Anal. Chem. 1996, 68, 1081-1088
Grating Light Reflection Spectroscopy for Determination of Bulk Refractive Index and Absorbance Brian B. Anderson, Anatol M. Brodsky, and Lloyd W. Burgess*
Department of Chemistry, University of Washington, Seattle, Washington 98195-1700
An optical sensing technique is described and evaluated for sensitivity to changes in refractive index and absorbance of model sample matrices. A binary dielectric/ metal transmission diffraction grating is placed in contact with a sample and utilized in reflection mode; thus, the light captured and analyzed does not pass through the sample. This particular condition creates thresholds at which a particular transmitted diffraction order is transformed from a traveling wave to an evanescent one. The positions of these thresholds depend upon the complex dielectric function of the sample, the period of the grating, and the wavelength and incident angle of light striking the grating. Experimental evidence directly supports the theoretical predictions regarding responses to both the real and imaginary portions of the refractive index: the reflection coefficient derivative wavelength peak position shifts linearly with changes in the real part of the refractive index, and the derivative peak amplitudes exhibit a square-root dependence on absorbance. Refractive index sensitivity to a series of ethanol/water solutions is demonstrated with detectable changes in index as small as 2 × 10-6. Absorbance sensitivity is shown via the differentiation of methylene blue samples having equivalent 1 cm path length absorbances between 0.459 and 244 AU. In a single reflection measurement, GLRS offers a large dynamic range for absorbance detection, allows simultaneous determination of bulk refractive index in optically dense media, and provides a platform for performing continuous process analysis.
The application of analytical techniques in chemistry has long relied on spectroscopic tools and measurements to obtain chemical information from an analyte matrix. The interaction of light with a sample medium is complex and information rich, but extracting the information can be difficult, especially in matrices that are heterogeneous or optically dense and thus not amenable to standard transmission measurements. One approach to this problem involves the use of reflective techniques originating from and including attenuated total reflection (ATR) spectroscopy, which has been utilized throughout the last half-century. The descendants of ATR use substrates with modified surfaces that rely on ATR or ATR-type phenomena to obtain spectroscopic information from a matrix; these techniques include surface plasmon resonance spectroscopy (SPRS) and surface-enhanced Raman spectroscopy (SERS). For example, 0003-2700/96/0368-1081$12.00/0
© 1996 American Chemical Society
surface plasmon generation on a silver grating with a polymer overlayer1 has been employed in gas sensing. However, because plasmon resonance techniques require the use of carefully controlled thicknesses and morphology of surface layers, they are restricted to the surfaces of a few specific metals (gold, silver, copper, etc.). The exceptional surface sensitivity associated with SPRS and SERS, and to some extent ATR itself, renders them difficult to employ in analytical applications where the determination of analyte concentration is precluded by the presence of surface-active interferrents. A number of recent advances in the area of optical chemical analysis have been based on the employment of high-quality diffraction gratings with periodicities on the order of the wavelength range of visible light. One of the first applications of such gratings in analytical technology utilized gratings as waveguide couplers in thin-film planar waveguide technology. For example, Tiefenthaler and Lukosz2-5 have demonstrated the feasibility of using integrated gratings embossed onto metal oxide thin-film waveguides as both differential refractometers and surfacesensitive immunosensors. Kuhn and Burgess6 have used a grating-tapped thin-film ion-diffused planar waveguide as an ATR element where the sensing mechanism included an interaction of the evanescent field with the cover solution. Along with their use as coupling elements, optical chemical sensors have employed gratings in the absence of a waveguiding layer. Following an old idea first proposed by Rytov,7 Sainov et al.8-12 utilized a binary metal diffraction grating, working in total internal reflection mode, as a sensor for determination of refractive index and absorbance in liquid samples. We describe an optical technique for analytical chemical sensing, termed grating light reflection spectroscopy (GLRS), based on the measurement of light intensity and phase reflected from a transmission diffraction grating which is in contact with a sample. This technique shares similarities with other reflectionbased techniques, such as SERS, SPRS, and ATR, in that incident (1) Jory, M. J.; Vukusic, P. S.; Sambles, J. R. Sens. Actuators B 1994, 17, 203209. (2) Lukosz, W.; Tiefenthaler K. IEE Conf. Pub. 1983, 227, 152. (3) Lukosz, W.; Tiefenthaler, K. Sens. Actuators 1988, 15 (3), 273-284. (4) Lukosz, W.; Tiefenthaler, K. Sens. Actuators 1988, 15 (3), 286-295. (5) Tiefenthaler, K.; Lukosz, W. J. Opt. Soc. Am. B 1989, 6 (2), 209-220. (6) Kuhn, K. J.; Burgess, L. W. Anal. Chem. 1993, 65 (10), 1390-1398. (7) Rytov, S. M.; Fabelynskii, I. L. Zh. Eksp. Teor. Fiz. USSR 1950, 20, 340 (in Russian). (8) Sainov, S.; Tonchtev, D. Opt. Lasers Eng. 1989, 10, 17-26. (9) Sainov, S.; Chernov, B.; Dushkina, N. Opt. Lasers Eng. 1993, 18, 297305. (10) Sainov, S. Appl. Opt. 1992, 31 (31), 6589-6591. (11) Sainov, S. Sens. Actuators A 1994, 45 (1), 1-6. (12) Dushkina, N.; Sainov, S. J. Mod. Opt. 1992, 39 (1), 173-187.
