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Numerical Approach to Modeling and Characterization of Refractive Index Changes for a Long-Period Fiber Grating Fabricated by Femtosecond Laser Akram Saad 1 , Yonghyun Cho 1 , Farid Ahmed 1 and Martin Byung-Guk Jun 2, * 1 2

*

Department of Mechanical Engineering, University of Victoria, Victoria, BC V8W 2Y2, Canada; [email protected] (A.S.); [email protected] (Y.C.); [email protected] (F.A.) School of Mechanical Engineering, Purdue University, West Lafayette, IN 47906, USA Correspondence: [email protected]; Tel.: +1-765-494-3376

Academic Editor: Dirk Lehmhus Received: 1 September 2016; Accepted: 2 November 2016; Published: 21 November 2016

Abstract: A 3D finite element model constructed to predict the intensity-dependent refractive index profile induced by femtosecond laser radiation is presented. A fiber core irradiated by a pulsed laser is modeled as a cylinder subject to predefined boundary conditions using COMSOL5.2 Multiphysics commercial package. The numerically obtained refractive index change is used to numerically design and experimentally fabricate long-period fiber grating (LPFG) in pure silica core single-mode fiber employing identical laser conditions. To reduce the high computational requirements, the beam envelope method approach is utilized in the aforementioned numerical models. The number of periods, grating length, and grating period considered in this work are numerically quantified. The numerically obtained spectral growth of the modeled LPFG seems to be consistent with the transmission of the experimentally fabricated LPFG single mode fiber. The sensing capabilities of the modeled LPFG are tested by varying the refractive index of the surrounding medium. The numerically obtained spectrum corresponding to the varied refractive index shows good agreement with the experimental findings. Keywords: refractive index; Gaussian beam; femtosecond laser; LPFG model; beam envelop; grating period; grating length; index sensor

1. Introduction Long-period fiber gratings (LPFGs) are fiber optic devices with various applications in optical communications and sensing systems [1]. Fiber gratings are produced by periodically altering refractive index inside the core region of an optical fiber [2]. Several techniques are available to produce LPFGs, such as those based on ultraviolet, CO2 , and femtosecond laser radiation, with the latter being the most versatile method [1,3–5]. LPFGs are usually fabricated with grating periods on the order of 100 micrometers to a millimeter. LPFGs are characterized by coupling light from a guided core mode into forward propagating cladding modes [5]. LPFGs are currently considered to be simple yet versatile devices for various sensing applications [6]. In-fiber long-period gratings have been demonstrated as effective and efficient tools for refractive index sensing [5,7]. Due to their ability to produce focused pulses, femtosecond laser systems have been recently used to fabricate LPFGs offering permanent refractive index increase in various glasses [8–11]. In such systems, light intensity is highly localized at the focal point resulting in inducing periodical refractive index modulation in the core of fiber without significantly affecting the cladding or the polymer coatings. LPFGs fabricated by femtosecond laser have superior aging characteristics and high resistance Materials 2016, 9, 941; doi:10.3390/ma9110941

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to thermal decay [9,11]. For effective fabrication, it is crucial to understand and model the fabrication process and characterize the index changes by femtosecond laser. The basic physical characteristic of optical waveguides is given by the distribution of the induced refractive index. From this distribution, all local parameters can be computed. Consequently, measurement of the refractive-index profiles provides valuable quality control during the development and production of the fiber optic components. Several technical approaches are reported to measure the refractive index profile of optical fibers [12–18]. However, the experimentally based approaches involve time-consuming cycles of design. Thus, computational methods and computerized modeling play a vital role in reducing the time and its associated costs as well as investigating novel phenomena that may not lend themselves to experimental procedures [19]. Mathematical models, the phase grating and Gaussian grating, have been extensively used in estimating the refractive index profile. However, the effect of the self-focusing is not explicitly reported in these models [20,21]. For design and fabrication purposes, self-focusing is of importance, as the modification of the beam must be taken into consideration [22]. The beam propagation method [23,24], nonlinear Schrödinger equation [25–27], and finite-difference time-domain method [28,29] have been employed to understand the temporal and spatial electromagnetic field variations in nonlinear bulk media including the self-focusing effect. By using a non-paraxial beam propagation method based on the Helmholtz equation, a non-paraxial treatment of the nonlinear process is reported to limit the reduction in beam size to the order of one optical wavelength, leading to multiple foci, beam breakup, and filament formation [23]. The multiple filamentation is reported to be suppressed for the circularly polarized beams in a Kerr medium using a scalar equation to describe self-focusing in the presence of vectorial and non-paraxial effects [26]. The finite-difference time-domain method has also been utilized to study the complex nonlinear phenomenon [28,29]. Given the rapid development of the computer industry, the finite element method has become an indispensable computation tool for solving engineering problems in electromagnetism. This numerical simulation method is established on the basis of variational principles, domain meshing, and interpolation function [30,31]. The refractive index profile induced by CO2 laser was calculated using an empirical formula based on numerically obtained thermal stresses [32]. A multi-layer thin lens self-focusing model that considers the refractive index and diffraction effects was proposed to analyze the self-focusing behavior of light beams [33]. The reported methods rely on mathematical functions to describe the refractive index profile independently of the laser intensity distribution within the focal volume [23–27,34]. Thus, with proper definition of the boundary conditions and full incorporation of the laser intensity field the finite element method can be utilized to account for the refractive index perturbation due to femtosecond laser. In this work, a finite element based approach that accounts for the refractive index dependency on femtosecond laser intensity is proposed. The robustness of the model is achieved by implementing the Beam Envelope Method to overcome the demanding computation criteria. In addition to the significance of characterizing the induced refractive index, recent advances in the field of guided-wave optics such as fiber optics have prompted the introduction of waveguide numerical modeling [35,36]. The most widely used numerical methods for photonics devices include Galerkin and moment methods, transfer matrix method, finite difference based methods, transmission line matrix methods, stochastic methods such as the Monte Carlo method, and finite element-based methods [9,37]. A general theoretical method for the transmittance calculation of ultraviolet-written gratings is reported [20]. The method is based on the coupled mode theory in step-index optical-fibers. The interactions between core mode and cladding modes in a cross-section of an LPFG are reported using an approach based on the coupled mode theory and transfer-matrix methods [11]. An empirical relation accounting for the induced refractive index for a uniform pure silica fiber experiencing uniaxial tension is reported [38]. In other work, the refractive index profile induced by CO2 laser is calculated using the aforementioned empirical formula based on numerically obtained thermal stresses [32]. An integration of a perfectly matched layer, perfectly reflecting the boundary, object meshing method, and boundary meshing method into the finite element model and eigenmode expansion method is

