Midinfrared absorbance spectra obtained from spatially inhomogeneous and finite samples often contain scattering effects characterized by derivative-like bands with shifted peak positions. Such features may be interpreted and accurately modeled by Fano theory when the imaginary part of the complex dielectric function is small and Lorentzian in nature—as is the case for many biological media. Furthermore, by fitting Fano line shapes to isolated absorbance bands, recovery of the peak position and pure absorption strength can be obtained with high accuracy. Additionally, for small and optically soft spherical scatterers, recovery of one or the other of constant refractive index or radius (given approximate knowledge of the other) is possible.
Asymmetrically distorted spectral peaks are often encountered in FTIR absorbance spectra—particularly noticeable in the Amide-I band, where a derivative-like line shape may be observed.1–3 Previously, it has been shown that these derivative features occur due to a nonlinear mixing between the absorbance and dispersive spectra on top of a smooth-varying broadband baseline.2–5 In such cases, identification of chemical bands in a scattering sample may be deceptive as the distorted bands frequently suffer from a shift in the peak position.2,3,5 Identification and correction of such distortions are critical for FTIR spectroscopy if one seeks a quantitative understanding of the sample under investigation.
A. Fano resonances
In atomic spectroscopy, interference between discrete state transitions and a background continuum of states results in a particular distortion of the line shape, characterized by a derivative-like appearance of an energy band. This interference was first described by Fano, who models the relative scattering cross section σ/σ0 for an energy E by the following equation:6
where q is an asymmetry parameter (often referred to as the Fano parameter) and the reduced energy ε is expressed in terms of the resonant peak energy Er and peak full width at half maximum ΓE as
The Fano parameter q introduces asymmetry in an otherwise Lorentzian band shape, as can be seen in Fig. 1 for various q. As q → 0, the peak becomes inverted, while finite values of q produce an asymmetric line profile with a shift in the peak position away from the resonance at ε = 0. The direction of peak shift is determined by the sign of q: positive q shifts the peak toward positive ε, while negative q shift toward negative ε (compare q = 1 and q = −2 in Fig. 1). At |q| = 1, the line shape becomes perfectly antisymmetric about ε = 0 (occasionally referred to as anti-Lorentzian in this paper). As |q| → ∞, however, the line shape approaches a symmetric Lorentzian centered about the resonance energy Er (ε = 0)
Such limiting behavior as q → ∞ can be seen in Fig. 1 for q = 100, where the line shape is Lorentzian in appearance.
While Fano theory was originally developed in the context of quantum spectroscopy,7 it has since been shown to be a general interference phenomenon applicable to a wide range of classical and quantum resonating systems.8–11 Generally speaking, Fano resonances occur due to a coupling between multiple oscillators; in many physical systems this can be interpreted as an interference between a resonance of interest and a “background process.”
B. Fano line shapes in FTIR spectroscopy
Asymmetric, derivative-like peaks are often observed in FTIR transmission and absorbance spectra, attributed to the “dispersion artefact,”12 which involves a distortion of the transmission spectrum due to reflectance line shapes.2,13 Absorbance spectra that contain such dispersion effects, however, also share these signature features and bear a striking resemblance to Fano line shapes. With this motivation, we evaluate whether observed spectral features may be interpreted as Fano resonances, where sharp resonating bands are coupled to a slowly varying background process. Above all, we evaluate whether these features can be accurately modeled with Fano theory and whether they can be corrected (i.e., related to a pure Lorentzian absorption line shape), and if so, to what extent and under what conditions.
Previous work has shown that very small spherical scatterers may exhibit “unconventional” Fano resonances.14 These Fano line shapes present as a function of the dielectric function, manifesting from interfering shape resonances and occurring in particles with large refractive indices in the absence of absorption. By contrast, our work focuses on absorbing scatterers with smaller refractive indices (applicable to typical biological media), and instead of presenting as a function of the dielectric function, as in Ref. 14, the resonances in our work appear as a function of wavenumber, as is appropriate for FTIR spectroscopy.
