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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 10 — Oct. 2, 2009
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Numerical analysis of the propagation properties of subwavelength semiconductor slit in the terahertz region

Xiao-Yong He  »View Author Affiliations


Optics Express, Vol. 17, Issue 17, pp. 15359-15371 (2009)
http://dx.doi.org/10.1364/OE.17.015359


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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

Gap (slit) surface polariton plasmons (GSPPs) waveguide is an important kind of subwavelength plasmonic devices and can be regarded as the metal-dielectric-metal (MDM) structure. Much research has been carried out to investigate the MDM structure in the visible [13

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. 11(4), 045708 (2009).

,19

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

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]

], infrared, and THz region [21

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 77(11), 113411 (2008).

24

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 3(3), 152–156 (2009).

]. MDM plasmonic waveguide demonstrates the merits of strong subwavelength localization of plasmon, weak dissipation, the possibility of single mode operation [25

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. 103(3), 034304 (2008).

], showing potential in many practical applications fields. For instance, MDM plasmonic structure can be used to conduct single-molecule analysis requiring pico- to nanomolar concentrations of fluorophore [26

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]

], fabricate optical tweezers for transporting micrometer or nanometer dielectric particles (water molecular or DNA molecules) [27

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]

]. The solute (alcohol, sugar) concentration of the aqueous solution can be determined by measuring the changes of the dielectric properties of liquids compressed between the metal plate [28

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]

]. The thickness and refractive index of the nanometer water-layer [29

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

] can also be acquired by comparing the dielectric constant from the empty guide with that from the waveguide containing the dielectric layer. With biological analyte compressed between the plates, MDM or modified MDM structure can also be utilized to develop SPPs biosensors by probing the interaction between analyte and THz waves [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]

].

Due to the fact that metals always show large dielectric constant in the THz region, the decay length of SPPs mode for metal-based MDM structure is very long, which weakens the SPPs mode and limits its practical applications. One of the possible solution methods is adopting active dielectric core materials [31

31. S. A. Maier, “Gain-assisted propagation of electromagnetic energy in subwavelength surface plasmon polariton gap waveguides,” Opt. Commun. 258(2), 295–299 (2006).

34

34. R. Mendis, “THz transmission characteristics of dielectric-filled parallel-plate waveguides,” J. Appl. Phys. 101(8), 083115 (2007).

]. In the THz region, the imaginary part of many dielectric material, such as doped semiconductor (GaAs), biological analyte containing water or other polar molecules, are always large, which is different from the case in the visible and infrared spectral region. To Investigate the propagation properties of MDM structure with complex dielectric constant core materials is very interesting and important. For the plasma frequencies within far-infrared, heavily doped semiconductors also show metallic characters in the THz region, and their dielectric constant are similar to that of metal (Ag and Au) in visible/UV spectral range. Heavily doped semiconductors have many merits and show more flexibility in fabricating waveguide devices. Furthermore, the SPPs modes on heavily doped semiconductor is significantly more sensitive to the dielectric layer than surface modes supported on a metal substrate [3

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. 93(24), 241115 (2008).

]. Indium Antimonide (InSb) is a kind of narrow gap semiconductor and has high intrinsic electron density at room temperature. Therefore, the propagation properties of GSPPs mode in the THz region based on heavily doped semiconductor slit have been shown and discussed. In addition, the effects of geometrical parameter, slit materials, and the core dielectric materials with complex refractive index on waveguide propagation properties have also been explored.

2. Theoretic Model and Research Method

The field component of transverse magnetic (TM) mode in the slit materials (|x|>w/2) can be written as [38

38. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocatities: Analysis of optical properties,” Phys. Rev. B 75(3), 035411 (2007).

]:
Exm(r,ω)=exp(α1,3|x|)×exp(i(βzωt)), Eq.
(5)
Hym(r,ω)=ωε0εmβexp(α1,3|x|)×exp(i(βzωt)), Eq.
(6)
Ezm(r,ω)=iα1,3β|x|xexp(α1,3|x|)×exp(i(βzωt)), Eq.
(7)
and in the slit (i.e. (|x|<w/2)),
Exd(r,ω)=2cosh(α2|x|)×exp(i(βzωt)), Eq.
(8)
Hyd(r,ω)=2ωε0ε2βcosh(α2|x|)×exp(i(βzωt)), Eq.
(9)
Ezd(r,ω)=2iα2βsinh(α2|x|)×exp(i(βzωt)), Eq.
(10)
where k is the wave vector, Eqs. (3)-(8) should be normalized with a common E0.

