Optical fluxes in coupled $\mathcal{PT}$-symmetric photonic structures

Optical fluxes in coupled $PT$ -symmetric photonic structures

Li Ge, Konstantinos G. Makris, and Lingxuan Zhang

Phys. Rev. A 96, 023820 – Published 9 August 2017

Abstract

In this work we first examine transverse and longitudinal fluxes in a $PT$ -symmetric photonic dimer using a coupled-mode theory. Several surprising understandings are obtained from this perspective: The longitudinal flux shows that the $PT$ transition in a dimer can be regarded as a classical effect, despite its analogy to $PT$ -symmetric quantum mechanics. The longitudinal flux also indicates that the so-called giant amplification in the $PT$ -symmetric phase is a subexponential behavior and does not outperform a single gain waveguide. The transverse flux, on the other hand, reveals that the apparent power oscillations between the gain and loss waveguides in the $PT$ -symmetric phase can be deceiving in certain cases, where the transverse power transfer is in fact unidirectional. We also show that this power transfer cannot be arbitrarily fast even when the exceptional point is approached. Finally, we go beyond the coupled-mode theory by using the paraxial wave equation and also extend our discussions to a $PT$ diamond and a one-dimensional periodic lattice.

Received 15 June 2017

DOI:https://doi.org/10.1103/PhysRevA.96.023820

Physics Subject Headings (PhySH)

PT-symmetric quantum mechanics Waveguide arrays

Atomic, Molecular & Optical

Authors & Affiliations

Li Ge^1,2,*, Konstantinos G. Makris³, and Lingxuan Zhang^1,4

¹Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, New York 10314, USA
²The Graduate Center, CUNY, New York, New York 10016, USA
³Department of Physics, University of Crete, P. O. Box 2208, 71003, Heraklion, Greece
⁴State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi'an 710119, China

^*li.ge@csi.cuny.edu

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Vol. 96, Iss. 2 — August 2017

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Figure 1
Asymmetric power oscillations in a $PT$ dimer. [(a) and (b)] Intensities $I_{a, b}$ in the two waveguides, where $c_{+} = 1, c_{-} = 0.7$ and $τ / J = 0.5$ and 0.99, respectively. The transverse flux in (b) is shown in (d). (c) $z_{1}$ as a function of $τ / J$ .
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Figure 2
(a) The values of “giant amplification” for $τ / J$ up to 0.99 and $z$ up to $10 J^{- 1}$ . Those at $τ / J = 0.4$ and 0.8 are highlighted by white dashed lines. [(b) and (c)] “Giant amplification” (dashed lines) compared with the exponential amplification in a single gain waveguide (solid lines). $τ / J = 0.99$ in (b) and 0.5 in (c).
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Figure 3
“Classical” $PT$ transition in a $PT$ dimer. [(a) and (b)] Real and imaginary parts of the two complete propagation constant $λ_{\pm}$ . [(c) and (d)] Spatial profiles of these two guided modes at $τ = 0.005$ and 0.01, respectively. The two waveguides with gain (“ $g$ ”) and loss (“ $l$ ”) are marked by the shaded regions, with width $w = 6$ and center-to-center distance $2 d = 10$ .
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Figure 4
Power oscillation and transverse flux in the $PT$ dimer shown in Fig. 3. [(a) and (b)] Wave intensities with $τ = 0.002$ and 0.005, respectively. $c_{+} = 2, c_{-} = 1$ in Eq. (35). The colormap is the same as in Figs. 1 and 1. [(c) and (d)] The corresponding transverse flux, which are bidirectional when $τ = 0.002$ and unidirectional when $τ = 0.005$ .
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Figure 5
(a) Schematic of a $PT$ diamond. Solid and dashed lines represent coupling strengths $J \in R$ and $K = 2 J$ , respectively. Filled and open dots show the two sublattices. The numbers label the positions of the waveguides in the wave functions. (b) Identical intensity pattern of the two modes with $λ \neq 0$ in the $PT$ -symmetric phase. The color scheme is the same as in Figs. 1 and 1. [(c) and (d)] Real and imaginary parts of the four eigenvalues as a function of the gain and loss strength. Two of them are always zero.
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Figure 6
(a) Transverse fluxes in the eigenmodes of the $PT$ diamond. The upper two and lower two curves show the fluxes through the upper and lower bridging waveguides [2 and 4 in Fig. 5], with the solid lines for $Ψ_{\pm}$ (their fluxes are identical) and the dashed lines for $Ψ_{2}$ . $K = 2 J$ as in Fig. 5. (b) Transverse fluxes when $τ = 3 J, c_{-} = 1$ , and $r = 2$ in Eq. (43), with the upper curves showing $J_{t, 12}$ (lighter green), $J_{t, 23}$ (darker green) and the lower curves showing $J_{t, 14}$ (lighter blue), $J_{t, 43}$ (darker blue). They are unidirectional in this case. (c) Power oscillations for the case shown in (b). (d) An example of generalized power oscillations when the input wave contains $Ψ_{1, 2}$ also. The color scale in (c) and (d) is the same as in Figs. 1 and 1.
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Physical Review A

covering atomic, molecular, and optical physics and quantum information

Optical fluxes in coupled $PT$ -symmetric photonic structures

Li Ge, Konstantinos G. Makris, and Lingxuan Zhang

Phys. Rev. A 96, 023820 – Published 9 August 2017

Abstract

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Physical Review A

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Optical fluxes in coupled PT-symmetric photonic structures

Li Ge, Konstantinos G. Makris, and Lingxuan Zhang

Phys. Rev. A 96, 023820 – Published 9 August 2017

Abstract

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Optical fluxes in coupled $PT$ -symmetric photonic structures