Abstract

We present a detailed quantum-statistical model of multimode far-infrared and submillimeter-wave astronomical interferometers. The scheme identifies explicitly the optical modes associated with each telescope and uses these to trace the quantum-statistical properties of the field from a source through the telescopes, through the beam combiners, and onto the detectors. The scheme can be used with any optical configuration, and elegant expressions result for the average rate at which photons are detected by the pixels of an imaging array, the mean-square fluctuations in the rates, and the correlations between the fluctuations in the rates of different pixels. Numerous extensions to the basic technique are possible.

© 2005 Optical Society of America

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  1. A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
    [CrossRef]
  2. J. E. Baldwin, C. A. Haniff, “The application of interferometry to optical astronomical imaging,” Philos. Trans. R. Soc. London, Ser. A 360, 969–986 (2002).
    [CrossRef]
  3. A. Quirrenbach, “Optical interferometry,” Annu. Rev. Astron. Astrophys. 39, 353–401 (2001).
    [CrossRef]
  4. J. D. Monnier, “Optical interferometry in astronomy,” 66, 789–857 (2003).
  5. D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
    [CrossRef]
  6. S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.
  7. S. Withington, M. P. Hobson, 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]
  8. S. Withington, M. P. Hobson, E. S. Campbell, “Modal analysis of astronomical bolometric interferometers,” J. Opt. Soc. Am. A 21, 1988–1995 (2004).
    [CrossRef]
  9. S. Withington, E. S. Campbell, M. P. Hobson, “A numerical procedure for simulating the behavior of multimode bolometric astronomical interferometers,” J. Appl. Phys. 97, 12490-1 to 24909-8 (2005).
    [CrossRef]
  10. J. Zmuidzinas, “Cramer–Rao sensitivity limits for astronomical instruments: implications for interferometer design,” J. Opt. Soc. Am. A 20, 218–233 (2003).
    [CrossRef]
  11. S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1233 (2001).
    [CrossRef]
  12. M. Feldmann, “Modes in general Gaussian optical systems using quantum-mechanics formalism,” J. Opt. Soc. Am. 61, 446–449 (1971).
    [CrossRef]
  13. A. Wünsche, “Quantization of Gauss–Hermite and Gauss–Laguerre beams in free space,” Quantum Semiclassic. Opt. 6, S47–S59 (2004).
    [CrossRef]
  14. D. Stoler, “Operator methods in physical optics,” J. Opt. Soc. Am. 71, 334–341 (1981).
    [CrossRef]
  15. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 639.
  16. E. Campbell, Department of Physics, University of Cambridge, UK (personal communication).
  17. S. Withington, G. Yassin, “Power coupled between partially coherent vector fields in different states of coherence,” J. Opt. Soc. Am. A 18, 3061–3071 (2001).
    [CrossRef]
  18. S. Withington, G. Yassin, “Analyzing the power coupled between partially coherent waveguide fields in different states of coherence,” J. Opt. Soc. Am. A 19, 1376–1382 (2002).
    [CrossRef]
  19. S. Withington, C. Y. Tham, G. Yassin, “Theoretical analysis of planar bolometric imaging arrays for THz imaging systems,” Proc. SPIE 4855, 49–62 (2002).
    [CrossRef]
  20. J. Zmuidzinas, “Thermal noise and correlations in photon detection,” Appl. Opt. 42, 4989 (2003).
    [CrossRef] [PubMed]
  21. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 503.

2005

S. Withington, E. S. Campbell, M. P. Hobson, “A numerical procedure for simulating the behavior of multimode bolometric astronomical interferometers,” J. Appl. Phys. 97, 12490-1 to 24909-8 (2005).
[CrossRef]

2004

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

A. Wünsche, “Quantization of Gauss–Hermite and Gauss–Laguerre beams in free space,” Quantum Semiclassic. Opt. 6, S47–S59 (2004).
[CrossRef]

2003

2002

J. E. Baldwin, C. A. Haniff, “The application of interferometry to optical astronomical imaging,” Philos. Trans. R. Soc. London, Ser. A 360, 969–986 (2002).
[CrossRef]

S. Withington, G. Yassin, “Analyzing the power coupled between partially coherent waveguide fields in different states of coherence,” J. Opt. Soc. Am. A 19, 1376–1382 (2002).
[CrossRef]

S. Withington, C. Y. Tham, G. Yassin, “Theoretical analysis of planar bolometric imaging arrays for THz imaging systems,” Proc. SPIE 4855, 49–62 (2002).
[CrossRef]

2001

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

A. Quirrenbach, “Optical interferometry,” Annu. Rev. Astron. Astrophys. 39, 353–401 (2001).
[CrossRef]

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1233 (2001).
[CrossRef]

1999

D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

1981

1971

Ade, P.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Ali, S.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Baldwin, J. E.

