- Email: [email protected]
Ag tri-layer gratings
Optics Communications 381 (2016) 24–29
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Beam focusing from double subwavelength slits surrounded by Ag/SiO2/Ag tri-layer gratings Wei Su a,n, Chong Zhou b, Gaige Zheng c, Xiangyin Li b a
Department of Mathematics and Physics, Hohai University, Changzhou Campus, Changzhou 213022, China Department of optical engineering, Nanjing University of Science and Technology, Nanjing, China c School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 2 May 2016 Received in revised form 4 June 2016 Accepted 22 June 2016
A silver(Ag)/SiO2/Ag tri-layer grating structure with double slits for beam focusing has been proposed. Compared with the metal/dielectric double-layer grating-based structure, the focusing efficiency of our proposed structure can be greatly enhanced. Numerical simulations using the finite-different time-domain (FDTD) method verify that the focal length and deflection angle can be controlled by adjusting the refractive indexes of dielectric mediums in the two slits. & 2016 Elsevier B.V. All rights reserved.
Keywords: Surface plasmon polaritons Beam focusing Grating
1. Introduction
2. Theory and structure
Surface plasmons (SPs) are coherent delocalized electron oscillations that exist along a metal/dielectric interface [1]. Since Ebbsen's discovery of high light transmission from subwavelength hole arrays [2], the investigations on subwavelength metallic structures have become hot [3–6]. One of the most meaningful applications on SPs is light focusing. Single metal layer or metal/ dielectric (MD) double-layer structures have been theoretically investigated for beam focusing [7–14]. But the focusing efficiencies of this kind of structures are not high. So many scholars did works on improving the focusing efficiency. For instance, Kim et al. obtained a high focusing efficiency by using the Ag/SiO2 multilayered zone plate [15]. In this paper, we extend the previous studies of beam focusing structures and propose the Ag/SiO2/Ag tri-layer grating structure with double slits. Compared with the Ag/SiO2 double-layer grating-based structure, the focusing efficiency can be greatly enhanced. In addition, simulation results based on two-dimensional (2D) FDTD method with perfect matching layer (PML) absorbing boundary conditions show that the plasmonic focusing property can be controlled by adjusting the refractive indexes of dielectric materials in the two slits.
Fig. 1 shows the schematic diagram of the proposed structure, which is composed of double slits in a metallic film with surface gratings on the exiting plane. Where w is the width of slit, hm1 and hm2 are the thickness of the substrate and the top metal layer of the grating, respectively. hd is the thickness of the dielectric layer sandwiched between metal layers, b and N are the ridge width and the period number of the grating at either side of the slits, respectively. Λ is the grating constant. a and d are the ridge widths of the grating and the interspacing between double slits, respectively. n1 and n2 are the refractive indexes of dielectric mediums in the two slits, respectively. The substrate of the whole structure is glass. The basic concept of the proposed structure is based on the interference modulation of surface plasmon polaritons (SPPs). In the optical regime, the dielectric functions of some noble metals can match with common dielectrics to sustain surface charge density oscillations, called SPPs. The structure proposed here is a metal/dielectric/metal (MDM) tri-layer grating. As we know, a light wave tends to confine itself mostly in a medium with higher refractive index than its adjacent ones. Therefore, for such MDM waveguide, the propagation constant β can be calculated as [16]:
(
tanh n
Corresponding author. E-mail address: [email protected] (W. Su).
http://dx.doi.org/10.1016/j.optcom.2016.06.062 0030-4018/& 2016 Elsevier B.V. All rights reserved.
)
β 2 − k 02εd hd /2 = −
εd β 2 − k 02εm εm β 2 − k 02εd
(1)
where k0 is the free space wave vector of optical wave, hd is
W. Su et al. / Optics Communications 381 (2016) 24–29
25
Fig. 1. Schematic diagram of the Ag/SiO2/Ag tri-layer grating structure with double slits.
