Multivariate optical computing using a digital micromirror device for fluorescence and Raman...

Multivariate optical computing using adigital micromirror device for

fluorescence and Raman spectroscopy

Zachary J. Smith,1 Sven Strombom,1 and SebastianWachsmann-Hogiu1,2,∗

1Center for Biophotonics Science and Technology, University of California, Davis, 2700Stockton Blvd. Suite 1400, Sacramento 95817, California, USA

2Department of Pathology and Laboratory Medicine, University of California, Davis, 4400 VStreet, Sacramento 95817, California, USA

∗[email protected]

Abstract: A multivariate optical computer has been constructed consistingof a spectrograph, digital micromirror device, and photomultiplier tube thatis capable of determining absolute concentrations of individual componentsof a multivariate spectral model. We present experimental results on ternarymixtures, showing accurate quantification of chemical concentrations basedon integrated intensities of fluorescence and Raman spectra measuredwith a single point detector. We additionally show in simulation that pointmeasurements based on principal component spectra retain the ability toclassify cancerous from noncancerous T cells.

© 2011 Optical Society of America

OCIS codes:(170.5660) Raman spectroscopy; (120.6200) Spectrometers and spectroscopicinstrumentation; (170.6280) Spectroscopy, fluorescence and luminescence.

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1. Introduction

Ramanand fluorescence spectroscopies are two tools that have long histories of being appliedto biological problems. Both Raman and fluorescence spectra have been used to discriminatebetween cancerous and noncancerous tissues [1–3]. The techniques have also been used exten-sively in applications ranging from bacterial identification [4, 5], hyperspectral imaging [6, 7],and optical tomography [8,9], to ultrasensitive analyte detection [10–12].

In many cases, the spectra themselves serve as stand-ins for more physical quantities of in-terest, such as concentration of a particular chemical or fluorescent probe. Even in cases whereexact concentrations of known analytes is not of interest, spectra are often decomposed intoa linear model using multivariate techniques such as principal components analysis (PCA) orvertex component analysis (VCA), and the concentrations of these multivariate components areused, for example, to descriminate between cells of a given type [13] or to false-color hyper-spectral images by presumed spectral constituency [14]. Even ordinary least squares analyseshave been shown to provide powerful insights into the chemical makeup of cells [15]. In anymultivariate technique, the original dataset contains many hundreds or thousands of individ-ual data points for each sample which are then compressed into a relatively smaller number ofhighly informative points representing contributions of components of an assumed linear modelto the original spectra.

In each case, the full spectral dataset can be seen not as an end in itself but as a windowto a different, more informative value obtained by treating the spectra as arising from a linearmodel of a relatively small number of components. In cases such as hyperspectral imaging andmultivariate analysis, there is a signal-to-noise price paid by dispersing the light across a CCDand recording a spectrum, rather than recording an integrated intensity value on a single pixel


(as would be done in a flow cytometer or some fluorescence imaging applications). This SNRpriceis particularly important in Raman spectroscopy where signal strength is often the limitingfactor in data acquisition speeds. In cases where the components of interest are knowna priori,either through careful measurement of several pure chemicals, or through prior decompositionof similar samples, this SNR penalty can be mitigated through the use of a multivariate opticalcomputer (MOC).

A multivariate optical computer is a device which, rather than recording spectra directly,performs dot products between spectra and arbitrary spectral functions, essentially recordingtheir projection onto a particular multivariate axis of interest. This is accomplished by passingthe light through an element whose transmission function represents the multivariate vector ofinterest (called the multivariate optical element), and measuring the total transmitted intensityon a point detector. Previous multivariate computers have primarily focused on utilizing in-tereference filters to form the multivariate optical elements [16, 17]. In these studies, havingthe multivariate optical element correctly model the spectral component of interest is a lim-iting factor [18], especially for sharp-featured curves typical of Raman spectra. Nevertheless,the principle of multivariate optical computing promises a significant signal to noise advantageover traditional detection of the full spectrum [19]. This can be understood as essentially aform of the well known Felgett’s advantage. This SNR advantage could then be parlayed intosignificant improvements in acquisition times, particularly for Raman spectroscopy.