J. E. Baldwin, C. A. Haniff, “The application of interferometry to optical astronomical imaging,” Philos. Trans. R. Soc. London, Ser. A 360, 969–986 (2002).
[CrossRef]

Campbell, E.

E. Campbell, Department of Physics, University of Cambridge, UK (personal communication).

Campbell, E. S.

S. Withington, E. S. Campbell, M. P. Hobson, “A numerical procedure for simulating the behavior of multimode bolometric astronomical interferometers,” J. Appl. Phys. 97, 12490-1 to 24909-8 (2005).
[CrossRef]

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

Feldmann, M.

Gear, W. K.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Haniff, C. A.

J. E. Baldwin, C. A. Haniff, “The application of interferometry to optical astronomical imaging,” Philos. Trans. R. Soc. London, Ser. A 360, 969–986 (2002).
[CrossRef]

Haynes, V.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Hobson, M. P.

S. Withington, E. S. Campbell, M. P. Hobson, “A numerical procedure for simulating the behavior of multimode bolometric astronomical interferometers,” J. Appl. Phys. 97, 12490-1 to 24909-8 (2005).
[CrossRef]

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

S. Withington, M. P. Hobson, 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]

Leisawitz, D.

D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 639.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 503.

Mather, J. C.

D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

Mauskopf, P.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Monnier, J. D.

J. D. Monnier, “Optical interferometry in astronomy,” 66, 789–857 (2003).

Moran, J. M.

A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Moseley, S. H.

D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

Murphy, J. A.

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1233 (2001).
[CrossRef]

Piccirillo, L.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Quirrenbach, A.

A. Quirrenbach, “Optical interferometry,” Annu. Rev. Astron. Astrophys. 39, 353–401 (2001).
[CrossRef]

Rossinot, P.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Stoler, D.

Swenson, G. W.

A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Tham, C. Y.

S. Withington, C. Y. Tham, G. Yassin, “Theoretical analysis of planar bolometric imaging arrays for THz imaging systems,” Proc. SPIE 4855, 49–62 (2002).
[CrossRef]

Thompson, A. R.

A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

Timbie, P.

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

Withington, S.

S. Withington, E. S. Campbell, M. P. Hobson, “A numerical procedure for simulating the behavior of multimode bolometric astronomical interferometers,” J. Appl. Phys. 97, 12490-1 to 24909-8 (2005).
[CrossRef]

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

S. Withington, M. P. Hobson, 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, C. Y. Tham, G. Yassin, “Theoretical analysis of planar bolometric imaging arrays for THz imaging systems,” Proc. SPIE 4855, 49–62 (2002).
[CrossRef]

S. Withington, G. Yassin, “Analyzing the power coupled between partially coherent waveguide fields in different states of coherence,” J. Opt. Soc. Am. A 19, 1376–1382 (2002).
[CrossRef]

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

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1233 (2001).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 639.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 503.

Wünsche, A.

A. Wünsche, “Quantization of Gauss–Hermite and Gauss–Laguerre beams in free space,” Quantum Semiclassic. Opt. 6, S47–S59 (2004).
[CrossRef]

Yassin, G.

S. Withington, C. Y. Tham, G. Yassin, “Theoretical analysis of planar bolometric imaging arrays for THz imaging systems,” Proc. SPIE 4855, 49–62 (2002).
[CrossRef]

S. Withington, G. Yassin, “Analyzing the power coupled between partially coherent waveguide fields in different states of coherence,” J. Opt. Soc. Am. A 19, 1376–1382 (2002).
[CrossRef]

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

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1233 (2001).
[CrossRef]

Zhang, X.

D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

Zmuidzinas, J.

Annu. Rev. Astron. Astrophys.

A. Quirrenbach, “Optical interferometry,” Annu. Rev. Astron. Astrophys. 39, 353–401 (2001).
[CrossRef]

Appl. Opt.

Astrophys. Space Sci.

D. Leisawitz, J. C. Mather, S. H. Moseley, X. Zhang, “The submillimeter probe of the evolution of cosmic structure (SPECS),” Astrophys. Space Sci. 269, 563–567 (1999).
[CrossRef]

IEEE Trans. Antennas Propag.

