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Proposal for 2-D High Precision Displacement Sensor Array
based on Micro-ring Resonators using Electromagnetically
Induced Transparency (Nanophotonic Device)
R. Yadipour [a], K. Abbasian [a], A. Rostami [a, b], Z. D. Koozehkanani [c] and R. Namdar [d]
a) Photonics and Nanocrystals Research Lab. (PNRL), Faculty of Electrical and Computer
Engineering, University of Tabriz, Tabriz 51664, Iran
b) School of Engineering Emerging technologies, University of Tabriz, Tabriz 51664, Iran
c) Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz 51664, Iran
d) Department of Physics, Azarbaijan University of Tarbiat-e-Moallem, Tabriz, Iran
Tel/Fax: +98 411 3393724
E-mail: [email protected]
Abstract- Array of ring resonators as a basic cell for detection of different physical quantities such as
displacement is used. Electromagnetically Induced Transparency (EIT) implemented by nanocrystals
doped in the ring resonators is used for enhancement of the capability of the ultra small displacement
detection. Different physical quantities such as temperature, mechanical strain, acoustic waves and
position of array of micro mirrors in the micro electromechanical systems can be measured using the
proposed basic cell too. We show that high precision measurement is available when EIT is used. Also,
using array of ring resonators, we show that the sensor array have high precision signal detection
capability compared single ring case. The proposed structure is investigated in detail and different
quantities for evaluation of the quality of the sensor are extracted. Finally we show that in the case of
nanocrystal doped ring resonators tolerances of the parameters in the proposed system have a little effect
on output quantities compared without nanocrystals.
Keywords- EIT, Displacement sensor, Ring resonator
1. Introduction
High-precision and integrated sensors have been studied recently generally in industry and biomedical
applications especially. Displacement sensing in Micro and Nano machines and also, ultrasound imaging
sensors for intravascular applications are two highly interested tasks for technology developers. Micro-
ring resonator is a suitable alternative for realization of different integrated tasks. Because of inherent
interesting properties of the ring resonators including high quality coefficient and easy to implement, it is
a good alternative for realization of sensor array especially in 2-D case. In this work we like present
framework of 2-D sensor array for detection of different physical quantities. For this purpose the transfer
function of the considered structure is studied and simulated results are discussed.
Combination of optical and mechanical systems named optomechatronics or optical micro
electromechanical systems (optical-MEMS) opened new insight to device design for making processing
blocks. Also, design strategy based on optical-MEMS is suitable approach for sensor design too. For
realization of ultra high precision systems in deep sub-micron or nano-scales one of important sensors is
displacement sensor. So, for realization of this sensor different methods have been used. Here, we are
going to review some of them and investigate advantages and disadvantages of the reported works. Some
of standard displacement sensing methods in technology community is used such as using variable
resistance as physical phenomenon for detection of the object displacement, using capacitance measuring,
using piezoelectric material, change of air gap in transformer which introduces variable induced voltage
and advanced version of this method for displacement measurement is linear variable differential
transformer (LVDT) [1]. This type of displacement sensors cover sub-micron to mm range of operation.
