Empirical Model of Dual Loss of Spiral Bend Optical Fiber

Communications in Control Science and Engineering (CCSE) Volume 3, 2015

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Empirical Model of Dual Loss of Spiral Bend Optical Fiber Feng Ni*1, Ruodan Ni2, Ben Buryar3 School of Materials Science and Engineering, Henan University of Science and Technology

1

Sannen Laboratory of Mechanics and Dynamics,

2, 3

Luoyang 471003, China nifeng@haust.edu.cn; 2niruodan@163.com; 3Buryar@163.com

*1

Abstract By a theoretical analysis, it was indicated that multiple losses of optical fiber transmission were suitable to superposition principle, i.e. the total loss of optical fiber transmission equaled the linear superposition of losses caused by various factors. On the basis of the superposition principle and the regression analysis of experimental data, a bend-torsion dual loss model of spiral bend optical fiber was proposed that the spiral bend loss of optical fiber was the superposition of pure bend loss and torsion loss. The pure bend loss was related to the real curvature radius and the number of spirals and the torsion loss was only related to the torsion rate of the spiral.

Keywords Optical Fiber Transmission; Multiple Loss Superposition; Spiral Bend Loss Model

Introduction The bend loss of optical fiber is one of the most important modulation methods of fiber optic sensor (FOS) technology. Compared to reflection and irradiation modulations, the bend loss modulation has better anti-noise ability with a closed light path and is applied widely in FOS technology[1]. Experimental researches along with theoretical models on the bend loss of optical fiber are the technological basic for developing the bending fiber sensor and have been investigated widely so far. The causes of optical fiber bend loss are complicated and manifold, so an exact theoretical model is often difficult to be built or too complicated in form to be used. Therefore, it is more significant practically to build a statistical model based on experimental data.

The earlier and the most fundamental works were finished by D. Marcuse[2] and K. Petermann[3] in 1976. They respectively gave out two different theoretical formulas to calculate the bend loss of optical fiber. And the formulas were all simpler than most of other results based on model coupling theories[4-7]. D. Marcuse simplified the structure of optical fiber as a dielectric core surrounded by an infinite cladding, assumed the refractive index distribution of the medium inside and outside of the core as a two value step function, used weak guidance approximation and acquired a bend loss formula as following[2].

2α =

C1 exp(−C 2 r ) r1 2

(1)

Where, α is the loss coefficient; C1 and C2 are constants related to characteristics of optical wave and fiber; r is curvature radius of optical fiber axis.

K. Petermann assumed the bend loss of optical fiber transmission to the result of power loss induced by converting of guided modes into radiation modes. A constant gradient of refractive index profile and random bends of fiber axis were considered. From a coupled-mode theory, using a quasi-guided mode instead of the radiation modes, the bend loss is calculated by taking account of the coupling between the guided and the quasi-guided modes. And the bend loss coefficient was expressed as following[3].

2α = C (1 r )

2

(2)

Where, C is a constant related to the characteristics of optical wave and fiber; (1 r )2 indicates the square mean of the

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Communications in Control Science and Engineering (CCSE) Volume 3, 2015

reciprocal of random curvature radius of fiber axis. Jun-Ichi Sakai gave out a similar result with an arbitrary refractive index profile acrossing the fiber[4].

The bend loss formulas given by D. Marcuse and K. Petermann are only for the case of pure curvature loss of optical fibers. In a practical transmission process of bend fibers, there are still oscillation phenomena caused by the coupling between the guided mode and other modes. The most typical one is the bend loss oscillation caused by the coupling between the guided mode of fiber core and the whispering-gallery mode of fiber cladding layer. It has been studied deeply by many people and the results are all the product of the pure curvature loss and an oscillation factor reflecting the coupling effects between different modes[5-7]. Although these theoretical analyses are mainly for single mode fibers, but it has been proved by researches that the bend loss of multimode fibers has a similar relation to the curvature radius as single mode fibers and the differences only exist in the constants determined by the characteristics of optical wave and fiber and the refractive index distribution across fiber section[8, 9].

