Abstract

A quasi-optical technique for characterizing micromachined waveguides is demonstrated with wideband time-resolved terahertz spectroscopy. A transfer-function representation is adopted for the description of the relation between the signals in the input and output port of the waveguides. The time-domain responses were discretized, and the waveguide transfer function was obtained through a parametric approach in the z domain after describing the system with an autoregressive with exogenous input model. The a priori assumption of the number of modes propagating in the structure was inferred from comparisons of the theoretical with the measured characteristic impedance as well as with parsimony arguments. Measurements for a precision WR-8 waveguide-adjustable short as well as for G-band reduced-height micromachined waveguides are presented.

© 2003 Optical Society of America

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2001 (2)

2000 (5)

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “THz waveguides,” J. Opt. Soc. Am. B 17, 851-863 (2000).
[CrossRef]

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6, 1122-1135 (2000).
[CrossRef]

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987-1989 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449-4451 (2000).
[CrossRef]

1999 (4)

R. W. McGowan, G. Gallot, and D. Grischkowsky, “Propagation of ultra-wideband, short Pulses of THz radiation through sub-mm diameter circular waveguides,” Opt. Lett. 24, 1431-1433 (1999).
[CrossRef]

U. L. Pen, “Application of wavelets to filtering of noisy data,” Philos. Trans. R. Soc. London, Ser. A 357, 2561–2571 (1999).
[CrossRef]

H. Krim, D. Tucker, S. G. Mallat, and D. Donoho, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225-2238 (1999).
[CrossRef]

S. Hadjiloucas, J. W. Bowen, J. W. Digby, J. M. Chamberlain, and D. P. Steenson, “Quasi-optical characterization of waveguides at frequencies above 100 GHz,” J. M. Chamberlain and P. Harrison, eds., Conference on Terahertz Spectroscopy and Applications, Munich, Proc. SPIE 3828, 357-365 (1999).
[CrossRef]

1998 (1)

D. Thompson, R. D. Pollard, and R. E. Miles, “One-port S-parameter measurements using quasi-optical multistate reflectometer,” Electron. Lett. 34, 1222-1224 (1998).
[CrossRef]

1997 (1)

G. F. Engen, “A (historical) review of the six-port measurement technique,” IEEE Trans. Microwave Theory Tech. 45, 2414-2417 (1997).
[CrossRef]

1992 (2)

K. J. Silvonen, “A general approach to network analyzer calibration,” IEEE Trans. Microwave Theory Tech. 40, 754-759 (1992).
[CrossRef]

I. Daubechies, S. Mallat, and A. S. Willsky, “Special issue on wavelet transforms and multiresolution signal analysis: introduction,” IEEE Trans. Inf. Theory 38, 529-531 (1992).

1991 (1)

H. J. Eul and B. Schiek, “A generalized theory and new calibration procedures for network analyzer self-calibration,” IEEE Trans. Microwave Theory Tech. 39, 724-731 (1991).
[CrossRef]

1989 (2)

B. Knudsen, G. F. Engen, and B. Guldbrandsen, “Accuracy assessment of the scalar network analyzer using sliding termination techniques,” IEEE Trans. Instrum. Meas. 38, 480-483 (1989).
[CrossRef]

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Pattern Anal. 11, 674-693 (1989).
[CrossRef]

1988 (1)

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[CrossRef]

1985 (1)

L. C. Oldfield, J. P. Ide, and E. J. Griffin, “A multistate reflectometer,” IEEE Trans. Instrum. Meas. 25, 198-201 (1985).
[CrossRef]

1981 (1)

S. M. Kay and S. L. Marple, Jr., “Spectrum analysis, a modern perspective,” Proc. IEEE 69, 1380-1419 (1981).
[CrossRef]

1979 (2)

C. A. Hoer, “Performance of a dual six port network analyzer,” IEEE Trans. Microwave Theory Tech. 27, 993-998 (1979).
[CrossRef]

G. F. Engen and C. A. Hoer, “Thru-Reflect-Line: an improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microwave Theory Tech. 27, 987-993 (1979).
[CrossRef]

1978 (1)

G. F. Engen, “Calibrating the six-port reflectometer by means of sliding terminations,” IEEE Trans. Microwave Theory Tech. 26, 951-957 (1978).
[CrossRef]

1977 (2)

M. P. Weidman, “A semi-automated six port for measuring millimeter-wave power and complex reflection coefficient,” IEEE Trans. Microwave Theory Tech. 25, 1083-1085 (1977).
[CrossRef]

