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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3279–3286
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Optical isolation with epsilon-near-zero metamaterials

Arthur R. Davoyan, Ahmed M. Mahmoud, and Nader Engheta  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3279-3286 (2013)
http://dx.doi.org/10.1364/OE.21.003279


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Abstract

We suggest a principle for isolation of circularly polarized waves in magnetically active extreme-parameter metamaterials. Using theoretical analysis and numerical simulations, we show that metamaterials with extreme parameters, such as epsilon-near-zero materials (ENZ), when merged with magneto-optical materials, become transparent for forward circularly polarized waves of a given handedness and opaque for backward propagating waves of the same handedness. We theoretically study two possible implementations of such hybrid materials: (1) the case of metal-dielectric stacks; and (2) rectangular waveguide near its cut-off frequency. We prove that these structures can be utilized as compact isolators for circularly polarized waves.

© 2013 OSA

1. Introduction

Many objects in nature ranging from molecules and biological species to astrophysical bodies exhibit different sensitivity to left- and right-handed circular polarizations of light [1

1. W. B. Sparks, J. Hough, T. A. Germer, F. Chen, S. DasSarma, P. DasSarma, F. T. Robb, N. Manset, L. Kolokolova, N. Reid, F. D. Macchetto, and W. Martin, “Detection of circular polarization in light scattered from photosynthetic microbes,” PNAS 106, 7816–7821 (2009). [CrossRef] [PubMed]

3

3. L. Kolokolova, W. Sparksm, and D. Mackowski, “Astrobiological remote sensing with circular polarization” in Polarimetric Detection, Characterisation and Remote Sensing, M. I. Mishchenko, Y. S. Yatskiv, V. K. Rosenbush, and G. Videen, eds. (Springer, Netherlands, 2011) pp. 277–294. [CrossRef]

]. Therefore the ability to filter and isolate circular polarizations is of practical importance for science and engineering. Progress with chiral heterostructures [4

4. M. Decker, M. W. Klein, M. Wegener, and S. Linden, “Circular dichroism of planar chiral magnetic metamaterials,” Opt. Lett. 32, 856–858 (2007). [CrossRef] [PubMed]

8

8. M. Schferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: Design principles for chiral plasmonic nanostructures,” Phys. Rev. X 2, 031010 (2012). [CrossRef]

], paved the way for the design and engineering of ultra-compact composite structures distinguishing left- and right-handed polarization states. Nevertheless, any chiral metastructure satisfying the Lorentz reciprocity theorem [9

9. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, Second Edition: Volume 8 (Course of Theoretical Physics) (Butterworth-Heinemann, 1984).

] is reciprocal, implying that interaction of the backward and forward propagation of the same circularly polarized states with such structures are identical and can not be distinguished. Imagine, for example, a right-handed circularly polarized (RCP) light incident normally in +z (forward) direction on a slab of chiral material. Its transmission and reflection may differ from left-handed circularly polarized (LCP) light incident from the same direction. At the same time due to reciprocity of the system RCP light propagating backward (in −z direction) experiences exactly the same transmission and reflection as forward propagating RCP, see schematic in Fig. 1(a). Hence, clearly the isolation and separation of forward and backward waves is not possible in such structures. It is worth mentioning here the works on asymmetric transmission with some types of chiral heterostructures [10

10. V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97, 167401 (2006). [CrossRef] [PubMed]

12

12. C. Menzel, C. Helgert, C. Rockstuhl, E.-B. Kley, A. Tunnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Phys. Rev. Lett. 104, 253902 (2010). [CrossRef] [PubMed]

], however these structures still satisfy the Lorentz reciprocity conditions and can not be considered as truly nonreciprocal systems with asymmetric transmission.

Fig. 1 Schematic of RCP forward and backward transmission through (a) reciprocal, (b) magneto-optical, and (c) magnetically active epsilon-near-zero slabs, respectively. (d) Transmittance of the forward and backward RCP plane waves through ENZ-MO slab (ε = 0 and α = 0.1) as a function of the slab thickness.

In this paper, we suggest a design approach for optical isolation for circularly polarized waves by utilizing both the epsilon-near-zero materials or structures and the magneto-optical materials. We show theoretically that magnetically active ENZ media is transparent for one of the circularly polarized waves, whereas it is opaque for another polarization handedness. We propose two realistic examples where such a scenario is feasible. Specifically, we study the case of a stack of alternating metal-dielectric slabs, and the case of a waveguide filled with magneto-optical media operating near its cut-off frequency. We demonstrate numerically that the isolation of circularly polarized states is possible in both of these example structures.

