Simulative study on integrated optical multimode waveguides with guided beams based on the system standardization of elements

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Ahmed Nabih Zaki Rashed*, Kausar Jahan, Ali Baig Mohammad, Shaik Hasane Ahammad,
Md. Amzad Hossain* and Farid Anwar Fawzy
Simulative study on integrated optical multimode
waveguides with guided beams based on the
system standardization of elements
https://doi.org/10.1515/joc-2022-0251
Received September 24, 2022; accepted December 10, 2022;
published online December 30, 2022
Abstract: This study has outlined simulative study on in-
tegrated optical multimode waveguides with guided beams
based on the system standardization of elements. Deflec-
tion of a cantilever are clarified with a point force at the
free end, due to the distributed weight at the beam, with a
mass at the free end, and at the free end under acceleration.
The bending of a double clamped beam under its distrib-
uted weight, deflection with the central mass double
clamped beam under its weight, the buckling of a double
clamped beam due to a compressive stress, out of plane
deflection (OPD) of a bent beam suspension, OPD of a folded
beam suspension, and OPD of a serpentine beam suspen-
sion are also clarified and reviewed. Dependence of cross
section of beam on torsion constant is outlined. The stiff-
ness ratio of lateral to vertical motion of hammock sus-
pension is also clarified. The design of a crab-leg suspension
and the dependence of the stiffness on thigh section of the
crab leg flexure and the design of a folded flexure
suspension and the dependence of stiffness on ratio of
column beam lengths are also reported.
Keywords: beam deflection; guided beams; MEMSolver
simulation; stiffness ratio; surface structure.
1 Introduction
There are many structures of beams [1–9]. These types in-
cludes cantilever, clamped beams, bent beams, folded
beams, serpentine beams, and guided beams [10–18]. The
cantilever types can be divided into end loading which the
cantilever beam (CM) with a point force at the free end can
be displayed and analyzed [17, 19–28]. The distributed load
which the bending of a CM under its own weight is inves-
tigated [29–41].Themassatfreeendwhichthebendingofa
CM with a mass at the free end and acceleration load which
the cantilever with end mass under acceleration is
demonstrated [42–53]. The clamped beam (CB) types can be
divided into the center loading which the bending of beam
forced is observed [54–74].
In this work, the distributed load which the bending
of a double CB due to its weight is outlined [75–83]. The
central mass can be divided into the distributed load
which the bending of a double clamped beam with a cen-
tral mass is clarified [84–93]. Acceleration load which the
double CB with a central mass and the buckling stress
which the buckling of a double CB is outlined [94–103]. The
bent beam which the design of a corner beam or bent
beam suspension is observed. The folded beam which the
design of a folded beam suspension for planar motion is
outlined [104–114]. The serpentine beam (SM) which the
design of a SM suspension is observed [115–128]. The torsion
bar which the design of a torsion bar beam suspension is
reported. The guided beam which the design of a guided
beam hammock suspension and guide beam are reported.
The crab leg flexurewhichthedesignofacrableg
suspension through the crab leg and the folded flexure
through the design of a folded flexure suspension
through the flexure are reported in the simulation results
[129–153].
*Corresponding authors: Ahmed Nabih Zaki Rashed, Electronics and
Electrical Communications Engineering Department Faculty of Electronic
Engineering, Menoufia University, Menouf 32951, Egypt,
E-mail: ahmed_733@yahoo.com. https://orcid.org/0000-0002-5338-1623;
and Md. Amzad Hossain, Institute of Theoretical Electrical Engineering,
Faculty of Electrical Engineering and Information Technology, Ruhr
University Bochum, 44801 Bochum, Germany; and Department of Electrical
and Electronic Engineering, Jashore University of Science and Technology,
Jashore 7408, Bangladesh, E-mail: mahossain.eee@gmail.com
Kausar Jahan, Department of ECE, Dadi Institute of Engineering and
Technology, Anakapalle, Andhra Pradesh, India, E-mail: kjahan@diet.edu.in
Ali Baig Mohammad, School of Electronics and Communication
Engineering, REVA University, Bangalore, India,
E-mail: alibaig.mohammad@reva.edu.in
Shaik Hasane Ahammad, Department of ECE, Koneru Lakshmaiah
Education Foundation, Vaddeswaram, Andhra Pradesh 522302, India,
E-mail: ahammadklu@gmail.com
Farid Anwar Fawzy, Electronics and Electrical Communications
Engineering Department Faculty of Electronic Engineering, Menoufia
University, Menouf 32951, Egypt,
E-mail: Faridanwar657fawzy@outlook.com
J. Opt. Commun. 2022; aop

