1. INTRODUCTIONThe terahertz (THz) frequency range, located midway be-tween microwaves and visible light, presents a new fron-tier containing an abundance of technical applicationsand fundamental research problems. Consequently, THzresearch solutions and techniques can come from eitheroptics or microwaves or in many cases unique combina-tions of both. Associated with THz investigations is theneed for enabling THz technology and techniques. How-ever, because of the developing nature of THz capability,commercial THz technology is minimal and one must relyon custom fabrication. At the present time, one-of-a-kindlaboratory, optoelectronic THz transmitters and receiversprovide an interim solution enabling both fundamentalresearch and applications studies.1,2
One outstanding problem has been the guided-wavepropagation of THz radiation and the associated efficientcoupling between guided and freely propagating THzwaves. Recently, efficient, broadband coupling of freelypropagating pulses of THz electromagnetic radiation intocircular metal waveguides has been demonstrated.3 Thiswork also reported the highest guided-wave performanceto date, obtained with a 240-mm-diameter stainless-steelwaveguide over the frequency range from 0.8 to 3.5 THzand with a power absorption coefficient of less than 1cm21. This absorption is much less than that of coplanarand microstrip transmission lines.3 However, for circu-lar waveguides the undistorted propagation of picosecondTHz pulses was shown not to be possible, because of theextreme dispersion near the cutoff frequencies overlappedby the pulse bandwidth covering many octaves in fre-quency.
A promising application of THz waveguides appears to
be surface-specific frequency-dependent absorption mea-surements with the use of THz time-domain spectroscopy(THz-TDS).2 From experimental measurements madeon the absorption of water, we believe that for a similarpolar molecule, THz-TDS has the sensitivity to character-ize the absorption of nanogram quantities of samples oftoxic or precious gases contained in a THz waveguide.
Here, we present a comprehensive experimental andtheoretical study of the application of quasi-optical tech-niques to couple freely propagating pulses of THz electro-magnetic radiation into both circular and rectangularmetal waveguides. We observe very dispersive, low-losspropagation over the frequency band from 0.65 to 3.5 THzwith frequency-dependent group velocities vg rangingfrom c/4 to c and phase velocities vp from 4c to c (wherevpvg 5 c2) for waveguides with typical diameters or rect-angular widths of 300 mm and lengths of 25 mm. Wepresent a complete classical waveguide theory calculationto obtain the coupling coefficients into the modes of thewaveguide for the incoming focused THz beam. Eventhough the input spectrum overlaps the cutoff frequenciesof more than 25 circular and 35 rectangular waveguidemodes, the calculated coupling coefficients and the experi-ments show that the linearly polarized incoming THzpulses significantly couple only into the TE11, TE12, andTM11 modes of the circular waveguide and the TE10 andTM12 modes of the rectangular guide. The propagation ofthe THz pulse through the guide is described as a linearsuperposition of the coupled propagating modes, eachwith a unique complex propagation vector. This pictureexplains in detail all the observed features of the THzpulse emerging from the waveguide. We demonstrateboth theoretically and experimentally that it is possible to
2000 Optical Society of America
852 J. Opt. Soc. Am. B/Vol. 17, No. 5 /May 2000 Gallot et al.
achieve TE10 single-mode coupling and propagation in asuitably sized rectangular waveguide for an incoming fo-cused, linearly polarized THz pulse with a bandwidth cov-ering many octaves in frequency and overlapping morethan 35 rectangular waveguide modes. Finally, to applythese THz waveguides to THz-TDS of dielectrics in thewaveguide, we present an analytic calculation for the ab-sorption and the dispersion of dielectric films within awaveguide.
For completeness we note that the reshaping of freelypropagating THz pulses by passage through thin andthick metal slits4,5 and dichroic metal filters6 has been ex-perimentally and theoretically studied. Hollow metallicwaveguide transmission has been reported up to 200GHz,7 where the waveguides were constructed with pho-tolithographic techniques. It should also be noted thatrectangular waveguides that cover a frequency range upto 220–325 GHz are commercially available on a special-order basis, but only for small lengths up to 25 mm.Such waveguides, designated as WR-3, have typical di-mensions of 850 mm 3 425 mm. In comparison, thewaveguide studies presented here cover the frequencyrange from 100 to 4000 GHz, representing a 1-order-of-magnitude increase in frequency.
The organization of this paper is as follows: First, wedescribe the experimental setup and the waveguide fabri-cation in detail, including multimode experimental re-sults. We then present the theory of the THz-TDS opto-electronic detection of beams propagating through THzwaveguides. This is followed by the theoretical and ex-perimental demonstration that a THz beam can becoupled into a single mode and propagate in a suitablysized rectangular waveguide. We conclude with a theo-retical study of some applications made possible by theuse of a single-mode propagation, in the domain of spec-troscopic measurements of waveguides with thin films.This is followed by a complete description in AppendixesA and B of the microwave theory used to obtain our cou-pling coefficients and complex propagation vectors. Thenew results pertaining to the THz waveguide and theTHz-TDS problem constitute the theory section in themain body of the manuscript.
