Plasmonics (2012) 7:191–199 DOI 10.1007/s11468-011-9293-5

Active Focal Length Control of Terahertz Slitted Plane Lenses by Magnetoplasmons

Bin Hu & Qi Jie Wang & Shaw Wei Kok & Ying Zhang

Received: 6 April 2011 /Accepted: 17 October 2011 /Published online: 3 November 2011 # Springer Science+Business Media, LLC 2011

Abstract Active plasmonic devices are mostly designed length. Various lenses, including one with two focal spots at visible frequencies. Here, we propose an active and a tunable lens for dipole source imaging, are realized terahertz (THz) plasmonic lens tuned by an external for the proposed structure, demonstrating the flexibility of magnetic field. Unlike other tunable devices where the the design approach. The proposed tunable THz plas- tuning is achieved by changing the plasma frequency of monic lenses may find applications in THz science and materials,theproposedactivelensistunedbychanging technology such as THz imaging. the cyclotron frequency through manipulating magneto- plasmons (MPs). We have theoretically investigated the Keywords Terahertz . Plasmonics . Lens . Magneto-optic dispersion relation of MPs of a semiconductor–insulator– systems semiconductor structure in the Voigt configuration and systematically designed several lenses realized with a doped semiconductor slab perforated with sub- Introduction wavelength slits. It is shown through finite–difference time–domain simulations that THz wave propagating Plasmonics has been enormously developed since the through the designed structure can be focused to a discovery of the extraordinary optical properties of surface small size spot via the control of MPs. The tuning plasmon polaritons (SPPs) [1, 2], coherent electron range of the focal length under the applied magnetic oscillations excited at the interface between a metal-like field(upto1T)is∼31, about 50% of the original focal material (real(εm)<0) and a dielectric (real(εd)>0), where ε represents the permittivity. Many plasmonic devices, such as lenses, collimators, waveguides, polarizers, and couplers were proposed and experimentally realized [3– B. Hu : Q. J. Wang (*) Division of Microelectronics, 11]. In recent years, in order to achieve ultra-miniaturized School of Electrical & Electronic Engineering, plasmonic circuits for high-speed computing and commu- Nanyang Technological University, nication, a range of active plasmonic devices have been 50 Nanyang Ave., investigated [12] in the visible range. The basic tuning Singapore 639798, Singapore e-mail: [email protected] mechanism of all these active tunable plasmonic devices is based on changing the permittivity of the dielectric Q. J. Wang material εd [12] through thermo-optic, electro-optic, and Division of Physics and Applied Physics, nonlinear optical effects [13–15] rather than changing ε . School of Physical and Mathematical Sciences, m ε ∼− w2 Nanyang Technological University, This is because m ne/ (ne is the free carrier density Singapore 637371, Singapore and w is the angular frequency of light) is difficult to be ∼ 22 −3 : modified due to the large ne ( 10 cm of metals), which S. W. Kok Y. Zhang makes the changes of the permittivity of a metal negligible Singapore Institute of Manufacturing Technology, 71 Nanyang Drive, by just varying the carrier density. However, it is Singapore 638075, Singapore interesting and desirable that εm of plasmonic devices 192 Plasmonics (2012) 7:191–199 can also be tuned so as to increase the design flexibility bifocal pattern and a tunable lens for dipole source imaging and device functionality. can also be realized by applying the external magnetic field With the increasing interests in the terahertz (THz) science to the proposed structure, demonstrating the flexibility and and technology [16–18], THz plasmonic devices have variety of the design approach. attracted a lot of attention recently. In these devices, highly doped semiconductors are often used as plasmonic materials to replace metals because they have similar dielectric Dispersion Relation of MPs in a SIS Waveguide properties (real(εm)<0, we use the same symbol εm to