The paper is organized as follows. In Section II we give a generalized concept for designing the ENZ-MO isolator for circularly polarized waves. We discuss two particular examples of such isolator in Section III. In particular, we consider a metal-dielectric stack and a waveguide near its cut-off, and show that both structures exhibit effective ENZ properties combined with gyrotropy allowing isolation of one of the circularly polarized states when magneto-optical activity is present. Section IV provides the conclusion.

2. Heuristic approach for the ENZ isolator deign

We begin our analysis with a brief insight into the properties of a conventional magneto-optical materials. Consider, for example, a slab of magneto-optical medium with a magnetization direction normal to the slab interface, see Fig. 1(b). In this case the MO relative permittivity can be described by an antisymmetric tensor [9

9. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, Second Edition: Volume 8 (Course of Theoretical Physics) (Butterworth-Heinemann, 1984).

]:
ε¯¯=(εmoiα0iαεmo000ε),
(1)
where εmo and ε are diagonal components of relative permittivity tensor, α > 0 is the off-diagonal component of the permittivity tensor responsible for the “strength” of magneto-optical activity of the medium. At optical frequencies, the parameter α is often very small and usually of the order of 10−2 or even smaller.

For plane waves propagating along the magnetization direction (i.e. in Faraday configuration) the eigenmodes in magneto-optical media are two circularly polarized waves: Ex ± iEy. Solving Maxwell’s equations for the circularly polarized waves propagating in the magneto-optical media in forward direction we get a well known relation for the corresponding wavevectors [9

9. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, Second Edition: Volume 8 (Course of Theoretical Physics) (Butterworth-Heinemann, 1984).

]: kLCP+=(ω/c)εmoα for left-handed circular polarized waves and kRCP+=(ω/c)εmo+α for right-handed circular polarized waves. To account for the backward propagating waves the time reversal operation (t → −t) needs to be performed. In this case the change of sign in front of α has to be accounted for in the wave dispersions: kLCP=(ω/c)εmo+α and kRCP=(ω/c)εmoα for backward LCP and RCP waves, respectively. Clearly, backward and forward propagating circular polarized waves of the same handedness have different wave numbers with differing dispersion (|k+| ≠ |k|) yielding difference in their transmittance and reflectance [9

9. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, Second Edition: Volume 8 (Course of Theoretical Physics) (Butterworth-Heinemann, 1984).

].

Exploiting the metamaterial concept, one can engineer metamaterials and structures in order to deliberately tailor the value of permittivity function [30

30. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nature Photon. 5, 523–530 (2011).

]. In Refs. [36

36. B. Edwards, A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev, Lett. 100, 033903–4 (2008). [CrossRef]

38

38. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–5 (2006). [CrossRef]

] it was shown that in properly designed structures, one can achieve ε → 0, and consequently very small phase progression (k → 0). Examples of such structures include waveguides near their cut-off frequency for the TE modes [36

36. B. Edwards, A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev, Lett. 100, 033903–4 (2008). [CrossRef]

], layered metal-dielectric stacks [38

38. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–5 (2006). [CrossRef]

] and natural materials at frequencies for which the real part of their permittivity is near zero (while its imaginary par is small) [39

39. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 43704379 (1972). [CrossRef]

]. By integrating such epsilon-near zero structures with magneto-optical materials it may be possible to design effective magneto-active media with nulled diagonal components in the permittivity tensor, i.e. εmo → 0 in Eq.(1). Assuming now that such a case is possible, we obtain following dispersion relations for circularly polarized wave eigenmodes:
forwardkLCP+=(ω/c)αandkRCP+=(ω/c)αbackwardkLCP=(ω/c)αandkRCP=(ω/c)α.
(2)

Assuming a lossless condition, as evident from Eq.(2), for α > 0 the forward propagating LCP waves and backward propagating RCP waves have imaginary wavevectors in such a medium, implying that the medium is opaque for these waves, while for the two other eigen-modes, the wave numbers are real, and thus the medium is transparent to these waves. Consequently, it would be possible to achieve two effects simultaneously: 1) isolate forward and backward propagating waves with the same handedness (see schematic in Fig. 1(c)), and 2) isolate LCP from RCP waves propagating in the same direction.