2 Models performance parameters
and discussions
Figure 1(a) clarifies the bending of a CB with a point force at
the free end. Where the cantilever length, width, and
thickness are 500 µm, 100 µm, and 5 µm, force at free end is
5 µN, Young’s modulus (YM) is 180 GPa. As the length of
cantilever increases, this results in the decrease of the
deflection of a cantilever.
The obtained results clarified the tip deflection is
1.111 µm, the maximum stress is 6 MPa, and the spring
Figure 1: Cantilever deflection variations against length. (a) Cantilever deflection with a point force at the free end. (b) Deflection of cantilever variations
versus cantilever length variations with a point force at the free end.
2A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

constant is 4.5 N/m. Figure 1(b) outlines the deflection of a
cantilever variations versus cantilever length variations
with a point force at the free end. It is demonstrated that the
deflection of a cantilever decreases with the increase of
cantilever length.
Figure 2(a) outlines the bending of a cantilever due to
the distributed weight of the beam. Where the length of
cantilever is 1000 µm, cantilever width is 500 µm, cantilever
thickness is 10 µm, the density is 2330 kg/m3, and Young’s
modulusis180GPa.Theclarified results show that the tip
deflection is 1.903 E-003 µm, the maximum stress is 6.850
E-003 MPa, the spring constant is 22.5 N/m. Figure 2(b)
shows the deflection of a cantilever variations against
cantilever length variations due to the distributed weight at
the beam. It is clear that the deflection of a cantilever de-
creases with the increase of cantilever length.
Figure 3(a) clarifies the bending of a cantilever due to a
mass at the free end. Where the cantilever length, width, and
thickness are 400 µm, 100 µm, and 5 µm, Mass at free end is
4E-06 kg, and YM is 180 GPa. The clarified results show that
the tip deflection is 4.46 µm, the maximum stress is
37.632 MPa, and the spring constant is 8.79 N/m. Figure 1(b)
Figure 2: Deflection against cantilever. (a) Deflection of a cantilever due to the distributed weight at the beam. (b) Deflection of cantilever variations
versus cantilever length variations.
A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides 3

outlines the deflection of a cantilever variations versus
cantilever length variations with a mass at the free end. It is
demonstrated that the deflection of a cantilever decreases
with the increase of cantilever length.
Figure 4(a) outlines the cantilever with a mass at the free
end under acceleration. Where the beam length, width, and
thickness are 400 µm, 100 µm, and 5 µm, mass at free end is
4E-06 kg, acceleration is 10 g, and YM is 180 GPa. It is clear
that from the results that the maximum deflection is
44.601 µm, the maximum stress is 376.32 MPa, and the
maximum strain is 2090.7 microstrains. Figure 4(b) outlines
the maximum stress versus acceleration for a cantilever
mass structure. The maximum stress increases as the
acceleration increases.
Figure 5(a) shows the deflection of a BC at both ends
with a force at the center. Where the beam length, width,
and thickness are 525 µm, 250 µm, and 5 µm, the force at
center is 100 µN, and Young’s modulus is 180 GPa. The
clarified results assured that the tip deflection is 0.1608 µm,
the maximum stress is 6.3 MPa, and the spring constant is
Figure 3: Cantilever deflection variations against Cantilever length. (a) Deflection of a cantilever with a mass at the free end. (b) Deflection of cantilever
variations versus cantilever length variations with a mass at the free end.
4A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

