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Proceedings ArticleDOI

Parameter-tolerant design of high contrast gratings

TL;DR: The tolerances of the grating dimensions have been precisely studied to develop a robust optimization algorithm with which high contrast gratings, exhibiting not only a high efficiency but also large tolerance values, could be designed.
Abstract: This work is devoted to the design of high contrast grating mirrors taking into account the technological constraints and tolerance of fabrication First, a global optimization algorithm has been combined to a numerical analysis of grating structures (RCWA) to automatically design HCG mirrors Then, the tolerances of the grating dimensions have been precisely studied to develop a robust optimization algorithm with which high contrast gratings, exhibiting not only a high efficiency but also large tolerance values, could be designed Finally, several structures integrating previously designed HCGs has been simulated to validate and illustrate the interest of such gratings

Summary (3 min read)

1. INTRODUCTION

  • High contrast gratings (HCG) are usually 1D photonic structures made of two materials with a large index step (∼ 2).
  • The coupling between the propagating modes and the outside waves as well as the resonances of the modes between the two slab interfaces can be tailored by adjusting the grating dimensions and the index contrast to give a wide range of optical properties.
  • 13 HCG mirrors have originally been developed for VCSEL devices : with only two subwavelength layers, they can surpass Bragg reflectors performances in term of high reflectivity.
  • The tolerance with respect to the errors of fabrication is an important aspect and has to be taken into account during the design of the structure to achieve a robust and efficient mirror.

2. HCG-VCSEL FABRICATION

  • The device is composed of five GaInAsSb quantum wells in a cavity composed by a Bragg mirror made from 24 pairs of AlAsSb/GaSb20 and high contrast grating on top .
  • To enhance the device performances, an electro-optic confinement solution has been introduce with the use of an oxide aperture.
  • Since oxidation of the antimonide alloy (AlSb) has not returned encouraging results for its application as a confinement layer, a metamorphic approach with oxidation of AlGaAs layer is used.
  • 21 A GaAs layer is thus grown on top of the Sb-based cavity of the VCSEL and the HCG can be obtained thanks to the high contrast between AlOx low index material (n ∼1.7) with low absorption22 and high index of GaAs (n ∼ 3.3).
  • This technological approach presents the advantage of using mature industrial process such as reactive ion etching (RIE) for the grating fabrication and wet oxidation for the AlOx layer.

3. VCSEL MIRROR REQUIREMENTS

  • Compared to other semiconductor lasers, VCSEL devices have a very thin gain region and require high Q cavities made from high reflectivity mirrors higher than 99.5% with typical bandwidth of 150 nm in the mid infrared.
  • Besides the mirror effect, a polarization selectivity is wanted to prevent polarization instabilities of the emitted beam which is an well known problem in VCSEL structures due to their circular symmetry.
  • The high reflectivity of 99.5% is chosen for the transverse magnetic (TM) polarization which is perpendicular to the grating slabs.
  • The transverse electric polarization (TE) is kept lower than 90% for the whole high reflectivity bandwidth to prevent any lasing for this polarization mode.

4. TECHNOLOGICAL CONSTRAINTS

  • The fabrication of the HCG structure is constrained by technological considerations.
  • The aluminum oxide layer thickness TA is limited between 300 and 400 nm around the optimum 23 (2k−1)λ/4 to prevent optical losses and delamination during the wet oxidation process.
  • The fill factor FF of the grating is limited to values between 35% and 55% by the e-beam lithography process and the RIE process used to ensure a well defined square shape pattern of the grating slabs.
  • Besides boundaries on the HCG dimensions, tolerances on the structure parameters have to be taken into account to ensure that the design is feasible and keep high performances despite the pitfalls of the manufacturing process.
  • The etching of the grating slab is the most critical step in the process flow since it is done without the use of a stop layer.

5. ROBUST OPTIMIZATION ALGORITHM

  • The HCG design can be done with a theoretical analysis of the Bloch mode resonances inside the grating,2 but the authors chose here to solve an inverse problem by computing the reflectivity of the mirror with a numerical analysis and adjust the grating parameters to increase its performances.
  • This method allows to design easily more complex structures which are not ideal HCG gratings and include the previously defined VCSEL requirements during the design process.