In Fig. 1(d) we plot the transmittance of the RCP wave in forward and backward directions through a slab made of such material with the variation of slab thickness. To be more specific we consider here the operating wavelength to be λ = 1.5μm, ambient medium is air and α = 0.1. Note that, α is usually smaller than this value, for example, for bismuth iron garnet, a commonly used magneto-optical material, α = 0.06 at λ = 1.55μm. However, here we intentionally consider slightly higher values of magneto-optical activity in order to stress and highlight the observed effects. We see that the transmission of backward RCP wave exponentially decays as the slab thickness increases, and for the 2μm thick slab it already falls below 1%. At the same time the slab acts as a Fabry-Perot resonator for the forward propagating RCP with a first transmission resonance at ∼ 2.3μm thick slab. Analogously the forward propagating LCP would be suppressed similarly to backward RCP. Hence, engineering a magnetically active ENZ slab would allow to achieve isolation of circularly polarized waves with a slab thickness comparable with the incident wavelength. In the following section we propose two possible realizations of such magneto-active ENZ isolators.

3. Design of ENZ isolator

In order to engineer the magneto-active ENZ media we follow the two methods developed in the area of ENZ metamaterials. In the first approach we consider a metal-dielectric stack [38

38. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–5 (2006). [CrossRef]

], shown schematically in Fig. 2(a). For the second method, a rectangular waveguide operating at the cutoff frequency of its TE mode is utilized [36

36. B. Edwards, A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev, Lett. 100, 033903–4 (2008). [CrossRef]

], see Fig. 2(b).

Fig. 2 Schematic of the magneto-active structures exhibiting effective epsilon-near-zero permittivity: (a) periodic metal-MO stack, and (b) rectangular metallic (PEC) waveguide with a square cross-section. Red arrow shows the direction of magnetization.

Metal-dielectric stack

We consider a stack of periodically alternating metal and magneto-optical slabs with thicknesses a and b, respectively (Fig. 2(a)). When the thickness of the slabs is much smaller than the wavelength of operation the effective-medium approach may be used [38

38. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74, 075103–5 (2006). [CrossRef]

]. Using the Cartesian coordinate systems shown in Fig. 2, the x and y components of the effective medium permittivity for an unbiased system, i.e. when α = 0, are given by: εxx = εyy = (met + MO)/(a + b). By properly choosing these parameters, we can achieve near-zero permittivity for εxx and εyy. Then according to the predictions made in Section II we expect that such a stack, when magnetically biased, behaves as an isolator for circularly polarized waves. To prove our predictions we perform full-wave numerical simulations with the CST Microwave StudioTM[40

40. CST Microwave Studio at www.cst.com

].

For the purpose of numerical demonstration here we consider the magneto-optical layers to be made of bismuth iron garnet with εMO = 6.25 and α = 0.1, and the metal to be lossless with relative permittivity ε = −12.5. In this case choosing a = (1/2)b we obtain zero effective epsilon for εxx and εyy. A plane wave with free space wavelength of 1.5μm is incident normally on a stack containing 64 unit cells. We choose the thickness of the metal layer to be a = 20nm and that of magneto-optical slab to be b = 40nm, so that the overall stack thickness of the entire structure is 3.84μm. Such choice of slab thicknesses allows the applicability of the effective medium approximation, i.e. a + bλ.

Figures 3(a) and 3(b) show instantaneous distributions of the electric field vector for the RCP and LCP forward propagating plane waves at the specific time instant (ωt = 0). We observe that RCP wave propagates through the stack and preserves its amplitude, Fig. 3(a), whereas LCP is almost completely reflected and only small fraction of it is tunneled trough the stack, Fig. 3(b). Note that, the incident LCP and its reflection from the stack form a standing circularly polarized wave. Instantaneously it looks like linearly polarized wave, however the polarization plane rotates with time (not shown here). To analyze the degree of polarization we plot the front view of the electric field vector, i.e. its projection on the plane transverse to the direction of propagation (xy plane) at the same specific time instant (ωt = 0), Figs. 3(c) and 3(d). Clearly, RCP preserves its polarization handedness with propagation, i.e. stays circular after the transmission through the stack, Fig. 3(c). At the same time reflected LCP, as was noted, forms an instantaneous linearly polarized state (which is rotating with time), Fig. 3(d). We note that the backward propagating RCP behaves similarly to the forward LCP, i.e., in particular it is reflected from the stack, indicating that the stack acts as an isolator for circularly polarized waves.

Fig. 3 (a) ( Media 1) and (b) ( Media 2) instantaneous electric field distribution for forward propagating RCP and LCP, respectively. (c) and (d) front views of the electric field for forward RCP and LCP, i.e. projections of the fields depicted in panels (a) and (b) on the (xy) plane, correspondingly.

