contrast analysis for a fiber-optic white-light interferometric system

Contrast analysis for a fiber-opticwhite-light interferometric system

Władysław A. Wozniak, Piotr Kurzynowski, Wacław Urbanczyk, and Wojtek J. Bock

The behavior of a system of fiber-optic white-light interferometric sensors is analyzed. Analyticalexpressions for contrasts of all the interference patterns that may occur at the setup’s output are derivedas a function of coupling coefficients between successive sensors. Two cases of exemplary systemsconsisting of highly birefringent optical fibers, one with nondichroic the other with some polarizing fibers,are analyzed. © 1997 Optical Society of America

1. Introduction

As a result of the intensive development of fiber-optictechniques and optoelectronics, a new generation ofsensors based on optical fibers has appeared.1,2

These sensors are more sensitive and allow one tomeasure many different physical quantities3–9 ~pres-sure, temperature, stress, and so on! simultaneously.The most important advantage of fiber-based sensorsis their high immunity to external influences ~such aselectromagnetic distortion!. The application of in-terferometric techniques further increases the sensi-tivity of optical measurements. The most effectivefiber–interferometric systems use low-coherencesources ~white-light interferometry!. A typical, sim-ple example of such a system is a setup built of awhite-light source, a sensor interferometer made ofoptical fiber, a receiving interferometer ~e.g., anotherfiber or Wollaston prism!, and some fibers used forleading the light between the source and the sensorand between the sensor and the receiving interferom-eter ~Fig. 1!. The sensor element is subjected to themeasured quantity and introduces a phase shift be-tween two eigenwaves of propagated light. Thisphase shift can be detected on the receiving inter-ferometer, in which controlled introduced changes ofphase shift must be possible. Because of the differ-

W. A. Wozniak, P. Kurzynowski, and W. Urbanczyk are with theInstitute of Physics, Technical University of Wrocław, WybrzezeWyspianskiego 27, 50-370 Wrocław, Poland. W. J. Bock is withthe Optoelectronics Laboratory, Universite du Quebec a Hull, CasePostale 1250, succ. B, Hull, Quebec, J8X 3X7, Canada.

Received 10 January 1997; revised manuscript received 7 July1997.

0003-6935y97y348862-09$10.00y0© 1997 Optical Society of America

8862 APPLIED OPTICS y Vol. 36, No. 34 y 1 December 1997

ent group delays introduced by the setup elements~not only by the sensor, but also by the linking fibers!many of the interference patterns can be raised anddetected only if the resultant group delay, responsiblefor the detected pattern, is smaller than the groupdelay of the broadband source. Pattern contrastsdepend on the alignment of all elements of the setup.A proper alignment of the setup should ensure highcontrast of elements from patterns, which includesthe information about measured quantity ~signal pat-terns!, while zeroing contrasts of other ~noise! pat-terns, or at least should ensure separation betweenthe signal and the noise patterns ~in general, some ofcross-interference patterns may overlap!.

The main advantages of white-light interferomet-ric systems are the possibility of absolute measure-ments, a significant reduction in the noise level, aninsensitivity to optical power fluctuations, and thepossibility of multiplexing a large number of sensorsin a measuring system. This new attractive branchof interferometry, also called fiber-optic low-coherence interferometry, is the subject of a paper inwhich a recent progress in this area is reviewed.1Numerous applications of fiber sensors with white-light interferometry have been reported,3–9 and sev-eral theoretical studies that describe their behaviorhave also been published. The main problem ofwhite-light interferometry systems, the appropriatechoice of group delays to avoid the overlapping ofcross-interference patterns and noise analysis, hasbeen studied.10–13 The dependence between con-trasts of interference patterns and coupling coeffi-cients or angular alignments of fibers in a setup alsohas been analyzed.14–16

In our recent papers we described the behavior ofsimple cases of white-light fiber-optic interferometricsystems. In Ref. 15 a system composed of sensing

and receiving interferometers separated by adichroic-type linking fiber was analyzed to allow us todetermine the minimum group imbalance necessaryto separate signal and noise patterns. The analyti-cal formulas for contrasts of all the patterns producedby all fibers ~also the cross-interference patterns!were derived. The contrast behavior was analyzedas a function of the alignment of all setup componentsand the dichroism of the linking fiber. We showed inRef. 15 that the minimum group imbalance may bedecreased to half for a certain alignment of the sys-tem. In Ref. 16 we analyzed the behavior of atemperature-compensated system of sensors, whichalso contained a compensating interferometer ~an-other fiber placed in the sensor position to compen-sate for temperature influences on the basic sensor!,that could be used to measure high pressures. Theanalytical expressions obtained for contrasts of allinterference patterns that occurred as a function ofcoupling coefficients between successive elements al-lowed us to find the optimal alignment of the systemand to determine alignment errors from measuredcontrasts of noise patterns. A simple alignment pro-cedure, based on derived formulas, was also pro-posed.16

