Application of cavity Maxwell Garnett theory in SPR based fiber optic sensor with porous alumina structure

The present manuscript describes the theoretical understanding of nanoporous alumina based fiber optic sensing structures. The Cavity Maxwell Garnett theory is used to calculate the dielectric functions of the proposed layer. The performance of the proposed sensing structure is evaluated in terms of its sensitivity towards change in the refractive index of the nearby medium. The sharpness of the resonance has also been calculated as an estimation of the performance parameters. It has been observed that the proposed structure is approximately 13 times more sensitive than the conventional fiber optic sensors. The study has further been extended by replacing the nanolayer of aluminum with the nanolayer of the gold. A comparative study has been provided in terms of the efficiency of the fiber optic probe. The effects of change in pore radius, thickness of the adsorbed medium and shell radius have also been studied.


Introduction
The application of surface plasmons in sensing is a rapidly growing field and is upgrading its performance in chemical sensing, environmental monitoring, foodborne screening, medical diagnostics and gas sensing continuously (Sharma et al. 2007;Piliarik et al. 2009;Sipova and Homola 2013;Homola 2008;Sharma and Mohr 2008). Surface plasmons are visualized as the coherent oscillations of free electrons on the metal dielectric interface (Verma and Gupta 2010). For the occurrence of surface plasmon resonance (SPR) between surface plasmon waves and evanescent waves a specific Kretschmann configuration is required (Chen et al. 2015;Fontana 2006;Kretschmann 1971). In fiber optic based SPR sensors small portion of cladding is removed from the middle portion of the fiber and then is coated with a thin layer of plasmonic material Ag or Au in general. Subsequently a polychromatic light is launched from one end of the fiber such that the light gets guided using total internal reflection phenomenon on the interface of its core and cladding (Srivastava and Gupta 2013). This has some additional advantages in remote sensing and online monitoring (Jha et al. 2008). Fiber optic with Bragg grating and long period grating written on its core are also being used for the enhancement of its performance parameters (Ma et al. 2014;Tripathi et al. 2008).
To foster the performance of fiber optic based SPR sensors, numerous studies have been carried out by using different metals for thin film deposition, nanostructures like nanorods, nanospheres, nanoporous structure and many others. An overlayer of an additional nanostructure ameliorates the performance of the conventional fiber optic based SPR sensors. The material for the selection of nanostructure should be chemically inert, non-degradable and thermally stable (Vojkuvka et al. 2012;Marsal et al. 2009). Alumina fulfils all these conditions and formation of nanoporous alumina structure is less expensive and quite easy. Synthesis of nanoporous material by using electro-chemical techniques have various optical applications in SPR, Raman spectroscopy, reflective interference and in waveguides (Santos et al. 2013). This technique has several advantages like highly ordered nanochannel array and have vertical alignment of cylindrical pores. Nanoporous anodic alumina (NAA) is an optimum example of this technique which has distinct chemical, optical and mechanical properties as well as good thermal stability and high surface to volume ratio (Gyurcsanyi 2008). NAA has self-ordered, homogeneous and honeycomb like structure. Its geometrical parameters like pore diameter, interpore distance and length of pore can be tuned by changing anodization condition like temperature, current, anodization voltage, electrolyte, composition of electrolyte and time. It has chemical inertness and ability to protect itself from self-degradation (Ghicov and Schmuki 2009;Lee et al. 2006;Santos et al. 2011). NAA have large surface area and density of pores and hence the molecules get attached with hydroxyl group using hydrogen bond which makes a physiosorbed layer. These adsorbed molecules make changes in effective refractive index of alumina layer which can be easily sensed by fiber optic based SPR sensor. Porous structure of NAA has high surface to volume ratio which helps in adsorption of more sensing medium. It has multiple applications in bio-sensing, filtration, solar cells, data storage, drug delivery, template synthesis, humidity and gas sensing (Kumeria et al. 2014;Losic and Lillo 2009;Lee et al. 2008a;Meng et al. 2005;Losic 2009). Further, to evaluate the optical properties of nanostructures the calculation of dielectric function is a major parameter for inhomogeneous mixture of dielectric materials. Several effective medium theories like Bruggeman, Monache, Landau-Lifshitz/Looyenga, Bergman representation and Cavity Maxwell Garnett effective medium theory can be used to calculate effective dielectric function (Bruggeman 1935a;Landau and Lifshitz 1961;Looyenga 1965;Maxwell Garnett 1904a;Monecke 1994;Bergman 1980). For nanoporous structure Cavity Maxwell Garnett theory is most suitable which considers the effect of three inhomogeneous materials. In our case we will consider the effect of 3 layers viz. metal layer, nanoporous alumina and air (Spanier and Herman 2000). In the present study, we proposed SPR based fiber optic sensor in which fiber core is coated with a nanolayer of aluminum metal with an additional nanoporous alumina layer coated on it. Additionally, since gold is also compatible to couple with nanoporous alumina layer (Evans et al. 2006), therefore the study has been extended to see the effect of replacing aluminum nanolayer with gold nanolayer. A comparison between the performance parameters of sensor is analysed for both aluminum and gold with porous alumina on to the top of it. The proposed structure with nanoporous alumina film enhances the sensitivity to approximately 13× to their conventional aluminum and gold based configurations. It was also observed that the gold based configuration has better sensitivity than aluminum based configuration though the sharpness of resonance is better in case of aluminum.