5.1 The optimization algorithm

  • An optimization algorithm has been used to automate the search of the most efficient mirror design in a same way that it can be done manually with a study of the evolution of the reflectivity of the mirror with respect to its dimensions.
  • The normalized bandwidth is multiplied by a Gaussian weighted average of the transverse magnetic reflection coefficients RTM of the bandwidth to ensure the centering around λ0.
  • Several optimization algorithm are available to search for the maximum of a function and the choice of the best algorithm is done with respect to the function characteristics.
  • PSO algorithm is based on a set of particles which are potential candidates sharing their knowledge of optimum positions when exploring the search space.
  • Secondly, a local velocity term moves the particle toward the local best position xlp known by the particle p.

5.2 The anti-optimization algorithm

  • Even if the optimization algorithm succeed in finding efficient HCG mirrors, a detailed study of the optimum found by the algorithm can show that this point is not realistic from a fabrication point of view due to its very low tolerance.
  • The vicinity of the point is defined in their case as the space ∆X = {∆Tg,∆FF,∆TA,∆Λ,∆TL} which represents the estimation of the errors that can be made during the fabrication process.
  • Thus, the minimum value of the MF function in the space X ±∆X corresponds to the worst case scenario of fabrication.
  • First, a memoization method has been introduced to store in memory the results of the MF (X) function for each point X met during the optimization.
  • By evaluating the robustness of only the best solutions during the optimization, the PSO algorithm has to compare solutions which have been tested in tolerance and solutions which have not yet been.

6. RESULTING DESIGNS

  • The execution of the robust optimization algorithm has been done to adjust the parameters of the GaAs/AlOx grating structure defined previously.
  • The boundary of the search space and tolerance requirements on the structure dimensions are summarized in Table 1 and come either from technological limitations as described previously for AlOx thickness TA and fill factor FF or chosen arbitrarily from empirical rules.
  • 24 The resulting mirror which dimensions are given in Table 2 exhibits a very high reflectivity and polarization selective bandwidth of 425 nm .
  • The tolerance evaluation of the parameters shows a good robustness with an optimum value well centered within large variation ranges.

7. CONCLUSION

  • An original robust optimization algorithm for high contrast grating design has been presented in this work which searches not only for the best solution but takes also into account the accuracy of the manufacturing technology.
  • The algorithm has shown its ability to design HCG mirrors for a VCSEL application at 2.3 µm while taking into account user-defined technological constraints.
  • The execution of the robust optimization algorithm returns not only efficient mirrors with 99.5% high reflectivity for 425nm large bandwidth but also fulfills the robustness requirements with large tolerance values of more than 10% on the most critical parameters.
  • This design method can be quickly adapted to design other high contrast grating structures for other manufacturing conditions by only adapting the figure of merit and the technological constraints.

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Parameter-tolerant design of high contrast gratings
Christyves Chevallier, Nicolas Fressengeas, Joel Jacquet, Guilhem Almuneau,
Youness Laaroussi, Olivier Gauthier-Lafaye, Laurent Cerutti, Frédéric Genty
To cite this version:
Christyves Chevallier, Nicolas Fressengeas, Joel Jacquet, Guilhem Almuneau, Youness Laaroussi, et
al.. Parameter-tolerant design of high contrast gratings. Photonics West, SPIE, Feb 2015, San Fran-
cisco, United States. pp.93720N-93720N-8, �10.1117/12.2081595�. �hal-01132683�

Parameter-tolerant design of high contrast gratings
Christyves Chevallier
a,b
, Nicolas Fressengeas
c
, Joel Jacquet
d
, Guilhem Almuneau
e
, Youness
Laaroussi
e
, Olivier Gauthier-Lafaye
e
, Laurent Cerutti
e
and Fr´ed´eric Genty
a
a
Sup´elec, Laboratoire Mat´eriaux Optiques, Photonique et Syst`emes EA 4423,
2 rue Edouard Belin, Metz, F-57070, France
b
Georgia Tech - CNRS, International Joint Research Lab, UMI 2958,
2-3 rue Marconi, Metz, F-57070, France
c
Universit´e de Lorraine, Laboratoire Mat´eriaux Optiques, Photonique et Syst`emes EA 4423,
2 rue Edouard Belin, Metz, F-57070, France
d
Captoor, F-57070 Chieulles, France
e
Laboratoire d’Analyse et d’Architecture des Syst`emes,
Centre National de la Recherche Scientifique,
7 Avenue du Colonel Roche, F-31077 Toulouse, France
f
Universit´e Montpellier 2; Institut d Electronique du Sud; UMR 5214 CNRS,
860, rue Saint Priest, F-34095 Montpellier Cedex 5, France
ABSTRACT
This work is devoted to the design of high contrast grating mirrors taking into account the technological con-
straints and tolerance of fabrication. First, a global optimization algorithm has been combined to a numerical
analysis of grating structures (RCWA) to automatically design HCG mirrors. Then, the tolerances of the grat-
ing dimensions have been precisely studied to develop a robust optimization algorithm with which high contrast
gratings, exhibiting not only a high efficiency but also large tolerance values, could be designed. Finally, several
structures integrating previously designed HCGs has been simulated to validate and illustrate the interest of
such gratings.
Keywords: High contrast grating, Near wavelength diffraction grating, Grating mirror, Robust optimization,
Tolerance analysis
1. INTRODUCTION
High contrast gratings (HCG) are usually 1D photonic structures made of two materials with a large index step
( 2). When the periodicity is in the range of the wavelength, diffraction is possible only into the 0
th
order and
only a few Bloch modes are propagative and can interact with each others. In the high contrast grating scenario,
the incoming lightwave is normal to the slab and the mode propagation is studied in a perpendicular direction to
the slab plane.
1, 2
The coupling between the propagating modes and the outside waves as well as the resonances
of the modes between the two slab interfaces can be tailored by adjusting the grating dimensions and the index
contrast to give a wide range of optical properties.
3
High contrast gratings have thus become a powerful tool for
many application such as high-Q resonators
4
and highly efficient sensors, planar lenses
5
and beam steering cite,
MEMS,
6, 7
broadband terahertz absorber with large angle tolerance.
8
But the most powerful application of HCG
is their ability to produce high reflectivity and large bandwidth mirrors
9–12
which is for instance a promising
solution for full color reflective display for e-book readers.
13
HCG mirrors have originally been developed for
VCSEL devices : with only two subwavelength layers, they can surpass Bragg reflectors performances in term of
high reflectivity. Moreover, due to their 1D symmetry, HCG can exhibit a high polarization selectivity which is
an important advantage to prevent polarization instabilities in the VCSEL emitted beam.
Further author information: (Send correspondence to Christyves Chevallier)
Christyves Chevallier: E-mail: cchevallier@georgiatech-metz.fr, Telephone: +33(0)387203945

Figure 1. Scheme of the mid infrared VCSEL with a bottom AlAsSb/GaSb DBR mirror and top GaAs/AlO
x
high contrast
grating mirror.
Air
T
g
T
L
n=3.3
Λ
n=1.66
AlOx
GaAs
L
e
L
f
GaAs
T
A
n=3.3
Figure 2. Scheme of the GaAs/AlO
x
mirror. The grating dimensions are adjusted by an optimization algorithm to exhibit
reflectivity higher than 99.5 % for a VCSEL application at 2.3
µ
m.
In this work, a HCG is designed to enhance VCSEL performances in the mid-infrared wavelength range.
For wavelength above 2 µm, VCSELs based on the antimony alloy are a very attractive light source to develop
and enhance numerous applications such as polluting gas sensing. However, due to the large thickness of the
device with mirrors as thick as 9µm, antimony alloy VCSEL emission is still limited at 2.6µm.
14, 15
High contrast
gratings are thus a promising new type of mirror to reduce the structure thickness, introduce polarization stability
and develop devices emitting at larger wavelengths. The design of the HCG is done here by taking into account
VCSEL mirror requirements through the use of an optimization algorithm. Since high contrast grating mirrors
have sub-micrometric dimensions with square-shaped patterns and the VCSELs require high quality mirrors, the
fabrication has to be accurately controlled in the nanometric range.
16
The tolerance with respect to the errors of
fabrication is an important aspect and has to be taken into account during the design of the structure to achieve
a robust and efficient mirror.
17–19
In this paper, we present the design of a GaAs HCG combined with an AlO
x
sublayer as low index material
to replace the top DBR of mid infrared GaSb-based VCSEL. In a first part, an optimization algorithm is used
to find the best dimensions of the HCG structure. Then, the tolerance of the geometrical parameters of the
optimum design with respect to the errors of fabrication are numerically investigated. In a second part, an
anti-optimization algorithm is combined to the optimization process to develop a robust optimization algorithm.
High contrast gratings are thus optimized to exhibit not only high efficiency but also large tolerance values.

