Extension of Babinet's Principle for Plasmonic Metasurfaces

Extension of Babinet’s principle for plasmonic metasurfaces Cite as: Appl. Phys. Lett. 119, 161103 (2021); doi: 10.1063/5.0065724 Submitted: 4 August 2021 .Accepted: 30 September 2021 . Published Online: 19 October 2021 J. D. Ortiz, 1 J. P. del Risco, 2,3 J. D. Baena, 2,a) and R. Marqu es 4 AFFILIATIONS 1 Faculty of Engineering, Universidad de San Buenaventura, Bogot a 110141, Colombia 2 Department of Physics, Universidad Nacional de Colombia, Bogot a 111321, Colombia 3 Department of Mathematics, Universidad Sergio Arboleda, Bogot a 111711, Colombia 4 Department of Electronics and Electromagnetism, Universidad de Sevilla, Sevilla 41012, Spain a) Author to whom correspondence should be addressed: [email protected]du.co ABSTRACT Babinet’s principle is widely applied in optics and has been useful for designing metasurfaces with dual behavior. Although this principle can be rigorously demonstrated for infinitely thin perfect conducting screens, it is not exact for any real screen. In fact, metals used in plasmonic metasurfaces are far from good conductors, and the thickness of samples is not negligible in comparison with the typical size of the patterned structure. In this paper, we propose an extension of Babinet’s principle valid for plasmonic metasurfaces by redefining the concept of complementary screens and finding impedance relations between such screens that ultimately leads to a simple relation between the transmission matrices of two complementary plasmonic metasurfaces. The theory is valid under the assumptions of the electroquasistatic approximation and plane waves in the far field. It may find applications in the design of optical plasmonic metasurfaces, nanocircuits, and nanoantennas. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0065724 The related topics of optical plasmonic metasurfaces, 1,2 nanocircuits, 3–6 and nanoantennas 7 have attracted much attention during the last two decades. Complementarity is a widely used concept in the diffraction theory that might be useful for new developments in these research areas. It is closely related to Babinet’s principle, which establishes certain duality relations between the fields scattered by two complementary screens. 8,9 In fact, Babinet’s principle has already been applied for designing complementary metasurfaces at microwaves, 10 terahertz, 11 and optical frequencies. 12–18 In those previous works, it was demonstrated that the transmittance of one metasurface is like the reflectance of its complementary counterpart, and vice versa, except for some degree of deviation. Aside from that, the duality between electric and magnetic fields has been confirmed in the near field for some specific complementary plasmonic structures. 19,20 However, while Babinet’s principle can be rigorously demonstrated for infinitely thin perfect conducting screens, 8,9 it is not exact for optical plasmonic screens for which neither the thickness is negligible nor the metals are good conductors. Nevertheless, very recent works 21–23 suggest that Babinet’s principle is still qualitatively valid for plasmonic structures. Would it be possible to extend Babinet’s principle for plasmonic screens, maybe by introducing some correction, and placing its application to these structures on more solid grounds? Along this Letter, we will show that this extension is possible. Namely, our extension of Babinet’s principle will be valid for complementary plasmonic metasurfaces composed of materials with high values of the modulus of the permittivity, which real part could be negative for regions filled with some plasmonic metal while positive in other regions filled with high permittivity dielectrics. We will not only provide a theoretical formula representing this extension of Babinet’s principle but also compare its performance against the predictions based on conventional Babinet’s principle. We will show that the agreement between theory and numerical simulations using our extension is greatly improved for three different shapes. Before extending this principle, it is worth to summarize the conventional form of Babinet’s principle which, under certain considerations, will promptly drive us to a useful corollary linking the transmission coefficients of two complementary periodic screens whose period is subwavelength (as usual for metasurfaces). Let us start assuming two complementary thin conducting screens, one might be called the original one while the other the complementary one, both screens orthogonal to the z-axis, and located at z¼0.Underthiscontext, complementary means that both screens have been etched with Appl. Phys. Lett. 