Electrically controlled Goos-Hänchen shift of a light beam reflected from the metal-insulator-semiconductor structure

Changyou Luo; Jun Guo; Qingkai Wang; Yuanjiang Xiang; Shuangchun Wen

doi:10.1364/OE.21.010430

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. LiNbO₃) 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_0.3Ga_0.7As layer, d₂ = 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 d₃ 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.

Fig. 1 Schematic of the electrically controlled Goos-Hänchen shift system. (a) Gold slab at the center of the top of an n + doped GaAs layer serving as control electrode and the cathode on top of all around the edge with ohmic contact. A voltage bias applied between the control electrode and ohmic contact controls the depletion depth. (b) Diagram of the Goos-Hänchen shift and internal detailed structure of the MIS system.

Download Full Size | PDF

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 = ε_{\infty} (1 - \frac{ω_{p}^{2}}{ω^{2} + i ω Γ}) = ε_{\infty} (1 - \frac{ω_{p}^{2}}{ω^{2} + Γ^{2}} + i \frac{ω_{p}^{2} Γ}{ω (ω^{2} + Γ^{2})}),

where 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_p are given by

ω_{p}^{2} = \frac{n q^{2}}{ε_{0} ε_{\infty} m^{*}},

and

Γ = \frac{1}{τ} = \frac{q}{μ m^{*}},

respectively, where 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 = 300K [29]:

m^{*} / m_{0} = 0.0635 + 2.06 \times 10^{- 22} n + 1.16 \times 10^{- 40} n^{2},

and

μ (n) = μ_{\min} + \frac{μ_{\max} - μ_{\min}}{1 + {(n / n_{r e f})}^{α}},

In Eqs. (4) and (5), n is the carrier density, here its unit is cm⁻³, n_ref is the reference electron concentration(When the electron concentration in GaAs is equal to n_ref, the electron mobility is half of sum of maximum and minimum of electron mobility), µ_min = 500 cm²/vs, µ_max = 9400 cm²/vs, n_ref = 6.0 × 10¹⁶ cm⁻³, and

α = 0.394

. Based on the above analysis, we can calculate the dielectric constant of the n-doped GaAs at

λ = 10

µm, n = 3.9 × 10¹⁷cm⁻³ and T = 300K.

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

W_{d e p l e t i o n} = {[\frac{2 ε_{G a A s} ε_{0}}{q n} (- ϕ_{s})]}^{1 / 2},

where the surface potential

ϕ_{s}

is related to the control electrode voltage

V_{g}

as follows [28, 30],

V_{g} = ϕ_{s} - \frac{ε_{G a A s}}{ε_{A l G a A s}} W_{b a r r i e r} {[\frac{2 q n}{ε_{G a A s} ε_{0}} | ϕ_{s} |]}^{1 / 2} + ϕ_{M S},

where

ϕ_{M S}

is the flat-band voltage (here

ϕ_{M S} = 1.2 V

). Note that dielectric constants we use here are the static values (

ε_{{Al}_{0. 3} {Ga}_{0. 7} As} = 12.05

and

ε_{G a A s} = 12.9

) [28], not the high frequency ones. From Eqs. (6) and (7), we can get the function of the depletion width on the voltage bias,and then can calculate the depletion thickness d₃ 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

λ = 10

µm is

ε_{Gold} =-2800+j791

[31].

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 $r (k_{y}, ω)$ for the reflected beam through the MIS structure. The transfer matrix of the jth layer can be expressed as [22, 32]

M_{j} (k_{y}, ω, d_{j}) = (\begin{matrix} \cos [k_{z}^{j} d_{j}] & i \sin [k_{z}^{j} d_{j}] / q_{j} \\ i q_{j} \sin [k_{z}^{j} d_{j}] & \cos [k_{z}^{j} d_{j}] \end{matrix}),

where ω is angular frequency of the probe beam on incident angles θ,

k_{y} = k \sin θ

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{ε_{j} k^{2} - k_{y}^{2}}

is the z component of the wave number in the jth layer,

q_{j} = k_{z}^{j} / k

,

d_{j}

is the thickness of the jth layer. The total transfer matrix for the MIS structure is given by

Q (k_{y}, ω) = M_{1} (k_{y}, ω, d_{1}) M_{2} (k_{y}, ω, d_{2}) M_{3} (k_{y}, ω, d_{3}) M_{4} (k_{y}, ω, d_{4})

. Therefore, the coefficients r is given by [22]

r (k_{y}, ω) = \frac{q_{0} (Q_{22} - Q_{11}) - (q_{0}^{2} Q_{12} - Q_{21})}{q_{0} (Q_{22} + Q_{11}) - (q_{0}^{2} Q_{12} + Q_{21})},

where

q_{0} = k_{z} / k

(here

k_{z}

is the z component of the wave number in vacuum), and

Q_{i j}

are the elements of the matrix

Q (k_{y}, ω)

.

