Abstract

The current literature on quantum key distribution is limited mainly to transmissions over fiber optic, atmospheric, or satellite links and is not directly applicable to underwater environments with different channel characteristics. In this paper, we analyze the quantum bit error rate (QBER) and secret key rate (SKR) performance of the well-known BB84 protocol in underwater channels. As a path loss model, we consider a modified version of the Beer–Lambert formula, which takes into account the effect of scattering. We derive a closed-form expression for the wave structure function to determine the average power transfer over a turbulent underwater path and use this to obtain an upper bound on QBER as well as a lower bound on SKR. Based on the derived bounds, we present the performance of the BB84 protocol in different water types including clear, coastal, and turbid and under different atmospheric conditions such as clear, hazy, and overcast. We further investigate the effect of system parameters such as aperture size and detector field of view on QBER and SKR performance metrics.

© 2020 Optical Society of America

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2019 (8)

E. Y. Zhu, C. Corbari, A. Gladyshev, P. G. Kazansky, H. K. Lo, and L. Qian, “Toward a reconfigurable quantum network enabled by a broadband entangled source,” J. Opt. Soc. Am. B 36, B1–B6 (2019).
[Crossref]

S. C. Zhao, X. H. Han, Y. Xiao, Y. Shen, Y. J. Gu, and W. D. Li, “Performance of underwater quantum key distribution with polarization encoding,” J. Opt. Soc. Am. A 36, 883–892 (2019).
[Crossref]

M. Lanzagorta and J. Uhlmann, “Assessing feasibility of secure quantum communications involving underwater assets,” IEEE J. Ocean. Eng. 99, 1–10 (2019).
[Crossref]

S. Zhao, W. Li, Y. Shen, Y. Yu, X. Han, H. Zeng, M. Cai, T. Qian, S. Wang, Z. Wang, Y. Xiao, and Y. Gu, “Experimental investigation of quantum key distribution over a water channel,” Appl. Opt. 58, 3902–3907 (2019).
[Crossref]

C. Q. Hu, Z. Q. Yan, J. Gao, Z. Q. Jiao, Z. M. Li, W. G. Shen, Y. Chen, R. J. Ren, L. F. Qiao, A. L. Yang, and H. Tang, “Transmission of photonic polarization states through 55-m water: towards air-to-sea quantum communication,” Photon. Res. 7, A40–A44 (2019).
[Crossref]

F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, “Characterization of an underwater channel for quantum communications in the Ottawa River,” Opt. Express 27, 26346–26354 (2019).
[Crossref]

J. Gariano and I. B. Djordjevic, “Theoretical study of a submarine to submarine quantum key distribution systems,” Opt. Express 27, 3055–3064 (2019).
[Crossref]

T. Wu, X. Ji, H. Zhang, X. Li, L. Wang, and X. Fan, “Rytov variance of spherical wave and performance indicators of laser radar systems in oceanic turbulence,” Opt. Commun. 434, 36–43 (2019).
[Crossref]

2018 (4)

M. Elamassie, F. Miramirkhani, and M. Uysal, “Performance characterization of underwater visible light communication,” IEEE Trans. Commun. 67, 543–552 (2018).
[Crossref]

F. Bouchard, A. Sit, F. Hufnagel, A. Abbas, Y. Zhang, K. Heshami, R. Fickler, C. Marquardt, G. Leuchs, and E. Karimi, “Quantum cryptography with twisted photons through an outdoor underwater channel,” Opt. Express 26, 22563–22573 (2018).
[Crossref]

M. Lopes and N. Sarwade, “Optimized decoy state QKD for underwater free space communication,” Int. J. Quantum Inf. 16, 1850019 (2018).
[Crossref]

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

2017 (2)

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

M. Elamassie, M. Uysal, Y. Baykal, M. Abdallah, and K. Qaraqe, “Effect of eddy diffusivity ratio on underwater optical scintillation index,” J. Opt. Soc. Am. A 34, 1969–1973 (2017).
[Crossref]

2016 (2)

M. Ibragimov, J. H. Lee, M. Kalyani, J. I. Namgung, S. H. Park, O. Yi, C. H. Kim, and Y. K. Lim, “CCM-UW security modes for low-band underwater acoustic sensor networks,” Wireless Pers. Commun. 89, 479–499 (2016).
[Crossref]

