just-focus 0.3.3
Vector electromagnetic field calculations in the focus of high NA microscope objectives.
Requires Python: >=3.11
Keywords: diffraction, electromagnetism, microscopy, richards-wolf

# Just Focus

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Just Focus is a Python package for computing vectorial electromagnetic fields in the focus of high numerical aperture microscope objectives.

## Quickstart

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.

```python
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)
```

## Installation

```console
pip install just-focus
```

### Extras

Install additional dependencies for making plots:

```console
pip install just-focus[plot]
```

Then you can use functions in the `leb.just_focus.plots` module to plot the inputs and results.

```python
from leb.just_focus.plots import plot_inputs

plot_inputs(inputs, pupil)

```

## Use

just-focus follows this workflow:

1. Define your input field in the pupil using `InputField`.
2. Define a pupil using `Pupil`.
3. Compute the focal field in the desired z-plane using the `Pupil.propagate` method.

`Pupil.propagate` returns an instance of a `FocalField` object which contains a complex 2D array for each field direction.

### InputField

Six parameters are required to construct a new `InputField`:

```python
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](https://doi.org/10.48550/arXiv.2211.06725). 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.

#### Common Input Fields

Three factory methods exist to compute commonly encountered input fields:

```python
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:

```python
Polarization.LINEAR_X
Polarization.LINEAR_Y
Polarization.CIRCULAR_LEFT
Polarization.CIRCULAR_RIGHT
```

Possible values for the `HalfmoonPhase` enum are:

```python
HalfmoonPhase.HORIZONTAL
HalfmoonPhase.VERTICAL
HalfmoonPhase.MINUS_45
HalfmoonPhase.PLUS_45
```

### Pupil

A `Pupil` instance is defined as follows:

```python
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.

#### Pupil.propagate

To compute the focal field at a given z plane, use:

```python
pupil.propagate(z_um, inputs, padding_factor=4)
```

where `z_um = 0` corresponds to the focal plane of the objective and`inputs` 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.

### FocalField

`Pupil.propagate` returns a `FocalField` instance which is defined as follows:

```python
@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 I
```

It 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

### Set up the development environment

Development requires [uv](https://docs.astral.sh/uv/).

After cloning this repo, run the following command from the project's root directory:

```console
uv sync --all-extras
```

This will create a virtual environment with the required dependencies in a folder named `.venv`.

### Tests

Just run `pytest` from the project's root directory:

```console
pytest
```

## Other Packages to Compute Vectorial Focal Fields

- PSF-Generator (Python) [https://github.com/Biomedical-Imaging-Group/psf_generator](https://github.com/Biomedical-Imaging-Group/psf_generator)
- InFocus (MATLAB) [https://github.com/QF06/InFocus](https://github.com/QF06/InFocus)
- Debye Diffraction Code (MATLAB and Python) [https://github.com/jdmanton/debye_diffraction_code](https://github.com/jdmanton/debye_diffraction_code)

## Resources

- 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](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/](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](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](https://doi.org/10.1017/CBO9780511813535)

Chapter 3 contains the derivation of the field at the focus of an aplanatic lens.