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Optical Fourier Transforms ofBirefringent FibresMaksymilian Pluta aa Central Optical Laboratory, Department of PhysicalOptics , ul.Kamionkowska 18, 03-805, Warsaw, PolandPublished online: 01 Mar 2007.

To cite this article: Maksymilian Pluta (1987) Optical Fourier Transforms of BirefringentFibres, Journal of Modern Optics, 34:11, 1451-1470, DOI: 10.1080/09500348714551361

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JOURNAL OF MODERN OPTICS, 1987, VOL . 34, NO . 11, 145 1-1470

Optical Fourier transforms of birefringent fibres

I. Polymeric textile fibres

MAKSYMILIAN PLUTACentral Optical Laboratory, Department of Physical Optics,ul.Kamionkowska 18, 03-805 Warsaw, Poland

(Received 3 April 1987 ; revision received 15 June 1987)

Abstract. A cylindrical birefringent fibre orientated diagonally between twocrossed polarizers can be regarded as a single-, double- or multiple-slit source oflight, depending on the refractive index of the medium which surrounds the fibre .When normally trans-illuminated by monochromatic light and observed througha polarizing microscope fitted with a condenser-slit diaphragm, the fibreproduces specific interference patterns in the exit pupil of the microscopeobjective . These patterns manifest themselves as optical Fourier transforms .Their properties and possible applications are discussed . The experiments wereperformed using polymeric textile fibres .

1 . IntroductionSix years ago it was noticed [1] that a cylindrical birefringent textile fibre can

produce specific interference patterns in the exit pupil of a standard polarizingmicroscope fitted with a subcondenser slit diaphragm, when the fibre occupies theposition of the object plane of the microscope objective and is trans-illuminated bycoherent light. The interference patterns are especially visible when monochromaticlight of continuously variable wavelength is used and the fibre is orientateddiagonally with respect to the crossed directions of light vibration of the sub-condenser polarizer and over-objective analyser . The orientation of the condenserslit is less important, but its preferable direction is parallel to the fibre axis ; however,the slit width should be sufficiently narrow in order to illuminate the fibre byspatially-coherent light .

It has been stated that the interference patterns observed in the objective exitpupil behave as the optical Fourier transforms . Some preliminary results of thisstatement were reported at the 1981 DGaO meeting in Bremen [1] and theInter/Micro-81 conference in Cambridge ; subsequently the experiments werecontinued and refined from a practical point of view, and the detailed results are nowpresented in this paper. Polymeric textile fibres in particular are considered, but thetheoretical discussion can be applied to other birefringent fibres as well . However,birefringent optical (or light-guiding) fibres are generally not textile fibres andremain beyond the scope of this paper .

2. Birefringent fibre as a bifocal cylindrical lensWhen surrounded by a medium of refractive index n', a birefringent fibre

orientated diagonally between two crossed polarizers acts as a specific bifocalcylindrical lens (figure 1), whose focal lines L 11 and L1 contain light vibrationsparallel (1I) and perpendicular (1) to the fibre axis (it is assumed that the fibre is madefrom polymeric material by spinning and drawing) . The focal lengths f jl and f,, i .e .

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11

Figure 1 . Birefringent fibre as a bifocal cylindrical lens, illustrating equations (1), (2) and(5)-(8) .

the distances from the fibre centre to the focal lines LI , and L1 , respectively, aredefined by

where 2r is the fibre diameter, n 11 and n1 are the refractive indices of the fibre for lightcomponents vibrating parallel and perpendicular to the fibre axis .

The above equations follow directly from the thick-lens formula

1 (n n') 1- 1 +(n-n')zt

(3)

f

ri r2

nr1r2

where n is the refractive index of the lens, r 1 and r2 are its radii, t is the lens thicknessand n' is the refractive index of a medium which surrounds the lens . For a cylindricalfibre of diameter 2r, the radius r1 =r and r 2 = -r (the sign minus is from theconvention of signs in geometrical optics), while t = 2r . Hence, the focal length fof anisotropic cylindrical fibre may be expressed as

f = 2r4(n nn,) n, .

