Phase induced transparency mediated structured beam generation in a closed-loop tripod configuration
aa r X i v : . [ phy s i c s . op ti c s ] J un Phase induced transparency mediated structured beam generation in a closed-looptripod configuration
Sandeep Sharma ∗ and Tarak N. Dey † Department of Physics, Indian Institute of Technology Guwahati, Guwahati- 781039, Assam, India (Dated: July 18, 2018)We present a phase induced transparency based scheme to generate structured beam patternsin a closed four level atomic system. We employ phase structured probe beam and a transversemagnetic field (TMF) to create phase dependent medium susceptibility. We show that such phasedependent modulation of absorption holds the key to formation of a structured beam. We use afull density matrix formalism to explain the experiments of Radwell et al. [Phys. Rev. Lett. 114,123603 (2015)] at weak probe limits. Our numerical results on beam propagation confirms that thephase information present in the absorption profile gets encoded on the spatial probe envelope whichcreates petal-like structures even in the strong field limit. The contrast of the formed structuredbeam can be enhanced by changing the strength of TMF as well as of the probe intensity. In weakfield limits an absorption profile is solely responsible for creating a structured beam, whereas inthe strong probe regime, both dispersion and absorption profiles facilitate the generation of highcontrast structured beam. Furthermore we find the rotation of structured beams owing to strongfield induced nonlinear magneto-optical rotation (NMOR).
PACS numbers: 42.50.Gy,42.65.-k, 42.50.Tx
I. INTRODUCTION
In the recent past, there has been a growing interestin the creation of structured beams due to their poten-tial applications in optical micromanipulation [1], quan-tum information processing [2], microtrapping and align-ment [3–6], and the biosciences [7]. Various techniquesfor the generation of such beams exists that uses conven-tional optical components like digital micro-mirror de-vice(DMD) [8], laser resonator [9, 10], axially symmetricpolarization element [11], porro-prism [12, 13] and spa-tial light modulator [14–16]. In most of the techniquesthe key feature is to coherently superpose two Laguerre-Gaussian(LG) beam modes with equal but opposite or-bital angular momentum(OAM), thus creating a struc-tured beam profile.A more intriguing approach for the generation of struc-tured beams has been developed by Radwell et. al. ina cold rubidium system [17]. They have used a singlephase structured light beam [18] and static magnetic fieldto form a closed-loop Hanle configuration [19] in a fourlevel atomic system. The relative phase difference be-tween the applied fields can drastically modify the Zee-man coherences of a closed-loop transition [20, 21]. Thephase-dependent Zeeman coherence is a basic ingredientin the control of optical dispersion, absorption, and non-linearity [22–29]. Manipulation of these coherences alongthe azimuthal plane is the main key behind spatially de-pendent electromagnetic transparency. Hence, an opaquemedium becomes transparent at certain angular positionsdue to the presence of a phase structured beam. Thus ∗ Electronic address: [email protected] † Electronic address: [email protected] controlling transparency in the transverse direction cre-ates a new avenue for the generation of the structuredbeam. Radwell et. al. [17] used a basic theoretical modelbased on Fermi-golden rule and provided an approximateexpression for the periodic variation of the absorptionprofile to demonstrate how the structured beam can beproduced. However to achieve good agreement with ex-periments, various transverse and longitudinal relaxationeffects must be incorporated in the propagation dynamicsof the light beam with an azimuthally varying polariza-tion and phase structure.In this paper, we provide a detailed theoretical expla-nation for the recent experiments on the generation ofstructured beams [17], based on full density matrix equa-tions. To facilitate these structured beam generation, weuse a homogeneously broadened four level atomic sys-tem driven by two orthogonal polarization componentsof probe beam as shown in Fig. 1. In order to createphase-dependent atomic coherences, we use a weak mag-netic field to couple the ground states. We start by deriv-ing an analytical expression for the probe susceptibilityin the weak field regime. However, numerical solutionsof density matrix equations at steady state limits is in-evitable to obtain the response of the medium at strongprobe field intensities. To illustrate the effect of phasedependent behaviour of the susceptibility on the probebeam propagation, we numerically study paraxial propa-gation equations. We find that the phase dependent ab-sorption creates petal like structures on the probe beam.The contrast of the generated structured beam can be en-hanced by increasing the coupling strength of the lowerlevel magnetic field. Furthermore we study the refractiveindex profile of two orthogonal polarisation componentsin the presence of strong probe field. A high contrastwaveguide and anti-waveguide like structure is achievedunlike in the case of the weak field regime. We exploitthese waveguide features to generate a diffraction con-trolled high contrast petal-like beam structure. Finallywe show the rotation of the generated petal-like beamstructure due to magneto optical rotation.The paper is organised as follows. In the next sec-tion, we introduce the theoretical model and derive theeffective Hamiltonian for a four-level closed-loop atomicconfiguration. In Sec. II.B, we adopt density matrix for-malism to study the evolution of the atomic populationand coherences. In Sec. II.C, we analytically derive thelinear atomic responses to the orthogonal polarizationcomponents of the probe field. In Sec. II.D, we describethe paraxial beam propagation equation for the spatialevolution of phase structured probe field. Next we pro-vide the results on azimuthally varying susceptibilitiesfor both weak and strong probe field regimes. Finallyour numerical results delineate the effect of the linearand nonlinear susceptibilities on the propagation dynam-ics of the probe beam. Sec. IV provides a summary ofour work.
