Zernike Polynomials and Optical Aberrations

Zernike polynomials are the standard basis for describing wavefront aberrations in optical systems. Janssen provides JAX-compatible implementations for generating and analyzing aberrations.

Mathematical Definition

Zernike Polynomials

Zernike polynomials are defined on the unit disk ($\rho \leq 1$):

$$ Z_n^m(\rho, \theta) = R_n^{|m|}(\rho) \cdot \begin{cases} \cos(m\theta) & m \geq 0 \ \sin(|m|\theta) & m < 0 \end{cases} $$

where $R_n^{|m|}(\rho)$ is the radial polynomial:

$$ R_n^{|m|}(\rho) = \sum_{s=0}^{(n-|m|)/2} \frac{(-1)^s (n-s)!} {s! \left(\frac{n+|m|}{2}-s\right)! \left(\frac{n-|m|}{2}-s\right)!} \rho^{n-2s} $$

First 15 Zernike modes arranged by radial order $n$ (rows) and azimuthal frequency $m$ (columns). Each mode represents a distinct aberration pattern.

Orthogonality

Zernike polynomials are orthogonal over the unit disk:

$$ \int_0^{2\pi} \int_0^1 Z_n^m(\rho, \theta) Z_{n’}^{m’}(\rho, \theta) \rho , d\rho , d\theta = \pi \epsilon_m \delta_{nn’} \delta_{mm’} $$

where $\epsilon_m = 2$ for $m = 0$ and $\epsilon_m = 1$ otherwise.

Indexing Conventions

Noll Indexing

The Noll index $j$ orders modes by radial degree, then azimuthal:

$j$	$n$	$m$	Name
1	0	0	Piston
2	1	1	Tip
3	1	-1	Tilt
4	2	0	Defocus
5	2	-2	Oblique astigmatism
6	2	2	Vertical astigmatism
7	3	-1	Vertical coma
8	3	1	Horizontal coma
9	3	-3	Vertical trefoil
10	3	3	Oblique trefoil
11	4	0	Primary spherical

Noll index ordering on the $(n, m)$ grid. The index increases left-to- right and bottom-to-top, with alternating sign for $m \neq 0$.

Converting Between Indices

from janssen.optics import noll_to_nm, nm_to_noll

# Noll index 11 -> (n=4, m=0) - spherical aberration
n, m = noll_to_nm(11)
print(f"Noll 11 = (n={n}, m={m})")

# (n=3, m=1) -> Noll index 8 - horizontal coma
j = nm_to_noll(3, 1)
print(f"(n=3, m=1) = Noll {j}")

Common Aberrations

Defocus (Z4)

Defocus represents axial displacement from the focal plane:

$$ W_{\text{defocus}}(\rho) = c_4 (2\rho^2 - 1) $$

Positive coefficient = focus behind detector.

Common aberrations: (a) defocus, (b) astigmatism, (c) coma, (d) spherical aberration. Color indicates phase deviation from flat wavefront.

Astigmatism (Z5, Z6)

Astigmatism causes different focal lengths along perpendicular axes:

$$ W_{\text{astig}}(\rho, \theta) = c_5 \rho^2 \sin(2\theta) + c_6 \rho^2 \cos(2\theta) $$

Coma (Z7, Z8)

Coma creates comet-shaped blur for off-axis points:

$$ W_{\text{coma}}(\rho, \theta) = c_7 (3\rho^3 - 2\rho) \sin\theta + c_8 (3\rho^3 - 2\rho) \cos\theta $$

Spherical Aberration (Z11)

Spherical aberration causes focus shift that depends on pupil radius:

$$ W_{\text{sph}}(\rho) = c_{11} (6\rho^4 - 6\rho^2 + 1) $$

Generating Aberrations

From Noll Coefficients

from janssen.optics import generate_aberration_noll

# Generate phase map with 0.1 waves of defocus and 0.05 waves of coma
coefficients = {
    4: 0.1,   # Defocus
    8: 0.05,  # Horizontal coma
}

phase_map = generate_aberration_noll(
    coefficients=coefficients,
    grid_size=(256, 256),
    pupil_radius=1.0,
    wavelength=632.8e-9,
)

From (n, m) Coefficients

from janssen.optics import generate_aberration_nm

# Same aberrations using (n, m) indexing
coefficients = {
    (2, 0): 0.1,   # Defocus
    (3, 1): 0.05,  # Horizontal coma
}

phase_map = generate_aberration_nm(
    coefficients=coefficients,
    grid_size=(256, 256),
    pupil_radius=1.0,
    wavelength=632.8e-9,
)

Applying Aberrations

To a Wavefront

from janssen.optics import apply_aberration

# Apply aberration to existing wavefront
aberrated = apply_aberration(
    wavefront=input_wavefront,
    coefficients={4: 0.1, 11: 0.02},
    pupil_radius=aperture_radius,
)

Effect on PSF

Effect of aberrations on point spread function. (a) Perfect, (b) defocus broadens symmetrically, (c) astigmatism creates elliptical blur, (d) coma creates asymmetric comet shape, (e) spherical creates halo.

Phase Analysis

RMS Wavefront Error

The RMS wavefront error quantifies aberration severity:

$$ \sigma_W = \sqrt{\frac{1}{\pi} \iint_{pupil} W^2(\rho, \theta) \rho , d\rho , d\theta} $$

from janssen.optics import phase_rms

rms_error = phase_rms(
    phase_map=phase,
    pupil_mask=pupil,
)
print(f"RMS error: {rms_error:.4f} waves")

Strehl Ratio

For small aberrations, the Strehl ratio is:

$$ S \approx \exp\left(-\left(\frac{2\pi \sigma_W}{\lambda}\right)^2\right) $$

The Maréchal criterion $S > 0.8$ requires $\sigma_W < \lambda/14$.

Zernike Decomposition

Fitting Coefficients

Given a measured wavefront, decompose into Zernike modes:

from janssen.optics import zernike_decomposition

# Fit first 15 Zernike modes
coefficients = zernike_decomposition(
    phase_map=measured_phase,
    pupil_mask=pupil,
    num_modes=15,
)

# coefficients[j] is the coefficient for Noll index j
print(f"Defocus: {coefficients[4]:.4f} waves")
print(f"Spherical: {coefficients[11]:.4f} waves")

Removing Low-Order Aberrations

Often piston, tip, and tilt are not of interest:

from janssen.optics import compute_phase_from_coeffs

# Reconstruct phase without first 3 modes
high_order_phase = compute_phase_from_coeffs(
    coefficients=coefficients,
    grid_size=(256, 256),
    start_index=4,  # Skip piston, tip, tilt
)

References

1. Noll, R. J. “Zernike polynomials and atmospheric turbulence” JOSA (1976)

2. Born, M. & Wolf, E. “Principles of Optics” Chapter 9 (1999)

3. Mahajan, V. N. “Zernike circle polynomials and optical aberrations of systems with circular pupils” Appl. Opt. (1994)