Abstract

A group of quantum dots can be designed to have a unique spectral emission by varying the size of the quantum dots (wavelength) and number of quantum dots (intensity). This technique has been previously proposed for biological tags and object identification. The potential of this system lies in the ability to have a large number of distinguishable wavelengths and intensity levels. This paper presents a communications system model for MxQDs including the interference between neighbouring QD colours and detector noise. An analytical model of the signal-to-noise ratio of a Charge-Coupled Device (CCD) spectrometer is presented and confirmed with experimental results. We then apply a communications system perspective and propose data detection algorithms that increase the readability of the quantum dots tags. It is demonstrated that multiplexed quantum dot barcodes can be read with 99.7% accuracy using the proposed data detection algorithms in a system with 6 colours and 6 intensity values resulting in 46,655 unique spectral codes.

© 2012 OSA

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W. R. Algar, M. Massey, and U. J. Krull, “The application of quantum dots, gold nanoparticles and molecular switches to optical nucleic-acid diagnostics,” Trends Anal. Chem. 28, 292–306 (2009).
[CrossRef]

2008 (1)

S. Chang, K. Yu, and J. Liu, “Advanced secure information retrieval technology for multiplayer information extraction,” J. Nanomater. 2008 (2008).
[CrossRef]

2007 (1)

S. V. Vaidya, M. L. Gilchrist, C. Maldarelli, and A. Couzis, “Spectral bar coding of polystyrene microbeads usign multicolored quantum dots,” Anal. Chem. 798520–8530 (2007).
[CrossRef] [PubMed]

2006 (1)

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

2001 (2)

M. Han, X. Ming, J. Z. Su, and S. Nie, “Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules,” Nat. Biotechnol. 19, 631–635 (2001).
[CrossRef] [PubMed]

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. K. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63205316, (2001).
[CrossRef]

2000 (1)

M. Konu, D. P. Fromm, H. F. Hamann, A. Gallagher, and D. J. Nesbitt, “Nonexponential “blinking” kinetics of single CdSe quantum dots: a universal power law behaviour,” J. Chem. Phys. 1123117–3120, (2000).
[CrossRef]

1998 (3)

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

W. Chan and S. Nie, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef] [PubMed]

O. I. Mi?i?, K. M. Jones, A. Cahill, and A. J. Nozik, “Optical, electronic, and structural properties of uncoupled and close-packed arrays of InP quantum dots,” J. Phys. Chem. B 1029791–9796, 1998.
[CrossRef]

Algar, W. R.

W. R. Algar, M. Massey, and U. J. Krull, “The application of quantum dots, gold nanoparticles and molecular switches to optical nucleic-acid diagnostics,” Trends Anal. Chem. 28, 292–306 (2009).
[CrossRef]

Alivisatos, A. P.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Asquith, S.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Bawendi, M. G.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. K. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63205316, (2001).
[CrossRef]

Bruchez, M.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Cahill, A.

O. I. Mi?i?, K. M. Jones, A. Cahill, and A. J. Nozik, “Optical, electronic, and structural properties of uncoupled and close-packed arrays of InP quantum dots,” J. Phys. Chem. B 1029791–9796, 1998.
[CrossRef]

Chan, W.

W. Chan and S. Nie, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef] [PubMed]

Chang, S.

S. Chang, K. Yu, and J. Liu, “Advanced secure information retrieval technology for multiplayer information extraction,” J. Nanomater. 2008 (2008).
[CrossRef]

S. Chang, M. Zhou, and C. P. Grover, “Information coding and retrieving using fluorescent semiconductor nanocrystals for object identification,” Opt. Express 12, 143–148 (2004).
[CrossRef] [PubMed]

Chen, F. F.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Chu, J. S. F.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Cooke, P.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Couzis, A.

S. V. Vaidya, M. L. Gilchrist, C. Maldarelli, and A. Couzis, “Spectral bar coding of polystyrene microbeads usign multicolored quantum dots,” Anal. Chem. 798520–8530 (2007).
[CrossRef] [PubMed]

de Feo, G.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Doctolero, M.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Eastman, P. S.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Empedocles, S. A.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. K. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63205316, (2001).
[CrossRef]

Fromm, D. P.

M. Konu, D. P. Fromm, H. F. Hamann, A. Gallagher, and D. J. Nesbitt, “Nonexponential “blinking” kinetics of single CdSe quantum dots: a universal power law behaviour,” J. Chem. Phys. 1123117–3120, (2000).
[CrossRef]

Gallagher, A.

