One-way light flow by spatio-temporal modulation

Mahmoud A. A. Abouelatta; Mahmoud A. A. Abouelatta; Mohamed A. Swillam; Artur R. Davoyan; Ahmed M. Mahmoud; Ahmed M. Mahmoud; Ahmed M. Mahmoud

doi:10.1364/OE.477167

1. Introduction

Reciprocity is a fundamental property of the wave equations that govern the propagation of light in a variety of systems, including lasers, integrated optical circuits, and fiber communications. The effect of reciprocity is manifested in the symmetry of light transmission in forward and backward directions of propagation [1]. In many applications, it is, however, desirable to break this fundamental symmetry, particularly when the optical input and output need to be isolated. Such symmetry breaking is possible with the use of magnetism [2–6], optical nonlinearities [7–12], and in systems with time-varying responses [13–16]. For example, nonreciprocal propagation of electromagnetic waves is possible in magnetized systems [2,17]. Hence, magnetization is widely used in many circulators (three-port devices that are used for optical isolation) [18]. In a waveguide, magnetism combined with a certain asymmetry magnetization may lead to a break in the symmetry of wave dispersion in forward and backward directions (i.e., f(ω,+k) ≠ f(ω,- k)). Similarly, such symmetry could also be broken by an electric bias [19,20]. Close to a cut-off, such symmetry breaking may result in different cut-off conditions [21] for forward and backward directions of propagation, which can be utilized to create the so-called “one-way” states [22,23]. However, in practice such one-way states are not achieved due to the presence of losses [21,24], or the nonlocal effects [25,26]. Fundamentally, no frequency band with one-way propagation exists in two-port magnetized systems. Nonlinearity presents another approach for reciprocity breaking. Diode-like optical systems have been proposed and demonstrated recently [7,27,28]. Nevertheless, such devices are vulnerable in instances when the power flows simultaneously in both propagation directions – the so-called effect of dynamic reciprocity [29]. Spatio-temporal modulation is another promising way for breaking reciprocity, and has been reported for a range of systems, including waveguides and metasurfaces [13,14,30–32]. In such systems, spatio-temporal modulation leads to unidirectional coupling between different guided and/or scattered modes of the system. In other words, it creates asymmetric coupling conditions when forward and backward directions of propagation experience different coupling. For example, in a waveguide for one propagation direction, no coupling is observed. The mode is transmitted without any perturbation, whereas in the other propagation direction, the mode couples to other modes in the system leading to an effective signal isolation. Notably, in this case light is transmitted in both propagation directions; isolation is achieved by signal conversion to a different frequency in one of the directions followed by a damping-filter. Reaching a complete and unconditional light propagation with full wave transmission in one propagation direction and no transmission in the other, to the best of our knowledge has not been demonstrated yet. Here, a novel phenomenon is introduced. The incident wave reflects on itself after being modulated for one direction of propagation, while it passes without alteration for the opposite direction of propagation. We make use of intermodal coupling in spatio-temporally modulated systems to induce coupling between modes with group velocities of opposite sign (i.e., between forward and backward propagating modes). We demonstrate that such coupling leads to a complete optical isolation in one of the propagation directions. More specifically, one-way light propagation is achieved in a two-port system (i.e., a two-terminal device such as a waveguide, as shown conceptually in Fig. 1), where light propagates in one direction only, and it gets fully reflected in the other. This effect is different from the previously studied spatio-temporally modulated systems, which exhibit a bi-directional response [13,14].

Fig. 1. Schematic of the proposed concept: A two-port network that allows electromagnetic energy, centered at a frequency ω₁, incident at one-port to be transmitted unperturbed to the other port, while reflecting the electromagnetic energy incident from the other direction back into the same port at a different center frequency (ω₂). ν_g, and ν_ph denote the group, and phase velocities, respectively. The inset represents the spatio-temporal modulation circuit that is designed to achieve the unidirectional intermodal transition.

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To elucidate our concept further, consider a two-port diagram for a linear system (e.g., a waveguide), as shown in Fig. 1. A spatio-temporally modulated signal is applied to the system. This modulation couples two modes in the system when the phase and energy matching conditions are satisfied. Since the modulation is spatio-temporally varying, the phase matching condition is different for the forward (+x) and backward (-x) propagation directions. We choose the parameters of the system in a way that the phase matching is satisfied only in the backward (–x) direction of propagation. In this case, the signal propagation in the forward (+x) direction is unperturbed and is transmitted through the waveguide. In the backward (-x) direction of propagation, the spatio-temporal modulation couples the eigen modes of the waveguide. In this case, the intermodal coupling with the use of the coupled mode theory can be described as follows in Eq. (1).

