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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 5 — Feb. 27, 2012
  • pp: 5762–5774
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Data detection algorithms for multiplexed quantum dot encoding

Kelly C. Goss, Geoff G. Messier, and Mike E. Potter  »View Author Affiliations


Optics Express, Vol. 20, Issue 5, pp. 5762-5774 (2012)
http://dx.doi.org/10.1364/OE.20.005762


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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

1. Introduction

Quantum dots (QDs) are semiconductor nanocrystals that have a broad optical absorption spectrum but a very narrow emission spectrum due to quantum confinement [1

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

]. The wavelength of the emission can be tuned by changing the composition and the size of the quantum dot. This means that quantum dots of the same material but varying sizes can be excited by the same monochromatic source but emit at different wavelengths. QDs have been used as a tagging agent in biomedical imaging [2

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

] and diagnostics [3

3. 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]

] as an alternative to traditional fluorescent dyes because of their narrow, symmetric and bright emission spectra, broad absorption spectra and resistivity to photobleaching [2

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

, 4

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

].

To increase the number of objects that can be given a unique QD marker or tag, Han et al. proposed multiplexed optical encoding of quantum dots [5

5. 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]

]. A ‘microbead’ acts as a host structure that contains designed quantities of QDs with different emission wavelengths. This creates unique emission spectra varying in colour and intensity. These microbeads, with different emission spectra, were used to tag biological samples.

The use of QD tagging for object identification will increase only if the technology can continue to increase the number of unique tags that can be fabricated and reliably identified. The established technology for object identification is the 1D barcode which can encode 12 decimal digits having 1012 unique codes. For chemical and biological multiplexed analysis, the leading technology uses conventional fluorescent dyes and their current product has 500 unique codes [6

6. Multiplexed Solutions For Life - Luminex Corporation, http://www.luminexcorp.com.

]. Within QD multiplexing for DNA analysis, the goal is to achieve a million unique codes to allow for the analysis of human genes [7

7. 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]

]. This target will only be achieved by a concerted effort both on the QD fabrication field and on the techniques used to identify the tags after deployment.

The work that has followed [5

5. 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]

] has primarily focused on the fabrication of multiplexed QD (MxQD) systems and their use in specific applications. While successfully creating a variety of QD host structures is clearly an important factor in determining the number of tags that can be used in a particular application, equally important is the detection method used to distinguish the tags. The study of these detection methods is much less mature. Basic detection schemes have focused on utilizing microscope systems with digital cameras [5

5. 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]

] and flow cytometry [8

8. 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]

] along with intensity detection software that needs to be manually trained by an operator [8

8. 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]

, 9

9. BD Biosciences, http://www.bdbiosciences.com.

]. However, the number of tags that can be reliably detected using these methods is on the order of 10.

The contribution of this paper is to design and analyze a multiplexed QD detection scheme that is based on digital communications fundamentals. The impairments experienced by a multiplexed QD system include interference between the emission spectra of different quantum dots and noise from both optical and electrical sources. This mirrors the inter-symbol interference (ISI) and additive noise experienced by conventional digital communications systems. Data detection algorithms for MxQD systems have been proposed to overcome the interference between spectra [10

10. 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]

] and others have explored methods to mitigate the effects of noise [7

7. 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]

, 11

11. 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]

]. This paper is the first to present data detection algorithms that mitigate both noise and ISI between wavelengths. Specifically, it demonstrate that the use of equalization and filtering communications techniques can significantly increase the number of QD tags that can be distinguished.

A secondary contribution of this paper is the first detailed signal and noise analysis of a MxQD detection system based on a charge coupled device (CCD) detector. Experimental and analytical results are presented that quantify the signal level and amount of noise in a CCD MxQD detector.

To date, the research that has focused on improving MxQD detection has not dealt with noise and the ISI that occurs between wavelengths in a unified manner. Chang et al. propose a deconvolution algorithm to overcome the ISI but do not consider the effect of noise [10

10. 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]

]. Eastman et al. [7

7. 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]

] recognize the effects of noise and attempt to overcome it by ensuring that QD micro-beads have the same total fluorescent intensity, thereby avoiding low intensity beads that may be overcome by noise. However, no attempt to quantify or mitigate the noise process itself is proposed.

