1. Introduction

Goos-Hänchen (GH) shift refers to the transverse shift between the actual reflected light beam and the geometrical optical reflected light beam, which was first discovered by Goos and Hänchen in 1947 [1]. In 1948, Artmann decomposed the incident beam into a series of monochromatic plane waves and proposed the stationary-phase method to calculate GH shifts [2]. After the stationary-phase method was proposed, many researchers applied this method to calculate GH shifts of Gaussian beams, because the Gaussian beam can be approximatively regarded as a superposition of a series of plane waves with different frequencies, amplitudes and weights [3–5]. In addition to the stationary-phase method, several methods can also be used to calculate the GH shift, such as the angular spectrum method [6–8]. The emergence of the GH shift has attracted lots of attention, and the GH shift phenomena in different materials have been studied in depth, such as the PT symmetric system [9,10], photonic crystals [11,12], Weyl semimetals [13,14] and metasurfaces [15–18]. According to the interesting properties of GH shifts in different materials, researchers have proposed applications based on GH shifts in many aspects, such as the biosensor [19], the temperature sensor [20] and the digital optical switch [21], which have important significance in the fields of optical sensing and optical communication.

In recent years, graphene, as a two-dimensional honeycomb structure whose electronic and optical properties are special and novel, has shown great significance in many areas [22–24]. The Fermi energy of the graphene is adjustable, many researchers have designed theoretical and experimental schemes to regulate the graphene conductivity [25–28], such as Ojeda-Aristizabal et al. fabricate a vertical thin-film barristor device, which can modulate the Fermi energy of graphene, tuned with an external gate voltage [28]. As a result, the graphene plays an important role in the regulation and the enhancement of GH shifts [29–32]. In addition to the graphene, the researches of GH shifts in various novel materials are also being carried out, such as epsilon-near-zero (ENZ) materials [33,34]. ENZ materials have an interesting property that the real part of their permittivity approaches to zero and the imaginary part remains a small value over certain wavelength range, and the permittivity between -1 and 1 is an important indicator of the ENZ region [35–37]. Regretful, only few researchers have studied the GH shift in the ENZ region where the change of the wavelength has a significant regulating effect on GH shifts [38]. Meanwhile, more and more artificial ENZ materials appear, one such example is black phosphorus which can achieve a polarization-dependent epsilon-near-zero behavior by tuning the charge density [39]. In addition, the indium tin oxide (ITO) is one of the artificial ENZ materials [40], which has shown potential applications in the field of the electrochemical or optical biosensors [41] and the metamaterial absorber [42]. Both of the graphene and ENZ materials have shown important prospects in the field of the GH shift researches. However, to the best of our knowledge, only few researchers have designed structures that combine graphene with ENZ materials to explore the GH shift phenomenon [43,44].

In this paper, we propose a graphene-$\mathrm {ITO/TiO_{2}/ITO}$ sandwich structure to explore the GH shifts in the ENZ region of the ITO, when a Gaussian beam is incident on the surface of the sandwich structure. Firstly, the Drude model is used to define the permittivity of the ITO, and the wavelength range whose module value of the permittivity is less than 1 is obtained, that is, the ENZ region. Then, we explore the effect of the wavelength and the Fermi energy on the sign and the magnitude of GH shifts in the ENZ region, in two cases of the zero and the non-zero conductivity of graphene. In both cases, we obtain the positive and the negative variation rules of GH shifts. Finally, according to the positive and the negative variation rules of GH shifts in two cases, we design a barcode encryption scheme based on the sign of GH shifts, which can achieve four groups of the coding state by regulating the wavelength and the Fermi energy.

2. Model and theory

We research the GH shift of a Gaussian beam impinging upon the graphene-$\mathrm {ITO/TiO_{2}/ITO}$ sandwich structure as shown in Fig. 1. $\epsilon _{air}$ and $\epsilon _{TiO_{2}}$ represent the relative permittivity of the air and the $\mathrm {TiO_{2}}$. The relative permittivity of the ITO, $\epsilon _{ITO}$ , is defined by using the Drude model: $\epsilon _{I T O}=\epsilon _{\infty }-\frac {\omega _{p}^{2}}{\omega ^{2}+i \mathit {\Gamma } \omega }$, where $\omega$, $\omega _{p}$, $\mathit {\Gamma }$ and $\epsilon _{\infty }$ refer to the angular frequency, the plasma frequency, the damping parameter and the permittivity at an infinite frequency, respectively [35]. $d_{ITO}$ and $d_{TiO_{2}}$ are the thickness of the ITO and the $\mathrm {TiO_{2}}$. Monolayer graphene is characterized according to the optical conductivity $\sigma$, which can be expressed as the following form based on the semiconductor theory [45,46]

