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- Photonics Research
- Vol. 6, Issue 4, A18 (2018)

Abstract

1. INTRODUCTION

The concept of parity-time (PT) symmetry in optics has attracted a great deal of attention following the first experimental demonstrations only a few years ago ^{[1,2]}. It suggests new prospects for the design of optical structures with symmetrically distributed regions of gain and losses; see review papers ^{[3–5]} and references therein.

Recently, PT-symmetric lasers were demonstrated experimentally in microrings ^{[6–9]}. Importantly, a single-mode laser operation was achieved in an initially multi-mode system through stronger mode discrimination, when the system operates in the PT-broken regime close to a PT-symmetry-breaking transition. Also, a single transverse-mode operation in a system of coupled microring lasers was demonstrated near the exceptional point ^{[10]}, and enhanced sensitivity ^{[11,12]} was realized. In a pair of coupled microdisk quantum cascade lasers, the reversal of generated power dependence ^{[13]} was identified in the vicinity of exceptional points (EPs), where spontaneous emission is enhanced ^{[14]}, and EPs were observed directly in photonic-crystal lasers ^{[15]}. Furthermore, a realization of PT-symmetry-based mode locking ^{[16]} and a PT-symmetric laser absorber ^{[17]} were theoretically proposed, and lasing and anti-lasing in a single cavity were demonstrated ^{[18]}. Recently, a statistical PT-symmetric lasing has been achieved in an optical fiber network incorporating semiconductor amplifiers ^{[19]}.

In this work, we develop the theoretical concept of a PT-symmetric fiber laser based on coupled cavities. We use a conventional ring cavity laser design and predict PT-symmetric and PT-broken lasing by means of tunable coupling of a ring cavity laser with a passive cavity.

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2. METHODS AND RESULTS

We present a diagram of the proposed PT-symmetric fiber ring laser in Fig.

Figure 1.Schematic of the proposed PT-symmetric fiber-ring laser, composed of two coupled fiber-ring cavities with gain and losses. The coupling between cavities is controlled by means of phase shifts

To describe the laser operation and associated PT transitions, we use a discrete transfer matrix model governing the evolution of the field amplitudes in both fiber cavities. Field evolution over a cavity round-trip in the first approximation is described by a transfer operator ^{[20]}.

We simplify Eq. (

We establish that the operator

The parity operator swaps the two fibers, which effectively interchanges the gain and phase coefficients since

The time-reversal operator

The signal propagation through the considered system can be fully described by the eigenvalues

We note that there is an exceptional point where the two eigenvalues become identical (

We present in Fig.

Figure 2.(a) Ratio of two linear mode eigenvalues,

The modes are in the PT-symmetric regime for

In the PT-broken regime when

We plot in Figs.

Essentially, the model with a constant gain as represented above describes an amplifier operating in the regime of small signal gain, not a laser. The laser is characterized by self-governing gain saturation that limits the total gain over the round trip to be equal to total round-trip losses ^{[21]}. In the following, we consider a simple model of gain saturation, where

For lasing to occur, small-amplitude fields need to be amplified, i.e., it is necessary to have

Figure 3.(a) Stationary regimes of laser operation with nonlinear gain saturation: no lasing (white background), pair of PT-symmetric laser modes (grey shading), or one mode in PT-broken regime (yellow shading). (b), (c) Characteristic mode amplification versus power for points A and B marked in (a) corresponding to different lasing regimes. Solid circles mark stable and the open circle marks unstable regimes with balanced gain and loss (zero mode amplification). Background shading marks PT-symmetric and broken regimes. Saturable gain parameter

A characteristic dependence of mode amplification on power in the PT-symmetric laser regime is presented in Fig.

Measurements of eigenvalues in a real laser are not straightforward. The most direct way to characterize the PT transition is to measure generated powers in both active and passive fiber cavities,

Figure 4.Power dependencies in stationary lasing regimes. (a) A ratio of power generated in passive and active cavities,

In a conventional laser, the higher the optical losses, the less is a generation power. We find that this is not always the case in a considered system of fiber laser with a PT symmetry. Indeed, when optical losses ^{[19]}.

Figure 5.(a) Dependence of generated power

We now analyze how the laser reaches its stationary state, i.e., the laser dynamics. To establish the general dynamical properties, we study the evolution of the following quantity:

Considering complex amplitudes

Thus, the following quantity remains invariant during evolution, both in PT and PT-broken regimes:

Accordingly, the relative phase is confined to one of two domains, depending on the initial conditions,

We show these regions with different shadings in Fig.

Figure 6.Dynamical properties of a PT-symmetric fiber laser. (a) Two trapping regions shown with shading according to Eq. (

The horizontal lines with

Stationary PT-symmetric modes can appear along the vertical lines at

3. CONCLUSION

In conclusion, we proposed and analyzed theoretically a PT-symmetric fiber-ring laser composed of two active and passive cavities with a cross-coupling element, which allows us to switch between PT-symmetric and broken regimes without using active modulation of gain/loss elements. We considered the effect of gain saturation at high powers and predicted that the system always converges to a stationary lasing state, while demonstrating bi-stable behavior in the PT-symmetric regime. We also revealed that the generated power nontrivially depends on the optical losses, as in PT-broken regime the lasing power increases for stronger losses, whereas lasing can be completely suppressed for intermediate loss levels between the PT-symmetric and PT-broken regions.

Acknowledgment

**Acknowledgment**. This work was supported by the Russian Science Foundation. S. V. Suchkov and A. A. Sukhorukov are supported by the Australian Research Council.

References

[16] S. Longhi. PT-symmetric mode-locking**. Opt. Lett., 41, 4518-4521(2016)**.

[17] S. Longhi. PT-symmetric laser absorber**. Phys. Rev. A, 82, 031801(2010)**.

[19] A. K. Jahromi, A. U. Hassan, D. Christodoulides, A. Abouraddy. Statistical parity-time-symmetric lasing in an optical fiber network**. Nat. Commun., 8, 1359(2017)**.

Sergey V. Smirnov, Maxim O. Makarenko, Sergey V. Suchkov, Dmitry Churkin, Andrey A. Sukhorukov. Bistable lasing in parity-time symmetric coupled fiber rings[J]. Photonics Research, 2018, 6(4): A18

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