Abstract

We present a comprehensive, spatiotemporal, modal theory of submillimeter-wave and far-infrared power detectors. The theory is based on the contraction of the coherence tensor of the light with another coherence tensor that incorporates all of the physics of the detector. The theory is extremely general and applies to detectors of any bandwidth, with light in any state of polarization and spatiotemporal coherence. The theory applies equally to quasi-monochromatic and pulsed systems. We show that the tensor associated with the detector is a measureable quantity and outline a procedure for its experimental determination. We derive expressions for the statistical properties of a detector’s output, including the correlations between the outputs of different detectors, say, in an array or interferometer. The theory provides a clear conceptual understanding of how any general detector couples to the modes of an optical system and thereby provides a powerful and flexible way of modeling the behavior of detectors and instruments.

© 2007 Optical Society of America

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    [CrossRef]
  13. S. Withington, G. Saklatvala, and M. P. Hobson, "Scattering coherent and partially coherent space-time pulses through few-mode optical systems," J. Opt. Soc. Am. A 23, 2775-2783 (2006).
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    [CrossRef]
  27. S. Withington, M. P. Hobson, and G. Saklatvala, "Quantum statistical analysis of multimode far-infrared and submillimeter-wave astronomical interferometers," J. Opt. Soc. Am. A 22, 1937-1946 (2005).
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2006

2005

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, "Opto-electronic pulsed THz systems," Semicond. Sci. Technol. 20, S121-S127 (2005).
[CrossRef]

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

S. Withington, M. P. Hobson, and G. Saklatvala, "Quantum statistical analysis of multimode far-infrared and submillimeter-wave astronomical interferometers," J. Opt. Soc. Am. A 22, 1937-1946 (2005).
[CrossRef]

S. Withington, E. S. Campbell, and M. P. Hobson, "A numerical procedure for simulating the behavior of multimode, bolometric, astronomical interferometers," J. Appl. Phys. 97, 124,909/1-124,909/8 (2005).
[CrossRef]

2004

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal foundations of close-packed optical arrays with particular application to infrared and millimeter-wave astronomical interferometry," J. Appl. Phys. 96, 1794-1802 (2004).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal analysis of astronomical bolometric interferometers," J. Opt. Soc. Am. A 21, 1988-1995 (2004).
[CrossRef]

S. Wang and X. Zhang, "Pulsed terahertz tomography," J. Phys. D 37, R1-R36 (2004).
[CrossRef]

P. J. B. Clarricoats, R. F. Dubrovka, and A. D. Olver, "High performance compact corrugated horn," IEEE Proc. Microwaves, Antennas and Propagation 151, 519-524 (2004).
[CrossRef]

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

2003

2001

1964

L. Mandel, E. C. G. Sudarshan, and E. Wolf, "Theory of photoelectric detection of light fluctuations," Proc. Phys. Soc. London 84, 435-444 (1964).
[CrossRef]

1963

R. Glauber, "Photon correlations," Phys. Rev. Lett. 10, 84-86 (1963).
[CrossRef]

R. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277-279 (1963).
[CrossRef]

1957

P. Felgett, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 956-957 (1957).
[CrossRef]

R. Q. Twiss and R. H. Brown, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 1128-1129 (1957).
[CrossRef]

R. M. Sillitto, "Correlation between events in photon detectors," Nature (London) 179, 1127-1128 (1957).
[CrossRef]

1956

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in 2 coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

E. M. Purcell, "The question of correlation between photons in coherent light rays," Nature (London) 178, 1449-1450 (1956).
[CrossRef]

1946

R. H. Dicke, "The measurement of thermal radiation at microwave frequencies," Rev. Sci. Instrum. 177, 268-275 (1946).
[CrossRef]

Ade, P. A. R.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Agnèse, P.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Allen, C. A.

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

André, P.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Benford, D. J.

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

Bock, J. J.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Bond, J. R.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Brown, R. H.

R. Q. Twiss and R. H. Brown, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 1128-1129 (1957).
[CrossRef]

Campano, B.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Campbell, E. S.