Ultrasound method is another interesting approach for displacement monitoring. In this method
ultrasound pulses applied to the object and backward wave is detected. Based on forward transmitting and
backward receiving waves the object displacement is calculated [1]. Since wavelength of the ultrasound
wave is high enough, so precision of the displacement measurement is low. Optical method is another
important technique for displacement measurement. This approach was discussed in [2-4]. In this method
two basic techniques are used. One of these methods based on the reflected back intensity. In this
technique variation of displacement converted to the level of intensity measured with high resolution
photodetectors. Interference of the forward and backward traveling waves and phase difference is another
criterion of the displacement measurement. In these methods displacement resolution can be increased to
near Pico meter ranges. In these sensors range of operation usually limited to micro meter range. Another
interesting method developed recently is based on ring resonators and used for high resolution
displacement measurement [5]. In this method usually a narrow gap is made on top part of the ring and
outgoing wave from this gap impact on object and reflected back to the ring. On the other hand object and
ring simultaneously introduce a resonant cavity. Displacement of object changes the oscillation and
resonance frequency of the complex system. So, measuring of output intensity for input light at given
wavelength is used for measurement of the displacement. Micro-ring resonator is a basic and important
device which is used recently more [6-14] as the building block for optical systems such as filters. So, in
this paper we have proposed array of nanocrystal doped micro resonators as an ultra-high precision
displacement sensor with equal spacing between adjacent resonators without coupling where each of them
are coupled to a waveguide. Thus the array is coupled to an input and output bus waveguides. There is a
basic problem with ring resonator. It is wide spectral shape which causes low precision in displacement
measurement. For this purpose in this paper, we present a new idea for improving this problem. Our
method is based on doping of ring resonator with 3-level atoms or nanocrystals. In this situation the
spectral profile of the ring resonators is decreased strongly and thus the precision of the sensor is
increased. For this proposal, we use quantum optical tools for description of optical properties of the
nanocrystals doped ring resonators [15]. In this proposal, we used 3-level atoms with given density. First,
we calculate the optical susceptibility and then using control field changing of the obtained susceptibility
is controlled. Obtained optical susceptibility is used for management of the guided wave and finally
optical output intensity is extracted. Since obtained optical susceptibility determine the resonance
frequency, so applied light in a given wavelength may have different output intensity in the output port.
So, ultra small narrowband spectral profile can be used for obtaining on-off behavior in the output for
small displacement. We show that our proposed method can measure well below nanometer and even
picometer range.
Also, in this work we consider 2-d array of ring resonators for developing imaging sensor especially in
intravascular applications for micro and nano machines. Our simulated results show that the proposed
idea works well.
Organization of the paper is as follows.
In section 2 mathematical backgrounds for theoretical description of the proposed system is presented.
Simulation results and discussion of the introduced structure are presented in section 3. Finally the paper
ends with a short conclusion.
2. Mathematical Background
In this section, mathematical background for description of the input-output relation of array of ring
resonators is presented. For doing this purpose, first, we consider a single ring resonator coupled to a
single mode optical waveguide which is illustrated in Fig. 1. This structure is used as a basic cell of
horizontal array.
Fig. 1. Schematic of the ring resonator as a basic cell of horizontal array
According to light propagation theory in linear and isotropic media the following relations are presented to describe
the input-output transfer function.
],1[1 2111 EjEE io κκγ −−−= (1)
],1[1 12114 iEjEE κκγ −−−= (2)
where γ1 and κ1 are coupler’s loss and the coupling coefficient respectively. By using the wave propagation inside
ring resonator and after some mathematical manipulation the following transfer function is obtained as follows.
Lj
L
Lj
L
i
o
ee
ee
E
E
βα
βα
κγ
κγγ
211
112
1
)1)(1(
)1)(1(1
−−−
−−−−= , (3)
where α and β are the ring (and fiber) loss coefficient and wave propagation vector respectively and L=2πr is the
total length of the ring. Now, we consider the following basic cell of vertical array as illustrated in Fig. 2.
Fig. 2. A single ring as basic cell of vertical array
Using mathematical manipulation, the transfer function of the ring resonator which is illustrated in Fig. 2 is obtained
as follows:
(3)
where γ1, γ2, κ 1 and κ 2 are coupler’s loss and coupling coefficients respectively. Also, α and β are loss coefficient
and propagating wave vector respectively.
Fig. 3. Two rings vertical array
Transfer function of two rings illustrated in Fig. 3 can be obtained similarly as follows.
, (4)
where , and are given with following relations.
(5)
(6)
(7)
Fig. 4. Three rings vertical array
Similarly the transfer function of three rings vertical array is obtained with following relation as well.
, (8)
where , , , and are given as follows respectively.
, (9)
(10)
(11)
(12)
(13)
Fig. 5. Two rings horizontal array
Now, we consider the horizontal case and the transfer function of two rings which is shown in Fig.4, is obtained
with some mathematical manipulations as follows as an example.
, (14)
where z1 is waveguide length between two couplers and α, β are loss and wave propagation vector of
propagating field in waveguide respectively. A1 and A2 are transfer functions of first and second rings
respectively which are given by following relations.
(14)
(15)
Fig. 6. Three rings horizontal array
Transfer function of three rings which is shown in Fig.5, is obtained with some mathematical
manipulations as well.
