Two-dimensional refractive index and birefringence profiles of a graded index bent optical fibre

Optical Fiber Technology 36 (2017) 115–124

Contents lists available at ScienceDirect

Optical Fiber Technology www.elsevier.com/locate/yofte

Regular Articles

Two-dimensional refractive index and birefringence profiles of a graded index bent optical fibre W.A. Ramadan, H.H. Wahba ⇑, M.A. Shams El-Din Physics Department, Faculty of Science, University of Damietta, 34517 New Damietta City, Egypt

a r t i c l e

i n f o

Article history: Received 25 July 2016 Revised 17 January 2017 Accepted 5 March 2017

Keyword: Refractive index profile of a bent GRIN fibre Two dimensional refractive index profile Digital holographic interferometric phase shifting Induced birefringence profile

a b s t r a c t A theory to recover refractive index profile of the bent graded index (GRIN) optical fibre, in core region, is proposed. This theory is applied to the bent GRIN optical fibre when it is located orthogonal in the light path of the object arm in digital holographic phase shifting interferometer; like Mach–Zehnder interferometer. In the experiment, the fibre is bent with two different bending radii and fixed on a microscope slide keeping it immersed in matching liquid. The produced phase shifted holograms, with the presence of the fibre, are recorded using an attached CCD camera. Two different processes controlling the index profile shape of the bent GRIN optical fibre are assumed. In the first process, a linear index variation is evolved from stresses in the direction of the bent radius. In the second one, there is a release of these stresses near the fibre surface, which depends on the fibre’s radius. This will affect the outer free surface of the cladding. Based on these assumptions, we are able to construct the index profile in two dimensions normal to the optical axis. We propose two functions to describe the refractive index profiles in cladding and the core regions of the bent GRIN optical fibre. The recorded phase shifted holograms are combined, reconstructed and analyzed to get the phase map of the bent GRIN optical fibre. Comparing the extracted optical phase differences with the calculated ones, a good agreement between them is found. This means that the used two dimensional proposed functions, which are describing cladding and the core indices profiles, are the most proper in this situation. Thus, we are able to determine a realistic induced birefringence profile inside the fibre which is generated by a bending operation, not only in the cladding but also in graded index core region as well. Ó 2017 Elsevier Inc. All rights reserved.

1. Introduction Optical fibres are widely used in fibre optic communications, data transmission and sensing devices [1–3]. The mechanical properties of the optical fibres are relevant to the sensing applications. This by expose them to different mechanical processes, such as the mechanical bending [4–6], twisting [5] and elongating through drawing [7], in the applications. These mechanical effects lead to an induced birefringence inside the optical fibre’s material. In the sensing applications, it is essential to characterize the optical and mechanical properties of the used optical fibres. It is important to detect the induced birefringence in some sensor applications [1–3]. Bending and twisting cause signal attenuation which is annoying in the fields of fibre optic communications and data transition [8,9]. The bending causes compression and elongation along the fibre radius around the neutral axis, which leads to an increase and a decrease of the fibre’s refractive index, respectively [5]. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (H.H. Wahba). http://dx.doi.org/10.1016/j.yofte.2017.03.005 1068-5200/Ó 2017 Elsevier Inc. All rights reserved.

Recently, bending has been employed to increase the surface strength of the fibres [10]. Many methods are able to detect the tiny changes of the refractive index [11] due to the mechanical effects in optical fibres [6,12]. Consequently, the mode field distributions inside the fibres have been revised [13–16]. In case of bending of a standard single mode optical fibre, one can reach a cut off signal at a curvature radius about 7.1 mm [12]. While, in case of bending of GRIN multimode optical fibre, the modes loss increase with decreasing curvature radius [17]. In many practical use of optical multimode fibre tight bends can be expected along the fibre’s path which causes an increase of bend loss. That is why a bend resistive multimode fibre was proposed. A ten loops of the proposed bend resistive multimode fibre were wrapped around a cylinder of 1.5 mm radius, bend losses below
0.2 dB were achieved [18]. The refractive index profile variation due to the bending of the optical fibre is quite interesting and difficult problem. One of the most accurate and sensitive tools, that is used to characterize the refractive index variation in fibres, is the light interferometry. Many attempts to solve this problem have been presented [6,12,19]. A theoretical study of treating pure bending