In the following Section 2, a MxQD system is described and the communications system model is presented. In Section 3 the filtering and equalization algorithms from communications system are proposed for the use of detecting MxQD tags. We then demonstrate the use of the system model and data detection algorithms in a case study of a MxQD object identification system. The case study includes a model of the expected optical signal power and detection noise power that is experimentally verified. Simulations are then performed to demonstrate the effectiveness of the data detection algorithms in Section 5.

2. System description

A MxQD system creates unique tags by designing groups of quantum dots that emit unique intensity levels at different wavelengths. This is described in Section 2.1. A key to the unified treatment of MxQD noise and ISI in this paper is the visualization of a MxQD optical signal as a time-domain digital communications signal. This abstraction is presented in Section 2.2.

2.1. Physical system

A MxQD tag is designed to have a unique spectral fluorescence by mixing different quantities of QDs that are designed to emit at different wavelengths, λi where i = 1 ...N. As a result, the tag will emit different intensities at different wavelengths. A finite number of intensity levels, L, will be used at each wavelength. As a result, the total number of possible unique tags would be LN – 1.

Ideally, all of the quantum dots would emit at exactly their designed wavelengths such that an example MxQD emission spectrum would look like Fig. 1(a) with no overlap between emissions at different λi.

Fig. 1 An example of the flow of the signal through a QD barcode system.

In practice the emission from QDs designed to emit at a particular wavelength, λi, will have a distribution that is approximately Gaussian as a function of λ with the distribution centered at λi [5

5. 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]

]. Figure 1(b) illustrates the summation of Gaussian distributions for 5 different colour QDs.

The linear superposition of the Gaussian distributions of QDs with different emission wavelengths is only possible if there is no energy transfer between quantum dots and no reabsorption. It has been shown that MxQD microbeads for biological applications can be fabricated such that the quantum dots are sufficiently separated to avoid energy transfer [5

5. 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]

]. For quantum dots that are evaporated on a surface, as in our case study presented in Section 4, energy transfer between quantum dots can occur. Introducing a buffering molecule to ensure physical separation of the quantum dots can mitigate this [12

12. O. I. Mićić, 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]

]. Additionally, quantum dots emitting different colours could be deposited in different piles on the card ensuring sufficient separation. In regards to re-absorption, while it has been observed to result in small changes in intensity values [11

11. 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]

], for quantum dots evaporated on surfaces with very small path lengths re-absorption is negligible [12

12. O. I. Mićić, 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]

]. For this paper, we move forward with the assumption that the total emission spectrum from the set of QDs is a linear superposition of the individual emission spectra of each colour of QDs.

As illustrated in Fig. 1(b), if the width of the Gaussian pulse is on the order of the separation between wavelengths, interference will occur between the QDs emitting at different wavelengths. This is illustrated in Fig. 1(b) and will be referred to in this paper as inter-symbol interference (ISI).

This interference will distort the intensities of the QD emissions and make it more difficult to distinguish the identity of the MxQD tag. The purpose of this paper is, in part, to present techniques to mitigate this interference.

A second source of distortion when identifying MxQD tags is the noise present in the optical intensity measurements collected by the tag reader. This noise originates both from thermal noise within the receiver as well as from optical sources. In the subsection that follows, this paper will present a communications systems abstraction for a MxQD system that captures the effects of interference between Gaussian distributed QD emission spectra and additive system noise.

2.2. Communications model

In order to capture the effects of both ISI and additive noise, the system can be represented as shown in Fig. 2. The system is divided into tag creation, the channel and tag reading.

Fig. 2 Communications systems model.

In tag creation, a series of Dirac delta impulses are created that are centered at λ1 ...λN. The area of δ(λλi) is given by Ii, which is proportional to the desired intensity for that QD group. This is an abstraction since, as noted in Section 2.1, it is not possible to fabricate QDs with this kind of ideal emission spectrum.

