Coalescence of Anderson-localized modes at exceptional point in random media 

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Limits of Siegert’s modes

Carrying on the example of the 1D slab, we plot the spatial evolution of the modes inside and outside the cavity in Fig. 2.8(b). Unlike hermitian modes, the non-hermitian modes have a complex frequency as shown in eq. (2.39). The imaginary part of the complex frequency, standing for the linewidth of the resonance, has to be negative to ensure temporal decay of the field intensity. The spatial amplitude of the modes is bounded inside the slab (see Fig. 2.8(b)). However, because of negative imaginary part of the complex frequency, the amplitude of all modes exponentially diverges outside the system (see Fig. 2.8(b)).
Physically, this exponential divergence corresponds to a wavefront excited at past times and propagating away from the system. The infinite energy can be understood as the accumulation of the energy radiated from the open system to the rest of the universe. This point stresses that space and time cannot be separately considered like in hermitian stationary cases: The spatial divergence is compensated by a temporal ”damping”. This spatial divergence of Siegert’s modes stands for the main limitation of Siegert states and requires specific mathematical investigations.

Biorthogonal formalism

In non-hermitian problems, the modes Φp are not orthogonal hΦp|Φqi 6= δpq (2.44) However, a projection operator is needed to develop linear algebra with non-hermitian modes. In non-hermitian problems it is possible to introduce a different product known as the biorthogonal product [54, 55]. The biorthogonal product relies on a very simple idea: The orthogonality of left and right eigenvectors of a linear operator. If we consider a non-hermitian matrix A, with eigenstates (λi, |Xii): ∀i A|Xii = λi|Xii.

Anderson-localized modes

We shown in section 2.2.1 how modes can be derived in any open system using the Siegert’s condition. In this section, we consider an open system with a disordered refractive index distribution, where the random scattering may lead to the spatial confinement of light. First, we briefly review the history of this physical effect known as Anderson localization. Then, using a 1D example, we show that modes can be extended or spatially localized, depending on the strength of the disorder. Finally, we summarize the different numerical methods that we have developed to compute these modes in disordered media.

A brief introduction to Anderson localization

In his seminal paper [56], Anderson was inspired by experiments performed by George Feher [57], where anomalous relaxation times of electron were observed in semiconductors. Using a quantum tight binding model of a lattice with a random potential in each site, he demonstrated that diffusion of electrons can go to a zero when disorder becomes important enough. In particular, this model has been used to explain why a metal can turn into an insulator when the density of impurities increases. In the eighties, the gap was bridged between quantum and classical waves. After an early prediction of existence of localized waves in classical systems [58], Anderson localization was demonstrated for classical waves in several experiments [59, 60, 61]. It is now recognized that Anderson localization originates from the interference between multiple scattering paths and plays also an essential role in classical wave physics.
A naive picture of localization mechanism is proposed in Fig. 2.9(a). We consider an incoming wave propagating in a 1D random potential. The wave is scattered each time it encounters a step in the random potential (see explanation in Chapter 1). The wave is spilt into a transmitted (forward-scattering) and a reflected wave (backscattering). The amplitude of the backscattering is triggered by the height of the step in the random potential.
The backscattered wave interferes with the incoming wave. If the wave encounters many steps of various amplitudes, the backscattering leads to a localization of the wave by constructive interference (see Fig. 2.9(a)). This spatial localization, known as Anderson localization, differs from trapping where light is confined because of presence of walls (see Fig. 2.9(b)). Localization is rather understood as the result of many reflections of moderate amplitude.

Modes in localized/weakly scattering regimes

As stated in section 2.3.1, the Anderson localization is triggered by the disorder. To emphasize this influence, we introduce disorder into an uniform 1D system and progressively increase its ”strength”. In this 1D example, we differentiate between two different kinds of modes resulting from Anderson localization.

Manipulation of modes via the dielectric permittivity

In this section we propose a general theory, which describes the evolution of modes in an open system in which scattering is modified. This approach relies on the biorthogonal formalism introduced in Chapter 2 and applied to modes of 2D dielectric open systems. We stress that this approach is not limited to disordered media but can be used for any open inhomogeneous dielectric system. First, we define modes of a non-hermitian system and recall the condition of the use of the biorthogonal formalism. Finally, we consider a modification of the system and investigate the evolution of modes by deriving a linear system.