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presented to correlate the design period to the spectral characteristics of superstructure fiber Bragg gratings [21]. Most of the reported FEM models solely consider a cross-section of the waveguide in Materials 2016, 9, 941 3 of 15 the analysis to avoid the daunting computation requirements [19,21,32,36,39–42]. Long-period fiber superstructure fiber Braggare gratings [21]. Mostby oftheir the reported FEMcompared models solely consider a cross-of the grating devices in particular characterized large sizes to the wavelength section the waveguide in themethods analysislack to avoid daunting computation requirements guided light.ofMoreover, the reported explicitthe coupling between the numerically induced [19,21,32,36,39–42]. Long-period fiber grating devices in particular are characterized by their large refractive index profiles with the proposed numerical models. In this work, a 2D finite element based sizes compared to the wavelength of the guided light. Moreover, the reported methods lack explicit model constructed in the COMSOL 5.2 Wave Optics Module is coupled with the proposed refractive coupling between the numerically induced refractive index profiles with the proposed numerical index model to fully describe the propagation of light through a long period grating single mode fiber models. In this work, a 2D finite element based model constructed in the COMSOL 5.2 Wave Optics in theModule frequency domain. The demanding computation requirements are overcome by using the beam is coupled with the proposed refractive index model to fully describe the propagation of light envelope method. Additionally, the single constructed tested as a refractive through a long period grating mode model fiber inis the frequency domain.index The sensor. demanding computation requirements are overcome by using the beam envelope method. Additionally, the

2. Intensity-Dependent Refractive Index Model constructed model is tested as a refractive index sensor.

Several optical waveguides and devices operate in a regime where the dielectric polarization of 2. Intensity-Dependent Refractive Index Model the material responds linearly to the electric filed of the light and the refractive index is independent Several of optical waveguides devices in operate in a regimeregime where the polarization of the intensity optical beams. and However, the nonlinear thedielectric dielectric responseofof the the material responds linearly to the electric filed of the light and the refractive index is independent material is nonlinear and the material refractive index is dependent on the optical filed intensity. the intensity of optical beams. However, in the nonlinear regime the dielectric response of the Thus,ofthe optical properties, behavior, and performance of the device become dependent on the material is nonlinear and the material refractive index is dependent on the optical filed intensity. optical beam intensity [43–45]. As the intense laser beam hits the core, the generated electric field Thus, the optical properties, behavior, and performance of the device become dependent on the perturbs the electron clouds around the nuclei leading to a significant refractive index dependency optical beam intensity [43–45]. As the intense laser beam hits the core, the generated electric field on laser intensity [46,47].clouds With progressive intensification the laser refractive power, noise the beam’s perturbs the electron around the nuclei leading to aofsignificant indexacross dependency cross-sectional intensity distribution can be amplified, resulting in the beam breaking up into several on laser intensity [46,47]. With progressive intensification of the laser power, noise across the beam’s self-focused filaments. The dielectric exhibits pulse intensity dependent index of refraction that is cross-sectional intensity distribution can be amplified, resulting in the beam breaking up into several givenself-focused by [48,49]: filaments. The dielectric exhibits pulse intensity dependent index of refraction that is given by [48,49]: n = no + γI (1) = nonlinear + where no is the nominal refractive index, γ is the refractive index coefficient, and(1)I is the no is the nominal refractive index, γ is the nonlinear refractive index coefficient, and I is the laser where intensity. laser intensity. The proposed numerical approach is carried out by implementing a 3D finite element model to The proposedindex numerical approach is carried out by implementing a 3D finite model to predict the refractive profile induced by femtosecond laser irradiation. Aelement fiber core irradiated predict the refractive index profile induced by femtosecond laser irradiation. A fiber core irradiated by a single shot of a pulsed laser can be modeled as a cylinder subject to predefined boundary by a single shot of a pulsed laser can be modeled as a cylinder subject to predefined boundary conditions using the Wave Optics Module built in COMSOL5.2 Multiphysics [47,50]. Based on the conditions using the Wave Optics Module built in COMSOL5.2 Multiphysics [47,50]. Based on the fabrication mechanism shown in Figure 1, the generated refractive index profile within a cylinder is of fabrication mechanism shown in Figure 1, the generated refractive index profile within a cylinder is a major interest. of a major interest.

Figure 1. Schematicillustration illustration of long-period fabrication. Figure 1. Schematic long-periodgrating grating fabrication.

The commercial single mode fiber used in this work is Corning SMF-28 (New York, NY, USA).

