Misalignment in imaging multifibers

M. E. Marhic, S. E. Schacham, and M. Epstein

A statistical approach is used to characterize image transmission through misaligned multifibers, i.e., differ-ently arranged at the input and output faces, leading to a spatially invariant line spread function. The re-sulting MTF is the product of the MTF for the aligned system and a function characterizing stochastic de-partures from alignment. The case of circular fibers with Gaussian misalignment is treated theoreticallyand is found to account satisfactorily for experimentally observed results.

Introduction

Transmission of images through optical fibers re-quires a multifiber arrangement, which can be obtainedby either the alignment of individual fibers or by fusionof step-index optical fibers. Another means of imagetransmission is through a single fiber with a gradedindex of refraction, but this case is not considered here.A multifiber, which is composed of individual small-diameter fibers, can be made flexible enough to beuseful in applications such as industrial inspection andmedical endoscopy. The multiple reflections encoun-tered in a step-index optical fiber cause its exit face tobe uniformly illuminated, averaging the input irra-diance. The image obtained through a multifiber isthus a matrix of dots much as a halftone photograph.The image transfer properties of such multifibers havebeen considered earlier. 1-5 The theoretical derivationsof all prior works were based on the assumption ofperfect alignment of the individual fibers in a multifiberor fiber bundle. However, the experimentally mea-sured values of contrast were substantially lower thanpredicted by theory, and this was. attributed to flaresand leakage of light through the fiber cladding,1-3 al-though the possibility of misalignment was also men-tioned.1 The multifibers used in these experimentswere composed of individual fibers of relatively largediameter (40 ,um or larger), which made it easier tofabricate a well aligned multifiber. On the other hand,commercial flexible fiberscopes, notably those used inendoscopy, are made of individual fibers approximately10 m in diameter, complicating the attainment of

The authors are with Northwestern University, Department ofElectrical Engineering & Computer Science, Evanston, Illinois60201.

Received 8 May 1978.0003-6935/78/1101-3503$0.50/0.©0 1978 Optical Society of America.

perfect alignment, especially in the case when structuresare obtained by using the economical technique of coilwinding. The resulting misalignments cause a reduc-tion in contrast.

In this paper the effects of misalignment on the av-erage modulation transfer function (MTF) in a multi-fiber are investigated, and relevant characteristic pa-rameters are suggested. Theoretical expressions for thetransfer functions are introduced by utilizing unex-pectedly short derivations. Experimental results usingindividually aligned flexible bundles and fused rigidmultifibers are described, and the effect of misalign-ment on the MTFs is discussed.

TheoryImaging optical multifibers are usually evaluated in

two modes, static or dynamic. When a sinusoidal testpattern is imaged onto the end-face of a well alignedmultifiber, the contrast obtained on the exit face usuallydepends on the relative orientation of the bar patternand the fiber array. Hence, the system under staticscanning is spatially variant,3 and it is not possible toascribe to it a unique MTF. In dynamic scanning,4 bothend-faces of the multifiber are oscillated in synchronismresulting in improved image resolution and the elimi-nation of the screenlike pattern formed by the indi-vidual fibers. Although this technique of imagetransmission can be described by a spatially invarianttransfer function, it is usually difficult to implement inpractice. Under certain conditions it is also possible toregard a well aligned multifiber as a spatially invariantsystem and to ascribe to it a unique MTF.3 In thispaper we extend this latter technique to the case of amisaligned multifiber, for which the fiber arrangementon the exit face is not identical to that on the entranceface. Considering N fibers of identical cross sections,centered at the points (xi,yi) in the input plane and

1 November 1978 Vol. 17, No. 21 / APPLIED OPTICS 3503

yI

X' , y''

Line Source ObservationSlit

Fig. 1. Coordinate systems in the input and output planes.

(x,y') in the output plane (Fig. 1), a multifiber will besaid to be aligned if and only if xi = x, and yi = y' for i= 1, 2,.. ., N. The fiber cross section is defined by theaperture function e (x,y) such that

e(x,y) = 1 for (x,y) on the cross section of a

fiber core centered at (0,0), (1)

= 0 otherwise.

