Eur. Phys. J. D (2014) 68: 55DOI: 10.1140/epjd/e2014-40246-1

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Controlling the Goos-Hanchen shift via quantum interference

Mojtaba Rezaeia and Mostafa Sahrai

Research Institute for Applied Physics and Astronomy, University of Tabriz, Tabriz, Iran

Received 17 April 2013 / Received in final form 17 November 2013Published online 21 March 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. The behavior of the Goos-Hanchen (GH) shifts of a probe beam reflected from or transmittedthrough a cavity with a fixed geometrical configuration is theoretically investigated. The effect of quantuminterference induced by incoherent pump and spontaneous emission upon the control of GH shifts is thendiscussed. In addition, the effect of the rate of an incoherent pump field and the intensity of coupling fieldon the behavior of GH shifts are presented.

1 Introduction

It is well-known that the totally reflected light beam is lat-erally shifted from the position predicted by geometricaloptics. This effect occurs because the reflections of a finitesized beam will interfere along a line transverse to the av-erage propagation direction, so each of its plane wave com-ponents undergoes a different phase shift. This phenomenais referred to as the Goos-Hanchen shift (GH shift) whichexperimentally observed by Goos and Hanchen [1,2]. Theshift is perpendicular to the direction of propagation, inthe plane containing the incident and reflected beams. TheGH shifts has some interesting applications in optical sens-ing [3], where it can be used to measure the beam angle,refractive index, and displacement, for surface and filmthickness studies [4]. It can also be used for probe irregu-larities, roughness, etc. on the surface of an isotropic spa-tially dispersive medium [5,6]. Therefore, possibilities tohave control on GH shifts behavior are very important fordifferent applications of this effect in nano-optics, acous-tics, quantum mechanics, plasma physics, nonlinear op-tics [7–10], and surface physics [11]. Various schemes withdifferent structures have been proposed to enhance lateralshifts [12–16]. This effect continues to be a topic of scien-tific research, for example in the context of nano-photonicsapplications. Merano et al. [17] studied the Goos-Hancheneffect experimentally for the case of an optical beam re-flecting from a metal surface (gold) at 826 nm. They founda negative lateral shift of the reflected beam in the plane ofincidence for a p-polarization and a smaller positive shiftfor the s-polarization case. Besides, the GH shift associ-ated with left-handed materials is of interest for potentialapplications. Berman [18] and Lakhtakia [19] studied ex-tensively the GH shifts at an interface between “normal”and left-handed media. Kong et al. [20] elaborated thelateral displacement of a Gaussian-shaped beam reflected

a e-mail: rezaei.photonics@gmail.com

from a grounded slab with simultaneously negative per-mittivity and permeability. It is also found that GH shiftsof the transmitted beam through a slab of left-handedmedium can be negative as well as positive [21].

Recently, Wang et al. [22] proposed a scheme for real-izing the manipulation on the GH shifts of a probe beamvia a coherent control field in a fixed geometrical config-uration. They proposed a cavity with two walls of somedielectric material and intracavity medium containing atwo level atomic medium whose dispersion and absorp-tion relation can coherently be controlled by the intensityof driving field. On the base of this study, many schemesfor controlling GH shifts of a probe beam reflected from ortransmitted through a cavity is introduced [23–25]. There-fore, the GH shifts of the reflected and transmitted probelight beam can be changed from positive to negative orvice versa, without changing the structure of the cavity,just by adjusting the controlling parameters such as de-tuning and intensity of the control fields. Note that thepositive or negative GH shifts in the reflected and trans-mitted light probe may correspond to positive or negativegroup index of the medium, respectively [23]. Sublumi-nal [26–28] and superluminal [29,30] light propagation, onthe other hand, in an atomic medium correspond to pos-itive and negative group index of the medium, which canbe coherently controlled via EIT [31,32].

In view of many proposals, we note that the controlof the GH shifts is proposed with changing the parame-ters controlling such as detuning, intensity of control fieldand so on. Therefore, it is worth studying the control andmanipulation of GH shifts without changing the param-eters controlling of a system. Nevertheless, the effect ofquantum interference on the GH shifts behavior has neverbeen investigated to our best knowledge. Now, intriguingquestion arises that what is the effect of quantum inter-ference on GH shifts. It is well-known that optical prop-erties of an atomic medium can be substantially modified

Page 2 of 10 Eur. Phys. J. D (2014) 68: 55

by quantum coherence and quantum interference inducedby laser fields [33]. This may open up new routes for co-herent control of the GH shifts via quantum interferenceand quantum coherence without changing the parameterscontrolling that is the main proposed of this paper.