S. Withington, G. Yassin, J. A. Murphy, “Dyadic analysis of partially coherent submillimetre-wave antenna systems,” IEEE Trans. Antennas Propag. 49, 1226–1233 (2001).
[CrossRef]

J. Appl. Phys.

S. Withington, E. S. Campbell, M. P. Hobson, “A numerical procedure for simulating the behavior of multimode bolometric astronomical interferometers,” J. Appl. Phys. 97, 12490-1 to 24909-8 (2005).
[CrossRef]

S. Withington, M. P. Hobson, 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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Philos. Trans. R. Soc. London, Ser. A

J. E. Baldwin, C. A. Haniff, “The application of interferometry to optical astronomical imaging,” Philos. Trans. R. Soc. London, Ser. A 360, 969–986 (2002).
[CrossRef]

Proc. SPIE

S. Withington, C. Y. Tham, G. Yassin, “Theoretical analysis of planar bolometric imaging arrays for THz imaging systems,” Proc. SPIE 4855, 49–62 (2002).
[CrossRef]

Quantum Semiclassic. Opt.

A. Wünsche, “Quantization of Gauss–Hermite and Gauss–Laguerre beams in free space,” Quantum Semiclassic. Opt. 6, S47–S59 (2004).
[CrossRef]

Other

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 639.

E. Campbell, Department of Physics, University of Cambridge, UK (personal communication).

A. R. Thompson, J. M. Moran, G. W. Swenson, Interferometry and Synthesis in Radio Astronomy (Wiley, 2001).
[CrossRef]

J. D. Monnier, “Optical interferometry in astronomy,” 66, 789–857 (2003).

S. Ali, P. Rossinot, L. Piccirillo, W. K. Gear, P. Mauskopf, P. Ade, V. Haynes, P. Timbie, “MBI: millimeter-wave bolometric interferometer,” in Proceedings of Experimental Cosmology at Millimeter Wavelengths (American Institute of Physics, 2002), Vol. 616, pp. 126–128.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 503.

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Figures (1)

Fig. 1
Fig. 1

The natural modes of the incoming source field are coupled to the natural modes of detector pixels r and r through probability amplitudes ϕ i j r and ϕ i j r . The operators b ̂ i r and b ̂ i r correspond to internal degrees of freedom. The external sensitivity of each pixel in each of its modes is characterized by Λ B i r .

Equations (87)