S. Withington, E. S. Campbell, and M. P. Hobson, "A numerical procedure for simulating the behavior of multimode, bolometric, astronomical interferometers," J. Appl. Phys. 97, 124,909/1-124,909/8 (2005).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal analysis of astronomical bolometric interferometers," J. Opt. Soc. Am. A 21, 1988-1995 (2004).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal foundations of close-packed optical arrays with particular application to infrared and millimeter-wave astronomical interferometry," J. Appl. Phys. 96, 1794-1802 (2004).
[CrossRef]

Cantalupo, C.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Chen, T. C.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Cheng, E.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Chervenak, J. A.

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Clarricoats, P. J. B.

P. J. B. Clarricoats, R. F. Dubrovka, and A. D. Olver, "High performance compact corrugated horn," IEEE Proc. Microwaves, Antennas and Propagation 151, 519-524 (2004).
[CrossRef]

Contaldi, C. R.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Cottingham, D. A.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Crawford, T. M.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Daub, M. D.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Devlin, M. J.

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Dicke, R. H.

R. H. Dicke, "The measurement of thermal radiation at microwave frequencies," Rev. Sci. Instrum. 177, 268-275 (1946).
[CrossRef]

Dicker, S. R.

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Downes, T.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Dubrovka, R. F.

P. J. B. Clarricoats, R. F. Dubrovka, and A. D. Olver, "High performance compact corrugated horn," IEEE Proc. Microwaves, Antennas and Propagation 151, 519-524 (2004).
[CrossRef]

Felgett, P.

P. Felgett, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 956-957 (1957).
[CrossRef]

Finkbeiner, F. M.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Fixsen, D. J.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Glauber, R.

R. Glauber, "Photon correlations," Phys. Rev. Lett. 10, 84-86 (1963).
[CrossRef]

R. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Goldstein, J. H.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Gomez, P. L.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Hanbury Brown, R.

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in 2 coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

Hobson, M. P.

S. Withington, G. Saklatvala, and M. P. Hobson, "Scattering coherent and partially coherent space-time pulses through few-mode optical systems," J. Opt. Soc. Am. A 23, 2775-2783 (2006).
[CrossRef]

G. Saklatvala, M. P. Hobson, and S. Withington, "Simulations of multimode bolometric interferometers," Mon. Not. R. Astron. Soc. 367, 1194-1200 (2006).
[CrossRef]

S. Withington, G. Saklatvala, and M. P. Hobson, "Partially coherent analysis of imaging and interferometric phased arrays: noise, correlations and fluctuations," J. Opt. Soc. Am. A 23, 1340-1348 (2006).
[CrossRef]

S. Withington, E. S. Campbell, and M. P. Hobson, "A numerical procedure for simulating the behavior of multimode, bolometric, astronomical interferometers," J. Appl. Phys. 97, 124,909/1-124,909/8 (2005).
[CrossRef]

S. Withington, M. P. Hobson, and G. Saklatvala, "Quantum statistical analysis of multimode far-infrared and submillimeter-wave astronomical interferometers," J. Opt. Soc. Am. A 22, 1937-1946 (2005).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal foundations of close-packed optical arrays with particular application to infrared and millimeter-wave astronomical interferometry," J. Appl. Phys. 96, 1794-1802 (2004).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal analysis of astronomical bolometric interferometers," J. Opt. Soc. Am. A 21, 1988-1995 (2004).
[CrossRef]

Holzapfel, W. L.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Horeau, B.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Irwin, K. D.

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Kuo, C. L.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Lange, A. E.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Logan, D.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Lueker, M.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Mandel, L.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, "Theory of photoelectric detection of light fluctuations," Proc. Phys. Soc. London 84, 435-444 (1964).
[CrossRef]

Mandel, V. L.

V. L. Mandel and E. Wolf, Coherence and Quantum Optics (Plenum, 1984).

Meyer, S. S.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Minier, V.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Moseley, S. H.

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

Newcomb, M.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Olver, A. D.

P. J. B. Clarricoats, R. F. Dubrovka, and A. D. Olver, "High performance compact corrugated horn," IEEE Proc. Microwaves, Antennas and Propagation 151, 519-524 (2004).
[CrossRef]

Page, L.

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

Pennec, J. L.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Perera, T.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Peterson, J. B.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Planken, P. C. M.

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, "Opto-electronic pulsed THz systems," Semicond. Sci. Technol. 20, S121-S127 (2005).
[CrossRef]

Pogosyan, D.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Purcell, E. M.