E. Diamanti, H. K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” NPJ Quantum Inf. 2, 16025 (2016).
[Crossref]

2015 (3)

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

P. Shi, S. C. Zhao, Y. J. Gu, and W. D. Li, “Channel analysis for single photon underwater free space quantum key distribution,” J. Opt. Soc. Am. A 32, 349–356 (2015).
[Crossref]

G. Han, J. Jiang, N. Sun, and L. Shu, “Secure communication for underwater acoustic sensor networks,” IEEE Commun. Mag. 53, 54–60 (2015).
[Crossref]

2014 (1)

2013 (1)

A. Caiti, V. Calabro, A. Munafo, G. Dini, and A. Lo Duca, “Mobile underwater sensor networks for protection and security: field experience at the UAN11 experiment,” J. Field Robot. 30, 237–253 (2013).
[Crossref]

2012 (1)

J. Martinez Mateo, D. E. Coronas, and V. Martn Ayuso, “Blind reconciliation,” Quantum Inf. Comput. 12, 791–812 (2012).

2011 (1)

M. Erol-Kantarci, H. T. Mouftah, and S. Oktug, “A survey of architectures and localization techniques for underwater acoustic sensor networks,” IEEE Commun. Surveys Tuts. 13, 487–502 (2011).
[Crossref]

2009 (1)

V. Scarani, H. Bechmann Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

2008 (1)

2006 (1)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

2003 (1)

J. H. Shapiro, “Near-field turbulence effects on quantum-key distribution,” Phys. Rev. A 67, 022309 (2003).
[Crossref]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

2001 (1)

T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[Crossref]

2000 (1)

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref]

1993 (1)

L. Andrews, S. Vester, and C. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661 (1991).
[Crossref]

1965 (1)

Abbas, A.

Abdallah, M.

Alléaume, R.

D. Elkouss, A. Leverrier, R. Alléaume, and J. J. Boutros, “Efficient reconciliation protocol for discrete-variable quantum key distribution,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1879–1883.

Andrews, L.

L. Andrews, S. Vester, and C. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[Crossref]

Andrews, L. C.

L. C. Andrews and L. C. Andrews, Special Functions of Mathematics for Engineers (McGraw-Hill, 1992).

L. C. Andrews and L. C. Andrews, Special Functions of Mathematics for Engineers (McGraw-Hill, 1992).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005), Vol. 152.

Baykal, Y.

Bechmann Pasquinucci, H.

V. Scarani, H. Bechmann Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, 1984), pp. 175–179.

Bouchard, F.

Boutros, J. J.

D. Elkouss, A. Leverrier, R. Alléaume, and J. J. Boutros, “Efficient reconciliation protocol for discrete-variable quantum key distribution,” in Proceedings of IEEE International Symposium on Information Theory (IEEE, 2009), pp. 1879–1883.

Boyd, R. W.

F. Hufnagel, A. Sit, F. Grenapin, F. Bouchard, K. Heshami, D. England, Y. Zhang, B. J. Sussman, R. W. Boyd, G. Leuchs, and E. Karimi, “Characterization of an underwater channel for quantum communications in the Ottawa River,” Opt. Express 27, 26346–26354 (2019).
[Crossref]

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (IEEE, 1984), pp. 175–179.

Cai, M.

Cai, W. Q.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Caiti, A.

A. Caiti, V. Calabro, A. Munafo, G. Dini, and A. Lo Duca, “Mobile underwater sensor networks for protection and security: field experience at the UAN11 experiment,” J. Field Robot. 30, 237–253 (2013).
[Crossref]

Calabro, V.

A. Caiti, V. Calabro, A. Munafo, G. Dini, and A. Lo Duca, “Mobile underwater sensor networks for protection and security: field experience at the UAN11 experiment,” J. Field Robot. 30, 237–253 (2013).
[Crossref]

Cao, Y.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Cerf, N. J.

V. Scarani, H. Bechmann Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Chang, X.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Chen, Y.

Chen, Y.-A.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Chuang, I.

M. A. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2002).

Corbari, C.

Coronas, D. E.

