Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal
The five-layer nanorod-mediated surface plasmon sensor with inhomogeneous liquid crystal layer was theoretically investigated. The reflectance as the function of the incident angle was calculated at different voltages applied to the liquid crystal (LC) for different analyte refractive indices. By c...
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Інститут фізики конденсованих систем НАН України
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irk-123456789-1570472019-06-20T01:29:20Z Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal Yarmoshchuk, Ye.S. Zadorozhnii, V.I. Reshetnyak, V.Yu. The five-layer nanorod-mediated surface plasmon sensor with inhomogeneous liquid crystal layer was theoretically investigated. The reflectance as the function of the incident angle was calculated at different voltages applied to the liquid crystal (LC) for different analyte refractive indices. By changing the LC director orientation one can control the position of the reflective dips and choose the one that is the most sensitive to the analyte refractive index. At the chosen angle of incidence, the analyte refractive index can be found from the reflectance value. The director reorientation effect is stronger when the prism refractive index is between ordinary and extraordinary refractive indices of the LC. In this case, the voltage increase and the prism refractive index decrease have a similar effect on the reflectance features. Теоретично дослiджено п’ятишаровий плазмонний сенсор з шаром наночастинок та неоднорiдним шаром рiдкого кристалу. Розраховано коефiцiєнт вiдбивання як функцiю кута падiння при рiзних значеннях показника заломлення аналiту та прикладеної до рiдкого кристалу (РК) напруги. Змiнюючи орiєнтацiю директора рiдкого кристалу, можна контролювати положення мiнiмумiв кривих вiдбивання i вибрати той, що найбiльш чутливий до показника заломлення аналiту. При вибраному кутi падiння показник заломлення аналiту може бути визначений по значенню коефiцiєнта вiдбивання. Вплив переорiєнтацiї директора сильнiший, коли значення показника заломлення призми є мiж звичайним i незвичайним показниками заломлення РК, в цьому випадку збiльшення напруги i зменшення показника заломлення призми мають подiбний вплив на форму кривої вiдбивання. 2018 Article Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal / Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13401: 1–10. — Бібліогр.: 34 назв. — англ. 1607-324X PACS: 42.79.Kr, 61.30.Gd, 78.15.+e, 73.20.Mf, 42.79.Pw, 87.85.fk DOI:10.5488/CMP.21.13401 arXiv:1803.11409 http://dspace.nbuv.gov.ua/handle/123456789/157047 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The five-layer nanorod-mediated surface plasmon sensor with inhomogeneous liquid crystal layer was theoretically investigated. The reflectance as the function of the incident angle was calculated at different voltages
applied to the liquid crystal (LC) for different analyte refractive indices. By changing the LC director orientation
one can control the position of the reflective dips and choose the one that is the most sensitive to the analyte
refractive index. At the chosen angle of incidence, the analyte refractive index can be found from the reflectance
value. The director reorientation effect is stronger when the prism refractive index is between ordinary and extraordinary refractive indices of the LC. In this case, the voltage increase and the prism refractive index decrease
have a similar effect on the reflectance features. |
| format |
Article |
| author |
Yarmoshchuk, Ye.S. Zadorozhnii, V.I. Reshetnyak, V.Yu. |
| spellingShingle |
Yarmoshchuk, Ye.S. Zadorozhnii, V.I. Reshetnyak, V.Yu. Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal Condensed Matter Physics |
| author_facet |
Yarmoshchuk, Ye.S. Zadorozhnii, V.I. Reshetnyak, V.Yu. |
| author_sort |
Yarmoshchuk, Ye.S. |
| title |
Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal |
| title_short |
Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal |
| title_full |
Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal |
| title_fullStr |
Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal |
| title_full_unstemmed |
Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal |
| title_sort |
numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157047 |
| citation_txt |
Numerical modelling of light propagation in surface plasmon resonance sensor with liquid crystal / Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13401: 1–10. — Бібліогр.: 34 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT yarmoshchukyes numericalmodellingoflightpropagationinsurfaceplasmonresonancesensorwithliquidcrystal AT zadorozhniivi numericalmodellingoflightpropagationinsurfaceplasmonresonancesensorwithliquidcrystal AT reshetnyakvyu numericalmodellingoflightpropagationinsurfaceplasmonresonancesensorwithliquidcrystal |
| first_indexed |
2025-07-14T09:23:13Z |
| last_indexed |
2025-07-14T09:23:13Z |
| _version_ |
1837613705125167104 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 1, 13401: 1–10
DOI: 10.5488/CMP.21.13401
http://www.icmp.lviv.ua/journal
Numerical modelling of light propagation in surface
plasmon resonance sensor with liquid crystal
Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak
Taras Shevchenko National University of Kyiv, Faculty of Physics,
2 Acad. Glushkov Ave., 03022 Kyiv, Ukraine
Received June 6, 2017, in final form October 4, 2017
The five-layer nanorod-mediated surface plasmon sensor with inhomogeneous liquid crystal layer was theo-
retically investigated. The reflectance as the function of the incident angle was calculated at different voltages
applied to the liquid crystal (LC) for different analyte refractive indices. By changing the LC director orientation
one can control the position of the reflective dips and choose the one that is the most sensitive to the analyte
refractive index. At the chosen angle of incidence, the analyte refractive index can be found from the reflectance
value. The director reorientation effect is stronger when the prism refractive index is between ordinary and ex-
traordinary refractive indices of the LC. In this case, the voltage increase and the prism refractive index decrease
have a similar effect on the reflectance features.