When a Gaussian beam of light, with wavelength of $\lambda$, incident upon the structure along the axis $z_{i}$ at an angle of $\theta$. The reflection coefficients of $p$ and $s$ polarizations can be calculated using the transfer matrix method [47]. The transfer matrix of the graphene-$\mathrm {ITO/TiO_{2}/ITO}$ sandwich structure is formulated as

Here $u=(p, s)$, and the transmission matrix of the beam passing through the dielectric $n$ and the dielectric $n$+1 is given by

The reflection coefficient for both of $p$ and $s$ polarizations can be obtained from the elements of the transfer matrix, namely

Here, the $R_{u}$ and $\phi _{u}$ mean the modulus value and the phase of reflection coefficients, respectively. The electric field of the incident Gaussian beam in the incident coordinate can be written in its angular spectrum representation as [48,49]

In order to obtain the GH shift, we need to calculate the center of mass of the Gaussian beam under the reflected frame, the expression as follows

After simplification and derivation, the GH shift is then defined as the following form [38]

 

Fig. 1. Schematic diagram of the graphene-$\mathrm {ITO/TiO_{2}/ITO}$ sandwich structure. $d_{ITO}$ means the thickness of the ITO, and $d_{ITO}=5\ \mathrm {nm}$. $d_{TiO_{2}}$ refers to the thickness of the $\mathrm {TiO_{2}}$, and $d_{TiO_{2}}=150\ \mathrm {nm}$. $\theta$ is the incident angle. $\epsilon _{air}$, $\epsilon _{ITO}$ and $\epsilon _{TiO_{2}}$ are the relative permittivities of the air, the ITO and the $\mathrm {TiO_{2}}$, the value of $\epsilon _{TiO_{2}}$ is 6.25.

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3. Results and discussions

First, we investigate the variation of the relative permittivity of the ITO material $\epsilon _{ITO}$ with the incident wavelength $\lambda$ as shown in Fig. 2. As $\lambda$ increases, the real and the imaginary parts of $\epsilon _{ITO}$ show up as decrease and increase, respectively. What noteworthy is that the absolute value of the real and the imaginary parts of $\epsilon _{ITO}$ are less than unity from 1220 nm to 1600 nm, which is the ENZ region. In other words, this is a region where ENZ effects are prominent and expect to have a great impact on GH shifts. Then, we select $\lambda$ of 1220 nm and assume that the conductivity of graphene $\sigma$ is zero to research the effect of the thickness of the ITO and the $\mathrm {TiO_{2}}$ on reflection phase $\phi _p$. Figure 3(a) describes the variation of $\phi _p$ with the incident angle $\theta$ and the thickness of the ITO $d_{ITO}$, in the case where $d_{TiO_{2}}$ is 100 nm. It can be concluded that the changing situation of $\phi _p$ with $\theta$ is more obvious when $d_{ITO}$ is small. Thus, we choose $d_{ITO}$ as 5 nm, which is regarded as the optimal thickness of the ITO, and discuss the effect of the thickness of the $\mathrm {TiO_{2}}$ on the reflection phase $\phi _p$, as shown in Fig. 3(b). We find that when $\mathrm {TiO_{2}}$ is about 100 nm and 150 nm, $\phi _p$ obviously changes in a relatively large range of $\theta$. In order to actually construct the sandwich structure better, we set $d_{ITO}$ to be 5 nm and $d_{TiO_{2}}$ to be 150 nm. Subsequently, we investigate the change of $\phi _p$ and $\phi _s$ with $\theta$ in the ENZ region of the ITO with the optimum thickness, and $\sigma$ is still set to zero. From Figs. 3(c) and (d), it can be clearly observed that $\phi _s$ changes very little with $\theta$ at a given working wavelength, while $\phi _{p}$ significantly changes with the increase of $\theta$ at different $\lambda$. And we can find an intriguing change rule from Fig. 3(d), when $\lambda$ is from 1220 nm to 1480 nm, $\phi _{p}$ is always positive and positively increases with the variation of $\theta$. Conversely, when $\lambda$ is from 1481 nm to 1600 nm, $\phi _{p}$ continues with negative value and holds negative growth. However, the abrupt change of $\phi _{p}$ from $-\pi$ to $\pi$ occurs (such as the position of $\theta =68.3^{\circ }$, $\lambda =1550$ nm) in the wavelength range of 1481 nm to 1600 nm. The abrupt change of $\phi _{p}$ is meaningless to GH shifts [32], and we ignore the abrupt area in this paper. Meanwhile, since $\phi _s$ is insensitive to $\theta$, we only discuss GH shifts in the case of $p$ polarization in this paper.

