Abstract

We study the effect of relative motion on a frequency-entangled-based ranging scheme. Two major puzzles arise, i.e., Doppler shift and ambiguity velocity. During condition of rapid relative motion, Doppler shift invalidates the measurement result of this scheme; while during condition of slow relative motion, the ambiguity velocity turns into a major limitation. If relative speed between targets and measurement platform exceeds the ambiguity velocity, an accumulated profile obtained by the coincidence measurement will be distorted, which causes a lower ranging accuracy. Theoretical analysis shows a time-varying delay can be introduced to solve the two major puzzles.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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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]
  15. Laser. Componets, “Single photon counting module count® blue series,” (Laser Componets, 2015), http://www.lasercomponents.com/fileadmin/user_upload/home/Datasheets/lcp/count-blue-series.pdf .
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  18. F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]

2014 (1)

2013 (1)

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

2012 (2)

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

J. Zhu, X. Chen, P. Huang, and G. Zeng, “Thermal-light-based ranging using second-order coherence,” Appl. Opt. 51(20), 4885–4890 (2012).
[Crossref] [PubMed]

2009 (2)

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

2008 (1)

2006 (2)

2005 (1)

2004 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

A. Valencia, G. Scarcelli, and Y. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85(13), 2655–2657 (2004).
[Crossref]

2002 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65(2), 022309 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum cryptographic ranging,” J. Opt. B Quantum Semiclassical Opt. 4(4), S413–S414 (2002).
[Crossref]

2001 (2)

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
[Crossref] [PubMed]

1989 (1)

J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989).
[Crossref]

1987 (1)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref] [PubMed]

1985 (1)

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of Time Delays in the Parametric Production of Photon Pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[Crossref] [PubMed]

Bai, Y.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Boitier, F.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

Cao, S.

Chen, X.

Coddington, I.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Dändliker, R.

Deibel, J.

Dong, R.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Fabre, C.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

Fang, C.

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

Friberg, S.

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of Time Delays in the Parametric Production of Photon Pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[Crossref] [PubMed]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65(2), 022309 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum cryptographic ranging,” J. Opt. B Quantum Semiclassical Opt. 4(4), S413–S414 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001).
[Crossref] [PubMed]

Godard, A.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

Han, X. C.

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

Holzwarth, R.

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of Time Delays in the Parametric Production of Photon Pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[Crossref] [PubMed]

Hou, F. y.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Huang, P.

Joo, K. N.

Kim, S. W.

Le Floch, S.

Lévêque, S.

Liu, T.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65(2), 022309 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum cryptographic ranging,” J. Opt. B Quantum Semiclassical Opt. 4(4), S413–S414 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001).
[Crossref] [PubMed]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65(2), 022309 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum cryptographic ranging,” J. Opt. B Quantum Semiclassical Opt. 4(4), S413–S414 (2002).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001).
[Crossref] [PubMed]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of Time Delays in the Parametric Production of Photon Pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[Crossref] [PubMed]

Nenadovic, L.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Newbury, N. R.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Nyberg, S.

Ou, Z. Y.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref] [PubMed]

Qu, X.

Quan, R.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Rarity, J. G.

J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989).
[Crossref]

Riles, K.

Rosencher, E.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

Salvadé, Y.

Scarcelli, G.

A. Valencia, G. Scarcelli, and Y. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85(13), 2655–2657 (2004).
[Crossref]

Schuhler, N.

Seward, S. F.

J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989).
[Crossref]

Shih, Y.

A. Valencia, G. Scarcelli, and Y. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85(13), 2655–2657 (2004).
[Crossref]

Swann, W. C.

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Tapster, P. R.

J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989).
[Crossref]

Valencia, A.

A. Valencia, G. Scarcelli, and Y. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85(13), 2655–2657 (2004).
[Crossref]

Walker, J. G.

J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989).
[Crossref]

Wong, F. N. C.

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001).
[Crossref] [PubMed]

Wu, H.

