Counter propagating Airy beams in photorefractive medium 

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Non conventional beams and nonlinear media

In this section we will detail and introduce more specific notions regarding this thesis such as non-conventional beams, nonlinear media, spatial solitons and imprinting in photorefractive crystals.

Non-conventional beams

In geometrical optics, the propagation of light is seen as series of rays propagating in strait lines from an emitting light source and through different optical devices that can change the direction of individual rays. From this standpoint, we can explain many properties such as mirrors, lenses, total internal-reflection or the formation of rainbows. This works as long as the illuminated objects are much larger than the wavelength of the illuminating beam. If an object is of the size of the wavelength, diffraction occurs and interesting phenomenons can take place. The Huygens-Fresnel principle states that every point of the wavefront is itself the source of spherical wavelets that interfere to form the propagating wavefront. In figure 1.5, the wavefront propagates in a medium of refractive index n0 (blue), at the speed v ˘ c/n0. At the interface, by considering every point of the wavefront as an emitter of spherical waves, the wavefront is reconstructed in the medium of refractive index n (green). The angle is due to the difference of propagating speed of the wave between the two media. The speed is slower in the second medium v ˘ c/n, so the reconstructed wavefront is tilted.
Using complex diffraction patterns, the transverse profile of a propagating wave-front can be reorganized and structured. In doing so we can create what is called non-conventional beams that give the impression of being diffraction-free because they preserve their profile along propagation. Bessel beams, Mathieu beams and Airy beams are examples of non-diffracting beams [12]. Figure 1.6 shows a plane wavefront travers-ing an axicon. The axicon divides the wavefront in two new wave fronts propagating at an angle. These two wavefront interfere constructively in the red area creating a Bessel beam. The Bessel beam preserves its profile along propagation and its propagation length depends on the plane waves transverse size. A common way to generate complex diffraction patterns is by using a spatial light modulator (SLM). This device formed by a matrix of small cells can be remotely controlled to change the wavefront of an incoming light beam. The beam will illuminate the SLM and by reflection or transmittance, the wavefront can be shaped to our choosing. This phase modulation can be different for each pixel of the SLM. Therefore, the resolution of the SLM will determine the number of points we use to reconstruct the wavefront using the Huygens-Fresnel principle.
Our group has taken interest in the most recent non-conventional beam, the Airy beam seen in figure 1.7. This beam has been extensively studied in the last twelve years since their first experimental observation in 2007 [13, 14]. The idea of the Airy beam originates from the field of quantum mechanics in 1979 with the work of Berry and Balazs [15]. The Airy beam has three interesting properties [16]; it is diffraction-free, propagates along a parabolic self-accelerating trajectory, and has self-healing properties [17]. Our work consists in studying the Airy beam, their generation, their nonlinear propagation and interactions in photorefractive crystal and possible applications such as waveguiding devices. In our case the advantage of studying non conventional beams is threefold. First they offer diverse ways to propagate light, for example inside a nonlinear photorefractive crystal, and can create complex waveguiding structures. Secondly they challenge the theoretical and numerical models used in anisotropcic photorefractive media, challenging our understanding of nonlinear effects. Finally they can help understand the transition dynamics from linear to nonlinear propagation. The Airy wave function is a spatial solution of the wave equation but it is also a temporal solution of the wave equation. By considering spatiotemporal dynamics, different beams that propagate undistorted in a 3D space-time environment can be derived from the Airy beam such as the Airy-Bessel beam [13, 18], or Airy light bullets [19, 20], and more [21, 22]. The combinations of temporal or spatial Airy beams or other non-conventional beams have greatly widened the possible applications for Airy beams in the field of photonics.

