Optics 060 · Diffraction, Fourier optics, and computational imaging

Diffractive Lens Workshop

How does a diffractive lens focus light, and what do finer phase steps improve? Zone spacing fixes the focal length of each diffraction order, and longer wavelengths focus closer, the opposite of a glass lens. Finer phase steps send more of the light into the main focus: two levels manage about 41 percent, and the closer the profile gets to a smooth kinoform, the closer it gets to all of it. More steps leave the color error unchanged.

Build an amplitude zone plate, a binary or multilevel phase relief and a continuous kinoform on one bench with a moving sensor. Record focal pixels through each finite ring-shaped pupil, then compare the multiple foci, the efficiency of each order, the relief depth and the chromatic shift, which runs opposite to that of a smooth refractive lens.

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 $\text{valid model regime}$

Model checked against analytic results

Diffractive Lens Workshop

Optic type (names on the 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 (raw power unchanged)

g

1

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

The model uses uniform scalar paraxial Fresnel and Hankel integrals; it does not solve Maxwell's equations or ray-trace a commercial Fresnel lens. The design wavelength is fixed at 532 nm, the spectral lines are synthetic, and the plates are ideal absorbers or lossless phase reliefs. Refractive indices are interpolated from SCHOTT line data. Power in each order, the scanned axial intensity and the power on the finite sensor are shown separately. Geometric magnifications are unequal and the display uses logarithmic gain; both are labeled.

Start with the essentials

Focus question: How does a diffractive lens focus light, and what do finer phase steps improve?
One-sentence intuition: One fixed diffractive plate can have many nominal foci: $f_m(\lambda)=\frac{f_0\lambda_0}{m\lambda}$ . A longer wavelength brings every positive-order focus closer. The power in an order is not its peak height on the axis. Negative orders diverge, and the zero order has no finite focus.

Core mathematical model

Zone layout of the fabricated plate

\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}

Every full period has the same area. Changing the whole number of periods changes the real aperture radius, not just how many rings are drawn. The design focal length moves every boundary. The source colour never changes the plate once it is made.

Glass relief adds 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 staircase stepping down toward the edge approximates the negative quadratic phase of a focusing lens, apart from a common phase that does not matter. The binary plate uses two levels; the continuous kinoform wraps the phase every full cycle. Thickness errors move power between orders, even though the nominal focal positions of the orders stay fixed.

Propagating the field through the real 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 integral runs over the exact annular regions of the finite circular pupil; no Lorentzian or Gaussian focus shape is assumed. The on-axis integral is analytic, and the same pupil gives the pixel averages off axis. The three source lines are mutually incoherent, so their intensities add. The reference is a separate propagation, not a reconstruction from selected pixels.

Power in each diffraction order

\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 hold for an ideal pupil of complete periods at the design wavelength and depth. A binary plate sends power into both converging and diverging odd orders. An amplitude plate also absorbs half the light and leaves a zero order. A higher-order focus is smaller, so it can reach a similar axial peak with less power.

Power on a finite sensor

\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 miss unfocused light and diffraction rings. The sum over infinitely many ideal orders is not what a finite window measures. Only fifteen orders are plotted, and the rest are not quietly counted as absorbed. Irradiance is scaled to the design peak of a perfect full pupil, never to the brightest pixel at the moment.

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}

With normal glass dispersion, red light focuses farther away in the smooth thin-lens reference but closer in the fixed diffractive element. The refractive index is interpolated from stated SCHOTT tables. The quartic-phase estimate flags truncation error in the Fresnel approximation at short distances; no vector or spherical-lens accuracy is claimed. Macroscopic refractive Fresnel grooves are outside this model.

Common difficulties

The name Fresnel does not tell you the mechanism

Typical misconceptionAll lenses with concentric grooves are the same, and the brightest focus holds all the transmitted power.

Better mental modelAmplitude blocking, phase wrapping and smooth refraction are different mechanisms. Order power, scanned intensity and the power a finite window actually catches are three different quantities, so compare them separately.

Run the experiment

01
Scene 1: Block alternate zones
Start with the amplitude plate. Move the sensor through the third-order focus and the primary focus, then look at the opaque alternate zones.
What to observe: The opaque rings block part of the incoming light. Several peaks appear along the axis, but the zero order has no finite focus.
02
Scene 2: Replace absorption with phase
Switch to binary phase. Look at the half-wave step in the glass, then reduce its depth to zero and watch the converging orders disappear.
What to observe: An ideal binary phase plate absorbs nothing, yet only about forty percent of the power goes into the primary order.
03
Scene 3: Quantization and chromatic focus
Switch to multilevel. Change the number of levels, try the continuous kinoform, then compare three colours with the smooth refractive reference.
What to observe: More steps concentrate the power into fewer orders. Changing the wavelength spreads it out again, and the focus shifts with colour in the opposite direction to smooth glass.