II. THEORETICAL FORMULATIONSA. Model
The system under consideration is shown in Fig. 1where the electric dipole allowed transitions | i ↔ | i and | i ↔ | i are coupled by two orthogonal polarisa-tions ˆ σ + and ˆ σ − of the probe field, respectively. Thusthe electric field propagation along the z -axis, containingboth orthogonal polarisations with carrier frequency ω p ,can be written as ~E ( ~r, t ) = (ˆ σ + E + ( ~r ) + ˆ σ − E − ( ~r )) e − i ( ω p t − k p z ) + c.c. , (1)where, E + ( ~r ) and E − ( ~r ) are the slowly varying envelopesof right and left circularly polarized probe fields, respec-tively. The wave number of probe field is denoted by k p . An arbitrary magnetic field ~B = B (cos θ ˆz + sin θ ˆx )is used to connect the electric dipole forbidden transi-tions | i ↔ | i and | i ↔ | i in order to form a closedloop system. Such a closed loop system exhibits interest-ing phase dependent behaviour of absorption and disper-sion. The longitudinal component of the magnetic field B cos θ induces the Zeeman shift between the states | i ,and | i whereas the transverse component B sin θ can beused to redistribute the population among the groundstates | i , | i , and | i . This level scheme has been re-alised experimentally in cold atomic Rb vapour wherethe ground levels | i = | S / , F = 1 , m F = − i , | i = | S / , F = 1 , m F = 0 i , | i = | S / , F = 1 , m F = 1 i ,and the excited level | i = | P / , F ′ = 0 , m ′ F = 0 i .In the presence of probe and magnetic fields, the Hamil-tonian of the system in the approximations of electric |1> |2> |3>|4> ∆ p γ γ γ
42 43 ω p σ ε σ + ε ω p + − − FIG. 1: (Color online) Schematic diagram of the four-levelclosed atomic system. The atomic transition | i ↔ | i and | i ↔ | i are coupled by left(ˆ σ − ) and right(ˆ σ + ) circularly po-larized component of the probe field,respectively, whereas theZeeman sub-levels | i , | i , and | i are coupled by a transversemagnetic field. γ i corresponds to the radiative decay ratesfrom excited state | i to ground states | i i where i ∈ , , dipole and rotating wave takes the following form H = H + H I + H B , (2a) H = ~ ω | ih | , (2b) H I = − ˆ D · ˆ E = − ~ ( | ih | g e − iω p t + | ih | g e − iω p t + H.c.) , (2c) H B = g F µ B ˆ F · ~B = ~ β L ( | ih | − | ih | ) + ~ β T ( | ih | + | ih | + H.c.) , (2d)where g = ~d + · ~ E + ~ e ik p z and g = ~d − · ~ E − ~ e ik p z , are the Rabi frequencies of the probe fields correspondingto the left and right circular polarizations, respectively.The magnitude of Zeeman shift and the coupling strengthbetween the ground levels are given by β L = g F µ B B cos θ and β T = g F µ B B sin θ/ √
2, respectively. We use follow-ing unitary transformation W = e − i ~ Ut where U = ~ ω p | ih | , to express the Hamiltonian in the time independent formas given below H I = W HW † = ~ ∆ p | ih | − ~ ( g | ih | + g | ih | )+ ~ β L ( | ih | − | ih | ) + ~ β T ( | ih | + | ih | ) + H.c. , (3)where ∆ p = ω p − ω is the probe detuning. B. Equation of motion
We now present the full density matrix formalism tostudy the experimental work by Radwell et al.[17]. Theclosed loop tripod system possesses various radiative andnon-radiative processes. To account for these incoherentdecay, we use following the Liouville equation˙ ρ = − i ~ [ H I , ρ ] + L ρ . (4)The second term in Eq.(4) represents radiative processesand non-radiative processes that can be determined by L ρ = L r ρ + L c ρ (5)with L r ρ = − X i =1 γ i | ih | ρ − | i ih i | ρ + ρ | ih | ) , L c ρ = − X j =1 3 X j = i =1 γ c | j ih j | ρ − | i ih i | ρ jj + ρ | j ih j | ) . The first term of Eq.