One-Port De-Embedding Technique for the Quasi ... - IEEE Xplore

IEEE SENSORS JOURNAL, VOL. 13, NO. 1, JANUARY 2013

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One-Port De-Embedding Technique for the Quasi-Optical Characterization of Integrated Components Sillas Hadjiloucas, Member, IEEE, Gillian C. Walker, and John W. Bowen

Abstract— We describe a one-port de-embedding technique suitable for the quasi-optical characterization of terahertz integrated components at frequencies beyond the operational range of most vector network analyzers. This technique is also suitable when the manufacturing of precision terminations to sufficiently fine tolerances for the application of a TRL de-embedding technique is not possible. The technique is based on vector reflection measurements of a series of easily realizable test pieces. A theoretical analysis is presented for the precision of the technique when implemented using a quasi-optical null-balanced bridge reflectometer. The analysis takes into account quantization effects in the linear and angular encoders associated with the balancing procedure, as well as source power and detector noise equivalent power. The precision in measuring waveguide characteristic impedance and attenuation using this de-embedding technique is further analyzed after taking into account changes in the power coupled due to axial, rotational, and lateral alignment errors between the device under test and the instruments’ test port. The analysis is based on the propagation of errors after assuming imperfect coupling of two fundamental Gaussian beams. The required precision in repositioning the samples at the instruments’ test-port is discussed. Quasi-optical measurements using the de-embedding process for a WR-8 adjustable precision short at 125 GHz are presented. The de-embedding methodology may be extended to allow the determination of S-parameters of arbitrary two-port junctions. The measurement technique proposed should prove most useful above 325 GHz where there is a lack of measurement standards. Index Terms— De-embedding, integrated component characterization, null-balance quasi-optical reflectometer, S-parameters.

I. I NTRODUCTION

A

T MICROWAVE frequencies, measurement of the scattering parameters, attenuation coefficients and characteristic impedances of waveguide components are usually carried out using 4-port or 6-port network analyzers [1]–[8]. The measured signals are modified by unwanted but unavoidable reflections introduced by connection to the test sample,

Manuscript received June 11, 2012; revised September 28, 2012; accepted October 2, 2012. Date of publication November 12, 2012; date of current version December 10, 2012. This work was supported in part by the EC through the TMR under Grant FMRX-CT96-0092 from the EU DGXII (Science, Research and Development). The associate editor coordinating the review of this paper and approving it for publication was Prof. Francis P. Hindle. The authors are with the School of Systems Engineering, University of Reading, Reading RG6 6AY, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSEN.2012.2226713

and these have to be removed by calibrating the network analyzer with at least three known terminations. Characteristic impedance is conventionally determined with reference to a standard impedance line. However, as the frequency is increased above 100 GHz, the measurement procedure becomes increasingly difficult. As the waveguide samples become smaller the repeatability of the measurement becomes increasingly dependent on the ability of the user to couple sufficient power to the waveguide ports and to accurately measure the power ratios between the analyzer’s ports under different calibration conditions. Furthermore, the manufacture of precision reference terminations becomes difficult. These aspects are particularly problematic for new types of monolithically fabricated waveguide pieces, for which waveguide-to-waveguide connectors may have yet to be developed. At these higher frequencies, alternative broadband quasioptical approaches for waveguide characterization are currently under investigation. Using a pair of waveguides of the same size and known lengths with one of their ends fitted with a conical feed horn and their other end shorted, co-polar broadband measurements can be performed at the sample arm of a polarizing Michelson type Dispersive Fourier Transform Spectrometer operated in reflectance mode, so that two interferograms are recorded. Upon time gating to remove the horn reflection, apodization and Fourier transformation, the resulting background and sample spectral signatures are ratioed and the complex reflection which is a direct measure of the propagation constant of the waveguide may be obtained. The technique has already been demonstrated for the characterization of short WR-10 waveguide samples [9] but with limited success due to the difficulty of inferring phase at 0 and 2π transitions in the presence of noise. Another broadband experimental procedure that utilizes a time domain coherent terahertz spectrometer instead of a continuous wave one may also be used for the characterization of micromachined waveguides of different lengths [10], [11]. Although there are inherent advantages in broadband measurements as the individual reflection signatures of the device under test are in the time domain and therefore can often be directly isolated and gated out, the frequency bands within which the device under test exhibits mono-mode operation must be identified. The interpretation of results of measured propagation constants in regions where the waveguides are over-moded is still a matter of investigation. Furthermore, the frequency resolution

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is limited by the maximum path-difference attainable with the interferometer and, although the bandwidth of such a system is much greater than that of a heterodyne receiver-based system, the frequency resolution is usually poorer. In addition, current limitations of the power output of the source require a long integration time to achieve sufficient signal to noise ratios. Using a single frequency source alleviates such problems but renders time-gating impossible. A de-embedding technique must be used under these circumstances. In the following sections we firstly give a brief description of a quasi-optical null-balance bridge reflectometer suitable for making vector reflection measurements. We go on to describe a 1-port de-embedding technique, especially suited to the characterization of terahertz integrated components. Using an error analysis, the precision of the technique is discussed. Oneport spot frequency quasi-optical measurements of the propagation constant for a WR-8 precision short are presented. The sensitivity of the measurements to misalignment between the instruments test port and the device under test is then analyzed. Finally the performance of the null-balance reflectometer is discussed taking into consideration current developments in vector network analyzers. II. N ULL -BALANCE R EFLECTOMETER AND THE D E -E MBEDDING T ECHNIQUE The layout of a null-balanced reflectometer and a possible implementation are illustrated in Figs. 1(a) and (b) (inset). It is a quasi-optical instrument, with freely propagating, single frequency beams travelling between the components. The network is constructed from wire grid polarising beam splitters and forms a bridge in which the vector reflection coefficient from the device under test (DUT) is balanced in amplitude and phase by the reflection from the movable rotating grid. Jones matrices (as shown in Fig. 1(a) may be used to relate the angle between the orientations of the reference beam grid that result in nulls to the amplitude reflection coefficient of the sample. Assuming a transmittance T and reflectance R of the incident polarized beam at each grid and a reflectance M in every mirror, we can write matrix expressions for the reference and sample arms of the interferometer respectively Cre f = R(45)Me− j kd R(θ )MR(45)T(0) [0 1]T (1a) Csample = T(−45)MSMT(45)T(0) [0 1]T where