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-1 Published under an exclusive license by AIP Publishing Applied Physics Letters ARTICLE scitation.org/journal/apl 04 June 2025 14:00:05 the same pattern but interchanging conductor and air regions. Moreover,letthesourcebeplacedinthez<0 half-space and emitting a plane wave that impinges normally onto the screen. Let us call Einc and Binc the electric and magnetic incident fields for the original screen, while for the complementary screen the incident fields E0 inc and B0 inc will be linked to the former as follows: E0 inc ¼cBinc and cB0 inc ¼Einc,beingcthe speed of light in vacuum. Then Babinet’s principle says 8,9 Etra cB0 tra ¼Einc; cBtra þE0 tra ¼cBinc;(1) where Etra and Btra are the electric and magnetic fields transmitted through the original screen while E0 tra and B0 tra have the same meaning but for the complementary problem. Although this is usually called a principle, it is an exact theorem when screens are planar, infinitely thin, and made of perfect conductor. 8,9 Assuming the source is very far from the screen, then the incident wave can be approximated as a plane wave. In general, the transmitted field can be expanded as an infinity sum of plane waves. Whenever the unit cell is much smaller than the wavelength, as usual for metasurfaces, only a plane wave remains far from the screen. By definition, the incident and transmitted electric fields can be connected one another through Etra ¼ tEinc for the original screen and E0 tra ¼ t0E0 inc for the complementary screen, where  tand  t0are the so-called transmission matrices. Similarly, reflection matrices could be defined by Eref ¼ rEinc and E0 ref ¼ r0E0 inc. They are just 2 2 square matrices, because they are linking fields that are transverse to the z-axis and taking into account both co-polar and cross-polar effects (i.e., they are, in general, nondiagonal matrices). Furthermore, as far as waves are approximated by plane waves, the magnetic fields can easily be obtained from the electric field by using the well-known formula B¼ð1=kcÞkE,wherek is the wavevector and kis the corresponding wavenumber. Alternatively, when kis parallel to the z-axis, this formula can also be expressed as follows: cBinc cBtra cBref cB0 inc cB0 tra cB0 ref 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; ¼01 10 !  Einc Etra Eref E0 inc E0 tra E0 ref 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; ¼  R Einc Etra Eref E0 inc E0 tra E0 ref 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; ;(2) where   Rrepresents a 90counterclockwise rotation around the z-axis. From (1) and (2),thedefinitionsof tand  t0introduced above and with the 2 2 identity matrix  1, it is a straightforward task to demonstrate the following corollary:  tþ  R t0  R1¼  1:(3) It is worth noting that, although a similar result has been reported in many papers in the scalar form tþt0¼1, this matrix form is more general because it also accounts for cases presenting cross-polarization effects. In principle, the transmission through one screen could be guessed from the transmission of its complementary screen by using (3). However, let us remind that (3) is only valid for screens of negligible thickness and made of perfect conductor, which, clearly, is not the case for plasmonic metasurfaces. Then, we need to replace it with another equation suitable for complementary plasmonic metasurfaces. Now, let us consider the structure shown in Fig. 1(a), which is a two-dimensional (2D) piecewise homogeneous structure filled with several media with isotropic relative permittivities ei; so, the structure is characterized by the relative permittivity piecewise function eðx;yÞ. If all regions are filled with high permittivities then, due to the continuity of the normal component of eEon the boundary, the normal component of Ecan be neglected inside the metasurface, and thus, Eðx;yÞ¼Exðx;yÞ^ xþEyðx;yÞ^ y. Considering the unit cell is much smaller than the wavelength and its thickness is also smaller than the skin depth of plasmonic regions, it is also possible to use the electroquasistatic (EQS) approximation: 24,25 the magnetic induction in Faraday’s law can be neglected for low