(4)

Substituting n=n ll and n=n1 yields equations (1) and (2) respectively for ananisotropic (double-refracting) fibre in which the directions of light vibrations areparallel and perpendicular to the fibre axis as shown in figure 1 .

f~~ --2r

n ll (1)4(n I ,

2r (2)f1 = 4(n1n1n) ,,

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Optical FTs of polymeric textile fibres

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Depending on the refractive index n', the focal lengths can be greatly varied . Themost interesting situation is when the fibre is surrounded by an air medium, whoserefractive index n' can be approximated to be equal to unity . In this case equations (1)and (2) take a simpler form :

will play a significant role. These and other parameters of the several birefringentpolymeric textile fibres which were examined are listed in table 1 .

3. ApparatusA standard, but slightly modified, polarizing microscope was used (figure 2) . The

modification consisted in adding a slit subcondenser diaphragm D, wedge inter-ference filter IF and a double-refracting Wollaston prism W. The prism W was notnecessary for observation of the optical Fourier transforms of birefringent fibres, butserved for measuring light wavelengths in real time . This was achieved via acalibration plot b(2), where b is the period of the interference fringes (or interfringespacing) produced by the Wollaston prism in monochromatic light extracted from amicroscope halogen lamp by means of the wedge interference filter IF . The lightwavelength 2 was varied by transverse translation of the filter IF .

In fact, the apparatus used as a combination of a standard polarizing microscopeand a simple double-refracting (or wavefront-shear) microinterferometer . When theoptical Fourier transforms were observed, the Wollaston prism W was of courseremoved from the path of light .

As shown in figure 2, the slit diaphragm D is located in the front focal plane of thesubstage condenser C, thus light incident on the birefringent fibre F consists ofparallel beams . Optical Fourier transforms (OFT) are observed, via a microscopeocular and Bertrand lens BL, in the exit pupil E of the microscope objective Ob . TheBertrand lens is a typical component (positive lens) of polarizing microscopes .Usually it serves, together with the microscope eyepiece (Oc, figure 2), forobservation of conoscopic images of birefringent microcrystals . These images arelocalized in the objective exit pupil (E) . However, in this instance this lens is used forobservation of the optical Fourier transforms of birefringent fibres as mentionedabove. Like the conoscopic images, these transforms are also localized in theobjective exit pupil (strictly speaking in the Fourier plane or the back focal plane ofmicroscope objective if the slit S of the condenser diaphragm D is coincident with thefront focal plane of condenser as shown in figure 2) . The conoscopic images ofbirefringent objects are however of quite another nature than the optical Fouriertransforms which are the subject of this paper .

f1=2r4(nn 11 (5)1) ,11 -

nlf1=2r (6)4(nl -1)

For the subject of this paper the fibre birefringence B defined as

B=n 11 -n1 , (7)

as well as the difference df in the focal lengths

df=f1 -f11 (8)

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Optical FTs of polymeric textile fibres

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1 45 5

Figure 2 . Schematic diagram of a microscope system used for observation and processing ofthe optical Fourier transforms of birefringent fibres .

The Bertrand lens BF can be removed from the path of light, and a normalmicroscopical image F' of the fibre F under study arises in the image plane II' of theobjective Ob; this image is observed through the eyepiece alone (Oc) .

The basic orientation of the polarizer P, analyser A and slit S of the condenserdiaphragm D, and the fibre under study F was as shown in the left-hand and right-hand sides of the diagram (figure 2), where SS denotes the direction of the condenserslit S, PP and AA are the directions of light vibration of the polarizer P and analyserA, respectively, and FF is the direction of the fibre axis .

4 . Single-slit optical Fourier transformsThese transforms are observed when the fibre birefringence B (see equation (7))

is very small . Consequently, the difference dJ in the focal lengths f 11 and f1 (seeequation (8)) is also very small and two focal lines L11 and Ll (see figure 1) arepractically unresolvable, they form therefore a single focal line, which manifestsitself as a narrow slit in an opaque thin screen illuminated by a parallel light beam . Ifthe focal line is extremely narrow, it can be treated as the one-dimensional Diracdelta function, whose Fourier transform is equal to unity . This means that thewavefront is a plane surface in the Fourier plane of the objective Ob (figure 2) and hasthe same amplitude over the objective exit pupil E (it is assumed that the objectivedoes not absorb the light across the pupil E) .

The situation described above is illustrated by figure 3 . This applies usually toundrawn cylindrical polymeric textile fibres (see table 1, fibre No . 1), whosebirefringence is normally weak and therefore permits us to obtain excellent Dirac

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(a)

M. Pluta

(b)

I= Imax sine ((p/2) = Imax sine (n8/A) .

