Theoretical Modelling of An End Face Reflection based Tapered Fiber Optic SPR Sensor with Different Probe Designs

Theoretical Modelling of An End Face Reflection based Tapered Fiber Optic SPR Sensor with Different Probe Designs

Rajneesh. K. Verma Department of Physics Central University of Rajastha Bander Sindri, Ajmer, India

Abstract – The theoretical modelling of a SPR based tapered fiber optic sensor is presented with the end face as a reflecting surface. The performance of the sensor is evaluated in terms of its sensitivity and detection accuracy with taper ratio. Apart from this, the effect of various taper profiles on to the sensitivity of the sensor is also studied. It has been found in the present study that the end face reflection technique reduces the size of the sensor probe and makes it miniaturized. The maximum sensitivity of the exponential taper profile makes it an optimum taper profile among the three profiles named linear, parabolic and exponential.

Keywords – Surface Plasmons, Fiber Optics, Tapered Fiber, Plasmons

1. INTRODUCTION

   Surface plasmons resonance (SPR) is a very promising tool

   exponential linear has also been studied and a comparative analysis has also been provided in terms of the best taper profile for the sensitivity among the rest of the two.

2. THEORETICAL MODEL

   The theoretical model is based on the principle of attenuated total reflection (ATR) with Kretschmann configuration. The plastic cladding from the middle portion of a step index multimode fiber is removed and the unclad portion is tapered. The unclad tapered region is then cut from the point where the waist diameter is minimum. The unclad tapered fiber region is then coated with a thin gold layer, which is further surrounded by sensing medium (Fig. 1).

   Sensing Region

   for the sensing of various biochemical and gaseus sensing [1- 12]. A vast number of researchers are engaged in this upcoming area to explore various possibilities in terms of making fiber optic probes for the continuous monitoring of changes occurring in biochemical, chemical and gasesus media [12-27]. Recently many theoretical models have been

   2i

   Cladding

   Core

   20

   Reflecting Surface

   proposed for the better sensing mechanisms such as localized surface plasmon based fiber optic sensors [3], use of conducting metal oxides is also an important study [10]-[14]. In the last year people have fabricated and characterized the fiber optic sensing probes for pH sensing [10] and H2S gas sensing [11]. Sensing of Phenolic compounds and Ammonia gas sensing has also been proposed [12]-[13]. Fiber optic grating is also being used in conjunction with spr to make a miniaturized fiber optic probe for sensing the refractive indecies [14]-[15]. Most recently people are also proposing certain models on optical waveguides which involves artificially structured materials such metamaterials as a SPR active material [16]. In this study a theoretical model has been proposed which involes the multimoded tapered optical fibers. Contrary to the conventional fiber optic probes, in the present probe the one of the end face is coated with a silver paste to make it reflecting. This probe proposed in the model will be miniaturized as well as it will work as a point sensor as its one end can be dipped directly in to the medium to which we are going to sense. The performance parameters such as sensitivity have also been evaluated in terms of the taper ratio. Effect of various taper profiles such as linear, parabolic and

   Gold layer

   L

   Figure 1: A typical SPR based fiber optic sensor with tapered probe.

   Different taper profiles can be generated during fabrication using Flame Brush technique as suggested by Birks and Li [23]-[24]. The method uses a point like heat source (a gas burner) which heats only a small section of the fiber at a time. This burner is made to travel at constant speed in an oscillatory manner along the distance of the fiber (to be tapered) so that in each cycle of oscillation every element of the fiber is heated identically. If the speed of the burner is large compared to the tapered elongation speed then a time averaged hot zone is set up in the fiber which satisfies the condition that the fiber section being heated is always cylindrical and is always heated uniformly. Allowing the hot zone length to change as taper elongation precedes any shape of the fiber taper can be generated. Three taper profiles namely linear (LP), parabolic (PP), and exponential linear

   (EP) have been studied in literature for a fiber optic evanescent field absorption sensor as shown in Fig. 2 [22]. We consider these profiles for SPR based fiber optic tapered sensor. The different parts of the sensor are discussed below.