frequencies while the displacement current density in the Ampe`re–Maxwell’s law is still kept because the low frequencies are compensated with the high values of permittivity. In this way, the problem is simplified to a 2D problem whose electric field satisfies $tðeEÞ¼0 and $tE¼0, where $t¼^ x@=@xþ^ y@=@y. In order to define the complementary problem, let us replace the permittivity eof the original structure with a new complementary permittivity, as shown in Fig. 1(b), defined by e0ðx;yÞ¼ C1 eðx;yÞ;(4) where C 1 is an arbitrary constant. It is worth noting that this definition of “complementarity” includes the conventional one as a particular case: when only two materials are used this constant can be chosen as C1¼e1e2, which means just an interchange of materials. Under this transformation, the fields inside the complementary structure can be obtained from those of the original problem as follows: E0¼C2e^ zE¼C2e   RE;(5) where C 2 is an arbitrary constant associated with the freedom of scaling solutions from linear equations. It can be straightforwardly demonstrated that $tðe0E0Þ¼0and$tE0¼0, by just replacing (4) and (5).Thisresultincludesasaparticularcasetheresultspreviously reported in Refs. 26–28 on the effective conductivity of 2D two phase composites, which can be obtained by simply replacing the relative permittivity eby ir=ðe0xÞ,whereris the spatial distribution of conductivity. FIG. 1. Illustration of the unit cells of two complementary metasurfaces: the “original” one (a) and the “complementary” one (b). Complementary permittivities and electric fields are related to Eqs. (4) and (5), respectively. Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-2 Published under an exclusive license by AIP Publishing 04 June 2025 14:00:05 In order to describe the response of the structure from a macroscopic point of view, let us define the average electric field Eave as well as the surface current density (coming from the displacement current) Jsinside the unit cell (u.c.) through the integrals Eave ¼1 Aððu:c: Eðx;yÞdx dy;(6) Js¼ixe0h Aððu:c: eðx;yÞEðx;yÞdx dy;(7) where Ais the area of the unit cell, his the thickness of the sample, and harmonic variation with time eixthas been assumed. We can now define the surface impedance of the structure,   Zs,asthematrix that relates the vectors in (6) and (7) in the following way: Eave ¼Eave;x Eave;y  ¼Zs;xx Zs;xy Zs;yx Zs;yy  Js;x Js;y  ¼  ZsJs:(8) By applying identical definitions (6) and (7) in the complementary metasurface, and using (4) and (5), it is readily shown that the complementary average electric field E0 ave and the complementary surface current density J0 sare related to the original ones as E0 ave ¼ C2 ixe0h   RJs;(9) J0 s¼ixe0hC 1C2   REave:(10) Yet, E0 ave and J0 sare related by the complementary surface impedance   Z0 sas: E0 ave ¼  Z0 sJ0 s, then, when we put together this definition with (8)–(10), we directly relate both surface impedances. With the vacuum wave number k¼xffiffiffiffiffiffiffiffiffi e0l0 pand the vacuum impedance Z0¼ffiffiffiffiffiffiffiffiffiffiffi l0=e0 p, the final result is   Z0 s  R  Zs  R1¼Z2 0 4K   1;(11) where K¼h2k2C1=4:(12) It recalls a previous theorem reported in Refs. 26–28 for the effective conductivity of 2D lattices of conductive cylinders, which can be deduced from (11) as a particular case by using C1¼e1e2and eiri=ðixe0Þ. It is also closely related to the property ZZ 0¼Z2 0=4 reported by Booker 29 and Deschamps 30 for perfectly conducting complementary screens, which can be obtained from (11) when K¼1, and the off diagonal components of the surface impedance matrix are zero. Now, let us use standard boundary conditions at the metasurface to solve the transmission matrix from the surface impedance matrix. First, by imposing the continuity of the tangential component of the electric field then Einc þEref ¼Eave ¼Etra,whichisequivalentto   1þ r¼ t. Second, due to the found surface electric current, the magnetic field must satisfy the following discontinuity relation: ^ zðBinc þBref BtraÞ¼l0Js, which is equivalent to   1  r t¼Z0   Z1 s t. In this way, we have got a system of two linear equations with two unknown matrices,   rand  t, that can be easily solved to get the transmission matrix  t¼  1þZ0 2   Z1 s  1 :(13) Reciprocally, the surface impedance is   Zs¼ðZ0=2Þ½ t1  11.Since it is an arbitrary choice what is the original or complementary screen, it is clear that we can write a relation