(1b)

− cos θ sin θ −1 0 sin2 θ , T(θ ) = 0 1 − cos θ sin θ cos2 θ − cos2 θ − cos θ sin θ R(θ ) = and T signifies the cos θ sin θ sin2 θ transpose operator. The variable path difference d is controlled by the translation stage upon which the rotatable grid holder rests, whereas the angle θ denotes the degree of rotation of the rotatable grid compared to a reference position. Upon recombination the detector of the two beams records a signal: D = R(0) Cre f + Csample . The modulated power at the detector P is given from 1 + sin 2θ R − j φ 2 − j kd e − e (2) P = P0 4 2 M=

where P0 is the source power output and the e− j kd and e− j φ are the corresponding phase delays in the two arms. When the instrument is balanced at a null, it gives a reflectance reading |R| = sin2 γ , where γ is half the angle between adjacent null orientations measured about the reference beam grid orientation that reflects all of the incident polarisation [12]. Two balancing procedures are performed, one with a reference (plane metallic mirror) at the test port and another one with the device under test. The subsequent introduction of the test sample at the sample arm results in a new reflection signature and a new phase delay which are accounted for at the reference arm with a new balanced position. The use of a null-balancing technique means that the measurement precision depends on the precision with which the null angles can be measured and is independent of source and detector fluctuations (which affect both the sample and reference beams in equal proportion). The source is modulated and the output from the detector, is fed into a lock-in amplifier for demodulation. Both the source and detector are fitted with corrugated feed horns to ensure maximum Gaussicity of the free-space propagating beams. In the implementation shown in Fig. 1(b), the beam at the sample location has a Gaussian transverse amplitude distribution with a 1/e amplitude halfwidth of 10 mm. An optical train of high density polyethylene lenses with a focal length of 10 mm are used throughout the two paths of the interferometer to re-focus the diffractively spreading beams. These have been omitted from Fig. 1(a) for the sake of clarity. If available, a quasi-optical isolator may be introduced in front of the source to reduce frequency pulling. Under manual operation, the user iteratively balances the instrument to a null by successive movements of the rotating grid holder and the rotating grid. The vector reflection coefficient of the DUT is directly related to the measured orientation and down-beam locations of the grid that are necessary to achieve a null. There is currently considerable interest in the development of integrated components based around photo-lithographically made or micro-machined waveguides [13], [14]. While a high repeatability of manufacture can be achieved, the quasi-optical characterisation of such waveguide components poses its own problems, particularly in the manufacture of terminations other than shorts and lengths of waveguide. The effects of imperfect coupling at integrated feed antennas and optics between the reflectometer and device under test must also be accounted for. Here we present a de-embedding procedure which may be used to determine the complex propagation constant of the waveguide based on four lengths of waveguide terminated with shorts with nominally identical integrated feed antennas. The reflections from these four test pieces are measured sequentially. It is assumed that the coupling between the reflectometer beam and the feed antenna is the same in all cases. While it is possible to construct null-balanced instruments that can measure vector transmission coefficients as well as reflection coefficients of 2-port test pieces [12], the method described here depends solely on measuring the vector reflection coefficient of 1-port test pieces. 2-port test pieces would consist of a range of lengths of waveguide with an antenna

HADJILOUCAS et al.: ONE-PORT DE-EMBEDDING TECHNIQUE

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Source chopper Vertical grid D

L2

L3

Movable rotating grid

R(0)

Detector 45° grid T(45)

ΓL 2 = −ξ 2l2

ΓL 4 = −ξ 2l4

2w0A

⎡ S11 S12 ⎤ ⎢S ⎥ ⎣ 21 S 22 ⎦

2wA

ΓSi meas

Reflector M

ΓSitrue

M

R(45)

ΓL1 = −1

ΓL3 = −ξ 2l3

R(θ )

T(0)

L4

d

2w0S 2wS z0S

(a)

Δ

z0A

(b)

Fig. 1. (a) Diagram of a null-balance reflectometer showing measurement setup. A de-embedding technique is used for estimating the S parameters at the instruments test port so that the propagation constant can be calculated. This requires the four lengths of antenna-waveguide test pieces to be sequentially placed at the test port. The projection of the coupling between the instruments test port and the device under test exaggerates alignment errors. A fundamental Gaussian beam-mode propagating through the system is assumed. (b) Inset: possible implementation of the null-balance bridge reflectometer using quasi-optical components from Thomas Keating Ltd.

at each end that would have to be aligned between a pair of lenses defining the test ports in a quasi-optical transmissometer. In general, the separation between the lenses in such a transmissometer set-up cannot be changed and so, as it would be necessary to measure a range of waveguide lengths, the distance between the feed antennas, and consequently the degree of coupling to the test beams, would be different for each test piece, causing an error in the measured S-parameters. The 1-port approach avoids this problem, the only requirement being the need to ensure that a constant separation is maintained between the reflectometer test port and the feed antenna on each test piece. Each 1-port test piece can be considered to be a length of waveguide terminated in a short and connected to the reflectometer by a 2-port device consisting of a further length of waveguide, the feed antenna and any additional optics necessary to improve coupling between the reflectometer and the feed antenna. The situation is illustrated in Fig. 1(a). For a reciprocal 2-port device with scattering matrix S the measured complex reflectance is related to the load reflectance τ Li by Si =