Linear and nonlinear media

When light illuminates a material, the light’s electric wave induces a movement of the electron charges inside the medium which results in the creation of dipoles and/or modifications of existing dipoles p(t) ˘ qd(t) (with q the displaced charge and d(t) the distance of displacement over time). Under the influence of the electric field, the dipoles oscillate and radiate their own electromagnetic field. For example, if the dipoles move in accordance with the light’s electric wave (ie. they oscillate at the same frequency and are in phase), they have no impact on the light’s electric field and the medium is considered transparent. On the contrary, if the electric field oscillates at a speed that the dipoles have difficulty matching, their oscillation amplitude is dampened, and the electric field is absorbed, the media appears opaque.
In optics, the terms linear and nonlinear media refers to the way polarization density p responds to the light’s electric field E inside the media. In the case of a linear response, the polarization density can be expressed p(t) ˘ †0´(1)E(t). In the case of a nonlinear response, the polarization density is more complex:
With †0 the electric permittivity of free space and ´ the electric susceptibility which characterizes the interactions of the nonlinear material. In the case of isotropic materials the susceptibility is the same in all direction. However, in the case of birefringent materials, the susceptibility will be related to preferential axes in the material, and be instead a susceptibility tensor. This is notably the case of photorefractive materials discussed further on.
In nonlinear media, if ´(2) 6˘0, the second term to the right of the equation represent-ing interactions of the order of the electric field squared can be considered. Therefore, by injecting two different light electric fields, their product will be able to interact within the crystal. This interaction is called three wave mixing. In function of the two electric field’s wavelengths, we obtain second-harmonic generation, sum-frequency generation or difference-frequency generation. This leads to applications such as optical parametric oscillators or phase-conjugated mirrors.

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NON CONVENTIONAL BEAMS AND NONLINEAR MEDIA

If ´(2) 6˘0 (respectively ´(3) 6˘0), ¢n is proportional to E (respectively E2) and the induced nonlinear effect is called the Pockels effect (respectively the Kerr effect). The fact that the refractive index variation varies with the electric field means that the refractive index will only change where light propagates. If the variation is positive (respectively negative) the propagation will undergo self-focusing (respectively defocusing).

Spatial solitons

In particular, if the focusing effect cancels out the diffraction of the propagating beam, self-trapped beams can propagate. These beams are called solitons and given the right nonlinear conditions, they preserve their profile when propagating over infinite distances. The name soliton was given to describe their particle-like nature, notably the way these solitary waves remained unchanged after collision [23]. They were first observed in 1966 in carbon disulfide [24] and the first observation of spatial solitons in dielectric media were done by Barthelemy et al. in 1985 in carbon disulfide (Kerr media) using high power light beams (» GW cm¡2) [25]. It had been established in 1965 that, in the case of Kerr circular solitons, over-focusing would tend to change the beam into filaments [26]. To avoid destructive over self-focusing of the propagating beam Barthelemy reduced the problem from three dimension to two dimensions. Instead of a three dimensional circular beam soliton he considered a two dimensional steady plane soliton, or (1+1)D soliton. The first spatial soliton in solid-state material was observed in 1990 using a single-mode planar glass waveguides [27] and in 1993 the first photorefractive soliton was observed, requiring less power to generate and propagate itself then the Kerr soliton [28, 29].
Due to the saturable nonlinear nature of photorefractive media, the photorefractive soliton contrarily to Kerr solitons can be stable in the two transverse dimensions enabling (2+1)D solitons to propagate. Due to saturation of the nonlinear focusing effect, the over-focusing observed in Kerr media is avoided in photorefractive media. The displaced charges relocate and the lensing effect instead of becoming stronger becomes wider [30, 31]. Therefore in photorefractive media, the particle-like interactions and collisions of soliton beams occur in three dimensions, and spatial solitons can spiral around each other [32]. The interactions in photorefractive media are therefore more rich and complex than in Kerr media where the interactions are in-plane attraction and repulsion [33]. Additionally, saturable nonlinearity means that inelastic collisions can be observed leading to soliton fusion or fission [30, 31]. The low power required to observe strong nonlinear effect, and the saturable nature of photorefractive crystals have made it our crystal of choice to study beam propagation.
(a) Linear propagation. (b) Nonlinear propagation in the presence of an external electric field leading to an OSS. (c), (d) Distribution of the refractive index change corresponding to (a) and (b) respectively. Arrows represent the Poynting vector.