If now the input face is illuminated by an irradianceI(x,y), the resulting irradiance I'(x',y') in the outputplane is given by3

NI'(x',y') e(x' - x',y' - y)

X 3 f I(x,y)e(x - xiy-yi)dxdy, (2)

an expression which is usually not spatially invariantbecause of the sampling effect of the fibers.6 To obtaina spatially invariant quantity this granularity must beaveraged in a number of ways, which are delineated inthe following steps.

(a) We set out to calculate the line spread functionfor the system and therefore assume I to be of theform

I(XY) = 6(x - Xo), (3)

so that

I'(x',y') = A e(x' - 4y'- y') e(xo - Xi,y - yi)dyi i _x

= A, e(x' - xi,y' - yi)g(xo - xi),

where we have introduced the chord function

g(xo - xi) =f e(xo - xiy -yi)dy,

which is just the length of the segment parallel to they direction passing through (xo,O) and limited by thecontour of the fiber centered at (xi,yi) (see Fig. 2). Theresponse to the line source given by Eq. (4) is in generalnot spatially invariant, and in particular it varies withy .

(b) A quantity independent of y' is obtained byintegration over y'. This corresponds to averaging I'

along the y direction by collecting light through a longnarrow slit in front of a large detector. We thus definethe average line spread function L (x',xo) as either

L(x',xo) = I'(x',y')dy'

=Lg(xo - xi) f e(x' - x',y' - y)dy'i o

or

L(x',xo) = E g(xo - xi)g(x' - x,). (6)

It may well be that L(x',xo) itself is not spatially in-variant (i.e., a function of x' - x0 only), if the fiberbundle or multifiber has a very regular arrangement, isideally aligned, and the line source and viewing slit areparallel to a remarkable axis of the pattern; for example,this would be the case for measurements parallel to anaxis of symmetry in an aligned hexagonal close-packedarray of circular fibers. 3

(c) We must now make assumptions about xi andx' to introduce further smoothing processes through thesummation over i. For a very large number of fibersintersected by the line source and viewing slit, we mayreplace the summation by an integral of the form

L(xl,xo) = fx f g (x0- x 1)

x g(x' - xl)pj(xi;xl)dxdx, (7)

where pj(x1;x)dxidx is the joint probability of findinga fiber centered between x1 and xl + dx, on the en-trance face and between xl1 and xl1 + dx 1 on the exit face.We can also express pj as

pj(xi;x;) = p(x1)pc(x1x 1), (8)

where p is the probability density for x1 alone, consid-ered as an independent random variable, and PC theconditional probability for xl knowing x1. Conse-quently L (x',xo) becomes

L(x',xo) = SSf g(xo - xl)g(x' - x')p(x)p,(xllx' 1)dxldx'j. (9)

We now assume that p(x ) 1, i.e., that the centers ofthe fibers are uniformly distributed in the x directionin the input plane. This situation can arise for twodifferent reasons: it can be the result of either an in-trinsic randomness in the fiber pattern, or a nonre-markable orientation of a well ordered fiber pattern

(4)

(5)B

KO

Fig. 2. The chord function g(xo - xi) is the length of segment AB.

3504 APPLIED OPTICS / Vol. 17, No. 21 / 1 November 1978

with respect to the line source. Furthermore, we as- ]Slitsume that Pc is spatially invariant, i.e., that

Pc(xilxi) = pA(x'-Xi). (10) IPhotomultiplier

With these assumptions L(x')xo) becomes

L(x',xo) = fg(xo - x1)pA(x! - xl)g(x' - xl)dxldx'l (11)

= f g(-xi)pA(x1 - xl)g(x' - xo - xl)dxldx'1

= .f g(-u)pA(v)g(x' - xo - u - v)dudv

= g(-x')*pA(x')ig(x' - 0), (12)which indicates that L(x',xo) is a double convolutionand a function only of the difference x' - x0, i.e., spa-tially invariant.

