Controlling Random Waves with Digital Building Blocks Based on Supersymmetry

Sunkyu Yu, Xianji Piao, and Namkyoo Park

Phys. Rev. Applied 8, 054010 – Published 6 November 2017

Abstract

Harnessing multimode waves allows high information capacity through modal expansions. Although passive multimode devices for broadband responses have been demonstrated in momentum or frequency domains, the difficulty in achieving collective manipulation of all eigenmodes has hindered the implementation of digital multimode devices such as switching. Here we propose building blocks for digital switching of spatially random waves based on parity-converted supersymmetric pairs of multimode potentials. We reveal that unbroken supersymmetric transformations of any parity-symmetric potential derive the parity reversal of all eigenmodes, which allows the complete isolation of random waves in the “off” state. With two representative solvable potentials, building blocks for binary and many-valued logics are then demonstrated for random waves: a harmonic pair for binary switching of arbitrary wave fronts and a Pöschl-Teller pair for multilevel switching which implements fuzzy membership functions. Our results realizing the transfer of arbitrary wave fronts between wave elements will lay the foundation of high-bandwidth data processing.

Received 21 March 2017

DOI:https://doi.org/10.1103/PhysRevApplied.8.054010

Physics Subject Headings (PhySH)

Integrated optics Quantum optics

Bound states Quantum correlations, foundations & formalism Schroedinger equation

Atomic, Molecular & Optical

Authors & Affiliations

Sunkyu Yu, Xianji Piao, and Namkyoo Park^*

Photonic Systems Laboratory, Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Korea

^*nkpark@snu.ac.kr

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Vol. 8, Iss. 5 — November 2017

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Figure 1
The operation principle of controlling random waves. (a) Conventional operation composed of multiplexer, demultiplexer, and multiple single-mode switches. (b) A switching building block for random waves with the modal-profile-independent operation.
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Figure 2
The building block for digital binary switching of random waves: the SUSY harmonic pair. (a) Off state from the parity-reversed decoupling between the harmonic potential (output) and its SUSY partner (input). (b) On state from the parity-aligned coupling (black dotted arrows) through the modulation of the output potential (red arrow $Δ ϵ = 2 {ϵ_{Δ}}^{1 / 2} / k_{0}$ ). The shapes of the eigenmodes are plotted on the potentials (red, even parity; blue, odd parity) in (a),(b). The configuration of the designed silicon waveguides is shown in (c) for the off state and (d) for the on state. The silicon refractive index is $n = 3.5$ at $λ_{0} = 2 π / k_{0} = 1500 nm$ . The “effective” relative permittivities for the original and SUSY-partner harmonic waveguide are $ϵ_{o} (x) = [10.38 - 7.48 \times 10^{- 4} {(k_{0} x)}^{2}]$ and $ϵ_{s} (x) = [10.32 - 7.48 \times 10^{- 4} {(k_{0} x)}^{2}]$ , respectively. The waveguide width $L = 24.84 μ m$ , and the distance between the waveguides is $d = 800 nm$ .
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Figure 3
Eigenmodes in the SUSY harmonic pair of silicon waveguides. (a) Effective modal indices $n_{eff}$ of each waveguide before the coupling. (b) Level spacing of $n_{eff}$ in the original waveguide. The black dashed line denotes $Δ n_{eff} \sim 8.5 \times 10^{- 3}$ . (c) Level splitting by the coupling between the switched waveguide and the SUSY-partner waveguide. The black dashed line is the averaged splitting. (d) The variation of $n_{eff}$ as a function of the refractive-index modulation $Δ n$ in the output waveguide. The $e$ state denotes the degeneracy from the decoupling due to the parity reversal between eigenmodes. The $f$ state (or $g$ state) represents the anticrossing (or crossing) of eigenvalues. The modal profiles ( $E_{x}$ ) for the states of $e$ , $f$ , and $g$ in (d) are shown in (e), (f), and (g), respectively. All of the results are obtained by the full-wave eigenmode simulation in comsol Multiphysics.
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Figure 4
Binary switching for point-source excitations in the SUSY harmonic pair. Light excitations (a),(b) at the center and (c),(d) $5 − μ m$ offset. (a),(c) Off state. (b),(d) On state. (e),(f) The reconstruction of the input wave front at the full coupling position in the output [white dashed lines in (a)–(d)]. The results are obtained by transfer-matrix calculations utilizing full-wave eigenmode simulations in comsol (Appendix pp2). All other parameters are the same as those in Figs. 2 and 2.
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Figure 5
Binary random wave switching in the SUSY harmonic pair: (a) off and (b) on states. (c) The reconstruction of the input wave front at the full coupling position in the output [dotted lines in (a),(b)]. (d) The power distribution in input and output waveguides at the full coupling position as a function of the modulation $Δ n$ . Error bar denotes the standard deviation for the ensembles of 400 random input waveform realizations. The results are obtained by transfer matrices linked with comsol eigenmode simulations (Appendix pp2). All other parameters are the same as those in Figs. 2 and 2.
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Figure 6
Multilevel switching in the SUSY Pöschl-Teller pair. (a) The off state from the decoupling between the Pöschl-Teller potential (output) and its SUSY partner (input). (b) One of the many-valued on states from the coupling (black dotted arrow) through the modulated output (red arrow). (c) Effective indices $n_{eff}$ of each waveguide before the coupling. (d) Level spacing of $n_{eff}$ in the original [blue-green lines, $ϵ_{o} (x) = 8.6 + 1.7 {sech}^{2} (5.5 \times 10^{- 2} k_{0} x)$ ] and SUSY [red-brown lines, $ϵ_{s} (x) = 8.6 + 1.6 {sech}^{2} (5.4 \times 10^{- 2} k_{0} x)$ ] potentials. (e) The 14th truth ( $Δ n = 9 \times 10^{- 3}$ ) and (f) zeroth truth ( $Δ n = 2.1 \times 10^{- 2}$ ) transparency for random wave incidences. (g) The transmission of each eigenmode (zeroth to 14th) in the output waveguide at the full coupling position as a function of the modulation $Δ n$ . Error bar denotes the standard deviation for the ensembles of 400 random samples. All other parameters are the same as those in Fig. 2.
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