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.

L. A. Lompré, A. L’Huillier, M. Ferray, P. Monot, G. Mainfray, and C. Manus J. Opt. Soc. Am. B 7(5) 754-761 (1990)

References

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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.