Nonlinear focal shift due to the Kerr effect for a Gaussian beam focused by a lens

Abstract

When a low-power, monochromatic Gaussian beam is focused by a thin lens in air and the waist of the beam is in the plane of the lens, there is a shift of the focus position if the waist of the beam is much smaller than the size of the lens. The point of maximum intensity relative to the geometrical focal point shifts closer to the lens. We show that for ultra-intense light beams, when the Kerr effect is unavoidable, there is a nonlinear focal shift. The nonlinear focus position shifts closer to the lens for laser powers below the critical power. To avoid the nonlinear focal shift below the critical power, the correct combination of Gaussian beam waist and focal system has to be used in the experimental setup. It will be shown that as the Fresnel number N w associated with the Gaussian beam radius increases, the nonlinear focal shift first increases and then begins to decrease.

Nonlinear focal shift in an ultraintense light beam focused by a lens in air

“…We are interested in the nonlinear behaviour of the beam for an input power equal to the critical power of air, P cr (air), which is the upper limit of the power range where the focusing of the beam was studied in reference. 1 Since P cr (air) > P cr (F S) we expect self-focusing in the beam as can be seen in Fig. 2 where the standard deviation of the intensity is shown for the linear propagation (red line) and for the nonlinear propagation (blue line) along 1cm in fused silica and for an input power equal to P cr (air), i.e., P 0 = P cr (air).…”

“…We will show in this paper, however, that this assumption is valid for the parameters of the beam and thin lens presented in reference. 1 So, as long as this approximation is satisfied and the size of the beam is smaller than the size of the lens, we have found that there is a nonlinear focal shift and that the nonlinear focal point moves further away from the lens as the input power increases up to the critical power. 1…”

“…We have modeled the focusing of a Gaussian beam by an ideal thin-lens when the waist of the Gaussian beam is located at the lens for an input power equal to and below to the critical power. 1 The critical power in air, P cr (air), is given by P cr (air) = 3.77λ 2 0 /8πn 1 n 2 , with λ 0 the wavelength of the Gaussian beam in vacuum and n 1 and n 2 the linear and nonlinear refractive indices of air, respectively. 2 In the modeling of the focusing of the Gaussian beam we have used a combination of two methods: 1) the Fourier transform (FT) to calculate the electric field from the lens to a plane, P , close to the geometrical focal point, F G , of the lens and 2) the nonlinear Schrödinger equation to propagate the beam along the optical axis and around the geometrical focal point.…”

“…We have called the combination of two methods as the hybrid method. In reference, 1 the Fourier transform neglects any nonlinear effect between the entrance pupil of the lens and the plane P so all the propagation of the light is linear in this region. The propagation from plane P and forward, using the nonlinear Schrödinger equation, takes into account the Kerr effect in air which is not negligible when the input power is close to the critical power, additionally, since the Kerr effect is proportional to the intensity of the beam, I, which is given by the ratio between the input power, P 0 , and the area of the beam, A, i.e., I = P 0 /A, the Kerr effect increases as the area of the beam decreases, i.e., around the focal region.…”

“…An analysis to estimate the region around the geometrical focal point where the Kerr effect is not negligible was presented in reference. 1 The assumption that the Kerr effect is negligible when the beam propagates through the thin lens is not obvious since the nonlinear refractive index of the Fused Silica glass, for example, is larger than the nonlinear refractive index of air. We will show in this paper, however, that this assumption is valid for the parameters of the beam and thin lens presented in reference.…”