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Photonics Research
Vol. 8, Issue 8, 1333 (2020)

Unidirectional reflection from an integrated “taiji” microresonator

A. Calabrese¹, F. Ramiro-Manzano², H. M. Price³, S. Biasi^4、*, M. Bernard⁵, M. Ghulinyan⁵, I. Carusotto⁶, and L. Pavesi⁴

Author Affiliations

¹Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Paris Sciences et Lettres, CNRS, Université de Paris, 75005 Paris, France

²Instituto de Tecnología Química (CSIC-UPV). Av. de los Naranjos, 46022 Valencia, Spain

³School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK

⁴Nanoscience Laboratory, Dipartimento di Fisica, University of Trento, Via Sommarive 14, 38123 Povo (TN), Italy

⁵Centre for Materials and Microsystems, Fondazione Bruno Kessler, Via Sommarive 18, 38123 Povo (TN), Italy

⁶INO-CNR BEC Center and Department of Physics, University of Trento, Via Sommarive 14, 38123 Povo, Italy

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DOI: 10.1364/PRJ.393070 Cite this Article Set citation alerts

A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, L. Pavesi. Unidirectional reflection from an integrated “taiji” microresonator[J]. Photonics Research, 2020, 8(8): 1333 Copy Citation Text

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Abstract

We study light transmission and reflection from an integrated microresonator device, formed by a circular microresonator coupled to a bus waveguide, with an embedded S-shaped additional crossover waveguide element that selectively couples counter-propagating modes in a propagation-direction-dependent way. The overall shape of the device resembles a “taiji” symbol, hence its name. While Lorentz reciprocity is preserved in transmission, the peculiar geometry allows us to exploit the non-Hermitian nature of the system to obtain high-contrast unidirectional reflection with negligible reflection for light incident in one direction and a significant reflection in the opposite direction.

(\begin{matrix} E_{in}^{L} \\ E_{out}^{L} \end{matrix}) = M (\begin{matrix} E_{out}^{R} \\ E_{in}^{R} \end{matrix}),

(1)

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M_{F R} = \frac{1}{τ_{R}} (\begin{matrix} 1 & ρ_{R} \\ ρ_{R} & 1 \end{matrix}) M_{F L} = \frac{1}{τ_{L}} (\begin{matrix} 1 & - ρ_{L} \\ - ρ_{L} & 1 \end{matrix}),

(2)

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M_{P j} = (\begin{matrix} e^{- i θ_{j}} & 0 \\ 0 & e^{i θ_{j}} \end{matrix}),

(3)

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M_{t a i j i} = (\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix}) = (\begin{matrix} \frac{1 - t_{1} t_{2} t_{3} e^{i γ p}}{t_{1} - t_{2} t_{3} e^{i γ p}} & - \frac{2 κ_{1}^{2} κ_{2} κ_{3} t_{3} e^{i γ (z_{2} + 2 z_{3} + z_{4})}}{(1 - t_{1} t_{2} t_{3} e^{i γ p}) (t_{1} - t_{2} t_{3} e^{i γ p})} \\ 0 & \frac{t_{1} - t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} \end{matrix}),

(4)

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(\begin{matrix} E_{out}^{L} \\ E_{out}^{R} \end{matrix}) = S (\begin{matrix} E_{in}^{L} \\ E_{in}^{R} \end{matrix}) .

(5)

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S_{t a i j i} = (\begin{matrix} r_{t a i j i}^{L} & t_{t a i j i}^{R} \\ t_{t a i j i}^{L} & r_{t a i j i}^{R} \end{matrix}) = (\begin{matrix} 0 & \frac{t_{1} - t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} \\ \frac{t_{1} - t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} & \frac{2 κ_{1}^{2} κ_{2} κ_{3} t_{3} e^{i γ (z_{2} + 2 z_{3} + z_{4})}}{{(1 - t_{1} t_{2} t_{3} e^{i γ p})}^{2}} \end{matrix}),

(6)

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t = \frac{τ_{R} τ_{L}}{A_{1} e^{- i Θ^{+}} - ρ_{R} ρ_{L} A_{4} e^{i Θ^{+}} + ρ_{R} A_{2} e^{i Θ^{-}} - ρ_{L} A_{3} e^{- i Θ^{-}}},

(7)

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r_{L} = \frac{ρ_{R} A_{4} e^{i Θ^{+}} - ρ_{L} A_{1} e^{- i Θ^{+}} - ρ_{R} ρ_{L} A_{2} e^{i Θ^{-}} + A_{3} e^{- i Θ^{-}}}{A_{1} e^{- i Θ^{+}} - ρ_{R} ρ_{L} A_{4} e^{i Θ^{+}} + ρ_{R} A_{2} e^{i Θ^{-}} - ρ_{L} A_{3} e^{- i Θ^{-}}},

(8)

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r_{R} = - \frac{ρ_{R} A_{1} e^{- i Θ^{+}} - ρ_{L} A_{4} e^{i Θ^{+}} + A_{2} e^{i Θ^{-}} - ρ_{L} ρ_{R} A_{3} e^{- i Θ^{-}}}{A_{1} e^{- i Θ^{+}} - ρ_{R} ρ_{L} A_{4} e^{i Θ^{+}} + ρ_{R} A_{2} e^{i Θ^{-}} - ρ_{L} A_{3} e^{- i Θ^{-}}} .

