Structuring a laser beam subject to optical Kerr effect for improving its focusing properties

Abstract

The use of laser beams made up of ultrafast pulses for the processing of materials can be bothered by the consequences of optical Kerr effect (OKE) cumulated by the propagation through optical devices (windows, laser crystal, prisms, lenses, …). The latter are mainly a reduction of the intensity in the focal plane accompanied by a distortion of the temporal and spatial pulse shape. We present a comparative study on such distortions for Gaussian, super-Gaussian and LG₁₀ (one central peak surrounded by a ring) beams. It is demonstrated that the LG₁₀ beam shows sensitivity to OKE which is smaller than that of the Gaussian LG₀₀, and super-Gaussian beams. As a result, the focusing performances of the LG₁₀ beam are quite superior to that observed with Gaussian or super-Gaussian beam: a higher on-axis intensity, a narrower intensity pattern, and a temporal shape and an energy fluence almost undistorted.

Appendices

Appendix

Kerr lens modelling for LG₀₀ and LG₁₀ beams

In the next, we will present a modelling of the Kerr lensing effect induced by LG₀₀ and LG₁₀ based on the Zernike polynomial decomposition. The propagating term of the collimated incident beam emerging from the Kerr medium is written $\exp \left[ {ikW(\rho )} \right]$, where $kW(\rho ) = - \Delta \varphi (\rho )$. The phase distribution $W(\rho )$ is usually called as wave aberration function (WAF) according to the terminology of optical aberration modelling [39]. The WAF is expanded as a linear combination of Zernike polynomials (ZP), $Z_{j}$, as follows

$$W(\overline{\rho }) = \sum\limits_{j = 1}^{\infty } {a_{j} Z_{j} (} \overline{\rho })$$

(17)

where the index j is a polynomial-ordering number, and $a_{j}$ the aberration coefficients. The normalised radial coordinate $\overline{\rho }$ is equal to $\rho /R$, where R is the radius of the unit circle. In the following we set R equal to 1.5 W (2.5 W) since the circle of radius 1.5 W (2.5 W) contains 99% of the incident LG₀₀ (LG₁₀) beam power. The Zernike polynomials $Z_{j} (\overline{\rho },\theta )$ are a set of orthogonal functions over the unit circle ($0 \le \overline{\rho } \le 1$). Table 2 gives the corresponding ZP’s [40].

Table 2 Zernike polynomials allowing to calculate the four-first non-zero aberration coefficients $a_{1}$, $a_{4}$, $a_{11}$ and $a_{22}$

For the calculation of the aberration coefficient $a_{j}$.we will take into account the amplitude profile $E(\rho )$ of the incident beam [41]:

$$a_{j} = \frac{{ - \int\limits_{0}^{1} {E(\overline{\rho })\Delta \varphi (\overline{\rho })Z_{j} (\overline{\rho })\overline{\rho }d\overline{\rho }} }}{{k\int\limits_{0}^{1} {E(\overline{\rho })\overline{\rho }d\overline{\rho }} }}$$

(18)

where $E(\overline{\rho })$ stands for $E_{1} (\overline{\rho })$, when the incident beam is a LG₀₀ and $E_{2} (\overline{\rho })$ for a LG₁₀:

$$E_{1} (\overline{\rho }) = E_{0} \times \exp \left[ { - 2.25\overline{\rho }^{2} } \right] \times \exp [ - t/\tau ],$$

(19)

$$E_{2} (\overline{\rho }) = E_{0} \times \left[ {1 - 9.68\overline{\rho }^{2} } \right]\exp \left[ { - 4.84\overline{\rho }^{2} } \right] \times \exp [ - t/\tau ].$$

(20)

The nonlinear phase shift $\Delta \varphi (\overline{\rho })$ stands for $\Delta \varphi_{1}$ and $\Delta \varphi_{2}$, given by Eqs. (8) and (9), and by taking into account the normalisation of the radial coordinate rewrite as

$$\Delta \varphi_{1} (\overline{\rho }) = \varphi_{0} \times \exp [ - 4.5\overline{\rho }^{2} ],$$

(21)

$$\Delta \varphi_{2} (\overline{\rho }) = \varphi_{0} \times [1 - 12.5\overline{\rho }^{2} ]^{2} \times \exp [ - 12.5\overline{\rho }^{2} ].$$

(22)

The sum in Eq. (17) is infinite, but is usually truncated. Here, we will work arbitrarily until j = 22, and because of the rotational symmetry of $\Delta \varphi$ most of coefficients $a_{j}$ are equal to zero except four of them: a₁, a₄, a₁₁ and a₂₂.

In the following, we will use the dimensionless aberration coefficients A_j which are expressed in unit of wavelength, and is defined as $A_{j} = a_{j} /\lambda$. By taking into account the expressions of $E(\overline{\rho })$ and $\Delta \varphi (\overline{\rho })$ we get finally.