(5) represents radiative decay fromexcited state | i to ground states | i i , and are labeled by γ i ( i ∈ , , ρ ij due to collision at a rate γ c . The dynamics of the atomic population and coher-ences for the closed loop tripod system can be obtainedby substituting the effective Hamiltonian (3) in the Liou-ville equation (4). Therefore, the following set of Blochequations can be conveniently written˙ ρ = γ ρ − iβ T ρ + iβ T ρ + ig ∗ ρ − ig ρ − γ c ρ + γ c ρ + γ c ρ , (7a)˙ ρ = iβ L ρ − iβ T ( ρ − ρ ) + iβ T ρ + ig ∗ ρ − γ c ρ , (7b)˙ ρ =2 iβ L ρ − iβ T ( ρ − ρ ) + ig ∗ ρ − ig ρ − γ c ρ , (7c)˙ ρ = − i (∆ p − β L ) ρ − iβ T ρ + ig ∗ ( ρ − ρ ) − ig ∗ ρ − (Γ + γ p ) ρ , (7d)˙ ρ = γ ρ − iβ T ( ρ − ρ ) − iβ T ( ρ − ρ )+ γ c ρ − γ c ρ + γ c ρ , (7e)˙ ρ = iβ L ρ − iβ T ρ − iβ T ( ρ − ρ ) − ig ρ − γ c ρ , (7f)˙ ρ = − i ∆ p ρ − iβ T ( ρ + ρ ) − ig ∗ ρ − (Γ + γ p ) ρ − ig ∗ ρ , (7g)˙ ρ = γ ρ − iβ T ( ρ − ρ ) + ig ∗ ρ + ig ρ + γ c ρ + γ c ρ − γ c ρ , (7h)˙ ρ = − i (∆ p + β L ) ρ − iβ T ρ + ig ∗ ( ρ − ρ ) − ig ∗ ρ − (Γ + γ p ) ρ . (7i)The remaining density matrix equations come from thepopulation conservation law P i =1 ρ ii = 1 and the complexconjugate expressions ˙ ρ ji = ˙ ρ ∗ ij . C. Probe susceptibility of a homogeneous medium
In this section, we calculate the linear response of theprobe field in a homogeneous medium. The probe field isto be weak enough to be treated as a perturbation to asystem of linear order under steady-state condition. Thisassumption leads us to get a good agreement of the recentexperiment results [17]. The perturbative expansion ofthe density matrix upto first order of probe field g i , ( i ∈ ,
2) can be expressed as ρ ij = ρ (0) ij + g ρ (+) ij + g ρ (+) ij , (8)where, ρ (0) ij is the solution in the absence of the probefield. The second and third terms in Eq.(8) denote first-order solutions of the density matrix elements for bothorthogonal polarizations at positive probe field frequency ω p . We now substitute Eq. (8) in Eqs (7) and equatethe coefficients of g and g . As a result, we obtain twosets of 12 coupled linear equations. Next, we solve thesealgebraic equations to derive the atomic coherences ρ (+) and ρ (+) . The off-diagonal density matrix elements ρ (+) and ρ (+) determine the linear susceptibility χ and χ of the medium at frequency ω p respectively. Hence themedium polarization induced by the probe field can beexpressed as χ (∆ p ) = N | d + | ~ ρ (+)41 , (9a) χ (∆ p ) = N | d − | ~ ρ (+)43 , (9b)with ρ (+)41 = N D g + β T N D g , (10) ρ (+)43 = β T N D g + N D g , (11)where N = β L (∆ p + i Γ )(∆ p + i Γ + β L ) ρ + β T (2 ρ − ρ ) − β T ( ρ (∆ p + i Γ ) + β L ρ (∆ p + i Γ + β L )) , (12) N =(∆ p + i Γ ) ( ρ − ρ ) + β L ρ − β T (2 ρ − ρ ) , (13) N = β L (∆ p + i Γ )(∆ p + i Γ − β L ) ρ + β T (2 ρ − ρ ) − β T ( ρ (∆ p + i Γ ) + β L ρ (∆ p + i Γ − β L )) , (14) D =( β L − β T )( β L + 2 β T − (∆ p + i Γ ) )(∆ p + i Γ ) ρ =( β T β L ) , (15) ρ =1 − β T β L ) . (16)Here N is the atomic density of the medium. The in-fluence of transverse magnetic field on the steady statepopulation in absence of the probe field is clearly seenfrom Eqs.(15) and (16). The phase dependent responseof the medium can be explored by considering the spatialinhomogeneity of the probe field. Thus the spatial struc-ture of the probe field for two orthogonal polarisationscan be expressed as g ( r, φ ) = g ( r ) e ilφ , (17) g ( r, φ ) = g ( r ) e − ilφ , (18)where l , φ and g ( r ) represents OAM, phase and trans-verse variation of the probe beam, respectively. Thephase dependent susceptibilities of the closed loop tri-pod system is given as χ = N | d + | ~ (cid:18) N D + β T e − ilφ N D (cid:19) , (19) χ = N | d − | ~ (cid:18) N D + β T e ilφ N D (cid:19) . (20)The above analytical expressions of the susceptibilitiesfor the transitions | i ↔ | i and | i ↔ | i display an in-sight on the physics behind the formation of structuredbeam profile. The transverse magnetic field β T , OAM l and transverse phase φ plays a crucial role in the ma-nipulation of the optical properties of closed loop tripodsystems. We adopt Gauss-Jordan elimination method tosolve linear algebraic Eq. (7) numerically at steady-statecondition of the density matrix for a probe field at higherintensities limits. D. Beam propagation equation with paraxialapproximation
