Breaking transmission symmetry without breaking reciprocity
in linear all-dielectric polarization-preserving metagratings

Transmission asymmetry in photonic systems, i.e. a transmission that depends on which side of the structure light is incident from, is a highly desirable property in modern photonics as it opens-up additional degrees of freedom for designed beam control across the EM spectrum. Transmission asymmetry of linearly-polarized light is generally regarded as synonymous to non-reciprocity and in its ideal form implies optical isolation; a property needed, for example, in lasers to protect the beam quality against back-reflections [1, 2]. Such a non-reciprocal behavior is typically onset by an external static magnetic field [1, 2, 3, 4] which however leads to devices with a large footprint. A weaker form of non-reciprocity [3, 5, 6] can be triggered by the intensity of the impinging electromagnetic (EM) wave in non-linear material systems that are asymmetric along the propagation direction [7, 8, 9, 10, 11, 12, 13, 14] (self-biasing non-reciprocity [3]). The capability of a strong transmission asymmetry in these non-linear systems, without necessarily isolation [3, 6], is certainly winning on the trade-off of more compact implementations.

A linear passive [15] route to transmission asymmetry is most certainly attractive due to its simplicity but appears counter-intuitive. However, many authors have reported transmission symmetry breaking of linearly polarized light without violating reciprocity in structures capable of rotating its polarization [16, 17, 18, 19, 20, 21, 22, 23]. The phenomenon stems from a planar structural asymmetry [24, 25] that is different at each of the structure’s faces. Specifically, in the aforementioned systems transmission is asymmetric because $S_{yx}^{+}\neq S_{yx}^{-}$, with $S_{yx}$ representing the respective scattering matrix element for light incident with $x$- linear polarization transmitted into the $y$- linear polarization channel (90 deg. polarization rotation), along the forward direction [($+$) superscript] and reverse direction [($-$) superscript] respectively. This inequality is consistent with reciprocity that requires $S_{yx}^{+}$ to be equal only to $S_{xy}^{-}$ but not necessarily to $S_{yx}^{-}$ [3].

Here, we explore an alternate reciprocal route to transmission symmetry breaking of linearly-polarized light without a rotation of its polarization in a class of metagratings. We aim to unveil a core general principle that can guide the design of practically realizable structures for transmission asymmetry relevant to applications from infrared to visible frequencies while showing its consistency with reciprocity. Our study focuses on a class of linear, passive, polarization-preserving metagrating (LPPPMG) systems. It is assumed the metagrating systems of this class comprise only isotropic optical materials and may include a substrate but would not include any elements made of photonic-band-gap (PBGs) [26, 27] or hyperbolic media [28]. While transmission asymmetry can be engineered by structuring media like PBGs which possess certain directions of forbidden propagation (e.g. see Ref. 29), these type of systems are out of the scope and discussions in this work.

Although some previous works have reported transmission asymmetry for linearly polarized light with systems falling under the LPPPMG class there is no clear consensus over the key underpinning mechanism. Specifically, an observed transmission asymmetry has been attributed to having structural elements of a different lateral periodicity [30], to a different number of diffraction orders at each side of the structure [31], or to the Raleigh-Wood anomaly [32] with dependence on the substrate’s dielectric, suggesting the excitation of surface waves as the key mechanism [33]. Other works [34, 35] allude to the periodic pitch being the crucial controlling parameter but with no or true explanation on how the periodic pitch relates to the cause of transmission symmetry breaking. Additionally, the observed transmission-asymmetry onset for the case of normally incident light was correlated to the emergence of two first-order Bragg diffracted beams in Ref. 36 but without an explanation as to why these Bragg beams [37, 38] enable the transmission symmetry breaking. Hence, these previous works neither have established a general consistent transmission symmetry breaking principle for linearly polarized light applicable to the LPPPMG class nor have they provided an explanation why the observed transmission symmetry breaking does not violate reciprocity. This will be the focus of this paper.