The commercialprovided single mode used inindicates this work isthe Corning SMF-28 (New The information by the fiber manufacturer that effective refractive indexYork, of theNY, fiberUSA). The information provided by the manufacturer indicates that the effective refractive index of the

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fiber is 1.4620 for a wavelength equal to 1550 nm. It is also mentioned in the specification sheet that the refractive index of the core is 0.36% higher than that of the cladding. To determine the nominal Materials 2016, 9, 941 of 15 refractive indices of the core and the cladding, a mode analysis is carried on a cross-section of4 the fiber. In thisisanalysis, the concept of reverse engineering is to be used, which means that the refractive index 1.4620 for a wavelength equal to 1550 nm. It is also mentioned in the specification sheet that the of therefractive core is toindex be parametrically obtain a discrete effective index each value the sweep. of the core is swept 0.36% to higher than that of the cladding. Tofor determine the of nominal An eigenvalue equation for core the electric E is derived Helmholtz and solved for the refractive indices of the and thefield cladding, a mode from analysis is carried equation on a cross-section of the fiber. Inλthis the concept of reverse engineering is to be used, which means that the refractive eigenvalue = −analysis, jβ [50,51]. index of the core is to be parametrically swept to obtain ω 2 2a discrete effective index for each value of ∇ × × E − n E=0 (2) (∇ ) the sweep. An eigenvalue equation for the electric field c2o E is derived from Helmholtz equation and solved for the eigenvalue λ = −jβ [50,51].

where E is the electric field, co is the speed of light, n is the refractive index, ω is the angular frequency, function and β is the propagation constant. (1) ∇ × (∇ × ) − =0 Since the amplitude of the electric field decays rapidly as a function of the radius of the cladding, thecladding’s speed of light, n is the refractive is the angularrefractive frequency,index where field E is the field,on co is the electric is electric set to zero the periphery. Figure 2 index, showsωthe effective function and β is the propagation constant. of the fiber as a function of the swept refractive index of the core. The obtained results indicate that a Since the amplitude the electric field as a function of the radius of the index cladding, value of the core’s refractiveofindex equals to decays 1.4649 rapidly corresponds to a targeted effective having the electric field is set to zero on the cladding’s periphery. Figure 2 shows the effective refractive a value of 1.4620. The confinement of the light in a single mode fiber having the aforementioned index of the fiber as a function of the swept refractive index of the core. The obtained results indicate refractive indices is shown in a surface plot depicted in Figure 3. The obtained refractive indices of that a value of the core’s refractive index equals to 1.4649 corresponds to a targeted effective index both the core the of cladding be used in the index model. having aand value 1.4620. are Thetoconfinement of intensity-dependent the light in a single refractive mode fiber having the The proposed intensity-dependent refractive model in this work is an extension of the 3D aforementioned refractive indices is shown in aindex surface plot depicted in Figure 3. The obtained modelrefractive presented in reference account forcladding femtosecond fabrication. The modeling scheme indices of both [50] the to core and the are tolaser be used in the intensity-dependent refractive index model. is carried out using the Beam Envelope method. As the electric field envelope has a slower spatial The proposed intensity-dependent refractive model thislaser workbeam is an extension of the 3D variation than the electric field, a coarse mesh can beindex used. Sinceinthe propagates essentially model presented in reference [50] to account for femtosecond laser fabrication. The modeling scheme in one direction, it is assumed that the electric field, E, of the wave can be written as [50]: is carried out using the Beam Envelope method. As the electric field envelope has a slower spatial − jβx Since the laser beam propagates essentially variation than the electric field, a coarse mesh used. E can = Ebe (3) 1e in one direction, it is assumed that the electric field, E, of the wave can be written as [50]:

where E1 is a slowly varying field envelope function. Inserting this electric field formulation (2) into the = Maxwell’s equations results in the following wave equation for the envelope function [47,50]: where E1 is a slowly varying field envelope function. Inserting this electric field formulation into the Maxwell’s equations results in the following wave equation for the 2 2envelope function [47,50]:

ω n E1 = 0 c20 = 0 −

(∇ − jβ) × ur−1 ((∇ − jβ) × E1 ) − (∇ −

)×

(∇ −

)×

wherewhere µr is μ the relative permeability, and c is the speed of light. r is the relative permeability, and0c0 is the speed of light.

Figure 2. The numerical refractiveindex indexofof the SMF. Figure 2. The numericaleffective effective refractive the SMF.

(4) (3)

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Figure 3. Visualization of the light intensity in the cross section of the SMF; the core’s index is 1.4649.

Figure 3. Visualization of the light intensity in the cross section of the SMF; the core’s index is 1.4649.

As shown in Figure 4, cylinder is subject to ansection incident beam approximated Gaussian Figure 3. Visualization of the the light intensity in the cross of the SMF; the core’s indexby is a1.4649.