In the present paper we discuss in detail the per-formance of a most complex system composed of anynumber of interferometric sensors, using as an exam-ple a setup consisting of highly birefringent ~HB! op-tical fibers. Analytical relations for the contrasts ofall interference patterns that may arise at the outputof the system were derived. Once these formulasare known, it is possible to find an optimal alignmentof the setup at which selected patterns reach maxi-mum contrast. Also, a system that contains somepolarizing fibers used as linking and separating ele-ments is considered. Our analysis can be easily ex-tended to a setup composed of fiber couplers or, in themost general case, to a white-light interferometricsystem based on any kind of sensor.

2. Calculation Method

As was mentioned in Section 1, the analysis carriedout in our paper can be valid for any white-lightinterferometric system; however, in our analysis weused as an example a system that contains HB opticalfibers. The described setup is built from many sec-tions of interconnected HB fibers, rotated ~by orien-tation of the main axis! relative to the precedingelements14–16 @Fig. 2~a!#. Another setup, consistingof conventional fibers connected by fiber couplers,which can be analyzed in a similar way, is presentedin Fig. 2~b!. The only difference is that in the secondcase the coupling ratios take over the functions of the

Fig. 1. Typical scheme of a white-light fiber-optic interferometricsystem.

rotating angles. Some of the fibers in both setupsplay the role of sensors; some of them can be treatedas linking ~or separating! media, leading the signalbetween the sensors and into the output, where de-tection takes place. However, all these fibers ~sen-sors and links! can produce interference patterns andcross patterns, provided that their coherence lengthis smaller than the coherence length of the source.Considering this, all possible contrasts of interfer-ence patterns of all the fibers should be analyzedwhen interferometers of this kind are designed.

A typical example of the system described above isa setup consisting of three or four HB fibers, de-scribed in our earlier papers,15,16 in which analyticalformulas for contrasts of interference patterns inthese relatively simple cases were derived. How-ever, more complicated setups can often incorporatemore than two sensing interferometers ~which detectseveral physical quantities at the same time or thesame quantity in different places! and more than twoleading elements ~which in some circumstances canalso play the role of active sensors!. A good exampleis a setup consisting of HB fibers in which, for con-structional or technological reasons ~or simply be-cause of the replacement of a defective part of a fiber!,the leading part is made from several sections. Inthis case a traditional analysis of the setup ~in par-ticular, the calculation of contrasts of interferencepatterns! by use of, for example, the Jones matrixformalism15,16 becomes an arduous task because ofthe rapidly growing number of necessary multiplica-tions. In our paper we use a more suitable ~forwhite-light interference! formalism that is based onthe transformation of the spectral coherence matrixK~v! ~where v denotes the frequency of light!. Thepropagation of spectral coherence matrix K~v!through the nth element of the setup is described bythe following equation17:

Kn~v! 5 Jn~v!Kn21~v!JnR~v!, (1)

where Jn~v! denotes a Jones matrix of the nth setupcomponents; Kn21~v! and Kn~v! are the spectral co-herence matrices before and after this nth compo-nent, respectively; and R denotes the Hermite

Fig. 2. Scheme of fiber-optic interferometric system made of ~a!HB fiber, ~b! single-mode fiber with fiber couplers ~FC!. Note thatthe orientation of the main axes of each fiber is rotated relative tothe preceding element.

1 December 1997 y Vol. 36, No. 34 y APPLIED OPTICS 8863

conjugation. The output intensity Iout can be ob-tained from the well-known formula17

Iout 5 *0

`

tr@Kout~v!#dv, (2)

where Kout~v! denotes the coherence matrix at theoutput of the system and tr is a trace of the matrix.The final formula for Iout can be presented in thefollowing form:

Iout 5 I#out 1 (p

@I#outVp cos~d#p!G~tp!#, (3)

where I#out is an average intensity of the output light;d#p is a phase difference between the fast and the slowwaves introduced by the pth element~s! of the setupfor central frequency v# 0 of the source; G~tp! denotes anormalized coherence function; tp is a group delayintroduced by the pth element~s! of the setup; and Vpis a contrast of the pth interference pattern, which isa coefficient and the method of calculating that be-comes the focus of this paper. Subscript p denotesall possible combinations of setup elements produc-ing respective patterns; for example, the symbols 1,13, 28 in the subscript of central phase differenced#1,13,28 denote a phase difference d#1,13,28 [ d#1 1 d#3 2d#8 introduced by the setup elements placed in posi-tions 1, 3, and 8, with the 8th element subtracting itsphase difference from the total phase difference,whereas the 1st and the 3rd add their phases to thetotal.