Analytical Chemistry, Vol. 68, No. 7, April 1, 1996 1081
light is reflected off of an interface in contact with the analytical matrix. ATR and SPRS rely exclusively on the attenuation of incident light, and thus no optical energy is redistributed into other scattering channels, where it might be detected optically. SERS techniques utilize the redistribution of the incident light into localized plasmons to increase coupling into the sample matrix to yield an enhanced Raman signal. The GLRS technique relies on the redistribution of incident optical energy at the grating, where changes in the diffraction efficiencies and phases of the reflected orders are directly related to sample dielectric changes. The main feature of GLRS is that it allows interrogation of the bulk properties of a sample without relying on the transmission of radiation through the sample. This feature is based on the fact that, at the specific values of combinations of parameters (thresholds), one of the diffracting waves is transformed from a traveling wave to an evanescent one. The characteristics of all reflected and transmitted waves, including the specular reflection, abruptly change at these thresholds. The analysis of corresponding changes in the reflected light allows for the easy separation of surface and bulk effects refractive index, and absorbance effects and, in addition, allows for the interrogation of optically dense and inhomogeneous matrices. These capabilities make this simple, robust, and easy-to-construct device an attractive analytical sensor. THEORY The GLRS theory can be developed as a specific realization of the general threshold theory of multichannel wave scattering.13 This section will outline the main elements of the theory, which will be described in detail elsewhere.14 For this application, the general theory as it applies to reflection and transmission through a planar dielectric interface will be analyzed. In the following derivation, n(ω), where n ) 0, 1, or 2, corresponds to the frequency-dependent dielectric constant of incident medium, substrate medium, and sample medium, respectively (see Figure 1); for mathematical simplicity, the light incidence plane is assumed to be perpendicular to the direction of the grating lines. δmcr is defined as the normal component of the mth (m ) 0, (1, (2, ...) diffracted transmitted diffracted order:
(
)
mcrλ 2 δmcr ) Re (ω) - sin θ + , λ ) 2πc/ω a (2)
The physical origin of these singularities can be interpreted as follows. The intensity I(rb,m), where br is the unit direction vector of propagation, of the light corresponding to the mth diffraction order in the substrate medium (2) is proportional to
I(b,m) r ∼ [E B(2)(b,m)E r B(2)*(b,m)] r for r1 > 0
(2)
where E B(2)(rb,m) is the complex amplitude of the mth diffracted transmitted beam and the asterisk denotes its corresponding complex conjugate. In the case of nonabsorbing media (Im(2)(ω) ) 0), ixδmr1 -ixδmr1 I(b,m) r e ) ∼ constant for δmcr g 0 (3a) cr ∼ (e
and
(1)
where the subscript cr refers to the critical order, the transmitted diffraction order undergoing the transformation from a traveling wave to an evanescent one. At a certain wavelength, the function δmcr is equal to zero for a given incident angle θ, grating period a, and real part of the sample dielectric function Re(2)(ω) (Re and Im denote the real and imaginary parts of the dielectric function). For the case of Im(2)(ω) < Re(2)(ω), we will observe specific features in the behavior of reflected light at δmcr ) 0, which we have termed “singularities” as the behavior of reflected light is discontinuous around δmcr ) 0 in the limit of infinite experimental resolution and perfect incident beam collimation. As the sample dielectric Re(2)(ω) is modulated due to changing composition, the characteristic wavelength changes in the incident medium (1) due to the corresponding shift in the singularity. (13) Newton, R. Scattering Theory of Waves and Particles; McGraw-Hill: New York, 1966; Chapter 17. (14) Anderson, B. B.; Brodsky, A. M.; Burgess, L. W. Threshold Effects in Light Scattering from a Binary Diffraction Grating. Submitted to Phys. Rev., 1995.
1082 Analytical Chemistry, Vol. 68, No. 7, April 1, 1996
Figure 1. Physical description of GLRS sensing mechanism. In this Figure, the m ) 1 transmitted diffraction order undergoes a transformation from a traveling wave to an evanescent one as the analyte matrix dielectric function, grating period, incident angle of light, and wavelength of light dictate according to eq 1. As the analyte dielectric function changes due to changing composition, the wavelength at which the transformation occurs, the GLRS threshold, modulates to higher wavelengths for larger values of the analyte dielectric function.