(c)

Figure 3 . Focal line (a), its optical Fourier transform (b), and conventional image (c) of aweakly birefringent textile fibre (see table 1, fibre sample No. 1) surrounded by airmedium. Objective magnifying power/numerical aperture : 40 x /0 . 65 .

delta functions (figure 3 (a)) if the fibre is surrounded by an air medium, while thecondenser slit S (figure 2) is sufficiently narrow and exactly parallel to the fibre axis .

It can be found that the objective exit pupil appears to be uniformly illuminated ifthe fibre focal line is not wider than 2/8A, where A is the numerical aperture of theobjective. The difference in light intensity at the centre and in the marginal zone ofthe exit pupil is then theoretically not greater than 25% . Consequently, figure 3 (b)shows that the width of the focal line which produced the optical Fourier transformsimilar to that of the Dirac delta function was equal to about 0-1 ,um . If, however, thecondenser slit S (figure 2) is orientated at an angle to the axis of the fibre F, the focalline is somewhat wider and becomes maximally wide when the slit S is at right anglesto the fibre axis, and a normal image F' (figure 3 (c)) of the fibre under study can beobserved if the microscope is then slightly defocused .

The intensity I of the focal line (figure 3 (a)) may approximately be expressed bythe formula

(9)

where (p and 6 are respectively the phase shift and the optical path difference betweenthe light components (11 and 1) due to the fibre birefringence (6=(P ;./21r) . As canbe seen, the maximum intensity I=I a ,, is obtained when 6=a/2 or, in general,6=(2m+1)A/2 (here m=0, ±1, ±2, ., .) . The optical path difference 6 producedby a birefringent fibre along its diameter is defined by the equation

6 = 2r(n 11 -n1) = 2rB .

(10)

As mentioned above, the birefringence B should be small to obtain an extremelynarrow focal line, which could be considered as the Dirac delta function .Consequently, the birefringent fibre must be relatively thick for obtaining b=)./2and thus the maximally intense focal line .

The birefringent fibres which produce extremely narrow focal lines permitted usto develop a simple double-refracting interferometer with variable direction of tilt oflaterally sheared wavefronts for testing microscope objectives [3] . The opticalsystem of the interferometer is similar to that shown in figure 2, but the Wollastonprism W is positioned in the image plane 11' of the objective under test (Ob), and an

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Optical FTs of polymeric textile fibres

1457

additional polarizer (polaroid) is installed between the objective Ob and the prismW. This additional polarizer (not shown in figure 2) is crossed with the subcondenserpolarizer P, hence its direction of light vibration is parallel to that of the analyser A .The birefringent fibre F is placed on the object stage as shown in figure 2 . TheWollaston prism W produces a lateral wavefront shear, while the fibre F producescontinuously variable direction and amount of tilt of laterally sheared wavefrontswhen the subcondenser slit S is slowly rotated around the condenser axis .Interference patterns deformed by the wave aberrations of the tested objective areobserved in the objective exit pupil .

For illustration, figure 4 shows the interference patterns of an aberration-freemicroscope objective . When the condenser slit S is exactly parallel to thebirefringent fibre F (figure 2) and the tested objective Ob is optimally focused on thefibre focal line, the interference pattern looks like that shown in figure 4 (a) ; nointerference fringes occur and the area where the sheared wavefronts overlap isuniform in brightness (or homogeneous in tint if white light is used) . If, however, thecondenser slit forms an angle o with the axis of the birefringent fibre, the uniform-field interference disappears and straight-line fringes occur in the overlapping areaof the sheared wavefronts (figure 4 (b)) ; the interference fringes are inclined withrespect to the direction of wavefront shear (S) . For a particular value of a the

(a)

Figure 4. Exactly focused shear-interference images of an aberration-free microscopeobjective, illustrating the double-refracting interferometer for testing microscopeobjectives, which uses a birefringent fibre for the variable direction of tilt of laterally-sheared wavefronts .

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M. Pluta

interference fringes become parallel to the direction of wavefront shear S, i .e . the axisof wavefront tilt (AT) is parallel to S (figure 4 (c)) .