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

E ̂ ( r , t ) = B ( i 2 ) e ̂ ( r , ν ) exp ( i 2 π ν t ) + ( i 2 ) e ̂ ( r , ν ) exp ( i 2 π ν t ) d ν ,
e ̂ ( r , ν ) = i m ± 1 m ( h ν ) 1 2 a ̂ i m ± ( ν ) Ψ i ± ( r , ν ) exp ( i k z ) .
e ̂ ( r 2 , ν ) = T ̿ ( r 2 , r 1 ) e ̂ ( r 1 , ν ) d 2 r 1 + e ̂ ( r 2 , ν ) ,
T ̿ ( r 2 , r 1 ) = i σ i U i ( r 2 ) V i * ( r 1 ) .
b ̂ ( ν ) = T a ̂ ( ν ) + b ̂ ( ν ) ,
a ̂ i ( ν ) = V i * ( r 1 ) e ̂ ( r 1 , ν ) d 2 r 1 ,
b ̂ i ( ν ) = U i * ( r 2 ) e ̂ ( r 2 , ν ) d 2 r 2 ,
b ̂ i ( ν ) = U i * ( r 2 ) e ̂ ( r 2 , ν ) d 2 r 2 .
[ a ̂ i ( ν 1 ) , a ̂ j ( ν 2 ) ] = δ i j I ̂ δ ( ν 1 ν 2 ) ,
[ b ̂ i ( ν 1 ) , b ̂ j ( ν 2 ) ] = σ i σ j [ a ̂ i ( ν 1 ) , a ̂ j ( ν 2 ) ] + [ b ̂ i ( ν 1 ) , b ̂ j ( ν 2 ) ] = I ̂ δ i j δ ( ν 1 ν 2 ) ,
[ b ̂ i ( ν 1 ) , b ̂ j ( ν 2 ) ] = I ̂ ( 1 σ i 2 ) δ i j δ ( ν 1 ν 2 ) .
E ̿ ( r , r ) = e ̂ ( r , ν 1 ) e ̂ ( r , ν 2 ) ,
E ̿ ( r 2 , r 2 ) = T ̿ ( r 2 , r 1 ) E ̿ ( r 1 , r 1 ) T ̿ ( r 1 , r 2 ) d 2 r 1 d 2 r 1 + E ̿ ( r 2 , r 2 ) ,
A i i = V i * ( r 1 ) E ̿ ( r 1 , r 1 ) V i ( r 1 ) d 2 r 1 d 2 r 1 ,
B i i = U i * ( r 2 ) E ̿ ( r 2 , r 2 ) U i ( r 2 ) d 2 r 2 d 2 r 2 ,
B i i = U i * ( r 2 ) E ̿ ( r 2 , r 2 ) U i ( r 2 ) d 2 r 1 d 2 r 1 ,
B = Σ A Σ + B .
A i j = ( h ν ) n ¯ i ( ν 1 ) δ i j δ ( ν 1 ν 2 ) ,
B = ( I Σ 2 ) ( h ν 1 ) n ¯ p ( ν 1 ) δ ( ν 1 ν 2 ) ,
n ¯ p ( ν 1 ) = 1 exp [ h ν 1 k T p ] 1
E ̿ ( r 2 , r 2 ) = i U i ( r 2 ) U i * ( r 2 ) ( 1 σ i 2 ) ( h ν 1 ) n ¯ p ( ν 1 ) δ ( ν 1 ν 2 ) .
E ̿ ( r , r ) = I ̿ e ̂ ( r , ν 1 ) e ̂ ( r , ν 2 ) ,
e ̂ n ( r , ν 1 ) e ̂ n ( r , ν 2 ) = g n ( r ) g n * ( r ) ( h ν 1 ) 1 2 ( h ν 2 ) 1 2 a ̂ n ( ν 1 ) a ̂ n ( ν 2 ) = g n ( r ) g n * ( r ) ( h ν 1 ) n n ¯ ( ν 1 ) δ ( ν 1 ν 2 ) ,
g n ( r ) g m * ( r ) d 2 r = δ m n .
E ̿ ( r , r ) = I ̿ n g n ( r ) g n * ( r ) ( h ν 1 ) n ¯ n ( ν 1 ) δ ( ν 1 ν 2 ) .
E ̿ ( r , r ) = I ̿ ( h ν 1 ) n ¯ ( r , ν 1 ) δ ( r r ) δ ( ν 1 ν 2 ) ,
n ¯ ( r , ν 1 ) = 1 exp [ h ν 1 k T ( r ) ] 1 .
E ̿ ( r , r ) = I ̿ ( h ν 1 ) [ n ¯ s ( r , ν 1 ) η + n ¯ a ( ν 1 ) ( 1 η ) ] δ ( r r ) δ ( ν 1 ν 2 ) ,
X i j = ( h ν 1 ) n ¯ i ( ν 1 ) δ i j δ ( ν 1 ν 2 ) ,
C i j = V i X V j ,
D i j = Σ i C i j Σ j .
Y = i j U i D i j U j + Y i δ i j ,
