2.1.

Sample Preparation

To verify the concept of oblique fibers, a two-layer tissue-mimicking phantom was constructed for use in the experiments. The phantom consisted of a $5\text{-}\mathrm{cm}$ -thick solid epoxy-based scattering phantom layer underneath a volume of Intralipid solution of controlled thickness. A 20% Intralipid solution (Sigma) was diluted to a final concentration of 2% w/v. The optical properties of Intralipid and the epoxy-based phantom were obtained by measuring reflectance and transmittance using a spectrometer with an integrating sphere and applying the inverse adding-doubling (IAD) method to calculate the absorption and scattering coefficients. The measurement procedure and the IAD method have also been described in the literature.²³

The absorption properties of Intralipid are dominated by water, for which the most intense spectral bands are located around $1199\mathrm{nm}$ , $1439\mathrm{nm}$ , and $1932\mathrm{nm}$ . The absorption coefficient of Intralipid and the epoxy-based phantom are given in Figs. 3a and 3b, respectively. The most prominent spectral bands of the epoxy-based material can be seen around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , $1392\mathrm{nm}$ , $1432\mathrm{nm}$ , $1476\mathrm{nm}$ , $1546\mathrm{nm}$ , $1677\mathrm{nm}$ , $1697\mathrm{nm}$ , $1721\mathrm{nm}$ , and $1916\mathrm{nm}$ . The strongest spectral features—those around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , and $1721\mathrm{nm}$ —are used here as specific signatures of the epoxy-based phantom.

Fig. 3

Absorption coefficients of materials composing a two-layer phantom: (a) epoxy-based layer; (b) 2% w/v Intralipid solution. The absorption properties of Intralipid are dominated by water, where the most intense spectral bands are located around $1199\mathrm{nm}$ , $1439\mathrm{nm}$ , and $1932\mathrm{nm}$ . The most prominent spectral bands of the epoxy-based material can be seen around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , $1392\mathrm{nm}$ , $1432\mathrm{nm}$ , $1476\mathrm{nm}$ , $1546\mathrm{nm}$ , $1677\mathrm{nm}$ , $1697\mathrm{nm}$ , $1721\mathrm{nm}$ , and $1916\mathrm{nm}$ . The most strong spectral features of the epoxy-based material around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , and around $1721\mathrm{nm}$ are marked with arrows and will be used through this article as the specific signature of the epoxy-based phantom.

The reduced scattering coefficients of the Intralipid solution and the epoxy-based phantom were on the order of $1{\mathrm{mm}}^{-1}$ , respectively, showing a typical wavelength dependency.

The two-layer phantom was assembled in the following way. A $125\text{-}\text{micrometer}$ -thick metal ring was positioned on the solid epoxy-based phantom and was filled with Intralipid solution. A coverslip ( $22\times 22\mathrm{mm}$ , $130\text{to}160\text{μm}$ thick, Esco) was put on top of the metal ring to ensure that the Intralipid layer was of a constant thickness. Optical contact between the illumination and the collection fibers was improved by using a drop of carbo-fluoride-based immersion fluid (FC-40, $n=1.29$ ), which is optically transparent in the near-infrared (NIR) spectral region, although it should be noted that the use of immersion fluid does not eliminate reflectance at interfaces between the glass-Intralipid and Intralipid-epoxy layer (Intralipid $n=1.45$ , $\text{glass}=1.5$ , $\text{epoxy}=1.56$ ). Figure 1 shows a schematic diagram of the assembled two-layer phantom with the optical fibers.

2.2.

Experimental Setup

A specialized unit for use in diffuse reflectance measurements of tissue (see Fig. 1) consisted of a sample stage, above which two optical fibers were mounted at a controlled height and center-to-center separation distance via microadjustment screws and a stepper motor. The fiber mounting unit setup was capable of maintaining a specific angular orientation of both optical fibers. Optical fibers of $600\text{-}\text{micrometer}$ core diameter were used in the experiment (Avantes BV). The center-to-center distance including the cladding was $820\text{μm}$ .

All measurements with the oblique fiber unit were performed using an NIR spectrometer, which included a stabilized halogen lamp source, a monochromator (SpectraPro-150, Acton Research Corporation), and a liquid nitrogen–cooled InGaAs detector (OMA V:1024-2.2, Princeton Instruments; see Fig. 2). The diffraction grating of the spectrometer provided dispersion of $37.69\mathrm{nm}∕\mathrm{mm}$ , which covers $965\mathrm{nm}$ spectral range on the detector. A long-pass filter with a cutoff wavelength of $1000\mathrm{nm}$ was used in the illumination path to reject wavelengths below $1000\mathrm{nm}$ and to prevent overlap of higher diffraction orders on the light detector.