Compared to pure bending, spiral bending not only has a higher sensitivity for its increased cycle number, but also has a more flexible design of sensors for its increased dimensions of parameters. But the investigations of optical fiber bend loss have so far been concentrated more on the pure bend case of optical fiber. An experimental research had investigated the loss of spiral bend optical fiber[10]. The number of spiral was used as a factor to modify Marcuse formula and Petermann formula. New loss formulas to be more suitable to the spiral bend fiber were proposed as called as “Modified Marcuse formula” 2α r =

C1 n exp(−C 2 r ) Lr 1 2

(3)

and “Modified Petermann formula” 2α r = Cnr −2 L .

Here, α r is the

(4)

α in formulas (1) and (2); and r is the real curvature radius and given by the formula as following. r =R+

h2 4π 2 R

(5)

Where, R is the spiral radius of optical fiber axis, h the pitch of spiral winding, n the number of spiral and L the length of curved part of fiber. −2 nr −1 2 exp(−C2 r ) was called Marcuse modified variable and nr Petermann modified variable. Fig. 1 shows the

relationships of the two variables to the spiral bend loss respectively. It can be seen that there is a good linear relationships between Marcuse modified variable with C 2 = 1 or Petermann modified variable and the spiral bend loss, i.e. the formulas (3) and (4) are suitable well, in the range of larger values. However the linear relationship is deviated at small values.

FIG. 1 RELATION OF MODIFIED VARIABLES AND LOSS[10]

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In this paper, it was proved by a theoretical analysis that the fiber transmission losses caused by multiple factors were suitable to superposition principle. The spiral bend loss of optical fiber was analyzed as two parts of pure bend loss and torsion loss and an empirical model of bend-torsion dual loss of spiral bend optical fiber was proposed with the regression analysis of experimental data. It was hoped to have some help to an elaborate theoretical modeling and more experimental researches.

Superposition of Multiple Losses of Fiber Transmission There are various factors causing light power losses in the process of optical fiber transmission. The loss coefficient α i of some single factor i is defined as following.

2α i = −

1 Li

∫

Li

0

dP 1 dl = − Pdl Li

∫

i Pout

Pini

i dP ln Pini − ln Pout = P Li

(6)

Where, i is the number of the factor causing light power loss; Li is the length of the part of optical fiber, on which the factor i acts; dP dl is the loss of unit length at some point along the axis of fiber; P is the power of transmission i are respectively the power of transmission light at the origin light at some point along the axis of fiber; Pini and Pout

and end point of the part of optical fiber, on which the factor i acts. Because light power transmitted through the optical fiber is a scalar quantity, the total loss should be the accumulation of various kinds of losses caused by all factors acting on the fiber when there were several loss factors acting on the light transmission process of optical fiber at the same time.

According to the relationship with each other, various transmission loss factors of optical fiber can be divided into two cases: tandem or parallel coexistence factors. The former is related to that various loss factors coexist successively along the optical path and the latter that various loss factors coexist simultaneously acting on the same part of optical path. In the case of tandem coexistence factors, it is assumed that the transmission loss of optical fiber could be divided into m kinds of different loss mechanisms, the light powers accumulating the losses one by one is Pin , P1 , P2 …, Pm−1 and Pout respectively from the origin to the end point. So the total efficiency of optical fiber transmission is T=

Pout P P P = 1 2  out .

Pin Pin P1 Pm −1

(7)

And the transmission loss coefficient of optical fiber is ln Pin − ln Pout L (ln Pin − ln P1 ) + (ln P1 − ln P2 ) + (ln Pm−1 − ln Pout ) . = L L L1 L2 = ⋅ 2α 1 + ⋅ 2α 2 +  + m 2α m L L L

2α =

(8)

While the total loss is 2αL = ln Pin − ln Pout

= (ln Pin − ln P1 ) + (ln P1 − ln P2 ) + (ln Pm−1 − ln Pout )

.

