Fizeau Interferometry for Bent Optical Fibers Characterization

Transcription

1 International Journal of Optics and Applications 015, 5(4): DOI: /j.optics Fizeau Interferometry for Bent Optical Fibers Characterization Fouad El-Diasty Department of Physics, Faculty of Science, Ain Shams University, Abbasia, Cairo, Egypt Abstract Bending causes reduction in propagating power by mean of coupling into radiation modes. Through the elastic-optic effect bending produces induced-birefringence. Birefringence disfigures all fiber parameters leading to a substantial mode conversion and pulse delay. Induced-birefringence is one of the key elements needed to use the optical fibers as sensors in variety of advanced applications. Reliable and successful construction and operation of all fiber-optic-based devices requires detailed knowledge of the underlying bending mechanism. The present review provides a comprehensive study for the role of bending on many linear and nonlinear fiber parameters and on all-optical switching. Fizeau interferometry is used to determine fiber refractive index profile and induced-birefringence with very high spatial index resolution. Keywords Optical fiber, Interferometry, Bending 1. Introduction Network designers have self-confident that the use of DWDM (dense wavelength division multiplexing that works at 1480 nm to 165 nm) or CWDM (coarse wavelength division multiplexing that utilizes the 170 nm to 1630 nm spectral band) techniques to achieve greater bandwidth over longer haul information distances is essential. Particularly, as the number of wavelengths increases, mainly at higher bit rates >>10 Gb/s, the chromatic dispersion (CD), polarization mode dispersion (PMD), spectral attenuation profile (SAP) and bending become critical parameters. Chromatic dispersion (CD) and wavelength for zero dispersion, λ zero, describe the variation of the fiber index of refraction with the transmitted wavelengths. Chromatic dispersion generates a broadening for the transmission pulse which creates distortion and increases the bit error rate ratio of the optical system. For instance, the long distance limit may be abridged from 640 km to less than 100 km as the bit rate increases from.5 Gb/s to 10 Gb/s. As the fiber bit rate increases up to 40 Gb/s, the maximum operating distance is 5 km [1]. Furthermore, the use of bent optical fibers as sensors depends on the curvature of a multi-mode optical fiber. Bending causes power propagating in guided modes to be lost by coupling into radiation modes and thus reduces the light power at the fiber output. At a constant bent radius, to * Corresponding author: fdiasty@yahoo.com (Fouad El-Diasty) Published online at Copyright 015 Scientific & Academic Publishing. All Rights Reserved maximize the mode losses, optical fibers with large diameters and low numerical apertures must be used. A decisive interest is that fiber becomes more sensitive to bending as the wavelength, λ, increases especially at λ 165 nm. For instance, a fiber with a bend radius R = 5 mm may result a loss of 0.3 db at 1550 nm. While at 165 nm, the fiber with R = 5 would suffer a large loss of db [1]. Bending of optical fiber creates also birefringence which induces polarization mode dispersion (PMD). The birefringence generates dual refraction indices, one for each orthogonal propagated polarization mode. Birefringence is due to photoelastic effect occurs from induced transverse asymmetric stresses distributed over the bent fiber region and/or to a waveguide geometrical effect arises as an alteration in the fiber cross-section geometry. Nevertheless the contribution from the waveguide geometry effect is 400 times smaller and therefore it can be negligible. The birefringence of the two opposing modes creates a time delay causing the transmission pulse to be broadened. Generally, it can be assumed that at bit rates higher than 10 Gb/s, PMD is one of the critical limiting factors for long length transmission links. All of the physical parameters of fibers used in optical communication systems, such as structure and index profile of the fiber (especially at bending conditions), should be systematically characterized in order to ensure optimum and reliable transmission network performance. Such accurate characterization may also provide information for future upgrade decisions. In the last three decades, the most reliable and accurate technique for fiber investigations is the interferometric one [, 3]. Through this review, the use of multiple-beam Fizeau fringe to resolve and infer the

2 104 Fouad El-Diasty: Fizeau Interferometry for Bent Optical Fibers Characterization structure and index profiles of the optical fibers with high index and spatial resolution is discussed. The deviation of many fiber key parameters together with the variation in refractive index of bent standard single-mode and graded-index fibers are argued.. Interferometric Fiber Characterization The experimental setup of the Fizeau interferometric technique that uses to characterize optical fibers is explained in details elsewhere [4]. A graded-index multi-mode optical fiber (GRIN) with a circular cross-section is introduced in a silvered liquid wedge interferometer. The lower silvered optical flat is just touch the circumference of the fiber. Fig. 1 shows an interferogram in the (x, z) plane for the cross-section of a GRIN optical fiber of radius r f having a cladding of constant refractive index n cl, a graded-index core of variable refractive index n c (r) and radius r c and a central dip of constant refractive index n d with radius r d. Figure 1. Digital micro-interferogram of Fizeau fringes crossing a straight GRIN fiber immersed in liquid where n L < n cl The fiber is immersed in a liquid of refractive index n L very close to n cl. A parallel beam of monochromatic light of wavelength λ is incident normal to the lower component of the wedge interferometer which has a small wedge angle, ε. If the fiber axis is chosen as the z-axis, so the mathematical equation of Fizeau fringes across this type of optical fiber in the region x 1 is taking the form [5]: δδδδ zz xx 1 λ = (nn cccc nn LL ) rr ff xx 1 + nn cc rr cc xx 1 [nn cc (rr dd ) nn dd ] rr dd xx 1 nn cc (rr cc rr dd ) αα cc rr dd xx1 rr cc xx 1 xx 1 + yy rr dd αα cc where nn cc = nn cc (rr dd ) nn cccc and α c is the shaping parameter controlling the shape of the core index profile. zz = λ nn LL dddd (1) fringe spacing and δz is the fringe shift. If there is no core-index central dip in the fiber, therefore rr dd xx 1 = 0 and Eq.(1) will take the form: rr cc xx 1 δδδδ λ = (nn zz xx 1 cccc nn LL ) rr ff xx 1 + nn cc rr cc xx 1 nn cc xx rr cc ααcc 1 + yy αα cc 0 is the dddd () Also, in case of no graded-index core, so the fiber takes the form of a single-mode or multi-mode step-index fibers considering the dip layer as a second core, (r d = r c ), of constant refractive index. In such case, n c = 0, n d = n c and n c (r d ) = n c (r c ) = n cl, therefore, E q. (1) will take the form: δδδδ zz xx 1 λ = (nn cccc nn LL ) rr ff xx 1 + (nn cc nn cccc ) rr cc xx 1 (3)