Key words: liquid crystal, surface plasmon resonance, sensor, nanoparticles, porous metal film
PACS: 42.79.Kr, 61.30.Gd, 78.15.+e, 73.20.Mf, 42.79.Pw, 87.85.fk
1. Introduction
Surface plasmons (SPs) are collective electron excitations that exist at the interface between metal
and dielectric. Surface plasmons play an important role in optical properties of metals, SPs are widely
used in various devices, such as sensors. The attenuated total reflection (ATR) method is one of the
ways to excite SPs [1, 2]. Widely used Kretschmann and Otto configurations are based on this method
[3–5]. The Kretschmann configuration contains a consecutively placed coupling prism, a thin metal
film and a dielectric layer (e.g., air). In the Otto configuration, a coupling prism and a metal layer are
separated by an air gap. In both configurations, the SP is excited at the metal-dielectric (air) interface. The
Otto configuration has the advantage of tunability by changing the air gap. When the surface plasmon
momentum and the tangential component of the photon momentum are equal, surface plasmon resonance
occurs. As a result, there is a dip in the reflectance. The dip position is sensitive to the refractive index
of the adjacent layer. Its shift can serve as the indicator of reactions that occur on the surface between
the metal and the investigated medium (analyte). This phenomenon is widely used in the development of
chemical and biological sensors [6, 7]. The effect of different sensor characteristics, for example metal
layer parameters, the prism refractive index, on the sensor sensitivity was presented in the paper [8].
At present, different modifications of sensors have been studied in order to enhance their sensitivity
and accuracy, reduce their size. In particular, long-range surface plasmon resonance (LRSPR) sensor with
sharp reflection spectrum was suggested (see reference [9] and reference within). This sensor contains a
prism, a dielectric layer, a metal layer and an analyte in sequence. During the last years, much attention
has also been paid to localized surface plasmon (LSP) sensors that are quite promising [10, 11]. The LSPs
excitation in the nanostructures leads to the field enhancement near the surface. As a result, the sensor
response will be stronger, with an increase of the contact surface. Sensor with the layer of periodic gold
nanowires [12] presents sensitivity enhancement that depends on the period of nanowires. Sensor with
porous metal film, where pores are filled with the analyte [13, 14], presents sensitivity enhancement by a
factor of about 1.5 [13] in comparison with standard SPR sensors. Theoretical and experimental research
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
13401-1
https://doi.org/10.5488/CMP.21.13401
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak
of nanorod-mediated sensor [15] with thin metal film and porous layer shows the sensitivity enhancement
by twofold, but its sensitivity depends from the analyte refractive index value: with the refractive index
increasing the prism replacing is needed.
Liquid crystal (LC) is an anisotropic medium so that the optical axis can be controlled by applied
electric or magnetic fields [16, 17]. SPs with liquid crystals were investigated in different geometries
[18, 19] and now LCs are a promising candidate for the development of active plasmonic devices [20]. In
this study, we consider a new sensor system with the additional nematic LC layer, which can be used for
tuning sensor characteristics. Contrary to [21], an electric field is applied to the LC cell that causes the
LC director reorientation. As a result, the LC director orientation and the LC dielectric tensor become
inhomogeneous and dependent on the distance to the cell substrate. In such a system, Fresnel equations
based methods [15, 21] become inapplicable and matrix computational methods are preferable. The study
aims to theoretically investigate the effect of the LC reorientation on the spectral reflectance properties
of the proposed sensor system.
