## Numerical analysis of the propagation properties of subwavelength semiconductor slit in the terahertz region

Optics Express, Vol. 17, Issue 17, pp. 15359-15371 (2009)

http://dx.doi.org/10.1364/OE.17.015359

Acrobat PDF (323 KB)

### Abstract

The propagation properties of terahertz (THz) waves passing through heavily doped semiconductor slit have been numerically investigated by using the transfer matrix method. The effects of geometrical parameters, carrier concentration, and dielectric materials filling in the slit have been considered. The contour for carrier concentration and slit width show that as slit width and carrier concentration decreases, the effective indices increase and the propagation lengths decrease. For the case of water filling in the slit, temperature has more effect on the imaginary part of propagation constant than the real part. Most of the energy stored in the slit is in the form of electric energy, which firstly decreases and then increases with the decreasing of slit width. It is expected that the semiconductor slit structure is very useful for the practical applications of THz waves in the fields of biological specimen analysis and medical diagnosis.

© 2009 OSA

## 1. Introduction

19. J. Lindberg, K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Spectral analysis of resonant transmission of light through a single sub-wavelength slit,” Opt. Express **12**(4), 623–632 (2004). [PubMed]

20. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. **6**(9), 1928–1932 (2006). [PubMed]

26. M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science **299**(5607), 682–686 (2003). [PubMed]

27. A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature **457**(7225), 71–75 (2009). [PubMed]

28. P. U. Jepsen, U. Møller, and H. Merbold, “Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy,” Opt. Express **15**(22), 14717–14737 (2007). [PubMed]

29. J. Q. Zhang and D. Grischkowsky, “Waveguide terahertz time-domain spectroscopy of nanometer water layers,” Opt. Lett. **29**(14), 1617–1619 (2004). [PubMed]

30. Y. B. Chen, “Development of mid-infrared surface plasmon resonance-based sensors with highly-doped silicon for biomedical and chemical applications,” Opt. Express **17**(5), 3130–3140 (2009). [PubMed]

## 2. Theoretic Model and Research Method

*k*is the wave vector, Eqs. (3)-(8) should be normalized with a common

## 3. Results and discussion

*i*[40], and 3.284 + 0.106

*i*[41], respectively; the carrier concentration of InSb is 8.0 × 10

^{15}cm

^{−3}; the radiation frequency is 1.0 THz. It could be found that as the slit width decreases, the effective indices increase and the propagation lengths decrease, which may result from the fact that the fraction of total electromagnetic energy of GSPPs mode residing in the InSb increases when slit width become smaller. The effective indices of GSPPs mode increase with the increasing of the real part of permittivity for dielectric materials filling in the slit, which may result from the fact that the fraction of GSPPs mode pushed into InSb layer increases. It could also be found from Fig. 2(a) that the propagation length is closely relate to the imaginary part of permittivity for dielectric materials filling in the slit. For metal MDM structure, the large propagation length in the THz region is one of the drawbacks to limit its application. It can be learned from our earlier publication [13] that the propagation length of GSPPs mode of heavily doped InSb is much smaller than that of metal structure. The propagation length could also be largely reduced by filling the slit with different dielectric materials, which has been shown in Fig. 2(a). For example, the propagation length are 1.28 × 10⁴

^{2}

30. Y. B. Chen, “Development of mid-infrared surface plasmon resonance-based sensors with highly-doped silicon for biomedical and chemical applications,” Opt. Express **17**(5), 3130–3140 (2009). [PubMed]

^{1}⁶cm

^{−3}; the water temperature is 292.3 K. As frequency increases, the effective indices and the propagation lengths decrease. The reason may come from the fact that the permittivity of water decreases with the increasing of frequency. The dielectric constant of water are 15.56 + 8.84

*i*, 7.57 + 6.16

*i*, 6.12 + 4.14

*i*, and 5.39 + 2.39

*i*with the corresponding frequency of 0.1 THz, 0.3 THz, 0.5 THz, and 1.0 THz. As shown above, the effective indices are mainly depended on the real part of dielectric materials filling in the slit. Therefore, the larger dielectric constant of water at lower frequency leads to larger effective index. The effects of temperature on the dispersive properties have been shown in Fig. 4(b), which manifests that the propagation length decreases with the increasing of temperature; the radiation frequency is 1.0 THz; the temperature are 278.8 K, 292.3 K, 315.0 K, and 366.7 K, respectively; their dielectric constant are 5.40 + 1.86

