Abstract

We describe a dispersion-enabled, ultra-low power realization of super-resolution in an integrated Mach-Zehnder interferometer. Our scheme is based on a Vernier-like effect in the coincident detection of frequency correlated, non-degenerate photon pairs at the sensor output in the presence of group index dispersion. We design and simulate a realistic integrated refractive index sensor in a silicon nitride on silica platform and characterize its performance in the proposed scheme. We present numerical results showing a sensitivity improvement upward of 40 times over a traditional sensing scheme. The device we design is well within the reach of modern semiconductor fabrication technology. We believe this is the first metrology scheme that uses waveguide group index dispersion as a resource to attain super-resolution.

© 2016 Optical Society of America

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2014 (2)

2013 (7)

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cramér-Rao bound via parity detection,” Phys. Rev. A 87(4), 043833 (2013).
[Crossref]

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. 37(2), 51–107 (2013).
[Crossref]

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N -photon interference patterns,” Phys. Rev. A 87(5), 053821 (2013).
[Crossref]

M. La Notte and V. M. N. Passaro, “Ultra high sensitivity chemical photonic sensing by Mach–Zehnder interferometer enhanced Vernier-effect,” Sens. Actuators B Chem. 176, 994–1007 (2013).
[Crossref]

2012 (2)

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

2011 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58(11), 945–953 (2011).
[Crossref]

2010 (5)

T. Claes, W. Bogaerts, and P. Bienstman, “Experimental characterization of a silicon photonic biosensor consisting of two cascaded ring resonators based on the Vernier-effect and introduction of a curve fitting method for an improved detection limit,” Opt. Express 18(22), 22747–22761 (2010).
[Crossref] [PubMed]

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101 (2010).
[Crossref]

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81(6), 063836 (2010).
[Crossref]

2009 (1)

2007 (3)

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15(23), 15377–15386 (2007).
[Crossref] [PubMed]

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007).
[Crossref] [PubMed]

2004 (1)

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

2002 (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

1993 (1)

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

1989 (1)

1986 (1)

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34(5), 3962–3968 (1986).
[Crossref] [PubMed]

1973 (1)

H. R. Philipp, “Optical Properties of Silicon Nitride,” J. Electrochem. Soc. 120(2), 295 (1973).
[Crossref]

1965 (1)

Agarwal, A.

Anisimov, P.

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

Anisimov, P. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Armenise, M. N.

C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. 37(2), 51–107 (2013).
[Crossref]

Arnold, S.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

Baets, R.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Bell, B.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

Berry, M.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

Bienstman, P.

Björk, G.

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N -photon interference patterns,” Phys. Rev. A 87(5), 053821 (2013).
[Crossref]

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81(6), 063836 (2010).
[Crossref]

Bogaerts, W.

Bourennane, M.

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81(6), 063836 (2010).
[Crossref]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

Burnett, K.

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

Cable, H.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

Campanella, C. E.

C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. 37(2), 51–107 (2013).
[Crossref]

Campanella, C. M.

C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. 37(2), 51–107 (2013).
[Crossref]

Chen, H.-Y.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Chen, Y.

Cheng, J.-Y.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Chiruvelli, A.

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58(11), 945–953 (2011).
[Crossref]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Ciminelli, C.

C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. 37(2), 51–107 (2013).
[Crossref]

Claes, T.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

T. Claes, W. Bogaerts, and P. Bienstman, “Experimental characterization of a silicon photonic biosensor consisting of two cascaded ring resonators based on the Vernier-effect and introduction of a curve fitting method for an improved detection limit,” Opt. Express 18(22), 22747–22761 (2010).
[Crossref] [PubMed]

Clark, A. S.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

Cohen, L.

Crespi, A.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

De Leonardis, F.

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

de Tullio, C.

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

Dell’Olio, F.

C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. 37(2), 51–107 (2013).
[Crossref]

Deshpande, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Dhakal, A.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Dovrat, L.

