=Paper=
{{Paper
|id=Vol-1638/Paper16
|storemode=property
|title=Diffraction of the Gaussian beam on layered lens and similar a conical and diffraction axicons
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper16.pdf
|volume=Vol-1638
|authors= Dmitry A. Savelyev
}}
==Diffraction of the Gaussian beam on layered lens and similar a conical and diffraction axicons ==
https://ceur-ws.org/Vol-1638/Paper16.pdf
Computer Optics and Nanophotonics
DIFFRACTION OF THE GAUSSIAN BEAM ON
LAYERED LENS AND SIMILAR A CONICAL AND
DIFFRACTION AXICONS
D.A. Savelyev1,2
1
Samara National Research University, Samara, Russia
2
Image Processing Systems Institute - Branch of the Federal Scientific Research Centre “Crys-
tallography and Photonics” of Russian Academy of Sciences, Samara, Russia
Abstract. In this paper we consider the possibility of replacing the diffraction
axicon and the conical axicon on the gradient lens with a linear variation of the
refractive index. Analytically and numerically using the finite-difference time-
domain method we performed a comparative study of the Gaussian beam dif-
fraction on diffraction mikro-axicon, conical axicon and gradient microlens
consisting of subwavelength layers. The parameters under consideration the
types of elements estimated in the depth of focus and a transverse dimension of
beam.
Keywords: diffraction optics, subwavelength structures, laser beams, diffrac-
tion axicon, layered lens, conical axicon, FDTD.
Citation: Savelyev DA. Diffraction of the Gaussian beam on layered lens and
similar a conical and diffraction axicons. CEUR Workshop Proceedings 2016;
1638: 117-124. DOI: 10.18287/1613-0073-2016-1638-117-124
Introduction
Environments with light propagates in curved paths are the subject of gradient optics
(GRIN - GRadient INdex) [1]. The flat surfaces of gradient lenses make them very
useful for collimating light from the end of single mode fiber and focusing of the
collimated beam to another single mode fiber [2]. Thus, light beams passing through
the gradient lens can be the use for better focusing [3-5].
When transmitting information over optical fibers easier connection between the fi-
bers do using gradient elements [6, 7], usually, such components are in some way
analogue of a lens [8, 9], which forms a short focus. Typically used two gradient ele-
ments with a sufficiently precise mutual agreement: one at the output, which scatters
the laser beams and one at the entrance, which collects the laser beam [10, 11].
One advantage of using the axicon is the formation of an extended focus [12, 13],
including subwavelength lateral size [14, 15]. The advantage of using a diffrac-tion
axicon before the conical axicon is in the relative simplicity of manufacturing, and in
the possibility of achieving, for this element of high numerical aperture values, inac-
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Computer Optics and Nanophotonics Savelyev DA. Diffraction of the…
cessible to the conical axicon due to total internal reflection [16-18]. An extended
focus [19] can be used to alleviate the requirements for alignment of the optical fiber
connection.
For connections required flat edge [20], and diffraction axicon has it. In this pa-per,
we consider particularly focusing Gaussian beams by using gradient optical ele-ments
[21, 22] and similar a conical and diffraction axicons. For the numerical simu-lation
of diffraction considered laser beams used finite-difference time-domain method
(FDTD) using high-performance computing [23].
Diffraction of the Gaussian beam
Under the linear change of the refractive index n ( r ) n0 (1 r ) the phase difference
is analogous to a conical axicon:
lin (r ) kn0 Lr , (1)
where n0 – the refractive index in the center, k 2 , is the wavelength, L – lens
thickness, α – parameter governing the rate of change of the refractive index. Let’s
define parameters of a conical axicon creating the same phase difference [14]:
ax ( r ) k ( nax 1) r ctg , (2)
In according to equations (1) and (2), we selected axicon angle:
n 1
arctg ax , (3)
Ln0
where nax – the refractive index of axicon material, β – a half of angle at the axicon
tip (Fig1(c)). H – axicon height:
R
H , (4)
tg
Let us consider the diffraction axicon (Fig.1 (b)). The phase difference between the
central ray and a ray extending from the center at a distance is equal:
диф k NA r , (5)
where NA – the numerical aperture of the axicon, r - radius of the axicon. Then, the
numerical aperture of axicon is
NA n 0 L , (6)
where n0 = 3,47 – the value of the central layer for the considered layered lens. Period
of axicon d is changed to the following law:
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Computer Optics and Nanophotonics Savelyev DA. Diffraction of the…
d , (7)
NA
Height axicon considered on the basis of the phase shift on π:
h 0, 21, (8)
k(n ax 1) 2(n ax 1)
when the refractive index nax = 3,47.
a) b)
c)
Fig. 1. The transverse structure (scheme) of the matched linearly layered lens (a), diffractive
axicon (b) and the conical axicon (c)
Simulation parameters: the wavelength = 1.55 microns, the size of the computation-
al 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. As the input laser radiation with the circular polariza-
tion we use the fundamental Gaussian mode. In the case of layered lens we use the
linearly changes of refractive index of lens: from n=3.47 in center to n=1.34 at the
lens edge. Let’s denote a lens width on propagation axis of the laser beam as L.
Numerical simulation was made using the computational cluster with the power of
775 GFlops. The cluster’s characteristics are the following: 116 cores, computing
nodes – 7 dual servers HP ProLiant 2xBL220c, RAM volume 112 Gbit.
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Computer Optics and Nanophotonics Savelyev DA. Diffraction of the…
We consider the half width at half intensity (FWHM) and depth of field (DOF). Fix a
lens width L = 1,55 with refractive index n = 3,47.The numerical results studies for
the axicon and the layered lenses with a corresponding α are given in Table 1.