J. Martinez Mateo, D. E. Coronas, and V. Martn Ayuso, “Blind reconciliation,” Quantum Inf. Comput. 12, 791–812 (2012).

Dai, H.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Diamanti, E.

E. Diamanti, H. K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” NPJ Quantum Inf. 2, 16025 (2016).
[Crossref]

Dini, G.

A. Caiti, V. Calabro, A. Munafo, G. Dini, and A. Lo Duca, “Mobile underwater sensor networks for protection and security: field experience at the UAN11 experiment,” J. Field Robot. 30, 237–253 (2013).
[Crossref]

Djordjevic, I. B.

Dušek, M.

V. Scarani, H. Bechmann Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

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J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Ren, J. G.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Ren, R. J.

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Richardson, C.

L. Andrews, S. Vester, and C. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[Crossref]

Richardson, T. J.

T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[Crossref]

Rodenburg, B.

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

Sánchez-Soto, L. L.

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

Sarwade, N.

M. Lopes and N. Sarwade, “Optimized decoy state QKD for underwater free space communication,” Int. J. Quantum Inf. 16, 1850019 (2018).
[Crossref]

Scarani, V.

V. Scarani, H. Bechmann Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

Shapiro, J. H.

J. H. Shapiro, “Near-field turbulence effects on quantum-key distribution,” Phys. Rev. A 67, 022309 (2003).
[Crossref]

Shen, W. G.

Shen, Y.

Shi, P.

Shu, L.

G. Han, J. Jiang, N. Sun, and L. Shu, “Secure communication for underwater acoustic sensor networks,” IEEE Commun. Mag. 53, 54–60 (2015).
[Crossref]

Shu, R.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Simon, C.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref]

Sit, A.

Slepian, D.

Sun, N.

G. Han, J. Jiang, N. Sun, and L. Shu, “Secure communication for underwater acoustic sensor networks,” IEEE Commun. Mag. 53, 54–60 (2015).
[Crossref]

Sussman, B. J.

Tang, H.

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Uhlmann, J.

M. Lanzagorta and J. Uhlmann, “Assessing feasibility of secure quantum communications involving underwater assets,” IEEE J. Ocean. Eng. 99, 1–10 (2019).
[Crossref]

Urbanke, R. L.

T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[Crossref]

Uysal, M.

M. Elamassie, F. Miramirkhani, and M. Uysal, “Performance characterization of underwater visible light communication,” IEEE Trans. Commun. 67, 543–552 (2018).
[Crossref]

M. Elamassie, M. Uysal, Y. Baykal, M. Abdallah, and K. Qaraqe, “Effect of eddy diffusivity ratio on underwater optical scintillation index,” J. Opt. Soc. Am. A 34, 1969–1973 (2017).
[Crossref]

Vaziri, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

Vester, S.

L. Andrews, S. Vester, and C. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[Crossref]

Wang, J.-Y.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Wang, L.

T. Wu, X. Ji, H. Zhang, X. Li, L. Wang, and X. Fan, “Rytov variance of spherical wave and performance indicators of laser radar systems in oceanic turbulence,” Opt. Commun. 434, 36–43 (2019).
[Crossref]

Wang, N.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Wang, S.

Wang, Z.

Weihs, G.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref]

Weinfurter, H.

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref]

Wu, T.

T. Wu, X. Ji, H. Zhang, X. Li, L. Wang, and X. Fan, “Rytov variance of spherical wave and performance indicators of laser radar systems in oceanic turbulence,” Opt. Commun. 434, 36–43 (2019).
[Crossref]

Xiao, Y.

Xu, Y.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Yan, Z. Q.

Yang, A. L.

Yi, O.

M. Ibragimov, J. H. Lee, M. Kalyani, J. I. Namgung, S. H. Park, O. Yi, C. H. Kim, and Y. K. Lim, “CCM-UW security modes for low-band underwater acoustic sensor networks,” Wireless Pers. Commun. 89, 479–499 (2016).
[Crossref]

Yin, J.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Yin, Y.-Y.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Yu, Y.

Yuan, Z.

E. Diamanti, H. K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” NPJ Quantum Inf. 2, 16025 (2016).
[Crossref]

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N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Zeilinger, A.

