Beam focusing

In this tutorial we simulate the focusing of a Gaussian beam by a thin lens and compare the result with the diffraction limit.

Theory

Gaussian beam

A Gaussian beam has the field distribution

$E(r) = E_0 \exp\left(-\frac{r^2}{w_0^2}\right)$

where $w_0$ is the waist radius — the distance from the axis at which the intensity drops by a factor of $e^2$ .

Diffraction limit

When a beam of waist $w_0$ is focused by a lens of focal length $f$ , the amplitude waist in the focal plane is

$w_{\text{focus}} = \frac{\lambda f}{\pi w_0},$

which corresponds to an intensity full width at half maximum of

$\text{FWHM} = w_{\text{focus}}\sqrt{2\ln 2}.$

This is the diffraction limit for a Gaussian beam.

Step 1: Imports and parameters


import math
import torch
import matplotlib.pyplot as plt
from svetlanna import SimulationParameters, Wavefront
from svetlanna.elements import ThinLens, FreeSpace, RoundAperture
from svetlanna.units import ureg
 
params = SimulationParameters.from_ranges(
    x_range=(-1.5*ureg.mm, 1.5*ureg.mm), x_points=2048,
    y_range=(-1.5*ureg.mm, 1.5*ureg.mm), y_points=2048,
    wavelength=632.8*ureg.nm,   # HeNe laser
)
 
w0 = 0.5*ureg.mm     # beam waist radius
f = 100*ureg.mm      # focal length
wavelength = 632.8*ureg.nm

The grid step is $\approx 1.5\ \mu\text{m}$ , so the focal spot is sampled by roughly 30 points — enough to measure its width reliably.

Step 2: Creating the Gaussian beam


wf = Wavefront.gaussian_beam(params, waist_radius=w0)
 
print(f"Shape: {wf.shape}")
print(f"Dtype: {wf.dtype}")
print(f"Max intensity: {wf.max_intensity:.3f}")

Output:


Shape: torch.Size([2048, 2048])
Dtype: torch.complex64
Max intensity: 1.000

Step 3: The optical setup

Aperture → lens → propagation to the focal plane.


# aperture, truncating the beam
aperture = RoundAperture(params, radius=1.2*ureg.mm)
 
# thin lens
lens = ThinLens(params, focal_length=f)
 
# propagation over one focal length
propagate = FreeSpace(params, distance=f, method="zpASM")

Step 4: Propagating through the setup


wf_after_aperture = aperture(wf)
wf_after_lens = lens(wf_after_aperture)
wf_focus = propagate(wf_after_lens)
 
print(f"Intensity in the focus: {wf_focus.max_intensity:.2e}")

Output:


Intensity in the focus: 1.53e+02

Step 5: Analysing the result


fwhm_x, fwhm_y = wf_focus.fwhm(params)
print(f"Measured FWHM: {fwhm_x*1e6:.2f} × {fwhm_y*1e6:.2f} um")
 
w_focus = float(wavelength) * float(f) / (math.pi * float(w0))
fwhm_theory = w_focus * math.sqrt(2 * math.log(2))
print(f"Theoretical FWHM: {fwhm_theory*1e6:.2f} um")

Output:


Measured FWHM: 45.43 × 45.43 um
Theoretical FWHM: 47.43 um

The 4 % difference comes from two sources: fwhm() counts only the grid points that lie strictly above half maximum (so it under-reports by up to one grid step), and the aperture truncates the tails of the Gaussian. Refining the grid brings the measurement closer to the analytical value.

