Just Focus is a Python package for computing vectorial electromagnetic fields in the focus of high numerical aperture microscope objectives.
Compute the field in the focal plane (z = 0.0) of a NA 1.4 oil immersion microscope objective assuming a linearly polarized, paraxial Gaussian beam with a waist size equal to the radius of the objective's back aperture. Use a hyperbolic tangent function to smooth the boundary of the stop and zero pad the mesh so that the final square mesh has 64 * 2^4 = 1024 samples in each direction.
import numpy as np
from leb.just_focus import InputField, Polarization, Pupil, Stop
mesh_size = 64
inputs = InputField.gaussian_pupil(
beam_center=(0.0, 0.0),
waist=1.0,
mesh_size=mesh_size,
polarization=Polarization.LINEAR_Y,
)
pupil = Pupil(
na=1.4,
refractive_index=1.518,
wavelength_um=0.561,
mesh_size=mesh_size,
stop=Stop.TANH,
)
results = pupil.propgate(0.0, inputs, padding_factor=4)pip install just-focusInstall additional dependencies for making plots:
pip install just-focus[plot]Then you can use functions in the leb.just_focus.plots module to plot the inputs and results.
from leb.just_focus.plots import plot_inputs
plot_inputs(inputs, pupil)just-focus follows this workflow:
- Define your input field in the pupil using
InputField. - Define a pupil using
Pupil. - Compute the focal field in the desired z-plane using the
Pupil.propagatemethod.
Pupil.propagate returns an instance of a FocalField object which contains a complex 2D array for each field direction.
Six parameters are required to construct a new InputField:
from leb.just_focus import InputField
input = InputField(
amplitude_x,
amplitude_y,
phase_x,
phase_y,
polarization_x,
polarization_y,
)All parameters should be 2D square arrays whose shape elements are powers of 2. The amplitude and phase arrays are of dtype np.float64, and the polarization arrays are of dtype np.complex128.
These inputs follow the implementation laid out by Herrera and Quinto-Su. Technically, they overspecify the field at the pupil in many "normal" cases. They are all required, however, to model a beam-shaping experiment where the x- and y-components of the field may be independently modulated in amplitude, phase, and polarization, such as setups with two SLMs and polarizing elements on two separate beam paths.
If all you want is to specify the amplitude and phase of the x- and y-components of the field at the pupil independently, set each of polarization_x and polarization_y to all ones. The elements of the resulting Jones vector describing the polarization at a point (x, y) in the pupil are then:
E_x = A_x / sqrt(A_x^2 + A_y^2)
E_y = A_y * exp(1j * (phi_y - phi_x)) / sqrt(A_x^2 + A_y^2)
where A_x, A_y, phi_x, phi_y are the amplitudes and phases in the x and y directions, respectively.
Alternatively, the relative phases may be determined by setting phase_x and phase_y to all zeros and setting the polarization arrays accordingly.
Three factory methods exist to compute commonly encountered input fields:
from leb.just_focus import HalfmoonPhase, InputField, Polarization
mesh_size = 64
gaussian = InputField.gaussian_pupil(
beam_center=(0.0, 0.0),
waist=1.0,
mesh_size=mesh_size,
polarization=Polarization.LINEAR_Y,
)
halfmoon = InputField.gaussian_halfmoon_pupil(
beam_center=(0.0, 0.5),
waist=2.0,
mesh_size=mesh_size,
polarization=Polarization.LINEAR_Y,
orientation=HalfmoonPhase.MINUS_45,
phase=np.pi,
phase_mask_center=(0.0, 0.0),
)
uniform = InputField.uniform_pupil(
mesh_size=mesh_size,
polarization=Polarization.CIRCULAR_LEFT,
)Coordinates and waist sizes are in units of normalized pupil coordinates, i.e. 0 is at the center and 1 is at the pupil edge.
Possible values for the Polarization enum are:
Polarization.LINEAR_X
Polarization.LINEAR_Y
Polarization.CIRCULAR_LEFT
Polarization.CIRCULAR_RIGHTPossible values for the HalfmoonPhase enum are:
HalfmoonPhase.HORIZONTAL
HalfmoonPhase.VERTICAL
HalfmoonPhase.MINUS_45
HalfmoonPhase.PLUS_45A Pupil instance is defined as follows:
from leb.just_focus import Pupil, Stop
pupil = Pupil(
na=1.4,
wavelength_um=0.561,
refractive_index=1.518,
focal_length_mm=3.3333,
mesh_size=64,
stop=Stop.TANH,
)The refractive index is that of the immersion medium. The incident beam is assumed to be incident from air (n = 1).
