Manipulation of lateral shift via driven cavity-optomechanical system

Optics Communications 450 (2019) 282–286

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Manipulation of lateral shift via driven cavity-optomechanical system Muhammad Musadiq, Salman Khan ∗ Department of Physics, COMSATS University Islamabad, Park road, Islamabad 45550, Pakistan

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Keywords: Electromagnetic optics Resonance Coherent optical effects

ABSTRACT ̈ We propose a scheme for the enhancement of Goos-Hanchen effect based on cavity-optomechanical system. The study provides an easy way for controlling the absorptive (dispersive) response of the system to a probe field, thereby enabling to manipulate lateral shifts in the output field of the cavity at the probe frequency. The scheme is experimentally realizable and substantial enhancement in lateral shift can be obtained by either adjusting frequency or strength of the drive field. Considerably larger lateral shifts of the order ranging from 20 to 1000 probe wavelength can be produced.

1. Introduction Lateral shift is one of the fascinating traits of light taking place at the interface of two different optical media. It can happen both in reflected and transmitted beams either in the forward or backward direction and is accordingly named positive or negative lateral shift. In the honor of its discoverers, this experimentally observed phenomenon ̈ is also known as Goos-Hanchen (GH) effect (shift) [1,2]. The GH effect can be explained through two different approaches. In one case, the incident beam is supposed to be a coherent superposition of a large number of plane waves, each hitting the interface at a slightly different incident angles [3–5]. In the other case, the expanded plane waves are considered to undergo through different phase changes right at the interface of the media [6–8]. The importance of the study of GH effect, under variety of conditions, can be realized from its potential applications in number of different fields. It works as a basic tool for precise measurement of refractive index of optical media, thickness of films and beam angle [9–11]. It is also a fundamental resource for constructing highresolution surface plasmon sensors [12], designing automatic optical switches [13] and optical temperature sensing devices [14]. The reliability of GH effect as a resource for different applications is pertinent to the amount of shift, which is not considerable in the earlier schemes suggested for producing it [15,16]. To obtain enhanced GH shift, other means including photonic bandgap, artificial media and defect modes in photonic crystals have been used [7,17–19]. In more recent studies, schemes are suggested that additionally provide the opportunity to manipulate GH shift of a probe field in a driven medium, enclosed inside a cavity, through a controllable laser field [20–23]. Although, quite promising, these schemes do not count the effect of cavity modes, which can get entangled with the modes of the enclosed medium. From experimental point of view, such an interaction between the modes of ∗

cavity and the enclosed driven medium must affect the net GH shift and should be theoretically taken into account. The effect of cavity coupling with the driven medium on GH shifts is very recently investigated and quite distinguishing results have been obtained [24]. The approach of this work is different from the schemes suggested previously for producing GH effect. The main idea is based on a driven cavity-optomechanical system, without enclosing a material medium, which is an essential part of previous schemes. Therefore, our scheme considerably simplifies the experimental setup. We show that the coupling of a single mode cavity with a mechanical oscillator and with a drive field can lead to enhanced GH shifts, in a controlled manner, in the output field of the cavity at the frequency of a weak probe field. Besides the simplified experimental setup, the scheme is very advantageous in producing considerably larger GH shifts, which cannot be obtained from previous schemes, based on different types of material media. 2. Physical model The schematic of our model is shown in Fig. 1. It consists a Fabry– Perot cavity with one movable wall connected to a mechanical resonator of mass 𝑚 (not shown). The walls, each of thickness 𝑑1 , are made up of nonmagnetic dielectric slabs and are separated by a distance 𝑑2 . The cavity is driven by a strong pump field of strength 𝐸𝑙 = √ 2𝜅𝑃𝑙 ∕(ℏ𝜔𝑙 ), where 𝑃𝑙 is power of the laser which couples the cavity to a mechanical resonator (not explicitly shown). The laser field falls normally from left on the wall of the cavity. A weak TE-polarized probe √ beam of strength 𝐸𝑝 = 2𝜅𝑃𝑝 ∕(ℏ𝜔𝑝 ) with power 𝑃𝑝 also falls on the left wall of the cavity, making an angle 𝜃 with the normal of the wall. 𝑆𝑟 and 𝑆𝑡 are the shifts in the reflected and transmitted output fields of the cavity at the probe frequency. We denote the frequencies of