Previous attempts at constructing flexible multivariate elements capable of studying sharp-featured spectra have utilized pixellated liquid-crystal-based spatial light modulators to varythe transmission of each wavelength in a dispersed spectrum [20, 21]. However, spatial lightmodulators have several important drawbacks, most important being the requirement that theyoperate on linearly polarized light. In this paper we present a multivariate optical computerbased upon a spectrograph and digital micromirror device (DMD) acting together as the multi-variate optical element. The combination of these two pieces of equipment results in an elementwhose spectral throughput can be programmed by changing the displayed pattern on the DMD.We use this device to quantify the concentration of fluorescent compounds dissolved in ethanolas well as mixtures of liquid samples by their Raman spectra. Finally, we simulate the perfor-mance of such a device in discriminating between cancerous and noncancerous T-cells usingthe multivariate optical computer to measure principle component scores.

2. Materials and methods

2.1. Mathematical principle of multivariate optical computing

Multivariate optical computing is related to the standard linear least squares modeling of spec-tra. Often times spectra from complex samples are assumed to be linear superpositions of asmall number of pure components, as shown in Eq. 1:

x = cP, (1)

wherex is an 1× k vector representing the spectrum from the sample,c is an 1× j vector ofconcentrations whileP is a j×k matrix of component spectra. Throughout this text we will uselower-case boldface fonts to denote vectors and uppercase boldface fonts to denote matrices.The least squares solution to the concentrations of each spectral component that make up thespectrumx is

c = xPT (PPT )−1, (2)

whereT represents the transpose operation. In our multivariate computer, the detection system,rather than measuring a full spectrum, represents the dot product between the spectrum and the


pattern displayed on the DMD, as shown in Eq. 3

m = xDT , (3)

whereD is ann×k matrix of 8-bit greyscale patterns, andm is a 1×n vector of measurements.One can trivially see from Eqs. 2 and 3 that lettingD equalP, the measurementsm can be usedas a direct measure of the concentration vectorc, the quantity of interest as follows,

c = m(PPT )−1. (4)

Alternatively, rather than projectingP onto the DMD, one could equivalently project the fullpseudoinverse,B, whereBT = PT (PPT )−1. Thus the measurementm would identically equalc.This has one important advantage: if one is only interested in one component of the model, theconcentration of componenti = ci, then one need only make a single measurement,mi = xBT

i ,whereBi is theith row of B. Contrast this with the case presented in Eq. 4 above, where onemeasurement must be made for each component of the model, regardless of whether they are ofinterest or not. The only downside to this technique is thatP can be positive definite, whileB isboth positive and negative. In order to measure the projection ofx ontoB, then, each row ofBmust be split into positive and negative components,B+ andB−. Thus, for each projection ontoB, two measurements must be made and combined. This will be discussed further in Section3.3.

2.2. Experimental system

The MOC system is shown schematically in Figure 1. Our system, aside from the detection

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Fig. 1: Schematic diagram of the MOC system. The excitation path is a light green line whiletheemission path is shown in dark green. Abbreviations as follows: BPF, band pass filter; DMD,digital micromirror device; L, lens; LBF, laser bandpass filter; PMT, photomultiplier tube.

arm, is a traditional Raman/fluorescence microscope. A laser illuminates a sample placed ona microscope stage, and the Stokes-shifted light is sent to the emission path by a filter cube


within the microscope. Different lasers were used for fluorescence and Raman studies. Thefluorescencestudies were performed using a 404 nm laser beam obtained by frequency doublinga 808 nm fundamental coming from a Ti:Sapphire system (Coherent Systems, Santa Clara, CA).Raman studies were performed using a 532 nm laser system (CrystaLaser, Reno, NV). In eachcase, the laser was bandpass filtered and directed into the microscope with steering mirrors. A3x telescope, composed of a 40 mm lens (L1) and a 120 mm lens (L2), expands the beam toa diameter of approximately 5 mm to more completely utilize the microscope objective’s fullaperture. In the focus of the telescope we place an optical chopper, which modulates the signalat 3 kHz for the purpose of lock-in detection of our emission signal. We used an Olympus IX-71inverted microscope equipped with a 10x objective with a numerical aperture of 0.3 (Olympus,Center Valley, PA) and beam cubes equipped with dichroic beamsplitters for 404 or 532 nmexcitation (Semrock, Rochester, NY). Residual light at the excitation wavelength was furtherattenuated by a second bandpass filter (passbands of 415 nm-477 nm for 404 nm excitation,and 540 nm-570 nm for 532 nm excitation, both from Semrock, Rochester, NY). The emissionlight is then focused into a 50 micron core optical fiber using a 40 mm focal length achromaticdoublet (L3) and delivered to our DMD-based detection system.