116

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

deformations as perturbations to step index single mode fibre with a circular core has been presented [20]. In this work a dielectric constant tensor components are given but it is difficult to be employed in case of interferometry applications. In the earliest work, a theoretical model to determine the refractive index profile of a bent single mode optical fibre has been proposed. This model is based on dividing the fibre’s cross section into a large number of slabs, each of them has a constant refractive index. An enhancement of the accuracy has been achieved by using the automated Fizeau interferometer [12]. Using the same theoretical approach, a development in interference image analysis has been presented by using digital holographic interferometric phase shifting method [19]. Again, the same theoretical approach has been developed to investigate the step index multimode bent optical fibre [21]. In these investigations [12,19,21], the maximum of compression and the minimum of tension are expected to be achieved at the two opposite surfaces of the fibre. The calculations of modes and macro–bending loss as a function of the bending radius of a fibre with a finite cladding, using beam propagating method, were presented [13–16]. In these studies, the refractive index change profiles of the bent optical fibres are considered to be a linear function. In fact, the previous experimental observations do not support such consideration [6,12,19,21]. This indicates that there is a surface impact on the refractive index profile that has not been taken into account in these calculations. This observation is well discussed and used in strengthening the fibre surface in an original work presented by Tomozawa et al. [22,23] and Lezzi et al. [24]. They explained that there is a release of generated stresses, due to bending, on the fibre surface. A confirmation of residual stress layers, formed on silica glass fibres by a surface stress relaxation process during bending, using FTIR reflection measurements was demonstrated [25,26]. In spite of the fact that the etching process is able to detect surface stress relaxation, it is a destructive technique [22]. In addition, they suggested different functions which are describing this stress release; one of them could be the exponential decay towards the fibre surface. An important step forward has been presented to investigate the refractive index profile of bent homogeneous optical fibre, considering the release of the stresses on the free fibre surface [27]. The refractive index profile is assumed to be featured by an exponential decay near the free surface of the fibre, and becomes linear on going to the fibre centre. The proposed refractive index function has been confirmed by comparing the experimental and calculated phase variations based on the proposed refractive index function. The experimental data have been obtained by placing a bent fibre in the object arm of Mach–Zehnder like interferometer. A 3D refractive index profile has provided and a figure of the cross sectional stresses distribution of a homogeneous bent fibre has been given [27]. In this paper, we developed the work of the homogeneous optical fibre bending by means, a much more complicated case is handled. In this case the GRIN optical fibre consists of cladding and core regions is investigated. The bending of the cladding region will be treated similar to bending of homogeneous fibre case [27]. The bending of the core region of a GRIN optical fibre leads to a refractive index variation. This variation is a direct influence of stress generated by the bending. It is quite common to describe the refractive index function of this region in terms of the shape parameter (a) and the refractive indices difference (Dn) at the core centre and the cladding. After bending, it is important to examine the parameters which are controlling the refractive index function. To realize this task, a modification of the previous theory [27] is considered to take into account the light path through the cladding and core regions. In addition, digital holographic phase shifting is employed to extract the optical phase differences across the bent GRIN optical fibre. The experimental data of optical phase differences are compared to the calculated ones. A good agreement

between them could be achieved when the appropriate parameters of the refractive indices functions are selected. A 3D refractive index and birefringence profiles of the bent GRIN optical fibre are illustrated. 2. Theory 2.1. Refractive index profiles of the graded index bent optical fibre The graded index optical fibre consists of a central graded index medium nco ðrÞ with core radius rc is described by the general function:


ao 12 r nco ðrÞ ¼ nc 1
2Dn ; rc

ð1Þ

where nc is the refractive index of the core centre, r distance from the fibre centre and

Dn ¼


2cl n2c
n 2 2nc

ð2Þ


cl is a constant refractive index of the cladding. ao is a constant n parameter, controlling the refractive index variation rate (shape parameter) of the graded index region [28]. In case of bending a GRIN optical fibre, with a bending radius R, the fibre cross section is divided into two halves by neutral plane, see Fig. 1-a. Expansion and compression stresses are generated on the both halves of the cross sections of the fibre, one of them is far from the bending centre and the other is near to the bending centre. These generated stresses will affect the refractive index profile of both cladding and the graded index core regions. Also, an induced birefringence is expected. When a bent GRIN optical fibre is immersed in a liquid of a suitable refractive index and situated in the path of the object arm of digital holographic interferometer, an optical phase change is observed, see Fig. 1-a. This change is a direct interpretation of the optical path change evolved due to the bent GRIN optical fibre presence in the light path. This provides information about the refractive index profile across the bent GRIN optical fibre. One can notice that the experimental observations of the phase variation is a result of two different processes which affect to a bent optical fibre. The first process is linearly dependent on the distance from the bending centre, which is dominant as we go towards the centre of the fibre. The second one is caused by the stresses release from the free surface [22,27]. This effect must have a decaying feature as we go towards the fibre centre. Also one can expect that, this effect will take place only on the free surface of the cladding. In this theory, we propose two dimensional functions to describe the refractive index profiles for both cladding and core regions. These functions are chosen to take into consideration the two processes described above. Regarding the multi–mode optical fibre, we can notice that there are two interfaces, the first one is the cladding–core interface and the second is the cladding–surrounding medium interface. The first interface is featured by the continuity of the refractive index value between cladding and core regions. Due to the rigidity of cladding material and fusion connection between cladding–core media, we assume that there is no stresses release through the core–cladding interface. Inversely, we demonstrate that there is a stresses release though the cladding–surrounding interface [27]. So, during the refractive index calculation of the cladding region, this stresses release must be considered. Let us define a certain number M of the incident rays on the bent GRIN optical fibre, starting from the fibre centre going towards the cladding–surrounding interface. The separate distance between these rays a is given by;

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

117

Fig. 1. (a) A 3D schematic diagram shows the relative position for the bent GRIN optical fibre, incident rays, bending radius, neutral plane and phase hologram. (b) A schematic diagram shows the different distances used in the theory, dc, d, kcl and kco. The incident angle h0 and the ray, in compression and expansion cases, are traced.

a¼

r cl ; M

ð3Þ

where r cl is the cladding radius. The distance between the axis passing through the fibre centre and the incident ray d takes the values;

a a a a a d ¼ ; a þ ; 2a þ ; 3a þ ;             ; ðM
1Þa þ 2 2 2 2 2

Z OPco ðdÞ ¼ 2

2.1.1. When d is smaller than dc As we indicated above, when d is less than dc the rays traverse both the cladding and core regions. The optical path in the bent fibre OPðdÞ of the transverse ray will be the summation of the path in the cladding region OPcl ðdÞ and the core region OPco ðdÞ. The infinitesimal change, with respect to the radius variation, in the metric distance along the ray path for an incident ray at a defined distance d from the fibre centre [25] and [26] is given by:

ð5Þ

rc kco

ð4Þ

With increasing M value a good accuracy can be achieved. We notice that, when we are close to the fibre centre, the rays will pass through the cladding and core regions. When d is superior to a certain value dc , the rays will pass only in the cladding region, see Fig. 1-b. dc is defined as the distance between the incident ray and the fibre centre which makes the nearest distance between the passing ray and the fibre centre equal to core radius rc . So, we have to predict dc value in advance, more information about calculating dc in following section.