Corresponding author. E-mail address: [email protected] (S. Khan).

https://doi.org/10.1016/j.optcom.2019.06.008 Received 10 April 2019; Received in revised form 31 May 2019; Accepted 3 June 2019 Available online 5 June 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

M. Musadiq and S. Khan

Optics Communications 450 (2019) 282–286

exact time dependent solutions to the set of these equations. However, steady-state solutions, which are exact in 𝐸𝑙 and correct only to the first order in 𝐸𝑝 , can be obtained through the following possible solution to each of these equations ⟨𝜒⟩ = 𝜒 𝑠 + 𝜒 − 𝑒−𝑖𝛿𝑡 + 𝜒 + 𝑒𝑖𝛿𝑡 ,

(8)

with 𝜒 being any of the expectation values in the above set of equations. For the expectation value of a given operator, 𝜒 𝑠 gives the steady-state solution when the cavity is driven only by 𝐸𝑙 and the rest two terms describe the steady-state solutions when additionally the probe field 𝐸𝑝 is also present in the cavity. Using Eq. (8) in Eqs. (5)–(7), comparing the constants and the coefficients of the corresponding exponents in each case, lead to a set of nine coupled equations. The solution to each 𝜒 𝑠 is given by 𝑞𝑠 =

Fig. 1. (Color online) Sketch of the optomechanical system coupled to a high quality cavity through radiation pressure. 𝑆𝑡 is the output of the cavity as a transmitted beam and 𝑆𝑟 is the output of the cavity as a reflected beam.

𝑔𝑚𝑐 |𝑐 𝑠 |2 𝑚𝜔2𝑚

𝐸𝑙

𝑐𝑠 =

𝜅 + 𝑖𝛥𝑐 −

𝑐− =

⟨𝑝⟩ , 𝑚

(5) (6)

𝜕𝑡 ⟨𝑐⟩ = −(𝜅 + 𝑖𝛥𝑐 )⟨𝑐⟩ + 𝑖𝑔𝑚𝑐 ⟨𝑐⟩⟨𝑞⟩ + 𝐸𝑙 + 𝐸𝑝 𝑒−𝑖𝛿𝑡 .

(7)

(12) (13)

The dimensionless quantity, ℰ𝑡 (𝜔𝑝 ), describes the response of the cavity-optomechanical system to the probe field and can be expressed as ℰ𝑡 (𝜔𝑝 ) = ℰ𝑡′ (𝜔𝑝 ) + 𝑖 ℰ𝑡′′ (𝜔𝑝 ), where the real and the imaginary parts describe dispersion and absorption properties of the cavityoptomechanical system, respectively [28]. The effect of cavity-optomechanical configuration on GH shift in the probe field can be investigated by first finding the reflection and transmission coefficients of the output field for the entire setup. This can be achieved through the approach of transfer matrix [29], which for the 𝑚th layer of a multi-layer setup is given by ( ) cos[𝑘𝑚 𝑖 sin[𝑘𝑚 𝑧 𝑑𝑚 ] 𝑧 𝑑𝑚 ]∕𝑞𝑚 , T𝑚 (𝑘𝑦 , 𝜔𝑝 , 𝑑𝑚 ) = (15) 𝑖 𝑞𝑚 sin[𝑘𝑚 cos[𝑘𝑚 𝑧 𝑑𝑚 ] 𝑧 𝑑𝑚 ] √ 𝜀𝑚 𝑘2 − 𝑘2𝑦 is the zwhere 𝑑𝑚 is thickness of the layer, 𝑘𝑚 = 𝑧 component, in the layer, of the wave number 𝑘 = 𝜔𝑝 ∕𝑐, with 𝑘𝑦 being its 𝑦 component in vacuum and 𝑞𝑚 = 𝑘𝑚 𝑧 ∕𝑘. The transfer matrix for the entire setup is constructed via matrix product of the layers’ matrices put in order following the sequence of the layers in the direction of light passage. In our case it becomes

