Dissymmetric, periodically nanostructured metal films can show non-reciprocal transmission of polarized light, in apparent violation of the Lorentz reciprocity theorem. The wave vector dependence of the extraordinary optical transmission in gold films with square and oblique subwavelength hole arrays was examined for the full range of polarized light input states. In normal incidence, the oblique lattice, in contrast to square lattice, showed strong asymmetric, non-reciprocal transmission of circularly polarized light. By analyzing the polarization of the input and the output with a complete Mueller matrix polarimeter the mechanisms that permits asymmetric transmission while preserving the requirement of electromagnetic reciprocity is revealed: the coupling of the linear anisotropies induced by misaligned surface plasmons in the film. The square lattice also shows asymmetric transmission at non-normal incidence, whenever the plane of incidence does not coincide with a mirror line.
© 2014 Optical Society of America
The Lorentz reciprocity theorem (LRT)  is a cornerstone of the electromagnetic theory of non-magnetic media that may be formulated as follows: if two incident beams of equal intensity and equal polarization but opposite direction emerge from a medium, the same polarization components of the light must have equal intensities. Arrays of periodic, nanostructured, metallic thin films appear to violate LRT in their nonreciprocal transmission of left and right circularly polarized light (CPL) [2–7]. Such materials are quite distinct from those that exhibit natural optical activity, the parity odd, time even differential refractivity (circular birefringence, CB) or absorption (circular dichroism, CD) of left and right CPL because natural optical activity does not depend on the direction of the wave vector . We presumed that the LRT in thin films must likewise be preserved, but that the mechanism whereby this preservation is achieved is masked in an incomplete polarimetric analysis. CD spectropolarimeters typically modulate the input polarization state of light. Complete polarimeters that also analyze the transmitted light polarization are employed infrequently. Here, we have measured the complete polarimetric response of oblique- and square-lattice metal hole arrays that display extraordinary optical transmission  at certain frequencies mediated by the excitation of surface plasmons (SPs) [10,11]. We focus on the changes in the polarization of light upon transmission through the holes so as to assay the full 4 × 4 Mueller scattering matrix, M, of each sample . The reciprocity of optical transmission can be characterized by studying the symmetry properties of the scattering matrix that describes the light-matter interaction. Such a matrix contains all of the information on how the medium behaves upon time reversal (wave vector reversal) [13,14]. Only in this way can we reckon why LRT is preserved even though the intensity of transmitted light is strikingly wave vector dependent.
2. Experimental details
Gold films (200μm × 200 μm × 250 nm) on fused silica slides were perforated by a focused ion beam (RaithionLiNE 35keV gallium beam, current of 50pA) to make 250 nm diameter holes. Two types of lattices were generated: a square array with lattice parameter a = 530 nm, and a oblique lattice with a = 530 nm, b = 730 nm and φ = 65° (Fig. 1). The total array size was 200 × 200 µm2 composed of 50 × 50 µm2 quadrants, the limit on the field of view for writing. The square lattice is mirror symmetric while the oblique lattice is not.
The spectroscopic Ms for the oblique and square lattice samples were measured in transmission at normal incidence using a Mueller matrix polarimeter . This instrument uses four different photoelastic modulators (Hinds Instruments) operating at different frequencies, two in the polarization state generator and two in the polarization state analyzer. The fifteen elements of a normalized M are simultaneously measured without moving parts in the instrument. The light source is a 150W Xe Arc lamp and the detector is a photomultiplier tube. To minimize stray light, two synchronized monochromators were used, one after the light source and one before the detector. Ms were measured over the wavelength range 320-840 nm. Collimated light was used for illumination and the transmitted beam was analyzed in the far-field.
The experiments were repeated by turning the backside of the films toward the light source, thus effectively changing the sign of the wave vector. The turn over was done by rotating the sample 180° around the y axis of the instrument system of reference. In the first pass, light was incident on the gold layer (forward configuration) and in the second pass, light entered the sample through the glass substrate (backward configuration). Ms were normalized to their m00 element, the total transmitted intensity, which accounts for reflectivity differences in the two configurations.
