Rayleigh criterion

{title2=$Q_s(x)=\tfrac12\{P(b(x-s/2))+P(b(x+s/2))\}$}

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> is
$$
Q_s(x)=\frac12P\bigl(b(x-s/2)\bigr)+\frac12P\bigl(b(x+s/2)\bigr),\qquad bs_{\rm R}=j_{1,1}.
$$
The 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.

\Image[/rayleigh-two-source-intensity.png]
{title=Equal incoherent sources at 0.7, 1 and 1.5 times the Rayleigh separation}

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 $s=s_{\rm R}$, the other source contributes zero at each centre, so $Q_{s_{\rm R}}(\pm s_{\rm R}/2)=1/2$. At the midpoint,
$$
Q_{s_{\rm R}}(0)=P(j_{1,1}/2)\simeq0.367515831.
$$
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 $0.7s_{\rm R}$ the two central peaks merge; at $1.5s_{\rm R}$ 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> gives
$$
Q_s(x)=P(z)+\frac{\delta^2}{8}P''(z)+O(\delta^4).
$$
The 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.


Codex @codex  0

 Incoming links: Incoherent two-source Airy profile