Giant and tunable Goos-H&#x00E4;nchen shift with a high reflectance induced by PT-symmetry in atomic vapor

Peng Han; Wenxiu Li; Yang Zhou; Shuo Jiang; Xiaoyang Chang; Anping Huang; Hao Zhang; Zhisong Xiao; Zhisong Xiao

doi:10.1364/OE.432082

1. Introduction

The Goos-Hänchen (GH) shift, experimentally discovered by Goos and Hänchen in 1947, refers to a displacement between the barycenters of incident and reflected light beams when total reflection occurs at the boundary of two different media [1]. The emergence of the GH shift was derived from the penetration of an evanescent wave into the forbidden transmission region of medium and was explained by stationary phase theory [2–4]. During the past decades, the investigations on GH shifts have attracted tremendous attention as it played pivotal roles in the researches on optical sensors [5,6], beam splitters [7,8], optical storage [9] and optical switches [10].

Generally, the GH shifts are only several times of wavelength which is difficult for experimental measurement. According to the stationary phase theory, the GH shift is proportional to the derivative of reflection coefficient phase to the incident angle [2]. Based on this principle, the Brewster angle effects [11,12] and the resonance effects [13–24] have been proposed to strengthen the GH effects in various materials and structures, including weak absorbing media [13], atomic gas media [14–16], waveguide structures [5,17,18], metal-dielectric multilayer [19] and one dimensional photonic crystal (1DPC) [20–25]. In these schemes, the structure of 1DPC can be artificially designed to achieve novel scattering properties which is not available in natural media. It has been shown that there is an obvious GH shift at the band gap edge of 1DPC [20]. When a defect layer was embedded into the 1DPC, the GH shifts were greatly enhanced at the defect mode owing to the electromagnetic wave localization [21]. Moreover, by exciting a guided mode or a Bloch surface mode in 1DPCs, giant GH shifts as large as three orders of magnitude of wavelength could be achieved in experiments [22,23]. However, the enhanced GH shifts in 1DPCs come with a rather low reflectance, which poses great challenges in experimental detection.

This dilemma can be resolved by introducing spatial balanced gain and loss into 1DPC, that is, constructing a non-Hermitian optical structures. Some investigations suggested that large GH shifts with a high reflectance could be realized in non-Hermitian photonic crystals [4,26–31]. Among these investigations, much attention were paid to the study of Parity-Time (PT) symmetric periodic optical structures [4,26–28]. PT symmetry refers to the invariance of a system under a combination of parity- and time-reversal operations [32,33]. Since the pioneering work of Bender and Boettcher, there has been increasing interests in non-Hermitian Hamiltonians with PT symmetry, which possess real eigenvalues in symmetric phase but complex eigenvalues in broken phase [32,33]. The optical PT symmetry requires the real and imaginary parts of the refractive index should be even and odd functions of spatial position [34–36]. Recent explorations revealed the general features of GH shifts in perfect PT-symmetric periodic structures [4]. But few works were devoted to the realistic physical scheme and setup for achieving large GH shifts with a high reflectance.

In this paper, we investigate the GH shifts of the probe light reflected from the atomic medium with PT-symmetric susceptibilities. The proposal is inspired by the earlier research about realization of complex PT symmetric periodic optical potentials (complex crystals) and unidirectional invisibility in helium atomic vapor [37]. We demonstrate that the complex crystal constructed in helium atomic vapor could be all-optically modulated to work near the exceptional point (EP) and coherent perfect absorption (CPA)-laser point. Within the reflection band, the GH shifts of probe light can be greatly enhanced, accompanied with a high reflectance. In the vicinity of CPA-Laser point, giant GH shift can be realized at the reflectance peak. In addition, the GH shifts could be flexibly manipulated in the configuration we proposed.

2. Realization of a complex crystal structure in atomic vapor

In this section, we introduce a scheme for realizing complex crystal structure in helium atomic vapor. The PT symmetry-breaking transition and the CPA-Laser effect of complex crystal are investigated.

2.1 Energy level structure of helium atoms and configuration

The energy level structure used to achieve complex crystal is depicted in Fig. 1(a). The resonant frequencies of |³S₁>↔|³P_0,1,2> is ω_J_0,J1,J2 when the magnetic field is not applied. The energy level structure in ³S₁↔³P₂ can be seen as a combination of two three-level Λ-type systems and a two-level system. One of the Λ-type systems is formed by |1>, |2>, |4>, the other is formed by |1>, |3>, |6 > . The two-level system is formed by |2> and |5 > . The π probe light and σ+ coupling light respectively couple all the transitions |i>↔|j > satisfying Δm=0 and Δm=1 in ³S₁↔³P₂. The frequencies of the π probe light ω_p and σ+ coupling light ω_c are chosen to satisfy ω_p–ω_c≈ω₂₁, ω₁₃, with ω_ij=|ω_i –ω_j| denotes the resonant frequency between level |i > and |j>. The detuning of the π probe lights and σ+ coupling light is Δ_p,ij=ω_ij–ω_p and Δ_c,ij=ω_ij–ω_c respectively. A static magnetic field B is used to induce the Zeeman effect, as shown in Fig. 1(b). In addition, two π control lights are used to induce spatially modulated stark shifts along x direction. The π control light 1 with frequency ω₁ simultaneously couple the transitions |8>↔|2> and |9>↔|3> in ³S₁↔³P₁, |³S₁, m=0>↔|³P₁, m=0> is not included owing to the selection rules. The π control light 2 with frequency ω₂ couples the transition |0>↔|1> in ³S₁↔³P₀. The detuning of the π control lights 1 and π control lights 2 is Δ_1,ij=ω_ij–ω₁ and Δ₂=ω₀₁–ω₂ respectively.