2 −S S ) S11 + Li (S21 11 22 . 1 − Li S22

ξ = e−a e− j kg

(5)

where λg , k g denote the guide wavelength and wavenumber respectively. Eq. 4 clearly has two solutions. However, only one will be found to give a physically realisable value for ξ , which must have a magnitude less than unity. The simultaneous equations can also be solved to give the S-parameters for the connecting 2-port S2 S3 + ξ S1 S2 − ξ S3 S1 − S3 S1 S2 + ξ S2 − ξ S3 − S1

= ± ξ 2 − 1 S1 − S3 S1 S2 − S3 S1 + S2 S3 − 2S2 × √ ξ ( S2 + ξ S2 − ξ S3 − S1)

S11 = S21

(3)

The terminating reflectance Li depends on the complex propagation constant of the waveguide ξ and the length li [Fig. 1(a)]. It is assumed that the shorts are loss-less. The unknown parameters in Eq. 3 can be found by terminating with four different load reflectances and solving the resulting ξ=

set of simultaneous equations. The terminal plane is defined by the location of the short in the shortest length of waveguide. It proves to be mathematically expedient to choose the lengths so that l2 = l3 /2 = l4 /3 and to work in length units of 2l2 . Then, the propagation constant per unit length is provided in (4) at the bottom of the page. The propagation constant ξ is related to the attenuation a and guide wavelength λg = 2π/k g through

S22 = −

S2 ξ + S2 − ξ S1 − S3 . ξ( S2 + ξ S2 − ξ S3 − S1 )

1 S4 S3 + S1 S2 −2 S3 S1 − S4 S2 + S1 S4 + S2 S3 2 S2 S3 + S2 S4 − S1 S4 − S2 S3 √ 2 2 2 2 2 2 − S2 S4 4 S4 S3 S2 + S2 S1 −3 S3 S2 +2 S1 S2 S3 −4 S4 S1 S2 −2 S4 S3 S1 −4 S3 S1 − S4 S3 +4 S3 S1 +3 S4 S1 1 ±2 S1 S3 + S2 S4 − S1 S4 − S2 S3

(6)

(7) (8)

(4)

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The S-matrix describes the absorption, scattering and coupling losses between the reflectometer and the test pieces. Eq. 5 is of indeterminate sign, and thus is of indeterminate phase by ±π. This may be resolved by introducing separate S-parameters for S21 and S12 , with appropriate elaboration of the equations, and measuring a further test piece of length 4l2 to determine the additional unknown. In practice, this sign is inconsequential if one’s goal is to determine ξ . III. E RROR A NALYSIS A full treatment of the precision of quasi-optical nullbalanced reflectometers for vector reflection coefficient measurements has been given in [12]. The error in reflectance, as a function of sample reflectance R, detector noiseequivalent power (NEP) at a bandwidth B related to the time constant of the lock-in amplifier used, source power P0 , and grid angle quantisation error σ γ Q , is given by √ √ σγ2 (9) σ R = 2 R cos(sin−1 R) 2σγ2 Q + 2 with √ −1 −1 2 σγ = ∓ sin ±4 P0 N E P B + 2R − 1

√ + ∓π/2 ± 2 sin−1 R 2 (10) where the upper and lower signs are used for reflectances above and below 0.5 respectively. The theoretical limits to the measurable sample reflectance Rlim , beyond which the nulls become unresolvable, are given by √ Rlim = 1 + sin(2θmax ) /2 ± 2 P0−1 N E P B (11) where θmax = 3π/4 for R < 0.5 and θmax = π/4 for R > 0.5. The error in the phase φ is a function of the total measurement errors in d and is given by σφ = kσdTotal where: 2 + σ2 + σ2 (12) σdT otal = 2σd2Q + σd0 dS d where k = 2π/λ is the free-space wavenumber, λ is the freespace wavelength, σd Q is the quantization error in measuring the down-beam location of the grid and the factor of 2 premultiplying this term is due to the fact that two measurements must be performed with and without the sample in place for the evaluation of the phase delay of the sample. The σd term represents the error in accurately replacing the metallic reflector with the sample (DUT) whereas the other two terms in Eq. 12 represent the NEP related error in d in the sample arm with and without the specimen present and are given from: √ 2P0−1 N E P B −1 1− (13) σd S = ± cos R2 √ σd0 = ± cos−1 1 − 2P0−1 N E P B . (14) These errors in amplitude and phase can be combined in quadrature to give an error in the vector reflection coefficient σ Si for each waveguide test piece (denoted by subscript i ) 1/2 2 2 σ Si = e− j φ (σ R )2 + − j Re− j φ (σφ )2 . (15) This is valid because σφ is small.

Fig. 2. Calculated percentage errors in the propagation constant ξ for a 3.3 dB m−1 WR-8 waveguide as a function of k g l, assuming σ d Q = 10 × 10−6 m, σ γ Q = 0.017 rd. The source power P0 is 10 mW, the NEP is 10−10 W Hz−1/2 W Hz−1/2 , and the post-detection bandwidth is 1 Hz. The inset shows a geometric representation of the locus of the errors in ξ . The circular region corresponds to amplitude and phase errors in the balancing procedure of the interferometer.

