Math 2.0

Optics, electromagnetic theory Definition

Beam propagation method

Builds on Helmholtz equationParaxial wave equationRefractive indexFourier transformFast Fourier transform

Definition

Let n(x,y,z)>0n(x, y, z) > 0 be a refractive index, let k0>0k_0 > 0 be the vacuum wave number, and let nˉ>0\bar{n} > 0 be a reference index. A scalar field ψ\psi solves the Helmholtz equation ∇2ψ+k02n2ψ=0\nabla^2 \psi + k_0^2 n^2 \psi = 0. Write ψ=u eiβz\psi = u \, e^{i \beta z} with β=k0nˉ\beta = k_0 \bar{n}, and drop the term ∂z2u\partial_z^2 u, which is small for a slowly varying envelope uu. This gives the paraxial wave equation ∂zu=i2β∇⊥2u+ik0(n2−nˉ2)2nˉu,∇⊥2=∂x2+∂y2.\partial_z u = \frac{i}{2\beta} \nabla_\perp^2 u + \frac{i k_0 (n^2 - \bar{n}^2)}{2 \bar{n}} u, \qquad \nabla_\perp^2 = \partial_x^2 + \partial_y^2. The beam propagation method is the numerical solution of this equation as an evolution problem in zz. It starts from a given input field u(x,y,0)u(x, y, 0) and computes uu on the planes z=mΔzz = m \Delta z, m=1,2,…m = 1, 2, \dots, one step at a time.

Remarks

Example: let V=k0(n2−nˉ2)/(2nˉ)V = k_0 (n^2 - \bar{n}^2)/(2 \bar{n}) on the plane zz. The split-step Fourier version takes the step u(z+Δz)=ℱ−1[e−i(kx2+ky2)Δz/(2β) ℱ[eiVΔzu(z)]],u(z + \Delta z) = \mathcal{F}^{-1}\left[ e^{-i (k_x^2 + k_y^2) \Delta z / (2 \beta)} \, \mathcal{F}\left[ e^{i V \Delta z} u(z) \right] \right], where ℱ\mathcal{F} is the Fourier transform in (x,y)(x, y), computed by the fast Fourier transform. Check: if n=nˉn = \bar{n} everywhere, then V=0V = 0, and each Fourier mode ei(kxx+kyy)e^{i(k_x x + k_y y)} solves the equation with the factor e−i(kx2+ky2)z/(2β)e^{-i(k_x^2 + k_y^2) z / (2\beta)}; so the step is exact. In general the error of the step comes from splitting the two terms, which do not commute.

Non-example: a structure with an abrupt end facet reflects part of the light backwards. The method drops ∂z2u\partial_z^2 u and so keeps only forward waves; it cannot compute the reflected wave.

Wide-angle versions replace the paraxial operator by better rational approximations, such as Padé approximants, of the one-way operator ∇⊥2+k02n2\sqrt{\nabla_\perp^2 + k_0^2 n^2}.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Big O notation, concept, no evidence levelBig O notationconceptConvolution, concept, no evidence levelConvolutionconceptDiscrete Fourier transform, concept, no evidence levelDiscrete Fourier tran…conceptDivisibility, definition, no evidence levelDivisibilitydefinitionFourier transform, concept, no evidence levelFourier transformconceptHelmholtz equation, concept, no evidence levelHelmholtz equationconceptParaxial wave equation, concept, no evidence levelParaxial wave equationconceptPrimitive root modulo n, concept, no evidence levelPrimitive root modulo…conceptPrimitive root of unity, concept, no evidence levelPrimitive root of uni…conceptRefractive index, concept, no evidence levelRefractive indexconceptCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCooley–Tukey FFT algorithm, statement, no evidence levelCooley–Tukey FFT algo…statementCoprime integers, definition, no evidence levelCoprime integersdefinitionBluestein's FFT algorithm, statement, no evidence levelBluestein's FFT algor…statementRader's FFT algorithm, statement, no evidence levelRader's FFT algorithmstatementChinese remainder theorem, statement, no evidence levelChinese remainder the…statementPrime-factor FFT algorithm, statement, no evidence levelPrime-factor FFT algo…statementFast Fourier transform, statement, no evidence levelFast Fourier transformstatementBeam propagation method, definition, no evidence levelBeam propagation meth…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Big O notation concept
  2. Convolution concept
  3. Discrete Fourier transform concept
  4. Divisibility definition
  5. Fourier transform concept
  6. Helmholtz equation concept
  7. Paraxial wave equation concept
  8. Primitive root modulo n concept
  9. Primitive root of unity concept
  10. Refractive index concept
  11. Congruence modulo n definition
  12. Greatest common divisor definition
  13. Cooley–Tukey FFT algorithm statement
  14. Coprime integers definition
  15. Bluestein's FFT algorithm statement
  16. Rader's FFT algorithm statement
  17. Chinese remainder theorem statement
  18. Prime-factor FFT algorithm statement
  19. Fast Fourier transform statement
  20. Beam propagation method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Beam_propagation_method
About this page
TagEJD5X520CQ
URIm2://EJD5X520CQ
Kinddefinition
WritingW0 stub
MSC78A40, 78M20, 65M06
DefinesBeam propagation method
UsesHelmholtz equation, Paraxial wave equation, Refractive index, Fourier transform, Fast Fourier transform
Changed3 November 2026

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