(4)

𝜕𝑡 ⟨𝑝⟩ = −𝑚𝜔2𝑚 ⟨𝑞⟩ − 𝛾𝑚 ⟨𝑝⟩ + 𝑔𝑚𝑐 ⟨𝑐 † ⟩⟨𝑐⟩,

(11)

,

The behavior of output field of the cavity at probe field frequency can be analyzed by employing the standard input–output theory [27]. It defines the output field operator 𝑐𝑜𝑢𝑡 (𝑡) in √ terms of input field operator 𝑐𝑖𝑛 (𝑡) and is given as 𝑐𝑜𝑢𝑡 (𝑡) = 𝑐𝑖𝑛 (𝑡) − 2𝜅 𝑐(𝑡). The transmission probability amplitude of the output field is defined in terms of the ratio of the amplitudes of the output to the input fields at the probe frequency, which takes the following form √ 𝐸𝑝 − 2𝜅 𝑐 − . (14) ℰ𝑡 (𝜔𝑝 ) = 𝐸𝑝

where 𝑐𝑖𝑛 (𝑡) is the quantum vacuum fluctuations of the cavity mode, † ′ characterized by the correlation [25] ⟨𝑐𝑖𝑛 (𝑡)𝑐𝑖𝑛 (𝑡 )⟩ = 𝛿(𝑡 − 𝑡′ ), with † † ′ mean values ⟨𝑐𝑖𝑛 (𝑡)𝑐𝑖𝑛 (𝑡 )⟩ = 0. The parameters 𝜅 and 𝛾𝑚 are the decays of the cavity and the resonator modes, respectively. Similarly, the parameter 𝜉(𝑡) represents the Brownian noise force with ⟨𝜉(𝑡)⟩ = 0 and obeys the temperature dependent correlation function [26] ⟨𝜉(𝑡)𝜉(𝑡′ )⟩ = ′ (𝛾𝑚 ∕2𝜋𝜔𝑚 ) ∫ 𝑑𝜔𝑒−𝑖𝜔(𝑡−𝑡 ) 𝜔[coth(ℏ𝜔∕2𝑘𝐵 𝑇 ) + 1], where 𝑘𝐵 is the Boltzmann constant and 𝑇 is the thermal bath temperature associated with the mechanical resonator. Under the mean-field approximation, ⟨𝑥𝑦⟩ = ⟨𝑥⟩⟨𝑦⟩, the equations for the mean values of the operators can be expressed as follows 𝜕𝑡 ⟨𝑞⟩ =

1 − 𝑁 [𝑀(𝛿) − 𝑀 ∗ (−𝛿)]

