Optics Toolbox — Diffractive Optical Element Engineering Calculators
Engineering calculators for spiral phase plates and π-plates. Size your focal ring, check how far off-design-wavelength you can run, and compare continuous against multilevel elements — before you order.
Vortex focal spot calculator
Focused ring dimensions for a Gaussian beam passing through a spiral phase plate of charge ℓ and an ideal lens. Computed by exact scalar diffraction (order-ℓ Hankel transform), not a fitted formula.
Beam & optics
Ideal thin lens, scalar paraxial model, focal plane. Diameters are measured where intensity crosses the selected fraction of the ring peak — check which convention your spec sheet uses before comparing numbers.
Results
Radial intensity at focus
— vortex ring (charge ℓ) · — ℓ = 0 Gaussian reference · ‑ ‑ threshold
Focal plane intensity maps
vortex, charge ℓ
ℓ = 0 Gaussian reference
each panel normalized to its own peak — shows shape, not the intensity ratio (see table and line plot for that). Scroll to zoom, drag to pan.
SPP surface profile
relief depth, charge ℓ
0 → 2π phase (one full 2π step). For ℓ>1 the ramp repeats in ℓ azimuthal sectors at the same physical depth — higher charge adds sectors, not depth. Idealized continuous relief.
Wavelength detuning & conversion efficiency
Can an SPP designed for one wavelength be used at another? This computes the phase depth error and the power converted into the intended vortex order — including the quantization penalty of multilevel elements. High charges are normally realised as a sectoral (height-reset) relief rather than a single tall ramp, which changes the detuning behaviour markedly.
Element & operation
Sectoral relief resets the physical depth every 2π, so the beam sees ℓ identical single-wavelength steps: the intended-order efficiency is sinc²(β−1), independent of ℓ, and stray power falls only onto orders m = ℓ·p (on-axis m = 0, m = 2ℓ, …). Single continuous ramp accumulates the full ℓ×2π depth, so the error scales with charge: efficiency is sinc²(ℓ(β−1)) — a much narrower tolerance at high ℓ, with stray power spread across every integer order near ℓ. The two coincide at ℓ = 1.
At operating wavelength
Efficiency vs. wavelength
— power in intended order m = ℓ · ● operating point · ‑ ‑ design λ
Focal plane intensity maps
intended order m = ℓ
total detuned field
each panel normalized to its own peak — shows structural changes at the operating wavelength. Scroll to zoom, drag to pan.
SPP surface profile
relief depth, charge ℓ
0 → 2π at the design wavelength (fixed physical relief). For ℓ>1 the ramp repeats in ℓ sectors at the same depth. The delivered phase at the operating λ differs — see table.
Continuous vs. multilevel SPP — far field
A staircase (multilevel) spiral phase plate diffracts part of the power into parasitic azimuthal orders. This tool computes the exact focal-plane field of an N-level element, decomposed into its harmonic orders, and compares it with a continuous-relief SPP of the same charge.
Beam & element
The multilevel field is built from its exact azimuthal harmonics m = ℓ·k, k ≡ 1 (mod N), each carrying a power fraction sinc²(k/N). Five dominant harmonics are included; the table shows where the power goes.
Power budget
Azimuthally averaged intensity at focus
— continuous SPP · — N-level SPP · same input power, same scale
Focal plane intensity maps
continuous
N-level
Scroll to zoom, drag to pan. Synchronized.
SPP surface profile (N-level)
relief depth, N-level
N-level quantized relief. Approximates the continuous phase profile using discrete 2π/N steps.
Near field propagation (Continuous vs. multilevel)
Computes the Fresnel diffraction pattern a short distance z after the spiral phase plate. See how the discrete phase steps of a multilevel element cause the vortex core to split and parasitic orders to dynamically emerge as the beam propagates.
Beam & element
The field is built from the exact azimuthal harmonics using numerical Fresnel integrals. At z = 0, the intensity remains a pure Gaussian, but the phase map reveals the discrete topological staircase structure.
Azimuthally averaged intensity
— continuous SPP · — N-level SPP · same input power, same scale
Intensity maps at distance z
continuous
N-level
Scroll to zoom, drag to pan. Synchronized.
Phase maps at distance z
continuous (phase)
N-level (phase)
Scroll to zoom, drag to pan. Synchronized.
Angular π-plate — TEM₀ₗ petal mode generator
An angular π-plate imposes an alternating binary 0/π phase pattern in azimuthal sectors. For an integer order ℓ, the element is divided into 2ℓ equal sectors. When placed at the beam waist, it converts a Gaussian beam into a petal-like TEM₀ₗ mode with 2ℓ lobes at the focal plane.
Beam & element
The far-field intensity is computed exactly by summing the complete spectrum of azimuthal harmonics (m = ±ℓ, ±3ℓ, ±5ℓ…) generated by the step transitions. The theoretical conversion efficiency into the primary ±ℓ fundamental mode pair is 8/π² ≈ 81.1%.
Results
Intensity cross-section at focus (through lobe peaks)
— angular π-plate · — no plate (Gaussian reference), same scale
Focal plane intensity map
angular π-plate, normalized to its own peak
no plate — Gaussian reference, own peak
Scroll to zoom, drag to pan. Synchronized.
Element phase profile
relief phase, alternating 0/π