Optics 064 · Diffraction, Fourier optics, and computational imaging

PSF–OTF–MTF Imaging Lab

A pupil-to-image bench that exposes contrast reversal and false sampled periods through consistent PSF, signed OTF, MTF and receiving pixels.

Incoherent optical cutoff / sensor Nyquist

\nu_c^{\rm inc},\nu_N

$0.50$

On-axis Strehl / aperture throughput

S_0,\eta_{\rm pupil}

$50\%$

Signed incoherent transfer / magnitude

\Re\mathrm{OTF},\mathrm{MTF}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

PSF–OTF–MTF Imaging Lab

Effective NA in air (changes real aperture)

\mathrm{NA}

0.10

Monochromatic wavelength

\lambda

550\,\mathrm{nm}

Edge-to-centre defocus waves (not RMS)

D

0

Orthogonal astigmatism coefficient in waves

A

0

Obstruction radius / aperture radius

\epsilon

0

Coherence mode (named buttons below)

c

0

Target index (named buttons below)

o

0

Integer cycles per field (sinusoid / chart bars)

k

24

Pixel doubling index (doubles each step)

b

0

Fixed linear display gain (not raw data)

G

1

A lens can turn a bright stripe dark without changing its period. Compare focus with the contrast-reversal preset: the signed OTF goes negative while MTF remains positive. Then enlarge the sensor pixels: an apparently coarser stripe can be an alias, not new specimen detail. One complex pupil generates the point response, transfer curves and simulated receiving image. Switch between incoherent intensity convolution and coherent amplitude filtering; the OTF curve is explicitly an incoherent diagnostic in both modes. Open the real iris, obstruct its centre, or combine defocus with astigmatism to split directional response. Calculated pupil phase and enlarged records stay outside the 3D bench.

Scalar paraxial monochromatic periodic imaging; synthetic targets and programmed phase, not measured lens performance. Discrete box averaging, fixed display gain and unequal apparatus scales. OTF normalization and aperture throughput are separate; no partial coherence, vector focus, noise, reconstruction or expert certification.

Physics tutorial

Audit the physics of PSF–OTF–MTF Imaging Lab

BackgroundEach incoherent object point has an intensity response determined by the complex pupil: $\mathrm{PSF}=\frac{|\mathcal F^{-1}P|^2}{\int|\mathcal F^{-1}P|^2\,d^2r}$ . Its integral is normalized separately from the light lost at an obstruction. A phase error redistributes energy instead of uniformly dimming every frequency.

Why it mattersCan a lens swap bright and dark—and can pixels invent a different stripe period?

Start with the essentials

Focus question: Can a lens swap bright and dark—and can pixels invent a different stripe period?
One-sentence intuition: The same point-response record gives both transfer functions: $\mathrm{OTF}=\mathcal F\{\mathrm{PSF}\},\qquad\mathrm{MTF}=|\mathrm{OTF}|$ . For our inversion-symmetric pupil the OTF is real. Defocus can make it negative, meaning contrast reversal; its magnitude discards that sign.

Core mathematical model

Programmed complex pupil, not an RMS slider

\begin{aligned}P&=\mathbf1_{\epsilon\le\rho\le1}e^{2\pi i w}\\w&=D\!\left(\rho^2-\frac{1+\epsilon^2}{2}\right)\\&\quad+A(u^2-v^2)\end{aligned}

Dimensionless pupil coordinates are frequency divided by the coherent cutoff. Defocus is edge-to-centre waves; the astigmatism coefficient has opposite signs on orthogonal axes. Piston is removed over the ideal annulus. Controls program phase at the selected wavelength; they are not fixed physical glass thickness.

Two different forward models

\begin{aligned}I_{\rm inc}&=\eta\,\mathrm{PSF}*I_{\rm ideal}\\I_{\rm coh}&=\left|\mathcal F^{-1}\{P\mathcal F[\sqrt{I_{\rm ideal}}]\}\right|^2\end{aligned}

Both use the inverted programmed target. Incoherent source points have no cross terms; coherent points do. The pupil area fraction controls incoherent throughput, whereas coherent energy is determined by the actual amplitude spectrum. The display uses fixed linear gain and clipping only after the raw calculation.

Cutoffs, phase sign and sensor sampling

\begin{aligned}\nu_c^{\rm coh}&=\mathrm{NA}/\lambda\\\nu_c^{\rm inc}&=2\mathrm{NA}/\lambda\\\nu_N&=1/(2a_{\rm pixel})\\\Delta z&=-2\lambda D/\mathrm{NA}^2\end{aligned}

Frequencies are cycles per micrometre in nominal unit-magnification image coordinates. Positive defocus phase corresponds to a sensor closer to the second lens. The relay uses the paraxial focal-plane approximation: sensor motion neglects its tiny magnification change. Sampling above Nyquist can fold frequencies; optical bandwidth is not automatically pixel bandwidth.

Normalized shape versus transmitted energy

\begin{aligned}\mathrm{OTF}(0)&=1\\\eta&=N_P/N_{\rm clear}\\S_0&=|\sum P|^2/N_P^2\\B&=\min(1,G I/I_0)\end{aligned}

The finite-grid on-axis Strehl compares the same aperture with and without phase error. It is not a general maximum-peak ratio or a small-error exponential approximation. The PSF map uses a separate labelled logarithmic display relative to that same unaberrated aperture peak. The raw centre cut and transfer curves never contain display gain.

Common difficulties

A nonzero MTF is not a promise of a faithful picture

Typical misconceptionStrehl is a multiplier for the entire MTF, and any visible stripe is correctly resolved.

Better mental modelNormalized OTF at zero frequency is one even when the on-axis point-response peak falls. Contrast can reverse, and sampled pixels can alias to a false period. Inspect the signed curve and compare the raw input, optical and detector cuts. In coherent mode, filter amplitude first and square afterwards; an incoherent OTF cannot predict its intensity transfer.

Run the experiment

01
Scene 1: In focus: faithful contrast
Compare focus and reversal, then tune the integer-cycle sinusoid by clicking the signed-transfer chart. Home restores the default cycle count.
What to observe: A narrowband source can be spatially incoherent. Monochromatic does not imply mutual coherence between different object points.
02
Scene 2: Defocus: bright becomes dark
Use the real-aperture view while changing NA and central obstruction. Combine defocus with astigmatism and compare horizontal and vertical signed curves.
What to observe: The pupil is sampled on a finite grid; PSF and image are periodic. This is a forward simulation of original synthetic targets, not a measured image or a reconstruction.
03
Scene 3: Pixels: false stripe period
Choose aliasing and enlarge pixels. Compare the actual sampled record with the optical image before sampling; then switch to coherent amplitude imaging.
What to observe: A central obstruction reduces incoherent throughput. With coherent axial illumination it also removes the DC field, so a uniform target can go dark.