 

Fig. 2. Dependence of the real and the imaginary parts of the ITO dielectric constant $\epsilon _{I T O}$ on the wavelength $\lambda$. For this figure, the plasma frequency of $\omega _{p}=2.65\times 10^{15}\ \mathrm {rad\ s^{-1}}$, the damping parameter of $\mathit {\Gamma }=2.05\times 10^{14}\ \mathrm {rad\ s^{-1}}$ and the permittivity at an infinite frequency of $\epsilon _{\infty }=3.91$.

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According to the positive and the negative signs of $\phi _{p}$ at different $\lambda$, the ENZ region is divided into two bands: 1220$-$1480 nm and 1481$-$1600 nm. GH shifts for the $p$ polarization within the range of two bands are studied respectively. In the case of $\sigma =0$, we plot Fig. 4(a) to describe the dependence of GH shifts on $\theta$ and $\lambda$ within the wavelength range of 1220$-$1480 nm. The sign of GH shifts is invariably positive and the peak value of GH shifts is quite large. And the peak value of GH shifts decreases first and then increases. Especially, the peak value of GH shifts significantly increases and obtains a giant peak value of GH shifts when $\lambda$ approaches to 1480 nm. In the case of $\lambda =1478\ \mathrm {nm}$, the GH shift can reach $2360\ \mu \mathrm {m}$, which is about $1596\ \lambda$, as shown by the dashed line in the figure. Meanwhile, the position of the angular resonance of the GH shift moves to the left as the wavelength increases. Figures 4(b) and 4(c) exhibit the GH shift and the reflection phase $\phi _{p}$ as a function of $\theta$ at the wavelength near 1480 nm, respectively. It can be obtained that when $\lambda$ is close to 1480 nm, $\phi _{p}$ is very sensitive to the variation of $\lambda$, and $\phi _{p}$ shows an obvious sharp variation. Furthermore, the sharp change in the $\phi _p$ moves to the left as the wavelength increases. Combining with the analytical expression of GH shifts, we can conclude that the huge GH shift and the movement of the resonance angle results from the sharp change of $\phi _p$. Figure 4(d) shows the impact of GH shifts with $\theta$ and $\lambda$ in the wavelength range of 1481$-$1600 nm. By observing Fig. 4(d), it can be found that the peak value of GH shifts decreases at first and increases later, and the peak value of GH shifts always keeps negative and giant. When $\lambda$ is close to 1481 nm, the GH shift is going to a huge value. Similarly, we plot Figs. 4(e) and 4(f) to explore the variation of GH shifts with $\theta$ at $\lambda$ near 1481 nm. As shown by the dashed line in Fig. 4(e), when the wavelength is 1482 nm, the peak value of the GH shift can reach $-2490\ \mu \mathrm {m}$, which is about $-1590\ \lambda$. Correspondingly, the dot-dashed line in Fig. 4(f) has a very obvious phase change. We can conclude from the above analysis that, when $\sigma$ is zero, the giant positive and negative GH shifts can be obtained by setting different $\lambda$, because of the obvious relection phase change.

 

Fig. 3. (a) Pseudo-color image of the reflection phase $\phi _{p}$ with the thickness of ITO $d_{ITO}$ and the incident angle $\theta$, for this figure, $d_{TiO_{2}}=100\ \mathrm {nm}$ and $\lambda =1220\ \mathrm {nm}$. (b) Pseudo-color image of the reflection phase $\phi _{p}$ with the thickness of $\mathrm {TiO_{2}}$ $d_{TiO_{2}}$ and the incident angle $\theta$, for this figure, $d_{ITO}=5\ \mathrm {nm}$ and $\lambda =1220\ \mathrm {nm}$. (c) Pseudo-color image of the reflection phase $\phi _{s}$ with the wavelength $\lambda$ and the incident angle $\theta$. (d) Pseudo-color image of the reflection phase $\phi _{p}$ with the wavelength $\lambda$ and the incident angle $\theta$.