   Figure 2: Three different taper profiles showing the linear, parabolic, and exponential-linear variation of core radius of taper with distance z from the

   C. Reflection Coefficient (Rp):

   We consider N-layer model to obtain the expression for amplitude reflection coefficient for p-polarized incident beam as shown in Fig. 3 [5].

   Figure 3: N-Layer model to determine the reflected light intensity.

   In the present study, N is equal to 3. The layers are assumed to be stacked along the z-axis. The arbitrary medium layer is defined by thickness dk, dielectric constant k, permeability k, and refractive index nk. The tangential fields at the first boundary Z = Z1 = 0 are related to those at the final boundary Z = ZN-1 by

   input end of the taper.

   U U

   1. Core of the optical fiber

      1 M

      N 1

      For theoretical modeling, the optical fiber considered is a step-index multimode plastic clad silica fiber. Sinced the

      V1

      VN 1

      (3)

      refractive index of the silica varies with the wavelength therefore the wavelength dependence of the refractive index (n1) of the silica core is given by Sellmeier dispersion relation:

      In Eq. (3) U1 and V1 are the tangential components of electric and magnetic fields at the boundary of first layer, respectively; UN-1 and VN-1 are the corresponding fields at the boundary of Nth layer and M, known as characteristic matrix of the combined structure, is given by

      M

      n1

      2

      a

      1 1

      2 b2

      2

      a

      2

      2 b2

      2

      a

      3

      2 b2

      N 1

      M

      M11

      M12

      1 2 3

      (1)

      M k

      k 2

      21 M

      22

      (4)

      a1, a2, a3, b1, b2, and b3 are the Sellmeier coefficients and

      is the wavelength of light used in m. The values of these coefficients are given in the table below [25]:

      with

      M

      cos k

      i sin k

      / qk

      k

      a1	0.6961663
      a2	0.4079426
      a3	0.8974794
      b1	0.0684043
      b2	0.1162414
      b3	9.896161

      where

      k iq

      sin k

      cos k

      (5)

      1/ 2

      n2 sin 2

      1/ 2

      q k cos

      k 1 1

   2. Dispersion relation for plasmonic material

   For the wavelength dependence of the dielectric

   k k

   k

   k

   (6)

   constant of the plasmonic material we have used the Drude model, given as

   2 n

   k k

   cosk

   zk

   • zk 1

     2dk

     k

   n2 sin 2

   1/ 2

   1

   1

   m mr

   • imi

     1

     2

     c

     2 i

     (7)

     p c (2)

     where p and c denote the plasma wavelength and collision wavelength, respectively. For gold, the following values of the plasma wavelength and collision wavelength are used: P= 1.6826×10-7 M AND C= 8.9342×10-6 M [26]

     and k is the angle of the ray normal to the kth interface. The amplitude reflection coefficient (rp) for p-polarized incident wave is given by

     p

     r M11 M12qN q1 M 21 M 22qN

     Exponential linear:

     z

     e L e1

     0

     M11

   • M12qN

     q1

     M 21

   • M 22qN

   (8)

   L

   z z

   e i 0

   (13)

   Finally, reflectance (Rp) for p-polarized light is

   2

   Rp rp

   Parabolic:

   2 z 2

   2 1/ 2

   (9)

   .

   P (z) i

   • L i

   0

   (14)

   1. Reflected Power

      The light is launched into the one end of the fiber through a broadband light source with proper optics. The other end is polished with silver paste to make it reflecting. The light rays that travels to the fiber gets reflected back due to this mirror end face. The end through which light is launched is connected with two way reflection probe. One end is used for

      If (1, 2) is the range of incident angles of the rays launched at the input end of the taper, then at a distance z, this range alters to [1(z), 2(z)] due to variation in core diameter in tapered region and the transmitted power at the output end of the fiber is given by

      light launching and other is used for collecting light back. This reflecting end is connected to a detection system to record the

