## Abstract

We proposed a scheme to manipulate the Goos-Hänchen shift of a light beam reflected from the depletion-type device via external voltage bias. It is shown that the lateral shift of the reflected probe beam can be easily controlled by adjusting the reverse voltage bias and the incidence angle. Using this scheme, the lateral shift can be tuned from negative to positive, without changing the original structure of the depletion-type device. Numerical calculations further indicate that the influence of structure parameters and light wavelength can be reduced via readjustment of the reverse bias. The proposed structure has the potential application for the integrated electronic devices.

© 2013 OSA

## 1. Introduction

When a total reflection happens at the interface between two media, there exists a tiny lateral shift between the totally reflected light beam and the incident light beam. This phenomenon is called Goos-Hänchen (GH) effect, which was discovered by Goos and Hänchen and theoretically explained by Artmann in the late 1948. It has since been broadly investigated both in theory [1–4] and in experiment [5–7]. In the last two decades, people have investigated the GH shift in various structures containing different kinds of media, such as a weakly absorbing semi-infinite medium [8, 9], photonic crystals [10], negative refractive media [11, 12], lossless dielectric slab [13], various level configurations quantum systems [14–17], the ballistic electrons in semiconductor quantum slabs or well [18, 19], and others.

The GH effect has many interesting applications. For example, in optical heterodyne sensors, it can be used to measure refractive index, beam angle, temperature, displacement, film thickness, and so on [20]. The permittivity and permeability of materials can also be characterized by the phenomenon of GH shift [21], due to the fact that the GH shift and the measured material parameters have a precise relationship. For the applications in sensor devices, the manipulation of the GH shift is vital. To this end, various schemes to control the GH shift by using the electric field or light field have been devised [22–25]. For example, modification of the electromagnetically induced transparency by using the coherent driving fields [22–24] and introduction of optically nonlinear material (i.e. LiNbO_{3}) into the guiding layer of the symmetrical metal cladding waveguide to control the lateral shift of the reflected beam by applying an external electric field [25, 26]. It should be noted that in these investigations, the configurations or devices for controlling GH shifts were made of cavity or symmetrical metal cladding waveguide. They can’t be integrated with optoelectronic circuits although their sizes can be nanometer level due to the absence of the semiconductor material. Moreover, the external control voltages were extremely gigantic (tens of thousands of volts) [25, 26]. In this paper, we proposed a metal-insulator-semiconductor (MIS) configuration to control the GH shift of the reflected beam by applying an external voltage. By modifying the depletion depth, the resonant conditions of the MIS system are expected to be changed dramatically. Therefore it is expected that the lateral shift of the reflected probe beam can be easily controlled by adjusting the external voltage. Our theoretical study displays that the GH shift can be thousands of wavelengths ($\approx 10$µm) and the external stimulating voltage can be within twenty volts. Importantly, our MIS structure is a type of depletion-type device and the control bias is low voltage, so the all control system can be integrated into the measuring circuit, thus the measuring apparatus based on GH effect will be simple and compact.

## 2. Model and theory

Figure 1 shows the schematic of the electrically controlled GH shift based on MIS structure system. The metal control electrode made of gold is placed at the center of the top of an n + doped GaAs layer and form a metal-semiconductor junction. An insulating barrier (i.e. undoped Al_{0.3}Ga_{0.7}As layer, *d _{2}* = 75nm) is included to reduce leakage current and control the depletion depth (see Fig. 1(a)). We consider a TE-polarized probe light beam (

*λ*= 10µm), which is incident on the metal control electrode of the MIS structure from vacuum with an angle

*θ*. As shown in Fig. 1(b), the MIS structure consists of four layers of materials: metal control electrode, insulating barrier, depletion region, and n + doped GaAs substrate. By applying a voltage bias to change the depletion depth

*d*in the substrate (the sum of the depletion width and n-doped substrate width is a fixed constant), we can tune the phase of the reflected light, and thus can obtain the dynamic adjustable GH shift.

_{3}Firstly, we analyze the dielectric properties respectively in the MIS structure system. We choose a Drude model to describe the permittivity of an n + GaAs substrate [27, 28]:

*ε*is the high frequency dielectric constant (

_{∞}*ε*= 10.86 for GaAs),

_{∞}*ω*is the angular frequency. Here, the relaxation frequency

*Γ*(i.e. damping term) and the plasma frequency

*ω*are given byandrespectively, where

_{p}*τ*is the scattering time,

*μ*is the electron mobility,

*m**is the electron effective mass,

*q*is the electron charge, and

*n*is the electron density.

We should note that the mobility *μ* and the electron effective mass *m** also vary with the carrier density *n*. We can obtain the theoretical values of the carrier concentration dependent effective mass and mobility of GaAs at *T =* 300*K* [29]:

*n*is the carrier density, here its unit is cm

^{−3},

*n*is the reference electron concentration(When the electron concentration in GaAs is equal to

_{ref}*n*, the electron mobility is half of sum of maximum and minimum of electron mobility),

_{ref}*µ*= 500 cm

_{min}^{2}/vs,

*µ*= 9400 cm

_{max}^{2}/vs,

*n*= 6.0 × 10

_{ref}^{16}cm

^{−3}, and $\alpha =0.394$. Based on the above analysis, we can calculate the dielectric constant of the n-doped GaAs at $\lambda =10$µm,

*n*= 3.9 × 10

^{17}cm

^{−3}and

*T =*300

*K*.