Fig. 1. (a) Energy level structure of helium atoms used to realize complex crystal structure. (b) Schematic of a probe light beam (red solid line) incident upon the helium atomic vapor cell. The magnetic field B and σ+ coupling lights are set to travel along y+ direction. The π control light 1 (2) illuminates the helium atomic vapor cell at fixed incident angles ±θ_c1 (±θ_c2). The total length of the helium atomic vapor cell is 800×Λ, with Λ is the length of a single period. G denotes the gain regions, L denotes the loss regions.

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By solving density matrix equations, the total susceptibility of the probe light is written as

(1)$$\chi (x )\textrm{ = }\frac{{{N_0}{{|{{\mu_{41}}} |}^2}}}{{{\varepsilon _0}\hbar }}{\chi _{41}} + \frac{{{N_0}{{|{{\mu_{63}}} |}^2}}}{{{\varepsilon _0}\hbar }}{\chi _{63}} + \frac{{{N_0}{{|{{\mu_{52}}} |}^2}}}{{{\varepsilon _0}\hbar }}{\chi _{52}}\textrm{,}$$

here μ_ji is the dipole moment of the transition |i>↔|j>, N₀ is the number density of the helium atoms. χ₄₁ and χ₆₃ denotes the susceptibilities of two three-level Λ-type systems respectively, χ₅₂ denotes the susceptibility of two-level system. Their expressions are given by

(2)$${\chi _{41}}\textrm{ = }\frac{{{\xi _1}({{\rho_1} - {\rho_2}} )}}{{{\omega _{2^{\prime}1^{\prime}}} - {\omega _p} + i{\gamma _{2^{\prime}1^{\prime}}}}} + \frac{{({{\rho_1} - {\rho_4}} )}}{{{\omega _{4^{\prime}1^{\prime}}} - {\omega _p} + i{\gamma _{4^{\prime}1^{\prime}}}}}\textrm{,}$$

(3)$${\chi _{63}}\textrm{ = }\frac{{{\xi _2}({{\rho_3} - {\rho_1}} )}}{{{\omega _{1^{\prime}3^{\prime}}} - {\omega _p} + i{\gamma _{1^{\prime}3^{\prime}}}}} + \frac{{({{\rho_3} - {\rho_6}} )}}{{{\omega _{6^{\prime}3^{\prime}}} - {\omega _p} + i{\gamma _{6^{\prime}3^{\prime}}}}}\textrm{,}$$

(4)$${\chi _{52}}\textrm{ = }\frac{{({{\rho_5} - {\rho_2}} )}}{{{\omega _{52}} - \Omega _{1,82}^2/{\Delta _{1,82}} + \Omega _{c,42}^2/{\Delta _{c,42}} - {\omega _p} + i{\gamma _1}}}\textrm{,}$$

with ρ_i denotes the population of |i>, ${\gamma _{2^{\prime}1^{\prime}}} = ({1 - {\xi_1}} ){\gamma _2} + {\xi _1}{\gamma _1}$, ${\gamma _{4^{\prime}1^{\prime}}} = ({1 - {\xi_1}} ){\gamma _1} + {\xi _1}{\gamma _2}$, ${\gamma _{3^{\prime}1^{\prime}}} = ({1 - {\xi_2}} ){\gamma _2} + {\xi _2}{\gamma _1}$, ${\gamma _{6^{\prime}3^{\prime}}} = ({1 - {\xi_2}} ){\gamma _1} + {\xi _2}{\gamma _2}$ and ${\xi _1} = \Omega _{c,42}^2/\Delta _{c,42}^2$, ${\xi _1} = \Omega _{c,61}^2/\Delta _{c,61}^2$. (Ω_1,ij)Ω_c,ij is the Rabi frequency of the π control light 1 (σ+ coupling light) interact with |i>↔|j>. Ω₂ is the Rabi frequency of the π control light 2. γ₁ is the decoherence rates between one sublevel of |³S₁> and one sublevel of |³P_0,1,2>, γ₂ is the decoherence rate among sublevels of |³P_0,1,2>. In addition, ω_2′1′=ω₄₁–Δ_c_,42+Δ_s₁, ω_4′1′=ω₄₁+Δ_s₂, ω_1′3′=ω₆₃–Δ_c_,61+Δ_s₃, ω_6′3′=ω₆₃+Δ_s₄ represent the resonant frequencies in dressed state picture, Δ_s_1,s2,s3,s4 represent the stark shifts caused by σ+ coupling lights and π control lights. The expressions of Δ_s_1,s2,s3,s4 are