Such a variable tilt of interfering wavefronts cannot be obtained if a conventionalslit (e .g . a slit ruled in a metallic thin film evaporated onto a glass slide) is used insteadof the birefringent fibre (the conventional object slit does not require an additionalpolarizer in the system shown in figure 2) . The use of the birefringent fibre radicallyimproves the ability of the interferometer of this kind . First of all, the sensitivity ofthe interferometer is higher and the interpretation of the lateral shearing interfer-ograms is easier when the tilt axis AT (figure 4 (c)) is parallel to the direction S ofwavefront shear .

The interference patterns such as shown in figure 4 correspond to those producedby an ideal microscope objective . Otherwise, if the objective suffers from a waveaberration, no uniform-field interference (figure 4 (a)) and no straight-line inter-ference fringes (figures 4 (b) and (c)) can be obtained . A detailed description of thisinterferometer can be found in [3] .

Let us now return to the basic subject of this paper . As mentioned earlier, aweakly birefringent fibre, such as No 1 in table 1, produces a single focal line similarto the Dirac delta function. This holds good if the fibre is surrounded by an airmedium. If, however, the same fibre is immersed in a liquid, its focal line becomeswider (figure 5 (a)) and a uniform distribution of light intensity is no longer obtainedin the objective exit pupil (figure 5 (b)) . The intensity distribution such as that shownin figure 5 (b) resembles the optical Fourier transform of a Gauss function ;consequently, the distribution of light intensity across the focal line (figure 5 (a)) isalso of the Gauss-function character . This optical Fourier transform is however lessinteresting and less useful than that shown in figure 3 (b) .

5. Double-slit optical Fourier transformsThe birefringent fibre discussed above produces two focal lines L11 and Ll

(figure 1), whose separation df is however so small (smaller than the light wavelengthA) that the focal lines cannot be resolved, and they therefore behave as a single focalline .

(c)Figure 5 . As in figure 3, but here the fibre is surrounded by a liquid medium of refractive

index n'= 1 .515 .

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Optical FTs of polymeric textile fibres

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Let us now consider a cylindrical fibre whose birefringence is significant and thudit can produce two focal lines separated from each other by a distance dj greater than2 (see fibres 2 to 5 in table 1) . Now, two focal lines, L 11 and Ll (see figure 1), act as twolight slits, one of which follows the other . The focal lines are mutually coherentacross their widths but they are incoherent along their lengths . Each pair of coherentpoints, P 11 and Pl (figure 6), produces spherical wavefronts E 11 and El whose radii ofcurvature are slightly different . The two wavefronts can interfere with each otherand produce an interference pattern with annular/circular fringes observed in theexit pupil E of the microscope objective Ob (figure 2) . Any other pair of coherentpoints, Q 11 and Ql (figure 6), gives rise to an individual interference pattern identicalwith that produced by light wavefronts emerging from the points P 11 and P1 . Allindividual interference patterns are mutually incoherent, they occupy the sameposition in the Fourier plane of the microscope objective and produce, by incoherentsuperposition, an intense resultant interference pattern of circular symmetry(figure 7). This pattern does not change its position in the exit pupil of the objectivewhen the birefringent fibre F (figure 2) is transversely or vertically translated . Thisproperty results from a well known theorem, which states that the Fourier transformdoes not translate if the object under the Fourier transformation is translated . Nochanges in the interference pattern were observed as the ideally-cylindrical fibre wasrotated about its own axis .

If the fibre is moderately birefringent (see fibre No . 2 in table 1), an OFT occurswhich consists of a single annular dark interference fringe . Its size (or diameter)depends on the wavelength 2 of light used . The light wavelength A can easily bevaried by using a source of white light and wedge interference filter IF (figure 2) .Starting from the short-wavelength region of this filter and passing continuouslytoward the long-wavelength region permits us to observe a fascinating flow of theinterference pattern and its annular fringes of consecutive orders . When one of thefringes appears in the marginal zone of the objective exit pupil for a blue light (figure7 (a)), the diameter of this fringe decreases with increasing A (figures 7 (b) and (c)),and eventually the annular fringe becomes reduced to a circular patch at the centre ofthe objective exit pupil (figure 7 (d)) . Further increasing the light wavelength repeatsthe above sequence of interference patterns . Overall number of these sequenceswithin the visible spectrum depends on the fibre birefringence B, namely it increaseswith increasing B (compare fibres 2, 3 and 4, as well as figures 7, 8 and 9) . Sometimes

L 11Figure 6 . Illustrating the origin of the circular or annular shapes of the optical Fourier

transform of two slightly vertically-separated focal lines (L I and L1 ) produced by abirefringent fibre surrounded by air medium .

