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The continuous emerging of novel manufacturing techniques, either top-down or bottom-up [1], enables a more and more precise control of the growth of structures at nanometric scale, which has motivated new and numerous applications in recent decades [2].

Particularly, the use of these novel manufacturing techniques for the fabrication of sensitive structures on fiber optic is very useful, since it allows a very precise design of the characteristics of the structure by adjusting a few parameters [3]. This has enabled the development of a wide variety of fiber optic devices, which have bridged the gap between optic and electronic devices in virtually all the disciplines, such as chemical, physical, biochemical or biological sensors [4]. These devices are based on different operating principles, such as fluorescence (see our previous blog post about fluorescence here), evanescent field (see our previous blog post about evanescent field here) or interferometry among others.

Interferometric devices have been commonly used with high coherence sources such as lasers (see our blog post about coherence here). However, the utilization of novel manufacturing techniques, such as the Layer-by-Layer electrostatic self assembly [2] permit to create ultra-thin structures of submicrometric thickness on the tip of the optical fiber, which will act as an interferometer [4]. These nanocavities are susceptible of being excited using a common broadband light source because the thickness of the nanocavities formed at the tip of the optical fiber are narrower than the coherence length of common broadband light sources, such as LED or SLED light sources (in the order of a few microns) [5].

Now, if we focus on a structure fabricated at the perpendicularly cleaved end of an optical fiber we can observe that it is quite similar to the geometry of a Fabry-Perot interferometer (FPI) formed by two flat and parallel mirror surfaces separated from each other by a cavity of nanometric thickness ($\inline&space;d$), as it is depicted in Figure 1 [6].

Figure 1: Schematic representation of a Fabry-Perot interferometer formed by a nanocavity fabricated at the perpendicularly cleaved end of an optical fiber

Therefore, the reflected waves at both mirrors ($R_{1}$ and $R_{2}$) will interfere each other in a constructive or destructive way as a function of the nanocavity size and the wavelength of the wave, among other parameters. Thus, for a given wavelength and optical constants it is possible to obtain the cavity length and vice versa, for a given cavity length and optical constants it is possible to obtain the wavelength that produces the interference at maximum or minimum attenuation. This phenomenon can be easily exploited for sensor fabrication provided that we have a nanocavity that modifies its size as a function of the measurand.

Before analyzing mathematically the optical system formed at the tip of the fiber, it is assumed that the coherence length of the incident light is greater than the equivalent optical length (optical path length), which enables the generation of the interference phenomenon. In addition, it is considered that $\inline&space;n_{2}>n_{1}>n_{3}$, something that will happen in most of the fiber optic sensing applications. As it was previously indicated, it is important to remark that in the case of nanometer thickness cavities it will be possible to use low coherence light sources or broadband light sources.

Then, it can be assumed that the phase shift of the incident light is $\inline&space;\pi$ radians when reflected in the first mirror ($\inline&space;n_{1}) and null when reflected in the second mirror ($\inline&space;n_{2}>n_{3}$) [7].

Following previous assumptions, an incident light beam of amplitude $E_{0}$ at the interface between the media Fiber optic/nanocavity will generate a refracted and a reflected beam. The refracted beam will be attenuated by $\inline&space;\sqrt{T_{12}}$, where $\inline&space;T_{12}$ is the transmission coefficient of the first mirror. The reflected beam will return to the optical fiber attenuated by $\inline&space;\sqrt{R_{1}}$  and with a phase offset of $\inline&space;\pi$ radians, with $\inline&space;R_{1}$ being the reflection coefficient first mirror, as it expressed in Eq. 1. In the same manner, the refracted beam generated earlier will generate a refracted and a reflected beam at the interface between the nanocavity and the external medium as it is represented in Figure 1.