Dowling, J. P.

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cramér-Rao bound via parity detection,” Phys. Rev. A 87(4), 043833 (2013).
[Crossref]

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

DuBois, B.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Eisenberg, H. S.

Erven, C.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

Fainman, Y.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101 (2010).
[Crossref]

Fedrizzi, A.

Gao, Y.

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

Ghosh, R.

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34(5), 3962–3968 (1986).
[Crossref] [PubMed]

Giannoccaro, G.

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

Gilchrist, A.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

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

Hardy, A.

He, J.-J.

Helin, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Herbst, T.

Holland, M. J.

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71(9), 1355–1358 (1993).
[Crossref] [PubMed]

Hong, C. K.

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34(5), 3962–3968 (1986).
[Crossref] [PubMed]

Hu, J.

Huver, S. D.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Ikeda, K.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101 (2010).
[Crossref]

Istrati, D.

Jansen, R.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Jennewein, T.

Jiang, X.

Kannan, S.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

Khoshsima, M.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

Kim, S.

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cramér-Rao bound via parity detection,” Phys. Rev. A 87(4), 043833 (2013).
[Crossref]

Kimerling, L. C.

Kothe, C.

C. Kothe, G. Björk, and M. Bourennane, “Arbitrarily high super-resolving phase measurements at telecommunication wavelengths,” Phys. Rev. A 81(6), 063836 (2010).
[Crossref]

La Notte, M.

M. La Notte and V. M. N. Passaro, “Ultra high sensitivity chemical photonic sensing by Mach–Zehnder interferometer enhanced Vernier-effect,” Sens. Actuators B Chem. 176, 994–1007 (2013).
[Crossref]

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

Lee, H.

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cramér-Rao bound via parity detection,” Phys. Rev. A 87(4), 043833 (2013).
[Crossref]

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58(11), 945–953 (2011).
[Crossref]

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Lee, K.-L.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Leyssens, K.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Li, M.

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

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

Lobino, M.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

Luine, J.

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5(4), 222–229 (2011).
[Crossref]

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

Malitson, I. H.

Mandel, L.

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34(5), 3962–3968 (1986).
[Crossref] [PubMed]

Marom, E.

Matthews, J. C. F.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

McMillan, A.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

Mousavi, M. Z.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Nagata, T.

T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007).
[Crossref] [PubMed]

Neal, C. R.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

Neutens, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Neville, A.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

O’Brien, J. L.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007).
[Crossref] [PubMed]

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

Okamoto, R.

T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007).
[Crossref] [PubMed]

Osellame, R.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

Ou, Z. Y.

R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, “Interference of two photons in parametric down conversion,” Phys. Rev. A 34(5), 3962–3968 (1986).
[Crossref] [PubMed]

Passaro, V. M. N.

M. La Notte and V. M. N. Passaro, “Ultra high sensitivity chemical photonic sensing by Mach–Zehnder interferometer enhanced Vernier-effect,” Sens. Actuators B Chem. 176, 994–1007 (2013).
[Crossref]

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

Peng, S.-W.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Peyskens, F.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Philipp, H. R.

H. R. Philipp, “Optical Properties of Silicon Nitride,” J. Electrochem. Soc. 120(2), 295 (1973).
[Crossref]

Plick, W. N.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Politi, A.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

Poppe, A.

Pregnell, K. L.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

Prevedel, R.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

Pryde, G. J.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

Ramponi, R.

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

Rarity, J. G.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

Raterman, G. M.

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit,” Phys. Rev. Lett. 104(10), 103602 (2010).
[Crossref] [PubMed]

Resch, K. J.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

Rottenberg, X.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Sasaki, K.

T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007).
[Crossref] [PubMed]

Selvaraja, S.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Seshadreesan, K. P.

K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, “Phase estimation at the quantum Cramér-Rao bound via parity detection,” Phys. Rev. A 87(4), 043833 (2013).
[Crossref]

Severi, S.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Shabbir, S.