Atomic coherence can be created by the coherent inter-action of a multi-level atomic system with coherent laserfields. In some recent studies, it has been demonstratedthat the optical properties of a multi-level atomic systemcan be controlled by coherent driving fields [34–40]. Insuch cases, quantum coherence and quantum interferenceare the basic mechanisms for modifying the optical re-sponse of the medium. Quantum coherence can also beencreated by the incoherent processes such as spontaneousemission and incoherent pumping field. Spontaneous emis-sion is usually believed to result in decay but not buildingthe coherence between atomic levels. However, it is shownthat the spontaneous emission can also be used for produc-ing atomic coherence as long as there are two closely lyinglevels. So, the radiation decay from common upper levelto two closely-lying lower levels (or from two closely-lyingupper levels to a common lower level) creates coherencebetween these levels [41,42].

In this investigation, we propose a new method to con-trol the GH shifts behavior via quantum interference in-duced by incoherent pumping field and decay processes.Recently, we (with collaborators) have investigated thedispersive and absorptive properties of a Λ-type atomicsystem with two fold lower-levels, and obtained a vari-ety of absorption spectrum that can be modified by thecontrolling parameters; we discussed the steady state be-havior of the system and demonstrated that the probedispersion, absorption and group index can be controlledby the quantum interference induced by incoherent pump-ing field and decay processes [43]. So, in this paper we usethis atomic medium as an intracavity and find that quan-tum interference induced by incoherent pumping field anddecay processes, has a major role on controlling GH shiftsof probe beams reflected from or transmitted through acavity. The behavior of GH shifts is quite different in thetwo distinct cases with or without the quantum interfer-ence effect. In particular, we can control the GH shiftsbehavior via quantum interference without changing theparameters controlling such as detuning, intensity of con-trol field and geometrical configuration of the structure,which is certainly favorable. This paper is organized asfollows; in Section 2, we present the model and equations.Atomic model of the intracavity medium is presented inSection 3. The results are discussed in Section 4, and theconclusion can be formed in Section 5.

2 Model and equations

Consider a simple model of cavity consisting three layerswith structure ABA. Two side layers A, acting as wallsof the cavity, are nonmagnetic dielectric slabs with thesame thickness dA and permittivity εA. Layer B is the in-tracavity medium with thickness dB and permittivity εB

(Fig. 1). The intracavity medium can be gas atoms [44,45],

Fig. 1. Schematic diagram of the GH shifts of the reflectedand transmitted light beams from a fixed cavity. The cavitycontains four-level Λ-type atoms. The red arrows display thecoupling and pumping beams and the blue ones display theprobe beam. Sr, St are the reflected and transmitted beamsGH shifts, respectively.

or can be some semiconductors [46,47]. According to ourmodel, intracavity medium consists of four-level gas atomsin Λ-configuration with fold lower levels. A probe beamwith angular frequency ωp is incident on a one side ofcavity from the vacuum with permittivity ε0 = 1. Thewave vector of incident probe beam makes an angle θwith the z-axis. This field is reflected back or transmit-ted through the cavity with a lateral shift (GH shift). Anincoherent pump field and a coupling field drive the four-level atomic system inside the cavity. Propagation of apulse through such cavity, a dispersive-absorptive system,has been described by Liu et al. [48]. Generally, the elec-tric and magnetic fields at two positions z and z + Δz inthe jth layer can be related to each other via a transfermatrix as [48]:

Mj(kj , ωp, Δz) =

⎛⎝ cos(kj

zΔz) isin(kj

zΔz)qj

iqj sin(kjzΔz) cos(kj

zΔz)

⎞⎠ , (1)

where kjz =

√εjk2 − k2

y is the z component of probe wavevector in the jth layer. ky is the y component of probewave vector in vacuum (k = ωp

c ). Here, c is the speedof light in vacuum, and parameter qj is defined as qj =kj

z

k · Δz is the thickness, and j represents the jth layer ofthe medium. Since the cavity has three layers, the totaltransfer matrix of the cavity is given by:

X (ky , ωp) = MA (ky, ωp, dA)MB (ky, ωp, dB)

× MA (ky, ωp, dA) . (2)

Here, X(ky, ωp) represents the total transfer matrix con-necting the fields at incident end and at the exit end.

Eur. Phys. J. D (2014) 68: 55 Page 3 of 10

The transmission coefficient t(ky, ωp) and the reflection co-efficient r(ky , ωp) can be obtained from the transfer matrixmethod [48],

t(ky, ωp) =2q0

[q0(X22 + X11)] − [q20X12 + X21]

, (3)

r(ky , ωp) =[q0(X22 − X11)] − [q2

0X12 − X21][q0(X22 + X11)] − [q2

0X12 − X21], (4)

where q0 =√

ε0 − sin2 θ. Parameter Xij(ky , ωp)(i, j = 1, 2) is the element of matrix X(ky , ωp).

Equations (1) to (4) show that the transmission and re-flection coefficients depend on the permittivity and thick-ness of the cavity walls, permittivity and thickness of theintracavity medium consisted of four-level atoms, and theangle incident of the probe field. The permittivity of in-tracavity medium εB depends on the susceptibility of thefour-level atoms inside the cavity, i.e.,

εB = 1 + χ, (5)

where χ = χ′ + χ′′. χ′ and χ′′

are the real and imagi-nary parts of χ, which represent the dispersion and theabsorption properties of the intracavity medium, respec-tively. The real and imaginary parts of susceptibility canbe controlled by the controlling parameters of the four-level atoms inside the cavity.