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Fig. 4. (a) Pseudo-color image of the GH shift $\Delta _{\mathrm {GH}}$ within the wavelength range of 1220$-$1480 nm. (b) Dependence of the GH shift $\Delta _{\mathrm {GH}}$ on the incident angle $\theta$ under the case of $\lambda =$1472 nm,1474 nm,1476 nm,1478 nm. (c) Dependence of the reflection phase $\phi _p$ on the incident angle $\theta$ under the case of $\lambda =$1472 nm,1474 nm,1476 nm,1478 nm. (d) Pseudo-color image of the GH shift $\Delta _{\mathrm {GH}}$ within the wavelength range of 1481$-$1600 nm. (e) Dependence of the GH shift $\Delta _{\mathrm {GH}}$ on the incident angle $\theta$ in the case of $\lambda =$1482 nm,1484 nm,1486 nm,1488 nm. (f) Dependence of the reflection phase $\phi _p$ on the incident angle $\theta$ in the case of $\lambda =$1482 nm,1484 nm,1486 nm,1488 nm.

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Next, in the case of $\sigma \ne 0$, we fix $\lambda$ and research the influence of $\sigma$ on GH shifts. What noteworthy is that the value of $\sigma$ can be changed by regulating $E_f$. Because the peak value of the GH shift is small when $\lambda$ is 1285 nm in Figure 4(a), we set $\lambda$ as 1285 nm, and draw a pseudo-color Fig. 5(a) to study the influence of $\theta$ and $E_f$ on GH shifts. Compared with the situation when $\sigma$ is zero, the GH shift can be significantly enhanced after adjusting $E_f$, and the larger peak value of GH shifts mainly occurs in the incident angle range of $67.9^{\circ }$ to $68.5^{\circ }$, and the sign of GH shifts remains positive or negative in the same Fermi energy range within this incident angle range. In order to explore the relationship between the sign of GH shifts and $\sigma$, we fix $\theta$ as $68^{\circ }$, and plot the change curves of GH shifts and $\sigma$ with $E_f$, as shown in Fig. 5(b). Comparison of the solid line and the dashed line curves can be found that they have the same variation trend with $E_f$, and the variation trend shows that it first decreases from the initial value to the negative peak value, and then increases to a positive value, what is slightly different is that the GH shift increases at a much faster rate to a positive value. And what important is that the decreasing and the increasing intervals of the two cruves basically coincide. Thus, it can be deduced that the imaginary part of $\sigma$ has a significant effect on the sign of GH shifts, i.e., the GH shift is negative when the imaginary part of $\sigma$ is the peak negative value, and the GH shift is positive when the imaginary part of $\sigma$ is a positive value. Meanwhile, we discover that the real part of $\sigma$ abruptly changes from a positive value to 0 with the increase of $E_f$, and when the real part of $\sigma$ is a positive value, the imaginary part of $\sigma$ decreases and gets the peak negative value at the abrupt point of the real part of $\sigma$. Thus, it is worth noting that the negative value of the GH shift can also be determined by the abrupt point of the real part of $\sigma$. Subsequently, we plot Fig. 5(c) and Fig. 5(d), which are pseudo-color diagrams of GH shifts and the imaginary part of $\sigma$ with $E_f$ and $\lambda$. By comparing Fig. 5(c) and Fig. 5(d), we can conclude that there is a close relationship between the sign of GH shifts and the imaginary part of $\sigma$ in the wavelength range of 1220 nm to 1480 nm. It can be found that the positive or the negative GH shift can be regulated by changing the imaginary part of $\sigma$ as the peak negative value or a positive value within this wavelength range. Meanwhile, by comparing the changing situation of GH shifts with $E_f$ in Fig. 5(c) with the changing situation of GH shifts with $\theta$ of $68^{\circ }$ in Fig. 4(a), it can be found that GH shifts can be effectively enhanced by adjusting $E_f$ within the wavelength range of 1220-1480 nm, and the positive and negative control of GH shift can be realized.

