Airy Pattern and the Rayleigh Resolution Limit

Derivation

Airy Pattern and the Rayleigh Resolution Limit

D-209 Home PU-206 Threads light · waves Depends on Fraunhofer Diffraction as a Fourier Transform

Statement

For a uniformly illuminated circular aperture of diameter \(D\), Fraunhofer diffraction of monochromatic light of wavelength \(\lambda\) produces the far-field intensity \(I(\theta)=I_0\left[\dfrac{2J_1(x)}{x}\right]^2\) with \(x=\dfrac{\pi D}{\lambda}\sin\theta\), an axially symmetric pattern (the Airy disk with surrounding rings) whose first zero lies at \(\sin\theta_1=1.22\,\lambda/D\). Applying the Rayleigh criterion — two incoherent point sources are just resolved when the central maximum of one falls on the first minimum of the other — gives the diffraction-limited angular resolution \(\theta_{\min}=1.22\,\lambda/D\).

Why it matters

The circular aperture is the aperture of nearly every real optical instrument: telescopes, microscopes, camera lenses, the human eye. The Airy pattern is therefore the universal point-spread function of a diffraction-limited imaging system, and the factor \(1.22\) — the first zero of \(J_1\) divided by \(\pi\) — sets the hard, wave-optical ceiling on how finely any such instrument can resolve detail. No amount of aberration correction can push past it; only a larger aperture or a shorter wavelength can.

The result also cleanly demonstrates how a two-dimensional Fourier transform with circular symmetry throws up Bessel functions where a slit throws up a sinc. Understanding where the \(1.22\) comes from — and when it does not apply — separates a memorised formula from a working command of physical optics.

Assumptions

Fraunhofer (far-field) regime.If \(N_F = a^2/\lambda z \not\ll 1\) the Fresnel quadratic phase across the aperture cannot be dropped, the field is no longer the plain Fourier transform of the aperture, and the pattern is a Fresnel diffraction pattern instead of a clean Airy disk. Uniform amplitude and phase over the aperture.Apodisation (a graded transmission) or a phase error (aberration, defocus) reshapes the transform: the rings redistribute, the first zero moves, and the tidy \(1.22\) is lost. Scalar diffraction, small angles.The scalar theory ignores the vector nature of the field; at very high numerical aperture the polarisation-dependent vector corrections matter and the point-spread function is no longer perfectly axisymmetric. Monochromatic, spatially coherent illumination of the aperture.A broadband source superposes Airy patterns of different scale (chromatic blur); partially coherent or extended-source illumination changes how two object points’ intensities add and modifies the resolution bookkeeping. Incoherent, equal-brightness point sources for the resolution step.If the two sources are mutually coherent their fields, not intensities, add; the combined pattern depends on their relative phase and the Rayleigh two-source condition no longer marks a fixed resolution.

Derivation

\[ U(\mathbf{q}) \;=\; C\!\int\!\!\int_{\text{aperture}} A(x,y)\,e^{-i\,\mathbf{q}\cdot\boldsymbol{\rho}}\;dx\,dy, \qquad \mathbf{q}=k\,(\sin\theta_x,\sin\theta_y),\; k=\frac{2\pi}{\lambda} \]

The prior result (Fraunhofer diffraction as a Fourier transform) gives the far-field amplitude as the 2D Fourier transform of the aperture transmission \(A\); \(\mathbf{q}\) is the transverse spatial frequency conjugate to aperture position \(\boldsymbol{\rho}=(x,y)\). A

\[ A(\boldsymbol{\rho}) = \begin{cases}1,&\rho\le a\\[2pt]0,&\rho> a\end{cases}\qquad a=\tfrac{D}{2} \]

Specialise to a clear circular aperture of radius \(a\): unit transmission inside, zero outside. This is the only place the circular geometry enters. A

\[ U(q) = C\int_0^{a}\!\!\int_0^{2\pi} e^{-i q \rho\cos(\phi-\alpha)}\,\rho\,d\phi\,d\rho \]

Adopt polar coordinates in the aperture, \(\boldsymbol{\rho}=(\rho,\phi)\), and in frequency space, \(\mathbf{q}=(q,\alpha)\) with \(q=|\mathbf{q}|=k\sin\theta\). The Jacobian supplies the factor \(\rho\). By the circular symmetry of \(A\) the result cannot depend on \(\alpha\). B

\[ \int_0^{2\pi} e^{-i q\rho\cos(\phi-\alpha)}\,d\phi \;=\; 2\pi\,J_0(q\rho) \]