$R_{1}=&space;\frac{\left(&space;n_{1}-n_{2}&space;\right)&space;^{2}}{&space;\left(&space;n_{1}+n_{2}&space;\right)&space;^{2}}&space;\:&space;\:&space;\:&space;\:&space;\:&space;\:&space;\:&space;\:&space;R_{2}=&space;\frac{\left(&space;n_{2}-n_{3}&space;\right)&space;^{2}}{&space;\left(&space;n_{2}+n_{3}&space;\right)&space;^{2}}\:&space;\:&space;\:&space;\:&space;\:&space;\:&space;\:&space;\:&space;(1)$

If we consider now the effects associated to the attenuation losses in the nanocavity, we can express the intensity of the optical field that reaches the nanocavity / external medium interface after crossing the nanocavity as:

$E_{o}\cdot\,\sqrt[]{T_{12}}&space;\cdot&space;e^{-&space;\alpha&space;d}&space;\cdot&space;e^{-j\frac{&space;\phi&space;}{2}}&space;\:&space;\:&space;\:&space;\:&space;(2)$

where $\inline&space;\alpha$ is the absorption coefficient of the medium, $\inline&space;d$ is the thickness of the nanocavity and $\inline&space;\phi$ is the round trip phase shift in the interferometer expressed in Eq. 3.

$\phi&space;=\frac{4&space;\pi&space;\cdot&space;n_{2}&space;\cdot&space;d}{&space;\lambda&space;}&space;\:&space;\:&space;\:&space;\:&space;(3)$

Thus, the optical field intensity reflected into the fiber can be obtained as the sum of all the reflections, as shown in Eq. 4.

$E_{R}=E_{0}&space;\left(&space;\begin{matrix}&space;-\:\sqrt[]{R_{1}}\\&space;+\:\sqrt[]{T_{12}}&space;\cdot&space;\sqrt[]{R_{2}}&space;\cdot&space;\sqrt[]{T_{21}}e^{-&space;\alpha&space;2d}&space;\cdot&space;e^{-j&space;\phi&space;}\\&space;+\:&space;\sqrt[]{T_{12}}&space;\cdot&space;\sqrt[]{R_{2}}&space;\cdot&space;\sqrt[]{R_{1}}&space;\cdot&space;\sqrt[]{R_{2}}&space;\cdot&space;\sqrt[]{T_{21}}e^{-&space;\alpha&space;4d}&space;\cdot&space;e^{-j2&space;\phi&space;}\\&space;+\:&space;\sqrt[]{T_{12}}&space;\cdot&space;\sqrt[]{R_{2}}&space;\cdot&space;\sqrt[]{R_{1}}&space;\cdot&space;\sqrt[]{R_{2}}&space;\cdot&space;\sqrt[]{R_{1}}&space;\cdot&space;\sqrt[]{R_{2}}&space;\cdot&space;\sqrt[]{T_{21}}e^{-&space;\alpha&space;6d}&space;\cdot&space;e^{-j3&space;\phi&space;}\\&space;\cdots&space;\\&space;\end{matrix}&space;\right)&space;\:&space;\:&space;\:&space;(4)$

Assuming without much error that $\inline&space;\sqrt{T_{12}}\cong\sqrt{T_{21}}\cong\sqrt{T_{1}}$ , taking out common factor and grouping the previous expression we can obtain the expression shown in Eq. 5.

$E_{R}=E_{o}&space;\left(&space;-\sqrt[]{R_{1}}+\sqrt[]{R_{2}}&space;\cdot&space;T_{1}&space;\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;\sum&space;_{n=0}^{\infty}&space;\left(&space;\sqrt[]{R_{1}&space;\cdot&space;R_{2}}e^{-&space;\alpha&space;2d}&space;\cdot&space;e^{-j&space;\phi&space;}&space;\right)&space;^{n}&space;\right)&space;\:&space;\:&space;(5)$

Considering that:

$\sum&space;_{n=0}^{\infty}a^{n}=\frac{1}{1-a}\;&space;\;&space;\;&space;\text{when}&space;\;&space;\;&space;\;&space;\vert&space;a&space;\vert&space;<1&space;\;&space;\;&space;\;&space;\text{and}\;&space;\;&space;\;&space;\vert&space;\sqrt[]{R_{1}&space;\cdot&space;R_{2}}\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;e^{-j&space;\phi&space;}&space;\vert&space;<1&space;\;&space;\;&space;\;&space;\;&space;(6)$

and that the energy conservation theorem must be fulfilled for total incident light power, which should be equal to the sum of all transmissions, reflections and absorptions produced, according to Eq. 6 (see also our previous post about transmission, reflection and absorption measurements here).

$A_{i}+T_{i}+R_{i}=1&space;\Rightarrow&space;T_{i}+R_{i}=1-A_{i}&space;\;&space;\;&space;\;&space;\;&space;(7)$

Where $\inline&space;A_{i}$, $T_{i}$ and $R_{i}$ represent the absorption, transmission and reflection respectively.

Then, considering Eq. 6 and Eq. 7 we can transform Eq. 5 into Eq. 8 as follows:

$E_{R}=E_{o}&space;\left(&space;\frac{-\;\sqrt[]{R_{1}}+\;\sqrt[]{R_{2}}&space;\cdot&space;\left(&space;1-A_{1}&space;\right)&space;\cdot&space;e^{-&space;\alpha&space;2d}e^{-j&space;\phi&space;}}{1-\;\sqrt[]{R_{1}R_{2}}&space;\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;e^{-j&space;\phi&space;}}&space;\right)&space;\;&space;\;&space;\;&space;(8)$

Regarding the reflected optical power, it is obtained from:

$I_{R}=\frac{1}{2}E_{R}E_{R}^{\ast}=&space;\ldots&space;=\frac{1}{2}E_{R}^{2}&space;\left(&space;\frac{R_{1}+R_{2}&space;\cdot&space;\left(&space;1-A_{1}&space;\right)&space;^{2}&space;\cdot&space;e^{-&space;\alpha&space;4d}-2\cdot\,\sqrt[]{R_{1}R_{2}}&space;\cdot&space;\left(&space;1-A_{1}&space;\right)&space;\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;cos&space;\phi&space;}{1+R_{1}R_{2}&space;\cdot&space;e^{-&space;\alpha&space;4d}-2\cdot\,\sqrt[]{R_{1}R_{2}}&space;\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;cos&space;\phi&space;}&space;\right)&space;\;&space;\;&space;(9)$

Given the intensity of the incident field $\inline&space;I_{0}=E_{0}^{2}/2$ we can express the reflected power coefficient of the Fabry-Pérot as the relationship between the reflected optical power and the incident optical power according to Eq. 10.

$R_{FP}=\frac{I_{R}}{I_{o}}=\frac{R_{1}+R_{2}&space;\cdot&space;\left(&space;1-A_{1}&space;\right)&space;^{2}&space;\cdot&space;e^{-&space;\alpha&space;4d}-2\cdot\,\sqrt[]{R_{1}R_{2}}&space;\cdot&space;\left(&space;1-A_{1}&space;\right)&space;\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;cos&space;\phi&space;}{1+R_{1}&space;\cdot&space;R_{2}&space;\cdot&space;e^{-&space;\alpha&space;4d}-2\cdot\,\sqrt[]{R_{1}&space;\cdot&space;R_{2}}&space;\cdot&space;e^{-&space;\alpha&space;2d}&space;\cdot&space;cos&space;\phi&space;}\;&space;\;&space;\;&space;(10)$

In order to obtain a reduced expression of reflectivity we can add some additional simplifications. For example, we can assume zero dispersion and absorption losses of the material ($\inline&space;A_{1}=0,&space;\alpha=0$) resulting in the expression of Eq. 11.

