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Super-structured D-shape fibre Bragg grating contact force sensor
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Superstructured DShape sensor
Christopher R. Dennison and Peter M. Wild
Department of Mechanical Engineering, University of Victoria, British Columbia, Canada
Corresponding Author:
Chris R. Dennison
University of Victoria
Department of Mechanical Engineering
P.O. Box 3055
Victoria, B.C.
V8W 3P6
Ph: (250) 853-3198
Fax: (250) 721-6051
e-mail: [email protected]
Abstract
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Super-structured D-shape fibre Bragg grating contact force sensor
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1. Introduction
In-fibre Bragg gratings (FBGs) can be configured as sensors for various parameters
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including displacement [1], strain [2], temperature [3], pressure [3], contact force [4],
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humidity [5], and radiation dose [6] among others. FBGs are an attractive alternative to
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other piezoelectric, resistive or other solid-state sensing technologies because they are:
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small (125 μm diameter and smaller), biocompatible, mechanically compliant, chemically
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inert, resistant to corrosive environments, immune to electromagnetic interference, and
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are capable of simultaneous multi-parameter sensing when suitably configured [1, 7-11].
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Moreover, multiple FBG sensors can be multiplexed along a single optical fibre thereby
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allowing spatially distributed measurements [12]. FBG-based contact force sensors
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typically comprise Bragg gratings in birefringent optical fibre.
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Birefringent fibre has different refractive indices along two orthogonal, or
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principal, directions that are commonly referred to as the slow (higher index) and fast
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(relatively lower index) axes. The difference in the refractive indices along the slow and
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fast axis is termed the birefringence. A Bragg grating in a birefringent fibre reflects two
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spectra, one polarized along the fast axis and one along the slow axis, with centre
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wavelength of each spectrum given by the Bragg condition for reflection [12]. The centre
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wavelength of each spectrum is a function of, among other parameters, the refractive
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index of the fast and slow axis.
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The physical mechanism causing sensitivity to contact force is stress-induced
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changes in birefringence [13] that are governed by the stress-optic effect [14]. Contact
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forces on the fibre induce stresses (or strains), the principal directions of which are
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uniform in the region of the fibre core. However, the magnitudes of the principal stresses
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are a function of the magnitude of the contact force [4]. Therefore, when the contact
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Super-structured D-shape fibre Bragg grating contact force sensor
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force magnitude changes, a predictable change in the principal stress magnitudes and,
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therefore, birefringence is induced. These force-induced changes in birefringence cause
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predictable changes in the spectrum reflected by the Bragg grating, including shifts in the
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Bragg spectra corresponding to the fast and slow axes of the fibre. Conversely, applied
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axial forces/strains and changes in fibre temperature cause relatively lower, in the case of
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certain fibres insignificant, changes in fibre birefringence [12]. Therefore, one advantage
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of Bragg grating sensors in birefringent fibres is that force measurements are not
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confounded by changes in axial strain and temperature. More specifically, D-shaped
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optical fibre has been shown to experience virtually no change in fibre birefringence with
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changes in fibre temperature [4] or changes in axial strain that can result from, among
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other influences, fibre bending [15].
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Bragg gratings in birefringent fibres possess higher sensitivity to contact force
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than gratings in non-birefringent fibres because they are typically constructed with stress
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concentrating features embedded in the fibre clad, or clad geometries, that increase force-
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induced changes in birefringence [12]. These features are generally referred to as stress-
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applying parts. Each commercially available birefringent fibre possesses unique stress-
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applying parts or clad geometry. Therefore, the contact force sensitivity of each type of
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birefringent fibre is unique. Contact force sensitivity is also a function of the orientation
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of the stress-applying parts relative to the direction of load [4]. Therefore, one limitation
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of birefringence-based contact force sensors is the necessity to control fibre orientation
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with respect to the contacting surface applying the force.
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A number of researchers have studied the relationship between contact force and
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birefringence for several types of birefringent fibres. Udd et al. (1996) [16] applied Bragg
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Super-structured D-shape fibre Bragg grating contact force sensor
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gratings in 3M, Fujikura and Corning birefringent fibres to simultaneously measure
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temperature and force-induced axial and transverse strain. Wierzba and Kosmowski
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(2003) [17] applied Side-Hole birefringent fibre to force measurements. Chehura et al.
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(2004) [4] investigated the force sensitivity of D-shape, Elliptical core, TruePhase,
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Panda, Bow Tie and Elliptical clad birefringent fibres as a function of fibre orientation.
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More recently, Abe et al. (2006) [18] measured force using chemically-etched Bow Tie
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and Elliptical clad birefringent fibres to understand the influence that reduction in fibre
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diameter has on fibre birefringence and force sensitivity.
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In all of the studies outlined above, fibre orientation relative to the contact force
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was controlled. However, there are applications for FBG-based contact force
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measurements where fibre orientation is difficult or impossible to control. One example
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application from the authors’ previous work is contact force measurements between the
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articulating surfaces of cartilage within intact human joints such as the hip [19]. Contact
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force measurements over the cartilage surfaces of the hip, and other articular joints, can
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indicate the mechanics of the joint, which is closely linked to the etiology of joint
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degeneration and osteoarthritis [20]. In an intact human hip, the contoured cartilage
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surfaces of the femoral head and acetabulum are covered by a synovial fluid layer and
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sealed by a fibrous joint capsule. The presence of synovial fluid within the joint prevents
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permanent fixation of sensors to cartilage surfaces with conventional adhesives or
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mechanical fixation. Because fixation is impossible, the orientation of fibre-based sensors
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with respect to the cartilage surfaces cannot be controlled. Moreover, because the joint
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capsule surrounds the entire joint and prevents optical access, it is impossible to orient
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Super-structured D-shape fibre Bragg grating contact force sensor
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sensors relative to cartilage surfaces using the techniques applied in the above
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applications of birefringent fibre.