Table 1. Boundaries of the search space and tolerance requirements for the robust optimization algorithm.
Parameter Minimum value Maximum value Minimum tolerance
T
g
500 nm 1100 nm T
g
> ±20 nm
F F 0.35 0.55 F F > ±0.02
T
A
300 nm 400 nm T
A
> ±50 nm
Λ 900 nm 1300 nm ∆Λ > ±3 nm
T
L
50 nm 1000 nm T
L
> ±1 nm
2. HCG-VCSEL FABRICATION
For a mid infrared emission between 2 and 3 µm, a VCSEL structure is grown by molecular beam epitaxy.
The device is composed of five GaInAsSb quantum wells in a cavity composed by a Bragg mirror made from
24 pairs of AlAsSb/GaSb
20
and high contrast grating on top (Figure 1). To enhance the device performances,
an electro-optic confinement solution has been introduce with the use of an oxide aperture. Since oxidation of
the antimonide alloy (AlSb) has not returned encouraging results for its application as a confinement layer, a
metamorphic approach with oxidation of AlGaAs layer is used.
21
A GaAs layer is thus grown on top of the
Sb-based cavity of the VCSEL and the HCG can be obtained thanks to the high contrast between AlOx low
index material (n 1.7) with low absorption
22
and high index of GaAs (n 3.3). This technological approach
presents the advantage of using mature industrial process such as reactive ion etching (RIE) for the grating
fabrication and wet oxidation for the AlOx layer. In this configuration, the GaAs grating layer is chosen to be
not completely etched to prevent delamination during the oxidation process and enhance mirror performances
16
as shown on Figure 2. The HCG presented here is thus made from a GaAs grating of thickness T
g
with a fill
factor F F on top of a GaAs sublayer of thickness T
L
and a low index AlOx layer of thickness T
A
.
3. VCSEL MIRROR REQUIREMENTS
Compared to other semiconductor lasers, VCSEL devices have a very thin gain region and require high Q cavities
made from high reflectivity mirrors higher than 99.5% with typical bandwidth of 150 nm in the mid infrared.
Besides the mirror effect, a polarization selectivity is wanted to prevent polarization instabilities of the emitted
beam which is an well known problem in VCSEL structures due to their circular symmetry. The high reflectivity
of 99.5% is chosen for the transverse magnetic (TM) polarization which is perpendicular to the grating slabs.
The transverse electric polarization (TE) is kept lower than 90% for the whole high reflectivity bandwidth to
prevent any lasing for this polarization mode.
4. TECHNOLOGICAL CONSTRAINTS
The fabrication of the HCG structure is constrained by technological considerations. The aluminum oxide layer
thickness T
A
is limited between 300 and 400 nm around the optimum
23
(2k 1)λ/4 to prevent optical losses and
delamination during the wet oxidation process. The fill factor FF of the grating is limited to values between
35% and 55% by the e-beam lithography process and the RIE process used to ensure a well defined square shape
pattern of the grating slabs.
Besides boundaries on the HCG dimensions, tolerances on the structure parameters have to be taken into
account to ensure that the design is feasible and keep high performances despite the pitfalls of the manufacturing
process. The etching of the grating slab is the most critical step in the process flow since it is done without the
use of a stop layer. The tolerances on the parameter T
g
and F F are thus wanted to be larger than T
g
> 20nm
and F F > 2%. The AlOx layer thickness is also a parameter which requires large tolerance values since the
thickness changes during the oxidation process and is not easily predictable. This tolerance is set to T
A
> 50nm
as it can be seen in Table 1 which summarized all the parameters limitations. The other dimensions T
L
or Λ
have been given empirical values but could be linked to technological requirements as well.

