1. Introduction
With the rapid development of high-energy and high-power laser technology, laser irradiation intensity can reach
(equivalent to field strength
, and peak power can be increased to the order of a petawatt (
). In the near future, laser power will reach 10 PW or even 100 PW, and laser intensity will exceed the order of
. This technology provides a powerful experimental means for the autonomous and controllable research of extreme physical state in the laboratory and in the research field of inertial confinement fusion (ICF) [
1,
2], which is a process where nuclear fusion reactions are initiated by high-power lasers and compressing a fuel target, typically in the form of a pellet that most often contains a mixture of deuterium and tritium. Meanwhile, the impact of the nonlinear optical effect of materials on the working performance of optical components has become one of the non-negligible considerations of scientists and engineers. When a high-intensity laser beam with non-uniform cross-sectional field distribution is incident on an optical component, it can cause the space-dependent change of refractive index of optical material due to the Kerr effect according to the theory of nonlinear optics. Consequently, the working performance of optical components with extremely high intensity laser incidence is different from that with relatively low intensity laser incidence [
3]. The optical lens is an important optical component that consists of a single piece of transparent material with curved sides that can focus or diverge an incident light beam by means of light refraction. Here, we focus on the numerical and quantitative analysis of the influence of Kerr effect on the focusing performance of a convex optical lens with extremely intense laser incidence.
In the literature, some traditional methods have been proposed to study the effect of optical nonlinearity on the focusing performance of optical components. As early as 1977, Censor presented a general method for describing self-focusing phenomena in various media by means of a geometrical-optics approach [
4]. In 2004, Subbarao discussed the paraxial lens approximation and self-focusing theory [
5]. In the same year, Kasparain and Wolf proposed a new ray-tracing scheme to simulate the nonlinear processes of ultra-short pulse propagation, including the self-focusing and self-guiding [
6]. They divided the propagation media into multiple thin layers and traced the evenly spaced light rays in the transverse plane to simulate the Kerr effect. In 2014, based on the work of Kasparain and Wolf [
6], Wei and Yan proposed a multilayer thin-lens self-focusing model to understand the formation and propagation characteristics of self-focusing beam spot inside nonlinear samples based on Fermat principle, which was utilized to calculate the light field distributions of nanoscale spots on the focal plane [
7]. However, the ray-tracing approaches based on geometric optics have inherent shortcomings since the light field distributions are obtained indirectly according to the relationship between the number density of light rays and light intensity, which is with limited simulation accuracy. By using the finite-difference time-domain (FDTD) method, Lee et al. analyzed the self-focusing effects in a nonlinear Kerr film and showed the formation of multiple filamentations. This method can discretely calculate the full-wave spatio-temporal distribution of optical fields, but its consumption of computer resource is also great and only micron-scale problems can be modeled and simulated [
8]. In 2011, Godoy-Rubio et al. presented an alternative wide-angle beam propagation method (BPM) to model nonlinear Kerr-type optical devices by using a reformulated Fourier-based complex Jacobi iterative technique [
9]. As compared with the FDTD method, BPM has improved convergence speed and reduced running time, and the nonlinear self-focusing processes of laser beams with beam sizes up to 10 wavelengths can be calculated. However, since the typical sizes of laser beams output from modern large laser facilities varies from a few millimeters to several decimeters, which are beyond the capabilities of both FDTD and BPM. As for the calculation of beam propagation in free space and lens focusing, many researchers have done relevant research in this area, and there were some mature calculation approaches can be used. In 2001, Delen and Hooker showed a fast Fourier transform (FFT)–based method for calculating the Rayleigh–Sommerfeld full diffraction integral [
10]. By comparing of that method with direct integration (DI) of the Rayleigh–Sommerfeld integral, they proved that the new method based on FFT was effective. In 2003, Mas et al. analyzed the fast algorithms using the Fourier transform and the fractional Fourier transform (FRT) [
11]. They also studied the calculation of Fresnel diffraction patterns under convergent illumination. Double-step Fresnel diffraction (DSF) is an efficient diffraction calculation method at the expense of computer memory and calculation time. In 2013, Okada et al. described a band-limited DSF method and mitigated the aliasing noise of the DSF method [
12]. In 2009, Matsushima et al. presented a novel method for simulating the free-space propagation of laser beams, which was an improvement of the angular spectrum method (AS) [
13]. They resolved the sampling problem of transfer function by limiting the bandwidth of the propagation fields. However, the above methods are only used to calculate the propagation of laser beams in free space without considering the Kerr effect.