In particular, this paper is organized as follows. In Sec. II we perform an all-angle transmission study through an all-dielectric paradigm of the LPPPMG class and show it can exhibit a high degree of transmission asymmetry without optical isolation [1]. We observe an abrupt phase transition from a symmetric-transmission to an asymmetric-transmission phase exactly at the Bragg critical wavelength where higher-order beams start to emerge for a certain incident angle. In Sec. III, we consider a multi-port-network framework and demonstrate that the emergence of at least one higher-order Bragg beam constitutes the necessary condition for transmission symmetry breaking in LPPPMG systems with no inversion symmetry along the propagation direction, while keeping reciprocity intact. Furthermore, in Sec. IV, we present the results of a numerical experiment on the metagrating paradigm of Sec. II with the Finite Difference Time Domain (FDTD) method [39], thus visualizing the mechanism of the transmission-symmetry breaking phenomenon discussed in Sec. III. Subsequently, in Sec. V, we discuss and elucidate on previous misconceptions regarding the transmission asymmetry phenomenon in the LPPPMG system class, as reported in some of the previous works discussed above. We further analyze all these aspects towards general design principles for practical set-ups. Finally, we present our conclusions in Sec. VI and discuss the relevance of this work to current and emerging photonic applications.

Diffraction gratings are widely-researched [37, 38] popular optical components. A diffraction grating embedded in vacuum with interfaces cut along its periodic direction with a pitch $a$ gives rise to higher-order reflected and transmitted beams (Bragg beams) for free-space wavelengths, $\lambdaup_{\textrm{free}}$, such that [27]:

for any integer $m\neq 0$, with $\uptheta_{\textrm{inc}}$ being the incident angle. Eq. (1) still yields the higher-order Bragg beams existing in vacuum even if the grating rests on a substrate at both the grating and substrate side; however, in practice if the substrate is thick and lossy some Bragg orders may disappear. The role of the substrate will be discussed more in Sec. V.

Here we consider a metagrating paradigm with its periodically-repeated building block being a three-log dielectric micropyramid lacking inversion symmetry along the propagation direction. The permittivity value of the dielectric, $\varepsilon$, is taken to be equal to 10.6, that is representative of semiconductors at mid-infrared frequencies. In the conceived metagrating design for either TM or TE incident light [40] polarization rotation is forbidden by virtue of the translational symmetry along one axis [41, 42, 43]. This means for either TM- or TE-polarized light, all reflected and transmitted beams from the metagrating, whether primary, $m=0$ in Eq. (1), or higher order, $m\neq 0$ in Eq. (1), will also be TM- or TE-polarized respectively.

We consider here the case of TM-polarized light [40] incident at an angle $\uptheta_{\textrm{inc}}$ on such meta-grating as depicted in Fig. 1(a). Fig. 1(b) depicts the design parameters of the micropyramid’s unit cell comprising the metagrating. We calculate the transmission with the transfer matrix method (TMM) [44, 45], for $\uptheta_{\textrm{inc}}=30$ deg. and show the respective results in Fig. 1(c) versus the free-space wavelength, $\lambdaup_{\textrm{free}}$. The red-dashed line, represents the transmission, ${\textrm{T}}{\ignorespaces\downarrow}$, for light coming from the top side, as in the left panel of Fig. 1(a), while the black-solid line represents the transmission, ${\textrm{T}}{\ignorespaces\uparrow}$, for light coming from the bottom side, as in the right panel of Fig. 1(a). Interestingly, we observe that ${\textrm{T}}{\ignorespaces\downarrow}$ and ${\textrm{T}}{\ignorespaces\uparrow}$ exactly match for long wavelengths, and become different below a critical wavelength, $\lambdaup_{\textrm{c}}$. To quantify such difference we introduce the transmission asymmetry, ${\textrm{T}}_{\textrm{asym}}$:

We observe at $\lambdaup_{\textrm{free}}>\lambdaup_{\textrm{c}}$, ${\textrm{T}}_{\textrm{asym}}$ is zero designating a symmetric response, while at $\lambdaup_{\textrm{free}}<\lambdaup_{\textrm{c}}$ we see that ${\textrm{T}}_{\textrm{asym}}$ varies from small to significant, taking values as large $\sim 75\%$ around $\lambdaup_{\textrm{free}}\sim$ 3.2 $\mu$m.

