Nonparaxial Gaussian beams

Abstract

When the waist size of a Gaussian beam becomes of the order of the wavelength of light, the beam does not satisfy the paraxial condition in which it is derived. In this paper, we define the lower bound to the waist size by showing that a Gaussian beam whose waist size is larger than this bound safely satisfies the paraxial condition. A beam which is Gaussian in form but violates the paraxial condition is called a nonparaxial Gaussian beam. We clarify the range of the waist size for which the first-order correction to this beam is effective. It is shown that a distinct value of the waist size exists for which the paraxial approximation completely fails and the first-order correction never works.

Airy beams beyond the paraxial approximation

“…(35) and (36) basically yield field wave forms consistent with the solution e iρ f = ρ of the Helmholtz equation for a point-like source, conforming indeed to the view that asymptotically the source functions, whatever they be, would be like point sources.…”

“…It is commonly used as the expansion parameter within the context of the approach to the non paraxial propagation through corrections at various orders to the paraxial form [34]. Roughly, values of f ≪ 1 designate the paraxial regime, whilst, following the analysis in [35], when f = 1 the paraxial 2 The reduced wavelength k − 1 = λ/2π plays within the optical context the role as the reduced Planck constant ℏ within the quantum mechanical context, as it rules the commutator between the "optical" position and momentum operators:…”

“…As is well known [43], they can be understood as the nonparaxial (within the approximation implied by (3)) version of the Fraunhofer diffraction formula [44]. According to (35) and (36), in the far zone, as in the near zone, the amplitudes of the transverse and longitudinal components of the electric field turn out to be for both beams roughly in the ratio: |E ⊥ / E ∥ | ≈ 1/f along any fixed direction, for which indeed ξ/ρ, η/ρ, and ζ/fρ are constant whilst ρ/f → ∞.…”

Laser Diode Beam Basics

“…Nemoto [11] shows that when ks 0 < 4 the paraxial Gaussian model deviates appreciably from the exact solution and that when ks 0 < 2 the paraxial Gaussian model differs considerably from the exact solution, where s 0 is the 1/e intensity radius of the beam waist and k = 2π/λ is the wave vector. s 0 can be converted to the more commonly used 1/e 2 intensity radius w 0 of the beam waist by s 0 = 0.368w 0 , then Nemoto's two conditions can be written, respectively, as w 0 \1:73k ð2:15Þ w 0 \0:87k ð2:16Þ…”

Vectorial Structure of a Phase-Flipped Gauss Beam in the Far Field

“…It has been known from previous reports [4,5,22] that, a PFG beam within larger f parameter may show more nonparaxial characteristics. To define a paraxial Gaussian beam, namely, f ≤ 0.18 should be required at the plane z = 0 [22]. It is also found that, for a PFG beam diffracting in free space, the paraxial approximation holds true if f ≤ 0.04 [4].…”