Waveguide at cut-off frequency

It is known that a metallic rectangular waveguide with the perfectly conducting walls, near its cut-off frequency for TE10 mode effectively behaves as an ENZ strcture [36

36. B. Edwards, A. Alu, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev, Lett. 100, 033903–4 (2008). [CrossRef]

]. We show that such a waveguide when filled with a magneto-optical material may also act as an ENZ isolator.

We consider a rectangular waveguide with a square cross-section (10μm×10μm) filled with bismuth iron garnet (εmo = 6.25 and α = 0.1), see Fig. 2(b). Note that the direction of magnetization in this case is along the waveguide, i.e. along the direction of mode propagation. The square cross section of the waveguide leads to the degeneracy of the lower order TE modes (TE01 and T E10). For the given waveguide dimensions both modes exhibit cut-off at 6THz (corresponding wavelength of 50μm).

When magneto-optical activity is introduced these degenerate states couple and form two new conjugate eigen-states ψ1 ± 2. Note that such coupling is possible only between orthogonal states existing simultaneously in the unbiased waveguide, for example, a single mode rectangular waveguide does not allow any rotating states. These new conjugate modes can be considered as analogues of LCP and RCP plane waves as the eigen-modes of the bulk media. Using this analogy we define these states as left-rotating and right-rotating eigen-modes respectively. The dispersion relations defining the eigenvalues of right and left rotating states are given as: k±=(ω2/c2)(ε±α)π2/a2, where a is the waveguide’s edge dimension. Hence near the cut-off frequency, when (ω/c)ε=π/a, we arrive to the conditions similar to that discussed in Section II. In particular, we see that near the cut-off frequency, one of the rotating states is not propagating, whereas the same rotating state propagates in the reverse direction.

To demonstrate this property of electromagnetic wave isolation inside the rectangular waveguide filled with the MO material, we perform a series of full-wave numerical calculations. In our simulations we assume that the waveguide walls are made of perfect electric conductor (PEC). We excite the waveguide with TE10 mode and study the corresponding field distribution and transmission characteristics. For magneto-optically active case the TE10 is decomposed into the counter-rotating eigen-modes supported by magnetized waveguide. Above the cut-off frequency both counter-rotating states exist, and due to the difference between their wavenumbers the phase between the TE10 and TE01 is accumulated with propagation along the waveguide, similarly to the Faraday effect in bulk media [9

9. L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, Electrodynamics of Continuous Media, Second Edition: Volume 8 (Course of Theoretical Physics) (Butterworth-Heinemann, 1984).

]. Below cut-off only one of the counter-rotating states exists, hence TE10 launched into the waveguide would generate this rotating state.

We trace the time variation of electric field at several different positions along the waveguide, i.e. along the central line, see Figs. 4(a) and 4(b). Fig. 4(a) shows the electric field variation above cut-off at 6.1 THz. We observe that in this case the polarization is highly elliptical, almost linear. Furthermore, we notice rotation of the polarization axis as we move along the waveguide as predicted earlier. Below the cut-off at 5.98THz, see Fig. 4(b), we observe propagation of purely circularly polarized state, which clearly manifests the isolation of one of the rotating states. Finally, we note that the small deviation from the linear polarization on Fig. 4(a) and modulation of the amplitude on Fig. 4(b) is due to small reflection appearing due to the numerical mismatch at the input and output ports of the waveguide.

Fig. 4 Electric field distribution along the waveguide at different cross sections and different time frames: (a) above cut-off at 6.1THz ( Media 3), and (b) below cut-off at 5.98THz ( Media 4).

4. Conclusion

We showed that magnetically active epsilon-near-zero structures acts as compact isolator for circularly polarized waves. We have demonstrated two cases for such media: a metal-dielectric stack and a rectangular waveguide near the cut-off frequency. Using full-wave numerical simulation, we have shown that in both structures the complete suppression of one of the circularly polarized states is possible.

Acknowledgments

This work is supported in part by the US Air Force Office of Scientific Research (AFOSR) grant number FA9550-10-1-0408.

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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 43704379 (1972). [CrossRef]

40.

CST Microwave Studio at www.cst.com

OCIS Codes
(230.3240) Optical devices : Isolators
(260.5430) Physical optics : Polarization
(260.2065) Physical optics : Effective medium theory

ToC Category:
Metamaterials

History
Original Manuscript: November 16, 2012
Revised Manuscript: January 19, 2013
Manuscript Accepted: January 21, 2013
Published: February 1, 2013

Citation
Arthur R. Davoyan, Ahmed M. Mahmoud, and Nader Engheta, "Optical isolation with epsilon-near-zero metamaterials," Opt. Express 21, 3279-3286 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3279


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