Antireflection layers in low-scattering plasmonic optics

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ScienceDirect Photonics and Nanostructures – Fundamentals and Applications 14 (2015) 101–105

Antireflection layers in low-scattering plasmonic optics夽 E.A. Bezus a,b,∗ , D.A. Bykov a,b , L.L. Doskolovich a,b a

Image Processing Systems Institute of the Russian Academy of Sciences, 443001 Samara, Russia b Samara State Aerospace University, 443086 Samara, Russia

Received 9 September 2014; received in revised form 18 February 2015; accepted 23 February 2015 Available online 3 March 2015

Abstract Antireflection layers for plasmonic optical elements analogous to conventional antireflection coatings are proposed and numerically investigated. It is shown that the average surface plasmon reflectance can be decreased by several orders of magnitude simultaneously with the suppression of the parasitic scattering of the surface plasmon energy. The application of the proposed approach to a binary plasmonic microlens array is considered as an example. The presented approach can be used for the design of other plasmonic elements working in transmission. © 2015 Elsevier B.V. All rights reserved. Keywords: Surface plasmon polariton; Antireflection coating; Parasitic scattering suppression; Electromagnetic optics; Rigorous coupled-wave analysis; Microlens array

1. Introduction Surface electromagnetic waves propagating in various dielectric and metal–dielectric structures have attracted considerable research interest during the recent years due to both their fundamental properties and potential applications [1]. Most studied are surface plasmon polaritons (SPP) propagating along the interfaces between metal and dielectric media. Another considered type of surface waves are the so-called Bloch surface waves (BSW) propagating at the interfaces between a photonic crystal and a homogeneous medium.

夽

The article belongs to the special section Metamaterials. Corresponding author at: Image Processing Systems Institute of the Russian Academy of Sciences, 443001 Samara, Russia. Tel.: +7 9272917395. E-mail address: [email protected] (E.A. Bezus). ∗

http://dx.doi.org/10.1016/j.photonics.2015.02.003 1569-4410/© 2015 Elsevier B.V. All rights reserved.

Several types of 2D optical elements for steering the surface wave propagation (e.g. for surface wave reflection and focusing) have been proposed, in particular, dielectric structures located directly on the surface of the metal or the photonic crystal [2–9]. Among others, prisms and lenses [2], Bragg gratings [3] and gradientindex optical elements [5] were proposed and experimentally demonstrated for SPP. Recently, 2D dielectric lenses for focusing BSW were also created [8,9]. The operating principle of most of these structures is based on the phase modulation of the incident surface wave. Along with the absorption losses intrinsic in plasmonic structures, parasitic scattering losses at the interfaces between different dielectric media are one of the major mechanisms decreasing the efficiency of such elements [10,11]. These losses are caused by the transverse field profile mismatch of the surface wave (SPP or BSW) across the interface and can reach up to 30% of energy at a single interface. Similarly to the conventional optical elements,

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(a)

(b)

lar

l

lar

εar ε 2

h2

εm

h1 lar

l

lar

ε1 ε d

εar ε 2

h2

εm

h1

ε1 ε d

Fig. 1. Considered structure geometries: a two-layer dielectric block without ((a), geometry A) or with ((b), geometry B) a continuous dielectric layer.

the efficiency of the 2D optical elements working in transmission can be also sufficiently decreased because of the partial reflection of the incident wave. In our previous works, we have proposed an approach for the design of low-scattering plasmonic optics based on the utilization of two-layer structures (dielectric–dielectric–metal plasmonic waveguides) made of isotropic materials [12–14]. The proposed approach is much simpler than other previously developed techniques based on the usage of anisotropic metamaterials [11,15]. It was shown that the approach presented in [12–14] allows decreasing the average parasitic scattering losses by an order of magnitude by means of partial matching of the transverse field profiles of the incident SPP and the plasmonic mode inside the element. In the present work, we propose and study numerically the antireflection layers (analogue of conventional antireflection coatings) for plasmonic optical elements with parasitic scattering suppression. As an application example, design of a binary plasmonic microlens array is considered. 2. SPP antireflection layers Let us first consider as a model problem the SPP propagation through a two-layer dielectric block with the length l, to which the antireflection layers can be added. Two structure geometries are studied (Fig. 1), the differences between them being the material of the antireflection layers (blocks) and the presence or absence of a continuous dielectric layer on the metal surface outside the block. SPP (Fig. 1(a), geometry A) or the mode of the dielectric–dielectric–metal plasmonic waveguide (Fig. 1(b), geometry B) is normally incident at a twodimensional structure (structure properties do not change in the direction perpendicular to the figure plane). Let us note that the continuous dielectric layer in geometry B