Structure details
In SPR based fiber optic sensors, surface plasmons can be excited using Kretschmann configuration in which a thin metal layer is deposited over the uncladded portion of the core (Conventional structure) and an aqueous sensing medium is place over this metal layer. However in the proposed structure we further coat this metal layer with porous alumina structure that will be surrounded by the medium to be sensed (Proposed structure) as shown in Fig. 1. A sharp dip in the transmitted spectra corresponding to the refractive index of the sensing medium is observed. The shift in dip position corresponding to the change in the sensing medium makes this configuration useful for sensing applications. In fiber based configuration, wavelength interrogation method is used in general (Homola 1997). To calculate the refractive index of fused silica fiber Sellemeir dispersion relation is used and the metal dielectric constant can be calculated by using Drude dispersion relation (Ghatak and Thyagarajan 1998;Mishra et al. 2016).
Further, in our proposed configuration we are using nanoporous aluminium oxide because of its various applications in molecular separation, energy generation, catalysis, electronics, photonics, sensors, storage and biosensors also (Jani et al. 2013). Nanoporous anodic alumina (NAA) fabrication is simple and economical with the process of self-ordering of nano-pores which do not even call for lithography and the end results are highly ordered (Ghicov and Schmuki 2791;Wehrspohn 2005;Ren et al. 2012). NAA has close packed structure of hexagonal arrays having nanopores aligned perpendicular to the surface of the substrate material. The pore geometry can be controlled by applied potential (Woo and Jae-Cheon 2010;Yi et al. 2009;Zhang et al. 2007), type of electrolyte (Friedman et al. 2007;Chu et al. 2006;Ono et al. 2005), composition of electrolyte (Ono et al. 2005), PH (Ono et al. 2005), current (Lee et al. 2008b(Lee et al. , 2010 and temperature (Sulka and Stepniowski 2009) to achieve the desired pore size and interpore distance. NAA has incredible properties like chemical and thermal stability, more surface area and hardness.