analogous to (13) for  t0and   Z0 s. Therefore, by using (11), we can establish a direct relation between both transmission matrices, which is  t0¼  R1  1 t ½   1ð1KÞ t  1  R:(14) It reproduces the corollary previously obtained from conventional Babinet’s principle, expressed in (3),onlywhenK¼1(seethesupplementary material for examples with K1) but, in general, K6¼ 1 and frequency dependent as indicated in (12). Therefore, Eq. (14) can be considered as an extension of Babinet’s principle for complementary plasmonic metasurfaces. It is worth to remind, however, that this formula was obtained under the assumption of the EQS limit, which is valid for screens with subwavelength periodicity, low variation of fields along the z-axis inside the metasurfaces, and high permittivities. Since these constrains are usually fulfilled by plasmonic metasurfaces operating at optical frequencies, this theory is expected to be useful for the analysis of these structures. To validate our theory, three couples of complementary plasmonic metasurfaces using different shapes will be numerically simulated. We will focus our attention in the infrared range of frequencies. For all cases, we will use silicon (Si) and silver (Ag) not only because they are extensively used in plasmonics but also because they both have high permittivity in the infrared range and, at the same time, they provide a big contrast of permittivity, which might be desirable for designing complementary structures. The relative permittivity of Si is almost constant within the considered frequency range, with a negligible imaginary part, and can be approximated as eSi 11:9. 31–33 On the other hand, the permittivity of Ag depends strongly on the frequency with a Drude–Lorentz behavior described by eAg ¼1x2 p= ðx2þixcÞ, where the plasma frequency is xp¼1:375 1016 rad/s and the collision frequency is c¼3:12 1013 s1; these parameters have been obtained from Ref. 34. As a first example, we will analyze the complementary screens shown in Figs. 2(a) and 2(b),whichare made of alternating Si and Ag bars. Periodic boundary conditions are used at the edges of the structures. Arbitrarily, (a) may correspond with the original problem while (b) with the complementary one. Following the same arguments of Ref. 6,fory-polarized (x-polarized) incident waves, it can be interpreted as a planar nanocircuit with series (parallel) connections of nanoinductors representing the Ag regions (negative permittivity below the plasma frequency) and nanocapacitors representing the Si regions (positive permittivity). By passing from the original to the complementary structure, each region switches from the nanoinductor to the nanocapacitor and vice versa. The complementary problem is defined not only by the interchange of materials but also by the 90rotation of the polarization state of the incident wave, which means that series (parallel) connections switch to a parallel (series) connection in the complementary problem. Under these assumptions, it can be shown that (11) is satisfied. In fact, we have numerically computed the transmission coefficients through both screens by using the commercial software CST Microwave Studio from 50 to 300 THz or, equivalently, from 1 to 6lm. The plots of Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-3 Published under an exclusive license by AIP Publishing 04 June 2025 14:00:05 Fig. 2 show simulated transmission coefficients (solid lines) as well as the curves derived from (3) (dashed lines) and (14) (dotted lines) by using the numerical result of the complementary screen. A very good agreement between our theory, i.e., (14), and the electromagnetic simulations has been found while the results coming from conventional Babinet’s principle, i.e., (3), show a significant deviation from the simulated results. Although Fig. 2(a) shows a big deviation in the phase above 250 THz, it is not actually relevant if we consider that the corresponding magnitude is near zero. (The phase is indefinite at zero.) It is also worth noting that, due to the mirror symmetries of the structure respect to the x-plane and y-plane, the off diagonal elements of the transmission matrices cancel out and then only the diagonal components are depicted in Fig. 2. The response of the former structure was poorly dispersive and dissipative. To check our theory for structures presenting strong dispersion and dissipation, which is the common situation for