dielectric layer thickness, εm and εd are the relative dielectric permittivities of metal and dielectric, respectively. When a transverse-magnetic (TM) electromagnetic wave illuminates from the bottom side, a part of light can pass through the slits directly, and the other can generate SPPs between metal layers. Consider the phases of the excited SPPs from the two slits first. For normal incidence, the relative phases at the two slits can be calculated as [9]:
φ1 + dksp = φ2 + 2Mπ ,
φ2 + dksp = φ1 +
( 2M + 1)π
The permittivities of the metals are determined by the LorentzDrude model [17]:
Z (μm)
8
(a)
4
0
∑
2 k = 0 ωk
(2)
8
(b)
I
-1
0
X (μm)
1
2
0
(3)
max
(c)
4
4
-2
− ω2 + jωΓk
where εr,1 is the dielectric constant at infinite frequencies, ωP is the plasmon frequency, and ωk, fk and Γk are the resonance frequency, strength and damping frequency, respectively, of kth oscillator. The Lorentz-Drude model uses K damped harmonic oscillators to describe the small resonances observed in the metal's frequency response.
where φ1 and φ2 correspond to the phases of SPPs at the entrance of the two slits, respectively. M is an integer. In this paper, the wavelength of the light is considered as 532 nm, so we can obtain the structure parameter d ¼470 nm.
8
fk ωP2
K
ε( ω) = εr, ∞ +
-2 II
-1
0
X (μm)
1
2
0
0
-2
-1
0
X (μm)
1
2
III
Fig. 2. The intensity distribution of the beam from the slits of the structure (a) without the left side grating, (b) without the right side grating, (c) with the complete grating. w¼ 90 nm, Λ ¼250 nm, hm1 ¼ 120 nm, hm2 ¼ hd ¼ 80 nm, a ¼b ¼80 nm, N¼ 6, λ ¼532 nm, d¼ 470 nm. The illustrations I, II and III are the simulation structures in (a), (b) and (c), respectively.
26
W. Su et al. / Optics Communications 381 (2016) 24–29
8
8
(a)
8
(b)
529 nm
0
Z (μm)
8
f = 3.22 μm
-1
0
X (μm)
1
2
-2
-2
-1
0
X (μm)
1
2
569 nm
4 f = 3.41 μm
0
-2
-1
0
X (μm)
1
2
max
(d)
639 nm
(e)
4 f = 3.29 μm
0
4 f = 3.39 μm
0 -2
Z (μm)
Z (μm)
4
(c)
normalized intensity
Z (μm)
510 nm
-1
0
X (μm)
1
2
0
width of a (nm)
Fig. 3. The intensity distribution of the focused beam from the subwavelength metal slits with (a) 40, (b) 80, (c) 120 and (d) 150 nm width of a. The other parameters are the same as in Fig. 2. The focal lengths are 3.22, 3.39, 3.41 and 3.29 μm, respectively. The FWHMs are 510, 529, 569 and 639 nm, respectively. (e) The normalized intensity at the focal spot as a function of the width of a.
3. Results and discussion Fig. 2 shows the schematic diagram of the focusing process. As we mentioned above, to form a focal spot, the lights radiating from each slit should interfere constructively. If we remove either side of the grating (Fig. 2(I) and (II)), SPPs can not be excited in the other slit, so there is no focal spot in the transmission region. For the complete structure as shown in Fig. 2(III), if the two slits are filled with the same medium, the phases of SPPs excited in the slits are the same, so the lights from each slits can interfere constructively and form a focal spot. For the structure we proposed here, the parameters of the grating between the two slits are significant. From Eq. (2), we can see that the distance between the two slits is an important parameter for calculating the phases of the excited SPPs from the two slits, which determines the transmission light can focus or not.