, (16)
where z1, z2, α and β are waveguide lengths between couplers, the loss coefficient and propagating wave
vector respectively. Also, A1, A2 and A3 are transfer functions of first, second and third ring resonator
respectively.
Finally the transfer function of considered 2-D matrix can be obtained with mathematical manipulation of
horizontal and vertical studied array relations.
Fig. 7. Two in two matrix array of rings
Transfer functions of the array of ring resonators illustrated in Fig. 7 are obtained with following relations.
(17)
, (18)
where , and are given by relations (5-7).
Fig. 8. Matrix array of rings
Transfer functions of three in three matrix array are obtained as follows.
(19)
(20)
, (21)
where , , , and are given by relations (9-13 ).
These ideas can be used in micro-ring arrays as illustrated in following Figure (micro ring with air gap,
see one basic block of Fig. 9), whose transfer function can be given as follows.
( )( )
( )( )
0 0
45
0 0
. . . .
. 2 .. 2 .4 2 4 21 1 1
.. ... 2 .. 2 .4 2 4 2
1 1
1 1 (1 ). .e .e .e .e .e .e .e .e
1 1 1 . .e .e .e .e .e .e .e .e
L j L L j L
x j xh j h
oi
j Lj L LL
i x j xh j h
K rE
EK r
α β α βα βα β
ββ ααα βα β
γ γ
γ
− − − −− ∆ − ∆− −
−− −−− ∆ − ∆− −
− − − −=
− − −
, (22)
where L, h, ∆x and r are ring length, fiber and free space lengths and the reflection coefficient of the
reflecting surface of the object, respectively. In the following an example is presented for monitoring of
position of array of micro-mirrors. In the following figure, we presented only four ring resonators as
displacement sensors. In the next section, we illustrate simulated result of each displacement ring
resonator. Also, the proposed structure can be used for high precision two-dimensional surface moving.
Fig. 9. Array of ring resonators for detection of position of array of Micro-mirrors
In this section array of ring resonators as building block of 2-D sensor matrix has been studied and analytical
transfer functions derived. In the next section simulation results are illustrated and discussed.
3. Simulation Results and Discussion
In this section, simulation results are divided into four cases as follows:
1. Normal case (ring resonator without 3-level doping)
2. Vertical arrays of ring resonators with EIT (ring resonator with 3-level doping)
3. Horizontal arrays of ring resonators with EIT (ring resonator with 3-level doping)
4. 2-D array of ring resonators with EIT (ring resonator with 3-level doping)
Also, in final part of this section effect of practical tolerances on output signals are investigated too.
1. Normal case (Without EIT) The following figure shows the transmission coefficient and phase of a single ring (basic cell of
horizontal array) without EIT versus differential wavelength. As we can see the transmission coefficient
sensitivity (wavelength displacement between minimum and maximum) is about 0.2 nm and phase
changes about 4π radians at resonance wavelength.
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x 10-9
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Co
ef.
Of
Sin
gle
Hori
z. B
asi
c C
ell
a
-2 -1.5 -1 -0.5 0 0.5 1 1.5
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Sin
gle
Ho
riz.
Ba
sic
Cel
l
b
Fig. 10. Transmission coefficient of single horizontal ring without EIT vs. differential wavelength
(a) Magnitude (b) Phase
0.15, L=2πr=5e-4 m,
EIT Case: Now, we are going to review mathematical method to describe effect of 3-level dopants, on
characteristics of the proposed sensor. Using Λ type 3-level nanocrystals in ring resonator, we show that
the resolution of the proposed sensor can be increased and tuned optically. Fig. 11 shows Λ type 3-level
particle schematic including probe and control fields and decay rates. In the model the control and probe
fields applied between levels 2-3 and levels 1-2 respectively. Due to applied electric field the optical
characteristics is changed and in the following brief theoretical calculation for description of the system
performance is presented. With strong control and weak probe laser pulses, the time evolution of
coherences between atomic levels are described by following equation [2]:
[ ],d i
Hdt
ρ ρ ρ= − − Γ , (23)
where H, ρ and Γ are the system’s Hamiltonian, density and decay rate matrixes respectively.
