Optics 060 · Diffraction, Fourier optics, and computational imaging

Diffractive Lens Workshop

Build an amplitude zone plate, binary or multilevel phase relief and continuous kinoform on one moving-sensor bench. Acquire focal pixels from their finite annular pupil and compare multi-focus, modal efficiency, fabricated depth and opposite chromatic shifts with a smooth refractive reference.

Nominal primary focus / actual sensor distance

f_1,z

$0.50$

Ideal transmission or primary-order power / finite-window power

T,\eta_1,P_{\rm window}

$50\%$

Physical aperture radius / full-wave relief depth

a,h_{2\pi}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Diffractive Lens Workshop

Optic type (named buttons below)

k

0

Design focal length at 532 nm

f_0

200\,\mathrm{mm}

Complete equal-area periods (changes aperture)

K

10

Incident wavelength (optic stays fixed)

\lambda

532\,\mathrm{nm}

Fabricated phase levels (multilevel only)

M

8

Actual relief depth / designed depth

s_h

1

Sensor travel / design focal length

z/f_0

1

Source: 0 monochromatic, 1 three colours

c

0

Display gain exponent (not raw power)

g

1

An opaque ring can help a plate focus: it removes light that would cancel the neighbouring zone. Compare amplitude and binary phase: the fabricated rings stay in the same places, but absorption becomes a half-wave step. Inspect the actual relief, then move the sensor through the third and primary foci. In multilevel, change the number of physical steps and compare ideal order powers with the separate axial scan: equal peak heights do not mean equal energy. Switch to three colours: red focuses closer for the diffractive lens, while the smooth refractive reference shifts the other way. Continuous kinoform is phase-wrapped; it is not a lighthouse Fresnel lens. The enlarged data row stays outside the apparatus.

Uniform scalar paraxial Fresnel/Hankel model, not Maxwell or commercial Fresnel-lens ray tracing. Fixed 532 nm design, synthetic lines, ideal amplitude absorption or lossless phase relief; SCHOTT line-index interpolation. Modal power, scanned axial intensity and finite sensor power are separate. Explicit unequal geometric magnification and logarithmic display gain.

Start with the essentials

Focus question: How do zone placement and phase quantization trade focal efficiency against chromatic behavior?
One-sentence intuition: A fixed diffractive plate can have many nominal foci: $f_m(\lambda)=\frac{f_0\lambda_0}{m\lambda}$ . Longer wavelengths bring each positive-order focus closer. Order power is not axial peak height; negative orders diverge and zero order has no finite focus.

Core mathematical model

One fabricated pupil

\begin{aligned}q&=\frac{r^2}{2\lambda_0 f_0},\quad 0\le q\le K\\a^2&=2K\lambda_0f_0\\h_{2\pi}&=\lambda_0/(n_0-1)\end{aligned}

Each full period has equal area. The integer period count changes the real aperture radius, not merely a decorative number of rings. Design focal length changes every boundary. Source colour never changes the fabricated plate.

Physical glass gives positive phase

\begin{aligned}t(r,\lambda)&=\exp\!\left[\frac{2\pi i}{\lambda}(n(\lambda)-1)h(r)\right]\\h_{\rm kin}&=s_hh_{2\pi}(1-\{q\})\\h_M&=\frac{s_hh_{2\pi}}M\left(M-1-\lfloor M\{q\}\rfloor\right)\end{aligned}

At the design wavelength the downward staircase approximates the negative quadratic focusing phase, apart from an irrelevant common phase. Binary uses two levels; the continuous kinoform wraps by one optical turn. Thickness errors redistribute power even though nominal modal focal positions stay fixed.

Propagate the actual rings

\begin{aligned}U(\rho,z)&=\frac{2\pi}{i\lambda z}e^{ikz+ik\rho^2/(2z)}\\&\quad\cdot\int_0^a t(r)e^{ikr^2/(2z)}J_0(k\rho r/z)r\,dr\\I&=|U|^2\end{aligned}

The finite circular pupil is integrated over exact annular domains, not an invented Lorentzian or Gaussian focus. Axial integration is analytic; the same pupil gives off-axis pixel averages. Three source lines add intensities because they are mutually incoherent. The reference is a separate propagation, not a reconstruction of selected pixels.

Modal power, not peak height

\begin{aligned}c_m&=\int_0^1t(u)e^{2\pi imu}\,du\\\eta_1^{\rm amp}&=\frac1{\pi^2}\\\eta_1^{\rm binary}&=\frac4{\pi^2}\\\eta_1^{(M)}&=\left[\frac{\sin(\pi/M)}{\pi/M}\right]^2\end{aligned}

These efficiencies apply to an ideal complete-period pupil at design wavelength and depth. Binary spends power in both converging and diverging odd orders; an amplitude plate also absorbs half and leaves zero order. A higher-order focus is smaller, so less modal power can still make a similar axial peak.

Finite sensor power

\begin{aligned}I_p&=\frac1{p^2}\int_{\rm pixel}|U|^2\,dA\\P_{\rm window}&=\frac{p^2\sum_p I_p}{\pi a^2 I_{\rm inc}}\\\sum_{m=-\infty}^{\infty}\eta_m&=\int_0^1|t(u)|^2\,du\end{aligned}

The small square sensor can lose unfocused light and diffraction rings. The ideal infinite modal budget is not the finite-window measurement. Only fifteen modes are plotted; the rest are not silently assigned to absorption. The irradiance reference is the design peak of a perfect full pupil, never the current brightest pixel.

Opposite chromatic shifts

\begin{aligned}f_1^{\rm DOE}&=f_0\lambda_0/\lambda\\f^{\rm ref}&=f_0\frac{n_0-1}{n(\lambda)-1}\\\epsilon_{\rm F}&\simeq\frac{ka^4}{8z^3}\end{aligned}

For normal glass dispersion red focuses farther in the smooth thin-lens reference, but closer in the fixed DOE. Refractive index uses declared SCHOTT table interpolation. The quoted quartic-phase estimate flags short-distance Fresnel truncation; no vector or spherical-lens accuracy is claimed. Macroscopic refractive Fresnel grooves are outside this model.

Common difficulties

Fresnel does not specify the mechanism

Typical misconceptionAll concentric grooved lenses are identical, and their brightest focus contains all transmitted power.

Better mental modelSeparate amplitude blocking, phase wrapping and smooth refraction. Compare modal power with scanned intensity and actual finite-window power; they are three different quantities.

Run the experiment

01
Scene 1: Block alternate zones
Start with amplitude. Move the sensor through third and primary focus, then inspect the opaque alternate zones.
What to observe: Opaque rings stop incident branches. Multiple axial peaks exist, but zero order has no finite focus.
02
Scene 2: Replace absorption with phase
Use binary phase. Inspect the physical half-wave step; reduce its depth to zero and observe the loss of converging orders.
What to observe: Ideal binary phase avoids absorption, yet primary-order power is only about forty percent.
03
Scene 3: Quantization and chromatic focus
Use multilevel. Change actual levels, try continuous kinoform, then compare three colours with the smooth refractive reference.
What to observe: More steps concentrate modal power; changing wavelength redistributes it and reverses chromatic shift relative to smooth glass.