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Tunable Three-Channel Mesoscopic Demultiplexer Based on Detuned Stubs

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Abstract

We study analytically and numerically the possibility of implementing a simple mesoscopic demultiplexer based on bound states in the continuum (BICs), induced transparency, and Fano resonances. The demultiplexer is made of a Y-shaped waveguide with an input line and two output lines where each output line contains two stubs grafted on two different sites in a U-shape far from the input line. The BICs are obtained for specific values of the lengths of the stubs of U-shaped structures and the segment separating them. By detuning the two stubs slightly in an appropriate way, BIC transforms into Fano or electromagnetic induced transparency (EIT) resonances. We give closed-form expressions of the geometrical parameters that allow a selective transfer of the given state in the first waveguide without perturbing the second waveguide. The effect of temperature on the transmission resonances is also examined using Landauer–Buttiker formula.

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Authors and Affiliations

  1. Laboratoire de Physique de la Matière et de Rayonnements, Département de Physique, Faculté des Sciences, Université Mohammed I, 60000, Oujda, Morocco

    Z. Labdouti, I. Quotane & E. H. El Boudouti

  2. Laboratoire Matériaux, Enérgie et Contrôle Système, Département de Physique, Faculté des Sciences et Techniques de Mohammedia, Université Hassan II, 28806, Casablanca, Morocco

    A. Mouadili

  3. Laboratoire Sciences et Technologies Avancées, Département de Physique, Faculté Polydisciplinaire de Larache, Université Abdelmalek Essaadi, 92000, Tétouan, Morocco

    T. Mrabti

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  1. Z. Labdouti
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  3. A. Mouadili
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  4. T. Mrabti
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  5. E. H. El Boudouti
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Correspondence to T. Mrabti.

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APPENDIX

APPENDIX

1.1 DEMULTIPLEXER BASED ON MIXED STRUCTURES

We consider a mixed demultiplexer with one input and two outputs composed of a cross-shaped and a U‑shaped structures, respectively (Fig. A1). The first output contains a cross structure made of two stubs of lengths d1 and d2 placed at the same position labeled 2 and connected to the input by the segment of length d5. The second output contains the U-shaped structure made of two stubs of lengths d3 and d4 placed, respectively, at the sites 3 and 4 at a distance d6 and d6 + d0 from input line 1. It’s worth mentioning that the cross resonator in the first output exhibits only EIT resonances, whilst the U-shaped resonator in the second output present both Fano and EIT resonances.

Fig. A1.
figure 10

(Color online) Schematic illustration of the Y-shaped demultiplexer based on combined structures of U- and cross-structures. The geometric parameters are defined in the text.

Full size image

In this section, the cross structure connected to the first output exhibits EIT resonance while the U-shaped structure in the second output presents the Fano resonance. By choosing appropriately the different lengths constituting the mixed demultiplexer, one can achieve a mixed filtering where the first output presents EIT resonance and the second output exhibits Fano resonance.

Therefore, the lengths of the segments and stubs should verify the following conditions:

$${{d}_{1}} = \frac{{{{d}_{0}}}}{2} - \frac{\delta }{2},$$
(31)
$${{d}_{2}} = {{d}_{3}} = {{d}_{4}} = \frac{{{{d}_{0}}}}{2} + \frac{\delta }{2},$$
(32)
$${{d}_{5}} = \frac{{{{d}_{0}}}}{4} + \frac{\delta }{4},$$
(33)
$${{d}_{6}} = \frac{{{{d}_{0}}}}{4},$$
(34)
$$d_{0}^{'} = {{d}_{0}} + 2\delta ,$$
(35)

where d0 = d1 + d2 is fixed.

Figure A2 shows the numerical results of the transmission coefficients T1 and T2 along the first and the second output, respectively, as well as the curves of reflection coefficient R at the input as a function of the dimensionless energy κ for three values of δ. It can be seen that the resonance of EIT type is formed at output 1 (red curves) while the Fano type is realized at output 2 (blue curves). Figure A2 exhibits the behavior of EIT resonance in the first output that occurs at the same reduced energy for different values of δ as d0 fixed. On the other side, the position of the Fano resonance in the second output varies as function of δ. For δ = 0.1 (Fig. A2a), the Fano resonance appears at the left side of the EIT resonance. Whilst, for δ = −0.1 (Fig. A2c), the Fano resonance arises at the right side of the EIT resonance. Moreover, the width of both resonances changes when the sign of δ changes. For δ = 0 (Fig. A2b), both resonances fall at the same dimensionless energy (κ = 2π) and appear with zero width, giving rise to BICs. These modes correspond to stationary waves in the cross and U stubs on both outputs and do not interact with the semi-infinite wires (continuum states) surrounding them.

Fig. A2.
figure 11

(Color online) The same as in Fig. 7, but for the structure depicted in Fig. A1.

Full size image

Similarly, one can get also EIT–EIT resonances; in this case, we should satisfy the following conditions:

$${{d}_{1}} = \frac{{{{d}_{0}}}}{2} - \frac{\delta }{2},$$
(36)
$${{d}_{2}} = \frac{{{{d}_{0}}}}{2} + \frac{\delta }{2},$$
(37)
$${{d}_{3}} = \frac{{{{d}_{0}}}}{2},$$
(38)
$${{d}_{4}} = \frac{{{{d}_{0}}}}{2} + \delta ,$$
(39)
$${{d}_{5}} = \frac{{{{d}_{0}}}}{4} + \frac{\delta }{4},$$
(40)
$${{d}_{6}} = \frac{{{{d}_{0}}}}{4},$$
(41)
$$d_{0}^{'} = {{d}_{0}} + \delta .$$
(42)

The numerical results are almost the same to those curves presented in Fig. 5. Therefore, we prefer to avoid giving the EIT–EIT figure.

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Labdouti, Z., Quotane, I., Mouadili, A. et al. Tunable Three-Channel Mesoscopic Demultiplexer Based on Detuned Stubs. Phys. Wave Phen. 31, 238–251 (2023). https://doi.org/10.3103/S1541308X23040064

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