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Paraxial beam propagation from Airy-type initial conditions via the Operator Method

I. Julían-Macías, M. A. Jácome-Silva, I. Ramos-Prieto, U. Ruiz-Corona, F. Soto-Eguibar, D. Sánchez-de-la-Llave, H. M. Moya-Cessa

Abstract

We employ quantum mechanical operator techniques to solve the equations of $(1+1)D$ and $(2+1)D$ for paraxial waves with initial conditions defined by Airy-type functions. In the first part, we find the expressions of $(1+1)D$ optical beams, considering initial conditions such as Airy, Airy-truncated, and Airy-Gaussian functions. Subsequently, we extended the analysis to $(2+1)D$ optical beams with initial conditions generated by the product of two Airy, two Airy-truncated and two Airy-Gaussian functions, providing a comprehensive study of multidimensional Airy beam propagation. To validate our theoretical derivations, we present both theoretical and experimental intensity profiles, showing excellent agreement between the two, illustrating the physical characteristics of these beams. Although these solutions have previously been obtained via the diffraction integral and thoroughly studied, the primary goal here is to demonstrate that the optical fields can be derived using quantum mechanical operator methods. Finally, we remark that this alternative approach offers an elegant and powerful framework for analyzing paraxial wave propagation.

Paraxial beam propagation from Airy-type initial conditions via the Operator Method

Abstract

We employ quantum mechanical operator techniques to solve the equations of and for paraxial waves with initial conditions defined by Airy-type functions. In the first part, we find the expressions of optical beams, considering initial conditions such as Airy, Airy-truncated, and Airy-Gaussian functions. Subsequently, we extended the analysis to optical beams with initial conditions generated by the product of two Airy, two Airy-truncated and two Airy-Gaussian functions, providing a comprehensive study of multidimensional Airy beam propagation. To validate our theoretical derivations, we present both theoretical and experimental intensity profiles, showing excellent agreement between the two, illustrating the physical characteristics of these beams. Although these solutions have previously been obtained via the diffraction integral and thoroughly studied, the primary goal here is to demonstrate that the optical fields can be derived using quantum mechanical operator methods. Finally, we remark that this alternative approach offers an elegant and powerful framework for analyzing paraxial wave propagation.
Paper Structure (12 sections, 53 equations, 3 figures)

This paper contains 12 sections, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Propagation of $(1+1)D$ paraxial beams using Airy-type functions as initial conditions
  3. Propagation of a paraxial beam with an Airy function as initial condition
  4. Propagation of a paraxial beam with an Airy-truncated function as initial condition
  5. Propagation of a paraxial beam with an Airy-Gaussian function as initial condition
  6. Propagation of $(2+1)D$ paraxial beams with Airy-type functions as initial conditions
  7. Propagation of paraxial beams with an Airy function as initial condition
  8. Propagation of paraxial beam with an Airy-truncated function as initial condition
  9. Propagation of paraxial beam with an Airy-Gaussian function as initial condition
  10. Conclusions
  11. Alternative way to obtain the propagated Airy beam applying the Hadamard lemma and the Baker-Campbell-Hausdorff formula
  12. Proof of the operator factorization formula

Figures (3)

  • Figure 1: Intensity distributions for the Airy ideal beam at three transverse planes: ($\mathrm{a}_1$) at $z = 0.0 \, \mathrm{m}$, ($\mathrm{b}_1$) at $z = 0.2 \, \mathrm{m}$ and ($\mathrm{c}_1$) at $z = 0.4 \, \mathrm{m}$. The corresponding experimental distributions are shown in ($\mathrm{a}_2$)-($\mathrm{c}_2$), the experimental parameters are $\lambda = 632.8 \, \mathrm{nm}$ and $T = 1 \times 10^{-4} \, \mathrm{m}$ all within a viewing window of $4 \, \mathrm{mm}$. The excellent agreement validates the algebraic operator approach as a powerful alternative to the standard diffraction integral. Note the preservation of the beam's structure despite the transverse displacement, illustrating the non-diffracting nature of the Airy wave packet in a laboratory setting.
  • Figure 2: Intensity distributions for the Airy-truncated beam at three transverse planes: ($\mathrm{a}_1$) at $z = 0.0 \, \mathrm{m}$, ($\mathrm{b}_1$) at $z = 0.2 \, \mathrm{m}$ and ($\mathrm{c}_1$) at $z = 0.4 \, \mathrm{m}$. The corresponding experimental distributions are shown in ($\mathrm{a}_2$)-($\mathrm{c}_2$), the experimental parameters are $T = 1 \times 10^{-4} \, \mathrm{m}$, $\alpha=0.05$ and $\lambda = 632.8 \, \mathrm{nm}$, all within a viewing window of $4 \, \mathrm{mm}$. Unlike the ideal case, the introduction of the decay factor ensures the beam is square-integrable and physically realizable, resulting in a spatially bounded intensity distribution.
  • Figure 3: Intensity distributions for the Airy-Gaussian beam at three transverse planes: ($\mathrm{a}_1$) at $z = 0.0 \, \mathrm{m}$, ($\mathrm{b}_1$) at $z = 0.2 \, \mathrm{m}$ and ($\mathrm{c}_1$) at $z = 0.2 \, \mathrm{m}$. The corresponding experimental distributions are shown in ($\mathrm{a}_2$)-($\mathrm{c}_2$), the experimental parameters are $T=1 \times 10^{-4} \, \mathrm{m}$, $g = 0.125 \times 10^{7} \, \mathrm{m}^{-2}$ and $\lambda = 632.8 \, \mathrm{nm}$, all within a viewing window of $4 \, \mathrm{mm}$. The Gaussian confinement factor provides a smoother energy decay compared to the exponential truncation, enhancing the beam's stability.