C. F. R. Caron and R. M. Potvliege, "Optimum conical angle of a Bessel–Gauss beam for low-order harmonic generation in gases," J. Opt. Soc. Am. B 16, 1377-1384 (1999)

We determine the conical half-angle at which a weak, loosely focused Bessel–Gauss beam of fixed focal intensity and confocal parameter is most efficient at generating a given harmonic in a gaseous target of arbitrary density profile. Simple analytical results are compared with fully numerical calculations for hydrogen, xenon, and rubidium. The variation of the conversion efficiency with the conical half-angle is shown to depend on the properties of the medium only through the macroscopic dispersion.

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${\chi}^{(q)}={\chi}^{(q)}(-q\omega ;\omega ,\dots ,\omega ),$${\chi}_{1}={\chi}^{(1)}(-\omega ;\omega ),$ and ${\chi}_{q}={\chi}^{(1)}(-q\omega ;q\omega ).$ We give the linear susceptibilities divided by 4π, i.e., the atomic polarizabilities. Numbers in parentheses are powers of ten.
From Ref. 13 (dressed susceptibilities at $1\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^{2}$).
From Ref. 30.
This paper.
Value given in Ref. 33 for $\mathrm{\lambda}=1060\mathrm{nm}.$
From Ref. 33.

The second column of the table gives $\mathcal{N}(z)$ for ${z}_{min}\u2a7dz\u2a7d{z}_{max}.$ In all cases, ${z}_{min}=-{z}_{max}$ and $\mathcal{N}(z)\equiv 0$ for $z\u2a7d{z}_{min}$ and $z\u2a7e{z}_{max}.$ The last column gives the density averaged over the full width at half-maximum of the density profile.

Table 3

Absolute Conversion Efficiency ${R}_{3}(\alpha )$
at $\alpha =0$ (Corresponding to an Incident Gaussian Beam) and at $\alpha ={\alpha}_{\mathrm{max}}$ as a Function of the Confocal Parameter ba

The harmonic is generated in an homogeneous xenon target of density ${\mathcal{N}}_{0}=2.0\times {10}^{18}\mathrm{atoms}/{\mathrm{cm}}^{3}$ and full width at half-maximum $L=0.5\mathrm{mm}.$ The wavelength of the incident beam is 355 nm. Numbers in parentheses indicate powers of ten.
The right-hand side of Eq. (28) is imaginary for this confocal parameter.

Tables (3)

Table 1

Atomic Susceptibilities Adopted in this Paper for Hydrogen, Xenon, and Rubidiuma

${\chi}^{(q)}={\chi}^{(q)}(-q\omega ;\omega ,\dots ,\omega ),$${\chi}_{1}={\chi}^{(1)}(-\omega ;\omega ),$ and ${\chi}_{q}={\chi}^{(1)}(-q\omega ;q\omega ).$ We give the linear susceptibilities divided by 4π, i.e., the atomic polarizabilities. Numbers in parentheses are powers of ten.
From Ref. 13 (dressed susceptibilities at $1\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^{2}$).
From Ref. 30.
This paper.
Value given in Ref. 33 for $\mathrm{\lambda}=1060\mathrm{nm}.$
From Ref. 33.

The second column of the table gives $\mathcal{N}(z)$ for ${z}_{min}\u2a7dz\u2a7d{z}_{max}.$ In all cases, ${z}_{min}=-{z}_{max}$ and $\mathcal{N}(z)\equiv 0$ for $z\u2a7d{z}_{min}$ and $z\u2a7e{z}_{max}.$ The last column gives the density averaged over the full width at half-maximum of the density profile.

Table 3

Absolute Conversion Efficiency ${R}_{3}(\alpha )$
at $\alpha =0$ (Corresponding to an Incident Gaussian Beam) and at $\alpha ={\alpha}_{\mathrm{max}}$ as a Function of the Confocal Parameter ba

The harmonic is generated in an homogeneous xenon target of density ${\mathcal{N}}_{0}=2.0\times {10}^{18}\mathrm{atoms}/{\mathrm{cm}}^{3}$ and full width at half-maximum $L=0.5\mathrm{mm}.$ The wavelength of the incident beam is 355 nm. Numbers in parentheses indicate powers of ten.
The right-hand side of Eq. (28) is imaginary for this confocal parameter.