Interference Fringes from a Wedged Etalon
A student illuminates a thin glass wedge with a helium-neon laser at 632.8 nm and observes straight, equally spaced fringes on a screen. The wedge angle is nominally 2.0 × 10⁻⁴ rad, and the glass index is 1.52. The student counts 24 bright fringes across a 12 mm span on the wedge surface and finds the spacing slightly irregular near one edge. They must decide whether the fringe count is consistent with the nominal wedge angle, and whether the irregularity indicates a defect or a measurement artifact. The laser linewidth is 1 pm and the coherence length is about 0.2 m.
1. Calculate the expected fringe spacing for the nominal wedge angle and compare it with the observed spacing.
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For a wedge in air with index n, adjacent bright fringes occur where the round-trip optical path changes by one wavelength. The condition gives fringe spacing Δx = λ / (2 n θ) for fringes viewed normal to the wedge surface. Plugging in: Δx = 632.8 × 10⁻⁹ / (2 × 1.52 × 2.0 × 10⁻⁴) = 632.8 × 10⁻⁹ / (6.08 × 10⁻⁴) ≈ 1.04 × 10⁻³ m, about 1.04 mm. Over 12 mm, that predicts about 11.5 fringes, not 24. The observed count is roughly double the prediction, which suggests either the wedge angle is about twice the nominal value (≈4 × 10⁻⁴ rad) or the student is counting both bright and dark fringes. Counting 24 bright fringes over 12 mm implies about 0.5 mm spacing, i.e., θ ≈ 4.2 × 10⁻⁴ rad.
2. The irregular spacing near one edge is about 15% off. Is this more likely a wedge defect or a coherence artifact? Justify with numbers.
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Coherence length is about 0.2 m, far larger than the path differences involved (a few wavelengths to tens of wavelengths across the wedge), so coherence is not the cause. A 15% local change in fringe spacing corresponds to a 15% local change in wedge angle or in the effective index/path. That is a plausible physical defect such as a non-flat surface or a localized thickness bump. The path difference at the 24th fringe is about 24 λ ≈ 15 µm of optical path, which is well within coherence. So the irregularity is more consistent with a surface defect than with coherence. The student should map fringe positions across the whole wedge to localize the defect.
3. If the student instead used a white-light source, how would the pattern change, and what would that reveal about the wedge?
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White light has a coherence length of only a few micrometers, so fringes would appear only near the region where the wedge optical path difference is within a few wavelengths of zero - a few colored fringes near the apex or contact line, washing out elsewhere. The number of visible fringes would be small (order 5–10), and their colors would vary with order. That behavior would confirm the wedge is thin near the contact edge and would let the student locate the zero-order line precisely. It would also show that the 24-fringe count with the laser depends on the long coherence of the laser, not on the wedge being ideal.