Abstract

The use of quantum correlations between photons to separate measure even- and odd-order components of polarization mode dispersion (PMD) and chromatic dispersion in discrete optical elements is investigated. Two types of apparatus are discussed which use coincidence counting of entangled photon pairs to allow sub-femtosecond resolution for measurement of both PMD and chromatic dispersion. Group delays can be measured with a resolution of order 0.1 fs, whereas attosecond resolution can be achieved for phase delays.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Kogelnik and R. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IVB: System and Impairments, I. Kaminow and T. Li, eds. (Academic Press, 2002), pp. 725–861.
  2. D. Andresciani, E. Curti, E. Matera, and B. Daino, “Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable,” Opt. Lett. 12, 844–846 (1987).
    [CrossRef] [PubMed]
  3. B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
    [CrossRef]
  4. C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,” Opt. Lett. 13, 155–157 (1987).
    [CrossRef]
  5. C. D. Poole, “Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling,” Opt. Lett. 14, 523–525 (1989).
    [CrossRef] [PubMed]
  6. D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).
  7. B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
    [CrossRef]
  8. S. Diddams and J. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13, 1120–1129 (1996).
    [CrossRef]
  9. P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. 1, 84–105 (2004).
    [CrossRef]
  10. D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
    [CrossRef]
  11. E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).
  12. M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
    [CrossRef] [PubMed]
  13. D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).
  14. O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
    [CrossRef] [PubMed]

2009 (1)

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

2004 (1)

P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. 1, 84–105 (2004).
[CrossRef]

2000 (1)

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
[CrossRef]

1999 (2)

E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

1996 (1)

1994 (1)

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

1989 (1)

1987 (2)

1982 (1)

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[CrossRef]

Andrekson, P. A.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Andresciani, D.

Bakhshi, B.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Bonato, C.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

Branning, D.

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
[CrossRef]

Brentel, J.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Costa, B.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[CrossRef]

Curti, E.

Daino, B.

Dauler, E.

E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).

Derickson, D.

D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).

Diddams, S.

Diels, J.

Giles, C. R.

Hansryd, J.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Jaeger, G.

E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).

Jopson, R.

H. Kogelnik and R. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IVB: System and Impairments, I. Kaminow and T. Li, eds. (Academic Press, 2002), pp. 725–861.

Karlsson, M.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Klyshko, D. N.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).

Kogelnik, H.

H. Kogelnik and R. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IVB: System and Impairments, I. Kaminow and T. Li, eds. (Academic Press, 2002), pp. 725–861.

Kolltveit, E.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Matera, E.

Mazzoni, D.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[CrossRef]

Migdall, A.

E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).

Migdall, A. L.

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
[CrossRef]

Minaeva, O.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

Muller, A.

E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).

Olsson, B. K.

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

Poole, C. D.

Puleo, M.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[CrossRef]

Rubin, M. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Saleh, B. E. A.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

Sergienko, A. V.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
[CrossRef]

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Shih, Y. H.

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

Simon, D. S.

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

Vezzoni, E.

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[CrossRef]

Williams, P.

P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. 1, 84–105 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Costa, D. Mazzoni, M. Puleo, and E. Vezzoni, “Phase shift technique for the measurement of chromatic dispersion in optical fibers using LED’s,” IEEE J. Quantum Electron. 18, 1509–1515 (1982).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. Bakhshi, J. Hansryd, P. A. Andrekson, J. Brentel, E. Kolltveit, B. K. Olsson, and M. Karlsson, “Measurement of the differential group delay in installed optical fibers using polarization multiplexed solitons,” IEEE Photon. Technol. Lett. 11, 593–595 (1999).
[CrossRef]

J. Opt. Fiber Commun. Rep. (1)

P. Williams, “PMD measurement techniques and how to avoid the pitfalls,” J. Opt. Fiber Commun. Rep. 1, 84–105 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Res. Natl. Inst. Stand. Technol. (1)

E. Dauler, G. Jaeger, A. Muller, and A. Migdall, “Tests of a two-photon technique for measuring polarization mode dispersion with subfemtosecond precision,” J. Res. Natl. Inst. Stand. Technol. 104, 1–10 (1999).