The channel in the model captures the effect of ISI, optical attenuation and additive system noise. The ideal impulses are first convolved with a Gaussian pulse with unit energy
w(λ)=(πσQD2)1/4exp[λ2/(2σQD2)]
(1)
which produces the MxQD emission spectrum that would be observed with a noise-free receiver. The unit energy Gaussian pulse is used to model the channel for simplifications within the communications model and corresponds to a multiplication factor that is accounted for when the MxQD is fabricated. Note that σQD2 controls the width of the QD emission and is proportional to the Full Width Half Maximum λ3dB), σQD2=λ3dB2(8ln(1/2))1.

After convolution with the Gaussian pulse, a factor 0 < A < 1 is applied to the signal to represent system attenuation and amplifier gain. A zero mean, additive noise source is then applied to represent noise from both optical and electronic sources.

It can be noted that the signal at the output of the channel block is analogous to a standard time domain pulse amplitude modulation (PAM) signal that is subject to additive noise [13

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

]. Therefore, the approach taken in this paper is to apply the same noise and ISI mitigation techniques to the MxQD spectrum in the wavelength domain that would be applied to a standard communications signal in the time domain. To the authors’ knowledge, this is the first time this approach has been used to improve the performance of a MxQD tag reader.

It is well established that the matched filter maximizes signal to noise ratio (SNR) in a communications system subject to additive white noise [13

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

]. Therefore, a filter with a response equal to Eq. (1) makes up the first stage of the MxQD read block in Fig. 2. Assuming that the wavelength values λ1 ...λN are uniformly spaced with separation Δλ, the output of matched filter is sampled at intervals Δλ. The ith sample at the output of the matched filter is equal to
y[i]=AIi+z[i]+n[i]
(2)
where z[i] is the ISI at sample i and n[i] is the additive noise component. The samples y[i] are then fed into a data detection block that attempts to mitigate the ISI, z[i], and produce a best guess as to the identity of the tag. This data detection scheme is discussed in more detail in Section 3.

3. Data detection

A MxQD tag can be represented by the vector I = [I1 I2 ... IN], where Ii is defined in Section 2.2 as being proportional to the desired intensity at wavelength λi. The purpose of the data detection block in Fig. 2 is to generate a best guess as to which vector is being represented by the MxQD emission spectrum and will be denoted by Î. In this paper, the performance of a data detection algorithm will be evaluated based on the probability of a correct read, P(Î = I).

The most basic form of data detection is to apply thresholds to the intensities observed at the different QD wavelengths, similar to [8

8. 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]

, 9

9. BD Biosciences, http://www.bdbiosciences.com.

]. Ideally, a detailed understanding of the ISI and noise processes corrupting the intensity levels would be used to calculate the threshold locations that minimize the overall probability of selecting the incorrect intensity level. In practice, the threshold levels are set manually [8

8. 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]

, 9

9. BD Biosciences, http://www.bdbiosciences.com.

]. Regardless of the method used to select the thresholds, it is important to note that the thresholds themselves do not reduce the amount of ISI or noise corrupting the signal.

Another approach to mitigating ISI is to utilize an equalizer on the samples y[i] at the output of the matched filter in Fig. 2. The equalizer is implemented as a discrete time finite impulse response filter. The nth tap in the equalizer filter is represented by g[n]. The samples at the output of the matched filter, y[n], are convolved with g[n] in order to mitigate the ISI present in y[n]. The output of the equalizer is then applied to a threshold detector where the thresholds are placed at the mid-point between the L possible transmitted intensity levels.

While the authors in [10

10. 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]

] do not account for additive system noise or use a matched filter, their deconvolution algorithm is equivalent to what is known as a zero forcing (ZF) equalizer. For the MxQD channel, the values of g[n] for a ZF equalizer solution can be determined by setting the discrete Fourier transform (DFT) of g[n] equal to the inverse of the DFT of Eq. (1) [13

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

]. This ensures that the convolution of g[n] and w[n] equals a discrete time delta function.

As mentioned in Section 1, this paper is the first to consider the effect of additive noise on MxQD system performance. The primary disadvantage of the ZF equalizer is that it has the potential to enhance noise. This occurs when the equalizer attempts to invert a portion of the channel with a deep null. Therefore, the ZF equalizer is typically not used in practice.

