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Opto-Electronics Review

Editor-in-Chief: Jaroszewicz, Leszek

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Volume 24, Issue 4

Issues

Influence of a thin metal layer on a beam propagation in a biconical optical fibre taper

K.A. Stasiewicz
  • Corresponding author
  • Institute of Applied Physics, Military University of Technology, ul. Gen. Kaliskiego 2, 00–908 Warsaw, Poland
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/ J.E. Moś
  • Institute of Applied Physics, Military University of Technology, ul. Gen. Kaliskiego 2, 00–908 Warsaw, Poland
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Published Online: 2016-10-17 | DOI: https://doi.org/10.1515/oere-2016-0027

Abstract

The paper presents results of a simulation of the plasmon effect achieved between a thin precious metal layer and a biconical optical fibre taper, manufactured on a standard single mode fibre. Gold, silver and titanium were used as a metal which fulfilled a cladding function for a small diameter structure. For simulation Mode Solution software was used on which modal and frequency analyses of a wavelength were provided in the range of 800–1700 nm. A displacement of a plasmon pick in dependence of thickness of a deposited precious layer for the highest plasmon effects was observed.

Keywords: optical fibre technology; tapered technology; plasmon effect; precious metal layer

1 Introduction

The development of optical fibre technology over the last century was significant. Fibre technology can be divided into two parts: telecommunication and non-telecommunication. The telecommunication part is a good compound and now many of guided researches are oriented on capacity increase, optimization or existing systems’ improvement. Development of a non-telecommunication area gives a lot of possibilities for manufacturing new advanced elements and arrangements, as well as discovering or developing new applications for the existing solution. One of the promising technologies is optical fibre taper technology [14]. This technique provides a small insertion loss to a propagating beam and is comparable with standard optical fibres’ dimension. The basic element, which is a bi-conical optical fibre taper, allows controlling light beam propagation in the structure. This element is a base for manufacturing advanced hybrid elements like polarization controllers [5], optical switchers [6], white lasers [7] or optical sensors [8,9]. Finally, the tapering technology is relatively easy and cheap. The most important fact about this technology is that it enables continuous monitoring of the light propagation through the tapering process [10]. Tapering process enables obtaining demanded diameters of a fibre structure without the necessity of a new fibre’s drawing. Process of tapering makes decrease the fibre diameter to the state in which a beam propagates in the whole taper structure and the surrounding air becomes a cladding. This effect gives possibility to change a boundary condition by replacing air with materials with different optical parameters like refractive index, chemical composition, extinction index, transparency. A very interesting phenomenon as surface plasmon resonance can be obtained by using a metal layer. For the last few years a research on manufacturing different optical sensors by using a metal layer as an active material has been carried out [1116]. Gupta worked on SPR effect in tapered optical fibres and described it mathematically. [11,13,16].

The research presented in this paper describes an influence of a boundary condition modification on propagation of the light parameters in a taper region. In this paper we present the simulation results of a thin, precious metal layer influence on spectral characteristics of attenuation and dispersion. These results give information about SPR effect which will be useful for design and manufacturing of advanced and hybrid optical fibre sensors. In Sect. II the main parameter of an optical fibre taper and beam propagation parameters like penetration depth and dispersion are described. In Sect. III optical characteristics of applied materials are presented. Section IV contains results of a simulation of a thin metal layer influence on dispersion and attenuation characteristics. Section V contains conclusions and additional information about development of the presented work.

2 Taper as an element allowing to control light beam penetration depth in a cladding and dispersion properties in a structure

In a standard or microstructured fibre light propagates along the structure of a core surrounded by a cladding – using material or geometrical properties. Such structure has on purpose to achieve as small propagation losses as possible. These properties require an adequate dimension of a core or cladding keeping an optical beam from propagating outside the fibre structure. There is no possibility to direct external influence on the beam. One of the ways allowing the light beam to propagate inside such structure is to reduce the fibre diameter to few microns.

A diameter of several micrometres can be acquired in an easy way in a process of fibre tapering. We can describe few methods of tapering including: elongation in a low pressure gas burning [10], CO2 laser [17,18], heating gravitation elongation [1], elongation in a filament fusion technology [19] or chemical etching [20]. All of these methods of fibre tapering are different from technological point of view (system of heating and elongation), but not for the output shape -the taper. The shape of manufactured tapers, as well as modification of beam propagation parameters in structure were widely described [10,11,21], so now we concentrate only on a possibility to control a penetration depth and dispersion properties.