As shown in Figure thex-direction cylinder is subject to an in incident beam approximated a Gaussian beam propagating in 4, the and polarizing the z-direction. To absorb thebyincident beamradiation propagating in x-direction and polarizing inmatched theincident z-direction. To absorb the conditions incident radiation without producing perfectto layer (PML) boundary are As shown inthe Figure 4, thereflections, cylinder isasubject an beam approximated by a Gaussian applied to the top and bottom of thematched cylinder to account forboundary the electricconditions filed distribution of theto the without producing reflections, a perfect layer (PML) are beam propagating in the x-direction and polarizing in the z-direction. To absorb theapplied incident incident Gaussian [19,20]. In addition tothe theelectric PML, the periphery of the cylinder is considered radiation without producing reflections, a perfect matched layer (PML) boundary conditions are top and bottom of thebeam cylinder to account for filed distribution of the incident Gaussian as a perfect electric conductor. For the model considered in this work, the beam waist, cylinder height, applied to the top and bottom of the cylinder to account for the electric filed distribution of the beam [19,20]. In addition to the PML, the periphery of the cylinder is considered as a perfect electric laser energy, repetition rate, width arePML, set tothe be periphery 0.50 μm, 8.30 μm, 0.50 μJ,is 1considered and incident Gaussian beam [19,20]. In pulse addition to the of the cylinder conductor. For thelaser model considered in this work, the beam waist, cylinder height, laserkHz, energy, laser 120 respectively. The top surface of the cylinder is meshed using a free triangular as a fs perfect electric conductor. For the model considered in this work, the beam waist,meshing cylinderpattern. height, repetition rate, pulse width are set to be 0.50 µm, 8.30 µm, 0.50 µJ, 1 kHz, and 120 fs respectively. The meshrepetition surface israte, then pulse sweptwidth acrossare the set entire model. the μm, beam0.50 envelope method, lasergenerated energy, laser to be 0.50 Using μm, 8.30 μJ, 1 kHz, and The top of the thesolution cylinder is meshed using aforfree triangular meshingelement pattern. generated it is surface found that is mesh-independent values of the maximum sizeThe ofpattern. the top 120 fs respectively. The top surface of the cylinder is meshed using a free triangular meshing meshsurface surfaceand is then swept across the entire model. Using envelope method, it is found themesh maximum size of the swept tothe be beam oneUsing third of the laser wavelength (λ) that The generated surfacemesh is then swept across themesh entire model. the beam envelope method, the solution is mesh-independent for values of thefor maximum element size of the top surface and the and the relight 0), respectively. it is found that range(x the solution is mesh-independent values of the maximum element size of the top maximum size of the swept one mesh third to ofbe theone laser wavelength (λ) and the(λ) relight surfacemesh and the maximum mesh mesh size of to thebe swept third of the laser wavelength rangeand (x0 ), therespectively. relight range(x0), respectively.

Figure 4. Schematic illustration of the model geometry and boundary conditions.

The surface plot4.of the refractive index profile (RIP) is illustrated in theconditions. multi-slice plot shown Figure Schematic illustration the model geometry boundary Figure 4. Schematic illustration ofofthe model geometryand and boundary conditions. in Figure 5a. The averaged refractive index over the entire inner cylinder for the conditions considered −4. To further explore the in this work is 1.4655 a refractive of 6 ×in10the The surface plot ofdemonstrating the refractive index profileindex (RIP) change is illustrated multi-slice plot shown The surface plot of the refractive index profile (RIP) is illustrated in the multi-slice plot from shown in growth of5a. theThe induced refractive index profile, isosurface plotcylinder with refractive indices ranging in Figure averaged refractive index overan the entire inner for the conditions considered Figure 5a. The averaged refractive index over the entire inner cylinder for the conditions considered the nominal 1.4649to the maximum refractive index value-of1.4663is. shown in Figure 5b. the As in this workvalueis 1.4655 demonstrating a refractive index change 6 × 10−4 To further explore

in this workofisthe 1.4655 demonstrating a refractive index change of 6refractive × 10−4 .indices To further explore growth induced refractive index profile, an isosurface plot with ranging from the growth the induced refractive index profile,refractive an isosurface plot with refractive indices ranging theof nominal value- 1.4649to the maximum index value1.4663is shown in Figure 5b. Asfrom the nominal value- 1.4649- to the maximum refractive index value- 1.4663- is shown in Figure 5b. As shown in the figure, averaging the refractive index within the inner cylinder seems to provide

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Materialsin 2016, 941 6 of 15 shown the9,figure, averaging the refractive index withinItthe inner be cylinder to provide reliable reliable characterization of the induced refractive index. should notedseems that the proposed model is characterization of the induced refractive It should be noted that the proposed model is limited limited to predict refractive index profilesindex. induced by peak powers that are lesser than three times shown in the figure, averaging the refractive index within the inner cylinder seems to provide reliable predict refractive index profiles induced by peak that than three times of theof the of theto critical power. This limitation is deemed toshould bepowers a result of are notlesser accounting for theiseffect characterization of the induced refractive index. It be noted that the proposed model critical power. This limitation is deemed to be a result of not accounting for the effect of the limited multimulti-photon ionization. The latter phenomenon has powers been reported be pronounced for of high to predict refractive index induced peak are to lesser than threelaser times thelaser photon ionization. The latterprofiles phenomenon hasbybeen reported tothat be pronounced for high powers powers [52]. The induced refractive index change due to the simulated fabrication conditions is to be critical power. Thisrefractive limitationindex is deemed to be a result not accounting for theconditions effect of the [52]. The induced change due to theofsimulated fabrication is multito be photon ionization. The latterscheme phenomenon beensensor. reported to be pronounced for high laser powers implemented in the modeling LPFG implemented in the modeling schemeofofthe thehas LPFG sensor.

[52]. The induced refractive index change due to the simulated fabrication conditions is to be implemented in the modeling scheme of the LPFG sensor.

