The transmitted intensity of the probe beam shown in panels (a) and (b) is 0.8 mm along the propagation length. The spatial inhomogeneity induced by the control beam in the refractive index of the medium has been used in focusing/defocusing [ 21 , 22 ], directing [ 23 , 24 ] and splitting [ 25 ] of the probe beam.
Theoretical background of atom-field interaction
Quantum description of matter
Using density matrix can overcome all the above disadvantages while using the wave function approach. The density matrix approach is a very robust technique to extract all information about the system without requiring a detailed expression of the wave function.
Classical theory for Electromagnetic fields
The term on the right side of the equation is the source term and represents the response of the medium. The first term on the left side of the equation describes the diffraction of the beam.
Polarization of the medium
In this thesis we use Eq. 1.24) to study the propagation of the different em-fields through the multi-level atomic system. We use the polarization expression of Eq. 1.26) to derive an expression of the medium sensitivity.
Atom-field interaction Hamiltonian
In the long time interaction limit, the Eq. 1.28) can be expressed in the following way. 1.29) and eliminate the integral sign to arrive at To see Cf. 1.43) invariant under the transformation as in Eq. 1.42), the scalar and vector potentials Φ(r, t) and A(r,t) must be replaced by the following gauge transformations.
Two level atomic system
Dynamics of a two-level atomic system
Here |ψi is the state of the two-level system and can be expressed as a linear combination of the two states |1i and |2i in the following way. 1.69). To study the evolution of atomic populations and coherence, we use the following Liouville equation.
Rabi oscillations
Figure 1.4 shows the variation of the real and imaginary part of the sensitivity with the disassembly of the probe ∆p. In the presence of a strong probe field, the FWHM is modified by the probe intensity.
Three level atomic system
- Steady state solution of density matrix elements
- Physical interpretation and analysis of EIT
- Atomic model system
- Dynamical equations for density matrix elements
- Solution of density matrix equations using perturbative approach 37
We further use this effective Hamiltonian in the Liouville equation to study the dynamics of the three-level Λ− system. In the dressed state picture, we first calculate the eigenvalues and eigenvectors of the effective Hamiltonian Eq. This leads to the increase in the width of the transparency window as in Fig.
Therefore, any modification of the dispersion can lead to a change in the refractive index of the medium.
Results and discussion
Susceptibility with homogeneous fields
Such modulation in the absorption of the probe due to the Kerr field is the key to the confinement and control of the light beam. Moreover, the formation of double transparency is generally due to Double-Dark resonance (DDR) effect, which can be well understood using the dressed state picture of the N-type system [79]. Therefore, with the increase of the Kerr field intensity, the separation between the |+′i and |−′i states increases.
Furthermore, the manipulation of the absorption and refractive index by the Kerr field is key to the formation of a high-contrast tunable optical waveguide, which will be discussed in the next section.
Susceptibility with inhomogeneous control field
2.3, that the refractive index reaches a maximum value in the higher intensity region of the control beam that forms the core of the atomic waveguide. However, an appropriate spatial profile of the Kerr beam can significantly improve the characteristics of the induced waveguide. The intensity distribution of the LG10 Kerr beam plays an important role in controlling the absorption and refractive index of the medium.
However, the non-zero intensity of the Kerr beam strongly contributes to the increase in the absorption of the medium towards the outer region.
Beam propagation dynamics
It is clear that the width of the probe beam increases to 4wp after propagation over a distance of 2 mm through the free space. In addition, the broadening of the probe beam can be controlled by manipulating the spatially varying refractive index of the medium. The characteristics of the probe beam propagation through the medium change dramatically in the presence of an LG Kerr beam.
The main reason for the low transmission of the probe beam is due to the absorption caused by the Kerr field.
Conclusion
Thus, controlling the transparency in the transverse direction creates a new avenue for structured beam generation. To illustrate the effect of the phase-dependent behavior of the sensitivity on the propagation of the probe beam, we numerically study the paraxial propagation equations. A high-contrast waveguide and anti-waveguide-like structure is achieved, unlike the case of the weak-field regime.
Finally, we show the rotation of the generated petal-like beam structure due to magneto-optical rotation.
Theoretical Formulations
- Model system
- Equations of motion for density matrix elements
- Probe susceptibility of a homogeneous medium
- Beam propagation equations
In this section we calculate the linear response of the probe field in a homogeneous medium. The perturbative expansion of the density matrix up to first order of probe field gi,(i∈1,2) can be expressed as. The phase-dependent response of the medium can be investigated by considering the spatial inhomogeneity of the probe field.
