Abstract

We investigate the polarization modulation properties of a variable-delay polarization modulator (VPM). The VPM modulates polarization via a variable separation between a polarizing grid and a parallel mirror. We find that in the limit where the wavelength is much larger than the diameter of the metal wires that comprise the grid, the phase delay derived from the geometric separation between the mirror and the grid is sufficient to characterize the device. However, outside of this range, additional parameters describing the polarizing grid geometry must be included to fully characterize the modulator response. In this paper, we report test results of a VPM at wavelengths of 350 μm and 3 mm. Electromagnetic simulations of wire grid polarizers were performed and are summarized using a simple circuit model that incorporates the loss and polarization properties of the device.

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2010 (1)

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

2009 (1)

2008 (1)

2006 (2)

D. T. Chuss, E. J. Wollack, S. H. Moseley, and G. Novak, “Interferometric polarization control,” Appl. Opt. 45, 5107–5117 (2006).
[CrossRef]

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

2004 (2)

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

G. Siringo, E. Kreysa, L. A. Reichertz, and K. M. Menten, “A new polarimeter for (sub) millimeter bolometer arrays,” Astron. Astrophys. 422, 751–760 (2004).
[CrossRef]

2003 (1)

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

2001 (1)

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

2000 (3)

V. V. Yatsenko and S. A. Tretyakov, “Higher order impedance boundary conditions for sparse wire grids,” IEEE Trans. Antennas Propag. 48, 720–727 (2000).
[CrossRef]

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69–85 (2000).
[CrossRef]

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

1999 (2)

K. Yasumoto and K. Yoshitomi, “Efficient calculation of free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

H. Shinnaga, M. Tsuboi, and T. Kasuga, “A millimeter polarimeter for the 45 m telescope at Nobeyama,” Publ. Astron. Soc. Jpn. 51, 175–184 (1999).

1998 (2)

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

J. Lagarias, J. Reeds, and M. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

1997 (1)

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

1996 (1)

R. L. Akeson, J. E. Carlstrom, J. A. Phillips, and D. P. Woody, “Millimeter interferometric polarization imaging of the young stellar object NGC 1333/IRAS 4A,” Astrophys. J. Lett. 456, L45 (1996).
[CrossRef]

1993 (1)

A. Harvey, “A quasi-optical universal polarizer,” Int. J. Infrared Millim. Waves 14, 1–16 (1993).
[CrossRef]

1991 (1)

M. Goldfarb and R. Pucel, “Modeling via hole grounds in microstrip,” IEEE Microwave Guided Wave Lett. 1, 135–137 (1991).
[CrossRef]

1987 (1)

N. Erickson, “A new quasi-optical filter: the reflective polarizing interferometer,” Int. J. Infrared Millim. Waves 8, 1015–1025 (1987).
[CrossRef]

1981 (1)

1980 (1)

W. Chambers, C. Mok, and T. Parker, “Theory of the scattering of electromagnetic waves by regular grid of parallel cylinder wires with circular cross section,” J. Phys. A: Math. Gen. 13, 1433–1441 (1980).
[CrossRef]

1977 (1)

1967 (1)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

1962 (1)

T. Larsen, “A survey of the theory of wire grids,” IEEE Trans. Microwave Theory Tech. 10, 191–201 (1962).
[CrossRef]

1960 (1)

S. Adachi and E. M. Kennaugh, “The analysis of a broad-band circular polarizer including interface reactions,” IEEE Trans. Microwave Theory Tech. 8, 520–525 (1960).
[CrossRef]

1954 (1)

J. R. Wait, “Reflection from a wire grid parallel to a conducting plane,” Can. J. Phys. 32, 571–579 (1954).
[CrossRef]

1941 (1)

Adachi, S.

S. Adachi and E. M. Kennaugh, “The analysis of a broad-band circular polarizer including interface reactions,” IEEE Trans. Microwave Theory Tech. 8, 520–525 (1960).
[CrossRef]

Ade, P. A. R.

Akeson, R. L.

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

R. L. Akeson, J. E. Carlstrom, J. A. Phillips, and D. P. Woody, “Millimeter interferometric polarization imaging of the young stellar object NGC 1333/IRAS 4A,” Astrophys. J. Lett. 456, L45 (1996).
[CrossRef]

Bastien, P.