Since these errors are independent, their variances may be added separately so that the overall error σξ in the propagation constant may be calculated from 4 ∂ξ 2 σ2Si . (16) σξ = ∂ Si i=1

The resulting error may be represented as a vector added to the vector representing the true value of the propagation constant. A geometric representation showing the corresponding error in the amplitude and phase of ξ is given in Fig. 2. The increase in errors for different waveguide lengths is similar to the ones discussed in [15], [16] with VNA measurement modalities. The calculated percentage errors in ξ as a function of a variety of parameters are presented in Figs. 3–7. All are calculated for the following representative measured set of connecting 2-port S-parameters: S11 = 0.036− j 0.309, S12 = −0.321 − j 0.709 and S22 = 0.354 + j 0.183. Standard WR-8 waveguide is assumed, with l = 1 mm for Figs. 3–7. IV. P ORT Q UASI -O PTICAL M EASUREMENTS In the following section, one-port spot frequency measurements of the propagation constant ξ are presented for a WR-8 precision adjustable waveguide short, using a quasioptical null-balance bridge reflectometer similar to the one shown in Fig. 1(a). The system uses tungsten wire polarising grid beam splitters, reflectors and a train of lenses to control the diffractive spreading of the beams. A corrugated feed horn fitted at the source ensures Gaussicity of the beam with minimal astigmatism. The corrugated feed-horn antenna (4.2 mm beamwidth) fitted to the front of the waveguide short facilitated the


Fig. 3. Calculated percentage errors in√the propagation constant ξ at 94 GHz, over a range of values of P0−1 N E P B for waveguide attenuation values of (a) 70 dB m−1 (b) 10 dB m−1 and 3.3 dB m−1 assuming σ d Q = 10 × 10−6 m, σ γ Q = 0.017 rd.

Fig. 4. Calculated percentage errors in the propagation constant ξ at 94 GHz for a waveguide attenuation of 3.3 dB m−1 , over a range of values of √ P0−1 N E P B for (a) σ γ Q = 0.017, (b) σ γ Q = 0.0085, and (c) σ γ Q = 0.0017 rd, assuming σ d Q = 10 × 10−6 m.

coupling of the radiation from free space to the waveguide. A backward-wave oscillator (BWO) tuneable from 125– 200 GHz having a power output of 20–40 mW and modulated at 5 kHz was used as a source, with the re-combined signal from the reference and sample arm of the reflectometer being detected using a liquid 4 He cooled cyclotron-enhanced hot electron bolometer (10−12 W Hz−1/2 NEP). A phase sensitive detector with a time constant set to 100 ms was used to demodulate the signal. The reflectometer was initially balanced in phase until a null was observed at the detector. The reflection coefficient of each of the four samples (feed-horn antenna plus waveguide with backshort adjusted to four different positions)

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Fig. 5. Calculated percentage errors in the propagation constant ξ at 94 GHz for a waveguide attenuation of 3.3 dB m−1 , over a range of values of √ P0−1 N E P B for (a) σ d Q = 10 × 10−6 , (b) σ d Q = 5 × 10−6 , and (c) σ d Q = 1 × 10−6 m, assuming σ γ Q = 0.017 rd.

Fig. 6. Calculated percentage errors in the propagation constant ξ at 94 GHz for a waveguide attenuation of 3.3 dB m−1 , over a range of values of σ γ Q assuming σ d Q = 10 × 10−6 m. The source power P0 is 10 mW, the NEP is 10−10 W Hz−1/2 and the post-detection bandwidth is 1 Hz.

was iteratively matched with the reflection coefficient of the rotating grid, the matching condition being identified by observing a null at the detector. The grid was rotated through 360° and the positions of the four nulls were recorded. The reflection coefficient R from each sample was calculated using R = sin2 γ where γ is the half angle between adjacent nulls. The phase of the reflectometer was then modulated by moving the rotating grid reference plane through several fringes using a computer-controlled positional indexing system, comprised of an encoder connected to a lead-screw and motor. The detectors’ output power was observed as a function of path-length difference and the reflectometer was balanced at a null. The initial balance should be sought with the reference

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IEEE SENSORS JOURNAL, VOL. 13, NO. 1, JANUARY 2013 10

(a)

Intensity (a.u)

5

(b)

0

(c) -5

(d)

minima -10

Phase (rd) Fig. 8. Typical output observed at the detector at 125 GHz after modulating the reference arm of the null-balance reflectometer with the antenna and waveguide short resting at the test port. The back-short was adjusted to four different positions (a) 0.2, (b) 0.4, (c) 0.6, and (d) 0.8 mm apart and the different depth of modulation observed are due to the impedance mismatch from the standing wave inside the waveguide. Fig. 7. Calculated percentage errors in the propagation constant ξ at 94 GHz for a waveguide attenuation of 3.3 dB m−1 , over a range of values of σ d Q assuming σ γ Q = 0.017 rd. The source power P0 is 10 mW, the NEP is 10−10 W Hz−1/2 and the post-detection bandwidth is 1 Hz.

grid as close as possible to a beam-waist plane in the reflectometer in order to avoid an over-estimate of the reflection coefficient and consequent de-embedding error due to beam dilutions at other down-beam balancing positions where the reflectometer would be asymmetric. Separate measurements of the half angle positions of the rotating grid corresponding to a null position in the reflectometer’s reflection signatures with the back-short position adjusted to 0, 0.2, 0.4, 0.6 and 0.8 mm are shown in Table 1. The table incorporates the corresponding phase delays observed in the reflectometer’s reference arm which are separately calculated from Fig. 8. These four measurements are then combined, so that the propagation constant as well as the S-parameters of the test port could be calculated. Similarly to most S-parameter de-embedding measurement techniques where the precision in estimating the waveguide propagation constant is improved when taking several measurements, in our case, several waveguide lengths which are multiples of the chosen unit length have been used. Redundancy ensures that sets of measurements with waveguide lengths that correspond to λg /4 are not included in the deembedding process as they would contain the largest errors. Such measurements can be easily identified and excluded from the data set used for the calculations of the propagation constant through an inspection procedure as they correspond to a decrease in the depth of modulation of the interferogram as the instrument moves through consecutive balanced positions in distance d. The saturation observed in two of the curves of Fig. 8 is a direct consequence of our lock-in settings which are optimized to resolve minima, they are of no consequence to our experimental precision as the goal is to identify the locations of local minima only (as shown by arrows in Fig. 8) and then read the associated half angle and phase delay values. In our system, the observed intensity modulation across

