Goos-Hänchen shifts of reflected terahertz wave on a COC-air interface

Qingmei Li; Bo Zhang; Jingling Shen

doi:10.1364/OE.21.006480

1. Introduction

In the case of total internal reflection the reflected wave appears a lateral displacement, named Goos-Hänchen (GH) shift, compared with the prediction of geometrical optics on the reflection interface. Renard’s energy flux theory [1] from conservation of energy and Artmann’s stationary phase theory [2] from Fresnel-Maxwell equations were put forward to explain this phenomenon, and they account for why GH shift happened in some extent. The time of interaction of the evanescent and real incident fields, traveling along the interface plays a major role in the GH effect. Due to the key role of the evanescent wave in studying some interesting physical phenomena such as optical tunneling, fast than light, the study of GH shift is a useful topic in finding the essence of these phenomena.

Research about GH shifts focus on attenuated reflection [3], partial reflection [4–6], and frustrated total internal reflection (FTIR) [7–10]. And there are many studies on the simple planer surface [11], prism surface [12, 13], and curved interface [14]. Absolutely, plentiful studies have been carried out on the topic, and have covered many optical regions including visible light [15, 16], infrared wave [17], microwave [13, 18], and terahertz wave [19–22]. Due to the source, detection and invisibility problems, experiments of measuring GH shift directly in terahertz region are difficulty and therefore the related work is less than in the other regions. M. T. Reiten and his colleagues from Oklahoma State University studied GH shift and the optical tunneling on a silicon prism surface in a terahertz time domain system [19–21]. The other related work came from J. J. Carey et al from University of Strathclyde [22], who investigated frustrated total internal reflection and found the noncausal time response in the air gap between two Teflon prisms. However, any report on direct measurement of reflected GH shift of terahertz wave has not been found. Here we extend previous investigations [19–22] to measure the reflected GH shift. The fact that the wave length in terahertz region is about 500 times longer than that of visible light makes us to get obvious GH shift in experiment. We chose a polymer material–Cyclo-Olefin Copolymer (COC) to be the reflected interface. Compared with the silicon, polymer as the material used in THz systems has such advantages as portable, ease of processing, low absorption, and low cost. The COC has been confirmed to be relatively a very good material to make terahertz elements [23, 24]. The study of GH shift happened on the COC-air interface make sense for applications of THz waves in sensor and power delivery systems.

In this work, two configurations are proposed to investigate GH shift in terahertz region. The first one includes a COC double-prism and the second contains a structure of COC prism-air-metal sheet. In the first configuration the investigations of simulation and experiments reveal the same tend behavior of both TM and TE polarized waves, which are introduced in detail in the next several sections. In the second configuration we used a metal sheet to replace the second prism for getting the metal reflection properties in order to carry out our further study on the periodic sub-wavelength structure metal sheet as a reflection surface.

2. Experimental details

The experimental system is illustrated in Fig. 1(a) . The terahertz wave, generated by a Backward Wave Oscillator (BWO), was a continuous source with the central frequency of 0.206 THz. By a spherical lens and a cylindrical lens, the THz wave was focused to a line with the width of 2mm, which is perpendicular to the edges of the prisms. Then, the wave penetrated into a symmetrical prism system, which consisted of two same prisms. The prisms are made of COC with a dimension of 17 × 17 × 10mm, and the refractive index of COC is measured about 1.54 in terahertz region, so the corresponding critical angle is 42°at 0.206 THz. And the reflected wave was detected by a pyroelectric detector. The second prism was placed on a stage to change the thickness of the air gap and keep the faces of the prisms parallel by moving the stage along the direction of the incident THz wave.

Fig. 1 (a) Schematic diagram of the experimental system, the polarization of the wave would be horizontal (b) or vertical (c) with or without the Polaroid sheets.

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Regarding the polarization, the original polarization from the BWO is horizontal or TE wave corresponding to our setup (Fig. 1(b)). We also changed it to vertical polarization or TM wave (Fig. 1(c)) by using two terahertz polaroids with the direction of 45° and 90°, respectively.

3. Theoretical simulation

Due to the difference of Fresnel reflection coefficients, the reflected GH effect differs for TE and TM wave. Artmann [1] derived the GH shift formula of different polarization from Fresnel-Maxwell equations by the stationary phase method:

D_{T E} = \frac{λ}{π n_{1}} \frac{\sin θ_{i}}{{[\sin^{2} θ_{i} - {(n_{2} / n_{1})}^{2}]}^{1 / 2}}

D_{T M} = {(\frac{n_{1}}{n_{2}})}^{2} \frac{λ}{π n_{1}} \frac{\sin θ_{i}}{{[\sin^{2} θ_{i} - {(n_{2} / n_{1})}^{2}]}^{1 / 2}}

where

θ

,

λ

are the incident angle and the wavelength in higher refractive index medium,

n_{1}

and

n_{2}

are the higher and lower refractive index of two medium, respectively.