(5)$${\Delta _{s1}} ={-} \Omega _{1,82}^2/{\Delta _{1,82}} - \Omega _{c,42}^2/{\Delta _{c,42}} + \Omega _2^2/{\Delta _2} + \Omega _{c,61}^2/{\Delta _{c,61}}\textrm{,}$$

(6)$${\Delta _{s2}} = \Omega _{c,42}^2/{\Delta _{c,42}} + \Omega _2^2/{\Delta _2} + \Omega _{c,61}^2/{\Delta _{c,61}}\textrm{,}$$

(7)$${\Delta _{s3}} ={-} \Omega _2^2/{\Delta _2} - \Omega _{c,61}^2/{\Delta _{c,61}} + \Omega _{1,93}^2/{\Delta _{1,93}} + \Omega _{c,73}^2/{\Delta _{c,73}}\textrm{,}$$

(8)$${\Delta _{s4}} = \Omega _{1,93}^2/{\Delta _{1,93}} + \Omega _{1,73}^2/{\Delta _{1,73}} + \Omega _{c,61}^2/{\Delta _{c,61}}.$$

A configuration is shown in Fig. 1(b), the probe light is incident on the surface of the atomic vapor cell, making an incident angle θ with x axis. The magnetic field B and σ+ coupling light are set to travel along y+ direction. The atomic vapor cell is illuminated on both sides by π control lights 1 (2) with fixed incident angles ±θ_c1(±θ_c2). Each pair of π control lights form a standing wave along x direction and take the form Ω_1,ij(x)=Ω_1,ij·cos(k_c_1x·x+φ), Ω₂(x)=Ω₂·cos(k_c_2x·x), k_c_1x and k_c_2x denote the x component of the π control light wave vector, φ is the phase difference between the control light 1 and control light 2. In this paper, the x components of the π control light wave vector are set to satisfy k_c_1x=k_c_2x=k_p/2. A pair of electro-optic modulators are placed on the beam path of two control lights 1 respectively. By adjusting the applied voltages of electro-optic modulators, the value of φ could be modified. The total length of the helium atomic vapor cell is L=800×Λ, with Λ=1.084μm is the length of a single period. The fabrication methods of chip-scale atomic device and atomic vapor cell with submillimeter thickness have been rapidly developed in the past decade [38,39].

In the following, we set |μ₄₁|²=7.67 × 10^–58(C·m)², |μ₅₂|²=|μ₆₃|²=0.75×|μ₄₁|², ω_J₂=2π×276732.18GHz, ω_J₀=ω_J₂+2π×31.91GHz, ω_J₁=ω_J₂+2π×2.29GHz, γ₁=5MHz, γ₂=0.01γ₁, B=−357Gauss, Ω_c,42=γ₁/2, ${\Omega _{c,61}} = \sqrt 3 {\Omega _{c,42}}$, ${\Omega _{c,73}} = \sqrt 6 {\Omega _{c,42}}$, ρ₀=0.35, ρ₁=0.35, ρ₄=0.30, ${\omega _c} = {\omega _{J\textrm{2}}} - 300{\gamma _1}$, ${\omega _1} = {\omega _{J\textrm{1}}}$, ${\omega _2} = {\omega _{J\textrm{0}}} + 100{\gamma _1}$ throughout this paper unless specified [37].

2.2 Realization of a complex crystal with PT-symmetric periodic optical potentials

The optical PT symmetry demands the real part of the susceptibility Re[χ(x)] should be an even function of spatial position, the imaginary part of the susceptibility Im[χ(x)] should be an odd function of spatial position [34–37]. We calculated the susceptibilities for probe light using Eq. (1) and demonstrate the numerical simulation results in Fig. 2. Figure 2(a) and Fig. 2(b) respectively plots Re[χ(x)] and Im[χ(x)] versus k_px/2 when the intensity of π control light 2 Ω₂ is set to different values. It is obvious that both Re[χ(x)] and Im[χ(x)] are periodically modulated along x direction, Re[χ(x)] and Im[χ(x)] are respectively an even and odd function of k_px/2, implying the complex crystal structure can be realized in helium atomic vapor. In addition, the modulation amplitudes of Re[χ(x)] and Im[χ(x)] could be manipulated by adjusting the intensity of π control light 2 Ω₂. This is quite important because the ratio (represented by σ) between the modulation amplitudes of Im[χ(x)] and Re[χ(x)] determines whether the system is below or above the EP of complex crystal [40,41], which will be discussed below.

