Coherent manipulation of giant birefringent Goos–Hänchen shifts by compton scattering using chiral atomic medium

Abstract

A four level chiral medium is considered to analyze and investigate theoretically the reflection/transmission coefficients of right circularly polarized (RCP) beam and left circularly polarized (LCP) beam as well as their corresponding GH-shifts under the effect of compton scattering. Density matrix formalism is used for calculation of electric and magnetic probe fields coherence. The polarization and magnetization are calculated from probes coherence terms in the chiral medium. The electric and magnetic susceptibilities as well as chiral coefficients are related with polarization and magnetization. The refractive indices of RCP and LCP beams under compton scattering effect is modified from the electric/magnetic susceptibilities, chiral coefficients, mass and charge of electron as well as compton scattering angle. The giant positive and negative birefringent Goos–Hänchen (GH) shifts in reflection and transmission beams are investigated in this manuscript under Compton scattering effect. The RCP and LCP beams obey the normalization condition $|R^{(+,-)}|+|T^{(+,-)}|=1$ at the interface of a lossy chiral medium of $|A^{(+,-)}|\simeq 0$ and a thin sheet of balsa wood under the effect of compton scattering angle, incident angle, probe field detuning, control field Rabi frequency, phases of electric and magnetic fields and phase of superposition states. Significant positive/negative giant GH-shifts in reflection and transmission beams are investigated. The results show potential applications in modification of cloaking devices, image coding, polarizing filters and LCD displays.

Introduction

The scattering of a photon by an electron or charged particle with change in wavelength is called Compton scattering or Compton shift¹. The Compton scattering has potential applications in optics^2,3,4, astronomy^5,6, nanotechnology⁷, condensed matter physics^8,9, quantum mechanics¹⁰, nuclear physics^11,12, plasma physics^13,14. The optical coherence phenomenon in which a light beam of linearly polarized light undergoes a deviation or lateral shift from its predicted optical path is called Goos–Hänchen shift/effect¹⁵ which is proportional to the penetration depth in comparison to the light wavelength¹⁶. The GH-shift was observed experimentally for the first time by Goos and Hänchen in 1947¹⁷. The GH-shift has gained an additional interest in recent years and has been studied theoretically and experimentally for a number of different media using different techniques. Artmann¹⁸ developed the stationary phase method while Renard¹⁹ introduced the concept of conservation principle to calculate the GH-shift theoretically. Giant GH-shifts have been modified and controlled in the reflected/transmitted beams from photonic crystals^20,21, dielectric slabs²², conductive medium²³, chiral medium²⁴.

Impressive efforts were made to modify and control the GH-shifts in reflection/transmission beams using various methods and techniques. Anas et al.²⁵ modified and controlled the GH-shift via coherent and incoherent fields. Muqaddar et al.²⁶ controlled the positive and negative GH-shifts via giant kerr nonlinearity. Zia et al.²⁷ controlled the GH-shift with the intensity of the probe field. Wen et al.²⁸ controlled the GH-shift via electromagnetically induced transparency (EIT). The medium in which the cross coupling of electric and magnetic field occurs is called chiral medium. The chiral medium exhibits an interesting and important optical property called birefringence, in which a linearly polarized light wave splits or resolves into two unequal parts of transmitted or reflected waves. If the y-component of electric field lags behind the x-component is called right circular polarized (RCP) light but if the y-component of electric field leads the x-component is called left circular polarized (LCP) light. The birefringent GH-shifts in reflection and transmission of a chiral atomic medium have also been investigated using various techniques. Sajid et al.²⁹ controlled the birefringent GH-shifts in the reflected and transmitted beams of a four level chiral medium under the strength of the driving fields. Akhlaq et al.²⁴ controlled the birefringent GH-shift in the reflection and transmission of a chiral medium via local field effects and Kerr nonlinearity. Idrees et al.³⁰ controlled the birefringent GH-shift in a four level chiral medium via probe and incoherent pump fields. The chirality has attracted the researchers’ interest due to its wide applications in image coding³¹, telecommunication³² and microwaves³³.

