Abstract

We explore a free-space polarization modulator in which a variable phase is introduced between the right- and left-handed circular polarization components and used to rotate the linear polarization of the outgoing beam relative to that of the incoming beam. In this device, the polarization states are separated by a circular polarizer that consists of a quarter-wave plate in combination with a wire grid. A movable mirror is positioned behind and parallel to the circular polarizer. As the polarizer–mirror distance is changed, an incident linear polarization will be rotated through an angle that is proportional to the introduced phase delay. We demonstrate a prototype device that modulates Stokes Q and U over a 20% bandwidth, from 77 to 94 GHz.

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  17. B. Maffei, G. Pisano, Ming Wah Ng, and Vic Haynes, “Millimetre wave photolithographic polariser beam impact,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1761–7165.
  18. G. Pisano, G. Savini, P. Ade, and V. Haynes, “A metal-mesh achromatic half-wave plate for use at submillimetre wavelengths,” Appl. Opt. 47, 6251–6256 (2008).
    [CrossRef]
  19. J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
    [CrossRef]

2012 (2)

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

D. T. Chuss, E. J. Wollack, R. Henry, H. Hui, A. J. Juarez, M. Krejny, S. H. Moseley, and G. Novak, “Properties of a variable-delay polarization modulator,” Appl. Opt. 51, 197–208(2012).
[CrossRef]

2011 (2)

J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
[CrossRef]

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

2010 (2)

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

S. A. Bryan, T. E. Montroy, and J. E. Ruhl, “Modeling dielectric half-wave plates for cosmic microwave background polarimetry using a Mueller matrix formalism,” Appl. Opt. 49, 6313–6323 (2010).
[CrossRef]

2009 (1)

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

2008 (2)

2006 (2)

2000 (1)

R. H. Hildebrand, J. A. Hildebrand, 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]

1979 (1)

G. P. Nanos, “Polarization of the blackbody radiation at 3.2 centimeters,” Astrophys. J. 232, 341–347 (1979).
[CrossRef]

1955 (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates—Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. A 42, 24–31 (1955).

1941 (1)

Ade, P.

Ade, P. A. R.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

G. Savini, G. Pisano, and P. A. R. Ade, “Achromatic half-wave plate for submillimeter instruments in cosmic microwave background astronomy: modeling and simulation,” Appl. Opt. 45, 8907–8915 (2006).
[CrossRef]

Benford, D. J.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Bennett, C. L.

J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
[CrossRef]

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Bierman, E. M.

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

Brosseau, C.

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

Bryan, S. A.

Chuss, D.

Chuss, D. T.

D. T. Chuss, E. J. Wollack, R. Henry, H. Hui, A. J. Juarez, M. Krejny, S. H. Moseley, and G. Novak, “Properties of a variable-delay polarization modulator,” Appl. Opt. 51, 197–208(2012).
[CrossRef]

J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
[CrossRef]

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

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

Dotson, J. L.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

R. H. Hildebrand, J. A. Hildebrand, 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]

Dowell, C. D.

R. H. Hildebrand, J. A. Hildebrand, 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]

Drouet d’Aubigny, C.

Eimer, J. R.

J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
[CrossRef]

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Fixsen, D. J.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Gault, A. C.

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

Golish, D.

Halpern, M.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Hanany, S.

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

Haynes, V.

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

G. Pisano, G. Savini, P. Ade, and V. Haynes, “A metal-mesh achromatic half-wave plate for use at submillimetre wavelengths,” Appl. Opt. 47, 6251–6256 (2008).
[CrossRef]

G. Pisano, M. W. Ng, V. Haynes, and B. Maffei, “A broadband photolithographic polariser for millimetre wave applications,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1748–1751.

Haynes, Vic

B. Maffei, G. Pisano, Ming Wah Ng, and Vic Haynes, “Millimetre wave photolithographic polariser beam impact,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1761–7165.

Henry, R.

Hildebrand, J. A.

R. H. Hildebrand, J. A. Hildebrand, 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]

Hildebrand, R. H.