2. System scheme
The investigated system is based on Kretschman geometry that consists of a glass prism, a thin metal
film and an analyte. Monochromatic p-polarized light is incident onto the interface between the glass
prism and the metal film. The plasmon wave vector depends on the dielectric constants of the metal (εm)
and analyte (εs) kp = 2π
λ
√
εmεs
εm+εs
. When the tangential component of the incident light wave vector is
equal to the plasmon wave vector, the plasmon is exited. The intensity of the reflected light is measured
by a detector. The analyte refractive index can be calculated from the angular position of the resonance
minimum. In [15], an anisotropic porous metal film was inserted between metal film and analyte for
the sensitivity increase. We add a layer of nematic liquid crystal to change the sensor characteristics
when it is already fabricated or during the measuring. The scheme of the investigated system is shown in
figure 1 . It consists of a glass prism with refractive index n1 (1), a nematic liquid crystal layer (2), a thin
metal (Ag) film (3) and an anisotropic porous metallic film (4) constituted by spheroidal nanoparticles,
deposited so that their rotation axis is tilted at an angle β to the Z-axis [15] (for example, using the
oblique-angle-deposition technique [13]). An analyte forms the last layer (5) and fills the spaces between
nanoparticles which increases the sensitivity of the system [15]. An external electric field is applied to
the LC cell in the Z direction. Monochromatic p-polarized light is incident at some angle θ onto the
interface between the glass prism and the LC layer. The reflectance angular spectrum was investigated at
different applied voltages.
Figure 1. Geometry of the investigated system.
13401-2
Light propagation in SPR sensor with LC
3. Theoretical model
3.1. Calculation method
To calculate the propagation of the obliquely incident light in a system with the inhomogeneosly
oriented LC layer (the LC director orientation depends on Z coordinate), the Berreman 4 × 4 matrix
method [22–26] was used. This method was developed for stratified anisotropic media [26, 27], which is
a more general approach compared to 2 × 2 matrix method and suitable for the oblique light incidence.
The method is based on the solution of the Berreman equation that is the first order differential equation
for the Berreman vector ψ
dψ(z)
dz
= ik0Q̂ψ(z), (3.1)
where ψ(z) =
(
Ex, η0Hx, Ey,−η0Hy
)
, η0 =
√
µ0/ε0, k0 = ω/c is the wave vector in vacuum, ε0 are µ0
are vacuum dielectric and magnetic constants. The matrix Q̂ depends on the components of the dielectric
tensor ε̂ and on the tangential component of the incident light wave vector. If the dielectric tensor does
not change in the region from z to z + ∆z then Q̂ also does not change in this region and the solution to
equation (3.1) is of the form ψ (z + ∆z) = exp
[
ik0Q̂(z)∆z
]
ψ(z) = P̂ (z,∆z)ψ(z). The matrix exponent
was calculated using faster Berreman method [23, 27] based on the Cayley-Hamilton theorem. The LC
cell was divided into 200 slabs with an approximately uniform director that, as it turned out, provides an
acceptable accuracy in our calculations. The reflectance was calculated as a ratio between the intensities
of the incident and the reflected waves using a technique from [24].
The following parameters were used for numerical simulation: the wavelength of the incident light
λ = 632.8 nm, the glass refractive index 1.51 and 1.57, the metal film thickness df = 40 nm, the diameter
of nanoparticles D = 30 nm and length l = 10 nm (oblate spheroid), the angle between rotation axis and
Z-axis β = 73◦, the volume fraction fm = 0.4.
3.2. Dielectric tensors
To calculate the reflectance using the Berreman method one needs to know the dielectric function of
each layer. For silver dielectric constant, we used the approximation ε(ω) = 1 − ω2
p
ω(ω+iγ) +
fω2
b
ω2
b−ω
2+iωΓb
,
where the last term is an approximation by Lorentzian tail of the contribution from interband electron
transitions [28], ωp is the plasma frequency, γ is the relaxation constant. This approximation is more
general than the one used in [15]. For silver film, it gives a result close to the Drude formula εm =
−18.37 + i0.47. For nanoparticles, the surface effect on the electron mean free path (L) is rather strong
and γ(L) is of the form γ(L) = γ0+Avf/L, where γ0 is the volume decay constant, vf is the Fermi velocity
of electrons, A is a parameter of the order of unity that accounts for the details of the scattering process.
The electron mean free path L was calculated as in [29].