E. M. Purcell, "The question of correlation between photons in coherent light rays," Nature (London) 178, 1449-1450 (1956).
[CrossRef]

Revéret, V.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Rodriguez, L.

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

Romer, A. K.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Ruhl, J.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Runyan, M. C.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Saklatvala, G.

Schouten, R. N.

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, "Opto-electronic pulsed THz systems," Semicond. Sci. Technol. 20, S121-S127 (2005).
[CrossRef]

Sharp, E. H.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Sillitto, R. M.

R. M. Sillitto, "Correlation between events in photon detectors," Nature (London) 179, 1127-1128 (1957).
[CrossRef]

Silverberg, R. F.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Stacey, G. J.

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

Staguhn, J. G.

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Sudarshan, E. C. G.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, "Theory of photoelectric detection of light fluctuations," Proc. Phys. Soc. London 84, 435-444 (1964).
[CrossRef]

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277-279 (1963).
[CrossRef]

Supanich, M. P.

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Torbet, E.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Twiss, R. Q.

R. Q. Twiss and R. H. Brown, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 1128-1129 (1957).
[CrossRef]

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in 2 coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

van Rijmenam, C. E. W. M.

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, "Opto-electronic pulsed THz systems," Semicond. Sci. Technol. 20, S121-S127 (2005).
[CrossRef]

Wang, S.

S. Wang and X. Zhang, "Pulsed terahertz tomography," J. Phys. D 37, R1-R36 (2004).
[CrossRef]

Wilson, G. W.

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

Withington, S.

S. Withington, G. Saklatvala, and M. P. Hobson, "Scattering coherent and partially coherent space-time pulses through few-mode optical systems," J. Opt. Soc. Am. A 23, 2775-2783 (2006).
[CrossRef]

G. Saklatvala, M. P. Hobson, and S. Withington, "Simulations of multimode bolometric interferometers," Mon. Not. R. Astron. Soc. 367, 1194-1200 (2006).
[CrossRef]

S. Withington, G. Saklatvala, and M. P. Hobson, "Partially coherent analysis of imaging and interferometric phased arrays: noise, correlations and fluctuations," J. Opt. Soc. Am. A 23, 1340-1348 (2006).
[CrossRef]

S. Withington, E. S. Campbell, and M. P. Hobson, "A numerical procedure for simulating the behavior of multimode, bolometric, astronomical interferometers," J. Appl. Phys. 97, 124,909/1-124,909/8 (2005).
[CrossRef]

S. Withington, M. P. Hobson, and G. Saklatvala, "Quantum statistical analysis of multimode far-infrared and submillimeter-wave astronomical interferometers," J. Opt. Soc. Am. A 22, 1937-1946 (2005).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal foundations of close-packed optical arrays with particular application to infrared and millimeter-wave astronomical interferometry," J. Appl. Phys. 96, 1794-1802 (2004).
[CrossRef]

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal analysis of astronomical bolometric interferometers," J. Opt. Soc. Am. A 21, 1988-1995 (2004).
[CrossRef]

S. Withington and G. Yassin, "Power coupled between partially coherent vector fields in different states of coherence," J. Opt. Soc. Am. A 18, 3061-3071 (2001).
[CrossRef]

Wolf, E.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, "Theory of photoelectric detection of light fluctuations," Proc. Phys. Soc. London 84, 435-444 (1964).
[CrossRef]

V. L. Mandel and E. Wolf, Coherence and Quantum Optics (Plenum, 1984).

Wollack, E. J.

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Woolsey, D.

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

Yassin, G.

Zhang, X.

S. Wang and X. Zhang, "Pulsed terahertz tomography," J. Phys. D 37, R1-R36 (2004).
[CrossRef]

Zmuidzinas, J.

Appl. Opt.

IEEE Proc. Microwaves, Antennas and Propagation

P. J. B. Clarricoats, R. F. Dubrovka, and A. D. Olver, "High performance compact corrugated horn," IEEE Proc. Microwaves, Antennas and Propagation 151, 519-524 (2004).
[CrossRef]

J. Appl. Phys.

S. Withington, M. P. Hobson, and E. S. Campbell, "Modal foundations of close-packed optical arrays with particular application to infrared and millimeter-wave astronomical interferometry," J. Appl. Phys. 96, 1794-1802 (2004).
[CrossRef]

S. Withington, E. S. Campbell, and M. P. Hobson, "A numerical procedure for simulating the behavior of multimode, bolometric, astronomical interferometers," J. Appl. Phys. 97, 124,909/1-124,909/8 (2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

S. Wang and X. Zhang, "Pulsed terahertz tomography," J. Phys. D 37, R1-R36 (2004).
[CrossRef]

Mon. Not. R. Astron. Soc.