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To avoid the challenge of controlling sensor orientation, a non-birefringent Bragg
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grating contact force sensor, with orientation-independent force sensitivity, was
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developed and applied ex vivo to intact cadaveric hips. This sensor addresses limitations
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of the current standard film-based sensors [21, 22] for contact force measurements in
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joints because it can be implanted while leaving the joint capsule and synovial fluid
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layers intact.
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The contact force sensor comprises a non-birefringent optical fibre containing a 1
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mm FBG that is coaxially encased within a silicone annulus and polyimide sheath
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(outside diameter 240 microns) [19]. Contact forces applied to the outside of the sheath
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cause compressive strain in both the sheath and silicone annulus surrounding the optical
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fibre. Because the silicone annulus is approximately incompressible, the annulus volume
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is conserved by tensile Poisson strain along the axis of the optical fibre containing the
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FBG. Furthermore, because the magnitude of Poisson strain is directly proportional to the
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contact force, Bragg wavelength shifts of the FBG within the non-birefringent fibre are
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directly proportional to contact force.
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This sensor exhibits orientation independence of sensitivity because of its coaxial
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configuration. However, like other sensors based on FBGs written in non-birefringent
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fibre, this sensor is also sensitive to both applied axial strain and temperature changes,
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which are two mechanical parameters that can confound force measurements on the
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contoured surfaces of articular joints. More recently, contact force sensors based on tilted
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Bragg gratings have been proposed that also exhibit orientation independence of
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Super-structured D-shape fibre Bragg grating contact force sensor
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sensitivity [23]. However, these approaches are also susceptible to confounding errors
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associated with axial strain, bending and temperature. (mention limited in multiplexing
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due to high spectral width?)
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The objective of the work reported here is to develop a birefringent Bragg
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grating-based contact force sensor hat addresses limitations of non-birefringent Bragg
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grating sensors, namely co-sensitivity to axial strain and temperature, and one limitation
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of Bragg grating sensors in birefringent fibres: the necessity to control sensor orientation.
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D-shape birefringent fibre is used in the sensor presented because it possesses negligible
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co-sensitivity to temperature and axial strain.
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2. Materials and methods
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2.1 Principles of FBGs, birefringence and D-shape optical fibre
FBGs are formed in optical fibres by creating a periodic variation in the refractive
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index of the fibre core [12, 24]. The length of the FBG and the magnitude and period of
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the variation in the refractive index determine the optical spectrum that is reflected by the
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FBG [12, 25]. When light spanning a broad range of wavelengths travels along the core
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of a non-birefringent fibre and encounters a Bragg grating (Figure 1a), a single-peaked
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spectrum of wavelengths is reflected. This spectrum is centered at the Bragg wavelength,
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B , which is given by:
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B  2n0
(1)
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where, as shown in Figure 1a,  is the spatial-period of the variation in the refractive
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index and, n0 , is the effective refractive index of the fibre core [12]. However, when light
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travels along the core of a birefringent optical fibre and encounters a grating, two single
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peaked spectra are reflected. One spectrum corresponds to light polarized along the slow
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axis (s) and the other corresponds to light polarized along the fast axis (f) (Figure 1b):
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Super-structured D-shape fibre Bragg grating contact force sensor
s  2ns
 f  2 n f
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(2)
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where the subscripts s and f denote the slow and fast axis, respectively. In general,
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because ns and n f differ in a birefringent fibre, the slow and fast axis Bragg wavelengths
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will be different, and the magnitude of the difference is termed the spectral separation.
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The spectral separation between the slow and fast axis Bragg wavelengths, s   f is
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given by:
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s   f  2(ns  n f )  2B
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where the refractive index difference between the slow and fast axis is referred to as the
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birefringence, B .
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(3)
For any optical fibre, the birefringence in the fibre core can be expressed as the
sum of three contributions [26]:
B  ns  n f  BG  BIS  BE
(4)
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where BG is the geometric contribution that, typically, is found only in optical fibres with
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asymmetric or elliptical cores; BIS is the internal stress contribution that, typically, is
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found in optical fibres with internal stress applying parts; and BE is the external
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contribution to fibre birefringence that is typically caused by externally applied contact
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forces. There are two main classes of birefringent fibre that are distinguished by the
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predominant birefringence contribution that exists in the fibre core: either internal stress-
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induced birefringence or geometric birefringence.
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In fibres with internal stress-induced birefringence, the difference in refractive
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index is created primarily by thermal residual stresses that are created within the fibre
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Super-structured D-shape fibre Bragg grating contact force sensor
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when the optical fibre cools from its drawing temperature to ambient temperature.
