temperature characteristics of fiber optic gyroscope sensing coils

ISSN 1064�2269, Journal of Communications Technology and Electronics, 2013, Vol. 58, No. 7, pp. 745–752. © Pleiades Publishing, Inc., 2013.Original Russian Text © A.M. Kurbatov, R.A. Kurbatov, 2013, published in Radiotekhnika i Elektronika, 2013, Vol. 58, No. 7, pp. 735–742.

745

1 INTRODUCTION

It is well known that the characteristics of a fiber�optic gyroscope (FOG) are strongly worsened underthe influence of time�varying thermal fields on itssensing coil [1]. These fields cause the FOG tempera�ture bias drift, or the Shupe effect (SE), which istreated as one of the FOG basic problems.

In Fig. 1, the FOG coil is depicted with a rectangu�lar cross section, and, four heat flows are shown in theremoved section. Let us call flows 3 and 5 radial andflows 4 and 6 axial. Below, we call the fiber–compoundmedium just the medium. In Fig. 1, the following coilgeometric parameters are also given: R1 and R2 are theinner and outer diameters, respectively, and h is theheight. Coils with small h are referred to as low ones,and coils with large h are referred to as high ones.

In papers [2–4], two equivalent expressions aregiven for the thermal drift:

(1a)

(1б)

where D and L are the coildiameter and fiber length, respectively; n is the fiberrefractive index; dn/dT is the fiber thermal sensitivity;Т(z, t) is the temperature distribution along the fiber attime moment t; and

1 The article was translated by the authors.

1

( ) ( ) ( ) ( )[ ]2

0

2 , , ,

L

t B z L T z t T L z t dzΔΩ = − − −∫ � �

( ) ( ) ( )0

2 , ,

L

t B L z T z t dzΔΩ = −∫ �

( ) ( )1 ,B DL n dn dT=

is the time partial derivative of this distribution. Here,we do not consider the contribution of fiber lengththermal changes, because it is small in conventionalquartz fibers. From (1à), it is seen that fiber sectionsthat are more distant from the fiber midpoint to SE[1–7] more substantially contribute than less distantones. From (1a) and (1b), it is also seen that two waysof reducing the SE influence are possible: (i) integra�tion paths specially chosen to minimize the integralsand (ii) reduction of integrands.

Physical realization of the first way consists in spe�cial fiber winding techniques [2, 4]. They considerablysuppress the SE influence; however, they still do notyield a necessary accuracy themselves [5].

The second way is thermal insulation and utiliza�tion of metallic screens [6]. Thermal insulation slowsdown the temperature field variations (reduces the val�ues and ), metallic screens acceleratetemperature smoothing within the coil (reduce thedifference . In combination withspecial fiber winding techniques, these steps allowreaching the necessary accuracy. For example, in [6],a coil with quadruple winding, air as a thermal insula�tor, and a copper carcass is described; the coil yieldsthe drift 0.04 (deg/h)/(°C/min). In [5, 7], a coil with adrift lower than 0.01 (deg/h)/(°C/min) is described; itis based on a quadruple winding [2–4], a carbon car�cass, and a compound that has high heat conductivity(unlike that from [6]) and is deposited on the fiber dur�ing its winding.

In this paper, we describe a combination of specialfiber winding techniques with thermal insulation and

( ) ( ), ,T z t T z tt∂

≡∂

�

2

( ),T z t� ( ),T L z t−

�

( ) ( ), ,T z t T L z t− −

� �

Temperature Characteristics of Fiber�Optic Gyroscope Sensing Coils1

A. M. Kurbatov and R. A. KurbatovThe Kuznetsov Research Institute of Applied Mechanics (a division of the Center for Ground�Based Space Infrastructure

Facilities Operation), ul. Aviamotornaya 55, Moscow, 111123 Russiae�mail: [email protected]

Received August 15, 2012

Abstract—In a 2D model, a fiber�optic gyroscope (FOG) temperature drift is theoretically investigatedunder the influence of temperature fields on its sensing coil for two techniques of fiber winding. The temper�ature field in the fiber cross section is calculated by means of the finite�difference method. It is establishedthat, for the FOG temperature drift reduction to the level of 0.01 deg/h, the coil size being retained smallenough, it is effective to combine the thermal insulation, metallic screens, and compound with high heat con�ductivity.