a full cycle can span over 5 orders of magnitude. Further improvement to the precision in inferring phase is possible by adopting an I-Q demodulation scheme of the interferometric output although this was not attempted here. Alternatively, by operating the instrument with a computer-controlled automatic positional indexing system and recording the detector output power over several complete cycles, it is possible to improve on the quantization error σ d Q in measuring the downbeam location of the grid beyond the limits imposed by the resolution of the positional indexing system (which in our set-up corresponds to a 100 μm path length √ difference, an equivalent phase delay error of 0.26 rd) to n where n is the number of points registered by the encoder over a single scan. Recording 4000 points over a distance of 0.2 m provides an effective encoder resolution of 1.68 μm which at 125 GHz is translated to a phase error of only 4 × 10−3 rd. Alternatively, fitting a sinusoidal function to the data to eliminate clipping from detector saturation as the instrument moves through unbalanced states and performing a fast Fourier transform of the fitted data after windowing it at multiple integers of a complete cycle, the complex part of the transform corresponding to the phase delay in each waveguide length can be determined with good accuracy. Similar improvements can be obtained in the grid angle quantization error σ γ Q , after recording the detector output power when the instrument is completely balanced in d over several complete cycles. In the following section an analysis based on the propagation of errors after assuming imperfect coupling of two fundamental Gaussian beams is presented. V. Q UASI -O PTICAL C OUPLING AND D E -E MBEDDING In the previous section, the use of an adjustable waveguide short meant that the coupling between the quasi-optical reflectometer and the waveguide under test could be kept constant eliminating the possibility of additional errors due to misalignment between the instrument’s test-port and the antenna-waveguide structure. However, during a set of measurements with different waveguide lengths, each with


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TABLE I M EASUREMENTS OF THE P ROPAGATION C ONSTANT |ξ | AND ATTENUATION FOR THE WR-8 S HORT W ITH THE BACK -S HORT P OSITION A DJUSTED TO 1–5 mm AT A R ANGE OF F REQUENCIES

Waveguide Length (mm)

φ (rd)±σφ

0.2

3.67±2.6

0.4

2.88±2.6

0.6

2.88±2.6

0.8

2.09±2.6

γ (degrees) ± σγ

γ1=347±1 γ2=53±1 γ3=168±1 γ4=234±1 γ1=344±1 γ2=55±1 γ3=164±1 γ4=236±1 γ1=341±1 γ2=57±1 γ3=164±1 γ4=238±1 γ1=344±1 γ2=56±1 γ3=165±1 γ4=239±1

R± σR

ΓSimeas

0.7034±0.016

ΓS1 =0.7034e–j3.67

0.6586±0.016

ΓS 2 =0.6586e–j2.88

0.6294±0.016

ΓS 3 =0.6294e–j2.88

0.6462±0.016

ΓS 4 =0.6462e–j2.09

its own antenna, the user sequentially places the waveguide pieces at the test-port of the instrument and alignment becomes an important issue. Simple beam-mode analysis shows that for the case where both the source S and receiving antenna-waveguide system A, are fundamental Gaussian beam-modes, U A (x, y, z) and U S (x, y, z), the relative amplitude of a coherently detected signal is given [17]–[19] by the overlap or coupling integral ∗ (17) U A |U S = U A (x, y, z)U S (x, y, z)d x d y c

evaluated over any convenient constant z-plane, C. If U A and U S are both normalized to unity power, i.e. and are both equal to 1, the maximum possible value for is 1 and this value is obtained if the modes U A and U S are co-axial and have equal and co-incident beamwaists, otherwise will be less than 1 indicating an imperfect match of incident signal to the receiving antennawaveguide system. For coaxial modes with non-co-incident beam waists of different size, it can be shown that the coupling coefficient expressed in terms of the beam-waist sizes w0 A and w0S and the separation of the beam waists = z 0 A − z 0S [see Fig. 1(a)] is

−1/2 (18) U A |U S axial = (kw0 )2 (k 2 w20 )2 + (k)2 √ 2 )/2 where w0 = w0 A w0S and w20 = (w02 A + w0S correspond to the geometric mean and arithmetic mean of the squares of the associated beam-waists respectively.

ξ ± σξ

α (dB m–1)

0.939±0.234

27.16±6.79

Furthermore, the corresponding phase term from the coupling integral gives a phase error term: φ = tan−1 (/kw20 ) −k. For beams with equal beam-waists, setting w0 A = w0S = w0 , −1/2 . Eq. (18) simplifies to U A |U S axial = 1 + (k/k 2 w02 )2 Close observation of Eq. 18 shows that the larger the value of kw0 , the greater the tolerance for . For laterally displaced modes at a distance d, where U A and U S have parallel but not coincident axes, the coupling integral is U A |U S lat = U A |U S 0 exp(−d 2 /de2 ) (19) 2 2 where de2 = 2 (k 2 w 0 )2 + (k)2 /(k 4 w0 ) and U A |U S 0 denotes the value for d → 0 and is given from

−1/2 U A |U S 0 = (kw0 )2 (k 2 w20 )2 . (20) In a similar manner, for rotationally displaced modes where U A and U S are inclined at a small angle θ , the coupling integral is U A |U S rot = U A |U S 0 exp(−θ 2 /θe2 ) (21) 4 where θe2 = 8w20 /k 2 w 0 . It is convenient to treat any misalignment in a uniform manner and therefore an alignment product A is defined so that A = U A |U S rot U A |U S lat U A |U S axial . (22) Furthermore, each 1-port test piece can be considered to be a length of waveguide terminated in a short and connected to the reflectometer by a 2-port device consisting of a further length of waveguide and the feed antenna. The situation is

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illustrated in Fig. 1 after replacing the complex reflectance Si with Sitrue = Si to indicate that we are assuming perfect coupling to that port. In order to take into account imperfect coupling and thus move to the terminal plane defined by the beam waist in the test port of the reflectometer, Eq. 3 must be modified taking into account an error term σ Si due to coupling losses so that Simeas = Sitrue − σ Si

(b)