(1)$$\textrm{i}\frac{\textrm{d}}{{\textrm{dx}}}\left[ {\begin{array}{c} a\\ b \end{array}} \right] = {\left[ {\begin{array}{cc} {{\mathrm{\sigma }_\textrm{a}}}&0\\ 0&{{\mathrm{\sigma }_\textrm{b}}} \end{array}} \right]^{ - 1}}\left[ {\begin{array}{cc} 0&\mathrm{\kappa }\\ \mathrm{\kappa }&0 \end{array}} \right]\left[ {\begin{array}{c} a\\ b \end{array}} \right]$$

where $\textrm{a}$ and $\textrm{b}$ denote the complex slowly varying amplitudes of the waveguide modes, respectively. $\mathrm{\kappa }$ represents the intermodal coupling strength [13,14]. Here, ${\mathrm{\sigma }_\textrm{i}} ={\pm} 1$ corresponds to the sign of the energy flow of a given mode, with ${\mathrm{\sigma }_\textrm{i}} ={+} 1$ corresponding to waves with the same direction for the group and phase velocities (i.e., ${\textrm{v}_\textrm{g}} = \frac{{\partial \mathrm{\omega }}}{{\partial \textrm{k}}} > 0$ and ${\textrm{v}_{\textrm{ph}}} = \frac{\mathrm{\omega }}{\textrm{k}} > 0$, respectively), and ${\mathrm{\sigma }_\textrm{i}} ={-} 1$ to a case with opposite directions for the group and phase velocities (i.e., ${\textrm{v}_\textrm{g}} = \frac{{\partial \mathrm{\omega }}}{{\partial \textrm{k}}} < 0$ and ${\textrm{v}_{\textrm{ph}}} = \frac{\mathrm{\omega }}{\textrm{k}} > 0$, also known as backward waves). In a conventional case of a spatio-temporally modulated system, both waves have collinear group and phase velocities, i.e., ${\mathrm{\sigma }_\textrm{a}} \times {\mathrm{\sigma }_\textrm{b}} = 1$. In this case, the eigen-values of the system are real-valued implying that the waves can propagate through the waveguide, and that the energy can flow in both ${\pm} \textrm{x}$ directions. In contrast, when one of the modes is a backward wave so that ${\mathrm{\sigma }_\textrm{a}} \times {\mathrm{\sigma }_\textrm{b}} ={-} 1$, the eigen-values become complex, which implies that the waveguide becomes impermeable to the waves. Consequently, the system acts as a one-way mirror (induced by spatio-temporal modulation) in the –x direction of propagation and is transparent in the $+ \textrm{x}$ direction of propagation. In other words, the system acts a complete isolator and operates in a conceptually new regime. We note that these dynamics are generic in nature and can be implemented in a wide range of spatio-temporally modulated systems with ${\textrm{v}_{\textrm{ph}}} \times {\textrm{v}_{\textrm{gr}}} < 0$, for example, negative index metamaterials [33], plasmonic systems [34], or photonic crystals with a properly designed band structure [35].

For the sake of concept demonstration, we study a simple two-dimensional system which only supports pure transverse magnetic (TM) modes, i.e., a metal-insulator-metal (MIM) plasmonic nanoscale waveguide (see Fig. 2(a)). It is known that such a waveguide for small slot widths exhibits eigen modes with a negative group velocity or a backward wave flow [34–36]. Moreover, such a waveguide also supports a forward mode (i.e., with a positive group velocity). A novel physical response is obtained when an interband transition is done between the forward mode and the backward mode in the thin MIM waveguide, shown in Fig. 2(a). After entering the modulation region, the incident wave reflects on the same port for one direction of propagation, whereas it passes without change for the opposite propagation direction. For comparison with previous works in the literature, Fig. 2(b) illustrates a common approach employed in spatio-temporally modulated systems. For such an approach, the interband transition is done between two forward modes (i.e., the sign of the group velocity is the same for both modes). The band structures for the waveguides in Figs. 2(a) and (b) are shown in Figs. 2(c) and (d) respectively. In contrast to the bi-directional response, shown in Fig. 2(b), where the group velocities of both the input (symmetric) mode and output (antisymmetric) mode have the same sign, one-way wave propagation is attained for the band structure, shown in Fig. 2(c). This stems from the fact that the input (symmetric) mode is coupled to an output (antisymmetric) mode with a group velocity of an opposite sign. More specifically, due to the momentum conservation, the incident symmetric mode “decays” into a reflected antisymmetric mode, as shown schematically in Fig. 2(a). Therefore, unidirectional optical isolation is achieved.

