Q.Obtain the equation for resolving power of an optical instrument.

Step 1. Even a perfect optical instrument images a point source not as a true point but as a small bright central disc (Airy disc) surrounded by faint rings, because the instrument's own finite circular aperture (of diameter $a$) diffracts the light passing through it; the central disc's half-angular spread is $a\sin\theta=1.22\lambda$, the factor 1.22 specific to a circular aperture (versus the plain 1 for a rectangular slit).

Step 2. When two point sources are imaged close together, their two Airy-disc patterns can overlap; if the overlap is too great, the two patterns merge into one indistinct blur that cannot be told apart from a single source.

Step 3. Rayleigh's criterion defines the two sources as 'just resolved' when the central maximum of one image's Airy pattern falls exactly on the first minimum of the other's diffraction pattern -- the borderline case where a shallow but still visible dip remains between the two overlapping peaks.

Step 4. Since the angular position of a pattern's own first minimum is exactly $\theta=1.22\lambda/a$ (from Step 1, for small angles), this angular value is also the minimum angular separation at which Rayleigh's criterion is satisfied for two objects: $\boxed{\theta=\dfrac{1.22\lambda}{a}}$. …

Apply Rayleigh's criterion (central maximum of one pattern on the first minimum of the other) using the cir …

• Forgetting the 1.22 factor specific to a circular aperture, and using the plain single-slit formula theta = lambda/a instead. …