(a)

(b)

Figure 5. (a) A multi-slice plot of the induced refractive index profile; (b) An isosurface plot

Figure 5. (a) A multi-slice(a)plot of the induced refractive index profile; (b) An(b) isosurface plot illustrating illustrating specified contours of the induced refractive index. specified contours of the induced refractive index. Figure 5. (a) A multi-slice plot of the induced refractive index profile; (b) An isosurface plot illustrating Grating specified Model contours of the induced refractive index. 3. Long-Period

3. Long-Period Grating Model

The geometry of theModel 2D model is illustrated in Figure 6. The in-fiber fabricated regions of the 3. Long-Period Grating pure silica single-mode fiber are is dealt with as rectangles with indices obtained from the pure The geometry of the 2D model illustrated in Figure 6. Therefractive in-fiber fabricated regions of the The geometry of the 2D model is illustrated in Figure 6. The in-fiber fabricated regions of the indexfiber model. a section a dielectric slab waveguide that isthe finite in silicarefractive single-mode are This dealtmodel with considers as rectangles withofrefractive indices obtained from refractive pure silica single-mode fiber are dealt with as rectangles with refractive indices obtained from the x and y directions. theafields drop exponentially the waveguide, the fields indexthe model. This model Because considers section ofoff a dielectric slaboutside waveguide that is finite in thecan x and y refractive index Thisto modelperiphery. considers Thus, a section dielectric is slab waveguide is finite in be assumed to bemodel. zero close the of airaboundary assumed bethat subject the directions. Because the fields dropthe offfields exponentially outside theoutside waveguide, theto fields canfields betoassumed the x and y directions. Because the drop off exponentially the waveguide, the can perfect electric conductor boundary condition. to be zero close totothe periphery. the air boundary subjecttotobethe perfect electric be assumed be zero close toThus, the periphery. Thus, the is airassumed boundarytoisbe assumed subject to the conductor boundary condition. perfect electric conductor boundary condition.

Figure 6. Geometrical depiction of the numerically modeled single mode fiber. Figure 6. Geometrical depictionof ofthe the numerically numerically modeled single mode fiber. Figure 6. Geometrical depiction modeled single mode fiber.

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The numerical numericalport portboundary boundary conditions x direction are utilized to model the guided The conditions in in thethe x direction are utilized to model the guided wave wave propagating in the x direction and calculate the generated spectrum. The generated propagating in the x direction and calculate the generated spectrum. The generated electric electric filed is filed is obtained by solving for theenvelope. beam envelope. The Electromagnetic Waves, Beam Envelopes obtained by solving for the beam The Electromagnetic Waves, Beam Envelopes physics physics interface is used to handle the propagation over distances that are many wavelengths interface is used to handle the propagation over distances that are many wavelengths long. Sincelong. the Since the wave propagates in one direction, the propagation in the x-direction wave propagates essentiallyessentially in one direction, the propagation constantconstant in the x-direction can be can be assigned to the modeled the nature the beam envelope method, assigned to the modeled regionsregions [47,50].[47,50]. Based Based on theonnature of theofbeam envelope method, the the meshing requirements of the LPFG model are expected to be computationally inexpensive, which meshing requirements of the LPFG model are expected to be computationally inexpensive, which means that that coarse coarse meshing meshing can can be Thus, aa mesh mesh convergence convergence study study is is essential essential for for the the means be utilized. utilized. Thus, numerically obtained solution to be mesh independent. Accordingly, a simulation study of a long numerically obtained solution to be mesh independent. Accordingly, a simulation study of a long period fiber various second order Lagrangian meshmesh element sizes issizes carried out. Theout. number period fibergrating gratingforfor various second order Lagrangian element is carried The of periods, grating grating length, and grating period period are set are to be and 435 µm, while number of periods, length, and grating set30, to 100, be 30, 100, and 435respectively, μm, respectively, the maximum element size issize varied. The The spectra of the assigned mesh sizes areare superimposed in while the maximum element is varied. spectra of the assigned mesh sizes superimposed Figure 7. The plotted results suggest that a maximum mesh size equal to the value of the wavelength in Figure 7. The plotted results suggest that a maximum mesh size equal to the value of the is adequate for the simulated to beresults mesh-independent. Consequently, one might notice that wavelength is adequate for theresults simulated to be mesh-independent. Consequently, one might a significant reduction in the meshing requirements has been achieved confirming that the beam notice that a significant reduction in the meshing requirements has been achieved confirming that envelope canmethod substantially alleviate the alleviate dauntingthe computational demands that solving forthat the the beam method envelope can substantially daunting computational demands full wave method would normally require. To investigate the spectral growth resulted from varying solving for the full wave method would normally require. To investigate the spectral growth resulted the number ofthe grating periods (α), various simulation studies are conducted. these studies, In both the from varying number of grating periods (α), various simulation studiesIn are conducted. these grating length andgrating gratinglength periodand are kept constant to are be 100 µm, and 435toµm shown studies, both the grating period kept constant berespectively. 100 μm, andAs435 μm in Figure 8, the spectral growth of an LPFG seems to evolve for various numbers of the grating periods respectively. As shown in Figure 8, the spectral growth of an LPFG seems to evolve for various around a of relatively constant attention dip, 1555 nm. To fully verify whether obtained numbers the grating periods around a relatively constant attention dip, the 1555numerically nm. To fully verify results using the BEM provide reliable numerical of the simulated a set of of whether the numerically obtained results using theinterpretation BEM provide reliable numericalspectrum, interpretation experiments with identical fabrication conditions are carried out. the simulated spectrum, a set of experiments with identical fabrication conditions are carried out.

Figure Figure 7. 7. Gradual Gradual growth growth of of LPFG LPFG in in the the modeled modeled single single mode mode fiber fiber for for different different number number of of periods periods using free triangular mesh. using free triangular mesh.