3 International Journal of Optics and Applications 015, 5(4): The resulting equations furnish the basis for the Fizeau interferometric characterization of structure and index profiles of the different types of optical fibers. 3. Bending Induces Mechanically Perturbation for Photoelastic Fiber Properties Using photo-elasticity with multiple-beam Fizeau-type interferometer, the light is divided into two rays, directing one ray to pass several times through the object under study (active ray), while the other ray (reference ray) is the direct one. The optical path of the active ray changes as it passes through the object. The fringe pattern, which is formed by the interference of the active and reference rays, contains information about the variation of the optical path of the active ray. The Fizeau fringes of a liquid wedge interferometer at transmission are straight-line bright fringes on a dark background. The fringes are parallel to the edge of the wedge (apex) with a constant inter-spacing z. Such fringes have equal optical thickness (the product of refractive index of the object and the object s metric thickness is constant). The variations in the object refractive index profile or its thickness or both produce shifts in the fringe position. The refractive index profile of an optical fiber is affected by the stresses caused by pure bending, and an induced-birefringence can be observed. Figure illustrates a system of Fizeau fringes crossing a straight single-mode fiber immersed in matching liquid. The core phase change is represented by a parabolic non-uniform central fringe shift [6]. The fringes cross the bent fiber are adjusted to be perpendicular to the fiber optic axis. To from the sharpest fringes across the fiber and to reveal correctly the induced birefringence in core and cladding, the phase lag between any successive beams must be suppressed. To reduce the phase lag producing the sharpest fringes a small interferometric gap thickness and hence small wedge angle are needed. As shown in Fig.3, the mechanical induced birefringence, β, in the fiber cladding is self-possessed of two index components. One of the components (n ) shows no fringe departure from the liquid fringe position. So as shown in Fig., the bend radius, R, does not affect this index component (ordinary one) which means also that n is equal to n L. On the other hand, the shifted fringe represents the extraordinary index component n of β depends absolutely on the curvature. Figure. Digital microinterferogram of Fizeau fringes crossing a straight single-mode fiber immersed in a matching liquid. The central shift is due to fiber core Figure 3. Digital microinterferogram in (x, z) plane for ordinary and extraordinary components of family of Fizeau fringes in the bent single-mode fiber

4 106 Fouad El-Diasty: Fizeau Interferometry for Bent Optical Fibers Characterization Considering a bent single-mode fiber immersed in matching liquid, a digital microinterferogram of the extraordinary index component of induced birefringence in the (x, z) plane is shown in Fig. 4. In the core region, asymmetry in the shape of core fringe providing asymmetric bell-like shape. The cladding fringe shift component follows inverted symmetric z-like shape curve with two opposite shift directions in the inner and outer halves of the cladding. Such representation for cladding fringe shift is due to the existence of radial expansion stress exerts in the convex side of the fiber where x > 0. Whereas at x < 0, there is a radial compression stress applied in all inner layers of the fiber concave side. The radial stress creates a strain that rearranges the intermolecular spacing and hence modifies the polarizability of the fiber materials to generate the induced-birefringence [6]. In the convex side, the expansion stresses are seen as a continues decrease in refractive index, n, with respect to n L. Moreover, the recorded steady increase for n with respect to n L in the concave side illustrates the gradual distribution of compression stresses. Furthermore, toward the smaller interferometric gap thickness (i.e., heading the apex) the fringe shift is considered positive with increasing n. Away from the wedge apex, the fringe shift is considered negative along with the direction of decreasing the index of refraction n [6]. Figure 4. Digital microinterferogram represents Fizeau fringe of parallel component of induced birefringence As explained before, macro-bending of optical fiber generates two opposite counterpart stress components. Such configuration provides the required conditions to determine the strength of optical fibers and to measure simultaneously some of the fiber elastic constants. As discussed before, the effect of mechanical bending is accounted for strain and or geometric effects. The strain effect is due changes in the fiber refractive-index, while the geometrical effect is due to the distortion of the fiber axis position [7]. In case of single-mode fiber, mathematical expressions were derived to evaluate both Poisson s ratio and a radial strain distribution profile of bent fibers from the measured fringe shift [8]. Accordingly, the Possion s ratio and the strain radial distribution profile of the cladding of bent fibers were successfully determined [8]. On the plane of fringe localization (x, z) close to the apex, where the fringe shift z at a distance x from the fiber axis, Poisson s ratio, v, was given by: λrrrr rr cccc xx 1 (ρρ 11 +ρρ 1 ) xxnn 3 zz(ρρ 11 +ρρ 1 ) vv = ρρ 1 where R is the radius of curvature, n here is the cladding refractive index, M is the interferometer magnification, and ρ 11, ρ 1 are strain-optic coefficients of the cladding material. Furthermore, the cladding radial strain tensor component ε x (z, R) is giving by [8]: εε xx = λmm rr cccc xx 1 xxρρ 1 (5) nn 3 zz(ρρ 11 +ρρ 1 ) RRRR(ρρ 11 +ρρ 1 ) The strain-optic coefficients for a silica fiber are ρ 11 = 0.1 and ρ 1 = 0.7 [9, 10]. From the maximum fringe shifts measured from microinterferograms and for four different radii of curvature, an average value for Poisson s ratio