2. EXPERIMENTAL SETUPThe experimental setup for broadband THz waveguides,shown in Fig. 1, consists of an optoelectronic transmitterand receiver along with beam-shaping and beam-steeringoptics. A detailed description of the entire system hasbeen previously published.1,2 The picosecond THz pulsesare generated as follows: 40-fs optical pulses at 820 nmfrom a KLM Ti:sapphire laser are focused onto the edge ofthe positive line of a coplanar strip line on semi-insulating GaAs, which is dc biased at 70 V. The opticalpulse creates an electron–hole plasma, and the subse-quent acceleration of the carriers generates a near-single-cycle electromagnetic pulse of THz radiation that is lin-early polarized along the direction of the bias field. Inthe standard THz-TDS setup, the sample is placed at theTHz beam waist at the confocal position between the twoparabolic reflectors. The THz beam system has symmet-ric confocal optics with respect to the center line of the
system between the transmitter and the receiver. Thissituation gives a unity, frequency-independent, powercoupling efficiency between the transmitter and the re-ceiver (see Eq. (2.16), Ref. 8). The transmitting antennais at the focus of the silicon lens, which collimates thefrequency-independent far-field pattern of the antennainto an approximately Gaussian beam with a 1/e beamwaist diameter of 5 mm and with a plane-wave phasefront. This is beam waist W1. This beam waist is inturn in the focal plane of the transmitter paraboloidalmirror, which focuses the THz beam to the second beamwaist W2 with beam diameters proportional to wave-length and with plane-wave phase fronts. In the stan-dard system, the THz optical train is identical (but re-versed) from this point on to the receiver. For thewaveguide experiments, an additional silicon lens isplaced at the beam waist W2, thereby producing a thirdfrequency-independent beam waist W3 with a 1/e ampli-tude waist diameter of 200 mm and a plane-wave phasefront. This third beam waist overlaps the waveguide en-trance face. The output face of the waveguide is again inthe focal plane of an identical silicon lens with identicalbut reversed optics to the receiver. In the absence of thewaveguide and with the two identical silicon focusinglenses moved so that their beam waists overlap, the en-tire system is again confocal and gives a frequency-independent, power coupling efficiency of unity betweenthe THz transmitter and receiver. With the waveguidein place, the total coupling efficiency between the trans-mitter and the receiver is determined by the product ofthe coupling efficiency of the incoming THz beam into thewaveguide, the transmission factor of the waveguide, andthe coupling efficiency of the output beam into the THzreceiver. Because the input and output faces of thewaveguide are in the focal planes (beam waist) of theidentical focusing lenses, the coupling efficiency of the in-coming THz beam is identical to the coupling efficiency ofthe output beam into the THz receiver. This coupling ef-ficiency is determined by evaluating the overlap integralbetween the THz beam waist and the modes of the wave-guide (see Eq. (2.1), Ref. 8).
The transmitted pulse is focused onto a polarization-sensitive, 10-mm dipole antenna on ion-implanted silicon-on-sapphire, which is photoconductively switched by asecond beam of 40-fs optical pulses, generating a dc cur-rent that is proportional to the instantaneous electric
Fig. 1. Optoelectronic THz-TDS system incorporating quasi-optical coupling to the THz waveguide. The generated THzpulse is linearly polarized in the plane of the figure and along theX axis at the waveguide entrance face.
Gallot et al. Vol. 17, No. 5 /May 2000/J. Opt. Soc. Am. B 853
field of the propagated THz pulse. By scanning the rela-tive delay between the detected THz pulse and the gatingoptical pulse, one obtains the entire time-dependent THzpulse, including both field and phase information.
All of the waveguides used in this experiment were fab-ricated in our laboratory by using commercially availablestock. The stainless-steel circular waveguide was madefrom the most precise metal tube available, a stainless-steel hypodermic needle. This long guide was 25 gaugeand was 24 mm long with a 240-mm diameter. The shortguide was also 25 gauge but had a measured 280-mm di-ameter and a length of 4 mm. Although the hypodermicneedles are the best commercial tubes available, theyhave significant inconsistency in inner-diameter dimen-sions and circular precision. To ensure no deformation ofthe tubes during the cutting process, we encapsulated thetube in hard mounting wax on a hot plate. Once encap-sulated, the tube was cut on a wire saw with low cuttingpressure. The mounting wax was removed by ultra-sounding in acetone, and the edges were carefullydeburred. This method is extremely important forsofter waveguide metals such as copper and brass. Com-pared with stainless steel, which has a conductivity of1.1 3 106 V21 m21, much longer waveguides of high-conductivity metals could be used, for example, copperand brass. Copper has a conductivity 36 times largerthan that of stainless steel, giving an absorption coeffi-cient six times smaller for Cu than for stainless steel.Brass is also a good choice, with a conductivity slightlyless than that of copper. However, for copper we couldnot obtain circular tubing without significant ellipticityand cross-section variations with length. Consequently,the complicated experimental results for copper guidesare not presented here.
The rectangular waveguides discussed here have beenfabricated by cutting grooves in brass plates. With theuse of a 250-mm jeweler’s saw blade, a precise groove iscut on a 25-mm-square, 6-mm-thick brass plate. Anotherbrass plate is tightly connected to the grooved plate tocomplete the waveguide. The entrance and exit sides ofthe guide are carefully milled to obtain high-precision flatand burr-free entrance and exit faces. To study the in-fluence of the waveguide dimension on the relative cou-pling efficiencies, we cut several waveguides of differentsize on the same 25-mm-long plate.
3. EXPERIMENTAL RESULTS FROMCIRCULAR WAVEGUIDESThe first experimental results were obtained with circularstainless-steel waveguides.3 A reference pulse is ob-tained by removing the waveguide and moving the twosilicon lenses to near contact with a 300-mm-diameter ap-erture placed between them at their common focus W3.Figure 2(a) shows the reference THz pulse, and Fig. 2(b)shows that the useful amplitude spectrum extends from0.1 to 4 THz. Figure 3(a) shows the measured transmit-ted pulse through the 24-mm-long, 240-mm-diameterstainless-steel circular waveguide. The incomingcoupled pulse, which has a duration of approximately 1ps, has been stretched to approximately 70 ps, with thehigh frequencies arriving earlier in time, corresponding to
negative chirp. The stretching and the consequent chirp-ing of the transmitted pulse of Fig. 3(a), compared withthe input pulse of Fig. 2(a), is due to the strong group-velocity dispersion of the waveguide. The waveguideacts as a dispersive delay line, where the lower frequen-cies travel more slowly than the higher frequencies. Fig-ure 3(b) shows the corresponding amplitude spectrum forthe waveguide. Note the sharp cutoff at 0.76 THz, whichcompares well with the theoretically calculated cutoff ofthe lowest-order waveguide mode TE11 of 0.76 THz (seeAppendix B). The spectrum presents unusual oscilla-tions starting at approximately 1.7 THz. As demon-strated in the theoretical section, these oscillations aredue to multimode propagation through the waveguide.