Xiao, J. J.

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

Xing, S.

Yang, H. J.

Zeng, G.

Zeng, G. H.

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

Zhang, F.

Zhang, S.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Zhang, T. y.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Zhang, Y.

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Zhao, J. K.

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

Zhu, J.

Adv. Space Res. (1)

F. y. Hou, R. Dong, R. Quan, Y. Zhang, Y. Bai, T. Liu, S. Zhang, and T. y. Zhang, “Dispersion-free quantum clock synchronization via fiber link,” Adv. Space Res. 50(11), 1489–1494 (2012).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

A. Valencia, G. Scarcelli, and Y. Shih, “Distant clock synchronization using entangled photon pairs,” Appl. Phys. Lett. 85(13), 2655–2657 (2004).
[Crossref]

Chin. Phys. Lett. (1)

J. J. Xiao, C. Fang, X. C. Han, J. K. Zhao, and G. H. Zeng, “Distance ranging based on quantum entanglement,” Chin. Phys. Lett. 30(10), 100301 (2013).
[Crossref]

J. Opt. B Quantum Semiclassical Opt. (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum cryptographic ranging,” J. Opt. B Quantum Semiclassical Opt. 4(4), S413–S414 (2002).
[Crossref]

Nat. Photonics (1)

I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3(6), 351–356 (2009).
[Crossref]

Nat. Phys. (1)

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

Nature (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced positioning and clock synchronization,” Nature 412(6845), 417–419 (2001).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Positioning and clock synchronization through entanglement,” Phys. Rev. A 65(2), 022309 (2002).
[Crossref]

Phys. Rev. Lett. (3)

V. Giovannetti, S. Lloyd, L. Maccone, and F. N. C. Wong, “Clock synchronization with dispersion cancellation,” Phys. Rev. Lett. 87(11), 117902 (2001).
[Crossref] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59(18), 2044–2046 (1987).
[Crossref] [PubMed]

S. Friberg, C. K. Hong, and L. Mandel, “Measurement of Time Delays in the Parametric Production of Photon Pairs,” Phys. Rev. Lett. 54(18), 2011–2013 (1985).
[Crossref] [PubMed]

Quantum Opt. (1)

J. G. Walker, S. F. Seward, J. G. Rarity, and P. R. Tapster, “Range measurement photon by photon,” Quantum Opt. 1(1), 75–82 (1989).
[Crossref]

Science (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Other (1)

Laser. Componets, “Single photon counting module count® blue series,” (Laser Componets, 2015), http://www.lasercomponents.com/fileadmin/user_upload/home/Datasheets/lcp/count-blue-series.pdf .

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Figures (5)

Fig. 1
Fig. 1 Schematic diagram of the FEB scheme
Fig. 2
Fig. 2 Accumulated profile in a motionless scenario
Fig. 3
Fig. 3 Accumulated profile in a scenario of relative motion
Fig. 4
Fig. 4 Accumulated profile in a scenario of relative motion
Fig. 5
Fig. 5 Schematic diagram of the improved FEB scheme

Equations (46)