*i*, 5.39 + 2.39

*i*, 5.30 + 3.15

*i*, and 6.00 + 4.54

*i*, respectively. As temperature increases, the real part of dielectric constant of water increases slowly, while the imaginary part of water increases seriously. This case is similar to the results given in Ref. 30

30. Y. B. Chen, “Development of mid-infrared surface plasmon resonance-based sensors with highly-doped silicon for biomedical and chemical applications,” Opt. Express **17**(5), 3130–3140 (2009). [PubMed]

*x*direction, the length along

*x*axis is normalized by

*w*, the slit materials is heavily doped InSb. The carrier concentration is 8.0×10

^{16}cm

^{−3}; the slit width is 20

*e*with respect to the value at the interface between InSb and dielectric core. As the permittivity of core dielectric materials increases, the propagation constant increases, the penetration depth decreases, leading to the fact that THz waves penetrate InSb more quickly. For example, the penetration depth are 1.44

## 4. Conclusions

## Acknowledgments

## References and links

1. | B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. |

2. | M. Lee and M. C. Wanke, “Design of n-type silicon-based quantum cascade lasers for terahertz light emission,” Science |

3. | T. H. Isaac, W. L. Barnes, and E. Hendry, “Determining the terahertz optical properties of subwavelength films using semiconductor surface plasmons,” Appl. Phys. Lett. |

4. | R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchi, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature |

5. | J. C. Cao, A. Z. Li, X. L. Lei, and S. L. Feng, “Current self-oscillation and driving-frequency dependence of negative-effective-mass diodes,” Appl. Phys. Lett. |

6. | J. T. Lü and J. C. Cao, “Coulomb scattering in the Monte Carlo simulation of terahertz quantum-cascade lasers,” Appl. Phys. Lett. |

7. | H. Li, J. C. Cao, J. T. Lü, and Y. J. Han, “Monte Carlo simulation of extraction barrier width effects on terahertz quantum cascade lasers,” Appl. Phys. Lett. |

8. | J. C. Cao, “Interband impact ionization and nonlinear absorption of terahertz radiation in semiconductor heterostructures,” Phys. Rev. Lett. |

9. | K. L. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature |

10. | J. A. Deibel, K. Wang, M. D. Escarra, and D. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express |

11. | X. Y. He, “Investigation of terahertz Sommerfeld propagation along conical metal wire,” J. Opt. Soc. Am. B |

12. | J. Q. Zhang and D. Grischkowsky, “Adiabatic compression of parallel-plate metal waveguides for sensitivity enhancement of waveguide THz time-domain spectroscopy,” Appl. Phys. Lett. |

13. | X. Y. He, “Comparison of the waveguide properties of gap surface plasmon in the terahertz region and visible spectra,” J. Opt. A, Pure Appl. Opt. |

14. | R. Mendis and M. Daniel, “Mittleman, “An investigation of the lowest-order transverse-electric (TE |

15. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

16. | S. I. Bozhevolnyi and J. Jung, “Scaling for gap plasmon based waveguides,” Opt. Express |

17. | H. Raether, “Surface plasmons on smooth and rough surfaces and on gratings,” (Springer, Berlin, 1988). |

18. | P. Neutens, P. V. Dorpe, I. D. Vlaminck, L. Lagae, and G. Borghs, “Electrical detection of confined gap plasmons in metal-insulator-metal waveguides,” Nat. Photonics |

19. | J. Lindberg, K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Spectral analysis of resonant transmission of light through a single sub-wavelength slit,” Opt. Express |

20. | J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. |

21. | T. H. Isaac, J. Gomez., J. R. Rivas, W. L. Sambles, Barnes, and E. Hendry, “Surface plasmon mediated transmission of subwavelength slits at THz frequencies,” Phys. Rev. B |

22. | Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. |

23. | R. M. Gelfand, L. Bruderer, and H. Mohseni, “Nanocavity plasmonic device for ultrabroadband single molecule sensing,” Opt. Lett. |

24. | M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics |

25. | K. C. Vernon, D. K. Gramontnev, and D. F. P. Pile, “Channel plasmon-polariton modes in V grooves filled with dielectric,” J. Appl. Phys. |