2. Analysis of the asymmetric transmission
Figure 2 shows the Ms corresponding to the oblique array for the forward and backward configurations, MF and MB, respectively. Both matrices are related by a transformation of reciprocity [13, 14], which can be expressed asFig. 2 is responsible for this difference. However, if instead of considering to the total intensity of transmitted light we analyze the intensity of the radiation that has preserved the same incoming polarization (the left or right circular polarization, LCLP and RCPL respectively) it is found that forward and backward beams carry exactly the same intensity of CPL as shown in Fig. 3(a) and 3(b) .
In the forward configuration, at around 750 nm, the oblique lattice transmits more intensity when it is illuminated with LCPL than RCPL. In the backward configuration, the opposite pertains; the array is more emissive with RCPL illumination, though the emission does not necessarily emerge as RCPL. In agreement with the Lorentz reciprocity theorem, both configurations transmit the same intensity of CPL. In earlier works on non-reciprocal transmission in simple instruments that only modulate the input between LCPL and RCPL, had the investigators placed a circular polarizer in the transmitted beams they would have detected the same intensity of CPL in the forward and backward configurations. This result is counter-intuitive, but it can be described as a polarimetric nonreciprocity that preserves Lorentz reciprocity.
Asymmetric transmission is consequence of M being non-reciprocal. A M is non-reciprocal when it is non Minkowski-symmetric, i.e. when it does not satisfy all the following six relations: m01 = m10, m02 = -m20, m03 = m30, m12 = -m21, m13 = m31 and m23 = -m32. Clearly the forward and backward matrices of Fig. 2 are non-reciprocal because the conditions m03 = m30 and m12 = -m21 are not satisfied. Non-reciprocal Ms are quite usual in optical systems made of sequences of optical elements, because the matrix product in general is not commutative. A classic example of such non-reciprocal response is an optical system composed of a polarizer followed by a retarder that is not equivalent to its reciprocal, a retarder followed by polarizer. However, of interest here is a more subtle manifestation of non-reciprocity, originating in a single medium without need of considering a train of different optical elements. The polarimetric description of light propagation through this medium is well described by differential analysis of its M [17,18]: L = lnM, where L is the matrix logarithm of M, proportional to the differential Mueller matrix (m) accumulated over a pathlength d: L = dm.
The accumulated matrix L traditionally delivers CD and CB that are defined in terms of the matrix elements as follows: CD = (L03 + L30)/2; CB = (L12-L21)/2where Lij is the element of ith-row and jth-column of L. Other effects such as horizontal linear dichroism  and birefringence, LD and LB respectively, and 45° linear dichroism and birefringence, LD’ and LB’, are given by: LD = -(L01 + L10)/2, LB = (L32-L23)/2, LD’ = -(L02 + L20)/2 and LB’ = (L13-L31)/2 . These six optical effects are integrated in three complex sets of generalized anisotropies : horizontal linear anistropy L = LB-iLD, 45° linear anisotropy L’ = LB’-iLD’ and circular anistropy C = CB-iCD. A Mueller matrix is non-reciprocal whenever there is coexistence between two or more of these three sets, in a process where the combination of two sets transforms into an “apparent” (non-reciprocal) contribution of the remaining third. For example, any medium with coexistence of L and L’ will lead to “apparent”, non-reciprocal C. This is the origin of non-reciprocal transmission of CPL in the oblique array of nanoholes. Notably, this non-reciprocity affects M but not L. In L all the optical effects are uncoupled in different matrix elements.
The effects of the combination of L and L’ can be quantified . It is possible to establish two non-reciprocal parameters, µ and ν, that depend only on L and L’ and that, respectively, determine how strong the asymmetric extinction and refraction of CPL will be.22, 23]. A nonvanishing ν is much more common effect, as it occurs whenever an eigenmode is at an intermediate position, i.e. not aligned or perpendicular to the horizontal and 45° axes.