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

τ max = t 2 t 1 =τ | max[c(τ)] =τ | max[G(τ)]
l dif =2 d 0 l reference =c×τ | max[G(τ)]
d 0 = 1 2 ( l reference +c×τ | max[G(τ)] )
|ψ= d ω ' f( ω ' ) | ω p /2+ ω ' s | ω p /2 ω ' i
Δ ω 2 =[ ω 2 (1+β)/(1β)] ω 2 =2 ω 2 β/(1β)
Δ ω 2 <2 ω 0 β/(1β)=2 ω 0 /[(1/β)1]
a 1 ( ω 1 )= a i ( ω 1 )exp(i ω 1 l/c) a 2 ( ω 2 )= χ 1/2 × a s (χ ω 2 )×exp{i ω 2 [2β(d/c)/(1β)2d/c]} = χ 1/2 a s (χ ω 2 )exp{i ω 2 [2d/(1β)c]}
E j () ( t j )= d ω j a j + ( ω j ) exp(i ω j t j ) E j (+) ( t j )= d ω j a j ( ω j ) exp(i ω j t j ),forj=1,2
P c = 0 T d t 1 0 T d t 2 ψ| E 1 () ( t 1 ) E 2 () ( t 2 ) E 2 (+) ( t 2 ) E 1 (+) ( t 1 )|ψ
P c ( t 1 , t 2 )G( t 1 , t 2 ) = | 0| E 1 + ( t 1 ) E 2 + ( t 2 )|ψ | 2
P c ( t 1 , t 2 )G( t 1 , t 2 ) = | d ω 1 d ω 2 0| a i ( ω 1 )×exp(i ω 1 l/c)×exp(i ω 1 t 1 )× χ 1/2 × a s (χ ω 2 )×exp{i ω 2 [2d/(1β)c]}×exp(i ω 2 t 2 )|ψ | 2 =χ | d ω 1 d ω 2 0| a i ( ω 1 ) a s (χ ω 2 )|ψ ×exp{i ω 1 l/c+i ω 2 [2d/(1β)c]}×exp[i( ω 1 t 1 + ω 2 t 2 )] | 2
0| a i ( ω 1 ) a s (χ ω 2 )|ψ = d ω ' f( ω ' )× χ ω 2 | s ω 1 | i | ω p /2+ ω ' s | ω p /2 ω ' i = d ω ' f( ω ' )×δ[χ ω 2 ( ω p /2+ ω ' )]δ[ ω ' ( ω p /2 ω 1 )] =f( ω p /2 ω 1 )δ( ω 1 +χ ω 2 ω p )
P c ( t 1 , t 2 ) | 0| E i + ( t 1 ) E s + ( t 2 )|ψ | 2 =χ | d ω 1 d ω 2 f( ω p /2 ω 1 )δ( ω 1 +χ ω 2 ω p ) ×exp{i ω 1 l/c+i ω 2 [2d/(1β)c]}×exp[i( ω 1 t 1 + ω 2 t 2 ) | 2 =χ | dωf(ω) ×exp(iω{ t 2 χ t 1 [2d/(1β)cχl/c]}) ×exp{i ω p /2×[2d/(1β)cχl/c+2(l/c t 1 )( t 2 χ t 1 )]} | 2 =χ | dωf(ω) exp(iω{ t 2 χ t 1 [2d(1+β)l]/((1β)c)}) | 2
a 2 ( ω 2 )= a s ( ω 2 )exp[i ω 2 (2d/c)] β0,χ1
G( t 1 , t 2 )= | d ω 1 d ω 2 0| a i ( ω 1 ) a s ( ω 2 )|ψ×exp[i ω 1 l/c+i ω 2 (2d/c)] ×exp(i ω 1 t 1 +i ω 2 t 2 ) | 2
0| a ^ i ( ω 1 ) a ^ s ( ω 2 )|ψ=f( ω p /2 ω 1 )δ( ω 1 + ω 2 ω p )
P c (τ,t) | d(ω) f(ω)exp(iω{τ[2 d 0 2v(t t 0 )l]/c}) | 2
Δ l dif = l dif ( t 0 ) l dif ( t 0 + T a )ξ× p d0
l dif ( t 0 )=2 d 0 l reference l dif ( t 0 + T a )=2( d 0 v× T a ) l reference
v ξ× p d0 2 T a
v ambiguity =max(v)= ξ× p d0 2 T a p d0 p limit
p d0 =c(τ | max[G(τ)] τ | max[G(τ)]/2 )