Fig. 2. (a) The schematic diagram of the MIM waveguide which illustrates the novel phenomenon, introduced in this work, for achieving one-way light flow. (b) The schematic diagram of the MIM waveguide with the conventional bi-directional response. The light grey color represents the dielectric (silicon), the dark grey color represents the modulated region, the black color represents the metal, the red (blue) color is for the input (output) mode, and the green color is for the modal TM profile (H_z). For both structures, the dielectric material is the silicon (ε_r = 12.25). For (a), the used parameters are a = 0.1 µm, L = 21.5 µm, ∂ε = 2, ω_p = 1.4 × 10¹⁵rad/s, γ = 0, and for (b) a = 1.7 µm, L = 40 µm, ∂ε = 1, ω_p = 5 × 10¹⁵rad/s, and γ = 0, where a, L, ∂ε, ω_p and γ are the dielectric thickness, the modulation length, the perturbation amplitude, the metal plasma frequency, and the metal collision frequency, respectively. (c) and (d) The TM band structure of the MIM waveguides, shown in Fig. 2(a) and (b), respectively. The red (blue) mode is the symmetric (antisymmetric) mode. The axes are normalized to the used dielectric thickness (a) in each case. The (I) point represents the point of the incident modal source, and the (II) point denotes the output wave after passing through the modulated waveguide.

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Next, we develop a model that details the modal interactions in such a plasmonic system. The spatio-temporal modulation is achieved by assuming that the permittivity of the medium varies as follows in Eq. 2.

(2)$$\mathrm{\varepsilon } = \,{\mathrm{\varepsilon }_\textrm{r}} + \,\partial {\varepsilon\,\cos}({\mathrm{\Omega }\textrm{t}\, - \,\Delta \textrm{kx}} )$$

where ε, ε_r, ∂ε, Ω, and Δ k represent the overall permittivity, the constant (dielectric) permittivity, the perturbation amplitude, the temporal and spatial modulation frequencies, respectively. As derived in Supplement 1 with the use of the coupled-mode theory, Eq. 3 is obtained.

(3)$$\frac{\textrm{d}}{{\textrm{dx}}}\left[ {\begin{array}{c} {{\textrm{a}_1}}\\ {{\textrm{a}_2}} \end{array}} \right] = \left[ {\begin{array}{cc} 0&{\textrm{i}\frac{\textrm{B}}{\textrm{A}}}\\ {\textrm{i}\frac{\textrm{D}}{\textrm{C}}}&0 \end{array}} \right]\left[ {\begin{array}{c} {{\textrm{a}_1}}\\ {{\textrm{a}_2}} \end{array}} \right]$$

(4)$$\textrm{A} = \mathop \smallint \limits_{ - \infty }^\infty ({\overrightarrow {{\textrm{E}_1}} \times {{\overrightarrow {{\textrm{H}_1}} }^\mathrm{\ast }} + {{\overrightarrow {{\textrm{E}_1}} }^\mathrm{\ast }} \times \overrightarrow {{\textrm{H}_1}} } ).\mathrm{\vec{n}\;\textrm{dl}}$$

(5)$$\textrm{C} = \mathop \smallint \limits_{ - \infty }^\infty \overrightarrow {{\textrm{E}_2}} \times {\overrightarrow {{\textrm{H}_2}} ^\mathrm{\ast }} + {\overrightarrow {{\textrm{E}_2}} ^\mathrm{\ast }} \times \overrightarrow {{\textrm{H}_2}} ).\mathrm{\vec{n}\;\textrm{dl}}$$

(6)$$\textrm{B} = \frac{{ - 1}}{2}{\mathrm{\varepsilon }_\textrm{o}}({{\mathrm{\omega }_2} - {\mathrm{\Omega }_{\textrm{mod}}}} )\partial \mathrm{\varepsilon }\mathop \smallint \limits_0^{\textrm{a}/2} \overrightarrow {{\textrm{E}_2}} .{\overrightarrow {{\textrm{E}_1}} ^\mathrm{\ast }}\,\textrm{dl}$$