Output intensity Iout can be rewritten as a sum ofthe average intensity I#out and the modulated in-tensity Iout,

Iout 5 (p

@I#outVp cos~d#p!G~tp!# 5 (p

@ip cos~d#p!G~tp!#,

(4)

where ip plays the role of an amplitude of oscillationsof a given pattern. Then the traditional definition ofthe contrast of the pth interference pattern can berewritten as

Vp 5 ipyI#out. (5)

Thus the procedure of calculating the contrast of thedesired pth pattern consists of computing the averageintensity I#out ~common for all possible patterns! andthe amplitude ip of this pattern.

It is easy to prove that in the case of N elementsthere are as many as ~3N 2 1!y2 1 1 possible combi-nations ~considering that there are three possibilitiesfor the presence of each element in every contrast:summing its phase difference; subtracting it; or itsnot taking part in the production of a given contrast,regarding the even character of the cosine function!.Moreover, to calculate one definite contrast, it is nec-essary to do calculations for all possible contrasts,because a full formula for output intensity Iout mustbe derived. This means that all the elements of thecoherence matrix K~v! have to be multiplied by all theelements of the Jones matrices Jn~v! of the setup


elements, even if most of them do not take part in theproduction of calculated contrast; that is, their groupdelays tn are higher than the group delay ts of thesource, and as a result they are zeroed from the nor-malized coherence functions G~tp!, in which tp con-tains the term tn. ~Note that p denotes acombination of elements, whereas s and n denote oneelement.! Remember that integrating with regardto the source coherence matrix, which finally leads tomany zeroing normalized coherence functions G~tp!,has to be done at the end of the calculations aftermany matrix multiplications, according to Eq. ~1!.In this paper, we greatly simplify the calculations onthe basis of the following: Although the influence ofthe spectral coherence matrix Kin~v! of the sourceshould be taken into account at the end of the com-putations @final integration, according to Eq. ~2!#, thefinal zeroing of some components of the elements ofthe spectral coherence matrix Kin~v! could be takeninto account during partial calculations @successivemultiplications, according to Eq. ~1!#. In otherwords, having information about the final ~after in-tegration! zeroing of the contributions of some setupelements to the ultimate formulas for contrasts, wemay neglect ~assume to be zero! some components ofKin~v! elements during the succeeding multiplication.This results in a significant reduction in the mathe-matical operations needed to obtain the final formu-las for the desired contrasts.

To formalize our considerations, let us introducesome notation. Our analysis is carried out by use of,for example, the setup of N interferometers made ofHB fibers. ~A common analysis could be performedfor a setup consisting of optical fibers and couplers;see Ref. 15.! Let us consider N fibers connected toone another and to HB fibers whose main axes arerotated by angle an relative to the preceding fiber@Fig. 2~a!#. Then the influence of each nth fiber onthe spectral coherence matrix K~v! can be describedby its Jones matrix Jn~v!,

Jn~v! 5 F cos an

2sin an exp@2idn~v!#sin an

cos an exp@2idn~v!#G .

(6)

Note that this form of the Jones matrix differs fromthe formula that can be found in the literature.16

This is because in the proposed method the system ofx, y coordinates is designated by the main axes ofsucceeding elements, and, in consequence, we rotatethe system of coordinates of the nth fiber during thetransformation of the light through the nth element@Fig. 2~a!#.

To yield the output intensity Iout, the spectral co-herence matrix Kout~v! at the output of the systemhas to be calculated according to Eq. ~1!. This equa-tion describes the formula for the transformation ofthe spectral coherence matrix K~v! after passagethrough the single nth fiber and should be appliedsuccessively.

Denoting the elements of matrix Kn~v!

Kn~v! 5 FKxx~n!~v!

Kyx~n!~v!

Kxy~n!~v!

Kyy~n!~v!G , (7)

one can obtain @from Eqs. ~1!, ~6!, and ~7!# the follow-ing formulas for the elements of matrix Kn~v! ex-pressed as a function of the elements of precedingmatrix Kn21~v!:

FKxx~n!

Kyy~n!G 5 0.5F1 1 cos~2an!

1 2 cos~2an!1 2 cos~2an!1 1 cos~2an!

GFKxx~n21!

Kyy~n21!G

1 0.5 sin~2an!F 121

121GFKxy

~n21!