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2D open dielectric medium

We first consider the general case of a finite-size dielectric medium in 2D space, with inhomogeneous dielectric constant distribution ǫ(r). In this chapter, for sake of notation compactness, we will consider the dielectric permittivity ǫ(r) rather than the index of refraction n(r) (ǫ(r) = n2(r)). The distribution of ǫ(r) is indifferently ordered or disordered. In the frequency domain, the electromagnetic field follows the Helmholtz equation: ΔE(r, ω) + ǫ(r)ω2E(r, ω) = 0

Table of contents :

I Light In Open Dielectric Media 
1 Light-matter interaction: Semiclassical description 
1.1 Light-Matter interaction: Light propagation in matter
1.1.1 Light at microscopic scale
1.1.2 Light at macroscopic scale
1.1.3 Propagation in dielectric media
1.2 Light scattering by a particle
1.2.1 Introduction
1.2.2 Scattering media
1.3 Light-Matter interaction: Matter excitation
1.3.1 Energy conversion transfer
1.3.2 A two-level atom in an electromagnetic field
1.3.3 Four-level atomic system
1.4 Summary
2 Modes in non-hermitian systems: The specific case of open random media 
2.1 Introduction to Modes in hermitian/non-hermitian systems
2.1.1 Stationary solutions of hermitian systems
2.1.2 Resonances of non-hermitian systems
2.1.3 Fingerprint of hermitian/non-hermitian systems
2.2 Modes in open system
2.2.1 Deriving modes in open media
2.2.2 Limits of Siegert’s modes
2.2.3 Biorthogonal formalism
2.3 Anderson-localized modes
2.3.1 A brief introduction to Anderson localization
2.3.2 Modes in localized/weakly scattering regimes
2.3.3 Numerical computation of modes
2.4 Summary
8 Table of Contents
II Passive Random Media 
3 Coalescence of Anderson-localized modes at exceptional point in random media 
3.1 Manipulation of modes via the dielectric permittivity
3.1.1 A 2D open dielectric medium
3.1.2 Modification of the permittivity
3.2 Application to Anderson-localized modes: Prediction of Exceptional Points
3.2.1 The 2D open disorder dielectric medium
3.2.2 Original modes and biorthogonal product
3.2.3 Exceptional Point between two Anderson-localized modes
3.2.4 FEM validation
3.3 A complex N-mode process
3.3.1 Mulimode process
3.3.2 Modes in the vicinity of an EP
3.3.3 Multiple EP and potential applications
3.4 Summary
4 Linear and non-linear Rabi oscillations of two-level systems resonantly coupled to an Anderson-localized mode 
4.1 A two-level system coupled to the electric field
4.1.1 Coupled levels
4.1.2 Oscillation of populations
4.1.3 Linear vs nonlinear polarization
4.2 Linear Rabi regime: Strong coupling
4.2.1 A two-level system coupled to a 2D Anderson-localized mode
4.2.2 Strong coupling and Rabi oscillations
4.2.3 Linear Rabi regime condition
4.3 Non-linear Rabi regime
4.3.1 A two-level atom in an Anderson-localized mode … with external excitation
4.3.2 Non-linear Rabi regime
4.3.3 Non-linear Rabi regime condition
4.4 Coexistence of both regimes in a realistic experiment in the temporal domain
4.4.1 Setup
4.4.2 Linear/Non-linear regimes in the transient regime
4.4.3 Numerical investigation
4.5 Summary
III Active Random Media 
5 Introduction to random laser: Basic concepts and experimental achievements
5.1 The ”photonic bomb” model
5.1.1 From conventional to random laser
5.1.2 A scattering process … with gain
5.2 The optofluidic random laser
5.2.1 Advantages of optofluidic devices
5.2.2 Fabrication process
5.2.3 1D optofluidic random laser
5.2.4 2D optofluidic random laser
5.3 An energetic model for random laser … an incomplete description
5.3.1 An energetic model
5.3.2 Incoherent random laser
5.3.3 Coherent random laser: Need of a modal description
5.4 Summary
6 Modes in random lasers: Below and Above threshold 
6.1 Below and Above threshold description of random lasers
6.1.1 Introduction
6.1.2 Modelling the random laser
6.2 Active mode, below threshold
6.2.1 Modal expansion
6.2.2 Broadband gain medium
6.2.3 Narrow gain medium
6.2.4 Numerical computation
6.3 Lasing modes, above threshold
6.3.1 Modal expansion
6.3.2 Perturbation expansion
6.3.3 Lasing modes
6.4 Summary
IV Control Of Random Lasers 
7 Adaptive pumping for the control of random lasers: Numerical investigation
7.1 Early achievements of local pumping
7.2 Taming random laser emission through the pump profile: Threshold optimization
7.2.1 Numerical system
7.2.2 Optimization in the localized regime
7.2.3 Optimization in the weakly scattering regime
7.3 Below threshold modal expansion
7.3.1 Principle of the below threshold pump profile optimization
7.3.2 Threshold optimization
7.3.3 Directivity optimization
7.4 Summary
8 Adaptive pumping for the control of random lasers: Experimental investigation
8.1 1D optimization
8.1.1 Experimental Setup
8.1.2 Optimization results
8.1.3 Optimization mechanism
8.2 2D optimization
8.2.1 Experimental setup
10 Table of Contents
8.2.2 Experimental results
8.2.3 Remarks and further work
8.3 Summary
Conclusion
Appendices
A Polarizability and susceptibility of a particle in 2D 
B Transfer Matrix Approach: Stationary and Travelling components 
C Perturbation Expansion of non-hermitian eigenvalue problem 
C.1 Perturbation expansion in the below threshold regime: Linear eigenvalue problem
C.2 Perturbative approach above threshold: non-linear case
D Optimization via Simplex Algorithm 
References

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