Experiments have been performed with several otherguides, including brass and nickel circular waveguides.Because of changes in alignment, the reference pulse var-ies between experiments, although the general features ofFig. 2(a) remain unaltered. Figure 4 shows the mea-sured THz (a) pulse and (b) spectrum transmittedthrough a 25-mm-long, 280-mm-diameter brass circular
Fig. 2. (a) Measured reference THz pulse and (b) relative am-plitude spectrum of the reference pulse. The small oscillationsapparent in (a) at approximately 6 ps are due to reflections be-tween the confocal silicon lenses.
854 J. Opt. Soc. Am. B/Vol. 17, No. 5 /May 2000 Gallot et al.
waveguide. A sharp cutoff at 0.67 THz is also observedin the brass guide, while for a 280-mm guide the cutoff iscalculated to be 0.65 THz.
The qualitative results obtained for the stainless-steel(Fig. 3) and brass (Fig. 4) circular waveguides are quitedifferent. For similar input pulses, the maximum ampli-tude of the pulse transmitted through the brass wave-guide is twice that from the stainless-steel waveguide.The temporal stretches, at 1/e, for the stainless-steel andbrass waveguides, respectively, are 20 and 40 ps. Thepulse emerging from the brass waveguide is thereforetwice as long as the one exiting from the stainless-steelwaveguide. These results demonstrate the significantlyreduced loss of the brass waveguide and are in goodagreement with our theoretical simulations. Precisequantitative results were not obtained because the shapeof the tubes is not constant and is not well defined.These imperfections lead to perturbations of the wave-guide modes and rotation of the THz polarization duringpropagation.
Fig. 3. (a) Measured THz pulse transmitted through a 24-mm-long, 240-mm-diameter stainless-steel waveguide. The inputpulse is shown in Fig. 2(a). (b) Amplitude spectrum of the mea-sured transmitted pulse.
4. INTRODUCTION TO THEORYTo study the behavior and the propagation of a subpico-second pulse of THz radiation in a metallic waveguide,the modern and current approach would be to directlysolve Maxwell’s equations by finite-difference time-domain numerical simulation.9 In such an approach, theelectric and magnetic fields are calculated through an it-erative application of the boundary conditions. Afterpropagation through the guide, the temporal pulse andspectrum are obtained, but what happens in the guide isnot intuitively clear. Quasi-optical methods combinedwith microwave theory are an analytic alternativewhereby the incoming wave of THz radiation at the en-trance to the waveguide is expanded as a sum of the sta-tionary field patterns of the modes of the waveguide.The waveguide modes have been studied for many yearsin microwave technology, so that the theory is well estab-lished (see Appendix A). In the case of optoelectronicallygenerated, subpicosecond pulses of THz radiation, the
Fig. 4. (a) Measured THz pulse transmitted through a 25-mm-long, 280-mm-diameter brass waveguide. The input has a dura-tion of approximately 1 ps. (b) Amplitude spectrum of the mea-sured transmitted pulse.
Gallot et al. Vol. 17, No. 5 /May 2000/J. Opt. Soc. Am. B 855
large bandwidth makes new phenomena appear, such asinterference between modes and exceptionally large dis-persion.
5. COHERENT POLARIZED TERAHERTZELECTRIC FIELD DETECTIONWe consider now the coherent detection of the THz radia-tion emerging from the waveguide.1,2 The electric fieldEt at the output of the guide can be expressed in terms ofits components with respect to the X and Y axes as
Et~x, y, z ! 5 Ex~x, y, z !ex 1 Ey~x, y, z !ey . (1)
In general, for any THz receiver with R(x, y, z, v) des-ignating the beam pattern that is collected with unitycoupling efficiency, the measured output field of the wave-guide would be given by the overlap integral of Et withR(x, y, z, v).8 Our receiver is optically identical to thetransmitter; it is linearly polarized along the X axis, andthe accepted beam profile at the focal plane (located at theoutput face of the waveguide) of the output silicon lens isthe same as that of the incoming THz beam to the wave-guide at the beam waist, namely, a Gaussian profileEg(x, y) with a frequency-independent 1/e amplitude di-ameter of 200 mm and a planar phase front. Conse-quently, R(x, y, z, v) 5 Eg(x, y) ex . Using the normal-mode expansion of Eq. (A6) from Appendix A, one obtainsthe expression for the detected signal Ed in terms of nor-mal modes designated by the subscript p:
Ed~v, t ! 5 (p
Ap Xptout exp@ivt 2 gp~v!z#, (2)
where
Xp 5 EESEtp • Eg exdS 5 EE
SEp,x EgdS (3)
and tout is the transmission factor out of the waveguidegiven in Eqs. (A10). The integration is over the outputface S of the waveguide. Because of this integral natureof the detected field, our receiver cannot detect modeswith zero overlap integral; odd modes are not detectable.The detected signal Ed(v, t) is expressed as the summa-tion of sinusoidal functions exp@iv t 2 ibp(v)z#, which isresponsible for additional oscillations in the temporal do-main and for interference fringes in the spectral domain.For instance, the power spectrum of the signal detected inthe presence of two modes is
uEdu2 5 c1~v!$1 1 c2~v!cos@b2~v! 2 b1~v!#z%, (4)
where c1 and c2 are constants depending on the particu-lar modes. Thus, as a result of the strong frequency de-pendence of the propagation constants, interference ap-pears where the two modes overlap the same frequencyrange, that is to say, for frequencies higher than the high-est cutoff frequency of the two modes. The interferencedescribed in Eq. (4) for the power spectrum is at first sur-prising, since the modes are orthogonal and linearly inde-pendent; however, for a receiver such as a bolometer,which detects the power rather than the electric field,there will be no interference in the power spectra.
Using Eqs. (2) and (A6), we compare the experimental(dots) and theoretical (solid curves) pulse shape and spec-trum of a THz pulse transmitted through a 4-mm-long,280-mm-diameter stainless-steel waveguide in Figs. 5(a)and 5(b), respectively. For the best fit the theory used awaveguide diameter of 270 mm, a beam diameter of 160mm, and the spectrum of the reference pulse, and it evalu-ated the coupling coefficients according to Appendix A.Analysis of this comparison shows that experimentally,the amplitude of the TM11 mode was approximately twicethat predicted. We ascribe this discrepancy to wave-guide imperfection. Under these circumstances we con-sider the agreement between theory and experiment to bequite satisfactory.
