This paper demonstrates the use of linear gradient lenses (a diverging lens and a converging lens) for subwavelength focusing of Gaussian beams. Numerical simulations have shown that increasing the length lens provides a more compact focal region in both the transverse and longitudinal directions.
The subject of the study gradient optics is phenomena associated with optical
effects in media with a gradual change of the refractive index [
Currently in the integrated and fiber optics there is a large variety of optical
waveguides with different properties. Planar waveguides and fibers along the profile
of the spatial distribution of the refractive index can be divided into two groups: with
step refractive index profile and with gradient profile [
One of the major advantages of the lenses belonging to the gradient optics is that the optical surfaces of gradient lenses may be flat, which is important for the inputoutput of radiation of the fiber.
It is known that high-order mode exhibit greater divergence, however, the use of
multimode method is one of the possible ways to increase the capacity of modern
communication systems [
Distribution and generation of free-space laser vortices studied quite well [
In this paper, numerically investigate the diffraction of Gaussian beams on a linear
gradient lens with different length. For the numerical simulation of diffraction of the
considered laser beams using finite-difference time-domain method (FDTD),
implemented in the software package Meep [
In the paper considers two types of laser beams, which retain their structure during propagation in the free space in the circular polarization: a fundamental Gaussian mode, mode Gauss-Laguerre (0,1).
Figure 1 shows the transverse structure of the considered optical trace element (diverging lens and the converging lens). Later in the linear gradient lenses, in which the refractive index decreases from the central part to the edges of the lens will be called converging lens. The refractive index gradient lenses varies linearly as: all considered 10 rings (width ), each of which had a their refractive index (from n = 3.47 on the center, until n = 1.5 on the edge of the lens). 0.8 0.6 0.4 0.2 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Simulation parameters: the wavelength = 1.55 microns, the size of the computational domain x, y,z[–4.5; 4.5]. The thickness of the absorbing layer PML ~ 0.65 (1 micron), the sampling step of space – /31, the sampling step of time – /(62c), where c is the velocity of light. Simulation time - 40 tacts (one for that meant the propagation time over a single wavelength), and was chosen so that the kind of laser radiation did not change with increasing simulation time (Figure 2).
The size of the focal spot on the FWHM considered a global maximum (maximum intensity). Table 1 shows the diffraction of a fundamental Gaussian mode and mode Gauss-Laguerre (0,1) on the converging lens when you change the length L. Considered the overall intensity.
The table shows both the diffraction pattern and the size of the focal spot, that with increasing length of the focusing lens has been increasing. If L = 0.25λ presence lens has almost no effect, then L = 2λ diffraction pattern has changed significantly: the size of the focal spot on the half-width at half intensity decreased by 2.67 times (from FWHMmax = 1.95λ for L = 0.25λ, to FWHMmax = 0.73λ for L = 2λ). Comparing 4 and 5 columns of Table 1, it can be seen that the depth of focus, decreased significantly with a slight reduction of the transverse dimension of the focal spot in both these cases. Increasing the length of the lens results in better focus, it is clearly seen when comparing the first and last column of Table 1 for the case of fashion Gauss - Laguerre (0.1). Note that FWHMmax also decreases with increasing length of the lens. Also the global maximum (max) also considered at the edge section of the optical element (out). FWHMout = 3.92λ FWHMout=0.48λ FWHMmax = 1.41λ FWHMmax = 0.76λ
Table 2 shows research of changes in the length L of the converging linear gradient lenses separately dedicated to the longitudinal component of the electric field. In addition to the global maximum (max) is considered the first maximum of the optical element (fmx). l ta e en od m m a n nd ia
s fu s
u e a hT G ) 1 , 0 ( e r r e u g a L s s u a G e d o M FWHMfmx = 1.22λ FWHMfmx = 0.33λ FWHMfmx = 0.36λ FWHMout = 0.44λ FWHMout = 0.4λ FWHMmax = 1.16λ FWHMmax = 0.9λ FWHMmax = 0.77λ FWHMmax = 0.58λ FWHMmax = 0.46λ
The longitudinal electric field component (component z) in the case of the fundamental Gaussian mode also becomes smaller with increasing length of the lens. Comparing Tables 1 and 2 for the case of mode Laguerre - (0,1), it should be noted that increasing the length of the lens, there is a significant enhancement z components, and the size of the focal spot begins to form mainly due to the longitudinal component of the electric field. This is particularly noticeable when analyzing the last column of table 1 and 2: for mode Gauss - Laguerre (0,1) size of the focal spot next to the central part of which contains the longitudinal component of the electric field is FWHMout = 0.4λ, while the total intensity of the light Spot contains a cross-polarized side lobes that broadens the spot size to FWHM = 0.48λ. In the plane of maximum intensity, the effect of cross-polarized components of the electric field is more significant (broadened to 0.76λ). Of special note is the case of L = 0.5λ, when the longitudinal component of the electric field near the plane of the first element in the maximum was obtained by the minimum size of the focal spot: FWHMfm = 0.33λ, but as you can see on the the graphs - by increasing the intensity of the side lobes.
Table 3 shows the result of diffraction of Gaussian beams at the diverging linear gradient lens when changing the length L, that is, on the lens, whose refractive index increases from the center to the edges. Such a lens can be used for the introduction of laser radiation into a hollow (annular) optical waveguide. The refractive index in this case, i have a similar case considered earlier: also examined 10 rings (width ), each 0.5 0.0 of which had a their refractive index. But they are changed in the reverse order of n = 1.5 in the middle, to n = 3.47 at the edge of the lens (Figure 1b).
Table 3 shows that the longitudinal component of the electric field using GaussLaguerre modes (0.1) in the plane of maximum intensity able to obtain a focal spot size smaller than the collecting lens, and in the case of L = 1.55λ size of the focal spot is the focal spot in the case of L = 2λ in Table 2. However, the intensity of the spots materially ~ 5 times less. 0.8 0.6
FWHMmax = 0.8λ FWHMmax = 0.63λ
FWHMmax = 0.46λ
Figure 3 is a graph showing the cross section of the components of z to LaguerreGaussian modes (0,1) (bottom row of Table 3). As seen from the graph, a reduction intensity of the focal spot accompanied by decrease intensity of side lobes.
In this paper using the FDTD method, was numerically research of circularly polarized Gaussian beams passing through a linear gradient lens, considered two options of lenses - converging and diverging.
It has been shown that increasing the length lens provides a more compact size of the focal region in all directions, both for total intensity and the longitudinal electric field component.
The minimum size of the focal spot was not obtained at the maximum intensity, but at the first maximum out of optical element. For modes Gauss-Laguerre (0,1) the minimum size of the focal spot on the FWHM for the longitudinal electric field component was 0.33 wavelength for L = 0.5λ; in the plane of maximum intensity using a diverging lens for z components able to obtain a focal spot size less than when using a converging lens (at falling intensity ~ 5 times).
The work was partially funded by the Russian Foundation of Basic Research Grant (№ 14-07-31079 mol_a) and Russian Federation Ministry of Education and Science.