G. Saklatvala, M. P. Hobson, and S. Withington, "Simulations of multimode bolometric interferometers," Mon. Not. R. Astron. Soc. 367, 1194-1200 (2006).
[CrossRef]

Nature (London)

R. Hanbury Brown and R. Q. Twiss, "Correlation between photons in 2 coherent beams of light," Nature (London) 177, 27-29 (1956).
[CrossRef]

E. M. Purcell, "The question of correlation between photons in coherent light rays," Nature (London) 178, 1449-1450 (1956).
[CrossRef]

P. Felgett, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 956-957 (1957).
[CrossRef]

R. Q. Twiss and R. H. Brown, "The question of correlation between photons in coherent beams of light," Nature (London) 179, 1128-1129 (1957).
[CrossRef]

R. M. Sillitto, "Correlation between events in photon detectors," Nature (London) 179, 1127-1128 (1957).
[CrossRef]

Phys. Rev.

R. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Phys. Rev. Lett.

E. C. G. Sudarshan, "Equivalence of semiclassical and quantum mechanical descriptions of statistical light beams," Phys. Rev. Lett. 10, 277-279 (1963).
[CrossRef]

R. Glauber, "Photon correlations," Phys. Rev. Lett. 10, 84-86 (1963).
[CrossRef]

Proc. Phys. Soc. London

L. Mandel, E. C. G. Sudarshan, and E. Wolf, "Theory of photoelectric detection of light fluctuations," Proc. Phys. Soc. London 84, 435-444 (1964).
[CrossRef]

Proc. SPIE

D. J. Benford, J. G. Staguhn, G. J. Stacey, L. Page, S. H. Moseley, K. D. Irwin, J. A. Chervenak, and C. A. Allen, "Design and fabrication of two-dimensional superconducting bolometeric arrays," in Proc. SPIE 5498, 647-658 (2004).
[CrossRef]

R. F. Silverberg, B. Campano, T. C. Chen, E. Cheng, D. A. Cottingham, T. M. Crawford, T. Downes, F. M. Finkbeiner, D. J. Fixsen, D. Logan, S. S. Meyer, T. Perera, E. H. Sharp, and G. W. Wilson, "A bolometer array for the SPEctral Energy Distribution (SPEED) camera," in Proc. SPIE 5498, 659-666 (2004).
[CrossRef]

D. J. Benford, S. R. Dicker, E. J. Wollack, M. P. Supanich, J. G. Staguhn, S. H. Moseley, K. D. Irwin, M. J. Devlin, J. A. Chervenak, and T. C. Chen, "A planar two-dimensional superconducting bolometer array for the Green Bank telescope," Proc. SPIE 5498, 208-219 (2004).
[CrossRef]

Rev. Sci. Instrum.

R. H. Dicke, "The measurement of thermal radiation at microwave frequencies," Rev. Sci. Instrum. 177, 268-275 (1946).
[CrossRef]

Semicond. Sci. Technol.

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, "Opto-electronic pulsed THz systems," Semicond. Sci. Technol. 20, S121-S127 (2005).
[CrossRef]

Other

V. Minier, V. Revéret, L. Rodriguez, J. L. Pennec, P. André, B. Horeau, and P. Agnèse, "CEA development of filled bolometer arrays at submillimeter wavelengths: A possible application to ground based telescopes at Dome C," European Astronomical Society Publication Series No. 14, 343-348 (2005).

J. B. Peterson, A. K. Romer, P. L. Gomez, P. A. R. Ade, J. J. Bock, J. R. Bond, C. R. Contaldi, D. Pogosyan, C. Cantalupo, M. D. Daub, W. L. Holzapfel, M. Lueker, M. Newcomb, D. Woolsey, C. L. Kuo, A. E. Lange, M. C. Runyan, J. Ruhl, J. H. Goldstein, and E. Torbet, "Imaging the cosmic microwave background with the arcminute cosmology bolometer array receiver," in Proceedings of the 216th International Astronomical Union Symposium (IAU) (IAU, 2005), p. 43-50.

