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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6480–6487
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Goos-Hänchen shifts of reflected terahertz wave on a COC-air interface

Qingmei Li, Bo Zhang, and Jingling Shen  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6480-6487 (2013)
http://dx.doi.org/10.1364/OE.21.006480


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Abstract

Goos-Hänchen (GH) shifts of terahertz wave reflected on the Cyclo-Olefin Copolymer (COC)-air interface was investigated in simulation and experiment. The relationship between the GH shifts with the incident angle and the frequency of incident wave were calculated to get a reference for the simulation and experiment. The reflected GH shift was measured on the COC-air interface when a terahertz wave with the frequency of 0.206THz was incident to a COC double-prism. By changing the thickness of the air layer we find experimentally and simulatively that the GH shift and the energy of the reflected wave increases with the increase of the air layer thickness. The study of GH shift can provide useful information for applications of THz waves in sensor and power delivery systems.

© 2013 OSA

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

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

] from conservation of energy and Artmann’s stationary phase theory [2

2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1–2), 87–102 (1948). [CrossRef]

] 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.

2. Experimental details

The experimental system is illustrated in Fig. 1(a)
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.
. 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.

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

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

] derived the GH shift formula of different polarization from Fresnel-Maxwell equations by the stationary phase method:
DTE=λπn1sinθi[sin2θi(n2/n1)2]1/2
(1)
DTM=(n1n2)2λπn1sinθi[sin2θi(n2/n1)2]1/2
(2)
where θ, λ are the incident angle and the wavelength in higher refractive index medium, n1 and n2are the higher and lower refractive index of two medium, respectively.

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
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.
. The black solid line was drawn at the center of the distribution in the assumption of symmetric THz beam profile.

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 dp can be easily derived as
dp=λ/4πsin2θn12n22
(3)
For the parameters in our simulation, θ=45,n1=1.54,n2=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 dpcan 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
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.
. 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.

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)
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).
and Fig. 6(a)
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).
, 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.

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.

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

1.

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

2.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten strahles,” Ann. Phys. 437(1–2), 87–102 (1948). [CrossRef]

3.

F. Pillon, H. Gilles, S. Girard, M. Laroche, R. Kaiser, and A. Gazibegovic, “Goos–Hänchen and Imbert–Fedorov shifts for leaky guided modes,” J. Opt. Soc. Am. B 22(6), 1290–1299 (2005). [CrossRef]

4.

R. P. Riesz and R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2(11), 1809–1817 (1985). [CrossRef]

5.

D. Müller, D. Tharanga, A. A. Stahlhofen, and G. Nimtz, “Nonspecular shifts of microwaves in partial reflection,” Europhys. Lett. 73(4), 526–532 (2006). [CrossRef]

6.

L. M. Zhou, C. L. Zou, Z. F. Han, G. C. Guo, and F. W. Sun, “Negative Goos-Hänchen shift on a concave dielectric interface,” Opt. Lett. 36(5), 624–626 (2011). [CrossRef] [PubMed]

7.

J. Broe and O. Keller, “Quantum-well enhancement of the Goos-Hänchen shift for p-polarized beams in a two-prism configuration,” J. Opt. Soc. Am. A 19(6), 1212–1222 (2002). [CrossRef] [PubMed]

8.

X. Chen, C. F. Li, R. R. Wei, and Y. Zhang, “Goos–Hänchen shifts in frustrated total internal reflection studied with wave-packet propagation,” Phys. Rev. A 80(1), 015803 (2009). [CrossRef]

9.

X. Liu, Q. Yang, P. Zhu, Z. Qiao, and T. Li, “The influence of Goos–H¨anchen shift on total reflection of ultrashort light pulses,” J. Opt. 12(3), 035214 (2010). [CrossRef]

10.

X. M. Liu and Q. F. Yang, “Total internal reflection of a pulsed light beam with consideration of Goos–Hänchen effect,” J. Opt. Soc. Am. B 27(11), 2190–2194 (2010). [CrossRef]

11.

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]

12.

M. Qu and Z. Huang, “Frustrated Total Internal Reflection: Resonant and Negative Goos-Hänchen Shifts in Microwave Regime,” Opt. Commun. 284(10-11), 2604–2607 (2011). [CrossRef]

13.

