# Modeling diffraction¶

Lentil uses Fourier transform-based algorithms to numerically model the propagation of an electromagnetic field through an optical system. The electromagnetic field is represented by a Wavefront object which stores the complex amplitude of the field at a discretely sampled grid of points. The optical system is represented by a set of user-defined Plane objects which the wavefront interacts with as it propagates through the system.

Note

This section of the User Guide assumes an undergraduate-level understanding of physical and Fourier optics. In-depth mathematical background and an extensive discussion of the validity of each diffraction approximation is available in 1.

## Numerical diffraction propagation¶

Lentil numerically models diffraction by propagating a Wavefront object through any number of Plane objects representing an optical system. This propagation always follows the same basic flow:

1. Create a new wavefront - A Wavefront represents a monochromatic, discretely sampled complex field that may propagate through space. By default, when a new Wavefront is constructed it represents an infinite plane wave (1+0j). Note that “infinite” in this case really just means that the wavefront field is broadcastable to any shape necessary.

>>> w1 = lentil.Wavefront(wavelength=650e-9)
>>> w1.field
1+0j
>>> w1.focal_length
inf

2. Propagate the wavefront through the first plane in the optical system - the planes that describe an optical system typically modify a propagating wavefront in some way. By multiplying a Wavefront and a Plane together, a new Wavefront is returned representing the the complex field after propagating through the plane:

>>> pupil = lentil.Pupil(amplitude=lentil.circle((256, 256), 120),
...                      pixelscale=1/240, focal_length=10)
>>> w2 = w1 * pupil


Note the complex field of w2 now clearly shows the effect of propagating through the circular aperture of pupil:

>>> plt.imshow(np.abs(w2.field), origin='lower')


Additionally, because w2 was propagated through a Pupil plane, it has inherited the pupil’s focal length:

>>> w2.focal_length
10


It is also possible to perform the multiplication in-place, reducing the memory footprint of the propagation:

>>> w1 *= pupil


Note

Additional details on the plane-wavefront interaction can be found in How a plane affects a wavefront.

3. Propagate the wavefront to the next plane in the optical system - the Wavefront object provides a number of methods to propagate between planes. The appropriate method should be chosen based on the plane types the wavefront is propagating between.

From

To

Method

Pupil

Image

propagate_image()

Image

Pupil

propagate_pupil()

Pupil

Pupil

N/A

Image

Image

N/A

Propagations are defined by the following attributes:

• pixelscale - the spatial sampling of the output plane

• npix - the shape of the output plane

• npix_prop - the shape of the propagation plane. See npix vs npix_prop for additional details.

• oversample - the number of times to oversample the output plane. See the section on user_guide.diffraction.sampling for more details.

For example, to propagate a Wavefront from a Pupil to an Image plane:

>>> w2.propagate_image(pixelscale=5e-6, npix=64, oversample=5)
>>> plt.imshow(w2.intensity**0.1, origin='lower')


Note

When propagating between like planes (pupil to pupil or image to image), no additional propagation step is required.

4. Repeat steps 2 and 3 until the propagation is complete - if multiple planes are required to model the desired optical system, steps 2 and 3 should be repeated until the Wavefront has been propagated through all of the planes.

### Performing propagations in-place vs. on copies¶

By default, all propagation operations operate on a Wavefront in-place. If desired, a copy can be returned instead by providing the argument inplace=False:

import matplotlib.pyplot as plt
import lentil

pupil = lentil.Pupil(amplitude=lentil.circle((256, 256), 120),
pixelscale=1/240, focal_length=10)

w1 = lentil.Wavefront(650e-9)
w2 = w1 * pupil
w3 = w2.propagate_image(pixelscale=5e-6, npix=64, oversample=5, inplace=False)

plt.subplot(121)
plt.imshow(w2.intensity, origin='lower')
plt.title('w2 intensity')

plt.subplot(122)
plt.imshow(w3.intensity**0.1, origin='lower')
plt.title('w3 intensity')


The steps outlined above propagate a single monochromatic Wavefront through an optical system. The example below performs the same operation for multiple different wavelengths and accumulates the resulting image plane intensity:

import matplotlib.pyplot as plt
import numpy as np
import lentil

pupil = lentil.Pupil(amplitude=lentil.circle((256, 256), 120),
pixelscale=1/240, focal_length=10)

wavelengths = np.arange(450, 650, 10)*1e-9
img = np.zeros((320,320))

for wl in wavelengths:
w = lentil.Wavefront(wl)
w *= pupil
w.propagate_image(pixelscale=5e-6, npix=64, oversample=5)
img += w.intensity

plt.imshow(img**0.1, origin='lower')


Keep in mind the output img array must be sized to accommodate the oversampled wavefront intensity given by npix * oversample.

Note

Each time wavefront.field or wavefront.intensity is accessed, a new Numpy array of zeros with shape = wavefront.shape is allocated. It is possible to avoid repeatedly allocating large arrays of zeros when accumulating the result of a broadband propagation by using Wavefront.insert() instead. This can result in significant performance gains, particularly when wavefront.shape is large.

The above example can be rewritten to use Wavefront.insert() instead:

for wl in wavelengths:
w = lentil.Wavefront(wl)
w *= pupil
w.propagate_image(pixelscale=5e-6, npix=64, oversample=5)
img = w.insert(img)


### npix vs npix_prop¶

Lentil’s propagation methods have two arguments for controlling the shape of the propagation output: npix and npix_prop.

npix specifies the shape of the entire output plane while npix_prop specifies the shape of the propagation result. If npix_prop is not specified, it defaults to npix. The propagation result is placed in the appropriate location in the (potentially larger) output plane when a Wavefront field or intensity attribute is accessed.

It can be advantageous to specify npix_prop < npix for performance reasons, although care must be taken to ensure needed data is not accidentally left out:

For most pupil to image plane propagations, setting npix_prop to 128 or 256 pixels provides an appropriate balance of performance and propagation plane size.

For image to pupil plane propagations, npix_prop must be sized to ensure the pupil extent is adequately captured. Because the sampling constraints on image to pupil plane propagations are typically looser, it is safest to let npix_prop default to the same value as npix.

### Discrete Fourier transform algorithms¶

Most diffraction modeling tools use the Fast Fourier Transform (FFT) to evaluate the discrete Fourier transform (DFT) when propagating between planes. While the FFT provides great computational and memory efficiency, high-fidelity optical simulations may require working with exceptionally large zero-padded arrays to satisfy the sampling requirements imposed by the FFT.

Lentil implements a more general form of the DFT sometimes called the matrix triple product (MTP DFT) to perform the Fourier transform to propagate between planes. While the MTP DFT is slower than the FFT for same sized arrays, the MTP DFT provides independent control over the input and output plane sizing and sampling. This flexibility makes the MTP DFT ideally suited for performing propagations to discretely sampled image planes where it is often necessary to compute a finely sampled output over a relatively small number of pixels.

The chirp Z-transform provides additional efficiency when transforming large arrays. Lentil selects the most appropriate DFT method automatically based on the plane size and sampling requirements.

#### Sign of the DFT complex exponential¶

Lentil adopts the convention that phasors rotate in the counter-clockwise direction, meaning their time dependence has the form $$\exp(-i\omega t)$$. While this is an arbitrary choice, it matches the choice made in most classic optics texts. The implications of this choice are as follows:

1

Goodman, Introduction to Fourier Optics.