Optics 065 · Diffraction, Fourier optics, and computational imaging

Phase Microscopy Lab

Can a transparent cell be seen without staining, and is the relief it seems to have real? It can. Zernike phase contrast, darkfield and DIC all turn the delay the cell gives the light into brightness differences, with no stain. The relief is not real height, though. DIC shading depends on direction, so the cell looks lit from one side, and phase contrast adds halos around edges.

Use a phase microscope with four imaging modes to make transparent patterns visible. See where halos and false relief appear, and follow the same complex specimen field all the way to the sensor.

Peak input phase / coherent cutoff

\phi_{\rm peak},\nu_c

$0.50$

Raw mean / empty-specimen background

\langle I\rangle,I_{\rm empty}

$50\%$

Raw intensity extrema

I_{\min},I_{\max}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Phase Microscopy Lab

Imaging mode (names on the buttons below)

m

1

Specimen pattern (names on the buttons below)

o

0

Signed optical path difference amplitude

d_0

120\,\mathrm{nm}

Monochromatic wavelength

\lambda

550\,\mathrm{nm}

Objective NA (changes the real opening)

\mathrm{NA}

0.18

Phase-ring delay (programmed at selected wavelength)

\delta

90^{\circ}

Phase-ring intensity transmission

T_{\rm ring}

0.20

Object-referred DIC shear distance

s

1.50\,\mathrm{\mu m}

DIC shear direction (clockwise from image right)

\alpha

0^{\circ}

DIC bias phase

\beta

60^{\circ}

Shared linear display gain (raw data unchanged)

G

1

A transparent cell can be nearly invisible in brightfield and appear once a thin phase ring is inserted. Compare four computed images of the same pure-phase specimen: the Zernike halo, the directional relief of DIC and weak darkfield scattering are different optical measurements. Reverse the ring phase or rotate the DIC shear, and the apparent shading changes while the specimen height stays the same. All records use one fixed display gain, and the raw center cross-sections are kept separately. The optical-path-difference map is a known input, not a recovered phase or a photograph of a real cell.

The specimen is synthetic, thin and purely phase, treated in the scalar paraxial approximation with periodic sampling and sixteen incoherent sources on a ring. DIC is modeled as ideal coherent lateral shear, not as a full commercial polarizing microscope. The unit-magnification inverting relay is drawn unfolded, and apparatus parts use unequal scales. Absorption, multiple scattering, camera noise and phase recovery are not included.

Physics tutorial

Audit the physics of Phase Microscopy Lab

BackgroundA thin transparent specimen changes the phase of the light passing through it but not its local intensity: $\phi=2\pi\,\mathrm{OPD}/\lambda,\qquad t=e^{i\phi}$ . The specimen is a known, programmed map of optical path difference (OPD), not a photograph. The same specimen is sent through four different optical setups.

Why it mattersCan a transparent cell be seen without staining, and is the relief it seems to have real?

Start with the essentials

Focus question: Can a transparent cell be seen without staining, and is the relief it seems to have real?
One-sentence intuition: Light from mutually incoherent illumination directions adds in intensity, not in amplitude: $I=\sum_j w_j|U_j|^2$ . Brightfield and Zernike phase contrast share the same ring of sixteen source points. Darkfield moves the direct illumination outside the angles the objective accepts. DIC uses an ideal polarized pair of fields on the axis instead, so its illumination is not the same partially coherent source.

Core mathematical model

The annular phase filter in the pupil

\begin{aligned}P(\boldsymbol\nu)&=\mathbf1_{|\boldsymbol\nu|\le\nu_c}\\&\quad\cdot\left[1+\left(\sqrt{T}e^{i\delta}-1\right)\mathbf1_{\rm ring}\right]\\\nu_c&=\mathrm{NA}/\lambda\end{aligned}

The ring is centred at sixty-five percent of the coherent pupil radius and has a full width of fifteen percent. The transmission is an intensity transmission, so the amplitude uses its square root. The ring phase is set for the chosen wavelength; it is not a fixed manufactured coating.