Process of tapering in a low pressure gas burner makes decrease the diameter of a core and cladding simultaneously. A change of the core-cladding boundary causes an increase of the mode field diameter (ω0). In the beginning, a part of the beam spot which propagates outside the core can be described as an evanescent field which does not propagate along the structure of a fibre. In a particular case, a beam spot covers the whole structure of a taper. From the Maxwell equation appears that a part of the beam propagates outside the guiding material (depth of penetration) (Fig. 1) [22ߝ24].

Course of a propagating optical beam in the standard optical fibre with the marked beam penetration depth [23].
Fig. 1

Course of a propagating optical beam in the standard optical fibre with the marked beam penetration depth [23].

The depth of beam penetration in the region of cladding for a chosen wavelength is given by the following equation [23] dp=λ2πncore2sin2θncladding2,(1)

where λ is the determined wavelength of light, θ is the angle of incident light on the core/cladding surface, ncore, ncladding are the refractive indices of core and cladding.

As a result of the tapering process, the penetration depth of a light beam increases more and more the cladding section to the point where the light beam is guided in a whole taper structure which remains a core and surrounding air becomes a cladding The boundary conditions and the refractive index are modified according to geometrical changes of the core and cladding diameters. The normalized frequency is written as [25] V=a2πcλncladding2nair2,(2)

where a is the radius of a taper, λ is the wavelength, ncladding is the taper refractive index, nair is the surrounding refractive index.

This effect of a light beam leaking an optical fibre over the structure determines the use of an additional material as a cladding. In this paper the results of simulation using gold, silver and titanium forming an additional thin layer as a cladding part are presented.

For description of an influence of a thin layer on propagation parameters, two kinds of coefficients were simulated: attenuation and dispersion. Attenuation characteristics were obtained as a ratio between input power and output power in a wide wavelength range [22,26]. Dispersion can be described as a dependence of a refractive index on a wavelength and is linked with a change of propagating impulse shape depending on its velocity in a fibre.

In the paper all research is focused on a chromatic dispersion. Such dispersion is set for two components: material dispersion and waveguide dispersion. First of them is associated with a dispersion property of the material from which the fibre is made. The waveguide dispersion is connected with the wavelength dependency on fibre geometry as well as on propagation constant [27,28]. The waveguide dispersion can be written as D=2πυg2λ2×dυgdω,(3)

where υg = dω/dβ is the group velocity: a> is the optical frequency, β is the propagation constant of light in the structure [28].

It should be noticed that in an optical fibre taper most of the guided power is concentrated in a very small region of a waist. Strong nonlinear effects including soliton generation, four-wave mixing, self-phase cross-phase modulation, Raman scattering and change of dispersion made by change of propagation condition [29,30] arise from this concentration of power in a small region. Effects can appear simultaneously. The result of the mentioned effects is a shift of dispersion characteristics towards shorter wavelengths which is shown in the further part of this paper.

3 Description of material layers

A refractive index of the absorption medium can be described by a formula which contains the real (n) and imaginary (κ extinction index) parts. The real part is connected with the phase velocity and the imaginary part influences the light attenuation.

n˜=n+iκ.(4)

The extinction index is a material constant and is important for description of an electromagnetic field attenuation which goes into the material. In our case the absorption material is one of the used precious metals. Light absorption in a metal material is linked with attendance of free electrons. Metals can be characterized as materials containing high density of free electrons (1028 m−3) which is coupled with an electric permeability in dependence on light frequency.

Phenomenon of a light propagation on a dielectric-metal boundary uses a surface plasmon polaritions’ effect (SPP). This effect can be described as a propagation of the electromagnetic field along a dielectric-metal boundary. Electric permittivity of a dielectric is higher than zero, however for the metal it is complex. For the dielectric-metal boundary there is a real part of dielectric constant with an opposite sign which enables SSP propagation. It is made by a density oscillation of electric charge on the medium boundary. Quantum of the oscillation is called a surface plasmon which is a transverse wave (TM) which sinks exponentially in a metal layer [31].

