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Plasmonic holographic imaging with
V-shaped nanoantenna array
Fei Zhou,1 Ye Liu,2 and Weiping Cai1,∗
1 Key Laboratory of Materials Physics, Anhui Key lab of Nanomaterials and Nanotechnology,
Institute of Solid State Physics, Chinese Academy of Sciences, Hefei, 230031, Anhui, China
2 Anhui provincial key lab of photonics devices and materials, Anhui Institute of Optics and
Fine Mechanics, Chinese Academy of Sciences, Hefei, 230031, Anhui, China
∗
[email protected]
Abstract: In this article, a novel method of holographic imaging with
Au nanoantenna array is presented. In order to obtain the plasmonic
holographic plate for a preset letter “NANO”, the phase distribution of the
hologram is firstly generated by the weighted Gerchberg-Saxton (GSW)
algorithm, and then 16 kinds of V-shaped nanoantennas with different
geometric parameters are designed to evenly cover the phase shift of 0 to
2π by finite-difference time-domain (FDTD) method. Through orienting
these nanoantennas according to the phase distribution of the hologram,
the plasmonic array hologram is obtained. Very good imaging quality is
observed with our nanoantenna array hologram plate. This method can be
used for holographic imaging of arbitrary shape, and may find potential
applications in holographic memory, printing and holographic display.
© 2013 Optical Society of America
OCIS codes: (240.6680) Surface plasmons; (090.2910) Holography, microwave; (310.6628)
Subwavelength structures, nanostructures.
References and links
1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830
(2003).
2. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193
(2006).
3. F. J. G. de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79, 1267–1290
(2007).
4. B. Lee, S. Kim, H. Kim, and Y. Lim, “The use of plasmonics in light beaming and focusing,” Prog. Quant.
Electron. 34, 47–87 (2010).
5. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen,
“Beaming light from a subwavelength aperture,” Science 297, 820–822 (2002).
6. P. Genevet, J. Lin, M. A. Kats and F. Capasso, “Holographic detection of the orbital angular momentum of light
with plasmonic photodiodes,” Nature Commun. 3, 1278 (2012).
7. I. Dolev, I. Epstein, and A. Arie, “Surface-plasmon holographic beam shaping,” Phys. Rev. Lett. 109, 203903
(2012).
8. N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with
phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
9. F. Aieta, P. Genevet, M. A. Kats, N. F. Yu, R. Blanchard, Z. Gahurro, and F. Capasso, “Aberration-free ultrathin
flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936
(2012).
10. P. Y. Chen and A. Alu, “Subwavelength imaging using phase-conjugating nonlinear nanoantenna arrays,” Nano
Lett. 11, 5514–5518 (2011).
11. M. Ozaki, J. Kato, and S. Kawata, “Surface-plasmon holography with white-light illumination,” Science 332,
218–220 (2011).
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Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
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12. M. C. King, A. M. Noll, and D. H. Berry, “A new approach to computer-generated holography,” Appl. Opt. 9,
471–475 (1970).
13. B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,”
Opt. Eng. 28, 629–637 (1989).
14. N. Masuda, T. Ito, T. Tanaka, A. Shiraki, and T. Sugie, “Computer generated holography using a graphics processing unit,” Opt. Express 14, 603–608 (2006).
15. Y. H. Chen, L. Huang, L. Gan, and Z. Y. Li, “Wavefront shaping of infrared light through a subwavelength hole,”
Light Sci. Appl. 1, e26 (2012).
16. R. W. Gerchber and W. O. Saxton, “A Practical algorithm for determination of phase from image and diffraction
plane pictures,” Optik 35, 237-246 (1972).
17. L. C. Thomson and J. Courtial, “Holographic shaping of generalized self-reconstructing light beams,” Opt. Commun. 281, 1217–1221 (2008).
18. N. Yoshikawa and T. Yatagai, “Phase optimization of a kinoform by simulated annealing,” Appl. Opt. 33, 863–
868 (1994).