A more common equalization method that attempts to balance ISI mitigation with noise enhancement is the minimum mean square error (MMSE) algorithm. The algorithm will minimize the mean squared error between I and Î = g[n]⊛y[n], where ⊛ indicates convolution. The equalizer taps that achieve this minimization can be calculated using the Weiner-Hopf equations [14

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

], g = R1p, where g = [g[1

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

] g[2

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

] ...] is the vector of tap values, R is the autocorrelation matrix of the matched filter output y[n] and the vector p, is the cross-correlation vector of the output Î and the desired output I. The cross-correlation vector includes information about the noise in the system and the resulting equalizer relaxes the zero ISI requirement to account for noise.

While equalizers are practical algorithms that can be implemented with reasonable complexity, maximum likelihood sequence detection (MLS) is the best possible detection method for a sequence of symbols that are subject to ISI [14

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

]. For this algorithm, the N samples of y[n] that correspond to the desired wavelengths are grouped into a vector y ∈ 𝒭N. For all possible I vectors, perfect knowledge of the Gaussian emission spectrum w(λ) and the channel attenuation A are used to create the LN – 1 possible points in 𝒭N that could appear at the matched filter output. The MLS algorithm then simply chooses the point that is the minimum Euclidean distance from y as the most likely transmitted vector.

4. Case study - quantum dot barcodes

In order to determine the maximum number of MxQD tags that can be distinguished in a practical system, it is important to realistically evaluate the impact of both ISI and system noise. The effects of the QD Gaussian emission spectrum that causes the ISI have been considered [7

7. 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]

, 10

10. 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]

] and algorithms to overcome them proposed. In contrast, to the authors’ knowledge, there has not been a detailed analytical or experimental evaluation of the additive system noise present in a MxQD system.

The contribution of this section is to present a QD barcode case study that analytically and empirically determines the amount of additive system noise present in a CCD-based reader. The noise levels are presented in terms of Signal-to-Noise Ratio (SNR), the ratio of the expected signal power and noise power, at the input to the matched filter. These SNR values will be incorporated into the system model described in Section 2 to determine a final estimate of the number of MxQD tags that can be distinguished in a realistic system.

In Section 4.1, the analytical model is presented and validated with an experimental prototype described in Section 4.2.

4.1. Barcode system model

Similar to the work by Chang et al., we consider a MxQD barcode application where the dots are deposited on an identification (ID) card [10

10. 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]

]. This system is shown in Fig. 3. It differs somewhat from Chang in that the QDs are illuminated with a free space laser. Once illuminated, the fluorescence of the QDs are read using a charge-coupled device (CCD) spectrometer, with a fiber aperture, which samples the optical intensity of the QD emissions as a function of wavelength.

Fig. 3 A barcode system for multiplexed quantum dots.

A detailed analytical model of the desired optical signal power is presented in Section 4.1.1. Section 4.1.2 describes the noise sources incorporated into our model and the resulting SNR model.

4.1.1. Desired optical signal power model

In this section, we present an analytical estimate of the MxQD signal power as a function of the input laser power. The desired signal in this system is the QD fluorescence. As mentioned in Section 2.1, we are assuming the QDs are sufficiently separated to avoid energy transfer and a short optical path to mitigate re-absorption. Let i represent the QDs designed to emit at λi. The proportion of optical power from these QDs measured at wavelength λ is denoted Pi(λ) and is equal to
Pi(λ)=αG,i(λ)αgαfT2Pf
(3)
where αG,i(λ) accounts for the Gaussian emission spectrum of the QDs, αg is grating loss, αf accounts for the amount of light coupled onto the CCD input fiber, T is the power transmittance of the air/fiber interface and Pf is the fluorescent power of the quantum dots.

The CCD within the spectrometer samples the QD emission spectrum as a function of wavelength. As noted in Section 2, QD emissions as a function of wavelength are Gaussian distributed. Therefore, αG,i(λ) in Eq. (3) is simply the integral of this Gaussian shape over λ ± δλ, where 2δλ is the wavelength resolution of the CCD device.