Moreover, we investigate the effect of the ridge width of the middle grating a on the focusing performance. Here, the values of the rest of the structural parameters are optimized to be w¼90 nm, Λ ¼ 250 nm, hm1 ¼120 nm, hm2 ¼80 nm, hd ¼80 nm, a¼ 80 nm, b¼ 80 nm, respectively. In addition, the incident wavelength λ is fixed at 532 nm, so the optimal value of the interspacing between double slits d can be calculated as 470 nm according to Eq. (2). The period number at both lateral grating N is chosen to be 6. Fig. 3 shows the intensity distribution of the focused beam from the slits with 40, 80, 120, 150 nm width of a. The focal lengths are 3.22, 3.39, 3.41 and 3.29 μm, respectively. Changes in ridge width a have little effect on the focal lengths. Their full width at half-maximum (FWHMs) at each focal length are 510, 529, 569 and 639 nm, respectively. FWHMs at each focal length are about one wavelength, which means that the focusing performance of the proposed structure is consistent with the diffraction limit. The
W. Su et al. / Optics Communications 381 (2016) 24–29
(a)
(b)
8
f = 2.78 μm
-1.8
0
X (μm)
4 f = 3.39 μm
0
1.8
-2
0
X (μm)
(d)
649 nm
Z (μm)
Z (μm)
Z (μm)
529 nm
4
0
(c)
8
484 nm
2
normalized intensity
8
27
f = 4.12 μm
4
0
-2.2
0
2.2
X (μm)
the number of periods (N)
Fig. 4. The intensity distribution of the focused beam from the slits with (a) 5, (b) 6 and (c) 7 periods at both lateral grating. The other parameters are the same as in Fig. 2. The focal lengths are 2.78, 3.39 and 4.12 μm, respectively. The FWHMs are 484, 529 and 649 nm, respectively. (d) The normalized intensity at the focal spot as a function of the period number at both lateral grating.
intensity distribution results show that the energy generating from the structure can focus within a small region. More details of the dependence of normalized intensity at the focal spot on width of a are shown in Fig. 3(e). By increasing a from 40 to 120 nm, the changes of the intensity is only a little. But the intensity decreases abruptly with further decrease or increase of a. The intensity decrease with very small or large a is due to the fact that a change in filling fraction can affect a lot the decoupling of the plasmon, thus affecting the intensity at the focal spot [18]. When a is 80 nm, the intensity is the highest. So 80 nm is the optimal value of the ridge width of grating between the two slits. With the optimal parameters, we change the period number at both lateral grating from 5 to 7, and find that the focal length can
Z (μm)
8
be enlarged by increasing the period number N. Fig. 4(a–c) show the intensity distribution of the focused beam from the slits with different period numbers at both lateral gratings. The focal lengths are 2.78, 3.39 and 4.12 μm, respectively. And their FWHMs at each focal length are 484, 529 and 649 nm, respectively. More details of the dependence of normalized intensity at the focal spot on period number at both lateral grating N are shown in Fig. 4(d). As we can see from the curve, when N is 6, the intensity is the highest. As we mentioned in the introduction part, scholars have already found that the MDM multilayer structures have a higher focusing efficiency than that of single metal or MD double-layer structures. Fig. 5(a) and (b) show the intensity distribution of the focused beam by MD and MDM grating structures, respectively. It
max 6 8
(a)
4
max
(b)
4
0 -2 I
-1
0
X (μm)
1
2
0
0
-2 II
-1
0
X (μm)
1
2
0
Fig. 5. The intensity distribution of the focused beam from the subwavelength metal slits by (a) MD and (b) MDM grating structures, respectively. The other parameters are the same as in Fig. 2. The illustrations I and II are the simulation structures in (a) and (b), respectively.
28
W. Su et al. / Optics Communications 381 (2016) 24–29
Z (μm)
8
8
(a)
4 f = 3.39 μm
max
(b)
4 f = 2.86 μm
0
0 -2
-1
0
X (μm)
1
2
-2
-1
0
X (μm)
1
0
2
Fig. 6. Intensity distribution of the focused beam from the double slits structure. (a) n1 ¼ n2 ¼ 1; (b) n1 ¼n2 ¼1.802. The structure parameters are the same as in Fig. 2.
visually shows the difference of the focusing property between the two structures. Illuminated by the light with the same intensity, it is obvious that the intensity at the focal spot of MD grating is much weaker than that of MDM grating. It is worth noting that the maximum value of the color bar in Fig. 5(a) is only one in six of that in Fig. 5(b). So our proposed MDM grating structure is much better than MD grating on the focusing property. By appropriately adjusting the focusing structural parameters, we find that the focal length and the deflection angle can be controlled by adjusting the refractive indexes of the dielectric mediums in the slits. The two beams can be generated and end up by forming a focal spot as interfering constructively. The intensity distributions of the radiated light from the proposed structure are shown in Fig. 6. The focal spots are generated under condition of constructive interference. With the refractive indexes of the slits are 1 and 1.802, the focal lengths are 3.39 and 2.86 μm,
Fig. 7. Phase difference between the fields exiting from the slits as a function of the refractive index difference between the two slits.