(23)

where σ Si is the error in the true Si due to coupling losses. In addition, for a given misalignment Simeas can be written in terms of a single-pass coupling coefficient A, as Simeas = A2 Sitrue where 1 ≥ A > 0 with the squared term arising from the double-pass nature of the measurements. For rotationally displaced modes σ Si =

(c)

∂ Simeas ∂ A ∂ Simeas σθ = σθ ∂θ ∂ A ∂θ ∂ A2 Sitrue ∂ A ∂A = σ θ = 2 A Sitrue σ θ. (24) ∂A ∂θ ∂θ

Over a restricted range of angles where the non-linear ∂ A/∂θ term may be approximated by the first term of its Taylor expansion series, it is possible to set σ Arot = (∂ A/∂θ ) σ θ . It follows that σ Si = 2 A Sitrue σ A so that the error in the calculated propagation constant due to rotationally displaced modes σξrot may be calculated from Simeas with four different waveguide lengths from: 4 ∂ξ σξrot = σ2Si (25) ∂ Simeas

(a)

Fig. 9. Propagation of error from the coupling coefficient A to στ si due to a rotational error in the alignment procedure of the test piece assuming w0S = 4 mm and (a) w0 A = 3 mm, (b) w0 A = 4 mm, and (c) w0 A = 6 mm at λ = 3 mm.

(c)

(b)

i=1

where explicit forms of the ∂ξ/∂ Si terms are shown elsewhere [20]. Similar arguments to those made in Eq. 24 can be made for small laterally and axially displaced modes so that again once the first term of their Taylor expansion series is taken, σ Alat = (∂ A/∂d)σ d and σ Aaxial = (∂ A/∂)σ respectively. The total error in the propagation constant can then be calculated from the sum of squares of these independent errors (26) σξtot = σξ2axial + σξ2lat + σξ2rot . From the above analysis it becomes evident that in order not to compromise the measurement precision, the waveguide alignment procedure should have tolerances comparable to the errors in the balancing procedure σ Si ≥ 2 A Sitrue σ A.

(27)

2 + σ A2 + σ A2 where σ A = σ Arot lat axial . In the following section we present theoretical errors in the coupling coefficient for a 100 GHz beam assuming beam-waist sizes of w0S = 4 mm, and w0 A = 3, 4 and 6 mm. The sensitivity of the error in the coupling coefficient due to a rotational positioning error of the test piece is given from σ Arot = (∂ A/∂θ) σ θ

= U A |U S 0 −2 (θ/θe ) exp −θ 2 /θe2 σ θ (28)

(a)

Fig. 10. Propagation of error from the coupling coefficient A to σ ξrot due to a rotational error in the alignment procedure of the test piece assuming w0S = 4 mm and (a) w0 A = 3 mm, (b) w0 A = 4 mm, and (c) w0 A = 6 mm at λ = 3 mm.

and this leads to an error in σ Si (Fig. 9) and an error in σ ξrot the propagation constant ξ via the σ ξt ot term. The percentage error in the actual value of ξ when |ξ | = 1.511 calculated using the σ Si values in Fig. 9 and Eqs. 25 and 28 are shown in Fig. 10. All errors are calculated for the same set of connecting 2-port S-parameters described earlier, which give a value of Sitrue = 0.281− j 0.678. As the alignment procedure of the test pieces at the test-port of the quasi-optical reflectometer requires the use of two microscopes with their eyepieces fitted with Vernier scales, the errors described in Fig. 10 would be typical when changing (repositioning) waveguide samples of different length during a set of sequential measurements.


119

(a) (b) (c) (a) (b) (c)

Fig. 11. Propagation of error from the coupling coefficient A to στ si due to a lateral error d in the alignment procedure of the test piece assuming w0S = 4 mm and (a) w0 A = 3 mm, (b) w0 A = 4 mm, and (c) w0 A = 6 mm at λ = 3 mm.

Fig. 13. Propagation of error from the coupling coefficient A to στ si due to a longitudinal (axial) error in the alignment procedure of the test piece assuming w0S = 4 mm and (a) w0 A = 3 mm, (b) w0 A = 4 mm, and (c) w0 A = 6 mm at λ = 3 mm.

(a) (a)

(b) (b)

(c)

(c) Fig. 14. Propagation of error from the coupling coefficient A to σ ξaxial due to a longitudinal (axial) error in the alignment procedure of the test piece assuming w0S = 4 mm and (a) w0 A = 3 mm, (b) w0 A = 4 mm, and (c) w0 A = 6 mm at λ = 3 mm.

Finally, the sensitivity of the error in the coupling coefficient due to axial positioning error of the test piece is given from Fig. 12. Propagation of error from the coupling coefficient A to σ ξlat due to a lateral error d in the alignment procedure of the test piece assuming w0S = 4 mm and (a) w0 A = 3 mm, (b) w0 A = 4 mm, and (c) w0 A = 6 mm at λ = 3 mm.

In a similar manner, the sensitivity of the error in the coupling coefficient due to lateral positioning error of the test piece is given from σ Alat = (∂ A/∂d) σ d

= U A |U S 0 −2 (d/de ) exp −d 2 /de2 σ d (29) and this leads to an error in σ Si (Fig. 11) and an error σ ξlat in the propagation constant. The percentage error in the actual value of ξ when |ξ | = 1.511 calculated using the σ Si values in Fig. 11 and Eqs. 25 and 29 is shown in Fig. 12.