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(a)

(c)

M. Pluta

(b)

(d)Figure 7 . Annular (a), (b) and (c) and circular (d) optical Fourier transforms produced by a

birefringent fibre surrounded by air medium (see fibre samples 2-5 in table 1) .

two annular dark fringes are simultaneously visible in the exit pupil of a microscopeobjective of high numerical aperture (greater than 0 . 65) if highly-birefringent fibresare examined .

6 . Measurement of fibre birefringenceOf particular interest to research and practical workers in the general field of

textile-fibre physics and technology is the measurement of the birefringence ofpolymer fibres . This quantity is determined by the physico-chemical properties ofthe polymer and the nature of the spinning and drawing operations, and it providesinformation about the degree of orientation of molecules of the fibrous material .Knowledge of this kind is greatly important for evaluating the effect of any chemicalor mechanical treatments of polymeric fibres .

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I

0 .5

0

Figure 8. Relation between light intensity I and light wavelength A, referred to the centre ofaxially-symmetrical optical Fourier transforms shown in figures 7 and 9, for differentbirefringent fibres listed in table 1 : PA-6, solid line ; PET, dashed line; and PET 5000dot-dashed line . The intensity I is given in arbitrary normalized units .

Polymeric fibres are frequently not uniform in their birefringence referred toconsecutive zones (layers) of the fibre . Sometimes the surface and subsurface zonesmanifest themselves as more birefringent than the inner or axial zones . We cantherefore speak of the birefringence profile across the fibre . It can be preciselydetermined by using more advanced micro-interferometric procedures which arelargely used in the studies of optical fibres . However, for textile fibre researchers andtechnologists it is usually sufficient to know the mean birefringence along the fibrediameter . This quantity is defined by equation (7) and is measured by means ofdifferent techniques of transverse micro-interferometry (see [4] and the papers citedthere). However, the optical Fourier transforms described in the previous sectionoffer us a new procedure for measuring the mean birefringence of cylindrical fibres,which removes some limitations of the techniques known to date. Moreover, thespectral dispersion of the mean birefringence can readily be determined .

It has experimentally been stated that each of the consecutive interferencepatterns shown in figures 7 and 9 includes quantitative information about the meanbirefringence B of the fibre . In particular, the patterns with a dark central patch(figures 7 (d), 9 (a), 9 (c), 9 (e), 9 (g) and 9 (z) occur when the optical path differencealong the fibre diameter 2r is given by

6 = 2r(n 11 -n1) = 2rB=m2,

(11)

where m is an integer referred to as the current interference order . On the other hand,the interference patterns whose centres are maximally bright refer to

b = 2r(n 11 -n1 ) = 2rB=-2'M A-

(12)

These relations between the interference patterns on the one hand and thebirefringence B and wavelength A on the other hand make it possible to determinethe spectral dispersion of the birefringence B(2) or discrete birefringences for givenlight wavelengths. The developed procedure uses monochromatic light withcontinuously variable wavelength .

Starting from a long wavelength permits us to select a particular wavelength A 1for which the centre of the interference pattern is maximally dark (figure 9 (a)) . This

Optical FTs of polymeric textile fibres

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1461

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1462 M. Pluta

(a)

(.f)

(b)

(K)

(c)

(h)

(d)

(i)

(e)

(1)

X10

Figure 9 . Illustrating the principle of the measurement of the spectral dispersion of fibrebirefringence by using axially-symmetrical optical Fourier transforms produced by thedrawn polymeric textile fibres (see fibre samples 4 and 5 in table 1) .

situation can be expressed as

6 1 =2rB1 =m 12 1 ,

(13)

where m 1 is an integer number referred to as the initial (or introductory) interferenceorder. Further decreasing the wavelength (by transverse sliding the wedge inter-ference filter IF as shown by the arrows in figure 2) leads to other particularwavelengths 22 < A 1 i 2 3 < 22, Al < A3 . . . . for which the centre of the interferencepattern becomes maximally bright (figure 9 (b)), dark (figure 9 (c)), bright(figure 9 (d)), etc . All these situations may be described as

bs= 2rBs=(m1+gs)2s=mss,

(14)

where s=2, 3, 4 . . . . (for s= 1 equation (13) holds good), and q s =0 . 5, 1, 1 . 5, . . . . It isself-evident that qs expresses the increment of the current interference orders ms withrespect to m 1 , i .e .

ms =m 1 +qs .