DOI: 10.1134/S1064226913060107

PHYSICAL PROCESSES IN ELECTRON DEVICES

746

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 7 2013

A.M. KURBATOV, R.A. KURBATOV

metallic screens. We consider the well�known quadru�ple winding [2–7] and the Malvern winding [8],which, as far as we know, previously was not investi�gated in the literature. In Section 1 of the study, thecalculation of thermal fields is described; in Section 2,the general features and realization techniques aredescribed for chosen windings; in Section 3, the SE incoils without thermal insulation and metallic screensis calculated; and in Section 4,the SE suppression withthe use of thermal insulation and metallic screens isdescribed.

1. CALCULATION OF THERMAL FIELDS

To calculate the thermal fields in the coil cross sec�tion, we proceed from the heat equation in cylindricalcoordinates [9]:

(2)

where κ, ρ and c are the heat conductivity, density andheat capacity at the point (r, z). The value κ/ρc is thethermal conductivity [10]. The medium is inhomoge�neous, because it contains a fiber, a polymer coating,and a compound. We replace this medium by a homo�geneous medium with parameters that are averagedover all of the aforementioned materials according totheir volume fraction.

Assume that the coil cross section is rectangular, so,the finite�difference method is suitable for solving Eq.(2). Let us introduce a uniform coordinate mesh with

( ) ( )

( ) ( )

( ) ( )

∂ρ∂

∂ ∂⎡ ⎤= κ⎢ ⎥⎣ ⎦∂ ∂∂ ∂⎡ ⎤+ κ⎢ ⎥∂ ∂⎣ ⎦

, , ( , , )

1 , , ,

, , , ,

r z c r z T r z tt

r r z T r z tr r r

r z T r z tz z

steps Δr and Δz in the radial and axial directions,respectively, and a time mesh with step Δt. Grid func�

tion = at time moment tn + 1 is deter�

mined from its values = at time momenttn with the help of Eq. (2) discretized according to thefollowing scheme [9, 11, 12]:

(3a)

(3б)

where

ξ is the parameter determining the type of discretiza�tion scheme (0 ≤ ξ ≤ 1). The index n + 1/2 correspondsto the time moment tn + 1/2 = (tn + 1 + tn)/2. Here, a 1Ddiscretization scheme from [11] is taken as a basis. Inthe calculation scheme determined by (3a) and (3b),tridiagonal linear systems are solved by means of thetridiagonal matrix algorithm (economical scheme)[12]. We put ξ = 1/2, corresponding to the Crank–Nicolson scheme, which, for values of Δt that are notsmall enough, yields a large temperature field at the

1,n

i jT +

( )1, ,i j nT r z t+

,n

i jT ( ), ,i j nT r z t

( )

( )

( )

+ + +

− +

+

+ + −

+ + +

⎡− = ξ − +⎣⎤ ⎡+ + − ξ⎦ ⎣

⎤− + + ⎦

1 1 1

1

2 2 2, , , 1, , 1, ,

21, 1, , 1,

, 1, , 1, 1,

1

,

n n n ni j i j r i j i j i j i j i j

n ni j i j r i j i j

n ni j i j i j i j i j

T T c ar T ar ar T

ar T c ar T

ar ar T ar T

( )

( )

( )