respectively. If the refractive indexes of the mediums in the two slits are different, the phase retardations of the light pass through the slits are different, the phase can be calculated as:
⎡ ⎛ 1 − β /k ⎢ i 0 φi = Re βi t + arg⎢ 1 − ⎜⎜ 1 /k + β ⎝ ⎢⎣ i 0
( (
( )
⎤
2
) ⎞⎟ exp 2jβ t ⎥, ( i )⎥ ) ⎟⎠ ⎥⎦
i = 1, 2 (4)
where the propagation constants in the slits can be calculated as:
(
tanh
)
βi2 − k 02εi w /2 = −
εi βi2 − k 02εm εm βi2 − k 02εi
,
i = 1, 2 (5)
where εi is the relative dielectric constant of the medium in the slit. According to Eqs. (4) and (5), we can obtain the phase differences between the two slits. Fig. 7 shows the phase differences as a function of the difference between the refractive indexes of mediums in the two slits. As a consequence of the phase differences depicted in Fig. 7, the beam exiting the slits is deflected. Fig. 8 is the intensity distribution when the right slit is filled with air and the left one with larger refractive index medium. Due to the refractive index difference and the phase difference, the beam exiting the slits is deflected. The deflection angles are determined based on the highest intensity spot. In Fig. 8(a), the refractive index of medium in the left slit is 1.4, the deflection angle is about 3.6°. Fig. 8(b) shows that if we further increase the refractive index of the medium in the left slit, the deflection angle will increase to a certain value 12°. Although the medium between the two metal layers we considered in this paper is SiO2, it is necessary to discuss the effect of the dielectric refractive index nd between the two metals on the deflection angle (As shown in Fig. 8(c)). We have considered 3 different refractive indexes which are 1.35, 1.54 and 1.75. Here, the right slit is filled with air and the
W. Su et al. / Optics Communications 381 (2016) 24–29
29
Fig. 8. (a) Intensity distribution of the focused beam from the double slits structure when the right slit is filled with air and the left one with the medium which the refractive index equals to 1.4. The other parameters are the same as in Fig. 2. (b) Angle formed by the beam with respect to the normal direction as a function of difference between the refractive indexes in the two slits. The structure parameters are the same as in Fig. 2. (c) The deflection angle as a function of the dielectric refractive index between the two metal layers.
left one with the medium which the refractive index equals to 1.4. From the curve, we can see that the deflection angle is from 1.5° to 3.5°. For higher refractive index, the intensity at the focal spot is very low, so we do not think about it.
4. Conclusion In summary, we have proposed an Ag/SiO2/Ag tri-layer grating structure with double slits for beam focusing. Compared with the metal/dielectric double-layer grating-based structure, the focusing efficiency of this MDM grating-based structure can be greatly enhanced. Numerical simulation results based on FDTD method indicate that the focal length and the deflection angle can be controlled by adjusting the refractive indexes of mediums in the slits. The focal length of the beam spot can be changed in the range of the order of one wavelength which is difficult to be achieved by the traditional focusing lens. We think it is useful for designing subwavelength optical systems.