By solving the Maxwell equation it is possible to find a relation between a surface plasmon wave number and frequency [13,16,32]: ksp=kx=ωcεmεdεm+εd,(5)

here c is the light speed in vacuum, ɛm and ɛd are the permittivity of a metal and dielectric, respectively.

The real part of a wave plasmon vector describes a surface plasmon phase velocity and the imaginary part describes the attenuation which is united with the width of the resonance peak [32].

Characteristics of the refractive index (n) and the extinction index (κ) vs. wavelengths for gold, silver and titanium metals are presented below in Fig. 2. As it can be seen, characteristics for gold and silver are similar [33,34]. Refractive indices for both of them are lower than 2, the extinction index increases in a linear way to above 16. So it can be seen that the extinction index is much higher than refractive index. Characteristics of refractive and extinction indices for titanium present a totally different shape. Both, refractive and extinction indices present similar value for the chosen wavelength and their changes are similar. The reason for which we have chosen precious metals was connected with their optical parameters like refractive and extinction indices, permittivity etc., as well as with a well-known technology for their deposition on an optical fibre which are necessary for further research.

4 Simulation of a mode field diameter for a taper waist with an external layer

This section describes results of the simulation of the mode field diameter modification (MFD) for light propagation in a biconical optical fibre taper with deposited three different metal layers (Fig. 3). All simulations were provided by the Mode Solution® software form Lumerical Solution Inc.

Material parameters of a fibre and taper like refractive indices of core and cladding used for simulation, as well as their diameters were chosen from the catalogue of material properties of a standard single-mode fibre SMF28 Corning [36]. For a taper only diameters of core and cladding changed, refractive indices remained unchanged.

A non-tapered fibre, a fibre tapered to 50% of SMF, 20% of SMF and 10% of original size of SMF were submitted to simulation. A simulated region was limited to the taper waist region in which the beam flows out from the dielectric. In the process of simulation it was assumed that for the fused taper, the diameters of core and cladding decrease in a linear way (Table 1).

Table 1

Diameter of core and cladding for simulated elements.

In Fig. 4 the simulation of a fundamental mode propagation in a structure described in Sect. II (mode analysis) is presented. In the picture a cladding, core and distribution of fundamental mode are shown.

In an untapered fibre, a light beam is bound in a core. In the process of tapering the structure diameter is reduced and the beam propagates not only in the core – some part of a light leaks to the cladding area. For a taper size of 10% of the original dimension of an SMF a diameter of the core is so small that it can be neglected in the propagation. A guided simulation was provided for the wavelength of 1550 nm from which the most valuable information according to the dimension of a Mode Field Diameter (MFD) and the effective refractive indies (neff) is presented in Table 2 and Table 3, respectively.

Refractive index n and extinction index κ for gold Au (a) [33], silver Ag (b) [34] and titanium Ti (c) [35].
Fig. 2

Refractive index n and extinction index κ for gold Au (a) [33], silver Ag (b) [34] and titanium Ti (c) [35].

Scheme of the optical fibre taper with a mark of main elements.
Fig. 3

Scheme of the optical fibre taper with a mark of main elements.

Table 2

MFD dependence on an optical fibre taper structure for a 1550 nm wavelength.

Simulation of a propagating light in an optical fibre taper structure for 100%, 50%, 20% and 10% of original dimension of a standard optical fibre.
Fig. 4

Simulation of a propagating light in an optical fibre taper structure for 100%, 50%, 20% and 10% of original dimension of a standard optical fibre.

Table 3

Effective refractive index dependence on an optical fibre taper structure for a 1550 nm wavelength.

As it can be seen, for a standard SMF, the fundamental mode is guided in a core. A small difference in an MFD value from the catalogue can be explained as the used difference in a boundary condition. For the other structures it can be observed that the fundamental mode propagates more and more in a cladding – increasing MFD to the size higher than the core diameter, as well as changing the effective refractive index which is going to be close to the cladding one. For a taper size of 10% of the original dimension of an SMF, the whole structure is going to be the core, and the cladding is made of surrounding air with a refractive index equalling 1. The value of effective refractive index of a taper structure is below the refractive index value of a cladding in a single mode fibre. In order to recognize theoretical assumption, the next simulation of a thin precious metal layer was applied. Three thicknesses of layers: 25 nm, 50 nm and 100 nm were chosen for the simulation. Selection of such thicknesses was suggested by literature [11,12,16] as the best optimized layer, as well as multiplying thickness to get the best solution enabled farther use in future research in our laboratory. In Table 4 there are presented results of the MFD of different precious metal layers with a combination of different thicknesses of the layers.