V. L. Mandel and E. Wolf, Coherence and Quantum Optics (Plenum, 1984).

Standard Test Procedures for Antennas (IEEE, 1979).

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Figures (3)

Fig. 1
Fig. 1

Pictorial representation of the coupled-mode model.

Fig. 2
Fig. 2

Surface of integration used to determine the integrated detector output with a long integration time. Only the shaded area contributes to the integral, so the integration range in the u direction can be extended to infinity.

Fig. 3
Fig. 3

Two-dimensional analog of the four-dimensional volume used to determine the covariance of the integrated outputs of two detectors. Only the shaded area contributes to the integral, and this area is proportional to the intersection of the two integration intervals.

Equations (65)

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

N ( t ) = 1 Z 0 S d 2 r E r ( r , t ) 2 ,
P = h ( T ) p ( T ) d T ,
X i j r ( r 1 , t 1 ; r 2 , t 2 ) E i r ( r 1 , t 1 ) E j r ( r 2 , t 2 ) ,
P = i j d t 1 d t 2 S d 2 r 1 S d 2 r 2 × V i j r ( r 1 , t 1 ; r 2 , t 2 ) X i j r ( r 1 , t 1 ; r 2 , t 2 ) ,
p ( T ) = i j d t 1 d t 2 S d 2 r 1 S d 2 r 2 × U i j r ( T , r 1 , t 1 ; r 2 , t 2 ) X i j r ( r 1 , t 1 ; r 2 , t 2 ) ,
V i j r ( r 1 , t 1 ; r 2 , t 2 ) = d T h ( T ) U i j r ( T , r 1 , t 1 ; r 2 , t 2 ) .
X i j ( r 1 , t 1 ; r 2 , t 2 ) E i ( r 1 , t 1 ) E j * ( r 2 , t 2 ) ,
E r ( r , t ) = 1 π 0 d ω d t cos ( ω t ) e i ω t E ( r , t ) .
V i j ( r 1 , t 1 ; r 2 , t 2 ) = 1 π 2 0 d ω 1 0 d ω 2 d t 1 d t 2 × e i ( ω 2 t 2 ω 1 t 1 ) cos ( ω 1 t 1 ) cos ( ω 2 t 2 ) V i j r ( r 1 , t 1 ; r 2 , t 2 )
U i j ( T , r 1 , t 1 ; r 2 , t 2 ) = 1 π 2 0 d ω 1 0 d ω 2 d t 1 d t 2 × e i ( ω 2 t 2 ω 1 t 1 ) cos ( ω 1 t 1 ) cos ( ω 2 t 2 ) U i j r ( T , r 1 , t 1 ; r 2 , t 2 )
P = i j d t 1 d t 2 S d 2 r 1 S d 2 r 2 × V i j * ( r 1 , t 1 ; r 2 , t 2 ) X i j ( r 1 , t 1 ; r 2 , t 2 ) ,
p ( T ) = i j d t 1 d t 2 S d 2 r 1 S d 2 r 2 × U i j * ( T , r 1 , t 1 ; r 2 , t 2 ) X i j ( r 1 , t 1 ; r 2 , t 2 ) .
X i j ( r 1 , t 1 ; r 2 , t 2 ) = k ψ i ( k ) ( r 1 , t 1 ) μ k ψ j ( k ) * ( r 2 , t 2 ) ,
V i j ( r 1 , t 1 ; r 2 , t 2 ) = l ϕ i ( l ) ( r 1 , t 1 ) λ l ϕ j ( l ) * ( r 2 , t 2 ) .
P = k l L ( k l ) λ ( l ) L ( k l ) * μ ( k ) ,
L ( k l ) = i d t S d r ψ i ( k ) * ( r , t ) ϕ i ( l ) ( r , t ) .
