Surface plasmon polaritons induced reduced hacking

Bakhtawar, Muhammad Haneef, and Humayun Khan

Lab of Theoretical Physics,Hazara University Mansehra,21300 KP,Pakistan

Keywords: surface plasmons,coherent control of atomic interactions with photons,reduced hacking,surface conductivity

1. Introduction

Are these modes capable of secure transmission of information, the reduced hacking? This is a well known phenomenon in which information can be hacked from external detectors through a trick of light.[1,2]The idea of reduced hacking was first introduced by McCallet al. using the concept of light splitting into slow and fast components.[3]This was experimentally demonstrated using optical fibers based on dispersion[4]as well as polarization control of a probe beam.[5]After the experimental demonstration of reduced hacking,several research groups put their efforts for the realization of photonic devices capable of achieving hacking.[6–8]It simultaneously attracted tremendous theoretical interest for its proper description and modification. For example, Liet al.[9]reported a reduced hacking scheme based on quantum destructive interference in a three-level warm atomic system where a reduced hacking was created in a long optical pulse propagating in the system. Wu and Wang demonstrated their reduced hacking scheme using the Fourier analysis method.[10]Chremmos presented a scheme where biconvex temporal gaps were created continuously in a medium having positive or negative dispersion.[11]In a previous work, our colleagues have discussed creation of the reduced hacking between the enhanced subluminal and superluminal pulse caused by the Doppler broadening in atomic medium.[12]

However, a question whether it is possible to construct a reduced hacking device that can work for surface plasmon polaritons(SPPs)arises. SPPs are electromagnetic modes that arise due to the interaction between light and collective oscillations of free electrons in a metal.They are bound to the surface of the conductor and prevent power to propagate away from the surface.[13,14]SPPs have generated remarkable interest for their ability to manipulate light at subwavelength scales,leading to nanophotonic devices with length scales much smaller than those achievable with ordinary light.[15,16]

So for,SPPs have found numerus applications in polarizers,sensors,photodetectors,spectroscopy,microscopy,metasurfaces,compact cameras,integrated imaging,and detection systems.[17–34]With these remarkable properties, SPPs can present some exciting opportunities for efficient reduced hacking. With this in mind,in this article we present a theoretical approach towards the reduced hacking based on SPPs. We consider the interface between a coherently driven four-level atomic medium and a metallic conductor where propagation of the SPPs is manipulated with conductivity of the metal and the control parameters of the atomic medium. The advantage of using a coherent treatment is that one has different controlling parameters in hand to manipulate the plasmonic modes.[35–37]We show that event cloaking is possible for the surface modes at the interface between the two media. Our results show improvement/development towards the plasmonstor based reduced hacking technology which may be of interest for the researchers working in either of the two fields,reduced hacking and surface science.

This paper is organized as follows.In Section 2,we define the geometry and present analytical expressions for describing both media above and below the interface. It is followed by the dispersion relation of surface plasmon together with the description for the reduced hacking. In Section 3,we present and discuss the results of the study. In Section 5,we summarize the results.

2. Model and analytical expressions

The schematic diagram of Fig. 1(a) shows the interface between a metallic conductor and an atomic medium along which SPPs are excited. The conducting medium is characterized by complex conductivityσ, complex electric permittivityεmand magnetic permeabilityμm. It is supposed that it obeys the source free Maxwell equations. When a probe field is incident on the coupled surface, it causes variations in the complex conductivityσand permittivityεmof the metal. The wave equations for the electric and magnetic field components inside the conducting medium can be written as

The solutions to Eqs. (1) and (2) are the model with plane wave time dependenceEm(r,t)=E0meikmr?ωtandBm(r,t)=B0meikmr?ωt, whereE0mandB0mstand for complex amplitudes for the electric and magnetic fields inside the metal.Further,kmrepresents the complex wave vector of the probe field and can be written as[36]

whereωis the angular frequency of the probe field.To include the phase dependence, the complex permittivity and permeability of the metal are written asεm=ε0εrmandμm=μ0μrm;ε0andμ0are the permittivity and permeability of the free space. The termsεrmandμrmrepresent complex dielectric and permeability constants. In polar formεrm=|εrm|eβ1,μrm=|μrm|eβ2,σ=|σ|eβ3. Here|εrm| and|σ| represent amplitudes of complex dielectric and permeability constants,|μrm|is the amplitude of complex conductivity, andβ1,2,3are corresponding phase of these quantities.Plugging these values in Eq.(3)we obtain

Sincekmis also a complex quantity and can be expressed askm=k1m+ik2m, the expressions for its real and imaginary parts are calculated by