Table 4

The MFD for the simulated structure with different metal thickness results for 1550 nm wavelength.

As it can be noticed in Table 4 in most cases MFD decreases when thickness of a layer and a diameter of the structure increase. In all cases the MFD is insignificantly smaller than the MFD value of a non-deposited structure what is corrected with the theoretical meaning [15,19], an additional layer does not allow to leak the beam out of its space. As it can also be noticed the difference in dispersion characteristics for different metal layers is an influence of optical parameters of metals like refractive or extinction indices which are associated with the metal permittivity.

A spectral analysis was provided for the wavelength range from 600 nm to 1700 nm which covers the first, second and third optical transmission window. Evaluated parameters for these simulations are dispersion and attenuation characteristics. These properties are the most important and interesting ones for design of a sensor.

In order to compare the simulation of a coated taper with a standard one, there was calculated a dispersion dependence on the wavelength for the SMF28 fibre from Corning with using the following formula [36,37] D(λ)=so4[λλo4λs]psnmkm,(6)

where λo is the zero dispersion wavelength from the range of 1304–1324 nm, So = 0.092 ps/nm2 · km describes a slope of the dispersion curve.

First there was provided a simulation for a standard SMF28 and a tapered structure 50% 20% and 10% of the original SMF size (Fig. 5). Comparison of the obtained results with the catalogue ones from Corning proved that boundary condition in our simulation are correct. As it can be observed with the decreasing size of the tapered structure the received curves are shifted into the shorter wavelength. This effect is a consequence of MFD as well as an effective refractive index change.

By comparison with characteristics of losses it can be seen that for a standard fibre and a taper structure in the range close to 1550 nm the attenuation is similar. For other wavelengths in a taper it can be observed a higher attenuation increasing to 2 dB/cm what is confirmed by the theoretical description point of view.

Dispersion characteristics for a real and tapered SMF to the size of 50%, 20%, 10% of the original SMF28,aswell astheir attenuation characteristics.
Fig. 5

Dispersion characteristics for a real and tapered SMF to the size of 50%, 20%, 10% of the original SMF28,aswell astheir attenuation characteristics.

4.1. Wavelength analysis of the proposed structure with gold layers

In the simulation we observe that deposition of gold layers increases the attenuation. Reducing the taper structure dimension and simultaneously increasing the layer thickness, a shift of the highest peak of attenuation towards the shorter wavelength is observed, as is shown in Fig. 6. Attenuation for the structure of 50% size of an original dimension of SMF possesses similar characteristics of attenuation as an uncoated structure. This is coupled with the MFD which is smaller than the diameter of a taper waist. For a wavelength close to 1.4 um it can be observed an increasing attenuation in comparison to the uncoated one. Considering the comparison of attenuation characteristics with dispersion ones it can be noticed that for a wavelength in which appears an amplification of attenuation with reference to the uncoated characteristics, a resonance peak in dispersion characteristics is also obtained. This peak is assembled from two parts, first when a dispersion increase described as a normal dispersion and after decreasing below zero described as an abnormal dispersion (Fig. 7).

Characteristics of attenuation for 50%, 20% and 10% of an original dimension SMF of the structure with gold layers.
Fig. 6

Characteristics of attenuation for 50%, 20% and 10% of an original dimension SMF of the structure with gold layers.

From data presented in Table 5 one can see that the highest dispersion change is provided by tapers with a 50 nm gold layer (3135.11 [ps/(nm · km)]). It should be mentioned that for a taper equal to the size of 50% of original dimension of SMF the change of dispersion for a gold layer possesses the smallest differences in results depending on thickness of these layers

Table 5

Resonance peak wavelengths and dispersion values for different taper structure with gold layers.

Dispersion characteristics for 50%, 20% and 10% of an original dimension SMF of the fibre structure with gold layers.
Fig. 7

Dispersion characteristics for 50%, 20% and 10% of an original dimension SMF of the fibre structure with gold layers.

Looking at changes of a wavelength for a characteristic dispersion peak it can be observed that with increasing of a gold layer thickness the peak is shifted towards the shorter wavelengths for taper structures (Fig. 8).