(9)

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\frac{E_{out}^{R}}{E_{in}^{L}} = t_{t a i j i}^{L} = t_{1} - \frac{κ_{1}^{2} t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} = \frac{t_{1} - t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} .

(A1)

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\frac{E_{1}}{E_{in}^{L}} = \frac{i κ_{1}}{1 - t_{1} t_{2} t_{3} e^{i γ p}},

(A2)

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\frac{E_{e 1}^{L}}{E_{in}} = - \frac{κ_{1} κ_{2} e^{i γ z_{1}}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} = i κ_{2} e^{i γ z_{1}} \frac{E_{1}}{E_{in}},

(A3)

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\frac{E_{e 2}^{L}}{E_{in}} = - \frac{κ_{1} t_{2} κ_{3} e^{i γ (z_{1} + z_{2})}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} = i κ_{3} t_{2} e^{i γ (z_{1} + z_{2})} \frac{E_{1}}{E_{in}} .

(A4)

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\frac{E_{out}^{L}}{E_{in}^{R}} = t_{t a i j i}^{R} = t_{1} - \frac{κ_{1}^{2} t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} = \frac{t_{1} - t_{2} t_{3} e^{i γ p}}{1 - t_{1} t_{2} t_{3} e^{i γ p}}, \frac{E_{out}^{R}}{E_{in}^{R}} = r_{t a i j i}^{R} = \frac{2 κ_{1}^{2} κ_{2} κ_{3} t_{3} e^{i γ (z_{2} + 2 z_{3} + z_{4})}}{{(1 - t_{1} t_{2} t_{3} e^{i γ p})}^{2}} .

(A5)

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\frac{E_{2}}{E_{in}^{R}} = - 2 \frac{i κ_{1} κ_{2} κ_{3} t_{3} e^{i γ (z_{2} + 2 z_{3} + z_{4})}}{{(1 - t_{1} t_{2} t_{3} e^{i γ p})}^{2}},

(A6)

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\frac{E_{e 1}^{R}}{E_{in}} = - \frac{κ_{1} κ_{3} t_{2} e^{i γ (z_{3} + z_{4})}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} + 2 \frac{κ_{1} κ_{2}^{2} κ_{3} t_{1} t_{3} e^{i γ (z_{1} + z_{2} + 2 z_{3} + z_{4})}}{{(1 - t_{1} t_{2} t_{3} e^{i γ p})}^{2}} = i κ_{3} t_{2} e^{i γ (z_{3} + z_{4})} \frac{E_{1}}{E_{in}} + i κ_{2} t_{1} e^{i γ z_{3}} \frac{E_{2}}{E_{in}},

(A7)

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\frac{E_{e 2}^{R}}{E_{in}} = - \frac{κ_{1} κ_{2} (t_{3}^{2} - κ_{3}^{2}) e^{i γ (z_{2} + z_{3} + z_{4})}}{1 - t_{1} t_{2} t_{3} e^{i γ p}} - 2 i \frac{κ_{1} κ_{2} κ_{3}^{2} t_{1} t_{2} t_{3} e^{i γ (z_{1} + 2 z_{2} + 2 z_{3} + z_{4})}}{{(1 - t_{1} t_{2} t_{3} e^{i γ p})}^{2}} = i κ_{2} (t_{3}^{2} - κ_{3}^{2}) e^{i γ (z_{2} + z_{3} + z_{4})} \frac{E_{1}}{E_{in}} + κ_{3} t_{1} t_{2} e^{i γ (z_{1} + z_{2})} \frac{E_{2}}{E_{in}},

(A8)

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M_{t a i j i} = \frac{1}{t_{t a i j i}^{L}} (\begin{matrix} 1 & - r_{t a i j i}^{R} \\ r_{t a i j i}^{L} & - \det [S] \end{matrix}) .

(A9)

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A. Calabrese, F. Ramiro-Manzano, H. M. Price, S. Biasi, M. Bernard, M. Ghulinyan, I. Carusotto, L. Pavesi. Unidirectional reflection from an integrated “taiji” microresonator[J]. Photonics Research, 2020, 8(8): 1333

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