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Typically, there is no geometric contribution to birefringence. The thermal residual
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stresses are created by different thermal expansion coefficients of the fibre clad, stress
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applying parts which are embedded in the clad, and core. When the fibre, which initially
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has uniform temperature throughout its cross section, cools from its drawing temperature
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to ambient temperature thermal residual stresses evolve because the clad, stress applying
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parts and core undergo differing volume contractions. The configuration of the stress
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applying parts results in a uniform principal stress field in the region of the core, which
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through the stress-optic effect [14] results in a uniform difference in refractive index, or,
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alternatively, uniform BIS throughout the core. In the context of contact force sensing,
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applied contact forces cause changes in the birefringence contribution BE and, therefore,
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the Bragg wavelengths associated with the fast and slow axis of the fibre. The
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mechanisms underlying changes in Bragg wavelengths will be detailed after the
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principles of D-shape fibre are discussed. Examples of fibres with stress-induced
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birefringence include Side-hole, PANDA, and Bow Tie. (Need more refs for this
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paragraph).
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Super-structured D-shape fibre Bragg grating contact force sensor
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Figure 1: a) schematic of D-shape fibre showing the clad, core and Bragg grating
comprising regions of modified refractive index, n spaced with period, Λ. b) schematic of
D-shape fibre showing variable names assigned to nominal major and minor outside
diameters of fibre (Dmajor and Dminor) and minimum core offset from clad, r; major and
minor diameters of the elliptical core (dmajor and dminor); and the directions of the fibre
axis, slow axis and fast axis.
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D-shape optical fibre (Figure 1a and 1b) is one example of the second class of
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birefringent fibre that is based on geometric birefringence. While the physics and
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mathematics behind geometric birefringence is beyond the scope of this work, a
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qualitative discussion of birefringence in D-shaped fibres with elliptical cores will be
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included.
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The majority of birefringence in D-shaped fibres is caused by the geometry of the
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fibre clad and elliptical core [27]. More specifically, the geometric birefringence of an
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Super-structured D-shape fibre Bragg grating contact force sensor
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optical fibre with an elliptical core is a function of the core location relative to the air
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surrounding the fibre, r (Figure 1b); the major and minor diameters of the elliptical core,
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dmajor and
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through the core; and the refractive indices of the clad, core and air surrounding the fibre
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[28]. The internal stress contribution to birefringence is negligible at the wavelengths
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used for Bragg grating sensing applications [29]. When the major axis of the fibre core is
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aligned as shown in Figure 1b, the fast axis of the fibre is normal to the flat-side of the D-
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shape clad (Figure 1b). The slow axis is orthogonal to the fast axis and is aligned with the
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major axis of the elliptical core (Figure 1b). In the context of force sensing, applied
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forces cause changes in the birefringence contribution, BE and, therefore, the Bragg
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wavelengths of the fast and slow axes.
dminor, respectively (Figure 1b); the wavelength of the light propagating
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There are two mathematical formulations that can be used to calculate changes in
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the Bragg wavelengths of the fast and slow axes as a function of applied contact forces.
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The first, or stress-optic, formulation relates changes in contact force-induced principal
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stresses to changes in fibre birefringence. The changes in birefringence can then be used
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to calculate spectral separation (Equation 3).
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The second, or strain-optic, formulation can calculate both spectral separation and
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the magnitudes of the Bragg wavelengths for the fast and slow axes. This formulation
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relates changes in force-induced strains to changes in the Bragg wavelength of the fast
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axis and slow axis, which can then be used to calculate spectral separation. Because
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experimental data for the D-shape sensor will comprise fast axis, slow axis and spectral
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separation data, we will apply the strain-optic formulation because it can be used to
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calculate Bragg wavelengths for the fast and slow axes and spectral separation.
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Super-structured D-shape fibre Bragg grating contact force sensor
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2.2 Strain-optic principles
As mentioned above, the centre wavelength of a Bragg grating changes as a function of
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the mechanical strains in the core of the optical fibre. In the case of birefringent optical
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fibre, the changes in the Bragg wavelength of the light polarized along the slow axis (x-
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axis) and fast axis (y-axis) are given by:


n2
s  s  z  s  pxz z  pxx  x  pxy y  
2


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

n 2f
 f   f  z   pyz z  pyx x  pyy y  
2


(5)
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where  denotes a change in Bragg wavelength;  denotes mechanical strain with the
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subscripts referencing the fibre coordinate system shown in Figure 1b; and
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pxz  pxy  p yx  p yz  0.252 and pxx  p yy  0.113 are strain-optic constants [12]. Note
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that the subscripts on the photoelastic constants are referred to the fibre coordinate
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system. To calculate the Bragg wavelength of the fast and slow axis for any state of strain
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the changes in Bragg wavelength (Equations 5) are added to the initial Bragg
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wavelengths (i.e. the Bragg wavelengths when the fibre is unstrained).
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To predict the contact force-induced changes in Bragg wavelength for the sensor
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presented in this work the equations relating contact force and mechanical strain must be
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established. To establish the equations, the sensor geometry and orientation to contact
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force is presented.