2 |𝑐 𝑠 |2 𝑖𝑔𝑚𝑐 ( ), 2 𝑚 𝜔𝑚 − 𝑖𝛾𝑚 𝛿 − 𝛿 2 1 . 𝑀(𝛿) = 𝜅 + 𝑖𝛥𝑐 − 𝑖𝛿 − 𝑖𝑔𝑚𝑐 𝑞 𝑠

(3) 2𝜅 𝑐𝑖𝑛 (𝑡),

𝑀(𝛿) [1 + 𝑁 𝑀 ∗ (−𝛿)] 𝐸𝑝

𝑁=

In Eq. (1), 𝑝 (𝑞) is the momentum (position) operator of the mechanical resonator, 𝑐 (𝑐 † ) is the annihilation (creation) operator of the cavity, 𝛥𝑐 = 𝜈 − 𝜔𝑙 is the cavity–pump detuning, 𝛿 = 𝜔𝑝 − 𝜔𝑙 is the probe– √ 𝜔 pump detuning and 𝑔𝑚𝑐 = 𝑑 𝑙 𝑚𝜔ℏ is the cavity–resonator coupling 2 𝑚 strength. Here the first two terms are the free Hamiltonian of the mechanical resonator and the cavity, respectively. The last three terms, respectively, describe the interaction of the cavity with the mechanical resonator, the pump field and the probe field. From Eq. (1), employing the respective commutation relations, it is easy to obtain the Heisenberg–Langevin equations for the operators of the mechanical resonator and the cavity mode. Incorporating the corresponding fluctuation and dissipation terms, these equations can be expressed as 𝑝 (2) 𝜕𝑡 𝑞 = , 𝑚 √

(10)

.

where

(1)

𝜕𝑡 𝑐 = −(𝜅 + 𝑖𝛥𝑐 )𝑐 + 𝑖𝑔𝑚𝑐 𝑐𝑞 + 𝐸𝑙 + 𝐸𝑝 𝑒−𝑖𝛿𝑡 +

2 |𝑐 𝑠 |2 𝑖𝑔𝑚𝑐 𝑚𝜔2𝑚

Similarly, solving the rest of equations, we find the solution for 𝑐 − as

cavity, mechanical resonator, pump and probe fields by 𝜈, 𝜔𝑚 , 𝜔𝑙 and 𝜔𝑝 , respectively. In a frame rotating at the frequency of pump field, the Hamiltonian describing the dynamics of the system can be expressed as follows [ 2 ] 𝑝 1 𝐻= + 𝑚𝜔2𝑚 𝑞 2 +𝛥𝑐 𝑐 † 𝑐 +𝑔𝑚𝑐 𝑐 † 𝑐𝑞+𝑖ℏ𝐸𝑙 (𝑐 † −𝑐)+𝑖ℏ𝐸𝑝 (𝑒−𝑖𝛿𝑡 𝑐 † −𝑒𝑖𝛿𝑡 𝑐). 2𝑚 2

𝜕𝑡 𝑝 = −𝑚𝜔2𝑚 𝑞 − 𝛾𝑚 𝑝 + 𝑔𝑚𝑐 𝑐 † 𝑐 + 𝜉(𝑡),

(9)

,

T(𝑘𝑦 , 𝜔𝑝 ) = T1 (𝑘𝑦 , 𝜔𝑝 , 𝑑1 ) T2 (𝑘𝑦 , 𝜔𝑝 , 𝑑2 ) T1 (𝑘𝑦 , 𝜔𝑝 , 𝑑1 ).

(16)

The reflection and transmission coefficients for the output field are given via matrix elements T𝑚𝑛 through the following relations

Note that the mean values of the Brownian and vacuum noises are set to zero. Owing to the time dependent exponent in Eq. (7), it is hard to find

𝑟(𝑘𝑦 , 𝜔𝑝 ) = 283

𝑞0 (T22 − T11 ) − (𝑞02 T12 − T12 ) 𝑞0 (T22 + T11 ) − (𝑞02 T12 + T12 )

,

(17)

M. Musadiq and S. Khan

Optics Communications 450 (2019) 282–286

Fig. 2. The real (solid black) and imaginary (dashed red) parts of ℰ𝑡 against (a) the detuning (𝜔𝑝 − 𝜔𝑙 )∕𝜔𝑚 for 𝐸𝑙 = 5.5 MHz. The inset shows its behavior for 𝐸𝑙 = 0. (b) Against the strength 𝐸𝑙 of the drive field for 𝛿 = 0.2 𝜔𝑚 . The values of the other parameters are set to 𝛥𝑐 = 𝜔𝑚 , 𝜅 = 0.02 𝜔2𝑚 , 𝜔𝑚 ∕(2𝜋) = 11 MHz, 𝛾𝑚 ∕(2𝜋) = 32 Hz, 𝑔𝑚𝑐 ∕(2𝜋) = 0.2 kHz.