6. SINGLE-MODE PROPAGATIONIn general, a single-mode, single-conductor waveguide ca-pable of propagating a subpicosecond THz pulse is notpossible, because the pulse bandwidth overlaps many oc-taves in frequency. However, this situation does not pre-clude the pulse propagating in a single mode of the wave-guide if coupling to only one mode can be performed.
Fig. 5. (a) Measured THz pulse (dots) transmitted through a4-mm-long, 280-mm-diameter stainless-steel waveguide and (b)amplitude spectrum (dots) of the measured transmitted pulse[Fig. 5(a)]. The solid curves are the theoretical predictions.
856 J. Opt. Soc. Am. B/Vol. 17, No. 5 /May 2000 Gallot et al.
The lower cutoff frequencies of the modes in a wave-guide are approximately distributed in frequency by oc-tave. If the bandwidth of the incoming signal is narrowenough (less than a factor of 2 in the frequency range), itis possible to choose the dimensions of the guide so thatthe bandwidth lies between the lowest cutoff frequencyand the next-higher-order mode. But for a wider band-width of many octaves, which is the case for optoelectroni-cally generated, subpicosecond pulses of THz radiation,the cutoff frequencies of many modes will be in the fre-quency range of the signal. Consequently, many higher-order modes can be excited. However, the relative am-plitudes of the coupling coefficients to these modes aresensitive to the polarization, the shape, and the phasefront of the incoming beam relative to the geometry of theguide. By changing the dimensions of the guide, theshape, or the polarization of the beam, one can favor thecoupling into select modes.
We performed numerical simulations of the mode pro-jection of our incoming Gaussian beam coupled into dif-ferent waveguides. The mode projection is determinedby decomposing the input field into transmitted and re-flected waveguide modes. The coefficients Ap and Bp forthe modes designated by the index p are defined by Eqs.(A7) from Appendix A and represent the transmitted andreflected mode amplitudes, respectively. These coeffi-cients are real because we consider coupling at the beamwaist, for which the field is planar. The relative trans-mitted power Pp into specific waveguide modes is given by
Pp~v!
P incident5 G
Zp~v!
Z0Ap
2~v!, (5)
where we refer to the fraction of the energy incident onthe aperture of the waveguide compared with the total in-cident energy as the geometric transmission, denoted G.The impedance of the medium outside the waveguide isdenoted Z0 , and Zp(v) is the waveguide impedance of themode, referring to either Zp
TE or ZpTM , defined in Eqs. (A4)
from Appendix A. The coupling efficiency of Eq. (5) canbe separated into frequency-dependent and frequency-independent terms by introducing a power transmissioncoefficient T(v) and the frequency-independent mode pro-jection (Ap 1 Bp) at the guide entrance calculated in Eqs.(A7). Then
Pp~v!
P incident5 G~Ap 1 Bp!2T~v!,
T~v! 5Zp~v!
Z0S Ap
Ap 1 BpD 2
54Z0Zp~v!
@Z0 1 Zp~v!#2 . (6)
Figure 6 shows the calculated mode projection squared(Ap 1 Bp)2 of the dominant five modes TE11, TE12, TE13,TM11, and TM12 for a circular guide as a function of theguide diameter. The solid curve labeled ‘‘sum’’ is the nor-malized sum of the squares of the mode projections in theabove five modes, where the normalized sum of thesquares of the projections for all the modes of the wave-
guide is equal to unity. It is found that TE11 remains thedominant mode over a wide range of guide diameters,with a maximum mode projection occurring for a diam-eter of 200 mm, equal to that of the incoming beam. Evenfor this optimal combination, the maximum projection forthe TE11 mode remains below 90%. The main reason forthis continued multimode coupling is that the linear po-larization of the incoming beam is not fully compatiblewith the boundary conditions of the circular waveguide,as can be seen from an inspection of the field pattern ofthe TE11 cylindrical mode shown below in Fig. 12 in Ap-pendix B. This can also be observed in Fig. 3(b), wherethe amplitude spectrum transmitted through a 24-mm-long, 240-mm-diameter waveguide shows an obvious in-terference pattern, in good agreement with the mode pro-jections presented in Fig. 6.
In contrast to the circular waveguides, inspection of thefield pattern of the TE10 mode of the rectangular wave-guide, displayed in Appendix B as Fig. 10, shows that thismode is fully compatible with a linearly polarized incom-ing wave. Figure 7 shows the mode projection squared(Ap 1 Bp)2 of the dominant four modes TE10, TE30, TE32,and TM12 of a rectangular guide as a function of the guidedimensions for the linearly polarized incoming plane-wave beam with a 1/e waist diameter of 200 mm. In Fig.7(a) the dimension b along the polarization of the beam iskept constant, equal to 280 mm, and the dimension a var-ies from 50 to 600 mm. The mode projection of the domi-nant mode TE10 presents a relatively broad maximum, re-maining below 80%, and therefore the propagation is notsingle mode. However, in contrast to circularwaveguides, the second dimension of the rectangularguide remains as a variable. Taking the best value forthe dimension a, we vary the dimension b, as shown inFig. 7(b), and observe that the TE10 mode projection canapproach unity if b is chosen small enough. For this con-dition the beam propagates through the guide as a singleTE10 mode. Of course, the limitation on b is the fractionof the incident power transmitted through the rectangu-lar aperture of the waveguide, G. A compromise betweenthe best dominant mode projection and the maximumtransmitted power needs to be found. For instance, witha rectangular guide of dimensions 280 mm 3 130 mm, G
Fig. 6. Mode projection squared (A 1 B)2 of a Gaussian beaminto a circular guide for the indicated modes.
Gallot et al. Vol. 17, No. 5 /May 2000/J. Opt. Soc. Am. B 857
is more than 80%, and the mode projection (Ap 1 Bp)2 ofthe beam is more than 98% into the dominant mode TE10.For any given beam size, it is possible to design a rectan-gular waveguide with very high coupling efficiency intothe dominant mode and simultaneously a good geometrictransmission of the aperture, as the relative mode projec-tions are due solely to the relative dimensions of the beamwaist and the waveguide. Because a change in the di-mensions of the guide affects the cutoff frequency, the fi-nal size of the waveguide must also take into account thebandwidth of the THz beam.