26. | M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science |

27. | A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature |

28. | P. U. Jepsen, U. Møller, and H. Merbold, “Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy,” Opt. Express |

29. | J. Q. Zhang and D. Grischkowsky, “Waveguide terahertz time-domain spectroscopy of nanometer water layers,” Opt. Lett. |

30. | Y. B. Chen, “Development of mid-infrared surface plasmon resonance-based sensors with highly-doped silicon for biomedical and chemical applications,” Opt. Express |

31. | S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. |

32. | M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express |

33. | R. Mendis, “Nature of subpicosecond terahertz pulse propagation in practical dielectric-filled parallel-plate waveguides,” Opt. Lett. |

34. | R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. |

35. | S. W. Gao, J. C. Cao, and S. L. Feng, “Waveguide design of long wavelength semiconductor laser based on surface plasmons,” Physica B |

36. | J. T. Lü and J. C. Cao, “Confined optical phonon modes and electron-phonon interactions in wurtzite GaN/ZnO quantum wells,” Phys. Rev. B |

37. | F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B |

38. | Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocatities: Analysis of optical properties,” Phys. Rev. B |

39. | J. A. Sánchez-Gil and J. G. Rivas, “Thermal switching of the scattering coefficients of terahertz surface plasmon polaritons impinging on a finite array of subwavelength grooves on semiconductor surfaces,” Phys. Rev. B |

40. | C. Rønne, P. O. Åstrand, and S. R. Keiding, “THz spectroscopy of liquid H |

41. | S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. |

42. | A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.3090) Physical optics : Infrared, far

(040.2235) Detectors : Far infrared or terahertz

(050.6624) Diffraction and gratings : Subwavelength structures

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: June 25, 2009

Revised Manuscript: July 24, 2009

Manuscript Accepted: July 24, 2009

Published: August 14, 2009

**Virtual Issues**

Vol. 4, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Xiao-Yong He, "Numerical analysis of the propagation properties of subwavelength semiconductor slit in the terahertz region," Opt. Express **17**, 15359-15371 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-17-15359