Battistelli, E. S.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Bersanelli, M.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Beunen, J. A.

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

Carlstrom, J. E.

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

R. L. Akeson, J. E. Carlstrom, J. A. Phillips, and D. P. Woody, “Millimeter interferometric polarization imaging of the young stellar object NGC 1333/IRAS 4A,” Astrophys. J. Lett. 456, L45 (1996).
[CrossRef]

Catalano, A.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Chambers, W.

W. Chambers, C. Mok, and T. Parker, “Theory of the scattering of electromagnetic waves by regular grid of parallel cylinder wires with circular cross section,” J. Phys. A: Math. Gen. 13, 1433–1441 (1980).
[CrossRef]

Chuss, D.

Chuss, D. T.

Conversi, L.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Costely, A. E.

Costley, A. E.

Davidson, J. A.

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

de Gregori, S.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

de Petris, M.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Dotson, J. L.

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

Dowell, C. D.

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

Drouet d’Aubigny, C.

Edwards, T.

T. Edwards, Foundations for Microstrip Circuit Design (Wiley, 1987).

Erickson, N.

N. Erickson, “A new quasi-optical filter: the reflective polarizing interferometer,” Int. J. Infrared Millim. Waves 8, 1015–1025 (1987).
[CrossRef]

N. Erickson, “A 0.9 mm heterodyne receiver for astronomical observations,” in IEEE MTT-S Intl. Microwave Symposium Digest (IEEE, 1978), pp. 438–439.

Ferreira, I.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Forbrich, J.

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

Goldfarb, M.

M. Goldfarb and R. Pucel, “Modeling via hole grounds in microstrip,” IEEE Microwave Guided Wave Lett. 1, 135–137 (1991).
[CrossRef]

Goldsmith, P. F.

P. F. Goldsmith, Quasioptical Systems (IEEE, 1998).

Golish, D.

Grammer, W.

E. Wollack, W. Grammer, and J. Kingsley, “The Boifot Orthomode Junction,” NRAO, ALMA Memo Series 425 (2002).

E. Wollack and W. Grammer, “Symmetric waveguide orthomode junctions,” in Proceedings of the 14th International Symposium on Space TeraHertz Technology, E. Walker and J. Payne, eds. (NRAO, 2003), pp. 169–176.

Grover, F.

F. Grover, Inductance Calculations (Van Nostrand, 1946).

Harvey, A.

A. Harvey, “A quasi-optical universal polarizer,” Int. J. Infrared Millim. Waves 14, 1–16 (1993).
[CrossRef]

Haynes, V.

Hildebrand, R. H.

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

Houde, M.

M. Krejny, D. Chuss, C. Drouet d’Aubigny, D. Golish, M. Houde, H. Hui, C. Kulesa, R. F. Loewenstein, S. H. Moseley, G. Novak, G. Voellmer, C. Walker, and E. Wollack, “The Hertz/VPM polarimeter: design and first light observations,” Appl. Opt. 474429–4440 (2008).
[CrossRef]

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

House, J.

Hui, H.

Hursey, K. H.

Inatani, J.

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

Jones, R.

Kasuga, T.

H. Shinnaga, M. Tsuboi, and T. Kasuga, “A millimeter polarimeter for the 45 m telescope at Nobeyama,” Publ. Astron. Soc. Jpn. 51, 175–184 (1999).

Kennaugh, E. M.

S. Adachi and E. M. Kennaugh, “The analysis of a broad-band circular polarizer including interface reactions,” IEEE Trans. Microwave Theory Tech. 8, 520–525 (1960).
[CrossRef]

Kingsley, J.

E. Wollack, W. Grammer, and J. Kingsley, “The Boifot Orthomode Junction,” NRAO, ALMA Memo Series 425 (2002).

Kovacs, A.

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

Krejny, M.

Kreysa, E.

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

G. Siringo, E. Kreysa, L. A. Reichertz, and K. M. Menten, “A new polarimeter for (sub) millimeter bolometer arrays,” Astron. Astrophys. 422, 751–760 (2004).
[CrossRef]

Kulesa, C.

Kushta, K.

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69–85 (2000).
[CrossRef]

Lagarias, J.

J. Lagarias, J. Reeds, and M. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Lamagna, L.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Lamb, J. W.

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

Landau, L.