By Fourier transformation we may now associate withthis invariant average line spread function L (x',xo) anaverage optical transfer function (OTF)

N = X exp( i27rfx')L(x xo)dx'

= -G(-f)G(f)P(f) exp(-i27rfxo)

= -G*(f)G(f)P(f) exp(-i2irfxo), (13)

where G (f) and P(f) are the Fourier transforms of g(x')and pA(x'), respectively. The unique average MTF ofthe system is then the magnitude of 1(f),

H(f) = I-K(f) = G(f)G*(f)|P(f)|. (14)

This compact result shows that the effect of mis-alignment on the average MTF of a multifiber simplyamounts to multiplying the average MTF H' = GG* ofthe corresponding perfectly aligned multifiber by afactor I PI containing the information about the sto-chastic departures from alignment. For fibers of knowncross section it is therefore possible to calculate H',measure H, deduce I P = H/H', and thereby charac-terize the misalignment in the system. This techniquecan easily be implemented in practice, and, to ourknowledge, it is the only quantitative method proposedto evaluate the misalignment of multifibers.

We now apply this analysis to treat the importantcase of circular fibers. Denoting the core diameter byd, the chord function is given by either

gc(x) = ( x2) /2 for x2 < d4 ~~~~4

or

= 0 otherwise. (15)

Since g is an even function, its Fourier transformis

G,(f) = 2 f ( - X2) cos(27rfx)dx. (16)

Making use of the classical integral,

C cos(a)(l - 42)1/

2dt =-Ji(a), (17)

Jo 2a

where J1 is the Bessel function of the first kind and ofthe first order, we obtain

G.(f) = [2J(rfd)]/(irfd), (18)

0 Imaging Multifiber

Objective ObjectiveSinusoidal

Test ChartLightSource

Fig. 3. Schematic outline of the measurement of MTF.

where a constant factor has been introduced to yieldGc(0) = 1.

We thus find that the average MTF H, of a perfectlyaligned arrangement of circular fibers is

H'(f) = [2Jl(irfd)]/(7rfd)} 2(19)

in agreement with the results obtained by other workersfor either static or dynamic scanning.3 5

To obtain a completely analytic model for the prob-lem, we now assume that the misalignment of the mul-tifiber is adequately described by the Gaussian distri-bution

pg(Xl - x) Xe -xI)] (20)

where r2 is the variance of the difference x 1 - x 1. Thisparticular form for pA is chosen because it matches theintuitively expected shape of the distribution, i.e.,maximum at zero displacement and smoothly droppingoff at large ones; in addition Gaussian distributions areoften encountered in stochastic problems, and our sit-uation is thus likely to be adequately described in thatfashion. A test for this hypothesis will be obtained inthe experimental section. By Fourier transformationof pg we finally obtain the (normalized) average MTFfor the multifiber with Gaussian misalignment

HC(f)[ 2Jl(fd)]2 exp(-27r 2 f2 a2 ). (21)

Experimental Results

Two types of imaging multifibers were considered:a flexible bundle of aligned fibers and a rigid fusedmultifiber. The flexible multifiber is obtained bywinding a threadlike fiber, approximately 10 Am in di-ameter, to form an orderly array. The resulting fiber-optic bundle is bonded at the ends only, so that the in-dividual fibers remain loose over most of the length ofthe bundle, which can therefore be quite flexible. Thequality of such a multifiber depends mainly on themethod used during the winding and alignment proce-dure. The fused multifiber is obtained in a two-stagedrawing process. First, the optical fiber is drawn downto a diameter of about 0.4 mm and is then assembledinto a preform consisting of 11,000 fibers about 50 cmor 60 cm long. Each fiber is carefully inspected forfaults and placed alongside the others. The multifiberpreform is then drawn down again to the desired size.

The experimental arrangement is shown in Fig. 3.The reduced image of a vertical bar pattern of sinuso-idally varying intensity is projected onto the entranceface of the multifiber by an objective lens. A similarlens forms an enlarged image of the exit face of the

1 November 1978 / Vol. 17, No. 21 / APPLIED OPTICS 3505

.9

.8

.7-

.6

.5

.4

.3 u MTF of Fexible Bundte

.2

.1 v I

Fig. 4.

multifiber ophotomultilwidth of whof a singlezontally, anrecorded.values on tlposition of of the respmen.