M. Konu, D. P. Fromm, H. F. Hamann, A. Gallagher, and D. J. Nesbitt, “Nonexponential “blinking” kinetics of single CdSe quantum dots: a universal power law behaviour,” J. Chem. Phys. 1123117–3120, (2000).
[CrossRef]

Gaponenko, S. V.

S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University Press, 1998).
[CrossRef]

Gilchrist, M. L.

S. V. Vaidya, M. L. Gilchrist, C. Maldarelli, and A. Couzis, “Spectral bar coding of polystyrene microbeads usign multicolored quantum dots,” Anal. Chem. 798520–8530 (2007).
[CrossRef] [PubMed]

Gin, P.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Gray, J. W.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Grover, C. P.

Hamann, H. F.

M. Konu, D. P. Fromm, H. F. Hamann, A. Gallagher, and D. J. Nesbitt, “Nonexponential “blinking” kinetics of single CdSe quantum dots: a universal power law behaviour,” J. Chem. Phys. 1123117–3120, (2000).
[CrossRef]

Han, M.

M. Han, X. Ming, J. Z. Su, and S. Nie, “Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules,” Nat. Biotechnol. 19, 631–635 (2001).
[CrossRef] [PubMed]

Haykin, S.

S. Haykin, Communication Systems 4th edition (John Wiley & Sons, Inc., 2001).

Hecht, E.

E. Hecht, Optics 4th edition (Pearson Education, Inc., 2002).

Janesick, J. R.

J. R. Janesick, Scientific Charge-Coupled Devices (SPIE Press, 2001).
[CrossRef]

Jones, K. M.

O. I. Mi?i?, K. M. Jones, A. Cahill, and A. J. Nozik, “Optical, electronic, and structural properties of uncoupled and close-packed arrays of InP quantum dots,” J. Phys. Chem. B 1029791–9796, 1998.
[CrossRef]

Konu, M.

M. Konu, D. P. Fromm, H. F. Hamann, A. Gallagher, and D. J. Nesbitt, “Nonexponential “blinking” kinetics of single CdSe quantum dots: a universal power law behaviour,” J. Chem. Phys. 1123117–3120, (2000).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics 7th edition (John Wiley & Sons, Inc., 1993).

Krull, U. J.

W. R. Algar, M. Massey, and U. J. Krull, “The application of quantum dots, gold nanoparticles and molecular switches to optical nucleic-acid diagnostics,” Trends Anal. Chem. 28, 292–306 (2009).
[CrossRef]

Lai, E. H.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Leatherdale, C. A.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. K. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63205316, (2001).
[CrossRef]

Li, S.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Liu, J.

S. Chang, K. Yu, and J. Liu, “Advanced secure information retrieval technology for multiplayer information extraction,” J. Nanomater. 2008 (2008).
[CrossRef]

Mahoney, W.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Maldarelli, C.

S. V. Vaidya, M. L. Gilchrist, C. Maldarelli, and A. Couzis, “Spectral bar coding of polystyrene microbeads usign multicolored quantum dots,” Anal. Chem. 798520–8530 (2007).
[CrossRef] [PubMed]

Massey, M.

W. R. Algar, M. Massey, and U. J. Krull, “The application of quantum dots, gold nanoparticles and molecular switches to optical nucleic-acid diagnostics,” Trends Anal. Chem. 28, 292–306 (2009).
[CrossRef]

Micic, O. I.

O. I. Mi?i?, K. M. Jones, A. Cahill, and A. J. Nozik, “Optical, electronic, and structural properties of uncoupled and close-packed arrays of InP quantum dots,” J. Phys. Chem. B 1029791–9796, 1998.
[CrossRef]

Ming, X.

M. Han, X. Ming, J. Z. Su, and S. Nie, “Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules,” Nat. Biotechnol. 19, 631–635 (2001).
[CrossRef] [PubMed]

Moronne, M.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Nesbitt, D. J.

M. Konu, D. P. Fromm, H. F. Hamann, A. Gallagher, and D. J. Nesbitt, “Nonexponential “blinking” kinetics of single CdSe quantum dots: a universal power law behaviour,” J. Chem. Phys. 1123117–3120, (2000).
[CrossRef]

Neuhauser, R. G.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. K. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63205316, (2001).
[CrossRef]

Nie, S.

M. Han, X. Ming, J. Z. Su, and S. Nie, “Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules,” Nat. Biotechnol. 19, 631–635 (2001).
[CrossRef] [PubMed]

W. Chan and S. Nie, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science 281, 2016–2018 (1998).
[CrossRef] [PubMed]

Nozik, A. J.

