### 2.1 Single fiber probe

We start from a single, bare optical fiber, which can be used as light source delivery and signal collection. This is the simplest form of optical fiber based probe, but of important practical usage [8]. When light is incident onto the sample, it will be subject to specular reflection due to refractive index mismatching at the interface and diffuse reflection due to scattering. To study the collection efficiency, it can be divided into two separate processes: (1) implementation of light transport model in the tissue and (2) light coupling between the tissue and the fiber probe. Light transport in tissue has been studied [10], which can be modeled using Monte Carlo simulations [11]. We will focus on the light coupling issues between the tissue and the fiber probe.

Assuming light is illuminated onto a semi-infinite tissue (which is always the case for *in vivo* or *ex vivo* measurement), the total intensity escaping the medium surface is [8],

{I}_{esc}={I}_{0}{R}_{sp}+{I}_{0}\left(1-{R}_{sp}\right){R}_{diffuse}

(1)

where *I*
_{0} is the incident light intensity illuminated on the medium surface. *R*
_{
sp
} is the specular reflection of the tissue surface due to index mismatching. *R*
_{
diffuse
} is a dimensionless factor, called the total diffuse reflectance. For contact measurement, the signal collected by the fiber is given by,

{I}_{collect}={I}_{0}{R}_{sp}+{I}_{0}\left(1-{R}_{sp}\right){\int}_{S}{\int}_{S}T\left(r,{r}^{\prime}\right)d{A}^{\prime}dA={I}_{0}{R}_{sp}+{I}_{0}\left(1-{R}_{sp}\right){R}_{collect}

(2)

where *S* is the cross section of the fiber core, *T*(*r*) is the transport factor from the fiber through the tissue to a position *r* on the surface. *dA* and *dA'* indicate the incremental aperture area for delivery and collection. For a single fiber, the collection fraction, *f*, is defined as [8],

f=\frac{{I}_{collect}-{I}_{0}{R}_{sp}}{{I}_{esc}-{I}_{0}{R}_{sp}}=\frac{{R}_{collect}}{{R}_{diffuse}}

(3)

where *I*
_{
esc
} represents the light escaped from sample surface (including specular reflection). The light collected, *R*
_{
collect
}, should be split into two parts: the light that enters the optical fiber with an angle smaller than the half-angle of the acceptance angle (*R*
_{
core
}), and the light that enters the optical fiber with an angle larger than the half-acceptance angle (*R*
_{
cladding
}). *R*
_{
core
} is guided to the detector by the fiber core, and the *R*
_{
cladding
} is lost by fiber from fiber cladding. Equation (3) can be reduced as *f* = *R*
_{
core
}/*R*
_{
diffuse
}. Both *R*
_{
core
} and *R*
_{
diffuse
} can be determined numerically by Monte Carlo simulations [11]. This is a reverse problem in that the collection efficiency is determined by measuring the collected signal divided by the total signal simulated from the sample surface. In reality, particularly for probe designing, people want to design the probe so that it can collect as much signal as possible, given the signal (*R*
_{
diffuse
}) from the surface is known (or constant).

As shown in Figure 1, if the fiber is not contact with the tissue, the illumination and collection efficiency must be accounted for. In this case, because the energy from the fiber surface equals to the energy illuminated on the tissue surface, the intensity on the tissue surface *I*
_{0} can be written as,

{I}_{0}={I}_{out}{r}^{2}cos\beta \u2215{\left(r+dtan\theta \right)}^{2}

(4)

where *I*
_{
out
} is the intensity out of the fiber surface. *r* is the radius of the fiber core, *d* is the distance from the fiber tip to the tissue surface (center of fiber core to the tissue surface along optical axis of the fiber). *θ* is the acceptance angle, determined by the numerical aperture of the fiber, *θ* = arcsin(*NA*/*n*
_{0}). If there is no water or other medium between fiber probe and tissue surface, *n*
_{0} is refractive index of air *n*
_{0} = 1. NA is the numerical aperture of the fiber. For the commonly used fiber with NA = 0.22, the acceptance angle is 12.7°. *β* is the tilt angle of the fiber probe. Notice that the intensity on the sample surface maximizes for normal illumination (*β* = 0), because the illumination surface is the smallest for any given probe-sample distance. In equation (4), we assume the light is uniformly illuminated on the tissue surface.

Note that all the mirror-reflected light from the tissue surface enters the fiber. For any point on the tissue surface, only the signal which hits the fiber core surface enters the fiber, and only the signal which has a smaller incident angle than the acceptance angle can transmit in the fiber. As a first-order approximation, the solid angle, Ω, that a signal can enter the fiber core is given by,

\text{\Omega}=1-{d}^{\prime}\u2215\sqrt{{d\prime}^{2}+{r}^{2}}

(5a)

where,

{d}^{\prime}=d+\left(r+dtan\theta \right)\frac{sin\beta tan\beta tan\theta}{{cos}^{2}\beta -{sin}^{2}\beta {tan}^{2}\theta}

(5b)

When the distance from the fiber probe to the tissue surface is 0, all the surface signal within the illumination area is collected, although some of the light is lost during propagation when the angle is larger than the acceptance angle. Note that when there is no tilt angle, *β* = 0, and *d'* = *d*. As the specular reflection does not provide any information about the tissue, we assume *R*
_{
sp
} = 0 for the following analysis. This is particularly useful for fluorescence and Raman measurement. Combining equations (1)-(5), one can obtain the signal that the fiber collects from the tissue surface, given by,

{R}_{collect}=\frac{{\left(r+{d}^{\prime}tan\theta \right)}^{2}{I}_{esc}}{{r}^{2}{I}_{out}}(1-\frac{{d}^{\prime}}{\sqrt{{d}^{\prime 2}+{r}^{2}}})

(6)

In Equation (6), *I*
_{
esc
} is determined by the tissue properties that can not affected by the fiber probe. *I*
_{
out
} is the illumination intensity from the fiber output which is determined by the laser power.