R. H. Hildebrand, J. A. Hildebrand, 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]

Hilton, G.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Hinderks, J.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Hindshaw, G.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Houde, M.

Hui, H.

Hyland, P. O.

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

Irwin, K.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Jackson, M. L.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Jah, M. A.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Jethava, N.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Jhabvala, C.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Jones, R.

Juarez, A. J.

Keating, B. G.

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

Kogut, A.

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

Kogut, A. J.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Krejny, M.

Kulesa, C.

Loewenstein, R. F.

Lowe, L.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Maffei, B.

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

B. Maffei, G. Pisano, Ming Wah Ng, and Vic Haynes, “Millimetre wave photolithographic polariser beam impact,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1761–7165.

G. Pisano, M. W. Ng, V. Haynes, and B. Maffei, “A broadband photolithographic polariser for millimetre wave applications,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1748–1751.

Malu, S. S.

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

McCullagh, N.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Melhuish, S.

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

Miller, T.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Mirel, P.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Montroy, T. E.

Moseley, S. H.

D. T. Chuss, E. J. Wollack, R. Henry, H. Hui, A. J. Juarez, M. Krejny, S. H. Moseley, and G. Novak, “Properties of a variable-delay polarization modulator,” Appl. Opt. 51, 197–208(2012).
[CrossRef]

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

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. 47, 4429–4440 (2008).
[CrossRef]

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

Nanos, G. P.

G. P. Nanos, “Polarization of the blackbody radiation at 3.2 centimeters,” Astrophys. J. 232, 341–347 (1979).
[CrossRef]

Ng, M. W.

G. Pisano, M. W. Ng, V. Haynes, and B. Maffei, “A broadband photolithographic polariser for millimetre wave applications,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1748–1751.

Ng, Ming Wah

B. Maffei, G. Pisano, Ming Wah Ng, and Vic Haynes, “Millimetre wave photolithographic polariser beam impact,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1761–7165.

Novak, G.

Pancharatnam, S.

S. Pancharatnam, “Achromatic combinations of birefringent plates—Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. A 42, 24–31 (1955).

Piccirillo, L.

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

Pisano, G.

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

G. Pisano, G. Savini, P. Ade, and V. Haynes, “A metal-mesh achromatic half-wave plate for use at submillimetre wavelengths,” Appl. Opt. 47, 6251–6256 (2008).
[CrossRef]

G. Savini, G. Pisano, and P. A. R. Ade, “Achromatic half-wave plate for submillimeter instruments in cosmic microwave background astronomy: modeling and simulation,” Appl. Opt. 45, 8907–8915 (2006).
[CrossRef]

G. Pisano, M. W. Ng, V. Haynes, and B. Maffei, “A broadband photolithographic polariser for millimetre wave applications,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1748–1751.

B. Maffei, G. Pisano, Ming Wah Ng, and Vic Haynes, “Millimetre wave photolithographic polariser beam impact,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1761–7165.

Rodriguez, S.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Rostem, K.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Ruhl, J. E.

S. A. Bryan, T. E. Montroy, and J. E. Ruhl, “Modeling dielectric half-wave plates for cosmic microwave background polarimetry using a Mueller matrix formalism,” Appl. Opt. 49, 6313–6323 (2010).
[CrossRef]

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

Savini, G.

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

G. Pisano, G. Savini, P. Ade, and V. Haynes, “A metal-mesh achromatic half-wave plate for use at submillimetre wavelengths,” Appl. Opt. 47, 6251–6256 (2008).
[CrossRef]

G. Savini, G. Pisano, and P. A. R. Ade, “Achromatic half-wave plate for submillimeter instruments in cosmic microwave background astronomy: modeling and simulation,” Appl. Opt. 45, 8907–8915 (2006).
[CrossRef]

Sharp, E.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Staguhn, J. G.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Timbie, P. T.

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

Tinbergen, J.

J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).

Tucker, C. E.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Vaillancourt, J. E.