To determine αf in Eq. (3), the light emitted from the QD pile is modeled as isotropic. Since the QDs rest on an opaque surface, the light will be radiating in a half sphere with surface area 2πrd2 where rd is the radius away from the QD barcode. Assuming the fiber is a sufficient distance from the QDs to be normal to the light, there is no need to account for the angle of incidence. Therefore, αf is just the area of the fiber relative to the area of the half sphere such that
αf=rf22rd2
(4)
where rf is the radius of the fiber core.

The transmittance factor T in Eq. (3) is the proportion of light transmitted through the air-fiber interface of the fiber connected to the spectrometer. Since the light is assumed to be normal to the interface, the transmittance is determined using Fresnel transmission coefficients [15

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

]. Note that T is squared in Eq. (3) to also account for the second air-fiber interface within the instrument.

Once the light leaves the fiber within the spectrometer, the αg factor in Eq. (3) accounts for the optical loss within the instrument. With no access to the optics within the spectrometer, we assume a typical setup consisting of two mirrors, with negligible loss, and an optical grating to physically separate light as a function of wavelength [16

16. Ocean Optics - Inventors of the World’s First Miniature Spectrometer, http://www.oceanoptics.com.

]. The optical power loss due to the grating is αg.

To determine Pf in Eq. (3), we can start by using αB to denote the fraction of the laser power, Plaser, incident on the ID card that illuminates the QDs. This is equal to the ratio of the QD pile area to laser beam area. The fraction of the incident power absorbed by the QDs is denoted αA. The quantum dots will then fluoresce a fraction of the absorbed light which is described by the quantum efficiency, ηQD. Therefore, the quantum dot fluorescence power is Pf = ηQDαAαBPlaser.

To estimate, αA, we can use Beer-Lambert’s law that states that the absorbance of light through a solution has a linear relationship with the extinction coefficient, ε, molar concentration, ρ, and length traveled by the light through the solution, l. The absorbance is the logarithmic ratio of incident light Pinc to transmitted light PT
A=log10(PincPT)=ερl
(5)
The amount of absorbed light is Pabs = PincPT = αAPinc; therefore, Eq. (5) can be rearranged to show that αA = 1 −10A.

To determine the value of A used to calculate αA, it is assumed that QD absorbance on the ID card and in solution is the same. To determine the absorbance per quantum dot in solution, the absorbance in a 1cm × 1cm cuvette is calculated by setting l = 1 cm. The number of quantum dots in the cuvette is calculated by multiplying the QD concentration by the volume through which light travels and Avogadro’s number, NA. The volume of solution, Vc, which the laser path travels through is the cross-section of the cuvette multiplied by the laser beam width of 1cm resulting in a total volume of Vc = 1cm3 = 1mL. The absorbance per quantum dot, AQD, is then the absorbance from Eq. (5) divided by the number of quantum dots in the cuvette
AQD=ερlρVcNA=εNA
The number of dots deposited on the ID card is approximated as NQD = ρVDNA, where VD is the volume deposited, so that
αA=110AQDVDNA
(6)

4.1.2. Signal to noise ratio

One of the contributions of this paper is the first model and experimental verification of the SNR of the detection system for MxQDs. In this section we present a model for the shot and thermal noise resulting from the detection process.

In a CCD, the incident photons are converted to electron-hole pairs that are separated by an applied electrostatic field [17

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

]. The charges build up within the pixel over the integration time, T. At the end of the integration time, the individual charge packets are moved along the array of pixels. At the end of the array, the charges are placed on a charge sensing capacitor to produce an output voltage. This voltage is then amplified and digitized.

SNR in this section will be defined at the output of this amplifier that would be at the input to the matched filter in Fig. 2. To begin, it is necessary to determine the mean and variance of the number of electrons gathered by the pixel at wavelength λ due to the QDs designed to emit at wavelength λi. The expected number of electrons can be written as
E{Ni(λ)}=ηPi(λ)λhcT
(7)
where η is the quantum efficiency of the CCD pixel, and Pi(λ)λT/hc is the number of photons incident on the CCD pixel. The term Pi(λ) is defined in Eq. (3) and λ/hc is the energy of a photon where h is Planck’s constant and c is the speed of light. To determine the variance in the number of electrons, σe,i2(λ), it is noted that the arrival of photons, and therefore the generation of electrons in the pixel, will follow a Poisson distribution [18

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

]. As a result, σe,i2(λ)=E{Ni(λ)}.