621.963 N/m. Figure 5(b) shows the beam clamped deflection
variations with the length beam variations based a force at
the center. As the beam length changes from 1 to 275 μm,
this results in the decreases of deflection of a beam expo-
nentially. But as the beam length changes from 275 to
525 μm, this results in the increases of deflection of a beam
exponentially.
Figure 6(a) outlines the bending of a double CB under its
distributed weight. Where the beam length, width, and
thickness are 835 µm, 125 µm, and 7.5 µm, the density is
2330 kg/m3, and YM is 180 GPa. The simulation results
assured that the tip deflection is 3.426 E-005 µm, the
maximum stress is 1.061 E-003 MPa, and the spring constant
is 260.872 N/m. As the length of beam varies from 1 to
425 μm, this results in the decreases of deflection of a
beam exponentially. But as the length of beam varies
from 425 to 835 μm, this results in the increases of deflection
of a beam exponentially.
Figure 7(a) clarifies the deflection of a double CB with a
central mass under its weight. Where the beam length,
width, and thickness are 400 µm, 100 µm, and 5 µm, the mass
length is 1500 µm, the mass width is 1500 µm, the mass
Figure 4: Maximum stress variations against acceleration variations. (a) Deflection of a cantilever at the free end under acceleration. (b) Maximum stress
versus acceleration for a cantilever mass structure.
A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides 5

thickness is 400 µm, density is 2330 kg/m3, and Young’s
modulus is 180 GPa. The simulation results indicates that the
deflection of the central mass is 0.292 µm, The maximum
stress is 4.934 MPa, and The spring constant is 70.312 N/m.
Figure 7(b) shows the stress on top of a double CB with a
central mass under its weight in the relation to beam length.
The stress decreases linearly with the increase of beam
length.
Figure 8(a) clarifies the buckling of a double CB due to a
compressive stress. Where the beam length, width, and
thickness are 400 µm, 125 µm and 3 µm, compressive stress
is 45 MPa, Young’s modulus is 180 GPa, and the simulation
results show that the buckling stress is 33.276 MPa, the
buckling force is 1.248E +004 µN, and the maximum
deflection is 1.678 µm. Figure 8(b). Deflection or Buckling of
a double CB due to a compressive stress in relation to beam
length. The Deflection or Buckling of a double CB due to a
compressive stress in relation to beam length is clarified
in Figure 8(b). The deflection increases exponentially with
the increase of beam length up to 200 μmandthen
Figure 5: Beam clamped deflection variations versus Cantilever length. (a) Deflection of a beam clamped (BC) at both ends with force at the center. (b)
Deflection of BC variations with the beam length variations at both ends with a force at the center.
6A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

decreases exponentially with the beam length varies from
200 to 400 μm.
Figure 9(a) outlines the design of a corner beam or bent
beam suspension. Where the length of a leg of the beam
length, width, and thickness are 40 µm, 5 µm, and 3 µm, YM is
180 GPa, and Poisson’s ratio (PR) is 0.3.
The results assured that the spring constant (SC) of
a single bent beam in Xaxis is 650.55 N/m, the SC of a
single bent beam in Yaxis is 650.55 N/m, and the SC of a
single bent beam in Zaxis is 35.13 N/m. The effect of beam
design on in plane stiffness of a bent beam is clarified in
Figure 9(b). The SC in x-direction decreases linearly with
(length/width) ratio of beam variations. Figure 10(a) out-
lines the design of a folded beam suspension for planar
motion. Where the first leg beam length is 50 µm, second leg
beam length is 30 µm, third leg beam length is 85 µm, beam
widthis15µm,beamthicknessis5µm,YMis180GPa,and
PR is 0.3.
Figure 6: Beam clamped bending variations versus beam length. (a) Bending of a double clamped beam under its distributed weight. (b) Deflection of
double CB variations with the beam length variations under its distributed weight.
A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides 7