Fig. 2. Plots of (a) Re[χ(x)] and (b) Im[χ(x)] versus k_px/2 when Ω_1,82(x)=Ω_1,93(x) = 0.765γ₁·cos(k_px/2−π/4), Ω₂(x)= Ω₂·cos(k_px/2), Ω₂=0.5γ₁ (black solid line), Ω₂=1.0γ₁ (blue solid line), Ω₂=1.5γ₁ (red solid line). The frequency of the probe light is ω_p=ω_J₂−99.9916γ₁.

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2.3 Symmetry breaking phase transition in complex crystals

For a complex crystal with sinusoidal optical potentials, the photonic band structures evolve from a symmetric phase with real gapped bands to a broken-symmetry phase with merged bands when value of σ is gradually increased [36,37,40,41]. The phase transition occurs at σ=1, where the photonic band gap at the Brillion zone boundary closes, giving rise to an EP [36,41]. To analyze the conditions for achieving EP, the expression of χ(x) can be rewritten in Fourier series expansion form [40,42]

(9)$$\chi (x )\textrm{ = Re[}\chi (x)\textrm{]} + i{\mathop{\rm Im}\nolimits} \textrm{[}\chi (x)\textrm{]} = \sum\limits_n {[{{\chi_{p0}} + i{\chi_{pi}} + {\chi_{p1}}\cos (n{k_p}x) + i{\chi_{p2}}\sin (n{k_p}x)} ]} \textrm{,}$$

where n is positive integer, χ_p1 and χ_p2 is the modulation amplitudes of Re[χ(x)] and Im[χ(x)] respectively, χ_p0 and χ_pi is the mean change of Re[χ(x)] and Im[χ(x)] respectively. Since the dynamic propagation of the probe light is only affected by the first order components when k_x approaches to the Bragg wave number, n=1 is taken into consideration, here k_x=k_p·cos(θ) [43,44]. The values of modulation amplitudes and mean change can be obtained by fitting the curves of χ(x) versus k_px/2 under specific conditions. The dependence of χ_p1, χ_p2, χ_p0 and χ_pi on Ω₂ is shown in Fig. 3. As depicted in Fig. 3(a) and Fig. 3(b), with the increase of Ω₂, the values of |χ_p1|(|χ_p2|) rises from 0.842×10⁻³ (0.790×10⁻³) to 2.150×10⁻³ (2.210×10⁻³), then decreases to 1.460×10⁻³ (1.910×10⁻³), whereas the values of the ratio σ=χ_p2/χ_p1 rises monotonically from 0.939 to 1.308. The EP is located at Ω₂=1.015γ₁, corresponding to χ_p1=χ_p2, σ=1. These results suggest, the complex crystal can be operated to work either below or above the EP by adjusting the intensity of π control light 2. This feasible method is realizable in experiments.

Fig. 3. Plots of the modulation amplitudes (a) χ_p1 and χ_p2, and mean change (c) χ_p0, (d) χ_pi versus the intensities of π control lights 2 Ω₂ with Ω_1,82(x)=Ω_1,93(x) = 0.765γ₁·cos(k_px/2−π/4). (b) Plots of the ratio σ=χ_p2/χ_p1 versus the intensities of π control lights 2 Ω₂, the red solid line is σ=1, the bottom region (green) denotes the PT-symmetric phase, while the top region (blue) denotes the PT-symmetry-broken phase. Other parameters are the same as in Fig. 2.

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The mean change of Re[χ(x)] and Im[χ(x)] is also important in realizing ideal PT-symmetric optical potentials. As shown in Fig. 3(c) and Fig. 3(d), with the increase of Ω₂, χ_p0 decreases monotonically from 1.784×10⁻³ to –1.473×10⁻⁴, χ_pi increases from –2.323×10⁻⁵ to –1.652×10⁻⁵ and then decreases to –1.710×10⁻⁵. The nonzero χ_p0 does not affects the realization of optical PT symmetry. Significantly, the values of χ_pi are always less than 0, indicating that a net gain is introduced in helium atoms. The perfect PT-symmetric optical potentials are destroyed owing to the unbalanced gain and loss. In fact, it is rather involved or even impossible to construct perfectly complex crystal with PT-symmetric optical potentials using atomic media [45–47]. Here, the degrees of imbalance (defined as χ_pi /χ_p2 in this work) for Im[χ(x)] is less than 2.9%, which does not affect the main results of this paper.