G. F. Engen, “The six port reflectometer: an alternative network analyzer,” IEEE Trans. Microwave Theory Tech. 25, 1075-1083 (1977).
[CrossRef]

1974 (1)

G. F. Engen, “Calibration technique for automated network analyzers with application to adapter evaluation,” IEEE Trans. Microwave Theory Tech. 22, 1255-1260 (1974).
[CrossRef]

1973 (1)

G. F. Engen, “Calibration of an arbitrary six-port junction for measurement of active and passive circuit parameters,” IEEE Trans. Instrum. Meas. 22, 295-299 (1973).
[CrossRef]

1971 (1)

G. F. Engen, “An extension to the sliding short method of connector and adapter evaluation,” J. Res. Natl. Bur. Stand. 75, 177-183 (1971).

1960 (1)

R. W. Beatty, G. F. Engen, and W. J. Anson, “Measurement of reflection and losses of waveguide joints and connectors using microwave reflectometer techniques,” IRE Trans. Instrum. 9, 219-226 (1960).
[CrossRef]

Anson, W. J.

R. W. Beatty, G. F. Engen, and W. J. Anson, “Measurement of reflection and losses of waveguide joints and connectors using microwave reflectometer techniques,” IRE Trans. Instrum. 9, 219-226 (1960).
[CrossRef]

Auston, D. H.

P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
[CrossRef]

Barasniuk, R. G.

Beatty, R. W.

R. W. Beatty, G. F. Engen, and W. J. Anson, “Measurement of reflection and losses of waveguide joints and connectors using microwave reflectometer techniques,” IRE Trans. Instrum. 9, 219-226 (1960).
[CrossRef]

Bowen, J. W.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

S. Hadjiloucas, J. W. Bowen, J. W. Digby, J. M. Chamberlain, and D. P. Steenson, “Quasi-optical characterization of waveguides at frequencies above 100 GHz,” J. M. Chamberlain and P. Harrison, eds., Conference on Terahertz Spectroscopy and Applications, Munich, Proc. SPIE 3828, 357-365 (1999).
[CrossRef]

Chamberlain, J. M.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

S. Hadjiloucas, J. W. Bowen, J. W. Digby, J. M. Chamberlain, and D. P. Steenson, “Quasi-optical characterization of waveguides at frequencies above 100 GHz,” J. M. Chamberlain and P. Harrison, eds., Conference on Terahertz Spectroscopy and Applications, Munich, Proc. SPIE 3828, 357-365 (1999).
[CrossRef]

Cronin, N. J.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

Daubechies, I.

I. Daubechies, S. Mallat, and A. S. Willsky, “Special issue on wavelet transforms and multiresolution signal analysis: introduction,” IEEE Trans. Inf. Theory 38, 529-531 (1992).

Davies, S. R.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

Digby, J. W.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

S. Hadjiloucas, J. W. Bowen, J. W. Digby, J. M. Chamberlain, and D. P. Steenson, “Quasi-optical characterization of waveguides at frequencies above 100 GHz,” J. M. Chamberlain and P. Harrison, eds., Conference on Terahertz Spectroscopy and Applications, Munich, Proc. SPIE 3828, 357-365 (1999).
[CrossRef]

Donoho, D.

H. Krim, D. Tucker, S. G. Mallat, and D. Donoho, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225-2238 (1999).
[CrossRef]

Dorney, S. T. D.

Engen, G. F.

G. F. Engen, “A (historical) review of the six-port measurement technique,” IEEE Trans. Microwave Theory Tech. 45, 2414-2417 (1997).
[CrossRef]

B. Knudsen, G. F. Engen, and B. Guldbrandsen, “Accuracy assessment of the scalar network analyzer using sliding termination techniques,” IEEE Trans. Instrum. Meas. 38, 480-483 (1989).
[CrossRef]

G. F. Engen and C. A. Hoer, “Thru-Reflect-Line: an improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microwave Theory Tech. 27, 987-993 (1979).
[CrossRef]

G. F. Engen, “Calibrating the six-port reflectometer by means of sliding terminations,” IEEE Trans. Microwave Theory Tech. 26, 951-957 (1978).
[CrossRef]

G. F. Engen, “The six port reflectometer: an alternative network analyzer,” IEEE Trans. Microwave Theory Tech. 25, 1075-1083 (1977).
[CrossRef]

G. F. Engen, “Calibration technique for automated network analyzers with application to adapter evaluation,” IEEE Trans. Microwave Theory Tech. 22, 1255-1260 (1974).
[CrossRef]

G. F. Engen, “Calibration of an arbitrary six-port junction for measurement of active and passive circuit parameters,” IEEE Trans. Instrum. Meas. 22, 295-299 (1973).
[CrossRef]

G. F. Engen, “An extension to the sliding short method of connector and adapter evaluation,” J. Res. Natl. Bur. Stand. 75, 177-183 (1971).