The simulation results assured that the SC of a single
beam in Xaxis is 1466.329 N/m, The SC of a single beam in Y
axis is 2960.526 N/m, and SC of a single beam in Zaxis is
206.045 N/m. Figure 10(b) clarifies out of plane deflection of a
folded beam suspension. The SC decreases linearly with
beam (length 2/length 1) [%] in all x,y, and zdirections.
Figure 11(a) shows the design of a serpentine beam. Where
the long arm length beam is 100 µm, the short arm length
beam is 20 µm, width beam is 15 µm, thickness beam is 9 µm,
YM is 180 GPa, and PR is 0.3.
The simulation results assured that the spring constant
in Xfor one beam is 262.861 N/m, The SC in Zfor one beam is
81.881 N/m, and The torsional spring constant for one
beam in Xaxis is 3.713E +005 µNµm/rad. Figure 11(b)
shows the dependence of stiffness on short leg of the
serpentine beam. The spring constant decreases linearly
Figure 7: Beam clamped deflection variations versus cantilever length. (a) Deflection of a double CB with a central mass under its weight. (b) Stress on
top of a double CB with a central mass under its weight in the relation to beam length.
8A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

with beam (length 2/length 1) [%] in all x,y, and zdirections.
Figure 12 clarifies the design of a torsion bar beam suspen-
sion. Where the beam length is 100 µm, longer shorter side of
cross section are 40 and 9 µm, applied torque is350,000 µNµm,
YM is 180 GPa, and PR is 0.3.
The results show that the torsional stiffness constant is
5.773E +006 µNµm/rad, and the maximum shearing stress is
377.209 MPa. The Figure also clarifies the dependence of
cross section of beam on torsion constant and the relation
between the spring constant and shorter/longer side length
is also clarified. As longer/shorter side length ratio increases,
this results in the increase of the spring constant. Figure 13
shows the design of a guided beam hammock suspension.
Where the beam length is 750 µm, the beam width is 5 µm,
the beam thickness is 15 µm, and YM is 180 GPa. The Stiffness
ratio of lateral to vertical motion of hammock suspension is
also clarified.
The results demonstrated that the SC in Xaxis is 3.2 N/m,
the spring constant in Yaxis is 18,000 N/m, and The SC in Z
axis is 28.8 N/m. The lateral shiftiness/out of plane shiftiness
Figure 8: Beam clamped buckling variations versus beam length. (a) Bucklingof a double CB due to a compressive stress. (b) Deflection or buckling of a
double CB due to a compressive stress in relation to beam length.
A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides 9

ratio increases as the beam width to thickness ratio also
increases. Figure 14 shows the design of a crab-leg suspen-
sion and the dependence of the stiffness on thigh section of
the crab leg flexure. Where the shin leg beam length is
150 µm, the shin leg beam width is 10 µm, the thigh leg beam
length is 100 µm, width of thigh leg of beam is 10 µm,
thickness of beam is 5 µm, YM is 180 GPa, and PR is 0.3.
The results assured that the SC of the suspension in X
axis is 746.667 N/m, the SC of the suspension in Yaxis is 1980
N/m, and the SC of the suspension in Zaxis is 89.471 N/m. The
SC decreases linearly in xand ydirections as thigh length
increases. Also the spring constant decreases exponentially
in zdirection as thigh length increases. Figure 15 outlines the
design of a folded flexure suspension and the dependence of
stiffness on ratio of column beam lengths. Where the length
of outer column beam is 100 µm, length of inner column
beam is 100 µm, width of column beam is 5 µm, length of
outer truss beam is 95 µm, length of inner truss beam is
95 µm, width of truss beam is 5 µm, beam thickness is 12 µm,
YM is 160 GPa, and PR is 0.3.
The results clarified that the spring constant of the
suspension in Xaxis is 306.736 N/m, the SC of the suspension
in Yaxis is 489.056 N/m, and the SC of the suspension in Zaxis
is 425.668 N/m. The spring constant decreases linearly for all
x,y,zdirections with the column beam length ratio [inner/
outer].
Figure 9: Beam bent suspension variations versus beam dimensions. (a) Out of plane deflection of a bent beam suspension. (b) Spring constant in
x-direction variations in relation to length/width ratio of beam.
10 A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

