In this talk, we use a KKE argument to show that the presence of dispersion alters the eigen-frequencies of coupled PT coupled microresonators, so that their eigen-frequencies are no longer complex conjugates after the EP

Benson, Trevor M.

Creagh, Stephen C.

Gradoni, Gabriele

Phang, Sendy

Sewell, Phillip

Vukovic, Ana

Nottingham eTheses

   Parity-Time coupled microresonators: Kramers-Kronig limitation   S. Phang1,*, A. Vukovic1, S. Creagh²,  T. M. Benson1, P. D. Sewell1, and G. Gradoni2  1.  George Green Institute for Electromagnetics, Faculty of Engineering, University of Nottingham, NG7 2RD Nottingham, UK 2.  School of Mathematical Sciences, University of Nottingham, NG7 2RD Nottingham, UK *sendy.phang@nottingham.ac.uk   New complex photonics structures, namely parity-time (PT)-symmetric structures, have inspired the innovative exploration of photonic structures with combined gain and loss. PT-symmetric  structures are identified by their signature exceptional point (EP); operation below this point yields a completely real eigenmode/frequency, but above this point the eigen-frequencies are complex conjugates [1], [2]. Several PT- symmetric structures, such as coupled waveguides and Bragg gratings, have been studied theoretically and experimentally, and exhibit exciting behaviour and components such as unidirectional loss-induced transparency, a coherent perfect absorber laser, and a photonic analogue of a topological insulator.    Fig. 1 Illustration of coupled micro-resonators, each with radius 𝑎, separated with a gap 𝑔 and surrounded by air. 𝜇𝑅𝐺 acts as an active resonator and 𝜇𝑅𝐿 as a lossy resonator,.   In passive structures, the relation between dispersion and losses is described by the Kramers-Kronig equations (KKE). Recently, the validity of those equations has been confirmed for PT-symmetric structures under a broadband excitation[3]. In this talk, we use a KKE argument to show that the presence of dispersion alters the eigen-frequencies of coupled PT coupled microresonators (Fig. 1), so that their eigen-frequencies are no longer complex conjugates after the EP. In a dispersive material the gain/loss spectra align to the resonant frequency only at a single frequency. Fig. 2(a) shows the real part of the eigen-frequencies of mismatched PT-coupled micro-resonators for material with gain/loss spectra centred at 𝜔𝜎 = 2𝜋(𝑓0 + 𝛿), for 𝛿 = 0.05 THz. It is noted that the coupled system no longer has an EP; the imaginary part of the eigen-frequencies splits early compared to the matched case, 𝛿 = 0. The difference in the real part of eigen-frequency for a fixed gain/loss parameter 𝛾𝜎 = 7.5 rad/ps, is denoted by Δ (see Fig. 2(a)) and is shown in Fig. 2(c). The figure shows that the operation of PT-coupled micro-resonators with practical parameters requires that the gain/loss spectra align with the resonant frequency of the resonator.      Fig. 2(a,b) Real and imaginary part of the eigen-frequencies of PT-coupled microresonators with mismatch between the material atomic transitional frequency and the resonant frequency where 𝜔𝜎 = 2𝜋(𝑓0 + 𝛿); (c) the difference of real part of eigen-frequency Δ (see Fig. 2(a)) for a fixed value of gain/loss parameter 𝛾𝜎 = 7.5 rad/ps and different mismatch parameter 𝛿. Operation in the (7,2) mode with resonant frequency 𝑓0(7,2)= 341.59 THz. 𝑎 = 0.24 𝜇m and 𝑔 = 0.24 𝜇m. For material model refer to [4].  References [1] C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity–time symmetry in optics,” Nat. Phys., vol. 6, no. 3, pp. 192–195, Jan. 2010. [2] K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Beam Dynamics in PT Symmetric Optical Lattices,” Phys. Rev. Lett., vol. 100, no. 10, p. 103904, Mar. 2008. [3] A. A. Zyablovsky, A. P. Vinogradov, A. V. Dorofeenko, A. A. Pukhov, and A. A. Lisyansky, “Causality and phase transitions in PT-symmetric optical systems,” Phys. Rev. A, vol. 89, no. 3, p. 033808, Mar. 2014. [4] S. Phang, A. Vukovic, H. Susanto, T. M. Benson, and P. Sewell, “Impact of dispersive and saturable gain/loss on bistability of nonlinear parity-time Bragg gratings.,” Opt. Lett., vol. 39, no. 9, pp. 2603–6, May 2014.  

Parity-Time coupled microresonators: Kramers-Kronig limitation

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Parity-Time coupled microresonators: Kramers-Kronig limitation

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