 @OPðdÞ 2nðd; rÞr ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @r 2 2 2 nðd;rÞ nðd; rÞ r 2
nðd; kÞ k

kcl is the nearest distance between the ray path through the cladding and the fibre centre, see Fig. 1-b, assuming that cladding is constituting the entire cross section of the bent fibre, we have

nco ðd; rÞr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr 2 2 2 2 nco ðd; rÞ r
nco ðd; kco Þ kco

ð8Þ

where kco is the nearest distance between the ray path and the fibre centre when it passes in core region, see Fig. 1-b. In Mach–Zehnder interferometer we have a reference arm and object arm. The sample is immersed in a liquid and inserted in the object arm. We get a straight lines fringes when the rays passing through the liquid (reference interference field). When the rays passing through the GRIN optical fibre, these straight lines fringes suffer shifts according to the optical path differences. These optical path differences, between the reference ray (passing in liquid) and the ray passing through the weakly gradient bent fibre in cladding OPDcl ðdÞ and core OPDco ðdÞ regions, are given by

Z OPDcl ðdÞ ¼ 2

r cl

rc

Z OPDco ðdÞ ¼ 2

rc kco

ðncl ðd; rÞ
nL Þncl ðd; rÞr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr 2 2 2 ncl ðd; rÞ r 2
ncl ðd; kcl Þ kcl ðnco ðd; rÞ
nL Þnco ðd; rÞr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr 2 2 2 nco ðd; rÞ r 2
nco ðd; kco Þ kco

ð9Þ

ð10Þ

The total optical path difference OPDðdÞ for a ray traversing the cladding and core regions is given by the summation of Eqs. (9) and (10), as,

OPDðdÞ ¼ OPDcl ðdÞ þ OPDco ðdÞ

ð11Þ

where k is the minimum distance between the ray path and fibre centre. Now, to get the total path length inside the fibre, we integrate equation (5) with respect to r, taking into consideration that the ray transverses the cladding region (r is varying from r c ! rcl ) and the core region (r is varying from 0 ! rc ), so

This means that we can get the entire optical path difference caused by the bent fibre. The optical phase difference uðdÞ can be presented in terms of the optical path difference OPDðdÞ as:

OPðdÞ ¼ OP cl ðdÞ þ OP co ðdÞ;

where k is the wavelength of the light used. A comparison between the calculated optical phase difference along the diameter of the fibre and the recovered experimental one, judges whether the refractive index function is correct or not. The calculation of OPDðdÞ, as obvious from Eqs. (9) and (10), demands the identification of two functions which are describing the refractive indices

where

OP cl ðdÞ ¼ 2

Z

r cl

rc

ncl ðd; rÞr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr 2 2 2 ncl ðd; rÞ r 2
ncl ðd; kcl Þ kcl

ð6Þ

ð7Þ

uðdÞ ¼

2p OPDðdÞ; k

ð12Þ

118

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

Fig. 2. Geometry of a digital holographic axes and planes systems.

in the cladding and core regions. These functions must consider the stresses, as previously discussed, in the cladding and core regions. For the cladding region, the function which is describing the refractive index of a bent homogeneous optical fibre has been presented in [27] in the form, rcl
r i dh ncl ðd; rÞ ¼
qnbf 1
e
ð rs Þ þ nbf R

ð13Þ

where q is the strain optical coefficient, nbf is the cladding bent–free fibre’s refractive index, R is the bending radius and r s is a proposed parameter to evaluate the distance from the fibre surface which affected by stresses release. This equation indicates that; the cladding refractive index will vary as the exponential decay begins from fibre free surface. As we go towards the fibre centre, the variation will depend linearly on the distance from the bending centre (d). In matching case, nbf is equal to the liquid refractive index nL . This means that the only two parameters are controlling the shape of the index profile are q and r s . The identification of these two parameters enables us to describe the index profile of the bent optical fibre in the cladding region. For the core region, we propose a modification on the refractive index profile (see Eq. (1)) to express the bending influence on the core region. We expect that: (i) the shape parameter will be changed linearly according to the value of d in the direction of the bend-

ing radius and (ii) the value Dn in Eq. (1) will vary, according to the value of ncl ðd; rc Þ at the boundary between cladding and core regions. Based on this expectation, we can define the refractive index function of the core region as,