Fig. 11. Schematic of 3-level particles
After some mathematical manipulation [1], the real and imaginary parts of optical susceptibility are given
as
( )
Ω−−∆++−=
4
2
31
2
313
0
2' µγγγγγ
ε
γχ
Z
N aba
, (24)
( )
Ω−−∆−+∆=
4"
2
31
2
331
2
0
2
µγγγγγε
γχ
Z
N aba
, (25)
where ( )2
21
2
22
21
2
4γγγγ µ +∆+
Ω−−∆=Z and υω −=∆ ab
, γ1 , γ2, γ3, µΩ and
aN are detuning of probe
frequency from resonance, decay rates of atomic levels population, Rabi frequency of control field and
density of doped nanocrystals. Based on basic and fundamental relations between optical susceptibility and
absorption coefficient and refractive index, we have the following relations.
"
2χ
κα = ,
2
'χδ nn = , (26)
Finally the propagating wave vector in ring resonator doped with 3-level particles is given in the following.
c
nυβ = , (27)
In the following illustrated simulation results, effect of the control field on performance of horizontal ring
arrangement is investigated. It is observed that, in wide frequency range of input signal, the considered
system is transparent and oscillatory nature of the transmission coefficient is increased [5]. In the
mentioned region the transmission coefficient sensitivity (wavelength displacement between minimum
and maximum) is modified to about 5 pm.
-5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Co
effi
cien
t
Normal
EIT
a
-5 -4.5 -4 -3.5 -3 -2.5 -2
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Normal
EIT
b
-3.015 -3.01 -3.005 -3
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
EIT
Normal
c
-3.015 -3.01 -3.005 -3
x 10-9
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
EIT
Normal
d
Fig. 12. Transmission coefficient vs. differential wavelength for normal and EIT cases of single horizontal ring (a)
Magnitude (b) Phase (c) Zoom in on the transmission coefficient (d) Zoom in on the phase
0.15, L=2πr=5e-4 m,
The similar situation is for the basic cell of vertical arrays, where in normal case the transmission
coefficient sensitivity (wavelength displacement between minimum and maximum) is about 0.2 nm and
phase rotates about 2π radians at resonance. In EIT case transmission coefficient sensitivity is modified to
about 5 pm as so.
-8 -6 -4 -2 0 2 4 6 8 10
x 10-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
f. o
f S
ingle
Ver
. B
asi
c C
ell
a
-8 -6 -4 -2 0 2 4 6 8 10
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Sin
gle
Ver
. B
asi
c C
ell
b
Fig. 13. Transmission of single vertical ring without EIT vs. differential wavelength, a) Magnitude and b) Phase
0.15, L=2πr=5e-4 m,
-4 -3.8 -3.6 -3.4 -3.2 -3 -2.8
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
Normal
EIT
a
-3.316 -3.314 -3.312 -3.31 -3.308 -3.306
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Co
effi
cien
t
EIT
Normal
c
-4 -3.8 -3.6 -3.4 -3.2 -3 -2.8
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Normal
EIT
b
-3.34 -3.33 -3.32 -3.31 -3.3 -3.29 -3.28
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
EIT
Normal
d
Fig. 14. Transmission coefficient vs. differential wavelength for normal and EIT cases of single vertical ring. (a)
Magnitude (b) Phase (c) Zoom in on the transmission coefficient (d) Zoom in on the phase
2. Vertical Array
In horizontal array our simulations show that by increasing the number of rings, the slope of transmission
coefficient increases. On the other hand in oscillatory region envelope of the transmission coefficient is
returned to minimum. Also, the phase variation increases to 4π times by increasing the number of rings at
resonance wavelength.
It is observed that sensitivity of the vertical arrays improves from 5 pm for single ring to 2 pm for three
rings where the transmission coefficient has one and three peaks for them respectively. Also, phase
rotation varies from 2π to 10π in same oscillation duration of wavelength.
-3.322 -3.32 -3.318 -3.316 -3.314 -3.312 -3.31 -3.308
x 10-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
Single
Two
Three
a
-3.322 -3.32 -3.318 -3.316 -3.314 -3.312 -3.31 -3.308
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Single
Two
Three
b
Fig. 15. Transmission of vertical array with fixed control field vs. differential wavelength. (a) Magnitude. (b) Phase
0.15, L1= L2= L3=5e-4 m,
Following figures show that oscillation wavelength varies with control field variation and oscillation wavelength
duration increases with increasing of control field.