Cavity Maxwell Garnett theory
Dielectric function is one of the major parameter to calculate optical properties like reflectance and transmittance (Spanier and Herman 2000). In the study of inhomogeneous mixture of dielectric materials which have dimensions small in comparison of wavelength the dielectric function has a notion of nanostructure of the dielectric material. The 2 effective models which are most used are Bruggeman (Bruggeman 1935b) and Maxwell Garnett theories (Maxwell Garnett 1904b). Maxwell Garnett model is further having two types one is Spherical Maxwell Garnett (SMG) and another is Cavity Maxwell Garnett (CMG). In SMG, spherical inclusion is filled with materials like metal oxides and air must be filled outside. In CMG, air inclusions are spherical and material is coated outside of it as depicted in Fig. 2. Afterward, various modification is done to these effective medium theories, because the voids are not always spherical sometimes these may be elongated and cylindrical. SMG model is less explored than CMG because of its undesired result of spectra (Spanier and Herman 2000). So further study was done on CMG in which the characteristics of damping of phonons on bulk surface are different from phonon characteristics into the bulk material. Weaver et al. (Weaver et al. 1973) and others (Sasaki et al. 1989) remodel the CMG model which consists of 3 dielectric materials in which 2 of them are related (like metal and metal oxide) and one is air. For different topology/geometry damping characteristics of phonon is different. Shell layer represents the sensing medium layer and its thickness, plasma frequency, rate of phonon damping affects the reflectance and transmittance of spectra. NAA consists three constituent materials in which alumina is considered as surrounding medium, sensing medium present on the walls of the pore and air is filled in open space of the pore (Koutsioubas et al. 2008). The effective contribution of these is given by following formula (Spanier and Herman 2000): where A c = ε s B, A s = (3 + 2ε s )B, ρ = r p /r s and where ε s = dielectric constant of shell (adsorbed molecules on the walls of the pore), r p pore radius, r s shell radius and f is the pore volume fraction, which is given for cylindrical pore and shell, and ε Al 2 O 3 is the dielectric constant of the aluminium oxide which is given as (Tabassum and Gupta 2017): (3) f = volume of pore volume of shell  Koutsioubas et al. fabricated NAA based structure with prism to design surface plasmon resonance-based sensor. Firstly, a thin metal nanofilm is deposited on the prism surface by using the vacuum evaporation with 1 × 10 −6 Torr base pressure and after that porous alumina is grown on it by anodization process. The results were explained for aluminum + NAA structure and added the suggestion that NAA structure can also be used to fabricate on gold nanofilm (Koutsioubas et al. 2008). Hotta et al. discussed the adsorption of Bovine serum albumin (BSA) on the nanoporous structure of alumina and reported remarkable shift in spectra due to its higher adsorption capacity. BSA have dimensions 4 × 4 × 14 nm 3 which are very tiny and can be penetrate into the nanoporous alumina structure very easily with pore diameter of ten to hundreds of nanometres. For different concentration of BSA solutions red shift is observed in recorded spectra (Hotta et al. 2012). Lau et al. explained that NAA provide more binding sites which increases the sensitivity. In this article, binding sites in a unit surface area were further increased by surface grafting of polymers. NAA structure is designed using the anodization process of aluminium film which forms close packed, uniform and cylindrical nanopores. Poly (γ-benzyl-L-glutamate) makes ester side chains which gives the advancement to use this in many practical applications. Firstly, the silane functionalization was done on NAA which gives the large density of surface groups for the polarization of N-carboxy anhydride. And after that optical properties are measured by using effective medium theory which depends on the sensing medium, nanostructure morphology and dielectric response of bulk material (Lau et al. 2009). All the experiments were done using SPR sensing technique on the prism, but we have modelled these structures on optical fiber which is more compact, convenient to use with a potential of remote sensing.

Performance parameters
The performance of the fiber optic SPR sensor is analyzed in terms of sensitivity, full width half maximum (FWHM) and the sharpness of resonance. In the wavelength interrogation mode, the resonance wavelength λ res corresponds with refractive index n s of sensing medium. Variation in the refractive index of sensing medium leads to the change in its resonance wavelength. Therefore, the sensitivity can be calculated as the ratio of change of resonance wavelength Δλ res , with the variation of refractive index of the sensing medium Δn s (Vikas and Verma 2018) The FWHM can be determined by calculating the full width at half maximum of the resonance curve (Δλ 0.5 ), expressed as: Sharpness of the resonance is defined as the reciprocal of FWHM.