Once the electrons are captured by the CCD pixel, they are converted to a voltage by being applied to a sense capacitor, C. The voltage across the capacitor is then amplified by a MOSFET amplifier [17

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

] with a gain of G. The amplifier also contributes additive thermal noise to the spectrometer output value. Let Yi(p) represent the voltage at the amplifier output for the pixel centered at wavelength λp due to the QDs designed to emit at λi. This voltage will be a function both of both Ni(λp) and the number of electrons added by the amplifier, NT, such that Yi(p)=Gqe(Ni(λp)+NT)/C where qe is electron charge. Assuming NT is zero mean with variance σT2, E{Yi(p)}=GqeE{Ni(λp)}/C and the variance of Yi(p) is σY2=G(qe/C)2(E{Ni(λp)}+σT2) due to the assumption of Poisson distributed photon arrivals.

The variance of the thermal noise from the amplifiers is calculated as [17

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

]:
σT2=4kT°BRout(GS)2
(8)
where k is Boltzmann’s constant, is the temperature of the device, B is the bandwidth, Rout is the output resistance of the amplifier, G is the amplifier gain and S is the sensitivity of the amplifier in units of volts per electron [V/e].

The resulting SNR for the spectrometer output is the squared mean over the variance
SNR=E{Yi(p)}2σY2=(GqeCηPi(λp)λphcT)2G(qeC)2(ηPi(λp)λhc+σT2)=(ηPi(λp)λphc)2T2ηPi(λp)λphcT+σT2
(9)

This first SNR model for MxQD systems includes the noise sources resulting from the detection system. Once other noise sources are characterized, such as re-absorption causing potential fluctuations in intensity values, these could be additional variances added to the denominator of Eq. (9). The communications model continues to be an accurate representation of the system and the MMSE and MLS data detection algorithms can be used to overcome this additional noise.

4.2. SNR experiments

In this section, experiments are utilized to verify the SNR model of the CCD spectrometer detection system developed in Section 4.1. An experimental setup similar to the prototype in [10

10. 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]

] is used.

The signal-to-noise ratio of the experimental data is calculated for the one pixel with the maximum quantum dot emission intensity. It is assumed that the mean of the 500 intensity measurements is proportional to E {Yi(p)} and that the variance of the measurements is proportional to σY2. Therefore, the experimental SNR is equal to the square of the mean of the intensity measurements divided by the variance of the measurements. These experimental values are shown in Fig. 4 versus integration time, T.

Fig. 4 Experimental Signal-to-Noise Ratio and Fitted Model.

To determine analytical SNR values to compare with these empirical SNR values, Eq. (9) is used. The values used to estimate Pi(λp), according to Eq. (3), are shown in Table 1. We anticipate the optical power loss to be −102.7 dB resulting in Pi(λ = 568 nm) = 1.08 × 10−12 W.

Table 1. Estimating the optical power incident on the CCD pixel

table-icon
View This Table

The other two unknown values in Eq. (9) are η and σT2. These values are varied to determine the best-fit to the empirical SNR measurements and provide factors that influence the numerator as well as both terms in the denominator. The best-fit algorithm minimizes the mean squared error between Eq. (9) and the data points in Fig. 4. This results in the values η = 0.02 and σT2=1.0×104 which, as shown in Fig. 4, result in an agreement between the theoretical and experimental SNR values with a difference of less than 1 dB. This close match between the model and experimental results demonstrates that our model accurately predicts the trend of the SNR of a CCD based MxQD detection system.

As a further verification of the model, it is useful to determine whether or not η = 0.02 and σT2=1.0×104 are reasonable values. Regarding η, we would expect a value closer to 0.1 for the relatively inexpensive spectrometer used. This amounts to an error of 7 dB which is likely due to residual uncertainty in the estimation of over 100dB of optical power loss. Part of this uncertainty is the ‘blinking’ affect observed in quantum dots [20

20. 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]

, 21

21. 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]

] that has not been considered in the model. Once characterized, the reduced signal intensity resulting from blinking quantum dots, αB, could be added as an additional source of lost optical power in Eq. (3).