2.3.

Data Acquisition and Processing

The optical spectra of the two-layer phantom were acquired and processed in the following way. First, the background signal was measured with no sample in place and no light falling on the detector: the main components of the background noise are thermal noise of the detector and thermal radiation from the environment. A signal was recorded during $20\mathrm{ms}$ with 2000 accumulations. This was followed by acquiring the spectrum of the light source back-reflected from a reflectance standard (Labsphere, certified reflectance standard).

Each acquired spectrum was subsequently corrected for the spectrally dependent setup throughput using the previously acquired spectrum of the light source. Wavelength calibration, i.e., a spatial relation between the detector pixels and a particular wavelength, was made by measuring the first and the second diffraction orders of HeNe laser and the diffuse back-reflectance spectrum of polystyrene, which is typically used for this purpose.²⁴

4.1.

Experimental Data

The aim of the first experiment was to demonstrate the dependency of the diffuse back-reflectance spectra on the angle of orientation of the illuminating fiber. In this experiment, the thickness of the Intralipid film positioned above the solid epoxy-based phantom was equal to $125\text{μm}$ . As mentioned earlier, the optical properties, namely—the absorption coefficient—of this two-layer phantom are layer dependent, which can be observed in the corresponding spectra (see Fig. 3).

The expectation was that an increase in the angle would result in the probing of a more shallow volume and therefore (1) an increase in the overall reflectance, (2) an increase in the signal at the position of the highest absorption bands, and (3) a decrease in the contribution from the epoxy phantom.

As is revealed by the spectra [see Figs. 4a, 4b, 4c, 4d ], it is indeed the case that overall reflectance increases with increasing angle. This is conclusive with the confinement of light to more superficial layers.

Fig. 4

Dependence of diffuse back-reflectance spectra of a two-layer phantom on the inward tilt of the illumination fiber. The angle of the illumination fiber was (a) $0\mathrm{deg}$ ; (b) $20\mathrm{deg}$ ; (c) $40\mathrm{deg}$ ; and (d) $60\mathrm{deg}$ . The thickness of the Intralipid layer was $125\text{μm}$ . All the spectra were acquired as described in Sec. 2.3 and were later normalized to unity. The spectral regions where the specific signature of the epoxy-based phantom is visible—namely, around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , and $1721\mathrm{nm}$ —are marked with dotted lines and arrows.

It is also shown that an increase in angle results in a decrease in water absorption, which can be observed as a decrease in absorption around $1200\mathrm{nm}$ and reduced saturation of the detector around $1439\mathrm{nm}$ and $1920\mathrm{nm}$ .

Second, the increase in angle results in first increased and, later, decreased definition of the spectral features of the phantom around $1200\mathrm{nm}$ and $1700\mathrm{nm}$ [compare Figs. 4b and 4c versus 4d, respectively]. This is because an increase in the angle of the illumination fiber results in a decrease in saturation of the detector caused by high absorption in the epoxy-based layer, thus revealing sharply the spectral features of the phantom. As the angle of the fiber increases further, the detected light is confined to the superficial layer (i.e., Intralipid), resulting in less pronounced absorption features of the epoxy layer.

In a second investigation, the effect of varying the angle of orientation of both fibers was compared with varying the orientation of only the illumination fiber, again with a fixed fiber-to-fiber separation distance of $820\text{μm}$ and Intralipid sample thickness of $125\text{μm}$ . The obtained spectra can be seen in Fig. 5 . The expectation here was a more pronounced effect than that obtained in the case of only a single obliquely oriented fiber.

Fig. 5

Comparison of the diffuse back-reflectance spectra of a two-layer phantom acquired using (a) perpendicular-oriented fibers; (b) $20\mathrm{deg}$ inward tilt of the illumination fiber; and (c) $20\mathrm{deg}$ tilt of both illumination and detection fibers. The thickness of the Intralipid layer was $125\text{μm}$ . All the spectra were acquired as described in Sec. 2.3 were later normalized to unity. The spectral regions where the specific signature of the epoxy-based phantom is visible—namely, around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , and $1721\mathrm{nm}$ —are marked with dotted lines and arrows.