The anisotropic porous metal film consists of the array of spheroidal nanoparticles. Since the rotation
axis is tilted at the angle β to the Z-axis, the dielectric tensor of the layer is of the form
ε̂4 =
©«
εx′ cos2 β + εz′ sin2 β 0 (εz′ − εx′) sin β cos β
0 εy′ 0
(εz′ − εx′) sin β cos β 0 εz′ cos2 β + εx′ sin2 β
ª®¬ , (3.2)
where εx′, εy′, εz′ are the principal values. The analyte fills the spaces between nanoparticles and the
formed layer is inhomogeneous. Such a structure can be considered as a homogeneous optical medium
with an effective dielectric constant that is different from dielectric constants of constituting materials.
In the Maxwell Garnett model, the effective dielectric constant ε of the system with a homogeneous
host and spherical inclusions can be obtained from the equation ε−ε2
ε+2ε2
= f1
ε1−ε2
ε1+2ε2
, where ε2, ε1 are host
and inclusions dielectric constants, respectively, f1 is the inclusions volume fraction [30]. The mixing
formula is modified for the case of the ellipsoidal particles. Considering the metal as the host [15], we
used the modified formula that is of the form
εi − εm
εm + Li(εi − εm)
= (1 − fm)
εs − εm
εm + Li(εs − εm)
, (3.3)
13401-3
Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak
where i = x ′, y′, z′, εm and εs are metal and analyte dielectric constants, fm is the metal volume fraction,
Li is the ellipsoid depolarization factor [31].
3.3. Equations for the LC director
The liquid crystal layer is located between the glass prism and the metal film in the system under
consideration. The LC director orientation at top and bottom substrates are parallel to the X-axis. When
the applied voltage is higher than the electric Frederiks transition threshold, the director rotates in the
X Z-plane. Its orientation can be described by the unit vector n = cos ϕ(z)ex + sin ϕ(z)ez , where ϕ(z) is
the rotation angle. Components of the LC dielectric tensor depend on the director orientation
ε̂LC =
©«
n2
e cos2 ϕ + n2
o sin2 ϕ 0
(
n2
e − n2
o
)
sin ϕ cos ϕ
0 n2
0 0(
n2
e − n2
o
)
sin ϕ cos ϕ 0 n2
e sin2 ϕ + n2
o cos2 ϕ
ª®¬ , (3.4)
where no and ne are ordinary and extraordinary refractive indices. The director orientation under the
applied voltage can be found byminimizing the LC free energy functional. The LC free energy density has
two components: elastic felastic = 1
2 K11 cos2 ϕ(ϕ′)2 + 1
2 K33 sin2 ϕ(ϕ′)2 and electric felectric = − 1
2ε0∆ε(E ·
n)2 = − 1
2ε0∆εE2 sin2 ϕ, whereK11 andK33 are Frank elastic constants,∆ε is a static dielectric anisotropy,
E is the electric field in the LC layer. By varying the free energy functional, the second order differential
equation was obtained(
K11 cos2 ϕ + K33 sin2 ϕ
)
ϕ′′ +
(
K33 − K11
)
sin ϕ cos ϕ(ϕ′)2 + ε0∆εE2 sin ϕ cos ϕ = 0. (3.5)
This equation must be accompanied by boundary conditions. We considered two variants of boundary
conditions: 1) strong anchoring at both surfaces (symmetric cell) ϕ(z = 0) = ϕ(z = h) = 0 with the
threshold voltage Uth = π
√
K11
ε0∆ε
; 2) strong anchoring at the top surface and very weak anchoring at
the bottom surface (non-symmetric cell) ϕ(z = 0), ϕ′(z = h) = 0 with the threshold voltage UthNS =
π
2
√
K11
ε0∆ε
. The electric field from equation (3.5) obeys the equation (∇∇∇ · D) = 0 that yields
d
dz
(ε0εzzE) = ε0
d
dz
{[
ε⊥ + ∆ε sin2 ϕ(z)
]
E
}
= 0 , (3.6)
where ε⊥ is the perpendicular component of the static dielectric tensor. It is convenient to introduce
the electric field potential E = −∇∇∇Φ. Then the potential satisfies the following equation and boundary
conditions
d
dz
{[
ε⊥ + ∆ε sin2 ϕ(z)
] dΦ
dz
}
= 0 , (3.7)
Φ(0) = 0, Φ(h) = U.
Table 1 shows the parameters of the chosen nematic LCs 5CB and E44. The LC cell thickness is h = 5 µm.
Table 1. LC parameters [17, 32, 33] and threshold voltages.