Kyx~n21!G , (8)

FKxy~n!

Kyx~n!G 5 Fexp~idn!

00

exp~2idn!G

3 H0.5Fcos~2an! 1 1cos~2an! 2 1

cos~2an! 2 1cos~2an! 1 1GFKxy

~n21!

Kyx~n21!G

2 0.5 sin~2an!F11

2121GFKxx

~n21!

Kyy~n21!GJ . (9)

~Note that for simplicity of notation we omit the de-pendence of the elements of matrices K and the phasedifference dn on v!.

We express these elements in a vector form to em-phasize the relationships between the important@from the point of view of Eq. ~2!# diagonal elementsKxx and Kyy of the spectral coherence matrix K~v! andthe unimportant ~unfortunately, only in the last ele-ment of the system! nondiagonal elements Kxy andKyx. However, as is shown in Eqs. ~8! and ~9!, thenondiagonal elements influence the diagonal ones inthe successive transformations and become impor-tant in the final calculations. Nevertheless, theequation that describes the propagation of nondiago-nal elements contains a term that includes dn, thephase difference introduced by the nth fiber. If onlythe group delay tn introduced by the nth fiber fulfillsthe condition

tn .. ts, (10)

~ts is a coherence time of the source!, then the finalintegration according to Eq. ~2! changes this terminto zero. The essence of our method consists of ne-glecting this finally zeroing term during partial cal-culations. To be precise, those terms can beneglected that contain all possible combinations~sums or differences! of group delays tp higher than ts@see the note about the special meaning of subscript pbelow Eq. ~3!#, since it could happen that, for exam-ple, tn, tm .. ts while tn 2 tm # ts ~or tn 1 tm # ts!.Anyway, if only the value of tn ~and all possible com-binations tp! can be estimated ~or assumed a prioriduring the calculation of selected contrast; see Section3!, the zeroing of the terms containing respective phasedifferences dp during the calculations ~except at theend of them! is well justified and leads to significantsimplification. How this method works in particularcases is described in the following sections.

3. Calculation of the Contrasts in the System withHighly Birefringent Fibers

Let us assume that only one polarization mode isexcited in the first ~n 5 1! lead-in fiber. This couldbe represented by the following form of the spectralcoherence matrix Kin~v! of the input light:

Kin~v! 5 Ein~v!F10

00G , (11)

where Ein~v! is a normalized spectral amplitude dis-tribution function of the source.

Our further calculations are illustrated in Fig. 3, inwhich all elements with group delays tn ~in everypossible combinations! that fulfill condition ~10! arewhite, while elements that produce visible interfer-ence patterns @nonzeroing coherence functions in Eq.~3!#, are black.

A. Average Intensity

Average intensity @Fig. 3~a!# stands for a constantcomponent of the output intensity of interference pat-terns, independent of variable group delays tp. Ac-cording to Eq. ~3!, output intensity Iout is equal toaverage intensity I#out when group delays tp betweenthe fast and the slow waves for every combination offibers are much higher than the coherence time ts ofthe source @condition ~10!#. Let us consider the effectof the nth HB fiber on the propagated light. Thetransformation of the spectral coherence matrix

Fig. 3. Calculation method. Positions of elements that producevisible contrasts ~black! among elements with group delays tn ..ts ~white! between the low-coherent source S and the analyzer A:~a! average intensity; ~b! detection of pattern produced by singlekth sensor; ~c! detection of cross pattern produced by adjacent kthand ~k 1 1!th sensors; ~d! detection of cross patterns produced byseparated kth and mth sensors.


Kn~v! of the light after passing through the nth fiberis performed successively according to Eq. ~1!. Nowwe use the vector form of notation in the spectralcoherence matrix K~v! iterative transformations pre-sented in Eqs. ~8! and ~9! to take advantage of ourmethod, which is described in Section 2. When av-erage intensity is calculated, the influences betweenthe diagonal and the nondiagonal elements of matrixK~v! disappear after the final calculation of outputintensity. This follows from the presence of matrix-contained phase differences dn in Eq. ~9! and, in con-sequence, leads to the zeroing of the respectivecoherence functions, because condition ~10! is fulfilledfor every dn ~and also for every possible p combinationsof dn!. Then, instead of zeroing these influences at theoutput of the setup, we may disregard them at the endof each element of the setup ~nondiagonal elements ofthe coherence matrix are assumed to be zero!. Thisleads to the following equations for the nonzeroing,diagonal components of matrix KN~v!:

FKxx~N!

Kyy~N!G 5 )

n51

N H0.5F1 1 cos~2an!1 2 cos~2an!