We note that the lateral shift in the reflected and trans-mitted light beams can be calculated by using stationaryphase theory [49,50] as:

Sr,t = − λ

2π

dφr,t

dθ(6)

that φr,t are the phases associated with the reflectionr (ky, ωp) and transmission t (ky, ωp) coefficients. So, thelateral shifts of the transmitted and reflected probe lightbeams can be expressed as [22]:

St = − λ

2π

1|t (ky, ωp)|2

{Re [t (ky, ωp)]

d (Im [t (ky , ωp)])dθ

−Im [t (ky, ωp)]d (Re [t (ky, ωp)])

dθ

}, (7)

Sr = − λ

2π

1|r (ky, ωp)|2

{Re [r (ky, ωp)]

d (Im [r (ky, ωp)])dθ

−Im [r (ky , ωp)]d (Re [r (ky, ωp)])

dθ

}, (8)

where the transmission coefficient and reflection coefficientare given by equations (3) and (4). For further discussion,we introduce the group index of the medium that is re-lated to the superluminal and subluminal behavior of lightpropagation through the medium. The group index of theatomic medium can be calculated by using the real partof susceptibility [29]. In fact, for superluminal light propa-gation the group index could be negative, while for sublu-minal light propagation it becomes positive. It should be

1

2r 1r

1γ

δ

pΩ

2γcΩ

3γcδ

2

3

4

Fig. 2. Energy levels diagram of a four-level Λ-type atomicsystem interacting with a coherent coupling field, incoherentpump field and a probe field.

noted that the behavior of the group index of the intracav-ity medium may be different from behavior of the groupindex of the cavity. Therefore, we discuss the relationshipbetween GH shifts and the group index of the cavity cor-responding to superluminal and subluminal light propa-gation. The group velocity of the reflected or transmittedlight beam is given by:

vr,tg =

L

τr,tg

, (9)

where L = 2dA + dB is thickness of the total cavity. Thesuperscripts r, t correspond to reflection and transmissionparts of the incident probe light probe. The correspondinggroup delay, i.e. τr,t

g , is then given by [51,52]:

τr,tg =

∂φr,t

∂ωp, (10)

so the cavity group index defined as:

nr,tg =

c

vr,tg

=c

L

∂φr,t

∂ωp. (11)

Relation (11) shows that the group index depends on thethickness of the cavity and the phase associated with thereflection and transmission coefficients. Therefore, the GHshifts depend on the sign of group index nr,t

g that can bepositive or negative depending on whether the group indexis positive or negative, respectively.

3 Atomic model of the intracavity medium

Now, we return to calculate the susceptibility of the in-tracavity medium, which is needed for examination thebehavior of GH shifts in the reflected and transmittedbeams. So, we consider four-level atoms inside the cav-ity with corresponding coupling, probe and incoherentpump fields to calculate the dispersion-absorption relationgiven by equation (5). The system we investigate here isa Λ-shaped four-level system with fold lower levels |1〉, |2〉and upper levels |3〉 and |4〉 as depicted in Figure 2. Boththe lower levels |1〉, |2〉, and upper level |3〉 are coupled

Page 4 of 10 Eur. Phys. J. D (2014) 68: 55

by a weak probe laser field. An incoherent pump field εpumps the system from two lower levels |1〉 and |2〉 toupper level |3〉 with the pumping rates r1 and r2. Corre-spondingly, upper level |3〉 decays to lower levels |1〉, |2〉,and |4〉 with the rates of γ1, γ2, and γ3, respectively. Inaddition, level |4〉 is coupled to the upper level |3〉 by acoherent coupling laser field. The interaction Hamiltonianof the system is given by:

H = H0 + H1 + H2 + H3 + H4, (12)

where the first term H0 stands free energy part, and H1

denotes the interaction between incoherent pump with lev-els |1〉, |2〉 and |3〉. The term H2 exhibits the interactionof the atom with the reservoir of vacuum oscillator, gov-erned by the decay processes from level |3〉 to levels |1〉, |2〉,and |4〉. H3 and H4 display the interaction of probe laserfield and coherent coupling field with atomic states, re-spectively. The detailed form of these terms can be writ-ten as:

H0 = �ω1 |1〉 〈1|+�ω2 |2〉 〈2|+�ω3 |3〉 〈3|+�ω4 |4〉 〈4| ,(13)

H1 = ℘12ε |1〉 〈3| + ℘23ε |2〉 〈3| + c. c. (14)

H2 = −�

∑k

[g(2)k e−iνkt |3〉 〈1| bk + g

(2)k e−iνkt |3〉 〈2| bk

+g(3)k e−iνkt |3〉 〈4| bk

]+ c. c. (15)

H3 = −�Ωp1e−iνpt |3〉 〈1| − �Ωp2e

−iνpt |3〉 〈2| + c. c. (16)

H4 = −�Ωce−iνct |3〉 〈4| + c. c. (17)

Here �ωi are the energies of the levels |i〉(i = 1, 2, 3, 4).It is assumed that incoherent pumping field has a broadspectrum with effective δ-like correlation, i.e.