U i j ( T , r 1 , t 1 ; r 2 , t 2 ) = U i j ( 0 , r 1 , t 1 T ; r 2 , t 2 T ) ,
F ̃ i ( r , ω ) = j S d 2 r S i j ( r ; r , ω ) E ̃ j ( r , ω ) ,
F i ( r , t ) = j d t S d 2 r T i j ( r , t ; r , t ) E j r ( r , t ) ,
T i j ( r , r ; t , t ) = 1 2 π d ω e i ω ( t t ) S i j ( r , r ; ω ) .
q ( r , t ) = i j α i j ( r ) F i ( r , t ) F j ( r , t ) ,
F i ( r , t ) = j d t S d 2 r K i j ( r , t ; r , t ) F j ( r , t ) ,
p ( T ) = i j d t S d 2 r α i j ( r ) g ( r , T t ) F i ( r , t ) F j ( r , t ) .
U i j r ( T , r 1 , t 1 ; r 2 , t 2 ) = k l d t S d 2 r × α k l ( r ) g ( r , T t ) T k i ( r , t ; r 1 , t 1 ) T l j ( r , t ; r 2 , t 2 ) ,
U i j r ( T , r 1 , t 1 ; r 2 , t 2 ) = k l m n d t d t 1 d t 2 S d 2 r S d 2 r 1 S d 2 r 2 α m n ( r ) g ( r , T t ) K m k ( r , t ; r 1 , t 1 ) T k i ( r 1 , t 1 ; r 1 , t 1 ) K n l ( r , t ; r 2 , t 2 ) T l j ( r 2 , t 2 , r 2 , t 2 ) .
P ( t ) = i j t τ 2 t + τ 2 d T d t 1 d t 2 S d 2 r 1 S d 2 r 2 × U i j * ( T , r 1 , t 1 ; r 2 , t 2 ) X i j ( r 1 , t 1 ; r 2 , t 2 ) .
P ( t ) = i j d T t τ 2 t + τ 2 d t 1 t τ 2 t + τ 2 d t 2 S d 2 r 1 S d 2 r 2 × U i j * ( T , r 1 , t 1 ; r 2 , t 2 ) X i j ( r 1 , t 1 ; r 2 , t 2 ) = i j t τ 2 t + τ 2 d t 1 t τ 2 t + τ 2 d t 2 S d r 1 S d r 2 × W i j * ( r 1 , t 1 ; r 2 , t 2 ) X i j ( r 1 , t 1 ; r 2 , t 2 ) ,
W i j ( r 1 , t 1 ; r 2 , t 2 ) = d T U i j ( T , r 1 , t 1 ; r 2 , t 2 ) .
Y i j ( r 1 ; r 2 , ω ) = d u e i ω u X i j ( r 1 , t ; r 2 , t + u ) ,
X i j ( r 1 , t 1 ; r 2 , t 2 ) = 1 2 π d ω e i ω ( t 2 t 1 ) Y i j ( r 1 ; r 2 , ω ) ,
Z i j ( r 1 ; r 2 , ω ) = d t 1 d t 2 e i ω ( t 2 t 1 ) V i j ( r 1 , t 1 ; r 2 , t 2 ) ,
P = 1 2 π i j d ω S d 2 r 1 S d 2 r 2 Z i j * ( r 1 ; r 2 , ω ) Y i j ( r 1 ; r 2 , ω ) .
P = 0 d ω P ω ,
P ( ω ) = i j S d 2 r 1 S d 2 r 2 Z i j * ( r 1 ; r 2 , ω ) Y i j ( r 1 ; r 2 , ω ) .
Z i j ( r 1 ; r 2 , ω ) = d u e i ω u W i j ( r 1 , r 2 ; t , t + u ) ,
W i j ( r 1 , t 1 ; r 2 , t 2 ) = 1 2 π d ω e i ω ( t 2 t 1 ) Z i j ( r 1 ; r 2 , ω ) .
P = 1 4 π 2 i j d ω 1 d ω 2 S d 2 r 1 S d 2 r 2 × t τ 2 t + τ 2 d t 1 t τ 2 t + τ 2 d t 2 e i ω 1 ( t 1 t 2 ) e i ω 2 ( t 2 t 1 ) × Z i j * ( r 1 , r 2 ; ω 1 ) Y i j ( r 1 , r 2 ; ω 2 ) .
P = τ 4 π 2 i j d ω 1 d ω 2 S d 2 r 1 S d 2 r 2 × d u e i u ( ω 2 ω 1 ) Z i j * ( r 1 ; r 2 , ω 1 ) Y i j ( r 1 ; r 2 , ω 2 ) = τ 2 π i j d ω S d 2 r 1 S d 2 r 2 Z i j * ( r 1 ; r 2 , ω ) Y i j ( r 1 ; r 2 , ω 2 ) .