+ + + +

− +

+ +

+ + −

+ +

− + +

⎡− = ξ − +⎣⎤ ⎡+ + − ξ⎦ ⎣

⎤− + + ⎦

1

1

1 1

1 2 1 1, , , , 1 , , 1 ,

1 2, 1 1, , , 1

2 2, , 1 , , 1 , 1

1

,

n n n ni j i j z i j i j i j i j i j

n ni j i j z i j i j

n ni j i j i j i j i j

T T c az T az az T

az T c az T

az az T az T

, 1, 1,

, 1, 1

2 2,i j i j i

i ji j i j i i

rar

r r−

−

− −

⎛ ⎞κ κ ⎛ ⎞= ⎜ ⎟⎜ ⎟κ + κ +⎝ ⎠⎝ ⎠

, , 1,

, , 1

2,i j i j

i ji j i j

az −

−

κ κ

=

κ + κ

2,r

tcrΔ

=

Δ2

,ztc

zΔ

=

Δ

h

R1

R2

1

2

3

z4

8

5

6

rO

R1 R2

h

7

Fig. 1. General diagram of a FOG coil and its cross section with indicated heat flow directions; 1 is a coil, 2 is the cross section,3 and 5 are radial heat flows, 4 and 6 are axial heat flows, 7 is the fiber–compound medium, 8 is the coil carcass, R1 and R2 arethe inner and outer coil radii, and h is the coil height.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 58 No. 7 2013

TEMPERATURE CHARACTERISTICS OF FIBER�OPTIC GYROSCOPE 747

points of large thermal conductivity jumps. However,the latter are absent in the medium; so, specifically forthe thermal drift, this scheme provides for rapid con�vergence in the case of time mesh refinement (stepreduction). The convergence is more rapid than thatfor ξ = 1 (Yanenko method [12]) by, at least order, anorder of magnitude.

As the boundary conditions (BCs) at r = R1,2, z = 0,and z = h (values R1,2 and h are shown in Fig. 1) we usea combination of the BCs of the second and thirdkinds (BC–2 and BC–3) [10]:

where α is the coefficient of heat exchange with theenvironment (BC–2 for α = 0), θr1,2(z, t) are radialexternal heat flows falling on the boundaries r = R1,2,θz0,h(r, t) are the axial external heat flows falling on theboundaries z = 0 and z = h (BC–3 for θr1,2 = θz0,h = 0),and Т(t) is the environmental temperature. Calculatedby the described scheme, the temperature field is usedin (1a) and (1b), providing for a temperature drift.

2. FIBER WINDING TECHNIQUES

Consider the quadruple winding (QW) [2–4] andwinding from [8]. They are illustrated in Figs. 2a, 2b,where the light propagates in one direction throughthe turns designated by filled circles and in anotherdirection through the turns designated by empty cir�

( ) ( )1,21,2

1,2( ) , ,r Rr RT r T T t r z t

==

κ∂ ∂ + α − = θ⎡ ⎤⎣ ⎦

( ) [ ] ( )0, 0,0,( ) , ,z h hz h

T z T T t z r t=

=

κ∂ ∂ + α − = θ

cles [2]. On the basis of Fig. 2b, we call the secondwinding a chess winding (its abbreviation CW shouldnot be confused with clockwise).

For the QW, the fiber midpoint is placed on the bot�tom left side, the first layer is wound in one direction,the second layer is wound in the opposite direction,the third layer is wound in the same direction as thesecond layer, the fourth layer is wound in the samedirection as the first layer, and so on. Each four layersare called quadrupoles, their number may be integer orhalf�integer. In the first case, the total lengths of filled�circle and empty�circle layers are equal to each other,unlike the second case. The latter means that the SEcan tend to a nonzero asymptotic value (similarly tothe dipole winding also described in [2–4]). Besidesthis, in the case of the half�integer number of quadru�poles, the winding is finished near the right wall, lead�ing to strong sensitivity to the axial temperature gradi�ent. Thus, number Nr of layers should be a multiple offour. As a result, for the QW, the fiber sections equidis�tant from its midpoint can be brought together; so,presently, the QW is regarded as the most suitable forhigh�performance FOGs [2–7].