19. R. D. Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,”
Opt. Express 15, 1913–1922 (2007).
20. M. Born and E. Wolf, Principles of Optics : Electromagnetic Theory of Propagation, Interference and Diffraction
of Light (Cambridge University Press, Cambridge ; New York, 1999), 7th ed.
21. A. Taflove and S. C. Hagness, Computational Electrodynamics : The Finite-Difference Time-Domain Method,
Artech House antennas and propagation library (Artech House, Boston, 2005), 3rd ed.
1.
Introduction
Plasmonic structures which can manipulate light at subwavelength scale have attracted considerable interests in recent years [1–3]. Many studies have shown that due to the interactions
between the surface plasmons (SPs) and the metallic nanostructure, the propagation of light
can be fully controlled [4–10]. For example, by using the groove-on-film structures, the wavefront can be reshaped, and a series of optical phenomena such as beaming, focusing, splitting
and beam shaping have been demonstrated theoretically and experimentally [4–7]. At the same
time, the phase of the wavefront can be tuned point by point with properly designed array of
nanoparticles. Based on this, anomalous reflection and refraction phenomena have been observed [8–10].
By combining these wavefront controlling technique with the concept of optical holography,
plasmonic holography has been realized by Ozakis group [11]. In their work, the holographic
plate is fabricated by exposure to the interference between object light and reference light, like
the way used in the traditional holography. Another way to realize holography is by using computer generated holograms and micro/nano-fabricating techniques. This way is more feasible,
and has received much attention [12–14]. Recently, this type of plasmonic holographic imaging
is realized by Chen’s group and Dolev’s group [7, 15]. However, in their work the amplitude
is binary modulated (using metal gratings), which causes the loss of details and results in the
mismatch between the original object and the image. In this article, we will realize plasmonic
holography with higher imaging quality by using nanoantenna array to modulate the phase of
the wavefront. In section 2, we will use the weighted Gerchberg-Saxton (GSW) algorithm to
generate optimized hologram of the preset letters “NANO”. In section 3, we will design suitable nanoantennas, which are the unit cells of our plasmonic array, to realize phase shift from
0 to 2π . In section 4, we will orient these nanoantennas according to the optimized hologram,
and realize high quality plasmonic holography. Finally, we will conclude this article.
2.
The design of the hologram using GSW algorithm
There are various methods of designing hologram for a preset shape. The simplest method is
setting a light source with the preset shape and recording the phase of electric field at the hologram plate. However, this method ignored the information of the amplitude of the electric field,
and hence caused low efficiency and uniformity of the image [16]. To overcome this weakness,
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Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
25 February 2013 / Vol. 21, No. 4 / OPTICS EXPRESS 4349
some hologram generation methods are proposed, such as Curtis-Koss-Grier algorithm [17],
simulated annealing [18] and Gerchberg-Saxton (GS) algorithm [16]. The GSW algorithm,
which is a variation of GS method with bias to the spatial uniformity, has been widely used in
the hologram generation for its good balance between efficiency and image quality [19].
To apply this method, we first discretize the preset shape into grids, and regard them as
(0)
M light sources with random phases θm . We calculate the wavefront in the hologram plane
of light radiated by these sources at first. The total electric field at the hologram plane is the
superposition of the electric fields radiated by every sources, so the phase of the electric field at
(k)
jth pixel (ϕ j , the superscript (k) denotes the kth iteration) is
(k)
(k)
(k)
,
(1)
ϕ j = arg ∑ wm exp i Δmj + θm
m
where Δmj is the phase delay between mth light source and jth pixel of the hologram given by
Δmj = λ2πf (x j xm + y j ym ), f is the focal length of the lens and λ is the wavelength of light, xm ,
ym , x j and y j are the coordinates of the mth source and the jth pixel, respectively; and wm is the
weight of the mth source, which is set to 1 in the first iteration.