LC K11,N K33,N ∆ε ε⊥ no ne Uth,V UthNS,V
5CB 0.64 · 10−11 0.10 · 10−10 13 6.7 1.5319 1.7060 0.7408 0.3704
E44 0.155 · 10−10 0.28 · 10−10 16.8 5.2 1.5239 1.7753 1.0141 0.50705
13401-4
Light propagation in SPR sensor with LC
4. Result and discussions
4.1. The prism refractive index 1.51
Figure 2 shows the light reflectance R without the LC director reorientation (planar geometry) for
5CB (a) and E44 (b) at different values of the analyte refractive index ns; the prism refractive index is
n1 = 1.51. Similar to the case [21] there are dips in the curves R(θ), that are caused by mixing of the
surface plasmons mode with the half-leak modes propagating in the LC layer. Plasmon resonance similar
to the one in [15] is observed at small ns values. There is also a dip on the right side of figure 2, its
position is fixed for the particular LC and the depth depends on the analyte refractive index ns. At the
incident angle that corresponds to the dip minimum, the analyte refractive index can be found by using
the reflectance value. The sensor sensitivity S to the analyte refractive index change is defined as the
ratio between the reflectance change and the analyte refractive index change dRmin/dns. For example,
in figure 2 (a) (5CB) for θmin = 88.3◦ the calculated sensitivity for ns = 1.33 − 1.34 is S = 4.11; in
figure 2 (b) (E44) for θmin = 84.6◦ S = 1.55. Sensitivity dependence on ns is shown in figure 3. It has a
maximum at some ns value, but when ns increases, sensitivity decreases. The maximum sensitivity for
5CB Smax = 4.57 is achieved at ns = 1.32, for E44 Smax = 1.93 at ns = 1.315.
In order to investigate the effect of the LC reorientation, director profiles were calculated for different
values of the applied voltage. Profiles obtained for two cases of boundary conditions are shown in figure 4.
Figure 5 shows the reflectance for symmetric LC cell at voltages 1.5, 2 and 3Uth. These curves are similar
to the curves in figure 2, but positions and depths of the last dips change depending on the applied
30 40 50 60 70 80 90
0.5
0.6
0.7
0.8
0.9
1.0
( )
ns
1.2
1.25
1.3
1.33
1.4
R
deg
30 40 50 60 70 80 90
0.5
0.6
0.7
0.8
0.9
1.0
1.2
1.25
1.3
1.33
1.4
R
deg
ns
(b)
Figure 2. Reflectance R versus the light incident angle θ without the LC director reorientation (planar
geometry) for 5CB (a) and E44 (b) at different values of the analyte refractive index ns, n1 = 1.51.
1.20 1.25 1.30 1.35 1.40 1.45
-4
-2
0
2
4
S
ns
1.5Uth, =82.6o
2Uth, =84.35o
3Uth, =82.35o
U=0 (a)
1.20 1.25 1.30 1.35 1.40 1.45
-4
-2
0
2
4
S
ns
1.5Uth, =85.65o
2Uth, =82.65o
3Uth, =84.6o
U=0 (b)
Figure 3. Sensitivity S versus the analyte refractive index ns for 5CB (a) and E44 (b) without reorientation
(U = 0) and at voltages 1.5, 2 and 3Uth, n1 = 1.51.
13401-5
Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
60
70
80
90
5CB E44
1.5Uth
2Uth
de
g
z/h
3Uth
( )
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
60
70
80
90
1.5UthNS
de
g
z/h
5CB E44
3UthNS
2UthNS
(b)
Figure 4. Director profiles under the applied voltage: strong anchoring at both surfaces (a), strong
anchoring at the top surface and very weak anchoring at the bottom surface (b).
60 65 70 75 80 85 90
0.5
0.6
0.7
0.8
0.9
1.0
R
deg
1.5Uth ns=1.30
1.5Uth ns=1.33
2Uth ns=1.30
2Uth ns=1.33
3Uth ns=1.30
3Uth ns=1.33 (a)
60 65 70 75 80 85 90
0.5
0.6
0.7
0.8
0.9
1.0
R
deg
1.5Uth ns=1.30
1.5Uth ns=1.33
2Uth ns=1.30
2Uth ns=1.33
3Uth ns=1.30
3Uth ns=1.33 (b)
Figure 5. Reflectance R versus the light incident angle θ for 5CB (a) and E44 (b) at voltages 1.5, 2 and
3Uth, n1 = 1.51, ns = 1.3, 1.33.
voltage. For 5CB [figure 5 (a)], the dips shift to smaller incident angles and their depths decrease. In
the case of E44 [figure 5 (b)], dips become deeper and stay in the same range of the incident angles.
