Two Dimensional Modeling of a Simple Fiber Optic Refractive Index Sensor using Finite Element Method (FEM)

Two Dimensional Modeling of a Simple Fiber Optic Refractive Index Sensor using Finite Element Method (FEM)

Virginia T. Santos, 1Alexandre R. Camara

1 Quantum Eletronic Department, Physics Institute, Rio de Janeiro State University

Rio de Janeiro, Brazil

2Clara Johanna Pacheco

2 Laboratory of Non-Destructive Testing, Corrosion and Welding, Rio de Janeiro Federal University

Rio de Janeiro, Brazil

Abstract Fiber optics are optical devices that are used in different areas in a large range of applications. The aim of this work is to show a model using Comsol Multiphysics® platform of a very simple fiber optic refractive index sensor that can be used as basis for the developing of different fiber optics sensors.

Keywords Fiber Optics; Fresnel Coeficcients; Refractive Index

1. INTRODUCTION

   Fiber optics [1,2] are optical devices usually made with silica (can be also made using plastic) that are used to guide light between two places. Nowadays these devices are largely used in many different areas. Due to advantages like possibility of remote sensing, low optical losses, immunity to radiofrequencies and low cost, fiber optics is a powerful tool used in telecommunications, industry and sensing (which is the focus of this work) for example. Specifically in the sensing area there are a huge number of applications e.g. to diagnose deseases [3], to monitor pressure [4,5], strain [6] and temperature [7,8] among others. Depending on the application it is necessary to modify one or more characteristic of the fiber optics to make it sensible to a quantity [9].

   This work shows a simulation of simple fiber optic refractive index sensor that uses the Fresnel reflectance and transmittance coefficients [10] to measure the refractive index of the medium that is surrounding the fiber endface. This model can be used as basis for the development of other fiber optics based sensors.

   The operation of this sensor is based on the change of the values of the Fresnel reflectance and transmittance coefficients caused by the variation of the refractive index of the external medium where the fiber endface is inserted. When light (that is considered here as a plane wave) propagates from a medium with refractive index (RI) equal to n1 to a medium with RI equal to n2 the incident wave is partially reflected in the interface between the two media and partially transmitted along the external medium. In according to the Fresnel Equations

   [8] the s-polarized (i.e. when the electric field is perpendicular to the plane of incidence) coefficients, rs and ts, and p-polarized (i.e. when the electric field is parallel to the plane of incidence), rp and tp are given by:

   rs = (n1cosi n2cost)*( n1cosi + n2cost)-1 (1) ts = (2n1cosi)*( n1cosi + n2cost)-1 (2) rp = (n2cosi n1cost)*( n2cosi + n1cost)-1 (3) ts = (2n2cosi)*( n2cosi + n1cost)-1 (4)

   where i and t are the incidence and transmission angles respectively, n1 is the refractive index of the incident medium (core of the fiber) and n2 is the refractive index of the transmission medium (external medium). The reflectance R and transmittance T are given by:

   R = r2 (5)

   T = t2 (6)

   Taking into account that the absorbances of the media are very low one can write that:

   R+T = 1 (7)

   In the case of the model shown in this work the change of the values of R and T are due to the change in the RI of the external medium, n2.

2. THE MODEL

   The physics behind the numerical model shown here is based on the Comsol Multiphysics® solution for the electromagnetic wave problem. The implementation of this model consists in a singlemode (SM) optical fiber with a 5m core and external diameter equal to 125m that is inserted in principle in an external medium which refractive index varies between 1.0 and 2.0 (figure 1). The refractive index of the cladding and the core of the fiber are ncore=1.4682 and ncladding=1.4615 resulting in a numerical aperture NA=0.14. A dense two-dimensional mesh is established over whole system and the problem solved in x and y-axis is wave equation as described below [11]:

   (r)-1(E) (0)2r E = 0 (8)

   where r is the relative permeability, E is the electric field, 0 is the wavenumber, r is the relative permittivity and 0 is the vaccum permittivity.

