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A diffraction-limited system has an optical response close to the ideal diffraction response for its specified optical pupil, wavelength and optical focus. Improving the optical aberrations further makes little difference to that response. This describes the optics: a poorly sampled or noisy detector can still deliver an image with much worse usable diffraction-limited resolution.

The optical pupil determines which wave amplitudes and wave phases contribute to an image point. In a scalar, paraxial approximation at the focal plane of an optical lens in air, write the pupil transmission as $T(u,v)e^{i\varphi (u,v)}$, after removing the ideal focusing phase. The amplitude spread function is proportional to its scaled Fourier transform:Here $(u,v)$ are pupil coordinates, $(x,y)$ are image coordinates, $f$ is focal length, and $\lambda$ is the wavelength in air. For a uniformly transmitting, unobscured circular optical pupil at ideal optical focus, the result is an Airy disk with surrounding rings. A blocked centre, support structures, or nonuniform transmission changes the ideal point spread function; “diffraction limited” does not specify one universal spot shape. The existing Fourier model of a Lyot coronagraph illustrates how pupil and focal masks deliberately alter this response.^[ref]

Optical aberrations add pupil wave phase errors. Defocus contributes a quadratic pupil wave phase in the paraxial approximation, redistributing light intensity from the best-focused spot. The Strehl ratio compares the peak with the ideal response for the same pupil and total radiant flux. A broad optical spectrum produces a weighted sum of wavelength-dependent point spread functions; chromatic aberration can additionally change optical focus with wavelength. Thus the operating conditions belong with any claim that a system is diffraction limited.

$\boxed{\text{(a): two-photon excitation; (b): one-photon/confocal excitation.}}$ The compact emitting region in (a) indicates that excitation itself is strongly localized. In (b), the illuminated path generates out-of-focus fluorescence as well as fluorescence near the focus. A confocal detection pinhole can reject much of that unwanted emission from the recorded image; it does not prevent the dye solution from emitting it.

In ordinary one-photon excitation, the local fluorescence-generation rate is approximately proportional to excitation light intensity $I$. In unsaturated two-photon excitation microscopy, it is proportional to $I^2$. Simultaneous absorption is appreciable only in the high-intensity focal region, generally produced using a pulsed laser. Two lower-energy photons jointly excite the fluorophore; the excitation wavelength is therefore typically substantially longer than in its one-photon experiment. This reduces out-of-focus excitation and permits detection without a confocal pinhole.^[ref]

For a small detection pinhole, the effective confocal point spread function is approximatelyHere each factor denotes an intensity response. Excitation and detection jointly select the focal region. Enlarging the pinhole increases collected signal but weakens axial rejection and changes the effective point spread function. In two-photon excitation microscopy with broad collection, the spatial response is dominated byThe pinhole product and excitation square are different physical mechanisms even when simplified mathematical profiles resemble one another.^[ref]

Compare both lateral and axial widths using the actual wavelengths and objective. For a diffraction-limited system, their characteristic scalings arewhere $\mathrm{NA}$ is numerical aperture and $n_{\mathrm{med}}$ is the medium's refractive index. Numerical prefactors depend on the width criterion and high-aperture optical model. Axial resolution is generally worse than lateral resolution. A small confocal pinhole improves axial sectioning and can narrow the lateral response, at the cost of photon throughput.

As a simple Gaussian function example, $h(x)=\exp [-x^2/(2{\sigma }^2)]$ has squared profile $h(x{)}^2=\exp [-x^2/{\sigma }^2]$, so its width is reduced by $1/\sqrt{2}$. However, the two-photon illumination spot itself is broader at its longer excitation wavelength. Two-photon excitation is not automatically higher-resolution than confocal microscopy, and ordinary two-photon sectioning is not intrinsically super-resolution. Compare measured point spread functions, pinhole settings, numerical apertures, scattering and optical aberrations, not just the fact that one profile is squared.

The original model figure below separates focal confinement from wavelength-dependent width. It is an illustrative one-dimensional Gaussian function calculation, not a reproduction or quantitative fit of the supplied photograph.

The Strehl ratio compares the peak of an observed point spread function with the ideal diffraction-limited system peak for the same pupil and total flux. It measures how strongly wavefront errors redistribute light away from the central peak.