of the cladding material of a single-mode fiber has been determined. The determined value of v is 0.17 ± 0.0 [8], which is in a good agreement with the reported value [9, 11] proven the consistency of the method. The study showed also the nonlinear mechanical nature of the cladding material where it is relatively easy to tensile rather than being compressed. For elastic strain (positive in tensile region), the actual stress strain relation in silica fibers that experience uniaxial tension is essentially nonlinear [1]. Fizeau interferometry was used through bending to analyze the nonlinear stress strain relationship of the fiber cladding material. An expression [13] was given to calculate the cross-sectional nonlinear Young s modulus E x at any radial distance x due to bending from the fringe shift of fringes crossing the cladding of a bent single-mode fiber where: (4) EE xx = EE ααλmm rr cccc xx 1 xxαααα 1 (6) nn 3 zz(ρρ 11 +ρρ 1 ) RRRR(ρρ 11 +ρρ 1 ) where E 0 is Young s modulus in the area of low strain, and α is the parameter of nonlinearity. In typical silica materials, E 0 = kg mm - and α = 6 [14]. Again, the study confirmed that the radial nonlinear asymmetric stress strain relation is due to asymmetric distribution of the compression and tensile stress profiles rather than a spatial shift in the fiber centroid (neutral axis). Twist is one of the imperfections usually present in any real, installed fiber. When fiber is twisted induced circular birefringence or optical activity is introduced. The magnitude of torsional shear stress, S s, produced by twisting in a glass optical fiber is related by the shear modulus of elasticity, µ, by means of this expression [15]. SS ss = μμμμ = μμ(rr cccc θθ/ll) (7) where τ is the shear strain, θ is the total angle of twist, and L is the fiber length subjected to twisting. The cross-sectional profile of shear modulus of elasticity, µ, of the cladding

5 International Journal of Optics and Applications 015, 5(4): material in terms of the fringe shift z as measured from an interferogram is given by [16]: µ = λmm rr cccc xx 1 8CC zz(xx/rrrr)(1+vv) where C is the relative stress-optic constant of the cladding material. Notice that C = C 1 -C where C 1 and C are the axial and transverse stress optic constants, respectively. At λ = nm, C 1 = mm /N while C = mm /N [11] and thus C = mm /N. Experimentally it was seen that, the shear modulus at the compressive side of the cladding is 4.3 ± N/mm, while its value at the tensile cladding side is 3.4 ± N/mm. This difference in the modulus value at both sides of the fiber cladding could be due to asymmetric distribution of tensile and compression stresses across the bent fiber. The modulus mean value is in good agreement with the literature [1, 17, 18]. The greater sensitivity of the interferometric method without need of sophisticated equipment allows a quantitative and a qualitative determination of shear modulus even for fiber samples having low strength behavior. Applying the forward scattering technique showed that the asymmetry in the shear modulus across the fiber cross-section results from a shift in the fiber centroid (neutral axis). This can account for the effect of the nonlinear behavior of Young s modulus of the fiber material. For an elastically deformed object, the photo elasticity through application of the stress-optic law is usually used to determine the states of interior stresses. The technique illustrates the distribution of fringes of equal stress, which in turn represents the reorientation in the polarization states of light traverses the deformed medium. Determination of the relative retardation of propagating polarized light waves and the reorientation of the principal axes leads to quantify of the states of stress in the elastically deformed object. In case of non-matching, Fizeau interferometry was used to measure the photoelastic properties of the fiber cladding material. Measuring the two fringe shift components of the fiber cladding z 1 (x) and z (x) which representing the changes in the indices n and n, the mechanical induced-birefringence, β, can be evaluated applying the formula [19]: ββ = nn nn = zz (xx) zz 1 (xx) λmm rr cccc xx 1 (9) 4 zz The relationship between the induced-birefringence, the refractive index of the strain-free straight fiber, n, the applied strain, s, and photo-elastic coefficient, ρ, is given by [0, 1]: ββ nn (8) = ρρρρ (10) Using the relation ss = rr cccc RR, thus the strain can be evaluated []. Hence, from the mechanical induced-birefringence, the photo-elastic coefficient ρ(r) represented by the fringe shifts can be given by [19]: ρ = zz (xx) zz 1 (xx) λmmrr rr cccc xx 1 4nnnn zz (11) Using matching immersion liquid offers zero fringe shift, z 1 (x), which represents the component of the cladding index, n. The slope of the linear relation between induced-birefringence β versus curvature R gave a value for the photo-elastic coefficient = 0.9. In spite of ρ for a fiber is slightly larger than that of the bulk glass (ρ ) [], the calculated value is in a very good agreement with the reported value for pure silica glass (ρ 3). Furthermore, with increasing the cladding refractive index the number of propagating modes decreases due to what is called mode cut-off. A mode is cut-off when its field in the cladding die away to be evanescent. Accordingly, the operational cut-off wavelength λ c is a significant parameter should be identified for each propagating mode. The cut-off wavelength is defined as: λ cc = ππππ VV (nn cccc nn cccc ) 1 (1) where a is the core radius and.405 represents the V-number (normalized waveguide parameter, or waveguide frequency) of a single-mode fiber that supports only the hybrid HE mode. In the tensile region with a radius of curvature ranging between 7.5 and 19 mm, an increase in the number of propagated modes is associated with a decrease in the cladding index. Owing to the index and the spatial resolution of the Fizeau method, three additional modes were calculated allowing them to be propagated [19]. Seeing for only a single guided propagated mode, the operating cut-off wavelength should enlarge to pay off the effect of decreasing the cladding index. 4. Bending and Propagating Modes Characteristics One of the important optical fiber parameters is the propagation constant Ω (it is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction). Propagation constant is the measure of modal power loss and the signal phase during transmission of optical signals inside the fiber and the signal phase difference. It is will known that more than 80% of the modal power resides within the core and only 0% of the propagated power leaks into the cladding. Accordingly, developments in fiber amplifiers are stimulated by studying the modes propagation in the fiber cladding [3]. The relation between core cladding index difference and the permissible propagation constants Ω i still represent an active point for future investigations [4-6]. In a core region directly next to the cladding, for a core having parabolic-index profile, propagation constants difference of two adjoining modes in the core is defined as [7]: Ω = Ω i Ω i+1 = nn (13) aa where n is the fractional refractive-index difference between core and cladding and is given as:

6 108 Fouad El-Diasty: Fizeau Interferometry for Bent Optical Fibers Characterization nn = nn cccc nn cccc nn (14) cccc Inside the core of a bent GRIN fiber, the variation of radius of curvature R versus the propagation constant difference of two adjacent modes as a function of n in both the tensile and compressed cladding regions is seen in Fig. 5 [6]. n is calculated at the core cladding interface for a radial distance x = a. In the compressed side and for R ranging from 4.7 to 11.8 mm -1 the propagation constant difference was increased from 4.8 to 4.39 mm -1. In the tensile side Ω was decreased from 4.60 to 4.4 mm -1. With continues increase in R, the propagation constant difference in the two sides approaches Ω = 4.4mm -1 of a strain-free straight fiber. The method provides encouraging experimental results when compared to theoretical values for the variation of the propagation constant difference between two adjacent propagated modes. Figure 5. Variation of propagation constant difference between two adjacent modes, Ω, versus radius of curvature, R If the propagated light comes across a macrobend fiber with R > a, the modal field is radially distorted away from in the direction of the bend [8]. As a result, the core modes are coupled to leaky higher-order cladding and as well as to backward-propagating modes. Such coupling generates losses can be divided into two components: (1) mode-conversion loss and () radiation loss both of them are increasing toward long wavelengths. The repeated loss of radiation from the distorted mode or the inability of mode tails to navigate the bend leads to a pure bend loss. In case of long lengths of spooled fiber, the pure bend loss dominates [8, 9]. Different numerical techniques are used to study radiation losses [30-33]. On the other hand, the sudden change in fiber curvature induces a modal distortion or transition loss [8]. This means the coupling of fundamental propagated guided modes into radiation modes [34]. The power is vanished if the length scale is very short for an adiabatic change in mode shape to take place. In telecommunication fiber systems, with lengths of fiber having many separated macrobends, multiple transition regions can contribute to an overall bend loss. For the design of a single-mode fiber, the major effect of curvature on the fundamental mode is to shift the radial distribution of the field outward a distance r d from the fiber axis causing a considerable loss [8]. In terms of the spot size r o, the field shift within a Gaussian approximation is given as [34]: rr dd = VV aa RR rr 0 aa 4 (15) where V is normalized waveguide parameter. In the case of weakly guided step-index optical fiber, the profile height parameter = nn cccc nn cccc nn cccc. Guided modes correspond to a propagation constant Ω has the range ππ nn λ cccc < Ω< ππ nn 0 λ cccc. 0 The value of the spot size r 0 is given by [34]: rr 0 = aa VV 1 (16) For a single-mode fiber, the fraction of power radiated, that is, the transition loss P, is given by [34]: PP = 1 exp rr dd r aa 0 RR rr 0 aa 6 VV 4 8 (17) In a perturbed index fiber profile, the field shift due to bending for the fundamental mode with the associated transition loss can be calculated using the value of r d which is determined from the measured shift of Fizeau fringes. Accordingly, a modal field shift between 0.44 and 0.1 µm was calculated for a range of R between 0.13 and mm -1 [8]. Within the same curvature range, a fraction for transition loss in the range of was estimated. Even with the slight deviation shown between experimental data and the theoretical prediction at R < 10 mm, the findings showed a good agreement with theory. The deviation can be attributed to a tendency toward nonlinear change for the induced-strain. The fraction of radiated modal power from the entire fiber loop of a length πr is defined by the ratio of lost radiated power to the input power. The fractional power loss per unit length is known as the power attenuation coefficient (γ). With weakly guiding fibers and with R less than a few centimeters, the radiated fraction of modal power cannot be ignored. For instance, in a bent single-mode fiber with a propagated light power 500 mw and R 13 mm an induced eternal damage would occur. The possible damage is due to an elevating temperature (easily go up to 1000 C or

7 International Journal of Optics and Applications 015, 5(4): more) that occurs when the power overflows the fiber and absorbed by its outer coating. This means that, the power attenuation coefficient due to bending is a vital parameter must be calculated. In cylindrically symmetric fibers, the total field can undergo a significant change in a distance equal to the beat length, z b, between a pair of modes. It is defined as [34]: zz bb = ππ Ω 1 Ω (18) where Ω 1 and Ω represent the two closest propagation constants For single-mode fibers, only Ω 1 exists, while Ω refers to a packet of radiation modes. Since the value of the beat length is function of n, so the fractional power loss per unit fiber length due to bending, or power attenuation coefficient, ζ, is defined as [34]: ζ = ππ aa aa RR 1 VV WW 1 AA UU eeeeee 4RR nnww3 3aaVV (19) where A is called the area factor and its value is 1.9 and W is a cladding parameter (bound modes) which is given by: WW = ππππ nn λ cccc nn cccc (0) The core parameter U is defined as: UU = VV WW (1) The beat length, z b, is given in terms of the relative index difference, n, as: zz bb = 4ππππ ( nn) 1 VV WW () Actually, the relative index difference, n, plays an important role in determining many of the fiber parameters, mainly the power attenuation coefficient and beat length. Table 1 summarizes the obtained data for attenuation coefficient, ζ, and beat length, z b, at different radius of curvature, R, and radial distance, x [35]. A good agreement is achieved between the measured values of both beat length and