Sensitivity dependences on the analyte refractive index at voltages 1.5, 2 and 3Uth are shown in figure 3 .
For 5CB at voltage 1.5Uth, sensitivity has a maximum Smax = 2.23 at ns = 1.31, for the other voltages, the
maximum sensitivities are as follows: Smax = 2.78 at ns = 1.315 (U = 2Uth) and Smax = 3.16 at ns = 1.31
(U = 3Uth). For E44, the maximum sensitivities are as follows: Smax = 2.23 at ns = 1.315 (U = 1.5Uth),
Smax = 2.41 at ns = 1.31 (U = 2Uth), Smax = 4.02 at ns = 1.315 (U = 3Uth). For E44, reorientation
causes the sensitivity increase, but for both LC, reorientation does not have a significant impact on the ns
value at which the sensitivity has a maximum (figure 3). In the case of the second boundary conditions,
reflectance and sensitivity curves look similar but the maximum sensitivities are smaller.
4.2. The prism refractive index 1.57
The prism refractive index 1.51 is smaller than ordinary (no) and extraordinary (ne) refractive indices
of the considered LCs. Since in this case reorientation does not have a significant impact on the maximum
sensitivity position, the reflectance was investigated at the prism refractive index value between no and
ne. Under this condition, LC reorientation changes the ratio between the prism and the LC layer refractive
indices. At some LC orientation, total internal reflection can occur which is analogy to the Otto geometry
[3, 17]. The reflectance for planar LC orientation at n1 = 1.57 is shown in figure 6. After some critical
angle (θcr = 77.3◦ for 5CB and θcr = 76.1◦ for E44), the reflectance reaches a maximum and becomes
13401-6
Light propagation in SPR sensor with LC
30 40 50 60 70 80 90
0.5
0.6
0.7
0.8
0.9
1.0
R
, deg
1.2
1.30
1.33
1.4
ns
(a)
30 40 50 60 70 80 90
0.5
0.6
0.7
0.8
0.9
1.0
(b)
ns
1.2
1.3
1.33
1.4
R
deg
Figure 6. Reflectance R versus the light incident angle θ for planar orientation for 5CB (a) and E44 (b)
and different values of the analyte refractive index ns, n1 = 1.57.
60 65 70 75 80 85 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
, deg
1.5Uth ns=1.30
1.5Uth ns=1.33
2Uth ns=1.30
2Uth ns=1.33
3Uth ns=1.30
3Uth ns=1.33 (a)
60 65 70 75 80 85 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
, deg
1.5Uth ns=1.30
1.5Uth ns=1.33
2Uth ns=1.30
2Uth ns=1.33
3Uth ns=1.30
3Uth ns=1.33 (b)
Figure 7. Reflectance R versus the light incident angle θ for 5CB (a) and E44 (b) at voltages 1.5, 2 and
3Uth, n1 = 1.57, ns = 1.3, 1.33.
constant R = 1. When n1 increases, the critical angle decreases. Since the reflectance dips are narrow
and have a small depth, we consider this case unsuitable for ns measuring.
We calculated the reflectance versus the incident angle at voltages 1.5, 2 and 3Uth for n1 from 1.53 to
1.63.When the n1 value is higher than 1.6, the reflectance dips are too narrow. For a detailed investigation,
the prism refractive index n1 = 1.57 was chosen, because it shows a better sensitivity and there exists a
real optical glass with close refractive index [34]. The reflectance at voltages 1.5, 2 and 3Uth [figure 4 (a)]
is shown in figure 7. At the voltage 1.5Uth, the reflectance has narrow dips similar to the planar case.
When the voltage increases, the dips become wider. Similar behaviour can be observed when the prism
refractive index changes at the fixed voltage: the dips become wider when n1 decreases. For curves in
figure 7, more than only last dips are suitable for measurements. Figure 8 shows sensitivity versus the
analyte refractive index at voltages 1.5, 2 and 3Uth for 5CB (a) and E44 (b). For 5CB, the maximum
sensitivities are as follows: Smax = 4.66 at ns = 1.355 (θ = 84.1◦, U = 1.5Uth) and Smax = 4.8 at
ns = 1.37 (θ = 85.9◦, U = 2Uth). For E44 the maximum sensitivities are as follows: Smax = 4.24 at
ns = 1.35 (θ = 81.6◦, U = 1.5Uth) and Smax = 4.87 at ns = 1.37 (θ = 85.8◦, U = 3Uth).