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

S_{r} = - \frac{λ}{2 π} \frac{d φ_{r}}{d θ},

where

φ_{r}

is the phase related to reflection coefficients

r

and

θ

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:

S_{r} = - \frac{λ}{2 π {| r |}^{2}} (Re (r) \frac{d Im (r)}{d θ} - Im (r) \frac{d Re (r)}{d θ}),

From Eq. (11), we can calculate the dependence of the lateral shift of the reflected probe beams on the external control voltage. In the following analysis, we take the fixed parameters for the MIS structure as follows: d₁ = 41 nm and d₃ + d₄ = 29.17µm.

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 × 10¹⁷cm⁻³. From Eqs. (1)-(5), we know that the dielectric constant of the n-doped GaAs is 10.313254 + j0.023701 at $λ = 10$ µm. Next we calculated the depletion region deepness d₃ 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 $φ_{r}$ related to the reflection coefficients on different incident angles. In Figs. 2(a)-2(c), $φ_{r}$ 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 $φ_{r}$ 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, $φ_{r}$ 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 $φ_{r}$ is the maximum on a particular incident angle, which also indicates that the lateral shift may be the maximum.

Fig. 2 Dependence of phase $φ_{r}$ on the incident angle $θ$ under different controlling voltage. (a) V_g = −12V, (b) V_g = −10V, (c) V_g = −7V, (d) V_g = −4V, (e) V_g = −1V and (f) V_g = 0V with d₁ = 41nm, d₂ = 900nm, d₃ + d₄ = 29.17µm, n = 3.9 × 10¹⁷cm⁻³ and λ = 10µm.

Download Full Size | PDF

Fig. 3 Dependence of lateral shift S_r on the incident angle $θ$ under different controlling voltage bias (a) V_g = −12V, (b) V_g = −10V, (c) V_g = −7V, (d) V_g = −4V, (e) V_g = −1V and (f) V_g = 0V, other parameters are the same as in Fig. 2.

Download Full Size | PDF

In Fig. 3, we plot the dependence of the lateral shift S_r on the incident angle $θ$ under different control voltage bias. It is shown that the GH shifts for the reflected probe beam are strongly dependent not only on V_g 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 V_g 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 S_r on V_g is due to the fact that the MIS internal depletion deepness and n-doped substrate width both vary with V_g (the sum of the depletion width and n-doped substrate width is a fixed constant). This variation in deepness with V_g 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.

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.

Fig. 4 Dependence of S_r on the wavelength of probe beam under different applied voltage for the incident angle (a) θ = 75°, (b) θ = 75.2°. Other parameters are the same as in Fig. 2.

Download Full Size | PDF

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 d₃ 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_0.3Ga_0.7As (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.

Fig. 5 Dependence of $S_{r}$ on the insulating barrier width under different reverse bias for the incident angle (a) θ = 75°, (b) θ = 75.2°. Other parameters are the same as in Fig. 2.

Download Full Size | PDF

Fig. 6 Dependence of Sr on the electron density in the n + doped GaAs under different applied voltage for the incident angle (a) θ = 75°, (b) θ = 75.2°.Other parameters are the same as in Fig. 2.

Download Full Size | PDF

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]:

{E_{x}^{(r)} |}_{z = 0} = {(1 / 2 π)}^{1 / 2} \int r (k_{y}) A (k_{y}) \exp (i k_{y} y) d k_{y},

and the electric field of the incident probe beam can be given by the integral form:

{E_{x}^{(i)} |}_{z = 0} = {(1 / 2 π)}^{1 / 2} \int A (k_{y}) \exp (i k_{y} y) d k_{y},

where

A (k_{y}) = (w_{y} / \sqrt{2}) \exp [- w_{y}^{2} {(k_{y} - k_{y 0})}^{2} / 4]

is the initial angular spectrum distribution of the Gaussian-shaped probe beam with an incident angle

θ

,

w_{y} = W / \cos θ, k_{y0} = k \sin θ

, and

W

is the half-width of the probe beam at the incident plane of 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]:

S_{r} = - \frac{\int_{- \infty}^{+ \infty} | r |^{2} A^{2} \frac{\partial φ_{r}}{\partial k_{y}} d k_{y}}{\int_{- \infty}^{+ \infty} | r |^{2} A^{2} d k_{y}},

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.

