Abstract

Quasi-optical techniques are used to efficiently couple freely propagating pulses of terahertz (THz) electromagnetic radiation into circular and rectangular metal waveguides. We have observed very dispersive, low-loss propagation over the frequency band from 0.65 to 3.5 THz with typical waveguide cross-section dimensions on the order of 300 µm and lengths of 25 mm. Classical waveguide theory is utilized to calculate the coupling coefficients into the modes of the waveguide for the incoming focused THz beam. It is shown that the linearly polarized incoming THz pulses significantly couple only into the TE11, TE12, and TM11 modes of the circular waveguide and the TE10 and TM12 modes of the rectangular guide. The propagation of the pulse through the guide is described as a linear superposition of the coupled propagating modes, each with a unique complex propagation vector. This picture explains in detail all the observed features of the THz pulse emerging from the waveguide. We demonstrate both theoretically and experimentally that it is possible to achieve TE10 single-mode coupling and propagation in a suitably sized rectangular waveguide for an incoming focused, linearly polarized THz pulse with a bandwidth covering many octaves in frequency and that overlaps more than 35 waveguide modes. Finally, to facilitate the application of these THz waveguides to THz time-domain spectroscopy of various configurations of dielectrics in the waveguide including surface layers, we present analytic results for the absorption and the dispersion of such layers.

© 2000 Optical Society of America

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References

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  1. M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
    [CrossRef]
  2. D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]
  3. R. W. McGowan, G. Gallot, and D. Grischkowsky, “Propagation of ultra-wideband, short pulses of THz radiation through sub-mm diameter circular waveguides,” Opt. Lett. 24, 1431–1433 (1999).
    [CrossRef]
  4. A. Nahata and T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–708 (1996).
    [CrossRef]
  5. J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, Jr., P. M. Fauchet, and R. Sobolewsky, “Spatiotemporal shaping of half-cycle terahertz pulses by diffraction through conductive apertures of finite thickness,” J. Opt. Soc. Am. B 15, 1399–1405 (1998).
    [CrossRef]
  6. C. Winnewisser, F. Lewen, J. Weinzierl, and H. Helm, “Transmission features of frequency-selective components in the far-infrared determined by terahertz time-domain spectroscopy,” Appl. Opt. 38, 3961–3967 (1999).
    [CrossRef]
  7. J. W. Digby, C. E. Collins, B. M. Towlson, L. S. Karatzas, G. M. Parkhurst, J. M. Chamberlain, J. W. Bowen, R. D. Pollard, R. E. Miles, D. P. Steenson, D. A. Brown, and N. J. Cronin, “Integrated micromachined antenna for 200 GHz operation,” in International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 561–654.
  8. J. Lesurf, Millimeter-Wave Optics, Devices and Systems (Hilger, Bristol, UK, 1990).
  9. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).
  10. P. A. Rizzi, Microwave Engineering, Passive Circuits (Prentice-Hall, Englewood Cliffs, N.J., 1988).
  11. N. Marcuvitz, Waveguide Handbook (Peregrinus, London, 1993).
  12. J. C. Slater, “Microwave electronics,” Rev. Mod. Phys. 18, 441–512 (1946).
    [CrossRef]
  13. J. C. Slater, Microwave Electronics (Van Nostrand, New York, 1950).

1999 (2)

1998 (1)

1996 (1)

A. Nahata and T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–708 (1996).
[CrossRef]

1990 (2)

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

1946 (1)

J. C. Slater, “Microwave electronics,” Rev. Mod. Phys. 18, 441–512 (1946).
[CrossRef]

Agrawal, G. P.

Bromage, J.

Fattinger, Ch.

Fauchet, P. M.

Gallot, G.

Grischkowsky, D.

Heinz, T. F.

A. Nahata and T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–708 (1996).
[CrossRef]

Helm, H.

Keiding, S.

Lewen, F.

McGowan, R. W.

Nahata, A.