v ambiguity = ξc(τ | max[G(τ)] τ | max[G(τ)]/2 ) 2 T a p d0 p limit
P c ( t 1 , t 2 ) E i () ( t 1 )[ E s () ( t 2 )+ E n () ( t 2 )][ E s (+) ( t 2 )+ E n (+) ( t 2 )] E i (+) ( t 1 )
P c ( t 1 , t 2 ) E i () ( t 1 ) E s () ( t 2 ) E s (+) ( t 2 ) E i (+) ( t 1 ) + [ E i () ( t 1 ) E i (+) ( t 1 ) E n () ( t 2 ) E n (+) ( t 2 ) =G( t 1 , t 2 )+ E i E n
σ= ( R s T a +2b T a ) 1/2
SNR= R s T a / ( R s T a +2b T a ) 1/2
T a =SN R 2 ( R s +2b)/ R s 2
p(τ)={ n 0 sin 2 [ω(τμ)]}/ [ω(τμ)] 2
p c (τ)={ n 0 sin 2 [ω(τμ)]}/ [ω(τμ)] 2 = n 0 sin 2 {ω[τ(2 d 0 l reference )/c]}/ {ω[τ(2 d 0 l reference )/c]} 2
p motion (τ,t)= n 0 sin 2 {ωτω[2 d 0 2v(t t 0 ) l reference ]/c} / {ωτω[2 d 0 2v(t t 0 ) l reference ]/c} 2
Δ l dif c = l dif ( t 0 ) l dif ( t 0 + T a ) c p d0 c
l dif ( t 0 ) l dif ( t 0 + T a ) c = l dif l dif ( t 0 + T a ) c =τ | max[G(τ)] τ reference = 2v T a c τ | max[G(τ)] τ | max[G(τ)]/2
τ reference =τ | max[G(τ)] 2v T a /c
τ reference τ | max[G(τ)]/2
δ l B (t)=v(t t 0 )
δ l A (t)= v add (t t 0 )
a Bob ( ω 2 )= χ 1/2 × a s (χ ω 2 )×exp{i ω 2 [2β( d 1 /c)/(1β) d 1 /c]} = χ 1/2 a s (χ ω 2 )exp[i ω 2 χ( d 1 /c)]
a 2 ' ( ω 2 )= χ add 1/2 × a Bob ( ω 2 / χ add ) ×exp{i ω 2 [2 β add ( d 2 /c)/(1+ β add )( d 2 /c)]} = (χ/ χ add ) 1/2 a s ( ω 2 χ/ χ add )×exp{i ω 2 [(χ d 1 + d 2 )/c χ add ]} = (χ/ χ add ) 1/2 a s ( ω 2 χ/ χ add ) ×exp{i ω 2 [(χ+1) d 0 +χδ l B (t)+δ l A (t)]/c χ add }
δ l A (t)= v add (t t 0 )=χδ l B (t)=χv(t t 0 )
χ add =(1+ β add )/(1 β add )=(1+χv/c)/(1χv/c) ={c+[v(1+β)/(1β)]}/{c[v(1+β)/(1β)]} =[c(cv)+v(c+v)]/[c(cv)v(c+v)] =( c 2 + v 2 )/ (cv) 2 =(1+ β 2 )/ (1β) 2
a 2 ' ( ω 2 )= (χ/ χ add ) 1/2 a s ( ω 2 χ/ χ add )×exp{i ω 2 [(χ+1) d 0 ]/c χ add } = (γ) 1/2 a s (γ ω 2 )×exp[i ω 2 (μ d 0 /c)]
γ=χ/ χ add =[(1+β)/(1β)]/[(1+ β 2 )/ (1β) 2 ]} =(1 β 2 )/(1+ β 2 )
P c ( t 1 , t 2 ) | d(ω) f(ω)exp{iω[ t 2 γ t 1 (μ d 0 l)/c]} | 2 = | d(ω) f(ω)exp{iω[ t 2 t 1 +(1γ) t 1 (μ d 0 l)/c]} | 2
P c (τ) | d(ω) f(ω)exp{iω[τ(μ d 0 l)/c]} | 2
d 0 = 1 μ {l+c×τ|max[G(τ)]}

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