Optics 030 · Imaging, instruments, and visual systems

Build a Microscope

Build an infinity-corrected microscope, trace separate Köhler field and aperture relays, and resolve a fixed pair of specimen points using 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 $\text{valid model regime}$

Ideal paraxial apparatus with a scalar sine-pupil diffraction inset. Real lens prescriptions, high-NA vector effects, specimen scattering and detector noise are outside the model.

Physics tutorial

Audit the physics of Build a Microscope

BackgroundAn infinity-corrected objective sends a focused specimen into a collimated space. A separate tube lens forms the real, inverted image with $m=-f_{\mathrm{tube}}/f_{\mathrm{obj}}$ . The eyepiece magnifies that existing image; it cannot supply spatial frequencies the objective never collected.

Why it mattersHow should objective, illumination, and detector share numerical aperture to resolve a tiny object?

Start with the essentials

Focus question: How should objective, illumination, and detector share numerical aperture to resolve a tiny object?
One-sentence intuition: The incoherent two-point Rayleigh reference is $d_R=0.61\lambda/\mathrm{NA}_{\mathrm{obj}}$ . Brightfield periodic resolution also depends on illumination aperture and coherence; its Abbe support estimate must not be confused with 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 relaxed-eye eyepiece at 25 mm. The minus sign records inversion. Object-space resolution remains unchanged when only objective focal length changes at fixed NA.

Two distinct resolution references

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}})}

Rayleigh describes two incoherent points. The Abbe expression gives the periodic brightfield support limit for the stated angular illumination; it is not a full partially coherent transfer function or contrast prediction.

Köhler conjugates

\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 maps every launch slope from one field point to the same inverted image point. The yellow family represents field diaphragm to specimen; the violet family represents source to aperture pupil. Those families are not interchangeable.

Computed scalar defocus

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 sums two independent point intensities and keeps the object-space field at 4 micrometres. Zero defocus reduces to the circular Airy pattern. The uniform sine pupil and scalar approximation omit polarization and high-NA vector effects.

Common difficulties

Empty magnification

Typical misconceptionIncreasing magnification must split two unresolved specimen points.

Better mental modelMagnification enlarges the same optical image. Compare the fixed-field card while changing focal length: improvement requires aperture, wavelength, focus or a different imaging method, not a larger display.

Run the experiment

01
Scene 1: Geometric magnification
In geometric magnification, vary objective focal length and compare specimen position, objective pupil and signed intermediate-image magnification. Keep NA fixed and check that the image card does not gain detail.
What to observe: Shorter objective focal length increases image magnification while the fixed object-space diffraction card remains identical at fixed NA, wavelength and defocus.
02
Scene 2: Köhler illumination
Open Köhler illumination. Move illumination NA and follow the violet source/pupil relay separately from the yellow field relay. Read the Abbe support estimate without interpreting it as the fluorescence PSF.
What to observe: Closing illumination aperture changes periodic brightfield support while leaving the independent incoherent point PSF unchanged.
03
Scene 3: Diffraction-limited resolution
In diffraction and focus, bring specimen defocus to zero, then vary objective NA and wavelength. The two fixed points either merge or separate. Drag the marked NA control on the bench for the same adjustment.
What to observe: Larger NA improves in-focus lateral resolution but makes the same specimen displacement produce stronger defocus. The on-axis intensity tracks that tradeoff.