Figure 2(a) shows the GH shift curves from the Eqs. (1) and (2) at the parameters of $θ_{i} = 45^{\circ}, n_{1} = 1.54$ , and $n_{2} = 1$ , from which one can see easily that the overall trends of the calculated curves of TE and TM waves are the same, that is to say, the GH shift reduces with the increase of the frequency of incident wave. In Fig. 2(b), when $λ = 1.456 m m$ ( $f = 0.206$ THz), $n_{1} = 1.54$ , $n_{2} = 1$ , the curves of GH shift versus the incident angle indicate that GH shift can reach a very large value around the critical angle and reduces to a constant as the angle toward to 90°. These results provide a reference to thefollowing simulation and experiments, the lower frequency and the critical angle of the incident wave are the best conditions to exhibit an obvious GH shift effect.

Fig. 2 Curves of the reflected GH shift versus the frequency and the incident angle for TE (black line) and TM (red line) waves. (a) The GH shift on the planar interface versus the frequency of the incident wave at the incident angle of 45°. (b) The GH shift versus the incident angle at the frequency of 0.206 THz.

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The COMSOL software is employed to simulate the field distribution of the reflection of the terahertz wave for TE polarization. In our simulation the refractive index of a double-triangle prism was set as 1.54, which is the same as the refractive index of COC, and the surroundings was designed as air. The terahertz wave with the frequency of 0.206 THz was incident from the left to the double-prism, and the GH shift was the displacement between the realistic (black solid) and geometrical line (black dashed), as shown in Fig. 3 . The black solid line was drawn at the center of the distribution in the assumption of symmetric THz beam profile.

Fig. 3 Field distribution of the reflection of the TE wave on the COC-air interface with varying the air layer thickness (t) between the two triangular prisms. (a) t = 3 mm, (b) t = 2 mm, (c) t = 1 mm, (d) t = 0.3 mm.

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In Fig. 3, the thickness of the air layer is gradually reduced from 3mm to 0.3mm as shown clockwise from up left to down left. It can be seen that the terahertz wave almost cannot transmit through the air layer to reach to the second prism until the thickness of the air layer is as small as 0.3mm. It is just the case of frustrated total internal reflection in which the evanescent wave has passed energy across the air layer into the third medium, the second prism, or the quantum tunneling has taken place. The penetration depth, determined by the wavelength, the incident angle, and the refractive index in the medium, is defined as the depth at which the intensity of the radiation inside the air falls to $1 / e$ of its original value at the surface. The depth $d_{p}$ can be easily derived as

d_{p} = λ / 4 π \sqrt{\sin^{2} θ n_{1}^{2} - n_{2}^{2}}

For the parameters in our simulation,

θ = 45^{\circ}

,

n_{1} = 1.54

,

n_{2} = 1

, the penetration depth was calculated as 0.27mm at 0.206THz, which is in agreement with the simulation results shown in Fig. 3. Furthermore, when we refer to reflection GH shift, the

d_{p}

can be thought to be the threshold value of the thickness of the air layer.

The value of GH shift increases gradually on the COC-air interface with the increase of the thickness of the air layer so long as it is larger than the threshold value. And while the air layer thickness is smaller than 0.27mm, the total internal reflection condition is no longer valid, that’s to say, the GH effect is broken. Most energy of the transmission field penetrates though the air gap into the second prism.

According to our second configuration to study the GH shifts influenced by varying the air layer thickness between a prism and a metal sheet, we simulated the field distribution of the reflection of TE wave as shown in Fig. 4 . The terahertz electric field distribution in Figs. 4(a)-(c), the case of air layer thickness larger than the penetration depth, indicates that the change trend of GH shift is consistent with the simulation result of our first configuration shown in Fig. 3. While for Fig. 4(d) the situation is different from the one in Fig. 3 due to the reflection of the metal sheet. The GH shift decreases with the decrease of the air layer thickness. However, when the prism is towards to the metal sheet, the GH shift decreases down to zero, revealing that the GH shift phenomenon is broken.

Fig. 4 Field distribution with varying the air layer thickness (t) between the triangular prism and the metal sheet. (a) t = 3 mm, (b) t = 2 mm, (c) t = 1 mm, (d) t = 0.3 mm.

