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Letter Vol. 47,No. 6 /15 March 2022 / Optics Letters 1375
Perfect diffractive circular metagrating for Bessel
beam transformation
Feng-Jun Li, Shuai Wang, Xiangping Li, AND Zi-Lan Deng∗
Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications, Institute of Photonics Technology, Jinan University,
Guangzhou 510632, China
*Corresponding author: zilandeng@jnu.edu.cn
Received 8 November 2021; revised 22 January 2022; accepted 29 January 2022; posted 31 January 2022; published 7 March 2022
Bessel beams, with their non-diffractive property, have
attracted great interest in recent years. Optical needle shap-
ing of Bessel beams is highly desired in many applications,
however, this typically requires low numerical aperture (NA)
bulky 4fconfocal systems incorporated with spatial light
modulators or round filters. Here, we employ a circular
dielectric metagrating for perfect Bessel beam transforma-
tion at a desired wavelength. The dielectric metagrating
exhibits a high transmissive diffraction efficiency (up to
75%) for a broadband (460 nm to 560 nm), wide-angle
range, and dual-polarization response, which is capable of
a high-performance transformation of Bessel beams with
arbitrary NAs. Our results show potential for special-beam-
required applications such as light storage, imaging, and
optical manipulation. © 2022 Optica Publishing Group
https://doi.org/10.1364/OL.448093
Non-diffractive Bessel beams have shown their unique advan-
tage in some advanced applications like microstructure process-
ing [1,2], optical imaging [3], and optical manipulation [4,5].
In physics, Bessel beams are a set of special solutions of a
free space scalar wave function which can be described by the
first-type Bessel function. The non-diffractive Bessel beams are
first proposed with the general scalar description in cylindrical
coordinates [6]:
E(r,φ,z)=E0exp(ikzz±ilφ)Jl(krr),(1)
where E0is the amplitude, the integer lrepresents the order
of the Bessel function as well as the topological number of
angular momentum, and krand kzare the transverse and lon-
gitudinal wavevectors, respectively. In this work, we mainly
study the zero-order Bessel beam (l=0). Equation (1) shows
that the transverse electric field pattern is related to the first-type
Bessel function and the intensity profile remains stable as the
beam travels along the z-coordinate, showing the non-diffractive
feature.
Since the ideal Bessel beams carry infinite energy in free
space, the proposal of quasi-Bessel beams provides a platform
to study the non-diffractive feature in a limited region. This
can be regarded as a superposition of multiple plane waves at
a fixed convergence angle. Using axicons is a typical way to
create quasi-Bessel beams efficiently in experiment. Note that
this convergence angle is dependent on the refraction angle θ
and the incident angle α, as shown in Fig. 1(a). Thus, with a
given axicon, the numerical aperture (NA) of the axicon can be
written as
NA =n2sin(θ−α)=n2sin θ−sin−1n2
n1
sin θ,(2)
where n1,n2are the refractive indexes of the axicon and outer
environment, respectively. Typically, the refractive index of the
axicons is 1.5 for glass and that of the environment is 1 for air.
Equation (2) shows that the NA of an axicon is strongly related to
the refraction angle θin air. However, considering the total inter-
nal refraction phenomenon in glass, the maximum refraction
angle is limited to 90°. Thus, the NA of an axicon cannot exceed
0.75. This property directly limits the minimum full width at
half maximum (FWHM) as well as the non-diffractive distance
Zmax for creating quasi-Bessel beams. In addition, the round tip
of axicons changes the phase distribution of its coherent transfer
function, which affects the transverse and longitudinal intensity
profile of quasi-Bessel beams [7].
To produce high-quality tunable quasi-Bessel beams with-
out the round-tip effect, an experimental setup of Bessel beams
transformation with a 4fconfocal system was proposed, as in
the upper panel of Fig. 1(d). With a spatial filter (opaque circular
obstacle) set in the confocal plane, the round-tip effect can be per-
fectly avoided [7]. However, this configuration inevitably results
in energy loss since the filter blocks the low spatial frequency
components of incident light.
Recently, flat optics based on a metasurface have been devel-
oped to allow full control of the incident wavefront [8–10],
which promises applications ranging from metalens [11–14],
holograms [15–17] to polarization-selective devices [17–19].
Though a meta-axicon with geometric phase has been reported
[20,21], the efficiency decreases rapidly as the NA increases.
A metagrating is an efficient wavefront-controlling metasur-
face with efficient large-angle diffraction capabilities [22–25].
