Tomography Methods for Vector Field Study Using

Tom ogr ap h y M eth od s for Vector F ield Stu d y Us in g Sp ace-Dis tr ib u ted Fib er Op tic Sen s or s with In tegr a l Sen s itivity

Downloaded by [University of California Santa Cruz] at 00:03 10 March 2013

YU. N. KULCHIN O . B. V ITRIK R. V . RO MA SHKO YU. S. PE TRO V O . V . KIRICHE NKO O . T. KAME NE V De partme nt of Physics Far E aste rn State Te chnical University Th e p ro blem o f to m ograph ic recon stru ctio n o f v ecto r ph ysical field s is stu d ied . Th is p ro blem can be so lv ed by u sin g fib er optic m easu rin g lin es (ML s) of specia l sh a pe. In th e case th at th e ML ou tp u t sign al is propo rtio n al to th e v ecto r’s pro jec tio n , th e ML m u st be sh a ped like a n arro w lo o p. Th is problem can be so lv ed by m ean s of th e in tegral th eorem . If an ML ou tp u t sign al is propo rtio n al to projec tio n o f a v ecto r d eriv ativ e w ith respect to ML directio n , th e ML w ith a step sh ape can be u sed. In th is case th e po ten tia l co m pon en t o f a v ecto r fie ld can be recon stru cted . Th is a ppro ach can be applied to research on d istrib u tio n s o f elec trom agn etic , d efo rm in g, a n d o th er v ecto r field s a n d can be u sed fo r d ev elo pin g system s to m o n ito r v ecto r p h ysical field s. K eywor d s

fibe r optic se nsor, tomography reconstruction, ve ctor fie ld

A traditional tomography is one of se veral e ffe ctive m e thods use d for studying scalar physical fields w 1, 2 x . At prese nt, rese arche rs are studying the problem of tomography reconstruction of vector fie lds w 3, 4 x . As is e vide nt from the se publications, the proble m can be solve d by fibe r optics. Howe ver, the de velope d m e thods have a num ber of disadvantage s that make the m difficult to apply in practice . For e xample , the m e thod w 3 x is de ve lope d for spe cific problem solution and doe s not make use of we ll-de ve lope d mathe matical principle s of classical tomography. As a re sult, the range of the me thod’s accuracy and the class of proble m s solved are not cle ar. The me asuring system w 4 x is ve ry com plex and require s pairs of se nsors with ide ntical se nsitivity to be used, which affects reliability and accuracy of the re sults. In the pre sent work we discuss some classes of proble ms that allow back-radon transformation w 1 x to be used for the study of ve ctor fields, and the simple me asuring sche me s and me thods for the se proble m solutions are de velope d.

Received 28 O ctobe r 1996; accepted 11 July 1997. Addre ss corresponde nce to O . B. V itrik, De partme nt of Physics, Far Eastern State Te chnical University, Pushkinskaya 10, V ladivostok, Russia 690600. E -mail: vitrik@ de lphin. marine.su

75 F ib er a n d In tegrated O p tic s, 17:75 ] 84, 1998

Copyright Q 1998 Taylor & Francis 0146-8030 r 98 $12.00 q .00

76

Yu . N . Ku lch in et a l.


Th e or y In the majority of case s a re sult of som e vector impact on a fibe r optical se nsor s re fe rred to he re as ``me asuring line ’’ or simply ``ML’ ’ . is a function of mutual orie ntation of a ve ctor and a fibe r axis w 4 x . Thus the re sults may be proportional to a proje ction of a vector both onto the optical fiber axis and onto the direction pe rpe ndicular to the axis w 3, 5, 6 x . The solution of the proble m of re construction of ve ctor fie ld parame te r distribution using the data re ce ive d from such MLs can be based on the use of the Gauss-Ostrogradsky and Stoke s the ore ms. Le t us consider ve ctor fie ld distribution in some two-dime nsional s 2-D . area D and an ML placed in this are a. Le t us assume that the ML’ s output signal is dire ctly proportional to a vector proje ction onto the fiber axis. s Such a situation occurs, for e xample , in the case of using some type s of fibe r optical sensors of magne tic w 5 x and e le ctric w 6 x fields.. In this case a phase shift of the light guide d by the fibe r can be pre sented as

u A

H

a

b

A 1s l . d l

s 1.