 

Fig. 5. (a) Pseudo-color image of the GH shift $\Delta _{\mathrm {GH}}$ with the Fermi energy $E_f$ and the incident angle $\theta$, in the case of $\mathrm {\lambda = 1285\ nm}$. (b) Dependence of the GH shift $\Delta _{\mathrm {GH}}$ and the conductivity of the graphene $\sigma$ on the Fermi energy $E_f$ in the case of $\mathrm {\theta = 68^{\circ }}$. (c) Pseudo-color image of the GH shift $\Delta _{\mathrm {GH}}$ with the Fermi energy $E_f$ and the wavelength $\lambda$ in the case of $\mathrm {\theta = 68^{\circ }}$. (d) Pseudo-color image of the imaginary part of the conductivity of graphene Im[$\sigma$] with the wavelength $\lambda$ and the Fermi energy $E_f$, in the wavelength range from 1220 nm to 1480 nm.

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In a similar manner, in the case of $\sigma \ne 0$, corresponding to the wavelength with a small peak value of GH shifts in Fig. 4(c), we set the wavelength as 1565nm, and depict a pseudo-color Fig. 6(a) to explore the change of GH shifts with $E_f$ and $\theta$. It can be observed that GH shifts can be significantly enhanced with the variation of $E_f$, and the large GH shifts are mainly obtained in the incident angle range of $67.9^{\circ }$ to $68.1^{\circ }$. And it is essential that the positive and negative inversion of the GH shift occurs with $E_f=0.565\ \mathrm {eV}$ as the critical point. For purpose of finding the relationship between the sign of GH shifts and $\sigma$, we fix $\theta$ as $67.9^{\circ }$, and plot the change curves of GH shifts and $\sigma$ with $E_f$, as shown in Fig. 6(b). We can easily find that the change rule of the real and imaginary parts of $\sigma$ with $E_f$ is consistent with that in Fig. 5(b), when the real part of $\sigma$ is positive, the imaginary part of $\sigma$ is negative and decreases. Comparing the solid line and the dashed line curves, we can find that when the imaginary part of $\sigma$ decreases from the initial value to the negative peak value, the GH shift basically keeps the same variation trend. But when the imaginary part of $\sigma$ increases from the peak negative value to a positive value, the GH shift only increases in a tiny range of $E_f$, then decreases to the peak negative value and subsequently increases rapidly to greater than 0. In other words, as $\lambda$ increases to more than 1480 nm, the imaginary part of $\sigma$ still has a great influence on the negative GH shift, but the positive GH shift is affected by the overall parameters and has no obvious relationship with $\sigma$. Then, we plot Fig. 6(c) and Fig. 6(d), which are pseudo-color diagrams of GH shifts and the imaginary part of $\sigma$ with $E_f$ and $\lambda$. It can be found that GH shifts can be effectively enhanced by adjusting $E_f$ within the wavelength range of 1481-1600 nm by comparing Fig. 6(c) with Fig. 4(d). By observing the two figures, we can see that the GH shift is negative when the imaginary part of $\sigma$ is negative, and the negative GH shift can also be determined by adjusting the real part of $\sigma$ to a positive value. And there also exist critical points making the sign of GH shifts invert within this wavelength range. The value of the critical point increases first and then decreases, among the maximum value of the critical point is 0.57 eV. Based on the above analysis, we can conclude that the positive or the negative GH shift can be regulated by adjusting $E_f$ to more than 0.57 eV or changing the imaginary part of $\sigma$ as a negative value.

 

Fig. 6. (a) Pseudo-color image of the GH shift $\Delta _{\mathrm {GH}}$ with the Fermi energy $E_f$ and the incident angle $\theta$, in the case of $\mathrm {\lambda = 1565\ nm}$. (b) Dependence of the GH shift $\Delta _{\mathrm {GH}}$ and the conductivity of the graphene $\sigma$ on the Fermi energy $E_f$ in the case of $\mathrm {\theta = 67.9^{\circ }}$. (c) Pseudo-color image of the GH shift $\Delta _{\mathrm {GH}}$ with the Fermi energy $E_f$ and the wavelength $\lambda$ in the case of $\mathrm {\theta = 67.9^{\circ }}$. (d) Pseudo-color image of the imaginary part of the conductivity of graphene Im[$\sigma$] with the wavelength $\lambda$ and the Fermi energy $E_f$, in the wavelength range from 1481 nm to 1600 nm.