This is always true if the spectral dispersion of the fibre birefringence is normal(anomalous dispersion is beyond the scope of this discussion) .

From equations (13) and (14) it follows that

Asm1 = qS B

sl 1 -2s,

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where

Optical FTs of polymeric textile fibres

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Ab=-=

__ B., __(n1 n±),

(17)Bs1 B1 (nll - n1)1

The coefficient Bs1 is determined by the spectral dispersion of the fibre birefringenceB(2) . However, from equations (13) and (14) it also follows that

B s1 =8 5 /b1 ;

(18)

the coefficient Bs1 may therefore be determined not only from the graph B(A) but alsofrom the graph of the spectral dispersion of the optical path difference 6(A) due to thefibre birefringence .

If a fibre does not suffer from the spectral dispersion of birefringence, thecoefficient Bs1 =1 and equation (16) takes the simpler form

m1=qs~1 - ,

which permits us to determine m 1 , and then the optical path difference 6 andbirefringence B, without any problems if A1 and As are known or measured .

Both natural and artificial textile fibres free from the spectral dispersion ofbirefringence are very rare . Nevertheless, the coefficient B s1 is only slightly higherthan unity for most polymeric textile fibres, and these are primarily within the scopeof this paper . Consequently, equation (19) may be used instead of equation (16), andthe initial interference order m 1 can be determined when the coefficient Bs1 is notexactly known a priori . In general, the approximate value for m 1 calculated fromequation (19) will be slightly greater than the true value given by equation (16) if thespectral dispersion B(2) is normal . For instance, if equation (19) gives m 1 = 7 .8 in thered light (compare with table 2), the true initial interference order m 1 may at most beequal to seven .

As mentioned earlier (see § 3), a double-refracting Wollaston prism (W, figure 2)was used for determining the light wavelengths A1, 22, 2 3, . . . in real time throughmeasuring the respective interfringe spacings b 1 , b 2 , b 3 , . . . of the empty interferencefield produced by this prism . The interfringe spacing b varies with the wavelength Aof monochromatic light as follows

2

2E 2D tan a 2(n,-n o ) tan a'

(20)

where e is the intersection angle of the interfering wavefronts (here this angle isexpressed in radians), D, n e and no are respectively the birefringence, extraordinaryrefractive index and ordinary refractive index of a uni-axial crystal (e.g. quartz) ofwhich the Wollaston prism is made, and a is the apex angle of this prism .

From equation (20) it follows that the wavelengths A1 and 2 (see equation (16))can be expressed as

2 1 =b 1 E 1 =b 1 2D1 tan a,

(21)

A,=b 5Es =b52D5 tan a .

(22)

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The initial interference order m 1 is now found to be

_

b5m1-q5 B D b b5 'sl is 1 -

where

D,= D1 (ne no)1

D S (ne - no)5

(23)

(24)

This coefficient results from the spectral dispersion of the Wollaston prismbirefringence and is always slightly smaller than unity . Consequently, the productB51D15 is closer to unity than Bs1 since this latter coefficient is usually slightly greaterthan unity . We can therefore assume B51D15^. 1, and equation (23) takes a simplerform

m1 =q5bs

(25)b 1 -b s

This formula is frequently more suitable than equation (19) for calculating the initialinterference order m 1 and usually gives only slightly greater or slightly smaller valuesfor m 1 than the exact formula (23) . Thus, an integer which is the nearest to the meanvalue m1 resulting from equation (25) is taken into account as the true initialinterference order m 1 .

The measurement of the interfringe spacings is carried out with the help of amicrometer screw (MS, figure 2) . This screw is linked with the transverse movement(p) of the double-refracting prism W . The microscope is provided with an ocularfitted with a pointer line onto which the centres of the interference fringes are guided .The measurement of multiple interfringe spacings, e.g. the distance d= 40b betweeninterference fringes of plus and minus twenty orders, rather than a single interfringespacing is recommended to obtain more accurate values for b .

The wavelengths 2 1, .~2' ).3, . . . are read out from the calibration plot of b(1,) . Thecalibration was performed very carefully using highly monochromatic light (He-Ne,Ar and He-Cd lasers) . The distance d=100b was measured with the help of themicrometer screw MS (figure 2) . Once plotted graph b(A) is valid for all experiments .For the quartz Wollaston prism this graph is almost linear over the visible spectrumand quite consistent with equation (20) .