M. Qu, Z. Huang, and G. Lu, “Investigation on Goos-Hänchen and Imbert-Fedorov shifts of the bounded microwave beam in a double-prism symmetry structure,” J. CUC 17(4), 5–10 (2010) (Science and Technology).

14.

H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: Phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett. 96(24), 243903 (2006). [CrossRef] [PubMed]

15.

H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett. 33(8), 794–796 (2008). [CrossRef] [PubMed]

16.

G. Y. Oh, D. G. Kim, and Y. W. Choi, “The characterization of GH shifts of surface plasmon resonance in a waveguide using the FDTD method,” Opt. Express 17(23), 20714–20720 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-23-20714. [CrossRef] [PubMed]

17.

C. W. Chen, Y. W. Gu, H. P. Chiang, E. J. Sanchez, and P. T. Leung, “Goos–Hänchen shift at an interface of a composite material: effects of particulate clustering,” Appl. Phys. B 104(3), 647–652 (2011). [CrossRef]

18.

J. Unterhinninghofen, U. Kuhl, J. Wiersig, H. J. Stöckmann, and M. Hentschel, “Measurement of the Goos–Hänchen shift in a microwave cavity,” New J. Phys. 13(2), 023013 (2011). [CrossRef]

19.

M. T. Reiten, K. McClatchey, D. Grischkowsky, and R. A. Cheville, “Incidence-angle selection and spatial reshaping of terahertz pulses in optical tunneling,” Opt. Lett. 26(23), 1900–1902 (2001). [CrossRef] [PubMed]

20.

M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3 Pt 2), 036604 (2001). [CrossRef] [PubMed]

21.

M. T. Reiten, “Spatially resolved terahertz propagation,” Oklahoma State University, Dissertation (2006).

22.

J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett. 84(7), 1431–1434 (2000). [CrossRef] [PubMed]

23.

A. Sengupta, A. Bandyopadhyay, B. F. Bowden, J. A. Harrington, and J. F. Federici, “Characterization of olefin copolymers using terahertz spectroscopy,” Electron. Lett. 42(25), 1477–1479 (2006). [CrossRef]

24.

M. Zhang, R. Pan, W. Xiong, T. He, and J. L. Shen, “A compressed terahertz imaging method,” Chin. Phys. Lett. 29(10), 104208 (2012). [CrossRef]

OCIS Codes
(120.5700) Instrumentation, measurement, and metrology : Reflection
(160.5470) Materials : Polymers
(240.7040) Optics at surfaces : Tunneling
(260.3090) Physical optics : Infrared, far

ToC Category:
Physical Optics

History
Original Manuscript: January 9, 2013
Revised Manuscript: February 14, 2013
Manuscript Accepted: February 14, 2013
Published: March 7, 2013

Citation
Qingmei Li, Bo Zhang, and Jingling Shen, "Goos-Hänchen shifts of reflected terahertz wave on a COC-air interface," Opt. Express 21, 6480-6487 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6480