Z i j = τ Z i j .
Z i j ( r 1 ; r 2 , ω ) = ϕ i ( r 1 , ω ) λ ( ω ) ϕ j * ( r 2 , ω ) ,
Y i j ( r 1 ; r 2 , ω ) = ψ i ( r 1 , ω ) μ ( ω ) ψ j * ( r 2 , ω ) ,
P = d ω λ ( ω ) μ ( ω ) S d 2 r ψ * ϕ 2 ,
Z i j ( r 1 ; r 2 , ω ) = δ ( r 1 r 2 ) δ i j λ ( r 1 , ω ) ,
P = d ω S d 2 r λ ( r , ω ) I ( r , ω ) ,
E i ( r 1 , t 1 ) E j * ( r 2 , t 2 ) E k ( r 3 , t 3 ) E l * ( r 4 , t 4 ) = E i ( r 1 , t 1 ) E j * ( r 2 , t 2 ) E k ( r 3 , t 3 ) E l * ( r 4 , t 4 ) + E i ( r 1 , t 1 ) E l * ( r 4 , t 4 ) E k ( r 3 , t 3 ) E j * ( r 2 , t 2 ) .
p ( a ) ( t α ) p ( b ) ( t β ) = i j k l d t 1 d t 2 d t 3 d t 4 S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 U i j ( a ) * ( t α , r 1 , t 1 ; r 2 , t 2 ) U k l ( b ) * ( t β , r 3 , t 3 ; r 4 , t 4 ) X i j ( r 1 , t 1 ; r 2 , t 2 ) X k l ( r 3 , t 3 ; r 4 , t 4 ) .
p ( a ) ( t α ) p ( b ) ( t β ) = i j k l d t 1 d t 2 d t 3 d t 4 × S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 [ U j i ( a ) * ( t α , r 2 , t 2 ; r 1 , t 1 ) X j k ( r 2 , t 2 ; r 3 , t 3 ) × U l k ( b ) * ( t β , r 4 , t 4 ; r 3 , t 3 ) X l i ( r 4 , t 4 ; r 1 , t 1 ) ] + p ( a ) ( t α ) p ( b ) ( t β ) ,
Cov [ p ( a ) ( t α ) p ( b ) ( t β ) ] = i j k l d t 1 d t 2 d t 3 d t 4 × S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 U j i ( a ) * ( t α , r 2 , t 2 ; r 1 , t 1 ) X j k ( r 2 , t 2 ; r 3 , t 3 ) U l k ( b ) * ( t β , r 4 , t 4 ; r 3 , t 3 ) X l i ( r 4 , t 4 ; r 1 , t 1 ) .
Cov [ p ( a ) ( t α ) p ( b ) ( t β ) ] = i j k l t α τ α 2 t α + τ α 2 d t 1 t α τ α 2 t α + τ α 2 d t 2 t β τ β 2 t β + τ β 2 d t 3 t β τ β 2 t β + τ β 2 d t 4 S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 W j i ( a ) * ( r 2 , t 2 ; r 1 , t 1 ) X j k ( r 2 , t 2 ; r 3 , t 3 ) W l k ( b ) * ( r 4 , t 4 ; r 3 , t 3 ) X l i ( r 4 , t 4 ; r 1 , t 1 ) .
Cov [ p ( a ) ( t α ) p ( b ) ( t β ) ] = 1 ( 2 π ) 4 i j k l d ω 1 d ω 2 d ω 3 d ω 4 S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 t α τ α 2 t α + τ α 2 d t 1 t α τ α 2 t α + τ α 2 d t 2 t β τ β 2 t β + τ β 2 d t 3 t β τ β 2 t β + τ β 2 d t 4 Z j i ( a ) * ( r 2 , r 1 , ω 1 ) e i ω 1 ( t 2 t 1 ) Y j k ( r 2 ; r 3 , ω 2 ) e i ω 2 ( t 3 t 2 ) × Z l k ( b ) * ( r 4 ; r 3 , ω 3 ) e i ω 3 ( t 4 t 3 ) Y l i ( r 4 ; r 1 , ω 4 ) e i ω 4 ( t 1 t 4 ) .