represent the permittivity of doped semiconductor) to those Before we compute the dispersion relation of MPs for SIS of metals in the visible [8, 19]. In addition, their dielectric structures, we first express the dielectric constant of a constant εm can be manipulated because the carrier density of doped semiconductor (in the THz regime, it functions as a 16 19 −3 doped semiconductors is relatively low (ne∼10 –10 cm ) metal) based on the Drude model in the Voigt configura- and can be easily tuned by changing temperature, doping, or tion, to see the effect of applying an external magnetic field. optical excitation [20]. Through these tuning schemes, some The 2D SIS structure is shown in Fig. 1a. We assume that active THz plasmonic devices have been proposed [21–23]. the structure is invariant in the y-direction, and the spatial However, all of these methods are based on modifying and temporal dependence of the fields is in the form of the carrier density ne, in other words, the plasma ∼exp[i(k·r-wt)]. The thickness of the insulator layer is w. 2 frequency wp (wp ∼ne). Another possible means of The propagation modes are TM polarized (i.e., with the changing εm is to vary the cyclotron frequency wc (as εm magnetic field component parallel to the y-direction) and is also a function of wc) by applying an external magnetic the magnetic field B is applied along the y-axis. Thus, the field B (wc ∼B)[24]. In this situation, the magneto- dielectric constant of the semiconductor can be expressed plasmons (MPs) are excited, which have different disper- as a tensor [24]: sion relations as compared to those of SPPs. For example, 2 3 "xx 0 "xz magnetic field modification can cause the semiconductor 6 7 " ¼ " : ð Þ material to become anisotropic in the Voigt and Faraday m 4 0 yy 0 5 1 configurations. Thus, MPs have several unique properties, "xz 0 "xx such as the non-reciprocal effect [25] and two propagation bands [27] for doped semiconductors in the Voigt If we do not consider the effects of holes and phonons, configuration. Although several structures including the which are negligible in the proposed structures, the single semiconductor–dielectric interface, thin film, and elements in the matrix can be calculated as: 2 3 periodic structures based on MPs have been studied [25– w 2ðÞw þ u 28], active devices are rarely reported. In addition, to the 4 hip i 5 "xx ¼ "1 1 ð2Þ 2 2 best of our knowledge, theoretical derivation of the wwðÞþ iu wc dispersion relation of MPs in a metal–insulator–metal (MIM) structure [or semiconductor–insulator–semiconductor (SIS) structure], an important and widely used configuration w 2w 2 hip c for various plasmonic devices [29–31], has not been "xz ¼i"1 ð3Þ wwðÞþ u 2 w 2 obtained yet. i c In this paper, we first theoretically investigate the dispersion relation of MPs in a SIS waveguide structure in 2 the Voigt configuration. As a design example, the derived wp " ¼ "1 1 ð4Þ dispersion relation is then applied to design an active THz yy wwðÞþ iu device—a slitted THz plasmonic lens—the focal length of which can be actively tuned by an external magnetic field. where w is the angular frequency of the incident wave, and 2 2 The lens is designed based on the diffraction theory by wp is the plasma frequency, evaluated by wp =nee / * * phase control of slits with different widths [32–34]. (m ε∞ε0), in which ne, e, and m are the density, the However, the phase retardation of each slit is modified by effective charge, and the effective mass of electrons, respec- manipulating MPs rather than SPPs in the proposed tively. ε∞ and ε0 are the high-frequency permittivity and devices. We find out through finite–difference time–domain vacuum permittivity, respectively. υ is the collision frequency (FDTD) numerical simulations that the focal length of the of free electrons, given by e/(μm*), and μ is the carrier * proposed lens can be tuned, the maximum tuning range of mobility. wc=eB/m is the cyclotron frequency, which can be which is ∼31 for a lens with a focal length of ∼61.A tuned by the applied external magnetic field B. Plasmonics (2012) 7:191–199 193