ix|δm|r1 -ix|δm|r1 I(b,m) r e ) ∼ e-2x|δm|r1 for δmcr < 0 cr ∼ (e
(3b) According to eqs 3a and 3b, the dependence of transmitted light on the distance from the surface has drastically changed near the point δmcr ) 0. At δmcr ) 0, the intensity I(rb,mcr) remains constant at infinite distance but begins to diminish with further decline of δmcr . In real systems this distance becomes finite even for δmcr ) 0 since the imaginary part of the dielectric constant is always greater than zero, resulting in a smoothing of the singular behavior near δmcr ) 0. Nevertheless, if
Im(2)(ω) < |Re((2)(ω) - (1)(ω))| the behavior of all reflected and transmitted beams with different m near the thresholds, where δmcr ) 0, will substantially change. This behavior can be described by the following expressions for reflection coefficients R(s,p)(m), for s- and p-polarized light, in the interval of parameters in the threshold region (where δm lies near zero):
The expressions 5 and 6 are correct for all diffraction orders m * mcr near the threshold order mcr. Of special interest is the behavior of the reflection coefficient for specular reflection, because it aids in the extraction of refractive index and absorbance information from the sample. In this case, the measurement of the relative reflection coefficients is desired: (s,p) R(s,p) /R ˜ (s,p) rel ) R (s,p) Figure 2. General solution forms to eq 5. The coefficients C2,3 determine the shape of the curves (a-d), where the position of the threshold is given at δmcr ) 0, denoted by the dashed lines. Combinations of coefficients: (a) C2 + C3, (b) -C2 - C3, (c) C2 C3, (d) -C2 + C3. These curves do not include the influence of the imaginary part of the dielectric function, which smooth the singular behavior at δmcr ) 0.
R(s,p)(m) (m) + C(s,p) (m) = C(s,p) 1 2
x / (xδ + (Im ) - δ ) + (m)x / (xδ + (Im ) + δ )
C(s,p) 3 as Im(2) f 0 (m) + ) C(s,p) 1
1
(2) 2
2
2
m
m
1
2
2
m
(2) 2
m
where R˜ (s,p) is the reflection coefficient for the first surface reflection depicted in Figure 1. Here, the reflecting surface is the air/dielectric interface, or a representative spectrum of the source intensity. The general functional form of the ratio dependence on the parameters of the analyte will remain the same as in eq 5 but with the complicated influence of the air/substrate reflections factored out. The dependence of the derivative of the relative reflection coefficient on δmcr is given by
dR(s,p) ) dδm
1
x
4 δm + (Im(2))2 2
xxδ
(C(s,p) 3
2
m
+ (Im(2))2 + δm -
xxδ
C(s,p) 2
{
C(s,p) (m)x|δm| for δm e 0 2 m * mcr (5) (s,p) C3 (m)x|δm| for δm > 0
(s,p) where coefficients C1,2,3 (m) are approximately constant near the thresholds. For m * 0, these coefficients are proportional to the difference in dielectric constant between the grating lines and the substrate material, and depending on the signs of the coefficients (s,p) C1,2,3 (m), we will have one of the behaviors of the reflection coefficients presented in Figure 2. Theory predicts that a separation of bulk and surface properties is possible with appropriate analysis of the reflected light. The positions of singularities δmcr ) 0 and the functional dependence on δm in the interval δm ∼ 0 are dependent only on the properties of the sample matrix at distances from the sample/grating interface greater than λ/Im(2). As a result, the position of the singularity, as well as the reflection coefficient’s and phase shift’s behaviors in the vicinity of this position, will be unaffected by surface fouling layers to a depth on the order of λ/Im(2). An important consequence of eq 5 arises from the specific dependence of R(s,p)(m) on the imaginary part of the dielectric function, Im(2)(ω). If
Im(2)(ω) , Re(2)(ω)
(6)
in all intervals of the parameters beyond the threshold regions, then the reflection coefficients R(s,p)(m) are almost independent of Im(2)(ω). However, in the vicinity of the thresholds, the influence of the imaginary part of Im(2)(ω) is substantial and affects the sharpness of the singularity behavior of R(s,p)(m), where the two terms on the right-hand side of eq 1 cancel each other. The described enhancement of the dependence of the reflection coefficient on the imaginary dielectric function opens, in principle, the possibility of measuring relatively small quantities of absorbing species, given appropriate experimental conditions, i.e., a sufficiently small difference between the sample and substrate dielectric functions. As the technique is not transmission based, it is possible to measure very highly absorbing matrices with the same device.