power attenuation coefficient, and the corresponding theoretical values. Table 1. Radius of curvature, R, attenuation coefficient, ζ, core radial distance, x and beat length, z b R (nm) ζ (µm -1 ) Ten. x (µm) z b (µm) Ten. z b (µm) Comp To explain the total electromagnetic field distribution and to characterize modes propagation through optical fibers two adjacent parameters are needed [36]. The first one describes the spatial steady state where the bound mode energies are guided through a non-absorbing waveguide. The second parameter is related to the radiation field where it describes the spatial transient of the transmitted signal. The combination of total internal reflection and the surface evanescent wave is very useful to analyze the field leakage into the cladding. It provides the upper limit benefits required for applications such as directional couplers. Since bending destroys total internal reflection, therefore the evanescent wave which describes the second parameter was studied in a bent single-mode fiber [36]. Fizeau interferometry was used to study phase propagation constant and decay constant of the evanescent field and their radial dispersion profiles in bent fiber as a function of R at the two standard operating wavelengths 1300 and 1550 nm [36]. For perpendicular polarization, the expression for an electric field E co propagating in the radial x-direction in terms of E cl in the cladding (lower refractive index medium) is given as: EE cccc = EE cccc ee δδδδ ee iiωxx (3) where δ is the decay constant or the spatial rate of decay for radiated field in the cladding and Ω is the phase propagation constant. The field associated with a propagated plane wave turn into evanescent and decay exponentially with increasing distance away from core-cladding interface. The evanescent wave propagates along the core-cladding interface with no average transport of power into the cladding. The amplitude is becoming negligible at a radial distance x = 1/δ into the cladding. Also, the decay in power varies exponentially with the inverse of the free space wavelength of the guided light [34]. The formulae that explain both δ and Ω as guidance parameters and as a function of angular frequency, ω, permeability of free space µ 0 (4π10-7 Hm -1 ) and permittivity of cladding ε and refractive indices are given by: δδ = ωω μμ 0 εε nn cccc nn sin θθ cc 1 (4) cccc Ω = ωω μμ 0 εε nn cccc sinθθ nn cc (5) cccc where θθ cc is the incident critical angle which is equal to sin 1 nn cccc. The permittivity of the cladding material, ε, is nn cccc expressed in terms of permittivity of vacuum, ε 0, as ε = Kε 0 where K is the dielectric constant and ε 0 = Fm -1. Distance x =1/δ is known as the skin or penetration depth. Variations of both the attenuation and propagation constants of the evanescent field with the radius of curvature are evaluated using multiple-beam Fizeau fringes technique [36]. The findings illustrated that the rate of change of the attenuation constant with radius of curvature (dδ/dr) at wavelength 1300 nm was in the compressed cladding side while in the tensile (convex) side dδ/dr = µm -1 /mm, respectively. At wavelength 1550 nm, dδ/dr = µm -1 /mm in the compressed (concave) cladding side, while in the tensile side dδ/dr = µm -1 /mm. Furthermore, the rate of change of the phase propagation constant, Ω, with radius of curvature (dω/dr) in the compressed side = and µm -1 /mm at wavelengths 1300 and 1550 nm, respectively. While in the tensile side dω/dr = and µm -1 /mm at wavelengths 1300 and 1550 nm, respectively. The results

8 110 Fouad El-Diasty: Fizeau Interferometry for Bent Optical Fibers Characterization elucidate that Fizeau interferometry is responsive to the microscopic deviation in the measured decay and propagation constants caused by induced perturbation due to macro-bending. In the tensile side, together δ and Ω increase with increasing R, leading to an increase in radiation losses, which confirm the inverse proportionality between the operating wavelength and propagation constants of the evanescent wave. This obtained discrimination between the optical response of tensile and compressed cladding regions is not accessible with other reported methods. The Fizeau method was also used to study the effect of compression on diminishing n in graded-index (GRIN). Such retreating led to a dissipation of energy and a substantial mode loss. At small R, the change in n due to bending strongly affects the fraction of propagating mode number. For instance, with R < 1 cm a huge fraction mode loss arises [37]. Moreover, it could lead to an incorrect evaluation to the modes dispersion and core-cladding interface loss. For the two-dimensional case with counting only one of the two possible polarization directions, the number of propagating modes, P, in GRIN as a function of n is giving by [38]: PP = ππππππ λ nn (6) For instance, if n = there are approximately 184 supported modes at nm for a straight GRIN fiber [37]. Taking into consideration that the total mode volume is proportional to the fiber core radius, a, and n, so in a bent fiber the fraction of modes P as a function of R is given by [38]: PP = aa (7) RR nn For a plane polarized light vibrating in a parallel direction to the fiber axis, a gradient in the fiber refractive index is seen leading to a mode loss [37]. Consider the case of matching n cl = n L taking into account the cladding index (= n L ) for perpendicular polarization is ten times less than the parallel one, so the expression for the induced birefringence, β, as a function of fringe shift z(x) is given by [16]: ββ = zz(xx)λ rr 4 zz cccc xx 1 (8) Since the gradient in refractive index is superimposed on the original refractive-index profile, therefore n in the compressed side is given by [37]: nn = nn cccc (nn 0 +β) nn (9) cccc while n in the stretched side of the fiber is given by: nn = nn cccc (nn 0 β) nn (30) cccc Notice that the fringe shift z(x) has a negative sign in the tensile side. Measuring z(x) due to bending at x = a and x = x max offers a direct evaluation for the induced change in n at the two cladding sides [37]. In addition, the mode losses P in GRIN multi-mode bent fiber as a function of R could be attained. Compression increases the cladding index leading to a decrease in n and in designed P. It leads also to dissipation of field energy and mode loss. At x = a, P in the compressed side changed from 0.46 (R = 4.7 mm) to 0.17 (R = 11.8 mm). At x = x max corresponds to the maximum z(x), P changed from 0.51 to 0.18 within the same range of R [37]. 