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Mechanism behind the Cavity Soliton Laser

Mon, 01 Jan 2018 00:00:00 +0000

Feedback configuration

The external cavity contains a two lens telescope, which is adjusted to infinity, i.e. the cavity is self-imaging and all transverse modes are degenerate. As a result, there is a 1:1 feedback for every point in the VCSEL onto itself. This ensures that self-localized solutions like spatial solitons can develop which do not dependent on transverse boundary conditions, but are stabilized by nonlinearity.

At the so-called Littrow frequency, the diffraction grating behaves as a conventional mirror (illustrated in panel a of the figure), i.e. all rays – independent of angle of incidence – are reflected such that there is a closed path in the cavity after one round-trip.

The setup is frequency-selective but in a peculiar way. If the frequency of the field is not at the Littrow frequency, the rays still return to the same location but at a different angle (illustrated in panel b of the figure). Since the resonance in the high Finesse VCSEL cavity is angularly selective, it will hence reject these rays, if the frequency detuning and thus the angular tilt is too high. We find that a stable soliton is still formed, if its frequency is slightly off the Littrow frequency. Then the wavefront of the returning beam is tilted (indicated by the black bar in panel b). From general soliton theory one expects that the soliton reacts to the wavefront tilt by a drift. Indeed we see a spatial shift depending on frequency, which is interpreted to result from an interplay of the grating-induced force and a pinning defect in the device. Under some conditions a localized excitation detaches and drifts (see Tanguy et al., PRA 2008, for details).

Mechanism of bistability

The origin of the bistability of solitons can be understood in the following way. Initially, the Littrow frequency is adjusted to be lower than the longitudinal resonance of the VCSEL cavity. All high order transverse modes of the VCSEL are at higher frequencies than the latter. Hence, there is a gap between grating frequency and VCSEL resonance, in which no linear states are allowed.

If now the intra-cavity power increases (e.g. due to a fluctuation), the carrier density decreases due to stimulated emission and due to the strong amplitude-phase coupling in semiconductors (described phenomenologically by Henry’s alpha factor) the refractive index increases. As a result, the longitudinal resonance of the VCSEL shifts to lower frequencies and becomes closer to the grating frequency. This increases in turn the intra-cavity power and hence there is positive feedback giving the possibility of bistability.

Mechanism of Spatial Instability

Mon, 01 Jan 2018 00:00:00 +0000

A laser beam with an initially homogeneous phase and amplitude distribution is incident on a medium whose refractive index depends on the light intensity. If there is a fluctuation in the refractive index distribution of the medium, the transmitted wave will be phase-modulated. During the propagation to the mirror and back diffraction couples the real and imaginary part of the field and thus

converts phase modulation into amplitude modulation. Since the medium is nonlinear it will react to this change of amplitude and positive feedback is possible if the conditions are such that extrema of the reflected field hit extrema of the modulation of the refractive index distribution with the correct phase. Under this condition a macroscopic modulation can emerge spontaneously from an infinitesimal small perturbation. The length scale of the conversion between phase and amplitude modulation and thus the characteristic wavelength of the pattern is given by the Talbot effect.

Optical Pumping

Mon, 01 Jan 2018 00:00:00 +0000

Optical pumping denotes the redistribution of population within atomic multiplets by state-selective optical excitation in an electronically excited state and the subsequent spontaneous emission. In most cases the considered multiplets consist of Zeeman substates. This allows to achieve huge nonlinear effects with very modest (down to micro Watts) power levels. The simplest case is the one of a J=1/2 to J’=1/2-transition with a two-fold degeneracy in both the ground and in the excited state.

Due to the selection rules for angular momenta circularly polarized light will couple only to one of the Zeeman substates of the ground state and thus the population of this level will be reduced. Since spontaneous emission occurs also into the unpumped sublevel, there will be a net accumulation of population in this sublevels. Optical pumping is particularly effective, if the population in the excited state is rapidly equalized between the Zeeman sublevels due to collisions with a buffer gas because then the spontaneous emission is isotropic. Note that the direction of the pumping will depend on the sign of the helicity of the pumping light. Linearly polarized light will not induce pumping because it contains sigma+ and sigma_ light of equal strength.

The induced population difference between the two Zeeman substates is called orientation and often denoted by w (normalized to [-1,1]). If the population of the excited state can be neglected, it obeys the following equation of motion:

The last term is the source term for the optical pumping. P+/_ denotes a pump rate which is proportional to the intensity of the sigma+, respectively sigma_ component. As mentioned above, the two components pump in different directions. Gamma denotes relaxation due to collisions and is very small (of the order of s-1). The damping term proportional to the pump rate represents saturation. The diffusion term models the thermal motion of the sodium atoms in the buffer gas atmosphere.