𝑡(𝑘𝑦 , 𝜔𝑝 ) =

2𝑞0 𝑞0 (T22 + T11 ) − (𝑞02 T12 + T12 )

,

frequency 𝛿 and develops additional sidebands at different frequencies around 𝛿 = 0 [31]. These sidebands results in the additional peaks in the response function of the cavity. One can see from the left inset, that only one peak in both profiles at 𝛿 = 𝜔𝑚 exists, whereas in the presence of drive field four peaks at different frequencies in each profile simultaneously arise, two each on either side of 𝛿 = 0. The peaks of the real part at the near resonance are more prominent than its peaks at far resonance. For imaginary part, the situation is reversed. Unlike the case in the left inset figure, the peaks in each profile take both positive and negative values. The right inset shows an enlarged view of one of the nearest resonant peaks, where the imaginary part becomes zero and the real part is maximized. In other words, a large refractive index with complete transparency can be obtained. Fig. 2(b) shows their behaviors against the strength of drive field for 𝛿 = 0.2 𝜔𝑚 , which dictates that the behavior of of output field at the probe frequency can be controlled by suitably tuning up the power of the drive field. The inset, in this case, is an enlarged view of the two profiles at the second resonant sideband, which shows that the refractive index and the transparency of the output field at probe frequency can be enhanced through the power of the drive field. Since the amount of GH shift of a probe field in a medium depends on the variation in its absorptive and dispersive properties, it is fair to expect that the cavity-optomechanical system will greatly affect GH shift in the probe field and this is exactly the case that we discuss next. To investigated the effect of drive field on the GH shifts 𝑆𝑟,𝑡 , we numerically simulate the analytical results both in the presence and in the absence of the drive field for different choices of the controllable parameters of the entire setup. In Fig. 3, the shifts in the two beams are plotted against the incident angle for different choices of the strength 𝐸𝑙 of the drive field with fixed detuning 𝛿. Fig. 3(a) shows their behavior in the absence of drive field. The shift in the reflected beam (black solid curve) can both be positive or negative and can be 20 times larger than the wavelength used. Irrespective of the choice of incident angle, the shift in the transmitted beam is always positive and can be of the order of 15 probe wavelength. This means, like schemes based on different atomic medium [20,21], cavity-optomechanical system with no material medium can be used to produce enhanced lateral shifts. To further highlight the advantage of our scheme over the different schemes based on the material medium, we plot 𝑆𝑟,𝑡 in the presence of the drive field in Fig. 3(b). In the figure, the left ordinate measures the values for 𝐸𝑙 = 4 MHz and the right one corresponds to 𝐸𝑙 = 5.2 MHz. It is clear that the drive field can modify the profiles of both reflected and transmitted shifts and can enhance them to 1000 order of probe wavelength. In fact, in the presence of drive field the cavityoptomechanical system behaves like a gain medium where resonant modes that lead to such a large shift (positive or negative) exist at certain incident angles.

(18)

where 𝑞0 = 𝑘𝑧 ∕𝑘 is the normalized 𝑧 component of wave number 𝑘 in vacuum. Now, for incident beam of relatively large space width, the shift (𝑆) in the reflected (𝑟) and transmitted (𝑡) beams under the stationary phase theory [3] are given by 𝑆𝑟,𝑡 = − =−