(7)
2 m
+ (Im(2))2 - δm) (8)
where the singularity appears as a peak in the plot of the derivative of the relative reflection coefficient. The use of the derivative of the reflection coefficient enables the refractive index of the sample to be determined directly from the peaks in the derivative spectra, where changes in the real and imaginary parts of the sample refractive index yield shifts in the position and height of the derivative peaks, respectively. DESCRIPTION OF EXPERIMENTS (A) Grating Fabrication and Characterization. Grating substrates consisted of 1 in. diameter, 3/16 in. thick fused silica disks (Esco), with standard photolithographic chrome mask layers deposited on the disks (Nanofilm Inc.) of 1000 Å thickness. A Lloyd’s mirror configuration15 was used to record interference fringes in a spin-coated positive photoresist from the spatially filtered, expanded, and collimated output of an argon ion laser. Each substrate was then developed and etched to the silica, forming a transmission grating on each. Two substrates were chosen for these studies, with measured grating periods of 829.4 ( 0.76 nm and 829 ( 1.1 nm. (B) Experimental Apparatus. The optical system used consisted of a dc tungsten-halogen source coupled into a 220 µm core optical fiber (Figure 3). The output of the fiber was collimated via a bulk optic achromat collimator, and the beam size was minimized using an iris aperture to reduce off-axis light incident on the grating. Incident polarization state was set via a Glan Taylor cube polarizer to either transverse electric (s) or transverse magnetic (p). Two reflections were captured via ball lens-coupled fiber optics. The reference beam was captured as a specular reflection off the first surface of the substrate or at the output of the collimator, removing a portion of the poorly collimated light at the limit of the clear aperture of the lens but not affecting the light incident on the grating. The grating/ substrate interface specular reflection (the zeroth-order diffracted beam from the grating) was collected to serve as the sample beam. (15) Mai, X.; Moshrefzadeh, R.; Gibson, U. J.; Stegeman G. I.; Seaton, C. T. Appl. Opt. 1985, 24, 3155.
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Figure 3. Experimental instrumentation for GLRS: (1) 32 V dc fiber bundle coupled tungsten-halogen source, (2) achromat fiber-optic collimator, (3) iris, (4) polarizer, (5) dual axis rotation stage, (6) McPherson monochromator with dual stacked fiber-optic input, (7) SBIG CCD camera, (8) 486 PC for data acquisition, (9) Teflon sample cell, (10) grating substrate, grating side facing sample cell, (11) sample fiber input into spectrometer, captures zeroth order reflection off of grating via ball lens assembly, (12) reference fiber for ethanol, absorbance, and incident angle dependence experiments, (13) reference fiber for temperature experiment, first surface reflection off of substrate is captured.
These two collection fibers were coupled into a McPherson 218 monochromator via aperture matching optics and adjustable slit as a stacked pair, and the dispersed images were vertically resolved onto a Santa Barbara Instruments Group ST-6 CCD camera placed in the image plane of the monochromator. The wavelength range of the spectrometer was selectable via a selection dial, and the bandwidth of the instrument was 77 nm at any one setting on the monochromator. The resolution of the spectrometer system was 1.9 nm for an entrance slit width of 500 µm. The grating substrate was mounted on a dual axis rotation stage with the incident angular resolution of 0.083° and an azimuthal angular resolution of 0.5° (see Figure 3). The sample was placed in a 2 mL Teflon sample holder, pressure fitted, and sealed with a Parafilm gasket against the grating side of the substrate; sample was introduced via a pipet. In order to test the GLRS sensor for its resistance to temperature fluctuations in the substrate material, the substrate was placed into a sandwich-type flow cell. A thermoelectric device (Melcor) was mounted between the face plate of the flow cell and the dielectric side of the grating substrate. The thermoelectric cooler (TEC) was a center hole type with a clear aperture of 14 mm. A flow cell gasket was fashioned out of optically clear silicone and placed in contact with the grating side of the substrate. Contained in this gasket was an RTD temperature transducer for TEC setpoint control. The resolution of temperature control for the TEC was 0.1 °C, and the range was 17-45 °C. Sample was introduced into the cell via a peristaltic pump using Teflon 0.5 mm i.d. tubing through the back of the flow cell back plate, thus placing the sample in contact with the grating. (C) GLRS Experimental Procedure. The refractive index of each ethanol/water solution and the deionized water was measured using an Abbe´ refractometer at a temperature of 28 °C at the outset of the refractive index GLRS experiments. Subse1084
Analytical Chemistry, Vol. 68, No. 7, April 1, 1996
quently, the refractive indices of the ethanol solutions were measured at temperatures of 25, 30, 35, and 40 °C, and the temperature dependence and dispersion were calculated from the refractive index at 589.3 nm and the compensator prism settings at each temperature. The absorbances of the first five methylene blue samples were measured on a Hewlett Packard 8540A UV/ vis diode array spectrometer in a cell fashioned from two fused silica plates and a thin Teflon spacer and then converted to the corresponding equivalent 1 cm. path length absorbances. The thin cell thickness was inferred from a comparison of the 1 cm cuvette measurements and thin cell measurements using the two least concentrated solutions. The sixth solution, ∼0.005 M, was placed between two silica plates with no spacer, and the cell length was approximated from the extinction coefficients calculated from the other absorbance measurements. In order to test the refractive index sensitivity of the sensor, the cleaned substrate (grating period 829.4 ( 0.76 nm) was mounted in the substrate/Teflon sample holder assembly, the incident angle set to 36.17° using the rough vernier, and the stage locked. The azimuthal angle was set to 0° and the azimuth stage locked. The polarization was initially set to p-polarization. The reproducibility of the sensor and optical system was tested using three replicate runs with four of the nine ethanol solutions in 10% ethanol increments, run in random order. Between samples, the cell