Using Mie theory, which models the interaction of electromagnetic radiation with an absorbing sphere, we show that Fano line shapes emerge in certain limiting cases, when scattering effects are sufficiently weak. For such situations, a link between an absorbing sphere’s physical size and constant refractive index can be established to the parameters of a Fano line shape that has been fit to the data. We also show that using our Fano approach, the bulk absorptivity, characteristic for an infinite medium, can be extracted from the line shapes measured for finite (spherical) samples.
Presented are results for various simulated spectra using Mie scattering theory, which have been fit with a Fano line shape. We establish that accurate recovery of the peak resonance shift for an isolated absorbance is possible using Fano theory. Additionally, extraction of the pure absorbance strength is obtained for small, optically soft spheres. Finally, we demonstrate how Fano modeling can be used for the recovery of the constant refractive index given an estimate of the particle size and vice versa.
One of the central goals of IR spectroscopy is the extraction of the spectral optical depth from measured transmission spectra. The spectral optical depth is a quantity that characterizes the absorption strength of an infinite medium without boundaries and interfaces. It is defined as the product of the molar extinction coefficient, , the chemical concentration, c, of the absorber, and the propagation distance, d, of the radiation inside of the medium,4,15,16 i.e.,
where , via the molar extinction coefficient , depends on the wavenumber . Inside of the medium, the intensity of the IR radiation propagates according to the (Boguer-) Beer-Lambert law,16 i.e.,
where is defined in Eq. (4). This shows that the optical depth is related to the transmittance
According to convention, in chemistry, biochemistry, and biophysics, the quantity of interest is the absorbance , defined as
Just like the optical depth, the absorbance refers to a bulk property of a given substance, independent of its geometry. Therefore, if intensity propagation measurements inside of the substance could be performed in the lab, the absorbance is obtained from the internally measured intensities and via Eqs. (6) and (8). However, such experiments are extremely difficult to perform, especially if one is interested in the absorption properties of microscopically small samples. Instead, one usually settles for measuring the transmission through thin films, single cells, or other finite-sized biological samples. This, however, immediately introduces interfaces between the embedding medium, containing the source of the IR irradiance, the sample, and the detector. These interfaces act as sources of exterior and interior reflection and scattering of IR radiation. Thus, the measured absorbance is no longer connected to the optical depth via the simple, linear relationship in Eq. (9), which holds only for the pure, bulk absorbance. In such cases, one defines the apparent absorbance
where is the measured intensity transmitted to the detector in the presence of interfaces and scattering and I0 is the irradiance of the sample. Thus, apparent absorbance is a good approximation of absorbance only if scattering effects on interfaces between embedding medium, sample, and detector can be neglected. Depending on the desired accuracy, even for the relatively small indices of refraction of typical biological materials, the scattering effects caused by interfaces may be too large for the apparent absorbance to be an acceptable approximation of the pure absorbance . The distorting effect of interfaces is particularly severe for approximately spherical microscopic samples, such as single cells, and is compounded by the existence of refraction and diffraction effects. Thus, the basic task of IR spectroscopy is to extract the pure absorbance from the measured, apparent absorbance . Our method, discussed in this paper, is to deconvolve the measured line shapes containing scattering, refraction, and diffraction effects by using Fano theory and thus to extract the pure absorbance from the measured, apparent absorbance . In order to prove the viability and power of our method, we model biological cells with spheres since for spheres an analytical theory, i.e., the Mie theory,15,17 is available that allows analytical insight into our methods and procedures.
A. Scattering on spheres
When electromagnetic radiation interacts with a dielectric sphere, it undergoes a form of scattering that is analytically solved by Mie theory. An approximate equation for the extinction efficiency18 Qext of an absorbing sphere of radius a and complex refractive index is given by van de Hulst as17
In the vicinity of chemical absorption bands, exhibits large and rapid variations as a function of . Away from absorption bands, however, may be approximated by a real, constant n0. For this constant refractive index n0, we define the background extinction efficiency QBG according to
Denote by Q′ ≡ Qext − QBG the removal of the background QBG from Qext, revealing the “interesting” spectral features (see Fig. 2). We then consider the behavior of Q′ for a Lorentzian line shape whose peak strength is f0, and the reduced spectral variable
is defined for a resonance about νr, with a peak width at half maximum in the same spirit as in Eq. (2). It is then shown in the Appendix that, to the first order in f0, a local expansion of Q′ about gives
and Φ1(ρr) and Φ2(ρr) are expressions involving sines and cosines of ρr [see Eqs. (A17) and (A18)]. To understand the behavior of Φ1(ρr) and Φ2(ρr), we further expand Eq. (17) to the second order in Δ
Here, the expansion leading to Eq. (19) requires that ρr is small, which restricts the expansion’s validity to a limited class of particles. However, we are only considering small ρr to illustrate our methods and procedures; the full local expansion for arbitrary ρr [Eq. (17)] is valid for all radii and indices of refraction for relevant biological tissues, provided the peak height f0 is small, is small, and a suitable background extinction QBG can be estimated.