4

σ Aaxial = (∂ A/∂)σ =

−k 4 w 0 k 4 w 40 + k 2 2

3/2 σ

(30)

and this leads to an error in σ Si (Fig. 13) and an error σ ξaxial in the propagation constant. The percentage error in the actual value of ξ when |ξ | = 1.511 calculated using the σ Si values in Fig. 13 and Eqs. 25 and 30 is shown in Fig. 14. As shown earlier, a mismatch in the source-antenna beamwaist ratio w0S /w0 A can have significant effects in the coupling coefficient and appropriate choice of this ratio might have to be made if misalignment errors are not to propagate to the estimation of ξ . Fig. 15 shows the propagation of error from the coupling coefficient A to σ Si due to a rotational error in the alignment procedure of the test piece assuming w0S = 4 mm and w0 A varying from 0.4 mm to 40 mm for a fixed small rotational error of (a) θ = 0.5°, (b) θ = 1° and (c)

120


(c)

(b) (a)

Fig. 15. Propagation of error from the coupling coefficient A to στ si due to a rotational error in the alignment procedure of the test piece assuming w0S = 4 mm and w0 A varying from 0.4 mm to 40 mm for a fixed small rotational error of (a) θ = 0.5°, (b) θ = 1° and, (c) θ = 3°, at λ = 3 mm.

(c)

(b) (a)

Fig. 16. Propagation of error from the coupling coefficient A to σ ξrot due to a rotational error in the alignment procedure of the test piece assuming w0S = 4 mm and w0 A varying from 0.4 mm to 40 mm for a fixed small rotational error of (a) θ = 0.5°, (b) θ = 1°, and (c) θ = 3°, at λ = 3 mm.

θ = 3°, at λ 3 mm. This error would lead to an error in the propagation constant in a manner similar to the one depicted in Fig. 16. VI. D ISCUSSION The results in Fig. 2 indicate that the correct choice of incremental waveguide length for the test pieces is important if errors are to be minimised. If l is chosen such that k g l is equal to a multiple of 2π, it proves impossible to obtain

meaningful results. This is a consequence of the phase delay in each successive test piece being identical. The behaviour can be, however, further complicated by constructive and destructive interference between reflections from the short and other reflections arising in the coupling region between the test piece and the reflectometer (e.g. from the feed antenna or connecting optics). In certain circumstances, the destructive interference may result in a reduction in the visibility of the fringes usually seen as the reference grid is moved along the axis of the beam. An attempt to balance the reflection coefficient phase would then be accompanied by a large error. The results in Fig. 3–7 indicate the dependence of the errors on fundamental instrument parameters and can be used to determine the level of improvement that can be had by increasing source and detector performance and reducing quantisation errors. Although in our measurement set-up we used a cooled detector, because the null-balance scheme was implemented using quasi-optical components originally configured for radiometric measurements, there is no need for such detector sensitivity. A simpler implementation such as the one shown in Fig. 1(b) depicting an electronic source and an in-waveguide crystal detector is adequate for de-embedding the propagation constant from S parameter measurements. It is also interesting to note the relative insensitivity of the errors to the waveguide absorption coefficient. An integral part to the proposed methodology is the associated requirement for good alignment of the test pieces with respect to the reflectometer’s test port. Variation in the location or orientation of the test piece feed antennas during a measurement set would result in variation in the degree of coupling to the test pieces. Similarly, the technique relies on a high degree of reproducibility in the test piece manufacturing process. These aspects are analogous to the connector repeatability considerations that arise in waveguidebased measurement systems. The treatment presented here has assumed that the short at the end of each waveguide piece is loss-less. Alternatively, the short may be replaced with any unknown but reproducible termination. In this case, it would be necessary to measure the reflection from 5 different length test pieces to solve for the further complex unknown that has been introduced into the system of equations. This technique would de-embed the terminating load. In many instances, the S-parameters for a waveguide structure are readily calculable from its dimensions and the propagation constants of its constituent waveguides. However, the technique presented here can be extended to determine the S-parameters of an arbitrary 2-port structure. The procedure would involve firstly determining the propagation constant for a given waveguide and the scattering matrix that describes its connection to the reflectometer using four test pieces, just as we have described above. Then the reflection coefficient of three test pieces, each of which would consist of identical copies of the unknown 2-port fed by the same antenna but with differing shorted lengths of waveguide at their output, would be measured. The terminating waveguides would be of the same size and type as those characterised in the


previous stage and so would act as three standard terminations. Likewise, the feed antennas would be identical to those used in the first stage and so would form part of the already characterised scattering matrix describing the connection to the reflectometer. The result of applying Eqs. 23 and 24 to the measured reflection coefficients would be to arrive at an overall scattering matrix incorporating, in cascaded form, the pre-characterised S-parameters describing the connection losses and the S-parameters of the 2-port under test. As the connection loss S-parameters would be known from the first stage, the unknown S-parameters could be determined. The levels of precision currently attained by vector network analyzers at high frequencies such as the Agilent 8510 are of the order of 0.005 dB in amplitude and 0.01 rd in phase. These values, however, assume in-waveguide measurements with almost perfect power coupling between the device under test and the instrument’s test port, and would be expected to become worse at a constant rate of approximately 2.5 × 10−3 dB per dB of power lost from a leaky connection [5]. Although comparisons between existing VNAs and quasi-optical measurements have been reported [21] they are not easy to perform without dielectric waveguide transmission lines [22] and the direct involvement of a Metrology laboratory such as the National Physical Laboratory (NPL). Advances in super-Gaussian horn designs [23] have the potential to further improve quasi-optical coupling to the test port of the interferometer, making such comparisons more fair. Without further enhancement from signal processing, a nullbalance quasi-optical reflectometer fitted with a grid with a quantisation error in the measured angle of 0.017 rd and a micrometer with a translational resolution of 2 μm or better (which is typical of many translation stages in the market) when coupled to a moderately powered source and a room temperature video detector assuming an observation time per point of no more than 0.1 s (post-detection bandwidth in the lock-in) it is possible to perform S parameter measurements to sufficient accuracy. From our analysis of the propagation of errors after taking into consideration errors due to imperfect coupling of two fundamental Gaussian beams it may be concluded that the associated propagation constant errors may be sufficiently small if good practice such as the use of microscopes with vernier scales to mark the position of the samples for each measurement is adopted. Furthermore, the methodology developed for assessing measurement errors between an instrument’s test port and the device under test when these are quasi-optically coupled is generic and of interest to existing users of VNAs considering performing measurements by launching their beams into free space. Although there is excellent support by manufacturers for in-waveguide measurements with VNAs (e.g., [24]–[27]), there is far less support on the methodologies associated to quasi-optical measurements and the present work addresses such need. The de-embedding technique proposed can also be suitably modified to account for different instrument topologies such as multi-state reflectometers [28] and may be used in analytical reconstruction formulas based on either impedance or propagation methods [29], [30] where the associated propagation matrices (Nicolson-Ross-Weir [31], [32], Stuchly-