Fig. 7 Numerical simulations of the reflected beam from the MIS structure under different controlling bias V_g = −1V for (a), (b) and (c), and V_g = −10V for (d), (e) and (f). The half-widths of the probe beam are W = 300λ for (a) and (d),W = 800λ for (b) and (e), and W = 1600λ for (c) and (f). The black and red curves denote the incident and reflected probe beams, respectively. Other parameters are the same as in Fig. 2.

Download Full Size | PDF

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.

Fig. 8 The lateral shifts versus beam half-width W under different controlling bias and incident angle. Other parameters are the same as in Fig. 2.

Download Full Size | PDF

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.

References and links

1. B. R. Horowitz and T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61(5), 586 (1971). [CrossRef]

2. M. McGuirk and C. K. Carniglia, “An angular spectrum representation approach to the Goos-Hänchen shift,” J. Opt. Soc. Am. 67(1), 103–107 (1977). [CrossRef]

3. R. H. Renard, “Total reflection: a new evaluation of the Goos-Hänchen shift,” J. Opt. Soc. Am. 54(10), 1190–1196 (1964). [CrossRef]

4. H. M. Lai, F. C. Cheng, and W. K. Tang, “Goos-Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3(4), 550–557 (1986). [CrossRef]

5. F. Bretenaker, A. Le Floch, and L. Dutriaux, “Direct measurement of the optical Goos-Hänchen effect in lasers,” Phys. Rev. Lett. 68(7), 931–933 (1992). [CrossRef] [PubMed]

6. E. Pfleghaar, A. Marseille, and A. Weis, “Quantitative investigation of the effect of resonant absorbers on the Goos-Hänchen shift,” Phys. Rev. Lett. 70(15), 2281–2284 (1993). [CrossRef] [PubMed]

7. O. Emile, T. Galstyan, A. Le Floch, and F. Bretenaker, “Measurement of the nonlinear Goos-Hänchen effect for gaussian optical beams,” Phys. Rev. Lett. 75(8), 1511–1513 (1995). [CrossRef] [PubMed]

8. W. J. Wild and C. L. Giles, “Goos-Hänchen shifts from absorbing media,” Phys. Rev. A 25(4), 2099–2101 (1982). [CrossRef]

9. H. M. Lai and S. W. Chan, “Large and negative Goos-Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. 27(9), 680–682 (2002). [CrossRef] [PubMed]

10. D. Felbacq, A. Moreau, and R. Smaâli, “Goos-Hänchen effect in the gaps of photonic crystals,” Opt. Lett. 28(18), 1633–1635 (2003). [CrossRef] [PubMed]

11. P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(6), 067603 (2002). [CrossRef] [PubMed]

12. X. Chen, R. R. Wei, M. Shen, Z. F. Zhang, and C. F. Li, “Bistable and negative lateral shifts of the reflected light beam from Kretschmann configuration with nonlinear left-handed metamaterials,” Appl. Phys. B 101(1-2), 283–289 (2010). [CrossRef]

13. C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. 91(13), 133903 (2003). [CrossRef] [PubMed]

14. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77(2), 633–673 (2005). [CrossRef]

15. S. Li, X. Yang, X. Cao, C. Xie, and H. Wang, “Two electromagnetically induced transparency windows and an enhanced electromagnetically induced transparency signal in a four-level tripod atomic system,” J. Phys. B 40(16), 3211–3219 (2007). [CrossRef]

16. Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99(12), 123603 (2007). [CrossRef] [PubMed]

17. Y. Chen, X. G. Wei, and B. S. Ham, “Optical properties of an N-type system in Doppler- broadened multilevel atomic media of the rubidium D2 line,” J. Phys. B 42(6), 065506 (2009). [CrossRef]

18. X. Chen, Y. Ban, and C.-F. Li, “Voltage-tunable lateral shifts of ballistic electrons in semiconductor quantum slabs,” J. Appl. Phys. 105(9), 093710 (2009). [CrossRef]