5. Effect of Bending on Some Fiber Guiding Parameters Bending stresses disintegrate the radial symmetry of the different fiber parameters of GRIN optical fiber. The induced perturbation in some of these parameters was evaluated by use of multiple-beam Fizeau fringes [37]. The variation in n in both the inner and outer regions of the bent fiber was measured with an error in the index is ± and spatial resolution 1.39 µm. The determined values of the acceptance angle, θ c, the numerical aperture, NA, and the V-number at two core radial x points (at core cladding boundary and at that corresponds to maximum fringe shift, z) measured from micro-interferograms at the both sides of the bent fiber were carried out and listed in Table [37]. In the compressed fiber region, for a system of different radius of curvature the acceptance angle is increased, the numerical aperture NA showed an increase and the V number is increased. In the same curvature range and at the tensile region [37], the acceptance angle is decreased while the numerical aperture NA is also decreased and the V number shows a decrease in its values. The obtained asymmetry is not due to a deformation in the circular fiber cross section to an elliptically deformed cross section under the effect of bending. But the asymmetry in the measured fiber parameters is ascribed to a shift in the fiber neutral axis (centroid), which is accounted for by the nonlinear mechanical behavior of Young s modulus of the fiber material [13]. Table. Radius of curvature, R, radial displacement, acceptance angle, θ c, numerical aperture, NA, and V-number R (mm) x (µm) o θ c Comp. Ten. NA Comp. Ten. V Comp. Ten. a x max a x max

9 International Journal of Optics and Applications 015, 5(4): Table 3. Radius of curvature, R, acceptance angle, θ c, numerical aperture, NA, dispersion per unit length, D/L, and modal dispersion, T R (mm) θ c o Comp. Ten. NA Comp. Ten. D/L (pskm -1 ) Comp. Ten. T (pskm -1 ) Comp. Ten In another study [39], the radial variation of parameters such as numerical aperture, the acceptance angle and dispersion per unit length of single-mode fibers undergoing bending were evaluated interferometrically. On the other hand, pulse broadening is initiated in optical fiber due to bending, where bending affects the amount of mode-coupling through fiber. Induced birefringence generates pulse broadening through polarization mode dispersion, T, (i.e., group delay difference between two propagated orthogonal polarized HE 11 modes) which was determined using multiple-beam Fizeau fringes in transmission [39]. The fiber parameters such as numerical aperture NA, acceptance angle θ c and dispersion per unit length D /L can be given by [40]: NA = nn cccc nn cccc, (31) 1 θθ cc = sin 1 NA, (3) (NA ) nn cccc cc DD LL = (33) Determining the change in nn cccc on each side of the bent cladding leads to evaluate the modal dispersion. To degrade the degree of polarization the use of a light source has short coherence length such as a superluminescent diode is recommended. The modal dispersion is given by [41]: TT = (1 cc) ββ + kk dddd (34) where c is the light velocity in free space and k is the free-space wavenumber. The second term in Eq. (34) is negligible compared with β in case of strain birefringence. Thus T is given by: TT = ββ (35) cc Equation (35) offers the differential group-delay which is the difference in the propagation time for two travelling polarization modes arise due to the two orthogonal indices, i.e. induced-birefringence. As listed in Table 3 and at radii of curvature (R = 6, 10, 1 and 16 mm), the gradual increase in the refractive index of single-mode fiber in the compressed side of the cladding provided an increase in the numerical aperture NA radial profile as the bent radius R changes from 6 to 16 mm [39]. Furthermore, in the tensile side as a result of the gradual decrease in the cladding index, a decrease in the numerical aperture radial profile is seen. An increase in the radial profile of acceptance angle θ c is seen in the compressed side with a decrease in the tensile side, as the curvature changes from 6 to 16 mm. The radial profiles of dispersion per unit length D/L showed an increase in the compressed cladding dddd side and a decrease in the tensile side with the same range of bend radii. The radial profiles of modal dispersion T showed decreases in the two fiber sides. The high bandwidth of single-mode fiber used in high-bit-rate optical communication systems is limited by material dispersion not only in the core but also in the cladding region [4]. In bent fibers short optical pulses ranging from ~ 10 ns to ~ 10 fs suffer from dispersion-induced pulse broadening defined as: RRλ 3 dd nn ππcc ddλ (36) When a light-emitting diode of 40 nm spectral width is used, a pulse broadening of 3.6 nskm -1 is generated [43]. Even for a GaAs laser of relatively narrow spectral width, the broadening may not be negligible. For maximum bandwidth, material dispersion (which could be described by the Cauchy s parameter of the fiber material) should be minimal at the operating wavelengths to balance the waveguide dispersion of opposite sign reducing pulse broadening. Therefore, dispersion measurements are of interest in the design of high bandwidth infrared optical fibers. Dispersion obeys, with a good approximation, the following empirical Cauchy dispersion relation [44]: nn = AA + BB λ + CC λ 4 (37) where A, B, and C are defined as Cauchy parameters of the investigated fiber material. However, in the high transparency region of the glass as in 1550 nm, the parameter C is very small and it can be ignored [43]. The physical significance of Cauchy parameters are given by the following equations: AA = 1 + NNee λ 8ππ εε 0 mmcc and BB = NNee λ 4 8ππ εε 0 mmcc (38) where N is the number of electron per unit volume, e the electron charge, m the mass of electron and ε 0 permittivity of free space. It is known that the normal refractive-index dispersion of optical dielectric constant in materials below the interband absorption edge is a function of oscillation energy E o and dispersion energy E d parameters, respectively [4, 43, 45]. E d is the electronic oscillator strength and it is a measure of the strength of interband optical transitions. E o is known as the single oscillator energy or the average electronic energy gap. The oscillator energy E o is related to the lattice oscillator strength, E l, where E o 1.5 E l. On other hand, E d values are related to the nearest neighbor cation coordination, anion valency, ionicity, and effective number of dispersion electrons [43, 45].