Results and discussion
In the present simulation, we have considered silica-based fiber core of diameter 600 μm and 0.5 cm of uncladded portion as a sensing region which is preferably placed in the middle of the fiber. To enhance the performance parameters of the sensor we put an overlayer of porous alumina on the nanofilm of metal layer. We present a comparison of the performance parameters of conventional aluminium based configuration (fiber + aluminum film + sensing medium) with the proposed version with an additional nanoporous alumina layer (fiber + aluminum film + nanoporous alumina + sensing medium). In another case, we consider conventional gold-based configuration (fiber + gold film + sensing medium) and compared with the proposed version (fiber + gold film + porous alumina + sensing medium) (Figs. 3 and 4). The holes in the nanoporous alumina are termed as shells of definite radius. After the adsorption of the sensing medium on to the inner walls of the shell the empty space is called as the pore. The difference between the shell radius and the pore radius is the thickness of the medium that has been adsorbed on to the walls of the shell. We have considered aluminum film thickness of 45 nm and gold film thickness as 50 nm. For both the cases we have examined the refractive indices of aqueous based sensing medium in the range of 1.333-1.337 with a difference of 0.001 (5 samples), since most of the bio-analytes are aqueous based and have similar refractive indices.
(6) FWHM=Δλ 0.5 (7) Sharpness of resonance = 1 FWHM Fig. 3 Magnified illustration of the nano-porous alumina surface before and after the formation of a thin layer on the pore walls from horizontal surface

Conventional structure
In Figs. 5 and 6, we observed 2 resonance dips corresponding to the refractive indices 1.333 and 1.337 in transmitted power for conventional aluminum and gold structures and the shift in resonance dip is quite small that leads to smaller sensitivity. We have calculated sensitivity for this configuration which is 2 μm/RIU and for conventional gold structure it is 3.5 μm/RIU. Sharpness of the resonance was also calculated for the conventional aluminium and gold configuration and is 0.4 μm −1 and 0.0769 μm −1 . Aluminum and gold are considered because aluminum have large sharpness of resonance and gold leads to higher sensitivity as reported in many research articles.

Proposed structure analysis
To realize the proposed structure the basic conventional structure is coated with an overlayer of porous alumina because its porous structure increases surface to volume ratio which may boost up the adsorption of sensing medium. From Figs. 3 and 4 (enlarge view) it can be seen that we are dealing with small feasible size of shells on porous alumina so that we can get large number of shells in a fix area and more sensing medium can get adsorbed on the walls of the shell. To plot the SPR curves for the proposed structure we have considered a fixed value of the shell radius as 0.0075 μm with the changing values of the pore radius (Figs. 7 and 8). Adsorbed layer thickness is the difference of shell radius and the pore radius. Further, in Fig. 9, the red shift in the resonance wavelength of the proposed configuration with aluminum layer can be seen corresponding to the change in the refractive index of sensing medium from 1.333 to 1.337. In this configuration the resonance condition gets satisfied at higher value of wavelength than the conventional structure. The conventional structure works with wavelength of visible region only whereas in the proposed configuration resonance wavelength extends in IR range also. We choose a wide wavelength range of light because the penetration depth in dielectric is higher at longer wavelength and surface plasmon wave penetration depth in metal is higher at lower value of wavelength (Maharana et al. 2013). This change in resonance dip with the variation of refractive index represents sensitivity of the proposed sensor. Similarly, when we place gold nanofilm at the place of aluminium we observe that resonance condition gets satisfied at even higher wavelength for the same refractive indices than in case of aluminum (Fig. 10). It is observed that red shift in transmitted spectra is more in case of gold than aluminum. The maximum sensitivity is observed for the pore radius 0.00573 μm with adsorbed layer thickness of 0.00177 μm and the sensitivity value in this case for aluminum and gold nanofilm is 26 μm/RIU and 46.25 μm/RIU respectively which is approximately 13× to their conventional structure (2 μm/RIU and 3.5 μm/RIU). This huge boost in the sensitivity is due to their larger shift in the resonance dip and creates a significant impact on the performance of the sensor. It is to be noted that the sensitivity is more in case of gold nanofilm than the aluminum nanofilm as in conventional case. The physical reason behind this enhanced value of sensitivity could be the large adsorption of the sensing medium owing to the large surface area of the nanostructure. In Fig. 11, the SPR curves observed for shell radius 0.0075 μm, pore radius 0.00573 μm and absorbed layer thickness 0.00177 μm which is also the case of maximum sensitivity for both aluminum and gold are shown. Careful observation of these SPR curves reveal that the aluminum nanofilm have sharp resonance dip than gold however the dip shift in gold is more than aluminum and is also at higher wavelength.

