In panel (b), the peak normalized intensity profile of the control beam is shown at different propagation distances z. 84 3.7 Image (a) shows the 3D intensity profile of the input control beam. shows the transmitted probe beam at the exit of a 1 cm long medium.
Theoretical background of atom-field interaction
Atom-field interaction Hamiltonian
We now discuss the motion of the electron in the nuclear and external fields and derive this joint Hamiltonian from a gauge invariant point of view.
Interaction Hamiltonian under gauge transformation
We now discuss the motion of the electron in the nuclear and external fields and derive this total Hamiltonian from a gauge invariant point of view. to satisfy local gauge or phase invariance. After simplification and rearrangement, the Eq. where ˆH is the total Hamiltonian and is given by Hˆ =−~2.
The electric dipole approximation
This interaction Hamiltonian was used to study the atom-field interaction in our upcoming problems. One of the simplest problems of atom-field interaction is a two-level atom model system.
Interaction of light with a system of two-level atoms
- The Liouville equation for a two-level atomic system
- Rabi oscillations in two-level system
- Dressed state analysis of the two-level atom
- Transient and steady state response of two-level atoms
- Linear and nonlinear susceptibility of the medium
The interaction energy of the atom-field system under dipole approximation is given by the interaction Hamiltonian ˆHI,. The first term of this equation corresponds to polarization due to the permanent dipole moment of the medium in the absence of an external electric field.
Interaction of light with a system of three-level atoms
- The Liouville equation for a three-level Λ-system
- Steady state solution of density matrix elements
- Linear and nonlinear susceptibility of the probe field
- Dressed state analysis of the three-level Λ-system
Therefore, multi-level systems are utilized to meet the need, i.e., high refractive index absorption extinction. Inserting the Hamiltonian effect from Eq. 1.98), we obtain the equations of motion for the elements of the density matrix, . Also, the slope of the normal dispersion curve is steepest near the center of the line.
This happens when the hyperfine splittings of the ground state coincide with the frequency difference between the two laser fields. The complex susceptibility given by Eq. 1.111) represents the response of the system to an applied field.
Maxwell’s wave equation in a dielectric medium
- The origin of paraxial diffraction and diffraction-limit
- Paraxial wave equation in free-space
- The Gaussian beam
- The Super-Gaussian beam
- Hermite-Gaussian beam
- Laguerre-Gaussian beam
The second term represents the dispersion and absorption of the beam in the medium. The transverse spatial profile of the electric field for the Hermite-Gaussian mode is given by. Thus, the solution of the paraxial Helmholtz equation in cylindrical coordinates (ρ, φ,z) leads to LaguerreGaussian modes.
The spatial profile of the strong control beam makes the probe sensitivity inhomogeneous along the transverse direction. Again, we observe a reduction in feature size by a factor of about 2 in the probe field.
Theoretical model and basic dynamical equations
- Equations of motion for density matrix elements
- Propagation equations for probe and control beams
- Medium susceptibilities of probe and control beams
- Transverse beam profiles
The tunings of the probe and control fields from the respective transition frequencies are defined as ∆1 =ω31−ω1 and ∆2. The second terms on the right-hand side are responsible for the scattering and absorption of both the control and probe beams. But first, to interpret the effect of the ray profiles on the propagation, we must
The initial peak amplitude and the width of the probe field are denoted by g0 and wp, respectively. The desired spatial profile of the probe beam can be generated using a spatial light modulator based on liquid crystals or coherent EIT media [137].
Numerical simulation and Results
Spatial modulation of linear and nonlinear susceptibility
This transparency window can be controlled via the external parameters and makes it possible to transfer the transverse distribution of the control field to the transmission profile of the probe field. This parabolic refractive index variation causes focusing of the probe field towards the center of the control field and also controls the propagation of the probe field along the propagation axis. Finally, we analyze the dependence of the probe receptivity on the transverse beam shape of the control field.
Fig.2.4 shows the transverse variation of the sensitivity of the probe field for a double peaked transverse spatial distribution of the control field obtained by setting m = 1 and n =0 in Eq. The generalization of these results to other spatial modes of the control field with different values of m,n is straightforward.
Propagation dynamics of probe and control beams
F2.6: Spatial intensity profile of the control field as a function of the transverse coordinate x after propagation through a 4 cm long atomic medium. CPT” as the initial profile of the control beam, and a plane wave for the initial profile of the probe beam. This distortion arises from diffraction and from the variation of the refractive index experienced by the control field.
F2.8: Transverse spatial intensity profile of the control and probe fields. a) shows the initial profile of the control field at the media entry. As a result, the cloned image is tightly focused in the transverse profile of the probe beam.
