Incoherent two-source Airy profile ($Q_s(x)=\frac{1}{2}\{P(b(x-s/2))+P(b(x+s/2))\}$)

Let $x$ measure position along the line joining two mutually incoherent optical point sources in the focal plane, with equal detected radiant fluxes and separation $s$. Extend the even Airy disk profile $P(u)$ to signed $u$, and put $b=\pi D/(\lambda f)$ as in Airy disk normalization and encircled energy. Relative to the peak of one source carrying the pair's total radiant flux, their summed light intensity isThe two-dimensional point spread functions underlying these line profiles all have the same total radiant flux. The line-profile integrals themselves need not equal one.

The dashed curves show individual contributions; the solid curve is their sum. All panels use the same brightness scale, and the vertical markers indicate the true source centres. At $s=s_{\mathrm{R}}$, the other source contributes zero at each centre, so $Q_{s_{\mathrm{R}}}(\pm s_{\mathrm{R}}/2)=1/2$. At the midpoint,The midpoint is therefore about $26.50\%$ below the source-centre peak. This is a calculation for this particular Airy disk and equal-incoherent-source convention, not a universal experimental threshold. At $0.7s_{\mathrm{R}}$ the two central peaks merge; at $1.5s_{\mathrm{R}}$ the valley is deeper. Changing separation continuously changes the profile, without a discontinuity at the Rayleigh criterion.

To see how information survives when the central peaks merge, set $z=bx$ and $\delta =bs$. A symmetric Taylor expansion givesThe linear terms cancel because the source positions are symmetric. At the midpoint, the Airy disk derivation gives $Q_s(0)=1-{\delta }^2/16+O({\delta }^4)$. Thus a close pair's shape differs from that of one source, but only quadratically at leading order. Detecting this small change becomes sensitive to noise and response calibration. For unequal flux fractions $\alpha$ and $1-\alpha$, the intensity instead becomes $\alpha P(b(x-s/2))+(1-\alpha )P(b(x+s/2))$; its valley and peak positions need not match the equal-source example.