Abstract

The Zenneck THz surface wave (Z-TSW) on metals is discussed with respect to its difficulty in generation and measurement. The spatial collapse of the extent of the Z-TSW evanescent field, upon the addition of a sub-wavelength dielectric layer on the metal surface, is explained by a simple model, in good agreement with exact analytical theory. Experimental measurements of the THz surface wave on an aluminum surface covered with a 12.5 µm thick dielectric layer have completely characterized the resultant single-mode dielectric layer THz surface wave (DL-TSW). The measured frequency-dependent exponential fall-off of the evanescent wave from the surface agrees well with theory. The DL-TSW frequency-dependent absorption coefficient, phase velocity, group velocity and group velocity dispersion have been obtained. These guided-wave parameters compare favorably with other guided wave structures.

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2008 (1)

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

2006 (3)

T.-I. Jeon and D. Grischkowsky, “THz Zenneck surface wave (THz surface plasmon) propagation on a metal sheet,” Appl. Phys. Lett. 88(6), 061113 (2006).
[CrossRef]

L. S. Mukina, M. M. Nazarov, and A. P. Shkurinov, “Propagation of THz plasmon pulse on corrugated and flat metal surfaces,” Surf. Sci. 600(20), 4771–4776 (2006).
[CrossRef]

Y. Zhao and D. Grischkowsky, “Terahertz demonstrations of effectively two-dimensional photonic bandgap structures,” Opt. Lett. 31(10), 1534–1536 (2006).
[CrossRef] [PubMed]

2005 (3)

J. O’Hara, R. Averitt, and A. Taylor, “Prism coupling to terahertz surface plasmon polaritons,” Opt. Express 13(16), 6117–6126 (2005).
[CrossRef] [PubMed]

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

J. 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).
[CrossRef]

2004 (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

2001 (1)

2000 (1)

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” Appl. Phys. Lett. 88, 4449–4451 (2000).

1991 (2)

M. Klopfleisch and U. Schellenberger, “Experimental determination of the attenuation coefficient of surface electromagnetic waves,” J. Appl. Phys. 70(2), 930–934 (1991).
[CrossRef]

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).
[CrossRef]

1989 (1)

D. L. Mills and A. A. Maradudin, “Surface corrugation and surface-polariton binding in the infrared frequency range,” Phys. Rev. B 39(3), 1569–1574 (1989).
[CrossRef]

1986 (1)

K. W. Steijn, R. J. Seymour, and G. I. Stegeman, “Attenuation of far-infrared surface plasmons on overcoated metal,” Appl. Phys. Lett. 49(18), 1151–1153 (1986).
[CrossRef]

1982 (2)

Z. Schlesinger and A. J. Sievers, “IR surface-plasmon attenuation coefficients for Ge-coated Ag and Au metals,” Phys. Rev. B 26(12), 6444–6454 (1982).
[CrossRef]

G. I. Stegeman and R. J. Seymour, “Surface plasmon attenuation by thin film overlayers in the far infrared,” Solid State Commun. 44(9), 1357–1358 (1982).
[CrossRef]

1981 (1)

E. S. Koteles and W. H. McNeill, “Far infrared surface plasmon propagation,” Intl. J. Infared Mil. Wav. 2(2), 361–371 (1981).
[CrossRef]

1979 (1)

D. L. Begley, R. W. Alexander, C. A. Ward, R. Miller, and R. J. Bell, “Propagation distances of surface electromagnetic waves in the far infrared,” Surf. Sci. 81(1), 245–251 (1979).
[CrossRef]

1951 (1)

S. S. Attwood, “Surface-Wave Propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
[CrossRef]

1907 (1)

J. Zenneck, “ Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
[CrossRef]

1899 (1)

A. Sommerfeld, “ Ueber die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Annalen der Physik und Chemie 303(2), 233–290 (1899).
[CrossRef]

Alexander, R. W.

D. L. Begley, R. W. Alexander, C. A. Ward, R. Miller, and R. J. Bell, “Propagation distances of surface electromagnetic waves in the far infrared,” Surf. Sci. 81(1), 245–251 (1979).
[CrossRef]

Armand, D.

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

Attwood, S. S.

S. S. Attwood, “Surface-Wave Propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
[CrossRef]

Averitt, R.