O. I. Mi?i?, K. M. Jones, A. Cahill, and A. J. Nozik, “Optical, electronic, and structural properties of uncoupled and close-packed arrays of InP quantum dots,” J. Phys. Chem. B 1029791–9796, 1998.
[CrossRef]

Nuttall, R.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

O’Brien, E.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Park, J. S.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Proakis, J. G.

J. G. Proakis and M. Salehi, Fundamentals of Communication Systems (Pearson Prentice Hall, 2005).

Ruan, W.

P. S. Eastman, W. Ruan, M. Doctolero, R. Nuttall, G. de Feo, J. S. Park, J. S. F. Chu, P. Cooke, J. W. Gray, S. Li, and F. F. Chen, “Qdot nanobarcodes for multiplexed gene expression analysis,” Nano Lett. 6, 1059–1064 (2006).
[CrossRef] [PubMed]

Salehi, M.

J. G. Proakis and M. Salehi, Fundamentals of Communication Systems (Pearson Prentice Hall, 2005).

Sedra, A. S.

A. S. Sedra and K. C. Smith, Microelectronic Circuits (Oxford University Press, 1998).

Sha, M. Y.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Shimizu, K. T.

K. T. Shimizu, R. G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W. K. Woo, and M. G. Bawendi, “Blinking statistics in single semiconductor nanocrystal quantum dots,” Phys. Rev. B 63205316, (2001).
[CrossRef]

Smith, K. C.

A. S. Sedra and K. C. Smith, Microelectronic Circuits (Oxford University Press, 1998).

Spurr, N. K.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Stubbins, M.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Su, J. Z.

M. Han, X. Ming, J. Z. Su, and S. Nie, “Quantum-dot-tagged microbeads for multiplexed optical coding of biomolecules,” Nat. Biotechnol. 19, 631–635 (2001).
[CrossRef] [PubMed]

Treadway, J. A.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Truong, A.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Uphoff, J.

H. Xu, M. Y. Sha, E. Y. Wong, J. Uphoff, Y. Xu, J. A. Treadway, A. Truong, E. O’Brien, S. Asquith, M. Stubbins, N. K. Spurr, E. H. Lai, and W. Mahoney, “Multiplexed SNP genotyping using the Qbead ™system: a quantum dot-encoded microsphere-based assay,” Nucleic Acids Res. 31, e43 (2003).
[CrossRef] [PubMed]

Vaidya, S. V.

S. V. Vaidya, M. L. Gilchrist, C. Maldarelli, and A. Couzis, “Spectral bar coding of polystyrene microbeads usign multicolored quantum dots,” Anal. Chem. 798520–8530 (2007).
[CrossRef] [PubMed]

Weiss, S.

M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science 281, 2013–2016 (1998).
[CrossRef] [PubMed]

Wong, E. Y.

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Figures (7)

Fig. 1
Fig. 1

An example of the flow of the signal through a QD barcode system.

Fig. 2
Fig. 2

Communications systems model.

Fig. 3
Fig. 3

A barcode system for multiplexed quantum dots.

Fig. 4
Fig. 4

Experimental Signal-to-Noise Ratio and Fitted Model.

Fig. 5
Fig. 5

Performance of all 4 data detection algorithms as a function of Signal-to-Noise Ratio (SNR) for a system with 6 colours and 6 intensity levels.

Fig. 6
Fig. 6

Performance of all 4 data detection algorithms as a function of the number of quantum dot colours, N, for a system with six intensity levels, L = 6 and a signal-to-noise ratio of 15dB.

Fig. 7
Fig. 7

Performance of all 4 data detection algorithms as a function of the number of intensity levels, L, for a system with six QD colours, N = 6, and a signal-to-noise ratio of 15dB.

Tables (1)

Tables Icon

Table 1 Estimating the optical power incident on the CCD pixel

Equations (10)

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w ( λ ) = ( π σ Q D 2 ) 1 / 4 exp [ λ 2 / ( 2 σ Q D 2 ) ]
y [ i ] = A I i + z [ i ] + n [ i ]
P i ( λ ) = α G , i ( λ ) α g α f T 2 P f
α f = r f 2 2 r d 2
A = log 10 ( P inc P T ) = ε ρ l
A Q D = ε ρ l ρ V c N A = ε N A
α A = 1 10 A Q D V D N A
E { N i ( λ ) } = η P i ( λ ) λ h c T
σ T 2 = 4 k T ° B R out ( G S ) 2
S N R = E { Y i ( p ) } 2 σ Y 2 = ( G q e C η P i ( λ p ) λ p h c T ) 2 G ( q e C ) 2 ( η P i ( λ p ) λ h c + σ T 2 ) = ( η P i ( λ p ) λ p h c ) 2 T 2 η P i ( λ p ) λ p h c T + σ T 2

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