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref]

Zeng, H.

Zhang, H.

T. Wu, X. Ji, H. Zhang, X. Li, L. Wang, and X. Fan, “Rytov variance of spherical wave and performance indicators of laser radar systems in oceanic turbulence,” Opt. Commun. 434, 36–43 (2019).
[Crossref]

Zhang, L.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Zhang, X.-X.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Zhang, Y.

Zhao, S.

Zhao, S. C.

Zhou, Y.-L.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Zhu, E. Y.

Zhu, Z.-C.

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Appl. Opt. (2)

IEEE Commun. Mag. (1)

G. Han, J. Jiang, N. Sun, and L. Shu, “Secure communication for underwater acoustic sensor networks,” IEEE Commun. Mag. 53, 54–60 (2015).
[Crossref]

IEEE Commun. Surveys Tuts. (1)

M. Erol-Kantarci, H. T. Mouftah, and S. Oktug, “A survey of architectures and localization techniques for underwater acoustic sensor networks,” IEEE Commun. Surveys Tuts. 13, 487–502 (2011).
[Crossref]

IEEE J. Ocean. Eng. (1)

M. Lanzagorta and J. Uhlmann, “Assessing feasibility of secure quantum communications involving underwater assets,” IEEE J. Ocean. Eng. 99, 1–10 (2019).
[Crossref]

IEEE Trans. Commun. (1)

M. Elamassie, F. Miramirkhani, and M. Uysal, “Performance characterization of underwater visible light communication,” IEEE Trans. Commun. 67, 543–552 (2018).
[Crossref]

IEEE Trans. Inf. Theory (1)

T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory 47, 599–618 (2001).
[Crossref]

Int. J. Quantum Inf. (1)

M. Lopes and N. Sarwade, “Optimized decoy state QKD for underwater free space communication,” Int. J. Quantum Inf. 16, 1850019 (2018).
[Crossref]

J. Field Robot. (1)

A. Caiti, V. Calabro, A. Munafo, G. Dini, and A. Lo Duca, “Mobile underwater sensor networks for protection and security: field experience at the UAN11 experiment,” J. Field Robot. 30, 237–253 (2013).
[Crossref]

J. Mod. Opt. (1)

L. Andrews, S. Vester, and C. Richardson, “Analytic expressions for the wave structure function based on a bump spectral model for refractive index fluctuations,” J. Mod. Opt. 40, 931–938 (1993).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

New J. Phys. (2)

S. Gröblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with qutrits,” New J. Phys. 8, 75 (2006).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

NPJ Quantum Inf. (1)

E. Diamanti, H. K. Lo, B. Qi, and Z. Yuan, “Practical challenges in quantum key distribution,” NPJ Quantum Inf. 2, 16025 (2016).
[Crossref]

Opt. Commun. (1)

T. Wu, X. Ji, H. Zhang, X. Li, L. Wang, and X. Fan, “Rytov variance of spherical wave and performance indicators of laser radar systems in oceanic turbulence,” Opt. Commun. 434, 36–43 (2019).
[Crossref]

Opt. Express (4)

Photon. Res. (1)

Phys. Rev. A (1)

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[Crossref]

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[Crossref]

T. Jennewein, C. Simon, G. Weihs, H. Weinfurter, and A. Zeilinger, “Quantum cryptography with entangled photons,” Phys. Rev. Lett. 84, 4729–4732 (2000).
[Crossref]

Quantum (1)

F. Bouchard, K. Heshami, D. England, R. Fickler, R. W. Boyd, B.-G. Englert, L. L. Sánchez-Soto, and E. Karimi, “Experimental investigation of high-dimensional quantum key distribution protocols with twisted photons,” Quantum 2, 111 (2018).
[Crossref]

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J. Martinez Mateo, D. E. Coronas, and V. Martn Ayuso, “Blind reconciliation,” Quantum Inf. Comput. 12, 791–812 (2012).

