## 1.

## Introduction

In single-fiber sensors, where a fiber is used to transmit both the excitation light and the signal from the analyte,^{1, 2} the signal from the analyte is often emitted or scattered on the outside surface of the fiber tip, i.e., the sample is in contact with the fiber probe, known as contact sensing. To increase the sensitivity of these sensors, it is desirable to increase the active surface area where the signal is generated and collected and to shape the probe in such a geometry that most of the signal light can be coupled into the transmission (confined) modes of the fiber. While multimode fibers can be used to increase the active surface area, it is not clear how the geometry of the fiber probe affects the total collectible optical power. In this paper, we analyze the total collectible optical power of various specially shaped multimode optical fiber probes for contact sensing. With the examples of spherical, parabolic, and cone-shaped fiber tips, we show that an optimal geometry exists for each type of end-polished fiber probes. This analysis can be used to guide the design and optimization of various fiber sensors, including fluorescence sensors,^{3} surface plasmon resonance sensors,^{4} surface enhanced Raman sensors,^{1, 2} etc.

## 2.

## Theoretical Model

The coupling efficiency from a source to an optical fiber has been calculated using geometrical approaches.^{5} However, most of the calculations on the shape-dependence of the fiber were calculated numerically using certain software, such as ZEMAX.

Consider a multimode fiber with its end polished to a well-defined (e.g., spherical) shape where the signal light is generated and coupled into the fiber (see Fig. 1). As the core diameter of the fiber is much larger than the wavelength of light, we use a geometrical optics model to study the total collectible optical power.^{5} For a long enough fiber, only those rays that satisfy the total internal reflection condition at the interface between the core and the cladding of the fiber can successfully reach the detector.

For a fiber probe with a specially shaped tip, we calculate the total collectible optical power as follows. Pick an arbitrary point on the fiber surface. Let $(x,y,z)$ be the local coordinate system, where the $z$ direction is normal to the local surface. In the case of a cylindrically symmetric fiber, such as a spherical one and the ones discussed later in this paper, we can have both the $x$ and $z$ axes in the cross-sectional plane containing the fiber axis. Consider an incident ray with the wave vector ${\mathbf{k}}_{\mathbf{0}}$ in air. In a local spherical coordinate system $(r,\theta ,\phi )$ , ${\mathbf{k}}_{\mathbf{0}}$ is written as

## Eq. 1

$${k}_{0x}={k}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\phi ,\phantom{\rule{1em}{0ex}}{k}_{0y}={k}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\phi ,$$If the angle between ${\mathbf{k}}_{\mathbf{1}}$ and ${z}_{0}$ is less than ${\theta}_{\mathrm{max}}$ , then this ray is confined inside the fiber core, where ${\theta}_{\mathrm{max}}$ is $(\pi \u22152)-{\theta}_{c}$ , with ${\theta}_{c}$ the critical angle for the total internal reflection at the interface between the fiber core and cladding. Supposing that the refractive indices of the core and the cladding are ${n}_{1}$ and ${n}_{2}$ , respectively, the confined rays must satisfy the following condition:

## Eq. 2

$${\mathbf{k}}_{1}\cdot {\mathbf{z}}_{0}\u2a7e\frac{2\pi {n}_{1}}{\lambda}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\theta}_{\mathrm{max}}=\frac{2\pi {n}_{1}}{\lambda}\cdot \frac{{n}_{2}}{{n}_{1}}=\frac{2\pi {n}_{2}}{\lambda}.$$Let $\alpha $ be the angle between ${z}_{0}$ and $z$ . In the $(x,y,z)$ coordinate, the unit vector in the ${z}_{0}$ direction can be written as ${\widehat{z}}_{0}=(\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha ,0,\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\alpha )$ , and Eq. 2 becomes:

## Eq. 3

$${k}_{1z}={k}_{1x}\phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha +{k}_{1z}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\alpha \u2a7e\frac{2\pi {n}_{2}}{\lambda}.$$Substituting Eq. 1 and the expression of ${k}_{1z}$ into Eq. 3, we can deduce:

## Eq. 4

$$\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\varphi \phantom{\rule{0.2em}{0ex}}\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\alpha +{({n}_{1}^{2}-{\mathrm{sin}}^{2}\phantom{\rule{0.2em}{0ex}}\theta )}^{1\u22152}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\alpha \u2a7e{n}_{2}.$$## Eq. 5

$$p(\alpha ,\beta )=\int \int T(\theta ,\phi )\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\mathrm{d}\theta \phantom{\rule{0.2em}{0ex}}\mathrm{d}\phi ,$$## Eq. 6

$$P=\int \int p[{x}_{0},{y}_{0},{z}_{0}({x}_{0},{y}_{0})]\mathrm{d}S=\int \int p[{x}_{0},{y}_{0},{z}_{0}({x}_{0},{y}_{0})]{(1+{z}_{0x}^{\prime 2}+{z}_{0y}^{\prime 2})}^{1\u22152}\mathrm{d}{x}_{0}\mathrm{d}{y}_{0},$$## 3.

## Results for Fiber Probes with Three Different Geometrical Shapes

Using typical multimode fiber parameters, i.e., the diameter of the core $D=62.5\phantom{\rule{0.3em}{0ex}}\mu \mathrm{m}$ and the refractive indices of the core and cladding ${n}_{1}=1.4800$ and ${n}_{2}=1.4546$ , the total optical power collectible by spherical, parabolic, and cone-shaped fiber tips are calculated and discussed in the following.

For a fiber probe with a spherical tip, the calculation result of the total collectible optical power [Eq. 6] is shown in Fig. 2. For simplicity, we have neglected any energy loss due to reflection, scattering, or absorption when the rays enter the fiber. Such loss can be included with the knowledge of the external medium. Furthermore, we have also assumed that each point on the outside surface of the fiber tip acts as a point source. This assumption can be modified with the knowledge of the specific radiation pattern of the light emitting∕scattering process involved in the sensing mechanism.

Notice that when $R\u2aa2r$ , the end surface becomes a flat plane. In this case, the active area where the signal is generated is the smallest compared with fibers having other shaped tips; however, the acceptance solid angle at any point is the largest. As the $R\u2215r$ value decreases, the active surface area increases; however, the acceptance solid angle decreases as we move away from the fiber axis. Figure 2 shows the collectible optical power for various radii of curvature normalized to that of a flat end. The results are obtained by numerically integrating Eq. 6 using MATLAB. As shown in Fig. 2, when $R\u2215r$ equals 1.4560, the collectible optical power reaches its maximum, which is 45.60% higher than that collectible by a fiber with a flat end.

The total collectible optical power using parabolic fiber tips is shown in Fig. 3. Again, using the flat tip as the reference, the total collectible optical power reaches its maximum when $d=0.008$ , i.e., the expression of the parabola is $z=-0.016{x}^{2}$ . As $d$ increases further, the tip of the fiber becomes sharper and the collectible optical power becomes smaller due to the decreased acceptance solid angles at points other than the vertex. For parabolic-shaped fiber probes, the maximum collectible optical power is 50.28% higher than that collectible with a flat-tip probe.

The total collectible optical power as a function of the half vertex angle of a cone-shaped fiber tip is shown in Fig. 4. When the half vertex angle of the cone tip is smaller than $36.899\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ , there is no collectable signal, because none of the rays entering from the side walls are confined. Notice that this is a result of the geometrical optics approximation for multimode fibers, which is no longer valid at the tip of a slowly tapered fiber. In this region, more accurate analysis needs to be carried out with the modal calculation of a tapered fiber. The total collectible optical power reaches its maximum, which is 119% larger than that collectible by a flat-ended fiber, when the half vertex angle equals $54.43\phantom{\rule{0.3em}{0ex}}\mathrm{deg}$ .

## 4.

## Conclusion

In summary, for contact sensing using an end-polished multimode optical fiber probe, we have developed a physical model to calculate and optimize the total collectible optical power using fiber tips with various geometrical shapes. The optimization parameters for fiber probes with spherical, parabolic, and cone-shaped tips are given. Compared to a flat-ended fiber probe, our results show that the total collectible optical power can be improved by a factor of 1.3 to 2.2 depending on the shape of the tips.

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