17-Phys-A7 Optics · December 2018

Question 9 of 10: Two-source interference and the Michelson interferometer

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 9: Two-source interference and the Michelson interferometer (10 marks)

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) — interference at the on-axis centre point. At $x_1=0$ the point is exactly equidistant from both sources, so the two waves arrive with zero path-length difference. But the sources themselves start $180^\circ$ out of phase, so the two arriving fields are $E_1=E_0\cos(\omega t)$ and $E_2=E_0\cos(\omega t+180^\circ)=-E_0\cos(\omega t)$ — equal in magnitude, opposite in sign, at every instant. Their sum is identically zero: $E_1+E_2=0$. This is destructive interference (a dark point) at the centre, the opposite of the usual in-phase two-source result, because the inherent $180^\circ$ source offset is not compensated by any path-difference phase at $x_1=0$.

Part (b) — first off-axis bright fringe.

Given. Source spacing $d=1$ mm, screen distance $L=30$ mm, $\lambda=0.5$ mm (as given), sources $180^\circ$ out of phase.

Find. $x_2$, $\theta_2$ of the first off-axis intensity maximum.

Approach. Because the sources are antiphase, the roles of "bright" and "dark" fringe conditions swap relative to the standard in-phase two-source formula: a path difference of $\Delta r=(m+\tfrac12)\lambda$ (half-integer multiples) now gives constructive interference (it adds the extra $180^\circ$ needed to bring the two waves back in phase), with $m=0$ giving the first off-axis bright fringe. Since $d$ and $L$ are comparable here (the angle is not small), $\Delta r$ is evaluated exactly from the two source-to-point distances rather than the small-angle approximation.

1. Exact path-difference condition. With the screen point at height $x$ and sources at $\pm d/2$, $\Delta r(x)=\sqrt{L^2+(x+d/2)^2}-\sqrt{L^2+(x-d/2)^2}$. Setting $\Delta r=\lambda/2=0.25$ mm and solving numerically for the smallest positive $x$ gives $\boxed{x_2=7.75\ \text{mm}}$, $\theta_2=\arctan(x_2/L)=\boxed{14.48^\circ}$.
2. Small-angle cross-check. The standard paraxial two-slit approximation $\Delta r\approx dx/L$ gives $x_2\approx\dfrac{L\lambda}{2d}=\dfrac{(30)(0.5)}{2(1)}=7.5$ mm, $\theta_2\approx14.0^\circ$ — close to, but a few percent below, the exact value, because at $\theta_2\approx14^\circ$ the small-angle approximation is only mildly stressed; the exact geometric result is reported as the primary answer.

Part (c) — Michelson interferometer operation. Light from the source strikes a $45^\circ$ beam splitter (a partially silvered mirror), which divides the beam into two paths of roughly equal intensity: one transmitted toward a fixed mirror $M_1$, the other reflected toward a mirror $M_2$ mounted on a precision (often micrometer-driven) translation stage. Each beam reflects off its mirror and returns to the beam splitter, where the two are recombined into a single output beam travelling toward the detector/screen. Because both beams originate from the same source and beam splitter, they are mutually coherent, and the recombined field's intensity depends on the phase difference accumulated over the two round-trip optical path lengths — primarily $2\Delta L$, twice the difference in mirror-to-splitter distance, plus any phase differences intrinsic to the mirrors or optical elements in each arm. Scanning $M_2$ sweeps this path-length difference continuously, producing alternating bright and dark fringes at the detector each time $2\Delta L$ changes by one wavelength — the basis of the instrument's use for precision length/displacement metrology, refractive-index measurement (inserting a sample in one arm), and, in the historical Michelson–Morley context, testing for a directional dependence of the speed of light.

Part (d) — interpreting the test-mirror fringe pattern. The observed fringes are described as straight, parallel, and equally spaced across the entire field of view. In a Michelson interferometer with a test mirror in place of $M_1$, this specific pattern (rather than curved, irregular, or locally distorted fringes) indicates that the test mirror surface is optically flat — free of local bumps, pits, or figure error over the tested aperture. Straight, evenly spaced fringes arise from a uniform, constant-rate path-length difference across the field, i.e. a small residual tilt (wedge) between the effective planes of $M_1$ (test) and $M_2$ (reference) with no additional local variation; any departure from flatness on the test mirror (a scratch, a dig, a low or high spot) would locally perturb the path length and bend or break the otherwise-straight fringes at that location. The fringe spacing further quantifies the tilt angle ($\Delta L$ per fringe $=\lambda/2$), while the fringes' straightness and regularity is itself the qualitative pass/fail evidence of surface flatness.