〈ε∗(t)ε(t′)〉 = Γδ(t − t′). (18)

Therefore, the spectrum of the field covers both lowerlevels simultaneously. So that one and the same fielddrives both transitions. The electric dipole moments ofthese transitions are characterized by �℘31 and �℘32. Theterm g

(1,2,3)k are the coupling constants between kth vac-

uum mode of frequency νk and the atomic transitions|3〉 ↔ |1〉, |3〉 ↔ |2〉, and |3〉 ↔ |4〉, respectively. bk(b†k)is the annihilation (creation) operator of a photon in kthvacuum mode. Here, Ωpj = �εp. �℘j3

2�(j = 1, 2) are the corre-

sponding Rabi-frequencies of the probe laser field in tran-sitions |3〉 ↔ |j〉, and Ωc = �εc. �℘43

2�is the Rabi-frequency of

the coherent coupling field in transition |3〉 ↔ |4〉, where �εc

and �εp denote the amplitude of the coherent coupling fieldand probe laser field, respectively. The density matrixequations of motion under rotating wave approximation

and in the rotating frame are:

ρ11 = − (γ1 + r1) ρ11 + r1ρ33 − 12η (

√r1r2) (ρ12 + ρ21)

+ i(Ω∗

p1ρ31 − Ωp1ρ13

),

ρ22 = − (γ2 + r2) ρ22 + r2ρ33 − 12η (

√r1r2) (ρ12 + ρ21)

+ i(Ω∗

p2ρ32 − Ωp2ρ23

),

ρ33 = r1ρ11 + r2ρ22 − (γ3 + r1 + r2) ρ33

+ η (√

r1r2) (ρ12 + ρ21) + i(Ωp1ρ13 − Ω∗

p1ρ31

)

+ i(Ωp2ρ23 − Ω∗

p2ρ32

) − iΩ∗c ρ34 + iΩcρ43,

ρ12 = −12η (

√r1r2) (ρ11 + ρ22) + η (

√r1r2 + 2

√γ1γ2) ρ33

−[12

(γ1 + γ2 + r1+ r2)−iΔ

]ρ12−iΩp2ρ13+ iΩ∗

p1ρ32,

ρ13 = iΩ∗p1

(ρ33 − ρ11) − iΩ∗p2

ρ12 −[(

12γ1 + 2r1 + r2

)

−i

(δ − 1

2Δ

)]ρ13 − iΩ∗

c ρ14 − 12η (

√r1r2) ρ23,

ρ14 =−iΩcρ13−[12(γ1 +γ3+ r1)−i

((δ− 1

2Δ

)− δc

)]ρ14

− 12η (

√r1r2) ρ24 + iΩ∗

p1ρ34,

ρ23 = iΩ∗p2

(ρ33 − ρ22) − iΩ∗p1

ρ21 − 12η (

√r1r2) ρ13

−[(

12γ2 + 2r2 + r1

)− i

(δ +

12Δ

)]ρ23 − iΩ∗

c ρ24,

ρ24 = −12η (

√r1r2) ρ14 − iΩcρ23

−[12

(γ2 + γ3 + r2) − i

((δ +

12Δ

)− δc

)]ρ24

+ iΩ∗p2

ρ34,

ρ34 = iΩc (ρ44 − ρ33) + iΩp1ρ14+ iΩp2ρ24

−[12(γ3+ r1+r2)+ iδc

]ρ34, ρ11 +ρ22+ ρ33+ρ44 =1.

(19)

The least equation expresses the conservation of probabil-ity for the closed four-level system. Here, γi = ℘3i ν3

3i

3π�ε0c3

(i = 1, 2) and γ3 = ℘34 ν334

3π�ε0c3 are the spontaneous de-cay rates in transition |3〉 → |i〉 and |3〉 → |4〉, respec-tively. ν3i, ν34 is the frequency deference between level |i〉,|4〉 and level |3〉, respectively. The terms ri = 2(℘2

3i

�2 )Γ(i = 1, 2) are the incoherent pumping rate. The detuningparameters are defined as:

δc = ω34 − νc, δp1 = ω13 − νp = δ − ω12

2

Eur. Phys. J. D (2014) 68: 55 Page 5 of 10

Fig. 3. Dependence of the both Sr (dashed line) and St (solid line) on the incident angle of probe light beam, (a) in the absenceof quantum interference, i.e. η = 0, (b) in the presence of quantum interference, i.e. η = 1. Other parameters are: γ1 = γ2 = 2γ,r1 = r2 = 0.2γ, γ3 = 0.1γ, γ = 1, δ = 0.183 γ, νp = 2π × 508.322 THz, Ωp1 = Ωp2 = Ωp = 0.01γ, Ωc = 1.5γ, δc = 0,dA = 0.1 μm, dB = 2.5082 μm, εA = 2.22, ω12 = 2γ.