whe re A l is ve ctor projection onto the ML axis and a and b are points crossing the ML and the borde r of are a D . Le t us assume that A s x, y . is a pote ntial field. The n integral E q. s 1 . represe nts a differe nce of potentials be twe e n points a and b , irre spe ctive of the ML shape w 7 x . In this case the me asure me nt re sults pre se nt a distribution of pote ntials on the borde r of area D , and the y are insufficie nt for reconstruction of pote ntial field distribution w 7 x . Howe ve r, the se data are sufficie nt for re construction of the rotor fie lds. Conside r the ML having the shape of a narrow rectangular loop as shown in Figure 1 a . An output signal of such an ML, according to e xpre ssion E q. s 1 . and Stokes the orem w 8 x , can be re corde d as

u s r, w . A

E As r .e d l s H L

S

s k ? rot A . d S f h 0

H

s x.

s k ? rot A . d x

s 2.

whe re e is a unit vector place d at a tange nt to the ML axis; L is an inte grating contour; r , w are polar coordinates de fining a position of the contour L ; S is the area cove red by L ; h 0 is the width of the loop; x is the axis paralle l to the long side of the loop; and k is a unit vector normal to flat area D . It was suppose d in e xpre ssion E q. s 2 . that the size of h 0 is small. Thus param e ters of the field cannot be conside rably change d in h 0 in the vicinity of any point of are a D . In E q. s 2 ., s k ? rot A . is a scalar value not de pe nde nt on mutual orientation of the ML and ve ctor A . The refore E q. s 2 . coincide s with radon transformation of this value for given r and w w 1 x . Thus it is possible to pe rform back-radon transformation of the u s r , w . function afte r scanning the polar coordinates. The re sult will be the distribution of rot A w 1 x , which can be conside re d as the proble m solution be cause the re constructe d distribution comple tely de scribe s a rotor fie ld w 8 x . An angle s D w . and radial s D r . scanning ste p de pe nds on the scanning sche me and the re solution of the re construction. In the case of the simple st paralle l scanning sche me w 1 x , the value of the resolution of d mi n s i.e ., the minimum transversal size of the compone nt of the function that has to be re constructe d . is belie ve d to be achieve d whe n D w s d mi n r s 2 d ., D r s d m in r 4, whe re d is the transverse size of the studied area w 1 x . The re are spe cial reconstruction me thods based on the the ory of incorre ct

T o m o gra p h y Meth o d s fo r Vecto r F ield Stu d y

77


proble m solution, which give s com parable re solution whe n D w s p r 3 % p r 4. E ve n this value e xce e ds the optimal angle step m e ntione d above and thus allows one to simplify the scanning procedure w 1 x . Howe ver, the se me thods can distort the function unde r reconstruction and m ust be use d with caution. Le t us now conside r a case whe n an output signal of the ML is directly proportional to a proje ction of a re se arche d ve ctor onto the dire ction normally orie nte d to the ML’ s axis s such is an e xample of the fibe r optical se nsor of the e le ctrical fie ld activated with pie zoe le ctric material w 3 x .. Conside ring that the pote ntial field and ML have a narrow loop shape s se e Figure 1 a ., one can de scribe the ML output signal according to E q. s 1 . and the Gauss-O strogradsky the ore m w 8 x formulate d for 2-D case, as follows:

u s r, w . A

E As r .n d l s H L

S

div A d S f h 0

H

s div A . d x

s 3.

s x.

whe re n is the unit vector located normal to the ML axis. O ne can se e that the output signal of such an ML prese nts the radon transformation from diverge nce of the fie ld. Thus, as in the pre vious case, having carried out a scanning procedure of the polar coordinate s and having applie d one of m e thods realizing back-radon transformation, it is possible to reconstruct the distribution of div A . This result is the solution of the proble m be cause the div A comple tely de scribes the distribution of a pote ntial fie ld w 8 x . Using both type s of ML s one be ing sensitive to longitudinal proje ctions and the othe r be ing se nsitive to normal proje ctions onto the ML axis ., one can solve

F igu r e 1. s a . A measuring line having the shape of a narrow loop. s b . Surface de formed by

the e ffe ct of a ve ctor field and me asuring line s of straight s curve 1 . and ste p s curve 2. shape . s c . A fibe r optic sensitive element deforme d by a ve ctor A s l 0 is initial position of optical fiber and l is position of the de formed fiber . .