We aim to understand what is the physical reason behind such a critical wavelength, $\lambdaup_{\textrm{c}}$, below which transmission symmetry breaking onsets. Clearly, polarization rotation enabling transmission asymmetry in various reciprocal systems [16, 17, 18, 19, 20, 21, 22, 23] cannot be the responsible mechanism since it is forbidden here [41, 42, 43]. To gain more insight, we calculate the transmission asymmetry, ${\textrm{T}}_{\textrm{asym}}$, versus the free-space wavelength and incident angles. We start from normal incidence with a step of 5 deg. until near-grazing incidence and show the results in Fig. 1(d). We observe an abrupt separation of the ($\uptheta_{\textrm{inc}}$, $\lambdaup_{\textrm{free}}$) space into two areas with corresponding zero, and non-zero values of ${\textrm{T}}_{\textrm{asym}}$.

If we look closer at these limiting $\lambdaup_{\textrm{free}}$ values for certain $\uptheta_{\textrm{inc}}$ at the boundary between these two aforementioned areas, we find that astonishingly, these are tracing the respective values where at least one higher-order Bragg diffracted beam emerges, as determined from Eq. (1). We represent the latter in Fig. 1(d) with the solid-white line. There may be more than one higher-order Bragg beam. Conversely, dashed-white and dotted-white lines represent respectively the $\lambdaup_{\textrm{free}}$ values at different $\uptheta_{\textrm{inc}}$, determined from Eq. (1), below which at least two or three higher-order Bragg beams emerge. Indeed, it is the emergence of the very first Bragg beam with $m=-1$ that abruptly changes the behavior of the system from having a symmetric to having an asymmetric transmission response. Such first-order Bragg beam is present on both sides of the structure of Fig. 1. We note in passing while Fig. 1 depicts the TM-polarization results, we have also verified that asymmetry abruptly kicks-in when the first Bragg beam with $m=-1$ emerges for the case of TE-polarized light as well [40].

In the following, we consider a general LPPPMG system and demonstrate why the emergence of at least one higher-order Bragg beam enables transmission asymmetry of linearly polarized light without violating reciprocity.

We present here an analogy between Bragg-diffracted beams at the surface of a LPPPMG system and networked ports. For gratings resting on a substrate the LPPPMG system would comprise both the grating and the substrate [46]. In other words, the ports are always defined in vacuum, where transmission is measured in experiments, with each higher-order Bragg beam contributing two ports, one for the reflection and one for the transmission channel. This means a two-port system, as seen in Fig. 2(a), would represent cases where no higher-order out-coupled Bragg beams may exist, such as the metagrating system for ($\lambdaup_{\textrm{free}}$, $\uptheta_{\textrm{inc}}$) values above the white solid line of Fig. 1(d). Conversely, a four-port system [Fig. 2(b)] describes cases where one higher-order Bragg beam emerges, such as the metagrating system for ($\lambdaup_{\textrm{free}}$, $\uptheta_{\textrm{inc}}$) values between the white solid and dashed lines of Fig. 1(d) while a six-port system corresponds to cases with two higher-order Bragg beams, such as the metagrating system for ($\lambdaup_{\textrm{free}}$, $\uptheta_{\textrm{inc}}$) values between the white dashed and dotted lines of Fig. 1(d) and so on and so forth.

We have that $[{\textrm{b}}]=S[{\textrm{a}}]$, with $[{\textrm{a}}]$, $[{\textrm{b}}]$ being column arrays contain the respective coefficients $a_{i}$ and $b_{i}$ yielding the power throughput of the incoming and outgoing beams at the $i^{th}$ port when their modulus squared is taken (see Fig. 2). $S$ is the scattering matrix, which must be symmetric for a reciprocal system [3, 2, 47], i.e. $S=S^{\textrm{T}}$, with T denoting the matrix transpose. Then, after adopting the notation $S_{ij}$ to represent the scattering matrix element from the $j^{th}$ to the $i^{th}$ port we obtain for the two-port system:

We see from Eq. (3) that reciprocity mandates ${\textrm{T}}{\ignorespaces\downarrow}$ and ${\textrm{T}}{\ignorespaces\uparrow}$ to be equal. Hence, transmission must be symmetric for LPPPMG systems where no higher-order Bragg beams emerge, which effectively behave as a two-port system. If additionally these systems are lossless, such as the metagrating paradigm of Fig. 1, reflection must also be symmetric. On the other hand, if these systems have a loss, gain or both, then they may exhibit an asymmetric reflection which however must be offset by an equivalent asymmetry in absorption or gain, thus resulting in a symmetric transmission (e.g. see Refs. [43, 48, 49, 50, 51, 52]).