Next we carry on a backward calculation from the hologram ϕ (k) to the sources, and we can
obtain the amplitude of the electric field at the position of the mth source:
(k)
Vm =
N
∑ exp
(k)
i ϕ j − Δmj .
(2)
j=1
(k+1)
Then we update the phases of the sources θm
(k+1)
weight wm
for every sources:
(k)
to the argument of Vm , and recalculate the
(k)
= arg Vm
(k+1)
θm
(3)
(k) (k+1)
(k−1) (k) wm
= wm
Vm Vm (4)
Successive application of Eqs. (1), (2), (3) and (4) comprises one iteration of the GSW algorithm. And usually, the convergence can be achieved in tens of iterations.
With the GSW algorithm, we design the hologram for a preset letter “NANO” shown in the
inset of Fig. 1(a). In our work, the wavelength of the incident light is 1550 nm, and the distance
between adjacent pixels is 750 nm. Figure 1(a) shows the designed hologram with 400 × 400
pixels. At the same time, Fig. 1(b) shows the corresponding image directly calculated from the
hologram by using the Fraunhofer diffraction formula [20]. From the image we can find that
the shape of the diffraction pattern is nearly the same as the preset shape, and the uniformity is
also very good.
We should notice that the hologram mentioned above contains continuous phase shifts from
0 to 2π . However, designing and fabricating a series of nanoantennas with continuous phase
shifts is impractical. So in our work, the phase shifts are limited to several equally spaced levels.
To decide the suitable number of phase levels (N), we study the relationship between N and the
image quality, which is characterized by the efficiency (e) and the relative standard deviation
(RSD) of the intensity:
e = ∑ Im
RSD =
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I0
(5)
m
(Im − I)
2
I
(6)
Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
25 February 2013 / Vol. 21, No. 4 / OPTICS EXPRESS 4350
NA
NO
y-axis (arb. unit)
1.0
(a)
1.0
(b)
0.5
0.0
0.5
-0.5
-1.0
-1.0
-0.5
0.0
0.5
1.0
0.0
x-axis (arb. unit)
Fig. 1. (a) The 400 × 400 hologram for the preset shape “NANO”; (b) The Fraunhofer
diffraction pattern of the hologram shown in Fig. 1(a).
Fig. 2. The efficiency and RSD of the holographic imaging when adapting different phase
shift levels. The dashed lines are the efficiency (black) and the RSD (red) for continuous
phase shifts.
The results are showed in Fig. 2. We can find that the efficiency is not very sensitive to N.
And when N is more than 8, the efficiency is close to 1, which indicates nearly all the energy
is in the preset region. We can also find that the RSD is approximately inverse proportional to
the number of levels, i.e. the uniformity of the intensity in the preset shape is higher for larger
N. In the following, we choose to use 16 levels in our design to achieve a balance in the image
quality and the difficulty of fabrication.
3.
Optical properties of V-shaped nanoantenna
According to Ref. [9], the V-shaped nanoantennas can tailor the phase of the scattered light
from 0 to 2π . Here, we also adopt the V-shape Au nanoantennas, which are placed on the Si
substrate and comprised of two nano-rods with semicircle ends, shown in the inset of Fig. 3(b).
We use finite-difference time-domain (FDTD) technique to simulate the phase shifts of the
scattered lights of the V-shaped nanoantennas. The simulation scheme is shown in Fig. 3(a).
The nanoantenna is placed in the xy-plane, and the incident light is along z-axis from bottom
to top. The angle between x-axis and the symmetric axis of the nanoantenna is 45°. We use an
x-polarized light to stimulate the nanoantenna, and record the y-component of the electric field
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Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
25 February 2013 / Vol. 21, No. 4 / OPTICS EXPRESS 4351
above the nanoantenna, which is perpendicular to the polarization of the incident light. This
arrangement assures that only the scattered light is recorded. Then we can obtain the phase
shifts by comparing the phase of the scattered light among different nanoantennas.