5. ROBUST OPTIMIZATION ALGORITHM
The HCG design can be done with a theoretical analysis of the Bloch mode resonances inside the grating,
2
but we
chose here to solve an inverse problem by computing the reflectivity of the mirror with a numerical analysis and
adjust the grating parameters to increase its performances. This method allows to design easily more complex
structures which are not ideal HCG gratings and include the previously defined VCSEL requirements during the
design process.
5.1 The optimization algorithm
An optimization algorithm has been used to automate the search of the most efficient mirror design in a same
way that it can be done manually with a study of the evolution of the reflectivity of the mirror with respect to
its dimensions.
24
To automate the search, optimization algorithms require to describe the problem with the use
of a mathematical function. A figure of merit M F has thus been defined to represent quantitatively the mirror
quality as a VCSEL application point of view :
MF =
λ
λ
0
1
N
λ
2
X
λ=λ
1
R
T M
(λ)g(λ) (1)
The MF function mainly represents the normalized high reflectivity bandwidth of the mirror, defined as
the wavelength range λ = λ
2
λ
1
around λ
0
where the reflectivity is larger than 99.5 % for TM mode and
below 90 % for the TE mode. The normalized bandwidth is multiplied by a Gaussian weighted average of the
transverse magnetic reflection coefficients R
T M
of the bandwidth to ensure the centering around λ
0
. Reflection
spectra of the mirror are computed by rigorous coupled wave analysis (RCWA)
25
for transverse magnetic and
transverse electric polarizations.
Several optimization algorithm are available to search for the maximum of a function and the choice of the
best algorithm is done with respect to the function characteristics. In our case, the MF function has many local
maximas and the use of Newton like optimization algorithm is not possible. The class of global optimization
algorithm is well suited to solve such problems. For instance, simulated annealing,
26
genetic algorithm
27
and
particle swarm
19, 28
have already shown their ability to design HCG efficiently. In this work, a particle swarm
optimization (PSO) algorithm have been chosen due to its ease of use and implementation. PSO algorithm is
based on a set of particles which are potential candidates sharing their knowledge of optimum positions when
exploring the search space. In our case, the search space is defined by the set of design parameters = {Λ, T
g
,
F F , T
A
, T
L
} and one point of this search space defines one design solution. The particles of the swarm move
inside the search space at each iteration with a velocity v
i,p
:
v
i,p
= v
i1,p
+ c
l
(x
l
p
x
i1,p
) + c
g
(x
g
p
x
i1,p
) (2)
The velocity v
i,p
at the iteration i of the particle p of the swarm is composed of 3 terms. Firstly, the inertia
of the particle is taken into account by keeping the velocity of the previous iteration v
i1,p
. Secondly, a local
velocity term moves the particle toward the local best position x
l
p
known by the particle p. Finally, the swarm
concept is created by sharing the best position of all particles thanks to the global best position x
g
p
which creates
a global velocity term. The parameters c
l
and c
g
are two weights for the local and global velocities which are
randomly chosen from a uniform law in the range [0, 2] at each particle move.
29
5.2 The anti-optimization algorithm
Even if the optimization algorithm succeed in finding efficient HCG mirrors, a detailed study of the optimum
found by the algorithm can show that this point is not realistic from a fabrication point of view due to its very
low tolerance.
30
The robustness can be enhanced manually after the optimization but this task becomes difficult
when done simultaneously on several parameters and does not ensure to combine the most efficient and robust
design solution. A robust solution corresponds to a design which keeps high performances for large fabrication
errors, in other terms, which keeps a high value of its figure of merit MF in the vicinity of the point in the search

References
More filters
Proceedings ArticleDOI
06 Aug 2002
TL;DR: A concept for the optimization of nonlinear functions using particle swarm methodology is introduced, and the evolution of several paradigms is outlined, and an implementation of one of the paradigm is discussed.
Abstract: A concept for the optimization of nonlinear functions using particle swarm methodology is introduced. The evolution of several paradigms is outlined, and an implementation of one of the paradigms is discussed. Benchmark testing of the paradigm is described, and applications, including nonlinear function optimization and neural network training, are proposed. The relationships between particle swarm optimization and both artificial life and genetic algorithms are described.

35,104 citations


Additional excerpts

  • ...The parameters cl and cg are two weights for the local and global velocities which are randomly chosen from a uniform law in the range [0, 2] at each particle move.(29)...

    [...]