In order to investigate the effect of azimuthally varyingsusceptibilities on both left and right polarized compo-nents of the probe beam, we use Maxwell’s wave equa-tions under slowly varying envelope and paraxial waveapproximations. The dynamics of the orthogonal po-larization components with Rabi frequencies g and g propagating along the z -direction can be expressed inthe following form: ∂g ∂z = i k p (cid:18) ∂ ∂x + ∂ ∂y (cid:19) g + 2 iπk p χ g , (21a) ∂g ∂z = i k p (cid:18) ∂ ∂x + ∂ ∂y (cid:19) g + 2 iπk p χ g . (21b)The terms within the parentheses on the right hand sideof Eq. (21a) and Eq. (21b) are account for transversevariation of the probe beam. These terms responsiblefor the diffraction either in the medium or in free space.The second terms on the right-hand side of Eq. (21a) andEq. (21b) leads to the dispersion and absorption of theprobe beam. III. RESULTS AND DISCUSSIONSA. Azimuthally varying susceptibility
We first study the effect of azimuthal phase on the ab-sorption of two orthogonal polarisation components ˆ σ ± of the probe field at weak intensity regime. The phasedependent susceptibilities χ and χ can be exploredby considering the amplitude of both the polarizationcomponents to be continuous wave with g i ( r ) = g =0 . γ, ( i ∈ , γ c is very negligible to be consistent with the experimentalresults for the cold atomic system [17]. The absorptionof right- and left-handed circular polarizations χ and χ are plotted against the azimuthal phase as shown inFig. 2. Results are presented in Fig. 2 for two differ-ent intensities of magnetic field. From Fig. 2, we findthat the absorption at the ˆ σ + transition oscillates pe-riodically. The periodic variation of this absorption canbe well explained by considering the perturbative expres-sion for the susceptibility as mentioned in Eq. (19). Theterm associated with the second fraction in the roundbracket of Eq. (19) leads to phase dependent responseof the medium. The 2 l factor in the exponential termdecides the number of transparency windows that canbe formed within a period. It is clear from Fig. 2 thatOAM l = 2 creates 4 transparency windows. The nar-rowing of the transparency window is a key mechanismto generate the high contrast periodic absorption struc-tures. It is evident from Fig. 2 that the sharp variationof transparency windows can be achieved by increasing θ value. The increment of θ returns the strength of TMF β T at higher values that leads to an increase in the pop-ulation of the ground states | i , and | i as shown by theEqs.(15) and (16). As a result, each polarisation com- φ -10123456 χ Im( χ ), θ = π /18 Im( χ ), θ = π /14 Re( χ ), θ = π /18 Re( χ ), θ = π /18 Im( χ ), θ = π /18 x10 -5 FIG. 2: (Color online) Real and imaginary part of suscepti-bilities χ and χ as a function of phase for different θ areplotted. The parameters are chosen as N = 10 atoms/cm ,Γ = 0 . γ , ∆ p = 0, β = 0 . γ , γ c = 10 − γ , g = 0 . γ and l = 2. ponent suffer more absorption due to the narrowing ofthe transparency window. Hence the strength of TMFand the azimuthal phase plays an important role in cre-ating a high contrast periodic absorption structure forthe probe field. Note that the absorption of the left-handed polarisation ˆ σ − is identical to the absorption ofthe right-handed polarisation ˆ σ + at ∆ p = 0 as shownin Fig. 2. The absorption of both circularly polarisedcomponents constitute the structure of the probe ab-sorption. The phase dependent absorption structure isa result of the coupling among the degenerate groundstates by a weak magnetic field. The degeneracy be-tween the ground states | i i , ( i ∈ , ,