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N -photon interference patterns,” Phys. Rev. A 87(5), 053821 (2013).
[Crossref]

Subramanian, A. Z.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Sun, P. C.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101 (2010).
[Crossref]

Sun, X.

Swillo, M.

S. Shabbir, M. Swillo, and G. Björk, “Synthesis of arbitrary, two-mode, high-visibility N -photon interference patterns,” Phys. Rev. A 87(5), 053821 (2013).
[Crossref]

Takeuchi, S.

T. Nagata, R. Okamoto, J. L. O’brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four-entangled photons,” Science 316(5825), 726–729 (2007).
[Crossref] [PubMed]

Tan, D. T. H.

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101 (2010).
[Crossref]

Tang, L.

Teraoka, I.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

Troia, B.

V. M. N. Passaro, C. de Tullio, B. Troia, M. La Notte, G. Giannoccaro, and F. De Leonardis, “Recent advances in integrated photonic sensors,” Sensors (Basel) 12(11), 15558–15598 (2012).
[Crossref] [PubMed]

Van Dorpe, P.

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

Vollmer, F.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

Wadsworth, W. J.

B. Bell, S. Kannan, A. McMillan, A. S. Clark, W. J. Wadsworth, and J. G. Rarity, “Multicolor Quantum Metrology with Entangled Photons,” Phys. Rev. Lett. 111(9), 093603 (2013).
[Crossref] [PubMed]

Wei, P.-K.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Weissman, Z.

White, A. G.

K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-Reversal and Super-Resolving Phase Measurements,” Phys. Rev. Lett. 98(22), 223601 (2007).
[Crossref] [PubMed]

Whittaker, R.

R. Whittaker, C. Erven, A. Neville, M. Berry, J. L. O’Brien, H. Cable, and J. C. F. Matthews, “Quantum-enhanced absorption spectroscopy,” 5 (2015).

Wildfeuer, C.

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

Wu, S.-H.

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Yu, F.

Zeilinger, A.

Analyst (Lond.) (1)

M. Z. Mousavi, H.-Y. Chen, S.-H. Wu, S.-W. Peng, K.-L. Lee, P.-K. Wei, and J.-Y. Cheng, “Magnetic nanoparticle-enhanced SPR on gold nanoslits for ultra-sensitive, label-free detection of nucleic acid biomarkers,” Analyst (Lond.) 138(9), 2740–2748 (2013).
[Crossref] [PubMed]

Appl. Phys. Lett. (3)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057 (2002).
[Crossref]

A. Crespi, M. Lobino, J. C. F. Matthews, A. Politi, C. R. Neal, R. Ramponi, R. Osellame, and J. L. O’Brien, “Measuring protein concentration with entangled photons,” Appl. Phys. Lett. 100(23), 233704 (2012).
[Crossref]

D. T. H. Tan, K. Ikeda, P. C. Sun, and Y. Fainman, “Group velocity dispersion and self phase modulation in silicon nitride waveguides,” Appl. Phys. Lett. 96(6), 061101 (2010).
[Crossref]

IEEE Photonics J. (1)

A. Z. Subramanian, P. Neutens, A. Dhakal, R. Jansen, T. Claes, X. Rottenberg, F. Peyskens, S. Selvaraja, P. Helin, B. DuBois, K. Leyssens, S. Severi, P. Deshpande, R. Baets, and P. Van Dorpe, “Low-Loss Singlemode PECVD Silicon Nitride Photonic Wire Waveguides for 532–900 nm Wavelength Window Fabricated Within a CMOS Pilot Line,” IEEE Photonics J. 5(6), 2202809 (2013).
[Crossref]

J. Electrochem. Soc. (1)

H. R. Philipp, “Optical Properties of Silicon Nitride,” J. Electrochem. Soc. 120(2), 295 (1973).
[Crossref]

J. Mod. Opt. (1)

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58(11), 945–953 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

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

JOSA B (1)

Y. Gao, P. Anisimov, C. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” JOSA B 27(6), 170–174 (2010).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

(a) MZI driven by frequency correlated SPDC photon pairs. A dichroic mirror and two photon counters are used to register coincident detections. (b) Evolution of the two photons state inside the interferometer. The combiner acts as a time-reversed splitter. The joint detection probability is given by the projector of the splitter output state onto the split state MZI>.