The dependence of the collection efficiency of single fiber probes on the probe distance is shown in Figure 2. It can be seen that contact measurement has the highest collection efficiency for single fiber probes. The collection efficiency also heavily depends on the diameter of the fiber core. This dependence is larger for smaller core-diameter fibers than for larger core-diameter fibers, e. g. the collection efficiency decreases to 20% of the contact measurement for a 50 μm fiber when the probe distance is 100 μm, while the collection efficiency decreases down to 20% of the contact measurement at a longer probe distance (1200 μm) for a 500 μm fiber. The collection efficiency also depends on the probe-sample angle. For any given probe-sample distance, the collection efficiency may be increased at a small tilt angle. The collection efficiency will eventually drop when the tile angle is too large. But this increase or decrease is minimal comparing with the probe-sample distance. In general, a tolerance of 15 degrees is acceptable for single-fiber probes.

### 2.2 Multi-fiber probe

For multi-fiber probes, the analysis is similar to those of single fiber probes. But because no specular reflection is collected in multi-fiber probes, *R*
_{
sp
} = 0. So the collection fraction in Eq. (3) can be rewritten as,

f=\frac{{I}_{collect}}{{I}_{esc}}=\frac{{R}_{collect}}{{R}_{diffuse}}

(7)

For contact measurement, the illumination area and the collection area are not overlapped. In this case, the signal is purely diffuse reflectance signal. Based on the distance of fibers, signal at different depth in the tissue can be acquired [5]. When increasing the probe distance from the tissue surface, the illumination and collection area may be overlapped. The overlapped area over the total illumination area are given by,

\frac{{S}_{collect}}{{S}_{illumi\phantom{\rule{2.77695pt}{0ex}}nation}}=\frac{4}{\pi}arccos\frac{r+{r}_{c}}{r+{d}^{\prime}tan\theta}-\frac{2\left(r+{r}_{c}\right)}{\pi {\left(r+{d}^{\prime}tan\theta \right)}^{2}}\sqrt{{\left(r+{d}^{\prime}tan\theta \right)}^{2}-{\left(r+{r}_{c}\right)}^{2}}

(8)

Combining equations (1), (4-5) and (7-8), one can obtain the signal that the fiber collects from the tissue surface in a multi-fiber probe, written as

{R}_{collect}=\frac{{(r+{d}^{\prime}tan\theta )}^{2}{I}_{esc}}{{r}^{2}{I}_{out}cos\beta}(1-\frac{{d}^{\prime}}{\sqrt{{d\prime}^{2}+{r}^{2}}})\times \frac{4arccos\frac{r+{r}_{c}}{r+{d}^{\prime}tan\theta}{\left(r+{d}^{\prime}tan\theta \right)}^{2}-2\left(r+{r}_{c}\right)\sqrt{{\left(r+{d}^{\prime}tan\theta \right)}^{2}-{\left(r+{r}_{c}\right)}^{2}}}{\pi {\left(r+{d}^{\prime}tan\theta \right)}^{2}}

(9)

In equation (9), *I*
_{
esc
} is determined by the tissue properties that can not be affected by the fiber probe. *I*
_{
out
} is the illumination intensity out of the illumination fiber, which is determined by the laser power. *r* is the radius of the fiber, *d'* is the equivalent fiber probe distance defined by equation (5b), *r*
_{
c
} is the distance between the illumination fiber and the collection fiber. *θ* is determined by the fiber NA. In the above analysis, we assume both the illumination and collection have same numerical apertures.

The collection efficiency of multi-fiber probes is shown in Figure 3. It can be seen that there is an optimal probe distance in multi-fiber probes. For a 100- μm fiber, the optimal probe distance is around 20 μm; for a 1000 μm fiber, the optimal probe distance is 700 μm. Comparing with contact measurement, the improvement of multi-fibers is over 5 fold. As shown in equation (9), the collection efficiency also depends on the distance between illumination and collection fiber. In the above analysis, the distance between illumination and collection fibers is assumed to be 10 μm. Figure 3b shows the collection efficiency of multi-fiber probes at different illumination and collection angle. It can be seen that the collection efficiency is increased, because there is more overlap of illumination area and collection area for a tilted multi-fiber probes.

### 2.3 Fluorescence measurement

In order to verify the analysis, we measured the fluorescence of a human finger skin *in vivo* using a multi-fiber probe configuration. The experiment is schematically shown in Figure 4. A laser with *λ* = 450 nm was used. The light was collimated, filtered by a band-pass filter (450 ± 1 nm) and coupled into a 400- μm delivery fiber. The light is delivered to the skin at a probe-sample distance of *d*. The laser power on skin is 2.5 mW, well below the ANSI standard. The fluorescence signal is collected by another six 400- μm fibers surrounding the excitation fiber. The collected fluorescence signal was collimated and filtered by a long pass filter (470 nm LP) and delivered to the spectrometer (USB2000+VIS-NIR, Ocean Optics, FL, USA). The probe distance can be adjusted so that the fluorescence at different probe distance can be recorded.