3) can be lifted inthe presence of a longitudinal magnetic field β L . Theprobe resonance condition ∆ p = 0 facilitates the red andblue shifted detuning by an amount of β L for each cir-cularly polarised component. Thus the refractive indexprofile for ˆ σ + component varies oppositely as the refrac-tive index profile for ˆ σ − component with a very smallmagnitude as shown in Fig. 2. This reverse nature of re-fractive index for both polarization components failed toresemble the wave-guided structure inside the medium.Hence the created structure of the probe beam suffers dis-tortion due to diffraction. In Fig. 3, we show the surfaceplot of χ as a function of transverse directions x and y . Two orthogonal axes x and y can be used to defineazimuthal phase φ = tan − ( y/x ). It can be seen fromFig. 3 that the medium becomes transparent at somespecific angular positions for the right-handed polarisa-tion. These angular positions can be defined by nπ/l where n can change from 0 to l . As a result of angulardependency, the absorption profile for the right-handedpolarisation shows fourfold symmetry with OAM, l = 2.A similar periodic absorption pattern is exhibited by theleft-handed circularly polarised component as mentionedin Eq. (20). Note that in absence of phase modulation,both polarisation components suffer from high attenua-tion. Thus at weak field limits, in a closed loop tripodsystem, the phase information of each polarisation com-ponent gets converted into the intensity information thatrenders transparent an otherwise opaque medium.We now discuss the response of the medium beyondthe weak field limits as shown in Fig.(4). For a relativelystrong probe field limit g = 0 . γ , the numerical solutionsof linear algebraic equations (7) are inevitable to ana-lyze the phase dependent susceptibility of the medium atsteady state condition. The oscillating amplitude of po-larization components reduces with increase in g . As aconsequence the population in the ground states | i , and | i gets depleted. The depletion of population in theseground states is the cause of width-broadening of thetransparency window. Surprisingly the refractive indexprofile of two orthogonal polarisation components mod-ify drastically as compared to the case in a weak fieldregime. It is evident from Fig. 4 that the gradient ofrefractive index is dependent on the detuning sign of po-larization components. The slope of the refractive indexattains its maximum around the transparency window FIG. 3: (Color online) Absorption pattern of the ˆ σ + polar-ization component is plotted against the two orthogonal axes x and y . Other parameters are same as in Fig. 2. φ -3-2-101234 χ x10 -5 Im( χ ) Im( χ ) Re( χ ) Re( χ ) FIG. 4: (Color online) The variations of real and imaginarypart of susceptibilities χ and χ as a function of phase forrelatively strong probe regime are plotted. The parametersare chosen as N = 10 atoms/cm , Γ = 0 . γ , ∆ p = 0, β = 0 . γ , θ = π/ γ c = 10 − γ and l = 2. and decreases gradually towards the wings for a red de-tuned right circular polarisation component. A convexlens like refractive index is formed for the red shifted po-larisation component whereas concave lens like refractiveindex is experienced by the blue shifted polarisation com-ponent. Thus by selecting detuning of two orthogonal po-larisations ˆ σ ± , leads to the formation of a waveguide [31]and an anti-waveguide [32] in the closed loop tripod sys-tem. Hence these waveguide/antiwaveguide structurescan lead to focusing/defocusing of polarisation compo-nents. These features are missing for the weak intensitylimits as the susceptibilities are independent of polarisa- (a)(b) FIG. 5: (Color online) Panel (a) and (b) depicts transmit-ted probe beam intensity in the transverse ( x − y ) plane for θ = π/
18 and π/
14, respectively. The intensity profile of theprobe beam is shown in the panel (a) and (b) after traversea distance of medium length 0 . mm . The mode, OAM andwaist of the Laguerre-Gaussian beam are m = 0, l = 2 and w p = 20 µ m, respectively at z = 0. Other parameters aresame as in Fig. 2. tion amplitude as shown in Eq. (19). Also the amplitudeof the refractive index is stronger here than in the weakfield limits. Hence a suitable choice of detuning of eachpolarization component at the strong field regime canlead to diffraction controlled petal like structured beamgeneration. B. Beam propagation dynamics