The expEThe flexibliin diametemade by AStamford, dred indivicI_ _-IIl ._

ordered, many such regions along the multifiber di-ameter are oriented differently to render a fiber ar-rangement of nearly uniformly random distribution.The measured values fit very well a curve obtained forthe computed MTF of a multifiber composed of circularfibers, indicating that the misalignment is negligible asis expected in a fused structure.

MATF / [2 J (rfd )/7rfd] \ ConclusionVTF of Fused Multifiber t The influence of random misalignment in imaging

, , , , | multifibers has been studied analytically, and the re-1/d .

2/d

3/d

4/d 5/d '

6/d sults show that the corresponding MTF is the product

of the ideal MTF and a function characterizing theMTF of flexible and fused imaging multifibers. misalignment statistics. This function can thus easily

be obtained by measuring the actual MTF with a sinegrating and a narrow slit in front of a detector and cal-culating the MTF of the ideally aligned multifiber. ByFourier transformation one obtains the probabilitydistribution function describing the displacement of the

n the photomultiplier. The light enters the fibers from one end to the other, fully characterizing thepther through a very narrow vertical slit the misalignment. Simple parameters, such as mean dis-

ich is a small fraction of the projected image placement, can then be calculated and used to describe

fiber. The test pattern is displaced honi- the degree of misalignment. The technique is appli-.d the contrast for different frequencies is cable to fibers of any shape, with arbitrary alignment

The spatial frequencies correspond to the statistics.Teexitface ofthemultifiber. Changing the The theory has been successfully tested on two

Lhe photomultiplier allows for the recording practical multifibers: a fused multifiber was found to

ose of different segments of the ec i exhibit negligible misalignment, while a flexible oneodf peci exhibited nearly Gaussian misalignment statistics.

rimental results are summarized in Fig. 4. We have thus shown that, in addition to losses in

e multifiber, a 60-cm long structure 5.5 mm contrast introduced by flares and leakage, the trans-containing 10-ym individual fibers, was mission of images through multifibers can be degraded

CMI (American Cystoscope Makers, Inc., by misalignment, and we have provided a practicalIonnecticut). Since there are several hun- method to evaluate precisely this effect. This tech-lual fibers across, the fiber arrangement can nique should find applications in the quantitative

tv+ he - rAnr- At.L characterization of imaging multifibers.LUe Well k1JJJru2111UUU try a U111111.11ly Ua> .s 1O-

bution. The squares in Fig. 4 represent the measuredvalues of the MTF, and the circles MTF/H' or IPI, bothsets with the corresponding ranges obtained in mea-surements of various segments of the multifiber crosssection. The latter set of points can be approximatedby computed values, solid curve, given by exp(-2ir 2f2 U2 )with a- = 0.51d, while the first set fits well the computedH; for d = 17 gm instead of the actual diameter of 10Aim. The only considerable deviation is noticed at thehighest measured spatial frequency of 0.6/d, where thecontrast is very much higher than expected. This dis-crepancy can be attributed to the method of the mul-tifiber fabrication. Since the winding of consecutivelayers of the multifiber is done in opposite directions,the illumination of a single line of fibers will cause twolines on the exit face, separated by 2d, to appear illu-minated.

The solid triangles in Fig. 4 represent measuredvalues of MTF for a 0.5-mm diam fused multifiberconsisting of 11,000 fibers, 4.2 Am in diameter. Al-though individual regions of the multifiber are very well

This work was performed under a research projectsupported by the National Institute of General MedicalSciences.

References1. N. S. Kapany, J. Opt. Soc. Am. 49, 779 (1959).2. H. Ohzu, T. Sawatari, and K. Sayanagi, J. Appl. Phys. (Jpn.) 4,

Suppl. 1, 323 (1965).3. R. Drougard, J. Opt. Soc. Am. 54, 907 (1964).4. N. S. Kapany, J. A. Eyer, and R. E. Klein, J. Opt. Soc. Am. 47,423

(1957).5. P. G. Roetling and W. P. Ganley, J. Opt. Soc. Am. 52, 99 (1962).

6. Since the final result is eventually normalized, we have omitted

proportionality constants in the rest of the analysis.

3506 APPLIED OPTICS / Vol. 17, No. 21 / 1 November 1978