Confocal Microscope Lab (2D): Pinhole, Optical Sectioning, Airy PSF and STED Resolution

Presets

Live readouts

Graphs

Lateral point-spread function: widefield, confocal, STED

Axial response and optical sectioning (thin sheet)

Pinhole trade-off: FWHM and light passed against pinhole size

Numerical checks

running...

How to use

Compare the three panels: widefield, confocal and STED. Try: (1) move the pinhole from 4 AU to 0.1 AU and watch the haze vanish and the beads sharpen (and the "light passed" curve drop); (2) use the focal plane height slider to look 1 to 2 micrometres above the test targets: the widefield image fills with haze while the confocal image stays dark; (3) raise NA or lower the wavelength and see the FWHM shrink; (4) enable STED and raise I / I_sat: pairs that were merged become resolved, but the signal falls; (5) read the table of resolved pairs in the readout. Space pauses, R resets.

The science

What this lab shows. Why a confocal microscope sees thin optical sections and slightly finer detail than a widefield microscope, how the pinhole trades resolution against light, and how STED depletion breaks the diffraction limit. Everything is computed from the point-spread function (PSF) of the objective.

1. The Airy pattern. A circular aperture of numerical aperture NA focuses light of wavelength λ into

h(r) = [ 2 J₁(v) / v ]², v = 2π NA r / λ

The first dark ring is at v = 3.8317, i.e. r = 0.61 λ / NA (Rayleigh); the full width at half maximum is 0.5145 λ / NA. Two points closer than about 0.61 λ / NA merge into one blob (Rayleigh criterion, 26.5 percent dip). Resolution therefore improves with shorter wavelength and higher NA (NA is limited by the immersion index n).

2. Confocal imaging. A point-like laser spot excites fluorescence with PSF h_ex, and the emission is detected through a pinhole conjugate to the focus, whose effective PSF is h_det convolved with the pinhole disc. The image of a point is the product

PSF_conf(r) = h_ex(r; λ_ex) · [ h_det(λ_em) ⊗ disc(a) ](r)

A pinhole of diameter D = 1 Airy unit has radius a = 0.61 λ_em / NA (the whole Airy disc). The convolution is evaluated exactly through the optical transfer function (a circular pupil autocorrelation) and the Fourier transform of the disc, 2 J₁(2π q a) / (2π q a). With a point pinhole the product is h² and the lateral FWHM falls to about 0.70 of the widefield value; a wide pinhole returns to the excitation PSF alone. The Stokes shift (emission longer than excitation) makes the detection PSF a little wider.

3. Axial behaviour and optical sectioning. In the paraxial (Lommel) approximation the defocused PSF is

h(v, u) = | 2 ∫₀¹ exp(-i u ρ²/2) J₀(v ρ) ρ dρ |², u = 2π NA² z / (n λ)

On the axis this is sinc²(u/4), so the widefield axial FWHM is 1.772 n λ / NA². A widefield microscope collects every plane equally: the signal from a uniform fluorescent sheet does not depend on its depth, which is why out-of-focus haze never goes away. With the pinhole the signal from the sheet is

S(z) = ∫ h_ex(r, z) · [ h_det(r, z) ⊗ disc(a) ] 2π r dr (normalised to z = 0)

which falls quickly with |z|. The width of S(z) is the optical section thickness. The graph on the right shows how a smaller pinhole improves both FWHM values but passes less light (the encircled energy 1 - J₀²(v) - J₁²(v) of the Airy pattern).

4. STED. A second, doughnut-shaped beam de-excites fluorophores by stimulated emission everywhere except at its central zero. A molecule is depleted with probability 1 - exp(-ln 2 · I / I_sat), so with a quadratic intensity near the zero, I(r) = ζ I_sat (r / r_d)², the effective excitation PSF is multiplied by exp(-ln 2 ζ r² / r_d²). For a Gaussian spot this gives exactly the square-root law

d = d₀ / √(1 + ζ) (with r_d = d₀ / 2), Abbe: d₀ = λ / (2 NA)

The lab uses r_d = λ_ex / (4 NA) and applies the depletion factor to the exact confocal PSF, so the numbers are close to, but not identical with, the square-root law. The cost is signal: the remaining fluorescence falls (shown as "STED signal kept") and the image becomes noisier for the same photon budget. STED here is lateral only.

5. The simulated images. Test targets (pairs of beads 110 to 700 nm apart), a helical filament running through 5 µm of depth and a cloud of out-of-focus beads are imaged by three virtual instruments. Each bead is drawn as a Gaussian with the FWHM computed above; out of focus the widefield spot spreads over a radius |z| tan α while the confocal image loses energy by S(z). Shot noise is Poisson-like. Each panel is auto-scaled like a detector gain setting.

Checks (run in the page). (1) The first zero of J₁ at 3.8317; (2) the Airy first-zero radius equals 0.61 λ / NA; (3) numerical widefield FWHM against 0.5145 λ / NA; (4) the point-pinhole confocal FWHM is about 0.70 of widefield; (5) the numerically integrated defocused PSF on axis equals sinc²(u/4) and conserves energy; (6) the Rayleigh dip of two Airy points is 0.735; (7) the widefield axial FWHM equals 1.772 n λ / NA²; (8) the STED square-root law, and that STED narrows the real PSF.

Simplifications. Scalar paraxial theory for the axial response (high-NA vector effects, refractive-index mismatch and aberrations are ignored); a single emission wavelength; a quadratic doughnut and an instantaneous, saturating depletion; no bleaching, no scanning-time or detector noise except shot noise; lateral STED only. The specimen is synthetic. Numbers are for teaching, not for instrument specification.