(9)

= 2α 1 L1 + 2α 2 L2 +  + 2α m Lm

In the case of m kinds of parallel coexistent loss factors, it is assumed that the interactions of different factors can be neglected, i.e. the first order approximations are suitable and total of light power losses can be written as dP ≈ dP1 + dP2 +  + dPm . Then the transmission loss coefficient of optical fiber is 2α = −

1 Pout dP 1 Pout dP1 + dP2 +  dPm ≈− ∫ = 2α 1 + 2α 2 +  2α m . ∫ P in L P L Pin P

(10)

While the total loss is

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Communications in Control Science and Engineering (CCSE) Volume 3, 2015

2αL ≈ 2α 1 L + 2α 2 L +  + 2α m L .

(11)

It is showed by formulas (8), (9), (10) and (11) that the total loss or the loss coefficient of multiple factors can be written or approximated as a linear sum of the corresponding items of various factors not only for the tandem coexistence factors but also for the parallel ones. This is so called the superposition characteristic of multiple losses of optical fiber transmission.

Regression Analysis of Multiple Loss of Spiral Bend Fiber Marcuse formula and Petermann formula are both obtained at the condition that the optical fiber is purely curved.

The curvature variable of pure bend is only the curvature radius when the length of deformational part is fixed.

But under the condition of spirally winding and the length of deformational part is fixed, the number of independent variables increases from one to two. With different spiral pitches, the optical fibers of the same curvature radius have different number of spiral. Apparently, besides the curvature radius, the density of spiral circles, i.e. the number of spiral with a unit deformational length, n L , is also a factor contributing to the transmission loss of the optical fiber. The reference [10] assumed that a multiplier should be put into the formulas obtained by Marcuse or Petermann, and the bend loss coefficient of spiral optical fiber should be expressed as formulas (3) and (4). However, these two formulas only reflect the bend loss caused by local curvature of fiber.

Except bending deformation, there is still torsion deformation on a spiral fiber. So the total transmission loss of spiral fiber includes not only the bend loss expressed by formulas (3) and (4) but also losses caused by torsion deformation or other factors, which should be relevant to the number of spiral, and here we assume its loss coefficient is 2α n = C 3 nν L .

(12)

The total transmission loss of a spiral optical fiber is the sum of bend loss, torsion loss and others, i.e. there is α = α r + α n +  , so its functional form could be expressed as 2αL = C1 nr −1 2 exp(−C 2 r ) + C3 nν + C 4

(13)

2αL = Cnr −2 + C3 nν + C 4

(14)

or

Where, C 4 represents other kinds of losses which were not taken into account of bend and torsion losses.

Making regression analysis according to these two formulas, good fitting results had been obtained. We rewrote the formulas (13) and (14) as following.

2αL = (C1 + C 3 )λ + C 4 ,

λ=

(15)

C3 C1 nr −1 2 exp(− C 2 r ) + nν .

C1 + C 3 C1 + C 3

2αL = (C + C 3 )σ + C 4 ,

σ=

(16)

C3 C nr − 2 + nν .

C + C3 C + C3

Here we call λ as Marcuse multi-analysis variable of spiral winding and σ as Petermann multi-analysis variable.

Fig. 2 shows that there are very good linear relationships between the spiral bend loss and these two variables respectively. The fitting values of the parameters are listed in Table 1.

TABLE 1 FITTING RESULTS OF EXPERIMENTAL DATA WITH MULTIPLE LOSS MODEL OF SPIRAL WINDING

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Formulas

Multi-analysis Variable

(13), (15)

λ

(14), (16)

σ

C

2.0807

C1

C2

C3

C4

ν

Regression Coefficient δ 2

5.1220

1

0.6035

-0.2469

0.5

0.9955

0.5574

-0.3039

0.5

0.9958

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FIG. 2 RELATION OF MULTI-ANALYSIS VARIABLE AND LOSS (DATA ARE THE SAME OF FIG.1 AND FROM REFERENCE [10])

Empirical Model of Dual Loss of Spiral Bend Fiber Having a deeper analysis, spiral bend deformation of optical fiber is composed of pure bend with the real curvature radius r and torsion deformation with a torsion rate (torsion angle on unit length of optical fiber) ω=

2πh 2πhn 2 . = 2 L2 h 2 + (2πR )

(17)

Imitating the formulas (13) and (14) above, we assume (18)

2αL = C1 nr −1 2 exp(−C 2 r ) + C3ω ν + C 4

or 2αL = Cnr −2 + C3ω ν + C 4 .