In this work, we try to analyze the influence of Kerr effect on the focusing performance of optical lens under the incidence of high-power laser beams with beam sizes up to 100 mm. For this purpose, Both the two-dimensional fast Fourier transform (2D-FFT) algorithm, and the angular spectrum algorithm are utilized according to the theory of Fourier optics. The presented calculation methodology was employed to reduce the computation resource for the focus of large laser beams and the Kerr effect is considered by adding phase term proportional to the Kerr effect in the 2D-FFT method to get the light field distributions in the nominal focal plane and in several planes adjacent to the focal plane. The results show that the different types of incident laser beams have different focusing characteristics for the same optical lens when the Kerr effect is taken into consideration.
2. Theoretical Background
According to the nonlinear optics, the Kerr effect is a nonlinear optical effect occurring when the highly intense light is interacting with crystals, glasses, gases or other media. Under the light field strength
of a high-power laser with angular frequency
, the total polarization of a nonlinear optical medium up to the third order is described by [
14]
where
and
are the first-order linear polarization and susceptibility, respectively;
and
are the third-order nonlinear polarization and susceptibility, respectively, which also represent the Kerr effect; and
is the permittivity of free space. By comparing the constitutive equation
and the polarization relationship
, we have the dielectric constant of a nonlinear optical material described by
Based on the formula of refractive index for nonmagnetic materials,
, we have the complete expression for the refractive index
of a Kerr medium
where the linear portion of refractive index
is given by
. when the nonlinear item
in the square root is much less than unit one, we obtain the approximate but simplified expression of refractive index
where
is the nonlinear coefficient. It can be seen that the refractive index of a Kerr medium is depending on the square of field magnitude
, which is proportional to the light intensity
of the incident laser beam.
Suppose that the optical lens is located on the transverse plane with coordinates
that is perpendicular to the optical axis. Then the phase transfer function of lens is
where
is the maximum thickness of lens,
is the thickness function of lens and
is the space-dependent refraction index of lens at position
. If the lens is considered as the superposition of three parts, as shown in
Figure 1, the total thickness function can be written as
which is the sum of three individual thickness functions. According to the geometric relationship illustrated in
Figure 1, the thickness function
of the left part is given by
where
is the radius of the left spherical surface of optical lens and
for a planar lens surface. Similarly, the right part has a thickness function given by
where
is the radius of the right spherical surface of optical lens and
for a planar lens surface. In this work, the values of
and
are considered relatively small as compared to the radii of the two spherical surfaces of lens, so that the following approximations hold true with sufficient accuracy [
15],
for
, respectively. The central component part is a flat glass with constant thickness,
. Finally, we get the total thickness function given by
where
. It is noted that the nominal focal length of optical lens,
, is defined by
Then, by combining Equations (4), (10), and (11), we obtain the transfer function of lens
where the spatial function
is proportional to the parameter
.
We further assume that the coordinates of the rear nominal focal plane of lens with
are
, and use the Fresnel diffraction integral formula to calculate the focal spots formed by the optical lens. In fact, the field pattern of focal spots in the rear nominal focal plane is given by
where
is the complex light field distribution of an incident laser beam in the front plane of the lens. On substituting Equation (12) into Equation (13), the quadratic phase factor within the integrand are canceled, and one obtains
Based on the light field distribution in the nominal focal plane given by Equation (14), we can further calculate the light field distribution in several transverse planes adjacent to the nominal focal plane according to the angular spectrum algorithm. As a matter of fact, we can define the transfer function of angular spectrum diffraction as
where
and
are the spatial frequency in the
and
directions, respectively. Then the laser transmission in free space from the transverse plane
to the transverse plane
with interval distance
can be calculated by the formula of angular spectrum,
where
and
represent the two-dimensional Fourier transform and inverse Fourier transform, respectively [
16,
17]. Then the light field distribution of focal spots in the transverse plane of interest with the specified interval distance
away from the nominal focal plane can be numerically calculated by applying the 2D-FFT and 2D-IFFT algorithms on Equation (16).