Normalized dispersion for the resonance peak for the simulated structure with different thickness of gold layers.
Fig. 8

Normalized dispersion for the resonance peak for the simulated structure with different thickness of gold layers.

4.2. Wavelength analysis of the proposed structure with silver layers

Dispersion and attenuation characteristics of silver layers deposition are similar to the structure with gold layers what results from similar characteristics of refractive and extinction indices (see Fig. 2). For higher attenuation in all analyzed wavelengths (Fig. 9) the formed peaks are related to dispersion peaks as is shown in Fig. 10. This effect can be a result of flatter characteristics of silver refractive index compared to gold. For the taper structure with 50% of a fibre original size not high increase of attenuation or dispersion characteristics can be also observed. This effect can be explained in this same way as for gold – light leaking for such structure is not high and still most of power propagates in a centre region.

Attenuation characteristics for 50%, 20% and 10% of the original dimension SMF of the structure with silver layers.
Fig. 9

Attenuation characteristics for 50%, 20% and 10% of the original dimension SMF of the structure with silver layers.

In Table 6 the values of resonance peak wavelengths and dispersion are presented. It can be noticed that for the taper structure equal to 50% of the original size, dispersion is on the same level of 20–25 [ps/ (nm · km)]. For the other taper structure the value of dispersion reaches even 4713.72 [ps/(nm km)].

Table 6

Resonance peak wavelengths and dispersion values for different taper structures with silver layers.

Dispersion characteristics for 50% 20% and 10% of the original dimension SMF of the structure with silver layers.
Fig. 10

Dispersion characteristics for 50% 20% and 10% of the original dimension SMF of the structure with silver layers.

Observing the position of the resonant peak it can be noticed that it is shifted towards shorter wavelengths with the increase of a silver layer (Fig. 11). For the structure of 50% of the standard size of a fibre the peak’s size can be skipped.

Normalized dispersion for resonance peak for simulated taper structures with different thickness of silver layers.
Fig. 11

Normalized dispersion for resonance peak for simulated taper structures with different thickness of silver layers.

4.3. Wavelength analysis of the proposed structure with titanium layers

Considering the analysis of attenuation characteristics for a tapered fibre with a titanium layer, it can be observed that the losses characteristics increase with the wavelength and decrease with a structure diameter (Fig. 12). Losses for the taper structure equal to 50% size of the original SMF are similar to the losses of non-deposited tapers (Fig. 5) Small changes in characteristics for each case are placed in the same position between 1.47–1.49 μm (Fig. 13). The peak that occurs in a dispersion characteristics is placed for the same wavelength as in the attenuation characteristics. In Figs. 12 and 13 characteristics for attenuation and dispersion are smoother than for gold and silver. Peaks for titanium start as a normal dispersion (increasing value) and, next, go to the anomaly dispersion (decreasing dispersion below zero) (Fig. 13). For previously analysed metals sequence of normal and abnormal dispersion was reversed. Dispersion characteristics for tapers deposited with titanium layers are smooth and possess only one peak in distinction of other simulated structures. This difference can be a result of optical properties (refractive and extinction indices) of each metal and of change of permittivity which are strictly linked with the refractive index.

Attenuation characteristics for 50%, 20% and 10% of the original dimension SMF of the structure with titanium layers.
Fig. 12

Attenuation characteristics for 50%, 20% and 10% of the original dimension SMF of the structure with titanium layers.

As it can be observed in Fig. 13, the resonance peaks for different thickness of a titanium layer, as well as for the tapered structure dimension, oscillate between 1.47 and 1.49 μm, only the value of dispersion changes. For the taper structure equal to 50% of the standard fibre, the difference between dispersion characteristics for a different thickness layer of deposited material is negligible. For the structure of 50 % size of an original SMF, when light propagates inside the structure of a taper waist, only the small part of power leaks out and the thickness of layers in fact influences only a no large way, the differences are of the order of 0.2-0.3 dB.For structures of 20% and 10% size of an original dimension of an SMF the thinnest layer in the lower way influences on light propagating in a taper waist. For the structures of 20% and 10% of the original dimension of an SMF thinner layers, to a lesser extent, affect the propagation of light in the taper area, what may be associated with the lack of impact of the titanium layers with light due to their optical properties as described in the first chapter (Fig. 2). Structures of 10% and 20% with a titanium layer do not have as high dispersion value as in other materials (Table 7).