B. Connection to Fano theory
We aim to model asymmetric absorbance bands using the approximations contained in Eqs. (17) and (19), with the purpose of extracting the pure absorbance A. As described in the Appendix, directly fitting to these equations is not possible: there exist five parameters, while only four may be considered independent. In particular, Eqs. (17) and (19) may only be determined up to a scaling factor as f0/Δ occurs in both terms. A simple replacement of f0/Δ with a generic scaling parameter would suffice; however, as the observed asymmetric bands resemble Fano resonances, the Fano model will instead be used to explicitly highlight the occurrence of these line shapes in Mie theory. We will show that the four Fano parameters may be predicted by Mie theory, and with them, explicit analytical results are obtained for the recovery of the absorbance A of chemical absorption bands under the conditions assumed in Eq. (19) (i.e., a small radius and constant refractive index). We therefore start by fitting a Fano line shape model to Eq. (19) and demonstrate that a connection is established between its model parameters and the scatterer’s radius and complex refractive index.
To fit Fano line shapes to transmission spectra, we rewrite Eq. (1) in terms of the dimensionless quantity [see Eq. (16)]. The factor σ0 serves as a scaling parameter that essentially captures the line shape’s strength. It will then prove useful to break it into the following form so that equivalence between the Fano parameters and scattering models can be easily made by inspection
With this form, a few observations are immediately apparent. For large |q|, the first term dominates and σ has the form of a symmetric Lorentzian. For finite q, a mixture of a Lorentzian and anti-Lorentzian results; when |q| = 1, the band shape is purely anti-Lorentzian. Furthermore, the necessity that σ is finite requires that as q2 → ∞, σ0 → 0 such that σ0(q2 − 1) → Const.
It can be seen by inspection, comparing the extinction Q′Δ expanded in Δ [Eq. (19)] with the Fano model [Eq. (20)], that the Fano parameters σ0 and q are related to the radius a, absorption strength f0, and constant refractive index n0 by the following two equations:
An additionally useful (although not independent) equation, obtained from the difference between the maximum and minimum of Q′exp [also taken to the second order in Δ; see Eq. (A16) of the Appendix], gives
Recovery of the pure absorbance A as defined in Sec. II from a Fano line shape in the case of an absorbing sphere can be achieved by noting that the absorbance strength at the resonance position —in the limit that scattering due to a constant refractive index is negligible (Δ → 0)—approaches19
Combining Eqs. (21)–(23), while considering only terms of order , gives an expression for A in terms of the Fano fit parameters σ0, q, , and
Furthermore, one may have a reasonable estimate of either the material’s radius a (e.g., from a bright-field measurement) or the material’s constant refractive index n0. For either such case, the known quantity may be used to recover the other unknown (either n0 or a) in terms of the Fano fit parameters by combining Eqs. (21) and (25) for A
Finally, the removal of QBG from Qext can be done exactly with simulated data, as the “background” Mie contribution [given by QBG in Eq. (14)] can be subtracted from the simulated data. In practice, however, ρ0 is unknown, and methods that estimate a suitable baseline must be employed. Such methods include extended multiplicative scatter correction (EMSC),3 which estimates the broad, smoothly varying contribution QBG to the spectrum. However, the primary necessity is that an appropriate broad background QBG must be found such that, as , (Qext − QBG) → 0 as accurately as possible.