Subject to the above literature review, the birefringent GH-shifts in reflection and transmission of a chiral medium have been controlled using different techniques but the birefringent GH-shifts in reflection and transmission of a chiral medium have not yet been controlled in the presence of compton scattering. We report and present a different theoretical approach to modify and control the birefringent GH-shifts in reflection and transmission of four-level chiral medium under the effect of compton scattering. The GH-shift in a four level chiral medium is controlled with the incident angle, probe field detuning, control field Rabi frequency, phases of electric and magnetic fields, phase of superposition states under the effect of compton scattering.

Model and its dynamics

Figure 1

Energy diagram of four level Chiral atomic system.

In Fig. 1, a four level chiral atomic medium is considered for the control of GH-shifts in reflection and transmission under compton scattering. The four level chiral atomic medium is placed inside a cavity of length $d_2$. The states $\left| 2\right\rangle$ and $\left| 4\right\rangle$ are coupled by electric probe field $E_p$ of Rabi frequency $\Omega _p$. The states $\left| 1\right\rangle$ and $\left| 3\right\rangle$ are coupled by magnetic probe field $B_m$ of Rabi frequency $\Omega _m$. The control field $E_c$ having Rabi frequency $\Omega _c$ couples the states $\left| 3\right\rangle$ and $\left| 4\right\rangle$. The states $\left| 1\right\rangle$ and $\left| 2\right\rangle$ are superposition states. The prepared superposition state of $\left| 1\right\rangle$ and $\left| 2\right\rangle$ is written as $\left| \Phi \right\rangle =\sqrt{x}\left| 2\right\rangle +\sqrt{1-x}e^{i\varphi _s}\left| 1\right\rangle$. The cavity of length $d_2$ is enclosed in another medium of thin sheet of balsa wood of small length $d_1$ and dielectric constant $\epsilon _1$. The self Hamiltonian of the proposed chiral atomic system is written bellow:

$$\begin{aligned} H_0=\sum ^{4}_{k=1}\hbar \omega _k \left| k\right\rangle \left\langle k\right| \end{aligned}$$

(1)

The interaction picture Hamiltonian for the chiral atomic system is written as:

$$\begin{aligned} H_I= & {} \sum ^4_{k=1}\hbar \left| k\right\rangle \left\langle k\right| -\frac{\hbar }{2}[\Omega _me^{-i\omega _m t}\left| 3\right\rangle \left\langle 1\right| +\Omega _pe^{-i\omega _pt}\left| 4\right\rangle \left\langle 2\right| +\Omega _ce^{-i\omega _ct}\left| 4\right\rangle \left\langle 3\right| +H.C] \end{aligned}$$

(2)

where the Rabi frequencies of electric and magnetic probe fields are $\Omega _p=\varrho _{42}e^{i\varphi _e}E_0/\hbar$ and $\Omega _m=\mu _{31}e^{i\varphi _m}B_0/\hbar$. Here $\varrho _{42}$ is the electric dipole moment between the states $\left| 2\right\rangle$ and $\left| 4\right\rangle$ and is related to the atomic decay as $\varrho _{42}=\sqrt{3\hbar \lambda ^3\gamma _e/8\pi ^2}$. Furthermore, $\mu _{31}$ is the magnetic dipole moment between the states $\left| 1\right\rangle$ and $\left| 3\right\rangle$ and is related to the atomic decay as $\mu _{31}=\sqrt{3c^2\hbar \lambda ^3\gamma _b/8\pi ^2}$. To calculate the coherence for electric and magnetic probe fields, the following density matrix equation is used:

$$\begin{aligned} \dot{\rho }=-\frac{i}{\hbar }[H_i,\rho ]-\sum \frac{1}{2}\gamma _{jk}( X^\dagger X \rho +\rho X^\dagger X-2X \rho X^\dagger ) \end{aligned}$$

