High Sensitive Chiral Molecule Detector Based on The Amplified Lateral Shift in Kretschmann Configuration Involving Chiral TDBCs


 We investigate a high sensitive chiral molecule detector based on Goos-Hanchen shift (S) in Kretschmann configuration involving chiral TDBCs. Fresnel equations and the stationary phase method are employed to calculate S. Due to the interaction between surface plasmon polaritons and chiral TDBCs, S with chiral TDBCs are amplified at near the resonant wavelengths of chiral TDBCs. Our calculation results show that although the difference between the resonant wavelengths of left and right TDBCs is 4.5nm, the difference of S with chiral TDBCs (ΔS) can reach to 400 times as the incident wavelength in certain conditions, which can be easily observed in experiments. There is an optimal thickness of the metal film to realize the largest difference of S between Kretschmann configurations with left TDBCs and right TDBCs. We also find that the positions of the largest S for the structures with left TDBCs and right TDBCs do not overlap. Furthermore, we discuss the oscillator strength f, which is mainly determined by TDBC concentration. We find that our proposed detector is quite sensitive with f. By changing f from 0.008 to 0.014 with the step of 0.002, the change of ΔS is no less than 5 times of the incident wavelength (2.9μm). Our proposed structure is very sensitive and has potential applications in experiments.

A chiral object does not have mirror planes or inversion symmetry [17]. Each of such non-superimposable mirror images is known as an enantiomer, and can be divided into left-and right-handedness, respectively. Chiral objects generally display similar physical properties to distinguish them [18]. SPPs and localized surface plasmons (LSPs) can enhance the light field in the near field . With the help of the enhancement of electric field, the chiral optical response can be significantly amplified in chiral systems in form of the circular dichroism (CD) spectra enhancement [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. If we use GH shift to replace the optical response such as the absorption and the extinction, the difference between the left molecules and the right molecules can be easily measured in experiment.
In this paper, we investigate a chiral molecule detector based on the lateral shift splitting/GH shift splitting in Kretschmann configuration involving chiral TDBCs by using classical methods. Kretschmann configuration is made up of a gold film sandwiched between the dielectric medium and left TDBCs or right TDBCs. Incident light is injected from the dielectric medium with an incident angle. Fresnel equations and the stationary phase method are employed to solve our proposed structure. The parameters of chiral TDBCs and Au film are changed in order to show the impact on the sensitivity of the molecule detector.

Calculation Model and Analysis
The sketch of our calculation model is shown in Fig. 1. An Au film with the thickness of d is sandwiched between the prism and chiral TDBCs. Incident chiral beam is injected from the dielectric medium with an incident angle θ. The dielectric constant function (ϵ 2 ) of Au is obtained in the Ref. [40]. The index of the dielectric medium ( √ ϵ 1 ) is chose as √ 3.6. Chiral TDBCs are modeled by a Lorentzian oscillator [29,30]: where, i means Left (L) or Right (R). ϵ ∞ = n 2 bg , and n bg = 1.33 is the index of the background. ω i is the resonant frequency of of TDBCs, and γ i is the damping constant of TDBC. f i is the oscillator strength, which depends on the molecular concentration. According to the experiment measurement of TDBC, we obtain that [29,30]. Incident chiral beam is composed of TE-polarized beam (E s ) and TM-polarized beam (E p ) as E s + E p e i±π/2 . In Kretschmann configurations, only TM-polarized beam can be used to generate SPPs. Here, we only consider TM-polarized beam is injected in our proposed system.
GH shift S is calculated by the classical method. According to the Fresnel equations of this kind of Kretschmann configurations, the reflection coefficients of TM-polarized are written as [7,12]: where the order of the layers 1, 2 and 3, counted from bottom to top and and, According to the stationary phase method, the lateral shift of the reflected beam S is given by [2,7]: where, ϕ r = ℑ(ln r) is the phase of r. Before we discuss the relation between the GH shift S and the incident wavelength λ, it should be mentioned that the incident angle θ is equal to the surface plasmon resonant angle θ SP P s [2,7]. Here, we adopt θ SP P s as the incident angle θ for each incident wavelength λ written as [41]: In the following, we calculate Kretschmann configurations with only LTDBCs and Right TDBCs to do the research on normalized GH shift S/λ and the difference of normalized GH shifts between Kretschmann configurations with LTDBCs and RTDBCs ∆S/λ.