L. Landau and E. Lifshitz, Electromagnetics of Continuous Media, Vol. 8 (Pergamon, 1960).

Larsen, T.

T. Larsen, “A survey of the theory of wire grids,” IEEE Trans. Microwave Theory Tech. 10, 191–201 (1962).
[CrossRef]

Leonardi, R.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Lifshitz, E.

L. Landau and E. Lifshitz, Electromagnetics of Continuous Media, Vol. 8 (Pergamon, 1960).

Loewenstein, R. F.

Lubin, P. M.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Manabe, T.

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

Maoli, R.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Marcuvitz, N.

N. Marcuvitz, Waveguide Handbook, MIT Rad. Labs. Series, Vol. 10 (McGraw-Hill, 1951).

Martin, D.

D. Martin, “Polarizing (Martin-Puplett) interferometric spectrometers for the near- and submillimeter spectra,” in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, 1982), Vol. 6, pp. 65–148.

Martin, D. H.

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

Meinhold, P. R.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Menten, K. M.

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

G. Siringo, E. Kreysa, L. A. Reichertz, and K. M. Menten, “A new polarimeter for (sub) millimeter bolometer arrays,” Astron. Astrophys. 422, 751–760 (2004).
[CrossRef]

Mok, C.

W. Chambers, C. Mok, and T. Parker, “Theory of the scattering of electromagnetic waves by regular grid of parallel cylinder wires with circular cross section,” J. Phys. A: Math. Gen. 13, 1433–1441 (1980).
[CrossRef]

Mok, C. L.

Moseley, S. H.

Murk, A.

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

Neill, G. F.

Novak, G.

M. Krejny, D. Chuss, C. Drouet d’Aubigny, D. Golish, M. Houde, H. Hui, C. Kulesa, R. F. Loewenstein, S. H. Moseley, G. Novak, G. Voellmer, C. Walker, and E. Wollack, “The Hertz/VPM polarimeter: design and first light observations,” Appl. Opt. 474429–4440 (2008).
[CrossRef]

D. T. Chuss, E. J. Wollack, S. H. Moseley, and G. Novak, “Interferometric polarization control,” Appl. Opt. 45, 5107–5117 (2006).
[CrossRef]

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

O’Neill, H.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Orlando, A.

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, Vol. 1 (Elsevier, 1998), Chap. 2, p. 15.

Parker, T.

W. Chambers, C. Mok, and T. Parker, “Theory of the scattering of electromagnetic waves by regular grid of parallel cylinder wires with circular cross section,” J. Phys. A: Math. Gen. 13, 1433–1441 (1980).
[CrossRef]

Parker, T. J.

Phillips, J. A.

R. L. Akeson, J. E. Carlstrom, J. A. Phillips, and D. P. Woody, “Millimeter interferometric polarization imaging of the young stellar object NGC 1333/IRAS 4A,” Astrophys. J. Lett. 456, L45 (1996).
[CrossRef]

Pisano, G.

Platt, S. R.

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

Pozar, D.

D. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2004).

Pucel, R.

M. Goldfarb and R. Pucel, “Modeling via hole grounds in microstrip,” IEEE Microwave Guided Wave Lett. 1, 135–137 (1991).
[CrossRef]

Reeds, J.

J. Lagarias, J. Reeds, and M. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Reichertz, L. A.

G. Siringo, E. Kreysa, L. A. Reichertz, and K. M. Menten, “A new polarimeter for (sub) millimeter bolometer arrays,” Astron. Astrophys. 422, 751–760 (2004).
[CrossRef]

Renbarger, T.

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

Savini, G.

G. Savini, P. A. R. Ade, J. House, G. Pisano, V. Haynes, and P. Bastien,“Recovering the frequency dependent modulation function of the achromatic half-wave plate for POL-2: the SCUBA-2 polarimeter,” Appl. Opt. 48, 2006–2013(2009).
[CrossRef]

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

Schleuning, D. A.

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

Seta, M.

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

Shinnaga, H.

H. Shinnaga, M. Tsuboi, and T. Kasuga, “A millimeter polarimeter for the 45 m telescope at Nobeyama,” Publ. Astron. Soc. Jpn. 51, 175–184 (1999).

Siringo, G.