(3)

where $X^\dagger$, X and $\gamma _{jk}$ represents raising (creation) operator, lowering (annihilation) operator and decays respectively. Using the density matrix equation, the following two coupled equations for electric and two coupled rate equations for magnetic coherence are calculated as:

$$\begin{aligned}{} & {} \overset{\cdot }{\overset{\sim }{\rho }}_{31}=\frac{i}{2}\Omega _m(\widetilde{\rho }_{33}-\widetilde{\rho }_{11})-\frac{i}{2}\Omega ^*_c\widetilde{\rho }_{41} -(\frac{1}{2}\gamma _b-\Delta _p)\widetilde{\rho }_{31} \end{aligned}$$

(4)

$$\begin{aligned}{} & {} \overset{\cdot }{\overset{\sim }{\rho }}_{41}=\frac{i}{2}\Omega _m\widetilde{\rho }_{43}-\frac{i}{2}\Omega _c\widetilde{\rho }_{31}-\frac{i}{2}\Omega _p\widetilde{\rho }_{21} +[i\Delta _p-( \gamma _e+\frac{1}{2}\gamma _1)]\widetilde{\rho }_{41} \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \overset{\cdot }{\overset{\sim }{\rho }}_{42}=\frac{i}{2}\Omega _p(\widetilde{\rho }_{44}-\widetilde{\rho }_{22})-\frac{i}{2}\Omega _c\widetilde{\rho }_{32} +[\Delta _p-\frac{i}{2}( \gamma _1+\gamma _e)]\widetilde{\rho }_{42} \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \overset{\cdot }{\overset{\sim }{\rho }}_{32}=(i\Delta _p-\frac{1}{2}\gamma _b)\widetilde{\rho }_{32}-\frac{i}{2}\Omega _m\widetilde{\rho }_{12} -\frac{i}{2}\Omega ^*_c\widetilde{\rho }_{42} \end{aligned}$$

(7)

Equations (4-5) are coupled equations for magnetic probe field coherence and equations (6-7) are for electric probe field coherence. Taking the electric and magnetic coupled coherence equations (4-7) in first order perturbation and superposition condition, the two coupled of electric probe and two coupled of magnetic probe coherences are solved by $Z=M^{-1}B$. For the small Rabi oscillation, the zero order terms are equal to zero, while for the weak probe electric and magnetic fields i.e $\Omega _p<<\Omega _c$ and $\Omega _m<<\Omega _c$ fields. The solution of equations (4-5) for magnetic probe coherence is written as:

$$\begin{aligned} \widetilde{\rho }_{31}^{(1)}=\frac{\Omega _p}{4W_1}\frac{\Omega ^*_c}{(\Delta _p-i(\gamma _1+\gamma _e)/2)}\widetilde{\rho }_{21}^{(0)}\ +\frac{\Omega _m}{2W_1}\widetilde{\rho }_{11}^{(0)} \end{aligned}$$

(8)

The solution of equations (6-7) for electric probe coherence is written as:

$$\begin{aligned} \widetilde{\rho }_{42}^{(1)}&=\frac{\Omega _m}{4W_2}\frac{\Omega _c}{(\Delta _p-i\gamma _b/2)}\widetilde{\rho }_{12}^{(0)}\ +\frac{\Omega _p}{2W_2}\widetilde{\rho }_{22}^{(0)} \end{aligned}$$

(9)

where

$$\begin{aligned} W_1= & {} \Delta _p-\frac{|\Omega _c|^2}{4(\Delta _p-i\frac{(\gamma _1+\gamma _e)}{2})}-\frac{i\gamma _b}{2} \end{aligned}$$

(10)

and

$$\begin{aligned} W_2= & {} \Delta _p-\frac{|\Omega _c|^2}{4(\Delta _p-i\frac{\gamma _b}{2})}-\frac{i(\gamma _e+\gamma _1)}{2} \end{aligned}$$

(11)

The polarization is calculated from electric probe coherence as:

$$\begin{aligned} P=N\varrho _{24}\widetilde{\rho }^{(1)}_{42} \end{aligned}$$

(12)

While the magnetization is calculated from magnetic coherence as:

$$\begin{aligned} M=N\mu _{13}\widetilde{\rho }^{(1)}_{31} \end{aligned}$$

(13)