The detection system is based on a design reported by Quyenet al. [22]. Light is coupledby optical fiber into a SpectraPro 2150i imaging spectrograph (Princeton Instruments, Trenton,NJ). However, rather than placing a CCD in the image plane of the spectrograph we reimagethat plane onto the digital micromirror device (Discovery 3000, Texas Instruments, Dallas,TX), using a 75 mm achromatic doublet with a 2 inch clear aperture. Depending on whethera pixel of the DMD is in the “on” or “off” state, it reflects light either to a photomultipliertube (H7826, Hamamatsu Photonics, Hamamatsu City, Japan) or to a beam dump, respectively.The light from the DMD is focused onto the large point detector and into the aperture of thebeam dump using identical 35 mm focal length lenses (L5 and L6). The current output of thePMT is passed through a current amplifier (Model 428, Keithley, Cleveland, OH) and then toa lock-in amplifier (SR510, Stanford Research Systems, Sunnyvale, CA) that uses the opticalchopper signal as a reference. Although the measurements could be performed without thelock-in amplification step, it provides highly efficient background light and noise rejection anddemonstrates one benefit of using a single point detector as opposed to an array detector.

Both the DMD and lock-in are controlled by the computer using a home-built interface run-ning in LabVIEW (National Instruments, Austin, TX). Although the DMD can only displaybinary patterns, because the DMD can display approximately 16500 independent patterns persecond, by integrating signal over several patterns effective 8-bit greyscale patterns can be ob-tained at 60 frames/sec. We note that more recent generations of the DMD chip are capableof even faster operation. Accessing the highest refresh speeds possible for the DMD was ac-complished using a dedicated daughter board connected to the DMD (ALP-3, DLInnovations,Austin, TX) that stores patterns in on-board RAM. Data is then fed to the DMD through anFPGA configured to act as a high speed data link.

2.3. Data processing

For the spectra shown in Figure 2, the spectra were corrected by removing a constant bias fromeach spectrum and then smoothed using a Whittaker smoother [23]. The Whittaker smootherseeks to minimize a cost function composed of data fidelity (measured by sum of squarederrors between the fit and the data) and data roughness (measured by a sum of the squareddiscrete derivative of the fit vector). The relative weight of the roughness term compared to thedata fidelity term in the cost function is given by a Lagrange parameter. For this work, we usedLagrange parameters of 3 for Raman spectra, and 10 for fluorescence spectra. For representationof these curves on the digital micromirror device, the curves were normalized to their maximum


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Fig.2: (a) Fluorescence spectra of perylene (blue), coumarin 1 (red) and coumarin 30 (magenta)dissolved in ethanol, taken with the DMD operating in scanning monochromator mode. (b)Raman spectra of toluene (blue), tetrahydrofuran (red), and benzene (magenta), taken with theDMD operating in scanning monochromator mode.

value, and scaled to span a range of 0 to 255, and then compressed from doubles to 8-bitvariables. The API included with the DMD (DLInnovations, Austin, TX) provides LabVIEWfunctions to convert the 8-bit data to instructions for displaying these patterns through rapiddisplay of many binary patterns.

For the multicomponent experiments, the obtained voltage values were transformed into con-centration predictions by multiplying the measurements by the normalized and scaled matrixPPT as discussed in section 2.1, Eq. 4. The resultant vector of concentrations must then berescaled to account for the different normalization constants employed in generating the matrixP. If we refer to our originally measured spectral components as a matrixS, and a vector ofnormalization constantsn, thenP = n×S, where we use× to represent an element-wise mul-tiplication (i.e. each row ofS is multiplied by one element inn). Since the true concentrations


we are trying to obtain arectrue= xST (SST )−1, (5)

it follows that the concentrations resulting from the the operation described by Eq. 4 differ fromctrueby 1/n, such that

ctrue= n×c, (6)

where again× means that each element ofc is multiplied by the corresponding element inn.All processing was done using in-house scripts running in MATLAB (The MathWorks, Natick,MA).