      L 2 ( z )

      dz

      2 N ( , z ) n 2 sin cos

      ref 1

      R d

      output signal. Consider the propagation of all the guided rays

      P

      (1 n 2 cos 2 )2

      launched in the fiber using a collimated source and a

      P 0 1 ( z ) 1

      microscope objective. The objective focuses the beam at the

      trans

      L 2 ( z )

      n 2 sin cos

      end-face of the fiber at axial point. The angular power distribution of rays guided in the fiber with as the angle of the ray with the normal to the core-cladding interface is given by [22]

      dz

      0

      1 ( z )

      1

      2

      (1 n1

      d

      cos 2 )2

      (15)

      n2 sin cos

      dP 1 d

      The expressions for 1(z) and 2(z) can be found by

      substituting the value of in Eq. (11). For 1(z),

      1

      1 n2 cos 2 2

      sin

      ncl

      n1 and for 2(z),

      2 . Further, Nref

      1 (10)

      To determine the effective reflected power, the reflectance (Rp) for a single reflection is raised to the power of the twice the number of reflections the specific propagating angle

      is the number of reflections the ray of angle undergoes in the fiber with radius (z) in the length L and is given as

      undergoes with the sensor interface (Due to reflecting end each ray retraces its path). The guided rays in the uniform core fiber enter the sensing tapered region as shown in Fig. 1. Assuming adiabatic (slow) tapering, if (z) is the taper radius

      Nref

      ( , z)

      L

      2(z) tan

      (16)

      at a distance z from the input end of the taper then the angle of the ray will transform into angle (z) in the tapered region and is given by [22]

   2. Sensitivity

   In any fiber optic SPR sensor based on wavelength interrogation, the light from a polychromatic source is launched into one of the ends of the fiber and the spectrum of the transmitted light received at the other end is recorded to

   1 i cos

   1 i 0

   know the resonance wavelength (the wavelength

   z cos

   z

   tan

   L

   (11)

   corresponding to minimum transmission) corresponding to the refractive index of the sensing layer. If sensing region has a

   In above equation, i and 0 are, respectively, the radii of the input (i.e. z = 0) and output end (i.e. z = L) of the taper. Further, the second term on right hand side is the taper angle () and L is the taper length. The taper radius varies with z in different fashion for different profiles given as [17]

   refractive index value of ns and m is the dielectric constant of the metal layer in contact with sensing region, then the surface plasmon resonance occurs when the following condition is satisfied

   Linear:

   2 2

   n 2

   1 / 2

   z n sin Re

   m s

   (z) i i 0

   1

   n 2

   L (12)

   m

   s

   (17)

   In Eq. (17), the expression on the left hand side is the propagation constant (Kinc) of the light incident at an angle while the right hand side represents the real part of surface

   plasmon propagation constant (KSP). In the SPR sensor based b. on wavelength interrogation, Fixed angles of incidence are

   used and resonance wavelength (res) is determined corresponding to refractive index of the sensing layer (ns). If the refractive index of the sensing layer is altered by ns, the resonance wavelength shifts by res. The sensitivity (Sn) of a SPR sensor with spectral interrogation is thus defined as [5]

   Sn

   res

   n

   s

3. RESULTS AND DISCUSSIONS

   (18)

   Various important parameters used in the present study have been taken as: numerical Aperture of the fiber = 0.22, diameter of the fiber core (2i) = 600 m, sensing region length (L) = 0.5 cm, gold layer thickness = 50 nm. In order to analyze the effect of taper ratio and taper profiles on the sensitivity, first of all we have to analyze the plasmonic resonance condition given by Eq. (17). As it is evident from Eq. (17), propagation constant (KSP) of surface plasmon wave (SPW) depends on the wavelength of the light as well as on the dielectric constant of metal layer (m) and refractive index (ns) of sensing region. Whereas, the propagation constant (Kinc) of the incident wave depends on the angle and wavelength of the light ray as well as on the refractive index of the fiber core. Figure 4a depicts the variation of KSP1 (i.e. for ns1 = 1.333), KSP2 (for ns2 = 1.343) and Kinc with wavelength for the case when no tapering of the fiber probe is done. The angle of the ray is taken at the middle of the corresponding angle (1, 2). As can be seen, KSP and Kinc curves cross each other at 613.86 nm for ns1 = 1.333 and at