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V. Scarani, H. Bechmann Pasquinucci, N. J. Cerf, M. Dušek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Science (1)

J. Yin, Y. Cao, Y. H. Li, S. K. Liao, L. Zhang, J. G. Ren, W. Q. Cai, W. Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (2017).
[Crossref]

Wireless Pers. Commun. (1)

M. Ibragimov, J. H. Lee, M. Kalyani, J. I. Namgung, S. H. Park, O. Yi, C. H. Kim, and Y. K. Lim, “CCM-UW security modes for low-band underwater acoustic sensor networks,” Wireless Pers. Commun. 89, 479–499 (2016).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Underwater BB84 QKD system under consideration.
Fig. 2.
Fig. 2. Effect of turbulence strength for different water types at night time with a full Moon on (a) QBER and (b) SKR.
Fig. 3.
Fig. 3. Effect of atmospheric condition in clear ocean with strong turbulence on (a) QBER and (b) SKR.
Fig. 4.
Fig. 4. Effect of FOV in clear ocean with strong turbulence under different atmospheric conditions on (a) QBER and (b) SKR.
Fig. 5.
Fig. 5. Effect of aperture size in clear ocean with strong turbulence under different atmospheric conditions on (a) QBER and (b) SKR.

Tables (1)

Tables Icon

Table 1. System and Channel Parameters

Equations (21)

