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In a double-slit arrangement, one slit lies in medium 2 (refractive index n2) and the other is at the interface between medium 1 (index n1, with n1 not equal to n2) and medium 2. The line joining the slits is perpendicular to the interface, with slit separation d (slit widths much less than d). A monochromatic parallel beam enters from medium 1. A detector far away in medium 2 lies at angle theta from the line of the slits, where theta equals the refraction angle of the beam. Considering the two nearly parallel rays reaching the detector, which statement(s) is/are correct?

1. The phase difference between the two rays is independent of d.
2. The two rays interfere constructively at the detector.
3. The phase difference between the two rays depends on n1 but is independent of n2.
4. The phase difference between the two rays varies with d and the angle of incidence, with theta the corresponding refraction angle.

Correct answer: The two rays interfere constructively at the detector.

Solution

The upper slit (at the interface) and lower slit (in medium 2) are separated by d perpendicular to the interface. The incident beam in medium 1 gives an extra optical path n1*d*sin(i) before the slits; the rays going to the detector at angle theta in medium 2 give an extra optical path n2*d*sin(theta). By Snell's law n1*sin(i) = n2*sin(theta), so these two contributions are equal and cancel, making the net path difference zero. Zero path difference means constructive interference at the detector. The phase difference does depend on both n1 and n2 (so statement 3 is wrong) and is not independent of d in general except through the Snell cancellation here; the unambiguously correct statement is that the two rays interfere constructively.

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