The dependence of the voltage bias on the depletion width in an n-doped GaAs can be obtained [28, 30],

*d*at a fixed applied voltage. We regard the depletion region as a lossless dielectric layer since free charge in the depletion region is almost zero. The dielectric constant of gold at $\lambda =10$µm is ${\epsilon}_{\text{Gold}}\text{=-2800+j791}$ [31].

_{3}Secondly, we calculate the GH shift ${S}_{r}$ for the reflected light. We can apply the standard characteristic matrix approach [32] to calculate the reflection coefficients $r({k}_{y},\omega )$ for the reflected beam through the MIS structure. The transfer matrix of the *j*th layer can be expressed as [22, 32]

*ω*is angular frequency of the probe beam on incident angles

*θ*, ${k}_{y}=k\mathrm{sin}\theta $ is the

*y*component of the wave number (

*k*=

*ω/c*) in vacuum, and

*c*is the light speed in vacuum,${k}_{z}^{j}=\sqrt{{\epsilon}_{j}{k}^{2}-{k}_{y}^{2}}$ is the

*z*component of the wave number in the

*j*th layer,${q}_{j}={k}_{z}^{j}/k$, ${d}_{j}$ is the thickness of the

*j*th layer. The total transfer matrix for the MIS structure is given by$Q({k}_{y},\omega )={M}_{1}({k}_{y},\omega ,{d}_{1}){M}_{2}({k}_{y},\omega ,{d}_{2}){M}_{3}({k}_{y},\omega ,{d}_{3}){M}_{4}({k}_{y},\omega ,{d}_{4})$. Therefore, the coefficients

*r*is given by [22]

*z*component of the wave number in vacuum), and ${Q}_{ij}$are the elements of the matrix$Q({k}_{y},\omega )$.

For a well-collimated probe beam with a sufficiently large width (i.e., with a narrow angular spectrum,$\Delta k\ll k$), according to the stationary phase theory [13, 33], the GH shift of the reflected beam can be calculated analytically using [22]:

where ${\phi}_{r}$ is the phase related to reflection coefficients $r$ and $\theta $ is the incident angle of the probe light. The reflection coefficients can be calculated using the standard characteristic matrix approach [22]. The GH shift in the reflected probe light can be written as:*d*= 41 nm and

_{1}*d*= 29.17µm.

_{3}+ d_{4}## 3. Numerical results and discussion

#### 3.1. The control of the GH shift with an applied voltage

Now we discuss the controlling effects of MIS structure system applied voltage bias on the lateral shift in detail on numerical calculation method. First we set the insulating barrier width *d2* to be 900nm, and the electron density in n + GaAS *n* = 3.9 × 10^{17}cm^{−3}. From Eqs. (1)-(5), we know that the dielectric constant of the n-doped GaAs is 10.313254 + j0.023701 at $\lambda =10$µm. Next we calculated the depletion region deepness *d _{3}* at different reverse bias from Eqs. (6) and (7). Then we calculated the lateral shift of MIS structure in accordance to Eqs. (8)-(11).

Figure 2 shows the dependence of phase ${\phi}_{r}$ related to the reflection coefficients on different incident angles. In Figs. 2(a)-2(c), ${\phi}_{r}$ is monotonic increasing with the incident angle, leading to the positive slope and hence the negative GH shifts, as shown in Figs. 3(a)-3(c). Moreover, it is clearly seen that the absolute value of the first derivative of ${\phi}_{r}$ is the maximum when the incident angle equal to a particular value under a certain voltage bias, in accordance with the Eqs. (10) and (11) which show that the absolute value of the lateral shift may be the maximum at the incident angle. Similarly, ${\phi}_{r}$ is monotonic decreasing with the incident angle in Figs. 2(d)-2(f), which leads to the positive GH shifts, as shown in Figs. 3(d)-3(f). The first derivative of ${\phi}_{r}$ is the maximum on a particular incident angle, which also indicates that the lateral shift may be the maximum.

In Fig. 3, we plot the dependence of the lateral shift *S _{r}* on the incident angle $\theta $ under different control voltage bias. It is shown that the GH shifts for the reflected probe beam are strongly dependent not only on

*V*but also on the incident angle. The lateral shift has different behaviors at different control voltage bias and incident angle. It is clear that the lateral shifts can be very large (positive or negative) at certain value of

_{g}*V*for the incident angle of the probe beam around 75°, as we predicted in Fig. 2. The largest lateral shift can be thousands of wavelengths. In Figs. 3(a)-3(c), the reflected probe beam suffers the large negative shift near the resonant condition of the MIS structure, while in Figs. 3(e)-3(f) it suffers large positive shift. The dependence of the lateral shift

_{g}*S*on

_{r}*V*is due to the fact that the MIS internal depletion deepness and n-doped substrate width both vary with

_{g}*V*(the sum of the depletion width and n-doped substrate width is a fixed constant). This variation in deepness with

_{g}*V*modifies the resonant condition of the MIS structure and we observe the manipulation effect of the control voltage on the lateral shift. Using this effect, without changing the initial structure of the MIS structure, we can easily manipulate the lateral shift of the reflected probe beam.