Next we illustrate how spatially dependent susceptibil-ity enables us to generate the structured probe beam. Forthis purpose, the transverse spatial profile of both the po-larisation components is to be in the Laguerre-Gaussianmode that can be written as g j ( r, z ) = g × w p w ( z ) × r √ w ( z ) ! | l | L lm (cid:18) r w ( z ) (cid:19) × e ± ilφ e − (cid:16) r w z ) (cid:17) e (cid:16) ikr R ( z ) (cid:17) e − i (2 m + l +1) tan − (cid:16) zz (cid:17) (22) r = p x + y φ = tan − (cid:16) yx (cid:17) . The indices m determine the shape of the probe field pro-file along the transverse directions. The radius of cur-vature and the Rayleigh length are defined as R ( z ) = z + ( z /z ), and z = πw p /λ , respectively. The beamwidth is varied with propagation distance z as w ( z ) = w p p z/z ) , where, w p is the beam waist at z = 0[30]. We adopt higher ordered split-step operator methodto numerically study the beam propagation Eq. (21).Fig. 5 shows the output intensity pattern of the probebeam at a propagation distance of 0 . I out = | g | + | g | . It is evident fromFig. 5 that the fourfold symmetry which exists in the ab-sorption profile of both the polarization components aremapped onto their transverse spatial profile. Also thisspatial profile assures that each polarization componentcarries OAM with units of ± ~ . Thus the value of OAMdictates the formation of structured probe beam with de-sired shape. Hence manipulation of the absorption profilealong the transverse direction forms the key idea behindthe structured probe beam generation. We also find thatthe transmission for the probe beam at a propagationdistance of 0 . θ = π/
18 and 62% for θ = π/
14 even at weak field limits. We further studyhow the magnetic field strength allows us to enhance thecontrast of the structured beam. A higher strength β T creates a high contrast periodic absorption profile as com-pared with a weak β T as shown in Fig. 2. This sharpvariation of absorption profile emulates a high contrastspatial probe beam profile as depicted in Fig. 6(b). Onthe other hand, Fig. 5(a) shows a low contrast spatialprobe beam for the weak field regime at a propagationdistance of z = 0 . FIG. 6: (Color online) Intensity variation of the probe intransverse ( x − y ) plane, after propagating through a mediumof 0 . g = 0 . γ , m = 0, l = 2and w p = 20 µ m, respectively. Other parameters are same asin Fig. 4. to the later. Fig. 4 exhibits the waveguide and anti-waveguide refractive index profile for the constituents ofthe probe beam at ∆ p = 0, which enhances the con-trast of the output pattern. The waveguide structureconfine the σ + polarization component whereas the σ − polarization component gets defocused due to the anti-waveguide structure. Moreover, the spreading of boththe polarization components in the azimuthal plane islimited by the width of the spatial transparency window.Hence at the strong regime both absorption and disper-sion profiles play important roles in improving the outputbeam pattern whereas the absorption profile is solely re-sponsible in the weak field case. The transmission of thestructured beam at a propagation distance of 0 . mm is found to be 66%. The increase in beam transmissionis due to waveguide induced focusing of the probe beamin the azimuthal plane. We also notice from Fig. 6 thatthe generated structure beam is rotated by an angle of10 ◦ . This rotation is attributed to strong field inducedNMOR. The rotation of the structured beam can be en-hanced by increasing the intensity of the probe and themagnetic field strengths [33]. Our approach opens upnew possibilities for generating high contrast structuredbeam in other closed loop systems that display narrowEIT resonances. The step variation of refractive indexaround the narrow transparency window is the main rea-son behind the formation of high contrast beams. Thusan atomic medium with buffer gas [34] and inhomoge-neously broadened atomic system [35] may be suitablecandidates for creating a diffraction controlled high con-trast structured beam. IV. CONCLUSION
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