Space-Time Cloaking based on Reﬂection, Transmission and Goos-H¨anchen Shift

: The birefringence characteristics of reﬂection and transmission of light pulse are explored in chiral medium driving by two probe electric and magnetic ﬁelds, while four control ﬁelds. The proposed medium show signiﬁcant birefringence behaviors in reﬂection/transmission and Goos-H¨anchen (GH) shifted beams. The birefringent GH-reﬂected and GH-transmitted pulses show signiﬁcant contribution in the space time gap creation for secure communication and invisibility. A space-time gap has been observed in the reﬂection, transmission and corresponding GH-shifted reﬂection and transmission birefringent beams. The theoretical results may be useful for noise free secure communication in invisible radar technology.


I. INTRODUCTION
Cloaking has recently played a significant role in the phenomenon of hiding events using electromagnetic fields. To attain spatial cloaking, the refractive index is controlled to flow light around the object from a probe, such that a "gap" in space is produced, and remains concealed [1][2][3]. Alternatively a "time cloak" or "temporal cloak" is the one which hide information over a finite period of time. The dispersion of material is manipulating in temporal cloaking to hide the occurrence of an event from the outside source. A new type of electromagnetic cloaking is under consideration, where the events can be hidden rather than objects termed as "Space-time cloaking". Space-time cloak is a means of controlling electromagnetic radiations in space and time in such a way that a collection of events, or happenings is hidden from the distant perceiver [4].
Space time cloaking manipulate and transform data from one to the receiver in secure form and based experimentally on temporal cloaking (TC) [4]. By upgrading time to equal basis with three dimensional space, space-time cloaks, that conceal region of space for a specific window of time can be imagined [4,5]. Although the full space-time cloak has not yet been implemented. The previous work has really shown pure temporal cloaking for spatially single mode lighting [6]. An event taking place in the gap is cloaked from the sensors and only a particular observer can be reached to the original information of light. This phenomenon leads to new successive possibilities in the field of quantum optics [7], such as it operates commonly for increasing or decreasing the speed of light without deviation or realignment of light [8]. Recent work on time cloaking observed a gripping approach to hide temporal events from an applied light field and also study intensity gaps in a probe beam. Experimental results show temporal cloaking of non-linear relation and large-speed optical data [9]. TC like phenomenon had also been reported in the last decade regarding conductivity problems [1]. Soon after origination, a lot of cloaking devices were carried out using electromagnetic meta-materials from microwave [2] to optical frequency range [3]. In a few years the field has reached to a high level of maturity due to its practical applications [10]. The idea of TC came out with the intention of hiding events rather than matter [11]. Firstly TC presentation was revealed in fiber optics setup [12]. Four-wave mixing was applied to impress a non-continuous linear frequency chirrup on a known segment of a monochromatic laser signal [13]. In addition, the acoustic cloaking, where an object can be hidden and transformed acoustically, is a practical application based on acoustic meta-materials [14].
In 1947 Goos and Hänchen showed experimentally the phenomenon of GH-Shift for the first time [15]. When an infinite plane wave is incident on the interface of two mediums, there occur a small shift between the incident and reflected wave called GH Shift [16] and is proportional to the depth of penetration comparable to the wavelength of the light [15,17]. This tiny shift occurs under the process of total internal reflection [18]. GH effect is studied theoretically and experimentally for different mediums having different refractive indices, and is observed in the case of electron [19,20] and neutron [5,21]. Allan and John [23] presented an easy derivation for total internal reflection grounded on energy considerations and Fresnel's equation. On the basis of these considerations GH-shift is observed at the interface of two semi-infinite dielectric media due to which this energy flows through the dielectric boundaries. GH effect can also be obtained by using energy flux conservation [24]. GH-Shift has a variety of applications in different fields such as acoustics, quantum mechanics, photonics, plasma physics, micro optics and nano-optics [25], i.e. optical wave-guide switches [26], ultra-layered structures [27], displacement sensors [28] and many more.
A large number of research work has been published on temporal, space and event cloaking using different techniques and formalism. In my knowledge no work is available in literature on space-time cloaking, in which five level atomic system having electric and magnetic probe fields and four control fields are involved to modify space-time cloaking based on reflection/transmission and corresponding GH-shifted beams. In this work we use the above formalism to modify event or space time cloaking for noise free secure communication to protect information hacking.