was rinsed and aspirated with copious amounts of deionized water and then rinsed twice with sample. Twenty scans of the array were taken for each sample at an integration time of 2 s, and a 20 scan dark current measurement was subtracted manually at the data analysis stage. Due to the sensitivity of the CCD array, signal-to-noise ratio requirements, and the CCD frame transfer rate, the time required for data acquisition was 140 s/sample. Identical experimental technique was used to run the nine ethanol test samples and two water samples, also in random order. The polarizer was then rotated to achieve s-polarization and the ethanol samples run again in random order. In order to test the incident angle dependence of the GLRS signal, data were acquired at incremental incident angles using the rotation stage micrometer vernier to determine the relative angular displacement. As the coupling angle was manually adjusted, the specular reflection collection optics were moved in order to collect the specular reflection. Two individual experiments were run with deionized water in the cell, with the angles incremented approximately 0.3° or 0.5°, yielding a range of incident angles from 33.53° to 38.42°. The system was also tested for absorbance sensitivity using six methylene blue solutions. The system was initially set to an incident angle of 37.11°, azimuthal angle of 0°, and p-polarization. The cell was cleaned with deionized water, rinsed with 0.1 M nitric acid, and rinsed again with water, and then the first sample was run. Subsequent samples were run, again preceded by a nitric acid rinse, water rinse, and two volumes of sample rinse. The methylene blue samples were run in random order at three different angles of incidence: 37.11°, 34.98°, and 30.62°. In order to test the effect of temperature fluctuations on the grating substrate and sample volume, the cleaned substrate (grating period of 829 ( 1.1 nm) was mounted in the temperaturecontrolled sandwich cell, the incident angle set to 36.5°, the azimuthal angle set to 0°, and the rotation stage locked with the polarization state set to p-polarization. Deionized water was pumped into the cell and then stopped, and the temperature of
the TEC set to 25 °C. The temperature transducer was in contact with the grating side of the substrate but not with the liquid in the cell, and the cell was allowed to stabilize for 5 min, although the TEC reached the setpoint in ∼15 s. Twenty scans of the array were taken for each subsequent temperature setting, and a total of five temperatures were recorded, 25, 30, 35, 37.5, and 40 °C, with stabilization achieved between each run. (D) Univariate Data Analysis. Each sample generated two 20 × 750 matrices of sample and reference data that were averaged to obtain 1 × 750 vectors of sample and corresponding reference spectra, which were then dark corrected. The high-frequency components of the signals which were unrelated to sample variations required the application of a 101 point smoothing filter to each temperature sample and reference spectrum. For all other experimental data, an 11 point smoothing filter was applied to each reference spectrum while the sample spectra were unaltered except for dark correction. Relative reflection spectra were calculated for each run by calculating the ratio of each sample spectrum to its corresponding reference spectrum. Due to the intensity variations in the angle measurements associated with translation of the coupling optics, each sample scan was normalized to a CCD pixel well away from modulation, where the intensity should be very similar from run to run. The reference scans were smoothed via an 11 point Savitsky-Golay smoothing filter, and then the ratio of each sample spectrum to its corresponding reference spectrum was calculated to obtain the relative reflection coefficient for each sample. The derivatives were calculated using a 71 point Savitsky-Golay first derivative filter, with the peak maxima located for comparison with theoretical predictions corrected for temperature and dispersion. The peak positions, grating period, and angle of incidence were used with eq 1 to generate a predicted absolute index at the wavelength of the singularity. Reference refractive index values were obtained from Abbe´ refractometer measurements and used for statistical comparison with the refractive index values from GLRS experimental data. This type of analysis was used to generate bias and prediction statistics for the univariate data analysis techniques, the bias calculated as the mean of the residuals for each experiment, and the precision calculated as the rms error with the bias subtracted for refractive indices predicted from eq 1. RESULTS AND DISCUSSION The data analysis techniques described above rely on the use of a smoothing derivative filter in the case of the univariate data analysis, as a derivative is calculated from experimental data for comparison with theory. The differences in the size filter applied to the temperature data versus that used in the later experiments result from optimization of the optical system. In the temperature experiments, the data contained more high-frequency components due to the nature of the optical fibers, coupling conditions, and the lower light levels on the detector, resulting in longer integration times and therefore larger dark current fluctuations. As comparison with theory was the prime objective, a large smoothing filter was justified in the calculation of each derivative, as the general shape of the sample and reference data allowed for the removal of high-frequency components without significant distortion of the line shape. A 71 point derivative filter was used because it represented a balance between distortion of the peak position and suppression of harmonics in the derivative. These harmonics are the result of utilizing a small filter size relative to the size of the features in
Figure 4. First derivative of GLRS relative reflection coefficients calculated for ethanol/water solutions, p-polarization. Ethanol was prepared from absolute ethanol (McCormick, Lot No. CO3512) and deionized water in the following concentrations: 1%, 5%, 10%, 15%, 20%, 25%, 30%, 35%, and 40% (v/v). The sodium D-line refractive index range for these samples is from 1.3321 for water to 1.3530 for 40% ethanol. The feature shifts to higher wavelengths for increases in refractive index due to increases in ethanol concentration, where the peak of the derivative is taken as the singularity.