In particular, the perfect diffraction metagrating enables all of
the incident power to be transferred to the −1st diffraction order
mode [26]. Here, we report a compact Bessel beam transformer
based on a perfect diffraction circular metagrating. The meta-
grating exhibits a high diffraction efficiency for broadband, large
incident angle tolerance and dual-polarization responses [22],
which enables the conversion of high-NA Bessel beams with
high performances in the visible region. All previous metagrat-
ings have been straight metagratings for manipulating the optical
0146-9592/22/061375-04 Journal ©2022 Optica Publishing Group
1376 Vol. 47, No. 6 /15 March 2022 /Optics Letters Letter
Fig. 1. (a) Schematic diagram of zero-order quasi-Bessel beams
generation with a glass axicon. The radius of incident Gaussian
beam w0, refraction angle θ, and the axicon angle αare labeled. (b)
Circular metagrating for zero-order quasi-Bessel beams steering.
(c) Unit cell of a straight metagrating for perfect diffraction design.
The pis 500 nm, wis 162 nm, and his 267 nm. (d) Comparison of
4fconfocal system and circular metagrating for the transformation
of quasi-Bessel beams. The two yellow curves indicate the intensity
profiles along the propagating direction for both methods.
diffraction in the Cartesian coordinate system. Here, we deal
with circular metagratings that manipulate optical diffractions in
the cylinder coordinate system, which provides a convenient and
efficient way for the transformation of Bessel beam. A schematic
diagram of the metagrating is shown in Figs. 1(b) and 1(c). The
unit cell of the metagrating simply consists of an identical tita-
nium dioxide (TiO2) nanorod with width w, height h, and the
period of metagrating p. The minimum feature size (162 nm)
is within the fabrication capability of standard electron beam
lithography.
In general, a metagrating acts like a grating that can control
the light at various incident angles, which indicates the internal
features of large-angle tolerance of the metagrating. However,
the generalized Snell’s law [9] predicts the emergence of a crit-
ical incident angle in the metagrating which mainly depends on
its period. When the incident angle is smaller than the critical
angle, the propagation wave will be converted to an evanescent
wave. Thus, to make sure that the designed metagrating works
in the visible region, the period is set as 500 nm. To realize
the perfect diffraction with ideally unitary diffraction efficiency,
the wand hparameters are carefully designed as 162 nm and
267 nm, respectively, by the rigorous coupled wave analysis
method. In general, the design principle is readily extended
to another frequency regime by optimizing the geometry and
material parameters of the metagrating.
We then demonstrate a circular metagrating based on a per-
fect diffraction metagrating to reshape the quasi-Bessel beams
directly. The principle is: the incident quasi-Bessel beams
act as a light cone illuminating the circular metagrating and
the incident beam will be transformed into a −1 diffraction
order channel to form a new quasi-Bessel beam. Note that
the high-efficiency conversion and strictly linear phase dis-
tribution of the circular metagrating promises superior beam
quality compared with a conventional axicon and 4fcombina-
tion system, as shown in Fig. 1(d). The diffraction angles can be
calculated by
θt=arcsin λ0
ntp−ni
nt
sin θi,(3)
where λ0is the vacuum wavelength, pis the period of the
metagrating, and ni,θi,nt,θtare the refractive indexes and prop-
agation angles of incident and transmission area, respectively.
Equation (3) can be easily proved by the generalized Snell’s law
with the certain phase gradient condition induced by the meta-
grating. Supposing ni,ntare equal to 1, the numerical aperture
of circular metagrating can be given by
NA =λ0
p−sin θi.(4)
As shown in Eq. (4), with a given incident wavelength and
period of the metagrating, the NA increases as the θidecreases.
To fully characterize the transformed quasi-Bessel beams, the
FWHM of zero-order Bessel beams is proposed to describe the
distance between two points of half-maximum intensity of the
center light spot. It can be simply written as
FWHM =0.358λ0
NA .(5)
In addition, the non-diffractive distance shown in Fig. 1(d),
which describes the length of the light needle of zero-order
Bessel beams, is given by
Zmax =w0·cot θt,(6)
where w0is the waist radius of incident quasi-Bessel beams.
Because of the weak dispersion of glass, the NA of the axicon
changes little in the visible region. Thus, the conventional quasi-
Bessel beam shaping focuses on changing the non-diffractive
distance or the FWHM with the 4fconfocal system. Herein,
the non-diffractive distance and the FWHM can be modulated
by incident angles and wavelength simultaneously. As shown in
Eqs. (3)–(6), for incident angle θi=30◦, increasing the wave-
length from 400 nm to 700 nm leads to a difference of 171% in
the FWHM and of 657% in the Zmax. Similarly, at the wavelength
λ0=530 nm, the difference of 457% in the FWHM and 968%
in the Zmax are caused by decreasing the incident angle from 60°
to 10°.
