Abstract

Analytical solutions are derived using a straightforward mathematical treatment that accurately describes the power of a ${\rm TEM}_{00}$ Gaussian beam transmitted through or around a finite far-field aperture after propagation though a nonlinear refractive medium. These equations are arranged as a series of transmitted power orders, which allow closed and eclipsing aperture ${{Z}}$-scan experiments to be analyzed analytically without requiring numerical simulation. It is shown with this formulism that the power eclipsing an obscuring aperture can be expressed through a linear relationship of the power transmitted through a closed aperture of the same dimensions, which means that the theoretical measurement accuracy of the two methods is the same. A study of the sensitivity of the solutions to nonlinear phase shifts up to ${{4}}\pi $ shows that the peak to valley change in normalized power transmission for both closed and eclipsing aperture decreases for phases beyond ${{3}}\pi {{/2}}$, which means for large phase shifts it is difficult to determine a nonlinear refraction coefficient from peak to valley transmission measurements alone. The solutions were experimentally verified with closed and eclipsing aperture ${{Z}}$-scan toluene measurements undertaken over a range of aperture radii, which agreed well with previously published empirical formulas. The solutions correctly model the observed changes and asymmetries, explained here through strong sample induced focusing, in the closed aperture ${{Z}}$-scan trace shape.

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