References [1] W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2004) 824. [2] F.J. García-Vidal, H.J. Lezec, T.W. Ebbesen, L. Martín-Moreno, Phys. Rev. Lett. 90 (2003) 213901. [3] S.I. Bozhevolnyi, V.S. Volkov, E. Devaux, T.W. Ebbessen, Phys. Rev. Lett. 95 (2005) 046802. [4] X. Luo, T. Ishihara, Appl. Phys. Lett. 84 (2004) 4780. [5] H.F. Shi, C.T. Wang, C.L. Du, X.G. Luo, X.C. Dong, H.T. Gao, Opt. Express 13 (2005) 6815. [6] K.L. Lee, C.W. Lee, P.K. Wei, Appl. Phys. Lett. 90 (2007) 233119. [7] S. Kim, Y. Lim, H. Kim, J. Park, B. Lee, Appl. Phys. Lett. 92 (2008) 013103. [8] A. Yanai, U. Levy, Opt. Express 17 (2009) 924. [9] G.G. Zheng, Y.Y. Chen, C.Y. Zhang, M. Lai, W. Su, Y.Z. Liu, Photonics Nanostruct. 10 (2012) 560. [10] A.E. Çetin, K. Güven, Ö.E. Müstecaplıoğlu, Opt. Lett. 35 (2010) 1980. [11] E.A. Bezus, L.L. Doskolovich, N.L. Kazanskiy, V.A. Soifer, S.I. Kharitonov, J. Opt. 12 (2010) 015001. [12] Y.T. Yu, H. Zappe, Opt. Express 19 (2011) 9434. [13] A.E. Çetin, A. Sennaroglu, Ö.E. Müstecaplıoğlu, Photonics Nanostruct. 8 (2010) 7. [14] K.Y. Kim, A.V. Goncharenko, J.S. Hong, K.R. Chen, J. Opt. Soc. Korea 15 (2011) 196. [15] H.C. Kim, H. Ko, M. Cheng, Opt. Express 17 (2009) 3078. [16] P. Lalanne, J. Hugonin, Nat. Phys. 2 (2006) 551. [17] B. Ung, Y.L. Sheng, Opt. Express 15 (2007) 1182. [18] K.V. Sreekanth, A.D. Luca, G. Strangi, Sci. Rep. 3 (2013) 3291.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Beam focusing from double subwavelength slits surrounded by Ag/SiO2/Ag tri-layer gratings Wei Su a,n, Chong Zhou b, Gaige Zheng c, Xiangyin Li b a
Department of Mathematics and Physics, Hohai University, Changzhou Campus, Changzhou 213022, China Department of optical engineering, Nanjing University of Science and Technology, Nanjing, China c School of Physics and Optoelectronic Engineering, Nanjing University of Information Science and Technology, Nanjing, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 2 May 2016 Received in revised form 4 June 2016 Accepted 22 June 2016
A silver(Ag)/SiO2/Ag tri-layer grating structure with double slits for beam focusing has been proposed. Compared with the metal/dielectric double-layer grating-based structure, the focusing efficiency of our proposed structure can be greatly enhanced. Numerical simulations using the finite-different time-domain (FDTD) method verify that the focal length and deflection angle can be controlled by adjusting the refractive indexes of dielectric mediums in the two slits. & 2016 Elsevier B.V. All rights reserved.
Keywords: Surface plasmon polaritons Beam focusing Grating
1. Introduction
2. Theory and structure
Surface plasmons (SPs) are coherent delocalized electron oscillations that exist along a metal/dielectric interface [1]. Since Ebbsen's discovery of high light transmission from subwavelength hole arrays [2], the investigations on subwavelength metallic structures have become hot [3–6]. One of the most meaningful applications on SPs is light focusing. Single metal layer or metal/ dielectric (MD) double-layer structures have been theoretically investigated for beam focusing [7–14]. But the focusing efficiencies of this kind of structures are not high. So many scholars did works on improving the focusing efficiency. For instance, Kim et al. obtained a high focusing efficiency by using the Ag/SiO2 multilayered zone plate [15]. In this paper, we extend the previous studies of beam focusing structures and propose the Ag/SiO2/Ag tri-layer grating structure with double slits. Compared with the Ag/SiO2 double-layer grating-based structure, the focusing efficiency can be greatly enhanced. In addition, simulation results based on two-dimensional (2D) FDTD method with perfect matching layer (PML) absorbing boundary conditions show that the plasmonic focusing property can be controlled by adjusting the refractive indexes of dielectric materials in the two slits.
Fig. 1 shows the schematic diagram of the proposed structure, which is composed of double slits in a metallic film with surface gratings on the exiting plane. Where w is the width of slit, hm1 and hm2 are the thickness of the substrate and the top metal layer of the grating, respectively. hd is the thickness of the dielectric layer sandwiched between metal layers, b and N are the ridge width and the period number of the grating at either side of the slits, respectively. Λ is the grating constant. a and d are the ridge widths of the grating and the interspacing between double slits, respectively. n1 and n2 are the refractive indexes of dielectric mediums in the two slits, respectively. The substrate of the whole structure is glass. The basic concept of the proposed structure is based on the interference modulation of surface plasmon polaritons (SPPs). In the optical regime, the dielectric functions of some noble metals can match with common dielectrics to sustain surface charge density oscillations, called SPPs. The structure proposed here is a metal/dielectric/metal (MDM) tri-layer grating. As we know, a light wave tends to confine itself mostly in a medium with higher refractive index than its adjacent ones. Therefore, for such MDM waveguide, the propagation constant β can be calculated as [16]:
(
tanh n
Corresponding author. E-mail address: [email protected] (W. Su).
http://dx.doi.org/10.1016/j.optcom.2016.06.062 0030-4018/& 2016 Elsevier B.V. All rights reserved.