Figure 5. First derivative of GLRS relative reflection coefficients calculated for ethanol/water solutions, s-polarization.
the data, which are in this case fairly large. The full width at half-maximum, on average, for the derivative peaks for all experiments except the temperature experiments, was ∼7 nm. A 7 nm wide feature in the data would correspond to a width of 68 pixels. Thus, a 71 point derivative filter would not significantly distort the feature. Refractive Index and Angle Variation Results. The derivatives of the reflection coefficients are shown in Figures 4 and 5 for the two respective orthogonal polarization states. The difference in the trend to negative or positive values of the peak maxima (minima) in Figures 4 and 5 may be explained by sign differences in the C(s,p) terms of eq 5 due to the polarization state dependence of those terms. Theory predicts that an increase in refractive index will result in a shift in the position of the singularity to longer wavelengths, yielding a peak shift in the derivative plots. This is borne out in the GLRS response to changes in sample refractive Analytical Chemistry, Vol. 68, No. 7, April 1, 1996
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Table 1. Precision and Bias Statistics for GLRS Prediction of Refractive Indexa bias, refractive precision (rms) index units refractive index units
experiment p-polarization, ethanol, dispersion-corrected reference values s-polarization, ethanol, dispersion corrected reference values p-polarization, ethanol, 589.3 nm reference values s-polarization, ethanol, 589.3 nm reference values angle variation, p-polarization
0.00089
0.00015
0.00071
0.00036
0.00040
0.00026
0.00053
0.00016
0.00120
0.00064
a Dispersion-corrected and sodium D-line refractive indices used as reference values.
Figure 6. Predicted versus reference values for GLRS determination of refractive index for ethanol and incident angle response. Peak positions were determined and used to calculate the refractive index using eq 1, where the angle of incidence, position of singularity, and grating period are known. Reference values were calculated from temperature- and dispersion-corrected Abbe´ refractometer measurements.
index for both polarizations. The asymmetry of the peaks at the base of each is due to the grating function modulation in regions away from the singularity. The absolute accuracy of the GLRS technique in determining relative index shifts is dependent upon the accuracy with which the peak position may be found. A 1 pixel shift in peak position corresponds to a 0.103 nm shift resulting in an absolute prediction accuracy of 1.2 × 10-4 refractive index units. It is important to note that this is a limitation placed on the system by the array detector, and the theoretical limit for refractive index is predicted to be several orders of magnitude lower. Figure 6 displays the GLRS-predicted versus reference refractive indices for the ethanol runs and for the incident angle experiments utilizing the peak position, grating period, and angle data to predict refractive index from the GLRS experimental data. The angle variation produced a range of GLRS predicted refractive indices corresponding to the wavelength-dependent dispersion of water at the corresponding singularity wavelengths. As the data were acquired at different times and on different days, ambient temperature changes affect the results. Temperature-corrected refractive index values and the temperature dependence of the dispersion were calculated for each ethanol solution, yielding a trend of increasing dispersion for increases in temperature and ethanol concentration. The range of dispersions was from ∂n/∂λ ) -3.73 × 10-5 for water at 25 °C to -3.81 × 10-5 for the 40% ethanol at 40 °C. The results agree with the linear dependence of the singularity on wavelength, refractive index, and angle of incidence and are consistent with the expected rms error from the absolute peak position determination when corrected for an initial bias and calculated as explained above. Bias is most likely introduced in the measurement of initial incident angle, where a 0.083° error in angle determination results in an index bias of 0.0011 refractive index units, which is on the order of the bias found in the bulk index experiments. The precisions for all refractive index determinations are between 0.0001 to 0.0007 index units (Table 1), which is consistent with the precision attainable in the determination of the peak positions and the angles of incidence. 1086 Analytical Chemistry, Vol. 68, No. 7, April 1, 1996
Table 2. RMSECV Values for PLS Calibration of Ethanol/Water Solution Refractive Indexa no. of PLS latent variables 1 2 3 4 5 6 7 8 9 10
RMSECV, refractive index units p-polarization s-polarization 0.000 4471 0.000 3861 0.000 1278 0.000 0763 0.000 0627 0.000 0413 0.000 0175 0.000 0018 0.000 0032 0.000 0039
0.000 2576 0.000 1583 0.000 0785 0.000 0488 0.000 0305 0.000 0240 0.000 0238 0.000 0238 0.000 0027 0.000 0054
a s- and p-polarization GLRS reflection coefficient data used as response matrix.