1 2 cos~2an!1 1 cos~2an!

GJFKxxin

KyyinG

5 0.5F1 1 P1,N

1 2 P1,N

1 2 P1,N

1 1 P1,NGFKxx

in

KyyinG

5 0.5F1 1 P1,N

1 2 P1,NGEin, (12)

where

Pk,1 5 )m5k

1

cos~2am!. (13)

Then coherence matrix Kout after the analyzer @for-mally the ~N 1 1!th element of the setup, necessaryto observe interference patterns! is given by

Kout 5 JAKNJAR 5 F0.5~1 1 P1,N11!Ein

000G , (14)

where JA represents the Jones matrix of the analyzer@with azimuth angle aN11 with respect to the last Nthfiber; see the note below Eq. ~6!#:

JA 5 Fcos aN11

0sin aN11

0 G . (15)

Finally, remembering that Ein is a normalized spec-tral amplitude distribution function of the source

* Ein~v!dv 5 1, (16)

one can obtain from Eq. ~2! the desired formula foraverage intensity:

I#out 5 0.5~1 1 P1,N11!. (17)

This formula shows immediately that average inten-sity I#out is equal to 0.5 if only one of the angles be-tween the component of the setup is equal to 645°.


B. Detection of Patterns Produced by a Single Sensor

Now we derive a formula for the contrast of the in-terference pattern produced by a single fiber placedin position k @Fig. 3~b!#. We assume that only thiskth element introduces a group delay tk smaller thanthe coherence time ts of the source. This means thatthe changes in intensity caused only by this fiber canbe detected at the output of the system. This alsomeans that the influences of the nondiagonal compo-nents of the coherence matrix on the diagonal ones inthe formula for the output intensity disappear whenpassing through every element except this kth one.And, as is described in Section 2, at the end of eachelement of the setup, we may disregard all of theseinfluences except the detected one.

The diagonal components of the spectral coherencematrix Kk21 of the light entering the considered kthelement are given, according to Eq. ~12!, by

FKxx~k21!

Kyy~k21!G 5 0.5F1 1 P1,k21

1 2 P1,k21GEin, (18)

while the nondiagonal components are assumed to bezero. Transforming this light through the detectedkth fiber, one can obtain a formula for the diagonaland the nondiagonal components of the coherencematrix at the output of the kth fiber

FKxx~k!

Kyy~k!G 5 0.5F1 1 P1,k

1 2 P1,kGEin, (19)

FKxy~k!

Kyx~k!G 5 20.5P1,k21 sin~2ak!F exp~idk!

exp~2idk!GEin. (20)

The spectral coherence matrix Kk11 of the lightafter the ~k 1 1!th element is again diagonal @disre-garding the phase terms that disappear owing to in-equality ~10! at the end of the setup#, with itsdiagonal vector expressed by the following formula:

FKxx~k11!

Kyy~k11!G 5 0.5F1 1 P1,k11

1 2 P1,k11GEin 2 0.5P1,k21

3 sin~2ak!sin~2ak11!cos~dk!F 121GEin. (21)

Calculating the spectral coherence matrix K overthe next elements of the setup from k 1 2 to N ~also,here each vector of diagonal components dependsonly on the preceding diagonal one! as

FKxx~N!

Kyy~N!G 5 )

n5k12

N H0.5F1 1 cos~2an!1 2 cos~2an!

1 2 cos~2an!1 1 cos~2an!

GJ3 FKxx

~k11!

Kyy~k11!G , (22)

then transforming it through the analyzer and calcu-lating output intensity Iout as

Iout 5 0.5I0~1 1 P1,N11! 2 0.5I0P1,k21

3 sin~2ak!sin~2ak11!Pk12,N11 cos~d#k!G~tk!, (23)

one can obtain this final formula for contrast Vk of theinterference pattern associated with the kth fiber inthe setup @see Eq. ~5!#:

Vk 5 UP1,k21 sin~2ak!sin~2ak11!Pk12,N11

1 1 P1,N11U

5 U P1,N11

1 1 P1,N11tan~2ak!tan~2ak11!U . (24)

The following conclusions can be drawn immedi-ately from Eq. ~24!:

• The contrast of the pattern produced by thechosen single kth fiber can reach a maximum valueequal to 1, whereas the contrasts of the patterns pro-duced by other single fibers are equal to 0; this can beachieved simply by adjustment of the angles betweenfibers when ai 5 0° or 690° ~i 5 1 . . . k 2 1, k 12 . . . N 1 1!, ak 5 645°, and ak11 5 645°. Notethat this alignment is equivalent to a setup consistingof only one fiber, in which both polarization modes areexcited and placed between two ~multielement! fibersthat play the role of lead-in elements.