Step 6: Visualisation


extent = [-1.5, 1.5, -1.5, 1.5]
fig, axes = plt.subplots(2, 3, figsize=(15, 10))
 
axes[0, 0].imshow(wf.intensity.cpu(), cmap='hot', extent=extent)
axes[0, 0].set_title('Input Gaussian beam')
axes[0, 0].set_xlabel('x, mm'); axes[0, 0].set_ylabel('y, mm')
 
axes[0, 1].imshow(wf_after_aperture.intensity.cpu(), cmap='hot', extent=extent)
axes[0, 1].set_title('After the aperture')
axes[0, 1].set_xlabel('x, mm')
 
axes[0, 2].imshow(wf_after_lens.phase.cpu(), cmap='twilight', extent=extent)
axes[0, 2].set_title('Phase after the lens')
axes[0, 2].set_xlabel('x, mm')
 
axes[1, 0].imshow(wf_focus.intensity.cpu(), cmap='hot', extent=extent)
axes[1, 0].set_title('Focal plane')
axes[1, 0].set_xlabel('x, mm'); axes[1, 0].set_ylabel('y, mm')
 
center, window = 1024, 60
axes[1, 1].imshow(
    wf_focus.intensity[center-window:center+window,
                       center-window:center+window].cpu(),
    cmap='hot',
)
axes[1, 1].set_title('Focal spot (zoom)')
 
profile = wf_focus.intensity[center, :].cpu()
x_um = params.x.cpu() * 1e6
axes[1, 2].plot(x_um, profile / profile.max())
axes[1, 2].set_xlim(-150, 150)
axes[1, 2].set_xlabel('x, um')
axes[1, 2].set_ylabel('Normalised intensity')
axes[1, 2].set_title('Profile in the focus')
axes[1, 2].grid(True)
 
plt.tight_layout()
plt.show()

Step 7: Effect of the aperture

How does the aperture size change the focus?


aperture_radii = [0.4, 0.6, 0.8, 1.2]  # mm
 
print("Aperture radius | FWHM     | Max I")
print("-" * 40)
for r in aperture_radii:
    aperture = RoundAperture(params, radius=r*ureg.mm)
    wf_out = propagate(lens(aperture(wf)))
    fwhm_x, _ = wf_out.fwhm(params)
    print(f"{r:.1f} mm          | {fwhm_x*1e6:5.1f} um | {wf_out.max_intensity:.3e}")

Output:


Aperture radius | FWHM     | Max I
----------------------------------------
0.4 mm          |  83.5 um | 3.436e+01
0.6 mm          |  60.1 um | 8.957e+01
0.8 mm          |  51.3 um | 1.309e+02
1.2 mm          |  45.4 um | 1.527e+02

A hard aperture much smaller than the beam acts as the dominant diffracting element: the spot broadens and the peak intensity drops. Once the aperture stops clipping the beam (radius $\gtrsim 2 w_0$ ), the focus converges to the diffraction limit.

Full code


import math
import torch
from svetlanna import SimulationParameters, Wavefront
from svetlanna.elements import ThinLens, FreeSpace, RoundAperture
from svetlanna.units import ureg
 
params = SimulationParameters.from_ranges(
    x_range=(-1.5*ureg.mm, 1.5*ureg.mm), x_points=2048,
    y_range=(-1.5*ureg.mm, 1.5*ureg.mm), y_points=2048,
    wavelength=632.8*ureg.nm,
)
 
w0, f = 0.5*ureg.mm, 100*ureg.mm
 
wf = Wavefront.gaussian_beam(params, waist_radius=w0)
aperture = RoundAperture(params, radius=1.2*ureg.mm)
lens = ThinLens(params, focal_length=f)
propagate = FreeSpace(params, distance=f, method="zpASM")
 
wf_focus = propagate(lens(aperture(wf)))
 
fwhm_x, fwhm_y = wf_focus.fwhm(params)
print(f"FWHM: {fwhm_x*1e6:.2f} × {fwhm_y*1e6:.2f} um")
print(f"Max I: {wf_focus.max_intensity:.2e}")

Conclusions

SVETlANNa reproduces diffraction-limited focusing accurately.
The measured spot size matches the analytical prediction once the grid is fine enough.
The aperture controls both the spot size and the peak intensity.

What next?

4f system — optical filtering
Phase grating on an SLM — programmable phase masks
Optical elements — every available element