Light propagation in PhC
EM propagation through a medium is dependent on the permittivity of the
medium. It is furthermore dependent on the RI of the material. As the RI
changes, the permittivity changes; as a result, the EM propagation also gets
impacted.
The propagation of light in the PhC (sensing element) is described by
Maxwell's electromagnetic (EM) equations as given below:
∇×Er,t=-μ(r)∂H(r,t)∂t,∇×Hr,t=σrEr,t+ϵ(r)∂E(r,t)∂t,∇Er,t=ρ(r,t)ϵ(r),∇Hr,t=0,
where E(r,t) is the time-varying electric field, H(r,t) is the
time-varying magnetic field, ϵr is the permittivity,
μr is the permeability, and σr is
the conductivity of the medium. Considering the propagation of an EM wave in
any medium, the equations in the Cartesian co-ordinate system for electric
and magnetic fields are given by
E=Exx^+Eyy^+Ezz^,H=Hxx^+Hyy^+Hzz^.
The Bloch–Floquet theorem states that an EM wave propagating in a varying
dielectric structure is modulated by the periodicity of the structure. The
periodic variation is given by
ϵr+p=ϵr,
where p is the period of the crystal. The EM field is given by
Ψr+p,t=Ψ(r,t)e-ik,p,
where Ψ(r,t) is the electric or magnetic field. K is the propagation
constant and a is the period of a crystal.
The proposed sensor uses the FDTD (finite difference time domain) algorithm,
which solves Maxwell's EM equations (Yee, 1996). For the proposed structure,
a Gaussian pulse is used as a source and the fields are updated at each point
of the Yee grid according to the finite difference Maxwell curl equations,
and the obtained output samples are normalized with respect to the input
signal.
Design of the two-dimensional PhC line defect.
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Salinity vs. RI of water. The figure shows variation of the RI with
a change in the salinity of water.
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Different types of sensors and target substances for detection.
Sensor type
Target substance
pH
Acids and bases
Oxidation reduction potential (ORP)
Redox active species
Electrical conductivity (EC)
Salts (TDS)
Turbidity
Particles
UV/Vis absorption
Aromatic substances
Photonic sensor
All substances
Shows variation of RI with % of salinity of water.
Salinity of water
RI of water
sample (%)
0.01
1.329701796
0.05
1.329701806
0.1
1.329701818
0.2
1.329701843
0.5
1.329701918
0.75
1.329701981
2.5
1.329702418
3.5
1.329702668
10
1.329704293
20
1.329706793
30
1.329709293
Transmitted spectrum for (a) 500 ppm saline water, (b) 52000 ppm
saline water, and (c) 35 000 ppm saline water. The above figures depict
transmission spectra with distinct shifts in peak frequencies for different
salinities. In (a) a nascent curve of the transmission spectrum until
20 % of the outcome is exited immediately, with a dip being followed.
These curves indicate that the light intensity drops abruptly around 25 %
of the frequency range of the input light wave. These lines of frequency
again tend to achieve maxima and do so at exactly 50 % of the frequency
spectrum. This is an indication of the highest possible absorption of the
intensity of an input light wave that may be a cause of polarization in the
vicinity of the waveguide of the proposed structure. This velocity of
intensity increase will again tend to become sluggish and abruptly embraces
an exponential decay, for which the trapping of light in the waveguide begins
to throw off a certain frequency of harmonic wave that tends to create a
disruptive interference of the travelling pulse of Gaussian mode. In (b) we
can observe that the salinity of water, being an analyte as compared against
(a), is increased in concentration by 300 %. Here the entire spectrum is
exactly the reciprocal of (a) in that the dip has happened in the first phase
of the frequency shift, while here the same has happened in the second. Also,
as against (a), the light intensity abruptly decreases before the center of
the frequency spectrum is achieved. Thereby the absorption and reflection
that have taken place before the identification of light intensity at the
output become noticeable; only after the frequency of the spectra are over
the central frequency of operation will the light intensity become noticeable
twice. The curve remains nascent for around 30 % of the applied frequency
and vigorously excites until 60 %. This excitement is immediately damped
with a scattering time of under 0.5 units of intensity and remains nascent
throughout. This signature curve of the transmission spectrum is incorporated
into the database of the application.
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Transmitted spectrum for water with various salinity levels. The
figure shows the overlapping of all the previous spectra to highlight the
shift in the frequency and amplitude.
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Workflow of the developed system.
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(a) Integrated device display with Raspberry Pi (red
circle), (b) the 2000 ppm salinity result (yellow circle), and
(c) the 1000 ppm salinity water result (blue circle).
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In a PhC, RI is periodically modulated where periodicity is in the order of
wavelength. PhCs are periodic structures of dielectric material which allow
the propagation of a certain frequency range of light (Joannopoulos et al.,
1995, 1997) and stop others (forbidden band gap). This unique behaviour of a
PhC is used to control the propagation of light (Meade et al., 1992). The
deviation of light in a lattice structure can be controlled by defect
engineering. The following Eq. (1) explains the movement of light in a PhC by
solving Maxwell's electromagnetic equation.
∇×1ϵ∇×H=ωC2H
H – the photon's magnetic field; ϵ – permittivity; C – speed
of light; and ω – angular frequency.
ϵ=n2
n – refractive index
As in Eq. (9), the permittivity of a medium (ϵ) changes as the
angular frequency of resonance (ω) changes. Equation (10) shows that
ϵ is dependent on RI and is the basis for using PhC as a sensor
(Liu and Salemink, 2012). Methods like the photonic band gap method, the
effective RI method, spectroscopy, and optical imaging are available (Fan et
al., 2008). Since input variations are significantly low, the sensitivity of
these methods is less (Nguyen et al., 2011).
The design and simulation of sensors is done by the MEEP tool. This is a FDTD
simulation software to model electromagnetic systems. To compute transmission
flux at each frequency “ω”, sampling of a continuous
electromagnetic field in a finite volume of space is done and is determined
by Eq. (11).
Pω=ReńEωX∗×HωXd2X
To calculate P(ω), the following steps are used in the MEEP tool.
Compute the integral of the Poynting vector P(t) for each time.
Fourier transform the value in no. 1.
Compute flux at the specified regions and frequencies.
Machine learning algorithm
Machine learning is an automated action in which improvement is done in the
future based on learning from the past. The key element of this is to devise
learning algorithms that do the learning automatically with minimum human
actions. The algorithm in machine learning allows the developed application
to come up with its own assessment based on supplied training data (Haung et
al., 2010).
The naive Bayes algorithm is a classification technique based on the Bayes
theorem with an assumption of independence among predictors (Rish, 2001). The
naive Bayes classifier assumes that the presence of a particular feature in a
class is unrelated to the presence of any other feature. Naive Bayes is known
for its simplicity that does better than other existing classification
methods. The Bayes theorem provides a way of calculating the posterior
probability P(c|x) from P(c), P(x) and P(x|c). This is defined in
Eq. (13):
Pc|x=Px|cP(c)P(x),Pc|x=Px1|cxPx2|cx…xPxn|cxPc.
P(c|x) is the posterior probability of a class (c, target) given the
predictor (x, attributes).
P(c) is the prior probability of a class.
P(x|c) is the likelihood, which is the probability of a predictor-given
class.
P(x) is the prior probability of a predictor.
Depending on various attributes, the algorithm based on the naive Bayes
theorem predicts the probability of different classes. This algorithm is used
to solve problems with multiple classes.