The optical properties of the medium now depend on the orientation and – via w=w(P) – on the intensity of the light field. If the linear absorption coefficient is alpha0 and the linear refractive index 1+n0, the nonlinear absorption coefficient is

and the nonlinear refractive index is

For circularly polarized light the vapor is bleached by the pump beam. For very high intensity, one sublevel will be completely empty, the orientation reaches one. Then the absorption drops to zero and the refractive index is one. The beam will essentially propagate as in vacuum. Note that an increase in optical density for the sigma+ component means a decrease for the sigma_ one and vice versa.

It turns out that the simple model of a homogeneously broadened J=1/2 to J’ =1/2-transition is not only of academic interest but a very appropriate description for the sodium D1-line if a buffer gas of sufficient pressure (typically 200 to 300 hPa argon or nitrogen) is introduced so that the homogeneous broadening is larger than the hyperfine splitting and the Doppler broadening. It is simple enough to allow for analytical investigations as well as extensive numeral studies on state of the art workstations. Since these are prerequisites for a thorough understanding of spatially extended nonlinear system the J=1/2 model is used in most of our theoretical studies.

Self-organization

Mon, 01 Jan 2018 00:00:00 +0000

What is the reason that an initially homogeneous system evolves spontaneously into a modulated, “structured” state?

This intriguing question arises in many sciences, in nature as well as in the laboratory. Spontaneous self-organization phenomena in space and time are also ubiquitous in optical systems in which intense laser beams interact with a nonlinear medium, i.e. a medium in which the optical properties (refractive index or absorption coefficient) depend on the intensity of the incident light. The interplay of spatial coupling by diffraction and nonlinearity is responsible for the pattern formation. It is highly fascinating that the properties of structures in such different regions of science like hydrodynamics, chemical reactions, gas discharges and optics possess remarkably universal aspects. This spontaneous emergence of nontrivial – often highly ordered – states is very common in spatially extended systems driven out of thermal equilibrium.

Optics is promoting the knowledge on these dissipative patterns by demonstrating phenomena not known before. Therefore the investigation of optical patterns is an important topic of interdisciplinary research using equipment of technical relevance and might on the other hand form the basis for future all-optical data processing.

Nonlinear effects occur in many media and in many different configurations. Previous investigations in “hot” sodium vapor established well controlled nonlinear optical systems as ideal candidates for investigating principles of self-organization. The experiments yielded first demonstrations of phenomena predicted to occur in a variety of model and experimental systems as well as unprecedented phenomena enriching Nonlinear Science (see here for details). Beside technical advantages (high optical quality, easy variation of parameters over a broad range, high resonant nonlinearity) the benefit of using an atomic vapour is that the equations governing the light-matter interaction can be derived directly from quantum mechanics via the density matrix approach.

The essential new ingredient in cold atoms is that refractive index modulations are not only due to internal degrees of freedom of the atoms but that opto-mechanical coupling can lead to density modulations: The modulated light field causes dipole forces on the cold atoms, which will respond by transverse bunching. Depending on parameters, the opto-mechanical coupling can trigger feedback effects either damping or enhancing the modulation and induce new coupled light-matter instabilities. Spontaneous bunching of atoms in the longitudinal direction (i.e., along the wavevector of the laser beam) due to recoil and dipole effects is known from collective atomic recoil lasing (CARL, see the investigations in the CNQO group). However, the two-dimensional patterns sought after are much richer than CARL and other one-dimensional longitudinal wavelength-scale density modulations known to develop in cold atoms. If longitudinal CARL effects turn out to co-exist with the transverse instability, the resulting 3D self-organization would be a novel and highly significant result.

Talbot Effect

Mon, 01 Jan 2018 00:00:00 +0000

The Talbot-effect is a near field ‘self-imaging’ effect generic for certain wave equations like the paraxial wave equation and the Schrödinger equation. Any spatial modulation of period Λ of a plane carrier wave is reproduced after the Talbot-length $z_R= 2Λ^2/λ$.

Moreover, starting with a pure phase modulation in a certain plane, after a distance of $z_T/4$ and $3z_T/4$ there are planes, in which the field has a pure amplitude modulation in first order.

Therefore, the resulting amplitude grating will be

in phase with the original refractive index grating after a propagation by $z_T/4$. The resulting wavelength of the pattern in a focusing nonlinear medium is $Λ_{foc} = \sqrt{4λd} $.
in anti-phase with the original refractive index grating after a propagation by $3z_T/4$ . The resulting wavelength of the pattern in a defocusing nonlinear medium is $Λ_{def} = \sqrt{4/3λd} $.

It should be cautioned that the real experimental system has two additional characteristic length scales in addition to the diffractive one given by the Talbot-effect. One is the diffusion length of the atomic motion. This will tend to suppress the instability for small wavelengths. The other is the overall size of the experimental system with is given by the diameter of the input beam.