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2.3 Super-structured D-shape fibre sensor and force/strain models
As shown in Figure 2a, the contact force sensor consists of a D-shaped birefringent fibre
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around which there is a super-structure that both transmits contact forces to the fibre and
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maintains the orientation of the sensor with respect to the contact force (Figure 2b). The
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super-structured D-shape fibre sensor, hereafter referred to as the sensor, comprises an
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Super-structured D-shape fibre Bragg grating contact force sensor
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outside Polyimide© sheath (SmallParts Inc., Miami FL, Ds = 165 micron, tW = 18 micron
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nominally) that contains a Nitinol© wire (Need the manuf. Details, Dw = 51 micron
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nominally) and a D-shape birefringent fibre (KVH Industries Inc., Middletown RI, Dmajor
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= 125 micron, Dminor = 76 micron, r = 14 micron, dmajor = 8 micron, dminor = 5 micron)
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with a 5 mm Bragg grating photo-inscribed in the fibre core (TechnicaSA, Beijing China,
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the grating details). The length, along the fibre axis (Figure 1b), of the Polyimide©
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sheath and Nitinol© wire is 12 mm at the center of which is the 5 mm Bragg. The
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clearance space (Figure 2a) between the sheath, wire and fibre is filled with a compliant
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adhesive (Dow Corning®, Midland MI, silicone 3-1753) that serves to maintain the
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alignment of the features comprising the super-structure. A Polyimide© alignment
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feature (Figure 2a) is affixed to the outside of the Polyimide© sheath by submerging the
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sheath in a heat curable Polyimide© solution (HD MicroSystems™, Parlin NJ, solution
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PI-2525) and curing it into the geometry shown. The length of the alignment feature is
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nominally the same as the length of the sheath (i.e. 12 mm). The width of the alignment
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feature was XXX (I actually made a bunch of them, how can I convey this).
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Contact forces, F, are applied to the outside of the sheath and are aligned with the
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internal features (e.g. wire and fibre) as shown in Figure 2b (alignment feature omitted
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for clarity). The upper force is transmitted through the sheath, wire, and onto the fibre at
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contact 1. The lower force is transmitted through the sheath and onto the fibre at contact
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2.
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The key difference between the sensor shown in Figure 2 and force sensors based
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on bare (i.e. not superstructured) D-shape fibre [4] is that the contact force transmitted to
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the flat side of the D-shape fibre of the super-structured sensor (Figure 2b) is
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Super-structured D-shape fibre Bragg grating contact force sensor
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concentrated by the wire. Conversely, force sensors using bare D-shape fibre distribute
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forces over the entire flat side of the D-shape clad (Figure 1). As will be shown,
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concentrating the contact force onto the fibre through the wire will result in increased
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force sensitivity of the super-structured D-shape sensor relative to force sensors based on
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bare D-shape fibre. This increase in force sensitivity is predicted by the equations,
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developed below, relating contact force to strain in the fibre core.
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Super-structured D-shape fibre Bragg grating contact force sensor
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Figure 2: a) schematic of super-structured D-shape sensor cross section showing
Polyimide© sheath, Nitinol© wire, D-shape fibre and Polyimide© alignment feature. b)
contact forces, F are applied to the outside of the sheath and are aligned with the internal
features of the sensor as shown. The upper contact force is transmitted through the
sheath, to the wire and to the fibre at contact 1. The lower force is transmitted through the
sheath to the fibre at contact 2.
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A plane elasticity model is applied to calculate the stresses and strains created in
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the D-shape fibre core as a result of applied contact forces. To calculate the stresses and
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strains, the D-shape fibre is approximated as an elastic half space [30] that is subjected to
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contact forces of magnitude F (N/mm, normalized to sensor length along the fibre axis).
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One of the key assumptions associated with elastic half space theory is that
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stresses and strains vanish at distances infinitely far from the point of force application.
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In the case of D-shape fibre, which is finite in size, this assumption cannot be satisfied.
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To address this seemingly irreconcilable difference between the boundary conditions of
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half space theory and the D-shape fibre sensor a composite half space model is proposed
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that accounts for the shape and boundary conditions of the super-structured D-shape fibre
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sensor.
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As shown in Figure 3a, elastic half space formulations of contact consist of a
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contact force applied at the origin of a large elastic continuum [30]. To satisfy boundary
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conditions, when x and z are large all stresses and strains vanish and static equilibrium is
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achieved by internal stresses that balance the external force. The stresses at any location
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in the elastic continuum are:
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Super-structured D-shape fibre Bragg grating contact force sensor
x2y
x  
 (x 2 +y 2 ) 2
2F
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y3
 (x 2 +y 2 ) 2
y  
2F
 xy  
2F
(6)
xy 2
 (x 2 +y 2 ) 2
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where  x ,  y ,  xy are the normal stresses aligned with the x and y directions and the
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shear stress that shears the xy plane [30].
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To address the difference in boundary conditions between the elastic half space
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(Figure 3a) and the D-shape fibre sensor (Figure 3b) Equations 6 are adapted to formulate
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the stresses for the composite half space.
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Figure 3: a) schematic showing contact force, F applied to elastic half space with zero
stress boundary condition at regions far from co-ordinate system origin. b) schematic
showing D-shape fibre cross section as a composite elastic half space. The core of the
fibre is located relative to two co-ordinate systems with origins at contact 1 and contact 2.
The co-ordinate of the core relative to contact 1 and contact 2 is x1, y1 and x’2, y’2,
respectively. c) schematic representation of half space model for a bare D-shape fibre
subjected to distributed contact force over the flat side of the clad (pressure, P) and
concentrated contact force on the round side of the clad.