We can now apply these principles to experimentallytest for the single propagating TE10 mode. The relativeamplitude spectra of THz pulses propagating throughthree different rectangular waveguides are presented inFig. 8. The brass waveguides have the following respec-tive dimensions (a 3 b): (a) 250 mm 3 800 mm, (b)250 mm 3 250 mm, and (c) 250 mm 3 125 mm. The cal-culated cutoff for the lowest-order mode TE10 is 0.6 THzfor the waveguide of Fig. 8(c). For this waveguide themodal characteristics are shown in Figs. 10 and 11 in Ap-pendix B. The transmitted spectrum presented in Fig.
Fig. 7. Mode projection squared (A 1 B)2 of a Gaussian beamwith the indicated polarization into a rectangular guide of di-mensions a 3 b for the modes TE10, TE30, TE32, and TM12. (a)The dimension a of the guide varies from 50 to 600 mm with b5 300 mm, and (b) the dimension b varies from 50 to 600 mmwith a 5 280 mm.
8(a) shows a dramatic interference pattern correspond-ing to the superposition of several modes during thepropagation. As shown above in Fig. 7, the reductionof the dimension b of the waveguide leads to reductionof the amplitude of oscillation [Fig. 8(b)] and even to acomplete disappearance of the oscillations within oursignal-to-noise ratio in Fig. 8(c). The propagation of theTHz pulse through this waveguide with dimensions250 mm 3 125 mm is effectively a single-mode propaga-tion. Qualitatively, the agreement between the ampli-tude of the oscillation found in the experimental spectrumof Fig. 8 and the amplitude obtained with the mode pro-jections is excellent. The oscillations in Figs. 8(a), 8(b),
Fig. 8. Relative amplitude spectra of a subpicosecond pulseof THz radiation after propagation through a rectangularbrass waveguide with different sizes of the guide. The dimen-sions of the guides (a 3 b) are (a) 250 mm 3 800 mm, (b)250 mm 3 250 mm, and (c) 250 mm 3 125 mm. The length ofthe guides is 25 mm.
858 J. Opt. Soc. Am. B/Vol. 17, No. 5 /May 2000 Gallot et al.
and 8(c) present modulations of 0.64, 0.45, and 0.06, re-spectively, compared with the theoretical modulations of0.67, 0.52, and 0.07, respectively. Here, the modulationis defined as the ratio of the total amplitude of the oscil-lation to the average value of the spectrum.
Figure 9 presents the measured THz pulse transmittedthrough the 25-mm-long, 250-mm 3 125-mm waveguidecorresponding to the spectrum of Fig. 8(c). The transmit-ted pulse shows regular oscillations with some irregularstructure on the trailing edge of the pulse that is due towater vapor in the path of the THz beam. Although thepulse has been significantly broadened by the group-velocity dispersion of the TE10 mode, the resulting 13-ps,1/e pulse width is significantly less than the correspond-ing 40-ps, 1/e pulse width for the brass circular guide asshown in Fig. 4. This is due to the difference betweensingle-mode and multimode propagation in thewaveguides and to the different dimensions.
7. WAVEGUIDE THz-TDSWe have demonstrated, in both theory and experiment,the feasibility of single-mode excitation of a rectangularTHz waveguide. We now present the theory that enablesthe application of this new THz technology to THz-TDS(Ref. 2) spectroscopic measurements of dielectrics in thewaveguide, which include thin-film and gas measure-ments. The developed theory obtains equations for themeasurement of the absorption and the dispersion of thedielectric media.
We now consider the case of waveguides partially filledwith a thin dielectric layer. In Appendix C it is shownthat the absorption coefficient in a layer within the wave-guide, ag,l , is related to that of the bulk dielectric, a l , by
ag,l 5 fa l
vl
vg, (7)
where the filling factor f is defined as the ratio of the en-ergy in the layer to the total energy within the guide.
Fig. 9. Measured THz pulse transmitted through a 25-mm-long,250-mm 3 125-mm rectangular brass waveguide.
This expression for the absorption coefficient is generaland applies to any distribution of dielectric in the guide.In most cases the group velocity in the layer material canbe approximate to the phase velocity c/nl , where nl is therefractive index of the layer material. For a thin layer,the group velocity of the guide is approximately that ofthe air-filled waveguide. Using Eqs. (A8), one obtainsthe absorption by propagation through the waveguidewith no conductive loss as
ag,l 5 fa l
1
nl@1 2 ~l/lc!2#1/2 . (8)
The absorption depends on the wavelength, increasing asthe wavelength approaches the cutoff wavelength. Thisenhancement of the bulk absorption of the layer can beunderstood by the behavior of the propagating fields nearthe cutoff. In a geometrical picture, the waves in theguide are reflected by the sidewalls of the metal wave-guide. The closer the wavelength is to the cutoff, the big-ger the angle is between the wave and the wall. Conse-quently, the effective length of travel is then increased bythe zigzag path, intensifying the effective absorption inthe guide.10
To obtain the total loss, we need to add the conductiveloss. For a thin dielectric layer, for which the group ve-locity is approximately equal to that of an empty guide,the conductive losses are unchanged by the presence ofthe dielectric layer. The total loss is then given by
aT 5 ag,l 1 aTE ~or aTM!, (9)
where ag,l is given by our Eq. (8) and aTE and aTM aregiven by Marcuvitz11 (Eq. 9, page 60, and Eq. 4, page 57,for a rectangular guide and Eq. 25, page 70, and Eq. 21,page 67, for a circular guide).
As an example for the rectangular waveguide, we canapply the filling factor of the mode TE10 to obtain the ab-sorption coefficient in the guide that is due to an absor-bent layer. The filling factors for rectangular and circu-lar waveguides with a thin dielectric layer adjacent totheir surfaces have been evaluated and are given in Ap-pendix C. By combining Eqs. (8) and (C10), we obtain
a10 51
nl3
a l
@1 2 ~l0 /lc!2#1/2
Dl
b1 aTE ~or aTM!. (10)
The first part is due to the absorption from the layer it-self, and the second part is due to the conductive loss inthe metal of the guide.