Dispersion characteristics for 50%, 20% and 10% of the original dimension SMF of the structure with titanium layers.
Fig. 13

Dispersion characteristics for 50%, 20% and 10% of the original dimension SMF of the structure with titanium layers.

Provided simulations unequivocally show that titanium layers change only value of a dispersion but not shift it. The fact that titanium influences only a value of dispersion is associated with the characteristic of refractive and extinction indices. Both of them possess similar course. Additionally, it can be observed that attenuation characteristic possesses smooth shape and one peak for a resonance wavelength. A layer of thickness of 50 nm gives the highest change of dispersion which can be caused by optical properties like refractive and extinction indices (imaginary and real part), reflectance and transmittance as well as absorption for chosen wavelength, and it is the best thickness to obtain the plasmon effect. The 100 nm thickness gives the least change of dispersion, so it can be explained that for these thicknesses the absorption process is meaningful.

Normalized dispersion for resonance peak for simulated taper structures with different thickness of titanium layers.
Fig. 14

Normalized dispersion for resonance peak for simulated taper structures with different thickness of titanium layers.

Table 7

Resonance peak wavelength and dispersion value for different taper structures with titanium layers.

5 Conclusions

Chosen well-known metals can be divided in two groups according to their optical parameters which are directly connected with the refractive and extinction indices. Gold and silver form the first group – in the dispersion characteristics an abnormal dispersion appears at first, then it changes into a normal dispersion for increased wavelengths. Gold and silver layers are most often used in biochemical and chemical sensors. As for sensors it is united with the fact that the metal layers shift the resonance peak in dependence on change of the refractive index. For gold it is linked with a higher real part of dielectric constant than in silver. For silver the imaginary part of dielectric constant is higher than for gold. Difference in the mentioned parameter is due to the fact that in silver there is a lower signal to noise ratio SNR. However, silver in comparison to gold possesses lower chemical stability. Gold is resistant to oxidation in a liquid or gas environment. In order to protect the silver layer in sensors, the complex structure consists of two metal layers – the silver layer and the gold layer should be used for protection.

The second group should include titanium in which dispersion is normal at first,, then it becomes abnormal. For titanium layers the SPR effect can be also observed, but there is no wavelengths shift in dependence on the layer thickness. Titanium possesses different optical properties like refractive and extinction indices, absorption etc. and can be used as a sensitivity enhancer or enhancing mechanical properties. Considering the simulation it is evident to see the difference in attenuation and dispersion characteristics for a taper with an external layer for the above groups of metals.

The investigation on individual materials presented in the article is an essential step in the development of appropriate technological processes – selecting thickness of layer and diameter of the taper waist for new sensors design and fabrication. As it can be noticed for the taper structure equal to 50% of the fibre original size , an increase of dispersion occurs for all kinds of simulated layers thicknesses and is within 20–80 [ps/(nm · km)]. Much higher difference in dispersion characteristics can be observed for tapers of a 20% size of an original dimension of SMF. Analyzing results of the influence of metal layers on light propagation in a fibre taper, silver gives the highest fluctuation even to 621 [ps/(nm · km)]. For titanium fluctuation of a dispersion is three times smaller (66–290 [ps/(nm · km)]). The most significant difference of dispersion characteristics can be observed in a taper structure with a 10% dimension of the original fibre size. The fluctuation of dispersion for different thickness of metal reaches even to 4713 [ps/(nm · km)] however for such a thin taper the structure losses are high. Considering the results of the simulation, it is noted that the optimal diameter is about the size of a 20% of an original dimension of an SMF fibre. Such a choice was coupled with the balance between the minimum losses and possible opportunities to control propagation of light inside the structure and to influence its parameters (polarization, phase modulation, etc.) The best results can be obtained for the thickness of a metal layer in the range of 25–50 nm in which the SPR effect is the most significant.

Acknowledgement

This work was supported by the Polish Ministry of Science and Higher Education the Statutory Activity PBS-654 of Military University of Technology in 2016.

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About the article

Published Online: 2016-10-17

Published in Print: 2016-12-01


Citation Information: Opto-Electronics Review, Volume 24, Issue 4, Pages 196–208, ISSN (Online) 1896-3757, ISSN (Print) 1230-3402, DOI: https://doi.org/10.1515/oere-2016-0027.

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