LightPipes Optical Toolbox

The LightPipes computer practical provides simulations of real optical problems for educational purposes. These simulations use advanced propagation routines to solve the Fresnel-Kirchhoff integral in two dimensions, providing realistic physical optics modeling.

How experiments work

• Interaction: Experiments are performed by changing parameters (such as wavelength, radius, or thickness) and measuring characteristic data, typically using a mouse.
• Goal: Students are tasked with deriving expressions for unknown parameters based on measurements.
• Randomization: Each simulation generates a random number to determine the specific parameter to be measured, which is used for verification in reports.
• Underlying Engine: The simulations utilize the LightPipes optical toolbox for propagation, apertures, lenses, and other optical components.

The constant $c$ is a parameter used in the ABCD matrix formulation to describe beam propagation. It can be determined using two primary methods:

Method 1: Heisenberg Uncertainty Relation

Using the uncertainty of impulse and position in the transverse direction, the constant is derived as:

$$c=\frac{\lambda}{4\sqrt{2}w_0}$$

Method 2: Physical Optics Comparison

By comparing geometric results with the physical optics solution for a round beam (where $z_R=\frac{\pi{w_0^2}}{\lambda}$), the constant is found to be:

$$c=\frac{\lambda}{\pi{\sqrt{2}w_0}}$$

Note: There may be slight discrepancies between these methods depending on the assumed geometry (e.g., slab vs. cylinder).

A Fresnel zone plate is a diffractive element composed of concentric rings that alternate between transparent and opaque.

• Focusing Mechanism: By blocking either the odd or even Fresnel zones, the plate acts as a focusing lens when a plane wave is incident upon it.
• Imaging: The plate can be configured to function as an imaging element.
• Resolving Power: The resolving power of a zone plate is determined by the size of its outermost ring ($\Delta r$). The relationship is defined as:

$$\mathit{w}=\frac{\lambda }{2\sin\theta}=\frac{\lambda }{2}\frac{2\Delta r}{\lambda }=\Delta r$$

Where $\mathit{w}$ is the resolving power and $\Delta r$ is the width of the outermost ring.

LightPipes models coherent optical devices by operating on a large data structure: a square two-dimensional array of complex amplitudes representing the optical field of a propagating light beam.

Simulation Workflow

1. Initialization: Generate an initial field (e.g., a plane wave) using a command like Begin.
2. Optical Elements: Apply functions that represent physical components like apertures (CircAperture), filters, beam-splitters, or lenses to the field.
3. Propagation: Use propagation models (e.g., Fresnel) to simulate how the light field travels through free space.
4. Combination: Combine multiple fields (e.g., after passing through different apertures) using commands like BeamMix.
5. Analysis: Calculate physical properties like intensity (Intensity) for visualization or measurement.

To model interferometers like Young's interferometer, you can split a field structure using IntAttenuator (which performs amplitude division) and then recombine the resulting fields using BeamMix. This allows you to process two separate beams (e.g., beams with different hole positions) and then mix them to observe interference patterns.

# Conceptual workflow for an interferometer
# 1. Split the field
field1, field2 = IntAttenuator(field, ...)

# 2. Process fields separately (e.g., shift holes)
# ...

# 3. Mix the beams
mixed_field = BeamMix(field1, field2)

Bessel beams are non-diffracting beams that maintain a narrow focus over large distances. There are several ways to generate them using LightPipes:

1. Using an Axicon: Convert a laser beam into a Bessel beam using an axicon lens.
2. Using an Annular Slit and Lens: Combine an annular slit with a positive lens or concave mirror.
3. Using a Poisson Spot: Illuminate a circular disk with a plane monochromatic beam. If the disk is positioned at the primary focus of a positive lens, the waves from the edge of the disk will be collimated, resulting in a non-diffracting Bessel beam.

In the paraxial approximation (small angles), a periodic system like a laser resonator can be described using a 1-pass propagation matrix $\mathbf{M} = \begin{bmatrix} A & B \ C & D \end{bmatrix}$.

A ray is defined by its position $y$ and angle $\theta$ as a vector $\mathbf{R} = \begin{bmatrix} y \ \theta \end{bmatrix}$.

Propagation through the system is calculated by matrix multiplication:

• After 1 roundtrip: $\mathbf{R_1} = \mathbf{M} \mathbf{R_0}$
• After $n$ roundtrips: $\mathbf{R_n} = \mathbf{M}^n \mathbf{R_0}$

For any physical system, the determinant of the ABCD matrix must be 1 ($AD - BC = 1$) due to the principle of ray reversibility.

# Conceptual representation of ray propagation
# R_n = M^n * R_0
import numpy as np

M = np.array([[A, B], [C, D]])
R0 = np.array([y0, theta0])
R_n = np.linalg.matrix_power(M, n) @ R0

When a monochromatic plane wave illuminates a disk (annular slit), the waves originating from the edges act as in-phase spherical waves. These waves interfere constructively to create a Poisson spot (also known as an Arago spot) near the optical axis.

Under certain conditions, this interference pattern can be modeled as a non-diffractive Bessel beam. The intensity distribution near the axis is approximately described by the squared zeroth-order Bessel function of the first kind:

I(r,z) ≈ I₀ J₀²( (2παr) / λ )

Where:

• α = a/r is the angle of the wavefront near the axis.
• 2a is the diameter of the disk.
• λ is the wavelength.
• r is the radial distance from the axis.
• z is the propagation distance.

The beam width w(z) is proportional to the distance z and is given by:

w(z) = (2.44 / π) * (λz / a)

Field structures in LightPipes are two-dimensional arrays of complex numbers. Memory usage scales with the grid dimension:

• 256 x 256: ~1 MB
• 512 x 512: > 4 MB
• 1024 x 1024: ~16 MB

While common PCs and laptops can handle grids up to 2000 x 2000 points, be aware that some commands require additional internal arrays, which increases the total memory footprint.