Opt. Lett. (3)

Phys. Rev. A (2)

M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, “Theory of two-photon entanglement in type-II optical parametric down-conversion,” Phys. Rev. A 50, 5122–5133 (1994).
[CrossRef] [PubMed]

D. Branning, A. L. Migdall, and A. V. Sergienko, “Simultaneous measurement of group and phase delay between two photons,” Phys. Rev. A 62, 063808 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

O. Minaeva, C. Bonato, B. E. A. Saleh, D. S. Simon, and A. V. Sergienko, “Odd- and even-order dispersion cancellation in quantum interferometry,” Phys. Rev. Lett. 102, 100504 (2009).
[CrossRef] [PubMed]

Other (3)

D. Derickson, Fiber Optic Test and Measurement (Prentice Hall, 1998).

H. Kogelnik and R. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IVB: System and Impairments, I. Kaminow and T. Li, eds. (Academic Press, 2002), pp. 725–861.

D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Classical (single-detector) white-light setup for finding total PMD.

Fig. 2
Fig. 2

Interferograms produced by apparatus of Fig. 1 for samples of different thicknesses. For a fixed thickness, the size of the shift may be used as a measure of the difference in phase velocities of the two polarizations.

Fig. 3
Fig. 3

Type A setup for measuring PMD parameters ΔααV αH and ΔββV βH .

Fig. 4
Fig. 4

Scanning over τ 1 while keeping τ 2 = 0. The horizontal shift of the minimum away from the origin determines Δα, while the depth of the dip determines Δk 0. The triangle function may lead either to a dip (as shown) or to a peak, depending on the sign of the cosine.

Fig. 5
Fig. 5

Scanning over τ 2 while keeping τ 1 = 0. In (a), a nonzero Δα shifts the envelope from its position for Δα = 0 in part (b). The size of the shift can be measured with accuracy on the order of 0.1 fs. In part (c), a nonzero Δk 0 shifts the locations of the peaks within the unshifted envelope. The size of the shift can be measured with accuracy on the order of .001 fs = 1 as.

Fig. 6
Fig. 6

Type B setup for finding even- and odd-order PMD.

Fig. 7
Fig. 7

The post-beam splitter delays corresponding to the four possible outcomes at the first beam splitter: one photon can go in each direction, with the vertical following the upper (a) or lower (b) path, both photons may follow the lower path (c), or both may follow the upper path (d). The second and third columns give the post-BS delays of the vertical and horizontal photons, respectively, while the final column gives the difference (vertical delay minus horizontal).

Fig. 8
Fig. 8

Effect of quadratic dispersion term Δβ on a pair of triangular peaks. The red curve is for Δβ = 0, the lower curves correspond to increasing values of Δβ for fixed Δα and Δk 0.

Fig. 9
Fig. 9

Setup with object and delays only after the first beam splitter.

Fig. 10
Fig. 10

Scan over τ for fixed τ 2, when there is an object only after the first beam splitter.

Equations (50)