Y i = U i ( I Σ i ) 2 U i ( h ν 1 ) n ¯ p i ( ν 1 ) δ ( ν 1 ν 2 ) ,
P c = Tr A B ,
A = U Λ A U ,
B = V Λ B V ,
P c = Tr A B = Tr U Λ A U V Λ B V
P c = Tr Φ Λ A Φ Λ B ,
P c = i j ϕ i j 2 Λ A j Λ B i ,
( n ¯ r ) τ = 1 τ 0 τ n ̂ r ( t ) d t ,
n ̂ r ( t ) = i n ̂ i r ( t ) = i b ̂ i r ( t ) b ̂ i r ( t ) ,
b ̂ i r ( t ) = B b ̂ i r ( ν ) exp [ j 2 π ν t ] d ν ,
( n ¯ r ) τ = 1 τ 0 τ B B i b ̂ i r ( ν 1 ) b ̂ i r ( ν 2 ) exp [ j 2 π ( ν 1 ν 2 ) t ] d ν 1 d ν 2 d t .
i b ̂ i r ( ν 1 ) b ̂ i r ( ν 2 ) = i j j ϕ i j r ( ν 1 ) ϕ i j r * ( ν 2 ) a ̂ j ( ν 1 ) a ̂ j ( ν 2 ) Λ B i r .
a ̂ j ( ν 1 ) a ̂ j ( ν 2 ) = n ¯ j ( ν 1 ) δ ( ν 1 ν 2 ) δ j j ,
i b ̂ i r ( ν 1 ) b ̂ i r ( ν 2 ) = Tr A n B r δ ( ν 1 ν 2 ) ,
( n ¯ r ) τ = B Tr A n B n d ν .
p r ¯ = B h ν Tr Y n Z r d ν ,
Δ n ¯ r ( t ) = n ̂ r ( t ) n ¯ r ,
Δ n ̂ r ( t ) = n ̂ r ( t ) n ¯ r .
( Δ n ̂ r ) τ = 1 τ 0 τ n ̂ r ( t ) d t n ¯ r ,
( Δ n ̂ r ) τ = 1 τ 0 τ n ̂ r ( t ) d t n ¯ r .
K τ r r = ( Δ n ̂ r ) τ ( Δ n ̂ r ) τ .
K τ r r = 1 τ 2 0 τ 0 τ n ̂ r ( t 1 ) n ̂ r ( t 2 ) d t 1 d t 2 n ¯ r n ¯ r .
K τ r r = 1 τ 2 0 τ ( τ δ t ) [ n ̂ r ( t + δ t ) n ̂ r ( t ) + n ̂ r ( t ) n ̂ r ( t + δ t ) ] d ( δ t ) n ¯ r n ¯ r ,
n ̂ r ( t + δ t ) n ̂ r ( t ) + n ̂ r ( t ) n ̂ r ( t + δ t ) = i n ̂ i r ( t + δ t ) i n ̂ i r ( t ) + i n ̂ i r ( t ) i n ̂ i r ( t + δ t ) .
n ̂ r ( t + δ t ) n ̂ r ( t ) + n ̂ r ( t ) n ̂ r ( t + δ t ) = i i B B B B d ν 1 d ν 2 d ν 3 d ν 4 × { exp [ j 2 π ( ν 1 ν 2 ) ( t + δ t ) ] exp [ j 2 π ( ν 3 ν 4 ) ( t ) ] + exp [ j 2 π ( ν 1 ν 2 ) ( t ) ] exp [ j 2 π ( ν 3 ν 4 ) ( t + δ t ) ] } × b ̂ i r ( ν 1 ) b ̂ i r ( ν 2 ) b ̂ i r ( ν 3 ) b ̂ i r ( ν 4 ) .
b ̂ i r ( ν 1 ) b ̂ i r ( ν 2 ) b ̂ i r ( ν 3 ) b ̂ i r ( ν 4 )
= i i j j k k ϕ i j r ϕ i j r * Λ B i r ϕ i k r ϕ i k r * Λ B i r
× a ̂ j ( ν 1 ) a ̂ j ( ν 2 ) a ̂ k ( ν 3 ) a ̂ k ( ν 4 ) .
a ̂ j ( ν 1 ) a ̂ j ( ν 2 ) a ̂ k ( ν 3 ) a ̂ k ( ν 4 )
= n ¯ j ( ν 1 ) [ n ¯ k ( ν 3 ) + 1 ] δ j k δ j k δ ( ν 1 ν 4 ) δ ( ν 2 ν 3 )
+ n ¯ j ( ν 1 ) n ¯ k ( ν 3 ) δ j j δ k k δ ( ν 1 ν 2 ) δ ( ν 3 ν 4 ) .
i i j k ϕ i j r 2 Λ B i r ϕ i k r 2 Λ B i r n ¯ j ( ν 1 ) n ¯ k ( ν 3 ) δ ( ν 1 ν 2 ) δ ( ν 3 ν 4 ) ,
2 [ B i j ϕ i j r 2 Λ B i r n ¯ j ( ν 1 ) d ν 1 ] [ B i k ϕ i k r 2 Λ B i r n ¯ k ( ν 3 ) d ν 3 ]
= 2 n ¯ r n ¯ r .