   To solve this problem some boundary conditions shall be adopted: (1) the external medium extends to the infinity in all directions; (2) the field is initially zero in all directions and then it is seted up to be y-axis linear polarized; (3) the size of the mesh shall be a fraction of the wavelength.

   Fig. 1. Geometry of the system used in the simulation.

   Here the light will propagate along the core of the fiber, from the left to the right, and until the core-external medium interface. When the light reaches this point it is partially reflected back to the core of the fiber and partially transmitted to the external medium. The incident electric field has the form:

   E = Ey e-ikx (9)

   Furthermore, the model is solved for different external refractive index that varies from 1.0 to 2.0 in steps of 0.05. This is interesting because this is the range that embraces substances like air, water, alcohol and many different gases for instance.

   The mesh used here produced 2547688 triangular elements and after all conditions described above are attended, the simulations produced the results shown in the next section.

3. RESULTS AND DISCUSSIONS

   The first interesting information that we can take from this model is how the light is transmitted from the core of the fiber to the external medium. If n2 is smaller than n1 then the interface acts as a negative lens (i smaller than t), diverging the incident rays as shown in figure 2. If n1

   = n2 then the incident rays are not deflected passing straight through the interface (figure 3). Finally, if n2 is higher than n1 then the rays are deflected with t smaller than i i.e. the interface acts as a positive lens converging the rays as one can see in figure 4. The relation between t

   and i is given by the Snells Law:

   n1 sin i = n2 sin t (10)

   Fig. 2. Intensity of light in the system when n1 > n2. Here n1 = 1.468 and n2 = 1.000.

   Fig. 3. Intensity of light in the system when n1 = n2. Here n1 = n2 = 1.468.

   Fig. 4. Intensity of light in the system when n1 < n2. Here n1 = 1.468 and n2 = 2.000.

   However not all incident light is transmitted from the

   fiber to the external medium. A fraction of the incident light is reflected back to the core of the fiber. The amount of the initial power of the light that is either transmitted or reflected to the fiber depends on the refractive index of the external medium as shown in figures 5 and 6.

   Fig. 5. Variation of the reflectance of the light in the core-external medium interface with the refractive index of external medium.

   It is possible to see that both reflectance and transmittance dependence on the external medium show a parabolic-like behavior. Furthermore, one can observe that when the refractive index of external medium has the same value of the refractive index of the core of the fiber no light is reflected. In other words, it means that in the light perspective there is no change of propagation medium. After that the transmittance decreases while the reflectance increases again.

   Fig. 6. Variation of the transmittance of the light in the core-external medium interface with the refractive index of external medium.

   As an example of specific application of this model

   0.0030

   00029

   Reflectance

   0.0028

   0.0027

   0.0026

   0.0025

   0 20 40 60 80 100

   o

   Temperature ( C)

   one can use it to simulate a sensor to monitor the temperature of water. Bashkatov and Genina [12] studied the dependence of the refractive index of water (nWATER) on the temperature for different wavelengths (), i.e. showing that the refractive index of water nWATER is a function of and temperature. If =632 nm the dependence of nWATER on temperature is shown in the figure 7:

   Fig. 8. Reflectance of light in the core of fiber-water interface versus temperature of water.

   0.9975

   0.9974

   Transmittance

   0.9973

   1.332

   Refractive Index

   1.328

   1.324

   1.320

   0.9972

   0.9971

   0.9970

   0 20 40 60 80 100

   o

   Temperature ( C)

   1.316

   n = (-1.15571*10 )t + (-4.45016*10 )t + 1.33329

   -6 2 -5

   0 20 40 60 80 100

   o

   Temperature ( C)

   Fig. 9. Transmittance of light in the core of fiber-water interface versus temperature of water.

   Figures 8 shows that the amount of light that is reflected back to the fiber in the core-water interface

   Fig. 7. Dependence on temperature of the refractive index of water[12].