78


the proble m of reconstruction of a comple x ve ctor field having both rotor and pote ntial compone nts. The re is anothe r class of proble m s that cannot be solved by me ans of integral the orem s. Le t us assume that the long optical fibe r inte rferome te r whose signal is proportional to e longation w 9 x is used as a sensor. The change of interfe rome ter le ngth can be caused by the de formation fie ld being distribute d in som e 2-D obje ct s se e Figure 1 b .. Le t the contour of this ML be de scribe d by som e e xpression y 0 s f s x0 .. Unde r de formation, e ach of the fiber e le me nts will take a ne w position s se e Figure 1 c .. Ne w coordinate s of the contour points can be calculate d as x s x0 q A xs x0 , y 0 ., y s y 0 q A y s x0 , y 0 ., whe re A xs x0 , y 0 ., A y s x0 , y 0 . are De scartes’ coordinates of the A s x0 , y 0 . ve ctor of displace me nt of points of the obje ct. Thus the e le me nt of le ngth of the ne w contour, while ne gle cting the quadric term s, can be e stimate d as dl s

X d x2 q

d y2

Ax

t

s dl0 1 q

x0

cos 2 w q

Ax y0

sin w

cos w q

Ay x0

cos w

sin w q

Ay y0

sin 2 w

/

X

whe re d l 0 s d x02 q d y 02 is the e le me nt of le ngth of the nonde forme d ML and cos s w . s d x0 r d l 0 , sin s w . s d x0 r d l 0 , and w are angle s be twe e n the x axis and the tange ntial line to the contour in a point of s x0 , y0 .. Thus a total change of contour le ngth cause d by de formation of the object can be calculated as

D Ls

H

d l0

s

H

d l0

s l0 .

s l0 .

Ax

t

x0 A l0

cos 2 w q

? e

t

Ax y0

q

Ay x0

/

sin w

cos w q

Ay y0

sin 2 w

/

whe re e s s i cos w q j sin w . is the unit ve ctor paralle l to the ML’ s axis and i , j are unit ve ctors parallel to De scartes’ axis. If the contour l 0 pre se nts a straight line , the last e xpression gives D L s A s a . ? e y A s b . ? e , whe re a and b are points crossing the contour and the object borde rs s se e Figure 1 b .. Thus the integral data dire ctly proportional to e longation will contain the information about displace me nt of borde rs only. The se data do not give inform ation about the distribution of the displace me nt vector inside the object. This me ans that nonrectiline ar contours for the ML’ s place me nt should be chose n to obtain the ne ce ssary information. Le t us conside r the contour of a ste p shape s se e Figure 1 b .. A ssuming short steps, one can conside r the vector A s x0 , y 0 . in the vicinity of e ach step as a constant. In this case the total change of contour le ngth caused by obje ct de formation can be pre sented as

D L s X2

H

s r.

t

Ax x0

q

Ay y0

/

dr s

X2

H

div A s x , y . d r

s 4.

s r.

whe re the curve r de fine s a position of the ste p contour s Figure 1 b . and, simultane ously, is a ne w inte grating contour in E q. s 4 . closing to rectiline ar. O ne