"


aðdÞ #1=2 r nco ðd; rÞ ¼ nc 1
2Dnðd; r c Þ ; rc

ð14Þ

where

Dnðd; r c Þ ¼

n2c
n2cl ðd; r c Þ 2n2c

ð15Þ

and

aðdÞ ¼ ao þ De d

ð16Þ

De being the slope of the linear relation between a (the shape parameter) and d (the distance between the incident ray and the bending centre). The other symbols are previously nominated. To identify completely the core region refractive index function, the correct values of nc , a0 and De are selected to provide a good agreement between the calculated and the experimentally recorded optical phase differences. Back to Eq. (9) and (10), one can realize that the values of kcl and kco are still unknown. In this case, Bouguer formula [30] can be

Fig. 3. Digital holographic interferometric phase shifting arrangement, 1: CW Laser source, 2: Spatial filter, 3: Polarizer, 4: Collimating lens, 5: Beam splitter, 6: Mirror, 7: Sample holder 8: Microscopic objective, 9: CCD camera, 10: PZT, 11: Two dimensional translation stage, 12: Microscope slide, 13: Immersion liquid, 14: Bent optical fibre, 15: Cylinder position and 16: fixing tab.

119

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

Here again, ncl ðd; rÞ has the definition given in the Eq. (13) and kcl is determined as it is previously explained with the aid of Eq. (19). Also, the phase difference is still given by Eq. (12). A MATLAB programme is prepared to perform the calculations as following: a suitable values for q, rs , nc , a0 and De are assumed to define completely the refractive index functions of ncl ðd; rÞ and nco ðd; rÞ. By using the functions in Eqs. (19) and (20) the values of dc , kco and kcl can be determined. The phase differences are determined by using Eqs. (11) and (12) when d is less than dc , but we use Eqs. (22) and (12) when d is greater than dc . We repeat this algorithm for 0 < d < r cl , with the variation of the parameters that describe the refractive index functions, until we get the best agreement between the experimental and calculated phase changes. This agreement is adjusted by evaluating standard deviation between the calculated and experimental data.

(a)

2.2. Digital holographic interferometric phase shifting method

0o 0.5π

Digital holographic interferometric phase shifting method is widely used to investigate the fibrous materials [19,31–33]. In this method, a set of five phase shifted holograms is recorded with known mutual phase shifts, starting from 0 in steps of p2 The intensity distributions of the recorded holograms are,

π 1.5π 2π

In ¼ Aðf; gÞ þ Bðf; gÞ cosðUðf; gÞ þ URn Þ;

n ¼ 1; 2; 3; . . . . . . . . . ð23Þ

where Aðf; gÞ and Bðf; gÞ are the additive and the multiplicative distortions and URn is the phase shift that is performed in the reference wave. So, a set of five linear equations are point wisely solved by the Gaussian least square method [31]. Then the complex wave field hðf; gÞ, which could express the combined phase shifted holograms, can be calculated [33],

(b) Fig. 4. Five phase–shifted holograms of bent optical fibre with bending radii (a) R = 5.3 mm, (b) R = 8 mm. The used laser beam is vibrating parallel to the optical fibre axis.

used to relate the radius, the incidence angles and the refractive index at any point on the light path, as

kcl ncl ðd; rÞ ¼ r ncl ðd; rÞ sin hr

ð17Þ

kco nco ðd; rÞ ¼ r nco ðd; rÞ sin hr

ð18Þ

where hr is the incident angle at the point r. By applying these formulas at the incident point with an angle of incident h0 on the outer surface of the fibre, we obtain;

kcl ncl ðd; rÞ ¼ r cl nL sin ho

ð19Þ

kco nco ðd; rÞ ¼ r cl nL sin ho

ð20Þ

By solving these two equations numerically, we can get the values of kcl and kco . In Eq. (19), when sin ho ¼ rdc we have kcl ¼ rc then cl

cl Þ dc ¼ ncl ðd;r r c . These values must be re-evaluated as the refractive nL

indices functions change.

ð21Þ

and OPDcl ðdÞ, described in Eq. (9), is still valid with the changing of the integration lower limit, thus;

Z

OPDcl ðdÞ ¼ 2

r cl

kcl

ðncl ðd; rÞ
nL Þncl ðd; rÞr qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr 2 2 2 ncl ðd; rÞ r 2
ncl ðd; kcl Þ kcl

þ i½7ðI4 ðf; gÞ
I2 ðf; gÞÞ

ð22Þ

ð24Þ

The Fresnel diffraction integral is used to reconstruct the com0 plex wave field b ðx0 ; y0 Þ in the image plane [31–33],

B0 ðx0 ; y0 Þ ¼

1 ik

Z Z

hðf; gÞu ðf; gÞ

expfiwbg dfdg b

ð25Þ

pffiffiffi02 where b ¼ d þ ðf
x0 Þ2 þ ðg
y0 Þ2 , w is the wave number 2p ðw ¼ k Þ and u ðf; gÞ is the complex conjugate of a numerical model of the reference wave. Fig. 2. shows the geometry of digital holographic axes in the object, the hologram and the image plans. The intensity distribution Iðx0 ; y0 Þ of the reconstructed field in the image plane is, 0

Iðx0 ; y0 Þ ¼ jb ðx0 ; y0 Þj2

ð26Þ

and the optical phase distribution Uðx0 ; y0 Þ cab be calculated by,



Uðx0 ; y0 Þ ¼ tan
1

2.1.2. When d is greater than dc In this case, the ray will transverse the cladding region only. The total OPDðd; rÞ will be;