A. Nahata and T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–708 (1996).
[CrossRef]

Radic, S.

Slater, J. C.

J. C. Slater, “Microwave electronics,” Rev. Mod. Phys. 18, 441–512 (1946).
[CrossRef]

Sobolewsky, R.

Stroud Jr., C. R.

van Exter, M.

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

D. Grischkowsky, S. Keiding, M. van Exter, and Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
[CrossRef]

Weinzierl, J.

Winnewisser, C.

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

A. Nahata and T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–708 (1996).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. van Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. 38, 1684–1691 (1990).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Rev. Mod. Phys. (1)

J. C. Slater, “Microwave electronics,” Rev. Mod. Phys. 18, 441–512 (1946).
[CrossRef]

Other (6)

J. C. Slater, Microwave Electronics (Van Nostrand, New York, 1950).

J. W. Digby, C. E. Collins, B. M. Towlson, L. S. Karatzas, G. M. Parkhurst, J. M. Chamberlain, J. W. Bowen, R. D. Pollard, R. E. Miles, D. P. Steenson, D. A. Brown, and N. J. Cronin, “Integrated micromachined antenna for 200 GHz operation,” in International Microwave Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 561–654.

J. Lesurf, Millimeter-Wave Optics, Devices and Systems (Hilger, Bristol, UK, 1990).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).

P. A. Rizzi, Microwave Engineering, Passive Circuits (Prentice-Hall, Englewood Cliffs, N.J., 1988).

N. Marcuvitz, Waveguide Handbook (Peregrinus, London, 1993).

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Figures (13)

Fig. 1
Fig. 1

Optoelectronic THz-TDS system incorporating quasi-optical coupling to the THz waveguide. The generated THz pulse is linearly polarized in the plane of the figure and along the X axis at the waveguide entrance face.

Fig. 2
Fig. 2

(a) Measured reference THz pulse and (b) relative amplitude spectrum of the reference pulse. The small oscillations apparent in (a) at approximately 6 ps are due to reflections between the confocal silicon lenses.

Fig. 3
Fig. 3

(a) Measured THz pulse transmitted through a 24-mm-long, 240-µm-diameter stainless-steel waveguide. The input pulse is shown in Fig. 2(a). (b) Amplitude spectrum of the measured transmitted pulse.

Fig. 4
Fig. 4

(a) Measured THz pulse transmitted through a 25-mm-long, 280-µm-diameter brass waveguide. The input has a duration of approximately 1 ps. (b) Amplitude spectrum of the measured transmitted pulse.

Fig. 5
Fig. 5

(a) Measured THz pulse (dots) transmitted through a 4-mm-long, 280-µm-diameter stainless-steel waveguide and (b) amplitude spectrum (dots) of the measured transmitted pulse [Fig. 5(a)]. The solid curves are the theoretical predictions.

Fig. 6
Fig. 6

Mode projection squared (A+B)2 of a Gaussian beam into a circular guide for the indicated modes.

Fig. 7
Fig. 7

Mode projection squared (A+B)2 of a Gaussian beam with the indicated polarization into a rectangular guide of dimensions a×b for the modes TE10, TE30, TE32, and TM12. (a) The dimension a of the guide varies from 50 to 600 µm with b=300 µm, and (b) the dimension b varies from 50 to 600 µm with a=280 µm.

Fig. 8
Fig. 8

Relative amplitude spectra of a subpicosecond pulse of THz radiation after propagation through a rectangular brass waveguide with different sizes of the guide. The dimensions of the guides (a×b) are (a) 250 µm×800 µm, (b) 250 µm×250 µm, and (c) 250 µm×125 µm. The length of the guides is 25 mm.

Fig. 9
Fig. 9

Measured THz pulse transmitted through a 25-mm-long, 250-µm×125-µm rectangular brass waveguide.

Fig. 10
Fig. 10

Electric field patterns of the dominant three modes in a rectangular waveguide.