Airy beams in nonlinear media

The propagation of non-conventional Airy beams in nonlinear media has been studied extensively in both Kerr media [34–36] and photorefractive media [35, 37–41] and to lesser extent in thermal nonlinear media [42]. The Airy beams in biased nonlinear media have interesting dynamics such as soliton-like behaviors [35, 37, 39–41] and interactions of co-and counter-propagating Airy beams [39–41]. With a refractive index gradient modifying the deflection of the propagating beam, in nonlinear medium, the deflection or acceleration of the Airy beam can be enhanced, reduced, or canceled [43]. Other interesting dynamics can be obtained using more complex index variations inside the photorefractive medium such as induced photonic lattices [44, 45].
Under focusing nonlinear conditions the Airy beam may split into a weak accelerating structure and a structure that has been named « off-shooting soliton » (OSS) and that propagates along the medium without transverse acceleration [46]. The first experimen-tal OSS was observed by Wiersma et al and the interactions between the photoinduced OSS and the accelerating beam have been studied [41], resulting in attraction, deflection and tightening effects of the OSS and interesting analogies with gravitational lensing and tidal forces.

Table of contents :

1 General Introduction 
1.1 Photonics
1.1.1 What is a Photon?
1.1.1.1 The MASER and the LASER
1.1.1.2 Fiber Optics
1.2 Non conventional beams and nonlinear media
1.2.1 Non-conventional beams
1.2.2 Linear and nonlinear media
1.2.2.1 Spatial solitons
1.2.2.2 Airy beams in nonlinear media
1.2.3 Imprinting in photorefractive media
1.2.4 Outline
2 Experimental generation of Airy beams 
2.1 An overview of Airy beams
2.1.1 What are Airy beams?
2.1.2 How are Airy beams generated?
2.2 Airy beam generation using LCOS SLM
2.2.1 LCOS SLM Technology
2.2.2 Influence of experimental conditions on the generation of Airy Beams
2.2.2.1 Border effects and pixelisation limit
2.2.2.2 Parameter range
2.3 Conclusions
3 Airy beam propagation in nonlinear media 
3.1 Physical concepts: Solitons and Photorefractive effect
3.1.1 Solitons
3.1.2 Photorefractive crystals
3.1.3 Photorefractive soliton
3.2 Finite Airy beam propagation in photorefractive media
3.2.1 Airy beams in photorefractive media
3.2.2 Experimental propagation of 1D Airy beam
3.2.3 Numerical analysis
3.3 Conclusion
4 2D Airy beam propagation in photorefractive media
4.1 Propagation dynamic and soliton formation in two-dimensional saturable nonlinear media
4.1.1 Experimental observation of 2D Airy beam propagation in photorefractive media
4.2 Self-bending of the OSS and time considerations
4.3 2D Airy beam propagation behavior and soliton existence curve
4.4 Conclusion
5 Counter propagating Airy beams in photorefractive medium 
5.1 Overview of beam interactions and interconnects in photorefractive media
5.2 Numerical Antisymmetric Airy beam propagation in nonlinear self-focusing conditions
5.2.1 Antisymmetric Airy beams interactions scheme
5.2.2 Optical interconnections for a transverse shift d=1
5.3 Stability of the photoinduced waveguides to input positions
5.4 Conclusion
6 Conclusion and Perspectives 
6.1 Conclusion
6.2 Perspectives
6.2.1 The use of nonconventional beams
6.2.1.1 Diverse ways to propagate light
6.2.1.2 Challenging the existing theoretical and numerical models
6.2.1.3 Transition dynamics from linear to nonlinear propagation
6.2.2 Off-shooting Soliton stability
6.2.3 Instability of counterpropagating waveguides
6.2.4 Experimental waveguiding of probe beams
6.2.5 Greater control of the focusing conditions

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