In addition, the Abbe´ refractometer measures refractive index to ( 0.0001 index units and requires the use of lookup tables to calculate the dispersion of a sample. Multivariate Analysis: Refractive Index Calibration. Partial least squares (PLS) calibration of refractive index for ethanol in water yields significantly better prediction errors than the determination of refractive index using peak positions and eq 1. A statistical model is constructed describing the variance that is correlated with a vector of reference values, in this case a vector containing the refractive indices of the ethanol solutions at 589.3 nm. The model is then checked via a leave-one-out cross validation and an rms error statistic (RMSECV) calculated. Table 2 summarizes the RMSECV values for the different numbers of latent variables used in constructing the PLS model on the GLRS ratio data. When three PLS factors are used in the model, the error in both the s- and p-polarization states is comparable with the measurement error associated with using the Abbe´ refractometer to obtain the reference values. The use of the full data set in calibration, as opposed to a single number obtained from the data, allows the calibration procedure to take into account the reflection differences due to refractive index away from the thresholds. Because there are nine ethanol solutions, with data taken in a static format, the maximum number of PLS factors that can be used to model the data is nine, as there is no continuous time axis with which to remove noncorrelated time-dependent drift. Using eight PLS factors for the p-polarization data gives an RMSECV of 1.8 × 10-6 index units, and using nine factors for the
Table 3. Methylene Blue in Water, Calculated Equivalent 1 cm Path Length Absorbances at Each Maximum
methylene blue concn, mol/L 5.127 × 10-3 1.022 × 10-3 4.996 × 10-4 1.000 × 10-4 4.996 × 10-5 1.000 × 10-5
absorbance maximum 1 in nm in AU
absorbance maximum 2 in nm in AU
599.2 605.2 606.7 610.3 610.7 615.3
661.7 661.2 661.3 661.3 661.8 662.3
244.14 43.83 24.30 4.620 2.464 0.459
106.35 28.30 17.42 5.422 3.300 .8345
Figure 8. Derivative peak position shifts versus concentration of methylene blue at three angles of incidence. Shifts in position at the highest concentrations of methylene blue indicate real refractive index changes due to the imaginary part of the dielectric function.
Figure 7. Derivative of relative reflection coefficients, p-polarization, for varying concentrations of methylene blue (Aldrich, recrystallized) in deionized water: 1.000 × 10-5, 4.996 × 10-5 , 1.000 × 10-4, 4.996 × 10-4, 1.022 × 10-3, and 5.127 × 10-3 M. Angles of incidence for GLRS measurements were chosen such that the singularity would occur at 605, 630, and 683 nm.
s-polarization data yields an RMSECV of 2.7 × 10-6 index units. Although these represent the true minima in the latent variable versus RMSECV curves, the large number of latent variables used indicates that the models are not compact. However, the facts that the third factor reaches the error limit imposed by the reference method and a 2 orders of magnitude decrease in the prediction error over the range of model sizes indicate that the GLRS technique is directly amenable to multivariate statistical analysis and calibration. Absorbance Results. Methylene blue was chosen as a model absorbing system due to its high solubility in water and unique absorbance properties at high concentrations. A shift in the absorbance maximum from 659 to 610 nm occurs with an increase in dye concentration due to a tautomeric reaction in solution.16 The equivalent 1 cm path length absorbances are tabulated for the six methylene blue concentrations in Table 3. The general shape of the absorbance data agrees with the predicted theoretical response. Figure 7 shows the absorbance response at the three different angles of incidence, where the angles of incidence are chosen to yield singularities in wavelength regions of different anomalous dispersion characteristics. The functional dependence of the peak height with concentration is seen for all three experiments, with a larger absorbance corre(16) Conn, H. J. Biological Stains, 6th ed.; Biotech Publications: Geneva, NY, 1953; p 111.
sponding to a larger peak amplitude decline in the derivative. A real refractive index change is occurring at the two lower angles of incidence due to the high concentrations of dye affecting the bulk index. This real index change is seen as a shift in the position of the singularity for high concentration of dye in Figure 7, and this results from the Kramers-Kronig relations between the real and imaginary parts of the refractive index, which predict large index shifts at wavelengths corresponding to shoulders of the absorption band for the highest concentration of dye (Figure 8). Figure 8 displays the dependence of the peak position on concentration for the three angles of incidence. The peak shift (and thus the index shift) is largest where the singularity occurs, at the high-wavelength shoulder of the absorbance band, and this is borne out experimentally as the singularity occurring at 683 nm produces the largest index shift with dye concentration. The singularity position of 605 nm yields a very small shift in index, consistent with the Kramers-Kronig relations, where 605 nm represents a position very close to the absorbance maximum at higher concentrations and thus the zero-crossing in index shift. At 630 nm, it is expected that an intermediate shift in position and height would occur, and this is borne out experimentally as the singularity is in a position where the absorbance maximum moves from the high-wavelength side of the singularity at low concentrations to the low-wavelength side at higher concentrations. The GLRS absorbance response is predicted to be a decline in the peak amplitude of the derivative of the reflection coefficient with increasing sample absorbance. The degree of peak height change is dependent upon the extinction coefficient of the dye, which is directly related to the imaginary portion of the dielectric function. The singularity at 683 nm represents a region in the absorbance spectrum where the extinction coefficient becomes smaller with concentration, yielding a smaller peak height decline relative to the singularity at 605 nm. For each of the angles of incidence, the amplitude change from a water baseline is calculated and plotted in Figure 9. Note that the x-axis is a logarithmic scale. A least-squares fit of the peak amplitude decline to the 1 cm equivalent path length absorbance measurements at 605 nm yields a power relationship given by the equation ∆(amplitude) Analytical Chemistry, Vol. 68, No. 7, April 1, 1996
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Figure 9. GLRS derivative peak amplitude changes for methylene blue solutions at three angles of incidence. Amplitude changes were calculated from the baseline peak height for water and plotted versus concentration of methylene blue.