and

δp2 = ω23 − νp = δ +ω12

2,

where

δ =(

ω23 + ω13

2− νp

)=

(δp1 + δp2

2

),

and Δ = δp2 − δp1 = ω12.The quantum interference terms containing the prod-

ucts of pump rates and decay constants are central toour discussion. They are denoted by ηp and ηs withηp = ηs = η. These factors are the normalized innerproduct of corresponding dipole matrix elements:

η =�℘31 · �℘32

|℘31| |℘32| = cosβ, (20)

where β is the alignment of the two dipole moments �℘31

and �℘32. In fact, η represents the strength of the inter-ference in spontaneous emission or incoherent pump field.Maximal coherence corresponds to parallel or antiparalleldipole moments, while zero coherence corresponds to or-thogonal dipole moments. These two extremes of maximaland minimal coherence deserve special attention. How-ever, intermediate values on the [–1, 1] internal are alsopossible. In a real experiment the parameter η can be de-termined by the intensity of applied fields. In fact, thecoefficient η depended on the angle between two electricdipole moments, which can be controlled by the intensityof applied fields.

Equations (19) can be solved to obtain the steady stateresponse of the intracavity medium.

Our main observable is the susceptibility as a responseof the intracavity atomic medium to the applied fields.Thus, the linear susceptibility can be written as [53]:

χ =2N

ε0 εp(�℘13ρ13 + �℘23ρ23), (21)

where N is the atomic density in the medium. Equa-tion (21) implies that the linear susceptibility χ is charac-terized by the coherence terms ρ13 and ρ23. These elementscan be determined by the steady state solution of equa-tions (19). At this point we use the fact that the probe fieldis weak compared to the coupling field. We keep the termsof all orders in the strong driving field, however, keepingonly the linear term in the probe field. Therefore, we useρ(0)11 = ρ

(0)22 = 1

2 , and ρ(0)ij = 0, (i, j = 1, 2, 3, 4). Solving

equations (19) to obtain density matrix elements ρ13 +ρ23

and substituting into equation (21) leads to dispersion andabsorption spectrum.

4 Results and discussion

In the following discussion, we investigate the behaviorof the GH shifts of the transmitted and reflected probebeams due to the quantum interference arising from inco-herent pump and spontaneous emission. It is well-knownthat quantum interference induced by incoherent pumpand decay processes may change the linear susceptibility.Therefore, according to equations (1)–(8), it is expectedthat the GH shifts for the reflected and transmitted probebeams can be controlled dramatically by the rate of quan-tum interference. In addition, the effect of the rate of in-coherent pumping fields, the effect of intensity of the cou-pling field and the role of phase fluctuations associatedwith the incoherent pump field on the behavior of the GHshifts of the transmitted and reflected probe beams arediscussed. Now, we present these results in Figures 3–8which show dependence of the GH shifts on the incidentangle of the probe beam. We consider γ1 = γ2 = 2γ,γ3 = 0.1γ and all the parameters are reduced to thedimensionless units though scaling by γ. Moreover, wechoose |℘31| = |℘32| = ℘, Ωp1 = Ωp2 = Ωp and all thefigures are normalized by 2N℘

ε0 εp= 1. First, the effect of

quantum interference mechanism arising from incoherent

Page 6 of 10 Eur. Phys. J. D (2014) 68: 55

Fig. 4. Dependence of the absolute values of (a) |r| and (b) |t| on the incident angle of the probe light beam in the absence ofquantum interference, i.e. η = 0. All the parameters are same as Figure 3a.

Fig. 5. Dependence of the absolute values of (a) |r| and (b) |t| on the incident angle of the probe light beam in the presenceof quantum interference, i.e. η = 1. All the parameters are same as Figure 3b.

pumping and decay processes on the behavior of the GHshifts of the transmitted and reflected probe beams aredisplayed in Figure 3. We find large GH shifts in the re-flected and transmitted probe beams at the certain in-cident angles that satisfy the resonance condition of thecavity. From Figure 3a is observed that in the absenceof quantum interference, η = 0, the GH shift of the re-flected probe beam become highly negative at the reso-nant angles, while the GH shift of the transmitted probebeam turn out to be positive in the same angles. So, wereach large negative GH shifts in the reflected probe beam(dashed line) and positive GH shifts in the transmittedprobe beam (solid line) at the resonant angles. In fact,the susceptibility in this case is weakly absorptive, whichleads to giant negative GH shifts in the reflected probebeam near resonance [54]. Figure 3b shows the behaviorof GH shifts of the reflected and transmitted probe beamsversus the incident angles θ for η = 1. We realize that theGH shifts for both the reflected and transmitted probebeam are positive for the whole range (0 to π/2 rad) ofthe incident angle. In this case the intracavity medium

is transparent for probe beam (subluminal wave propaga-tion), which leads to positive GH shifts for both reflectedand transmitted probe beams.