In Eqs.(4)–(6),the constant terms are defined as

Ifnmris the refractive index of the conducting medium, thenkm=k0nmr, wherek0=ω/c=2π/λis the free space wave number of the incident light. Using the above relations the refractive index of the conducting medium is calculated by

Further we assume a four-level atomic whose energy level diagram as shown in Fig.1(b). It can be experimentally realized in four energy levels in the D lines of87Rb atom.[38,39]The same configuration was used recently for the manipulation of SPP’s solitary waves[40]as well as for subluminal and superluminal SPP propagation.[36]Here,a probe fieldEphaving Rabi frequency?pis coupled with states|2〉and|3〉. A control fieldE1having Rabi frequency?1is applied between the states|1〉and|3〉while the control fieldE2of Rabi frequency?2couples the states|2〉and|4〉. The spontaneous decay rates between the coresponding states are denoted byγ31,γ32,γ41andγ42as shown in the figure.

Fig.1. (a)The interface between a metal and the atomic medium where SPPs are generated. (b)Energy diagram of four-level atomic system.

The self energy part of Hamiltonian of this atomic configuration is

Here,?p=d23Ep/,?1=d31E1/and?2=d24E2/;d23,d31,andd24are the dipoles matrix elements between the states|i〉and|j〉.?1,2,prepresent the detuning frequencies of the corresponding fields.

The master density matrix equation used for the dynamical solution of the system is given by[41]

whereQ?(Q)is the raising(lowering)operator. After straight forward algebraic manipulation,the coupling equations for the given system can be obtained as follows:[40]

whereω0is the central frequency of the probe field. Next, a propagating SP at the interface can be expressed as

whereSi(t)is taken Gaussian pulse have widthτ0in time domain while

in frequency domain.

Delay or advancement of the SP is caused by the phase shift in frequency domain. Therefore, ifS(t)is the propagating SP, a phase shift e±iω?t(sp)is created due to the fact that delay or advancement of the SP is caused nearly at the central timet0. To create a time window in the SP transmission spectrum,the transfer function at the interface of two media can be obtained as

The transmission spectrumStr(?p)of the pulseSi(?p)is written as

which can be further written in time domain via the Fourier transform

This is a piece-wise continuous function and therefore,phase shift theorem is applied toStr(t) which results in the subsequent ranges[42]

This relation shows that the transmission frequency spectrum is zero in the time intervalt0??t(sp)＜t ＜t0+?t(sp). This interval is the created time gap which is double of the pulse delay or advance time, i.e., 2?t(sp), where the information can be transmitted without any detection. If any eventSe(t)is transmitted within this gap, then the expression becomesSe(t)Str(t)=0, which shows that the eventSe(t) has no effect on the transmission frequency spectrum and can be safely transmitted without hacking. To close the time gap another interface is required whose transfer function is complex conjugation of the original transfer function. This will reverse the process and remove the time gap.

3. Results and discussion

In this section we present and discuss the results obtained on the basis of the analytical expressions of the previous section. The parametersκ1= i?p?(γ32+γ42)/2,κ2=i(?1??p)+(γ32+γ31+γ42+γ41)/2,κ3=i(?2??p),κ4=i(?1+?2??p)+(γ31+γ41)/2 andκ5=4(κ1κ2+κ3κ4).Propagation of the surface waves,their absorbtion,dispersion,group index and delay/advance time are investigated at the interface of the proposed media. Time gap in the delay or advancement of the SPP is also manipulated. For numerical results, we adopt a spontaneous decay rate ofγ=1 MHz and scale other parameter of the atomic medium with thisγ. Other common parameters assumed for the study are?1,2= 0γ,?1=3.5γ,?2=0.5γandμrm=1.Moreover,the decay terms are taken to beγi j=2γ.[38]The phases of the complex quantities related to the conducting medium are kept asβ1=0,β2=π/3 andβ3=π/4. The conductivity of the metal is expressed in units of S/m.

Fig. 2. Absorption and dispersion behavior of surface plasmon polariton versus conductivity of the metal such that ?p =0γ (solid line),?p=0.5γ (dashed line)and ?p=0.8γ (dotted line).

The probe field, coupling the states|2〉and|3〉of the atomic medium, decays evanescently along the interface and excites SPs. Absorption and dispersion behaviors of the surface modes are plotted against conductivity of the conducting medium as shown in Fig.2. The imaginary Im(ksp)shows absorption and real part Re(ksp) shows dispersion spectrum of the SPP. A single maximum of the absorption and minimum of the dispersion spectra are noticed nearσ=2 S/m at specific parameters of the proposed media. However, the peaks in the dispersion curves can be shifted to higher values of the metal’s conductivity as seen from dashed and dotted curves.With increasing the conductivity further, the dispersion spectrum finally becomes saturated and the surface mode acquires a fixed character.