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### References

- B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1(1), 26–33 (2002).
- M. Lee and M. C. Wanke, “Design of n-type silicon-based quantum cascade lasers for terahertz light emission,” Science 316, 64–65 (2007). [PubMed]
- T. H. Isaac, W. L. Barnes, and E. Hendry, “Determining the terahertz optical properties of subwavelength films using semiconductor surface plasmons,” Appl. Phys. Lett. 93(24), 241115 (2008).
- R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchi, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417, 154–157 (2002).
- J. C. Cao, A. Z. Li, X. L. Lei, and S. L. Feng, “Current self-oscillation and driving-frequency dependence of negative-effective-mass diodes,” Appl. Phys. Lett. 79(21), 3524–3526 (2001).
- J. T. Lü and J. C. Cao, “Coulomb scattering in the Monte Carlo simulation of terahertz quantum-cascade lasers,” Appl. Phys. Lett. 89(21), 211115 (2006).
- H. Li, J. C. Cao, J. T. Lü, and Y. J. Han, “Monte Carlo simulation of extraction barrier width effects on terahertz quantum cascade lasers,” Appl. Phys. Lett. 92(22), 221105 (2008).
- J. C. Cao, “Interband impact ionization and nonlinear absorption of terahertz radiation in semiconductor heterostructures,” Phys. Rev. Lett. 91(23), 237401 (2003). [PubMed]
- K. L. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [PubMed]
- J. A. Deibel, K. Wang, M. D. Escarra, and D. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006). [PubMed]
- X. Y. He, “Investigation of terahertz Sommerfeld propagation along conical metal wire,” J. Opt. Soc. Am. B 26(9), A23–A28 (2009).
- J. Q. Zhang and D. Grischkowsky, “Adiabatic compression of parallel-plate metal waveguides for sensitivity enhancement of waveguide THz time-domain spectroscopy,” Appl. Phys. Lett. 86(6), 061109 (2005).
- X. Y. He, “Comparison of the waveguide properties of gap surface plasmon in the terahertz region and visible spectra,” J. Opt. A, Pure Appl. Opt. 11(4), 045708 (2009).
- R. Mendis and M. Daniel, “Mittleman, “An investigation of the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide for THz pulse propagation,” J. Opt. Soc. Am. B 26(9), A6–A13 (2009).
- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [PubMed]
- S. I. Bozhevolnyi and J. Jung, “Scaling for gap plasmon based waveguides,” Opt. Express 16(4), 2676–2684 (2008). [PubMed]
- H. Raether, “Surface plasmons on smooth and rough surfaces and on gratings,” (Springer, Berlin, 1988).
- P. Neutens, P. V. Dorpe, I. D. Vlaminck, L. Lagae, and G. Borghs, “Electrical detection of confined gap plasmons in metal-insulator-metal waveguides,” Nat. Photonics 3(5), 283–286 (2009).
- J. Lindberg, K. Lindfors, T. Setala, M. Kaivola, and A. T. Friberg, “Spectral analysis of resonant transmission of light through a single sub-wavelength slit,” Opt. Express 12(4), 623–632 (2004). [PubMed]
- J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Lett. 6(9), 1928–1932 (2006). [PubMed]
- T. H. Isaac, J. Gomez., J. R. Rivas, W. L. Sambles, Barnes, and E. Hendry, “Surface plasmon mediated transmission of subwavelength slits at THz frequencies,” Phys. Rev. B 77(11), 113411 (2008).
- Y. Todorov, A. M. Andrews, I. Sagnes, R. Colombelli, P. Klang, G. Strasser, and C. Sirtori, “Strong light-matter coupling in subwavelength metal-dielectric microcavities at terahertz frequencies,” Phys. Rev. Lett. 102(18), 186402 (2009). [PubMed]
- R. M. Gelfand, L. Bruderer, and H. Mohseni, “Nanocavity plasmonic device for ultrabroadband single molecule sensing,” Opt. Lett. 34(7), 1087–1089 (2009). [PubMed]
- M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3(3), 152–156 (2009).
- K. C. Vernon, D. K. Gramontnev, and D. F. P. Pile, “Channel plasmon-polariton modes in V grooves filled with dielectric,” J. Appl. Phys. 103(3), 034304 (2008).
- M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003). [PubMed]
- A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009). [PubMed]
- P. U. Jepsen, U. Møller, and H. Merbold, “Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy,” Opt. Express 15(22), 14717–14737 (2007). [PubMed]
- J. Q. Zhang and D. Grischkowsky, “Waveguide terahertz time-domain spectroscopy of nanometer water layers,” Opt. Lett. 29(14), 1617–1619 (2004). [PubMed]
- Y. B. Chen, “Development of mid-infrared surface plasmon resonance-based sensors with highly-doped silicon for biomedical and chemical applications,” Opt. Express 17(5), 3130–3140 (2009). [PubMed]
- S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006).
- M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12(17), 4072–4079 (2004). [PubMed]
- R. Mendis, “Nature of subpicosecond terahertz pulse propagation in practical dielectric-filled parallel-plate waveguides,” Opt. Lett. 31(17), 2643–2645 (2006). [PubMed]
- R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007).
- S. W. Gao, J. C. Cao, and S. L. Feng, “Waveguide design of long wavelength semiconductor laser based on surface plasmons,” Physica B 337(1-4), 230–236 (2003).
- J. T. Lü and J. C. Cao, “Confined optical phonon modes and electron-phonon interactions in wurtzite GaN/ZnO quantum wells,” Phys. Rev. B 71(15), 155304 (2005).
- F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B 44(11), 5855–5872 (1991).
- Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocatities: Analysis of optical properties,” Phys. Rev. B 75(3), 035411 (2007).
- J. A. Sánchez-Gil and J. G. Rivas, “Thermal switching of the scattering coefficients of terahertz surface plasmon polaritons impinging on a finite array of subwavelength grooves on semiconductor surfaces,” Phys. Rev. B 73(20), 205410 (2006).
- C. Rønne, P. O. Åstrand, and S. R. Keiding, “THz spectroscopy of liquid H2O and D2O,” Phys. Rev. Lett. 82(14), 2888–2891 (1999).
- S. Kohen, B. S. Williams, and Q. Hu, “Electromagnetic modeling of terahertz quantum cascade laser waveguides and resonators,” J. Appl. Phys. 97(5), 053106 (2005).
- A. K. Azad, Y. Zhao, and W. Zhang, “Transmission properties of terahertz pulses through an ultrathin subwavelength silicon hole array,” Appl. Phys. Lett. 86(14), 141102 (2005).

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