serves as a protective layer preventing the degradation of the metal surface. Fig. 1 shows the dielectric blocks with two attached antireflection layers on the left and right sides having equal length lar . Dielectric blocks without antireflection layers and with the left antireflection layer only were also simulated. For both geometries, the following parameters similar to the parameters of the examples considered in our previous works [13,14] were used in the simulations: free-space wavelength λ0 = 800 nm, εm = −24.06 + 1.51i (corresponds to Au) [16], εd = 1, ε1 = 1.452 , ε2 = 1.72 , and h2 = 1 ␮m. The thickness of the first layer h1 was equal to 62 nm and 38 nm for geometries A and B, respectively. In both cases, the h1 value was chosen to minimize average parasitic scattering losses in the case of SPP transmission through a dielectric block without antireflection layers. Block length was varied from 0 to 1 ␮m. For geometry A, εar = ε1 = 1.452 , and for geometry B, εar = 1.32 (the latter was chosen to provide a nearly optimal value of the effective refractive index of the plasmonic mode in the antireflection layer as described below). Antireflection layer lengths lar were calculated from the following expression: lar =

λ

, 4neff,ar

(1)

where neff,ar = Re{neff,ar } is the real part of the effective refractive index (propagation constant normalized by the wave number) of the plasmonic mode in the antireflection layer. Eq. (1) is identical to the expression defining the thickness of conventional antireflection coatings. For geometry A neff,ar = 1.5175 + 0.0046i, and for geometry B neff,ar = 1.3811 + 0.0044i. In the latter case, the neff,ar value is close to the optimal value neff,ar,opt = 1.3642 which can be found from the expression:  neff,ar,opt = neff,inc neff,block , (2) where neff,inc and neff,block are the real parts of the effective refractive indices of the incident SPP (or plasmonic mode) and the plasmonic mode propagating inside the block, respectively. Eq. (2) is also identical to the equation defining the optimal refractive index of a conventional antireflection coating providing minimal reflection possible. The values of lar found from Eq. (1) are equal to 132 nm and 145 nm for geometries A and B, respectively. Let us note that since the studied reflection and scattering suppression effects are not resonant, the used parameters are not specific and reflection and scattering suppression similar to the results presented below can be

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Fig. 2. Transmittance (T, solid lines) and reflectance (R, dashed lines) of SPP ((a), geometry A) or plasmonic mode ((b), geometry B) vs. dielectric block length without antireflection layers (blue lines) or with one (green lines) or two (red lines) antireflection layers. The case without scattering suppression and antireflection layers (black lines) is shown for comparison. (For interpretation of the references to color in text, the reader is referred to the web version of this article.)

achieved for different wavelengths and combinations of the materials of the structure provided that the thickness h1 and length lar are chosen properly. Fig. 2(a) shows the calculated SPP transmittance T and reflectance R vs. the block length for geometry A. The cases of zero, one or two antireflection layers were considered. SPP transmittance and reflectance values for the case of a one-layer dielectric block (h1 = 0, no scattering suppression) without antireflection layers are also shown for comparison. The simulation was carried out using our in-house implementation of the rigorous coupled-wave analysis method (RCWA) extended to aperiodic diffraction problems [17]. Average values of SPP transmittance (T ) and reflectance (R) are shown in Table 1. According to Table 1, the average SPP transmittance increases by 0.05 when two antireflection layers Table 1 Average SPP transmittance (T ) and reflectance (R) values for geometry A. Number of antireflection layers

h1 (nm)

T

R

0 0 1 2

0 62 62 62

0.68 0.81 0.83 0.86

0.09 0.09 0.06 0.02

Table 2 Average plasmonic mode transmittance (T ) and reflectance (R) values for geometry B. Number of antireflection layers

h1 (nm)

T

R

0 0 1 2

0 38 38 38

0.68 0.87 0.90 0.93

0.09 0.07 0.03 4 × 10−5

are added to the structure with the scattering suppression (maximum transmittance enhancement in this case amounts to 0.15). At the same time, average reflectance is reduced by 4.5 times. The average transmittance enhancement that can be obtained by simultaneous addition of scattering suppression and antireflection layers to a one-layer dielectric block is 0.18 (maximal value: 0.37). Fig. 2(b) shows the transmittance and reflectance of the plasmonic mode vs. the block length for geometry B. The corresponding average values are given in Table 2. The average transmittance increase in this case equals 0.06 (maximum transmittance enhancement amounts to 0.13), the average reflectance being decreased by three orders of magnitude (by more than 1500 times). Such a decrease in reflectance is achieved by the proper choice of the material εar which provides the effective refractive index of the plasmonic mode in the antireflection layers which is close to the optimal value defined by Eq. (2). The average increase in transmittance as compared to a one-layer dielectric block is 0.18, and the maximum increase is 0.39. Let us note that without antireflection layers and scattering suppression (h1 = 0) geometries A and B become the same and correspond to a one-layer dielectric block on the metal surface. Hence, black lines in Fig. 2(a) and (b) are identical. 3. Plasmonic element example: a binary microlens array Let us now consider as an example a periodic binary microlens array for focusing SPP (periodicity direction being the x axis) in geometry A with the following parameters: period d = 8λ = 6.4 ␮m, focal length f = 0.45d = 2.88 ␮m. The SPP is propagating along the z axis and is normally incident on the lens array. The zone