Putting the values of $\widetilde{\rho }^{(1)}_{31}$ and $\widetilde{\rho }^{(1)}_{42}$ from the equations (8-9) in equations (12-13), we have

$$\begin{aligned} P=\alpha _{EE}E_0+\alpha _{EB}B_0 \end{aligned}$$

(14)

$$\begin{aligned} M=\alpha _{BE}E_0+\alpha _{BB}B_0 \end{aligned}$$

(15)

where $B_0=\mu _0(M+H)$

$$\begin{aligned} \alpha _{BB}=\frac{N\mu ^2_{31}\widetilde{\rho }^{(0)}_{11}}{\hbar W_1}, \alpha _{EE}=\frac{N\varrho ^2_{42}\widetilde{\rho }^{(0)}_{22}}{\hbar W_2} \end{aligned}$$

(16)

$$\begin{aligned} \alpha _{BE}=\frac{N\mu _{31}\varrho _{42}|\Omega _c|e^{i(\varphi _m-\varphi _e)}\rho ^{(0)}_{21}e^{-i(\varphi _s+\varphi _c)}}{2\hbar W_1[\Delta _p-\frac{i(\gamma _1+\gamma _e)}{2}]} \end{aligned}$$

(17)

$$\begin{aligned} \alpha _{EB}=\frac{N\mu _{31}\varrho _{42}|\Omega _c|e^{i(\varphi _e-\varphi _m)}\rho ^{(0)}_{21}e^{i(\varphi _s+\varphi _c)}}{2\hbar W_1[\Delta _p-\frac{i(\gamma _1+\gamma _e)}{2}]} \end{aligned}$$

(18)

The electric and magnetic polarizations in the form of chiral coefficients are given by the equations (14-15), while electric and magnetic susceptibilities are defined as $P_e=\epsilon _0\chi ^{(1)}_eE+\frac{\xi _{EH}H}{c}$ and $M=\chi ^{(1)}_mH+\frac{\xi _{HE}H}{\mu _0c}E$. Comparing the above two polarization and magnetization, we obtain the first order complex electric and magnetic susceptibilities as well as chiral coefficients for this atomic system, which are written below: The complex electric susceptibility of the first order for the proposed system is:

$$\begin{aligned} \chi _e=\frac{\alpha _{EE}+\mu _0(\alpha _{EB}\alpha _{BE}-\alpha _{BB}\alpha _{EE})}{\epsilon _0(1-\mu _0\alpha _{BB})} \end{aligned}$$

(19)

Similarly, the complex magnetic susceptibility of the first order for the proposed system is:

$$\begin{aligned} \chi _m=\frac{\mu _0\beta _{BB}}{1-\mu _0\beta _{BB}} \end{aligned}$$

(20)

While the chiral coefficients for the proposed system are:

$$\begin{aligned} \xi _{HE}=-\frac{c\mu _0\beta _{BE}}{1-\mu _0\beta _{BB}}, \xi _{EH}=\frac{c\mu _0\beta _{EB}}{1-\mu _0\beta _{BB}} \end{aligned}$$

(21)

The chiral medium makes the incident ray birefringent. The divided ray has refractive indices $n^{(-)}_r$ and $n^{(+)}_r$ corresponding to left and right circular polarization (LCP and RCP) beams. The expression for complex refractive index is written as:

$$\begin{aligned} n^{(\pm )}_r= & {} \sqrt{(1+\chi _m)(1+\chi _e)-\frac{(\xi _{HE}+\xi _{EH})^2}{4}}\pm i\frac{(\xi _{EH}-\xi _{HE})}{2} \end{aligned}$$

(22)

The real and imaginary parts of complex refractive index are related to velocity and amplitude attenuation respectively. The refractive index under compton scattering effect is written bellow:

$$\begin{aligned} n^{(\pm )c}_{r}=\frac{n^{(\pm )}_r}{1+\frac{\hbar \omega n^{(\pm )}_r}{m_0c}(1-\cos (\theta _c))} \end{aligned}$$

(23)

Figure 2

Reflection, transmission coefficients and corresponding GH-shifts in reflection transmission for left and right circularly polarized beams under compton scattering effect vs control field Rabi frequency $\Omega _c/\gamma$ and compton angle $\theta _c$.