-3.318 -3.316 -3.314 -3.312 -3.31 -3.308 -3.306 -3.304 -3.302 -3.3
x 10-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Co
effi
cien
t
Single
Two
Three
a
-3.318 -3.316 -3.314 -3.312 -3.31 -3.308 -3.306 -3.304 -3.302
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Single
Two
Three
b
Fig. 16. Transmission of vertical arrays with fixed control field vs. differential wavelength
(a) Magnitude (b) Phase
0.15, L1= L2= L3=5e-4 m,
3. Horizontal Array In following figures it is illustrated that increasing of the control field extends the oscillation region. Also,
in horizontal array it can be shown that by increasing the number of rings the transmission coefficient is
decreased and find the minimum value while it is at maximum for single ring in out of oscillation region.
On the other hand in oscillatory region the envelope of transmission coefficient is returned to minimum
value. Also, the phase rotation increases 2π times by increasing the number of rings at oscillation
wavelength.
-3.7 -3.6 -3.5 -3.4 -3.3 -3.2 -3.1 -3 -2.9
x 10-9
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Co
effi
cien
t
Single
Two
Three
a
-3.41 -3.4 -3.39 -3.38 -3.37 -3.36 -3.35 -3.34
x 10-9
-4
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Single
Two
Three
b
Fig. 17. Transmission of horizontal arrays with fixed control field vs. differential wavelength (a) Magnitude (b)
Phase
0.15, L1= L2= L3=5e-4 m,
-4 -3.5 -3 -2.5
x 10-9
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
Single
Two
Three
a
-3.41 -3.4 -3.39 -3.38 -3.37 -3.36 -3.35 -3.34
x 10-9
-4
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Single
Two
Three
b
Fig. 18. Transmission of horizontal arrays with fixed control field vs. differential wavelength (a) Magnitude (b) Phase
0.15, L1= L2= L3=5e-4 m,
4. Matrix Array
In the following results it is shown that the transmission coefficient and phase of matrix arrays have
characteristics of both horizontal and vertical arrays except the first one which, in fact, is a vertical array.
-3.322 -3.32 -3.318 -3.316 -3.314 -3.312 -3.31 -3.308
x 10-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Co
effi
cien
t
Single
2x2
3x3
-3.326 -3.324 -3.322 -3.32 -3.318 -3.316 -3.314 -3.312 -3.31 -3.308 -3.306
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Single
2x2
Three
a b
Fig. 19. First transmission of matrix arrays with fixed control field vs. differential wavelength a) Magnitude b)
Phase
0.15, L1= L2= L3=5e-4 m,
-3.32 -3.318 -3.316 -3.314 -3.312 -3.31 -3.308 -3.306 -3.304 -3.302 -3.3
x 10-9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
Single
2x2
Three
a
-3.318 -3.316 -3.314 -3.312 -3.31 -3.308 -3.306 -3.304 -3.302
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
Single
2x2
Three
b
Fig. 20. First transmission of matrix arrays with fixed control field vs. differential wavelength a) Magnitude b)
Phase
0.15, L1= L2= L3=5e-4 m,
Following figures show that the second transmission coefficient of matrix arrays varies between 0 and
about 0.23 and phase of transmission can’t complete its rotations as horizontal and vertical array cases.
-3.322 -3.32 -3.318 -3.316 -3.314 -3.312 -3.31 -3.308
x 10-9
0
0.05
0.1
0.15
0.2
Wavelength Shift
Tra
nsm
issi
on
Coef
fici
ent
2 x 2
3 x 3
a
-3.32 -3.315 -3.31 -3.305
x 10-9
-3
-2
-1
0
1
2
3
wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
2 x 2
3 x 3
b
Fig. 21. Second transmission of matrix arrays with fixed control field vs. differential wavelength
(a) Magnitude (b) Phase
0.15, L1= L2= L3=5e-4 m,
-3.32 -3.315 -3.31 -3.305 -3.3
x 10-9
0
0.05
0.1
0.15
0.2
Wavelength Shift (m)
Tra
nsm
issi
on
Co
effi
cien
t
2 x 2
3 x 3
a
-3.32 -3.315 -3.31 -3.305 -3.3
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
2 x 2
3 x 3
b
Fig. 22. Second transmission of matrix arrays with fixed control field vs. differential wavelength
(a) Magnitude (b) Phase
0.15, L1= L2= L3=5e-4 m,
It is illustrated that the third transmission coefficient of matrix arrays has decreased to about 0.14. Also, it
can be seen that not only horizontal and vertical array specification contributions on third transmission
vary by magnitude of the control field but also oscillation wavelength position and duration varies too. It
can be seen that phase rotations aren’t complete as so.
