
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

66206
10.1038/s41598-024-66206-7
Article
Coherent manipulation of giant birefringent Goos–Hänchen shifts by compton scattering using chiral atomic medium
Haq Zia Ul 1
Ahmad Iftikhar 1
Bacha Bakht Amin 1
Akgül Ali aliakgul00727@gmail.com

23
Hassani Murad Khan mhassani@gu.edu.af

4
1 https://ror.org/012xdha97 grid.440567.4 0000 0004 0607 0608 Department of Physics, University of Malakand, Chakdara Dir(L), Pakistan
2 https://ror.org/00hqkan37 grid.411323.6 0000 0001 2324 5973 Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut, 11022801 Lebanon
3 https://ror.org/05ptwtz25 grid.449212.8 0000 0004 0399 6093 Department of Mathematics, Art and Science Faculty, Siirt University, 56100 Siirt, Turkey
4 https://ror.org/0075h8406 grid.448871.6 0000 0004 7386 4766 Department of mathematics, Ghazni University, Ghazni, Afghanistan
6 9 2024
6 9 2024
2024
14 2082114 2 2024
28 6 2024
© The Author(s) 2024
2024
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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(+,-)|≃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.

Subject terms

Environmental sciences
Materials science
Mathematics and computing
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pmcIntroduction

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 optics2–4, astronomy5,6, nanotechnology7, condensed matter physics8,9, quantum mechanics10, nuclear physics11,12, plasma physics13,14. 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 crystals20,21, 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 d2. The states 2 and 4 are coupled by electric probe field Ep of Rabi frequency Ωp. The states 1 and 3 are coupled by magnetic probe field Bm of Rabi frequency Ωm. The control field Ec having Rabi frequency Ωc couples the states 3 and 4. The states 1 and 2 are superposition states. The prepared superposition state of 1 and 2 is written as Φ=x2+1-xeiφs1. The cavity of length d2 is enclosed in another medium of thin sheet of balsa wood of small length d1 and dielectric constant ϵ1. The self Hamiltonian of the proposed chiral atomic system is written bellow:1 H0=∑k=14ħωkkk

The interaction picture Hamiltonian for the chiral atomic system is written as:2 HI=∑k=14ħkk-ħ2[Ωme-iωmt31+Ωpe-iωpt42+Ωce-iωct43+H.C]

where the Rabi frequencies of electric and magnetic probe fields are Ωp=ϱ42eiφeE0/ħ and Ωm=μ31eiφmB0/ħ. Here ϱ42 is the electric dipole moment between the states 2 and 4 and is related to the atomic decay as ϱ42=3ħλ3γe/8π2. Furthermore, μ31 is the magnetic dipole moment between the states 1 and 3 and is related to the atomic decay as μ31=3c2ħλ3γb/8π2. To calculate the coherence for electric and magnetic probe fields, the following density matrix equation is used:3 ρ˙=-iħ[Hi,ρ]-∑12γjk(X†Xρ+ρX†X-2XρX†)

where X†, X and γjk 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:4 ρ∼·31=i2Ωm(ρ~33-ρ~11)-i2Ωc∗ρ~41-(12γb-Δp)ρ~31

5 ρ∼·41=i2Ωmρ~43-i2Ωcρ~31-i2Ωpρ~21+[iΔp-(γe+12γ1)]ρ~41

6 ρ∼·42=i2Ωp(ρ~44-ρ~22)-i2Ωcρ~32+[Δp-i2(γ1+γe)]ρ~42

7 ρ∼·32=(iΔp-12γb)ρ~32-i2Ωmρ~12-i2Ωc∗ρ~42

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-1B. For the small Rabi oscillation, the zero order terms are equal to zero, while for the weak probe electric and magnetic fields i.e Ωp<<Ωc and Ωm<<Ωc fields. The solution of equations (4-5) for magnetic probe coherence is written as:8 ρ~31(1)=Ωp4W1Ωc∗(Δp-i(γ1+γe)/2)ρ~21(0)+Ωm2W1ρ~11(0)

The solution of equations (6-7) for electric probe coherence is written as:9 ρ~42(1)=Ωm4W2Ωc(Δp-iγb/2)ρ~12(0)+Ωp2W2ρ~22(0)

where10 W1=Δp-|Ωc|24(Δp-i(γ1+γe)2)-iγb2

and11 W2=Δp-|Ωc|24(Δp-iγb2)-i(γe+γ1)2

The polarization is calculated from electric probe coherence as:12 P=Nϱ24ρ~42(1)

While the magnetization is calculated from magnetic coherence as:13 M=Nμ13ρ~31(1)

Putting the values of ρ~31(1) and ρ~42(1) from the equations (8-9) in equations (12-13), we have14 P=αEEE0+αEBB0

15 M=αBEE0+αBBB0

where B0=μ0(M+H)16 αBB=Nμ312ρ~11(0)ħW1,αEE=Nϱ422ρ~22(0)ħW2

17 αBE=Nμ31ϱ42|Ωc|ei(φm-φe)ρ21(0)e-i(φs+φc)2ħW1[Δp-i(γ1+γe)2]

18 αEB=Nμ31ϱ42|Ωc|ei(φe-φm)ρ21(0)ei(φs+φc)2ħW1[Δp-i(γ1+γe)2]

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 Pe=ϵ0χe(1)E+ξEHHc and M=χm(1)H+ξHEHμ0cE. 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:19 χe=αEE+μ0(αEBαBE-αBBαEE)ϵ0(1-μ0αBB)

Similarly, the complex magnetic susceptibility of the first order for the proposed system is:20 χm=μ0βBB1-μ0βBB

While the chiral coefficients for the proposed system are:21 ξHE=-cμ0βBE1-μ0βBB,ξEH=cμ0βEB1-μ0βBB

The chiral medium makes the incident ray birefringent. The divided ray has refractive indices nr(-) and nr(+) corresponding to left and right circular polarization (LCP and RCP) beams. The expression for complex refractive index is written as:22 nr(±)=(1+χm)(1+χe)-(ξHE+ξEH)24±i(ξEH-ξHE)2

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:23 nr(±)c=nr(±)1+ħωnr(±)m0c(1-cos(θc))

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 Ωc/γ and compton angle θc.