)
β 2 − k 02εd hd /2 = −
εd β 2 − k 02εm εm β 2 − k 02εd
(1)
where k0 is the free space wave vector of optical wave, hd is
W. Su et al. / Optics Communications 381 (2016) 24–29
25
Fig. 1. Schematic diagram of the Ag/SiO2/Ag tri-layer grating structure with double slits.
dielectric layer thickness, εm and εd are the relative dielectric permittivities of metal and dielectric, respectively. When a transverse-magnetic (TM) electromagnetic wave illuminates from the bottom side, a part of light can pass through the slits directly, and the other can generate SPPs between metal layers. Consider the phases of the excited SPPs from the two slits first. For normal incidence, the relative phases at the two slits can be calculated as [9]:
φ1 + dksp = φ2 + 2Mπ ,
φ2 + dksp = φ1 +
( 2M + 1)π
The permittivities of the metals are determined by the LorentzDrude model [17]:
Z (μm)
8
(a)
4
0
∑
2 k = 0 ωk
(2)
8
(b)
I
-1
0
X (μm)
1
2
0
(3)
max
(c)
4
4
-2
− ω2 + jωΓk
where εr,1 is the dielectric constant at infinite frequencies, ωP is the plasmon frequency, and ωk, fk and Γk are the resonance frequency, strength and damping frequency, respectively, of kth oscillator. The Lorentz-Drude model uses K damped harmonic oscillators to describe the small resonances observed in the metal's frequency response.
where φ1 and φ2 correspond to the phases of SPPs at the entrance of the two slits, respectively. M is an integer. In this paper, the wavelength of the light is considered as 532 nm, so we can obtain the structure parameter d ¼470 nm.
8
fk ωP2
K
ε( ω) = εr, ∞ +
-2 II
-1
0
X (μm)
1
2
0
0
-2
-1
0
X (μm)
1
2
III
Fig. 2. The intensity distribution of the beam from the slits of the structure (a) without the left side grating, (b) without the right side grating, (c) with the complete grating. w¼ 90 nm, Λ ¼250 nm, hm1 ¼ 120 nm, hm2 ¼ hd ¼ 80 nm, a ¼b ¼80 nm, N¼ 6, λ ¼532 nm, d¼ 470 nm. The illustrations I, II and III are the simulation structures in (a), (b) and (c), respectively.
26
W. Su et al. / Optics Communications 381 (2016) 24–29
8
8
(a)
8
(b)
529 nm
0
Z (μm)
8
f = 3.22 μm
-1
0
X (μm)
1
2
-2
-2
-1
0
X (μm)
1
2
569 nm
4 f = 3.41 μm
0
-2
-1
0
X (μm)
1
2
max
(d)
639 nm
(e)
4 f = 3.29 μm
0
4 f = 3.39 μm
0 -2
Z (μm)
Z (μm)
4
(c)
normalized intensity
Z (μm)
510 nm
-1
0
X (μm)
1
2
0
width of a (nm)
Fig. 3. The intensity distribution of the focused beam from the subwavelength metal slits with (a) 40, (b) 80, (c) 120 and (d) 150 nm width of a. The other parameters are the same as in Fig. 2. The focal lengths are 3.22, 3.39, 3.41 and 3.29 μm, respectively. The FWHMs are 510, 529, 569 and 639 nm, respectively. (e) The normalized intensity at the focal spot as a function of the width of a.
3. Results and discussion Fig. 2 shows the schematic diagram of the focusing process. As we mentioned above, to form a focal spot, the lights radiating from each slit should interfere constructively. If we remove either side of the grating (Fig. 2(I) and (II)), SPPs can not be excited in the other slit, so there is no focal spot in the transmission region. For the complete structure as shown in Fig. 2(III), if the two slits are filled with the same medium, the phases of SPPs excited in the slits are the same, so the lights from each slits can interfere constructively and form a focal spot. For the structure we proposed here, the parameters of the grating between the two slits are significant. From Eq. (2), we can see that the distance between the two slits is an important parameter for calculating the phases of the excited SPPs from the two slits, which determines the transmission light can focus or not.