As a result, the GH shift of the reflected probe beamchanges from large negative values to positive values whenrate of quantum interference changes from 0 to 1, whilethe GH shift of the transmitted probe beam is positivefor two values η (=0, 1). Specially, two interesting resultswere presented in the investigation. One the huge lateralshift in resonant angles and second all-optical controlla-bility of these shifts from positive to negative and viceversa. According to equations (7) and (8), the behavior ofthe transmission and reflection coefficients have the lead-ing role in the magnitude of the lateral shifts. The na-ture of these results lays on susceptibility of the atomsinside the cavity that is driven by one coherent and twoincoherent fields. We have discussed the behavior of thesusceptibility for such a system in reference [43]. Control-ling of the susceptibility via quantum interference leadsto all-optical control of the refractive index of the layer.Thereby, the transmission and reflection coefficients are

Eur. Phys. J. D (2014) 68: 55 Page 7 of 10

Fig. 6. Dependence of the both Sr (dashed line) and St (solid line) on the incident angle of probe light beam in the absenceof quantum interference (a) Ωc = 0.5 γ, (b) Ωc = 1.5 γ and (c) Ωc = 2.5 γ. Other parameters are same as Figure 3a.

related to the susceptibility of the atoms. This way, a steepchange in magnitude and slope of the susceptibility causesto change in the magnitudes of the lateral shifts accordingto equations (7) and (8).

The group indexes (ng) of the cavity corresponding tothe reflected and transmitted probe beams are calculatedusing equation (11). In a large negative GH shift at inci-dent angle θ = 0.597 rad, the group index of the cavity isnr

g = −13.51. Correspondingly, the cavity group index forthe transmitted probe beam nt

g at the same incident angleis nt

g = 1.25, which is positive. In the case of η = 1 and atincident angle θ = 0.597 rad, the group index of the cavityis nr

g = 1.07 for reflected probe beam and is ntg = 1.31 for

transmitted probe beam, which both are positive. For thisgiven incident angle and in the case of η = 0, the GH shiftof the reflected probe beam is negative, while it becomespositive for the transmitted probe beam. It is found thatthe cavity group index in the case of η = 0, is negativewhenever the GH shift is negative. We emphasize thatnegative (positive) GH shifts appear only when the groupindex of the total cavity becomes negative (positive). Thisis to say that the GH shift for superluminal case (groupindex < 0) is negative, while it becomes positive for sublu-minal case (group index >). In addition, the negative GHshift of superluminal case only occurs for certain choice

of the incident angle and not for the whole range from 0toπ/2 rad.

Now, we plot the absolute values of the reflection andtransmission coefficients in the absence of quantum inter-ference (Fig. 4) and in the presence of quantum interfer-ence (Fig. 5). We observe dips in the reflection and trans-mission curves, which correspond to the angles satisfyingthe resonance condition. In comparing Figures 4 and 5with Figure 3, we find that large negative and positiveGH shifts appear at the resonant angles.

In Figures 6 and 7, we display plots of the GH shifts inthe reflected and transmitted probe beams versus probebeam incident angle for various intensities of the coherentcoupling field. In the absence of quantum interference, i.e.η = 0, the GH shift in the reflected probe beam is largenegative at resonant angles for all values of coherent cou-pling field intensities (Fig. 6). Physically, for an intensecoherent coupling field, the population will be trapped inupper levels |3〉 and |4〉 leading to reduction of absorptionin transition (|1〉+|2〉) → |3〉. Therefore, we have huge neg-ative GH shifts in the reflected probe beam [54]. However,for η = 1, the GH shift of the reflected probe beam can beswitched from large negative values to positive values asthe intensity of coupling field change from 0.5 γ to 1.5 γor 2.5 γ (Fig. 7). In this case, when the value of Ωc is fur-ther increased, the susceptibility becomes gain, which also

Page 8 of 10 Eur. Phys. J. D (2014) 68: 55

Fig. 7. Dependence of the both Sr (dashed line) and St (solid line) on the incident angle of probe light beam in the presenceof quantum interference (a) Ωc = 0.5 γ, (b) Ωc = 1.5 γ and (c) Ωc = 2.5 γ. Other parameters are same as Figure 3b.

Fig. 8. Dependence of the both Sr (dashed line) and St (solid line) on the incident angle of probe light beam in the presenceof quantum interference (a) r1 = r2 = 0.2 γ and (b) r1 = r2 = 6.2 γ. Other parameters are same as Figure 3b.

leads to large GH shifts (negative or positive) near reso-nance [55]. So, the GH shift of the reflected probe beamcan be controlled from large negative values to positivevalues or vice versa by intensity of coupling field just inthe presence of quantum interference. Finally, the effectof incoherent pumping rate in the case of η = 1 on thebehavior of the GH shifts in the reflected and transmit-ted probe beams is displayed in Figure 8. The GH shift

of the reflected probe beam changes from positive val-ues to large negative values when the rate of incoherentpump field changes from r = 0.2 γ to r = 6.2γ. We notethat for a weak incoherent pumping rate (Fig. 8a), theGH shift of the reflected probe beam is positive at cer-tain incident angles of the probe beam corresponding tosubluminal light propagation, while for strong incoherentpumping rate (Fig. 8b); the GH shift of the reflected probe

