Approximation for the Rayleigh Resolution of a Circular Aperture

Abstract

Rayleigh's criterion states that a pair of point sources are barely resolved by an optical instrument when the central maximum of the diffraction pattern due to one source coincides with the first minimum of the pattern of the other source. As derived in standard introductory physics textbooks,1 the first minimum for a rectangular slit of width a is located at angular position θ = sin−1 (λ/a) for light of wavelength λ. If the angular separation of the two sources is small, we can use the small-angle approximation sin θ ≈ θ to conclude that the resolution is θmin = λ/a for a rectangular aperture. On the other hand, for a circular aperture of diameter D, the limiting angle is shown in optics texts2 to be θmin = 1.22 λ/D. The derivation of the numerical prefactor of 1.22 involves finding the zero of a Bessel function and is beyond the reach of introductory physics students. Consequently, elementary texts simply pull that prefactor out of thin air. The purpose of the present paper is to briefly explain why we expect a prefactor larger than unity and to make simple estimates of its value, using only algebra.