M6 formalism – generalization of the laser beam quality factor M2 to the 3D domain

Alexander Brodsky; Natan Kaplan

doi:10.1515/aot-2020-0007

Article

M⁶ formalism – generalization of the laser beam quality factor M² to the 3D domain

Alexander Brodsky and Natan Kaplan

Published/Copyright: May 16, 2020

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From the journal Advanced Optical Technologies Volume 9 Issue 4

Abstract

Here we define a theoretical basis for the generalization of the beam quality factor M² to three-dimensional (3D) space, which we call M⁶ formalism. The formalism is established through the use of examples of multifocal and Axicon optical systems to illustrate discrete and continuous axial beam shaping, respectively. For the continuous case, we expand the definition of the Rayleigh range to incorporate a quality factor having both axial and transverse components Madd2 and M². Using geometrical ray tracing simulations, a proportion factor C is found to empirically describe the axial quality factor Mz2 of an optical setup including an Axicon and a paraxial focusing lens with a Gaussian single mode input beam. Using our M⁶ formalism depth of focus (DOF) ranges are calculated for higher M² beams, and are shown to be in good agreement with the simulated DOF range, demonstrating the usefulness of the M⁶ formalism for the design of real optical systems.

Keywords: beam parameter product; beam shaping; Bessel-like beam; coherence; diffractive multifocal lens; laser beam quality; M2; multimode laser; optical entropy; Rayleigh range

6 Appendix

6.1 Generalized DOF formula

The generalized DOF formula is derived from the well-known Gaussian waist size equation:

(8)ωz=ω01+(ZZR)2

where ω_z, and ω₀ are beam sizes in distance Z from focal plane.

We define p factor as the intensity drop from its maximum in the focal plane, and it is proportional to the beam squared beam size (area).

(9)p=1−ω02ωz2, ω02ωz2=1−p

Expressing Z from (8)

(10)Z2=ZR2(ωz2ω02−1)

Substituting p in (9)–(10)

(11)Z2=zR2(11−p−1)

Z is only half DOF range Δ_p, and it is proportional to refractive index n.

Using (3) the final expression for DOF is:

(12)Δp=n⋅Madd2+M2⋅2ZR011−p−1

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Received: 2020-03-05

Accepted: 2020-04-06

Published Online: 2020-05-16

Published in Print: 2020-09-25

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Articles in the same Issue

https://doi.org/10.1515/aot-2020-0007

Keywords for this article

beam parameter product; beam shaping; Bessel-like beam; coherence; diffractive multifocal lens; laser beam quality; M2; multimode laser; optical entropy; Rayleigh range