Optics 030 · Imaging, instruments, and visual systems

Build a Microscope

What sets the smallest detail a microscope can resolve? The objective's numerical aperture, which measures how wide a cone of light it collects, sets the smallest detail it can resolve. The illumination should fill a matching cone; Köhler illumination is the standard way to set that up. Extra magnification only enlarges what the objective has already resolved, and the detector's pixels must then be fine enough to sample it.

Build an infinity-corrected microscope, trace the separate Köhler relays for field and aperture, and resolve a fixed pair of specimen points in a computed scalar diffraction image.

Signed image and visual magnification

m, M_{\mathrm{visual}}

$0.50$

Point and periodic resolution references

d_R, p_{\mathrm{Abbe}}

$50\%$

Focus and on-axis intensity

\Delta z, S_0

$0.00\pi$

Model regime $\text{valid model regime}$

The apparatus is ideal and paraxial, and the inset shows scalar diffraction from a sine pupil. Real lens prescriptions, vector effects at high NA, specimen scattering and detector noise are not included.

Start with the essentials

Focus question: What sets the smallest detail a microscope can resolve?
One-sentence intuition: For two incoherent points, the Rayleigh reference separation is $d_R=0.61\lambda/\mathrm{NA}_{\mathrm{obj}}$ . The finest periodic pattern that brightfield can resolve also depends on the illumination aperture and coherence. That Abbe estimate is a different quantity from this point-source criterion.

Core mathematical model

Objective and eyepiece magnification

m=-\frac{200\,\mathrm{mm}}{f_{\mathrm{obj}}},\qquad M_{\mathrm{visual}}=m\frac{250\,\mathrm{mm}}{25\,\mathrm{mm}}

The tube lens is fixed at 200 mm, and the eyepiece, used with a relaxed eye, at 25 mm. The minus sign means the image is inverted. At fixed NA, changing only the objective focal length leaves the resolution in the specimen unchanged.

Two different measures of resolution

d_R=\frac{0.61\lambda}{\mathrm{NA}_{\mathrm{obj}}},\qquad p_{\mathrm{Abbe}}=\frac{\lambda}{\mathrm{NA}_{\mathrm{obj}}+\min(\mathrm{NA}_{\mathrm{ill}},\mathrm{NA}_{\mathrm{obj}})}

The Rayleigh criterion applies to two incoherent points. The Abbe expression gives the finest period that brightfield can pass for the given range of illumination angles. It is not a full partially coherent transfer function, and it does not predict contrast.

Conjugate planes in Köhler illumination

\mathbf P(f)\mathbf L(f)\mathbf P(2f)\mathbf L(f)\mathbf P(f)=\begin{pmatrix}-1&0\\0&-1\end{pmatrix}

Each unfolded 4f relay sends every ray from one field point, whatever its launch slope, to the same inverted image point. The yellow rays image the field diaphragm onto the specimen; the violet rays image the source onto the aperture pupil. The two sets of conjugates are not interchangeable.

Scalar defocus of a point image

I(r,z)=\left|2\int_0^1\rho J_0(k\mathrm{NA}\,r\rho)\exp\!\left[\mathrm i kz\left(\sqrt{1-\mathrm{NA}^2\rho^2}-1\right)\right]\mathrm d\rho\right|^2

The image card adds the intensities of two independent points and keeps a fixed field of 4 micrometres in the specimen. At zero defocus the result is the Airy pattern of a circular pupil. The uniform sine pupil and the scalar approximation leave out polarization and the vector effects of high NA.

Common difficulties

Empty magnification

Typical misconceptionIncreasing magnification must split two unresolved specimen points.

Better mental modelMagnification only enlarges the same optical image. Watch the fixed-field card while you change the focal length. Seeing more detail takes a change of aperture, wavelength or focus, or a different imaging method; a larger display does not help.

Run the experiment

01
Scene 1: Geometric magnification
In the geometric magnification scene, change the objective focal length and compare the specimen position, the objective pupil and the signed magnification of the intermediate image. Keep NA fixed and check that the image card gains no detail.
What to observe: A shorter objective focal length raises the image magnification. With NA, wavelength and defocus fixed, the diffraction card for the fixed specimen field stays exactly the same.
02
Scene 2: Köhler illumination
Open Köhler illumination. Change the illumination NA and follow the violet source-to-pupil relay apart from the yellow field relay. The Abbe estimate you read there is not the fluorescence PSF.
What to observe: Closing the illumination aperture changes the finest period that brightfield can pass, but the PSF of an independent incoherent point stays the same.
03
Scene 3: Diffraction-limited resolution
In the diffraction-limited resolution scene, set the specimen defocus to zero, then change the objective NA and the wavelength. The two fixed points merge or separate. Dragging the marked NA control on the bench does the same thing.
What to observe: A larger NA sharpens the lateral resolution in focus, but the same specimen shift then causes stronger defocus. The on-axis intensity shows this tradeoff.