R. H. Hildebrand, J. A. Hildebrand, 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]

Voellmer, G.

Voellmer, G. M.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Walker, C.

Wollack, E.

Wollack, E. J.

D. T. Chuss, E. J. Wollack, R. Henry, H. Hui, A. J. Juarez, M. Krejny, S. H. Moseley, and G. Novak, “Properties of a variable-delay polarization modulator,” Appl. Opt. 51, 197–208(2012).
[CrossRef]

J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
[CrossRef]

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

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

Zeng, L.

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Appl. Opt. (6)

Astrophys. J. (1)

G. P. Nanos, “Polarization of the blackbody radiation at 3.2 centimeters,” Astrophys. J. 232, 341–347 (1979).
[CrossRef]

IEEE Microw. Wireless Compon. Lett. (2)

A. C. Gault, E. M. Bierman, P. O. Hyland, B. G. Keating, S. S. Malu, and P. T. Timbie, “Measurements of a cryogenic linear polarization modulator for mm-wavelengths,” IEEE Microw. Wireless Compon. Lett. 99, 1 (2012).

G. Pisano, S. Melhuish, G. Savini, L. Piccirillo, and B. Maffei, “A broadband W-band polarization rotator with very low cross polarization,” IEEE Microw. Wireless Compon. Lett. 21, 127–129 (2011).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. Conf. Ser. (1)

P. A. R. Ade, D. T. Chuss, S. Hanany, V. Haynes, B. G. Keating, A. Kogut, J. E. Ruhl, G. Pisano, G. Savini, and E. J. Wollack, “Polarization modulators for CMBPol,” J. Phys. Conf. Ser. 155, 012006 (2009).
[CrossRef]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Achromatic combinations of birefringent plates—Part II. An achromatic quarter wave plate,” Proc. Indian Acad. Sci. A 42, 24–31 (1955).

Proc. SPIE (1)

D. T. Chuss, P. A. R. Ade, D. J. Benford, C. L. Bennett, J. L. Dotson, J. R. Eimer, D. J. Fixsen, M. Halpern, G. Hilton, J. Hinderks, G. Hindshaw, K. Irwin, M. L. Jackson, M. A. Jah, N. Jethava, C. Jhabvala, A. J. Kogut, L. Lowe, N. McCullagh, T. Miller, P. Mirel, S. H. Moseley, S. Rodriguez, K. Rostem, E. Sharp, J. G. Staguhn, C. E. Tucker, G. M. Voellmer, E. J. Wollack, and L. Zeng, “The primordial inflation polarization explorer (PIPER),” Proc. SPIE 7741, 77411P (2010).
[CrossRef]

Publ. Astron. Soc. Pac. (1)

R. H. Hildebrand, J. A. Hildebrand, 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]

Rev. Sci. Instrum. (1)

J. R. Eimer, C. L. Bennett, D. T. Chuss, and E. J. Wollack, “Vector reflectometry in a beam waveguide,” Rev. Sci. Instrum. 82, 086101 (2011).
[CrossRef]

Other (4)

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

J. Tinbergen, Astronomical Polarimetry (Cambridge University, 1996).

G. Pisano, M. W. Ng, V. Haynes, and B. Maffei, “A broadband photolithographic polariser for millimetre wave applications,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1748–1751.

B. Maffei, G. Pisano, Ming Wah Ng, and Vic Haynes, “Millimetre wave photolithographic polariser beam impact,” in PIERS 2012 Kuala Lumpur Proceedings (2012), pp. 1761–7165.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic representations of ideal polarization modulators on the Poincaré sphere are shown. The green line shows the axis connecting the two polarization states between which a phase delay, δ , is introduced. Blue arrows indicate the modulation path on the sphere. Red arrows indicate static phase delays. Deviations from ideal behavior are generally caused by finite bandwidth, biattenuance, differential reflection, etc. An example Mueller matrix for each architecture is shown on the right for free-space implementations (i.e., A and B are in transmission; for C and D, the Mueller matrices are in reflection).

Fig. 2.
Fig. 2.