Begley, D. L.

D. L. Begley, R. W. Alexander, C. A. Ward, R. Miller, and R. J. Bell, “Propagation distances of surface electromagnetic waves in the far infrared,” Surf. Sci. 81(1), 245–251 (1979).
[CrossRef]

Bell, R. J.

D. L. Begley, R. W. Alexander, C. A. Ward, R. Miller, and R. J. Bell, “Propagation distances of surface electromagnetic waves in the far infrared,” Surf. Sci. 81(1), 245–251 (1979).
[CrossRef]

Bolívar, P. H.

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Bradberry, G. W.

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).
[CrossRef]

Coutaz, J.-L.

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Garet, F.

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

Gómez Rivas, J.

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Grischkowsky, D.

T.-I. Jeon and D. Grischkowsky, “THz Zenneck surface wave (THz surface plasmon) propagation on a metal sheet,” Appl. Phys. Lett. 88(6), 061113 (2006).
[CrossRef]

Y. Zhao and D. Grischkowsky, “Terahertz demonstrations of effectively two-dimensional photonic bandgap structures,” Opt. Lett. 31(10), 1534–1536 (2006).
[CrossRef] [PubMed]

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

J. 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).
[CrossRef]

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” Appl. Phys. Lett. 88, 4449–4451 (2000).

Janke, C.

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Jeon, T.-I.

T.-I. Jeon and D. Grischkowsky, “THz Zenneck surface wave (THz surface plasmon) propagation on a metal sheet,” Appl. Phys. Lett. 88(6), 061113 (2006).
[CrossRef]

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

Klopfleisch, M.

M. Klopfleisch and U. Schellenberger, “Experimental determination of the attenuation coefficient of surface electromagnetic waves,” J. Appl. Phys. 70(2), 930–934 (1991).
[CrossRef]

Koteles, E. S.

E. S. Koteles and W. H. McNeill, “Far infrared surface plasmon propagation,” Intl. J. Infared Mil. Wav. 2(2), 361–371 (1981).
[CrossRef]

Kurz, H.

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Maradudin, A. A.

D. L. Mills and A. A. Maradudin, “Surface corrugation and surface-polariton binding in the infrared frequency range,” Phys. Rev. B 39(3), 1569–1574 (1989).
[CrossRef]

Martín-Moreno, L.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

McNeill, W. H.

E. S. Koteles and W. H. McNeill, “Far infrared surface plasmon propagation,” Intl. J. Infared Mil. Wav. 2(2), 361–371 (1981).
[CrossRef]

Mendis, R.

R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26(11), 846–848 (2001).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” Appl. Phys. Lett. 88, 4449–4451 (2000).

Miller, R.

D. L. Begley, R. W. Alexander, C. A. Ward, R. Miller, and R. J. Bell, “Propagation distances of surface electromagnetic waves in the far infrared,” Surf. Sci. 81(1), 245–251 (1979).
[CrossRef]

Mills, D. L.

D. L. Mills and A. A. Maradudin, “Surface corrugation and surface-polariton binding in the infrared frequency range,” Phys. Rev. B 39(3), 1569–1574 (1989).
[CrossRef]

Mukina, L. S.

L. S. Mukina, M. M. Nazarov, and A. P. Shkurinov, “Propagation of THz plasmon pulse on corrugated and flat metal surfaces,” Surf. Sci. 600(20), 4771–4776 (2006).
[CrossRef]

Nazarov, M.

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

Nazarov, M. M.

L. S. Mukina, M. M. Nazarov, and A. P. Shkurinov, “Propagation of THz plasmon pulse on corrugated and flat metal surfaces,” Surf. Sci. 600(20), 4771–4776 (2006).
[CrossRef]

O’Hara, J.

Pellemans, H. P. M.

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Pendry, J. B.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Sambles, J. R.

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).
[CrossRef]

Saxler, J.

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Schellenberger, U.

M. Klopfleisch and U. Schellenberger, “Experimental determination of the attenuation coefficient of surface electromagnetic waves,” J. Appl. Phys. 70(2), 930–934 (1991).
[CrossRef]

Schlesinger, Z.