2.4 Coherent perfect absorption-laser point in a PT-symmetric complex crystal

In the region of broken-symmetry phase (σ>1), a complex crystal exhibits a self-dual spectral singularity, at which the PT-symmetric system could behave simultaneously as a coherent perfect absorber and a laser [48–50]. Therefore, the self-dual spectral singularity is also referred to as the CPA-Laser point, which is manifested as divergence of the reflection coefficient for probe light [48–50]. According to the coupled-mode equation analysis in appendix, the CPA-Laser point can be estimated by solving M₂₂(ω_p) = 0 under Bragg condition [4], the expression of the corresponding critical value of σ_c is

(10)$${\sigma _c} = \sqrt {1 + {{\left( {\frac{{2{\pi^2}}}{{k_p^2\mathrm{\Lambda }L{\chi_{p1}}}}} \right)}^2}} \textrm{,}$$

Eq. (10) shows, the critical value σ_c depends on the modulation amplitude χ_p1 and will be altered as the value of Ω₂ changes. The dependence of σ and σ_c on Ω₂ is shown in Fig. 4. In the range of 1.015γ₁<Ω₂<1.200γ₁, σ rises monotonically from 0.998 to 1.068, whereas the critical value σ_c for realizing CPA-Laser condition rises monotonically from 1.043 to 1.046, σ grows faster than σ_c. For the probe light with frequency ω_p, the CPA-Laser point can be attained when Ω₂ approaches to 1.140γ₁.

Fig. 4. Plots of σ and σ_c versus the intensity of the π control lights 2 Ω₂. Other parameters are the same as in Fig. 2.

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3. Goos-Hänchen shifts in complex crystals

In this section, the GH shifts of probe light reflected from the complex crystals are investigated near the EP and the CPA-Laser point.

3.1 Method to calculate the reflection coefficient and Goos-Hänchen shift

The propagation dynamics of probe light in Fig. 1(b) is characterized by scalar Helmholtz equation

(11)$$\frac{{{\partial ^2}{E_z}}}{{\partial {x^2}}} + \frac{{{\partial ^2}{E_z}}}{{\partial {y^2}}} + \frac{{\omega _p^2\varepsilon (x )}}{{{c^2}}}{E_z} = 0\textrm{,}$$

with ε(x) = 1+χ(x), c is the light speed in vacuum. As k_x approaches to the Bragg wave number of complex crystal π/Λ, the solution to Eq. (11) takes the form of

(12)$${E_z}(x )\simeq u(x )\textrm{exp}[{{{ix\pi } / \mathrm{\Lambda }} + i{k_y}y} ]+ v(x )\textrm{exp}[{{{ - ix\pi } / \mathrm{\Lambda }} + i{k_y}y} ]\textrm{,}$$

here u(x) and v(x) respectively denotes the amplitude of forward (from left to right) and backward (from right to left) propagating light wave inside the complex crystal, k_y is the y component of the probe light wave vector. For left and right side incident probe light, the reflection coefficient is r_L=E_b(−L/2)/E_f(−L/2) and r_R=E_f(+L/2)/E_b(+L/2) respectively, E_f(−L/2), E_b(+L/2) and E_b(−L/2), E_f(+L/2) are amplitudes of the incident and reflected light outside the complex crystal.

The expression of reflection coefficient can be obtained using transfer matrix method (TMM) [4]. For a complex crystal containing N periods, each period is divided into 100 parallel layers with identical thickness d, the dielectric constant of l-th layer ${\varepsilon _l} = 1 + {\chi _l}$ is regarded as a constant. The transfer matrix of l-th layer is

(13)$${m_l} = \left[ {\begin{array}{cc} {\textrm{cos}({k_z^ld} )}&{{{i\textrm{sin}({k_z^ld} )} / {{q_l}}}}\\ {i{q_l}\textrm{sin}({k_z^ld} )}&{\textrm{cos}({k_z^ld} )} \end{array}} \right]\textrm{,}$$

here $k_z^l = \sqrt {{\varepsilon _l}k_p^2 - k_p^2\textrm{si}{\textrm{n}^2}\textrm{(}\theta \textrm{)}} $ is the z component of the probe light wave vector in l-th layer, ${q_l} = k_z^l/{k_p}$. The total transfer matrix is m=(m_unit)^N with $m_{unit}=m_1 \ldots m_l \ldots m_{100}$. The expression of reflection coefficient is given by [4,14,15]

(14)$$r({{k_y},{\omega_p}} )= \frac{{\textrm{cos}(\theta )({{m_{22}} - {m_{11}}} )- ({\textrm{co}{\textrm{s}^2}(\theta ){m_{12}} - {m_{21}}} )}}{{\textrm{cos}(\theta )({{m_{22}} + {m_{11}}} )- ({\textrm{co}{\textrm{s}^2}(\theta ){m_{12}} + {m_{21}}} )}}.$$

Here, when the probe light is incident from left side and right side, the corresponding reflection coefficient r_L and r_R is respectively obtained by substituting the phase of π control light 1 $\varphi ={-} \mathrm{\pi }/4$ and $\varphi ={+} \mathrm{\pi }/4$ into Eq. (14), because χ(x) with $\varphi ={-} \mathrm{\pi }/4$ equals to the complex conjugate of χ(x) with $\varphi ={+} \mathrm{\pi }/4$ [37].