Figure 3

Reflection, transmission coefficients and corresponding GH-shifts in reflection transmission for left and right circularly polarized beams under compton scattering effect vs probe field detuning $\Delta _p/\gamma$ and angle of incidence $\theta _i$.

Figure 4

Reflection, transmission coefficients and corresponding GH-shifts in reflection transmission for left and right circularly polarized beams under compton scattering effect vs phase of control field $\varphi _c$ and phase of superposition state $\varphi _s$.

Figure 5

Reflection, transmission coefficients and corresponding GH-shifts in reflection transmission for left and right circularly polarized beams under compton scattering effect vs phase of electric field $\varphi _e$ and phase of magnetic field $\varphi _m$.

The probe beam propagating toward the medium can be reflected back and can be transmitted through the birefringent medium. The reflection and transmission birefringent beams in a more explicit form are written as:

$$\begin{aligned} R^{(\pm )}=\frac{\cos \alpha ^{\pm }(k^2_0-k^2_1)k_1\psi ^{\pm }\sin 2\alpha _1+Q_3\sin \alpha ^{\pm }}{k_1\psi ^{\pm }Q_1\cos \alpha ^{\pm }+Q_2\sin \alpha ^{\pm }} \end{aligned}$$

(24)

$$\begin{aligned} T^{(\pm )}=\frac{2ik_0\psi ^{\pm }k^2_1}{k_1\psi ^{\pm }Q_1\cos \alpha ^{\pm }+Q_2\sin \alpha ^{\pm }} \end{aligned}$$

(25)

The GH shift in the reflection and transmission probe are written in explicit form as:

$$\begin{aligned} S^{\pm }_r= & {} -\frac{\lambda }{2\pi |R^{\pm }|^2}[Re(R^{\pm })\frac{\partial }{\partial \theta _i}(Im(R^{\pm }))-Im(R^{\pm }) \frac{\partial }{\partial \theta _i}Re(R^{\pm })] \end{aligned}$$

(26)

$$\begin{aligned} S^{\pm }_t= & {} -\frac{\lambda }{2\pi |T^{\pm }|^2}[Re(T^{\pm })\frac{\partial }{\partial \theta _i}(Im(T^{\pm }))-Im(T^{\pm }) \frac{\partial }{\partial \theta _i}Re(T^{\pm })] \end{aligned}$$

(27)

Results and discussions

The results are presented for the reflection/transmission coefficients of left circularly polarized (LCP) and right circularly polarized (RCP) beams and their corresponding GH-shifts in reflection/transmission. All the parameters are taken in atomic units. The decay rate “$\gamma$” is taken as 13.6 GHz and is assumed as a unit ($\gamma$= 1GHz) to compare the other frequency parameters with it. The other parameters are $\epsilon _1=1.22$ (dielectric constant of Balsa wood), $d_{1}=1.5\lambda$, $d_{2}=10.5\lambda$ and $L=2d_1+d_2$, $k_{0,1}=\sqrt{\epsilon _{0,1}-\sin ^2\theta _i}$, $\alpha ^{\pm }=\frac{2\pi }{\lambda _p}d_{2}k_2^{\pm }$ and $\alpha _{1}=\frac{2\pi }{\lambda _p}d_{1}\sqrt{\epsilon _{1}-\sin ^2\theta _i}$ and $\lambda =5869A^{0}$.

