Optics 065 · Diffraction, Fourier optics, and computational imaging

Phase Microscopy Lab

A four-mode phase microscope: make transparent patterns visible, expose halo and false relief, and follow the same complex specimen field into the receiving 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 $\text{valid model regime}$

Validated scientific apparatus

Phase Microscopy Lab

Imaging mode (named buttons below)

m

1

Specimen pattern (named 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 (not raw data)

G

1

A transparent cell can hide in brightfield and appear when a thin phase ring is inserted. Compare four calculated images of the same pure-phase specimen: Zernike halo, directional DIC relief and weak darkfield scattering are different optical measurements. Reverse the ring phase or rotate the DIC shear; apparent shading changes without changing specimen height. All records use one fixed display gain, with raw centre cuts kept separate. The OPD map is a known input, never a recovered phase or a biological photograph.

Scalar paraxial thin pure-phase synthetic specimen, periodic sampling and sixteen incoherent annular sources; ideal coherent lateral-shear DIC, not a full commercial polarized microscope. Unfolded unit-inverting relay and unequal apparatus scales. No absorption, multiple scattering, camera noise or phase recovery.

Start with the essentials

Focus question: Can a transparent cell become visible without staining—and is its apparent relief real?
One-sentence intuition: Mutually incoherent illumination directions add intensities, not amplitudes: $I=\sum_j w_j|U_j|^2$ . Brightfield and Zernike use the same sixteen-point annulus. Darkfield moves direct illumination outside objective acceptance. DIC instead uses an ideal on-axis polarized pair; it is not the same partially coherent illumination.

Core mathematical model

A real annular phase filter

\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 with fifteen-percent full width. The transmission slider is INTENSITY transmission; amplitude uses its square root. Ring phase is programmed at the selected wavelength, not a fixed manufactured coating.

Ideal DIC: interfere two shifted fields

\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}

Equal splitting and an ideal crossed analyzer give the quarter factor relative to total incident intensity. The shear vector points clockwise from image right. Translation uses Fourier phase ramps, not rounded pixel shifts. Filtering occurs 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 phase-only limit assumes the objective passes the relevant field. The differential approximation requires small shear relative to feature scale. The full model retains the exponential specimen field even outside the weak-phase regime; larger OPD does not obey a universal linear contrast law.

Darkfield exclusion and honest 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 snapped outer annulus illuminates the same pure-phase object. No accepted direct field means zero empty-specimen intensity. Display gain multiplies all four records equally; it is not exposure physics, camera noise, automatic normalization or a phase reconstruction. The finite periodic cell and sixteen source directions can introduce 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 result from diffracted light also entering the phase ring. DIC depends on lateral OPD differences along its shear direction: turn the shear and apparent shading changes. A uniform phase offset is invisible in every mode. No single intensity record is labeled as phase recovery.

Run the experiment

01
Scene 1: Nearly invisible brightfield
Begin with brightfield, then choose 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 rear-focal phase plate are conjugate pupil planes, not specimen-image planes. Opening NA changes the physical stop and matched ring.
02
Scene 2: Phase ring: contrast and halo
Use the wide phase disk to see halo and shade-off. Make the ring fully transmitting with zero delay: its phase-contrast image must become brightfield.
What to observe: Darkfield records only light scattered into the objective. Empty-specimen direct illumination ends at an opaque pupil region, never behind the absorbing detector.
03
Scene 3: DIC: direction is not height
Choose DIC and rotate shear by a quarter turn. Then select darkfield and use the explicitly labeled exposure button; only display gain changes, not raw intensity.
What to observe: All recorded images use one fixed linear gain and clipping. The OPD colour diagnostic has a fixed signed range; neither is a measured biological specimen.