Figure 3 shows group index and delay/advance time of the SPP. The plots show that group index and delay/advance time is positive at low values of the metal conductivity and negative at higher values and become saturated at increasing the conductivity further, which is in agreement with the previous results. At resonance?p=0γin the atomic medium,the group index is?637.5 at conductivity of the metalσ=3.96 S/m and is noted to be?272 and?80 at the probe detuning frequencies?p=0.5γand?p=0.8γ,respectively,for the same value of conductivity. This shows that the plasmon delays and advances during its propagation along the interface. The value of advance time at the interface at resonance is?20×10?8s while at?p=0.5γthe same value of conductivity is 10×10?8s. At?p=0.8γdelay time of 5×10?8s is noted. Furthermore, the mode propagation is noted to change between the superluminal and subluminal propagation by varying the probe detuning in the atomic medium at the specific value of conductivity in the metallic medium.

Fig.3. Group index and group delay/advance time of surface plasmon polariton versus conductivity of the metal. Other parameters are kept the same as in Fig.2.

In Fig.4,we plot the normalized intensities of the generated plasmonic pulse,its transmission at the interface and the final output pulse versus time. The timetis normalized by the probe field widthτ0,which is assumed to be 25 ns in this case. The upper plot (red curve) shows the initially excited plasmonic pulse in time domain. The transmission is denoted by the three curves blue,green and purple,in the central portion of this figure,for different values of probe field detunings.Note that a temporal window can be created for the plasmonic pulse which splits for an arbitrary time during its transmission. The front part of the pulse speeds up while its back part slows down,the so-called superluminal and subluminal propagation. Time gaps of 400 ns, 200 ns and 100 ns are generated at the advance/delay times of 20×10?8s, 10×10?8s and 5×10?8s for probe field detuning?p=0γ,?p=0.5γand?p=0.8γ, respectively. This shows that the time gaps can be conveniently controlled by adjusting different parameter related to the atomic medium. These time gaps provide the necessary durations for an event to occur and can be made hidden from external detectors. The initial form of the pulse can be restored by reversing the whole process. In other words,when the time gape is closed the original information can be restored as shown in the final plot,showing output SPP pulse intensity.

Fig.4. The normalized SPP pulse,its transmission and output intensities versus t/τ0 such that τ0=25 ns,?p=0γ(blue solid line),?p=0.5γ(green dashe line)and ?p=0.8γ (purple dotted line).

In Fig. 5 we show how the normalized intensities of the generated SPP pulse,its transmission through the interface and the output pulse intensities can be manipulated with different probe field widths.Here,the probe field is assumed to be resonantly coupled and conductivity of the metal is kept constant atσ=3.96 S/m.We find that the time gap decreases as the probe width is increased,and vice versa,as shown in the central part of Fig. 5. The transmission pulse intensity remains zero for long duration of time when the pulse width is kept small and hence a comparatively larger time gap is created. The time gap is closed by the reverse process and the initial SPP pulse is obtained in its original form as shown in the lower plot.This shows that the time gap for the event cloaking can be well manipulated with varying width of the probe field. Moreover,as the properties of the SPPs are shown to depend on conductivity of the metal(Fig.2),the cloaking phenomenon can also be controlled by conductivity of the metal.

Fig.5. The normalized SPP pulse,its transmission and output intensities versus t/τ0 such as ?p=0γ,τ0=25 ns(blue solid line),τ0=35 ns(green dashed line),and τ0=45 ns(purple dotted line).

4. Conclusions

In summary we have investigated reduced hacking based on SPPs propagating along the interface between a four-level atomic medium and a metallic conductor. We have modified the SPPs propagation with conductivity of the metal. Further,a plasmonic pulse is noticed to delay and advance in time during its propagation at the interface. This delay and advance of the pulse lead to the creation of the time gap for reduced hacking.Temporal gaps of the orders of nanoseconds are measured for reduced hacking of plasmonic waves.These gaps are noted to be further modified with conductivity of the metal, detuning frequency and width of the probe field. The time gaps are also closed by the reverse process. Our results may have potential applications in nanoscale-sized devices for storing or sending secure information. Comparisons with the previous methods and published articles are present. In this work, we have modified the reducing hacking by surface Plasmon polariton at the interface of atomic and metallic media. In the methods of other works the frequencies are used to modify the cloaking. Here the metal conductivity along with frequencies plays an important role for the modification reducing hacking.Further we have used here piece wise continuous function for SPPs at the interface. A nano second time gap is control for SPPs,which is very large as compared to its decay.