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Fig. 3. |Ey |2 distributions 10 nm above the metal surface within one period of the binary microlens array for SPP with scattering suppression: without antireflection layers (a) and with two antireflection layers (b). Lens ridges are shown with solid white lines.

boundaries along the x axis can be found by quantizing the eikonal function of the lens  Ψ = f − f 2 + x2 (3) into 2 levels (0 and λspp /2, where λspp = λ/neff,inc ). Assuming that the middle point of the period is the point x = 0, the following coordinate values that determine the location of the binary lens ridges can be obtained from Eq. (3):   jλspp xj = + f − f 2 , j = ±1, ±2, . . . (4) 2 Maximum j value can be found from the condition xj ≤ d/2. The length of the lens ridges in the SPP propagation direction l can be derived from the following equation: l(neff,block − neff,inc ) + mlar (neff,ar − neff,inc ) =

λspp , 2 (5)

where m is the number of the antireflection layers in the structure (0, 1 or 2). Eq. (5) ensures that the optical path difference between SPP transmitted through adjacent lens zones is equal to λspp /2. Taking into account Eq. (1), we obtain from Eq. (5):   m(neff,ar − neff,inc ) λspp l= 1− . (6) 2(neff,block − neff,inc ) 2neff,ar According to Eq. (6), for the considered example l = 572 nm at m = 0, l = 479 nm at m = 1, and l = 385 nm at m = 2. Fig. 3 shows the calculated distributions of the intensity of the main SPP electric field component I = |Ey |2 10 nm above the metal surface within one period of the microlens array at m = 0 (a) and at m = 2 (b). For comparison, similar distribution for the case of m = 0, h1 = 0 (no SPP scattering suppression) is shown in Fig. 4. For

Fig. 4. |Ey |2 distribution 10 nm above the metal surface within one period of the binary microlens array for SPP without scattering suppression and antireflection layers. Lens ridges are shown with solid white lines.

the structures with parasitic scattering suppression, maximum focal intensity Imax is 8.0 at m = 0 (Fig. 3(a)), 8.5 at m = 1, and 9.1 at m = 2 (Fig. 3(b)). Thus, the addition of antireflection layers to the structure with scattering suppression leads to the increase in the focal intensity by 14%. Without scattering suppression (at h1 = 0) and antireflection layers, Imax = 6.8 (Fig. 4). As compared to this structure, the focal intensity in Fig. 3(b) is increased by more than 30%. In the authors’ previous work [14], it was shown that the SPP refraction in dielectric structures with parasitic scattering suppression is quantitatively well described by conventional Fresnel equations for TE-polarized plane wave. As the refractive indices of the materials in the plane wave model, the effective refractive indices of the plasmonic modes in the corresponding regions are used. Thus, let us compare the SPP and plane wave diffraction on a binary microlens array. Fig. 5 shows the |Ey |2 distribution within one period of a microlens array for TE-polarized plane wave. Structure parameters correspond to the plasmonic microlens array shown in

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Acknowledgement This work was funded by Russian Science Foundation grant 14-19-00796. References

Fig. 5. |Ey |2 distribution within one period of a binary microlens array for TE-polarized plane wave corresponding to the plasmonic structure shown in Fig. 3(b). Lens ridges are shown with solid white lines.

Fig. 3(b). The distributions in Figs. 3(b) and 5 have the same structure, the focal intensity in Fig. 5 amounts to 9.4. Let us note that in the plane-wave case the transition from m = 0 to m = 2 (i.e. the addition of two antireflection coatings) leads to the increase in the focal intensity by 14%, exactly as in the SPP case. Thus, the SPP diffraction on structures with scattering suppression is close to the diffraction of TE-polarized plane wave. 4. Conclusions In the present work, we have proposed and numerically studied antireflection layers for surface plasmon polaritons. It was shown that the reflection can be decreased simultaneously with parasitic scattering suppression of SPP. For the considered examples, the total SPP transmittance enhancement as compared to dielectric structures without scattering suppression and antireflection layers reaches 0.4. Average SPP reflectance can be decreased by three orders of magnitude. As an example, a binary microlens array for focusing SPP was considered. The addition of antireflection layers to the structure with scattering suppression led to an almost 15% increase in the focal intensity. As compared to the structure without antireflection layers and scattering suppression, the focal intensity was increased by more than 30%. The results may be extended to the case of Bloch surface waves and find application in the design of 2D optical elements for surface waves working in transmission.

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