1

(a) Insulator (b) Vacuum c=0

Semiconductor Semiconductor 0.8 d =0.5 k c p m m 0.6 c=1 p p 0.56

0.5 Ex 0.4 0.4

BBHy p 0.3 z 0.2 0.2 I 0.1 II III 0 0 0.5 1 1.5 y /kp w w 0 x x x 0 0.5 1 1.5 2 2.5 3 2 2 /kp

Fig. 1 a A semiconductor–insulator–semiconductor structure in a Electromagnetic wave in vacuum and magnetoplasmons (MPs) under Voigt configuration. The propagation mode is TM polarized, and the the magnetic field of wc=0, wc=0.5wp, and wc=1wp are denoted by the magnetic field B is applied along the y-axis. The thickness of the green, black, blue, and red lines, respectively. The inset shows the insulator is denoted by w. b Dispersion relation of MPs in the structure enlarged view of the low-frequency region. The insulator width is set for InSb–air–InSb material. The electronic dissipation is considered. as w=0.11p

For an SIS structure, we can express the magnetic and In region (III), (x>w/2) electric field components Hy and Ex in the semiconductor − and insulator as follows: In region (I) ( w/2

ðIÞ ik1x ik1x Hy ¼ Ae þ Be ð5Þ

k ðÞ = ðÞib" þ " k e 2 x w 2 E ðÞIII ¼ D xz xx 2 ð10Þ k z w" ðÞ" 2 þ " 2 ðIÞ 1 ik1x ik1x i 0 xx xz Ez ¼ ðAe Be Þð6Þ w"0"d In region (II), (x<−w/2) where β is the propagation constant along the z- 2 2 2 2 2 2 2 direction, κ1 =β −k0 εd, κ2 =β −k0 εV. εV=εxx+εxz / ð Þ k ð þ = Þ II ¼ 2 x w 2 ð Þ Hy Ce 7 εxx is the bulk Voigt dielectric constant. A, B, C,andD are the undetermined coefficients. Employing the bound- k ðÞþ = ðÞb" " k 2 x w 2 − ðÞII ¼ i xz xx 2 e ð Þ ary conditions at the two interfaces (x= w/2 and x=w/2) Ez C 2 2 8 iw"0ðÞ"xx þ "xz yields,

2 3 ðÞb"xz i"xxk2 k1 k = ðÞb"xz i"xxk2 k1 k = ! 6 þ ei 1w 2 e i 1w 2 7 6 ðÞ" 2 þ " 2 " ðÞ" 2 þ " 2 " 7 A 6 xx xz d xx xz d 7 ¼ ð Þ 4 5 0 11 ðÞb"xz þ i"xxk2 k1 k = ðÞb"xz þ i"xxk2 k1 k = B þ i 1w 2 i 1w 2 2 2 e 2 2 e ðÞ"xx þ "xz "d ðÞ"xx þ "xz "d

The nontrivial solution of these linear equations requires the vanishing of the coefficients determinant, and then we can derive the following relation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "#  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 b2 " 2 " b " k " " b " V k0 tanh w b2 " k2 1 þ d V 0 þ d xz þ 2 d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 ð12Þ d 0 " b2 " 2 " " b2 " 2 " V dk0 V xx dk0 V b2 " 2 dk0 194 Plasmonics (2012) 7:191–199

Equation 12 is the dispersion relation of TM-polarized result, the phase retardation of the propagation mode in a MPs in the Voigt configurations for the SIS structure. It slit is increased with the increase of the magnetic field. should be noted that, when w→∞, and B=0, Eq. 12 becomes the dispersion relations of surface MPs in a semi- infinite structure [25] and plasmons in an MIM structure Tunable THz Plasmonic Lenses [29], respectively.