Z. Schlesinger and A. J. Sievers, “IR surface-plasmon attenuation coefficients for Ge-coated Ag and Au metals,” Phys. Rev. B 26(12), 6444–6454 (1982).
[CrossRef]

Seymour, R. J.

K. W. Steijn, R. J. Seymour, and G. I. Stegeman, “Attenuation of far-infrared surface plasmons on overcoated metal,” Appl. Phys. Lett. 49(18), 1151–1153 (1986).
[CrossRef]

G. I. Stegeman and R. J. Seymour, “Surface plasmon attenuation by thin film overlayers in the far infrared,” Solid State Commun. 44(9), 1357–1358 (1982).
[CrossRef]

Shkurinov, A.

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

Shkurinov, A. P.

L. S. Mukina, M. M. Nazarov, and A. P. Shkurinov, “Propagation of THz plasmon pulse on corrugated and flat metal surfaces,” Surf. Sci. 600(20), 4771–4776 (2006).
[CrossRef]

Sievers, A. J.

Z. Schlesinger and A. J. Sievers, “IR surface-plasmon attenuation coefficients for Ge-coated Ag and Au metals,” Phys. Rev. B 26(12), 6444–6454 (1982).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “ Ueber die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Annalen der Physik und Chemie 303(2), 233–290 (1899).
[CrossRef]

Stegeman, G. I.

K. W. Steijn, R. J. Seymour, and G. I. Stegeman, “Attenuation of far-infrared surface plasmons on overcoated metal,” Appl. Phys. Lett. 49(18), 1151–1153 (1986).
[CrossRef]

G. I. Stegeman and R. J. Seymour, “Surface plasmon attenuation by thin film overlayers in the far infrared,” Solid State Commun. 44(9), 1357–1358 (1982).
[CrossRef]

Steijn, K. W.

K. W. Steijn, R. J. Seymour, and G. I. Stegeman, “Attenuation of far-infrared surface plasmons on overcoated metal,” Appl. Phys. Lett. 49(18), 1151–1153 (1986).
[CrossRef]

Taylor, A.

Ward, C. A.

D. L. Begley, R. W. Alexander, C. A. Ward, R. Miller, and R. J. Bell, “Propagation distances of surface electromagnetic waves in the far infrared,” Surf. Sci. 81(1), 245–251 (1979).
[CrossRef]

Yang, F.

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).
[CrossRef]

Zenneck, J.

J. Zenneck, “ Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
[CrossRef]

Zhang, J.

J. 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).
[CrossRef]

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

Zhao, Y.

Annalen der Physik (1)

J. Zenneck, “ Über die Fortpflanzung ebener elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie,” Annalen der Physik 328(10), 846–866 (1907).
[CrossRef]

Annalen der Physik und Chemie (1)

A. Sommerfeld, “ Ueber die Fortpflanzung elektrodynamischer Wellen längs eines Drahtes,” Annalen der Physik und Chemie 303(2), 233–290 (1899).
[CrossRef]

Appl. Phys. Lett. (5)

T.-I. Jeon and D. Grischkowsky, “THz Zenneck surface wave (THz surface plasmon) propagation on a metal sheet,” Appl. Phys. Lett. 88(6), 061113 (2006).
[CrossRef]

K. W. Steijn, R. J. Seymour, and G. I. Stegeman, “Attenuation of far-infrared surface plasmons on overcoated metal,” Appl. Phys. Lett. 49(18), 1151–1153 (1986).
[CrossRef]

J. 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).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” Appl. Phys. Lett. 88, 4449–4451 (2000).

T.-I. Jeon, J. Zhang, and D. Grischkowsky, “THz Sommerfeld wave propagation on a single metal wire,” Appl. Phys. Lett. 86(16), 161904 (2005).
[CrossRef]

C. R. Phys. (1)

M. Nazarov, F. Garet, D. Armand, A. Shkurinov, and J.-L. Coutaz, “Surface Plasmon THz waves on gratings,” C. R. Phys. 9(2), 232–247 (2008).
[CrossRef]

Intl. J. Infared Mil. Wav. (1)

E. S. Koteles and W. H. McNeill, “Far infrared surface plasmon propagation,” Intl. J. Infared Mil. Wav. 2(2), 361–371 (1981).
[CrossRef]