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

G. Siringo, E. Kreysa, L. A. Reichertz, and K. M. Menten, “A new polarimeter for (sub) millimeter bolometer arrays,” Astron. Astrophys. 422, 751–760 (2004).
[CrossRef]

Stebor, N. C.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Sternberg, S.

S. Sternberg, Group Theory and Physics (Cambridge University, 1994).

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tait, G.

Tretyakov, S. A.

V. V. Yatsenko and S. A. Tretyakov, “Higher order impedance boundary conditions for sparse wire grids,” IEEE Trans. Antennas Propag. 48, 720–727 (2000).
[CrossRef]

Tsuboi, M.

H. Shinnaga, M. Tsuboi, and T. Kasuga, “A millimeter polarimeter for the 45 m telescope at Nobeyama,” Publ. Astron. Soc. Jpn. 51, 175–184 (1999).

Ulrich, R.

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

Vaillancourt, J. E.

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

Villa, F.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Villela, T.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Voellmer, G.

Wait, J. R.

J. R. Wait, “Reflection from a wire grid parallel to a conducting plane,” Can. J. Phys. 32, 571–579 (1954).
[CrossRef]

Walker, C.

Ward, J. M.

Williams, B.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Wollack, E.

M. Krejny, D. Chuss, C. Drouet d’Aubigny, D. Golish, M. Houde, H. Hui, C. Kulesa, R. F. Loewenstein, S. H. Moseley, G. Novak, G. Voellmer, C. Walker, and E. Wollack, “The Hertz/VPM polarimeter: design and first light observations,” Appl. Opt. 474429–4440 (2008).
[CrossRef]

E. Wollack, W. Grammer, and J. Kingsley, “The Boifot Orthomode Junction,” NRAO, ALMA Memo Series 425 (2002).

E. Wollack and W. Grammer, “Symmetric waveguide orthomode junctions,” in Proceedings of the 14th International Symposium on Space TeraHertz Technology, E. Walker and J. Payne, eds. (NRAO, 2003), pp. 169–176.

Wollack, E. J.

Woody, D. P.

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

R. L. Akeson, J. E. Carlstrom, J. A. Phillips, and D. P. Woody, “Millimeter interferometric polarization imaging of the young stellar object NGC 1333/IRAS 4A,” Astrophys. J. Lett. 456, L45 (1996).
[CrossRef]

Wright, M.

J. Lagarias, J. Reeds, and M. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Wuensche, C. A.

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Wylde, R. J.

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

Yasumoto, K.

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69–85 (2000).
[CrossRef]

K. Yasumoto and K. Yoshitomi, “Efficient calculation of free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

Yatsenko, V. V.

V. V. Yatsenko and S. A. Tretyakov, “Higher order impedance boundary conditions for sparse wire grids,” IEEE Trans. Antennas Propag. 48, 720–727 (2000).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Sect. 9.7.

Yoshitomi, K.

K. Yasumoto and K. Yoshitomi, “Efficient calculation of free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. (1)

G. Siringo, E. Kreysa, L. A. Reichertz, and K. M. Menten, “A new polarimeter for (sub) millimeter bolometer arrays,” Astron. Astrophys. 422, 751–760 (2004).
[CrossRef]

Astrophys. J. (1)

C. D. Dowell, R. H. Hildebrand, D. A. Schleuning, J. E. Vaillancourt, J. L. Dotson, G. Novak, T. Renbarger, and M. Houde, “Submillimeter array polarimetry with Hertz,” Astrophys. J. 504, 588 (1998).
[CrossRef]

Astrophys. J. Lett. (1)

R. L. Akeson, J. E. Carlstrom, J. A. Phillips, and D. P. Woody, “Millimeter interferometric polarization imaging of the young stellar object NGC 1333/IRAS 4A,” Astrophys. J. Lett. 456, L45 (1996).
[CrossRef]

Can. J. Phys. (1)

J. R. Wait, “Reflection from a wire grid parallel to a conducting plane,” Can. J. Phys. 32, 571–579 (1954).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

M. Goldfarb and R. Pucel, “Modeling via hole grounds in microstrip,” IEEE Microwave Guided Wave Lett. 1, 135–137 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

V. V. Yatsenko and S. A. Tretyakov, “Higher order impedance boundary conditions for sparse wire grids,” IEEE Trans. Antennas Propag. 48, 720–727 (2000).
[CrossRef]