For a thin layer in the waveguide of length L, the de-tected amplitude spectrum El(v) is proportional toexp(2aT L), whereas the reference amplitude spectrumEempty(v) for the air-filled waveguide is proportional toexp@2aTE(or aTM)L#. The ratio of the amplitude spec-trum of the waveguide with the layer to the reference am-plitude spectrum gives exp(2ag,lL). Finally, with the useof Eq. (8), the absorption coefficient of the layer a l is ob-tained from the ratio of the two amplitude spectra Eemptyand El by
Gallot et al. Vol. 17, No. 5 /May 2000/J. Opt. Soc. Am. B 859
a l 5 21
fL F1 2 S l0
lcD 2G1/2
lnU El~v!
Eempty~v!U. (11)
A. Measurement of the Absorption from a Thin Layerin a WaveguideThe above result can be compared with traditional meth-ods, where the layer covers a mirror and where thechange of absorption from the layer is detected. In thiscase, with a layer of the same thickness, the amplitudeabsorption is given by exp(2al 3 2Dl), where the factor 2is due to the fact that the layer is traveled through twice.If now we take a guide of length L, the absorption is thenexp(2a10L). For a wavelength much smaller than thewavelength cutoff, the ratio of the effective lengths isgiven by
Gwg 5a10 L
a l 3 2Dl5
1
2nl3
L
b. (12)
This ratio is very sensitive to the index of the layer.For an index close to unity, Gwg may be much bigger than1. For example, for a waveguide where L 5 100 mm andb 5 125 mm and for a layer with an index nl 5 2, Gwg isequal to 50. It is possible in this waveguide case to mea-sure absorption with samples 50 times less absorbentthan is possible with traditional single-layer reflection.On the contrary, for a high index, Gwg is small, and thistechnique is not as effective. However, for the study ofsingle molecular layers or chains with nl ' 1 this tech-nique can yield extremely large sensitivity enhance-ments.
B. Determining the Dielectric Constant of a ThinDielectric Layer in the WaveguideThrough comparison of the phase of the electric field, inboth an air-filled guide and one in which a thin dielectriclayer is present, it is possible to determine the dielectricconstant of the layer. We consider a rectangular wave-guide with a thin layer adjacent to the waveguide surfaceand a polarization of the THz beam inside the waveguideorthogonal to the surface of the layer. The phase of theelectric field is determined by the propagation constant ofthe guide, such that E(v) 5 uE(v)uexp(2ibL). For anonabsorbing dielectric, we can then compare a sampleand a reference spectrum to obtain
argS El
EemptyD 5 ~bempty 2 b l!L 5 S 2p
lg,empty2
2p
lg,lDL ,
(13)
where arg represents the argument (angle) of the complexratio. Marcuvitz has presented the following approxi-mate expression for lg,l for the dominant (approximatelyTE10) mode11:
lg,l 'l0
F 1
1 2Dl
b S 1 21
e lD 2 S l0
2a D 2G 1/2 , (14)
where e l is the relative dielectric constant of the layer, aand b are the dimensions of the rectangular guide, and Dl
is the dielectric thickness. With this result an expansionin terms of the dielectric thickness leads to the first-orderexpression
bempty 2 b l 5p~1/e l 2 1 !
bl0F1 2 S l0
2a D 2G1/2 Dl. (15)
This result, together with Eq. (13), leads to an expressionfor the relative dielectric constant of the layer:
e l 51
1 1 argF El~v!
Eempty~v!G l0b
pLDl F1 2 S l0
2a D 2G1/2 . (16)
8. CONCLUSIONSWe have demonstrated the efficacy of quasi-optical tech-niques to efficiently couple freely propagating pulses ofTHz radiation into submillimeter circular and rectangu-lar waveguides with typical diameters and edge dimen-sions of 300 mm and with lengths of 25 mm. We observedlow-loss, very dispersive propagation through thesewaveguides over the frequency band from 0.65 to 3.5 THzwith frequency-dependent group velocities vg rangingfrom c/4 to c and phase velocities vp from 4c to c. Thelow loss, inversely proportional to the square root of theconductivity, would enable propagation lengths muchlonger than the previous 25-mm lengths to be demon-strated. Here, the experimental limitation is the avail-ability of longer lengths of precision metal tubing to beused as waveguides.
Even though our input spectrum overlaps the cutofffrequencies of more than 25 waveguide modes, the lin-early polarized incoming THz pulses significantly coupleonly into five modes for the circular waveguides and fourmodes for the rectangular waveguides. Using classicalwaveguide theory, we obtain the coupling coefficients intothe modes of the waveguides for the incoming focusedTHz beam, where the propagation of the THz pulsethrough the waveguide is described as a linear superposi-tion of the coupled propagating modes, each with a uniquecomplex propagation vector. We demonstrate that thissuperposition of the propagating modes explains in detailall of the observed features of the THz pulse emergingfrom the waveguide.
Through our understanding of the coupling of the in-coming linearly polarized, focused THz pulse with aplane-wave phase front to the waveguide modes, we showthat it is possible to design a rectangular waveguide forwhich the THz pulse couples to only a single mode of thewaveguide. We have experimentally demonstrated thisconclusion by observing single-mode propagation over thefrequency range from 0.7 to 4 THz for an optimal250-mm 3 125-mm rectangular brass waveguide 25 mmlong. These results are significant in that they enableTHz pulse propagation in the waveguide with a single-valued, analytic propagation vector.
Such single-mode propagation makes possible the ap-plication of waveguide THz time-domain spectroscopy(THz-TDS). To facilitate these applications, we have de-
860 J. Opt. Soc. Am. B/Vol. 17, No. 5 /May 2000 Gallot et al.
veloped a waveguide theory for thin-film measurementsand have shown that an enhancement of measurementsensitivity of up to 50 times that of a comparative single-pass reflective measurement is feasible. The waveguideTHz-TDS technique also appears to be ideal for the THz(far-infrared) study of precious or hazardous gases, sincesuitable spectra should be possible with nanogram quan-tities of material.
