Abstract

In this paper, we report on a novel method to control the coherence behavior in a sum frequency generation interferometer powered by two independent pump lines. At the output of the interferometer, the two incoherent fringe patterns must be superimposed to maximize the contrast. The first step consists in canceling the differential group delay. The second one uses the phase control on one pump to synchronize the fringe patterns. This innovative method is experimentally demonstrated with a setup involving a 1544 nm signal and two pump lines around 1064 nm leading to a converted signal around 630 nm. It can be easily extended to a greater number of pump lines.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref] [PubMed]
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2017 (1)

2016 (2)

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

2015 (3)

2013 (2)

J.-T. Gomes, L. Grossard, D. Ceus, S. Vergnole, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Demonstration of a frequency spectral compression effect through an up-conversion interferometer,” Opt. Express 21(3), 3073–3082 (2013).
[Crossref] [PubMed]

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

2008 (2)

2006 (1)

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

2005 (2)

C. Langrock, E. Diamanti, R. V. Roussev, Y. Yamamoto, and M. M. Fejer, “Highly efficient single-photon detection at communication wavelengths by use of upconversion in reverse-proton-exchanged periodically poled LiNbO3 waveguides,” Opt. Lett. 30(13), 1725–1727 (2005).
[Crossref] [PubMed]

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Asobe, M.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Baek, B.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Barh, A.

Baudoin, R.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Ceus, D.

Cheng, Y.-H.

Darré, P.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

P. Darré, L. Szemendera, L. Grossard, L. Delage, and F. Reynaud, “Effect of spectral sampling on the temporal coherence analysis of a broadband source in a SFG interferometer,” Opt. Express 23(20), 25450–25461 (2015).
[Crossref] [PubMed]

Delage, L.

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

P. Darré, L. Szemendera, L. Grossard, L. Delage, and F. Reynaud, “Effect of spectral sampling on the temporal coherence analysis of a broadband source in a SFG interferometer,” Opt. Express 23(20), 25450–25461 (2015).
[Crossref] [PubMed]

J.-T. Gomes, L. Grossard, D. Ceus, S. Vergnole, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Demonstration of a frequency spectral compression effect through an up-conversion interferometer,” Opt. Express 21(3), 3073–3082 (2013).
[Crossref] [PubMed]

Diamanti, E.

Dou, X.

Farrington, C.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

Fejer, M. M.

Gerrits, T.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Gisin, N.

R. T. Thew, H. Zbinden, and N. Gisin, “Tunable upconversion photon detector,” Appl. Phys. Lett. 93, 071104 (2008).
[Crossref]

Gomes, J.-T.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

J.-T. Gomes, L. Grossard, D. Ceus, S. Vergnole, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Demonstration of a frequency spectral compression effect through an up-conversion interferometer,” Opt. Express 21(3), 3073–3082 (2013).
[Crossref] [PubMed]

Grossard, L.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

P. Darré, L. Szemendera, L. Grossard, L. Delage, and F. Reynaud, “Effect of spectral sampling on the temporal coherence analysis of a broadband source in a SFG interferometer,” Opt. Express 23(20), 25450–25461 (2015).
[Crossref] [PubMed]

J.-T. Gomes, L. Grossard, D. Ceus, S. Vergnole, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Demonstration of a frequency spectral compression effect through an up-conversion interferometer,” Opt. Express 21(3), 3073–3082 (2013).
[Crossref] [PubMed]

Harrington, S.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Herrmann, H.

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

J.-T. Gomes, L. Grossard, D. Ceus, S. Vergnole, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Demonstration of a frequency spectral compression effect through an up-conversion interferometer,” Opt. Express 21(3), 3073–3082 (2013).
[Crossref] [PubMed]

Jia, X.

Kato, Y.

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

Kurimura, S.

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

Langrock, C.

Lita, A. E.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Marsili, F.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Maruyama, M.

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

Migdall, A. L.

Mirin, R. P.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Miyazawa, H.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Nakajima, H.

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

Nam, S. W.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Nishida, Y.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Pan, J. W.

Pedersen, C.

Pelc, J. S.

Polyakov, S. V.

Reynaud, F.

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

P. Darré, L. Szemendera, L. Grossard, L. Delage, and F. Reynaud, “Effect of spectral sampling on the temporal coherence analysis of a broadband source in a SFG interferometer,” Opt. Express 23(20), 25450–25461 (2015).
[Crossref] [PubMed]

J.-T. Gomes, L. Grossard, D. Ceus, S. Vergnole, L. Delage, F. Reynaud, H. Herrmann, and W. Sohler, “Demonstration of a frequency spectral compression effect through an up-conversion interferometer,” Opt. Express 21(3), 3073–3082 (2013).
[Crossref] [PubMed]

Roussev, R. V.

Scott, N. J.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

Shangguan, M.

Shaw, M. D.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Shentu, G.

Silberhorn, C.

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

Sohler, W.

Solomon, G. S.

Stern, J. A.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Sturmann, J.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

Suzuki, H.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Szemendera, L.

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

P. Darré, L. Szemendera, L. Grossard, L. Delage, and F. Reynaud, “Effect of spectral sampling on the temporal coherence analysis of a broadband source in a SFG interferometer,” Opt. Express 23(20), 25450–25461 (2015).
[Crossref] [PubMed]

Tadanaga, O.