OPDðdÞ ¼ OPDcl ðdÞ;

hðf; gÞ ¼ ½4I1 ðf; gÞ
I2 ðf; gÞ
6I3 ðf; gÞ
I4 ðf; gÞ þ 4I5 ðf; gÞ

 0 Imjb ðx0 ; y0 Þj 0 Rejb ðx0 ; y0 Þj

ð27Þ

Finally, the optical phase distribution can be processed to get the optical phase differences due to the bent graded index optical fibres. 3. Experiment Digital holographic phase shifting arrangement, Mach–Zehnder like interferometer, is used in the experimental investigations of the graded bent optical fibres. Fig. 3. shows digital holographic

120

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

2 (a)

(b)

3 4 5 6

50 m

50 m

7 8

(c)

(d)

50 m

Fig. 5. (a), (c) The enhanced reconstructed interference phase modulo 2p, for bending radii R = 5.3, 8 mm at reconstruction distances
8.3 and
38.6 mm, respectively. (b), (d) the interference phase distribution after background subtraction of (a) and (c).

(a)

50 m

(b)

50 m

Fig. 6. The enhanced reconstructed interference phase modulo 2p, (b) the interference phase distribution after background subtraction where the used laser beam is vibrating perpendicular to the optical fibre axis.

phase shifting set-up [19,32,33]. A He–Ne laser source of wavelength k ¼ 632:8 nm is used. The incident laser beam is filtered and collimated. The graded index optical fibre is a DRAKA OM1 fibre with core diameter 62:5  2:5 lmand cladding diameter 125  1 lm, according to the specifications of the supplier. The numerical aperture is 0:275  0:015. The GRIN optical fibre is circled around a cylinder with a proper radius, the bending radius R, and is glued from the fibre ends. The bent fibre is fixed to a microscopic slide, immersed in liquid of a matching refractive index and covered by another very thin microscopic slide cover. The prepared sample is attached to a 2D translation stage to select a proper position of it through the object arm of the set-up. The desired polarization state is

selected by using a polarizer crossed by laser beam. Two identical microscope objectives are used in the object and the reference arms. The object microscope objective magnifies the optical field of the sample. The reference microscope objective is used to adjust the field curvature. The phase shifts are controlled by a Piezo mounted mirror with respect to the reference wave. The holograms are recorded using a CCD camera with pixel pitch Df  Dg ¼ 4:65 lm  4:65 lm and pixel numbers 1392  1040. 4. Results and discussion The bending effect on a graded index optical fibre with a diameter 125 lm is investigated. A graded index optical fibre has been

121

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

15.70

Experimental Data, R =5.3 mm Calculated Data, R =5.3 mm

(a)

12.56

Phase Shift Rad

9.42

6.28

3.14

0.00

-3.14

-6.28 -60

-45

-30

-15

0

15

30

45

60

r m

15.70

Experimental Data, R =8 mm Calculated Data, R =8 mm

(b)

12.56

9.42

Phase Shift Rad

bent with two bending radii 5.3 and 8 mm with tolerance ±0.5 mm. The bent fibre is immersed in a matching liquid of refractive index 1.46. The prepared sample is attached to the object arm in the digital holographic arrangement. The produced phase shifted holograms are recorded using the CCD camera. A set of five phase shifted holograms is recorded for each bending radius. The effect of bending can be observed when the component of the polarized light is vibrating in parallel direction with respect to the optical fibre axis. In this case, Fig. 4-a, b shows a set of five mutual phase shifted holograms, starting from 0 to 2p in steps of p2 and R ¼ 8 mm, respectively. The recorded phase shifted holograms are combined and reconstructed, applying the procedures which are explained in Section 2.2. In addition, the reconstructed optical interference maps are filtered by using an appropriate algorithm [19]. Fig. 5-a, c shows the enhanced reconstructed interference optical phase differences maps of the recorded phase shifted holograms for bending radii R = 5.3, 8 mm at reconstruction distances -8.3 and -38.6 mm, respectively, while the incident light is selected to vibrate parallel to the optical fibre axis. The subtraction method [17,30] is employed to extract the resultant optical phase differences due to the presence of the bent graded index optical fibre for the two bending radii, see Fig. 5-b, d. In addition, one can observe the effect of bending in Figs. 4 and 5. The upper halves in these figures indicate the negative fringes shifts due to the extended region of the bent fibre whereas the lower halves indicate positive fringes shifts in the compressed region, especially in the cladding with respect to the optical fibre axis. No effect of the bending when the incident light is vibrating perpendicular to the optical fibre axis is observed [6,12]. Fig. 6-a represents the enhanced reconstructed interference optical phase differences map, selecting the perpendicular vibrating component of light. The extracted optical phase differences due to the bent fibre are shown in Fig. 6-b. One can notice that the optical phase differences, caused by the fibre bending, and are nearly undetectable, indicating that, no effect of bending as shown in Fig. 6. The upper and lower halves of the bent fibre in this figure show no fringes shifts due to the extended and compressed regions of the bent fibre, especially in the cladding region. In Fig. 7-a, b, the extracted optical phase differences, due to the bent fibre, are in good agreement with the calculated ones according to the proposed theory. The mean values of the experimental data, which represent the phase differences, are calculated using 580 rows of pixels through the phase maps of size 380  580 pixels [34]. In this case, Fig. 7-a, b) represents the calculated optical phase differences across the bent fibre as spread data points at bending radii R ¼ 5:3 mm and 8 mm. The parameters of the suggested refractive index functions in the cladding and core areas are accurately selected to provide minimum standard deviation [29]. Fig. 7 displays an excellent agreement between the experimental optical phase differences data and the estimated ones; continuous data line. The coincidence between the two optical phases is confirmed by getting the minimum standard deviation. The minimum deviation value between the two phases is ±0.001 Rad. This could be devoted to the uncertainty of the determined parameters, given in Table 1 By repeating the calculations with different values of