The electric and magnetic energy densities in the dielectric core are [38

38. Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocatities: Analysis of optical properties,” Phys. Rev. B 75(3), 035411 (2007).

]:
uei(x,ω)=14ε0εi|εm(ω)εiexp(kmT/2)cosh(kiT/2)|2|A|2(|cosh(kix)|2+|kiksinh(kix)2|), Eq.
(11)
umi(x,ω)=14ε0|ωεm(ω)ckexp(kmT/2)cosh(kiT/2)|2|A|2(|cosh(kix)|2), Eq.
(12)
while the electric and magnetic energy densities in the heavily doped semiconductor are:

uem(x,ω)=14ε0Re((ωεm(ω))ω)|Aexp(km|x|)|2(1.0+|kmk|2), Eq.
(13)
umm(x,ω)=14ε0|Aωεm(ω)ckexp(km|x|)|2, Eq.
(14)

where km=β2εmk02, ki=β2εdk02, and the factor of A should be chosen to make the following Eq. (38):

Uei+Umi+Uem+Umm=1, Eq. (15) in which Uei and Umi are the total electric energy and magnetic energy in the dielectric core, Uem and Umm are the total electric energy and magnetic energy in the metal, which could be acquired by integrating energy density above Eqs. (11)-(14).

neff=Re(β)/k0, Eq.
(21)
L=[2Im(β)]1. Eq.
(22)

3. Results and discussion

Figure 2(a)
Fig. 2 The effective indices and propagation lengths of GSPPs mode versus slit width for different dielectric materials, the radiation frequency is 1.0 THz; the carrier concentration of InSb is 8.0×1016 cm−3. (a) The dielectric materials filling in the slit are air, polyethylene, water, and GaAs, respectively. (b) The dielectric materials filling in the slit is GaAs, the real part of dielectric constant keeps 3.4, the imaginary part are 0.0, 0.02, 0.05, 0.10, 0.20 and 0.50, respectively.
shows the effective indices and propagation lengths of GSPPs mode with different core dielectric materials. The dielectric materials filling in the slit are air, low-density polyethylene, water, and GaAs, with the corresponding refractive index of 1.0, 1.51 [42

42. 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).

], 2.375 + 0.502i [40

40. C. Rønne, P. O. Åstrand, and S. R. Keiding, “THz spectroscopy of liquid H2O and D2O,” Phys. Rev. Lett. 82(14), 2888–2891 (1999).

], and 3.284 + 0.106i [41

41. 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).

], respectively; the carrier concentration of InSb is 8.0 × 1015cm−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

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. 11(4), 045708 (2009).

] 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 × 10⁴μm and 2.06 × 102 μm for air and GaAs filling in the slit (the slit width is 100 μm). The absorbing core dielectric materials lead to the propagation length of SPPs reducing significantly, which is according with the results in Ref. 3

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. 93(24), 241115 (2008).

. The GSPPs mode can be better confined by filling dielectric materials (water or GaAs) in the slit, which is very interesting and important for the application of semiconductor made MDM structure. Figure 2(b) displays that the effects of imaginary part of permittivity of core dielectric materials on the dispersive properties, the dielectric material filling in the slit is GaAs. The dielectric constant of GaAs can be changed with carrier concentration, shown in Eq. (19). The real part of dielectric constant of GaAs keeps constant because it changes litter with the changing of carrier concentration, the imaginary part of dielectric constant of GaAs are adopted as 0.0, 0.02, 0.05, 0.10, 0.20, and 0.50, respectively. It also could be found from Fig. 2(b) that the real part of effective indices change litter, while the propagation lengths decrease noticeably with the increasing of imaginary part of dielectric materials filling in the slit. The reason is as follows. As the imaginary part of permittivity of dielectric materials filling in the slit increases, the imaginary part of effective index of GSPPs mode increases, leading to the propagation length increasing.