Fig. 2
Fig. 2

Time-correlated output of the MZI. The cascaded output (red thick line) is enclosed by the Vernier-like curve (black thick line) defined in (4). The output intensity for the single probes as measured by single photon detectors are represented by solid and dashed gray thin lines. Representing the single probes as phasors whose position is regulated by (2) and (3), the Vernier-like envelope is defined as the single photon peaks go from being out of phase (left diagram) to in-phase (right diagram) as the wavelength of the SPDC source is scanned.

Fig. 3
Fig. 3

(a) Scheme of the integrated, silicon nitride based, MZI sensor considered. The inset shows a simulated mode profile of the waveguides. (b) Calculated FSRs for the two probes (left scale) and super-resolving factor GSPDC (right scale) for a measurement arm length of 4mm. The ideal working range (central green shaded area) is located between a region with very short FSR that would require an unrealistically narrow input bandwidth (left shaded area) and one where the FSR diverges to very large values (right shaded area), yielding very broad output peaks and a Vernier-like envelope that falls outside the chosen bandwidth of our waveguide system. The expected GSPDC at the chosen working point (vertical gray dashed line, Lr = 3.8mm) is approximately 45.

Fig. 4
Fig. 4

Numerical results for the Vernier-like joint measurement. (a) Output curves for the MZI at rest and for a refractive index shift Δn = 10−4 are shown. The wavelength responsivity is 71.5μm/RIU, opposed to the responsivity of a regular, single wavelength readout of 1.5μm/RIU. Central peaks in the envelope function before and after the index shift are highlighted. (b) Comparison between the peak wavelength responsivity for the integrated MZI driven by a single probe (dashed green line) and the Vernier-like, SPDC-based responsivity in the same device (orange full line).

Equations (9)

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Δ λ MZI = λ ¯ Δn - λ ¯ = λ ¯ L s Δn ΔL eff,λ ΔL eff,λ = L s ( n s,λ n s,λ λ λ ) L r ( n r,λ n r,λ λ λ ),
I SPDC = | 1 2 ψ MZI |( | 0, λ 1 + e i φ 1 | 1, λ 1 )( | 0, λ 2 + e i φ 2 | 1, λ 2 ) | 2 = cos 2 [ φ 1 2 ] cos 2 [ φ 2 2 ], φ i = π λ i ( L s ( n s, λ i +Δn ) L r n r, λ i );i=1,2
λ 2 = ( 1 λ p - 1 λ 1 ) -1 ,
cos 4 [ ( φ 1 + φ 2 ) 4 ] ; sin 4 [ ( φ 1 + φ 2 ) 4 ] .
Δ λ SPDC =( λ ¯ 1 2 λ p ) Δn L s ΔL eff, λ 1 ΔL eff, λ 2 .
FSR= λ 2 ΔL eff,λ ,
Δ λ SPDC = G SPDC FS R 1 L s Δn λ ¯ 1 G SPDC = Δ λ SPDC Δ λ MZI =( λ ¯ 1 λ p ) FS R 2 FS R 2 ( λ p λ ¯ 1 λ p ) 2 FS R 1 ,
δ( Δ λ SPDC ) 1 4 FS R 1 FS R 2 ( λ ^ 2 2 λ ¯ 1 2 )FS R 1 SNL,
LoD SPDC LoD MZI 1 4 FS R 2 ( λ ^ 2 2 λ ¯ 1 2 )FS R 1 FS R 1 .

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