6.28

3.14

0.00

-3.14

-6.28 -60

-45

-30

-15

0

15

30

45

60

r m Fig. 7. The phase differences across fibre diameter for two bending radii (a) R = 5.3 mm, (b) R = 8 mm.

these parameters and recording the corresponding standard deviation, a direct dependence of the experimental data deviation and for each of the selected parameters can be got. Thus, we are able to define the tolerance of the selected parameters, see Table 1. Note that, a given in Eq. (16), which includes both of a0 and D e still have a tolerance given as that in reference [35]. The bent optical fibre is divided around the neutral plane into two halves. The inner half, toward the bending centre, suffers a compression stress. While the outer half, outward the bending centre, suffers a tensile stress. The induced birefringence in the bent optical fibre is dependent on the strain optic coefficient q. In Table 1, the calculated values of q are in a quite agreement with the published value (q ¼ 0:22) [36]. The values of release distance constant r s mentioned in Table 1 are selected during the calculation to give the best fit of the experimental phase. The release

Table 1 The controlling parameters of the proposed refractive index function for the two bending radii. Bending radius [mm]

Strain optic coefficient q ± 0.0024

Release distance constant rs ± 0.32 [lm]

Shape parameter a0 ± 0.063

De [lm
1]

5.3

Compression Expansion

0.215 0.217

5.5 5.3

2.1 2.0


0.001 0.0095

8.0

Compression Expansion

0.215 0.217

4.3 4.1

2.0 1.95


0.0005 0.009

122

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124 1.495

Birefrengence, R=5.3 mm Birefrengence, R=8.0 mm

0.010 1.490

ll

n R =5.3 mm ll

n R =8.0 mm n (bent free)

0.005

1.480

Birefrengence

n Refractive index

1.485

1.475 1.470

0.000

-0.005

1.465 1.460

-0.010

1.455

-60 -60

-45

-30

-15

0

15

30

45

60

r m Fig. 8. The calculated refractive index profiles, fitted to our experimental phase shift differences for the two bending radii, in comparison with refractive index profile of the bent free fibre. The used laser beam is vibrating parallel to the optical fibre axis.

1.495

n R =5.3 mm per n R =8.0 mm per

1.490

n Refractive index

1.485 1.480 1.475 1.470 1.465 1.460 1.455 -60

-45

-30

-15

0

15

30

45

60

r m Fig. 9. The calculated refractive index profiles, fitted to our experimental phase shift differences for the two bending radii. The used laser beam is vibrating perpendicular to the optical fibre axis.

distance constant is inversely proportional to the bending radius, moreover the stresses are reduced with large bending radius [23,27]. These release distance values have asymmetrical properties, where it is larger in the case of compression than that in the case of expansion. This asymmetric behaviour could be raised from increasing and decreasing the fibre outer surface area in the cases of expansion and compression respectively. Asymmetric distributions of the generated stresses and the induced birefringence, with respect to the central fibre axis, have been demonstrated in previous studies [6,22,27]. The controlling shape parameter aðdÞ, of the core refractive index, is a function of a0 and De, see Eq. (16). The value of a0 is selected according to the information provided by the manufacturer and the value of De is selected to recompense the bending effect on the core region. The values of the parameters in Table 1 were used to get the refractive index profiles at the two bending radii R ¼ 5:3 mm and 8 mm. Fig. 8 displays these refractive index profiles in the plane of bending and the incident light is selected to vibrate parallel to the optical fibre axis. Fig. 9 shows

-40

-20

0

20

40

60

r m Fig. 10. The calculated birefringence profiles, for the two bending radii. The used laser beam is vibrating parallel to the optical fibre axis.

the calculated refractive index profiles for the two bending radii, when light is vibrating perpendicular to the optical fibre axis. One can notice that, in Fig. 9 the refractive index still constant in the cladding region and follows the standard profile in the core region. So we can consider this profile to be the same of the straight optical fibre (without bending) for both light polarization components. On light of this consideration, one can compare the profiles of the GRIN optical fibre, with and without bending, to the parallel vibrating component of the polarized light. Subtracting the two refractive index profiles, for the two polarization states of the used light, we obtain the induced birefringence [30]. Refereeing to Fig. 1-a, it is obvious that the bending operation tacks place in XZ plane. Let us clarify that the neutral plane will be located in YZ plane, see Fig. 1-a. Based on the pervious explanation, due to bending, we get a positive birefringence in compressed region and negative in expanded region. Both of the two regions are located in XZ plane, front and behind the natural plane. Fig. 10 shows these birefringences of the two bending radii in the plane of bending. Now, it is possible to get a 2D refractive index profile shape by drawing Eqs. (13) and (14) with d values changing from
rcl to +r cl and r from 0 to rcl . A complete two dimensional refractive index profile of the bent GRIN fibre cross section is demonstrated in Fig. 11-a. This figure is a result of applying the proposed method on the bent GRIN fibre with bending radius of R ¼ 5:3 mm when the light beam is vibrating parallel to the optical fibre axis. It explains how the refractive index varies from the neutral plane to maximum variation at the plane of bending which is quite interesting. Also, it is clear that the refractive index is decreasing in the elongated half and increasing in the compressed half of the bent GRIN fibre. While Fig. 11-b represents the two dimensional refractive index distributions, selecting laser beam to vibrate perpendicular to the optical fibre axis. The induced birefringence across the bent fibre is calculated and presented in two dimensional distribution, see Fig. 11-c. Fig. 12-a, b demonstrates the cross section refractive index distributions in two dimensions of bending radius R ¼ 8 mm, while the incident light is selected to vibrate parallel and perpendicular to the optical fibre axis, respectively. Fig. 12-c shows the cross sectional distribution of the induced birefringence in two dimensions at R ¼ 8 mm.These figures are impressive because they clarify the effect of stress on the refractive index of the bent GRIN fibre. A notable stress release is observed on the bent GRIN fibre cladding region circumference. From Figs. 11-c and 12-c, we realize that the induced birefringence in the core region behaves quasi linearly.