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Figure Figure 8. 8. Gradual Gradual growth growth of of LPFG LPFG in in the the modeled modeled single single mode mode fiber fiber for for different different number number of of periods periods using mapped mesh. using mapped mesh. Figure 8. Gradual growth of LPFG in the modeled single mode fiber for different number of periods

mapped mesh. 4. Spectralusing Growth Experimental Validation 4. Spectral Growth Experimental Validation In this section, the Experimental numerically obtained are to be experimentally verified. The refractive 4. Spectral Growth Validationspectra In this section, the numerically obtained spectra are to be experimentally verified. The refractive index change in the fiber core was carried out using femtosecond laser (Spectra-Physics ultrafast Ti: In thisinsection, the numerically obtained spectra arefemtosecond to be experimentally verified. The refractive index change the fiber core was carried out using laser (Spectra-Physics ultrafast Sapphire Spectra-Physics, Clara,out CA, USA) with a pulse width of 120 femtosecond indexlaser, change the fiber core Santa was Santa carried using (Spectra-Physics ultrafast Ti:and a Ti: Sapphire laser,inSpectra-Physics, Clara, CA,femtosecond USA) withlaser a pulse width of 120 femtosecond centerSapphire wavelength of 800 nm. The optical fiber was firmly mounted on a computer-controlled laser, Spectra-Physics, Santa CA,fiber USA) with a pulse width of on 120afemtosecond and a3-axis and a center wavelength of 800 nm. TheClara, optical was firmly mounted computer-controlled stagecenter for alignment with submicron precision. The laser beam was reflected by a dichroic mirror and of 800with nm. The optical fiber was firmly mounted on a computer-controlled 3-axis stagewavelength for alignment submicron precision. The laser beam was reflected by a3-axis dichroic stage to forthe alignment withlens submicron precision. The laser beam was reflected by ainto dichroic mirrorcore. and The delivered objective (Numerical aperture: 0.55) and then focused the fiber mirror and delivered to the objective lens (Numerical aperture: 0.55) and then focused into the fiber to the objective (Numericalby aperture: 0.55) and then focused into the mirror. fiber core. LPFGdelivered fabrication process can lens be monitored the CCD camera above the dichroic TheThe period core. LPFG The LPFG fabrication process can be monitored by the CCD camera above the dichroic mirror. fabrication process can be monitored by the CCD camera above the dichroic mirror. The period and length of refractive index modulation of the fabricated LPFGs are 435 μm and 100 μm The period and length of refractive index modulation of the fabricated LPFGs are 435 µm and 100 and length refractive index modulation of the fabricated LPFGs areBBS-1550) 435 μm and μm µm respectively. Theoffiber was coupled with a broad-band light source (AFC and100 a spectrum respectively. The fiber was coupled with a broad-band light source (AFC BBS-1550) and a spectrum respectively. The fiber was coupled with a broad-band light source (AFC BBS-1550) and a spectrum analyzer (PHOTONETICS Walics, PHOTONETICS INC., Peabody, MA, USA) to monitor the growth analyzer (PHOTONETICS Walics, PHOTONETICS Peabody,MA, MA, USA) to monitor the growth analyzer (PHOTONETICS Walics, PHOTONETICS INC., INC., Peabody, USA) to monitor the growth of LPFG during fabrication. A schematic illustration of the experimental setup is shown in Figure 9. of LPFG during fabrication. A schematicillustration illustration of setup is shown in Figure 9. 9. of LPFG during fabrication. A schematic of the theexperimental experimental setup is shown in Figure

Figure Schematicillustration illustration of setup. Figure 9. 9. Schematic of the theexperimental experimental setup.

Figure 9. Schematic illustration of the experimental setup.

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The spectral growth of the inscribed fiber with an attenuation dip of 19 dB at 1555 nm is shown The spectral growth of the inscribed fiber with an attenuation dip of 19 dB at 1555 nm is shown in Figure 10. The results plotted in Figures 8 and 10 clearly state that a good agreement between the in Figure 10. The results plotted in Figures 8 and 10 clearly state that a good agreement between the numerically obtained results and the experimental findings is achieved. Thus, the beam envelope numerically obtained results and the experimental findings is achieved. Thus, the beam envelope technique is is toto bebeimplemented design an anLPFG LPFGrefractive refractiveindex indexsensor. sensor. technique implementedininthis thiswork workto to numerically numerically design However, designofof proposed model necessitates investigation of the However,full full numerical numerical design thethe proposed model necessitates furtherfurther investigation of the proper proper selection of the grating period and length. selection of the grating period and length. 5

0 = 20 = 22 = 24 = 26 = 28 = 30 = 32 = 34 = 36 = 38 = 40 = 42

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Wavelength(nm) Figure 10.10.Gradual fiber for for different differentnumber numberofofperiods. periods. Figure Gradualgrowth growthof ofLPFG LPFG in in SMF SMF fiber