Numerical simulations, fitting, and analysis were written and conducted in the numerical computing environment MATLAB.20 Simulations of absorption from a spherical scatterer were produced using the van de Hulst approximation Eq. (11). Although the size parameter 17 depends only on the product of a and , we choose to discern between particle size and resonance wave number to ground the analysis on a practical footing. We choose to simulate a sphere with an isolated absorption band at . The choice of is consistent with observations of Fano line shapes in the Amide-I band. Given the resonance wave number , we vary the sphere’s radius such that ρr [Eq. (18)] is between about 0.1 and 2, which gives an indication for which conditions these methods are and are not valid.
Extinction spectra are simulated for 100 spheres with refractive indices and radii ranging from 1.1 to 1.5 and 0.6 µm to 2 µm, respectively. With an absorption band at , this corresponds to size parameters of approximately 0.6–2.01 near the resonance. The restriction of small ρr in deriving the approximated Q′Δ [Eq. (19)] means that the applied methods for many of these particles (particularly as n0 becomes larger than about 1.2) are not expected to be valid.
A complex dielectric function is modeled in MATLAB20 using wave numbers ranging from 1000 cm−1 to 4000 cm−1, with ≈3 cm−1 resolution, as
where we have chosen a strength fdiel = 0.1249, width , and peak position The complex refractive index is then calculated as , resulting in a complex refractive index whose imaginary component ni has a peak strength of f0 ≈ 0.06. The value of fdiel (and hence f0) was chosen to bring the absorbance of a bulk measurement with an optical depth d ∼ 10 µm [using Eq. (9)] just under unity. The real part of fluctuates with a small amplitude in the immediate vicinity of 1; we add various values of Δ such that the constant refractive indices fall between about n0 = 1.1 and 1.5 for each of the particles.
|.||Simulated parameters .||Fano fit parameters .||Expansion errors (%) .|
|Label .||n0 .||a .||.||σ0(×10−3) .||q .||Q′exp .||Q′Δ .|
|.||Simulated parameters .||Fano fit parameters .||Expansion errors (%) .|
|Label .||n0 .||a .||.||σ0(×10−3) .||q .||Q′exp .||Q′Δ .|
Next, fits are made to the simulated spectra, which provide the Fano fit parameters: σ0, q, , and . Fitting is performed using a least-squares gradient descent algorithm in MATLAB.20 Fano fits and extracted Lorentzian line shapes for select simulations are shown in Fig. 4. The examples selected in Figs. 3 and 4 encapsulate the average and extreme cases of the results.
The error in recovered resonant positions for various constant refractive indices n0 are shown in Fig. 5. We also estimate the constant refractive index n0 for each particle using Eq. (26), with the assumption that the radius is known. We show how these estimates of n0 compare with exact values in Fig. 6 as a function of q (left) and of the size parameter (right).
Finally, the percent error in extracted Lorentzian peak heights for select simulations is shown in Fig. 7 (solid markers). This can be compared with the absorbance strengths that would be obtained by naïvely subtracting the anti-Lorentzian component of the Fano line, shown in Fig. 7 (hollow markers). Removal of the dispersive component gives an apparent absorbance
It is reasonable that the derivative line shape that arises from electromagnetic interference can also be understood as interference between a broadband continuum of states and a discrete absorption band as it occurs in Fano theory. In the case of Mie scattering, the broadband continuum of states can be attributed to the dielectric interactions that produce the constant (in the midIR region) refractive index. This is evident from Eq. (19): as n0 becomes large, the line shape gains its characteristic Fano-like asymmetry.
The goal to extract the pure absorbance from Fano line shapes arises from the desire to recover quantitative chemical understanding from observations. This is desirable, especially when conducting imaging experiments where spatially resolved chemistry can provide critical insight for the question at hand. For example, there are extensive infrared imaging efforts based on cancer diagnosis and evaluation.21–23 The protein bands cannot be used for evaluation in these cases since tissues at edges can lead to scattering signatures that can modify the peak frequency and absorption band line shape. However, proteins provide large signatures that could prove useful if they could be accurately recovered. Furthermore, extracting the bulk absorbance from the measured Fano profiles is imperative for 3-dimensional infrared imaging; standard tomographic reconstruction algorithms assume bulk absorption, i.e., the validity of Beer’s law, at a fundamental level.24
Based on our simulated data, we first compare the accuracy of the expansions Q′exp and Q′Δ. For weak absorption bands, Q′exp should be accurate for arbitrary radii and refractive indices. Figure 3 (left) shows the accuracy of the expansion, and Table I verifies that the simulations are within 10% error. However, the small radii and refractive index expansion (Fig. 3, right) Q′Δ clearly breaks down when the refractive index exceeds n0 ≳ 1.2.
