What are the conditions for bright and dark fringes in reflected light from a thin film?

In one line

In reflected light from a thin film, brightness occurs when \(2\mu t\cos r = (2n-1)\lambda/2\) and darkness when \(2\mu t\cos r = n\lambda\).

Explained simply

Two waves that meet "crest on crest" add up to a bright fringe; "crest on trough" cancel to a dark one. Because one reflected ray has an extra half-wave shift, the usual rules flip: a film that would be bright for transmitted light is dark for reflected light, and vice versa.

How it works

1. Effective path difference in reflected light: \(\Delta = 2\mu t\cos r - \lambda/2\).
2. Maximum (bright): \(\Delta = n\lambda\), so \(2\mu t\cos r - \lambda/2 = n\lambda\), giving \(2\mu t\cos r = (2n+1)\lambda/2\), \(n = 0, 1, 2, \dots\) (equivalently \((2n-1)\lambda/2\) with \(n = 1, 2, \dots\)).
3. Minimum (dark): \(\Delta = (2n+1)\lambda/2\), which gives \(2\mu t\cos r = (n+1)\lambda\); renaming, \(2\mu t\cos r = n\lambda\), \(n = 0, 1, 2, \dots\).
4. Transmitted light: no extra \(\lambda/2\), so bright when \(2\mu t\cos r = n\lambda\) and dark when \(2\mu t\cos r = (2n+1)\lambda/2\). The reflected and transmitted patterns are complementary.
5. When \(t \to 0\), \(\Delta = -\lambda/2\), so a very thin film looks dark in reflected light (like the top of a soap film just before it bursts).

Key formula

Reflected, bright: \(2\mu t\cos r = (2n-1)\dfrac{\lambda}{2}\), \(n = 1,2,3\dots\). Reflected, dark: \(2\mu t\cos r = n\lambda\), \(n = 0,1,2\dots\). Symbols as before; \(n\) is the order of the fringe.

Quick example

Find the minimum thickness of an oil film (\(\mu = 1.4\)) that appears dark for \(\lambda = 560\) nm at normal incidence. Dark: \(2\mu t = n\lambda\); the smallest non-zero thickness uses \(n = 1\): \(t = 560/(2\times1.4) = 200\) nm. For it to look bright instead, \(2\mu t = \lambda/2\), \(t = 100\) nm.

Common mistakes

• Using the transmitted-light conditions for reflected light.
• Taking \(n = 0\) for the minimum non-zero thickness in a dark-fringe problem.
• Forgetting \(\cos r = 1\) at normal incidence, or using \(\cos i\).

How MU asks it

Usually part of the thin-film derivation (5 marks) or as a numerical: "find the smallest thickness for which the film appears dark/bright" (3 to 5 marks). Write both conditions clearly, mention the \(\lambda/2\) reflection shift, and for numericals state which \(n\) you chose and why.