_{g}#### 3.2. Influence of structure parameters and wavelength

Now, we analyze numerically whether the lateral shift can be controlled or not when the wavelength of the incident beam changes slightly and MIS structure is fixed. Taking into account when the wavelength changes slightly but the change of dielectric constant in n + GaAs is a primary factor, because dielectric constants of metal electrodes in this small change range (see Fig. 4) almost unchanged, we plot the dependence of *S _{r}* on the wavelength of probe beam under different applied voltage and incident angle in Fig. 4. It is shown that the lateral shift will be greatly reduced if the wavelength slightly increased or decreased when the lateral shift peak. However, it can run up to a new peak at another reverse bias if the variety on wavelength is suffered.

When the incident angle and the wavelength of the probe beam are fixed, there are some factors affecting the lateral shift. First, the applied voltage is the main factor as we analyzed before (Fig. 3). Second, the insulated barrier thickness is another necessary factor for it changes depletion region width *d _{3}* in accordance with the Eqs. (6) and (7). Third, of course the electron concentration in the n + doped GaAs is the other vital factor, which not only can affect the depletion width, but also affect the permittivity of n + GaAs.

The undoped Al_{0.3}Ga_{0.7}As (insulated barrier) layer can be grown by molecular beam epitaxy [28]. However, perhaps there has been a deviation on an order of 0.5nm. We plot the dependence of *S _{r}* on the insulating barrier width under different reverse bias for the incident angle θ = 75°in Fig. 5(a) and for θ = 75.2°in Fig. 5(b). We can know that the lateral shift also will be greatly reduced if the insulating barrier width slightly varied. However, the lateral shift can run up to a new peak too at another reverse bias. Due to technical reasons or in different light conditions, the electron concentration in n + doped GaAs will change a little. Figure 6 shows that there always exists an appropriate reverse bias at which the lateral shift is very large when the electron density in the n + doped GaAs varied slightly. The above analysis shows that there exists greatly influence of the structure parameters and wavelength on the controlling effect, but the influence can be greatly reduced via adjustment of the reverse bias.

#### 3.3. Numerical simulations of GH shift in real system

In the above discussions, we got the results based on the stationary-phase theory [13, 33], in which the incident probe beam is assumed to be a well-collimated beam. When the incident probe beam has a finite width (i.e. a Gaussian profile), as discussed in the following, we will show that our results are still available in the real system. According to the method described in [22], we can simulate the reflection process of the Gaussian-shaped probe beam passing through the MIS structure system under applied voltage bias. At the plane of *Z =* 0, the electric field of the reflected probe beam can be written as [22, 34]:

*Z =*0.

Equations (10) and (11) are only approximate for a well-collimated probe beam with a sufficiently large width, and for the finite beam, the lateral shift can be obtained by the following equation [10, 35]:

Figure 7 shows the numerical simulation results of the TE polarized reflected light beams form the MIS structure under different controlling voltage bias for different half-width *w*. It is also shown that the lateral shift of the reflected probe beam can be controlled to be positive or negative with the external voltage bias.

We calculated the lateral shift according to Eq. (14) with different half-width of the probe beam under some incident angles and reverse bias, the results on lateral shifts versus different beam width were plotted in Fig. 8. Numerical simulations for a Gaussian-shaped incident beam have demonstrated the validity of the stationary phase method. The numerical results are in good agreement with the theoretical results when the beam waist of the incident beam is sufficiently large wide, that is, the incident beam is well-collimated. The discrepancy between theoretical and numerical results is due to the distortion of the reflected light beam (see Fig. 7(a) and 7(d)), especially when the beam waist is narrow. In this case, there is another angular GH shifts on the loss dielectric surface [36–39]. We only study the spatial lateral shift of well-collimated beam in this paper, ignoring the angular GH shift. The angular GH shift in the experimental work should be considered.

## 4. Conclusion

In this paper, we have proposed a scheme to realize the manipulation of the lateral shift of a light probe beam via an external control bias on the MIS structure. By adjusting the reverse bias at a particular incident angle, the depletion width in the doping n + GaAs substrate can be changed, so the lateral shift is dynamically tuned. When the structure parameters and light wavelength are slightly changed, the influence on the controlling effect can be greatly reduced by adjusting the reverse bias.

## Acknowledgments

This work is partially supported by the National 973 Program of China (Grant No. 2012CB315701), the National Natural Science Foundation of China (Grant No. 11004053), the China Postdoctoral Science Foundation (Grant No. 2012M511365), the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20120161120013), and the Young Teacher Development Plan of Hunan University.

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