Figure 2(a) shows the −1st-order diffraction efficiency T−1of
the metagrating with incident angle θi=30◦for both TM and
TE polarization. For TM polarization, the T−1reaches 100%
at the wavelength λ0=530 nm, showing a perfect diffraction
feature. Meanwhile, the maximum T−1for TE polarization is
as high as 90% at the wavelength λ0=570 nm. This highly
efficient performance at both TM and TE polarization distin-
guishes most metagratings from phase-gradient metasurfaces,
which only work efficiently at certain polarization. The mag-
netic field distribution at the perfect diffraction point [Fig. 2(a)]
reveals that once the perfect diffraction occurs, there is only
one diffractive propagation channel, with other transmission and
reflection channels simultaneously suppressed.
The broadband and large-angle tolerance characteristics of
this perfect diffraction metagrating are illustrated in Figs. 2(c)
and 2(d). We sweep the incident angles and wavelengths simul-
taneously to calculate the −1st-order diffraction efficiency of
the metagrating with the perfect diffraction parameters. The red
dash lines mark the broadband line of TM and TE polarization.
Interestingly, both are located at θi=30◦. In addition, the cr itical
incident angles emerge when the ratio of incident wavelength to
structure period is greater than unity. The value of the critical
angles can be calculated by Eq. (3) satisfying the condition of
lim
θi→θc
θt=π
2, where θcrepresents the critical angle.
Letter Vol. 47,No. 6 / 15 March 2022/ Optics Letters 1377
Fig. 2. (a) The −1st-order diffraction efficiency and the field
distribution of a perfect diffraction metagrating in TM and TE
polarization. (b) Intensity profile of TM and TE polarized quasi-
Bessel beams steering with circular metagrating. (c), (d) 2D sweep
by varying the incident wavelength and incident angle of a perfect
diffraction metagrating in TM polarization and TE polarization,
respectively.
The incident angle tolerance increases when the wavelength
increases. However, the average diffraction efficiency within
varying incident angles declines generally when the wavelength
deviates from the perfect diffraction point. Thus, we set the wave-
length λ0=550 nm to further study of the circular metagrating
in quasi-Bessel beam transformers.
The intensity profiles of quasi-Bessel beam with a 30-
µm-diameter circular metagrating under radially (TM) and
azimuthally (TE) polarized beams calculated by the finite-
difference time-domain (FDTD) method are shown in Fig. 2(b).
It shows that the peak diffraction efficiency occurs at θt=34◦,
λ0=530 nm for TM polarization; and at θt=38◦,λ0=570 nm
for TE polarization, which are consistent with the theoretical
values calculated by Eq. (3). With perfect diffraction at radial
polarization, the incident quasi-Bessel beam is perfectly trans-
formed to a new quasi-Bessel beam without reflection, whereas
the azimuthally polarized quasi-Bessel beam suffers from small
reflective wave interference and has a slightly lower efficiency.
The large incident angle tolerance gives the potential of
long–short light needle conversion in quasi-Bessel beams.
Figures 3(a)–3(e) exhibit this highly efficient conversion perfor-
mance calculated by the FDTD method with the 30-µm-diameter
circular metagrating at radial polarization. As for the radially
polarized quasi-Bessel beams, the conversion efficiency remains
above 80% within the converging angles from 20°to 50°, while
the efficiency decreases to 40% at 10°, as predicted. The length
of the light needle can be described with the non-diffractive dis-
tance Zmax. In our case, the waist radius of incident quasi-Bessel
beams is 5 µm and the Zmax of output quasi-Bessel beams can
continuously change from 2 µm to 15 µm when the incident angle
varies from 10°to 50°. In addition, the real parts of the radial Er
components at the center of the converted quasi-Bessel beam are
shown in Figs. 3(f)–3(j), which correspond to Figs. 3(a)–3(e),
respectively. Obviously, since the non-polarization conversion of
a circular metagrating, the hollow property of radially polarized
quasi-Bessel beams maintains after conversion. In addition, our
circular metagrating enables the NA of the output quasi-Bessel
beams to cover the range of 0.92 to 0.33 within the input NA
from 0.17 to 0.76, largely exceeding the NA range (<0.75) of the
bulky axicon system. This feature means that a circular meta-
grating can realize continuous high output NA conversion of
Fig. 3. (a)–(e) XoZ plane profile of the radially polarized quasi-
Bessel beams transformation by circular metagrating with varying
non-diffractive distances (or the converging angles θi) of incident
quasi-Bessel beams at 530 nm. The circular metagrating locates at
the white dash lines. The light needles are marked by green dash
frames and (f)–(j) the real part of radial Ercomponents at the center
of converted quasi-Bessel beam. The propagation direction is from
bottom to top. Scale bar =500nm.
quasi-Bessel beams by varying the input NA without any adjust-
ments of the metagrating. Note that due to the same cylindrical
symmetry for zero- and higher-order quasi-Bessel beams, the
circular metagrating enables efficient conversion for any order
quasi-Bessel beams.