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Based on the conclusion that the positive and the negative GH shifts can be controlled in cases of $\sigma =0$ and $\sigma \ne 0$, we propose a barcode encryption scheme based on the positive and the negative GH shift. The positive GH shift represents “1”, and the negative GH shift represents “0”. Next, according to the sign of GH shifts in the two cases of the zero and the non-zero conductivity of graphene, and we achieve zero and non-zero graphene conductivity by the means of changing the Fermi energy, which is tuned with external gate voltage, in a graphene-$\mathrm {ITO/TiO_{2}/ITO}$ sandwich structure. We can get four groups of the coding state, the specific coding rule as shown in Table 1, and we apply instances to implement the barcode encryption scheme. At first, based on the variation rule between the sign of GH shifts and $\lambda$ in the case of $\sigma =0$, we set $\lambda$ as 1450 nm to obtain “1” at the first position of the coding state. Notably, when $\lambda =1450\ \mathrm {nm}$, we can achieve the zero graphene conductivity by regulating $E_f$ as 0.514 eV deduced from Eq. (1). Then, in the case of $\sigma \ne 0$, according to the variation rule between the sign of GH shifts and the imaginary part of $\sigma$ within the wavelength of 1220$-$1480 nm, we can obtain “0” or “1” at the second position of the coding state by setting the imaginary of $\sigma$ as the peak negative value or a positive value. We can infer from the Eq. (1) that the imaginary part of $\sigma$ obtain the peak negative value when $E_f=0.428\ \mathrm {eV}$. Thus, we can achieve the coding states of “0 1” and “1 1” by setting $\lambda$ as 1450 nm first and then adjusting external gate voltage to make $E_f$ as 0.514 eV, 0.428 eV and 0.750 eV, as shown in Fig. 7(a). Similary, we set $\lambda$ as 1570 nm to obtain “0” at the first position of the coding state, and the zero graphene conductivity can be obtained by regulating $E_f$ as 0.475 eV deduced from Eq. (1). Next, in the case of $\sigma \ne 0$, we can obtain “0” or “1” at the second position of the coding state by setting the imaginary part of $\sigma$ as a negative value or setting $E_f$ more than 0.57 eV. As a result, we can achieve the coding states of “0 0” and “1 0” by setting $\lambda$ as 1570 nm first and then adjusting $E_f$ as 0.475 eV, 0.420 eV and 0.650 eV by the means of adjusting external gate voltage, as shown in Fig. 7(b). In this instance, we simply achieve four different coding states by adjusting $\lambda$ and $E_f$.

 

Fig. 7. (a) The variation of GH shifts with the incident angle $\theta$ in the case of $E_f$ = 0.428 eV, 0.514 eV, 0.750 eV with the wavelength of 1450 nm. (b) The variation of GH shifts with the incident angle $\theta$ in the case of $E_f$ = 0.420 eV, 0.475 eV, 0.650 eV with the wavelength of 1570 nm.

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Table 1. Four groups of the coding state based on the sign of GH shifts

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4. Conclusion

In conclusion, we propose a graphene-$\mathrm {ITO/TiO_{2}/ITO}$ sandwich structure, which can adjust the positive and the negative signs of GH shifts in two cases of $\sigma =0$ and $\sigma \ne 0$ via dynamically adjusting the wavelength $\lambda$ or the Fermi energy $E_{f}$, and design a barcode encryption scheme based on the positive and the negative signs of GH shifts. The research results show that in the case of $\sigma =0$ , the signs of GH shifts are significantly regulated by $\lambda$ in the ENZ region (from 1220 nm to 1600 nm). The signs of GH shifts are positive when $\lambda$ is less than 1480 nm, and keep negative when the wavelength is more than 1480 nm. In the case of $\sigma \ne 0$, the influence of $\sigma$ on GH shifts is studied under the wavelength range of 1220$-$1480 nm and 1481$-$1600 nm, and in two bands, the positive and the negative regulation rules of GH shifts are obtained. Finally, based on the positive and the negative regulation rules of GH shifts in two cases of $\sigma =0$ and $\sigma \ne 0$, four coding states “0 0”, “0 1”, “1 0” and “1 1” are coded by adjusting $\lambda$ and $E_f$. Our research provides a theoretical foundation for the design of optical switching devices and coding devices based on GH shifts.