According to the stationary phase theory, the GH shift S_GH is given by

(15)$${S_{GH}} ={-} \frac{{d\phi }}{{d{k_y}}} ={-} \frac{{{\lambda _p}}}{{2\pi }}\frac{1}{{\textrm{cos}(\theta )}}\frac{{d\phi }}{{d\theta }}\textrm{,}$$

here ϕ is the phase of reflection coefficient calculated using Eq. (14), λ_p is the wavelength of the probe light [14,15].

3.2 Goos-Hänchen shifts at the exceptional point

Firstly, we investigate the GH shifts for probe light when the complex crystal is working at EP. The coupled-mode theory (CMT) is used to analyze the numerical simulation results. Based on the results in section 2.3, Fig. 5(a) gives the dependence of the reflectance on incident angles when Ω_1,82(x)=Ω_1,93(x) = 0.765γ₁·cos(k_px/2±π/4), Ω₂(x) = 1.015γ₁·cos(k_px/2), the black and violet solid curve is the reflectance of left and right side incident probe light calculated using Eq. (14) respectively, whereas the red and yellow dashed curves are the predictions based on coupled-mode equation analysis. The reflectance of left side incident probe light is greatly enhanced inside the reflection band, whereas the reflectance of right side incident probe light is diminished in the entire reflection band. Since we are interested in large GH shift with a high reflectance, the GH shifts should be probed from the left side. The corresponding normalized GH shifts for left side incident probe light is shown in Fig. 5(b), the black solid curve is the normalized GH shifts calculated using Eq. (15), the red dashed curve is the prediction based on coupled-mode equation analysis. As can be seen from Fig. 5(b), the GH shifts are much larger than the incident probe light wavelength. Within the reflection band, the large GH shifts are almost independent of incident angles, accompanied with a high reflectance. Beside the large GH shifts inside the high-reflection band, there are five peaks in the curve of GH shifts owing to the Bragg resonances [26,42], the GH shifts are positive which is caused by the net gain in complex crystals [42]. At these resonance points, the reflectance is weak.

Fig. 5. Plots of the (a) reflectance $|{r_{L,R}}{|^2}$ and (b) the normalized GH shifts S_GH/λ_p versus the incident angles θ when Ω_1,82(x)= Ω_1,93(x) = 0.765γ₁·cos(k_px/2±φ), Ω₂(x) = 1.015γ₁·cos(k_px/2). In (a), the black and violet solid curves are the reflectance obtained by TMM, calculated using Eq. (14) with $\varphi ={-} \pi /4$ and $\varphi ={+} \pi /4$, whereas the red and yellow dashed curves are the predictions based on CMT. In (b), the black solid curve is the GH shift calculated using Eq. (15) with $\varphi ={-} \pi /4$, the red dashed curve is the prediction based on CMT. The blue dashed curve denotes S_GH/λ_p=1387. The white area correspond to the reflection band. Other parameters are the same as in Fig. 2.

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As shown in Fig. 5, the results obtained by TMM fit well with the predictions based on CMT, hence the behaviors of reflectance and GH shifts can be explained through CMT. The asymmetric reflection (|r_L|²≠|r_R|²) between left and right side incident probe light is caused by the unbalanced coupling coefficients between forward and backward propagating light waves inside the complex crystal [51]. For left side incident probe light, the two counter-propagating light waves form destructive interferences in the loss segments and constructive interferences in the gain segments, leading to an enhanced reflectance. On the contrary, the multiple interference inside the complex crystal suppresses the reflection when the probe light is incident from right side [51,52]. As the EP (σ=1) is approached, the reflectance of right side incident probe light is nearly zero, the asymmetrical reflection becomes most pronounced [51].

The enhancement of GH shifts can be explained through the concept of penetration depth for probe light [3,4]. According to coupled-mode equation analysis, in the region of 0<σ<σ_c, the penetration depth of probe light is a monotonous increasing function of σ, resulting in enhanced GH shifts in complex crystal (σ≠0). In particular, at EP i.e., σ=1, the penetration depth of incident photons equals to L/2 and is independent of the incident angles. Therefore, in the configuration we proposed, the penetration depth should be 400λ_p at EP, the corresponding GH shift is 1387λ_p and almost unchanged inside the reflection band, as shown in Fig. 5(b). Large GH shifts with a high reflectance is achieved inside the reflection band when the PT-symmetric complex crystal system is working at the EP.