Chapter conclusion
- Atomic model system
- Dynamical equations for density matrix elements
- Solution of density matrix equations under perturbative approach
- Beam propagation equations
This is because the propagation dynamics of probe field is dependent on the diffraction and dispersive properties of the medium. The diffraction and dispersion properties of the atomic medium can be manipulated using proper spatially inhomogeneous control field. We use Liouville equation to include the coherent and incoherent processes of the atomic system.
Therefore, the steady state solutions of the density matrix equations can be written in the following expansion form. The second and third terms indicate the solutions at positive and negative frequencies of the probe field, respectively.
Results and Discussions
- Susceptibility with homogeneous fields
- Susceptibility with inhomogeneous control field
- Beam propagation dynamics
- Spatial optical switching
The plots are shown against the transverse axis coordinate x of the control beam for the y = 0 plane. This increase will reduce the transmission of the probe beam and therefore its visibility appears to be limited. This contrast enhancement causes strong focusing of the probe beam towards the center of the two peaks of the control field.
The results for the spatial evolution of the control and probe profiles throughout the environment are shown in Fig.3.5. Fig.3.7 shows the radial distribution of the limited Sparrow input control beam (at z= 0) and the output probe beam at z = 1 cm.
Chapter conclusion
The guidance and steering of an optical beam is made possible on the basis of a refractive index of the medium. The high refractive index together with reinforcement allows the probe beam to deflect when launched at the wings of the pump beam. The control field parameters such as detuning and intensity can be used to control the transmission intensity and width of the deflected probe beam.
The bright and dark areas of the pump field profile cause a high (cladding) and low (core) index of refraction of the probe field, which leads to the formation of antiwave structures within the medium. The atomic coherence decay rate is defined as Γαβ = 1. Substituting the interaction Hamiltonian of Eq. 4.2) and the Liouvillian matrix of Eq. 4.4), the equations of motion for the four-level atomic system can be described as
Probe Susceptibility for hot atomic medium
In the next section we obtain the analytical expression for the linear sensitivity of the probe field in a compact form assuming equal decay rates of excited states, that is, γ13 = γ23 = γ14 = γ24 = γ/2. While for a hot atomic system, the thermal motion of the atoms causes an inhomogeneous broadening of the atomic spectra. The thermal velocity v of the atom can be incorporated into the sensitivity expression (4.27) by introducing velocity-dependent field detunements ∆j(v)= ∆j−kjv, j∈ {1,2,3}.
The term kjv is the Doppler shift experienced by an atom with a velocity component v in the direction of propagation of the field beam. The atomic velocity distribution is assumed to follow the Maxwell-Boltzmann distribution P(kv)d(kv)= 1.
Beam propagation equations and beam profiles
The LGPmbeam exhibits a dark spot in the center and a bright profile in the annular region. Therefore, the phase modulation imposed on the probe beam due to the spatially varying pump beam is effective over the entire length of the medium. The initial peak amplitude and field width of the probe are denoted by g0 and wp and a is the initial location of the center of the probe beam along the x-direction.
We have chosen the initial intensity of the probe beam so that it is absorbed into the medium without pumping and control fields. The integer values of f determine the input profile of the probe beam: either a Gaussian (f = 1) or a super Gaussian (f > 1).
Results and Discussions
Spatial modulation of the probe field susceptibility
The gradient of the refractive index is dependent on the sign of the control field tuning. In redshifted control field tuning, the slope of the spatial refractive index reaches a maximum at the center of the line and gradually decreases towards the wings. Therefore, the refractive index gradient allows us to focus or defocus the probe beam towards the center of the pump beam.
The position-dependent refractive index and gain both increase in the bright region while decreasing in the dark region of the donut-shaped pump beam. Therefore, the shape of the pump beam profile can be effectively transferred to the transmitted probe beam.
Numerical simulation of paraxial beams equations
- Optical beam steering
- Optical beam splitting
- Optical beam cloning
- Arbitrary image cloning
The bright region of the pump beam tends to refract the probe beam into it and consequently increases the amplitude of the probe beam. As a result, the probe beam is directed towards the region of high pump intensity and remains confined there. At the input side of the medium, the probe beam is fired in the dark region of the double Gaussian pump beam, as shown in Figure 4.5.
Therefore, the transverse pattern of the pump beam can be effectively transferred to the probe beam. The cloned probe beam also experiences focusing effects at the high intensity regions of the pump beam.
Chapter conclusion
The spatial profile of the pump beam causes a transverse modulation in the refractive index as well as in the gain profiles of the probe beam. Assuming that the operator ˆS is independent of z, an. the exact solution of equation A.2) for a small propagation distance ∆z is given by z. Therefore, we can write down the approximate solution of the equation. A.7) The exponential containing operator ˆS in Eq. A.7) is evaluated in the real space domain.
The accuracy of the split-step Fourier method can be further improved by repeated application of the BCH formula of Eq. This method is known as symmetrized split-step Fourier method because of the symmetric. form of the exponential operators in Eq.