metasurfaces as well as metamaterials based on resonating particles, the transmission coefficients through a screen made of split rings and its complementary screen have been computed in the range from 15 to 65 THz. Their unit cells and their geometrical parameters are fully specified in Fig. 3. For (a), a split ring of Ag is embedded in a layer of Si while materials appear interchanged in (b). The plots of Fig. 3 show the simulated transmission coefficients (solid lines) as well as the curves derived from (3) (dashed lines) and (14) (dotted lines). It can be appreciated how the curves obtained with our theory by using (14) agree with the simulations much better than (3), which comes from conventional Babinet’s principle. Once again, we can argue that off diagonal components of  t, i.e., t xy and t yx , are canceled out because of the mirror symmetry respect to the y-plane. As a last example, in order to demonstrate the validity of our theory for cases that present cross-polarization effects, we decided to rotate the previous split rings by 45as shown at the top of Fig. 4.This figure also shows the simulated results (solid lines) as well as the curves derived from (3) (dashed lines) and (14) (dotted lines) within the range between 15 and 65 THz. The diagonal components of the transmission matrix t xx and t yy exactly match each other because of the mirror symmetry respect to the diagonal direction. However, neither the x-plane nor the y-plane are mirror planes, thus, in this case, a nonzero response is observed for the off diagonal components t xy and t yx which, by the way, are also equal one each other. As in the previous FIG. 3. Transmission coefficients (magnitude and phase) through two different split ring metasurfaces made of silver (Ag) and silicon (Si). The geometrical parameters are a¼250 nm, r0¼100 nm, g¼10 nm, w¼30 nm, and h¼25 nm. Plots show simulated results (solid lines) as well as curves derived from (3) (dashed lines) and (14) (dotted lines). FIG. 4. Transmission coefficients (magnitude and phase) through two different split ring metasurfaces made of silver (Ag) and silicon (Si), where the split rings are oriented at 45respect to the lattice axes. The geometrical parameters are a¼250 nm, r0¼100 nm, g¼10 nm, w¼30 nm, and h¼25 nm. Plots show simulated results (solid lines) as well as curves derived from (3) (dashed lines) and (14) (dotted lines). FIG. 2. Transmission coefficients (magnitude and phase) through two different parallel strip gratings made of silver (Ag) and silicon (Si). Their unit cells are shown on top where the geometrical parameters are w1¼50 nm, w2¼10 nm, and h¼25 nm. Plots show simulated results (solid lines) as well as curves derived from (3) (dashed lines) and (14) (dotted lines). Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-4 Published under an exclusive license by AIP Publishing 04 June 2025 14:00:05 examples, our theory summarized in (14) approaches the simulations better than (3) (conventional Babinet’s principle). In conclusion, we have extensively analyzed the concept of complementarity and the applicability of Babinet’s principle to plasmonic metasurfaces. The definition of complementary screens is generalized in such a way that the conventional definition, involving only the interchange of two materials, is contained as a particular case. This analysis leads to an extension of Babinet’s principle summarized in (14). We numerically demonstrated that our theory represents a substantial improvement for three different geometries. Aside from that, we feel that it is quite general and can be applied to other complementary plasmonic metasurfaces operating in the optical range. Furthermore, the relation between complementary impedances shown in (11) may find applications for designing complementary nanocircuits and nanoantennas operating in the infrared range. This will be explored in an upcoming work. See the supplementary material for information related to the chance of recovering the validity of conventional Babinet’s principle or formula (3) by choosing a suitable thickness for which K1ina wide range of frequencies. This work has been supported by the Ministerio de Ciencia e Innovaci on with EU FEDER Funds (Project No. TEC2017-84724P), by the Colombian Government through COLCIENCIAS (Project No. 1101-521-29389), and the Universidad de San Buenaventura (Project No. CBI013-012-2018). 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