5. Varying thethe Grating Length 5. Varying Grating Lengthand andGrating GratingPeriod Period AsAsthe LPFG sensor reliestheupon the introduction a periodic the fabrication fabrication ofofanan LPFG sensor chieflychiefly relies upon introduction of a periodicof modulation modulation mainlybyachieved bymodification permanent ofmodification the ofrefractive index of the the core, mainly achieved permanent the refractiveof index the core, characterizing characterizing the fabrication variables such as the grating period (Δ), and grating length (L) are of a fabrication variables such as the grating period (∆), and grating length (L) are of a major importance to major importance to fabricate reliably design and fabricate LPFG sensors.the The effectperiod of varying grating reliably design and LPFG sensors. The effect of varying grating of the the modeled LPFGofinthe this work is LPFG illustrated in work Figureis11. The plotted results11. clearly show that for each period period modeled in this illustrated in Figure The plotted results clearly show there is a unique The wavelength. numerical results indicate that as the gatingthat period that for each period resonant there is a wavelength. unique resonant The numerical results indicate as the increases, theincreases, resonant curve tends to get narrower suggesting a grating period equal gating period the resonant curve tendsand to deeper, get narrower andthat deeper, suggesting that a to 435period µm is an optimal choice increase sensitivity. Thethe effect of numerically grating equal to 435 μmtoispotentially an optimal choicethe to sensing potentially increase sensing sensitivity. assigning grating lengths different is shown grating in Figurelengths 12. In this plot, three sets of12. numerical studies The effect ofdifferent numerically assigning is shown in Figure In this plot, three are carried with assigned number of grating periods of 15, 30, and 45 periods while the grating sets of numerical studies are carried with assigned number of grating periods of 15, 30, and 45 periods period, ∆, is fixed at a value of 435 Asof stated in Figure 12a,b, setting 12a,b, the number periods while the grating period, Δ, is fixed at aµm. value 435 μm. As stated in Figure settingof the number to be 15 and 30 seems to provide wider and shorter attention dips. The plotted results shown in of periods to be 15 and 30 seems to provide wider and shorter attention dips. The plotted results Figure 12c suggest that fabricating the LPFG model with 42 periods causes the attention dips to rapidly shown in Figure 12c suggest that fabricating the LPFG model with 42 periods causes the attention surge, forming narrower valleys as the grating length reaches a value of 100 µm, whereas further dips to rapidly surge, forming narrower valleys as the grating length reaches a value of 100 μm, increments in the grating length seem to have an insignificant impact on the spectrum. Thus, in this whereas further increments in the grating length seem to have an insignificant impact on the work, the values of the grating period, grating length, and number of periods are 435 µm, 100 µm, and spectrum. Thus, in this work, the values of the grating period, grating length, and number of periods 42 periods respectively. are 435 μm, 100 μm, and 42 periods respectively.

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dB dB dB dB

dB dB

dB dB

10 of 15 10 10 of of 15 15

Figure Figure 11. Transmission spectrum of an an LPFG modeled in in SMF 11. Transmission Transmission spectrum spectrum of LPFG numerically numerically modeled SMF with with various various grating grating periods; μm respectively. periods; the the number number of of grating grating periods periods and and the the grating grating length length are are 42 42 and and 100 100 µm μm respectively. respectively.

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L=25 L=25 m m L=50 L=50 m m L=75 L=75 m m L=100 L=100 m m L=125 L=125 m m L=150 L=150 m m

L=25 L=25 m m L=50 L=50 m m L=75 L=75 m m L=100 L=100 m m L=125 L=125 m m L=150 L=150 m m

L=25 L=25 m m L=50 L=50 m m L=75 L=75 m m L=100 L=100 m m L=125 L=125 m m L=150 L=150 m m

Wavelength(nm) Figure Figure 12. The numerically acquired spectrum for various grating lengths; the grating period isis435 435 μm. Figure 12. 12.The Thenumerically numericallyacquired acquiredspectrum spectrumfor forvarious variousgrating gratinglengths; lengths;the thegrating gratingperiod periodis 435μm. µm. The numbers of periods in (a), (b), and (c) are 15, 30, and 42 respectively. The numbers of periods in (a), (b), and (c) are 15, 30, and 42 respectively. The numbers of periods in (a–c) are 15, 30, and 42 respectively.

6. 6. Experimental Experimental Validation Validation of of the the Modeled Modeled Refractive Refractive Index Index LPFG LPFG Sensor Sensor In In this this section, section, the the sensing sensing principles principles of of an an LPFG LPFG refractive refractive index index sensor sensor are are to to be be numerically numerically presented and experimentally validated. Due to the coupling of fiber optic core modes presented and experimentally validated. Due to the coupling of fiber optic core modes into into the the

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6. Experimental Validation of the Modeled Refractive Index LPFG Sensor In this section, the sensing principles of an LPFG refractive index sensor are to be numerically Materials 2016, 9, 941 11 of 15 presented and experimentally validated. Due to the coupling of fiber optic core modes into the cladding mode, the transmission valley appears. The phase matching between forward-propagating core mode cladding mode, the transmission appears. The phase matching between forward-propagating and cladding mode in an LPFG isvalley governed by [6,18]: core mode and cladding mode in an LPFG h is governed by [6,18]: i cl (m)

( ) λres == ncore ∆ e f f ((λres ))− −ne f f ((λres )) ∆

(5) (4)

( )

