Supersymmetric laser arrays

R. El-Ganainy, Li Ge, M. Khajavikhan, and D. N. Christodoulides

Phys. Rev. A 92, 033818 – Published 11 September 2015

Abstract

We introduce the concept of supersymmetric laser arrays that consist of a main optical lattice and its superpartner structure, and we investigate the onset of their lasing oscillations. Due to the coupling of the two constituent lattices, their degenerate optical modes form doublets, while the extra mode associated with unbroken supersymmetry forms a singlet state. Singlet lasing can be achieved for a wide range of design parameters, either by introducing stronger loss in the partner lattice or by pumping only the main array. Our findings suggest the possibility of building single-mode, high-power laser arrays and are also important for understanding light transport dynamics in multimode parity-time symmetric photonic structures.

Received 8 December 2014

DOI:https://doi.org/10.1103/PhysRevA.92.033818

Authors & Affiliations

R. El-Ganainy^*

Department of Physics, Michigan Technological University, Houghton, Michigan 49931, USA

Li Ge

Department of Engineering Science and Physics, College of Staten Island, CUNY, Staten Island, New York 10314, USA and The Graduate Center, CUNY, New York, New York 10016, USA

M. Khajavikhan and D. N. Christodoulides

College of Optics & Photonics–CREOL, University of Central Florida, Orlando, Florida 32816, USA

^*ganainy@mtu.edu

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Vol. 92, Iss. 3 — September 2015

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Images

Figure 1
A schematic of possible physical realizations of supersymmetric laser arrays using (a) a waveguides platform and (b) optical resonator configurations.
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Figure 2
Conceptual demonstration of singlet lasing using a supersymmetric array. (a) A schematic of the individual spectra of an optical array and its superpartner that does not share the singlet state (with highest frequency). (b) The spectrum of the combined system. It consists of a set of doublets and one singlet. When optical loss is added to the superpartner lattice, supermode coupled-mode theory (SCMT) predicts that the singlet state (dashed line) will exhibit the lowest lasing threshold.
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Figure 3
(a) Eigenvalue spectrum of a supersymmetric array when $κ^{'} / \tilde{κ} = 0.2$ and $γ / \tilde{κ} = 0.06$ , where $\tilde{κ}$ is the uniform coupling coefficients between the cavities of the original array, $κ^{'}$ is the coupling constant between the innermost channels of the main lattice and its superpartner, while $γ$ is the uniform optical loss coefficient of the superpartner structure $A_{2}$ . Results obtained using exact diagonalization of the discrete system and from supermode coupled-mode theory (SCMT) are compared on the same figure. (b) Temporal evolution of an initial arbitrary optical intensity distribution in this SUSY array. As expected, only the fundamental eigenmode of $A_{1}$ survives as time lapses. Only the optical cavities of the array $A_{1}$ are depicted.
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Figure 4
The effect of the disorder on the maximum possible gain value before the second mode starts to lase, when the disorder is introduced to the coupling coefficients only or the resonant frequency only, are indicated by the blue dotted (middle) line and the green dashed line, respectively. The combined effects are shown by the red solid curve. The figure of merit is defined as the relative spacing between the gain threshold of the first and second lasing modes in the disordered case compared to the ideal case, i.e., $\frac{{(g_{1}^{t h} - g_{2}^{t h})}_{disorder}}{{(g_{1}^{t h} - g_{2}^{t h})}_{ideal}}$ . The system still displays a good range of single-mode operation at disorder levels of up to $10 %$ of the value of $\tilde{κ}$ . The current fabrication technique allows for building waveguide or cavity arrays with such accuracy.
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Figure 5
(a) Increased range of single-mode operation due to nonlinear spatial hole burning interactions. The system is the same as in Fig. 3, and we have considered a uniform loss $γ_{0} = γ / 5$ in all cavities. The top solid line shows the singlet state, and the other two solid lines show the doublet pair next to it in frequency. The dotted line indicates the effective gain of the lower-loss one of them in the absence of nonlinear interactions. The vertical dashed line marks the threshold of the singlet state. (b) Spatial profile of the lower-loss doublet state when $g_{s} = 0$ (top) and $γ$ (bottom).
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Figure 6
(a) Eigenvalue distribution of a supersymmetric array made of nine resonators (five of which belong to $A_{1}$ ) designed to eliminate the second-order eigenmode of $A_{1}$ from the spectrum of $A_{2}$ . In this example, $κ^{'} / \tilde{κ} = 0.2$ and $γ / \tilde{κ} = 0.5$ . The doublet state indicated by the dashed line reaches the lasing point before the singlet state, in contrary to what one would expect. This counterintuitive result is confirmed in (b), where an initial state made of an equal superposition of the five eigenmodes of $A_{1}$ (middle panel) evolves to the doublet eigenstates (upper panel) instead of the singlet (lower panel). Panel (c) presents the time-dependent dynamics for the projection coefficients $C_{n} (t)$ , defined by $\vec{φ} (t) = \sum_{n = 1}^{N} C_{n} (t) {\vec{V}}_{n}$ in the main array. $C_{n} (t = 0) = 1 / \sqrt{5}$ for all $n$ .
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