Moreover, we investigate the effect of the ridge width of the middle grating a on the focusing performance. Here, the values of the rest of the structural parameters are optimized to be w¼90 nm, Λ ¼ 250 nm, hm1 ¼120 nm, hm2 ¼80 nm, hd ¼80 nm, a¼ 80 nm, b¼ 80 nm, respectively. In addition, the incident wavelength λ is fixed at 532 nm, so the optimal value of the interspacing between double slits d can be calculated as 470 nm according to Eq. (2). The period number at both lateral grating N is chosen to be 6. Fig. 3 shows the intensity distribution of the focused beam from the slits with 40, 80, 120, 150 nm width of a. The focal lengths are 3.22, 3.39, 3.41 and 3.29 μm, respectively. Changes in ridge width a have little effect on the focal lengths. Their full width at half-maximum (FWHMs) at each focal length are 510, 529, 569 and 639 nm, respectively. FWHMs at each focal length are about one wavelength, which means that the focusing performance of the proposed structure is consistent with the diffraction limit. The
W. Su et al. / Optics Communications 381 (2016) 24–29
(a)
(b)
8
f = 2.78 μm
-1.8
0
X (μm)
4 f = 3.39 μm
0
1.8
-2
0
X (μm)
(d)
649 nm
Z (μm)
Z (μm)
Z (μm)
529 nm
4
0
(c)
8
484 nm
2
normalized intensity
8
27
f = 4.12 μm
4
0
-2.2
0
2.2
X (μm)
the number of periods (N)
Fig. 4. The intensity distribution of the focused beam from the slits with (a) 5, (b) 6 and (c) 7 periods at both lateral grating. The other parameters are the same as in Fig. 2. The focal lengths are 2.78, 3.39 and 4.12 μm, respectively. The FWHMs are 484, 529 and 649 nm, respectively. (d) The normalized intensity at the focal spot as a function of the period number at both lateral grating.
intensity distribution results show that the energy generating from the structure can focus within a small region. More details of the dependence of normalized intensity at the focal spot on width of a are shown in Fig. 3(e). By increasing a from 40 to 120 nm, the changes of the intensity is only a little. But the intensity decreases abruptly with further decrease or increase of a. The intensity decrease with very small or large a is due to the fact that a change in filling fraction can affect a lot the decoupling of the plasmon, thus affecting the intensity at the focal spot [18]. When a is 80 nm, the intensity is the highest. So 80 nm is the optimal value of the ridge width of grating between the two slits. With the optimal parameters, we change the period number at both lateral grating from 5 to 7, and find that the focal length can
Z (μm)
8
be enlarged by increasing the period number N. Fig. 4(a–c) show the intensity distribution of the focused beam from the slits with different period numbers at both lateral gratings. The focal lengths are 2.78, 3.39 and 4.12 μm, respectively. And their FWHMs at each focal length are 484, 529 and 649 nm, respectively. More details of the dependence of normalized intensity at the focal spot on period number at both lateral grating N are shown in Fig. 4(d). As we can see from the curve, when N is 6, the intensity is the highest. As we mentioned in the introduction part, scholars have already found that the MDM multilayer structures have a higher focusing efficiency than that of single metal or MD double-layer structures. Fig. 5(a) and (b) show the intensity distribution of the focused beam by MD and MDM grating structures, respectively. It
max 6 8
(a)
4
max
(b)
4
0 -2 I
-1
0
X (μm)
1
2
0
0
-2 II
-1
0
X (μm)
1
2
0
Fig. 5. The intensity distribution of the focused beam from the subwavelength metal slits by (a) MD and (b) MDM grating structures, respectively. The other parameters are the same as in Fig. 2. The illustrations I and II are the simulation structures in (a) and (b), respectively.