3. Results and discussion

3.1. Fluorescence experiments

Spectra of three solutions of fluorophores (perylene, coumarin 1, and coumarin 30) dissolvedin ethanol are shown in Figure 2a, and were collected by turning individual columns of DMDpixels on in sequence, with each wavelength measurement consisting of an average of 10 volt-age readings from the lock-in amplifier set with a time constant of 1 second, current amplifierset with a gain of 1×106 and a control voltage on the PMT of 0.7. The spectra have been nor-malized and scaled to range between 0 and 255 for display on the DMD, as discussed above.The sharp falling edge at approximately 465 nm is due to the falling edge of the bandpass filterused to reject the laser line. Laser power on the sample during these measurements was 35microwatts.

Using these three spectra as the display patterns for the DMD, we used our multivariatecomputer to predict chemical concentrations of mixtures of these fluorophores in varying con-centrations. We first created stock solutions of each fluorophore and from these three solutionswe created two measurement solutions, one consisting only of coumarin 1 and coumarin 30(solution A), and one consisting only of perylene and coumarin 30 (solution B), where the mo-lar concentration of coumarin 30 is identical for each solution. For each measurement, 1 mLof solution A was pipetted out from the stock and placed in a parafilm-covered measurementchamber (A7816, Invitrogen, Carlsbad, CA), measured 5 times (placing and replacing the sam-ple between measurements to randomize any placement-depending signal variations), and thendiscarded. Following this, 1 mL of solution B is added to solution A, changing the concentra-tion of perylene and coumarin 1 within solution A, and the process was repeated for a totalof 9 samples. A final 10th sample was a measurement of 1 mL of solution B. The results areshown graphically in Figure 3. The x-axis in Figure 3 represents individual samples created us-ing the method described above, with the solid lines showing the nominal concentration of eachanalyte within each sample. Individual points (circles, stars and triangles) represent the meanpredicted values for each analyte within each sample, with the error bars being the standarddeviation across the 5 measurements. The dashed lines represent the estimated accuracy of thepipettes which limit the ultimate accuracy of our reference concentrations. Because of the hugeincrease in signal gained by putting photons from the entire spectrum onto a single detector,the laser had to be attenuated with an OD 2 filter to prevent saturation of the lock-in amplifier.Thus these measurements were conducted with only 350 nanowatts of power at the sample. Theresults clearly show the ability of the DMD-based MOC to accurately quantify concentrationsof fluorophores using only a small amount of power at the sample. This could significantly aidapplications where photobleaching due to high laser powers is an issue. We also anticipate thatthis system could be used in point-scanning hyperspectral fluorescence imaging to perform di-rect fluorescence unmixing and using the gain in SNR to reduce pixel dwell times and speed upoverall image acquisition.


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Fig. 3: Concentration predictions for a 3 component mixture of fluorophores, measured withanmultivariate optical computer. Individual points represent mean MOC measurements, solidlines represent the nominal concentration of each analyte within each sample, and dashed linesrepresent error due to accuracy of pipettes used to create samples.

3.2. Raman experiments

Similar to the fluorescence experiments, pure spectra of toluene, benzene, and tetrahydrofuranwere acquired first, shown in Figure 2b, using 40 mW of 532 nm light, using the same lock-inamplifier parameters as above. Multicomponent mixtures were also generated in an analogousmanner to fluorescence samples. However, due to the hazardous nature of the chemicals used inthis study, a sample chamber was constructed consisting of a coverslip cemented to a threadedbrass fitting into which a teflon-taped threaded stopper could be screwed to prevent evapora-tion of the sample during measurement. Two solutions were created, one with only toluene andtetrahydrofuran (solution A), and one with only tetrahydrofuran and benzene (solution B), withthe concentration of tetrahydrofuran kept constant across the two solutions. 1 mL of solutionA was placed within the sample chamber and measured identically to the fluorescence exper-iments. After measurement, 100 microliters of solution B was added to the 1 mL of solutionA already in the sample chamber. Due to the fact that the sample chamber can only hold ap-proximately 1.6 mL of liquid, samples 1 through 5 were obtained by progressively adding B toA, while samples 6 through 10 were obtained by progressively adding A to B. The results areshown graphically in Figure 4, with similar labeling as in Figure 3. Once again we attenuatedthe laser by a factor of 100 for these measurements, resulting in a power of 400 microwatts onthe sample. The system is clearly able to accurately quantify the concentration of each chemicalin the mixture. The accuracy of the concentration predictions is slightly lower here than in thefluorescence experiments, but we believe that this is primarily due to the fact that the laser weused for this experiment has a slow variation in output power of approximately±5%, whichdirectly affects the accuracy of the predictions.