Thus the spatial structure of the probe field for two orthogonal polarizations can be expressed as
Results and Discussions
Phase dependent susceptibility
3.3 (Color online) The absorption pattern of the σˆ+polarization component is plotted against the two orthogonal axes x and y. Note that the absorption of the left-handed polarization σˆ− is identical to the absorption of the right-handed polarization σˆ+ at. The slope of the refractive index reaches its maximum around the transparency window and gradually decreases towards the wings for a red detuned right circular polarization component.
Also the amplitude of the refractive index is stronger here than in the weak field boundaries.
Formation of structured beam patterns
We further study how the magnetic field strength allows us to enhance the contrast of the textured beam. Thus, the contrast enhancement of the structured beam is possible by using an appropriate magnetic field strength. As shown in Figure 3.6, the probe intensity output pattern shows the same quadruple symmetric patterns as in the case of a weak field.
The increase in beam transmission is due to waveguide-induced focusing of the probe beam in the azimuth plane.
Conclusion
Furthermore, the spread of both polarization components in the azimuthal plane is limited by the width of the spatial transparency window. The rotation of the structured beam can be increased by increasing the intensity of the probe and the strength of the magnetic field [114]. Our approach opens up new possibilities for generating a high-contrast structured beam in other closed-loop systems that exhibit narrow EIT resonances.
Thus, an atomic medium with a buffer gas [115] and inhomogeneously extended atomic system [116] can be a suitable candidate for creating a diffraction-controlled high-contrast structured beam.
Appendix
- Model
- Equation of motion
- Probe susceptibility of a homogeneous medium
- Paraxial beam propagation equations
Therefore, this counter-intuitive sequence leads to Table 3.3 The value of the atom populations estimated with the counter-intuitive approach. Next, we investigate the effect of the spatially varying control field on medium sensitivity. Furthermore, the phase-dependent characteristics of the medium can be well understood by taking into account spatial variations of the probe field components.
The spatial structure of the two orthogonal components of the probe field is expressed as.
Results and Discussions
Azimuthally varying susceptibility
The control beam intensity distribution in the transverse plane shows the structure of two petals symmetrically distributed along the x-axis [113]. Therefore, the spatial profile of the control beam plays a decisive role in creating an asymmetric absorption pattern. In addition, the asymmetry in the absorption profile can be controlled by the intensity and width of the control beam.
Proper control and manipulation of absorbance asymmetry can be helpful in creating high-contrast structured beams.
Beam propagation dynamics
The spatial profile of the probe components at the input of the medium is expressed as. To study the propagation dynamics of the probe beam components, we numerically solve the propagation Eq. 4.19b) using the Fourier split-step operator method. Therefore, the control beam acts as a phase-selective tool to control and manipulate the spatial properties of the structured probe beam.
Moreover, the low-intensity sidelobes in the structured probe can be made negligible by choosing the appropriate width and intensity of the control beam.
Conclusion
We assume that the light and dark areas in the structured beam are of equal width. Thus, for a probe of width wp, the width of each bright or dark area is given by the ratio of the beam circumference to the total number of dark and bright areas. Therefore, as the OAM value increases, the width of the spot or petal decreases, as can be seen in Fig.
Manipulation of structured beams possessing narrow petal features with a spatially dependent control beam is the key to forming various patterns of asymmetric structured beams.
Appendix
- Model
- Theory
- Susceptibility of the medium
- Pulse propagation equation
In the coated continuum picture, the wave function of the system is a coherent superposition of two coupled states|1i,|2i and the coated continuum |ǫi as given by. The time-dependent Hamiltonian of the system according to the electric dipole approximation can be written as The dynamics of the coupled atom-molecule system is governed by the following governing equation.
We assume that the lifetime of the resonance would be in the millisecond regime, which is long enough compared to the excited state lifetime (usually in the microsecond regime).
Results and discussion
As Gc increases, the transparency window widens, leading to a decrease in the slope of the dispersion, as shown in Figure 5.2. However, with the increase of Γ, the width of the absorption profile increases, because the probability of transitions from ground state |1i to the two dressed states |+i and|−i increases. This increase in the width of the absorption profile results in a decrease in the width of the transparency window along with an increase in absorption.
Therefore, the product of the time delay and the spectral width of the transparency window, i.e. DBP, decreases with increasing Gc.
Conclusion
We found that the contrast of the structured beam can be controlled by changing the strength of both probe and TMF. We found that the EIT can be manipulated by changing the strength of the control field as well as the width of the Feshbach resonance. The future scope of the current thesis is not only limited to the propagation of optical fields in an atomic medium, but can also be extended to other coherent media such as molecular or solid state medium.
Closed-loop atomic systems show very interesting features where the properties of the medium can be modified by the phase of the interacting optical fields.