(b)
Phase shift (rad)
Si substrate
Incidence
3.0
Phase delay
Amplitude
3π
2
2.5
2.0
π
1.5
π
1.0
2
θ
0
w
Au rod
L
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
0.5
Amplitude (arb. unit)
3.5
2π
(a)
0.0
Antennas Index
Fig. 3. (a) Scheme of the simulation to get the phase shifts for nanoantennas with different geometric parameters. The nanoantennas are placed on a Si substrate, and the angle
between the symmetric axis of each nanoantenna and x-axis is 45°. The incidence is xpolarized, and the y-component of the scattered electric field is recorded. (b) The phase
shifts and amplitudes of the scattered electric field for nanoantenna #1 to #16.
Table 1. The arm lengths and angles for the 16 nanoantennas. The arm widths of all nanoantennas are 50 nm.
Antenna No.
Length (nm)
Angle (degree)
1, 9
175
39
2, 10
167
43
3, 11
156
47
4, 12
150
60
5, 13
140
71
6, 14
134
76
7, 15
125
85
8, 16
111
115
We calculated the phase shifts of varies of V-shaped nanoantennas with different lengths (L)
and angles (θ ), and find 8 V-shaped nanoantennas with phase shifts from 0 to π . The nanoantennas with phase shifts from π to 2π can be obtained simply by flipping these 8 nanoantennas with
respect to x-axis, since the y-components of the scattered electric field of the flipped nanoantennas are just opposite to the original ones, which means an additional phase shift of π [8]. The
geometrical parameters of the 16 nanoantennas which we use in the following of this article
are showed in Table 1. At the same time, the amplitudes (black squares) and phase shifts (red
circles) of the scattered light are also depicted in Fig. 3(b). We can see that the phase shifts
increases linearly with the No. of the nanoantennas, and the amplitudes of the scattered lights
for all V-shape nanoantennas are within 3.44 ± 0.20. The evenly spaced phase shifts and near
constant scattering amplitudes will guarantee the good performance of the nanoantenna array
hologram.
4.
Nanoantenna array hologram
In the above discussions, we solved the two prerequisites required by our nanoantenna array hologram: the hologram generated by GSW method and 16 nanoantennas with equally
increasing phase shifts from 0 to 2π and constant scattering amplitude. And in this section,
we will build the nanoantenna array hologram. However, the simulation of an array containing 400 × 400 nanoantennas is very memory and time consuming. But fortunately, due to the
unique property of holography, only a small piece of the hologram still has the ability to reconstruct the whole image. The hologram shown in Fig. 4(a) is 1% of the whole hologram shown in
Fig. 1(a), which only has 40 × 40 pixels. We also calculated its Fraunhofer diffraction pattern,
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Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
25 February 2013 / Vol. 21, No. 4 / OPTICS EXPRESS 4352
(a)
(b)
1.0
1.0
0.8
y-axis (arb. unit)
0.5
0.6
0.0
Normalized Intensity
which is shown in Fig. 4(b). We can see that although the pixels are much less than the whole
hologram, the preset “NANO” shape can still be clearly recognized.
In the following, we will build the holographic plate with the 16 kinds of nanoantennas. We
use nanoantennas #1 to #16 to realize the phase shifts in the hologram. Take the hologram
in the red square in Fig. 4(a) for example, the phase shifts of the pixels at the top-left corner
are 11π /8, 5π /4, 5π /8, . . . , so we put nanoantennas #12, #11, #6, . . . in the corresponding
positions, respectively. Part of the nanoantenna array are also shown in the figure detailedly.
The distance between adjacent nanoantennas is 750 nm, which can ensure that the interaction
among the nanoantennas is weak enough to be neglected [8].