10 11 Fouad El-Diasty: Fizeau Interferometry for Bent Optical Fibers Characterization According to the single oscillator model [4, 43, 45], the dispersion of refractive index of a glass material can be described by the modified Sellmeier equation which is given as: nn 1 = EE 0EE dd EE 0 EE (39) dd where the dispersion energy parameter is related to the fiber structure [45, 46]. If the light wavelength is much shorter than the phonon resonance, the formula that correlates lattice energy E l to n associated with the oscillation energy and dispersion energy is given as [43, 46]: nn 1 = EE 0EE dd EE ll EE 0 EE EE (40) where E is photon energy. At longer wavelengths, where EE EE 0, Eq. (40) can take the following simplified form: nn 1 = EE dd EE ll EE 0 EE (41) According to Wemple [4] the material dispersion M(λ) is given by: MM(λ) = EE dd EE EE ll λ nn (4) nnλ 3 where the energies are given in ev and λ is in µm. In high-speed >> 40 Gbit/s and high capacity long-haul communication networks operate with standard single-mode fibers, macrobends in optical fiber cannot be avoided. So, precise information of bending-induced dispersion for the preceding fiber parameters is required to preserve the bit error rate as low as To design long distance optical transmission systems have low pulse broadening, maximum bandwidth and hence the upper limit information capacity, material dispersion should be least to balance the waveguide dispersion at the system operating wavelengths. Multiple-beam Fizeau fringes at transmission was used to determine the variations in radial profiles of Cauchy s parameters, oscillation energy, dispersion energy and lattice energy due to effect of bending in single-mode fiber [43]. In the tensile side with compression to the case of straight single-mode fiber (R = ), the A parameter of Cauchy s formula of the cladding material is decreased from ± to [43]. The B parameter increased at R = 5 mm from to µm with an error The variation in material dispersion, M, for a straight fiber was 3.7 to 810 ± 0.1 psec/km nm with the increase in wavelength in range nm. In bent free fiber, with an increase in same wavelength range, the material dispersion, M, showed an asymmetric decrease where M decreased from to be psec/ km nm in the tensile side, but in the compressed side M decreased from to psec/km nm. For bent fiber (R = 5 mm), M showed an asymmetric decrease in the two fiber sides [43]. The variations in the oscillation energy is from E o = 5.8 ± 0.01 ev to E o = 5.69 ev in the tensile cladding side, whereas in the compressed side E o = 5.93 ev. The dispersion energy decreased from E d = 5.78 ± 0.01 ev (R = ) to E d = 5.59 ev in tensile side but in compressed side it increased to reach E d = 6.0 ev. The lattice energy is changed at the two sides of the bent fiber with respect to a straight fiber from E l = 0.49 ± 0.00 ev to E l = 0.45 ev in the tensile side and E l = 0.43 ev at the compressed side, respectively [43]. 6. Kerr Nonlinearity of Bent Fiber for All-optical Switching Interaction of high power lasers with matter in the field of optical fiber illustrates the way to applications in nonlinear optics. The applications are stimulated Raman and Brillouin scattering [47-49], induced birefringence [50], parametric fourwave mixing [51, 5], and self-phase modulation [53, 54]. Soliton-like pulses can be stand in optical fibers. The interplay between the dispersive and the nonlinear effects leads to ultrashort optical pulse generation [55, 56] and pulse compression (pulses of 6 fs) [57, 58]. The cross phase modulation and soliton switching are photonics-based devices used in telecommunication technologies [4]. Maxwell s equation that depicts the electric field E of the propagated light waves inside optical fibers takes the form: EE 1 EE = μμ PP LL cc tt 0 + μμ PP NNNN tt 0 tt (43) where µ 0 is permeability of vacuum and P L and P NL are the linear and nonlinear polarizations, respectively. Intensity-dependent refractive index n is the origin of nonlinear refraction needed for many nonlinear optical devices. The total index of refraction NN as a function of light intensity, I, and linear index of refraction of a straight fiber, n 0, is given by: NN = nn 0 + nn II (44) In silica fibers n is small by at least orders of magnitude compared to other nonlinear materials [4]. Due to the intrinsic low loss (< 0.15 db/km) and the small core radius (~ 4 µm), as in single-mode fibers, nonlinear effects in optical fibers can be seen even at relatively low laser power levels [47]. Kerr effect in optical fiber arise from the contribution of third-order optical susceptibility χ (3) through the relation [59]: nn = 1ππ χ (3) (45) nn 0 where n is in esu. The third-order susceptibility χ (3) (in esu) of a material can be estimated from refractive index, n, applying Miller s rule where [60]: χ (3) = nn 1 4ππ (46) Therefore, materials having larger n induces larger χ (3). Investigation have been carried out on bent single-mode optical fibers to study the induced changes in some nonlinear parameters such as third-order susceptibility χ (3) and second-order refractive index n [47]. For this purpose, multiple-beam Fizeau interferometry (with its high spatial and index resolutions) was used at the standard operating

