CUI De-dong(崔得東)，HAO Chong-yang(郝重陽)

(Northwestern Polytechnical University，Xi'an 710072 Shaanxi，China)

Introduction

The optical fiber image guidance system is an inner loop in missile control system，and its main functions are searching，locking-on and tracking the target artificially or automatically，meanwhile，it outputs the angle rate of sight of missile to target for the missile control.In this process，the image of targets and background acquired by the seeker and the tracking error calculated by the image tracker are transmitted bidirectionally through an optic fiber［1］.Differing from the common communication optical fiber，the guidance optical fiber is wound on a bobbin being mounted near the missile's aft，and is paid out in high velocity accompany the missile.It is used in mal-conditions，thus，its key parameters，such as consecutive length，strength，losses etc，are all claimed strictly.In this paper，we discuss the additional losses mainly.

1 Mechanisms and Model of Optical Fiber Additional Losses

When the light wave propagates in an optical fiber，its intensity reduces gradually with the increase of the fiber's length.This kind of attenuation of light wave is caused by the optical fiber's losses.

The reasons causing the losses include the absorption，scattering and radiation.The absorption losses consist of the inherent absorption and impurity absorption.The scattering losses result from the inherent scattering and imperfect structure scattering［2］.

Some of above factors depend upon the material，structure and production process of the fiber，such as the inherent absorption，impurity absorption including transition metal and hydroxyl absorptions，inherent scattering including Brillouin，Rayleigh and Raman scatterings，and some imperfect structure scattering，such as imperfect structure scattering of the interface between core and clad.The losses caused by these factors are confirmed after production of the optical fiber，we call them as the intrinsic losses.The losses caused in the winding process of the guidance optical fiber，are called as the additional losses.They mainly include the macro bend loss，micro bend loss and the losses caused by low temperature.They are discussed in this paper.

1.1 Macro Bend Loss

Being limited by the diameter of missile，the diameter of bobbin can not be too large.Thus，when the guidance fiber is wound on the bobbin，the macro bending loss occurs.

By using WKB method，the bended fiber can be equivalent to a strait fiber with varied refraction index.We can derive the formula of macro bend loss by using Schrodinger equation［3］

1.2 Micro Bend Loss

The random micro bend of the fiber's axis may occur due to external factors，and the so called micro bend loss is the loss caused by energy exchange among the propagation modes in the bended fiber.The guidance optical fiber is wound on a bobbin in multi-layers with a certain tensile force to guarantee that the fiber can be released without knots and breaks.The hands of spirals in adjacent layers are inverted，thus，there are many cross effects among the layers，as shown in Fig 1.At the cross point，the random bend of fiber's axis occurs due to interlayer pressure，which result in the energy coupling between the basic mode and radiant mode，thereby attenuating the light energy.The main factors causing the light energy in the winding process include pre-tensile force，elastic deformation of optical fiber structure caused by pressure between adjacent layers，change of outside environment，etc.

1.3 Low Temperature Losses

Fig.1 Cross effect between adjacent layers of optical fiber

The fiber's optic characteristics will change under low temperature，and the additional losses will occur.The main factors include the axial compression strain caused by great difference of thermal expansion coefficient between the fiber core and the polymer clad，and the variation of the lateral pressure between the fiber's core and clad.

Usually，the model shown in Fig.2 can be used to analyze the fiber's deformations［4］.

Fig.2 Deformation model of optical fiber

The following total force balance equation［4］can be used to calculate the fiber's deformationW.

whereE0andI0=πd2/4 are the Yang's modulus of optical fiber and the moment of geometry inertia，F(xiàn)fis the axial force，is the lateral force caused by different mechanisms，i.e.，lateral shrink of the clad and increase of lateral stress between layers at the cross region.

whereais the fiber's radius，as shown in Fig.2，Eb(T)andαb(T)are Yang's modulus and thermal expanding coefficient of fiber's coating，ΔT=T－Trefis the difference between actual temperature and refer-ence temperature.Suppose that the deformationW1caused by lateral shrink of the clad is a sine wave with spatial frequency

wherelis a constant representing the length.In the deformed optical fiber，the spatial frequency is unknown.We can denote the deformation as

By using equation(2)and(3)，the initial deformation(W1)nof the optical fiber can be written as

The deformationW2of the optical fiber in the cross region under low temperature can be analyzed similarly［5］.In the wound fiber，the lateral buffer forcecaused by the tensile force of winding and the radial stress between layers mainly depends on the Yang's modulus of the clad，and the modulus can increase under low temperature.The fiber's deformation tends to be bigger because the clad material becomes harder.