As for the CW, the fiber midpoint is at point 2 inFig. 2b, and two fiber halves are wound in the oppositedirections to the vertical edges. After reaching theseedges (the first layer) fiber sections are wound awayfrom these edges to baffle 1; then, a transition of fibersections into other winding halves divided by baffle1 takes place. Next, the fiber is again wound to verticaledges; then, again to baffle 1; after that, there is one

r

z

r

z

(а) (б)

1

2

Fig. 2. FOG coil cross section with (a) QW and (b) CW. Filled and empty circles denote the turns along which the light propagatesin opposite directions; 1 is a baffle and 2 is the fiber midpoint.

748



more transition to other winding halves, and so on.This winding type is much easier to implement thanthe QW due to smooth fiber transitions from one layerto another (near vertical edges and through baffle 1 inFig. 2b).

For the CW, two modifications are possible: with asingle outer layer (Fig. 2b), when fiber ends are situ�ated near vertical edges, and without it (fiber ends aretogether near baffle 1). It is clear that, in the case of thehigh regularity of fiber layers forming, the CW is prin�cipally nonsusceptible to radial thermal flows and toaxial flows with equal powers (symmetrical axial tem�perature gradient).

However, for the CW, unlike the QW, fiber sectionsequidistant from the fiber midpoint are not placedtogether; so, the sensitivity to single�sided axial ther�mal flow should take place (asymmetrical axial tem�perature gradient). One may think that the SE herewill be as large as for ordinary winding (OW), which ismade sequentially from lower layers to upper ones [2–4], but this is not so. Indeed, the CW part situatedunder the outer layer can be divided into pairs of layers(Fig. 2b). Two layers of each pair give a contribution tothe SE with the same signs, but the contributions fromdifferent layer pairs will be of the variable sign. On theone hand, each following pair consists of sections

more distant from the fiber midpoint, than the previ�ous pair; so, there is no interpair SE cancellation. Onthe other hand, this allows us to expect that, here, theSE, at least, will be not as large, as for the OW.

Thus, the QW places together the fiber sectionsequidistant from the fiber midpoint, thus, ensuringmaximally close physical conditions for them. In thecase of the CW this is not so, but this winding yields anoriginal way of summing the SE throughout individualfiber turns, yielding a small resulting thermal drift.

3. CALCULATION OF THE THERMAL DRIFT

For the SE calculation, we set radial heat flows forthe QW ([4, 13]) and a single�sided axial flow for theCW (the worst situations for them). As will be shownbelow, a low coil (see above) is better for both windingtechniques.

Assume that the medium is incorporated into ametallic carcass of invar with 1�mm�thick walls. Thethermal parameters of the rest materials correspond to[4] and are listed in Table 1. Assume that the heat flowhas a power which warms up the wall the closest to itscarcass approximately by 10°С in 10 min. In Figs. 3aand 3b, the SE time dependences are shown for theouter and inner radial heat flows at 2R1 = 30 mm,L = 1000 m, the fiber diameter 80 µm, the fiber coat�ing diameter 160 µm, the diameter of the fiber withcompound d = 200 µm, and the number of fiber layersalong the r axis Nr = 92. For the outer radial flow, theSE peak value is 125 deg/h; for the inner flow, it is only0.37 deg/h. Such difference is due to the fact that theouter layers are most distant from the fiber midpoint,and vise versa for the inner layers [4].

Note, that, under the axial heat flow, the SEextreme value is 3.1 deg/h due to the thermal conduc�tivity jump at the carcass/medium boundary, which

Table 1

Material Thermal conductivity, m2/min

Invar 2.18 × 10–4

Quartz fiber 5.08 × 10–5

Fiber coating 7.8 × 10–6

Potting compound 7.8 × 10–6

0

–20

–40

–60

–80

–100

–120

–1403210

t, s

(a)ΔΩ, deg/h0.05

0

–0.05

–0.10

–0.15

–0.20

–0.35

–0.403210

t, s

(б)ΔΩ, deg/h

–0.25

–0.30

Fig. 3. Temperature drift for QW in the presence of (a) external and (b) internal radial heat flows.



distorts the thermal field. In the absence of this jump,the QW sensitivity to axial heat flow is small.