When verifying the value of σT2, it is necessary to estimate the values in Eq. (8). The specification sheet for the CCD array provides two bandwidths and we assume the largest bandwidth of B = 1 × 106 Hz. The temperature of the device is approximated at room temperature, = 293K. The output impedance is approximated at 2 kΩ [22

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

]. The remaining two unknowns are more difficult to estimate as there is a large range of possible values. For scientific CCDs, the range of sensitivity values range from 1 to 20μV /e. To match the predicted model of σT2=1.0×104 the gain would have to be in the range of 12 to 25dB. This range of reasonable gain and sensitivity values that result in a matched σT2 further confirms that the model and experimental data match.

5. Data detection simulations

With a SNR model that has been empirically verified in Section 4.2 and a communications systems model for an MxQD system provided in Section 2.2 that accounts for both noise and ISI, it is now possible to realistically estimate the number of distinguishable MxQD tags.

Simulations are used to determine the probability of a correct read, P(I = Î). A series of MxQD tags are created with intensities at each of the N wavelengths randomly selected from the L possible levels. Noise and ISI is applied to each tag, as described in Section 2.2, and then a data detection algorithm, as described in Section 3, is used to determine the tag intensity levels. The number of correct reads and misreads are recorded and the simulation is run until 500 misreads are counted to ensure a statistically significant estimate of P(I = Î). The simulations cover a range QD emission wavelengths from 490 nm to 620 nm.

When different QD colours emit at different intensities, they are received at different SNRs. Therefore, SNR in these simulations is defined as the average received SNR for all L intensity levels at the input to the matched filter, where desired signal energy is defined using the intensity of light emitted over the entire Gaussian pulse, not just at the desired wavelength.

While the fabrication of QDs with narrow emission spectra has been demonstrated [24

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

], in this paper the variance of the Gaussian pulse is set such that the Full Width Half Maximum (FWHM) is 20 nm to match the QDs that are currently commercially available.

To observe the effect of variable SNR, Fig. 5 shows P(I = Î) plotted versus SNR. The number of colours is set to N = 6 and L = 6 intensity levels to match what Han predicted to be possible when MxQD was first proposed in [5

5. 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]

]. The SNR values are incremented starting at 5dB until the performance reaches an acceptable level. The simulations indicate that a system could accurately decoded the MxQD 99.7% of the time with a SNR = 15dB. For the barcode system implemented experimentally in this work, this level of SNR would require an integration time of less than 10ms which is more than adequate.

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.

When comparing the different detection algorithms in Fig. 5, a sufficient accurate read rate is achieved when the equalizers or MLS detectors are used. With the traditional threshold detector and SNR = 15dB, the MxQD tags are only accurately read 11% of the time.

In order to determine the feasible data density for this system, it is important to observe the effect of varying both the number of colours and intensity levels for a realistic SNR value. We use the set of system parameters, (SNR = 15 dB, N = 6, L = 6), as a starting point and then independently increase the number of QD colours, N, and the number of intensity values, L, to observe the effects on accurate reads. Figure 6 shows P(I = Î) plotted versus increasing number of QD colours and Fig. 7 for increasing number of intensity levels.

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 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.

These simulations indicate that increasing the number of intensity levels has less effect on the readability than increasing the number of QD colours used. As a consequence, the number of MxQD tags can easily be increased by increasing the number of intensity levels as the proposed detection algorithms continue to be effective. Since the range of the QD emissions is fixed, as the number of QD colours increases the inter-symbol interference increases. Therefore the MLS detection algorithms and MMSE equalizers are better able to overcome the increase in interference in the presence of noise.

6. Conclusion

In this paper, we propose data detection algorithms for MxQD systems that are known to overcome the inter-symbol interference and additive noise. A case study is performed on a CCD-based detection system for a MxQD barcode technology. Analytical models are developed for the expected signal and noise power and verified with a reasonable prototype. With knowledge of the SNR, we simulate all four data detection algorithms and demonstrate that the proposed detection algorithms provided significant increases in the ability to accurately identify MxQD tags. We confirm that the proposed detection algorithms enable a MxQD barcode system to store and read 6 QD colours and 6 intensity levels as predicted by [5

5. 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]

] resulting in 46,655 unique spectral codes given that a SNR = 15 dB is achieved.