Transmission, dB

is the resonant wavelength, core is the effective index of the core mode, and cl (m) is the where λ where λres res is the resonant wavelength, ne f f is the effective index of the core mode, and ne f f is the effective index of the mth-order mth-order cladding cladding mode. mode. Due to absorption and scatterings, the cladding modes tend to escape from the fiber into the surrounding medium mediummaking makingLPFGs LPFGssuperior superiorrefractive refractive index sensing tools. stated in Equation surrounding index sensing tools. AsAs stated in Equation (5), (5), the resonant wavelength a function effectiveindices indicesofofthe thecore coreand andcladding cladding modes modes as the resonant wavelength is aisfunction of of thethe effective grating period period [6,18,20]. As the effective indices of the cladding modes depend on the well as the grating likely to to shift shift the the value value of of resonant resonant wavelength. wavelength. In the ambient, a change in the ambient index is likely proposed numerical model, the variation variation of the refractive index of the medium is achieved by conducting a parametric sweep of the refractive index values on the regions enclosing the SMF domains. The The numerical numerical results results are are plotted plotted in in Figure Figure 13, 13, showing showing a left shift of the transmission spectrum as the value of the swept refractive index increases. To verify the numerical variations of the refractive index of the medium on the light spectrum, sets of experiments are carried out at room temperature. In each set of the experiments, experiments, the the fabricated fabricated grating grating is is immersed immersed in in aa glycerin glycerin solution. solution. The gradual variation of the glycerin concentration is set such that the refractive index varies from 1.3307 to 1.4474. The spectral behaviour behaviour of varying the refractive index of the solution is captured and recorded using the spectrum analyzer. The effect of varying the refractive index of the medium on the spectrum for the LPFG inscribed in pure silica core SMF is shown in Figure 14, where a shift of the attention dip to the left gradually gradually occurs as the the index index increases. increases. As shown in Figure 15, the numerical and experimental spectral shift of the transmission valley increases non-linearly as the medium refractive refractiveindex index increases. the refractive indices in thisa maximum work, a maximum medium increases. ForFor the refractive indices tested tested in this work, deviation deviationthe between theand numerical and experimental is the found to of be0.175%, in the which order confirms of 0.175%, which between numerical experimental is found to be in order adequate confirms adequate between the numericalfindings. and experimental findings. agreement betweenagreement the numerical and experimental

Figure function of of medium medium refractive refractive index. index. Figure 13. 13. The The numerically numerically obtained obtained spectral spectral growth growth as as aa function

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55 00 -5 -5 Air Air 1.3307 1.3307 1.3478 1.3478 1.3634 1.3634 1.3767 1.3767 1.3917 1.3917 1.4063 1.4063 1.4204 1.4204 1.4378 1.4378 1.4474 1.4474

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Wavelength(nm) Wavelength(nm)

Figure 14. 14. Experimentally Experimentally obtained obtained shift of the transmission transmission valley valley when when the the LPFG LPFG is is exposed exposed to to Experimentally obtained shift shift of of the the transmission valley when the LPFG is exposed Figure solutions of of different different refractive refractive indices. indices. solutions

Figure 15. Numerical Numerical vs. experimental experimental refractive index index characterization of of an LPFG LPFG sensor. Figure Figure 15. 15. Numerical vs. vs. experimental refractive refractive index characterization characterization of an an LPFG sensor. sensor.

7. Conclusions Conclusions 7. 7. Conclusions Two finite finite element element models models constructed constructed in in the the COMSOL5.2 COMSOL5.2 Wave Wave Optics Optics Module Module and and correlating correlating Two Two finite elementindex models constructed in the COMSOL5.2 Wave Optics Module andLPFG correlating the induced refractive profile generated by a femtosecond laser inscription to an model the induced refractive index profile generated by a femtosecond laser inscription to an LPFG model the induced refractive index profile generated by a femtosecond laser requirements inscription to of an the LPFG model of a single-mode fiber are presented. The demanding computational proposed of a single-mode fiber are presented. The demanding computational requirements of the proposed of single-mode fiber areby presented. demanding computational requirements of the proposed FEamodels models are overcome overcome using the theThe beam envelope method. method. The The first first model is is aa 3D 3D model and is is FE are by using beam envelope model model and FE models are overcome by using the beam envelope method. The first model is a 3D model and is capable of characterizing the intensity-dependent refractive index of the core. The induced refractive capable of characterizing the intensity-dependent refractive index of the core. The induced refractive capable of characterizing the intensity-dependent indexslab. of the core. Themodel induced refractive index is is deployed deployed in an an LPFG LPFG model portrayed portrayed as asarefractive a 2D 2D dielectric dielectric The LPFG simulates the index in model slab. The LPFG model simulates the index is deployed in an LPFG model portrayed as a 2D dielectric slab. The LPFG model simulates spectral distribution generated from the interaction between the light and the SMF. The analysis of spectral distribution generated from the interaction between the light and the SMF. The analysis of the spectral distribution generated in from the interaction between thevalues light and thenumber SMF. The analysis the simulation results conducted this work suggests that the of the of grating the simulation results conducted in this work suggests that the values of the number of grating of the simulation resultsand conducted in thisare work suggests thatμm, the respectively. values of theThe number of grating periods, grating period, period, gratinglength length 42,435, 435,and and100 100 spectral growth periods, grating and grating are 42, μm, respectively. The spectral growth periods, grating period, and grating length are 42, 435, and 100 µm, respectively. The spectral of the LPFG model resulting from setting the averaged laser energy to be 0.50 μJ for various numbers of the LPFG model resulting from setting the averaged laser energy to be 0.50 μJ for various numbers of periods periods is is experimentally experimentally validated validated using using identical identical inscribing inscribing conditions. conditions. The The numerically numerically of

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growth of the LPFG model resulting from setting the averaged laser energy to be 0.50 µJ for various numbers of periods is experimentally validated using identical inscribing conditions. The numerically obtained results exhibit good agreement with the experimental data. Additionally, the refractive index of the domains surrounding the modeled SMF is varied to account for various glycerin solution concentrations. Non-linear shifts of the numerically acquired spectra for various refractive indices are observed and successfully verified by experimentally immersing the gratings in a glycerin container. Acknowledgments: The authors acknowledge the support of the Korea Carbon Capture and Sequestration Research and Development Center for this work. Author Contributions: Akram Saad conceived and designed the experiments based on the proposed numerical approach; Yonghyun Cho and Farid Ahmed performed the experiments; Akram Saad analyzed the data; Martin Jun contributed reagents/materials/analysis tools; Akram Saad wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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