1 τ 2 0 τ ( τ δ t ) 2 n ¯ r n ¯ r d ( δ t ) = n ¯ r n ¯ r ,
i i j k B B 2 cos [ 2 π ( ν 1 ν 2 ) δ t ] ϕ i j r ϕ i k r * Λ B i r ϕ i k r ϕ i j r * Λ B i r n ¯ j ( ν 1 ) [ n ¯ k ( ν 2 ) + 1 ] d ν 1 d ν 2 .
K τ r r = i i j k B B ϕ i j r ϕ i k r * Λ B i r ϕ i k r ϕ i j r * Λ B i r n ¯ k ( ν 1 ) [ n ¯ k ( ν 2 ) + 1 ] 1 τ 2 0 τ ( τ δ t ) 2 cos [ 2 π ( ν 1 ν 2 ) δ t ] d ( δ t ) d ν 1 d ν 2 ,
1 τ 2 0 τ ( τ δ t ) 2 cos [ 2 π ( ν 1 ν 2 ) δ t ] d ( δ t ) 1 τ δ ( ν 1 ν 2 )
K τ r r = 1 τ B i i j k ϕ i j r ϕ i k r * Λ B i r ϕ i k r Λ i j r * Λ B i r n ¯ j ( ν 1 ) [ n ¯ k ( ν 1 ) + 1 ] d ν 1 .
K τ r r = 1 τ B j k Ψ k j r Ψ k j r * n ¯ j ( ν ) [ n ¯ k ( ν 1 ) + 1 ] d ν ,
Ψ r = Φ r Λ B r Φ r , Ψ r = Φ r Λ B r Φ r .
K τ r r = 1 τ B Tr ( B r A n B r A a ) d ν ,
K τ r r = 1 τ B Tr ( B r A n B r A a ) d ν ,
( Δ p r ) τ ( Δ p r ) τ ¯ = 1 τ B ( h ν ) 2 Tr ( Z r Y n Z r Y a ) d ν ,
( Δ p r ) τ ( Δ p r ) τ ¯ = 1 τ B ( h ν ) 2 Tr ( Z r Y n Z r Y a ) d ν .
A ̂ ( r , t ) = B 1 2 π ν 1 2 [ e ̂ ( r , ν ) exp ( i 2 π ν t ) + e ̂ ( r , ν ) exp ( i 2 π ν t ) ] d ν ,
E ̂ ( r , t ) = A ̂ ( r , t ) t = B ( i 2 ) e ̂ ( r , ν ) exp ( i 2 π ν t ) + ( i 2 ) e ̂ ( r , ν ) exp ( i 2 π ν t ) d ν ,
[ E ̂ m ( r 1 , t 1 ) , A ̂ m ( r 2 , t 2 ) ] ,
[ E ̂ m ( r 1 , t 1 ) , A ̂ m ( r 2 , t 2 ) ] = B { Q * ( r 1 , r 2 ) exp [ i k ( z 1 z 2 ) ] + Q ( r 1 , r 2 ) exp [ i k ( z 1 z 2 ) ] } × h ν 2 1 2 π ν ( 2 i ) cos [ 2 π ν ( t 1 t 2 ) ] δ m , m d ν ,
Q ( r 1 , r 2 ) = i ψ i ( r 1 , ν ) ψ i * ( r 2 , ν ) .
[ E ̂ m ( r 1 , t 1 ) , A ̂ m ( r 2 , t 2 ) ] = Q ( r 1 , r 2 ) δ m m ( i ) × B cos [ 2 π ν ( t 1 t 2 ) k ( z 1 z 2 ) ] + cos [ 2 π ν ( t 1 t 2 ) + k ( z 1 z 2 ) ] d ν .
[ E ̂ m ( r 1 , t 1 ) , A ̂ m ( r 2 , t 2 ) ] = δ ( r 1 r 2 ) δ m m i 2 × { δ [ ( t 1 t 2 ) + ( z 1 z 2 ) c ] + δ [ ( t 1 t 2 ) ( z 1 z 2 ) c ] } .
[ E ̂ m ( r 1 , t 1 ) , A ̂ m ( r 2 , t 2 ) ] = i ϵ 0 1 ( 2 π ) 3 × ( δ i j k i k j k 2 ) exp [ i k t ( r t 1 r t 2 ) ] exp [ k 3 ( z 1 z 2 ) ] cos [ 2 π ν ( t 1 t 2 ) ] d 3 k ,
[ E ̂ m ( r 1 , t 1 ) A ̂ m ( r 2 , t 2 ) ] = Q ( r 1 , r 2 ) δ m m ( i ) ϵ 0 1 c × B cos [ 2 π ν ( t 1 t 2 ) k ( z 1 z 2 ) ] + cos [ 2 π ν ( t 1 t 2 ) + k ( z 1 z 2 ) ] d ν ,
Q ( r 1 , r 2 ) = 1 ( 2 π ) 2 exp ( i k t r t ) d k t .

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