3.3 Goos-Hänchen shifts near the CPA-laser point

When the intensity of π control light 2 is further enhanced i.e., Ω₂>1.015γ₁, the penetration depth of the probe light is larger than L/2 in broken-symmetry regime (σ>1). As the CPA-Laser point is approached, the incident probe photons in amplifying modes spend more time in gain segments than the loss segments, it can be expected that the GH shifts will be greatly enhanced at the reflectance peak owing to the divergence of the reflection coefficient for probe light [4,49]. Based on the results in section 2.4, Fig. 6, Fig. 7 plots the reflectance and the corresponding normalized GH shifts versus the incident angles when Ω₂=1.130γ₁, 1.135γ₁, 1.137γ₁ (Ω₂=1.138γ₁, 1.140γ₁, 1.145γ₁). The reflectance angular spectra curves become narrower and sharper as the critical value σ_c is approached. Consequently, the corresponding GH shift is greatly enhanced, giant GH shifts as large as ±10⁵ times of the wavelength can be achieved in theory. Different from the GH shift enhanced at the reflectance dip in conventional passive optical system, in the configuration we proposed, the maximum GH shift is exactly located at the reflectance peak which is more suitable for detection.

Fig. 6. Plots of (a), (c), (e) the reflectance |r|² and (b), (d), (f) normalized GH shifts S_GH/λ_p versus the incident angles θ when Ω_1,82(x)=Ω_1,93(x) = 0.765γ₁·cos(k_px/2−π/4), Ω₂(x)=Ω₂·cos(k_px/2), ω_p=ω_J₂−99.9916γ₁, (a) and (b) refers to the case of Ω₂=1.130γ₁, (c) and (d) refers to the case of Ω₂=1.135γ₁, (e) and (f) refers to the case of Ω₂=1.137γ₁. Other parameters are the same as in Fig. 5.

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Fig. 7. Plots of (a), (c), (e) the reflectance |r|² and (b), (d), (f) normalized GH shifts S_GH/λ_p versus the incident angles θ when Ω_1,82(x)=Ω_1,93(x) = 0.765γ₁·cos(k_px/2−π/4), Ω₂(x)=Ω₂·cos(k_px/2), ω_p=ω_J₂−99.9916γ₁, (a) and (b) refers to the case of Ω₂=1.138γ₁, (c) and (d) refers to the case of Ω₂=1.140γ₁, (e) and (f) refers to the case of Ω₂=1.145γ₁. Other parameters are the same as in Fig. 5.

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Besides achieving a giant GH shift with a high reflectance, the GH shift can also be manipulated. As can be seen from Fig. 6 and Fig. 7, in the region of 1.130γ₁<Ω₂<1.145γ₁, the peak value of the GH shift first reaches the positive maximum value, and then dramatically switches to the negative maximum value. The GH shifts of probe light could be manipulated by only adjusting the intensity of control light 2. Therefore, the GH shifts are detectable and tunable in a large range, which is suitable for optical sensors and optical beam manipulation in photonic crystals-based devices. Note that the expression of Eq. (15) is an approximate result because the incident probe light is considered to be ideal plane wave in stationary phase theory [2,14,15]. In practice, the results may be deviated from the stationary phase analysis due to the distortion of the reflected light beam [21]. To achieve such a high multiple of GH shifts in practice is challenging owing to the effect of fluctuations and noise. But, the appearance of the maximum GH shift at the reflectance peak near the CPA-Laser point is a robust phenomenon.

4. Conclusions

In summary, we have demonstrated the realization of complex crystal with PT-symmetric optical potentials in helium atomic vapor. It is found that large GH shifts can be obtained within the high-reflection band when the complex crystal is operated at the EP. In the vicinity of the CPA-Laser point, giant GH shifts could be obtained at the reflectance peak. In addition, by only adjusting the intensity of the control light, the behavior of the GH shifts can be flexibly controlled in the configuration we proposed, which has potential applications in the design of photonic crystal-based devices.

Appendix: explanation for the behavior of reflectance and GH shifts in complex crystals

In this appendix, the behaviors of reflectance and GH shifts of the probe light are explained using CMT. Substituting the expression of susceptibility [χ(x)= χ_p0+i·χ_pi+χ_p1·cos(nk_px)+i·χ_p2·sin(nk_px)] in Eq. (9) and the solutions in Eq. (12) into the scalar Helmholtz equation in Eq. (11), keeping only the synchronous terms, the coupled-mode equations can be obtained

(16)$$i\frac{{du\textrm{(}x\textrm{)}}}{{dx}} ={-} \delta u\textrm{(}x\textrm{)} - {\kappa _1}v\textrm{(}x\textrm{),}$$

(17)$$i\frac{{dv\textrm{(}x\textrm{)}}}{{dx}} = \delta v\textrm{(}x\textrm{)} + {\kappa _2}u\textrm{(}x\textrm{),}$$

with δ=k_x–[π/Λ–k_p(χ_p0+i·χ_pi)] is the detuning, ${\kappa _1} = \textrm{[}k_p^2{\chi _{p1}}\Lambda \textrm{(}1 - \sigma \textrm{)]/}4\pi$ and ${\kappa _2} = \textrm{[}k_p^2{\chi _{p1}}\Lambda \textrm{(}1 + \sigma \textrm{)]/}4\pi$ is the coupling coefficients between forward and backward propagating light waves. On the left and right boundaries, the amplitudes of incident and reflected probe light are related through a 2×2 transfer matrix [E_f(+L/2), E_b(+L/2)]^T=M(ω_p)[E_f(–L/2), E_b(–L/2)]^T, the superscript T denotes transposition. The elements of the transfer matrix M(ω_p) are