J. Appl. Phys. (2)

M. Klopfleisch and U. Schellenberger, “Experimental determination of the attenuation coefficient of surface electromagnetic waves,” J. Appl. Phys. 70(2), 930–934 (1991).
[CrossRef]

S. S. Attwood, “Surface-Wave Propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (4)

J. Saxler, J. Gómez Rivas, C. Janke, H. P. M. Pellemans, P. H. Bolívar, and H. Kurz, “Time-domain measurements of surface plasmon polaritons in the terahertz frequency range,” Phys. Rev. B 69(15), 155427 (2004).
[CrossRef]

Z. Schlesinger and A. J. Sievers, “IR surface-plasmon attenuation coefficients for Ge-coated Ag and Au metals,” Phys. Rev. B 26(12), 6444–6454 (1982).
[CrossRef]

D. L. Mills and A. A. Maradudin, “Surface corrugation and surface-polariton binding in the infrared frequency range,” Phys. Rev. B 39(3), 1569–1574 (1989).
[CrossRef]

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).
[CrossRef]

Science (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
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Solid State Commun. (1)

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[CrossRef]

Surf. Sci. (2)

L. S. Mukina, M. M. Nazarov, and A. P. Shkurinov, “Propagation of THz plasmon pulse on corrugated and flat metal surfaces,” Surf. Sci. 600(20), 4771–4776 (2006).
[CrossRef]

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Other (12)

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Figures (16)

Fig. 1
Fig. 1

Cross-section detail of the surface wave setup.

Fig. 2
Fig. 2

(a) Intensity curve of surface wave and diffracted wave field for λ0 = 600 μm. (b) Field overlap of the bare metal surface wave field (curve of yellow area) and diffracted field.

Fig. 3
Fig. 3

A proposed large-scale TSW coupling arrangement.

Fig. 4
Fig. 4

Calculated collapse of the evanescent field due to the dielectric film. (a) Bare aluminum surface. (b) Aluminum surface coated with 12.5 μm polyethylene film (n = 1.5). Solid lines show exp [-β0y] (cos β0y) dependence. Dashed line (red on-line) shows exp [-β0y] dependence for 0.5 THz, giving the 153 mm fall-off distance.

Fig. 5
Fig. 5

Theoretical equivalence of slab waveguide structure, if the conductivity in (a) is infinite, then its field distribution is equivalent to the upper half of (b); (c) Ey field profile of coated surface. (n = 1.5) film thickness 12.5 µm; (d) Ey field of the dielectric slab waveguide, TM0 mode at 0.5 THz; Simple model vs. exact theory (e): Exponential fall-off constant; (f) ratio of propagation constant k = 2π ne0 to free space wave vector k0 = 2π /λ0.

Fig. 6
Fig. 6

2D schematic of the system setup.

Fig. 7
Fig. 7

Assembling drawing of surface wave sample.

Fig. 8
Fig. 8

Detail of the THz surface wave setup.

Fig. 9
Fig. 9

3D view of THz receiver of the dashed box 2.

Fig. 10
Fig. 10

(a) TSW pulse on coated surface without block. (b) TSW pulse on coated surface with block (c) The freely propagating diffraction wave given by subtraction of TSW pulse (b) from TSW pulse (a). Inserts show corresponding spectra.

Fig. 11
Fig. 11

THz surface wave pulses measured at the surface and at 0.60 mm above surface. (a): Before compensating the time delay caused by receiver movement. (b): After compensation.

Fig. 12
Fig. 12

Time domain DL-TSW waveforms measured at different distances from surface.

Fig. 13
Fig. 13

Overlapped view of the frequency dependent field fall-off measurement (black) of the coated metal surface with the corresponding theoretical field fall-off (red). Distance from the surface is given in mm and frequency in THz.

Fig. 14
Fig. 14

Experimental and theoretical field fall-off at selected frequencies.

Fig. 15
Fig. 15

Comparison of the experimental (dots) and theoretical (solid line) time domain curves. The insert shows the input pulse with a peak to peak amplitude of 65 pA.

Fig. 16
Fig. 16

(a) The amplitude absorption coefficient due to the dielectric (lower line) and metal (upper line); (b) Ratio of phase velocity vp over c (upper line) and ratio of group velocity vg over c (lower line).

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