(19)

Where, C 4 represents other kinds of losses which were not taken into account of pure bend and torsion losses.

Making regression analysis according to these two formulas, better fitting results than that in above section can be obtained. We may rewrite the formulas (18) and (19) as following.

2αL = (C1 + C3 )κ + C4 ,

κ=

(20)

C3 C1 ων . nr −1 2 exp(− C2 r ) + C1 + C3 C1 + C3

2αL = (C + C3 )θ + C4 ,

θ=

(21)

C C3 ων . nr − 2 + C + C3 C + C3

Here we call κ as Marcuse bend-torsion variable of spiral winding and θ as Petermann bend-torsion variable. Fig. 3 shows that there are better linear relationships between the spiral bend loss and these two variables respectively than that in Fig. 2. The fitting values of the parameters are listed in Table 2. It is interested that the values of parameter C 4 are nearly zero for two formulas of (20) and (21). This implies that there is almost not any other loss item included in the spiral bend loss of optical fiber excepting pure bend loss and torsion loss. So it is reasonable to model spiral bend loss of optical fiber with two factors of pure bend and torsion loss.

TABLE 2 FITTING RESULTS OF EXPERIMENTAL DATA WITH BEND-TORSION DUAL LOSS MODEL OF SPIRAL WINDING

Formulas

Dual analysis Variable

(18), (20)

κ

(19), (21)

θ

C

2.1815

C1

C2

C3

C4

5.3778

1

1.4472

2×10-7

1.1930

-9×10

-16

ν

Regression Coefficient δ 2

0.5

0.9942

0.5

0.9950

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FIG. 3 RELATION OF BEND-TORSION VARIABLE AND LOSS (DATA ARE THE SAME OF FIG.1 AND FROM REFERENCE [10])

Conclusions It was indicated by the theoretical analysis that the total loss of optical fiber transmission caused by multiple factors was suitable to superposition principle, i.e. the total loss of optical fiber transmission equaled the linear superposition of losses caused by various factors.

On the basis of the superposition principle and the regression analysis of experimental data, a bend-torsion dual loss model of spiral bend optical fiber was proposed that the spiral bend loss of optical fiber was the superposition of pure bend loss and torsion loss. The pure bend loss was related to the real curvature radius and the number of spirals and the torsion loss was only related to the torsion rate of the spiral.

ACKNOWLEDGMENT

Financial support came from No.0111040400, No.200410464002 and No.200510464022 of the natural science foundation of Henan province and No.2005ZD004 of the research foundation of Henan university of science and technology.

REFERENCES

[1]

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[2]

Marcuse D. “Curvature loss formula for optical fibers.” J. Opt. Soc. Am. 66(1976) No. 3: 216-220.

[3]

K. Petermann. “Fundamental mode microbending loss in graded-index and W fibres.” Optical and Quantum Electronics 9 (1977): 167-175.

[4]

Jun-Ichi Sakai. “Microbending Loss Evaluation in Arbitrary-Index Single-Mode Optical Fibers. Part I: Formulation and General Properties.” IEEE Journal of Quantum Electronics QE-16(1980) No. 1: 36-44.

[5]

Hagen Renner. “Bending Losses of Coated Single-Mode Fibers: A Simple Approach.” Journal of Lightwave Technology 10(1992) No. 5: 544-551.

[6]

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[7]

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[8]

Zhe-ming Zhao, Cheng-hua Sui. “Measurement and analysis of characteristic of bend loss in multi- mode fiber.” Optical Instruments (in Chinese) 27(2005) No.5: 29-32.

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[9]

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Ross T. Schermer, and James H. Cole. “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment.” IEEE Journal of Quantum Electronics 43(2007) No. 10: 899-909.

[10] Ruo-Dan Ni, Xue Song, Li-Juan Qian, Si-Hui Wang, Ben Buryar. “Experimental research on spiral macrobend loss of multimode fibers.” Applied Mechanics and Materials 423-426: 2321-2329.

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