In the visible range, when B=1 T, wc is in the order of Then we design a MPs lens by Eq. 12. The slitted lens 11 −1 10 s for metals, which is negligible as compared to wp structure is shown in Fig. 2. We consider a symmetric 2D (∼1016 s−1) of metals in the visible (w ∼1015 s−1). InSb slab surrounded by air and perforated with 2N-1 slits Therefore, the effect of an applied magnetic field on arranged on the x-y plane. The thickness of the slab is h; the changing the dielectric constant εm cannot be clearly lens width is d; and the width of the i-th slit is denoted by observed. However, the effect becomes obvious for some wi where the first slit is in the center. The incident plane doped semiconductors with low effective mass where wc wave is TM-polarized with frequency f. Each slit can be 13 −1 13 −1 ∼10 s is comparable with wp ∼10 s in the THz considered as an SIS waveguide structure in Fig. 1a with range (w ∼1012 s−1). finite length in z-direction. Therefore, according to Eq. 12, To calculate the dispersion relation of Eq. 12, we choose different slit widths and magnetic fields provide different InSb as the semiconductor material. The corresponding propagation constants and also the phases. In the following parameters of InSb at room temperature are given by m*= design procedures, we set the thickness of the slab as h= μ 0.014m0 (m0 is the free electron mass in vacuum), μ=7.7× 300 m and the incident frequency is f=1 THz (i.e., the 4 2 −1 −1 16 −3 μ ε − 10 cm V s , ne=1.1×10 cm , and ε∞=15.68 [35]. wavelength in free space is 300 m), thus V= 43.6+15.4i ε − Therefore, the plasma frequency wp is 12.6 THz. The for B=0 T and V= 10.9+4.27i for B=1 T. In order to meet results are depicted in Fig. 1b, where kp and 1p are defined the phase retardation of a 2D lens [31, 32], we select the ’ as kp=wp/c, and 1p=2πc/wp. We should mention that when widths of each slit on principle of the Fresnel s equations: an external magnetic field is applied, the MPs band splits the phase retardation of a single slit can be expressed as — into two bands the low-frequency and high-frequency bands, similar to surface MPs [25]. Here, we do not show t t eia Δ8 ¼ arg 01 12 ð13Þ the high-frequency band because we only use the low- i2a 1 r10r12e frequency region in our design. It is found from the figure that, when the magnetic field is increased, the dispersion where α=k0hneff, r10=(n1−neff)/(n1+neff), r12=(n2−neff)/ curve moves toward lower frequency side. For a certain (n2+neff), t01=1−r10, t12=1+r12. neff=β/k0n1 is the effec- frequency, the propagation constant increases with the tive refractive index, and n2 are the refractive indices of the increase of the magnetic field, as shown in the inset of materials on and below the slit, respectively. It should be Fig. 1b, applicable for lower frequencies below the point at noted that although Eq. 12 corresponds to the structure w/wp=0.56 (indicated by a dotted line in Fig. 1b). As a invariant in the z-direction, it gives good agreement of the

f

wN wN-1 w2 w1 w2 wN-1 wN z BB h ...... d y x

fl Focus

Fig. 2 Schematic structure of the proposed tunable THz plasmonic focusing can be realized by modulating those phases. The thickness of lens. The structure consists of an InSb slab tunable by an external the slab is denoted by h. The width of the i-th slit from the middle to magnetic field B and perforated with 2N-1 sub-wavelength slits. THz the two sides is wi. fl denotes the focal length of the lens. The external waves, indicated by the red arrows, are incident onto the slab from the magnetic field B is applied along the y-axis in order to change the top surface. They have different phase retardations after the slits, thus focal length Plasmonics (2012) 7:191–199 195

8 conclude that the change of the phase retardation of B=0 T 4 B=0.5 T narrower slits is more prominent than that of wider slits B=0.8 T 3 under the same magnetic field. For example, as shown in 6 B=1 T 2 the inset of Fig. 3, the phase retardation Δ8 increases 4π fortheslitwidthof1μm, while it increases only less than