(7)$$\textrm{D} = \frac{{ - 1}}{2}{\mathrm{\varepsilon }_\textrm{o}}({{\mathrm{\omega }_1} + {\mathrm{\Omega }_{\textrm{mod}}}} )\partial \mathrm{\varepsilon }\mathop \smallint \limits_0^{\textrm{a}/2} \overrightarrow {{\textrm{E}_2}} .{\overrightarrow {{\textrm{E}_1}} ^\mathrm{\ast }}\,\textrm{dl}$$

where A and C in Eqs. (4) and (5) are proportional to the input and output power flux, respectively. $\mathrm{\vec{n}}$ represents the direction which is normal to the line of integration (the x-direction in Fig. 2), and dl represents the differential element for the line integration (dy for Fig. 2). B and D, which are given in Eqs. (6), and (7), respectively, quantify the coupling strength between the input and output modes. Using the initial conditions (a₁(0) = 1 and a₂(0) = 0), the solution to Eq.3 is a₁ = cos(κx) and a₂ = sin(κx), and κ is equal to ($\sqrt {\frac{{\textrm{BD}}}{{\textrm{CA}}}} $), while |a_n |² is the photon number flux of the n^th mode. In this context, κ represents two different physical quantities for the two MIM structures, shown in Figs. 2(a) and (b). The difference between the response of the structures, shown in Figs. 2(a) and (b), stems from the sign of output flux (C). To illustrate, for the MIM waveguide, depicted in Fig. 2(b), this value is positive. Hence, κ represents the phase mismatch between the coupled even and odd modes. L_c denotes the coherence length (L_c=$\frac{\mathrm{\pi }}{\mathrm{\kappa }}$), i.e., the minimum required modulation length for full mode conversion within the structure. On the other hand, based on our analytical derivation, it can be shown that for some selected values of the MIM parameters, the output flux (C) is negative, which is the case for Fig. 2(a). As a result, the value of κ is imaginary. In this case, such a value is connected to the decay rate of the input mode.

Furthermore, based on the derived equations, the coherence length (L_c) is equal to 40.1 µm for the structure, depicted in Fig. 2(b), while the decay coefficient (κ) for the structure, shown in Fig. 2(a), is equal to 0.4894 µm⁻¹. Consequently, based on our analytical model, the power is expected to drop to 1% of the input value, within a modulation length which is equal to around 4.7 µm.

To verify the analytical derivation, we perform numerical simulations using the finite difference time domain (FDTD) method. For the structures, shown in Figs. 2(a) and (b), the incident source spectrum is a nearly sinusoidal wave with a central frequency ω₁ = 0.0133(2πc/a) and 0.2266(2πc/a), respectively. The coherence length is numerically verified by the FDTD simulations for the structure, depicted in Fig. 2(b), and is found to be equal to 40 µm. Such a value is in a great agreement with the predicted value from our analytical model (i.e., L_c= 40.1 µm). The one-way light propagation is depicted in Fig. 3(a) for the structure shown in Fig. 2(a). It can be seen that the incident symmetric mode passes without any perturbation for one incidence direction, while it decays as it passes through the modulation region (the black rectangle) for the other direction of incidence. The bi-directional light propagation for the structure, depicted in Fig. 2(b), is also shown in Fig. 3(b) for the sake of comparison. It is worth mentioning here that a zero loss is assumed only to demonstrate that this decay of the input mode is due to the novel phenomenon we introduce and is not owing to any metallic losses. This decay stems from the fact that the incident mode is coupled to an antisymmetric mode flowing backward with a central frequency (ω₂ = 0.03(2πc/a)). The spectrum of the output wave for the structures, depicted in Fig. 2(a) and (b), are shown in Fig. 3(c) and (d), respectively, where for one direction of incidence, the output wave spectrum is identical to the input wave spectrum (i.e., the no-mode-conversion case). Conversely, for the other direction of incidence, the input symmetric mode is converted into a reflected and transmitted antisymmetric mode with a central frequency ω₂ = 0.03(2πc/a) and 0.3966(2πc/a) for the waveguides, shown in Fig. 2(a) and (b), respectively. To the best of our knowledge, this one-way response, shown in Fig. 3(a), in which a guided mode in a lossless medium substantially decays into another reflected guided mode only for one direction of incidence, has not been reported in the literature. Such a one-way response could be achieved in any system supporting backward waves, and it is not limited to the example that we used for illustration (i.e., the MIM waveguide), which has practical limitations (i.e., high losses and narrow bandwidth).