Results and Discussion
First, we choose f i =0.01 and d=20nm to calculate the reflection spectrum R and normalized GH shifts S/λ with both LTDBCs and RTDBCs, respectively. The incident angle is considered as θ SP P s . The results are shown in Fig. 2. Furthermore, the difference of normalized GH shifts ∆S/λ is also plotted in Fig. 2.
Due to the interactions between SPPs and chiral TDBCs, normalized GH shifts S/λ are amplified near the resonant frequencies of chiral TDBCs as shown in Fig. 2(a). They are 24 times of LTDBCs and 10 times of RTDBCs as the incident wavelength. Due to the difference resonant wavelength, the positions of the maximum S for LTDBCs (λ=577nm) and RTDBCs (λ=582nm) do not overlap. As Fig. 2(b) shown, the difference of normalized GH shifts ∆S/λ can reach to 24 at λ=577nm. It seems that the reflected beam with LTDBCs moves a lot, but the one with RTDBCs hardly moves at λ=577nm. This phenomenon can be easily measured in experiment, and LTDBCs and RTDBCs are easily distinguished. Also, the working wavelength can be taken as λ=582nm.
Then, we decrease the thickness d of Au film to show how ∆S/λ changes. The thickness d is changed from 20nm to 50nm with the step of 10nm. The results are shown in Fig. 3. For each thickness of the proposed system, S i /λ is also calculated and shown in the inserted picture.  As shown in Fig. 3, the working wavelength of our proposed structure is changed by the thickness of Au film d. At the same time, the value of S i /λ decreases. The dissipation of Au film increases sharply with the thickness of Au film increasing, while the decoupling conversion rate decreases. It means that there are more SPPs joining into the interaction between chiral TDBCs and Au film with the increase of Au film thickness. According to the analysis above, we can change the working wavelength by tuning the thickness of Au film. With the increase of d, the working wavelength has a red shift. Although there are more SPPs interact with chiral TDBCs, the intensity of ∆S/λ is reduced by the large dissipation. Comparing the intensity of ∆S/λ, we also find that with the changes of d, there is an optimal thickness to make ∆S/λ largest. We pick the maximum values and the positions of ∆S/λ for both LTDBCs and RTDBCs by changing d with the step of 0.5nm. The results are shown in Fig. 4.
As shown in Fig. 4(a), there is a little difference between the optimal thicknesses for LTDBCs and RTDBCs. The largest ∆S/λ is larger than 100, which Finally, we show the impact of the oscillator strength f on ∆S/λ. We change a small step of 0.002 to show the sensitivity of our proposed structure. f is changed from 0.08 to 0.014, and ∆S/λ versus λ are shown in Fig. 5. S i /λ is also inserted in the figures for each f .
As Fig. 5 shown, the peaks and the dips of ∆S/λ have a slight shift (within 5nm) due to a quite small change of f . The values of the peaks and the dips in ∆S/λ changes a lot. This is because that f has great influences on ϵ 3 and θ SP P s . Hence, the value of ∆S/λ changes a lot. We list the values of the peaks and the dips in Tab. 1. We find that although f changes a little 0.002, the difference of GH shifts between peak values or dip values is no less than 5 times of the incident wavelength about 2.9µm. This difference is easy to observe in experiment.

Summary
In summary, we investigate a high sensitive chiral molecule detector based on GH shift in Kretschmann configuration involving chiral TDBCs. Fresnel equations and the stationary phase method are employed to calculate the lateral shift/GH shift. Due to the interaction between SPPs and the chiral TDBCs, the GH shifts in our proposed structures with two kind of chiral TDBCs are amplified at different wavelength. GH shift can reach to 24 times as the incident wavelength, and be easily observed in experiment. The working wavelength of the detector can be tuned by the thickness of Au film. We also find there is an optimal thickness to make ∆S largest, which is over 100 as the incident wavelength and can be easily observed in experiments. Furthermore, we discuss the oscillator strength f and find that our proposed detector is quite sensitive with f . By changing 0.002 for f , the change of ∆S is no less than 5 times of the incident wavelength, which is about 2.9µm at our working wavelength. Our proposed structure is very sensitive and has potential applications in experiments.

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Authors Contributions
Gang Song and Song Wang provide the idea. All the authors involve to prepare the manuscript. Song Wang and Gang Song have the same contributions to this work.

Funding
This research was supported by the Science and Technology Nova Plan of Beijing City (No. Z201100006820122), the Fundamental Research Funds for the Central Universities.