increase (1 ) 0.5π the slit width of 50 μm when the magnetic field

() 4 0 increases from 0 to 1 T. 0 10 20 30 40 50 1 Slit width ( m) Then, we design a lens with focal length of fl=7 for B= 2 0, the slit positions and widths are depicted in Fig. 4a. The total width of the lens and slit number are chosen as d= 2 mm and 2N-1=41, respectively. This is to make the slit width increases monotonously from the middle slit to the 0 0 10 20 30 40 50 sides of the slit lens, thus making the phase curve of the Slit width ( m) lens a parabolic shape with the increase of the magnetic field as shown in Fig. 4b. When an external magnetic field Fig. 3 Phase retardation of a single slit under external magnetic field is applied and strengthened, the phase curve at the slab of different intensity. The incident frequency is 1 THz, and the thickness of the slab is 300 μm. The inset shows the corresponding increase of Δ8 output surface will be more convex (see Fig. 4b) because against the slit width for a magnetic field from 0 to 1 T the increases of the phase retardation of the inner slits are larger than that of the outer slits. This means, in analogous phase retardation in Eq. 13 with the FDTD [36] simulation to a conventional lens, the focal length of the proposed lens in our height–finite structure. For example, when the slit is reduced with the increase of the magnetic field. length is h=300 μm, for the slit width w is 10, 20, 30, 40, Then, we calculate the distributions of magnetic field μ Δ8 π π π π 2 and 50 m, equals 1.12 , 0.61 , 0.43 , 0.33 , and intensity |Hy| of the electromagnetic field by FDTD 0.27π, respectively, calculated by Eqs. 12 and 13, while the simulations. In the simulations, we set a uniform cell of corresponding FDTD-simulated results are 1.07π, 0.51π, Δx=Δz=0.5 μm in the semiconductor slab region (includ- 0.34π, 0.27π, and 0.23π, respectively. Therefore, Eq. 12 ing the slits) and a nonuniform cell in other places. The can be used in a lens design. results are shown in Fig. 5. The magnetic field intensity Since phase retardation is the key factor in the design, distributions of the lens for B=0 T and B=1 T are shown in we study the impact of the external magnetic field on Δ8 the panels a and b of Fig. 5, respectively. From the first. The curves of Δ8 versus slit width under several dependence of the focal length on the intensity of applied magnetic field intensities are plotted in Fig. 3. It shows that magnetic field, which is depicted in Fig. 5c, we find that it for a single slit, the phase retardation becomes larger when decreases by 31 (from 5.921 to 2.871). By applying a an external magnetic field is applied, which can be also magnetic field larger than 1 T, an even wider tunable range indicated in Fig. 1b. The reason is that with the increase of of the focal length can be achieved. We also note that there the external magnetic field, the negative real part of bulk is a deviation of the calculated focal length (5.921) from the dielectric constant εV increases, resulting in an increase of designed one (71), and some strong side lobes appear. By the penetration depth of THz wave into the semiconductor changing the parameters of the structure, we find that the [37]. This leads to an increase of the effective refractive deviation is only dependant on the lens width d and the index of the slit, thus resulting in larger phase retardation. designed focal length. This effect is similar to traditional In addition, from the results in Fig. 3, we can also lenses [38], thus we believe that the deviation is due to the

Fig. 4 Design of a tunable THz (a) (b) plasmonic lens with focal 60 3 length of 71 and d=2 mm, N=21. a The slit position and 50 2.5 corresponding slit width of the designed structure. b The m) 40 2 relative phase retardation of th ( d the slits under magnetic fields i 30 1.5 etardation ( ) of 0, 0.5, 0.8, and 1 T r

lit W 20 1 B=0 T S B=0.5 T

10 Phase 0.5 B=0.8T B=1 T 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 x (mm) x (mm) 196 Plasmonics (2012) 7:191–199

Fig. 5 Magnetic field modula- tion on the THz plasmonic lens. (a) (b) 2 a, b are |Hy| distributions of the structure when the external magnetic field is 0 and 1 T, respectively. c FDTD-calculated focal length when the external magnetic field is increased λ λ from 0 to 1 T z/ z/

B=0 B=1 Tesla Tesla

x(mm) x (mm)

(c) 6 λ 5

4

cal length/ 3 Fo

2 0 0.2 0.4 0.6 0.8 1 B(Tesla) diffraction effects for the Fresnel number (about 1.6 for our magnetic field. The reason is that for a larger lens width, structure) is very small. the difference of the side slit width and middle slit width is In order to further study the impact of the magnetic field greater. Therefore, the change of the phase curve on the exit on the focal length, we change the designed parameters plane of the slab is larger when an external magnetic field is such as the lens width, the slit number, and designed focal applied. Although a large focal length change can be length, as shown in Fig. 6. It is found that, compared with obtained by a wide lens width, it should be noted that the slit number and designed focal length, the slit width d has a slits in the middle of a wider lens are much narrower, which greater impact on the focal spot shift by the external causes a relatively smaller transmittance. Therefore, one

(a) (b) (c) 6 5 5 λ λ λ 4 4 4 3 3 2 2 2 Focal spot shift/ Focal spot shift/ 0 1 Focal spot shift/ 1 1 1.5 2 2.5 15 20 25 7 8 9 10 d (mm) N Designed focal length/λ