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Matuszewsky [33], Palaith-Chang [34] equations) are derived using Mason diagrams based on standard Graph-Theoretic approaches. Currently, agreed standards (MIL-DTL-85/3C and IEC 60153-2) for TRL and LRL measurements exist up to 325 GHz only. While there are proposals [35], [36] to define waveguide sizes above 325 GHz, and there is an associated IEEE Working Group (P1785) for deciding upon the waveguide sizes and the associated flanges, advances in sources and detectors enable measurements to be performed at much higher frequencies and the current work addresses such need. Advances in micro-machining and photolithographic techniques, as well as in numerically controlled three-dimensional printer prototyping coupled to a metallization step as well as the wider proliferation of electromagnetic wave simulation software packages such as HFSS and FEKO are enabling several high frequency designs to be rapidly considered and optimized. The proposed measurement protocol is simple to implement and can directly benefit the above developments. For example, the utility of the technique could be further explored in future studies through measurements of substratemounted antenna impedances and radiation loss, after allowing for additional measurements for the full characterization of the non-reciprocal 2-ports where S12 = S21 or in on-wafer network analysis [37] or N-port [27] measurements. The work should also be of general interest to the sensing community e.g. when there are uniform refractive index transitions across the beam aperture in well-defined planes in optical fibre based systems or in de-embedding reflections from acoustic impedance mismatch across different layers.

VII. C ONCLUSION Although there are currently several manufacturers providing VNA solutions above 100 GHz, e.g., Agilent, Rohde & Schwarz, Anritsu and specialist manufacturers such as ABmm can deliver an impressive measurement dynamic range of more than 40 dB even at frequencies as high as 1 THz, these solutions are expensive when full vector measurements are needed. The proposed de-embedding methodology obviates the need for purchasing a high frequency VNA and the associated TRL/LRL calibration kits or the need for developing new precision terminations and calibration kits de novo for frequencies above 325 GHz prior to engaging in the characterization of micro-machined or photo-lithographically made components at their prototyping stage. It enables vector reflection coefficient measurements to be performed at any frequency by simply using a coherent source, a video detector, a rotation stage mounted on a translation stage and a few quasi-optical components assembled in a null-balance bridge configuration. The current contribution is, therefore, of much relevance to the metrology of quasi-optical measurement techniques which are much needed for the wider proliferation of THz technology [38]–[56]. Furthermore, the de-embedding technique may also be adapted for use in other measurement modalities across the sensing community.

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Sillas Hadjiloucas (M’94) received the B.Sc. (Hons.) and M.Phil. degrees in pure and applied biology from the University of Leeds, Leeds, U.K., in 1989 and 1992, respectively, and the Ph.D. degree in cybernetics from the University of Reading, Reading, U.K., in 1996. He was with the Greek Army in 1997, and was twice appointed as an EC TMR Post-Doctoral Research Fellow working on THz Instrumentation as part of the INTERACT project in U.K., and the Deutsches Zentrum Für Luft-und Raumfahrt, DLR, Berlin. In January 2000, he was appointed a Lecturer of systems engineering, University of Reading. His areas of expertise include instrumentation and measurement across the optical, IR, THz, and microwave parts of the spectrum, the study of ultrafast phenomena using femtosecond lasers, and the application of control theory and system identification techniques to spectrometry and systems biology. He has authored or co-authored papers for AIP, IoP, IEEE, and OSA journals and conferences, and is currently serving as Secretary of the IoP Instrument Science and Technology Group. Dr. Hadjiloucas is a Committee Member of the Dielectrics Group, chairing the 2013 IoP Dielectrics Conference at Reading, and a fellow of the Institute of Measurement and Control.

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Gillian C. Walker was born in Dewsbury, U.K., in 1978. She received the M.Phys. (Hons.) degree in physics from Magdalen College, Oxford, U.K., in 2000, and the Ph.D. degree in medical physics from the University of Leeds, Leeds, U.K., in 2004. She is an AHRC/EPSRC Funded Research Fellow in the U.K. based Science and Heritage Program and is based at the School of Systems Engineering, University of Reading, Reading, U.K. She has authored or co-authored more than 25 academic publications. She specializes in terahertz spectroscopy and imaging of matter. Her current research interests include the application of terahertz technologies for the investigation and preservation of cultural heritage.

John W. Bowen was born in Malvern, U.K., in 1963. He received the B.Sc. degree in physics from Queen Mary College, University of London, London, U.K., in 1985, and the Ph.D. degree from the University of London in 1993 for his work on techniques for wideband millimetre wave spectrometry. He took up a lectureship with the University of Reading, Reading, U.K., in 1993, and is currently a Senior Lecturer of cybernetics with the School of Systems Engineering, Reading, U.K. He has authored over 100 academic publications in the field of millimeter wave and terahertz technology. His current research interests include the development of terahertz systems and techniques, quasi-optics, terahertz spectroscopy of biological systems, and terahertz applications in art and archaeology. Dr. Bowen is a Chartered Physicist. He is a member of the Institute of Physics, the Optical Society of America, the Society of Photo-Optical Instrumentation Engineers, and the European Optical Society. He was the recipient of the 1989 National Physical Laboratory Metrology Award for his invention of a wideband millimeter wave noise source.