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References

  1. R. H. Renard, “Total reflection: A new evaluation of the Goos-Hänchen shift,” J. Opt. Soc. Am.54(10), 1190–1196 (1964). [CrossRef]
  2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten strahles,” Ann. Phys.437(1–2), 87–102 (1948). [CrossRef]
  3. F. Pillon, H. Gilles, S. Girard, M. Laroche, R. Kaiser, and A. Gazibegovic, “Goos–Hänchen and Imbert–Fedorov shifts for leaky guided modes,” J. Opt. Soc. Am. B22(6), 1290–1299 (2005). [CrossRef]
  4. R. P. Riesz and R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A2(11), 1809–1817 (1985). [CrossRef]
  5. D. Müller, D. Tharanga, A. A. Stahlhofen, and G. Nimtz, “Nonspecular shifts of microwaves in partial reflection,” Europhys. Lett.73(4), 526–532 (2006). [CrossRef]
  6. L. M. Zhou, C. L. Zou, Z. F. Han, G. C. Guo, and F. W. Sun, “Negative Goos-Hänchen shift on a concave dielectric interface,” Opt. Lett.36(5), 624–626 (2011). [CrossRef] [PubMed]
  7. J. Broe and O. Keller, “Quantum-well enhancement of the Goos-Hänchen shift for p-polarized beams in a two-prism configuration,” J. Opt. Soc. Am. A19(6), 1212–1222 (2002). [CrossRef] [PubMed]
  8. X. Chen, C. F. Li, R. R. Wei, and Y. Zhang, “Goos–Hänchen shifts in frustrated total internal reflection studied with wave-packet propagation,” Phys. Rev. A80(1), 015803 (2009). [CrossRef]
  9. X. Liu, Q. Yang, P. Zhu, Z. Qiao, and T. Li, “The influence of Goos–H¨anchen shift on total reflection of ultrashort light pulses,” J. Opt.12(3), 035214 (2010). [CrossRef]
  10. X. M. Liu and Q. F. Yang, “Total internal reflection of a pulsed light beam with consideration of Goos–Hänchen effect,” J. Opt. Soc. Am. B27(11), 2190–2194 (2010). [CrossRef]
  11. 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]
  12. M. Qu and Z. Huang, “Frustrated Total Internal Reflection: Resonant and Negative Goos-Hänchen Shifts in Microwave Regime,” Opt. Commun.284(10-11), 2604–2607 (2011). [CrossRef]
  13. M. Qu, Z. Huang, and G. Lu, “Investigation on Goos-Hänchen and Imbert-Fedorov shifts of the bounded microwave beam in a double-prism symmetry structure,” J. CUC17(4), 5–10 (2010) (Science and Technology).
  14. H. Schomerus and M. Hentschel, “Correcting ray optics at curved dielectric microresonator interfaces: Phase-space unification of Fresnel filtering and the Goos-Hänchen shift,” Phys. Rev. Lett.96(24), 243903 (2006). [CrossRef] [PubMed]
  15. H. G. L. Schwefel, W. Köhler, Z. H. Lu, J. Fan, and L. J. Wang, “Direct experimental observation of the single reflection optical Goos-Hänchen shift,” Opt. Lett.33(8), 794–796 (2008). [CrossRef] [PubMed]
  16. G. Y. Oh, D. G. Kim, and Y. W. Choi, “The characterization of GH shifts of surface plasmon resonance in a waveguide using the FDTD method,” Opt. Express17(23), 20714–20720 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-23-20714 . [CrossRef] [PubMed]
  17. C. W. Chen, Y. W. Gu, H. P. Chiang, E. J. Sanchez, and P. T. Leung, “Goos–Hänchen shift at an interface of a composite material: effects of particulate clustering,” Appl. Phys. B104(3), 647–652 (2011). [CrossRef]
  18. J. Unterhinninghofen, U. Kuhl, J. Wiersig, H. J. Stöckmann, and M. Hentschel, “Measurement of the Goos–Hänchen shift in a microwave cavity,” New J. Phys.13(2), 023013 (2011). [CrossRef]
  19. M. T. Reiten, K. McClatchey, D. Grischkowsky, and R. A. Cheville, “Incidence-angle selection and spatial reshaping of terahertz pulses in optical tunneling,” Opt. Lett.26(23), 1900–1902 (2001). [CrossRef] [PubMed]
  20. M. T. Reiten, D. Grischkowsky, and R. A. Cheville, “Optical tunneling of single-cycle terahertz bandwidth pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.64(3 Pt 2), 036604 (2001). [CrossRef] [PubMed]
  21. M. T. Reiten, “Spatially resolved terahertz propagation,” Oklahoma State University, Dissertation (2006).
  22. J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection?” Phys. Rev. Lett.84(7), 1431–1434 (2000). [CrossRef] [PubMed]
  23. A. Sengupta, A. Bandyopadhyay, B. F. Bowden, J. A. Harrington, and J. F. Federici, “Characterization of olefin copolymers using terahertz spectroscopy,” Electron. Lett.42(25), 1477–1479 (2006). [CrossRef]
  24. M. Zhang, R. Pan, W. Xiong, T. He, and J. L. Shen, “A compressed terahertz imaging method,” Chin. Phys. Lett.29(10), 104208 (2012). [CrossRef]

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