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Fig. 4: Concentration predictions for a 3 component mixture of Raman scatterers, measuredwith an multivariate optical computer. Individual points represent mean MOC measurements,solid lines represent the nominal concentration of each analyte within each sample, and dashedlines represent error due to accuracy of pipettes used to create samples.

3.3. Simulation of T-cell sorting

In the above sections we created samples composed of known concentrations of pure compo-nents, and the patterns projected on the DMD represented physical spectra of these compo-nents. However, the technique is considerably more general. It is often the case that althoughthe exact chemical constituents of a sample may not be knowna priori, the spectra from manysamples can be analyzedpost hoc to determine meaningful regression vectors through partialleast squares, principal components analysis, or other multivariate techniques. In this simula-tion we take Raman spectra from 90 individual T lymphocytes, evenly split between normaland neoplastic cells. The details of this dataset and its acquisition can be found in [24]. A repre-sentative spectrum from a single cell is shown in blue in the top left panel of Figure 5. The fullRaman dataset is submitted to principal components analysis. As described in [24], cancerousand noncancerous cells can be separated based on the first and second principal componentscores. In the bottom panel of Figure 5, we plot each cell as a point in a two dimensional spacewhose coordinates are given by their first two principal component scores, showing the separa-tion between cancerous and noncancerous cells. Magenta stars and green circles represent theresults of the principal components decomposition using the full spectra as reported in [24],with the stars representing normal cells and circles representing Jurkat cells (an immortalizedneoplastic T-cell line). Notice that with very few exceptions, the normal and neoplastic cellscluster into distinct regions of the principal component space.

If instead of measuring the spectra themselves we use the multivariate optical computer to re-turn the scores directly, we can make faster measurements less sensitive to noise. The principalcomponents model can be thought of as a linear model representing a mean centered spectraldataset. In other words,

xmc = cP, (7)

where nowP are the principal components of the dataset, andxmc = x− x, wherex is one meanspectrum of a dataset. If, as in Section 2.1, we letBT = PT (PPT )−1, the least squares solution


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Fig. 5: Simulation of measuring principal component scores using a multivariate optical com-puter. Upper left: a representative raw spectrum measured by a traditional CCD-based spec-trometer (blue) and that spectrum corrupted with 10 times larger Poisson noise (red). Upperright: the first two rows of matrixB, calculated from the dataset in [24]. Lower panel: Starsrepresent measurements on normal cells, while circles represent measurements on Jurkat T-cells. Magenta and green points are scores computed in software using full spectra. Red andblue points are simulated results obtained by projecting spectra onto the first two rows of thepseudoinverse matrix using data that would correspond to the red curve in the upper left panelif measured using a traditional spectrometer.

for this situation can be written as

c = xmcBT = xBT − xBT . (8)

As discussed above in Section 2.1, if we projectB onto the DMD, then the measurements willbe

m = xBT = c+ xBT . (9)

Assuming we have made many full-spectra measurements on equivalent samples in the pastin order to generate our principal component model in the first place, we have access to theterm xBT and can computec, the principal component scores, for any component of interest.However,B is in general both positive and negative. In order to represent this, we must split eachcomponent of theB matrix into its positive components,B+, by setting all negative values tozero, and its negative components,B−, by setting its positive values to zero. Thus, to determinethe concentration of theith principal component,ci, for each sample, we first measurem+

i ,the projection ofx ontoB+

i , whereBi is theith row of B. Next, we measurem−i = xB−T

i . Wethen construct the “true” measurement,mi = m+

i −m−i . By then subtracting off the computed

quantity xBTi we have directly measured the score of principal componenti for a particular

sample with a high signal to noise.To illustrate the robustness of this method to noise, we simulated a very weak spectrum en-

tering our spectrometer. If we assume that the dominant noise source is shot noise, the spectrum


