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Opt Lett. Author manuscript; available in PMC 2012 March 16.

Published in final edited form as:

PMCID: PMC3306185

NIHMSID: NIHMS361598

Department of Biomedical Engineering and Medical Physics Program, Duke University, Durham, North Carolina 27708, USA

We present an analytical method that yields the real and imaginary parts of the refractive index (RI) from low coherence interferometry measurements, leading to the separation of the scattering and absorption coefficients of turbid samples. The imaginary RI is measured using time-frequency analysis, with the real part obtained by analyzing the non-linear phase induced by a sample. A derivation relating the real part of the RI to the non-linear phase term of the signal is presented along with measurements from scattering and non-scattering samples that exhibit absorption due to hemoglobin.

Substantial efforts have been made to quantitatively measure absorption of biological chromophores *in vivo* for diagnosis [1–3]. Unfortunately, several drawbacks have been encountered, primarily dealing with the limitation of the number of physical parameters that may be independently measured and determined using analytical models. For example, in spectroscopic optical coherence tomography (SOCT), some promising results for quantifying absorption *in vivo* have been reported [2–4], but these have been limited owing to the fact that SOCT measures the total attenuation coefficient and the absorption and scattering contributions have not been separated without *a priori* information. Consequently, in order to assess absorption, models must include additional parameters, such as packing factors and anisotropic coefficients, which results in low confidence intervals, thereby hindering quantification [5]. Rather than using attenuation, some researchers have used phase information as means of obtaining concentrations of chromophores [6]. These methods, however, are often confined to thin, transparent samples and thus are not well suited for *in vivo* measurements.

In this letter, we analyze the dispersion of turbid samples using low coherence interferometric (LCI) signals, in combination with time-frequency (TF) analysis, to retrieve the real and imaginary parts of the RI independently of scattering contributions and without *a priori* information. Using a supercontinuum broadband light source, we demonstrate that this method yields wide-band absorption spectral profiles with high spectral resolution, and wavelength-dependent RI profiles. By isolating the contribution due to absorption, the wavelength-dependent scattering coefficient is also obtained.

To understand how the real and imaginary parts of the RI may be obtained, consider a Michelson interferometer with a reference field described as *E _{r}(ω)=S(ω)exp[i(ω/c_{0})2z_{r}]*, and a sample field returned by

$${E}_{s}(\omega )=\sum _{m}\sqrt{{I}_{s}^{(m)}(\omega )}\xb7{e}^{i(\omega /{c}_{0})2{z}_{d}}{e}^{i(\omega /{c}_{0})n(\omega )2({z}_{s}^{(m)}-{z}_{d})},$$

(1)

where *z _{r}, z_{d}, z_{s}* are the distances from the beamsplitter to the reference mirror, dispersive medium, and scatterer, respectively;

$$\stackrel{~}{I}(\omega )=2\sqrt{{I}_{s}(\omega ){I}_{r}(\omega )}\xb7{e}^{i(\omega /{c}_{0})2({z}^{\prime}-dn(\omega ))}=2{I}_{r}(\omega ){e}^{-{\mu}_{\mathit{tot}}(\omega )d}\xb7{e}^{i(\omega /{c}_{0})2({z}^{\prime}-dn(\omega ))},$$

(2)

where *I _{r}* =

$$\stackrel{~}{I}(\omega )=2{I}_{r}(\omega ){e}^{-{\mu}_{\mathit{tot}}(\omega )d}\xb7{e}^{i(\omega /{c}_{0})2({z}^{\prime}-dn({\omega}_{0}))}\xb7{e}^{-i(\omega /{c}_{0})2d\mathrm{\Delta}n(\omega )}.$$

(3)

As Eq. 3 describes, the measured signal contains three parts: The first part modulates the intensity, which yields spectroscopic information and, if scattering is negligible, the imaginary part of the RI, κ*=cμ _{a}/(2ω),* may be directly measured. The second part linearly modulates the phase of the interference signal, which can be processed to yield depth resolved information by simply taking a Fourier transform, as is commonly done in spectral domain OCT. Lastly, the third term describes dispersion due to the sample. This term shows a non-linear modulation of the phase, which can degrade the resolution of the axial information in OCT, but also contains the wavelength-dependent changes in the real part of the RI. Previous efforts to quantify absorption [2–4] and scattering [9], have only used the intensity and the linear phase term and have ignored the information contained in the dispersive term.