Conversely, for the four-port system [case of Fig. 2(b)], taking into account that there is no input power in some ports, we obtain:

Hence, for a LPPPMG system behaving as a four-port system, such as the metagrating system of Fig. 1 when one high-order Bragg beam emerges, transmission will be in general asymmetric with an asymmetry equal to:

This is because while reciprocity requires $S=S^{\textrm{T}}$ and thus $S_{21}$ to be equal to $S_{12}$, it does not require $S_{41}$ and $S_{32}$ to be equal. In a similar manner we find that transmission is generally asymmetric for a LPPPMG system with two higher-order Bragg beams, behaving as a six-port system, and so on and so forth.

We note in passing that the higher-order Bragg channels on one of the structure’s sides may be engineered to have no output, e.g. like in the case of systems studied in Ref. 31. In the latter cases, the higher-order channels on one of the structure’s side are inactive/closed, i.e. effectively non-existent with a zero corresponding scattering matrix element $S_{ij}$. Engineering this behavior is rather challenging, with designs being generally less practical, especially in the infrared and optical spectrum, without necessarily a higher asymmetry performance. We discuss these type of cases in more detail in Sec. V-C. The important take away of this section is, as Eq. (5) attests, that it just takes the emergence of only one higher-order Bragg beam to break transmission symmetry, whether the corresponding channels of such higher-order beam are open on either or both sides of the structure. However, in general, wherever any higher-order Bragg beams emerge these will act as open channels on both sides of the structure, like in the case of our metagrating paradigm of Fig. 1.

In the following, we perform a first-principle numerical experiment with the metagrating of Fig. 1 at $\lambdaup_{\textrm{free}}=\lambdaup_{2}$, demonstrating how transmission symmetry breaking occurs by means of the first-order transmitted Bragg beams ($m=-1$) that can out-couple from both structure sides, albeit with different strengths, $|S_{41}|^{2}$ and $|S_{32}|^{2}$, respectively.

We simulate the metagrating’s response to a TM quasi-monochromatic Gaussian beam [53] launched at 30 deg. with the two-dimensional (2D) Finite-Difference Time Domain Method (FDTD) [39] with Perfect Matched Layer (PML) boundary [54, 55] conditions (translational symmetry is taken out-of-the plane of incidence). We perform the FDTD experiment for the two indicative wavelengths $\lambdaup_{\textrm{1}}$ and $\lambdaup_{\textrm{2}}$, designated in Fig. 1(c), and show the respective results in Figs. 3(a) and 3(b) for incidence from both sides of the micropyramid-array. The micropyramid-array is represented as a white rectangle that has its tip side coinciding with the top rectangle side and its base side coinciding with the bottom rectangle side. The quantity depicted in Fig. 3 is $|\tilde{\textrm{S}}_{y}\rvert$, with $\tilde{\textrm{S}}_{y}$ being the normalized time-averaged $y$-component of the Poynting vector, calculated in FDTD at steady-state by averaging the $y$-component of the Poynting vector over a wave period, $T_{\textrm{per}}$, and then dividing this by the corresponding value of the source maximum.

For $\lambdaup_{\textrm{free}}=\lambdaup_{\textrm{1}}$ we observe in Fig. 3(a) only one reflected and transmitted beam; their respective strengths appear the same irrespective of the side the incident beam is launched from. To check this further, we take the $|\tilde{\textrm{S}}_{y}\rvert$ power throughput collected over the line-segment port (dashed line) and normalize this by dividing it with the respective power throughput collected from an equivalent line-segment port at the same $y$ distance from the source, in the absence of the metagrating structure. This process yields respectively for the top and bottom panels of Fig. 3(a) $|S_{21}|^{2}$, and $|S_{12}|^{2}$; reciprocity mandates these to be equal; to no surprise $|S_{21}|^{2}$ and $|S_{12}|^{2}$ are found to be equal in the FDTD experiment [56].

On the other hand, for $\lambdaup_{\textrm{free}}=\lambdaup_{\textrm{2}}$ we observe two reflected and transmitted beams; this is expected from Eq. (1) which predicts that at this wavelength and launch angle a Bragg beam of order $m=-1$ would emerge. It is interesting that we see in Fig. 3(b) that while the strength of the primary transmitted beams ($m=0$) appear the same, whether the beam is launched from the top or bottom side, the ones corresponding to Bragg order $m=-1$ have visibly very different strength. By performing, the detailed calculation of the power throughput through the dashed and dotted line ports, as we did for the case of Fig. 3(a), we confirm the initial visible observation. Indeed, the power throughput of the primary transmitted beams, corresponding to $|S_{21}|^{2}$ and $|S_{12}|^{2}$ are equal [56].