   It is possible to observe that the dependence of nWATER on the temperature (t) of water when =632 nm (it was chosen because it is the wavelength of He-Ne laser) can be described by a parabola which function is:

   nWATER = 1.33329 4.45016*10-5 t 1.15571*10-6 t2 (11)

   To show that the purposed model can be used to simulate a temperature sensor the value of n2 is made equal to nWATER given by equation 11. Figures 8 and 9 show the reflectance and transmittance of the light in the interface core-water in different temperatures respectively:

   increased 15.12% when the temperature of water grows from 0oC to 100oC (consequently the transmittance of light decreased as shown in figure 9). If the source of light has enough intensity (some tens of miliwatts) 15.12% of increase in the intensity of reflected light is easily measured by a detector e.g. an InGaAs photodetector, a power meter or a spectrum analyzer.

4. CONCLUSIONS

In this work we purposed a very simple fiber optic refractive index sensor built using the finite elements method (FEM) of Comsol Multiphysics® platform that is based on the difference in the amount of light that is reflected and/or transmitted in the core-external medium interface that can be used as basis to modelling many other fiber optics sensors. Using this model one can sense different substances, once in general different substances have different refractive indices, or perform indirect measurements of an entity (e.g. temperature or concentration of a solute) that causes a variation in the refractive index of a specific substance. This work showed that the purposed model is useful to identify the temperature of water once a variation of 100oC in the temperature of water produced an increase of 15.12% for light that is reflected back to the fiber in the core-water interface.

ACKNOWLEDGMENT

Alexandre Camara thanks to Luis Pereira from Comsol and João Manoel Pereira from PUC-Rio for the discussions. Virginia Santos thanks to Martin Silva and Diego González for the discussions.

REFERENCES

1. J. Crisp, Introduction to Fibre Optics, Newnes, Oxford, 2001.

2. A. Nechibvute and C. Mudzigwa, Modelling Of Optical Waveguide Using COMSOL Multiphysics, Int. J. Eng. Res. Technol., vol 2 (5), pp. 1663-1667, May 2013.

3. A. R. Camara, P. M. P. Gouvêa, A. C. M. Dias, A. M. B. Braga, R.

   F. Dutra, R. E. de Araujo and I. C. S. Carvalho, Dengue immunoassay with an LSPR optical fiber sensor, Optics Express, vol. 21, issue 22, pp. 27023-27031, August 2013.

4. F. C. Fávero et al. Hydrostatic Pressure Sensing with High Birefringence Photonic Crystal Fibers, Sensors (Basel, Switzerland), vol 10 (11), pp. 96989711, November 2010.

5. Wenhui Wang, Nan Wu, Ye Tian, Christopher Niezrecki, and Xingwei Wang, "Miniature all-silica optical fiber pressure sensor with an ultrathin uniform diaphragm," Opt. Express, vol. 18, pp. 9006-9014, April 2010.

6. O. B. Frazao, J. M. Baptista, and J. L. Santos, Temperature- independent strain sensor based on a Hi-Bi Photonic Crystal Fiber Loop Mirror, Sens. Journal, vol 7 (10), pp. 14531455, October 2007.

7. B. Larrión, M. Hernánez, F. J. Arregui, J. Goicoechea, J. Bravo, and I. R. Matías, Photonic crystal fiber temperature sensor based on quantum dot nanocoatings, J. Sens. vol 2009, pp. 93247, April 2009.

8. F. C. Favero, R. Spittel, F. Just, J. Kobelke, M. Rothhardt, and H. Bartelt, "A miniature temperature high germanium doped PCF interferometer sensor," Opt. Express, vol. 21, pp. 30266-30274, 2013.

9. A. R. Camara, O. Tarasenko and W. Margulis, Study of thermally poled fibers with a two-dimensional model, Optics Express, vol. 22, issue 15, pp. 27023-27031, July 2014.

10. E. Hecht, Optics,3rd edition, Addison Wesley Longman, 1998.

11. www.comsol.com

12. A. N. Bashkatov and E. A. Genina, Water refractive index in dependence on temperature and wavelength: a simple approximation , Proc. SPIE 5068, Saratov Fall Meeting 2002: Optical Technologies in Biophysics and Medicine IV, October 2003