Next, results of fitting Fano line shapes to simulations are shown in Fig. 4. Particles with small radii a and constant refractive indices n0 have a symmetric Lorentzian appearance whose peak is nearly centered on the resonance frequency . Figure 4(a) shows, for a small particle of radius a = 0.76 µm and constant refractive index n0 = 1.1, that the magnitude of q is relatively large in order to model the near Lorentzian symmetry. As the particle size doubles to a = 1.5 µ m, Fig. 4(b) displays how scattering effects begin to present ever so slightly, and the Fano q parameter decreases in magnitude to q = −9.3.
The effect of scattering becomes much more prominent as one increases the constant refractive index. Figures 4(c) and 4(d) show the line shapes when the constant refractive index is increased to n0 = 1.3. When increased even further to n0 = 1.5, Figs. 4(e) and 4(f) show that the line shapes begin to appear nearly anti-Lorentzian and q2 → 1 as required to make the leading term in the Fano Eq. (20) zero.
B. Bulk-absorption limit
As our primary motivation is to recover the pure absorbance in the bulk-absorption limit [Eq. (24)], i.e., in the limit where no boundaries or interfaces are present, three quantities are necessary to reconstruct a Lorentzian line shape. These include the resonance position , the pure absorbance strength A [see Eq. (24)], and the width at half maximum .
Recovery of is reliable to better than 0.2% for nearly all particles considered in our simulations. Figure 5 shows for select constant refractive indices n0 that as the size parameter and n0 increase, the percent error in the peak position of the spectrum containing scatter climbs to as much as 1.5%. The parameters from the Fano fitting, however, reduce the error to under 0.1% for most cases.
While recovery of the Lorentzian width from the Fano fitting falls within 2% error for large |q| > 6 (i.e., when the peak is mostly Lorentzian), our algorithm experiences difficultly as the peak becomes more antisymmetric. This is seen in Fig. 8, where the error is scattered to as much as 6% when |q| < 6. It is unclear whether this error is due to the precision of our algorithm or if there may be an inherent limitation on the ability to extract q and simultaneously.
Finally, an important result from the expansion of Q′Δ [Eq. (19)] is that simply removing the dispersive (anti-Lorentzian) component does not lead to the correct absorbance strength when Mie scattering occurs. Comparing the Lorentzian term in Q′Δ [which is captured by the Fano parameters σ0(q2 − 1) in Eq. (21)] with the bulk absorbance in Eq. (24), the apparent absorbance is reduced (to second order in Δ) by an amount
Figure 7 demonstrates an improvement of the pure absorbance strength compared to a Lorentzian construction that simply uses the height of the original asymmetric line shape. Such naïve reconstruction produces an error reaching over 95% as the size parameter and constant refractive index increase. Using the corrective Eq. (25), the recovered absorbance strengths improve to under 10% for all simulated particles.
C. Connecting Fano parameters with radius and refractive index
Recovery of the constant refractive index n0 given knowledge of the particle radius a is shown in Fig. 6, where dark markers signify calculations that fall within 1% of the exact value and light markers signify calculations that fall outside 1% error.
The requirement that ρr is small in deriving the approximation Q′Δ in Eq. (19) restricts the class of particles for which the analysis is valid to constant refractive indices n0 ≲ 1.2. However, while Fig. 6 shows that estimates of n0 using Eq. (26) are most accurate for very small constant refractive indices, there is still reasonable accuracy for the larger refractive indices where many biological media fall (between 1.3 and 1.525).
These limitations are due in part to the expansion Q′Δ [Eq. (19)] which, because it assumes small ρr, does not hold as n0 becomes significant. In particular, Eqs. (21) and (22) are accurate to only . Future development could take into account higher orders of Δ when expanding Q′exp, thus improving the accuracy of Eqs. (21) and (22).