Let us give a physical explanation of the results.While the coil is warmed, the function in themedium exhibits the following behavior. At first, thisfunction is zero (heat did not pass through the car�cass); then, it becomes nonzero successively in the firstlayers, but it is nonuniformly distributed over themand sharply grows in time. Thus, these layers give sub�stantially different contributions to the SE; so, as aresult, it is large enough (Fig. 3a). However, then thefunction involves more and more layers anddecreases in time, but the principal moment is that itbegins to be more and more uniformly distributed overthe medium. The latter means rather uniform sum�ming of SE contributions from individual quadru�poles, so that the SE value in Fig. 3a quickly goesdown.

Thus, it follows that, for the QW, it is not importantwhat number of winding layers will be after the firstlayers, which form the SE extremum in Fig. 3a.Besides this, the higher the coil, the larger the numberof turns within a layer; i.e., the larger the contributionof each layer to the SE. This means that the low coil ismore advantageous.

In the presence of the radial heat flow from theinside to the outside (Fig. 3b), the SE has a negativeextremum, then, reverses the sign (as in [4]), and,next, tends to zero after 4 min. This slow decaying wasnot observed in [4, 13], where the SE after the sign

T�

T�

reversal tends to zero within a few seconds. In ourcase, such a long decaying is due to fast heat branchingthrough the side carcass walls to layers more distantfrom the fiber midpoint (and their involvement fromthe medium lateral sides), in comparison with directheat wave penetration into the medium after passing thecarcass wall that is the closest to the heat flow. Note that,in the absence of side walls, our calculation results are ingood agreement with the data from [4, 13].

Thus, the QW indeed does not provide for coilacceptable performance itself.

Figure 4 illustrates the SE for the CW under thesingle�side axial heat flow with the same power as inthe case of the QW. It is seen that the SE extremum forthe CW with an outer layer is seven times smaller thanthe absolute value of the SE extremum for the CWwithout an outer layer, which is in good agreementwith the data from [8]. This is explained by the factthat the total contribution of layer pairs that precedethe outer layer to the SE is close to the contribution ofthe outer layer, but has the opposite sign. As it is seenfrom Fig. 4, in both cases, the absolute value of the SEextremum appears to be much smaller than that in Fig. 3a.Thus, although in the CW, equividistant sections arenot put together, the SE values are much smaller thanfor the QW and OW.

Note that raising of compound heat conductivityby 10 times leads to the absolute value of the SE extre�mum for the CW in the presence of an outer layer isreduced by 3 times, and, in the absence of an outerlayer, it i reduced only by 5 deg/h. This is probably due

0

–10

–20

–3015129630t, s

(b)

4

3

2

01512963t, s

(а)

1

ΔΩ, deg/h

Fig. 4. Temperature drift for CW (a) with and (b) without an outer layer.

750



to the fact that, near the outer layer, the temperaturefield distortion due to the jump of the medium andcarcass’s thermal parameters now is smaller. That iswhy the SE contribution of the outer layer becomescloser in the absolute value (and retains the oppositesign!) to the contribution of the rest layers. However, itis clear that, in itself, the CW, similarly to the QW, doesnot provide for coil acceptable performance either.

4. THERMAL INSULATIONAND METALLIC SCREENING

For coil protection from thermal fields one canthermally insulate it, for example, by an air layer, as itis described in [6]. Howeve,r in this case, convectionheat flows can be formed; therefore, for thermal insu�lation, it is better to use a solid medium. As an insula�tor, let us consider foamed polyurethane, which has alow heat conductivity (0.03 W/(m K)), high heatcapacity (1500 W/(sec kg K)), and high enough den�sity (200 kg/m3). According to the calculation, for theSE to be not not higher than a value of 0.01 deg/h forboth winding cases, a layer with a thickness of severaltens millimeters is required, which strongly increases

the coil size. The situation is a little better for the lowercoil (Nr = 112) and for a heat conductive compound,but this is still not enough.