The reflectance for non-symmetrical cell at voltages 1.5, 2 and 3UthNS is shown in figure 9. The
curves are similar to figure 7 but R(θ) dips are deeper than in the symmetric case. Sensitivity versus the
analyte refractive index at voltages 1.5, 2 and 3UthNS is shown in figure 10 for 5CB (a) and E44 (b). For
5CB, maximum sensitivities are as follows: Smax = 5.3 at ns = 1.35 (θ = 78.2◦, U = 1.5UthNS) and
Smax = 4.67 at ns = 1.335 (θ = 77.7◦, U = 2UthNS). For E44, maximum sensitivities are as follows:
Smax = 5.34 at ns = 1.335 (θ = 76.3◦, U = 1.5UthNS), Smax = 5.23 at ns = 1.335 (θ = 73.75◦,
13401-7
Ye.S. Yarmoshchuk, V.I. Zadorozhnii, V.Yu. Reshetnyak
1.20 1.25 1.30 1.35 1.40 1.45
-4
-2
0
2
4
6
ns
S
1.5Uth =79.9o
1.5Uth =84.1o
2Uth =79.55o
2Uth =85.9o
Uth =75.45o
Uth =80.9o
(a)
1.20 1.25 1.30 1.35 1.40 1.45
-4
-2
0
2
4
6
S
ns
1.5Uth =81.6o
2Uth =79.75o
3Uth =77.25o
3Uth =85.8o
(b)
Figure 8. Sensitivity S versus the analyte refractive index ns for 5CB (a) and E44 (b) at voltages 1.5, 2
and 3Uth, n1 = 1.57.
60 65 70 75 80 85 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
, deg
1.5UthNS ns=1.3
1.5UthNS ns=1.33
2UthNS ns=1.3
2UthNS ns=1.33
3UthNS ns=1.3
3UthNS ns=1.33 (a)
60 65 70 75 80 85 90
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R
, deg
1.5UthNS ns=1.3
1.5UthNS ns=1.33
2UthNS ns=1.3
2UthNS ns=1.33
3UthNS ns=1.3
3UthNS ns=1.33 (b)
Figure 9. Reflectance R versus the light incident angle θ for 5CB (a) and E44 (b) at voltages 1.5, 2 and
3UthNS, n1 = 1.57, ns = 1.3, 1.33.
1.20 1.25 1.30 1.35 1.40 1.45
-4
-2
0
2
4
6
S
ns
=78.2o
=75.65o
=74.3o
=77.7o
=74.25o
=78.7o
(a)
1.20 1.25 1.30 1.35 1.40 1.45
-4
-2
0
2
4
6
ns
S
=73.95o
=76.3o
=73.75o
=77.4o
=70.8o
=75.3o
=81.15o
(b)
Figure 10. Sensitivity S versus the analyte refractive index ns for 5CB (a) and E44 (b) at voltages 1.5, 2
and 3UthNS (black, gray and light gray, respectively), n1 = 1.57.
U = 2UthNS) and Smax = 5.1 at ns = 1.35 (θ = 81.15◦, U = 3UthNS). Thus, the highest sensitivity
values were obtained for non-symmetrical cell at n1 = 1.57.
In spite of the case n1 = 1.51, in the case n1 = 1.57, themaximum sensitivity position shifts depending
on the voltage and the selected dip (for the selected angle of incidence). By varying the voltage, one can
choose a better sensitivity depending on the analyte. In the sensor considered in [15], to measure the
13401-8
Light propagation in SPR sensor with LC
analyte refractive index higher than 1.33, one should take the prism with a higher refractive index, for
example 1.78 for ns = 1.45. Since the effects of the LC director reorientation and the prism refractive
index variation are similar, it is appropriate to use LCs instead of replacing the prism. Although we did
not obtain a sensitivity increase at ns = 1.45, further optimization of system parameters and LC geometry
can enhance the situation.
5. Conclusions
In this study we theoretically investigated the nanorod-mediated surface plasmon sensor with an
inhomogeneous LC layer. Using the Berreman method, the light reflectance from multilayer system was
calculated as a function of the incident angle for 5CB and E44 LCs at two types of boundary conditions.
The reflectance has dips that could be explained by mixing of plasmon mode and half-leaky modes that
propagate in the LC layer. Calculations were carried out at two values of the prism refractive index. In
the case when the prism refractive index equals 1.51, for both boundary conditions there is a dip whose
position is constant for the chosen LC configuration and its depth depends on the analyte refractive
index. The analyte refractive index can be found from the reflectance value at the incident angle that
corresponds to this reflectance dip. The reflectance at different applied voltages shows that the LC director
reorientation allows one to control the dip position. Therefore, by changing the applied voltage, one can
control the position of the reflective dips and arrange the measurements in the most convenient way. In
the case of the non-symmetrical boundary conditions, a stronger reorientation effect was not obtained.