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

k ( Ω 0 ± ω ) = k 0 ± α ω + β ω 2 ± γ ω 3 +
k 0 = k ( Ω 0 ) , α = d k ( ω ) d ω | ω = Ω 0 ,
β = 1 2 ! d 2 k ( ω ) d ω 2 | ω = Ω 0 , γ = 1 3 ! d 3 k ( ω ) d ω 3 | ω = Ω 0 ,
k ( Ω 0 + ω ) = k e v e n ( ω ) + k o d d ( ω ) ,
k e v e n ( ω ) = k 0 + β ω 2 + 𝒪 ( ω 4 ) ,
k o d d ( ω ) = α ω + γ ω 3 + 𝒪 ( ω 5 ) .
k H ( Ω 0 ± ω ) = k H 0 ± α H ω + β H ω 2 +
= k H , e v e n ( ω ) + k H , o d d ( ω )
K V ( Ω 0 ± ω ) = k V 0 ± α V ω + β V ω 2 +
= k V , e v e n ( ω ) + k V , o d d ( ω ) ,
Δ k 0 = k V 0 k H 0 , Δ α = α V α H , Δ β = β V β H .
Δ φ l Δ k 0 , Δ A l Δ α , Δ B l Δ β ,
Φ ( ω ) = sinc ( 1 2 τ ω ) ,
( A H ( ω ) A V ( ω ) ) d ω ,
( A H ( ω ) 0 ) d ω .
( 0 1 1 0 ) ( A H ( ω ) 0 ) d ω = ( 0 A H ( ω ) ) d ω .
J 0 = A H ( ω ) ( e i k ( ω ) d 1 e i k ( ω ) ( d 2 + δ ) ) d ω ,
J 0 = A H ( ω ) 2 ( e i k ( ω ) d 1 + e i k ( ω ) ( d 2 + δ ) e i k ( ω ) d 1 + e i k ( ω ) ( d 2 + δ ) ) d ω ,
I = | J 0 | 2 = | A H ( ω ) | 2 [ 1 + cos ( k ( ω ) ) ( Δ d δ ) ] d ω .
J 0 = A H ( ω ) ( e i [ k ( ω ) d 1 + k H ( ω ) l ] e [ i k ( ω ) ( d 2 + δ ) + k V ( ω ) l ] ) d ω .
I = | J 0 | 2 = | A H ( ω ) | 2 [ 1 + cos ( ω c ( Δ d δ ) Δ k ( ω ) l ) ] d ω .
| Ψ = d ω Φ ( ω ) a ^ H ( Ω 0 + ω ) a ^ V ( Ω 0 ω ) | 0 ,
E ^ 1 ( + ) ( t 1 ) = 1 2 d ω { a ^ H ( ω 1 ) e i k H ( ω 1 ) l 1 + a ^ V ( ω 1 ) e i [ k V ( ω 1 ) l 1 + ω 1 τ 1 ] } e i ω 1 t 1
E ^ 2 ( + ) ( t 2 ) = 1 2 d ω { a ^ H ( ω 2 ) e i k H ( ω 2 ) ( l 1 + l 2 ) + a ^ V ( ω 2 ) e i [ k V ( ω 2 ) ( l 1 + l 2 ) + ω 2 ( τ 1 + τ 2 ) ] } e i ω 2 ( t 2 + τ ) .
G ( 2 ) ( t 1 , t 2 ) = | 0 | E 1 ( + ) ( t 1 ) E 2 ( + ) ( t 2 ) | Ψ | 2 | 2
R c ( τ 1 , τ 2 ) = T / 2 T / 2 d t 1 d t 2 G ( 2 ) ( t 1 , t 2 ) .
R c ( τ 1 , τ 2 ) = R 0 { 1 + C M ( τ 1 , τ 2 ) } ,
M ( τ 1 , τ 2 ) = 1 2 d ω | Φ ( ω ) | 2 e i [ Δ k ( ω ) Δ k ( ω ) ] l 1 2 i ω τ 1 × { e i Δ k ( ω ) l 2 + i ( Ω 0 ω ) τ 2 + e i Δ k ( ω ) l 2 i ( Ω 0 + ω ) τ 2 }
= d ω | Φ ( ω ) | 2 × cos { [ Δ k ( ω ) Δ k ( ω ) ] l 1 + Δ k ( ω ) l 2 + 2 ω τ 1 + ( Ω 0 + ω ) τ 2 } ,
M ( τ 1 , τ 2 ) = d ω | Φ ( ω ) | 2 { cos  [ Δ k o d d ( ω ) ( 2 l 1 + l 2 ) + ω ( 2 τ 1 + τ 2 ) ] cos [ Δ k e v e n ( ω ) l 2 + Ω 0 τ 2 ] sin [ Δ k o d d ( ω ) ( 2 l 1 + l 2 ) + ω ( τ 1 + τ 2 ) ] sin [ Δ k e v e n ( ω ) l 2 + Ω 0 τ 2 ] } .