   648.46 nm for ns2 = 1.343. Thus, a shift (res) of 34.60 nm in resonance wavelength is observed for ns = 0.010. Now, the next task is to estimate the effect of taper profile (Eqs. 12-

   14) and taper ratio (i/o ) on the shift (res) for the same value of ns. Figure 4b shows the variation of KSP1, KSP2, Kinc (LP), Kinc (PP) and Kinc (EP) with wavelength for taper ratio of 3.0. The curves corresponding to KSP1 and KSP2 are similar to those in Fig. 4a.

   a.

   Figure 4: Variation of propagation constants for SPW and incident wave with wavelength for (a) Taper ratio = 1.0 and (b) Taper ratio = 3.0.

   A noticeable change in Fig. 4b can be seen in terms of three- fold variation of Kinc corresponding to three different taper profiles. This happens due to two reasons: one, the angular range (1, 2) is different for different taper profiles, and second, the dependence of both 1 and 2 on the taper ratio (Eq. 11). Consequently, the curves corresponding to KSP1 and KSP2 will intersect these three Kinc curves at six different points. In Fig. 4b, the above intersections corresponding to LP, PP, and EP have been shown with the help of two each solid, dashed, and dotted arrows, respectively. Table below shows the points ofintersection and corresponding shift (res) for three taper profiles and three taper ratios.

   It is quite visible from Fig. 4b and from Table above that for a given ns, maximum shift (i.e. res) is obtained for EP whereas minimum shift is found for PP. For LP, the corresponding shift is found to be in between the two. The similar trend was observed for all profiles at several angles in the corresponding range (1, 2) and at several taper ratios between 1.0 and 2.2. This observation prepared the ground to check the present case when all the guided rays are launched together in the tapered fiber.

   The calculations for transmitted power for a given refractive index of the sensing region, using Eq. (15), were

   carried out with all the guided rays launched separately for all the three taper profiles with different taper ratios. From the plot of SPR spectrum resonance wavelength (res) was determined. To appreciate the shift in resonance wavelength (i.e. res) due to the change in the refractive index (ns) much more precisely, the refractive index change (ns) was narrowed by taking the values of ns1 and ns2 as 1.333 and 1.335, respectively, (i.e., ns = 0.002). As was expected from above discussion, the shift (res) was found to be maximum for EP and minimum for PP with LP lying in between the two. The sensitivity of the sensor was calculated as defined by Eq. (18). Figure 5 gives the corresponding variation of the sensitivity (in m per RIU) with taper ratio for all the three taper profiles. The figure shows that the sensitivity increases with an increase in taper ratio for any taper profile. Further, the sensitivity is maximum for exponential-linear profile (EP). The minimum sensitivity is observed for parabolic profile (PP). For the present set of design parameters mentioned above, a significant sensitivity enhancement of more than 80% in comparison to untapered fiber optic SPR sensor is observed at a taper ratio of 3.0 for exponential-linear taper profile (EP).

   Figure 5: Variation of sensitivity (in m per RIU) with taper ratio for three different taper profiles.

   Apart from this explanation of the sensitivity enhancement with taper profile and taper ratio in terms of the shifting of resonance point (i.e., KSP = Kinc), another physically logical reason can be given in terms of evanescent wave absorption. By tapering the fiber we intend to broaden the angular range from (1, 2) to (1, 2), because lower value of angle means more spreading of the evanescent field into the region adjacent to fiber core. Hence the coupling between evanescent wave and SPW will become stronger and sensitivity is bound to increase due to tapering. Now, as far as the effect of taper profile is concerned, the angular range (1,

   2) in tapered region will be maximum for EP and minimum for PP according to Eqs. (13) and (14). This is why sensitivity is found to be maximum for EP. Furthermore, this improved coupling between evanescent wave and SPW due to tapered fiber causes SPR curve to broaden. However, the degree of SPR curve broadening is much lesser than the sensitivity enhancement with taper ratio and taper profile.

4. CONCLUSIONS