• The contrasts of the pattern produced by the cho-sen K-independent fibers placed in positions ki ~i 51 . . . K! can be maximized while the other contrastsare equal to 0. This maximum contrast reaches val-ues at least as high as ~K 2 1!K21yKK. This isachieved by adjustment of all angles in the input andin the output of selected fibers to aki

5 aki115 0.5

arccos@~K 2 1!yK#1y2 and the other angles to 0° or 690°.

C. Detection of Cross Patterns Produced by AdjacentSensors

In this section a formula for the contrast of a commoninterference pattern produced by two adjacent fibers@Fig. 3~c!# is derived. Let us consider a system of Nfibers in which only two adjacent elements the kthand the ~k 1 1!th in the system are taken into accountand in which they introduce a sum ~tk 1 tk11! ordifferential ~tk 2 tk11! group delay smaller than thecoherence time ts of the source. Since the contrast ofthe interference pattern associated with an individ-ual fiber was calculated in Subsection 3.B, let usassume for convenience that individual group delaystk and tk11 are much higher than ts.

The diagonal components of the spectral coherencematrix at the end of the ~k 1 1!th fiber are given byEq. ~21!. The nondiagonal component cannot be ne-glected in this case, and it is calculated according toEqs. ~8! and ~9!:

The spectral coherence matrix Kk12 of the lightafter the ~k 1 2!th element is now diagonal, with itsdiagonal components expressed by the following for-mula:

FKxx~k12!

Kyy~k12!G 5 20.5P1,k21 sin~2ak!cos2~ak11!sin~2ak12!

3 cos~dk11 1 dk!F 121GEin

1 0.5P1,k21 sin~2ak!sin2~ak11!sin~2ak12!

3 cos~dk11 2 dk!F 121GEin. (26)

The procedure for the calculation of the coherencematrix transformation over the next elements of thesetup and the output intensity is the same as de-scribed in Subsection 3.B. This leads to a formulafor the contrast of two interference patterns associ-ated with the sum @Vk,1~k11!# or the difference@Vk,2~k11!# of phase differences introduced by adja-cent fibers:

Vk,1~k11! 5

UP1,k21 sin~2ak!cos2 ak11 sin~2ak12!Pk13,N11

1 1 P1,N11U , (27)

Vk,2~k11! 5

UP1,k21 sin~2ak!sin2 ak11 sin~2ak12!Pk13,N11

1 1 P1,N11U , (28)

which for clarity can be presented in the followingform:

Vk,6~k11! 5 U P1,N11

1 1 P1,N11tan~2ak!

3 0.5@sec~2ak11! 6 1#tan~2ak12!U , (29)

where 1 refers to a sum of the phase shifts while 2refers to a difference of the phase shifts.

Note @cf. Eqs. ~24! and ~29!# that two adjacent fibersproduce interference patterns with the same con-trasts as given by an equivalent single fiber with aninput angle ak, an output angle ak12, and a contrastmodification coefficient D equal to

D 5 usec~2ak11! 6 1u. (30)

As one can see from Eqs. ~27! and ~28!, for ak11 5 0°,

FKxy~k11!

Kyx~k11!G 5 20.5P1,k21 sin~2ak!F cos2 ak11 exp@i~dk11 1 dk!# 2 sin2 ak11 exp@i~dkn11 2 dk!#

cos2 ak11 exp@2i~dk11 1 dk!# 2 sin2 ak11 exp@2i~dk11 2 dk!#GEin, (25)

where terms that produce average intensity andterms correlated with individual phase differenceswere omitted.

the contrast of the pattern associated with the sum ofthe phase shifts is maximal and equal to 1, whereasthe contrast of the pattern associated with the differ-


ence of the phase shifts is minimal and equal to 0.This is obvious because condition ak11 5 0° means thatthe directions of the first eigenvectors in both consid-ered fibers are parallel. For ak11 5 90° ~the firsteigenvector in the kth fiber is perpendicular to the firsteigenvector in the ~k 1 1!th fiber!, the maximum valueequal to 1 is reached by the contrast associated withthe difference of the phase shifts, whereas the contrastof the pattern associated with the sum of the phaseshifts is minimal and is equal to 0. Finally, whenak11 5 645°, both contrasts are equal to 0.5.