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4. Results and discussion

In the experiment, the reflected terahertz intensity distribution was measured by scanning the pyroelectric detector with the step of 0.5mm as shown in Fig. 5(a) and Fig. 6(a) , respectively, for two different polarization cases. From both Fig. 5(a) and Fig. 6(a) one can see that the beam profile is not symmetry as we predicted. The deformation may come from the error of the two lenses before the terahertz beam entering the prism. More deformation can be seen in Fig. 6(a) since two polaroids were used in the TM case. According to the terahertz intensity distribution the GH effect can be determined by extracting the peak locations of the experimental curves, or the location is proportional to the GH shift of the terahertz beam. The measured GH shifts versus air layer thickness is plotted as in Fig. 5(b) with the blue line, and the simulated results are also shown in Fig. 5(b) in black. It is clear that the changing trend of the two curves is in agreement with each other. From both Figs. 5(a) and 5(b) we can see that the reflection intensity and the GH shift increase up to a saturation value with the increase of the air layer thickness by moving the second prism apart from the first one.

Fig. 5 (a) The reflected terahertz intensity (TE) versus location of the beam for the different thickness of the air gap. (b) The value of the GH shift versus air layer thickness by simulation (black) and experiment (blue).

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Fig. 6 (a) The reflected terahertz energy (TM) versus location of the beam for the different thickness of the air gap. (b) The value of the GH shift versus air layer thickness by simulation (black) and experiment (blue).

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As the polaroid sheets were inserted into the experimental setup, the polarization of the incident wave was changed to be vertical, and in this case the measured terahertz beams are present in Fig. 6(a). The GH shift extracted from the related curves is shown in Fig. 6(b), which has the same meaning as the curves in Fig. 5(b). The measured curve is consistent with the simulated one, when the air layer thickness varies from 0.3mm to 3mm.

From the simulation and experimental results shown in Figs. 5(b) and 6(b) it can be seen that GH shift increases with the increase of the air gap thickness and tend to saturation. This result is consistent with our knowledge about the physical mechanism of the GH shift. As the interaction of the evanescent and the incident fields play a major role in the GH effect, the value of GH shift is affected by the dimension and the energy of the incident wave, and also by the energy of the evanescent wave. With the increase of the thickness of the air gap the incident wave was not changed in our experiment while the energy of the evanescent wave became larger with the tunneling weakened and finally tend to a stable value, which is just the case of plateaus in Figs. 5(b) and 6(b). Furthermore, the value of the GH shift from both the simulation and the measurement is less than that of the prediction in Fig. 2 by the Eqs. (1) and (2). The reason may come from both the hypothesis in the simulation and the precision limitation of the experiments, such as the instability of the terahertz source caused by the perturbation of the chopper frequency.

In the following we investigated the GH shift influenced by the width of the air layer between a prism and an aluminum sheet, and the experimental procedures were the same as in the case of double prisms. Figure 7 shows the GH shift of both TE and TM terahertz intensity varying with the separation of the prism and the Al sheet, and reveals that the GH effect has almost the same variation tendency as the case of double prisms with the air layer thickness. The result shown in Fig. 7 also indicates that TM wave has more GH effect than TE wave and the polarization of the incident wave determines the values of GH shift. Here the overall results found by experiments are consistent in trends with the calculated and simulated curves, shown in Figs. 2(a) and 4, respectively. The difference of the reflected GH shift between the double prisms and the one prism plus an aluminum mainly take place in the thinner air layer case because of the quite different transmission properties of the COC prism and the aluminum sheet, and the optical tunneling can take place in the first configuration. The phase shifts with the different FTIR boundary conditions should also be different but it can only be investigated in a THz-TDS system instead of here in our BWO source and pyroelectric detector setup.

Fig. 7 (a) Sketch of the second configuration consists of a COC triangular prism and an aluminum sheet. (b) The GH shift of the reflected terahertz beam versus the air layer thickness between the prism and the Al sheet.

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5. Conclusions

In summary, we have simulated the field distribution on the COC-air interface, and consequently operated the experiments to measure the GH shifts upon two total internal reflection configurations in the terahertz region. We found that the GH effect always exists when a total internal reflection takes place, whether the third medium is a prism with a high transmissivity or an aluminum sheet. Furthermore, in the case where the air layer thickness is larger than penetration depth, the GH shift increases with the increase of the air layer thickness until the saturation is reached. Certainly, the polarization of the incident wave affects the value of GH shift, and TM wave has a larger value of GH shift than that of TE wave.

The double-prism structures or the prism-metal sheet assembly discussed in this paper is commonly used in a terahertz setup and therefore the results can be as references for explaining some phenomena in terahertz experiments. The study of GH effect on the COC-air interface makes sense for COC used to make terahertz devices. Further work should be done to investigate the properties of terahertz propagation such as the speed of light in the situation of frustrated total internal reflection. Certainly, these kinds of investigation must be carried out in a Terahertz TDS.

Acknowledgments

The authors thank Dr. Qu Min for the useful discussions. This work was supported by the Nature Science Foundation of Beijing under Grant No. 4102016.

References and links

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Goos-Hänchen shifts of reflected terahertz wave on a COC-air interface

Abstract

1. Introduction

2. Experimental details

3. Theoretical simulation

4. Results and discussion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (3)

Optics Express