However, the power though-put of the transmitted Bragg beams with order $m=-1$, corresponds to $|S_{41}|^{2}$ and $|S_{32}|^{2}$, for incidence from the top and bottom side respectively. Reciprocity does not require these to be equal and our numerical experiment reveals that in fact they do turn out to be significantly different. It is that significant difference that caused the significant transmission asymmetry we observed in Fig. 1(c). Now that we have calculated with the FDTD method the scattering matrix elements $|S_{21}|^{2}$ and $|S_{12}|^{2}$ for case of Fig. 3(a) and $|S_{21}|^{2}$, $|S_{12}|^{2}$, $|S_{41}|^{2}$ and $|S_{32}|^{2}$ for the case of Fig. 3(b), we can use them to calculate ${\textrm{T}}{\ignorespaces\downarrow}$ and ${\textrm{T}}{\ignorespaces\uparrow}$ from Eqs. (3) and (4). We do this and show the respective results with filled circles and open diamonds in Fig. 1(c); a very excellent agreement is observed with the TMM results.

To recap, our FDTD experiment validated our hypothesis that the transmission asymmetry of linearly polarized light in the metagrating system stems from the introduction of additional transmission channels via higher-order Bragg beams.

We briefly discussed in the introduction previous works that have reported a transmission asymmetry of linearly polarized light incident on a LPPPMG structure. Indeed, there was apparent lack of consensus in these previous reports regarding the key responsible mechanism for such a transmission asymmetry in this class of systems. In the previous sections, we have unveiled with our systematic all-angle calculations and the multi-port network framework that the only necessary condition is the emergence of at least one higher-order Bragg beam in a LPPPMG system that lacks inversion symmetry along the propagation direction. In this section, we discuss in more detail how such previous misconceptions arose while correlating the results of these previous works with our findings here. We analyze different aspects of previous reported structures, while highlighting interpretation mistakes. The discussion below emphasizes how identifying correctly the core mechanism of transmission asymmetry of linearly polarized light in LPPPMG systems as stated above is crucial towards transferable design principles for practical implementations across the EM spectrum.

With all-angle transmission calculations and the aid of a multi-port network analysis, we showed that a reciprocal route to transmission asymmetry of linearly-polarized light without rotation of its polarization is to excite at least one higher-order Bragg diffracted beam on either or both sides of a LPPPMG structure [46] that has broken inversion symmetry along the propagation direction. We have further demonstrated this core mechanism in a metagrating paradigm with a numerical experiment. Moreover, we established that asymmetric transmission can occur even when the same number of diffracted beams are present at both sides of the structure, contrary to previous reports asserting the need for a different number [31]. We further disproved the conclusion of Ref. 32 and showed that the onset wavelength of the transmission asymmetry is actually independent on the presence of a substrate. Finally, our results and analysis revealed that complex designs incorporating structural elements of different pitch are neither necessary nor have an apparent advantage.

Thus, this work offers transferable design insights for transmission symmetry breaking of linearly polarized light across the EM spectrum with simple LPPPMG systems that lack inversion symmetry along the propagation direction. The mechanism uncovered here for transmission asymmetry would be applicable to linear passive periodic metasurfaces [57] as well, however the contribution from polarization rotation that is possible in these systems would need to be accounted as well. We believe this work will further inspire non-linear self-biased non-reciprocal [3, 12] metasurfaces and metagratings that are highly relevant to applications like integrated infrared photonics [58], self-isolating lasing architectures [59] and directional second-harmonic generation [60, 61]. Other interesting asymmetric effects that have been reported involve the lack of transmission mirror symmetry with respect to the surface normal caused by the excitation of topological surface waves [62], i.e. a transmission that is different for incident angles $\uptheta$ and $-\uptheta$. Hence, the metagrating systems reported here may further inspire functional designs for highly versatile light flow control with both mirror and propagation-direction transmission asymmetry.

We would like to thank the UNM Center for Advanced Research Computing, supported in part by the National Science Foundation, for providing the high performance computing resources used in this work.