Fig. 6 Focal spot shift caused by an external magnetic field increase changes as 11, 16, 21, and 26, when the designed focal length and lens from 0 to 1 T as a function of various lens parameters. a Lens width width are fixed at fl=101, d=2.5 mm; c Designed focal lengths changes as d=1, 1.5, 2, and 2.5 mm, when the designed focal length changes as 71,81,91,101, and the length width and slit number are and slit number are fixed at fl=101,2N-1=21, respectively; b N fixed as d=2 mm, 2N-1=41 Plasmonics (2012) 7:191–199 197

Fig. 7 Bifocal pattern produced by an external magnetic field. 50 (a) The parameters are same as the (c) lens in Fig. 4, except d=1 mm, 40 N=31. a The slit position and

corresponding slit width of (m) the lens. b Phase retardation of h 30 the slits under different external λ

2 z/ magnetic fields. c |Hy| distri- tWidt 20 bution under a magnetic field i 2 Sl of B=0T.d |Hy| distribution under a magnetic field of 10 B=0.7 T B=0Tesla 0 -1 0 1 x (mm) x (mm)

8 (b) (d) 6 λ

() 4 B=0 T z/ B=0.5 T 2 B=0.7 T B=1.0 T B=0.7Tesla 0 -1.5 -1 -0.5 0 0.5 1 1.5 x (mm) x (mm) should choose appropriate parameters in the design of THz intervals: (0, 0.9 mm) and (0.9, 1.5 mm). We plot the magnetic field-tunable lenses. corresponding slit phase retardations under different We have demonstrated that a monotonically decreased magnetic fields, as shown in Fig. 7b. When the magnetic focal length with a single spot can be realized by a field increases, phase retardations of the two intervals monotonically increased slit width from the middle to the grow asynchronously. Therefore, a discontinuous jump two sides of the lens in the presence of an external magnetic appears between the two intervals (see B=0 and 0.7 T for 2 field. Next, we investigate a lens with two monotonically comparison). We depicted the calculated |Hy| distribution increased sets of slits. We set the lens width and slit number of the transmitted wave at B=0 T and B=0.7 T (where the as d=3 mm and N=31, as shown in Fig. 7a. The designed largest jump appears) in panels c and d of Fig. 7, focal length remains 71. The structure has two monotone respectively. It shows that when the magnetic field is

Fig. 8 Magnetic field-tunable lens for a dipole source imaging. The designed object and image distances are both 31. The source is an electric dipole with the electric field vibration direction perpendicular to the z-axis, which is represented by a white circle on the top portion of the 2 two figures. a |Hy| distribution under a magnetic field of B=0T. The obtained image distance 2 is 2.251. b |Hy| distribution under a magnetic field of B=0.8 T. The obtained image B Tesla B Tesla distance is 0.991 198 Plasmonics (2012) 7:191–199 increased to 0.7 T, we can achieve two focal spots along 6. Fu Y, Liu Y, Zhou X, Xu Z, Fang F (2010) Experimental 1 investigation of superfocusing of plasmonic lens with chirped the z-axis. The corresponding focal lengths are 5.91 and – 1 circular nanoslits. Optics Express 18:3438 3443 2.7 ,respectively. 7. Yu N, Fan J, Wang QJ, Pflugl C, Diehl L, Edamura T, Yamanishi Last but not least, we design a magnetic field-tunable M, Kan H, Capasso F (2008) Small-divergence semiconductor lens for a dipole source imaging to demonstrate the variety lasers by plasmonic collimation. Nature Photon 2:564–570 of the proposed THz lenses. The source is an electric dipole 8. Yu N, Wang QJ, Kats MA, Fan JA, Khanna SP, Li L, Davies AG, Linfield EH, Capasso F (2010) Designer spoof surface plasmon with the electric field vibration direction perpendicular to structures collimate terahertz laser beams. Nature Materials the z-axis. The designed object distance and image distance 9:730–735 are both 31. The distributions of magnetic field intensity for 9. Yu N, Wang QJ, Pflugl C, Diehl L, Capasso F, Edamura T, Furuta the external magnetic field of B=0 T and B=0.8 T are S, Yamanishi M, Kan H (2009) Semiconductor lasers with integrated plasmonic polarizers. 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