On the contrary, if one tries to suppress the thermaldrift with the help of a copper carcass, then, its thick�ness also should be several tens of millimeters,although the drift time duration here is smaller by twoor three 2–3 orders of magnitude.

In [6], a combined application of the QW, thermalinsulation (air), and metallic screening of the coilwound on a copper carcass is described. Let us showthat, combining the thermal insulation, metallicscreens, and a heat conductive compound, for the CW,one can reach a required accuracy, while keeping thecoil size small enough. We consider only a low coil(Nr = 112) with the compound thermal conductivity#10 times larger than in Table 1.

Consider a coil with the cross section shown in Fig. 5.The coil contains a Permalloy screen, which is also amagnetic shield, foamed polyurethane and copperlayers, a carcass with a baffle, and a medium. The geo�metric parameters of the coil and layers are summa�rized in Table 2. In Fig. 6, the SE behavior is shown forthis coil. It is seen that the CW with and outer layerprovides for the accuracy 0.01 deg/h at a rather smallcoil dimension (even with a 3�mm�thick Permalloyscreen). Note that a similar structure (but with 2.5�mm�thick radial insulator and 1.5�mm�thick lateralthermal layers) yields for the QW the extreme SE value~0.1 deg/h, which is larger by an order of magnitudethan for the CW with an outer layer but smaller by anorder of magnitude than for the CW without an outerlayer.

It is also clear that metallic screens reduce thethickness of ths lateral layer of the thermal insulator tothe value 2.5 mm; however, even this thin layer plays animportant role, because, without it, the SE extremumincreases more than by an order of magnitude. Besides

1

2

3

4

5

r

z

Fig. 5. FOG coil cross section; 1 is a Permalloy screen, 2 is a thermal insulator, 3 is copper, 4 is a carrying carcass, 5 is a medium.

Table 2

Parameter Value, mm

First (inner) fiber layer diameter 30

Coil outer diameter 86.4

Coil height 25.8

Permalloy screen width 3

Thermal insulation lateral side layers width 2.5

Thermal insulation radial layers width 1.5

Copper layers width 0.5



this, the SE time duration appears to be shorter by twoorders of magnitude than in the above case, corre�sponding to thermal insulation alone. This could beexplained by the fact that the Permalloy layer effec�tively branches the heat to the coil lateral wall oppositeto the one where the heat flow falls, a circumstancethat is critical for the CW.

Besides this, the copper layer yields additionallyreduces the SE value by a factor of four, which is prob�ably due to more rapid equalization of axial heat flows .In the considered coil, it is also possible to depositcopper onto the inner carcass walls in each of the tworegions containing the fiber. However, according tothe calculation, this technique does not lead to a sub�stantial change in the SE value.

Note in addition that the increase of the compoundthermal conductivity, in turn, yields the two�fold SEvalue reduction. Thus combined application of theCW fiber, thermal insulation, metallic screens, and aheat conductive compound allows reaching the FOGcoil low temperature sensitivity, relatively smalldimensions of the coil being retained.

5. ON THE SHUPE EFFECTOF THE SECOND KIND

The purely thermal drift considered above is usuallycalled as SE of the first kind or SE�1 [5], because, in[5, 6, 14], a conception of the SE of the second kind

1

(SE�2) is introduced. The latter effect is considered tomean the FOG drift due to thermal mechanicalstresses. These arise from the difference of the thermalexpansion coefficients (TECs) of coil composingmaterials. Due to the photoelastic effect, these stressesinduce fiber refractive index variations in time in addi�tion to the value dn/dT, which is responsible for SE–1.