Standard integral representation of the zeroth-order Bessel function of the first kind, \(J_0(u)=\frac{1}{2\pi}\int_0^{2\pi}e^{-iu\cos\psi}d\psi\); the \(\alpha\)-shift drops out over the full period. This is where circular symmetry produces Bessel rather than sinc behaviour. B

\[ U(q) = 2\pi C\int_0^{a} J_0(q\rho)\,\rho\,d\rho \]

Substitute the angular result back; only the radial integral remains. A

\[ \int_0^{a} J_0(q\rho)\,\rho\,d\rho \;=\; \frac{a}{q}\,J_1(qa) \]

Use the Bessel recurrence \(\frac{d}{du}\!\left[u\,J_1(u)\right]=u\,J_0(u)\). Put \(u=q\rho\), so \(\int_0^{a}J_0(q\rho)\rho\,d\rho=\frac{1}{q^2}\int_0^{qa}u\,J_0(u)\,du=\frac{1}{q^2}\big[u J_1(u)\big]_0^{qa}=\frac{a}{q}J_1(qa)\), since \(uJ_1(u)\to0\) as \(u\to0\). C

\[ U(q) = 2\pi C\,\frac{a}{q}\,J_1(qa) \;=\; \pi a^2 C\,\frac{2J_1(qa)}{qa} \]

Collect factors and rewrite so the aperture area \(\pi a^2\) appears explicitly; the dimensionless combination \(qa\) is the natural variable. A

\[ x \equiv qa = k a\sin\theta = \frac{2\pi}{\lambda}\cdot\frac{D}{2}\sin\theta = \frac{\pi D}{\lambda}\sin\theta \]

Define the dimensionless radial coordinate \(x\) using \(k=2\pi/\lambda\) and \(a=D/2\). This packages wavelength, aperture size and observation angle into a single variable. A

\[ I(\theta) = |U|^2 = I_0\left[\frac{2J_1(x)}{x}\right]^2, \qquad I_0 = |\pi a^2 C|^2 \]

Intensity is the modulus squared of the amplitude. The normalisation is fixed by the on-axis limit \(\lim_{x\to0}\frac{2J_1(x)}{x}=1\) (from \(J_1(x)\simeq x/2\)), so \(I(0)=I_0\) is the central peak intensity. B

\[ J_1(x_1)=0 \;\Rightarrow\; x_1 = 3.8317\ldots \;=\; 1.2197\,\pi \]

The bright core (Airy disk) is bounded by the first dark ring, the first positive zero of \(J_1\). Its numerical value, \(x_1/\pi=1.2197\), is the origin of the famous factor. B

\[ \frac{\pi D}{\lambda}\sin\theta_1 = x_1 \;\Rightarrow\; \sin\theta_1 = \frac{x_1}{\pi}\,\frac{\lambda}{D} = 1.22\,\frac{\lambda}{D} \]

Set \(x=x_1\) in the definition of \(x\) and solve for the angular radius \(\theta_1\) of the first dark ring. A

\[ \boxed{\;\theta_{\min} \;=\; \theta_1 \;\approx\; 1.22\,\frac{\lambda}{D}\;} \qquad (\sin\theta\approx\theta) \]

Rayleigh criterion: two equal incoherent point sources are just resolved when the central peak of one Airy pattern coincides with the first zero of the other, i.e. their angular separation equals \(\theta_1\). For the small angles typical of instruments, \(\sin\theta\approx\theta\). A

Result

\[ I(\theta)=I_0\left[\frac{2J_1(x)}{x}\right]^2,\quad x=\frac{\pi D}{\lambda}\sin\theta \qquad\Longrightarrow\qquad \theta_{\min}=1.22\,\frac{\lambda}{D} \]

Reading. The far field of a round aperture is a bright central Airy disk carrying about \(83.8\%\) of the energy, ringed by faint concentric maxima. The disk’s angular radius, to the first dark ring, is \(1.22\,\lambda/D\). Two point objects closer together than this on the sky merge into a single blur: their combined intensity profile loses its central dip. Resolution therefore improves with a bigger aperture (\(D\!\uparrow\)) or a shorter wavelength (\(\lambda\!\downarrow\)) — the two levers every optical designer pulls.

Units check. \(x=\frac{\pi D}{\lambda}\sin\theta\) is (length/length)×(dimensionless) = dimensionless, as a Bessel argument must be. \(\theta_{\min}=1.22\,\lambda/D\) has units (length/length) = radians, a pure number, correct for an angle. \(I_0\) carries all the dimensional intensity; the bracket \([2J_1(x)/x]^2\) is dimensionless and equals \(1\) on axis.