APPENDIX A: FIELDS IN THE WAVEGUIDETo clarify the notation used in this paper and to facilitatecomparison with the many different treatments in the lit-erature, we present the formulation used for our calcula-tions of the waveguide coupling and propagation dis-cussed in the main body of this paper. Here, we definethe propagating electric (E) and magnetic (H) fields of thewave in the form
E 5 E0~x, y !exp~ivt 2 gz !,
H 5 H0~x, y !exp~ivt 2 gz !. (A1)
The complex propagation constant g can be defined interms of the real wave propagation constant b and at-tenuation constant a by
g 5 a 1 ib. (A2)
It is convenient to introduce the waveguide and cutoffwavelengths lg and lc , respectively, through
b 52p
lg5
2p
lF1 2 S l
lcD 2G1/2
, (A3)
where l is the wavelength in the dielectric medium fillingthe guide, with index of refraction nd.
We separate the electric and magnetic fields of thepropagating wave into transverse and longitudinal com-ponents Et , Ht and Ez , Hz , respectively. The solutionsof Eqs. (A1) that satisfy the boundary conditions comprisethe transverse electric (TE) modes, with Ez 5 0, and thetransverse magnetic (TM) modes, with Hz 5 0. Thetransverse components of E and H are related as12
ZTEHtTE 5 k 3 Et
TE , ZTE 5Z0
nd
lg
lfor TE,
ZTMHtTM 5 k 3 Et
TM , ZTM 5Z0
nd
l
lgfor TM, (A4)
which defines the characteristic impedance of the twotypes of solutions, where Z0 5 Am0 /e0 is the impedanceof the vacuum k is the unit vector in the longitudinal (z)direction. The modes are normalized according to
EES
uEtpu2dS 5 Z2EESu Htpu2dS
5 ZEES
k • ~Etp 3 Htp* !dS 5 1. (A5)
There are an infinite number of solutions of the waveequation, each of type TE or TM, corresponding to a par-ticular cutoff wavelength. In the general case, each TEor TM mode can be identified by an integer number p.
In practical examples, such as rectangular and circularguides, p refers to a couple of identifying integers:p [ (m, n). However, for conciseness we often use thenotation p to also include the distinction between TE andTM modes.
The TE and TM modes form a complete and orthogonalbasis set for describing the electromagnetic fields withinthe waveguide.12,13 We can therefore determine a uniqueexpansion of the incoming electromagnetic field in termsof the normal modes. Each mode propagates freely in theguide, without interacting with the other modes, with aconstant cross section, and with the complex propagationconstant gp responsible for the absorption and the disper-sion of the mode. Assuming that the amplitude of thetransverse electric field for the pth mode is ApEtp for thewave propagating in the 1z direction and BpEtp for thatin the 2z direction, the field E of the propagating wave inthe waveguide is given by
E 5 exp~ivt !(p
Etp
3 @Ap exp~2gp • z ! 1 Bp exp~gp • z !#
1 kEzp@Ap exp~2gp • z ! 2 Bp exp~gp • z !#. (A6)
The amplitudes Ap and Bp of each mode are obtained byprojection of the transverse components of the incomingelectromagnetic field over the transverse pattern of themodes12,13:
Ap 1 Bp 5 EES
~Et • Etp* !dS,
Ap 2 Bp 5 ZpEES
~Ht • Htp* !dS, (A7)
where S denotes an integration over the waveguide crosssection. We can now analyze separately the propagationof the modes. Simple expressions for the phase velocityvf and the group velocity vg of a particular mode in theabsence of dispersion of the dielectric medium in theguide are given by
vf 5v
F1 2 S l
lcD 2G1/2 , vg 5 vF1 2 S l
lcD 2G1/2
(A8)
with v 5 1/Aem. Equations (A8) lead to the simple rela-tion vfvg 5 v2, for which, in an empty guide, vf isgreater than c, while vg is naturally less than c. Thephase and group velocities for rectangular and circularwaveguides are presented below in Figs. 11(b) and 13(b),respectively.
At the input or the output of the waveguide, the changein impedance between free space and the guide resultsin reflections. The amplitude reflection and transmiss-ion coefficients are defined by r 5 B/(A 1 B) andt 5 A/(A 1 B). The amplitude and power reflection co-efficients at the input, r in and R in , and at the output, routand Rout , for the TE and TM modes are given by13
r in~v! 5 2rout~v! 5Z~v! 2 Z0
Z0 1 Z~v!,
Gallot et al. Vol. 17, No. 5 /May 2000/J. Opt. Soc. Am. B 861
R in 5 Rout 5 ur inu2 5 uroutu2, (A9)
where Z(v) refers to ZpTE or Zp
TM as defined in Eqs. (A4)for the TE and TM modes, respectively. The amplitudeand power transmission coefficients are also given by13
t in~v! 52Z~v!
Z0 1 Z~v!, tout~v! 5
2Z0
Z0 1 Z~v!,
T in 5Z0
Z~v!t in
2 , Tout 5Z~v!
Z0tout
2 . (A10)
APPENDIX B: RECTANGULAR ANDCIRCULAR WAVEGUIDESThe analytic expressions for the modes of rectangular andcircular metallic hollow waveguides can be found inMarcuvitz.11 The electric field patterns for the dominantthree modes in a rectangular guide for a linearly polar-ized, plane-wave Gaussian input beam are presented inFig. 10. For the rectangular waveguide with horizontaland vertical dimensions a 3 b, the cutoff wavelength isthe same for both the TEmn and TMmn modes and is de-termined by the two integers m and n:
lc 51
F S m
2a D 2
1 S n
2b D 2G1/2 . (B1)
However, the characteristic impedances of the TE andTM modes are different and are given by
ZTE 5Z0
ndY F1 2 S l
lcD 2G1/2
,
ZTM 5Z0
ndF1 2 S l
lcD 2G1/2
, (B2)
where nd is the refractive index of the bulk medium in thewaveguide, and l is the wavelength in the bulk medium.