Water is the major ingredient in biological materials and shows a large imaginary part of dielectric constant in the THz region. THz waves can be used to diagnose cancer or tumors by measuring water content in tumors or cancer, which contain a larger amount of water. Because the changes of propagation constant caused by THz waves with analyte is proportional to the refractive index change [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]

], subwavelength dimensions biological analyte can be investigated by using SPPs biosensors based on MDM structure. It could also be found from Eq. (17) that the dielectric constant of water is closely relative to temperature and frequency, which is different from the case in the visible and near infrared spectral region [13

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. 11(4), 045708 (2009).

]. The effective indices and propagation length for the case of water filling in the slit at different frequencies have been shown in Fig. 4(a)
Fig. 4 The GSPPs mode of effective indices and propagation lengths versus slit width, the dielectric material filling in the slit is water with different radiation frequencies and temperatures, the doping concentration of InSb is 8.0×1016 cm−3. (a) The water temperature is 292.3 K, the radiation frequencies are 0.1 THz, 0.3 THz, 0.5 THz, and 1.0 THz, respectively. (b) The water temperature are 278.8 K, 292.3 K, 315.0 K, and 366.7 K, respectively.
; the carrier concentration of InSb is 8.0 × 101⁶cm−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 + 8.84i, 7.57 + 6.16i, 6.12 + 4.14i, and 5.39 + 2.39i 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 + 1.86i, 5.39 + 2.39i, 5.30 + 3.15i, and 6.00 + 4.54i, 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]

, which displays that the solute (sucrose, alcohol) concentration has larger effect on the imaginary part of refractive index than that of real part. The possible reason maybe that as temperature increases, the motion of water molecular and the friction between them increases, the imaginary part of permittivity of water increasing, leading to the propagation length dropping. Therefore, temperature has more effects on the propagation length than the effective indices.

Figure 5(a)
Fig. 5 The field distribution of electric component of GSPPs mode along the x-direction. The slit material is heavily doped InSb with carrier concentration of 8.0×1016 cm−3, the radiation frequency is 1.0 THz. (a) The dielectric materials filling in the slit are air, polyethylene, water, and GaAs, respectively, the slit width is 20 μm. (b) The slit width are 1.0 μm, 10.0 μm, 20.0 μm, 50.0 μm, 100.0 μm, and 200.0 μm, respectively, air is filled in the slit.
shows the field distribution of the electric component along x direction, the length along x axis is normalized by w, the slit materials is heavily doped InSb. The carrier concentration is 8.0×1016 cm−3; the slit width is 20 μm; the radiation frequency is 1.0 THz; the dielectric materials filling in the slit are air, polyethylene, water, and GaAs, respectively. It can be found that the GSPPs mode shows the maximum at the metal-dielectric interface for different slit widths. Furthermore, as the refractive indices of dielectric materials filling in the slit increase, the modes decrease more quickly, which means that mode can be better confined with the increasing refractive index of the core dielectric materials. The penetration depth can be defined by the distance where the absolute value of Ex field decreased by a factor 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 μm, 0.96 μm, 0.60 μm, and 0.44 μm for the case of air, polyethylene, water and GaAs filling in the slit, respectively. The normalized mode distribution at different slit width have been shown in Fig. 5(b), the slit width are 1.0 μm, 10.0 μm, 20.0 μm, 50.0 μm, 100.0 μm, and 200.0 μm, air is filled in the slit. Additionally, Fig. 5(b) displays that the mode can be well confined in the wide slit. The reason may be as follows. As slit width decreases, there are more THz waves penetrating into InSb, leading to the effective indices increasing, shown in Fig. 3. This phenomenon can be well explained by the ratio of penetration depth to slit width, which are 1.447, 0.145, 0.072, 0.029, 0.014, and 0.007 with corresponding for slit width of 1.0 μm, 10.0 μm, 20.0 μm, 50.0 μm, 100.0 μm, and 200.0 μm, respectively.

Fig. 6 The electric and magnetic energy densities in the metal and slit versus frequency for different slit width are shown in Fig. 6(a)-6(d), respectively. Air is filled in the slit; the carrier concentration of InSb is 8.0×1016 cm−3; the slit width are 1 μm, 2 μm, 10 μm, 20 μm, 50 μm, and 100 μm, respectively.

4. Conclusions

Acknowledgments

This work is supported by the Doctoral Funding of Henan University of Technology (2007BS044).

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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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