K. Yasumoto and K. Yoshitomi, “Efficient calculation of free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

T. Manabe, J. Inatani, A. Murk, R. J. Wylde, M. Seta, and D. H. Martin, “A new configuration of polarization-rotating dual-beam interferometer for space use,” IEEE Trans. Microwave Theory Tech. 51, 1696–1704 (2003).
[CrossRef]

T. Larsen, “A survey of the theory of wire grids,” IEEE Trans. Microwave Theory Tech. 10, 191–201 (1962).
[CrossRef]

S. Adachi and E. M. Kennaugh, “The analysis of a broad-band circular polarizer including interface reactions,” IEEE Trans. Microwave Theory Tech. 8, 520–525 (1960).
[CrossRef]

Infrared Phys. (1)

R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

Int. J. Infrared Millim. Waves (2)

A. Harvey, “A quasi-optical universal polarizer,” Int. J. Infrared Millim. Waves 14, 1–16 (1993).
[CrossRef]

N. Erickson, “A new quasi-optical filter: the reflective polarizing interferometer,” Int. J. Infrared Millim. Waves 8, 1015–1025 (1987).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Phys. A: Math. Gen. (1)

W. Chambers, C. Mok, and T. Parker, “Theory of the scattering of electromagnetic waves by regular grid of parallel cylinder wires with circular cross section,” J. Phys. A: Math. Gen. 13, 1433–1441 (1980).
[CrossRef]

New Astron. Rev. (2)

A. Catalano, L. Conversi, S. de Gregori, M. de Petris, L. Lamagna, R. Maoli, G. Savini, E. S. Battistelli, and A. Orlando, “A far infrared polarimeter,” New Astron. Rev. 10, 79–89 (2004).
[CrossRef]

R. Leonardi, B. Williams, M. Bersanelli, I. Ferreira, P. M. Lubin, P. R. Meinhold, H. O’Neill, N. C. Stebor, F. Villa, T. Villela, and C. A. Wuensche, “The cosmic foreground explorer (COFE): a balloon-borne microwave polarimeter to characterize polarized foregrounds,” New Astron. Rev. 50, 977–983 (2006).
[CrossRef]

Proc. SPIE (1)

G. Siringo, E. Kreysa, A. Kovacs, K. M. Menten, and J. Forbrich, “Beginning of operation of the polarimeter for the large APEX bolometer camera (LABOCA),” Proc. SPIE 7741, 774108 (2010).

Prog. Electromagn. Res. (1)

K. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” Prog. Electromagn. Res. 29, 69–85 (2000).
[CrossRef]

Publ. Astron. Soc. Jpn. (1)

H. Shinnaga, M. Tsuboi, and T. Kasuga, “A millimeter polarimeter for the 45 m telescope at Nobeyama,” Publ. Astron. Soc. Jpn. 51, 175–184 (1999).

Publ. Astron. Soc. Pac. (3)

M. Houde, R. L. Akeson, J. E. Carlstrom, J. W. Lamb, D. A. Schleuning, and D. P. Woody, “Polarizing grids, their assemblies, and beams of radiation,” Publ. Astron. Soc. Pac. 113, 622–638 (2001).
[CrossRef]

R. H. Hildebrand, J. A. Davidson, J. L. Dotson, C. D. Dowell, G. Novak, and J. E. Vaillancourt, “A primer on far-infrared polarimetry,” Publ. Astron. Soc. Pac. 112, 1215–1235(2000).
[CrossRef]

D. A. Schleuning, C. D. Dowell, R. H. Hildebrand, S. R. Platt, and G. Novak, “Hertz, a submillimeter polarimeter,” Publ. Astron. Soc. Pac. 109, 307–318 (1997).
[CrossRef]

SIAM J. Optim. (1)

J. Lagarias, J. Reeds, and M. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Other (15)

E. Wollack, W. Grammer, and J. Kingsley, “The Boifot Orthomode Junction,” NRAO, ALMA Memo Series 425 (2002).

E. Wollack and W. Grammer, “Symmetric waveguide orthomode junctions,” in Proceedings of the 14th International Symposium on Space TeraHertz Technology, E. Walker and J. Payne, eds. (NRAO, 2003), pp. 169–176.