Eur. Phys. J. D (2014) 68: 55 Page 9 of 10

beam is large negative at the same angles of the probebeam corresponding to superluminal light propagation.For further information, if we use a probe beam with amuch narrow width respect to the width of the incoherentpumping fields, the GH shifts of the reflected and trans-mitted probe beams will not be affected by the randomphase fluctuations of the incoherent pumping fields. More-over, in this paper, the intensity of the incoherent pumpingfields is not so large to cause to a considerable broaden-ing in spontaneous emission lines and a shift in resonancefrequency. The effect of incoherent pumping phase fluctu-ations on line broadening and resonance frequency shiftare discussed in more details in references [56–58].

We repeat that in this paper we have investigatedthe behavior of the GH shifts of a well-collimated probebeam with a narrow angular spectrum, Δk � k. Thismeans that our results have been calculated using sta-tionary phase theory. However, it has already been shownthat the results are still valid in the real system when theprobe beam has a finite width (typically, with a Gaussianprofile) [22]. As a realistic example and for experimen-tal realization, we choose Sodium D1 transition [57]. Sothe transitions |1〉 ↔ |3〉 and |2〉 ↔ |3〉 correspond to3 2s1/2(F = 2) → 3 2p1/2(F = 2) and 3 2s1/2(F = 1) →3 2p1/2(F = 2) transitions, respectively. The coupling fieldcan also be tuned to 3 2s1/2(F = 2) → 3 2p1/2(F = 2)transition with deferent mF . For such a case the decayrate and the dipole moment are γ = 2π × 9.76 MHz and℘ = 2.1 × 10−29 cm, respectively. The detailed configura-tion of the level structures can be found in reference [59].

5 Conclusion

In conclusion, we have investigated the behavior of the GHshifts in the reflected and transmitted probe light beamsin the presence of quantum interference. It is shown thatlarge negative GH shifts in the reflected light beam canbe switched to positive GH shifts when quantum interfer-ence rate changes from 0 to 1. This implies that the GHshifts of the reflected probe beam can be controlled viaquantum interference. The GH shifts of the transmittedlight beam are positive for two values of η (=0, 1). In ad-dition, it is found that the behavior of GH shifts just inthe presence of quantum interference, i.e. η = 1, can becontrolled via the incoherent pumping rate and the inten-sity of coupling field. However, in the absence of quantuminterference, i.e. η = 0, the intensity of coupling field andthe rate of incoherent pumping field don’t change the GHshifts from negative to positive or vice versa. We may em-phasize that switching of GH shifts from positive to neg-ative or vice versa is possible via the incoherent pumpingrate and the intensity of coupling field when quantum in-terference is included. We note that negative and positiveGH shifts in the reflected beam, corresponding to superlu-minal and subluminal reflection of the probe light beam,respectively. So, the group index of the cavity is negativeas well as positive for the reflected beam. The transmit-ted probe beam, however, exhibits only positive GH shifts

for incident angle from 0 to π/2 rad. So, the group indexof the cavity is always positive for the transmitted probelight beam.

References

1. F. Goos, H. Hanchen, Ann. Phys. 436, 333 (1947)2. F. Goos, H. Hanchen, Ann. Phys. 440, 251 (1949)3. T. Hashimoto, T. Yoshino, Opt. Lett. 14, 913 (1989)4. N.J. Harrick, Phys. Rev. Lett. 4, 224 (1960)5. J.L. Birman, D.N. Pattanayak, A. Puri, Phys. Rev. Lett.

50, 1664 (1983)6. M. Merano, J.B. Gotte, A. Aiello, M.P. van Exter, J.P.

Woerdman, Opt. Express 17, 10864 (2009)7. H.K. Lotsch, Optik (Stuttgart) 32, 116 (1970)8. H.K. Lotsch, Optik (Stuttgart) 32, 189 (1970)9. H.K. Lotsch, Optik (Stuttgart) 32, 299 (1971)

10. H.K. Lotsch, Optik (Stuttgart) 32, 553 (1971)11. N.J. Harrick, Phys. Rev. Lett. 4, 224 (1960)12. R.H. Renard, J. Opt. Soc. Am. 54, 1190 (1964)13. H.K. Lotsch, J. Opt. Soc. Am. 58, 551 (1968)14. Y. Huang, B. Zhao, L. Gao, J. Opt. Soc. Am. A 29, 1436

(2012)15. T. Tamir, H.L. Bertoni, J. Opt. Soc. Am. 61, 1397 (1971)16. R. Talebzadeh, A. Namdar, Appl. Opt. 51, 6484 (2012)17. M. Merano, A. Aiello, G.W. ‘t Hooft, M.P. van Exter, E.R.