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

Fig. 11. The refractive index cross section distribution (a) the laser beam is vibrating parallel to the optical fibre axis, (b) the laser beam is vibrating perpendicular to the optical fibre axis and (c) the cross section birefringence, at R = 5.3 mm.

123

Fig. 12. The refractive index cross section distribution (a) the laser beam is vibrating parallel to the optical fibre axis, (b) the laser beam is vibrating perpendicular to the optical fibre axis and (c) the cross section birefringence at R = 8 mm.

This supports our expectation of having no stress release at the cladding–core interface.

5. Conclusion In this work, we demonstrate that evaluating the refractive index of a bent graded index optical fibre in a core region is possible. This task is quite difficult for these reasons; (i) when a graded index optical fibre is bent the core and cladding regions suffer a compression and an expansion stresses that applied on the fibre

cross section opposite sides, in the direction of bending radius. (ii) Existence of a release of these stresses on the fibre free surface of the cladding region. The resultant phase variation, for rays traversing the bent GRIN optical fibre, gives a qualitative and implicit information about the refractive indices of the fibre profile. The proposed refractive index profile, of the bent GRIN optical fibre, consider the effect of these stresses, not only on the homogeneous cladding region but also on the graded index core region. Using digital holographic interferometer and comparing

124

W.A. Ramadan et al. / Optical Fiber Technology 36 (2017) 115–124

the calculated optical phase variations to the experimental ones, we are able to confirm the correction of the chosen refractive indices functions. For the first time the 3D refractive indices of the bent GRIN fibre in core region, with two bending radii, are shown. A quite interesting figure of the induced birefringence is given in core region. We think that this way of handling the problem of determining the refractive index profile of the bent GRIN optical fibre is quite interesting, since it can detect stresses generated along fibre diameter in the direction of bending radius in a non-destructive manor. Unprecedented, this method enables us to assess the impact of bending on the parameters controlling the refractive index of the graded index core region. We found that the refractive index value of the core centre is unchanged, whereas Dn value is varying according to the variation of cladding refractive index caused by bending, and then the shape parameter aðdÞ varies linearly with linear stresses which are generated along the fibre diameter. Finally, the point to be addressed here, is that the proposed method has the advantage of providing the change in refractive index profile parameters of the core region aðdÞ and Dn during bending, which must be taken into consideration during the determination of the propagation modes of the launched signals in GRIN bent optical fibres. Acknowledgment The authors would like to express their gratitude to the staff members of ‘‘Bremen Institute For Applied Beam Technology” for their support. Also, many thanks to Professor A. A. Hamza for his useful discussions and encouragement. References [1] A. Méndez, T.F. Morse, Specialty Optical Fibers, 1st ed., Academic Press Elsevier Science Publishing Co Inc., USA, 2007. [2] Y. Shizhuo, B.R. Paul, T.S. Yu Francis, Fiber Optic Sensors, 2nd ed., CRC Press, by Taylor & Francis Group, LLC, USA, 2008, 978-1-4200-5365-4. [3] Y. Fan, G. Wu, W. Wei, Y. Yuan, F. Lin, X. Wu, Fiber-optic bend sensor using LP21 mode operation, Opt. Exp. 20 (2012) 26127–26134. [4] A. Roberts, K. Thorn, M.L. Michna, Determination of bending-induced strain in optical fibers by use of quantitative phase imaging, Opt. Lett. 27 (2002) 86–88. [5] H. Tai, R. Rogowski, Optical anisotropy induced by torsion and bending in an optical fiber, Opt. Fiber Tech. 8 (2002) 162–169. [6] F. El-Diasty, Characterization of optical fibers by two- and multiple-beam interferometry, Opt. Lasers Eng. 46 (2008) 291–305. [7] Y. Park, Tae-J. Ahn, Y.H. Kim, Won-T. Han, Un-C. Paek, D.Y. Kim, Measurement method for profiling the residual stress and the strain-optic coefficient of an optical fiber, Appl. Opt. 41 (2002) 21–26. [8] D.E. Vogler, A. Lorencak, J.M. Rey, M.W. Sigrist, Bending loss measurement using a fiber cavity ringdown scheme, Opt. Lasers Eng. 43 (2005) 527–535. [9] S. Makouei, M.S. Oskouei, A. Rostami, Study of bending loss and mode field diameter in depressed inner core triple-clad single-mode optical fibers, Opt. Commun. 280 (2007) 58–67. [10] M.J. Matthewson, C.R. Kurkjian, S.T. Gulati, Strength measurement of optical fibres by bending, J. Am. Ceram. Soc. 69 (1986) 815–821. [11] A.A. Hamza, T.Z.N. Sokkar, A.M. Ghander, M.A. Mabrouk, W.A. Ramadan, On the determination of the refractive index of a fiber. II. Graded index fiber, Pure Appl. Opt. 4 (1995) 161–177.