The absorption coefficients of the TEmn and TMmnmodes for a rectangular waveguide that are due to the fi-
Fig. 10. Electric field patterns of the dominant three modes in arectangular waveguide.
nite conductivity of the metal are given by Marcuvitz11
(Eq. 9, page 60, and Eq. 4, page 57). The absorption andthe phase and group velocities of the dominant three
Fig. 11. (a) Field absorption and (b) phase and group velocitiesfor the dominant three modes in the air-filled 250-mm3 125-mm rectangular brass waveguide.
Fig. 12. Electric field patterns of the dominant three modes in acircular waveguide.
862 J. Opt. Soc. Am. B/Vol. 17, No. 5 /May 2000 Gallot et al.
modes for a 250-mm 3 125-mm brass waveguide are pre-sented in Figs. 11(a) and 11(b), respectively.
For a circular waveguide, the cutoff frequencies of theTEmn and TMmn modes are determined by the nth rootsof the mth-order Bessel function and of the derivative ofthe mth-order Bessel function.11 The characteristic im-pedances are the same as those for the rectangular guide[Eqs. (A4)]. The electric field patterns of the dominantthree modes in a circular guide, for a linearly polarized,plane-wave Gaussian input beam, are presented in Fig.12.
The absorption of a circular waveguide that is due tothe finite conductivity of the metal has been calculated byusing the expression given by Marcuvitz11 (Eq. 25, page70, and Eq. 21, page 67). The absorption coefficient andthe phase and group velocities of the dominant threemodes for a 240-mm-diameter stainless-steel waveguideare presented in Figs. 13(a) and 13(b), respectively.
APPENDIX C: ABSORPTION FROM ALAYER WITH THE WAVEGUIDEWe consider the absorption of waveguides partially filledwith a thin dielectric layer by using Poynting’s theorem,which states the conservation of electromagnetic energyas13
Fig. 13. (a) Field absorption and (b) phase and group velocitiesfor the dominant three modes in a 240-mm-diameter stainless-steel waveguide.
]W
]z1
1
vg
]W
]t5 2
v
Q
W
vg, (C1)
where W is the total energy per unit length, defined by in-tegration of the energy density over the cross section ofthe waveguide, vg is the group velocity, and Q is definedby analogy to oscillating circuits. To explain Q, we canrewrite Eq. (C1) by using the reduced time t* 5 t2 z/vg to obtain
]W
]t*5 2
v
QW. (C2)
The factor Q corresponds to the energy loss in an elemen-tary cell traveling in the guide with the same velocity vgas the flow of energy, so that the energy in the cell de-creases as exp(2vt* /Q). If Wl and Wg are defined as theenergy per unit length in a cross section of the layer andthe empty remaining space of the waveguide, respec-tively, the total energy W is given by W 5 Wl 1 Wg . Wewrite the conservation of energy, for both the layer andthe empty space, as
vg
]Wl
]z5 2
v
Qg,lWl 2 r12Wl 1 r21Wg , (C3)
vg
]Wg
]z5 r12Wl 2 r21Wg , (C4)
where r12 and r21 are the energy-transfer rates betweenthe layer and the empty space and Qg,l is the energy lossin the layer within the waveguide. Summing Eqs. (C3)and (C4) and defining the filling factor f such that Wl5 fW, we obtain
vg
]W
]z5 2
v
Qg,lWl 5 2
vf
Qg,lW 5 22ag,lvgW. (C5)
In Eq. (C5) we have defined the absorption constant ag,lof the waveguide so that W decreases as exp(22ag,l z).We also define the bulk absorption for the material in thelayer by a l so that the energy in the bulk of the materialdecreases as exp(22al z). Applying Poynting’s theoremto the bulk of the material constituting the layer, wherethe energy flow in the bulk material travels with the ve-locity vl , one obtains a relationship between Ql and a l :
vl
]W
]z5 2
v
QlW 5 22a lvlW. (C6)
Assuming that Ql 5 Qg,l then leads to the relationship
ag,l 5 fa l
vl
vg. (C7)
We have calculated expressions for the filling factor of athin dielectric layer on the surface of rectangular and cir-cular air-filled waveguides. For a rectangular guide ofdimensions a 3 b with a dielectric layer of thickness Dland refractive index nl on the side of dimension a, the fill-ing factors fmn and gmn for the TEmn and TMmn modes,respectively, are
Gallot et al. Vol. 17, No. 5 /May 2000/J. Opt. Soc. Am. B 863
fmn 5Wl,TE
WTE5
dmdn
nl2
m2lc2
8a2
Dl
b,
gmn 5Wl,TM
WTM5
1
nl2
n2lc2
2b2
Dl
b, (C8)
where
dp 5 H 1 if p 5 0
2 if p Þ 0with p 5 m or n. (C9)
For example, the filling factor of the TE10 mode is
f10 5Dl
bnl2 , (C10)
corresponding to the ratio of the cross-section area of thelayer to the total cross-section area of the guide, multi-plied by the reciprocal of the squared index of the layer,which is proportional to the energy density in the layer.
For an air-filled circular guide of radius a and a layer ofthickness Dl at the surface of the guide, consideration ofthe energy densities in the guide and the layer leads tothe TE and TM filling factors, fmn and gmn , respectively;
fmn 51
nl2
pdm
4p2a2
m2lc2 2 1
Dl
a,
gmn 5dm
nl2 F Jm8 ~2pa/lc!
Jm11~2pa/lc!G 2 Dl
a.
(C11)
ACKNOWLEDGMENTSWe acknowledge careful reading of this manuscript andmany helpful and stimulating suggestions by R. AlanCheville, John O’Hara, and Jiang-Quan Zhang. RajindMendis was especially helpful through his many discus-sions of the waveguide solutions of Maxwell’s equationsand in particular for his help in clarifying the boundaryconditions for the dielectric layer. The work was par-tially supported by the National Science Foundation andthe U.S. Army Research Office.
Address correspondence to D. Grischkowsky at the lo-cation on the title page or by phone, 405-744-6622; fax,405-744-9198; or e-mail, [email protected].
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