An important difference is observed when comparing traces (b) and (c), when either one [trace (b)] or both [trace (c)] fibers are oriented at an angle of $20\mathrm{deg}$ . There is more reflectance in the case when both fibers are oriented obliquely and less saturation of the signal at the $1439\mathrm{nm}$ and $1920\mathrm{nm}$ water band than in the case of the single obliquely oriented fiber. Furthermore, the contribution of the epoxy phantom is further decreased compared to the situation described by trace (a). This confirms our hypothesis that by obliquely orienting two fibers simultaneously, even greater depth selectivity can be achieved, i.e., the spectral features of more superficial layers can be measured in absence of (or with reduced interference from) deeper layers. The results of this experiment indicate the advantages of a dual obliquely oriented fiber arrangement, as opposed to a singular obliquely oriented fiber, such as described in Ref. 15.

In a third experiment, we compared an obliquely oriented fiber pair, set at an angle of $25\mathrm{deg}$ outward, with a dual perpendicular arrangement [see Fig. 1b]. The aim was to compare the effect of directing the light into the sample such that the light cones did not overlap or deliberately confine light to shallow layers. It was previously postulated that this could result in a deeper overall penetration capability, with better-defined spectral features of the deeper layers of the sample due to the reduction of interference from the more superficial layers.¹⁵

In Fig. 6, the results of a perpendicular arrangement [trace (a)] can be seen plotted against the outwardly obliquely oriented fiber pair [trace (b)]. The significance of the result is in the degree of selectivity in probing depth by only obliquely orienting the fibers: in the case of the dual perpendicular arrangement, a more superficial layer is probed, which is observed as a dominant contribution of Intralipid with little contribution of the epoxy phantom below. In the case of dual outward orientation, a deeper layer, i.e., the epoxy phantom, is probed (seen as an increase in the sharp, phantom-specific spectral features around $1200\mathrm{nm}$ ). This effect is visible only in spectral regions having a low absorption coefficient (e.g., around $1200\mathrm{nm}$ ) because outside of these regions, the photons are already being absorbed in more superficial layers. Note that the data was normalized to unity and shifted such that it could be displayed clearly and without overlapping.

Fig. 6

Comparison of the diffuse back-reflectance spectra of a two-layer phantom acquired using (a) perpendicular-oriented fibers; and (b) $25\mathrm{deg}$ outward tilt of both illumination and detection fibers. The thickness of the Intralipid layer was $125\text{μm}$ . All the spectra were acquired as described in Sec. 2.3 and normalized to unity. The graphs are shifted along the $y$ axis for clarity. The spectral regions where the specific signature of the epoxy-based phantom is visible—namely, around $1190\mathrm{nm}$ , $1140\mathrm{nm}$ , and $1721\mathrm{nm}$ —are marked with dotted lines and arrows.

4.2.

Monte Carlo Simulations

4.2.1.

Dependence of the mean penetration depth and variation thereof on fiber angle

The results of the Monte Carlo simulations were used to calculate the mean penetration depth, the variation in the mean penetration depth, and the number of scattering events of the detected photons for the two-layer phantom, as a function of fiber angle and wavelength.

The resulting spectra and the corresponding estimated parameters are given in Table 1 and Figs. 7 and 8 , respectively. The data presented in Table 1 demonstrate the following trends:

• • As the inward tilt of an illumination fiber increases, the maximal penetration depth, the mean penetration depth, and the spread in the penetration decrease (see Table 1).

• • With an increase in the inward tilt of an illumination fiber, the distribution of the number of the scattering events of the detected photons is shifted toward lower values (see Fig. 8).

• • For a large fiber angle of $60\mathrm{deg}$ , the mean penetration depth is restricted to the Intralipid layer, i.e., to $125\text{μm}$ at $1459\mathrm{nm}$ , with minor penetration into the epoxy phantom at $1187\mathrm{nm}$ and $1697\mathrm{nm}$ (see Table 1).

• • An increase in the inward angle of the illumination fiber results in a decrease in the variation of the penetration depth—for example, for a $60\mathrm{deg}$ inward angle, we observed an approximately twofold decrease in the variation of the penetration depth when compared to an angle of $0\mathrm{deg}$ (see Table 1).

Fig. 7

Dependence of diffuse back-reflectance spectra of a two-layer phantom on the inward tilt of the illumination fiber, which was (a) $0\mathrm{deg}$ ; (b) $20\mathrm{deg}$ ; (c) $40\mathrm{deg}$ ; and (d) $60\mathrm{deg}$ . The spectra were obtained using Monte Carlo simulation. The thickness of the Intralipid layer was $125\text{μm}$ . The data were normalized to unity.

Fig. 8

The number of photon scattering events in the two-layer phantom for (a) perpendicular-oriented fibers; and (b) $60\mathrm{deg}$ inward orientation of the illumination fiber. The data were obtained by analyzing the photon tracks resulting from the Monte Carlo simulation at $1459\text{-}\mathrm{nm}$ wavelength. Note that unscattered photons result from reflectance at the interface between glass and immersion fluid (see Fig. 1).