Finally, by fitting multiple Fano line shapes to overlapping spectra, our methods may be extended to more general cases where absorbance bands are not well-isolated, such as Amide-I bands, which typically have neighboring Amide-II and phospholipid bands in biological materials. While, in principle, Fano theory applies to isolated bands, overlapping bands may receive a similar treatment since the background process is the same for both resonances. Similarly, nonLorentzian bandshapes, such as those due to many closely packed resonances that cannot be resolved individually (e.g., the Amide-I peak, which often contains a mixture of many proteins), will not fit the Fano line-shape model exactly but are still described by Fano theory. This is because we have shown that for weak resonances (as we noted are typical for biological media in the infrared), a first-order expansion of the imaginary component and its Kramers-Kronig transform are sufficiently accurate (see Fig. 3, left). The problem is therefore linear, and nonisolated and nonLorentzian bands can be treated as a sum of independent Fano processes.
We have shown that Fano line shapes in FTIR transmission and absorption spectra can occur due to Mie scattering as the complex electromagnetic interference produces a spectrum that includes contributions from both the real and imaginary components of the refractive index. As absorbance peaks are typically Lorentzian in nature, this superposition is equivalent to the Fano line shape for weakly scattering particles. Fano line shapes cause both a shift in the peak frequency position along with a change in absorbance strength.
While we have focused on spherical scatterers in this report, Fano line shapes are not restricted to this particular geometry. Any general first-order mixing of the real and imaginary refractive indices is equivalent to Fano line shapes for sufficiently weak Lorentzian absorbance bands. As scattering occurs due to a changing refractive index, we attribute the occurrence of Fano line shapes in the infrared regime to interference between sharp molecular vibration resonances with a background continuum process.
We also presented that, by fitting Fano line shapes to simulated spectra, in general, recovery of the peak shift can be found to within 0.1% in most cases when the peaks are isolated. Applying our fitting algorithm to spectra from small spherical scattering particles with various constant refractive indices, we demonstrated a link between the Fano fit parameters and the particle’s radius and constant refractive index. In particular, if one has a reasonable estimate of either the radius or constant refractive index, one may obtain the other quantity with reasonable accuracy. We also demonstrated that recovery of the bulk absorbance is possible for spherical particles.
In many cases, a reasonable estimate of the particle size and refractive index will be known (e.g., by the manufacturer or bright-field measurements), which would immediately inform whether one is working within the limitations presented in Sec. IV. However, the method presented may still be useful in cases where these values are either unavailable or are outside the range of validity, where the full Mie theory is required. In this case, the task of fitting directly to the Mie model requires knowledge of the dielectric function, along with the material’s size and refractive index. As we discussed in Sec. IV B and illustrate in Fig. 4 (left), the Fano line shape is a very accurate model for determining the Lorentzian parameters, without suffering from the limitations imposed on the size and refractive index. The limitation on refractive index and particle radius occur due to the limited accuracy of the expansion when moving from Eqs. (17)–(19). The method presented therefore allows one to rapidly extract the resonant position and width without resorting to a full implementation of the Mie equations. From there, if one wished, an iterative forward calculation using full Mie theory could be performed using the Lorentzian features that were extracted from the Fano method as a starting place. This would effectively reduce the search space from (3N + 1) parameters to (N + 1), where N are the number of isolated Lorentzian peaks in the spectrum.
We also reemphasize that while it would be advantageous to simply fit directly to Eqs. (17) and (19) to obtain the full set of parameters, this is not possible since these equations deceptively contain only four independent parameters (see the Appendix). Since the Fano formalism also contains four parameters, there is no loss of information when using this model.
Finally, we stress that an experimental evaluation of our methods is very challenging as it requires manufacturing an experiment where an isolated homogeneous sphere containing an isolated absorption band is embedded in an infinite, nonabsorbing homogeneous medium whose refractive index is closely matched with that of the sphere’s. However, if the effects of a finite medium are forgivable, experiments where microspheres are embedded in a nonabsorbing medium (e.g., potassium bromide) would be a natural starting point. Unfortunately, however, for many standard microspheres [e.g., polystyrene, silica, or polymethyl methacrylate (PMMA)], the requirement of well-isolated absorbance bands remains difficult to satisfy. Therefore, the next logical step is to generalize the methods presented to the more realistic situation of overlapping absorbance bands. As such a generalization will necessarily deviate from the strict Fano model, those situations fall outside the scope of this article, where emphasis on the Fano mechanism will be reduced.