-3.32 -3.315 -3.31 -3.305 -3.3
x 10-9
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
6.58 e7
16.45 e7
a
-3.32 -3.315 -3.31 -3.305 -3.3 -3.295
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
6.58 e7
16.45 e7
b
Fig. 23. Third transmission of matrix arrays with variable control field vs. differential wavelength
(a) Magnitude (b) Phase
(b) 0.15, L1= L2= L3=5e-4 m,
The effect of refractive index variation on transmission of basic cells in both normal and doped cases is
illustrated in Figs. 24-27.
In Figs. 24-26 it can be seen that resonance wavelength displacement is about 0.5 nm for 0.0005 change
of the refractive index in normal case of basic cells. On the other hand, in doped case, it is shown that
resonance wavelength displacement is about 4 pm for the same change of refraction index in Figs. 25-27.
-4 -3 -2 -1 0 1 2
x 10-9
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
1.5
1.5005
1.501
a
-4 -3 -2 -1 0 1 2
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
1.5
1.5005
1.501
b
Fig. 24. Transmission of normal horizontal basic cell vs. differential wavelength for different refraction index
(a) Magnitude (b) Phase
-3.08 -3.06 -3.04 -3.02 -3 -2.98 -2.96
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
1.5
1.5005
1.501
a
-3.32 -3.3 -3.28 -3.26 -3.24 -3.22 -3.2
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
1.5
1.5005
1.501
b
Fig. 25. Transmission of doped horizontal basic cell vs. differential wavelength for different refraction index
(a) Magnitude (b) Phase
-4 -3 -2 -1 0 1 2
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
1.5
1.5005
1.501
a
-4 -3 -2 -1 0 1 2
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
1.5
1.5005
1.501
b
Fig. 26. Transmission of normal vertical basic cell vs. differential wavelength for different refraction index
(a) Magnitude (b) Phase
-3.4 -3.38 -3.36 -3.34 -3.32 -3.3 -3.28
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift (m)
Tra
nsm
issi
on
Coef
fici
ent
1.5
1.5005
1.501
a
-3.4 -3.38 -3.36 -3.34 -3.32 -3.3 -3.28
x 10-9
-3
-2
-1
0
1
2
3
Wavelength Shift (m)
Ph
ase
of
Tra
nsm
issi
on
1.5
1.5005
1.501
b
Fig. 27. Transmission of doped vertical basic cell vs. differential wavelength for different refraction index (a)
Magnitude (b) Phase
Finally effect of displacement on the transmission coefficient for different displacement values in both normal and
EIT cases of micro-ring resonators are illustrated in Fig. 28. It is shown that the EIT case is so sensitive compared
normal case.
-2.91 -2.9 -2.89 -2.88 -2.87 -2.86 -2.85 -2.84 -2.83 -2.82 -2.81
x 10-9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Wavelength Shift [m]
Tra
nsm
issi
on
Coef
fici
ent
1pm
0.1nm
10nm
1nm
5nm
10nm,Normal
Fig. 28. Wavelength displacement vs. displacement in EIT and Normal Cases for different displacement values
In this section different aspect of the proposed structure for operation as displacement or sensing other
quantities were considered. It was shown that application of EIT in ring resonator has critical effect on
increasing the sensitivity of the proposed sensor.
4. Conclusion In this paper sensor matrix based on array of ring resonators has been presented. It has been shown that application
of nanocrystals such as 3-level atoms in ring resonator strongly improves displacement measurement even less than
5 pm. The proposed structure can be used to measure displacement of 2-D micro mirror array. The proposed
structure can be used as 2-D imaging sensor for ultrasound signals especially in intravascular case.
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