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The sensor (Figure 2) is approximated using the boundary conditions shown in
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Figure 3b. We assume that the contact forces applied to the top and bottom of the sheath
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Super-structured D-shape fibre Bragg grating contact force sensor
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(Figure 2b) are transmitted to the D-shape fibre. To calculate the stresses at any location
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in the D-shape fibre, we assume that the state of stress in the fibre is due to a stress field,
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given by Equations 6 due to the contact 1 and a stress field, also given by equations like
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6, due to contact 2. Because we assume linear elasticity, we apply superposition to find
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the total state of stress due to these contacts as:
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x  
x 22 y'2 
2 F  x12 y1
+
  (x12 +y12 ) 2 (x 22 +y'22 ) 2 
y  

y'32
2 F  y13
+
 2 2 2
2
2 2 
  (x1 +y1 ) (x 2 +y'2 ) 
 xy  
x 2 y'22 
2 F  x1 y12
+
  (x12 +y12 ) 2 (x 22 +y'22 ) 2 
(7)
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In the region of the centre of the fibre core, where the majority of light is transmitted, the
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dimensions x are small relative to the dimensions y and Equations 7 can be simplified as:
x  0
y  
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2F  1
1 
  
  y1 y'2 
(8)
 xy  0
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Based on Equations 8, in the region of the centre of the fibre core there is negligible shear
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and there is one principal stress. The simple state of stress can be used with Hooke’s Law
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for plane strain [30] to obtain strains:
 (1  ) y
1
(1  2 ) x  (1  ) y 
E
E
(1  2 ) y
1
y 
(1  2 ) y  (1  ) x 
E
E
2(1  )
 xy 
 xy  0
E
x 
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



(9)
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Super-structured D-shape fibre Bragg grating contact force sensor
345
where E and  are the Young’s modulus and Poisson ratio of the silica glass of the fibre,
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77 GPa and 0.17, respectively [29] (check these values and the reference). When
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Equations 7 are substituted into Equations 9, the resulting expressions for strain are
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functions of contact force, F, and the co-ordinates of the core relative to contact 1 and
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contact 2. For the sensor presented in Figure 2, y1 and y’2 are constant: 14 microns and 62
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microns, respectively.
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For the purpose of eventual comparison of the contact force sensor presented in
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this work and contact force sensors based on bare D-shape fibre, we have also developed
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a half-space model to estimate the stresses and strains in a bare D-shape fibre (Figure 3c).
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In this model, contact force on the flat side of the clad is distributed, shown as contact
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pressure, P, while contact force on the opposite side of the fibre remains concentrated.
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We apply similar reasoning as that described for Figure 3b to obtain the stresses in the
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region of the fibre core as:
x 
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y 
P

P

 xy  0
  sin  
  sin   
2F  1 
  y'2 
(10)
359
360
where P is the applied pressure (N/mm2) and α is the included angle (i.e. 2.73 radians for
361
D-shape fibre) that is subtended by the pressure. The boundary conditions shown in
362
Figure 3b do not completely satisfy those of half-space theory because the pressure
363
boundary condition extends to dimensions comparable to the fibre diameter.
Nevertheless, Equations 10 can serve to approximate the state of stress and be used to
17
Super-structured D-shape fibre Bragg grating contact force sensor
364
365
gain insight into increases/decreases in sensitivity between the sensor presented here and
366
those based on bare D-shape fibre. The state of stress given by Equations 10 can be
367
substituted into Hooke’s Law for plane strain to obtain the strains in a similar form to
368
those presented as Equations 9.
369
The force-induced strains for the sensor presented here (Equations 9), and those
370
for a bare D-shape sensor, can then be substituted into Equations 5 to calculate the Bragg
371
wavelength shifts for the fast and slow axis for the sensor presented in this work and that
372
of a bare D-shape fibre subjected to contact force.
373
The process of Bragg wavelength shift calculation described above was
374
completed for contact forces ranging from 0 N/mm to 2 N/mm to predict force
375
sensitivity. Equations 5 were also used to calculate the axial strain sensitivity, for the
376
contact force sensor, as described below.
377
As will be described in detail below, the contact force sensor was tested
378
experimentally for sensitivity to axial strain. The experimental set-up for this test
379
consisted of the force sensor affixed to the top of an aluminum cantilever. Prescribed
380
deflections were applied to the cantilever, and the fast and slow axis Bragg wavelengths
381
were monitored. To estimate axial strain at the sensor, equations from classical beam
382
theory [31] were used to calculate the axial strain at the location along the beam of the
383
force sensor. To verify that results from this experiment agreed with strain-optic theory
384
and the known behavior of D-shape fibres subjected to axial strain [15], the axial strains,
385
and corresponding transverse Poisson strains, for the cantilever beam were used with
18
Super-structured D-shape fibre Bragg grating contact force sensor
386
Equations 5 to predict the Bragg wavelength shifts. Verification involved comparison of
387
the experimentally measured wavelength shifts to the strain-optic model predicted
388
wavelength shifts.
389
2.4 Sensor calibration protocols
390
A single force sensor was constructed and calibrated for sensitivity to contact-force, axial
391
strain and temperature in three separate calibration protocols.
392
The apparatus used for force calibration was identical to that used in a previous
393
study (Figure 4a) [19]. The force sensor and a support cylinder of identical outside
394
diameter were subjected to contact force between two metrology gauge blocks (Class 0,
395
24.1 mm × 24.1 mm, steel, Mitutoyo Can., Toronto, ON). The bottom gauge block
396
supported the force sensor and support cylinder and the top block applied contact forces.
397
Both gauge blocks were constrained by guide blocks allowing motion only in the
398
direction of load application.