Table 1. Diffraction of Gaussian beam on a layered lens, diffraction and conical axicon
Type α = 0,13 α = 0,12 α = 0,11 α = 0,1
of
elem
ent
Layered lens
DOF = 3,2λ DOF = 2,85λ DOF = 3,2λ DOF = 3,4λ
FWHM = 0,67λ FWHM = 0,68λ FWHM = 0,67λ FWHM = 0,68λ
Diffraction axicon
DOF = 2,55λ DOF = 3,23λ DOF = 3,77λ DOF = 4,36λ
FWHM = 0,71λ FWHM = 0,75λ FWHM = 0,77λ FWHM = 0,84λ
Сonical axicon
DOF = 1.36λ DOF = 1.77λ DOF = 2.17λ DOF = 2.81λ
FWHM = 0.78λ FWHM = 0.76λ FWHM = 0.75λ FWHM = 0.76λ
Reducing the parameter α for a layered lens increases the length of the light segment
with a substantially constant radius of the light spot. A separate case with α = 0.12,
where the observed change in the overall diffraction patterns and reducing the depth
of focus.
For diffraction axicon situation is as follows: reduction in α (that means reducing the
numerical aperture) also leads to an increase in the length of the light segment. No
cases like the case α = 0.12 for a layered lens. And also we see expected focal spot
size increases.
For the conical axicon also decrease α (which is equivalent to an increase of the angle
β) leads to elongation of the light segment. But in this case also seems certain number
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Computer Optics and Nanophotonics Savelyev DA. Diffraction of the…
α = 0.11, where the focal spot is minimal. Reduction of α leads to a broadening of the
beam.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-5 -4 -3 -2 -1 0 1 2 3 4 5
Fig. 2. Diffraction of Gaussian beam on a layered lens with changing L (α = 0.11), the intensi-
ty: L = 1.55λ (black line), L = 1.75λ (gray line)
Table 2. Results of numerical simulation when changing the height of the layered lens
α L=λ L = 1.55λ L = 1.75λ L = 2λ
0.11
DOF = 3.7λ DOF = 3.2λ DOF = 2.94λ DOF = 1.9λ
FWHM = 0.89λ FWHM = 0.67λ FWHM = 0.78λ FWHM = 0.73λ
0.12
DOF = 3.73λ DOF = 2.85λ DOF = 2.64λ DOF = 0.93λ
FWHM = 0.85λ FWHM = 0.68λ FWHM = 0.87λ FWHM = 0.79λ
When comparing rows of Table 1 it should be noted that the use of a layered lens
provides a more narrow size of the focal spot, and when the value of α is higher than
0.12, and more extended focal light segment. Let us consider in more detail the lay-
ered lens effect in changing its length along the axis of propagation of the laser beam
for two cases mentioned earlier: α = 0.11 and α = 0.12 (Table 2). We consider the
FWHM at the point of maximum intensity.
Table 2 shows that the increase in length of the lens leads to a reduction of the focal
length for DOF. Increasing the lens length of an increase of the numerical aperture,
only makes sense to a certain value (Figure 2). Table 2 shows that the increase in the
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length of more than 1.55λ lens reduces the depth of focus at a constant value of
FWHM.
Consider the change in the height of the diffraction axicon in case α = 0.12, i.e. at a
numerical aperture of NA = 0.64. We varied the refractive index n. The height of the
respective axicon considered on the basis of the phase shift at π by the formula (8).
Numerical simulation result is shown in Table 3.
Table 3. Diffraction of Gaussian beam on diffraction axicon with a change of the refractive
index n
n = 3,47 n = 2,25 n = 1,68 n = 1,46
Intensity
DOF = 3,23λ DOF = 3,05λ DOF = 2,21λ DOF = 2,25λ
FWHM = 0,75λ FWHM = 0,71λ FWHM = 0,68λ FWHM = 0,68λ
Decrease in the refractive index and simultaneously increase axicon relief leads to a
reduction of the lengths of light segment. However, after a certain limiting value (n =
1.68) DOF begins to increase again. It is also worth noting the reduction in the size of
the focal spot with a decrease in the index of refraction of the axicon. However, it
should be noted that after reaching a limiting value of the refractive index (in this
case, when n = 1.68) of the focal spot size is stabilized and becomes comparable to
the previously discussed case of layered lenses.
Conclusion
Analytically and numerically using the finite-difference time-domain method we per-
formed a comparative study of the diffraction of Gaussian beam by diffraction micro-
axicon and conical axicon, and gradient micro-lens consisting of sub-wavelength
layers. The parameters under consideration the types of elements estimated on the
depth of focus and a transverse dimension of beam.
Studies have shown that layered lens with linear variation of the refractive index has
an advantage over diffraction axicon with the same numerical aperture, as it allows to
form a narrower focal lengths. Increasing the numerical aperture of the axicon reduces
the focal spot formed by them, but it is accompanied by a reduction of the light seg-
ment lengths. With a value of more than α = 0.12 (numerical aperture of more than
0.64) was obtained more extended light length segment for a layered lens.
By reducing the thickness of the layered lens is extended light segment and increases
its width in the plane of maximum intensity along the propagation axis. After a certain
point, in our case 1.55λ, there is the stabilization of transverse dimension with short-
ening the length of a segment.
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Computer Optics and Nanophotonics Savelyev DA. Diffraction of the…
Studies on the reduction of the refractive index of the diffraction axicon show that
after reaching a limiting value of the refractive index (in this case, when n = 1.68) the
focal spot size is stabilized and not decreases.
Acknowledgment
This work was supported by the Russian Foundation for Basic Research (grants 16-
07-00825a) and by the Ministry of Education and Science of Russian Federation.
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