The sensor sensitivity to the analyte refractive index change was calculated as the ratio between the
reflectance change and the analyte refractive index change. For E44, reorientation causes the sensitivity
increase, but for 5CB, the sensitivity of homogeneous LC is higher. For both LCs, reorientation does
not have a significant impact on the ns value at which the sensitivity has a maximum. The case of the
value of the prism refractive index being between ordinary and extraordinary refractive indices of the
LCs was also studied. This case presents an interest because the LC reorientation can change the ratio
between the refractive indices of the layers and the effect of the LC director reorientation should be
pronounced stronger. At n1 = 1.57 without voltage, there is a region in the reflectance spectrum with
constant reflectance R = 1 that disappears when the voltage higher than the threshold is applied. In this
case, the number of dips suitable for measurement increases, and when the voltage increases, the dips
become wider. A decrease of the prism refractive index with a constant voltage has the same effect on
the reflectance features. The improvement in the proposed measurement scheme is due to the fact that
the voltage change and the resulting reorientation of the LC is easier than replacing the prism. When
n1 = 1.57, the LC reorientation allows one to shift the maximum sensitivity. Such a sensor can be used
for measuring in two steps. The first step is to measure the region of the analyte refractive index, the
second one is to choose the voltage and make a more accurate measurement. A better sensitivity was
obtained at n1 = 1.57 in the case of non-symmetrical boundary conditions. Our results can be used for
designing tunable sensors.
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Числове моделювання поширення свiтла в сенсорi на основi
поверхневого плазмонного резонансу з рiдким кристалом
Є.С. Ярмощук, В.I. Задорожний, В.Ю. Решетняк
Київський нацiональний унiверситет iменi Тараса Шевченка, Фiзичний факультет,
просп. Академiка Глушкова, 2, 03022 Київ, Україна
Теоретично дослiджено п’ятишаровий плазмонний сенсор з шаром наночастинок та неоднорiдним ша-
ром рiдкого кристалу. Розраховано коефiцiєнт вiдбивання як функцiю кута падiння при рiзних значеннях
показника заломлення аналiту та прикладеної до рiдкого кристалу (РК) напруги. Змiнюючи орiєнтацiю ди-
ректора рiдкого кристалу, можна контролювати положення мiнiмумiв кривих вiдбивання i вибрати той,
що найбiльш чутливий до показника заломлення аналiту. При вибраному кутi падiння показник заломле-
ння аналiту може бути визначений по значенню коефiцiєнта вiдбивання. Вплив переорiєнтацiї директора
сильнiший, коли значення показника заломлення призми є мiж звичайним i незвичайним показниками
заломлення РК, в цьому випадку збiльшення напруги i зменшення показника заломлення призми мають
подiбний вплив на форму кривої вiдбивання.
Ключовi слова: рiдкий кристал, поверхневий плазмонний резонанс, сенсор, наночастинки, металева
пориста плiвка
13401-10
https://doi.org/10.2217/nnm.11.117
https://doi.org/10.1007/s00216-004-2708-9
https://doi.org/10.1364/OL.32.001902
https://doi.org/10.1016/j.photonics.2009.03.003
https://doi.org/10.1063/1.3081031
https://doi.org/10.1364/AO.48.004637
https://doi.org/10.1007/978-1-4612-2692-5
https://doi.org/10.1080/09500348914551061
https://doi.org/10.1063/1.3242363
https://doi.org/10.3390/ma7021296
https://doi.org/10.1080/15421406.2015.1032080
https://doi.org/10.1364/JOSA.62.000502
https://doi.org/10.1364/JOSAA.5.001554
https://doi.org/10.1364/JOSAA.6.001657
https://doi.org/10.1063/1.369638
https://doi.org/10.1103/physreve.51.1191
https://doi.org/10.1364/OE.16.001186
https://doi.org/10.1063/1.1587686
https://doi.org/10.1002/9783527618156
https://doi.org/10.1109/JDT.2005.853357
https://doi.org/10.1063/1.1767289
https://refractiveindex.info/?shelf=glass&book=BAK1&page=SCHOTT
Introduction
System scheme
Theoretical model
Calculation method
Dielectric tensors
Equations for the LC director
Result and discussions
The prism refractive index 1.51
The prism refractive index 1.57
Conclusions
|