Algorithms have been developed for removing dispersion inherent in an optical system or due to a sample, based on removing the phase terms that are non-linear with respect to ω [10]. One such algorithm, presented by Zhu et al. [10], fits the unwrapped phase of the signal to a line of the form = (ω/*c _{0})L*−

Let us consider the quantification of hemoglobin (Hb) concentration for two cases. In the case where the scattering coefficient is negligible (case A), the real and imaginary parts of the RI may be calculated simultaneously. This case is highly idealized and only serves as a base of comparison for case B, where scattering is not negligible. In case A, *μ _{tot} = μ_{a}* =

To demonstrate the validity of this approach, LCI signals from scattering and non-scattering Hb phantoms were analyzed. The system used a typical Michelson interferometer geometry, where broadband light from a supercontinuum source (Fianium, Eugene, OR), is focused onto the sample. A series of filters are used to shape the source spectrum, thus delivering a total power of ~1.5mW, with a spectral range from ~450nm to 700nm, onto the sample (see inset of Fig. 2). Scattered light returned from the sample is mixed with a reference field and relayed to a spectrometer for detection. With the center wavelength λ* _{0}=*575nm, spectral resolution δλ=0.2nm, and bandwidth

Measured change in the real part of the refractive index of the Hb phantoms without scattering (case A) (a), and with scattering (case B) (b). The theoretical change in the real part of the refractive index of Hb, with a concentration of 40 g/L, is also **...**

First, the absorption due to the sample is analyzed to obtain the imaginary part of the RI. For case A, fully oxygenated Hb (Sigma-Aldrich, St. Louis, MO), diluted to 40 g/L in water, was used. For case B, 10.75% by volume of the aqueous solution in the phantom was made up of 10%-Intralipid (IL), which possesses a scattering coefficient four orders of magnitude greater than its absorption coefficient. The solutions were then placed between microscope slides with spacers used to set the sample thickness to 400 μm. For comparison, the total Hb absorption from these samples is approximately equivalent to a 100 μm thick sample of whole blood (150 g/L Hb) or a few millimeters of tissue (~2–7 g/L). Depth resolved spectral profiles were acquired from the interferometric data using the dual window method as described in [4]. To obtain concentration values, individual Hb spectra were acquired and normalized by a reference spectrum obtained from the average of 10 acquisitions of a pure water sample. The logarithm of the ratio was taken, yielding, *ln(I/I _{0}) =* −

Measured cumulative absorption (a) and total attenuation coefficient (b) of the Hb phantoms without scattering (case A) and with scattering (case B). The theoretical absorption with *d* = 400μm and *C =* 40 g/L (a), and absorption coefficient with **...**

Next, the non-linear phase term was analyzed to determine the real part of the RI. After dispersion effects inherent in the system and resulting from a water phantom were accounted for, the residual phase, *Δ*, from Hb phantoms, was obtained using the method described above. Then, using a subtractive KK relation [8], *Δn* values, which depend linearly on concentration, were determined; thus allowing assessment of Hb concentrations via a linear least squares fitting method. Here, a limited bandwidth was used to avoid regions of low intensity, due to the source, as seen in the inset of Fig. 2, and due to regions of substantial Hb attenuation at the lower wavelengths, as seen in Fig. 1. Based on analysis of the nonlinear phase, we obtained *C* = 39.80 +/− 1.00 g/L for case A, and *C* = 41.09 +/− 1.93 g/L for case B; both are in excellent agreement with the known Hb concentration. The measured and theoretical *Δn* values are shown in Fig. 2(a) for case A and in Fig. 2(b) for case B. Note that the theoretical and measured real RI profiles are in good agreement for both cases; however, some discrepancies are observed at the edges of the range of analysis for case B, which need further investigation. From these results, the imaginary part of the RI, and hence *μ _{a}*, may be determined independently of scattering. Since the calculated concentrations for both cases are approximately equal, the absorption coefficient for case B (not shown) is the same as

The scattering coefficient of the scattering sample (case B) can now be determined from the imaginary part of the RI. The obtained *μ _{a}* is subtracted from the total attenuation coefficient,

In conclusion, we have derived a relation that yields the real part of the RI from the non-linear phase term of interferometric signals in LCI. Using this relation, in combination with TF analysis, the absorption and scattering coefficients may be separated for samples that both scatter and absorb light, without *a priori* information. We validated the approach by analyzing the composition of Hb phantoms with and without scattering present. The results show that this method may help overcome the issues that currently prevent depth-resolved interferometric techniques from quantifying absorption in turbid samples. Further, this method may be readily extended to SOCT for imaging, in addition to allowing quantification of chromophores in blood and tissue *in vivo.*

This research has been supported by grants from the National Institutes of Health (NIH) (NCI 1 R01 CA138594-01).

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