79

can se e that E q. s 4 . coincide s with radon transformation from the div A s x, y . function. This is the function that will be re constructed by using back-radon transformation. This le ads to the solution of a problem of reconstruction of the displace me nt field pote ntial compone nt w 8 x . In orde r to re construct a rotor compone nt of a fie ld, additional data are require d. In the proce ss of solving the problem in que stion, a numbe r of approximations are used. The refore a te st to verify the me thod de scribed is re quire d. This te st was pe rforme d by compute r simulation of a numbe r of distributions of a flat obje ct’s displace me nt field w As x, y .x . All we re assume d as pote ntial s rot A s 0 .. In the first e xample the diverge nce of A was a constant inside some circle of radius R ; outside the circle , div A s 0 s se e Figure 2 a .. A numbe r of MLs having a ste p shape we re simulate d to m e asure integral data s se e Figure 2 c .. The ir position was chose n to pe rform a proce dure of tom ographic paralle l scanning of the obje ct. The change s of the ML’ s le ngth cause d by the de forming field we re nume rically calculated and use d as the data for tomographic reconstruction of div A . The re sults of this value re construction and vector field distribution corre sponding to this value are pre se nte d in Figure s 2 b and 2 d . The re sults obtaine d for some othe r e xample s are prese nte d in Figure 3. In this figure the initial distributions of the fie ld dive rge nce s Figure s 3 a and 3 c . and the results of the diverge nce reconstruction s Figures 3 b and 3 d . are shown. The quantity r in the figure s de note s the corre lation coe fficient betwe e n initial and re constructed distributions. O ne can see that initial and reconstructe d functions are in good agre e me nt. The distinctions be twe e n initial and re constructed distributions can be e xplaine d by an e rror appe aring in the proce ss of realizing back-radon transformation.

F igu r e 2. s a , c . Initial and s b , d . reconstructed distribution of displaceme nt field dive rgence

and distribution of displace me nt vector; ML is one of the me asuring line s.


80


F igu r e 3. s a , c . Initial and s b , d . reconstructed distribution of pote ntial ve ctor field diver-

gence for different cases.

E xp er im e n t An e xpe rime nt was pe rforme d only for the case of re construction of distribution of a longitudinal displace me nt vector of a 2-D e lastic plate . The e xpe rim e ntal se tup s Figure 4 . contains a rectangular plate s labele d 5 ., size 29 = 350 = 4 m m, stretche d on a re ctangular skele ton. A microme tric shifte r s labe le d 7 . was use d for longitudinal displace me nt of a faste ning point s labe le d 6 .. Thus all othe r points of the plate we re also shifted in the same direction. The calculation of the re sulting distribution of the displacem e nt ve ctor s Figure 5 a . was carrie d out according the re sults of pre vious work w 7 x . This vector fie ld contains both pote ntial and rotor compone nts.

F igu r e 4. The e xpe rimental setup s 1, He -Ne lase r; 2, lense s; 3, single-mode optical fibe r; 4,

me asuring line ; 5, de forme d plate; 6, point of fastening; 7, microme tric shifte r; 8, reductor; 9, potentiomete r; 10, photode tector; and 11, two-coordinate autoplotte r. .



81

F igu r e 5. s a . Re searche d ve ctor fie ld and s b . deforme d plate .

The distribution of the fie ld’s pote ntial compone nt that must be reconstructe d in the e xpe rime nt is indicate d in Figure 6 a . The ML pre se nts a two-mode inte rferome te r with attache d mode filte r w 9 x , pe rmitting one to transform the change of interferome te r le ngth into a proportional optical signal. Be ing fixed on the de forme d surface , the ML has the same de formation shifts as the plate. The single ML with a step shape is use d to pe rform the proce dure of tomographic parallel scanning of the plate w 1 x . In orde r to re alize the scanning proce dure , this line was place d ste p by ste p in all ne ce ssary positions. The numbe r of steps of radial scanning was e qual to 10 and that of angle scanning was e qual to 4, which gives sufficie nt re solution whe n using Kulchin e t al.’ s me thod w 10 x . The re sult of re construction of the distribution of the vector field dive rge nce is pre se nte d in Figure 7 b as compared with the calculated distribution s Figure 7 a .. A corre lation coe fficient betwe e n calculate d and re constructe d distributions is e qual to 0.94. Re ceived distribution of dive rge nce de fine s a place of source s of the de formation and also allows one to calculate the pote ntial compone nt of the obje ct de formation fie ld by using the results of Arse nin w 7 x , as is shown in Figure 6. O ne can see that the initial and the reconstructed distributions are in good agree me nt.

