Mask

Free space "type": "mask"

A mask is a thin plane with an amplitude transmission and a delay you write as formulas. It models apodisers, pinholes, spiral phase plates, fork holograms, blazed and binary gratings, a spatial light modulator's pattern, and an ideal lens or objective.

The model

What it does to the field

A mask multiplies the field by its transmission:

\[E_\text{out}(x, y) = A(x, y)\; e^{-i 2\pi D(x, y)}\; E_\text{in}(x, y),\]

with \(A\) the amplitude and \(D\) the delay, in waves. The variables are:

\(x\): along the mask, in the bench plane, mm;
\(y\): out of the plane, mm;
\(r = \sqrt{x^2 + y^2}\) and \(\theta = \operatorname{atan2}(y, x)\);
\(\lambda\): the light’s wavelength, in mm.

A mask that is round about the beam keeps the field round. Anything else makes it a 2D field.

What it does to rays

Rays take \(A\) and \(D\) where they cross. They also bend by the slope of the delay in the plane, as a grating or a lens would bend them:

\[\Delta(\sin\vartheta) = \lambda\,\frac{\partial D}{\partial x}.\]

So a phase lens focuses rays, and a grating’s slope steers them. The Gaussian beam’s pilot takes the lens the delay’s curvature makes, in the plane and out of it. A jump in the delay, a binary grating’s step, has no slope and bends nothing. A jump of whole waves, a blazed grating’s or a kinoform’s reset, is no jump at all.

Formulas

A formula can use + - * / ^ (a power, which binds tighter than a leading minus), the numbers pi and e, and these functions:

Functions
`sin cos tan asin acos atan atan2`	trigonometry
`exp log sqrt abs pow`
`min max floor ceil round sign hypot`
`mod(a, b)`	\(a\) modulo \(b\), always positive
`step(v)`	1 where \(v \ge 0\), else 0
`rect(v)`	1 where \(\lvert v\rvert \le 1/2\)
`circ(v)`	1 where \(v \le 1\)
`sinc(v)`	\(\sin(\pi v)/(\pi v)\)
`gauss(v)`	\(e^{-v^2}\)

Formulas are parsed into a tree and compiled. Only these names exist in them, so a mask cannot run any other code, whoever wrote the bench.

Presets

Preset	Amplitude	Delay
Gaussian apodiser	`exp(-(r/2)^2)`	0
Pinhole, 1 mm	`circ(r/0.5)`	0
Annulus	`step(r-1)*step(2-r)`	0
Phase lens, f = 100 mm	1	`-r^2/(2*100)/lambda`
Axicon, 10 mrad	1	`-0.01*r/lambda`
Vortex, charge 1	1	`theta/(2*pi)`
Fork hologram, 0.1 mm	1	`mod(x/0.1 + theta/(2*pi), 1)`
Blazed grating, 0.1 mm	1	`mod(x/0.1, 1)`
π step (TEM₀₀ to HG₁₀)	1	`0.5*step(x)`
Binary phase grating, 0.2 mm	1	`0.5step(sin(2pi*x/0.2))`
Amplitude grating, 0.2 mm	`step(sin(2pix/0.2))`	0

An ideal lens of focal length \(f\) has the delay of a perfect converging sphere, -(sqrt(r^2+f^2)-f)/lambda, with \(f\) written as a number in mm. It focuses without aberration at any numerical aperture, so it stands in for a microscope objective whose prescription you don’t have.

Sampling

Where the formula jumps inside a grid sample, the sample takes the mean transmission over its footprint, amplitude and phase alike, not the value at its centre. A hard edge is then anti-aliased, as a stop’s edge is. A mask whose detail is finer than about four samples is flagged rather than quietly smoothed: raise the source’s grid (popN) to resolve it.

Settings

Field	Default	Meaning
`length`	`25.4`	mm aperture.
`amp`	`"1"`	amplitude transmission as a formula in x (mm along the mask), y (mm out of the plane), r, theta, lambda (mm) and pi; e.g. "1" or "circ(r/2)".
`delay`	`"0"`	delay in waves as a formula in the same variables. An ideal lens of focal length f mm: "-(sqrt(r^2+f^2)-f)/lambda" (write f as a number).
`preset`	`"custom"`	leave "custom".

From the tool's own tables, file format version 1: the names are those in a .jekray file.

Assumptions

The mask is thin: it multiplies the field in one plane, with no thickness and no angle dependence.
A delay is in waves at the light’s own wavelength, as an SLM writes it. A real glass phase plate’s delay scales with \(1/\lambda\); write the formula with lambda to model that.

Stop, for plain openings.
Source: higher-order modes, for beams a mask can make.

JEKray documentation, for file format version 1. Something wrong or unclear? Tell us at [email protected].