Although standard methods exist for correction of entire spectral regions,2,3,5 we believe that understanding scattering phenomena for small spheres with isolated resonances in terms of Fano theory may set the stage for corrective methods that may generalize to more complicated spectra and possibly to arbitrary morphology.
This work was supported by the US NSF under Award No. CHE-1508240.
APPENDIX: EXPANSION DERIVATION
When a plane electromagnetic wave (whose wave number ) interacts with a dielectric sphere of complex refractive index and radius a, the coherent interference between the incident and scattered waves results in an extinction Qext at a detector in the far field, approximated by van de Hulst17 as
The imaginary part of the refractive index, , is a dimensionless quantity inherent to the medium and is related to the real component of the refractive index, , through the Kramers-Kronig transformation
Here, n0 ≥ 1 is assumed constant (in the midinfrared region), and P denotes the Cauchy principal value required because of the singularity in the integrand (see, e.g., Refs. 26 and 27). It is also assumed that the refractive index surrounding the spherical medium is air, approximately equal to unity.
Concealed in Qext [Eq. (A1)] is an oscillating “background” spectrum where resonating features are superimposed. Away from resonances (that is, where ), this background QBG appears as
Band resonances in FTIR spectra typically1 (p.23) take the shape of a Lorentzian L, centered on the resonance position as
where the reduced spectral dimension is defined for the peak width at half maximum . While technically the dielectric function follows the Lorentzian model, the complex refractive index is related to the complex dielectric function (for a nonmagnetic material) as
and for small peaks, it is sufficient to approximate the refractive index to be composed of Lorentzian line shapes
We therefore write as a Lorentzian of height f0, and its Kramers-Kronig transform nr − n0 = K[ni(ν)] as
Denote by Q′ ≡ Qext − QBG the removal of the background QBG from Qext, revealing the “interesting” spectral features (see Fig. 2). We then consider the behavior of Q′ for a Lorentzian line shape .
where Δ ≡ n0 − 1, and is the Kramers-Kronig transform of ni as defined on the right-hand side of Eq. (A4). Indeed ni and K[ni] are typically very small, and Δ is typically less than 0.5 in the infrared region for dilute biological media.4,13,28 In such cases, expanding Q′ (using the computer algebra system SageMath29) to the linear order in f0 followed by a first-order expansion of the numerator gives
Assuming that , the factor in Eq. (A13) approximates to , and the extinction is approximated as
where Φ1 and Φ2 are related to Ψ1 and Ψ2 by
The numerator was expanded to the linear order so that Q′exp has the expected boundary conditions (i.e., Q′ → 0 as ). Note also that the same expression for Q′exp may be derived by assuming a solution of the form ϕ(z) = (ϕ1 + ϕ2z)/(1 + z2) and equating coefficients of a linear expansion in both ϕ and Q′.
An additionally useful result is the difference in Q′exp at its maximum and minimum values and . Using SageMath,29 an expression for this difference is obtained and expanded to the second order in Δ, giving
Assuming again that , Eq. (A19) is approximated to as
It is noted that while Eq. (A20) contains five parameters (f0, , , a, and Δ), only four are independent. Observe that only terms containing the product of radius a and relative refractive index Δ appear in ρr, while f0 and Δ only ever occur in both terms as the ratio f0/Δ.
Finally, investigating Q′exp for small, optically soft particles, a further expansion of Eq. (A16) to the second order in Δ gives
In the geometric-optics limit and for a wide enough detector aperture, the extinction efficiency Qext is related to the transmittance T = I/I0 via T = 1 − (g/G)Qext, where g = πa2 is the geometric cross section of the sphere, a is the radius of the sphere, and G is the detector aperture (see Refs. 15 and 27 for details).
For simplicity, it is assumed that the detector’s aperture is set tightly about the particle, so that its geometrical area G is close to the particle’s cross-sectional area g. In this case the transmittance is directly related to the extinction T = 1 − (g/G)Qext = 1 − Qext (see Refs. 15 and 27). In the case that g ≠ G, the scaling factor g/G (determined from the experimental setup) must be included in the derivation of Eq. (24). The results in Eqs. (26) and (27) are, however, independent of the g/G scaling factor.