F igu r e 6. Distribution of the ve ctor fie ld’ s potential compone nt: s a . theore tical and s b .

receive d from proce ssing of e xpe rime ntal results.

82


F igu r e 7. Distribution of re se arche d vector fie ld divergence : s a . theoretical and s b . e xperi-


me ntal.

C on clu s ion s In the prese nt work we show the possibility of tomographic re construction of vector fie ld param e ter distribution by using nonre ctiline ar fibe r MLs. If the ML’ s output signal is proportional to the inte gral from the proje ction of a re se arche d vector, it is possible to use the ML in a narrow loop shape . In this case the proble m can be solve d by using inte gral the ore ms. If the ML’ s output signal is proportional to the inte gral from the proje ction of a ve ctor de rivative with re spe ct to the ML direction s e .g., the proble m of a de forme d 2-D object., it is possible to use the ML having a step shape , and the potential compone nt of a vector field can be re constructed. The step-by-step scanning proce ss used in the pre se nt work is slow, of course. Howe ver, for purpose s of practical application, one can use a me asuring ne twork containing a sufficie nt numbe r of MLs to allow the system to be ope rate d in re al time , as was proposed for scalar fie lds w 11 x . The m e thod sugge sted in this work can be wide ly applie d for inve stigating distributions of e le ctromagne tic, de forming, and othe r vector fie lds and can be come a basis for systems use d to monitor ve ctor physical fields.

R efer en ce s 1. Natte rer, F. 1986. Th e m ath em atic s o f co m pu terized to m ograph y. New York: John Wile y. 2. Le vin, G. G., and G. N. V ishnyakov. 1989. O ptic al to m o graph y s in Russian .. Moscow: Radio i svyaz. 3. Gine vskiy, S. P., O . I. Kotov, V . M. Nikolaev, and V . Yu. Pe trunkin. 1995. Using me thods of reconstructing proce ssing tomography in optical fiber sensors s in Russian . . Kv an tov aya E lec tron ika 22 s 10 . :1013 ] 1018. 4. Kulchin, Yu. N., O . B. V itrik, O . T. Kamene v, O . V . Kiriche nko, and Yu. S. Pe trov. 1995. Re construction of ve ctor physical fie lds by optical tomography method. Q u an tu m E lec tron . 25 s 10 .. 5. Busurin, V . I., and Yu. R. Nosov. 1990. O ptica l fib er sen sors : ph ysical basis, calc u la tio n an d applica tio n problem s s in Russian .. Moscow: Ene rgoatomizdat. 6. Bokne rt, K., and F. Nehring. 1989. Fibe r-optics se nsing of voltage by line inte gration of e le ctric field. O pt. L ett. 14 s 5. :290 ] 291. 7. Arsenin, V . Ya. 1974. Meth od s of m ath em a tic al ph ysics an d specia l fu n ctio n s s in Russian . . Moscow: Nauka. 8. Korn, G., and T. Korn. 1978. Ma th em a tic al h an d boo k. Ne w York: McGraw-Hill. 9. Kulchin, Yu. N., O . B. V itrik, O . G. Maxae v, O . V . Kiriche nko, O . T. Kamenev, and Yu. S. Pe trov. 1996. Ne w me thod of multimode fibe r inte rfe rome te rs signal proce ssing. Proc. of 17th Congre ss of the Inte rnational Commission for O ptics, for Science and New Te chnology, Part II, Tae jon, Kore a. SPIE 2778.


83

10. Kulchin, Yu. N., O . B. V itrik, O . V . Kirichenko, Yu. S. Pe trov, and O . T. Kame ne v. 1995. The lase r tomography method using minimum of projection for biological obje ct structure study. L aser Bio l. 4 s N3 .:679 ] 684. 11. Kulchin, Yu. N., O . B. V itrik, O . V . Kiriche nko, and Yu. S. Pe trov. 1995. Multidime nsional signal proce ssing by using fiber optic distributed measuring ne twork. Q u an tu m E lec tron . 23:444 ] 447.