0.4
-0.5
-1.0
-1.0
0.2
-0.5
0.0
0.5
1.0
0.0
CCD
y-polarizer
incidence
y-axis (arb. unit)
focal plane
(d)
1.0
1.0
0.5
0.8
0.6
0.0
Normalized intensity
x-axis (arb. unit)
(c)
0.4
-0.5
0.2
x-polarizer
-1.0
-1.0
-0.5
0.0
0.5
x-axis (arb. unit)
1.0
0.0
Fig. 4. (a) The 40 × 40 hologram, and the part of the corresponding nanoantenna array;
(b) The Fraunhofer diffraction pattern of the hologram shown in (a); (c) The simulation
scheme; (d) The intensity distribution in the focal plane simulated by FDTD technique.
Then the plasmonic holographic imaging with above holographic plate is studied by FDTD
method. The simulation arrangements are shown in Fig. 4(c). The stimulating light whose wavelength is 1550 nm is from the bottom to the top of the nanoantenna array, and the image is in
the focus plane of a positive lens. We record the near-field electric field distribution above
the nanoantenna array. By using near-to-far-field transformation, we can obtain the angular distribution of the scattered electric field [21]. Then the electric field distribution in the focus plane
of lens can be simply calculated by the lens equation for collimated light
x = f · tan (θ ) · cos (ϕ )
y = f · tan (θ ) · sin (ϕ ) .
(7)
Finally, we obtain the intensity distribution in the focal plane of the lens. The result is shown
in Fig. 4(d). We can see that the quality of the image is surprisingly good, and agree very well
with the results directly calculated from the hologram using Fraunhofer diffraction equations
(shown in Fig. 4(b)). This indicates that the design of our plasmonic holographic plate with
nanoantenna array is feasible. Additionally, it should be noticed that in our simulations we
only have 40 × 40 nanoantennas in the array due to the limitation of our computer hardware.
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Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
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However, with recent micro/nano fabricating techniques such as UV lithography and electron
beam lithography (EBL), nanoantenna array with much larger size can be fabricated, and holographic imaging with higher quality can be achieved. Currently, the major drawback of this type
of plasmonic holography is the relatively low total efficiency, which is about 0.2%. The low total efficiency is mainly caused by two reasons: One is the low fill factor of the nano-antennas
array, so most part of the incidence is not scattered by the nano-antennas; another reason is the
low coupling coefficient between the two perpendicular polarizations. Based on this, several
methods to improve the total efficiency are implied, such as: decreasing the distance between
adjacent nano-antennas, increasing the scattering cross-section of the nano-antennas, and enhancing the coupling coefficient between different polarizations.
5.
Conclusion
In this article, we demonstrated a novel high-quality holographic imaging with nanoantenna array. By using the GSW algorithm, we designed a hologram of a preset shape “NANO” with high
efficiency and good uniformity. We also give 16 kinds of V-shaped nanoantennas with phase
shifts ranged from 0 to 2π . By orienting these nanoantennas, we obtain the holographic plate
for the preset shape “NANO”, which contains 40 × 40 nanoantennas. We calculated the electric
field distribution of the holographic plate under plane-wave illustration by using FDTD technique and near-to-far field transformation. A clear “NANO” shape appears in the focal plane.
This simulation result agrees very well with the result by the Fraunhofer diffraction equation
directly from the hologram, which indicates the hologram is implemented by nanoantenna array
correctly and effectively. And since our method allows more levels, the phases of the wavefront
is tuned more precisely, and higher imaging quality is obtained. We believe this technique can
be helpful in beam shaping, holographic printing and holographic detection.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant Nos.
11204317, 50831005, 11104282), and the China Postdoctoral Science Foundation (Grant No.
2012M511429).
#181987 - $15.00 USD
(C) 2013 OSA
Received 19 Dec 2012; revised 23 Jan 2013; accepted 25 Jan 2013; published 12 Feb 2013
25 February 2013 / Vol. 21, No. 4 / OPTICS EXPRESS 4354