that would be detected on a CCD, given the simulated number of photons per wavelength chan-nel, is shown as the red curve in the upper left panel of Figure 5. Despite the fact that fewphotons are falling on each wavelength channel, the total number of photons across all wave-length channels is still relatively large. Therefore, our measurements will have a better signal tonoise, since they integrate a large fraction of the incident photons into a single bucket. A carefulcomparison of the signal to noise ratios in each case is provided in Appendix A. We simulatedprincipal component values calculated using our optical computer based on Eq. 9 and project-ing the noisy data onto the first two rows ofB, shown in the upper right panel of Figure 5. Thecalculated principal component scores are shown in the bottom panel of Figure 5 as red starsand blue circles. In the limit of no noise, the measurements from the CCD and optical computerwould lie on top of each other. The added noise moves the optically computed values awayfrom their true locations. However, as is clear from the figure, despite decreasing the numberof photons by 100 times compared to the original data, with a corresponding 10-fold increasein shot noise, the estimated principal component scores are still accurate enough to permit dis-crimination between cancerous and noncancerous cells. This analysis also does not include thepotential of lock-in amplification to improve the signal to noise ratio even further, as discussedin Appendix A. However, even in this case, the separation between the normal and neoplasticcells within the principal component space is clearly maintained. Thus, we expect such a sys-tem could be used to substantially speed up Raman-activated cell sorting or high-throughputscreening.

4. Conclusions

We have reported on the construction and validation of a multivariate optical computer usinga digital micromirror device and spectrometer as the multivariate optical element. By project-ing grayscale patterns onto the DMD, we were able to record dot products between samplespectra and the displayed pattern that were used to quantify chemical concentrations withinternary mixtures. Although previous instruments have been constructed using multilayer stacksand liquid-crystal-based spatial light modulators, the DMD has several key advantages. Multi-layer stacks offer a compact and rugged design, but suffer from being difficult to manufactureand being completely inflexible once constructed. With respect to spatial light modulator-basedMOCs, because the DMD is increasingly used within consumer electronics, the cost of a DMDchip is approximately 2 orders of magnitude less expensive than a spatial light modulator whilemaintaining higher throughput due to the polarization and wavelength insensitivity of the de-vice.

We further explored the possibility of separating normal from neoplastic cells based on prin-cipal component scores measured directly using the MOC. Despite reducing the number of pho-tons measured by 100, the direct measurements of the component scores do not differ greatlyfrom those computed from measurements of full spectra. Neoplastic and normal cells still clus-ter into distinct groups within the principal component space. Since our mixture experimentsindicated that there is an increase in signal of approximately 100 times using the projected spec-tra versus a single pixel of a spectrum, and given the simulation results presented in Figure 5,we expect that our MOC system could determine if an individual cell was normal or neoplasticapproximately 100 times faster than a traditional dispersive-based system (with the caveat thatthe relevant principal components and mean spectrum must be knowna priori).

Given our results, we can also provide an order-of-magnitude estimate of detection limits forour current configuration based on the results obtained thus far. For the fluorescence experi-ments we were easily able to quantify micromolar concentrations of fluorophores in solution.Given that we attenuated our laser beam by 2 orders of magnitude, used a moderate objectiveNA of 0.3, and a control voltage on the PMT of 0.7, we estimate that there is likely 4 to 5


orders of magnitude improvement possible in our signal strength, giving us at least nanomolarsensitivity to fluorophores. Given a diffraction limited focal volume of a few femtoliters, thiscorresponds to nearly the single molecule limit. For Raman scattering the results are similar. Welikely have 4 to 5 orders of magnitude of signal improvement, which would give us micromolarsensitivity, depending on the cross sections of the molecules of interest.

Beyond considerations of signal-to-noise, in a case where signal strength is not of primaryconcern, it is worth noting that the speed of the single point detectors used in this device permitdetection much more rapidly than an equivalent CCD-based system. As CCDs are typicallylimited to a few kilohertz frame rates, while as discussed in Appendix A, the PMT used herecan detect signals as fast as 600 MHz.

Extending this work to measurements of biological samples, including sorting and hyper-spectral imaging applications, are directions our group is actively pursuing.

