BeamletOptics.jl

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Gaussian beamlet ​

Lasers are common devices in modern optical laboratories. Modeling their propagation through an optical setup can be of interest when planning new experiments. Geometrical ray tracing struggles to capture the propagation of a laser beam correctly, since it can not inherently capture the wave nature of e.g. the Gaussian beam.

The electric field of the spatial Gaussian mode can be calculated analytically using the BeamletOptics.electric_field function:

BeamletOptics.electric_field Method
julia
electric_field(r, z, E0, w0, w, k, ψ, R) -> ComplexF64

Computes the analytical complex electric field distribution of a stigmatic TEM₀₀ Gaussian beam which is described by:

Arguments

  • r: radial distance from beam origin

  • z: axial distance from beam origin

  • E0: peak electric field amplitude

  • w0: waist radius

  • w: local beam radius

  • k: wave number, equal to 2π/λ

  • ψ: Gouy phase shift (defined as !)

  • R: wavefront curvature, i.e. 1/r (radius of curvature)

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The evolution of this field through an optical system can be modeled e.g. by the ray transfer matrix formalism using the complex -factor [9, pp. 27]. A Julia-based implementation of this approach can be found in ABCDMatrixOptics.jl. However, in the case of this package another approach will be used.

Complex ray tracing ​

In 1968 an internal publication at Bell Labs by J. Arnaud introduced the concept of complex rays wherein three geometrical beams can be used to model the propagation of a Gaussian in fundamental mode through a symmetric optical system, i.e. without the Gaussian obtaining astigmatism and/or higher-order abberations. This method is analoguos to the ray transfer matrix based -method [10].

Without extensions of the original method, the following key assumptions must be met such that this method can be applied

  • all (complex) beams of the Gaussian in question must intersect the same optical elements

  • the optical elements are large compared to the beam (waist)

  • the paraxial approximation must hold for each beam

  • the Gaussian may not be clipped by hard apertures

  • Lagrange invariant must be fulfilled

Various versions of this approach have been implemented under different names in commercial software, most notably FRED, Code V and QUADOA, as well as in open source software, e.g.

  • Raypier - based on Cython, maintenance status not known

  • Poke - based on Zemax API and Python, maintained by J. Ashcraft et al. [11]

  • IfoCAD - maintenance status not known, refer to Wanner et al. [12]

This package implements the above method via the stigmatic GaussianBeamlet and the AstigmaticGaussianBeamlet, the latter of which is presented in more detail in the Astigmatic polarized beamlets chapter.

Stigmatic beamlets ​

The GaussianBeamlet implements the BeamletOptics.AbstractBeam interface and can be used to model the propagation of a monochromatic Gaussian (-mode) through optical system where all optics lie on the optical axis, e.g. no tip and/or tilt dealignment, and abberations can be neglected. It is represented by a chief (red), waist (blue) and divergence (green) beam. See below how these beams are placed in relation to the envelope of the Gaussian beam.

BeamletOptics.GaussianBeamlet Type
julia
GaussianBeamlet{T} <: AbstractBeam{T, Ray{T}}

Ray representation of the stigmatic Gaussian beam as per J. Arnaud (1985). The beam quality M2 is fully considered via the divergence angle. The formalism for the beam parameter calculation is based on the following publications:

Jacques Arnaud, "Representation of Gaussian beams by complex rays." Appl. Opt. 24, 538-543 (1985)

and

Donald DeJager and Mark Noethen, "Gaussian beam parameters that use Coddington-based Y-NU paraprincipal ray tracing," Appl. Opt. 31, 2199-2205 (1992)

Fields

  • chief: a Beam of Rays to store the chief ray

  • waist: a Beam of Rays to store the waist ray

  • divergence: a Beam of Rays to store the divergence ray

  • λ: beam wavelength in [m]

  • w0: local beam waist radius in [m]

  • E0: complex field value in [V/m]

  • parent: reference to the parent beam, if any (Nullable to account for the root beam which has no parent)

  • children: vector of child beams, each child beam represents a branching or bifurcation of the original beam, i.e. beam-splitting

Additional information

Beam parameters

Parameters of the beam, e.g. or , can be obtained through the gauss_parameters function.

Astigmatism and abberations

It is assumed, but not forbidden, that the optical system contains non-flat or non-parabolic beam-surface-interactions that cause the beam to obtain astigmatism or higher-order abberations. These can not be represented by the GaussianBeamlet.

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A GaussianBeamlet can be constructed via:

BeamletOptics.GaussianBeamlet Method
julia
GaussianBeamlet(position, direction, λ, w0; kwargs...)

Constructs a Gaussian beamlet at its waist with the specified beam parameters.

Arguments

The following inputs and arguments can be used to configure the beamlet:

Inputs

  • position: origin of the beamlet

  • direction: direction of the beamlet

  • λ: wavelength of the beamlet in [m]. Default value is 1000 nm.

  • w0: beam waist (radius) in [m]. Default value is 1 mm.

Keyword Arguments

  • M2: beam quality factor. Default is 1

  • P0: beam total power in [W]. Default is 1 mW

  • z0: beam waist offset in [m]. Default is 0 m

  • support: Nullable support vector for the construction of the waist and div rays

Additional information

Waist offset

The z0 keyword arg. can be used in order to spawn a beam where the waist is not located at the specified position, but rather at an offset z0 in [m] along the chief ray axis.

Support vector

In order to calculate the basis vectors required for the beamlet construction, a random orthogonal vector is chosen. If results fluctuate due to the randomness of this vector, make sure to specify a fixed orthogonal support vector.

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Obtaining the beam parameters ​

Once a GaussianBeamlet has been traced through an optical system, several parameters might be of interest for further analysis. In order to relate the traced geometrical beams/rays to the Gaussian parameters, the publications of Arnaud, Herloski et al. and DeJager et al. are used [10, 13, 14]. Consider the following system where a Gaussian beam with arbitrary parameters has been traced through a lens using the approach outlined in the Complex ray tracing section.

The user can obtain parameters such as the beam waist radius, the radius of curvature and more using the gauss_parameters and/or waist_parameters functions. Below the local waist radius and curvature   have been calculated for the example above.