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 Δp/γ and angle of incidence θi.

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 φc and phase of superposition state φs.

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 φe and phase of magnetic field φm.

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:24 R(±)=cosα±(k02-k12)k1ψ±sin2α1+Q3sinα±k1ψ±Q1cosα±+Q2sinα±

25 T(±)=2ik0ψ±k12k1ψ±Q1cosα±+Q2sinα±

The GH shift in the reflection and transmission probe are written in explicit form as:26 Sr±=-λ2π|R±|2[Re(R±)∂∂θi(Im(R±))-Im(R±)∂∂θiRe(R±)]

27 St±=-λ2π|T±|2[Re(T±)∂∂θi(Im(T±))-Im(T±)∂∂θiRe(T±)]

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 “γ” is taken as 13.6 GHz and is assumed as a unit (γ= 1GHz) to compare the other frequency parameters with it. The other parameters are ϵ1=1.22 (dielectric constant of Balsa wood), d1=1.5λ, d2=10.5λ and L=2d1+d2, k0,1=ϵ0,1-sin2θi, α±=2πλpd2k2± and α1=2πλpd1ϵ1-sin2θi and λ=5869A0.

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 Ωc/γ and compton angle θc for the fixed values of phase of electric field φe=π/3, phase of magnetic field φm=π/4, phase of control field φc=π/3, phase of superposition states φs=π/3, angle of incidence θi=π/4 and probe field detunung Δp=2γ. 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(+,-)|≃0. The reflection and transmission of LCP and RCP beams are the exponential inreasing/decreasing functions of control field Rabi frequency Ωc/γ and fluctuated slowly with compton angle θc. 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 (Ωc, θc)= (2.7γ, 0.4 rad), the values of the RCP coefficient in the reflection and transmission are |R+|≃0.9 and |T+|≃0.1 while the values of LCP coefficient in the reflection and transmission are |R-|≃0.8 and |T-|≃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 40λ and LCP beams is calculated to 30λ. Similarly, the maximum negative value of GH-shifs for the RCP beam in reflection and transmission is calculated to -20λ while for the LCP beam it is calculated to -10λ 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 Δp/γ and angle of incidence θi under compton scattering effect for the fixed values of the phases of coupled driving fields as taken in Fig. 2, compton scattering angle θc=π/4 and Ωc=2γ. 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 (Δp, θi)= (1γ, -1.5γ, 1.5rad) in reflection while at points (Δp, θi)= (2γ, -2γ, 1.5rad) in transmission. At a point (Δp=2.5γ, θi=0.2rad), the values of reflection and transmission and absorption coefficients of RCP beam are |R+|≃0.6, |T+|≃0.4 and |A(+)|≃0. Similarly at a point (Δp=0γ, θi=0.2rad), the values of reflection, transmission and absorption coefficients of LCP beam are |R-|≃0.6, |T-|≃0.4 and |A(+)|≃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 80λ in reflection while to 90λ in transmission. Also the maximum value of negative GH-shift for RCP and LCP beams in the reflection and transmission are calculated to -50λ 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 φc and phase of superposition states φs under compton scattering effect for fixed values of phase of electric field φe=π/3, phase of magnetic field φm=π/4, angle of incidence θi=π/3, compton scattering angle θc=π/4, probe field detuning Δp=0γ and control field Rubi frequency Ωc=2γ. The RCP and LCP beams in the reflection and transmission are rapidly oscillating functions of phase of control field φc and weakly oscillating function of phase of superposition states φs. The maximum splitting for RCP and LCP beams is reported at φc= ( 1 rad, 3.5 rad ) and φs=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(+,-)|≃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 ±40λ while for the LCP beam in the reflection and transmission is calculated to ±20λ 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 φe and magnetic field φm for fixed values of phase of control field φc=π/3, phase of superposition states φs=π/4, compton scattering angle θc=π/4, angle of incidence θi=π/3, probe field detuning Δp=0γ and control field Rubi frequency Ωc=2γ. The RCP and LCP beams in reflection and transmission are strongly oscillating functions of phases φe and φm. 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 ±30λ while ±20λ 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 ±70λ 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.

Appendix

ψ±=(nr(±)c)2-sin2θiQ1=2ik0k1cos2α1+(k02+k12)sin2α1Q2=k12(k02+(ψ±)2)cos2α1-(k14+k02(ψ±)2)sin2α1-ik0k1(k12+(ψ±)2)sin2α1Q3=k12(k02-(ψ±)2)cos2α1+(k14-k02(ψ±)2)sin2α1

Author contributions

Zia Ul Haq (ziaulhaq_1982@yahoo.com): Formal analysis, Investigation, Initial draft, software, witting and conceptualization. Bakht Amin Bacha (aminoptics@gmail.com): Supervision, Review, software. Ali Akgül (aliakgul00727@gmail.com): Review, Validation, Edition, investigation and project administration. Iftikhar Ahmad : Investigation,conceptualization and supervision of the project. Murad khan Hassani (mhassani@gu.edu.af): Review, investigation, supervision and project administration. This work was carried out in the centre for computational material sciences, University of Malakand, Chakdara KPK Pakistan.

Data availibility

The corresponding author will provide the available data on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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