Limiting cases

On axis, \(x\to0\): \(2J_1(x)/x\to1\), so \(I\to I_0\); the pattern is smooth and peaked, no singularity at the centre.
Slit limit (recover 1D): a long narrow rectangular aperture replaces \(2J_1(x)/x\) by \(\operatorname{sinc}\)-type behaviour with first zero at \(\sin\theta=\lambda/D\); the geometry factor drops from \(1.22\) to \(1.00\).
Large \(x\) (outer rings): \(J_1(x)\sim\sqrt{2/\pi x}\cos(x-\tfrac{3\pi}{4})\), so the ring intensities fall off as \(I\propto x^{-3}\) — the rings dim rapidly.
\(\lambda/D\to0\) (geometric-optics limit): \(\theta_{\min}\to0\); diffraction vanishes and a point source images to a point, recovering ray optics.
Microscope form: writing \(\theta\) through numerical aperture \(\mathrm{NA}=n\sin\alpha\) converts the angular limit into the linear resolution \(d\approx0.61\,\lambda/\mathrm{NA}\) (same \(1.22\), halved).

Breaks when

Near field / Fresnel regime. When the observation distance is not large compared with \(a^2/\lambda\) (Fresnel number \(N_F\gtrsim1\)), the quadratic phase term cannot be dropped, the far-field Fourier relation fails, and there is no clean Airy disk to define \(1.22\).
Aberrated or apodised aperture. A phase error (defocus, coma, spherical aberration) or a non-uniform transmission redistributes the rings and shifts the first zero; the Strehl-degraded point-spread function no longer has its first null at \(1.22\,\lambda/D\).
Coherent or non-uniform illumination of the two sources. The Rayleigh step assumes incoherent, equal-brightness points whose intensities add. Mutually coherent sources add amplitudes and the resolvability depends on relative phase; unequal brightness lets a bright source swamp a faint neighbour well inside \(1.22\,\lambda/D\).
Very high numerical aperture. At large \(\sin\alpha\) the scalar, small-angle theory breaks; vector diffraction gives a polarisation-dependent, slightly non-circular focal spot and corrects the numerical prefactor.
Sub-diffraction (super-resolution) techniques. Rayleigh is a criterion for a single passive linear aperture; methods that exploit prior knowledge, photon statistics, or fluorophore switching (deconvolution, STED, PALM/STORM) legitimately localise structure far below \(1.22\,\lambda/D\).

Failure modes

Using \(a\) where \(D\) belongs (or vice versa). The Bessel argument is \(x=\pi D\sin\theta/\lambda=k a\sin\theta\); mixing radius and diameter drops or inserts a factor of 2 and turns \(1.22\) into \(0.61\).
Quoting the sinc first zero. Writing \(\theta=\lambda/D\) (the single-slit result) for a round aperture — forgetting that the circular geometry’s first zero is at \(3.83\), not \(\pi\).
Squaring \(J_1\) at the wrong place. Forgetting that intensity is \(|U|^2\), so the amplitude \(2J_1(x)/x\) must be squared; the zeros are the same but the ring intensities and the \(83.8\%\) encircled-energy figure require the square.
Reading \(J_1(x)/x=0\) at \(x=0\). \(x=0\) is a removable point where \(2J_1/x\to1\), the central maximum — not a dark ring.
Confusing angular and linear resolution. \(1.22\,\lambda/D\) is an angle; the disk radius in the focal plane is \(1.22\,\lambda f/D=1.22\,\lambda F_\#\), and the microscope form uses \(0.61\,\lambda/\mathrm{NA}\).
Treating Rayleigh as a physical law. It is a conventional \(26\%\)-dip criterion; the true information limit depends on signal-to-noise, and Sparrow’s or a fitted criterion can differ.

Discussion

The appearance of \(J_1\) is not an accident of special functions but the direct fingerprint of circular symmetry under Fourier transformation. A slit’s aperture function factorises into two one-dimensional top-hats, each transforming to a sinc; a disk cannot be factorised, and the angular integral instead folds into a Bessel function — the natural “radial harmonic” of the plane. The whole difference between the slit’s first zero at \(\pi\) and the disk’s at \(3.83\) is the difference between \(\operatorname{sinc}\) and \(J_1(x)/x\). This is a recurring theme: the symmetry of the boundary dictates the special function that describes the field.