3 Conclusions
We have outlined the review study on different guided
beams surface structure mechanics in MEMS by using
MEMSolver simulation. The design of a crab-leg suspension
and the dependence of the stiffness on thigh section of the
crab leg flexure is clarified. The design of a folded flexure
suspension and the dependence of stiffness on ratio of col-
umn beam lengths is demonstrated. Besides the design of a
guided beam hammock suspension, the design of a serpen-
tine beam and the design of a torsion bar beam suspension
are presented. The design of a folded beam suspension for
planar motion, the design of a corner beam or bent beam
suspension, the buckling of a double CB due to a compressive
Figure 10: Plane deflection variations versus folded beam dimensions. (a) Out of plane deflection of a folded beam suspension. (b) Dependence of
stiffness on mid-section of the folded beam.
A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides 11

Figure 11: Plane deflection variations versus short leg of serpentinelength. (a) Out of plane deflection of a serpentine beam suspension. (b) Dependence
of stiffness on short leg of the serpentine beam.
Figure 12: Dependence of cross section of
beam on torsion constant.
12 A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

stress, the deflection of a double CB with a central mass
under its weight, and the bending of a double clamped beam
under its distributed weight are demonstrated.
Author contributions: All the authors have accepted
responsibility for the entire content of this submitted
manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no
conflicts of interest regarding this article.
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Figure 13: Stiffness ratio of lateral to vertical
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Figure 15: Spring constant versus column
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46. Amiri IS, Kuppusamy PG, Rashed ANZ, Jayarajan P,
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47. Amiri IS, Rashed ANZ, Jahan S, Paul BK, Ahmed K. Polar polarization
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48. Sivaranjani S, Sampathkumar A, Rashed ANZ, Sundararajan TVP,
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49. Amiri IS, Rashed ANZ, Yupapin P. High-Speed transmission circuits
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50. Amiri IS, Rashed ANZ, Jahan S, Paul BK, Ahmed K, Yupapin P. Technical
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51. Amiri IS, Rashed ANZ. Signal processing criteria based on electro-optic
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52. Amiri IS, Rashed ANZ, Yupapin P. Pump laser automatic signal control
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53. Amiri IS, Rashed ANZ, Parvez AHMS, Paul BK, Ahmed K. Performance
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54. Amiri IS, Rashed ANZ, Yupapin P. Z Shaped like resonator with crystal
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55. Amiri IS, Rashed ANZ, Yupapin P. Influence of device to device
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56. Eid MMA, Amiri IS, Rashed ANZ, Yupapin P. Dental lasers applications
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57. Amiri IS, Rashed ANZ, Yupapin P. Comparative simulation study of
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61. Amiri IS, Rashed ANZ, Rahman Z, Paul BK, Ahmed K. Conventional/
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63. El-Hageen HM, Alatwi AM, Rashed ANZ. Silicon-Germanium dioxide
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66. El-Hageen HM, Kuppusamy PG, Alatwi AM, Sivaram M, Yasar ZA,
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67. El-Hageen HM, Alatwi AM, Rashed ANZ. High-speed signal processing
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68. El-Hageen HM, Alatwi AM, Rashed ANZ. Laser measured rate
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70. Eid MMA, Rashed ANZ, Shafkat A, Ahmed K. Fabry Perot laser
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75. Alatwi AM, Rashed ANZ. Conventional doped silica/fluoride glass
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16 A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides

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125. Bulbul AAM, Rashed ANZ, El-Hageen HM, Alatwi AM. Design and
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18 A.N.Z. Rashed et al.: Study on integrated optical multimode waveguides