𝜆𝑝 𝑑 𝜑𝑟,𝑡 2𝜋 𝑑 𝜃 𝜆𝑝 2𝜋|𝑟(𝑘𝑦 , 𝜔𝑝 ), 𝑡(𝑘𝑦 , 𝜔𝑝 )|2

[ ] [ ] 𝑑 Im 𝑟(𝑘𝑦 , 𝜔𝑝 ), 𝑡(𝑘𝑦 , 𝜔𝑝 ) Re 𝑟(𝑘𝑦 , 𝜔𝑝 ), 𝑡(𝑘𝑦 , 𝜔𝑝 ) 𝑑𝜃 [ ] [ ] 𝑑 Re 𝑟(𝑘𝑦 , 𝜔𝑝 ), 𝑡(𝑘𝑦 , 𝜔𝑝 ) } − Im 𝑟(𝑘𝑦 , 𝜔𝑝 ), 𝑡(𝑘𝑦 , 𝜔𝑝 ) , (19) 𝑑𝜃 where Re (Im) stands for real (imaginary) part and 𝜑𝑟,𝑡 represents the phase of the corresponding coefficient. We link the permittivity of the cavity-optomechanical system to the quantity ℰ𝑡 of the output field and define it as 𝜀2 = 1 + ℰ𝑡 (𝜔𝑝 ). The parameters related to the drive and probe fields enter into 𝑆𝑟,𝑡 via 𝜀2 , therefore, the strength of the drive field and its detuning with the mode of cavity and probe fields affect the shifts in both the transmitted and reflected components of the output field, at probe frequency, of the entire setup. The forms of the analytical relations for the 𝑆𝑟,𝑡 are very complex and lengthy. It is difficult to understand the effects of different parameters on GH shift directly from these relations. To this end, we rely on the numerical simulations of the analytical results and present them with related discussion in the next section. ×

{

3. Results and discussion For numerically analysis of our results, we use the parameters’ values from an experiment which demonstrated normal mode splitting in the sideband limit under the condition 𝜔𝑚 ≫ 𝜅 [30]. We work in the upper sideband where the drive field is detuned above the cavity resonant with 𝛥𝑐 = 𝜔𝑚 . The values of other parameters are: 𝑚 = 480 pg, 𝛾𝑚 ∕(2𝜋) = 32 Hz, 𝜔𝑚 ∕(2𝜋) = 11 MHz, 𝜅 = 0.02 𝜔𝑚 , 𝑔𝑚𝑐 ∕(2𝜋) = 0.2 kHz. Before analyzing GH shifts, we first briefly discuss how the driveprobe fields detuning 𝛿 and the strength 𝐸𝑙 of the drive field affect the dispersive and absorptive properties of the probe field. The real and imaginary parts of ℰ𝑡 are plotted against the normalized detuning 𝛿∕𝜔𝑚 and 𝐸𝑙 in Fig. 2. Fig. 2(a) shows their behaviors against 𝛿∕𝜔𝑚 in the presence of drive field (𝐸𝑙 = 5.5 MHz). In comparison to their behaviors in the absence of drive field (the left inset figure), the drive field considerably modifies both the real and imaginary parts of ℰ𝑡 (𝜔𝑝 ). In fact, the radiation pressure, arising from the presence of both drive and probe fields, set the mechanical oscillator in motion at 284

M. Musadiq and S. Khan

Optics Communications 450 (2019) 282–286

Fig. 3. The behavior of 𝑆𝑟 (solid black) and 𝑆𝑡 (dashed red) with 𝛿 = −1.15 𝜔𝑚 against the incident angle 𝜃 for (a) 𝐸𝑙 = 0, (b) 𝐸𝑙 = 4 MHz (left ordinate) and 𝐸𝑙 = 5.2 MHz (right ordinate). The values of the other parameters are the same as in Fig. 2.

Fig. 4. The behavior of |𝑟(𝑘𝑦 , 𝜔𝑝 )| (solid black) and |𝑡(𝑘𝑦 , 𝜔𝑝 )| (dashed red) against 𝜃 for (a) 𝐸𝑙 = 4 MHz, (b) 𝐸𝑙 = 5.2 MHz. The values of the other parameters are the same as in Fig. 2(b).