The number \(1.22=3.8317/\pi\) is thus a pure consequence of geometry, independent of \(\lambda\) and \(D\). Everything physical — how good your telescope is, how small a feature your microscope resolves — is carried by the ratio \(\lambda/D\); the \(1.22\) merely says “round aperture.” This clean separation of a dimensionless geometric constant from a physical scale is characteristic of diffraction problems and is worth internalising.

The Rayleigh criterion itself is a human convention layered on top of the physics. When the central peak of one Airy pattern sits on the first zero of the other, the summed intensity shows a central dip of about \(26\%\) below the flanking peaks — historically judged “just resolvable” by eye. Nothing in the wave equation singles out this configuration; the Sparrow criterion (peaks merge to a flat top) is tighter, and with high signal-to-noise one can fit two overlapping Airy profiles and localise their centroids far more precisely. The criterion is best read as a convenient, instrument-independent benchmark, not a fundamental barrier.

More deeply, the encircled-energy result — that \(83.8\%\) of the transmitted power falls inside the first dark ring, with the remaining fraction distributed over the rings as \(1-J_0^2(x_1)-J_1^2(x_1)\) evaluated at successive zeros — reflects that the Airy amplitude is the Hankel transform (the radial Fourier transform) of a top-hat. The same Hankel/Bessel machinery governs the modes of circular waveguides and the eigenfunctions of a circular drum: the zeros of \(J_1\) that set the dark rings are cousins of the zeros of \(J_0\) that set a drumhead’s nodal circles. Diffraction, waveguide cutoff and vibrational modes are one mathematical family viewed through different physical windows.

Common misconceptions. The \(1.22\) is not a property of light or of the medium — it is the first zero of \(J_1\) divided by \(\pi\), fixed purely by the round shape of the aperture. And resolving power is set by the aperture diameter, not by magnification: enlarging a blurred image only enlarges the blur. “Empty magnification” beyond the diffraction limit adds no information.

Worked examples

Two stars viewed with a telescope of aperture \(D=0.20\ \text{m}\) at \(\lambda=550\ \text{nm}\). Find the angular resolution and the smallest separation resolvable at the Moon’s distance \(r=3.84\times10^{8}\ \text{m}\).

Symbolic form first: \(\theta_{\min}=1.22\,\lambda/D\), and the transverse separation is \(s=r\,\theta_{\min}\). A

\[ \theta_{\min}=1.22\times\frac{5.50\times10^{-7}\ \text{m}}{0.20\ \text{m}}=3.36\times10^{-6}\ \text{rad}\;(\approx0.69'') \]

Insert numbers; \(\lambda\) and \(D\) both in metres so the ratio is dimensionless, result in radians (arcsec via \(\times 206265\)). A

\[ s=r\,\theta_{\min}=(3.84\times10^{8})(3.36\times10^{-6})=1.29\times10^{3}\ \text{m} \]

Arc length at distance \(r\); small-angle, so \(s=r\theta\). A

\[ \theta_{\min}\approx3.4\times10^{-6}\ \text{rad}\ (0.69''),\qquad s\approx1.3\ \text{km} \]

Reading. This 20 cm telescope, if diffraction-limited, separates features about \(1.3\) km apart on the Moon. Real ground-based seeing (turbulence, \(\sim1''\)) usually dominates, so the aperture limit is only reached from space or with adaptive optics.

Units check. \(\lambda/D\) dimensionless → \(\theta\) in radians; \(r\,\theta\) is m×rad = m. Consistent.

A camera lens at \(f/2.8\) (\(F_\#=2.8\)) photographs at \(\lambda=550\ \text{nm}\). Find the Airy disk radius in the focal plane and the diffraction-limited line resolution in line pairs per millimetre.

Focal-plane radius to first zero: \(r_{\text{disk}}=1.22\,\lambda f/D=1.22\,\lambda F_\#\), since \(F_\#=f/D\). Resolvable pitch \(\approx r_{\text{disk}}\). B

\[ r_{\text{disk}}=1.22\,\lambda F_\#=1.22\,(5.50\times10^{-7}\ \text{m})(2.8)=1.88\times10^{-6}\ \text{m}=1.88\ \mu\text{m} \]

Insert numbers; \(F_\#\) dimensionless, so \(r_{\text{disk}}\) inherits the units of \(\lambda\). A

\[ \text{resolution}\approx\frac{1}{r_{\text{disk}}}=\frac{1}{1.88\times10^{-3}\ \text{mm}}\approx5.3\times10^{2}\ \text{lp/mm} \]