The focal length of an objective may be computed from the ratio between the corresponding tube lens focal length and its magnification. For example, a 100x Nikon objective will have a focal length of 2 mm because Nikon tube lenses have focal lengths of 200 mm, and 200 mm / 100 = 2 mm. The focal length used here is the focal length of the objective for a sample in air, i.e. the distance from the principle plane where the paraxial marginal ray from an object located at infinity intersects the optical axis in air. It is not already multiplied by the refractive index of the immersion medium, which is the convention used in Herrera and Quinto-Su and the textbook by Novotny and Hecht. The convention used in this package puts the location of the focus at a distance n * f from the principle reference sphere in sample space. This is consistent with the well-known formula R = f * NA for the radius of the back aperture of the objective. See the Resources section below for more information.
The stop parameter determines whether and how the aperture should be softened to reduce artifacts from the fast Fourier transform. Possible values are:
Stop.UNIFORM
Stop.TANH
A uniform stop is a pupil with a discontinuous edge. Stop.TANH softens this edge with a hyperbolic tangent function as introduced by Leutenegger, et al. in the Resources section below.
To compute the focal field at a given z plane, use:
pupil.propagate(z_um, inputs, padding_factor=4)where z_um = 0 corresponds to the focal plane of the objective andinputs is an InputField instance.
padding_factor describes the amount by which the input field will be zero-padded before computing the fast Fourier transforms. If the linear size of an input field array is N, then the padded array will be of size N * 2^padding_factor in each dimension. This will also be the size of the resulting focal field arrays.
Pupil.propagate returns a FocalField instance which is defined as follows:
@dataclass(frozen=True)
class FocalField:
field_x: NDArray[Complex]
field_y: NDArray[Complex]
field_z: NDArray[Complex]
x_um: NDArray[Float]
y_um: NDArray[Float]
def intensity(self, normalize: bool = True) -> NDArray[Float]:
I = np.abs(self.field_x)**2 + np.abs(self.field_y)**2 + np.abs(self.field_z)**2
if normalize:
return I / np.max(I)
return IIt has five parameters: three, 2D complex arrays representing the field in each direction and two, 1D arrays representing the x- and y-coordinates in the focal region.
In addition, there is an intenstiy helper method that computes the intensity from the fields.
Development requires uv.
After cloning this repo, run the following command from the project's root directory:
uv sync --all-extrasThis will create a virtual environment with the required dependencies in a folder named .venv.
Just run pytest from the project's root directory:
pytest- PSF-Generator (Python) https://github.com/Biomedical-Imaging-Group/psf_generator
- InFocus (MATLAB) https://github.com/QF06/InFocus
- Debye Diffraction Code (MATLAB and Python) https://github.com/jdmanton/debye_diffraction_code
- I. Herrera and P. A. Quinto-Su, "Simple computer program to calculate arbitrary tightly focused (propagating and evanescent) vector light fields," arXiv:2211.06725 (2022). https://doi.org/10.48550/arXiv.2211.06725.
This manuscript describes the specific numerical implementation of the vectorial field propagation algorithm used here.
- K. M. Douglass, "Coordinate Systems for Modeling Microscope Objectives," (2024). https://kylemdouglass.com/posts/coordinate-systems-for-modeling-microscope-objectives/
This blog post explains how to set up the various coordinate systems and numerical meshes for evaluating the results of the Richards-Wolf model for high NA objectives.
- M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser. Fast focus field calculations. Opt. Express 14, 11277-11291 (2006). https://doi.org/10.1364/OE.14.011277
This manuscript was the first to describe the calculation of vectorial focal fields using the fast Fourier transform.
- L. Novotny and B. Hecht, "Principles of Nano-Optics," Cambridge University Press, pp. 56 - 66 (2006). https://doi.org/10.1017/CBO9780511813535
Chapter 3 contains the derivation of the field at the focus of an aplanatic lens.