19. X. Chen, X. J. Lu, Y. Wang, and C. F. Li, “Controllable Goos-Hänchen shifts and spin beam splitter for ballistic electrons in a parabolic quantum well under a uniform magnetic field,” Phys. Rev. B 83(19), 195409 (2011). [CrossRef]

20. T. Hashimoto and T. Yoshino, “Optical heterodyne sensor using the Goos-Hänchen shift,” Opt. Lett. 14(17), 913–915 (1989). [CrossRef] [PubMed]

21. X. Hu, Y. Huang, W. Zhang, D. K. Qing, and J. Peng, “Opposite Goos-Hänchen shifts for transverse-electric and transverse-magnetic beams at the interface associated with single-negative materials,” Opt. Lett. 30(8), 899–901 (2005). [CrossRef] [PubMed]

22. L. G. Wang, M. Ikram, and M. S. Zubairy, “Control of the Goos-Hänchen shift of a light beam via a coherent driving field,” Phys. Rev. A 77(2), 023811 (2008). [CrossRef]

23. S. Ziauddin, S. Qamar, and M. S. Zubairy, “Coherent control of the Goos-Hänchen shift,” Phys. Rev. A 81(2), 023821 (2010). [CrossRef]

24. S. Ziauddin and Qamar, “Control of the Goos-Hänchen shift using a duplicated two-level atomic medium,” Phys. Rev. A 85(5), 055804 (2012). [CrossRef]

25. Y. Wang, Z. Q. Cao, H. G. Li, J. Hao, T. Y. Yu, and Q. S. Shen, “Electric control of spatial beam position based on the Goos-Hanchen effect,” Appl. Phys. Lett. 93(9), 091103 (2008). [CrossRef]

26. X. Chen, M. Shen, Z. F. Zhang, and C. F. Li, “Tunable lateral shift and polarization beam splitting of the transmitted light beam through electro-optic crystals,” J. Appl. Phys. 104(12), 123101 (2008). [CrossRef]

27. C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1995).

28. Y. C. Jun, E. Gonzales, J. L. Reno, E. A. Shaner, A. Gabbay, and I. Brener, “Active tuning of mid-infrared metamaterials by electrical control of carrier densities,” Opt. Express 20(2), 1903–1911 (2012). [CrossRef] [PubMed]

29. Sadao Adachi, Properties of Group-IV, III–V and II–VI Semiconductors (Wiley, 2005).

30. R. F. Pierret, Semiconductor Device Fundamentals (Addison Wesley, 1996).

31. M. A. Ordal, L. L. Long, R. J. Bell, S. E. Bell, R. R. Bell, R. W. Alexander Jr, and C. A. Ward, “Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared,” Appl. Opt. 22(7), 1099–20 (1983). [CrossRef] [PubMed]

32. N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(44 Pt 2B), 046607 (2002). [CrossRef] [PubMed]

33. A. M. Steinberg and R. Y. Chiao, “Tunneling delay times in one and two dimensions,” Phys. Rev. A 49(5), 3283–3295 (1994). [CrossRef] [PubMed]

34. L. G. Wang, H. Chen, and S. Y. Zhu, “Wave propagation inside one-dimensional photonic crystals with single-negative materials,” Phys. Lett. A 350(5-6), 410–415 (2006). [CrossRef]

35. C. F. Li, “Unified theory for Goos-Hänchen and Imbert-Fedorov effects,” Phys. Rev. A 76(1), 013811 (2007). [CrossRef]

36. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008). [CrossRef] [PubMed]

37. M. Merano, A. Aiello, M. P. van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3(6), 337–340 (2009). [CrossRef]

38. M. Merano, N. Hermosa, A. Aiello, and J. P. Woerdman, “Demonstration of a quasi-scalar angular Goos-Hänchen effect,” Opt. Lett. 35(21), 3562–3564 (2010). [CrossRef] [PubMed]

39. A. Aiello, M. Merano, and J. P. Woerdman, “Duality between spatial and angular shift in optical reflection,” Phys. Rev. A 80(6), 061801 (2009). [CrossRef]

Electrically controlled Goos-Hänchen shift of a light beam reflected from the metal-insulator-semiconductor structure

Abstract

1. Introduction

2. Model and theory

3. Numerical results and discussion

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

3.2. Influence of structure parameters and wavelength

3.3. Numerical simulations of GH shift in real system

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (14)

Optics Express