II. DYNAMICS OF THE SYSTEM
A five level chiral atomic system is under consideration for controlling space-time cloaking through GH-shifted beams as shown in the Fig.1. The lower energy level |1 is coupled to energy levels |2 and |4 with control fields E 1,2 having Rabi frequencies Ω 1,2 respectively. The level |1 is also coupled to energy level |3 with magnetic field B having Rabi frequency Ω m . Likewise the energy levels |2 and |4 are coupled to level |5 with controlled fields E 3,4 having Rabi frequencies Ω 3,4 . The level |3 is coupled to level |5 with probe field E p having Rabi frequency Ω p . The decay rates γ ij between two states and detuning of driving fields ∆ i are shown in the figure. To discuss the system dynamics the following interaction picture Hamiltonian is used FIG. 2: Group refractive index n (±) g vs probe field detuning ∆p/γ and control field Rabi frequencies |Ω1,2|/γ, such that (a) Ω1 = 2γ, 4γ, 6γ, 8γ (b) Ω2 = 2γ, 4γ, 6γ, 8γ. The external fields are tuned to their angular frequencies and atomic state resonance frequencies as: 13 − ω m of the system. The following Von-Neumann equation of density matrix is to measure coherence between two states [29][30][31]: where δ and δ † are the lowering and raising operator respectively for the atomic decay Γ j=1,2,3,4,p,m . By putting After large algebraic manipulation the following coupled rate equations are obtained Initially the state |1 , |2 , |3 , and |4 , are taken in superposition state as The zero order density matrix is In the above mentioned equations Ω m and Ω p are taken in first order perturbation, while Ω 1 , Ω 2 Ω 3 and Ω 4 are taken in all orders. From superposition states, ρ 52 = 0, We can use the following relation to solve coupled rate equation [32,33]: where Z(t) and B denotes column matrices, and A denotes 4x4 matrix. Using eq.8 the coherence ρ 35 and ρ 31 are obtained in the following forms: The polarization and magnetization equation can be written from coherence P = N σ 35 ρ 35 and M = µ 31 N ρ 31 . Plugging B = µ 0 (H + M ), Ω m = µ 31 B/h and Ω p = ρ 35 E/h in polarization as well as magnetization equation we obtain the following.
where 5 and Q are given in appendix. The polarization and magnetization in the form of electric and magnetic susceptibility and chiral condiments are available in literature as: Comparing eqs.11 and eqs.13, while eqs.12 and eqs.14 the following electric and magnetic susceptibilities and chiral coefficients are obtained: The complex refractive index of chiral medium is calculated from eqs. (15)(16)(17)(18) and can be written as: The chiral coefficient contribute additional terms ξ EH and ξ HE to refractive index. If ξ HE,EH = 0 the medium will be left handed or right handed. It is dependent on r = 1 + χ e and µ r = 1 + χ m . If both r and µ r are positive the refractive index will be positive. This medium is called positive refracted medium or conventional medium or right handed medium. If both r and µ r are negative the refractive index will be negative. This medium is called negative refracted medium or left handed medium. If ξ HE,EH = 0 the medium is chiral. It may be conventional or negative refracted medium with the need of positive and negative r and µ r . These terms show circular birefringence in the medium having refractive index n (+) r and n (−) r . The birefringent reflection and transmission are written as: The corresponding left and right circularly polarized GH-shifts in reflection and in transmission beams are written below: The input pulse is taken in the Gausian form as: where The reflection and transmission birefringent pulses are written as: The GH-shifted birefringent pulses are written as:

III. RESULTS AND DISCUSSIONS
In this section we present and explain our main outcomes. The parameter γ is considered to be 36.1GHz and other parameters are scaled to this γ. In our numerical simulations atomic units are used, whileh, µ 0 and 0 are considered to be unity. Those parameters which are same for all the plots are: Ω 3,4 = 1γ, γ 1,2,3 = 1γ, γ 4 = 0.2γ, γ m = 0.01γ, γ p = 2γ, ∆ 1,3,4,m = 0γ, ϕ 1 = π/3, ϕ 2 = π/4, ϕ 3,4 = 0,φ 1,3 = π/6, φ 2,4 = 0,φ 5 = π/4, θ = π/3, α 1 = 0.4, α 2 = 0.1, In FIG.4 the graphs are traced for reflection and corresponding GH-shift in reflection of birefringent beam under controlled field of Rabi frequencies |Ω 1,2 |/γ. The reflection beams oscillate between the value of -1 to 1 at incident angle shows the intensity of transmission with greater amplitude while the second one shows the transmission with very less amplitude. Once again the effect is observed with comparable intensity peak for reflection of the birefringent beams having the same intensities range and space time gap as that of FIG.6a. The incident angle is supposed here θ i = π/6 as shown in FIG.6c. In FIG.6d the birefringent transmitted beam have intensities from 0 to 0.0001, which having less intensity as compared to the reflected beams of FIG.6c. In all these cases no amplitude of pulses is occured in the space time gaps. Beside this |Ω 1 | and |Ω 2 | are taken 5γ and 8γ for all the plots respectively.
In FIG.7 the graphs are traced for birefringent GH-reflection and GH-transmission pulses intensities vs time and space coordinates normalized to pulse width τ 0 and W . The birefringent GH-reflected pulses have different intensities. The intensity of right peak is maximum while the intensity of the left peak is very less at incident angle θ i = π/6. There occurs a space time gap, where the event can be hidden from the outside sources as shown in the FIG.7a. The birefringent GH-transmitted pulses intensities with the same parameters are shown in FIG.7b. Both the plots (a,b) have the same intensity range i.e. from zero to 0.4. The reflected pulse is splitted through the medium into two intensities, where the intensity of one peak is greater than the other at incident angle θ i = π/3 and the intensity range is from zero to 1 as shown in FIG.7c. In the same manner the birefringent GH-transmitted beams with broaden peaks, having minimum left pulse and maximum intensity ranges from 0 to 0.1 as shown in the FIG.7d. |Ω 1 | and |Ω 2 | taken taken 8γ and 5γ for all plots of this figure respectively.

IV. CONCLUSIONS
The birefringent characteristics of reflection and transmission of light pulse are explored in a five-level chiral atomic medium. Two probe electric and magnetic fields, while four control driving fields are considered in the system to modify the birefringent response of the medium. The proposed medium shows significant birefringence behaviors in the reflection/transmission and GH-shifted beams. The birefringence behavior of the beams is controlled and modified with intensities of control fields, probe and control field detunings, the phases variation of control fields and phases of superposition states. The birefringent GH-reflection and GH-transmission pulses intensities are controlled for space time gaps. A space-time gap has been observed in reflection, transition and corresponding GH-shifted reflection and transmission birefringent beams. The amplitude or intensity of the electromagnetic pulse is zero in this space-time gap where an external event remain undetected. The space time gap is important for secure communication between different systems and invisibility objects. The theoretical results may be useful for noise free secure communication in an invisible radar technology.