R. W. Beatty, G. F. Engen, and W. J. Anson, “Measurement of reflection and losses of waveguide joints and connectors using microwave reflectometer techniques,” IRE Trans. Instrum. 9, 219-226 (1960).
[CrossRef]

Eul, H. J.

H. J. Eul and B. Schiek, “A generalized theory and new calibration procedures for network analyzer self-calibration,” IEEE Trans. Microwave Theory Tech. 39, 724-731 (1991).
[CrossRef]

Gallot, G.

Griffin, E. J.

L. C. Oldfield, J. P. Ide, and E. J. Griffin, “A multistate reflectometer,” IEEE Trans. Instrum. Meas. 25, 198-201 (1985).
[CrossRef]

Grischkowsky, D.

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

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “THz waveguides,” J. Opt. Soc. Am. B 17, 851-863 (2000).
[CrossRef]

D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6, 1122-1135 (2000).
[CrossRef]

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987-1989 (2000).
[CrossRef]

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449-4451 (2000).
[CrossRef]

R. W. McGowan, G. Gallot, and D. Grischkowsky, “Propagation of ultra-wideband, short Pulses of THz radiation through sub-mm diameter circular waveguides,” Opt. Lett. 24, 1431-1433 (1999).
[CrossRef]

Guldbrandsen, B.

B. Knudsen, G. F. Engen, and B. Guldbrandsen, “Accuracy assessment of the scalar network analyzer using sliding termination techniques,” IEEE Trans. Instrum. Meas. 38, 480-483 (1989).
[CrossRef]

Hadjiloucas, S.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

S. Hadjiloucas, J. W. Bowen, J. W. Digby, J. M. Chamberlain, and D. P. Steenson, “Quasi-optical characterization of waveguides at frequencies above 100 GHz,” J. M. Chamberlain and P. Harrison, eds., Conference on Terahertz Spectroscopy and Applications, Munich, Proc. SPIE 3828, 357-365 (1999).
[CrossRef]

Hoer, C. A.

C. A. Hoer, “Performance of a dual six port network analyzer,” IEEE Trans. Microwave Theory Tech. 27, 993-998 (1979).
[CrossRef]

G. F. Engen and C. A. Hoer, “Thru-Reflect-Line: an improved technique for calibrating the dual six-port automatic network analyzer,” IEEE Trans. Microwave Theory Tech. 27, 987-993 (1979).
[CrossRef]

Ide, J. P.

L. C. Oldfield, J. P. Ide, and E. J. Griffin, “A multistate reflectometer,” IEEE Trans. Instrum. Meas. 25, 198-201 (1985).
[CrossRef]

Jamison, S. P.

G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “THz waveguides,” J. Opt. Soc. Am. B 17, 851-863 (2000).
[CrossRef]

S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76, 1987-1989 (2000).
[CrossRef]

Karatzas, L. S.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

Kay, S. M.

S. M. Kay and S. L. Marple, Jr., “Spectrum analysis, a modern perspective,” Proc. IEEE 69, 1380-1419 (1981).
[CrossRef]

Knudsen, B.

B. Knudsen, G. F. Engen, and B. Guldbrandsen, “Accuracy assessment of the scalar network analyzer using sliding termination techniques,” IEEE Trans. Instrum. Meas. 38, 480-483 (1989).
[CrossRef]

Krim, H.

H. Krim, D. Tucker, S. G. Mallat, and D. Donoho, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225-2238 (1999).
[CrossRef]

Mallat, S.

I. Daubechies, S. Mallat, and A. S. Willsky, “Special issue on wavelet transforms and multiresolution signal analysis: introduction,” IEEE Trans. Inf. Theory 38, 529-531 (1992).

Mallat, S. G.

H. Krim, D. Tucker, S. G. Mallat, and D. Donoho, “On denoising and best signal representation,” IEEE Trans. Inf. Theory 45, 2225-2238 (1999).
[CrossRef]

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Pattern Anal. 11, 674-693 (1989).
[CrossRef]

Marple Jr., S. L.

S. M. Kay and S. L. Marple, Jr., “Spectrum analysis, a modern perspective,” Proc. IEEE 69, 1380-1419 (1981).
[CrossRef]

McGowan, R. W.

McIntosh, C. E.