Journal ArticleDOI
TL;DR: S 4 is described, a free implementation of the Fourier modal method (FMM), which has also been commonly referred to as rigorous coupled wave analysis (RCWA), for simulating electromagnetic propagation through 3D structures with 2D periodicity and the design aspects that allow S 4 to be a flexible platform for these types of simulations are detailed.

619 citations


"Parameter-tolerant design of high c..." refers background in this paper

  • ...But the most powerful application of HCG is their ability to produce high reflectivity and large bandwidth mirrors which is for instance a promising solution for full color reflective display for e-book readers.(13) HCG mirrors have originally been developed for VCSEL devices : with only two subwavelength layers, they can surpass Bragg reflectors performances in term of high reflectivity....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a non-periodic pattern of the grating surface is proposed to give full control over the phase front of reflected light while maintaining a high reflectivity, which could have a substantial impact on a number of applications that depend on low-cost, compact optical components.
Abstract: Sub-wavelength dielectric gratings have emerged recently as a promising alternative to distributed Bragg reflection dielectric stacks for broadband, high-reflectivity filtering applications. Such a grating structure composed of a single dielectric layer with the appropriate patterning can sometimes perform as well as 30 or 40 dielectric distributed Bragg reflection layers, while providing new functionalities such as polarization control and near-field amplification. In this Letter, we introduce an interesting property of grating mirrors that cannot be realized by their distributed Bragg reflection counterpart: we show that a non-periodic patterning of the grating surface can give full control over the phase front of reflected light while maintaining a high reflectivity. This new feature of dielectric gratings allows the creation of miniature planar focusing elements that could have a substantial impact on a number of applications that depend on low-cost, compact optical components, from laser cavities to CD/DVD read/write heads.

561 citations

Journal ArticleDOI
TL;DR: In this article, a novel subwavelength grating that has a very broad reflection spectrum and very high reflectivity is presented. But the design is scalable for different wavelengths. And it does not support monolithic integration of optoelectronic devices at a wide range of wavelengths from visible to far infrared.
Abstract: We report a novel subwavelength grating that has a very broad reflection spectrum and very high reflectivity. The design is scalable for different wavelengths. It facilitates monolithic integration of optoelectronic devices at a wide range of wavelengths from visible to far infrared.

495 citations


"Parameter-tolerant design of high c..." refers methods in this paper

  • ...An optimization algorithm has been used to automate the search of the most efficient mirror design in a same way that it can be done manually with a study of the evolution of the reflectivity of the mirror with respect to its dimensions.(24) To automate the search, optimization algorithms require to describe the problem with the use of a mathematical function....

    [...]

Journal ArticleDOI
TL;DR: High Contrast gratings (HCGs) as mentioned in this paper are a class of planar optics with a large refractive index contrast, which can be designed top-down based on intuitive guidelines.
Abstract: A new class of planar optics has emerged using subwavelength gratings with a large refractive index contrast, herein referred to as high-contrast gratings (HCGs). This seemingly simple structure lends itself to extraordinary properties, which can be designed top-down based on intuitive guidelines. The HCG is a single layer of high-index material that can be as thin as 15% of one wavelength. It can be designed to reflect or transmit nearly completely and with specific optical phase over a broad spectral range and/or various incident beam angles. We present a simple theory providing an intuitive phase selection rule to explain the extraordinary features. Our analytical results agree well not only with numerical simulations but also experimental data. The HCG has made easy fabrication of surface-normal optical devices possible, including vertical-cavity surface-emitting lasers (VCSELs), tunable VCSELs, and tunable filters. HCGs can be designed to result in high-quality-factor (Q) resonators with surface-normal output, which is promising for wafer-scale lasers and optical sensors. Spatially chirped HCGs are shown to be excellent focusing reflectors and lenses with very high numerical apertures. This field has seen rapid advances in experimental demonstrations and theoretical results. We provide an overview of the underlying new physics and the latest results of devices.

451 citations

Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "Parameter-tolerant design of high contrast gratings" ?

This work is devoted to the design of high contrast grating mirrors taking into account the technological constraints and tolerance of fabrication. Then, the tolerances of the grating dimensions have been precisely studied to develop a robust optimization algorithm with which high contrast gratings, exhibiting not only a high efficiency but also large tolerance values, could be designed.