.
e) Why physical optics was needed. Geometrical optics cannot account for interference fringes, diffraction (bending of light around edges and through apertures comparable to $\lambda$), or polarization; all three are wave phenomena that only Maxwell's equations reproduce.
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, 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, and $n_1\sin\theta_1=n_2\sin\theta_2$.
i) Wavelength/frequency table (any two rows, using $c=f\lambda$ with $c\approx3.00\times10^{8}\ \text{m/s}$).
| Band | Wavelength range | Frequency range |
|---|---|---|
| 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 |
j) Ray ↔ plane wave. In the short-wavelength limit, a wave's phase fronts (surfaces of constant phase) become locally planar, and a geometrical-optics ray is defined as the trajectory everywhere normal to those wavefronts — i.e. the ray marks the local direction of propagation of the plane wave.
k) 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.
l) Numerical aperture. $\mathrm{NA}=n\sin\theta_{\max}$, where $\theta_{\max}$ is the half-angle of the widest cone of rays 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.
m) Dispersion. The dependence of a material's refractive index on wavelength, $n=n(\lambda)$ — the physical origin of a prism splitting white light into colours, since each $\lambda$ refracts by a different amount.
n) Spatial coherence. The degree to which the phase of a wave is correlated between two different points across a wavefront at the same instant — i.e. whether light leaving different points of a source can still produce stable interference fringes.
o) Temporal coherence. The degree to which the phase of a wave at one point is correlated with itself at a later time — governed by the source's spectral bandwidth, and setting the coherence length/time over which stable interference is possible.
p) Refractive index from wavelength. $n=\lambda_0/\lambda$ (frequency is unchanged crossing into a medium, so $v=f\lambda=c/n$ gives $\lambda=\lambda_0/n$).
q) Frequency in a medium. $\nu=\nu_0$ — frequency is set by the source and is invariant as light crosses into a medium (only the wavelength and phase velocity change, via $\lambda=\lambda_0/n(\nu)$).
r) Interference vs. diffraction. Interference is the superposition of a small number of discrete coherent waves (e.g. two beams), while diffraction is the superposition of a continuum of secondary wavelets from every point of a wavefront or aperture (Huygens–Fresnel); the distinction is one of convenience/degree rather than kind — a double slit is usually called interference, an N-slit grating or single aperture, diffraction.
s) Fraunhofer diffraction. The far-field limit, in which the source and/or observation point are effectively at infinity (or a lens is used to image infinity), so the diffracted wavefronts arriving at the observation plane are effectively plane waves and the pattern is the Fourier transform of the aperture function.
t) Fresnel diffraction. The near-field regime, where the observation plane is close enough to the aperture that wavefront curvature cannot be neglected; the diffraction integral keeps quadratic phase terms that Fraunhofer diffraction discards.