3. Calculation Method and Parameter Selection
The Fresnel diffraction integral formula given by Equation (14) is usually too difficult to be analytically calculated since the integrand may be complicated. Thus, the numerical methods are often applied. We can rewrite Equation (14) into the form of two-dimensional Fourier transform,
To numerically calculate the double integral of two-dimensional Fourier transform in Equation (17), the efficient algorithm based on the two-dimensional fast Fourier transform (2D-FFT) is utilized. For this purpose, we first spatially discretize the front plane
of optical lens, make the 2D-FFT manipulation, and then apply the variable substitutions
and
to obtain the light field distribution in the focal plane
. Assume that the sizes of computational domain in the front lens plane are
, and the gird numbers of the two-dimensional discrete mesh are
with discrete grid sizes
and
. Then, the discrete parameters in the spatial domain and the frequency domain satisfy the relationship
It can be seen that the discrete grid sizes in the focal plane, and , are determined by the sizes of computational domain, and , rather than the grid numbers, and .
In practice, the of computational sizes
and
are determined by two factors—the waist size of incident laser beam and the grid fineness of focal plane. In order to avoid the aliasing errors in the 2D-FFT manipulation, it is necessary to ensure that the spectral components of the original function satisfy the Nyquist sampling conditions [
18,
19]. That is, the highest spectral component of
should satisfy the conditions,
and
. The requirements can be utilized to choose the proper grid sizes,
and
. Then the grid numbers,
and
, can be determined. For the incidence of an axisymmetric laser beam, the square computational domain with square discrete grids is set, so that
and
.
In the following calculations, the nominal focal length and thickness of the focusing lens made by fused silica glass are assumed to be
and
, respectively. The wavelength of incident laser beams is
in vacuum, which are usually applied in large laser facilities for realizing ICF. At this wavelength, the linear portion of refractive index of fused silica glass is
[
20], the nonlinear refractive-index coefficient of light intensity is
cm
2/W [
21], and the nonlinear refractive-index coefficient of squared electric-field magnitude is
. Moreover, the loss coefficient of fused silica is commonly between
and
, depending on the purity of material. Thus, the absorption effect and loss of optical lens is ignored in this work.
In this work, three classical types of incident laser beams are considered. In most cases, a laser emits light in the form of a laser beam often close to a fundamental Gaussian beam, where the transverse profile of the optical intensity of the beam can be described with a Gaussian function and the variation of beam size can be very small for beams with large width. Thus, the first type is the fundamental Gaussian beam, whose electric field is given by
where
is the maximum field amplitude at the center of beam waist,
is the radius of beam waist,
is the Rayleigh length,
is the beam radius at z,
is the radius of curvature of the beam’s wavefront at
, and
is the Gouy phase shift. Therefore, we can substitute Equation (19) into Equation (17) to calculate the light field distribution of focal spots in the nominal focal plane of lens.
The second type are the Hermite-Gaussian beams. A Hermite-Gaussian beam is one of the higher-order solutions of the paraxial Helmholtz equation in Cartesian coordinate system. The expression of light field for the
th order Hermite-Gaussian laser beam is written as
where
is the Gouy phase shift for the Hermite-Gaussian beams,
and
are the
th and
th Hermite polynomials [
16]. It should be noticed that the Hermite-Gaussian beams degenerate to the fundamental Gaussian beam when
.
The third type are the Laguerre-Gaussian beams. A Laguerre-Gaussian beam is one of the high-order solutions of the paraxial Helmholtz equation in cylindrical coordinate system. The light field of the
th order Laguerre-Gaussian beam is given by
where
is the Gouy phase shift, and
is the associated Laguerre polynomial. A Laguerre-Gaussian beam of
mode is specified by two mode indices, the angular mode index
and the transverse radial mode index
. When
, the Laguerre-Gaussian beams degenerate to the fundamental Gaussian beam. When the angular mode index
is greater than zero, the light field has an azimuthal phase change of
, which results in a phase singularity in the field and a node in the intensity at the center of the beam. Thus, the laser beams propagating in Laguerre-Gaussian modes may have orbital angular momentum and multiply connected topology.