[12] T.Z.N. Sokkar, M.A. El-Morsy, H.H. Wahba, Automatic fringe analysis of the induced anisotropy of bent optical fibers, Opt. Commun. 281 (2008) 1915– 1923. [13] H. Vendeltorp-Pommer, J.H. Povlsen, Bending loss and field distributions in a bent fibre calculated with a beam propagating method, Opt. Commun. 75 (1990) 25–28. [14] J. Yamauchi, M. Ikegaya, H. Nakano, Analysis of bent step-index optical fibres by the beam propagation method, IEE Proc. J. 139 (1992) 201–207. [15] S.J. Garth, W.M. Henry, J.D. Love, Limitations in the perturbation analysis of bent finite-clad fibres and waveguides, Opt. Quant. Electron. 27 (1995) 15–33. [16] R.T. Schermer, J.H. Cole, Improved bend loss formula verified for optical fiber by simulation and experiment, IEEE J. Quant. Electron. 43 (2007) 899–909. [17] D. Gloge, Bending loss in multimode fibers with graded and ungraded core index, App. Opt. 11 (1972) 2506–2513. [18] D. Donlagic, A low bending loss multimode fiber transmission system, Opt. Exp. 17 (2009) 22081–22095. [19] H.H. Wahba, Th. Kreis, Digital holographic interferometric characterization of bent optical fibers, J Opt. A: Pure Appl. Opt. 11 (2009) 105407–105411. [20] J. Sakai, T. Kimura, Birefringence and Polarization Characteristics of SingleMode Optical Fibers under Elastic Deformations, IEEE J. Quant. Electron. (1981) 1041–1051, QE-17. [21] T.Z.N. Sokkar, W.A. Ramadan, M.A. Shams El-Din, H.H. Wahba, S.S. Aboleneen, Bent induced refractive index profile variation and mode field distribution of step-index multimode optical fiber, Opt. Lasers Eng. 53 (2014) 133–141. [22] M. Tomozawa, J.-W. Hong, R.W. Hepburn, Y.-K. Lee, Y.-L. Peng, Surface relaxation of silica glass—permanent deformation of SiO2 optical fibres at low temperatures, Phys. Chem. Glasses 43C (2002) 503–507. [23] M. Tomozawa, P.J. Lezzi, R.W. Hepburn, T.A. Blanchet, D.J. Cherniak, Surface stress relaxation and resulting residual stress in glass fibers: a new mechanical strengthening mechanism of glasses, J. Non-Cryst. Solids 358 (2012) 2650– 2662. [24] P.J. Lezzi, Q.R. Xiao, M. Tomozawa, T.A. Blanchet, C.R. Kurkjian, Strength increase of silica glass fibers by surface stress relaxation: a new mechanical strengthening method, J. Non-Cryst. Solids 379 (2013) 95–106. [25] P.J. Lezzi, M. Tomozawa, R.W. Hepburn, Confirmation of thin surface residual compressive stress in silica glass fiber by FTIR reflection spectroscopy, J. NonCryst. Solids 390 (2014) 13–18. [26] P.J. Lezzi, M. Tomozawa, An overview of the strengthening of glass fibers by surface stress relaxation, Int. J. Appl. Glass Sci. 6 (2015) 34–44. [27] W.A. Ramadan, H.H. Wahba, M.A. Shams El-Din, Two-dimensional refractive index and stresses profiles of a homogenous bent optical fibre, Appl. Opt. 53 (2014) 7462–7469. [28] A.W. Snyder, J. Love, Optical Waveguide Theory, Chapman and Hall, London, 1983. [29] A.A. Hamza, M.A. Mabrouk, W.A. Ramadan, M.A. Shams-Eldin, Determination of GR-IN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre, Opt. Commun. 200 (2001) 131– 138. [30] M. Born, E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, 1999, 7th (expanded) edition. [31] Th. Kreis, Handbook of Holographic Interferometry, Optical and Digital Methods, Wiley-VCH, Weinheim, 2005. [32] K. Yassien, M. Agour, C.V. Koyplow, H. Dessouky, On the digital holographic interferometry of fibrous materials, I: optical properties of polymer and optical fibers, Opt. Lasers Eng. 48 (2010) 555–560. [33] H.H. Wahba, Reconstruction of 3D refractive index profiles of PM PANDA optical fiber using digital holographic method, Opt. Fiber Tech. 20 (2014) 520– 526. [34] M.A. Shams El-Din, H.H. Wahba, Investigation of refractive index profile and mode field distribution of Optical Fibers using digital holographic phase shifting interferometric Method, Opt. Commun. 284 (2011) 3846–3854. [35] H.H. Wahba, Th. Kreis, Characterization of graded index optical fibers by digital holographic interferometry, Appl. Opt. 48 (2009) 1573–1582. [36] K. Oh, Un-C. Paek, Silica optical fiber technology for devices and components: design, fabrication, and international standards, John Wiley & Sons Inc, Hoboken, New Jersey, 2012.