Sensitivity
Koutsioubas et al. fabricated a sensor with shell radius of 0.0075 μm and adsorbed layer thickness of 0.002 μm (Koutsioubas et al. 2008). With taking that as a reference we also modelled a sensor with same shell radius of 0.0075 μm and optimized the value of adsorbed layer thickness to check the performance parameters. It was observed that 0.00177 μm adsorbed layer have highest value of sensitivity. Pore radius can be measured by taking the difference of shell radius and adsorbed layer thickness which is 0.00573 μm in this case. Figure 12 shows the structural change with increase in pore radius for a fix value of shell radius and the corresponding decrease in the thickness of adsorbed layer of sensing medium. In Fig. 13, we have shown a comparison between the proposed configurations of aluminum and gold with variation of pore size in the nanoporous structure of alumina. The fix value of shell radius is 0.0075 μm and the pore radius varies from 0.00573 to 0.0061 μm with decreasing thickness of the adsorbing layer. In proposed configuration, gold base structure has maximum sensitivity of 46.25 μm for pore radius 0.00573 μm and Fig. 12 Structural change with variation in pore radius and thickness of sensing medium Fig. 13 Comparison of Sensitivity for Al film + porous alumina and Au film + porous aluminum structures with pore radius minimum sensitivity 4.5 μm/RIU, corresponding to the pore radius 0.0061 μm. In the case of aluminum based proposed configuration, maximum sensitivity is 26 μm/RIU for pore radius 0.00573 μm and minimum sensitivity is 2.75 μm/RIU corresponding to pore radius 0.0061 μm. It has been observed that for the further increase in the radius of pore, sensitivity becomes nearly equal to their conventional structure. It can be concluded that the more is the thickness of adsorbed layer better is the sensitivity. If we further increase the adsorbed layer thickness beyond 0.00177 μm the resonance dip falls in mid infrared region where the dispersion relation for metals given by Drude model is not valid and if thickness of adsorbed layer is less than 0.00153 μm then sensitivity becomes approximately equal to their conventional configuration. Therefore, if the adsorption of the sensing layer on to the walls of the shell is very weak the sensitivity value will be poor as can been seen from Table 1.

Sharpness of resonance
The sharpness of resonance is dependent on full width at half maximum of the SPR curve. In Sect. 4.2 we have discussed that the gold configuration have broader resonance dip than aluminum configuration and hence it is expected from equation 7 that the sharpness of resonance for gold-based configuration is more than aluminum. From Fig. 14, it can be seen that for proposed aluminum configuration slope for the sharpness of resonance of aluminum is much higher than that of the gold based structure. In the proposed aluminum configuration, the highest sharpness of resonance is 250 μm −1 for pore radius 0.00593 μm and lowest sharpness of resonance is 30.3030 μm −1 for pore radius 0.00573 μm corresponding to the fix shell radius 0.0075 μm. For proposed gold configuration, highest sharpness of resonance is 55.55 μm −1 for the pore radius 0.00593 μm, and lowest sharpness of resonance is 3.4364 μm −1 for pore radius 0.00573 μm, for the same value of shell radius 0.0075 μm. Hence it can be concluded that the better is the adsorption of the sensing layer on to the walls of the shell poor will be the sharpness of the resonance. The numerical values of the sharpness of the resonance can be seen from Table 2.