$R_{FP}=\frac{I_{R}}{I_{o}}=\frac{R_{1}+R_{2}-2\cdot\:\sqrt[]{R_{1}R_{2}}&space;\cdot&space;cos&space;\phi&space;}{1+R_{1}&space;\cdot\!&space;R_{2}-2\cdot\:\sqrt[]{R_{1}R_{2}}&space;\cdot&space;cos&space;\phi&space;}&space;\;&space;\;&space;\;&space;(11)$

Considering the particular case of standar optical fiber and external medium liquid or air ($\inline&space;n_{2}>n_{1}>n_{3}$) enables to assume, with some error, that $\inline&space;R_{1},R_{2}\ll&space;1$, reducing the prior expression to Eq. 12. [7]

$R_{FP}&space;\approx&space;R_{1}+R_{2}-2\cdot\,\sqrt[]{R_{1}&space;\cdot&space;R_{2}}&space;\cdot&space;cos&space;\phi&space;\;&space;\;&space;\;&space;\;&space;\;&space;(12)$

In view of the previous equation it can be deduced that the maximum and minimum attenuation response will occur when $\inline&space;\cos&space;\phi=-1$ or when $\inline&space;\cos&space;\phi=1$ respectively. Taking into account Eq. 3, the wavelength at the maximum and minimum attenuation can be expressed as a function of the of the nanocavity and will comply:

$d_{max-max}&space;\approx&space;\frac{&space;\lambda&space;}{2n_{2}}&space;\left(&space;\phi&space;_{max-max&space;}=2&space;\pi&space;\right),&space;\;&space;\;&space;\;&space;d_{max-min}&space;\approx&space;\frac{&space;\lambda&space;}{4n_{2}}&space;\left(&space;\phi&space;_{max-min&space;}=&space;\pi&space;\right)&space;\;&space;\;&space;\;&space;\;(13)$

Previous equation reveals the relation between the nanocavity size ($\inline&space;d$) and the interference wavelength as it was mentioned at the beginning of this article. This allows to fabricate optical fiber sensors that rely on this phenomena and to use broadband light sources, which is theoretically demonstrated in the interference pattern of Figure 2. This image shows the variation of the optical power as a function of the wavelength and the nanocavity size (number of bilayers) and has also been validated experimentally for a device that is later used as pH sensor [8].

Figure 2: Theoretical evolution of the optical reflectance from a Fabry–Perot interferometer as the distance between the two mirrors is varied, using the model shown in Eq. (10)

[1] J. Chen, “Novel patterning techniques for manufacturing organic and nanostructured electronics” Ph.D. Thesis, Massachusetts Institute of Technology, Dept. of Materials Science and Engineering (2007).

[2] F. J. Arregui, Sensors Based on Nanostructured Materials, Springer Berlin, Heidelberg, 2009.

[3] C. R. Zamarreño, I. R. Matias, F. J. Arregui, “Nanofabrication techniques applied to the development of novel optical fiber sensors based on nanostructured coatings”           IEEE Sensors Journal, vol. 12(8), pp. 2699-2710, 2012.

[4] C. Elosua, F. J. Arregui, I. Del Villar et al., “Micro and nanostructured materials for the development of optical fibre sensors,” Sensors, vol. 17(10), pp. 2312, 2017

[5] Y. Deng, D. Chu, “Coherence properties of different light sources and their effect on the image sharpness and speckle of holographic displays,” Sci. Rep., vol. 7, pp. 5893, 2017.

[6] F. J. Arregui, I. R. Matias, “OpticaL fiber nanometer-scale Fabry-Perot interferometer formed by the ionic self-assembly monolayer process,” Optics Letters, vol. 24(9), pp. 596-598, 1999.

[7] F. L. Pedrotti, Introduction to Optics, London: Prentice Hall, 2017.

[8] C. R. Zamarreño, J. Goicoechea, I. R. Matias, F. J. Arregui, “Utilization of white light interferometry in pH sensing applications by means of the fabrication of nanostructured cavities,” Sensors and actuators B, vol. 138(2), pp. 613-618, 2009.

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