It is easy to prove that in the general case of Kadjacent fibers placed in positions from k to k 1 K 21, a formula for the contrasts of the common inter-ference patterns can be expressed as

V 5 U P1,N11

1 1 P1,N11tan 2ak~0.5!K21

3 )i51

K21

@sec 2ak1i 6 1#tan 2ak1KU , (31)

where indices in V were omitted owing to complicatednotation. The same remark about equivalency isvalid in the case with a modified contrast modifica-tion coefficient.

D. Detection of Cross Patterns Produced by SeparatedSensors

Consider a system of N fibers in which only two ele-ments, the kth and the mth one, separated by at leastone fiber ~m . k 1 1!, are taken into account @Fig. 3~d!#,and they introduce a sum ~tk 1 tm! or a differential ~tk2 tm! group delay smaller than coherence time ts of thesource. Also, in this case, we assume for conveniencethat individual group delays tk and tm are much higherthat ts. ~The contrast of the interference pattern as-sociated with an individual fiber was calculated inSubsection 3.B.! To calculate the respective con-trasts, the procedure presented in Subsection 3.B ~thedetection of a single fiber! was applied twice: the firsttime to calculate the transmission of the spectral co-herence matrix through the kth fiber ~the diagonalmatrix at the input of the kth element and at theoutput of the ~k 1 1!th element! and the second timethrough the mth fiber ~the diagonal matrix at the inputof the mth element and at the output of the ~m 1 1!thelement!. The final formula for the diagonal vector ofthe spectral coherence matrix contains only terms thatinclude functions of a sum or of a difference of phasedifferences dk and dm. This leads to the following ex-pression for the contrasts of the common interferencepatterns produced by two separated fibers:

Vk,6m 5 0.5U P1,N11

1 1 P1,N11tan~2ak!

3 tan~2ak11!tan~2am!tan~2am11!U , (32)

where 1 refers to a sum of the phase differenceswhile 2 refers to a difference of the phase differences.


Note that both ~sum and differential! contrasts asso-ciated with two separated fibers take the same value,unlike the contrasts associated with two adjacent fi-bers.

It is easy to generalize this analysis to a case of thedetection of the interference patterns produced by Kseparated fibers placed in positions ki ~i 5 1 . . . K!.The formula for the contrasts of these patterns is asfollows:

V 5 ~0.5!K21U P1,N11

1 1 P1,N11)i51

K

tan~2aki!tan~2aki11!U ,

(33)

where indices in V were omitted as in Eq. ~31!.Note that Eq. ~33! can be rewritten for convenience

in the following form:

V 5 ~0.5!K21US1 1 P1,N11

P1,N11DK21

)i51

K

VkiU , (34)

which means that the contrast produced by K sepa-rated fibers can be calculated as a product of thecontrasts produced by individual fibers ~as in Subsec-tion 3.B! with a certain contrast modification coeffi-cient. Therefore, for example, in the case of thedetection of interference patterns produced by sev-eral fibers in interaction with the last ~receiving! fiber~whose phase shift can be adjusted arbitrarily!, themaximization of this contrast consists of the maximi-zation of the contrasts produced by selected individ-ual fibers, with the fixed contrast being produced bythe receiving fiber. Note that the maximum value ofthe contrast presented in Eq. ~34! decreases with in-creasing number K of interfering fibers, which is de-scribed by the term ~0.5!K21.

E. Detection of Cross Patterns Produced by SeparatedGroups of Sensors

The most general case is the detection of the inter-ference cross patterns produced by K-separatedgroups of adjacent fibers; the ith group consists of miadjacent elements placed in positions from ki to ki 1mi 2 1. Thus the contrast is given by

V 5 ~0.5!K21U P1,N11

1 1 P1,N11~0.5!•i51

K ~mi21! )i51

K

3 Htan~2aki! )

j51

mi21

@sec~2aki1j! 6 1#tan~2aki1m!JU ,

(35)

@indices have been omitted as in Eqs. ~31! and ~33!#.

4. Calculation of the Contrasts in Setup with PolarizingFibers

Assume that some fibers in the setup are of the di-chroic type, with the ratio of attenuation of orthogo-nally polarized modes equal to 0 ~perfectly polarizingfibers!. A 3M polarizing fiber available on the mar-ket allows one to control the attenuation ratio of po-

larization modes simply by varying the length of thisnew fiber and thus to reach the desired polarizationratio. The same contrast calculation procedure asthe one described in Section 2 can be applied to a casein which one or more polarizing fibers are present inthe setup. Below we present the final formulas forthe average intensity and contrasts of the interfer-ence patterns produced by some fibers in a setupconsisting of N fibers from which K fibers ~placed inpositions ki, i 5 1 . . . K! are of the polarizing type.