Ideal DIC: two shifted fields interfere

\begin{aligned}U&=\mathcal F^{-1}\{P_{\rm clear}\mathcal F[t]\}\\I_{\rm DIC}&=\frac14\left|e^{i\beta/2}U(\mathbf r+\mathbf s/2)\right.\\&\quad\left.-e^{-i\beta/2}U(\mathbf r-\mathbf s/2)\right|^2\end{aligned}

An equal split and an ideal crossed analyzer give the factor of one quarter relative to the total incident intensity. The shear direction is measured clockwise from the right of the image. The fields are shifted with Fourier phase ramps, not by rounding to whole pixels. The pupil filter acts before the two fields interfere.

Why the shadow is directional

\begin{aligned}I_{\rm DIC}/I_0&=\sin^2\!\left(\frac{\beta+\Delta\phi}{2}\right)\\\Delta\phi&\approx\mathbf s\cdot\nabla\phi\\I_{\rm empty}/I_0&=\sin^2(\beta/2)\end{aligned}

This pure-phase limit assumes the objective passes the field that matters. The gradient approximation needs a shear that is small compared with the features. The full model keeps the exponential specimen field even when the phase is not weak, and at larger OPD the contrast follows no general linear law.

Darkfield blocks the direct light; gain only changes the display

\begin{aligned}|\boldsymbol\nu_{\rm DF}|&\approx1.2\nu_c>\nu_c\\B&=\min(1,G I/I_0)\\|t|^2&=1\end{aligned}

A discrete outer ring of illumination lights the same pure-phase object. The objective accepts no direct light, so an empty specimen gives zero intensity. The display gain scales all four images equally; it does not represent exposure physics, camera noise, automatic normalization or a phase reconstruction. The finite periodic cell and the sixteen source directions can cause discretization artifacts.

Common difficulties

Visible contrast is not a map of specimen height

Typical misconceptionEvery phase microscope shows true thickness; a brighter halo means more material.

Better mental modelZernike halos and shade-off appear because some diffracted light also passes through the phase ring. DIC responds to how the OPD changes along the shear direction, so rotating the shear changes the apparent shading. A uniform phase offset is invisible in every mode. No single intensity image is presented as a recovered phase.

Run the experiment

01
Scene 1: Nearly invisible brightfield
Start with brightfield, then switch to phase contrast without changing the specimen. Reverse the ring phase and compare the same raw centre cut.
What to observe: The annular condenser aperture and the phase plate in the back focal plane sit in conjugate pupil planes, not in planes conjugate to the specimen image. Raising the NA changes the physical stop and its matching ring.
02
Scene 2: Phase ring: contrast and halo
Use the wide phase disk to see the halo and shade-off. Then make the ring fully transmitting with zero delay, and the phase-contrast image turns back into brightfield.
What to observe: Darkfield records only light scattered into the objective. With no specimen, the direct illumination stops on an opaque region of the pupil and never passes through the absorbing detector.
03
Scene 3: DIC: direction is not height
Choose DIC and rotate the shear by a quarter turn. Then select darkfield and press the labeled exposure button. It changes only the display gain, not the raw intensity.
What to observe: All recorded images share one fixed linear gain and clipping level. The OPD colour map has a fixed range of positive and negative values. Neither shows a measured biological specimen.

Phase Microscopy Lab

Audit the physics of Phase Microscopy Lab

Core mathematical model

The annular phase filter in the pupil

Ideal DIC: two shifted fields interfere

Why the shadow is directional

Darkfield blocks the direct light; gain only changes the display

Common difficulties

Visible contrast is not a map of specimen height

Run the experiment

Scene 1: Nearly invisible brightfield

Scene 2: Phase ring: contrast and halo

Scene 3: DIC: direction is not height

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Diffraction, phase and surface structure