M ( τ 1 , τ 2 ) = d ω | Φ ( ω ) | 2 cos [ Δ k o d d ( ω ) ( 2 l 1 + l 2 ) + ω ( 2 τ 1 + τ 2 ) ] × cos [ Δ k e v e n ( ω ) l 2 + Ω 0 τ 2 ] .
M ( τ 1 , τ 2 ) = 2 π τ cos [ Δ k 0 l 2 + Ω 0 τ 2 ] Λ [ Δ α ( 2 l 1 + l 2 ) + ( 2 τ 2 + τ 2 ) τ ] .
d ω sinc 2 ( a ω ) cos  ( ω τ ) = π a Λ ( τ 2 a ) ,
Λ ( x ) = { 1 | x | , | x | 1 0 , | x | > 1
R c ( τ 1 , τ 2 ) = R 0 { 1 + cos [ Δ k 0 l 2 + Ω 0 τ 2 ] Λ [ Δ α ( 2 l 1 + l 2 ) + ( 2 τ 1 + τ 2 ) τ ] } .
Φ ( ω ) = Φ ( ω ) .
Δ τ p r e τ V τ H = Δ α l 1 + τ 1 .
Δ τ p r e + Δ τ p o s t = 0
Δ τ t o t a l τ V τ H = Δ α l 1 + τ 1 = Δ τ p r e .
Δ τ t o t a l = Δ α ( l 1 + l 2 ) + ( τ 1 + τ 2 ) .
M ( τ 1 , τ 2 , τ ) = d ω | Φ ( ω ) | 2 e 2 i ω τ 1 e 2 i Δ k o d d ( ω ) l 1 × { 1 e i [ ( k V ( Ω 0 + ω ) k V ( Ω 0 ω ) ) l 2 + 2 ω ( τ + τ 2 ) ] + e i [ ( k H ( Ω 0 + ω ) k V ( Ω 0 + ω ) ) l 2 ( Ω 0 + ω ) τ 2 ] + e i [ ( k V ( Ω 0 ω ) k H ( Ω 0 ω ) ) l 2 + ( Ω 0 ω ) τ 2 ] e i [ ( k H ( Ω 0 + ω ) k H ( Ω 0 ω ) ) l 2 + 2 ω τ ] e i [ ( k H ( Ω 0 + ω ) + k V ( Ω 0 ω ) ) l 2 + 2 Ω 0 τ + ( Ω 0 ω ) τ 2 ] e i [ ( k H ( Ω 0 ω ) + k V ( Ω 0 + ω ) ) l 2 + 2 Ω 0 τ + ( Ω 0 + ω ) τ 2 ] + e i [ ( k H ( Ω 0 + ω ) k H ( Ω 0 ω ) k V ( Ω 0 + ω ) + k V ( Ω 0 ω ) ) l 2 2 ω τ 2 ] } .
k V ( Ω 0 + ω ) = k V 0 + α V ω
k H ( Ω 0 + ω ) = k H 0 + α H ω
Δ k ( Ω 0 + ω ) = Δ k 0 + Δ α ω
k V , o d d = α V ω k V , e v e n = k V 0 k H , o d d = α H ω k H , e v e n = k H 0 Δ k o d d = Δ α ω Δ k e v e n = Δ k 0
d ω sinc 2 ( a ω ) e i ω ( τ + c ) = π a Λ ( τ + c 2 a )
M ( τ 1 , τ 2 , τ ) = 2 π τ { Λ ( 2 ( τ 1 + Δ α l 1 ) τ ) + 4 Λ ( 2 τ 1 + Δ α ( 2 l 1 + l 2 ) τ 2 τ ) sin ( k 0 V l 2 + Ω 0 ( τ + τ 2 ) ) sin ( k 0 H l 2 Ω 0 τ ) Λ ( 2 ( τ 1 + Δ α l 1 + α V l 2 + τ + τ 2 ) τ ) Λ ( 2 ( τ 1 + Δ α l 1 α H l 2 τ ) τ ) + Λ ( 2 ( τ 1 + Δ α ( l 1 + l 2 ) + τ 2 ) τ ) } .
τ 1 = Δ α l 1 , 1 2 [ τ 2 Δ α ( 2 l 1 + l 2 ) ] , ( α V l 2 + Δ α l 2 + τ + τ 2 ) , α H l 2 + τ Δ α l 2 , τ 2 Δ α ( l 1 + l 2 ) .
R ( τ , τ 2 ) = R 0 { 2 + 4 Λ ( Δ α l 2 τ 2 τ ) sin [ k 0 V l 2 + Ω 0 ( τ + τ 2 ) ] sin [ k 0 H l 2 Ω 0 τ ] Λ ( 2 ( α V l 2 + τ + τ 2 ) τ ) Λ ( 2 ( α H l 2 + τ ) τ ) + Λ ( Δ α l 2 + τ 2 τ ) . }
τ = α V l 2 τ 2 and τ = α H l 2 ,

Metrics