A. Average Intensity

In the considered case, average intensity I#out is givenby

I#out 5 )i51

K

@0.5~1 1 Pki11,ki11!# , (36)

which is similar to Eq. ~17! obtained in Subsection3.A. However, the maximum value of average in-tensity in this case is ~0.5!K, and it quickly decreaseswith an increasing number of polarizing fibers.

B. Detection of Patterns Produced by a Single Sensor

Let us assume that the interference pattern producedby only one fiber placed in position j between the~ki!th and the ~ki11!th polarizing fiber is detected atthe output of the setup. Thus the formula for therespective contrast can be presented as

Vki,j,ki115 UPki11,ki11

tan~2aj!tan~2aj11!

1 1 Pki11,ki11

U , (37)

which is almost the same as Eq. ~24! but with modi-fied indices in the terms Pk,l. Note that only theangles between fibers placed between the ~ki!th andthe ~ki11!th polarizing fibers appear in Eq. ~37!.This means that the contrast of the interference pat-tern produced by a single fiber depends only on thetransformation of the light between the polarizingfibers that limit this fiber, and, in consequence, itdoes not depend on the alignments of the fibers beforeand after the considered polarizing fibers. Thus, ifthe setup is not optimally adjusted, the degradationof the considered contrast depends only on misalign-ments of the fibers placed between two polarizingfibers.

C. Detection of Cross Patterns Produced by AdjacentSensors

To calculate the contrast of the interference patternsproduced by two adjacent fibers placed in positions jand j 1 1 between the ~ki!th and the ~ki11!th polar-izing fibers, one should carry out a similar analysis asin Subsection 3.C, taking into account the remarks

made in Subsection 4.B. This leads immediately tothe following formula:

Vki,j,6~ j11!,ki115 U Pki11,ki11

1 1 Pki11,ki11

tan~2aj!

3 0.5@sec~2aj11! 6 1#tan~2aj12!U ,

(38)

which can be simply generalized to a case of K adja-cent fibers.

D. Detection of Cross Patterns Produced by SeparatedSensors

The next problem to solve is the computation of thecommon contrast for the interference pattern pro-duced by two separated fibers placed, for example, inpositions j and m ~m . j 1 1!. When there are fibersof a different kind present in the setup ~polarizingfibers that are used to separate the detected nonpo-larizing fibers!, this problem splits into two cases:In the first case, both detected fibers are placed be-tween the same polarizing fibers @~ki!th and ~ki11!thto keep the notation used in the previous sections#.In the second case, they are separated by one or morepolarizing fibers. In the first case, one can easilytransform Eq. ~32! into the following form:

Vki,j,6m,ki115 0.5U Pki11,ki11

1 1 Pki11,ki11

tan~2aj!tan~2aj11!

3 tan~2am!tan~2am11!U . (39)

One can show that, in the second case, when thedetected jth and mth separated fibers are placed be-tween two different pairs of polarizing fibers ~ki , j ,ki11, k1 , m , kl11, i Þ l !,

Vki,j,ki11,k1,m,kl115 0.5UPki11,ki11

tan~2aj!tan~2aj11!

1 1 Pki11,ki11

U3 UPkl11,kl11

tan~2am!tan~2am11!

1 1 Pkl11,kl11

U ,

(40)

which, taking Eq. ~38! into account, can be rewritteninto the following form:

Vki,j,ki11,k1,m,kl115 0.5Vki,j,ki11

Vkl,m,kl11. (41)

5. Conclusions

A complete contrast analysis for a complex ~consist-ing of any number of elements! fiber-optic white-lightinterferometric system based on spectral coherencematrix formalism has been carried out. This hasbeen done for a setup of white-light interferometricsensors made from HB fibers, but the analysis is validfor other versions of this measurement system. Theanalysis can be easily made, for example, for a setup


consisting of fiber couplers ~here coupling ratios takethe role of rotation angles between the HB fibers!. Acomplete analysis ~but without proof ! also has beencarried out for a case in which perfectly polarizedfibers, which can be used as leading and separatingelements, are present in the setup. The derived an-alytical formulas for the contrasts of all possible in-terference patterns produced by a setup consisting ofany number of sensors allow one to find the optimalalignment for this setup. These formulas let us si-multaneously maximize contrasts of selected pat-terns while minimizing those of other patterns andalso allow us to adjust all the elements of the systemproperly. Finally, the results of our study let usestimate the possible number of detected sensorswhen designing white-light interferometric systems,depending on the possibility of detecting specific vis-ibilities of desired patterns and taking into accountthe permitted noises and the sensitivity of the detec-tors.

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contrast analysis for a fiber-optic white-light interferometric system

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