11 International Journal of Optics and Applications 015, 5(4): wavelengths 1300 and 1550 nm and at R ranging from 5 to 11 mm. The asymmetric variations of χ (3) and n (due to the likely nonlinear response of the Young s modulus of fiber material) versus R and radial distance x were observed and calculated. Regards to the straight fiber, the findings showed that the electronic susceptibility increases as the radius of curvature decreases on the compressed side. In the tensile side, the susceptibility decreases with decreasing radius of curvature. The radial profile of second-order refractive index n showed the same trend of χ (3) versus R and x. For straight fiber at λ = 1300 nm the cladding χ (3) and n were ± esu and ± esu, respectively. For λ = 1550 nm, the two parameters were χ (3) = esu and esu, respectively [47]. In the tensile side of cladding and at λ = 1300 nm, the induces asymmetry in χ (3) at R = 5 mm was esu, whereas on the compressed side with the same R the value of χ (3) = esu [47]. At the same conditions, bending induces also asymmetry in the value of n, where in the tensile cladding side n was esu. In the compressed side it was esu. At R = 5 mm and λ = 1550 nm, the cladding χ (3) and n on the tensile side were esu and esu, respectively [47]. On the compressed cladding side χ (3) and n were esu and esu, respectively. For straight fiber at λ = 1300 nm the core χ (3) and n were esu and , respectively. But for λ = 1550 nm, the two nonlinear parameters were χ (3) = esu and esu, respectively [47]. Asymmetry of both χ (3) and n radial profiles due to bending were determined versus R in the core region using Fizeau interferometry [47]. In the tensile side and at λ = 1300 nm and R = 5 mm, the core χ (3) asymmetry was χ (3) = esu and esu on the compressed side. The asymmetry in n was esu in the compressed side while it was esu in the tensile side [47]. At λ = 1550 nm and R = 5 mm, in the tensile side the core χ (3) = esu while it was esu in the compressed side. The asymmetry in n was esu in the tensile side and esu in the compressed side [47]. The obtained asymmetries in χ (3) and in n were attributed to induced asymmetric variation in n with bending and to nonlinear response of the Young s modulus of the fiber material [13]. In accordance with the variation in the input laser power, optical switching-based fiber devices for ultrafast signal processing depend mainly on intensity-induced branching and instability of the propagated modes [6]. Examples of these devices are nonlinear directional couplers [61], nonlinear tapered waveguides [6], X junctions [63], and Y-splitters [64]. The simplest nonlinear switching device is the direction coupler. Through the induced changes in n of cores by the input laser power, the strength of coupling between the propagating modes of two weakly interacting cores can be controlled. Switching in other nonlinear optical devices is conducted by the emission of self-guided beams (solitons) into the cladding [65]. The technique is based on instability-induced threshold behavior of nonlinear surface waves in the waveguide with a nonlinear self-focusing cladding. The relative high laser power required for switching represents the main disadvantage for the practical realization of these types of devices [6]. Modifying the waveguide geometry to enhance the induced instability of the propagated modes is one of the available solutions by using a small-size device such as optical fibers. Optical fibers acquire large nonlinearity through the inherent effective combination of small core area (small mode area) and long interaction propagation distances [66]. The central idea for all-optical processing or all-optical switching of beams is controlling the modal phase change and hence modes coupling. Such modal phase change designs the interference conditions between multiple guided beams. Thus through macro bending of single-mode waveguides [67] or optical fibers [68], the mode coupling and hence the all-optical switching can be enhanced. Depending upon whether real or virtual electronic states are involved in the optical excitation process, Kerr nonlinearity is classified into nonresonant- and resonant-based nonlinear effects. The dispersive nonlinearity or the nonresonant type is due to anharmonicity in the restoring force of the bounded electrons. It exposes itself through the real part of third-order susceptibility, Re χ (3). In contrast, the absorptive nonlinearity or resonant type represents the imaginary part of Kerr nonlinearity, Im χ (3). Such type of absorptive nonlinearity is attributed to a nuclear contribution. The two nonlinear optical excitation processes (in esu) are given by [69]: Re χ (3) = ε 0 ccnn nn (47) Im χ (3) = εε 0 cc λnn σσ (48) The value of the two-photon absorption coefficient, σ, is (1.9 ± 0.) cm/w for fused silica, whereas for germanosilicate glass it is (4 ± 3) cm/w [70]. Applying the empirical relation of Boling [71], n (in esu units) can be determined using n d lays between 1.4 < n < 17, where: nn = κ nn dd 1 GG 5 (49) 4 Parameter G is known as Abbe dispersion number while κ is a constant equals 391 [71]. The Abbe dispersion number is given by: ππ GG = (nn dd 1) (nn FF nn CC ) (50) where n d, n F and n C are the linear refractive indices at the yellow He d-line of the following standard wavelengths; λ d = µm, λ F = µm, and λ C = µm. The n d and G values of the studied single-mode bent optical fiber are available in reference [43] Dispersive or Real Third-order Susceptibility In the case of bent free fiber with R = and at λ = 1300 nm,