B iogr ap h ie s Yu r i N . K u lc h in graduated from the Moscow E ngine ering Physics Institute s ME PhI . in 1976. From 1976 to 1979 he worke d in the Automatic and Control Proce ssing Institute of V ladivostok. From 1982 to 1988 he was he ad of the Physics department of the Far Eastern Polyte chnic Institute s FE PI . in V ladivostok. He received a Doctor of philosophy de gree and a Doctor of scie nce degre e in radiophysics and laser physics from ME PhI in 1982 and 1991, respective ly. Since 1991 Dr. Kulchin has be en a profe ssor at the Physics departme nt of the Far Easte rn State Te chnical Unive rsity s FE STU . . Now he is also vice pre side nt of FE STU and a scientific advise r for the O ptoele ctronic laboratory. Pre se ntly, the range of his inte re sts include s optical data proce ssing, fiber optic se nsors and their application in laser inte rfe rometry, tomography, and optical neural ne tworks. Dr. Kulchin is me mber of SPIE . Ole g B . Vi tr ik graduated from the Moscow Engine e ring Physics Institute s ME PhI . in 1986. Since 1986 he has worke d in the Far Easte rn State Te chnical University s FE STU . . In 1988 he won the Second Popov Socie ty Award for young re se arche rs. He received a Doctor of philosophy de gre e in radiophysics from ME PhI in 1990. In 1991 Dr. V itrik be came an assistant te ache r, and since 1995 he has be en an associate profe ssor in the Physics department of FE STU , as well as a se nior researche r in the O ptoele ctronic laboratory. In 1996 Dr. V itrik was awarded the Primorye Gove rnor Prize for re search deve lopment in the Primorye Province . Pre sently the range of his inte re sts include s optical data proce ssing, fibe r optic se nsors and the ir application in laser inte rfe rometry, tomography, and statistical me thods in optics. Dr. V itrik is a member of O SA. Rom an V. Rom as h ko graduate d from the Moscow E ngine ering Physics Institute in 1995. Subse que ntly he worked as a scie ntific re se arche r in the O ptoe le ctronic laboratory of the Far E astern State Te chnical Unive rsity s FE STU .. Since Se ptember 1995, he has be en a post-graduate stude nt at FE STU with a specialty in lase r physics. He was awarde d the Stipend of the Pre side nt of Russia for post-graduate students. His scie ntific inte re sts include lase r physics, fibe r-optic se nsors, optical information proce ssing, and neural ne tworks. Yu r i S. Petr ov graduated from the Far Easte rn State Polytechnic Institute in 1989. He is curre ntly a scientific re searche r in the O ptoele ctronic laboratory of the Far Eastern State Te chnical Unive rsity. His scie ntific interests include fibe r-optic se nsors technology and optical information proce ssing.

Ole g V. K ir ich en ko graduated from the Moscow Engine ering Physics Institute in 1992. He the n worke d as a scientific re se arche r in the O ptoe lectronic laboratory of the Far Easte rn State Te chnical University. A post-graduate stude nt of the Far Easte rn State Te chnical Unive rsity since Se pte mber 1992, his spe cialty is lase r physics. He re ce ives the Stipend of the Pre side nt of Russia for post-graduate stude nts. He was also awarded se cond place for his scientific poste r at the O ceans ’ 95 confe re nce s San Die go, CA, USA .. His scie ntific intere sts include lase r physics, fiber-optic sensors, optical information proce ssing, and ne ural networks.

84



Ole g T. K am en ev graduated from the Moscow Engine e ring Physics Institute in 1993. He subse que ntly worke d as a scie ntific re searche r in the O ptoe le ctronic laboratory of the Far Easte rn State Te chnical University. Since Septembe r 1993 he has be en a post-graduate student of the Far Easte rn State Te chnical University with a specialty in laser physics. He was awarde d the Stipe nd of the Pre sident of Russia for post-graduate stude nts. His scie ntific inte re sts include lase r physics, fiber-optic se nsors, optical information proce ssing, and ne ural networks.