The complete polarimetric analysis derived from L reveals that the oblique hole array has a remarkable natural activity, manifested through CD and CB [Figs. 4(a) and 4(b)]. Obviously, these effects remain unchanged for forward and backward propagation. At the same time, the sample is rich in linear effects that, when combined together, lead to a non-reciprocal contribution [Figs. 4(c) and 4(d)]. The achiral square lattice, measured analogously as a control, did not yield appreciable CD, CB, µ or ν nor any linear anisotropies, confirming its anticipated, isotropic response when the electric vector of the light is parallel to the sample plane.
The non-reciprocaltransmission of CPL light in planar 2D nanostructures has been previously studied in pioneering papers by Zheludev and associates [2–4, 24]. They correctly asserted that non-reciprocal transmission does not violate the LRT but they did not solve the complete polarimetric analysis. In our experiments, symmetric, wave-vector invariant CB and CD coexist with non-reciprocal contributions that arise due to combinations of linear effects . Complete polarimetric analysis is essential here because it can distinguish CD from µ and, ultimately, reveal the mechanism that permits the oblique array to asymmetrically transmit CPL without breaking the LRT (Fig. 5). Likewise, it is worth to remind that, although this plasmonic nanomaterial is showing rather large non-reciprocal “effects”, such type of optical response is not a consequence of any new optical phenomena and it is a quite ubiquitous in absorbing media with low symmetry.
Extraordinary optical transmission in nanohole arrays is a process whereby incident light launches SPs that propagate in the array until scattered at the far interface as photons with characteristic polarizations . The SPs are longitudinal, compressive electron density waves with directions of propagation in the gold interfaces parallel to the electric field of the exciting photons. In the far field, a plane wave is reconstructed from the scattered radiation from each hole. Both the propagation velocity  and radiative decay rates depend on the plasmon momentum. CPL, as opposed to linearly polarized light, can excite SPs in any in-plane direction. At either metal-dielectric interface, the square array supports degenerate SPs with a relative orientation of 90° that will be excited with equal probability by CPL [Fig. 6(a)]. Because the modes have equal momenta, and identical temporal evolutions, the superpositon of the radiation by this degenerate pair of SPs is coherent and reconstructs the input light polarization. The response is isotropic for light at normal incidence .
Mode degeneracy is lifted in the oblique array [Fig. 6(b)]. Now, the two most closely spaced distances correspond to [-1,1] of 697 nm and [0,1] of 730 nm [w and v, respectively in Fig. 6(b)]. It is well known that peak broadening is related to the hole size [29,30], and in our sample the hole diameter is more than four times bigger than the difference between these characteristic lengths. Therefore, at some frequencies non-degenerate SPs associated with similar lattice constants (e.g. SPw and SPv) can be excited. For CPL, these two anisotropic SP modes, that have a misalignment of ∠wv = 43.5° [Fig. 6(b)] will be active at the same time. Their associated linear retardations and linear extinctions will couple together and their combined effect, manifest as a circular extinction. When the wave vector is reversed or the sample is turned over there is a change in the system of reference that is manifested in the exchange of 45° and 135° axes [Fig. 6(c)]. As LB’ and LD’ are two linear effects defined over these axes their sign is immediately changed and this sign change is invariably transferred to µ [Eq. 2(b)]. Herein, lies the etiology of the non-reciprocity.