By using compressive rod stability theory in thematerial dynamic，the lateral force of the optical fiber can be denoted as

whereWfandWbare the deformations of core and clad in the cross region respectively.They are the interference sources of the optical fiber due to radial stress between layers produced by the tensile force when winding the optical fiber.The parametertis the thickness of clad，fis the elasticity constant of the clad，its definition is

The deformation at cross region is

whereLhas the equivalent order of magnitude to the diameter of the optical fiber.Using the similar method，we can obtain the result of the second mechanism.

Under low temperature，(W2)ntends to be bigger due to the increase of Yang's modulusEb(T)mainly，andαnis known.Whereas，for the initial interfere outside the cross region，the deformation(W1)nis sensitive to the spatial frequency change.Aslis unknown，αncan not be used.W1andW2have the similar relation to the temperature.In this model，there is an increment for the deformation and additional losses of the inner optical fiber.It is what we hoped，because the stress model adopted here predicts that the radial force and radius have an inverse ratio relationship.

wherehμνis the couple coefficient fromμmode toνmode，α'μis the intrinsic loss coefficient of modeμ.Obviously，it is a function of axial dimension of optical fiber.

Suppose that the external interfereδ(x，z)is introduced through refractive index.where

is the distribution index of the optical fiber without interference，n1，aand Δ are axial refractive index，radius of core and difference of refractive indexes between core and cladding.Thus，the couple coefficient can be expressed as

kμν(z)is the transfer probability，and

whereEμis the transverse component of electric field.The constants in equation(16)can be obtained from the orthogonal condition.k0，ω，μ0，P0are wave length in vacuum，frequency，magnetic conductivity in vacuum and incidence power，respectively.Assume that the deviation of ideal optical fiber and micro interfered optical fiber is less than the wave length propagating in the fiber，that is，?1.

The fiber's deformation can be transformed through refractive indexX'=X－W2，using the parabola refractive in equation(13)，the micro interfereδ(x，z)can be denoted as［5］

Now，we can calculate the power coupling coefficient［6］

Forμth mode group，the power coupling equation(12)can be written as

The equation(19)is usually used to calculate the additional losses caused by the deformation of optical fiber under low temperature.

2 Simulation and Experiment Results

Take the optical fiber of 1 000 m in length，andλ=1.31 μm，a=4.5 μm，nc1=1.493，R=50 mm，K=90.The experiment losses for standard bar winding are given in Fig.3.

Taking the fiber's diameterd=250 μm，clad radiusa'=62.5 μm，core radiusa=4.5 μm，refractive indexnc1=1.493，core's axial refractive indexnc0=1.471，thermal expansion coefficientαf=5.5 ×10－7/℃，Yang's modulusEf=7.17 ×1011dynes/cm2，geometric inertiaI0=1.2 × 10－9cm4，coat's thermal expansion coefficientαb=1.0 ×10－5/℃ ，coat's Yang's modulus under 60℃，40℃，25℃，－20℃，－40℃and －60℃ are 5.19×109dynes/cm2，6.23×109dynes/cm2，6.65 ×109dynes/cm2，1.06 ×1010dynes/cm2，1.53×1010dynes/cm2and 1.85×1010dynes/cm2，and the winding tension force is 300 g，spatial frequencyαn= π/2d，we have the simulation result and experiment data of fiber additional loss under low temperature，as shown in Fig.4.The experiment data are the average of three test values under each temperature.

Fig.3 Fiber bending loss

It can be seen that the additional losses of bended optical fiber are inversely proportional to the bending radius.The simulation results coincide better with the experiment data.The additional loss fits well in the high temperature range，but have bigger error in low temperature range，due to the estimating error of Yang's modulus of coating materials in low temperature range is greater than that in high temperature rang.But，the results are all in the dynamic rang of the bidirectional transmission system.

Fig.4 Simulation and experiment results of fiber additional loss under low temperature

3 Conclusions

In proper conditions，the analytical model established in this paper can estimate and analyze the additional losses of optical fiber in the bending and under low temperature with a certain tensile force.It has practical value in the fiber guidance engineering applications.

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