d ω Y i j ( r 1 ; r 2 , ω ) e i ω ( t 2 t 1 ) = e i ω 0 ( t 2 t 1 ) ω 0 Δ ω 2 ω 0 + Δ ω 2 d ω Y i j ( r 1 ; r 2 , ω ) ,
P = τ 2 π i j k l S d 2 r 1 S d 2 r 2 Z i j * ( r 1 ; r 2 , ω 0 ) Y i j 0 ( r 1 ; r 2 ) ,
Cov [ P ( a ) P ( b ) ] = τ 2 4 π 2 i j k l S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 Z j i ( a ) * ( r 2 ; r 1 , ω 0 ) × Y j k 0 ( r 2 , r 3 ) Z l k ( b ) * ( r 4 ; r 3 , ω 0 ) Y l i 0 ( r 4 ; r 1 ) ,
Y i j 0 ( r 1 ; r 2 ) = ω 0 Δ ω 2 ω 0 + Δ ω 2 d ω Y i j ( r 1 ; r 2 , ω ) .
Cov [ P ( a ) P ( b ) ] = τ 2 π i j k l d ω S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 Z j i ( a ) * ( r 2 ; r 1 , ω ) Y j k ( r 2 ; r 3 , ω ) Z l k ( b ) * ( r 4 ; r 3 , ω ) Y l i ( r 4 ; r 1 , ω ) ,
Var ( P ) = τ 2 π ( ω η ) 2 Δ ω [ η n 0 ( 1 + η n 0 ) ]
P = κ ω 0 n ,
Cov [ P ( a ) , P ( b ) ] = τ 2 π d ω i j k l S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 × Z j i ( a ) * ( r 2 ; r 1 , ω ) Y j k ( r 2 ; r 3 , ω ) Z l k ( b ) * ( r 4 ; r 3 , ω ) × [ Y l i ( r 4 ; r 1 , ω ) + ω κ ( ω ) δ l i δ ( r 4 r 1 ) ]
Cov [ P ( a ) , P ( b ) ] = τ 2 π d ω i j k l S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 × Z j i ( a ) * ( r 2 ; r 1 , ω ) Y j k ( r 2 ; r 3 , ω ) Z l k ( b ) * ( r 4 ; r 3 , ω ) × Y l i ( r 4 ; r 1 , ω ) + τ 2 π d ω κ ( ω ) ω i j k l S d 2 r 1 S d 2 r 2 × Z i j ( a ) * ( r 1 ; r 2 , ω ) Y i j ( r 1 ; r 2 , ω ) δ a b ,
n = τ 2 π κ ω 0 i j k l S d 2 r 1 S d 2 r 2 Z i j * ( r 1 ; r 2 , ω 0 ) Y i j 0 ( r 1 ; r 2 ) .
Cov [ P ( a ) , P ( b ) ] = τ 2 4 π 2 i j k l S d 2 r 1 S d 2 r 2 S d 2 r 3 S d 2 r 4 × Z j i ( a ) * ( r 2 ; r 1 , ω 0 ) Y j k 0 ( r 2 ; r 3 ) Z l k ( b ) * ( r 4 ; r 3 , ω 0 ) × Y l i 0 ( r 4 ; r 1 ) + τ 2 π κ ω 0 i j k l S d 2 r 1 S d 2 r 2 Z i j ( a ) * ( r 1 ; r 2 , ω 0 ) Y i j 0 ( r 1 ; r 2 ) δ a b .
P ( ω ) = λ ( ω ) ω n ( ω ) ,
Y i j ( r 1 ; r 2 , ω ) = δ ( r 1 r ) δ ( r 2 r ) δ ( ω ω 0 ) δ i k δ j k .
Y i j ( r 1 ; r 2 , ω ) = δ ( ω ω 0 ) [ δ ( r 1 r 3 ) δ ( r 2 r 3 ) δ i k δ j k + δ ( r 1 r 4 ) δ ( r 2 r 4 ) δ i l δ j l + δ ( r 1 r 3 ) δ ( r 2 r 4 ) × e i ϕ δ i k δ j l + δ ( r 1 r 4 ) δ ( r 2 r 3 ) e i ϕ δ i l δ j k ] ,
P = 1 2 π [ Z k k ( r 3 ; r 3 , ω 0 ) + Z l l ( r 4 ; r 4 , ω 0 ) + Z k l ( r 3 ; r 4 , ω 0 ) cos ϕ ] ,

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