Table 1

Comparison of the mean penetration depths and the variation in the mean penetration depth [see Eqs. 1, 2] for the two-layer tissue-mimicking phantom as obtained using Monte Carlo simulations for different fiber angles and different wavelengths. Inward and outward fiber tilt is noted as positive and negative angles, respectively. The thickness of the Intralipid layer was constant and equal to 125μm .

Angle (deg)	Mean penetration depth (μm)	Variation in penetration depth (μm)
$1187\mathrm{nm}$
0	533	340
20	374	308
40	317	275
60	191	219
$-10$	526	410
$-25$	540	386
$-50$	538	462
$1459\mathrm{nm}$
0	415	320
20	283	275
40	221	203
60	100	148
$-10$	428	347
$-25$	456	396
$-50$	432	386
$1697\mathrm{nm}$
0	422	283
20	302	259
40	268	212
60	145	148
$-10$	457	319
$-25$	456	323
$-50$	451	332
$-10$	520	398
$-25$	538	418
$-50$	577	493

Thus, the general results of these simulations support the assumption made earlier—namely, that by varying the angle of orientation of a fiber, the depth to which a sample is probed can be varied accordingly. Moreover, an increase in inward angle of an illumination fiber also results in a decrease in a variation of the penetration depth. The changes in light interaction with the sample when using an obliquely oriented illumination fiber are also characterized by a decreased number of scattering events, as shown in Fig. 8.

The spectral data obtained using Monte Carlo simulations (see Fig. 7), qualitatively support our experimental results presented in Sec. 4.1 (see Fig. 4). Namely, (1) as the fiber angle increases, the contributions of both water and epoxy phantom in the spectrum decrease, (2) the total amount of light back-reflected from the sample increases, and (3) there is no significant contribution of the epoxy phantom visible in the spectrum at $60\mathrm{deg}$ fiber angle [see Figs. 4d and 7d], which manifests as a diminished epoxy-specific spectral signature at $1200\mathrm{nm}$ and its total absence at around $1700\mathrm{nm}$ . In order to better distinguish the differences in contribution as separate components, a more rigorous analysis can be performed such as the chemometric techniques employed by Pham and Berger ^{26, 27}

4.2.2.

Comparison of a variable-angle oblique arrangement with a variable-distance perpendicular arrangement

Our next step was to test whether the effect of using an inward oblique arrangement could be approximated to a decrease in fiber-to-fiber separation distance in the case of a perpendicular fiber arrangement. Our approach was to verify that the spectral intensity for the obliquely oriented arrangement could be matched to that of the perpendicular arrangement. In particular, the aim was to determine the fiber-to-fiber separation distance required in the perpendicular arrangement for match in light intensity (outside strong absorption bands) with a single obliquely oriented arrangement at a fixed fiber-to-fiber separation distance.

For this purpose, we used Monte Carlo simulations to simulate the spectral response of a single-layer Intralipid-based phantom of $5\mathrm{mm}$ thickness. The calculations were performed for a fiber diameter of $100\text{μm}$ . The former was done to simplify the shape of the spectral response and as a result the interpretation thereof, while the latter decreased the spread in the photon paths.

In the first case, the illumination fiber was positioned at a fixed $500\text{-}\text{μm}$ distance with respect to the detection fiber, but at varying angles of 10, 20, 30, 40, 50, and $60\mathrm{deg}$ . In the second case, two perpendicular fibers were used at varying distances.

Figure 9 shows an example of the spectrum: the intensity obtained using an inward obliquely oriented arrangement at an angle of $60\mathrm{deg}$ and fiber-to-fiber separation distance of $500\text{μm}$ is matched (outside of strong absorption bands) to the intensity obtained by using a perpendicular oriented arrangement with a fiber-to-fiber separation distance of $200\text{μm}$ .

Fig. 9

Comparison of the diffuse back-reflectance spectra of a homogeneous Intralipid-based phantom obtained using (a) an obliquely oriented arrangement at $60\text{-}\mathrm{deg}$ inward tilt with $500\text{-}\text{micrometer}$ fiber-to-fiber distance and (b) a perpendicular arrangement with $200\text{-}\text{micrometer}$ fiber-to-fiber distance. The spectra were obtained using Monte Carlo simulation. $100\text{-}\text{micrometer}$ -diameter fibers were used in this simulation. The data were normalized to unity.