For LG₀₀:

$$A_{j} = \frac{{ - \varphi_{0} \int\limits_{0}^{1} {\overline{\rho }\exp [ - 6.75\overline{\rho }^{2} ]} Z_{j} (\overline{\rho })d\overline{\rho }}}{{2\pi \int\limits_{0}^{1} {\overline{\rho }\exp [ - 2.25\overline{\rho }^{2} ]} d\overline{\rho }}}.$$

(23)

For LG₁₀:

$$A_{j}^{\prime } = \frac{{ - \varphi_{0} \int\limits_{0}^{1} {\overline{\rho }[1 - 12.5\overline{\rho }^{2} ]^{3} \exp [ - 18.75\overline{\rho }^{2} ]} Z_{j} (\overline{\rho })d\overline{\rho }}}{{2\pi \int\limits_{0}^{1} {\overline{\rho }[1 - 12.5\overline{\rho }^{2} ]\exp [ - 6.25\overline{\rho }^{2} ]} d\overline{\rho }}}.$$

(24)

The integrals in Eq. (23) and (24) are solved numerically using a FORTRAN routine based on the numerical integrator dqdag from the International Mathematics and Statistical Library (IMSL). The results are given in Table 3:

Table 3 Dimensionless aberrations coefficients $A_{1}$,$A_{4}$,$A_{11}$ and $A_{22}$ normalised by the nonlinear axis phase shift $\varphi_{0}$ associated with the Kerr phase aberration $\Delta \varphi_{1} (\rho )$ or $\Delta \varphi_{2} (\rho )$, when the input is a LG₀₀ or a LG₁₀ beam, respectively

It has been already shown [42] that it is possible to deduce an equivalent focal length $\sqrt 3 R^{2} /(12\lambda A_{4} )$, noted $f_{K0}$, for LG₀₀ input, and $f_{K1}$, for LG₁₀ input, from the aberration coefficients $A_{4}$ and $A_{4}^{^{\prime}}$:

For LG₀₀:

$$f_{K0} = \frac{{4.48W^{2} }}{{\lambda \varphi_{0} }} \times \exp [2(t/\tau )^{2} ],$$

(25)

For LG₁₀:

$$f_{K1} = \frac{{ - 84W^{2} }}{{\lambda \varphi_{0} }} \times \exp [2(t/\tau )^{2} ].$$

(26)

The first remark that could be made from the above is that the Kerr lensing effect induced by a LG₀₀ (LG₁₀) is a converging (diverging) lensing effect which has a tendency to shift the best focus point toward (away) the lens. In addition, Eqs. (25) and (26) show that $\left| {f_{K1} } \right| > > \left| {f_{K0} } \right|$ and that explains the focal shift behaviour illustrated in Fig. 3.

The position $z_{F0}$ ($z_{F1}$) of the best focus when the input is a LG₀₀ (LG₁₀) beam in the framework of the above Kerr lens modelling is given by

$$\frac{1}{{z_{F0} }} = \frac{1}{{f_{L} }} + \frac{1}{{f_{K0} }}\,{\text{and}}\,\frac{1}{{z_{F1} }} = \frac{1}{{f_{L} }} + \frac{1}{{f_{K1} }}.$$

(27)

The results are shown in Fig. 17, for t = 0, which displays the best focus position $z_{F0}$ and $z_{F1}$, resulting from the above Kerr lens modelling for LG₀₀ and LG₁₀ input, and position $z_{\max }$ calculated from the on-axis intensity distribution. The agreement between the Kerr lens modelling ($z_{F0}$ and $z_{F1}$) and the position $z_{\max }$ of the best focus is not perfect but satisfactory at the very least concerning the trend in focus shift due to optical Kerrr effect (Fig. 22).

Fig. 22

Comparison of the best focus position obtained (i) by solving the diffraction integral ($z_{\max }$) and (ii) by the Kerr lens modelling based on the aberration coefficients characterising the Kerr phase profile induced by LG₀₀ and LG₁₀ beams ($z_{F0}$ and $z_{F1}$)

Validity of Fresnel–Kirchhoff integral for ultrashort pulses

The well-known Fresnel–Kirchhoff integral given by Eq. (5) assumes that the incident wave is monochromatic, and this is rigorously not the case when the incident beam is an ultrashort pulse. The time variation of the electric field associated with the laser pulse takes the following form

$$E(t) = \exp \left[ { - t^{2} /\tau^{2} } \right] \times \exp (i\omega_{0} t).$$

(28)

The spectral content $E(\omega )$ is deduced from the following Fourier transform

$$E(\omega ) = \tau \sqrt \pi \times \exp \left[ { - \pi^{2} \tau^{2} (\omega - \omega_{0} )^{2} } \right].$$

(29)

The spectral width (at half-maximum) is equal to $\Delta \omega = \sqrt {\ln 2} /(\pi \tau )$. If one considers a central wavelength $\lambda_{0} = 1064nm$ we obtain the width $\Delta \lambda$ of the spectral distribution

$$\frac{\Delta \lambda }{{\lambda_{0} }} = \frac{{0.149 \times 10^{ - 15} }}{\tau }.$$

(30)

If one takes into account Eq. (4), we can estimate the variation $\Delta \varphi_{0}$ of the nonlinear on-axis phase shift which is the key parameter of the diffracted field distribution:

$$\frac{{\Delta \varphi_{0} }}{{\varphi_{0} }} = \frac{\Delta \lambda }{{\lambda_{0} }} = \frac{{0.149 \times 10^{ - 15} }}{\tau }.$$

(31)

The following table (Table 4) shows that for a pulse duration $\tau > 100fs$, the resulting variation of the nonlinear on-axis phase shift is too small for having a significant influence on the diffracted field. As a result, in these conditions we can use the classical Fresnel–Kirchhoff integral for calculating the pattern in plane $z = f_{L}$.

Table 4 Relative variations of the nonlinear phase shift due to the spectral content of the pulse of duration $\tau$

In the case where the pulse duration $\tau$ is too short, the application of Eq. (4) is not valid and the calculation of the diffracted field should need to take into account the finite spectrum.