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Ten Brummelaar, T. A.

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

Thew, R. T.

R. T. Thew, H. Zbinden, and N. Gisin, “Tunable upconversion photon detector,” Appl. Phys. Lett. 93, 071104 (2008).
[Crossref]

Thomay, T.

Tidemand-Lichtenberg, P.

Usui, Y.

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

Vayshenker, I.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Vergnole, S.

Verma, V. B.

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Wang, C.

Xia, H.

Xia, X.

Yamamoto, Y.

Zbinden, H.

R. T. Thew, H. Zbinden, and N. Gisin, “Tunable upconversion photon detector,” Appl. Phys. Lett. 93, 071104 (2008).
[Crossref]

Zhang, J.

Zhang, Q.

Appl. Phys. Lett. (2)

R. T. Thew, H. Zbinden, and N. Gisin, “Tunable upconversion photon detector,” Appl. Phys. Lett. 93, 071104 (2008).
[Crossref]

S. Kurimura, Y. Kato, M. Maruyama, Y. Usui, and H. Nakajima, “Quasi-phase-matched adhered ridge waveguide in LiNb O3,” Appl. Phys. Lett. 89, 191123 (2006).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched device using continuous phase modulation of χ(2) grating and its application to variable wavelength conversion,” IEEE J. Quantum Electron. 41(12), 1540–1547, (2005).
[Crossref]

Monthly Notices of the Royal Astronomical Society (1)

L. Szemendera, P. Darré, R. Baudoin, L. Grossard, L. Delage, H. Herrmann, C. Silberhorn, and F. Reynaud, “In-lab ALOHA mid-infrared up-conversion interferometer with high fringe contrast @λ = 3.39μm,” Monthly Notices of the Royal Astronomical Society 457(3), 3115–3118, (2016).
[Crossref]

Nature Photonics (1)

F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93 % system efficiency,” Nature Photonics 7(3), 210–214 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

P. Darré, R. Baudoin, J.-T. Gomes, N. J. Scott, L. Delage, L. Grossard, J. Sturmann, C. Farrington, F. Reynaud, and T. A. Ten Brummelaar, “First On-Sky Fringes with an Up-Conversion Interferometer Tested on a Telescope Array,” Phys. Rev. Lett. 117, 233902 (2016).
[Crossref] [PubMed]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 2008).

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Figures (6)

Fig. 1
Fig. 1 Experimental setup. OPM: optical path modulator, FDL: fiber delay line, C: 50/50 coupler, WDM: wavelength division multiplexing, WG: waveguide, F: filters, L: lens.
Fig. 2
Fig. 2 Power spectral density transfer through the non-linear SFG process when two pump lines converts two different spectral samples centered on the infrared signal frequencies νsi.
Fig. 3
Fig. 3 Spectral phase transfer of the infrared signal through the non-linear process. The spectral compression leads to a discontinuity of the spectral phase when expressed as a function of νc.
Fig. 4
Fig. 4 Synchronized envelopes without fringe pattern synchronization. A: fringe acquisition window.
Fig. 5
Fig. 5 Synchronized envelopes with fringe pattern synchronization.
Fig. 6
Fig. 6 Measured contrast as a function of the voltage V1 applied on the phase modulator integrated on one optical path linked to the pump source. This contrast evolution is obtained after canceling the differential group delay τg. The dashed orange curve is the best theoretical fit using Eq. (13) where τg(x0) = 0.

Tables (1)

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Table 1 General overview of the different spectral phase contributions

Equations (16)

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ν s + ν p = ν c ,
B ( ν ) = sinc 2 ( ν Δ ν acc ) .
B si ( ν s ) = B ( ν s ν si ) ,
B ci ( ν c ) = B ( ν c ν ci ) .
Φ i = Φ si ( ν s ) + Φ pi ( ν pi ) + Φ ci ( ν c ) ,
Φ si = a 0 si + a 1 s × ( ν s ν si ) .
Φ si = a 0 si + a 1 s × ( ν c ν ci ) ,
Φ pi = K i V i ,
Φ ci = a 0 ci + k 0 i x + ( a 1 c + k 1 x ) × ( ν c ν ci ) ,
Φ i = a 0 ti ( x , V i ) + a 1 t ( x ) × ( ν c ν ci ) .
a 1 t ( x ) = 2 π τ g ( x ) .
I ( x , V 1 , V 2 ) = i = 1 N = 2 ( B × ( ν c ν ci ) { 1 + exp [ j 2 π τ g × ( ν c ν ci ) ] exp [ j a 0 ti ( x , V i ) ] } d ν c ) .
C ( x , V 1 , V 2 ) = Triangle ( Δ ν acc τ g ( x ) π ) | cos ( a 0 t 2 ( x , V 2 ) a 0 t 1 ( x , V 1 ) 2 ) | .
τ g ( x 0 ) = 0 Triangle ( Δ ν acc τ g ( x 0 ) π ) = 1 .
| cos ( a 0 t 2 ( x 0 , V 2 ) a 0 t 1 ( x 0 , V 1 ) 2 ) | = 1 .
a 0 t 2 ( x 0 , V 2 ) a 0 t 1 ( x 0 , V 1 ) = 0 modulo 2 π .

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