Optics 037 · Imaging, instruments, and visual systems

Smartphone Camera Module

Build a fixed-sensor wide, periscope or macro module. Real lens autofocus and lateral OIS couple to an opaque folding mirror, a computed Airy exposure image, exact pixel integration and signed motion transfer before sampling.

Field / focused spacing / magnification

\theta_h,v,|m|

$0.50$

Airy diameter / optical cutoff / Nyquist

d_A,\nu_c,\nu_N

$50\%$

Lens shift / residual pixels / thickness

\delta_{\mathrm{pk}},A_{\mathrm{res}}/p,T

$0.00\pi$

Model regime $\text{valid model regime}$

Ideal finite-conjugate thin lens and 550 nm central incoherent circular-pupil tests; one full sinusoidal shake cycle. Not a commercial lens prescription or actuator controller. Aberrations, Bayer processing, noise, rolling shutter and electronic cropping are omitted.

Start with the essentials

Focus question: How do real folding, autofocus, diffraction, pixels and lens-shift stabilization compete inside a thin camera module?
One-sentence intuition: For small angular shake, ideal lens translation cancels image motion with $A_{\mathrm{res}}=v\alpha_{\mathrm{pk}}(1-g)$ . It cannot undo diffraction or recover information aliased by coarse pixels. The cards integrate one complete shake cycle; the apparatus shows its peak snapshot.

Core mathematical model

Finite autofocus and field

v=\frac{fs}{s-f},\quad m=-\frac{v}{s},\quad \theta_h=2\arctan\frac{w}{2v}

Use image distance, not focal length, in the finite-distance chief-ray field proxy. The lens moves while the 6.4 by 4.8 mm sensor stays fixed. Aperture changes the real opening, not the glass curvature.

A genuine mirror fold

\mathbf d_r=\mathbf d_i-2(\mathbf d_i\cdot\hat{\mathbf n})\hat{\mathbf n}

The finite circular mirror turns the axial ray through 90 degrees. Missed mirror, blocked lens aperture and out-of-sensor rays terminate at their first rejection. The open wireframe is an 8 mm thickness budget, not an opaque phone case; the budget includes mounts.

Finite-conjugate diffraction

N_{\mathrm{eff}}=\frac{v}{2a}=N(1+|m|),\quad d_A\simeq2.44\lambda N_{\mathrm{eff}},\quad \nu_c=\frac{1}{\lambda N_{\mathrm{eff}}}

At 550 nm, the unit-total-flux circular-pupil point image broadens with effective f-number. The diameter is its first-zero diameter, not pixel resolution. One fixed physical window and square-root display gain preserve comparisons; the card is not a photon-throughput measurement.

Circular-pupil contrast transfer

H_{\mathrm{opt}}(t)=\frac{2}{\pi}\left(\arccos t-t\sqrt{1-t^2}\right),\quad t=\frac{|\nu|}{\nu_c}\le1

For ideal incoherent imaging, the pupil autocorrelation gives this optical transfer, with zero beyond cutoff. The cyan MTF omits pixels and shake; the green curve includes them and displays the magnitude of their signed transfer.

Pixel aperture before sampling

H_{\mathrm{pix}}(\nu)=\frac{\sin(\pi\nu p)}{\pi\nu p},\quad \nu_N=\frac{1}{2p}

Each square pixel integrates intensity before sampling at its center. Four independent infinite sinusoidal inputs have fixed frequencies and a fixed window. Above Nyquist a carrier can masquerade as a lower frequency; surviving contrast is not recovered detail.

Lens-shift OIS and a complete exposure cycle

\delta(t)=gf\alpha_{\mathrm{pk}}\sin\omega t,\quad H(\nu)=H_{\mathrm{opt}}H_{\mathrm{pix}}J_0(2\pi\nu A_{\mathrm{res}})

Derive residual image motion from the translated thin-lens ray kick at finite conjugates. Averaging the sinusoidal displacement over a full cycle gives the Bessel factor; its sign is retained in the images. This is an ideal gain model, not a commercial actuator or DeepOIS implementation.

Common difficulties

More pixels or perfect OIS guarantees more detail

Typical misconceptionMake pixels smaller or turn compensation to full gain and all lost information returns.

Better mental modelOIS removes only the modeled angular motion. Optical cutoff remains, and sampling has its own limit. Real aberrations, Bayer processing, noise, rolling shutter and controller bandwidth are deliberately outside this ideal model.

Run the experiment

01
Scene 1: Compact wide module
Compare wide and periscope. Find the opaque fold and follow the gold ray to the same fixed sensor. Change focal length and watch both real geometry and thickness.
What to observe: All hardware uses one millimetre scale; only the incoming ray segment is trimmed. The enlarged cards are central tests, not complete sensor photographs.
02
Scene 2: Opaque periscope fold
Choose macro and move object distance. Watch the lens carriage and effective diffraction scale rather than a fake zooming photograph.
What to observe: The central test windows have an unvignetted circular pupil. Off-axis illustrative rays may miss the finite fold mirror; the wave cards do not claim full-field aberration or vignetting accuracy.
03
Scene 3: Macro autofocus and OIS
Set OIS gain to zero and then one at the same shake. Next vary aperture and pixel pitch separately. Compare the point image, held pixels and Nyquist line.
What to observe: The sampled tiles keep their mean intensity and transfer sign. Exposure is not auto-normalized to make faint or aliased stripes look like resolved detail.