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Miles, R. E.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
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L. C. Oldfield, J. P. Ide, and E. J. Griffin, “A multistate reflectometer,” IEEE Trans. Instrum. Meas. 25, 198-201 (1985).
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J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
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U. L. Pen, “Application of wavelets to filtering of noisy data,” Philos. Trans. R. Soc. London, Ser. A 357, 2561–2571 (1999).
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J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
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P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quantum Electron. 24, 255-260 (1988).
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J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
[CrossRef]

S. Hadjiloucas, J. W. Bowen, J. W. Digby, J. M. Chamberlain, and D. P. Steenson, “Quasi-optical characterization of waveguides at frequencies above 100 GHz,” J. M. Chamberlain and P. Harrison, eds., Conference on Terahertz Spectroscopy and Applications, Munich, Proc. SPIE 3828, 357-365 (1999).
[CrossRef]

Thompson, D.

D. Thompson, R. D. Pollard, and R. E. Miles, “One-port S-parameter measurements using quasi-optical multistate reflectometer,” Electron. Lett. 34, 1222-1224 (1998).
[CrossRef]

Towlson, B. M.

J. W. Digby, C. E. McIntosh, G. M. Parkhurst, B. M. Towlson, S. Hadjiloucas, J. W. Bowen, J. M. Chamberlain, R. D. Pollard, R. E. Miles, D. P. Steenson, L. S. Karatzas, N. J. Cronin, and S. R. Davies, “Fabrication and characterization of micro-machined rectangular waveguide components for use at millimeter wave and terahertz frequencies,” IEEE Trans. Microwave Theory Tech. 48, 1293-1303 (2000).
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D. Grischkowsky, “Optoelectronic characterization of transmission lines and waveguides by THz time-domain spectroscopy,” IEEE J. Sel. Top. Quantum Electron. 6, 1122-1135 (2000).
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I. Daubechies, S. Mallat, and A. S. Willsky, “Special issue on wavelet transforms and multiresolution signal analysis: introduction,” IEEE Trans. Inf. Theory 38, 529-531 (1992).

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L. C. Oldfield, J. P. Ide, and E. J. Griffin, “A multistate reflectometer,” IEEE Trans. Instrum. Meas. 25, 198-201 (1985).
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Figures (16)

Fig. 1
Fig. 1

Experimental setup for waveguide characterization.

Fig. 2
Fig. 2

(a) Time-domain reflection signatures for five different WR-8 waveguide lengths 1 mm apart and (b) for four reduced-height micromachined waveguides 5λg, 6λg, 7λg, and 8λg long.

Fig. 3
Fig. 3

Photograph of micromachined waveguide with antennas at both ends.

Fig. 4
Fig. 4

Copolar far-field patterns for the integrated antenna at 197 GHz.

Fig. 5
Fig. 5

Backshorted micromachined reduced-height waveguide structures of different lengths coupled to exponentially flared antenna structure, as used for the experiments.

Fig. 6
Fig. 6

(a) Consecutive phase measurements with the first backshort position (1 mm) used as a background interferogram and backshort positions at 2 mm, 3 mm, 4 mm, and 5 mm treated as samples; (b) phase measurements for reduced-height micromachined waveguides with the 5λg sample used for the background interferogram and consecutive lengths as samples. Dashed curves represent simulated results assuming the TE10 mode only propagating through the structure.

Fig. 7
Fig. 7

Block diagram of the wavelet-transform filtering procedure. hd and gd are low-pass and high-pass decomposition filters, respectively, whereas hr and gr are their reconstruction counterparts.

Fig. 8
Fig. 8

(a) Original time-domain signature of the backshort and (b) the wavelet-transform filtered signal.

Fig. 9
Fig. 9

(a) Calculated transmission coefficient of a unit-length and a two-unit-length waveguide for the backshort and (b) confidence levels (one sigma) of the identification procedure.

Fig. 10
Fig. 10

Measured (thin curve) time-domain response for the d3/d2 case and predicted (thick curve) response with a fourth-order model.

Fig. 11
Fig. 11

Residual statistics for the d3/d2 case, with a fourth-order model. The autocorrelation values (a) of the modeling residual [k] and (b) the cross correlation between the modeling residual and the input signal. The dashed curves are the bounds of the region in which the residual statistics should be with a 68% confidence level (one sigma) for a conveniently chosen model structure.

Fig. 12
Fig. 12

(a) Calculated transmission coefficient of a waveguide of unit length of reduced height and (b) confidence levels (one sigma around the d5/d6 curve) of the identification procedure.

Fig. 13
Fig. 13

Pole-zero diagram for the d3/d2 backshort case, with a fourth-order model. The over-parameterized sixth-order model in the inset is also shown for illustrative purposes.