Sensitivity
After the optimization of adsorbed layer thickness, we kept this thickness constant 0.0018 μm (by rounding off 0.00177 μm, the last optimized value) and varied the shell radius in the range of 0.00765 μm to higher shell radius and pore radius varied in the range of 0.00585-0.00705 μm (Fig. 15). We found that the highest sensitivity of the proposed aluminum configuration was 23.25 μm/RIU and for gold configuration it was 41.5 μm/ RIU, corresponding to pore radius 0.00585 μm and shell radius 0.00765 μm. Least sensitivity of proposed aluminum and gold configurations were 2.5 μm/RIU and 4.25 μm/RIU corresponding to pore radius of 0.00705 μm and shell radius of 0.00885 μm as shown in Fig. 16 (Table 3). Further increase of pore and shell radius gave approximately similar sensitivity to their conventional structure and the nanoporous alumina layer is no longer useful. It is to be noted that in porous alumina structure, with the increase of shell radius for a fix area and interpore distance the number of pores will be less and in the case of small pore sizes the number of pores will be large. Larger the number of pores means more material get adsorbed on the walls of pore leading to higher value of the sensitivity.

Sharpness of resonance
The sharpness of resonance of proposed configurations has also been evaluated for fix thickness of adsorbed layer of sensing medium corresponding to increasing pore and shell radius. It can be seen from Fig. 17, the slope of the proposed aluminum configuration is much higher than that of the gold configuration because aluminum has sharper SPR dip than gold. The maximum sharpness of resonance was found to be 222.22 μm −1 and 74.07 μm −1 for proposed aluminum and gold configuration corresponding to pore and shell radius of 0.00705 μm and 0.00885 μm respectively. The numerical values can be read from Table 4.

Trade-off between sensitivity and sharpness of resonance
It can be noticed from the above discussion that there is a tradeoff relationship between performance parameters i.e. between sensitivity and the sharpness of the resonance. For the same value of pore radius we did not to get both high sensitivity and sharpness of resonance. To locate an optimum value of the pore radius for a reasonably good value of both the parameters we have plotted in Fig. 18, both the performance parameters with pore radius. From the intersection point of both the curves we got an optimum value of pore radius where both sensitivity and sharpness have moderately high value. The pore radius value of 0.00623 μm gave the optimum point where we got the sensitivity and sharpness of the resonance as 7.9 μm/RIU and 81.2865 μm −1 respectively for aluminum based configuration. Similar to the case of aluminum, we located the optimum pore radius for the proposed gold configuration where both the sensitivity and sharpness of resonance can provide a moderately high value. From the Fig. 19, it can be seen that the pore radius 0.006434 μm the moderately high values of sensitivity are 9 μm/RIU and sharpness of resonance 16.608 μm −1 respectively.

Fig. 17
Sharpness of resonance variation with pore radius corresponding to fix sensing layer thickness

Conclusions
A rigorous analysis of the porous alumina based fiber optic sensor has been deliberated. A comparative analysis between the aluminum and gold based structures revealed that the sensitivity is better in case of gold based configuration as compared to aluminum based configuration. However in both the configurations, the sensitivity values are approximately 13 times higher than their conventional counterparts. Further, the sharpness of the resonance has also been studied in detail and a trade-off relationship between Fig. 18 Optimum value of pore radius for reasonably good sensitivity and sharpness of resonance for Al + porous alumina Fig. 19 Optimum value of pore radius for reasonably good sensitivity and sharpness of resonance for Au + porous alumina the sensitivity and sharpness of the resonance has been observed. The optimum value of the pore radius has been located for both the configurations.