Fig. 11
Fig. 11

(a) Field absorption and (b) phase and group velocities for the dominant three modes in the air-filled 250-µm×125-µm rectangular brass waveguide.

Fig. 12
Fig. 12

Electric field patterns of the dominant three modes in a circular waveguide.

Fig. 13
Fig. 13

(a) Field absorption and (b) phase and group velocities for the dominant three modes in a 240-µm-diameter stainless-steel waveguide.

Equations (48)

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Et(x, y, z)=Ex(x, y, z)eˆx+Ey(x, y, z)eˆy.
Ed(ω, t)=pAp Xptout exp[iωt-γp(ω)z],
Xp=SEtp·Eg eˆxdS=SEp,x EgdS
|Ed|2=c1(ω){1+c2(ω)cos[β2(ω)-β1(ω)]z},
Pp(ω)Pincident=G Zp(ω)Z0Ap2(ω),
Pp(ω)Pincident=G(Ap+Bp)2T(ω),
T(ω)=Zp(ω)Z0ApAp+Bp2=4Z0Zp(ω)[Z0+Zp(ω)]2.
αg,l=fαl vlvg,
αg,l=fαl 1nl[1-(λ/λc)2]1/2.
αT=αg,l+αTE(orαTM),
α10=1nl3 αl[1-(λ0/λc)2]1/2Δlb+αTE(orαTM).
αl=-1fL1-λ0λc21/2 lnEl(ω)Eempty(ω).
Γwg=α10 Lαl×2Δl=12nl3Lb.
argElEempty=(βempty-βl)L=2πλg,empty-2πλg,lL,
λg,lλ011-Δlb1-1l-λ02a21/2,
βempty-βl=π(1/l-1)bλ01-λ02a21/2Δl.
l=11+argEl(ω)Eempty(ω) λ0bπLΔl1-λ02a21/2.
E=E0(x, y)exp(iωt-γz),
H=H0(x, y)exp(iωt-γz).
γ=α+iβ.
β=2πλg=2πλ1-λλc21/2,
ZTEHtTE=k×EtTE,ZTE=Z0ndλgλforTE,
ZTMHtTM=k×EtTM,ZTM=Z0ndλλgforTM,
S|Etp|2dS=Z2S| Htp|2dS=ZS k·(Etp×Htp*)dS=1.
E=exp(iωt)pEtp×[Ap exp(-γp·z)+Bp exp(γp·z)]+kEzp[Ap exp(-γp·z)-Bp exp(γp·z)].
Ap+Bp=S (Et·Etp*)dS,
Ap-Bp=ZpS (Ht·Htp*)dS,
vϕ=v1-λλc21/2,vg=v1-λλc21/2
rin(ω)=-rout(ω)=Z(ω)-Z0Z0+Z(ω),
Rin=Rout=|rin|2=|rout|2,
tin(ω)=2Z(ω)Z0+Z(ω),tout(ω)=2Z0Z0+Z(ω),
Tin=Z0Z(ω)tin2,Tout=Z(ω)Z0tout2.
λc=1m2a2+n2b21/2.
ZTE=Z0nd 1-λλc21/2,
ZTM=Z0nd1-λλc21/2,
Wz+1vgWt=-ωQWvg,
Wt*=-ωQW.
vg Wlz=-ωQg,lWl-r12Wl+r21Wg,
vg Wgz=r12Wl-r21Wg,
vg Wz=-ωQg,lWl=-ωfQg,lW=-2αg,lvgW.
vl Wz=-ωQlW=-2αlvlW.
αg,l=fαl vlvg.
fmn=Wl,TEWTE=δmδnnl2m2λc28a2Δlb,
gmn=Wl,TMWTM=1nl2n2λc22b2Δlb,
δp=1ifp=02ifp0withp=morn.
f10=Δlbnl2,
fmn=1nl2πδm4π2a2m2λc2-1Δla,
gmn=δmnl2Jm(2πa/λc)Jm+1(2πa/λc)2 Δla.

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