When it comes to the high-NA beam transformation, the
polarization properties of quasi-Bessel beams can be very dif-
ferent from the low NA cases. Generally, the vectorial Debye
theory is widely used in the analysis of vectorial point spread
function (PSF) in high-NA focal strategies. In our work, we pro-
pose a method of using vectorial Debye theory to calculate the
electrical field distribution of the center light spot of high-NA
quasi-Bessel beams. The vectorial Debye theory gives the full
descriptions of the electric field of an x-polarized quasi-Bessel
beams propagating along the zdirection:
−→
E(r2,ψ,z2)=πi
λ[(I0+I2cos 2ψ)−→
i+I2sin 2ψ−→
j+2iI1cos ψ−→
k],
(7a)
⎧
⎪
⎪
⎪⎨
⎪
⎪
⎪
⎩
I0=∫β
αP(θ)sin θ(1+cos θ)J0(kr2sin θ)exp(−ikz2cos θ)ϕ(θ)dθ
I1=∫β
αP(θ)sin2θJ1(kr2sin θ)exp(−ikz2cos θ)ϕ(θ)dθ
I2=∫β
αP(θ)sin θ(1−cos θ)J2(kr2sin θ)exp(−ikz2cos θ)ϕ(θ)dθ
,
(7b)
where the adding phase factor ϕ(θ)is the correction of the orig-
inal spherical wave approximation as the quasi-Bessel beam is
regarded as the superposition of two plane waves. The integra-
tion interval α,βshown in Equation (7b) is related to the radius
of quasi-Bessel beams and the expression of the correction
term. Notice that the corrections correspond to the transmit-
tance function of the metagrating. In fact, the phase factor ϕ(θ)
mainly changes the electrical field distribution along the propa-
gating direction of the quasi-Bessel beams compared to the PSF
calculated with spherical wave approximation. Additionally,
the reduction of the integration interval affects the horizontal
intensity profile of quasi-Bessel beams.
Equation (7a) also indicates that the depolarization effect
appears when the high-NA quasi-Bessel beam transform is con-
sidered. As the orthogonal field components appear, the shape
1378 Vol. 47, No. 6 /15 March 2022 /Optics Letters Letter
Fig. 4. Comparison of the square of the electric fields from the-
oretical calculation using (a)–(d) the FDTD method and (e)–(h)
Equations (7a) and (7b). (a), (e) Total intensity, (b), (f) |Ex|2, (c),
(g) |Ey|2, (d), (h) |Ez|2for the x-polarized quasi-Bessel beam in
high-NA steering.
of the center light spot is no longer dominant. The series of
electric fields in the high-NA x-polarized quasi-Bessel beam
transform are calculated by the FDTD method and shown in
Figs. 4(a)–4(d). With the zero-order quasi-Bessel beams, the
|Ex|2component gives a circular shape in the center region.
Since the |Ez|2component is distributed symmetrically along
the y-axis and has a numerical value comparable to the |Ex|2
component, the shape of the center region becomes elliptical
with the long axis along the x-axis. In general, the direction of
the long axis is dependent on the polarized direction and this
elliptical shaping only happens in the case of linear polarization
incidence. Moreover, these numerical results are in good agree-
ment with the theoretical results computed by Eqs. (7a) and (7b)
as shown in Figs. 4(e)–4(h).
In summary, we demonstrate a circular dielectric metagrating
for perfect Bessel beam transformation. The high transmissive
diffraction efficiency results in a broadband, wide-angle range
and dual-polarization response. In particular, the perfect diffrac-
tion feature is achieved in simulation and provides a possibility
of perfect conversion in radial polarization. Highly efficient
long–short light needle transformation of quasi-Bessel beams
is demonstrated with such circular metagratings. The continu-
ous high-NA conversion of quasi-Bessel beam is also achieved
by simply varying the input NA without any adjustments of
the metagrating. Finally, we study the polarization properties in
high-NA quasi-Bessel beams transformation and successfully
applied the vectorial Debye theory to predict the space-variant
polarization with high NA. Based on these appealing features,
the circular metagrating can sufficiently be regarded as a superior
alternative to the conventional bulky system in a quasi-Bessel
beam transform setup.
Funding. National Key Research and Development Program of China
(2021YFB2802003); National Natural Science Foundation of China
(62075084); Guangdong Provincial Innovation and Entrepreneurship
Project (2016ZT06D081); Basic and Applied Basic Research Founda-
tion of Guangdong Province (2020A1515010615); Fundamental Research
Funds for the Central Universities (21620415); Guangzhou Science and
Technology Program (202102020566).
Disclosures. The authors declare no conflicts of interest.
Data availability. No data were generated or analyzed in the presented
research.
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