) 0.0000295 (abs)0.530, indicating a qualitative agreement with theory, as the derivative amplitude in eq 9 is proportional to the inverse square in regions very close to the singularity. The error in determining the peak amplitude was 0.000015 units, and this corresponds to an error in absorbance of 0.28 AU. Thus, the limit of detection for absorbance is 0.83 AU, with a dynamic range extending to ∼250 AU. Temperature Results. The temperature data revealed that the technique is independent of substrate refractive index changes due to temperature. This is predicted by theory in that the planar nature of the substrate cancels its effect on the singularity position. As most reflection techniques rely on the difference between the refractive indices of the sample matrix and the crystal or element used as the reflecting surface, changes in temperature affect the measurements by inducing shifts of the indices of both the crystal and the sample. Since GLRS is not an ATR technique, theory predicts that changes in the temperature of the system would not affect the bulk refractive index measurement. This agreement with theory is reflected in the peak position shift with temperature. A bulk refractive index shift of -0.0020 ( 0.0001 refractive index units for a 15 °C temperature change is demonstrated, corresponding to a ∂n/∂T of -1.3 × 10-4/°C, in good agreement with the literature value17 of -1.17 × 10-4/°C for water. CONCLUSIONS This paper demonstrates the direct experimental verification of a new optical technique based on the thresholds observed in the diffraction of light from a transmission grating. A key attribute that makes this technique attractive for direct application is the ability to obtain bulk dielectric properties of a sample in a single, non-ATR reflection from a binary transmission grating in the visible portion of the spectrum. The deconvolution of the real part from the imaginary part of the dielectric function is readily (17) CRC Handbook of Chemistry and Physics, 70th ed.; CRC Press: Boca Raton, FL, 1989; p E-384. (18) Anderson, B. B.; Brodsky, A. M.; Burgess, L. W. Application of Grating Light Reflection Spectroscopy for Analytical Chemical Sensing J. Process Anal. Chem., in press.
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performed, as shown in the absorbance data. The large absorbance dynamic range and the tunability of the technique to specific analysis parameters make the GLRS sensor particularly suited to optically dense samples and heterogeneous matrices. Sample conditioning or preprocessing is not required for measurements using GLRS, making this a viable tool for continuous process analysis that could be easily implemented with appropriate optical and instrumental design. In addition, as there are no material constraints placed on GLRS, the grating/substrate materials may be chosen for a given environment, whereas ATR and SPR are constrained in their material choices to high-index substrates and precise metal film thickness and composition, respectively. The ability to predict refractive index absolutely from the data, without reference or calibration, corroborates the theoretical development, as the positions of the thresholds in relation to their dependence on bulk dielectric properties have been shown experimentally as well as theoretically. Furthermore, as the data are first order in nature, first-order calibration techniques may be employed to build models that readily predict bulk index to 1 × 10-6 index units from a single-hit, nontransmissive measurement. Most refractive index measurement schemes rely on the transmission of light through the sample and the measurement of a critical angle or laser beam displacement, are differential in nature, and often achieve several orders of magnitude better sensitivity than the data presented in this paper. The uniqueness of GLRS is that it allows for the use of white light and the measurement of shifts in wavelength to obtain refractive index and the deconvolution of refractive index and absorbance effects directly from reflected light. The limiting factors in predicting absolute refractive index lie in the design of the incident light launch optics and the spectroscopic system, as well as the precision with which the angle of incidence and grating period are determined. The future of this project will include investigation of the optical interaction of the GLRS sensing mechanism with scattering samples. Preliminary data indicate that GLRS will be a very effective tool in obtaining particle size and distribution data from very heterogeneous matrices,18 as well as extending the GLRS sensor to the infrared and UV portions of the spectrum. In addition, the threshold phenomenon demonstrated will also occur in the phase relationship between the incident and reflected light. As the threshold is reached, a dramatic shift in phase is expected and should yield 2-3 orders of magnitude improvement in real and imaginary dielectric sensitivity. The tunability of the technique and the general nature of the wave phenomenon will allow the application of GLRS to difficult optical sensing problems and analogous applications in acoustics and particle diffraction techniques. ACKNOWLEDGMENT The authors thank the Center for Process Analytical Chemistry, Nanofilm, Inc. for fabrication assistance and Security Pacific Bank for additional funding. Received for review December 5, 1995. Accepted January 26, 1996.X AC951177S X
Abstract published in Advance ACS Abstracts, March 1, 1996.