399
Contact forces were applied by compressing a calibrated spring with a manual
400
screw-follower (not shown in Figure 4a). These forces were transmitted through a pre-
401
calibrated load cell (445 N capacity, ±0.1% FS non-repeatability, Futek Inc., Irvine, CA),
402
fixed to the top gauge block, connected to a data-acquisition system implemented in
403
LabView™ (version 8, National Instruments Inc., Austin, TX). The force was transmitted
404
through the top gauge block, through the sensor and support cylinder and finally to the
405
bottom gauge block. Applied contact forces in all calibrations ranged from 0 N mm−1 to,
406
nominally, 2 N mm−1. This range of forces was chosen because it brackets the range of
407
forces experienced in the hip during walking [57]
19
Super-structured D-shape fibre Bragg grating contact force sensor
408
409
410
411
412
413
414
Figure 4: a) schematic showing relevant features of contact force calibration apparatus.
Force is applied by compressing a calibrated spring that is in contact with a load cell and
top gauge block. Forces are transmitted through the force sensor and support cylinder,
and to the bottom gauge block. b) schematic showing forces applied to force sensor
without alignment feature, and c) with alignment feature.
415
To allow comparison of sensor sensitivity for a sensor without and with an
416
alignment feature, calibration was completed first for a sensor without an alignment
417
feature (Figure 4b), and subsequently with the alignment feature (Figure 4c). For the
418
sensor without the alignment feature, sensor orientation was controlled with an external
419
fibre rotator while contact forces were applied. The alignment feature was then added to
420
the sensor and calibration was repeated without external orientation control. One
421
calibration was performed for the sensor without an alignment feature, while three were
422
performed for the sensor with the alignment feature. Bragg wavelength shifts of the fast
20
Super-structured D-shape fibre Bragg grating contact force sensor
423
and slow axis were measured using an optical spectrum analyzer (ANDO AQ6331,
424
Tokyo, Japan) as described below.
425
Bragg wavelengths were demodulated by directing light from a broad C-band
426
light source (AFC-BBS1550, Milpitas, California) into a linear polarizer (PR 2000, JDS
427
Uniphase, Milpitas, California) and then into one of the input channels of a 3 dB optical
428
coupler (Blue Road Research). The light was then directed via the coupler to the FBG in
429
the test fiber, and the reflected spectrum was directed back through the optical coupler
430
and into the optical spectrum analyzer. The polarization of the light was adjusted to
431
illuminate either the fast or the slow axis of the fiber by using the tuning facilities of the
432
linear polarizer. The demodulation scheme described is similar to that presented by
433
numerous researchers [3, 32, 33].
434
The sensitivity to force, for both the slow and the fast axes, was calculated by
435
using linear regression as the slope in the recorded data (slope ± standard deviation for
436
slope), in terms of wavelength shift versus applied force. Sensitivity was also calculated
437
based on the spectral separation of the slow and fast axes (i.e., the difference between the
438
slow and the fast axis sensitivities).
439
In the axial strain protocol, the force sensor was affixed to the top surface of an
440
aluminum cantilever (Figure 5, w = 25 mm, h = 0.25 mm, L = 45 mm, and x”s = 10 mm)
441
using a UV curable adhesive (Norland Products Inc., Cranbury NJ, Blocking adhesive
442
107). To create axial strains in the cantilever, prescribed deflections, δ, were created at
443
the end of the cantilever. The prescribed deflections were measured using a mechanical
444
indicator (Mitutoyo Corp., JP, Indicator 2046F, 10 mm range), while Bragg wavelengths
445
of the fast and slow axis were recorded. As described in Section 2.3, the axial strains
21
Super-structured D-shape fibre Bragg grating contact force sensor
446
transmitted to the sensor from the cantilever were estimated from the deflections using
447
classical beam equations [31] by assuming that the tensile strains at the sensor/beam
448
interface (Section A-A, Figure 5) were transmitted to the sensor. The estimated axial
449
strains ranged from 0 με to 70 με. The protocol was conducted twice. The sensitivity to
450
strain, for both the slow and the fast axes and based on spectral separation, was calculated
451
by using linear regression as the slope in the recorded data, in terms of wavelength shift
452
versus strain.
453
454
455
456
457
458
Figure 5: schematic showing relevant feature axial strain calibration apparatus. The force
sensor was affixed to the top of the cantilever (Section A-A). Prescribed deflections were
applied to the end of the cantilever (x” = L).
459
Controls Inc., Winona MN, Model CascadeTEK, controller series 982). The force sensor
460
and a pre-calibrated thermocouple (Omega Engineering Inc., Stamford CT, Type-T),
461
interrogated with a thermocouple amplifier (Omega Engineering, Super MCJ), were fixed
462
to a glass slide (VWR International™, 76 mm by 26 mm by 1 mm) and placed within the
Temperature calibration was performed within a controllable oven (Watlow
22
Super-structured D-shape fibre Bragg grating contact force sensor
463
oven. The temperature, as reported by the thermocouple, of the force sensor was
464
increased from 30 ˚C to 60 ˚C, in 5 ˚C increments, while the Bragg wavelengths of the
465
fast and slow axis were recorded. This temperature range was chosen because it spans a
466
larger range than that experienced in most ex vivo experiments with cadaveric material.
467
temperature The protocol was repeated three times. The sensitivity to temperature, for
468
both the slow and the fast axes and based on spectral separation, was calculated by using
469
linear regression as the slope in the recorded data, in terms of wavelength shift versus
470
temperature.