The topology for the TPR compared with the VPM is shown. The TPR consists of a circular polarizer placed in front of and parallel to a movable mirror and introduces a variable phase delay between the orthogonal circular polarization states.

Fig. 3.
Fig. 3.

The metal-mesh wave plate is shown (A) along with a close-up view of its surface (B). The prototype TPR is shown in C.

Fig. 4.
Fig. 4.

Test setup used to validate the VPM is shown (left). The grid–mirror separation is given by A, and the overall displacement of the TPR is given by B. The rotational coordinate system used for the polarization measurements is shown on the right.

Fig. 5.
Fig. 5.

The measured q and u spectra are shown for a grid–mirror separation of 1143 μm. The dashed lines are the corresponding instrument models for q and u . The shaded region corresponding to each of the Stokes parameters illustrates the effect of a ± 15 μm uncertainty in the grid–mirror separation.

Fig. 6.
Fig. 6.

The response of the TPR to an incident q (vertically polarized) signal is shown. The measured q and u response for the integrated 77–94 GHz response is superposed on the integrated response expected from theory.

Equations (16)

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

Q 2 + U 2 + V 2 = constant .
J ¯ TPR = J ¯ WP ( θ , β ) J ¯ VPM ( 0 , δ ) J ¯ WP ( θ , β ) ,
J ¯ WP ( θ , β ) = R ¯ ( θ ) J ¯ WP ( 0 , β ) R ¯ ( θ ) = ( e i β 2 cos 2 θ + e i β 2 sin 2 θ i sin 2 θ sin β 2 i sin 2 θ sin β 2 e i β 2 sin 2 θ + e i β 2 cos 2 θ ) .
R ¯ ( θ ) = ( cos θ sin θ sin θ cos θ ) ,
J ¯ WP ( 0 , β ) = ( e i β 2 0 0 e i β 2 ) .
J ¯ VPM ( 0 , δ ) = ( e i δ 2 0 0 e i δ 2 ) ,
J ¯ TPR 1 , 1 = e i δ 2 ( e i β cos 4 θ + e i β sin 4 θ + 2 cos 2 θ sin 2 θ ) e i δ 2 sin 2 2 θ sin 2 β 2 , J ¯ TPR 1 , 2 = i sin 2 θ sin β 2 [ e i δ 2 ( e i β 2 cos 2 θ + e i β 2 sin 2 θ ) + e i δ 2 ( e i β 2 sin 2 θ + e i β 2 cos 2 θ ) ] , J ¯ TPR 2 , 1 = i sin 2 θ sin β 2 [ e i δ 2 ( e i β 2 cos 2 θ + e i β 2 sin 2 θ ) + e i δ 2 ( e i β 2 sin 2 θ + e i β 2 cos 2 θ ) ] , J ¯ TPR 2 , 2 = e i δ 2 ( e i β cos 4 θ + e i β sin 4 θ + 2 cos 2 θ sin 2 θ ) + e i δ 2 sin 2 2 θ sin 2 β 2 .
J ¯ TPR ( π 4 , π 2 ) = i ( sin δ 2 cos δ 2 cos δ 2 sin δ 2 ) .
D = J ¯ TPR D J ¯ TPR .
D = I σ 0 ¯ + Q σ 1 ¯ + U σ 2 ¯ + V σ 3 ¯ ,
= I ( 1 0 0 1 ) + Q ( 1 0 0 1 ) + U ( 0 1 1 0 ) + V ( 0 i i 0 ) .
M ¯ TPR ( π 4 , π 2 ) = ( 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ 0 0 0 0 1 ) .
Q = Q cos δ U sin δ , U = Q sin δ U cos δ
q ( d ) = H ( d ) f q V ( d ) H ( d ) + f q V ( d ) ,
u ( d ) = D + ( d ) f u D ( d ) D + ( d ) + f u D ( d ) .
q ( model ) = Δ λ cos δ ( λ ) d λ , u ( model ) = Δ λ sin δ ( λ ) d λ .

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