A. Appendix: Signal to noise of multivariate optical computing measurements

Although signal to noise of a measurement is highly system dependent, we can make an orderof magnitude estimate of the improvement in signal to noise of the constructed system versusother configurations or versus a traditional CCD-based spectrometer. Because the describedsystem sees its greatest advantage when signal strengths are low, consider a signal where 10photons are incident on a single wavelength element (pixel) of a CCD in a time of 0.1 seconds.Assuming the CCD has a quantum efficiency of 90%, the signal will produce 9 photo-electronswith a variance due to shot noise of 3. The photo-electrons are digitized by an A/D converterwith a gain of 2 and a read noise of approximately 5 counts, giving a total signal of 18 countswith a noise variance of 8. Each pixel of the recorded spectrum (and thus the spectrum as awhole) would have a SNR of approximately 1.6.

Now consider a case where a multivariate optical computing system is described above, but,as in [22], we simply integrate the output of the single point detector over a period of 0.1seconds. Quyenet al. accomplish this with an integrating PMT, where the electrons from theanode are stored at a capacitor for a specified integration time, and then the voltage across thecapacitor is read. In this case, assuming 1024 pixels on the DMD, each receiving on average 10photons per 0.1 seconds, with a reflectivity of the DMD of 80% and a pattern that on averagereflects 50% of the photons incident on the DMD, the total number of photons incident onthe photomultiplier tube would be 4096. Our detector has a peak quantum efficiency of 21%meaning that the total number of photoelectrons produced is 860, with a variance of 29 dueto shot noise. The photomultiplier tube has a dark current of 3 nA, corresponding to 1.8×1010 electrons per second at the anode. Assuming the dominant source of noise in the PMT isthermionic emission of electrons at the cathode, and given that our detector has a gain of 0.6million, the rate of emission of dark electrons at the cathode is 30 kHz. Thus, in 0.1 seconds,3000 photoelectrons will be emitted with a variance of 54. Assuming the voltage reading andsubsequent A/D conversion in the PMT is noiseless, the total signal will be 860 counts with avariance of 83 due to Poisson noise, giving an SNR of 10, or nearly a 10 times improvementcompared to using a CCD, with the obvious caveat that full spectral information is lost.

We next consider a case where rather than use an integrating PMT, we take the current outputof the PMT and send it through a lock-in amplifier. In this case we once again have 4096 photonsincident on the PMT, of which 21% are detected, leading to 860 photoelectrons with a varianceof 29. After a gain of 0.6 million this produces a signal current of .86 nA with a variance of.029 nA. Once again assuming that the 3nA of noise is due to a 30 kHz dark electron emissionrate, the variance on the dark current is 0.054 nA, to give a signal of 0.86 nA with a variance of0.083 nA. This is then sent through a current amplifier with a noise density of 1 nV/

√Hz with

anamplification of 1 MV/A, giving a signal of 860 microvolts with a noise of 83 microvolts.


The detector has a rise time of 1.5 nanoseconds, giving it a maximum response frequency of666MHz. Assuming the noise on our signal is white, this means that the power spectral densityof the shot noise is 3.2 nV/

√Hz with an additional 1 nV/

√Hz from the amplifier. If we send the

signalinto a lock-in amplifier, we can reject a large fraction of this noise by placing a narrowbandwidth filter on the signal around the reference frequency. WIth a 0.1 second time constantset on the lock-in amplifier, the width of the lock-in’s filter is 2.5 Hz. Therefore, the total noisevoltage leaking through the lock-in is 6.6 nV. The lock-in itself has about 11 nV of noise inthis bandwidth, giving a total signal of 860 microvolts, with a noise of 17.6 nV, for an SNRof about 48000. The lock-in amplifier, by reducing the noise by a factor of close to 5000, hasdramatically improved the signal to noise ratio of the measurement, enabling extremely precisemeasurements and thus higher accuracy when predicting concentrations of components of thespectral model.

Acknowledgements

This work was funded by NSF award IDBR 0852891. Part of this work was also funded by theCenter for Biophotonics Science and Technology, a designated NSF Science and TechnologyCenter managed by the University of California, Davis, under Cooperative Agreement No.PHY0120999. We gratefully acknowledge Agilent Technologies, Inc. for their loan of the DMDchip, and we thank Gerry Owen and Chris Coleman of Agilent for helpful discussions. We alsothank James W. Chan for sharing the T-cell data utilized in this paper and Kaiqin Chu for helpfuldiscussions.


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