Eliel, J.P. Woerdman, Opt. Express 15, 15928 (2007)18. P.R. Berman, Phys. Rev. E 66, 067603 (2002)19. A. Lakhtakia, Electromagnetics 23, 71 (2003)20. J.A. Kong, B.-I. Wu, Y. Zhang, Phys. Lett. 80, 2084 (2002)21. X. Chen, C.-F. Li, Phys. Rev. E 69, 066617 (2004)22. L.-G. Wang, M. Ikram, M.S. Zubairy, Phys. Rev. A 77,

023811 (2008)23. Ziauddin, S. Qamar, M.S. Zubairy, Phys. Rev. A 81,

023821 (2010)24. Ziauddin, S. Qamar, Phys. Rev. A 85, 055804 (2012)25. Ziauddin, S. Qamar, Phys. Rev. A 84, 053844 (2011)26. L.V. Hau, S.E. Harris, Z. Dutton, C.H. Behroozi, Nature

397, 594 (1999)27. M. Haas, C.H. Keitel, Opt. Commun. 216, 385 (2003)28. M.M. Kash, V.A. Sautenkov, A.S. Zibrov, L. Hollberg,

G.R. Welch, M.D. Lukin, Y. Rostovtsev, E.S. Fry, M.O.Scully, Phys. Rev. Lett. 82, 5229 (1999)

29. L.J. Wang, A. Kuzmich, A. Dogariu, Nature 406, 277(2000)

30. A. Dogariu, A. Kuzmich, L.J. Wang, Phys. Rev. A 63,53806 (2001)

31. M.O. Scully, Phys. Rev. Lett. 67, 1855 (1991)32. O. Budriga, Eur. Phys. J. D 66, 137 (2012)33. Z. Ficek, S. Swain, Quantum Coherence and Interference;

Theory and Experiments (Springer, Berlin, 2004)34. S.Y. Zhu, M.O. Scully, Phys. Rev. Lett. 76, 388 (1996)35. E. Paspalakis, S.Q. Gong, P.L. Knight, Opt. Commun.

152, 293 (1988)36. W.-H. Xu, J.-H. Wu, J.-Y. Gao, Eur. Phys. J. D 30, 137

(2004)37. M. Sahrai, H. Tajalli, K.T. Kapale, M.S. Zubairy, Phys.

Rev. A 70, 23813 (2004)38. S. Ghosh, J. Opt. Soc. Am. B 30, 2450 (2013)39. A.S. Zibrov, M.D. Lukin, L. Hollberg, D.E. Nikonov, M.O.

Scully, H.G. Robinson, V.L. Velichansky, Phys. Rev. Lett.76, 3935 (1996)

Page 10 of 10 Eur. Phys. J. D (2014) 68: 55

40. Hongjun Zhang, Yueping Niu, Hui Sun, Jian Luo,Shangqing Gong, J. Phys. B 41, 125503 (2008)

41. A. Imamouglu, Phys. Rev. A 40, 2835 (1989)42. M. Sahrai, A. Maleki, R. Hemmati, M. Mahmoudi, Eur.

Phys. J. D 56, 105 (2010)43. M. Sahrai, R. Nasehi, M. Memarzadeh, H. Hamedi, J.B.

Poursamad, Eur. Phys. J. D 65, 571 (2011)44. K.J. Boller, A. Imamoghlu, S.E. Harris, Phys. Rev. Lett.

66, 2593 (1991)45. J.E. Field, K.H. Hahn, S.E. Harris, Phys. Rev. Lett. 67,

3062 (1991)46. M. Lindberg, R. Binder, Phys. Rev. Lett. 75, 1403 (1995)47. M. Artoni, G.C. La Rocca, F. Bassani, Europhys. Lett. 49,

445 (2000)48. N.H. Liu, S.Y. Zhu, H. Chen, X. Wu, Phys. Rev. E 65,

046607 (2002)49. K. Artmann, Ann. Phys. 2, 87 (1984)

50. C.-F. Li, Phys. Rev. Lett. 91, 133903 (2003)51. E. Dopel, J. Mod. Opt. 37, 237 (1990)52. O.F. Siddiqui, M. Mojahedi, G.V. Eleftheriades, IEEE

Trans. Ant. Propag. 51, 2619 (2003)53. M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge

University Press, Cambridge, 1997)54. L.G. Wang, H. Chen, S.Y. Zhu, Opt. Lett. 30, 2936 (2005)55. Y. Yan, X. Chen, C.F. Li, Phys. Lett. A 361, 178 (2007)56. S. Sultana, M.S. Zubairy, Phys. Rev. A 49, 438 (1994)57. A.H. Toor, S.-Y. Zhu, M.S. Zubairy, Phys. Rev. A 52, 4803

(1995)58. B. Hall, M. Lisak, D. Anderson, R. Fedele, V.E. Semenov,

Phys. Rev. E 65, 035602 (2002)59. E.S. Fry, X. Li, D. Nikonov, G.G. Padmabandu, M.O.

Scully, A.V. Smith, F.K. Tittel, C. Wang, S.R. Wilkinson,S.-Y. Zhu, Phys. Rev. Lett. 70, 3235 (1993)