471
472
473
3. Results
Figure 6 shows spectra recorded using the optical spectrum analyzer for both the fast and
474
slow axis. As shown, the Bragg spectra of both the fast and slow axis are symmetric
475
about their initial Bragg wavelength when the sensor is not subjected to contact force. As
476
force is applied to the sensor, both the fast and slow axis Bragg wavelengths shift to
477
longer wavelengths while retaining symmetry. The symmetry exhibited by the fast and
478
slow axis spectrums, when the sensor is subjected to force, indicates that the strains
479
experienced along the Bragg grating are uniform along the grating length. Conversely,
480
non-uniform strains over the grating length would manifest in increased width, along the
481
wavelength axis, of the Bragg spectrums and decreased peak light intensity [25]. As
482
shown in Figure 6, the slow axis Bragg wavelength experiences greater shift than the fast
483
axis. Therefore, as described below, the slow axis Bragg wavelength exhibits greater
484
sensitivity to contact force than the fast axis (Figure 7 and Table 1).
485
23
Super-structured D-shape fibre Bragg grating contact force sensor
486
487
488
489
490
491
492
493
Figure 6: Bragg spectra for the fast and slow axis recorded using the optical spectrum
analyzer. Contact forces applied to sensor cause shifts to longer Bragg wavelengths. The
symmetry of the spectra indicate that strains along grating are uniform while the relative
shifts of the slow and fast axis indicate that the slow axis has greater force sensitivity
than the fast axis.
494
wavelengths obtained from the half-space/strain-optic model, while Figure 7b shows the
495
measured Bragg wavelength shifts for the contact force sensor without the alignment
496
feature. As shown, for the super-structured sensor, the model predicted (Figure 7a) and
497
measured (Figure 7b) slopes, or sensitivities to contact force, match to within 1.8 % and
Figure 7a shows the predicted wavelength shifts of the fast and slow axis Bragg
24
Super-structured D-shape fibre Bragg grating contact force sensor
498
13.2 % (relative to experimental slope) for the slow and fast axis, respectively. Relative
499
to a bare D-shape sensor, the modeled slow and fast axis sensitivities for the super-
500
structured sensor are 125.4 % and 3.5% greater than those modeled for a bare D-shape
501
sensor.
502
25
Super-structured D-shape fibre Bragg grating contact force sensor
503
504
505
506
507
508
509
510
511
512
513
514
515
Figure 7: a) Predicted sensitivities of the fast and slow axis Bragg wavelengths obtained
from the half-space/strain-optic model for both the super-structured sensor and a bare Dshape sensor. b) measured wavelength shifts of the fast and slow axis Bragg wavelengths
for the sensor without the alignment feature. Sensitivities are reported as slope from
regressions (dashed lines). Error bars shown are not clearly visible at given scale but
convey non-repeatability of contact force measurements (±0.02 N/mm) and observed
non-repeatability of wavelength shift measurements (mean of ±1 pm for all data points
presented) from optical spectrum analyzer.
Table 1: Summary of experimental results for slow and fast axis sensitivity to contact
force, axial strain and temperature of contact force sensor. Sensitivities reported obtained
using linear regression and are reported as the best-fit slope ± the standard deviation on
slope. Correlation coefficients, r2, also obtained from regression calculations.
Sensitivity (pm/measurand unit)
Calibration protocol
Measurand unit
Trial
Slow axis
r†
2
Fast axis
r†
2
Spectral separation
force, without
alignment feature
(N/mm)
1
255.9±21.1
0.96
61.66±7.98
0.91
194.2
force, with alignment
feature
(N/mm)
1
2
3
mean
228.6±3.98
232.1±4.30
234.3±4.06
231.7
0.99
0.99
0.99
0.99
62.42±3.71
62.48±4.05
61.57±5.36
62.16
0.98
0.98
0.96
0.97
166.2
169.6
172.7
169.5
axial strain, with
alignment feature
(micro-strain)
1
2
mean
1.100±0.0786
1.130±0.0827
1.115
0.97
0.96
0.97
1.091±0.0996
1.129±0.0240
1.11
0.95
0.99
0.97
0.009‡
0.001‡
0.005‡
degree Celsius
1
2
3
mean
8.435±0.156
8.451±0.126
8.408±0.144
8.431
0.99
0.99
0.99
0.99
9.577±0.310
9.527±0.309
9.548±0.310
9.551
0.99
0.99
0.99
0.99
-1.142
-1.076
-1.140
-1.119
temperature, with
alignment feature
notes:
516
517
518
† r2 is abbreviated notation for linear correlation coefficient from least squares regression
‡ only one significant digit reported because of relatively low magnitude compared to slow and fast axis sensitivity
Figure 8 shows typical measured Bragg wavelength shifts for the contact force
519
sensor with the alignment feature (trial 1, results also noted in Table 1). As shown by the
520
results in Table 1 (force, with alignment feature), both the slow and fast axis sensitivities
521
are nearly constant over the three trials in which the sensor aligns itself (experimental
522
configuration shown in Figure 4c. The lowest and highest sensitivities over the three
523
trials deviate from the tabulated mean sensitivities for the slow and fast axis by only
524
-1.3% and 1.1% for the slow axis and -0.9% and 0.5% for the fast axis (Table 1). The
26
Super-structured D-shape fibre Bragg grating contact force sensor
525
lowest and highest spectral separation sensitivities deviate from the mean (i.e. 169.5
526
pm/(N/mm)) by only -1.9% and 1.9% (Table 1).
