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\b[$\boxed{\text{(a): two-photon excitation; (b): one-photon/confocal excitation.}}$] 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.<Two-photon laser scanning fluorescence microscopy by Winfried Denk and others>[pp. 73–75, quadratic excitation, optical sectioning and pulsed illumination]{ref}
For a small detection pinhole, the effective confocal <point spread function> is approximately
$$
\boxed{h_{\rm conf}(\mathbf r)\propto h_{\rm exc}(\mathbf r)h_{\rm det}(\mathbf r)}.
$$
Here 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 by
$$
\boxed{h_{2p}(\mathbf r)\propto[h_{\rm exc,2p}(\mathbf r)]^2}.
$$
The pinhole product and excitation square are different physical mechanisms even when simplified mathematical profiles resemble one another.<Confocal microscopy (chapter manuscript) by Brad Amos and others>[Confocal formation and conjugate focal selection, Fig. 3; discussion of excitation and detection responses]{ref}
\b[Compare both lateral and axial widths using the actual wavelengths and objective.] For a <diffraction-limited system>, their characteristic scalings are
$$
\delta_{xy}\sim\frac{\lambda}{\mathrm{NA}},\qquad
\delta_z\sim\frac{n_{\rm med}\lambda}{\mathrm{NA}^2},
$$
where $\mathrm{NA}$ is <numerical aperture> and $n_{\rm med}$ 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 $1/\sqrt2$. However, the two-photon illumination spot itself is broader at its longer excitation <wavelength>. \b[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.
\Image[/past-exam-of-the-systems-biology-course-of-the-university-of-cambridge/2024/paper-2-optical-response.png]
{title=Local excitation nonlinearity and wavelength-dependent point-spread widths}
{description=Left: squaring the same normalized Gaussian illumination narrows the local generation profile. Right: a longer-wavelength illumination profile can remain broader after squaring than an illustrative confocal excitation-times-detection response. All widths are synthetic; peak normalization does not equate photon throughput or excitation dose.}
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