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- Photonics Research
- Vol. 6, Issue 5, A31 (2018)

Abstract

1. INTRODUCTION

Photonic device designers have been toying with the idea of using gain and loss in linear ^{[1]} and nonlinear ^{[2]} optical systems in order to produce unidirectional couplers or low intensity switches, in that order, for a long time. The advent of parity time (^{[3,4]}, the idea that non-Hermitian Hamiltonians invariant under space–time reflection might possess a real spectrum, brought a new structure to these approaches. Two ideas made use of this quantum mechanical tool to describe a planar slab waveguide ^{[5]} and coupled optical structures with symmetric gain and losses ^{[6]}. The latter became seminal for the field of ^{[7,8]}.

The quintessential ^{[9–13]}. It can be understood as a finite, nonunitary realization of the Lorentz group ^{[14]} allowing for three types of propagation dynamics: periodic field amplitude oscillation with amplification, as well as linear and exponential field amplitude amplification. The first case corresponds to the ^{[14]} and nonlinear systems ^{[7]}.

We are interested in a natural extension of the nonlinear ^{[15]}. The nonlinear four-waveguide array is known to produce an asymmetric distribution of optical power ^{[16]}, to possess continuous families of nonlinear modes ^{[17]}, and it is not ^{[18]}. The absence of exceptional points, for homogeneous arrays of dimension four, was discussed using linear arrays of four subwavelength waveguides ^{[19]}, while the possibility to restore ^{[20]}.

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Here, we will construct a slightly more general system where the standard ^{[21]}. Then, we diagonalize our coupled mode model in the cyclic group basis to show that its dynamics is consistent with those of uncoupled effective dimers with variable couplings. We discuss the eigenvalues of these effective dimers and show that homogeneously distributed necklaces with even numbers of copies always have at least a pair of imaginary eigenvalues and

Figure 1.Sketch for necklaces formed by the repetition of identical

2. LINEAR MODEL AND ITS UNDERLYING SYMMETRY

The coupled mode formalism for a tightly bound necklace formed by ^{[15]} by allowing different intra(inter)-dimer couplings,

The coupled mode equations in Eq. (

The mode-coupling matrix is represented by the operator ^{[7,14]}, the Lorentz group in

In addition, our necklace is invariant to rotations of

Note that the adjoint of the shift operator is given by the relation

This underlying symmetry will also manifest in the field mode amplitude vector, Eq. (

3. PT SYMMETRY AND PROPAGATION IN THE LINEAR DIMER

We can take advantage of the underlying symmetries of our necklace. It is well known that the Fourier matrix ^{[22]}, diagonalizes the cyclic group shift operator into the so-called clock matrix,

Note that we have obviated the dimension subindex

Thus, we can suggest a change of vector basis provided by the Fourier matrix basis, and recover effective propagation dynamics, described by a block-diagonal mode-coupling matrix, where each diagonal block describes a

Here, it is straightforward to calculate the eigenvalues of these matrices, and realize that different intra- and inter-dimer complex couplings can restore the ^{[15]}. This phase can be restored using inhomogeneous intra- and interdimer couplings, ^{[23]}, suggesting a possible stabilizing role for such couplings.

In homogeneously coupled arrays, the block

Figure 2.Real (solid lines) and imaginary (dashed lines) parts of eigenvalues as a function of the gain to interdimer coupling ratio,

The block diagonal, ^{[24]}, where we have used the cardinal sine function, ^{[25]}, Finally, it takes purely imaginary values in the broken symmetry regime, providing exponential amplification and attenuation of the field amplitudes,

4. DIMER OUTPUT REPLICATION

Cyclic symmetry has a curious signature. Whenever such a symmetry is present, we can address specific block diagonal elements and produce

It is well known that outside the ^{[7,14]}, dictated by the parameters of the corresponding effective dimer

This type of dimer output port replication could be observed in any physically plausible realization of our necklace. For example, let us consider the case of laser inscribed waveguides in silicon, c:Si, for telecommunication wavelength, ^{[26,27]}. We use the commercial finite element software COMSOL to simulate necklaces of two and three identical copies of a passive

Figure 3.(a) Waveguide geometry used in the finite element simulation and electric field input for replication of the (b) first and (c) second effective dimers. In (b) and (c), red corresponds to an in-phase beam and blue corresponds to a

The two-dimer system can be described by the coupled mode equation, where the imaginary part of the propagation constant,

This array can be mapped to a necklace composed by two

In the homogeneous coupling configuration, addressing the first effective dimer provides oscillations of the renormalized power with approximated propagation constant

Figure 4.Renormalized power propagation,

For the sake of completeness, we also simulate a necklace of three dimers in the homogeneous and inhomogeneous configurations. The intra- and inter-dimer distances are the same as those used for the four-waveguide necklace. In the homogeneous configuration, these distances are

Figure 5.(a) Waveguide geometry used in the finite element simulation and electric field input for replication of the (b) first and (c) second effective dimers. In (b) and (c), red corresponds to an in-phase beam, orange corresponds to a

In the homogeneous case, we find effective coupling values

Figure 6.Renormalized power propagation,

5. CONCLUSIONS

We have presented a group theoretical analysis for the mode-coupling description of light propagating through so-called necklaces of

We compared our effective mode-coupling analysis with finite element simulations to obtain good agreement in all cases studied. Our analytic results and computational simulations show that it is possible to use impinging light to switch between the different effective dynamics provided by a necklace of

As a final note, we want to emphasize that port replication is an intrinsic characteristic of the underlying cyclic symmetry. Thus, we could replace the

Acknowledgment

**Acknowledgment**. D. J. Nodal Stevens acknowledges financial support from Red de Tecnologías Cuánticas through the program Verano de Investigación Científica and the Photonics and Mathematical Optics group hospitality. B. M. Rodríguez-Lara acknowledges support from the Photonics and Mathematical Optics Group at Tecnologico de Monterrey and Consorcio en Óptica Aplicada through CONACYT FORDECYT #290259 project grant.

References

[4] C. M. Bender. Introduction to PT-symmetric quantum theory**. Contemp. Phys., 46, 277-292(2005)**.

[7] J. D. Huerta Morales, J. Guerrero, S. Lopez-Aguayo, B. M. Rodríguez-Lara. Revisiting the optical PT-symmetric dimer**(2016)**.

[20] S. Longhi. PT phase control in circular multi-core fibers**. Opt. Lett., 41, 1897-1900(2016)**.

[25] T. Kato**. Perturbation Theory for Linear Operators(1995)**.

D. J. Nodal Stevens, Benjamín Jaramillo Ávila, B. M. Rodríguez-Lara. Necklaces of PT-symmetric dimers[J]. Photonics Research, 2018, 6(5): A31

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