(18)$${M_{11}}({{\omega_p}} )\textrm{ = }M_{22}^\ast ({{\omega_p}} )= \left[ {\textrm{cosh}({\xi L} )+ i\frac{\delta }{\xi }\textrm{sinh}({\xi L} )} \right]\textrm{exp}({ - i\delta L} )\textrm{,}$$

(19)$${M_{12}}({{\omega_p}} )= i\frac{{{\kappa _1}}}{\xi }\textrm{sinh}({\xi L} )\textrm{exp}({ - i\delta L} )\textrm{,}$$

(20)$${M_{21}}({{\omega_p}} )={-} i\frac{{{\delta ^2} + {\xi ^2}}}{{\xi {\kappa _1}}}\textrm{sinh}({\xi L} )\textrm{exp}({i\delta L} )\textrm{,}$$

with $\xi = \sqrt {{\kappa _1}{\kappa _2} - {\delta ^2}}$, the reflection coefficient is given by

(21)$${r_L} ={-} \frac{{{M_{21}}({{\omega_p}} )}}{{{M_{22}}({{\omega_p}} )}} = i\frac{{{\kappa _2}\textrm{sinh}({\xi L} )}}{{\xi [{\textrm{cosh}({\xi L} )- i({{\delta / \xi }} )\textrm{sinh}({\xi L} )} ]}}\textrm{,}$$

(22)$${r_R} = \frac{{{M_{12}}({{\omega_p}} )}}{{{M_{22}}({{\omega_p}} )}} = i\frac{{{\kappa _1}\textrm{sinh}({\xi L} )\exp ({ - 2i\delta L} )}}{{\xi [{\textrm{cosh}({\xi L} )- i({{\delta / \xi }} )\textrm{sinh}({\xi L} )} ]}}.$$

Eq. (21) and Eq. (22) clearly shows, in complex crystals (σ≠0), the asymmetrical coupling coefficient (κ₁≠κ₂) between forward and backward propagating light waves lead to the asymmetrical reflection coefficient in left and right side. Hence, the reflectance is enhanced for left side incident probe light but diminished for right side incident probe light. This asymmetry becomes most pronounced at EP [51]. At exact Bragg resonance (i.e., δ=0, the net gain in complex crystal is neglected), the expression of GH shift is given by

(23)$${S_{GH}}\textrm{ = } - {{d{\varphi _L}} / {d{k_y} = }}\textrm{2tan}(\theta )\rho = \frac{{\textrm{tan}(\theta )}}{{\sqrt {{\kappa _1}{\kappa _2}} }}\textrm{tanh}\left( {\sqrt {{\kappa_1}{\kappa_2}} L} \right)\textrm{,}$$

(24)$$\rho = \frac{1}{2}\left( {\frac{{\partial {\varphi_L}}}{{\partial {k_x}}}} \right) = \frac{1}{{2\sqrt {{\kappa _1}{\kappa _2}} }}\textrm{tanh}\left( {\sqrt {{\kappa_1}{\kappa_2}} L} \right)\textrm{,}$$

with φ_L=π/2+atan[(δ·sinh(ξL))/(ξ·cosh(ξL))] is the phase of reflection coefficient calculated using Eq. (21), ρ is the penetration depth of the probe light. The expressions in appendix are similar to those derived in Ref. [4]. As can be seen from Eq. (24), in the region of 0<σ<σ_c, the penetration depth rises monotonically. Hence, within the reflection band, the GH shifts for complex crystal (σ≠0) is always larger than for the absorption-free 1DPC (σ=0). At the EP, the reflection coefficient phase becomes φ_L=π/2+(k_x–π/Λ)·L, the penetration depth of the probe light equals to L/2 and is independent of incident angles.

Funding

National Natural Science Foundation of China (11804017, 51872010, 61975005); Beijing Academy of Quantum Information Sciences (Y18G28).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Giant and tunable Goos-Hänchen shift with a high reflectance induced by PT-symmetry in atomic vapor

Abstract

1. Introduction

2. Realization of a complex crystal structure in atomic vapor

2.1 Energy level structure of helium atoms and configuration

2.2 Realization of a complex crystal with PT-symmetric periodic optical potentials

2.3 Symmetry breaking phase transition in complex crystals

2.4 Coherent perfect absorption-laser point in a PT-symmetric complex crystal

3. Goos-Hänchen shifts in complex crystals

3.1 Method to calculate the reflection coefficient and Goos-Hänchen shift

3.2 Goos-Hänchen shifts at the exceptional point

3.3 Goos-Hänchen shifts near the CPA-laser point

4. Conclusions

Appendix: explanation for the behavior of reflectance and GH shifts in complex crystals

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (24)

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