527
528
529
530
531
532
533
534
Figure 8: measured wavelength shifts of fast and slow axis Bragg wavelengths for contact
force sensor with alignment feature (trial 1 in Table 1). Error bars not clearly visible at
given scale but convey non-repeatability of measurements conveyed in caption for Figure
7b.
Figure 9a shows the strain/strain-optic model predicted wavelength shifts for the
535
fast and slow axis as a function of axial strain along the Bragg grating. As shown, the
536
model predicted sensitivities of the fast and slow axis differ by only 0.008% of the mean
537
sensitivity of the fast and slow axis (i.e.1.24475 pm/microstrain). The measured changes
538
in Bragg wavelength versus axial strain (Figure 9b) also show nearly identical sensitivity
539
of the fast and slow axis (also shown in Table 1, axial strain, with alignment feature). The
540
mean difference between the fast and slow axis sensitivities reported for trial 1 and 2,
541
shown in Table 1, is only 0.18%.
542
27
Super-structured D-shape fibre Bragg grating contact force sensor
543
544
545
546
547
548
549
Figure 9: a) Predicted sensitivities to axial strain of the fast and slow axis Bragg
wavelengths obtained from the strain/strain-optic model. b) measured (trial 2) wavelength
shifts of the fast and slow axis Bragg wavelengths for the sensor with the alignment
feature. Sensitivities are reported as slope from regressions (dashed lines). Error bars
shown are not clearly visible at given scale but convey non-repeatability of axial strain
values (±1.69 micro-strain) and observed non-repeatability of wavelength shift
28
Super-structured D-shape fibre Bragg grating contact force sensor
550
551
552
553
measurements (mean of ±1 pm for all data points presented) from optical spectrum
analyzer.
554
protocol (Table 1, temperature, with alignment feature). Like the results for axial strain
555
calibration, the fast and slow axis sensitivities to temperature were nearly identical. For
556
example, as shown by the results based on spectral separation (Table 1), the spectral
557
separation sensitivity was only 12.4 % of the mean of the fast and slow axis sensitivities
558
(i.e. 8.99 pm/degree Celsius).
559
560
561
562
563
564
565
566
567
568
569
Figure 10 shows typical results obtained during the temperature calibration
Figure 10: measured (trial 1) fast and slow axis Bragg wavelengths versus temperature
for the contact force sensor with the alignment feature. Sensitivities are reported as slope
from regressions (dashed lines). Error bars shown are not clearly visible at given scale
but convey non-repeatability of thermocouple measurements (±0.1 degrees Celsius) and
observed non-repeatability of wavelength shift measurements (mean of ±1 pm for all data
points presented) from optical spectrum analyzer.
29
Super-structured D-shape fibre Bragg grating contact force sensor
570
571
4. Discussion
The contact force sensor presented in this work has desirable characteristics when
572
compared to other birefringence-based force sensors and the authors’ previous work [19]
573
including: increased sensitivity to contact force relative to bare-Dshape fibre and other
574
birefringent fibres; near-constant force sensitivity without external orientation control;
575
and negligible co-sensitivity to extraneous axial strain and temperature changes. Insights
576
gained from the half-space/strain-optic model, with reference to sensor design features,
577
can be used to elucidate the causes of these benefits.
578
As shown by the results plotted in Figure 7, the slow axis Bragg wavelength of
579
the super-structured sensor has sensitivity 125.4 % greater than the slow axis of a bare D-
580
shape fibre. Conversely, the fast axis sensitivities of the super-structured and bare D-
581
shape sensors are nearly identical, and differ by only 3.5%. The principal cause of the
582
increase in slow axis sensitivity is a 273% increase, relative to the case of bare D-shape
583
fibre, in compressive stresses,  y , that result from concentrating contact forces at contact
584
1 with the Nitinol© wire. Conversely, fast axis sensitivities of the super-structured and
585
bare D-shape sensors match to within 3.5% because the state of stresses in the two
586
sensors (Equations 8 for super-structured and Equations 10 for bare fibre) lead to
587
mechanical strains in the core that cause nearly identical fast axis wavelength shifts for a
588
given contact force.
589
Experimental measurements reinforce the model-predicted increases in sensitivity
590
for the slow axis of the super-structured sensor. The half-space/strain-optic model
591
predicted slow axis sensitivity is 260.6 pm/(N/mm), which is 45% higher than previously
592
reported maximum slow axis sensitivity for bare D-shape fibre (i.e. 180 pm/(N/mm) [4]).
593
The model-predicted slow axis sensitivity of the bare D-shape fibre is 115.6 pm/(N/mm),
30
Super-structured D-shape fibre Bragg grating contact force sensor
594
which is much lower than the previously sensitivity (180 pm/(N/mm)). We suspect the
595
primary cause for the discrepancy between the model predicted and previously reported
596
slow axis sensitivity is differences in boundary conditions for compressive stresses,  y ,
597
between the half-space model and actual bare D-shape sensors. As shown in Figure 3c,
598
compressive stresses are equivalent to the contact pressure, P, along the entire flat of the
599
D-shape clad, which contradicts one of the assumptions of half-space theory: stresses
600
vanish at regions remote from the center of contact.
601
602
Experimental measurements also reinforce model predicted values for the fast
axis Bragg wavelength for both the super-structured and bare D-shape sensors.
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
5. Conclusions
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Super-structured D-shape fibre Bragg grating contact force sensor
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Super-structured D-shape fibre Bragg grating contact force sensor
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33