According to [5, 14], a feature, differing SE–2from SE–1, is the presence of a drift even in the casewhen time�varying temperature is uniformly distrib�uted over the coil cross section. This is due to the factthat mechanical stresses in this situation at any timemoment (unlike the temperature) are distributed overthe coil cross section nonuniformly. This means thatthe SE�2 time duration can be much longer than thatof SE�1. Note, however, that, in [5, 6, 14], the case ofthe QW is considered. In this situation, SE�2 disap�pears only after temperature equilibrium distributionover the coil cross section is settled; i.e., (forexample, as a result of equalization of the external heatflow by the convective heat exchange with the envi�ronment). At the same time, for the CW, it is obviousthat SE–2 disappears not only for the uniform fielddistribution over the coil cross section, but even for itssymmetrical distribution over it. Thus, the above�described ways of quick temperature equalization overthe coil cross section for tyhe CW should also reducethe SE�2 time duration.

0T =�

,T�

0

–0.2

–0.4

–0.6

–0.8543210

t, s

ΔΩ, deg/h

0.010

0.005

0

–0.005543210

t, s

ΔΩ, deg/h(a)

(b)

Fig. 6. Temperature drift in the case of CW for the coil cross section, shown in Fig. 5 for CW (a) with and (b) without an outerlayer.

752



The quantitative calculation of SE�2 requires anadditional investigation. Note that, as certain addi�tional practical measures preventing from SE�2, in[5, 7], it is suggested applying a carbon coil carcass(low TEC, high heat conductivity) and its advantageover an aluminum carcass with a higher TEC is dem�onstrated. Due to exactly this reason, we have consid�ered above the carcass of invar (low TEC), whose rel�atively low thermal conductivity can be compensatedfor by the copper layer that has also been consideredabove.

CONCLUSION

According to quantitative 2D modeling of SE�1(purely temperature drift), of various fiber windingtechniques for FOG sensing coil, the chess windingwith a single outer layer [8] provides for the best resultsfor the coil temperature sensitivity, leaving behind eventhe quadruple winding, which is now regarded as thebest for high–accuracy FOGs. Besides this, accordingto qualitative consideration, the CW should be advanta�geous in terms of the time duration of SE�2 induced bythermal stresses. Note that, here, we have assumed thatfiber turns are placed with ideal regularity, whereas theirdisplacements and coil asymmetry, unavoidable inpractice, can lead to and additional drift.

REFERENCES

1. D. Shupe, Appl. Opt. 19 (5), 654 (1980).

2. P. B. Ruffin, C. M. Lofts, C. C. Sung, and J. L. Page,Opt. Eng. 33, 2675 (1994).

3. C. M. Lofts, P. B. Ruffin, M. Parker, and C. C. Sung,Opt. Eng. 34, 2856 (1995).

4. F. Mohr, J. Lightwave Technol. 14, 27 (1996).

5. A. Cordova, D. J. Bilinski, S. N. Fersht, et al., USPatent No. 5371593 (6 Dec. 1994).

6. O. Tirat and J. Euverte, Proc. SPIE 2837, 230 (1996).

7. A. Cordova, R. Patterson, J. Rahn, et al., Proc. SPIE2837, 207 (1996).

8. A. Malvern, US Patent No. 5465150 (7 Nov. 1995).

9. A. A. Samarskii and E. S. Nikolaev, Numerical Methodsfor Grid Equations (Birkhäuser Verlag, Boston, 1989).

10. A. V. Lykov, The Theory of Heat Conduction (VysshayaShkola, Moscow, 1967) [in Russian].

11. A. N. Tikhonov and A. A. Samarskii, Equations ofMathematical Physics (Pergamon, Oxford, 1963).

12. V. F. Formalev and D. L. Reviznikov, Numerical Meth�ods (Fizmatlit, Moscow, 2006) [in Russian].

13. K. Hotate and Y. Kikuchi, Proc. SPIE 4204, 81 (2001).

14. F. Mohr and F. Schadt, Proc. SPIE 5502, 410 (2004).

1

SPELL: 1. Shupe, 2. integrands

temperature characteristics of fiber optic gyroscope sensing coils

Engineering