For all other cases, the required fiber-to-fiber separation for the perpendicular arrangement, which gives the same spectral intensity as an obliquely oriented fiber arrangement at fixed separation distance, is shown in Table 2 . Note that the correspondence between an inward tilt of the fiber and the shift in the separation distance between the perpendicular fibers is calculated for the given scattering and absorption coefficient of the sample as well as for the given refractive indices of the sample and surrounding medium. Yet, in general, the required decrease in the fiber-to-fiber distance in the case of a perpendicular arrangement will follow a cosine law with respect to the angle corresponding to the inward tilt, as supported by the data shown in Table 2.

Table 2

Comparison of the fiber-to-fiber separation distances for the oblique arrangement at varying inward tilt and the perpendicular arrangement and varying fiber-to-fiber distance.

Inward tilt (deg)	Fiber-to-fiber distance foroblique arrangement (μm)	Fiber-to-fiber distance forperpendicular arrangement (μm)
60	500	200
50	500	220
40	500	260
30	500	310
20	500	360
10	500	550

The corresponding mean penetration depths at the wavelengths outside as well as within the strong absorption band ( $1136\text{-}\mathrm{nm}$ and $1459\text{-}\mathrm{nm}$ wavelengths, respectively) are given in Table 3 .

Table 3

Comparison of the mean penetration depth outside and within the strong absorption band for the oblique arrangement at varying inward tilt and the perpendicular arrangement and varying fiber-to-fiber distance.

Mean penetration			Mean penetration
Inwardtilt (deg)	Fiber-to-fiberdistance forobliquearrangement (μm)	1136nm	1459nm	Fiber-to-fiberdistance forperpendiculararrangement (μm)	1136nm	1459nm
60	500	265	109	200	262	88
50	500	285	109	220	283	89
40	500	291	112	260	308	93
30	500	338	115	310	321	97
20	500	365	116	360	351	103
10	500	367	112	550	334	110

The data presented in Fig. 9 and Table 2 confirm that indeed the effect of using an inward oblique arrangement can be approximated to a decrease in fiber-to-fiber separation distance in the case of a perpendicular fiber arrangement. Note that a significant reduction of fiber-to-fiber separation between perpendicular fibers is required to achieve the same intensity as yielded when using a single obliquely oriented arrangement.

Furthermore, although this approximation is valid in the regime of weak or no absorption $({\mu }_a⪡{\mu }_s^{\prime })$ , it does not hold completely for the regions of high absorption. This observation is based on the data presented in Fig. 9 and Table 3. Namely, as can be seen in Fig. 9, while outside the strong absorption band there is an intensity match between the spectra obtained using an obliquely and perpendicular oriented fiber arrangement, no such match is observed at the position of the strong absorption band, i.e., around $1440\mathrm{nm}$ . In particular, absorption was slightly higher for the case of the oblique arrangement. This mismatch in the intensity was highest for the largest angle of the inward tilt (in our case, $60\mathrm{deg}$ ) and decreased with decreasing inward tilt. Table 3 shows the mean penetration depths for the obliquely and perpendicularly oriented cases calculated both outside and within the strong absorption—namely, at $1136\mathrm{nm}$ and $1459\mathrm{nm}$ . It can be seen that the effect of using an inward oblique arrangement can indeed be approximated to a decrease in fiber-to-fiber separation distance, in the case of a perpendicular fiber arrangement—there is a good correspondence in mean penetration depth between the oblique and the perpendicular arrangements at $1136\mathrm{nm}$ , i.e., outside the strong absorption band. However, there is a mismatch between the calculated mean penetration depths corresponding to the $1459\text{-}\mathrm{nm}$ wavelength. The penetration depth is larger for the inward oriented oblique configuration when compared to that of the perpendicular arrangement. This difference is more pronounced at larger angles, e.g., the corresponding difference is around 1% for the $10\mathrm{deg}$ angle and reaches 20% for angles of 50 and $60\mathrm{deg}$ . This increase in penetration depth corresponds to an increase in light absorption in the case of the oblique arrangement, as shown in Fig. 9. This effect can be accounted for as follows. First, by launching light into the sample under an oblique angle, the virtual isotropic point source, to which the beam can be approximated, is located closer to the surface. In other words, light is more confined to the superficial layers (see Ref. 2). The larger the angle, the more confined the light is to the surface of the sample. Therefore, to achieve an intensity match between an obliquely and a perpendicularly oriented arrangement, one needs to significantly decrease the fiber-to-fiber separation of the perpendicular-oriented fibers.

As a result, the larger fiber-to-fiber separation distance in the case of the inward obliquely oriented arrangement yields a longer average photon trajectory. This in turn increases the probability of an absorption event, which results in stronger light absorption within the absorption band (see Fig. 9).