The oblique metallic nanohole array can hold two sets of SP modes, one at the air-metal interface and another at a glass-metal interface. By turning over the sample, the excitation of modes by the incoming light (in-coupling) and the radiation (out-coupling) switch from one interface to the other. It could be argued that the observed asymmetric transmission is a direct consequence of exchanging the in-coupling and out-coupling interfaces. However, this serial excitation, which also takes place for the square lattice, is not responsible of the non-reciprocal optical effect. An interpretation can be secured in the lattice geometry of one interface, as we have described above. The physical process must be understood as the coupled action of two linear plasmon modes that creates a “mixed” circular eigenmode for light propagation regardless of the interface in which they originate. In agreement with this description, µ and ν predominate between a 650 nm and 800 nm, the spectral region where there is an asymmetric non-reciprocal transmission of light in the range of the characteristic length parameters of the two coupled plasmon modes shown in Fig. 6(b).
4. Further results of the square lattice at oblique incidence
The case of oblique incidence in lattices of nanoholes has rarely been treated experimentally. In this configuration there is an in plane component of the incident wave vector that provides additional momentum for SP coupling in this direction. Even at relatively small angles of incidence the polarimetric response of the square lattice becomes drastically different from normal incidence, and in fact when tilted around certain lattice direction it can show asymmetric transmission of CPL in a similar way the oblique lattice does.
For these experiments we have varied the angle of incidence (θ) between 0° and 30° and the azimuthal tilt angle (α) between 0° and 360° [Fig. 7(a)] thereby characterizing the polarimetric response through the measurement of the M of the square lattice in a sizable region of k-space at several wavelengths (see appendix A). The results of these measurements have been plotted in polar coordinates, where α is the polar angle and the radius is the angle of incidence θ [Fig. 7(c)]. Although the polarimetric response of the nanohole array changes from one wavelength to another, the polar symmetry of every element of M is preserved in all the spectrums, as it depends only on the symmetry of the lattice of holes and the orientation of the sample.
At a given θ, angle α determines the projection of the lattice sampled by the optical wave [Fig. 7(b)]. Therefore, the experiments could be described approximately as rectangular, rhombic, or oblique 2D lattices of nanoholes excited at normal incidence. As anticipated from the results above, asymmetric transmission of CPL (i.e. a nonvanishing µ) through the square array is only possible for oblique projections as it is shown in the polar plots of Fig. 7(c); that is when plane of incidence does not correspond with a mirror line . When α = 0°, 90° and when α = 45°, 135° the square lattice projects, respectively, rectangular and rhombic arrays that contain mirror lines thus giving a completely reciprocal polarimetric response. Opposing sides of any mirror line must show opposite signs of CD and µ because the projected lattices are enantiomorphous, identified as D-oblique and L-oblique in Fig. 7(b).
By studying the extraordinary asymmetric transmission of light in sub-wavelength hole arrays with a complete polarimeter for the first time, we have discovered mechanisms that enable non-reciprocal transmission while preserving Lorentz reciprocity. This brings new possibilities for the control of the polarization information in SPs. We have shown that engineered, plasmonic nanostructured materials based on arrays of holes distributed over an oblique lattice can be useful to encode and decode information-rich SPs, since they are not only sensitive to the intensity, frequency, and polarization of the optical signal but also to the k-vector direction. Understanding the intrinsic differences in the plasmonic response for 2D lattices is a first step in the development of plasmonic crystallography to localize and control light in sub-wavelength regions.
Appendix A: k-space mapping of the square array at different wavelengths
This appendix shows in Figs. 8, 9, 10, 11, 12, and 13 the results of the Mueller matrix k-space mapping at different wavelengths. To elaborate these polar (θ, α) plots the angle of incidence (θ) has been varied between 0° and 30° in steps of 1° and the azimuthal tilt angle (α) between 0° and 360° in steps of 2°.
This work was supported by the US National Science Foundation (CHE-0845526, DMR-1105000), by the Israel Science Foundation grant no. 172/10 and by the James Frank program on light-matter interaction. O. A. acknowledges a Marie Curie Fellowship within the European Union's Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n° . The authors are grateful to Dr. A. Tzukernik, Dr. Y.Lilach, M. Eitan and A. Hazzan for sample fabrication. B.M.M. was supported by The Tel Aviv University Center for Nanoscience and Nanotechnology.
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