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

= { 0 i f 0 '' w a s c h o s e n + 90 i f 1 '' w a s c h o s e n ,
= { 45 i f 0 '' w a s c h o s e n + 45 i f 1 '' w a s c h o s e n .
n N = n B 2 + n D = I d c Δ t + 1 2 π R d A Δ t λ Δ λ ( 1 cos ( Ω ) ) 2 h p c l i g h t ,
R d ( λ , z d ) = R d ( λ , 0 ) exp ( k z d ) ,
Q B E R = P r ( e r r o r ) P r ( s i f t ) ,
Q B E R n N ( 1 μ + μ e η n S l ) n S l 2 μ e η n S l + 2 n N ( 1 μ + μ e η n S l ) ,
l = exp [ ς L ( d 2 θ L ) T ] ,
μ = 8 F π 0 1 e ( W ( d 1 x , L ) 2 ) ( arccos ( x ) x 1 x 2 ) J 1 ( 4 x F ) d x ,
W ( ρ , L ) = 8 π 2 k 2 L 0 1 0 [ 1 J 0 ( κ ζ ρ ) ] Φ ( κ ) κ d κ d ζ ,
Φ ( κ ) = 0.18 ( α t h 2 χ T ω 2 ) ( ε κ 11 ) 1 3 π [ 1 + g κ 2 3 ] ( ω 2 exp ( a κ 4 3 b κ 2 ) + d r exp ( c κ 4 3 d κ 2 ) ω ( d r + 1 ) exp ( e κ 4 3 f κ 2 ) ) .
W ( ρ , L ) = 1.44 π k 2 L α t h 2 χ T ε ( 1 / 3 ) 0 1 n = 1 ( 1 ) n 1 ( ρ ζ ) 2 n ( n ! ) 2 2 2 n d ζ 0 κ 2 n 8 3 exp ( a κ 4 3 b κ 2 ) d κ + 1.44 π k 2 L ( α t h 2 χ T ω 2 ) d r ε ( 1 / 3 ) 0 1 n = 1 ( 1 ) n 1 ( ρ ζ ) 2 n ( n ! ) 2 2 2 n d ζ 0 κ 2 n 8 3 exp ( c κ 4 3 d κ 2 ) d κ 1.44 π k 2 L ( α t h 2 χ T ω ) ( d r + 1 ) ε ( 1 / 3 ) 0 1 n = 1 ( 1 ) n 1 ( ρ ζ ) 2 n ( n ! ) 2 2 2 n d ζ 0 κ 2 n 8 3 exp ( e κ 4 3 f κ 2 ) d κ + 1.44 π k 2 L α t h 2 χ T ε ( 1 / 3 ) g 0 1 n = 1 ( 1 ) n 1 ( ρ ζ ) 2 n ( n ! ) 2 2 2 n d ζ 0 κ 2 n 2 exp ( a κ 4 3 b κ 2 ) d κ + 1.44 π k 2 L ( α t h 2 χ T ω 2 ) d r ε ( 1 / 3 ) g 0 1 n = 1 ( 1 ) n 1 ( ρ ζ ) 2 n ( n ! ) 2 2 2 n d ζ 0 κ 2 n 2 exp ( c κ 4 3 d κ 2 ) d κ 1.44 π k 2 L ( α t h 2 χ T ω ) ( d r + 1 ) ε ( 1 / 3 ) g 0 1 n = 1 ( 1 ) n 1 ( ρ ζ ) 2 n ( n ! ) 2 2 2 n d ζ 0 κ 2 n 2 exp ( e κ 4 3 f κ 2 ) d κ .
0 1 n = 1 ( 1 ) n 1 ( ζ ρ ) 2 n ( n ! ) 2 2 2 n 0 κ 2 n 8 3 exp ( a κ 4 3 b κ 2 ) d κ d ζ = 1 2 b 5 6 Γ ( 5 6 ) [ 1 2 F 2 ( 5 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] + 5 a 3 432 b 7 6 Γ ( 5 6 ) [ 1 2 F 2 ( 7 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] 1 2 a b 1 6 Γ ( 1 6 ) [ 1 2 F 2 ( 1 6 , 1 2 ; 1 , 3 2 , ρ 2 4 R ) ] 5 a 3 5184 a b 11 6 Γ ( 1 6 ) [ 1 2 F 2 ( 11 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] + 1 4 a 2 b 1 2 Γ ( 1 2 ) [ 1 2 F 2 ( 1 2 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] a 3 320 a 2 b 13 2 Γ ( 1 2 ) [ 1 2 F 2 ( 5 2 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] ,
0 1 n = 1 ( 1 ) n 1 ( ζ ρ ) 2 n ( n ! ) 2 2 2 n 0 κ 2 n 8 3 exp ( a κ 4 3 b κ 2 ) d κ d ζ 1 2 b 5 6 Γ ( 5 6 ) Γ ( 3 2 ) Γ ( 4 3 ) Γ ( 1 2 ) Γ ( 11 6 ) Γ ( 7 3 ) ( ρ 2 4 b ) 5 6 = 0.4194 ρ 5 3 .
0 1 n = 1 ( 1 ) n 1 ( ζ ρ ) 2 n ( n ! ) 2 2 2 n 0 κ 2 n 2 exp ( a κ 4 3 b κ 2 ) d κ d ζ = 1 2 b 1 2 Γ ( 1 2 ) [ 1 2 F 2 ( 1 2 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] + a 3 48 b 3 2 Γ ( 1 2 ) [ 1 2 F 2 ( 3 2 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] 1 2 a b 1 6 Γ ( 1 6 ) [ 1 2 F 2 ( 1 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] + 7 a 3 1728 a b 13 6 Γ ( 1 6 ) [ 1 2 F 2 ( 13 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] + 1 4 a 2 b 5 6 Γ ( 5 6 ) [ 1 2 F 2 ( 5 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] 11 a 3 1728 a 2 b 17 6 Γ ( 5 6 ) [ 1 2 F 2 ( 17 6 , 1 2 ; 1 , 3 2 , ρ 2 4 b ) ] .
0 1 n = 1 ( 1 ) n 1 ( ζ ρ ) 2 n ( n ! ) 2 2 2 n 0 κ 2 n 2 exp ( a κ 4 3 b κ 2 ) d κ d ζ 1 2 b 1 2 Γ ( 1 2 ) Γ ( 1 2 ) ( ρ 2 4 b ) 1 2 = 0.5 ρ .
W ( ρ , L ) = 1.44 π k 2 L ( α t h 2 χ T ω 2 ) ε 1 3 ( 1.175 η K 2 / 3 ρ + 0.419 ρ 5 3 ) × ( ω 2 + d r ω ( d r + 1 ) ) .
Q B E R n o n = 2 n N n S μ 0 l + 4 n N ,
Q B E R 2 n N μ n S l + 4 n N ,
R = 1 ( 1 + f ) h ( Q B E R ) ,
f = 1 R c h ( Q B E R t h ) ,
R 1 ( 1 + 1 R c h ( Q B E R t h ) ) × h ( n N ( 1 μ + μ e η n S l ) n S l 2 μ e η n S l + 2 n N ( 1 μ + μ e η n S l ) ) .