L. Landau and E. Lifshitz, Electromagnetics of Continuous Media, Vol. 8 (Pergamon, 1960).

F. Grover, Inductance Calculations (Van Nostrand, 1946).

T. Edwards, Foundations for Microstrip Circuit Design (Wiley, 1987).

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

N. Marcuvitz, Waveguide Handbook, MIT Rad. Labs. Series, Vol. 10 (McGraw-Hill, 1951).

N. Erickson, “A 0.9 mm heterodyne receiver for astronomical observations,” in IEEE MTT-S Intl. Microwave Symposium Digest (IEEE, 1978), pp. 438–439.

D. Martin, “Polarizing (Martin-Puplett) interferometric spectrometers for the near- and submillimeter spectra,” in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, 1982), Vol. 6, pp. 65–148.

C. Brosseau, Fundamentals of Polarized Light (Wiley, 1998).

S. Sternberg, Group Theory and Physics (Cambridge University, 1994).

D. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2004).

P. F. Goldsmith, Quasioptical Systems (IEEE, 1998).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Sect. 9.7.

E. D. Palik, Handbook of Optical Constants of Solids, Vol. 1 (Elsevier, 1998), Chap. 2, p. 15.

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Figures (10)

Fig. 1.
Fig. 1.

The VPM consists of a polarizing grid placed in front of and parallel to a planar mirror. The polarization parallel to the grid wires is reflected by the grid. The orthogonal linear polarization passes through the grid and is reflected off the mirror. The two components are recombined at the output port with a relative phase delay that is dependent upon the grid-mirror separation, d. The wire grid spacing or pitch is indicated by g.

Fig. 2.
Fig. 2.

In the limit the wavelength is large compared to the wire pitch, the VPM can be modeled by two independent circuits. An inductive circuit is used for the polarization component having the electric field parallel to the grid wires (top). A capacitive circuit is used to model the polarization component having the electric field perpendicular to the grid wires (bottom).

Fig. 3.
Fig. 3.

The geometry for the general polarizing grid lying in the x-y plane is shown.

Fig. 4.
Fig. 4.

The Marcuvitz circuit parameters are plotted as functions of the geometric filling fraction, 2a/g. The vertical gray line is shown at 2a/g=1/π. The shaded region indicates the region in which the Marcuvitz approximation ceases to hold [22]. Triangles indicate typical values of 2a/g for grids constructed for submillimeter and millimeter use.

Fig. 5.
Fig. 5.

The unit cells used for the simulation in both the inductive (left) and capacitive (right) modes are shown. The large arrow in each case indicates the direction of the incident electric field in the simulation. “Perfect-E” and “Perfect-H” indicate the use of perfect electric and magnetic mirrors, respectively, on the boundaries of the unit cell. Plane wave illumination of the unit cell with these boundary conditions allows representation of an infinite grid for λ>2g.

Fig. 6.
Fig. 6.

(Top) The grid loss factor, η, is plotted as a function of the geometric filling factor for a wire grid. Both HFSS and Microstripes simulations are shown for the cases of the electric field perpendicular to and parallel to the grid wires. The filling factors for typical polarizing grids are plotted for comparison. (Middle) The reactance for the inductive mode is shown as a function of the geometric filling factor. A useful interpolation function is Xn0.511.19ln(2πa/g)+0.53(ln(2πa/g))2+0.11(ln(2πa/g))3. For comparison, the Marcuvitz model is shown, as well as the measured inductance for the grid. (Bottom) The normalized reflection phase of the inductive mode is plotted as a function of geometric filling factor for the HFSS simulation. In this case, we find ϕn13.545(2a/g). In each of the panels, the condition 2a/g=1/π is denoted by a dashed vertical line. Triangles indicate typically manufactured grid geometries.

Fig. 7.
Fig. 7.

The main elements of the experimental setup for the 350 μm test are shown [2]. Radiation from a blackbody source is polarized by a wire grid polarizer having wires oriented at an angle of 45° with respect to the plane of the page. A chopper modulates the intensity of the signal. The radiation is collimated prior to being processed by the VPM. Upon exiting the VPM, the radiation is relayed to the Hertz cryostat. Inside, the radiation passes through the HWP and the bandpass filter before being diplexed into two orthogonal linear polarizations. A bolometer detects the signal in each polarization.

Fig. 8.
Fig. 8.