Acknowledgments

The authors thank Dr. David Cramb from the Department of Chemistry at the University of Calgary for generously supplying us with samples of quantum dots and discussions on calculating the absorbance factor. The authors also thank the Natural Sciences and Engineering Research Council of Canada and Alberta Ingenuity Foundation for their financial support.

References and links

1.

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

2.

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

3.

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]

4.

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

5.

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]

6.

Multiplexed Solutions For Life - Luminex Corporation, http://www.luminexcorp.com.

7.

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]

8.

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]

9.

BD Biosciences, http://www.bdbiosciences.com.

10.

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]

11.

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]

12.

O. I. Mićić, 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]

13.

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

14.

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

15.

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

16.

Ocean Optics - Inventors of the World’s First Miniature Spectrometer, http://www.oceanoptics.com.

17.

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

18.

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

19.

Invitrogen by Life Technologies, http://www.invitrogen.com.

20.

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]

21.

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]

22.

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

23.

Evident Technologies, http://www.evidenttech.com.

24.

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

OCIS Codes
(040.1520) Detectors : CCD, charge-coupled device
(070.4790) Fourier optics and signal processing : Spectrum analysis
(160.2540) Materials : Fluorescent and luminescent materials

ToC Category:
Detectors

History
Original Manuscript: September 23, 2011
Revised Manuscript: February 8, 2012
Manuscript Accepted: February 16, 2012
Published: February 24, 2012

Virtual Issues
Vol. 7, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Kelly C. Goss, Geoff G. Messier, and Mike E. Potter, "Data detection algorithms for multiplexed quantum dot encoding," Opt. Express 20, 5762-5774 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5762


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References

  1. S. V. Gaponenko, Optical Properties of Semiconductor Nanocrystals (Cambridge University Press, 1998). [CrossRef]
  2. M. Bruchez, M. Moronne, P. Gin, S. Weiss, and A. P. Alivisatos, “Semiconductor nanocrystals as fluorescent biological labels,” Science281, 2013–2016 (1998). [CrossRef] [PubMed]
  3. 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]
  4. W. Chan and S. Nie, “Quantum dot bioconjugates for ultrasensitive nonisotopic detection,” Science281, 2016–2018 (1998). [CrossRef] [PubMed]
  5. 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]
  6. Multiplexed Solutions For Life - Luminex Corporation, http://www.luminexcorp.com .
  7. 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]
  8. 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]
  9. BD Biosciences, http://www.bdbiosciences.com .
  10. S. Chang, M. Zhou, and C. P. Grover, “Information coding and retrieving using fluorescent semiconductor nanocrystals for object identification,” Opt. Express12, 143–148 (2004). [CrossRef] [PubMed]
  11. 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]
  12. O. I. Mićić, 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. B1029791–9796, 1998. [CrossRef]
  13. J. G. Proakis and M. Salehi, Fundamentals of Communication Systems (Pearson Prentice Hall, 2005).
  14. S. Haykin, Communication Systems 4th edition (John Wiley & Sons, Inc., 2001).
  15. E. Hecht, Optics 4th edition (Pearson Education, Inc., 2002).
  16. Ocean Optics - Inventors of the World’s First Miniature Spectrometer, http://www.oceanoptics.com .
  17. J. R. Janesick, Scientific Charge-Coupled Devices (SPIE Press, 2001). [CrossRef]
  18. E. Kreyszig, Advanced Engineering Mathematics 7th edition (John Wiley & Sons, Inc., 1993).
  19. Invitrogen by Life Technologies, http://www.invitrogen.com .
  20. 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]
  21. 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. B63205316, (2001). [CrossRef]
  22. A. S. Sedra and K. C. Smith, Microelectronic Circuits (Oxford University Press, 1998).
  23. Evident Technologies, http://www.evidenttech.com .
  24. S. Chang, K. Yu, and J. Liu, “Advanced secure information retrieval technology for multiplayer information extraction,” J. Nanomater.2008 (2008). [CrossRef]

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