In Fig. 2, the plots are traced for reflection, transmission coefficients and corresponding GH-shifts in reflection transmission of RCP and LCP beams under compton scattering versus control field Rabi frequency $\Omega _c/\gamma$ and compton angle $\theta _c$ for the fixed values of phase of electric field $\varphi _{e}=\pi /3$, phase of magnetic field $\varphi _m=\pi /4$, phase of control field $\varphi _c= \pi /3$, phase of superposition states $\varphi _s=\pi /3$, angle of incidence $\theta _i=\pi /4$ and probe field detunung $\Delta _p= 2\gamma$. The coefficients of LCP and RCP beams increase in the reflection while decrease in the transmission beams. The RCP and LCP beams of the reflection and transmission obey the normalization condition $|R^{(+)}|+|T^{(+)}|+|A^{(+)}|=1$ and $|R^{(-)}|+|T^{(-)}|+|A^{(-)}|=1$. The absorption in RCP and LCP beams is controlled to zero such as $|A^{(+,-)}|\simeq 0$. The reflection and transmission of LCP and RCP beams are the exponential inreasing/decreasing functions of control field Rabi frequency $\Omega _c/\gamma$ and fluctuated slowly with compton angle $\theta _c$. Due to compton scattering, the multiple scattering occur. At different angles of scattering from the medium, the reflection of light enhances from the medium but not due to the incident point. Therefore, the GH-shift enhances in reflection with compton scattering angle. Also with compton effect, the multiple scattered light emerge at different points from the medium during transmission, so the GH-shift also enhances in transmission. At the point ($\Omega _c$, $\theta _c$)= ($2.7\gamma$, 0.4 rad), the values of the RCP coefficient in the reflection and transmission are $|R^{+}|\simeq 0.9$ and $|T^{+}|\simeq 0.1$ while the values of LCP coefficient in the reflection and transmission are $|R^{-}|\simeq 0.8$ and $|T^{-}|\simeq 0.2$ as shown in Fig. 2a,b. The GH-shifs in reflection and transmission of LCP and RCP beams are also oscillating function of control field Rabi frequency and compton scattering angle. Positive and negative giant GH-shifts for LCP and RCP beams are reported in reflection and transmission. The maximum positive value of GH-shift in reflection and transmission of RCP beam is calculated to $40\lambda$ and LCP beams is calculated to $30\lambda$. Similarly, the maximum negative value of GH-shifs for the RCP beam in reflection and transmission is calculated to $-20\lambda$ while for the LCP beam it is calculated to $-10\lambda$ as shown in Fig. 2c,d. The oscillation of GH-shifts in reflection and transmission is vary large with Rabi frequency as compared to fluctuation with compton scattering angle so it is not visible with compton scattering. This fluctuation of GH-shifts in reflection and transmission with compton scattering angle is shown in 2D graph bellow Fig. 2c,d of in Fig. 2c–I,d–I.

In Fig. 3, the plots are traced for reflection, transmission coefficients and corresponding GH-shifts in reflection transmission of RCP and LCP beams versus probe field detuning $\Delta _p/\gamma$ and angle of incidence $\theta _i$ under compton scattering effect for the fixed values of the phases of coupled driving fields as taken in Fig. 2, compton scattering angle $\theta _c=\pi /4$ and $\Omega _c= 2\gamma$. The LCP and RCP beams in reflection and transmission fluctuated smoothly with the angle of incidence, while increases and decreases exponentially with probe field detuning and obey the normalization condition. The swamping occurs for RCP and LCP beams at points ($\Delta _p$, $\theta _i$)= ($1\gamma$, $-1.5\gamma$, 1.5rad) in reflection while at points ($\Delta _p$, $\theta _i$)= ($2\gamma$, $-2\gamma$, 1.5rad) in transmission. At a point ($\Delta _p=2.5\gamma$, $\theta _i=0.2 rad$), the values of reflection and transmission and absorption coefficients of RCP beam are $|R^{+}|\simeq 0.6$, $|T^{+}|\simeq 0.4$ and $|A^{(+)}|\simeq 0$. Similarly at a point ($\Delta _p=0\gamma$, $\theta _i=0.2 rad$), the values of reflection, transmission and absorption coefficients of LCP beam are $|R^{-}|\simeq 0.6$, $|T^{-}|\simeq 0.4$ and $|A^{(+)}|\simeq 0$ and obey the normalization condition as shown in Fig. 3a,b. Positive giant GH-shifts for RCP and LCP beams in the reflection and transmission are reported. The maximum values of positive GH-shifts for RCP and LCP beams are calculated to $80\lambda$ in reflection while to $90\lambda$ in transmission. Also the maximum value of negative GH-shift for RCP and LCP beams in the reflection and transmission are calculated to $-50 \lambda$ as shown in the Fig. 3c,d.