A, The q and u values were measured as a function of grid-mirror separation at 350 μm. B, Measurements of the VPM at 3 mm are also shown. The transmission-line model is in close agreement with the infinite-wavelength approximation at this frequency; however, the difference between the two is measurable.

Fig. 9.
Fig. 9.

The setup for the 3 mm VPM transmission test is shown. Radiation is emitted from port 1 of the HP 8510 with a vertical polarization. The radiation is then reflected off of the grid and collimated by an ellipsoidal mirror. At this point, the radiation is reflected off of the VPM. An identical ellipsoidal mirror follows, and a folding flat directs the beam into a second feed that is attached to port 2 of the HP 8510. A 90° twist is added to the waveguide to change the sensitivity from the vertically polarized state to the horizontally polarized state.

Fig. 10.
Fig. 10.

The modeled VPM reflection phase delay is shown for single frequencies at 350 and 3000 μm for the grid models obtained above. In the limit g/λ1, a sinusoidal form for Stokes U is observed. As this condition is relaxed, the VPM reflection phase delay differs from the free-space grid-mirror delay. The legend on the right shows the parameters corresponding to each of the curves.

Tables (1)

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Table 1. Fit Results for the 350 μm Measurements

Equations (33)

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U=Ucosδ(d)+Vsinδ(d).
δ(d)=4πdλcosθ,
(1+iXLCiXL+RL(2+iXLCiXL+RL)iXLC1iXL+RL1+iXLCiXL+RL).
1iXCC+(2+XCXCC)RC(iXC+iXCC+(2+XCXCC)RC(2RC+iXCC)iXC2+XCXCC(2+XCXCC)RC+(1+XCXCC)iXCC).
J¯=(Γ00Γ).
D=I(1001)+Q(1001)+U(0110)+V(0ii0).
J¯(d)=(cosϕsinϕsinϕcosϕ)(1001)(ΓTM(θ,d)cosϕΓTE(θ,d)sinϕΓTM(θ,d)sinϕΓTE(θ,ϕ)cosϕ)=(ΓTM(θ,d)cos2ϕΓTM(θ,d)sin2ϕ(ΓTE(θ,d)+ΓTE(θ,d))cosϕsinϕ(ΓTM(θ,d)+ΓTM(θ,d))cosϕsinϕΓTE(θ,d)sin2ϕΓTE(θ,d)cos2ϕ).
XLC=gλ(2πag)2,
XL=gλ{lng2πa+m=1(1m2(gλ)21m)}.
XCC=λg{2(g2πa)2A2},
XC=λg{2(λg)2(g2πa)2A114(2πag)21A2}1,
A1=1+12(2πaλ)2{ln(g2πa)+34+m=1(1m2(gλ)21m)},
A2=112(2πaλ)2{ln(g2πa)114}+(2πag)2{124m=1(m12m(gλ)2m2(gλ)2)}.
R=|S11|2=|11+2ZL|=1(1+2RL)2+(2XL)2,
ϕr=πarctan((2ZL)1+(2ZL))=πarctan(2XL1+2RL).
T=|S21|2=|2ZL1+2ZL|2=(2RL)2+(2XL)2(1+2RL)2+(2XL)2,
AR=4RL=η4RSZ0=η4πδλ,
4RSZ0=4ωμ/2σμ/ϵ=4πδλ.
ri(αj)=([riin-phase(αj)]2+[riquad(αj)]2)12,
ti(αj)=([tiin-phase(αj)]2+[tiquad(αj)]2)12.
R(αj)=1Ni=1Nri(αj),
T(αj)=1Ni=1Nti(αj).
f=jR(αj)jT(αj).
S(αj)=R(αj)fT(αj)R(αj)+fT(αj).
S(α)=ϵ1(qcos4α+usin4α)+ϵ2cos2α+ϵ3sin2α.
(ϵ2ϵ3)=(cos2χsin2χsin2χcos2χ)(ϵ2ϵ3).
D¯=0J¯(ν)D¯(ν)J¯(ν)ψn(ν)dν,
H(d)Δν|S12H(d,ν)|2,
V(d)Δν|S12V(d,ν)|2.
H(d)=12[IH+UHcosδ(d)+VHsinδ(d)],
V(d)=12[IVUVcosδ(d)VVsinδ(d)].
f=IHIV.
u(d)=H(d)fV(d)H(d)+fV(d).

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