Fig. 5. Behavior of 𝑆𝑟 (solid black) and 𝑆𝑡 (dashed red) (a) against 𝛿∕𝜔𝑚 for 𝜃 = 0.915 rad and 𝐸𝑙 = 6 MHz. (b) against 𝐸𝑙 for 𝜃 = 0.915 rad and 𝛿 = −1.15 𝜔𝑚 .

Next, we want to observe the effect of the resonant modes on

angles of probe field at which the GH shifts (positive and negative) in Fig. 3(b) are large.

reflection and transmission coefficients of the output field at the probe

Keeping in view the dependence of the absorptive and dispersive properties of the system on the detuning 𝛿 and strength 𝐸𝑙 of the drive field (Fig. 2), it is also important to investigate the effect of these parameters on the shifts in the two beams. To this end, we plot 𝑆𝑟,𝑡 against 𝛿∕𝜔𝑚 in Fig. 5(a) and 𝐸𝑙 in Fig. 5(b). It is observed that the modification of the absorptive and dispersive properties or change in resonant conditions of the system, due to variation in these parameters,

frequency. In Fig. 4, we plot the absolute values of these quantities against the incident angle of the probe field. The parameters values for the two sub-figures are, respectively, set to the ones used for the left and right ordinates in Fig. 3(b). In each case, both the reflection and transmission curves show peaks at angles satisfying the resonant conditions. The largest peaks, in each case, correspond to the incident 285

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considerably affect the shifts in the transmitted and reflected output beams of the cavity at the probe frequency at a fixed incident angle of the probe field. As both these parameters can be adjusted through the control drive field, without modifying the rest structure of the setup and shifts (positive and negative) ranging from small to considerably larger amount both in transmitted and reflected output fields of the cavity at probe frequency can be obtained, therefore, the scheme may prove very useful in variety of applications.

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4. Conclusions We have presented a scheme for manipulation of GH shifts, in the output field at probe frequency of a cavity-optomechanical system, with experimentally realizable values of the system’s parameters. Employing the standard input–output theory, we have obtained analytical relation for the response function of the output field at the probe frequency of the cavity-optomechanical system. Using stationary phase theory, the results for transmitted and reflected GH shifts are obtained by using transfer matrix approach. As the absorptive and dispersive properties at the probe frequency of the setup can be controlled through two different parameters of the drive field; the drive-probe fields detuning and power of the drive field, therefore, such manipulation of the response function through either of the two parameters, in turn, enables to modify the transmitted and reflected profiles of GH shifts, and that is exactly what we have shown in this work. With no material medium inside the cavity, the working of the setup is considerably simple as compared to the previous schemes based on different types of material media. Both small and very large positive and negative GH shifts are obtained just by adjusting the drive-probe field detuning or power of the drive field. A GH shift the order of 1000 probe wavelength is large enough to be obtained from any of the previously studied schemes. References [1] F. Goos, H. Hänchen, Ein neuer und fundamentaler versuch zur totalreflexion, Ann. Phys. 1 (1947) 333. [2] F. Goos, H. Hänchen, Neumessung des strahlversetzungseffektes bei totalreflexion, Ann. Phys. 5 (1949) 251. [3] K. Artmann, Berechnung der Seitenversetzung des totalreflektierten Strahles, Ann. Phys. 437 (1948) 87–102. [4] C. Imbert, CaLculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam, Phys. Rev. D 5 (1972) 787–796. [5] B. Zhao, L. Gao, Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites, Opt. Express 17 (2009) 21433–21441. [6] V. Shadrivov, A.A. Zharov, Y.S. Kivshar, Giant Goos-Hänchen effect at the reflection from left-handed metamaterials, Appl. Phys. Lett. 83 (2003) 2713–2715. [7] L.G. Wang, S.Y. Zhu, Giant lateral shift of a light beam at the defect mode in one-dimensional photonic crystals, Opt. Lett. 31 (2006) 101–103.

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