Fig. 14
Fig. 14

Pole-zero chart of three propagating modes in an antenna-coupled reduced-height waveguide when a waveguide length of 5λg is used as a background and a waveguide length of 6λg is used as a sample (the limits of the 68% confidence loci for the zeros are represented as ellipses in a thick curve). Results after ratioing the 8λg to the 6λg waveguide lengths are also shown (in this case the confidence loci for the zeros are represented as ellipses in a thin curve).

Fig. 15
Fig. 15

Calculation of the effective (multimoded) magnitude of the propagation constant for a micromachined waveguide of unit length λg.

Fig. 16
Fig. 16

Multiple reflections due to impedance mismatch between air–antenna and antenna–waveguide interfaces.

Tables (1)

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Table 1 Mode Characteristics Calculated with the ARX Model for the d5/d6 Data Set a

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

U(jω)=H(jω)G12(jω)P(jω),
Y(jω)=H2(jω)G22(jω)P(jω).
H(jω)=Y(jω)U(jω).
y[k]+a1y[k-1]++anay[k-na]
=b1u[k-nk]+b2u[k-nk-1]++bnbu[k-nk-nb+1]+[k],
H(z)=Y(z)U(z)=z-nkb1+b2z-1+ bnbz-nb+11+a1z-1+ anaz-na=z-nkN(z)D(z).
H(z)=z-nki=1n+mwi1+fiz-11+ci1z-1+ci2z-2,
φ(ω)=tan-1[Im(H(jω))/Re(H(jω))].
Hˆ(jω)=H(ω)exp[-jϕ(ω)],
λcmn=2abm2ba+n2ab.
H(jω)=i=1m+nwi{exp[-αi(ω)2d]exp[jφi(ω)]}.
Wf(a, b)=t=0N-1f(t)ψa,b(t),
ψa,b(t)=1a ψt-ba,
z=exp[Ts(-ζωn±jωn(1-ζ2)],
S0(jω)=P(jω)R12(jω),
S1(jω)=P(jω)T122(jω)R23(jω),
S2(jω)=P(jω)T122(jω)R232(jω)R12(jω),
S3(jω)=P(jω)T122(jω)T232(jω)H(jω),
S4(jω)=P(jω)T122(jω)T232(jω)R23(jω)H2(jω).
S0(jω)S1(jω),S2(jω)S1(jω),S3(jω)S2(jω),S4(jω)S3(jω)
H(jω)=S5(jω)S4(jω)1R23(jω)
H(z)=i=1n+mwi1+fiz-1(1-pi1z-1)(1-pi2z-1),
H(z)|z=exp(jωTs)
=i=1n+mwi1+fiexp(-jωTs)1+ci1exp(-jωTs)+ci2exp(-2jωTs)
=i=1n+m(αi+jβi)=A exp(jϕ),
H(jω)=i=1n+mwi[1+ficos(ωTs)]-jfisin(ωTs)[1+ci1cos(ωTs)+ci2cos(2ωTs)]-j[ci1sin(ωTs)+ci2sin(2ωTs)],
A=i=1m+nαi2+i=1m+nβi2,
ϕ=a tani=1m+nβi/i=1m+nαi,
αi=wi[1+ficos(ωTs)][1+ci1cos(ωTs)+ci2cos(2ωTs)]+fisin(ωTs)[ci1sin(ωTs)+ci2sin(2ωTs)][1+ci1cos(ωTs)+ci2cos(2ωTs)]2+[ci1sin(ωTs)+ci2sin(2ωTs)]2,
βi=wi[1+ficos(ωTs)][ci1sin(ωTs)+ci2sin(2ωTs)]-fisin(ωTs)[1+ci1cos(ωTs)+ci2cos(2ωTs)][1+ci1cos(ωTs)+ci2cos(2ωTs)]2+[ci1sin(ωTs)+ci2sin(2ωTs)]2.
θ=[-a1-a2-anab1b2bnb]T
Φ(N-na)×(na+nb)=y[na]y[na-1]y[1]u[na]u[na-1]u[na-nb+1]y[na+1]y[na]y[2]u[na+1]u[na]u[na-nb+2]y[N-1]y[N-2]y[N-na]u[N-1]u[N-2]u[N-nb]
y^(N-na)×1=[yˆ[na+1]yˆ[na+2]yˆ[N]]T,
y(N-na)×1=[y[na+1]y[na+2]y[N]]T.
S=(ΦTΦ)-1(y-yˆ)T(y-yˆ)(N-na)-(na+nb),

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