Fig. 3. (a) and (b) The distribution of the magnetic field (H_z) for the structures, shown in Figs. 2(a) and (b), respectively, based on the FDTD method, where A and B represent the reflection and transmission regions, respectively. The modulated waveguide region is enclosed by the black rectangle (see Visualization 1, Visualization 2 , Visualization 3, and Visualization 4 for illustration). Different plot scales have been used in the upper and lower part in (a), however, the modulated regions have the same length. (c) and (d) The normalized spectrum of output wave for the structures, depicted in Figs. 2(a) and (b), respectively. The normalized output wave spectrum is the same as that of input wave spectrum for the no-mode-conversion cases, while it shifts to a higher frequency for the mode-conversion cases. Each output wave is normalized to its maximum value.

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In summary we have introduced a novel concept for one-way propagation and optical isolation. These results can pave the way for one-way light propagation in several areas in photonics such as optical interconnects, optical routing, laser physics, optical communication, and systems on a chip.

2. Methods

The used dielectric (silicon material) is modelled with a constant relative permittivity (ε_r = 12.25). The metal is modelled by the following Drude’s equation.

(8)$${\mathrm{\varepsilon }_\textrm{D}} = {\mathrm{\varepsilon }_\infty } - \frac{{\mathrm{\omega }_\textrm{p}^2}}{{\mathrm{\omega }({\mathrm{\omega } + \textrm{i}{\mathrm{\gamma }_\textrm{D}}} )}}$$

Zero collision frequency (γ = 0) for the metal is used in both structures. This is to verify that the decay of the power in the isolated direction, as shown in Fig. 3(a), is solely due to the introduced phenomenon, and is not due any inherent losses in the system. For the two structures, shown in Figs. 2(a) and (b), the used plasma frequencies are (ω_p = 1.4 × 10¹⁵) and (ω_p = 5 × 10¹⁵), respectively. The band structures, shown in Figs. 2(c) and (d), are calculated using Lumerical Mode eigenmode solver. The FDTD simulations in Figs. 3(a) and (b) are executed using Lumerical FDTD software package using a time dependent material C++ plugin (i.e., a cos function), and a script that creates a spatio-temporally variant permittivity based on this plugin. Stabilized perfectly matched layer (PML) boundary conditions are used (more than 100 layers), and a stability factor of around 0.1 is necessary. The simulation region is extended so that all of the energy goes outside the structure, and then, the simulation is intentionally stopped before the energy reaches the boundaries since if that happens, a divergence would occur due to the reflections from the boundaries.

Acknowledgments

A.R.D. acknowledges partial support of the AFOSR award no. FA9550-22-1-0036 and the Hellman Society of Fellows. M.A.A.A. carried out numerical simulations. A.M.M. and A.R.D. conceived the idea, suggested the designs and performed the analytical derivations. All authors discussed the theoretical and numerical aspects and interpreted the results. All authors contributed to the preparation and writing of the manuscript. A.M.M. pursued this work during his time as a research associate at the Nanophotonics Research Lab (NRL) led by M.A.S. within the Department of Physics, American University in Cairo. A.M.M. was then appointed as a visiting assistant professor at the Electronics and Communications Engineering Department at the American University in Cairo. He then joined the school of Engineering and Applied Sciences, Nile University where he is currently an assistant professor. M.A.S. continued to support the project with computational resources throughout the process.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Name	Description
Supplement 1	Supplemental Document
Visualization 1	Animation of the field distribution of magnetic field (Hz) for the case shown in fig. 3(a) (top panel)
Visualization 2	Animation of the field distribution of magnetic field (Hz) for the case shown in fig. 3(a) (bottom panel)
Visualization 3	Animation of the field distribution of magnetic field (Hz) for the case shown in fig. 3(b) (top panel)
Visualization 4	Animation of the field distribution of magnetic field (Hz) for the case shown in fig. 3(b) (bottom panel)

One-way light flow by spatio-temporal modulation

Abstract

1. Introduction

2. Methods

Acknowledgments

Disclosures

Data availability

Supplemental document

References

Supplementary Material (5)

Data availability

Cited By

Figures (3)

Equations (8)

Optics Express

Mahmoud A. A. Abouelatta	https://orcid.org/0000-0003-3930-3802
Artur R. Davoyan	https://orcid.org/0000-0002-4662-1158
Ahmed M. Mahmoud	https://orcid.org/0000-0002-5784-9502