Zernike Polynomial Definition (OSA/ANSI)
Orthonormal over the unit disk ρ ≤ 1 in polar coordinates (ρ, θ):
Orthonormality: ∫|Z_n^m|² ρ dρ dθ / π = 1. Any wavefront W(ρ,θ) = Σ c_n^m Z_n^m; RMS WFE = √(Σ c²).
Mode selector
Z_n^m (OSA, units λ)
Radial cross-section at θ=0
Wavefront from Zernike Expansion
Any wavefront over a circular aperture: W(ρ,θ) = Σ c_n^m · Z_n^m(ρ,θ), coefficients in waves (λ). RMS WFE = √(Σ c²). Strehl ratio (Maréchal approximation):
S ≈ exp[−(2π · σ_W)²]Strehl > 0.8 → diffraction-limited (Rayleigh criterion). The PSF is computed via 2D FFT of the pupil function: P(ρ,θ) = A(ρ) · exp[i·2π·W(ρ,θ)].
Zernike expansion coefficients
Wavefront W(ρ,θ) (λ)
PSF = |FT{P}|²
SLM Phase → Far-Field Intensity
A spatial light modulator (SLM) imprints a programmable phase φ(x,y) on an input Gaussian beam. In the focal plane of a lens, the field is the Fourier transform of the modulated input:
E_ff(u,v) = FT{ E_in(x,y) · exp[iφ(x,y)] } → I_ff = |E_ff|²Helical phase φ = l·θ converts a Gaussian to a Laguerre–Gaussian LG₀ˡ donut (for l ≠ 0), carrying orbital angular momentum l·ℏ per photon.
SLM Parameters
Input intensity |E_in|²
SLM phase φ (mod 2π)
Far-field intensity |FT{E}|²
Laguerre–Gaussian LG₀ˡ beams
Carry OAM l·ℏ per photon. At the beam waist (p=0): |LG₀ˡ| ∝ (r√2/w)^|l| · exp(−r²/w²) Intensity ring at r = w√(|l|/2). Larger ring for larger |l|.
1. Input: Gaussian beam
2. Phase: $\varphi = l\cdot\theta$ (helical/vortex phase)
3. Far field $\approx$ LG$_0^l$ (donut for $l\neq0$)
Blazed grating $l\cdot\theta + k\cdot x$: shifts donut off-axis, separates OAM orders spatially
| l | Name | Far-field shape |
|---|---|---|
| 0 | Gaussian (LG₀⁰) | Central Airy peak |
| ±1 | First-order vortex | Single donut ring |
| ±2 | Second-order vortex | Larger donut ring |
| ±3 | Third-order vortex | Even larger ring |
| 0 + defocus | Defocused Gaussian | Broadened central peak |
| 0 + astigmatism | Astigmatic Gaussian | Elongated / elliptical |
| 0 + grating | Blazed diffraction | Off-axis Gaussian spot |
Zernike Pyramid, all modes at a glance
Every Zernike polynomial Z_n^m arranged in the standard pyramid (OSA/ANSI order). Columns index the azimuthal frequency m, rows the radial order n. Cosine modes (m > 0) on the right, sine modes (m < 0) on the left. Blue = negative wavefront, white = zero, red = positive (RdBu colormap).
Gerchberg-Saxton (GS) Algorithm — iterative phase retrieval
Given a desired far-field intensity |T(u,v)|² (e.g. a reconfigurable tweezer array), GS finds the SLM phase pattern φ(x,y) that produces it. The algorithm alternates between the SLM plane and the target plane, enforcing known amplitude constraints in each:
GS Parameters
Target intensity
SLM phase φ (mod 2π)
Reconstructed far-field
Convergence: normalised RMSE vs iteration
Algorithm details
Step 1 (SLM plane): Combine input Gaussian amplitude A(x,y) with
current phase estimate φₖ(x,y).
Step 2 (FFT → target plane): Propagate the field to the focal plane
using a 2D discrete Fourier transform.
Step 3 (target constraint): Replace the amplitude with √T(u,v) while
keeping the retrieved phase arg(FT{E}).
Step 4 (IFFT → SLM plane): Propagate back; extract new phase φₖ₊₁.
Typically converges in 20–50 iterations. Efficiency (fraction of power in target spots)
reaches 70–90% for typical arrays; adding a blazed grating shifts the 1st order off-axis
to separate it from undiffracted light.