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W. Su et al. / Optics Communications 381 (2016) 24–29
Z (μm)
8
8
(a)
4 f = 3.39 μm
max
(b)
4 f = 2.86 μm
0
0 -2
-1
0
X (μm)
1
2
-2
-1
0
X (μm)
1
0
2
Fig. 6. Intensity distribution of the focused beam from the double slits structure. (a) n1 ¼ n2 ¼ 1; (b) n1 ¼n2 ¼1.802. The structure parameters are the same as in Fig. 2.
visually shows the difference of the focusing property between the two structures. Illuminated by the light with the same intensity, it is obvious that the intensity at the focal spot of MD grating is much weaker than that of MDM grating. It is worth noting that the maximum value of the color bar in Fig. 5(a) is only one in six of that in Fig. 5(b). So our proposed MDM grating structure is much better than MD grating on the focusing property. By appropriately adjusting the focusing structural parameters, we find that the focal length and the deflection angle can be controlled by adjusting the refractive indexes of the dielectric mediums in the slits. The two beams can be generated and end up by forming a focal spot as interfering constructively. The intensity distributions of the radiated light from the proposed structure are shown in Fig. 6. The focal spots are generated under condition of constructive interference. With the refractive indexes of the slits are 1 and 1.802, the focal lengths are 3.39 and 2.86 μm,
Fig. 7. Phase difference between the fields exiting from the slits as a function of the refractive index difference between the two slits.
respectively. If the refractive indexes of the mediums in the two slits are different, the phase retardations of the light pass through the slits are different, the phase can be calculated as:
⎡ ⎛ 1 − β /k ⎢ i 0 φi = Re βi t + arg⎢ 1 − ⎜⎜ 1 /k + β ⎝ ⎢⎣ i 0
( (
( )
⎤
2
) ⎞⎟ exp 2jβ t ⎥, ( i )⎥ ) ⎟⎠ ⎥⎦
i = 1, 2 (4)
where the propagation constants in the slits can be calculated as:
(
tanh
)
βi2 − k 02εi w /2 = −
εi βi2 − k 02εm εm βi2 − k 02εi
,
i = 1, 2 (5)
where εi is the relative dielectric constant of the medium in the slit. According to Eqs. (4) and (5), we can obtain the phase differences between the two slits. Fig. 7 shows the phase differences as a function of the difference between the refractive indexes of mediums in the two slits. As a consequence of the phase differences depicted in Fig. 7, the beam exiting the slits is deflected. Fig. 8 is the intensity distribution when the right slit is filled with air and the left one with larger refractive index medium. Due to the refractive index difference and the phase difference, the beam exiting the slits is deflected. The deflection angles are determined based on the highest intensity spot. In Fig. 8(a), the refractive index of medium in the left slit is 1.4, the deflection angle is about 3.6°. Fig. 8(b) shows that if we further increase the refractive index of the medium in the left slit, the deflection angle will increase to a certain value 12°. Although the medium between the two metal layers we considered in this paper is SiO2, it is necessary to discuss the effect of the dielectric refractive index nd between the two metals on the deflection angle (As shown in Fig. 8(c)). We have considered 3 different refractive indexes which are 1.35, 1.54 and 1.75. Here, the right slit is filled with air and the
W. Su et al. / Optics Communications 381 (2016) 24–29
29
Fig. 8. (a) Intensity distribution of the focused beam from the double slits structure when the right slit is filled with air and the left one with the medium which the refractive index equals to 1.4. The other parameters are the same as in Fig. 2. (b) Angle formed by the beam with respect to the normal direction as a function of difference between the refractive indexes in the two slits. The structure parameters are the same as in Fig. 2. (c) The deflection angle as a function of the dielectric refractive index between the two metal layers.
left one with the medium which the refractive index equals to 1.4. From the curve, we can see that the deflection angle is from 1.5° to 3.5°. For higher refractive index, the intensity at the focal spot is very low, so we do not think about it.
4. Conclusion In summary, we have proposed an Ag/SiO2/Ag tri-layer grating structure with double slits for beam focusing. Compared with the metal/dielectric double-layer grating-based structure, the focusing efficiency of this MDM grating-based structure can be greatly enhanced. Numerical simulation results based on FDTD method indicate that the focal length and the deflection angle can be controlled by adjusting the refractive indexes of mediums in the slits. The focal length of the beam spot can be changed in the range of the order of one wavelength which is difficult to be achieved by the traditional focusing lens. We think it is useful for designing subwavelength optical systems.
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