One resolvable line pair per Airy radius (Rayleigh-style estimate); convert metres to millimetres. B

\[ r_{\text{disk}}\approx1.9\ \mu\text{m},\qquad \text{resolution}\approx530\ \text{lp/mm} \]

Reading. At \(f/2.8\) the Airy disk is under \(2\ \mu\text{m}\); on a sensor with \(4\ \mu\text{m}\) pixels the lens out-resolves the sensor. Stopping down to \(f/16\) grows the disk to \(\sim11\ \mu\text{m}\) and diffraction visibly softens the image — the origin of the “diffraction-limited aperture” in photography.

Units check. \(1.22\,\lambda F_\#\): length × dimensionless = length. \(1/\text{length}=\) lp/mm after unit conversion. Consistent.

Problems

Show from \(J_1(x)\simeq x/2 - x^3/16\) for small \(x\) that \(2J_1(x)/x\to1\) as \(x\to0\), and find the leading correction to the central intensity.
Solution
\(\dfrac{2J_1(x)}{x}=\dfrac{2}{x}\left(\dfrac{x}{2}-\dfrac{x^3}{16}+\cdots\right)=1-\dfrac{x^2}{8}+\cdots\). Squaring, \(I/I_0=\left(1-\tfrac{x^2}{8}\right)^2\approx1-\dfrac{x^2}{4}+\cdots\). So on axis \(I=I_0\), and the peak falls off quadratically, \(I\approx I_0(1-x^2/4)\), near \(x=0\).
The Hubble Space Telescope has \(D=2.4\ \text{m}\). Compute its diffraction-limited angular resolution at \(\lambda=500\ \text{nm}\), in radians and arcseconds.
Solution
\(\theta_{\min}=1.22\,\lambda/D=1.22\times\dfrac{5.0\times10^{-7}}{2.4}=2.54\times10^{-7}\ \text{rad}\). In arcseconds: \(2.54\times10^{-7}\times206265=0.052''\). About \(0.05\) arcsec — roughly the width of a coin seen from hundreds of kilometres.
A microscope objective has numerical aperture \(\mathrm{NA}=0.95\) in air. Using \(d=0.61\,\lambda/\mathrm{NA}\), find the smallest resolvable separation at \(\lambda=450\ \text{nm}\), and state why oil immersion (\(n=1.52\)) helps.
Solution
\(d=0.61\times\dfrac{4.5\times10^{-7}}{0.95}=2.9\times10^{-7}\ \text{m}=0.29\ \mu\text{m}\). Oil immersion raises \(\mathrm{NA}=n\sin\alpha\) up to \(\sim1.4\) because \(n\) increases, shrinking \(d\) to \(\sim0.20\ \mu\text{m}\): a higher-index medium lets the objective collect steeper (higher spatial-frequency) rays, improving resolution.
For a circular aperture, the fraction of total energy inside a circle of radius \(x\) (in the pattern) is \(1-J_0^2(x)-J_1^2(x)\). Evaluate the encircled energy inside the first dark ring \(x_1=3.8317\), given \(J_0(3.8317)=-0.4028\), \(J_1(3.8317)=0\).
Solution
Encircled fraction \(=1-J_0^2(x_1)-J_1^2(x_1)=1-(-0.4028)^2-0^2=1-0.1622=0.8378\). So \(\approx83.8\%\) of the transmitted energy lies within the central Airy disk, the standard result.
The dark-adapted human eye has a pupil \(D\approx6\ \text{mm}\). At \(\lambda=550\ \text{nm}\), find the diffraction-limited angular resolution, convert to a linear separation at a reading distance of \(0.25\ \text{m}\), and compare with the \(\sim1'\) (arcminute) resolution actually achieved.
Solution
\(\theta_{\min}=1.22\,\lambda/D=1.22\times\dfrac{5.5\times10^{-7}}{6\times10^{-3}}=1.12\times10^{-4}\ \text{rad}\). In arcminutes: \(1.12\times10^{-4}\times(180/\pi)\times60=0.38'\). Linear separation at \(0.25\ \text{m}\): \(s=r\theta=0.25\times1.12\times10^{-4}=2.8\times10^{-5}\ \text{m}=28\ \mu\text{m}\). The eye’s realised resolution (\(\sim1'\)) is about \(2.6\times\) coarser than the diffraction limit, because photoreceptor spacing on the retina and optical aberrations, not diffraction, are the binding constraint at this pupil size.