# Product of Two Laguerre–Gaussian Beams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}beam. For a particular case of pLG beams described by Laguerre polynomials with azimuthal numbers n − m and n + m, an explicit expression for the complex amplitude in a Fourier plane is derived. Similar to conventional LG beams, the pLG beams can be utilized for information transmission, as they are characterized by orthogonal azimuthal numbers and carry an orbital angular momentum equal to their topological charge.

## 1. Introduction

## 2. Theoretical Background

_{0})

^{2}]

^{1/2}is the Gaussian beam radius, γ(z) = arctan(z/z

_{0}), γ(z) is the Gough phase, z

_{0}is the Rayleigh range, and k is the wavenumber of light. The Rayleigh range of the beam from Equations (1) and (2) is given by

_{0}) and far field (z >> z

_{0}).

_{0}/z)

^{2}] being the curvature radius of the wavefront.

^{2}beam, is given by:

_{f}{ } is the Fourier transform. A comparison of relationships for complex amplitudes in the source plane (Equation (8)) and at the focus of a spherical lens (Equation (9)) shows that they are identical up to a constant. In the Fresnel diffraction zone, the (LG)

^{2}beam is described by a finite superposition of conventional LG beams, which are similar to Equation (7) but have different coefficients:

_{0}/z, Γ(x) is the gamma function. In the general case, Equation (10) is consistent with Equation (7) because in both cases, the complex amplitude is expressed via a finite sum of conventional LG beams. The difference between Equations (7) and (10) is that the latter only contains LG beams with even radial indices. Equation (10) also suggests that at p = 0 (zero-valued radial index), the intensity pattern of the LG beam is a single ring, as ${L}_{0}^{n}(x)=1$, so in the sum in Equation (10), only the first term is retained, meaning that the beam from Equation (1) with a squared amplitude is preserved upon free-space propagation.

^{2}beam is seen to occur.

## 3. Numerical Simulation

^{1/2}w, and not w, the Rayleigh range of beam from Equation (1) is twice that of beam from Equation (3). Shown in Figure 1 are intensity and phase patterns at the Rayleigh range for beam from Equation (3), i.e., at kw

^{2}/2 = z

_{0}/2. The patterns were numerically simulated using Equations (1) and (3) in the source plane and using a Fresnel transform at the half Rayleigh range.

^{2}/2, these diverge not by a factor of 2

^{1/2}≈ 1.41, but by a factor of just (5/4)

^{1/2}≈ 1.12 (Figure 1a,c,e,g). In the meantime, the beam from Equation (3) expands by a factor of just about 1.5.

## 4. Conclusions

^{2}beam. The proposed pLG beams have been expressed as the superposition of a finite sum of conventional LG beams. For the (LG)

^{2}beams, an explicit Fourier transform has been derived. A particular case of pLG beams whose Laguerre polynomials are described by specially tailored azimuthal indices n − m and n + m has been analyzed, and their Fourier transform has been deduced in an explicit form. The pLG beams are promising for optical communication applications [22,24].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{k}using Equation (A3). Then, instead of Equation (A6), we obtain:

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**Figure 1.**Patterns of intensity (columns 1 (

**a**,

**e**,

**i**) and 3 (

**c**,

**g**,

**k**)) and phase (columns 2 (

**b**,

**f**,

**j**) and 4 (

**d**,

**h**,

**l**), dark—0, white—2π) from conventional LG beams (Equation (1)) of the orders (p, m) = (2, 1) (row 1) and (q, n) = (3, 4) (row 2) and from a pLG beam (Equation (3)) of the order (p, q, m, n) = (2, 3, 1, 4) (row 3) in the source plane z = 0 (columns 1 and 2) at the half Rayleigh range z = z

_{0}/2 (columns 3 and 4) for the following parameters: wavelength λ = 532 nm, waist radius w = 0.5 mm. The scale bar in all pictures is 1 mm. The topological charge was measured along a dashed circle in the phase patterns. Shown in the bottom row in (

**i**,

**k**) are intensity profiles.

**Figure 2.**Intensity patterns of a pLG beam (Equation (3)) of the orders (p, q, m, n) = (1, 4, 1, 4) (row 1(

**a**–

**d**)), (p, q, m, n) = (2, 3, 2, 3) (row 2 (

**e**–

**h**)), (p, q, m, n) = (3, 3, 1, 1) (row 3 (

**i**–

**l**)) in four different planes: in the source plane z = 0 (column 1), at the half Rayleigh range z = z

_{0}/2 (column 2), at the Rayleigh range z = z

_{0}(column 3), and in the far field z = 10z

_{0}(column 4). Other parameters are the following: wavelength λ = 532 nm, waist radius w = 0.5 mm. The scale bar is 1 mm (columns 1–3) and 10 mm (column 4). The yellow curve below each figure shows the intensity profile.

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**MDPI and ACS Style**

Kotlyar, V.V.; Abramochkin, E.G.; Kovalev, A.A.; Savelyeva, A.A.
Product of Two Laguerre–Gaussian Beams. *Photonics* **2022**, *9*, 496.
https://doi.org/10.3390/photonics9070496

**AMA Style**

Kotlyar VV, Abramochkin EG, Kovalev AA, Savelyeva AA.
Product of Two Laguerre–Gaussian Beams. *Photonics*. 2022; 9(7):496.
https://doi.org/10.3390/photonics9070496

**Chicago/Turabian Style**

Kotlyar, Victor V., Eugeny G. Abramochkin, Alexey A. Kovalev, and Alexandra A. Savelyeva.
2022. "Product of Two Laguerre–Gaussian Beams" *Photonics* 9, no. 7: 496.
https://doi.org/10.3390/photonics9070496