In Fig. 4, the plots are traced for reflection, transmission coefficients and corresponding GH-shifts in reflection transmission of RCP and LCP beams versus phase of control field $\varphi _c$ and phase of superposition states $\varphi _s$ under compton scattering effect for fixed values of phase of electric field $\varphi _{e}=\pi /3$, phase of magnetic field $\varphi _{m}=\pi /4$, angle of incidence $\theta _i=\pi /3$, compton scattering angle $\theta _c=\pi /4$, probe field detuning $\Delta _p= 0\gamma$ and control field Rubi frequency $\Omega _c= 2\gamma$. The RCP and LCP beams in the reflection and transmission are rapidly oscillating functions of phase of control field $\varphi _c$ and weakly oscillating function of phase of superposition states $\varphi _s$. The maximum splitting for RCP and LCP beams is reported at $\varphi _c$= ( 1 rad, 3.5 rad ) and $\varphi _s$=0.5rad in the reflection and transmission. The RCP and LCP beams in the reflection and transmission obey the normalization condition while taking the absorption coefficient $|A^{(+,-)}|\simeq 0$ as shown in Fig. 4a,b. Again positive and negative giant GH-shifts for RCP and LCP beams are reported in the reflection and transmission. The maximum values of positive and negative GH-shifts for the RCP beam in the reflection and transmission is calculated to $\pm 40\lambda$ while for the LCP beam in the reflection and transmission is calculated to $\pm 20\lambda$ as shown in Fig. 4c,d.

In Fig. 5, the plots are traced for reflection, transmission coefficients and corresponding GH-shifts in reflection transmission of RCP and LCP beams under compton scattering versus phases of electric field $\varphi _e$ and magnetic field $\varphi _m$ for fixed values of phase of control field $\varphi _c=\pi /3$, phase of superposition states $\varphi _s= \pi /4$, compton scattering angle $\theta _c =\pi /4$, angle of incidence $\theta _i=\pi /3$, probe field detuning $\Delta _p= 0\gamma$ and control field Rubi frequency $\Omega _c= 2\gamma$. The RCP and LCP beams in reflection and transmission are strongly oscillating functions of phases $\varphi _e$ and $\varphi _m$. The RCP and LCP beams in the reflection and transmission are oscillated between the values zero and 0.6 and obey the normalization condition as shown in Fig. 5a,b. The maximum value of positive and negative GH- shifts is measured for the RCP beam to $\pm 30\lambda$ while $\pm 20\lambda$ for the LCP beam as shown in Fig. 5c,d.

Conclusion

A four level chiral medium was considered to analyze and investigate theoretically the reflection and transmission of right and left circularly polarized beams under the effect of compton scattering. The Goos Hanchens shifts in reflection and transmission were also studied with compton scattering effect. Density matrix formalism was used for calculation of electric and magnetic probe fields coherence. The polarization and magnetization were calculated from probes coherence terms in the chiral medium. Electric and magnetic susceptibility and chiral coefficients were related with polarization and magnetization. The refractive indices of RCP and LCP beams under compton scattering effect was modified from the electric and magnetic susceptibility, chiral coefficients, mass and charge of electron as well as compton scattering angle. The reflection and transmission and correspond GH-shifts in the reflection and transmission beams were plotted against controls fields Rabi frequencies, deriving fields detuning, incident and compton scattering angles. The GH-shifts in reflection and transmission are strong variation functions of controls fields Rabi frequencies, deriving fields detuning, incident and compton scattering angles. Positive and negative GH-shifts were investigated for RCP and LCP beams in the chiral medium under compton scattering effect along with coupled deriving fields parameters. Gaints positive and negative GH-shifts in reflection and transmission for RCP and LCP beams in the chiral medium under compton scattering effect were reported. The maximum average value of positive and negative GH-shifts in reflection and transmission beams were measured to $\pm 70\lambda$ in the presence of compton scattering. The modified Giant GH-shifts in reflection and transmission of RCP and LCP beams have useful applications in cloaking devices, optical heterodyne sensors technology, nano-optoelectronic sensors, image coding, optical switches, optical storage devices and telecom industry.