The practical performance of the OFT technique described above will beillustrated in detail in a separate paper ; here only the results of an exemplarymeasurement performed on a polyester (PET) textile fibre are given (see table 2 andfigure 10). This and other fibres were also measured, for comparison purposes, usinga standard micro-interferometric method [5] . No discrepancy between the twomethods was observed ; the OFT method is, however, incomparably better for thestudy of highly-birefringent fibres immersed in an air medium .

There is another advantage to emphasize ; namely that the measurement processor testing procedure using the OFT method is very suitable for fully automaticoperation since the optical Fourier transform is invariable when the fibre understudy changes its position by its inevitable or possible translation in the field of viewof the microscope objective .

It is, however, worthwhile noting that the annular or circular dark fringes (seefigures 7 and 9) take an oval or elliptical shape when the condenser slit S (figure 2)

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Optical FTs of polymeric textile fibres

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ACa

0 .200

0 .190

0460

Figure 10. Graphical representation of the results of an exemplary measurement of thespectral dispersion of birefringence B of a PET textile fibre (sample No . 4 in table 1) byusing its axially-symmetrical optical Fourier transforms (compare with figure 9 andtable 2) .

forms an angle a with the fibre axis . This deformation is caused by a tilt of thewavefronts I II and El (figure 6) when the slit S is not parallel to the fibre axis . If theangle a increases, the ellipticity mentioned above also increases . Nevertheless, smallangles o are quite tolerable and do not introduce any errors into the measurement ofthe fibre birefringence .

Moreover, it is important to note that the shape of the optical Fourier transforms(figures 7 and 9) is regular if the cylindrical fibre does not suffer from local opticalinhomogeneities and/or geometrical irregularities . Otherwise, the annular andcircular dark fringes are deformed (figure 11) . It has been stated that the OFTmicroscopy discussed here is a very useful tool and highly sensitive method for thequality control of cylindrical polymeric textile fibres and for the detecting and rapidassessment of their optical and geometrical microdefects .

7 . Multiple-slit optical Fourier transformsTable 3 presents data for the same polymeric textile fibres as table 1, but the fibre

samples are now immersed in liquids whose refractive indices n' approximate thefibre refractive indices n and n1 . The focal lengths f l andf1 are actually relativelylong. Consequently, the distance df between the focal lines L 11 and L1 is also long (seefigure 1) . No optical Fourier transforms of the focal lines are therefore observed inthe exit pupil E of the microscope objective Ob of an optical system as shown infigure 2 . Instead, we can observe an optical Fourier transform of the fibre itself. Thecylindrical birefringent fibre now behaves as a multiple-slit object consisting of two,three, four, and even more parallel slits of different width arranged as a line grating ina plane perpendicular to the objective axis . In fact, these slit components formed bythe fibre are the bright interference fringes of polarized light . They are separatedfrom each other by dark fringes . All these fringes are localized in or near the objectplane II of the microscope objective Ob (figure 2), and their width and position varywhen the light wavelength is changed (increased or decreased) . Two fringearrangements are however predominant : first, two relatively-wide bright fringes,

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Optical FTs of polymeric textile fibres

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Figure 11 . Irregular optical Fourier transforms of a cylindrical polyester textile fibre whichsuffers from local optical inhomogeneities and/or geometrical irregularities .

Table 3 . Focal lengthsf and fl , and their difference df of the same textile fibres as in table 1,11but the fibres are immersed in liquids of refractive index n' close to n I , and n1 .

separated by a central dark fringe, appears for particular wavelengths (figure 12 (a)) ;second, a wide bright fringe surrounded by two dark fringes covers the central zoneof the fibre for other particular wavelengths of monochromatic light (figure 12 (c)) .In the first instance we observe a specific optical Fourier transform in the exit pupilof the microscope objective (figure 12 (b)) ; this transform is similar to the Younginterference pattern. Clearly another optical Fourier transform is observed(figure 12 (d)) if the fibre appears in the object plane as shown in figure 12 (c) . As canreadily be seen, figure 12 (d) resembles the optical Fourier transform of a single andrelatively-wide slit described by a rectangular function A(x), where A is the lightamplitude and x is the coordinate perpendicular to the fibre axis in the object plane ;

Samplenumber

Name offibre n'

f11

(µm)fl(µm)

df(µm)

Type ofOFT

1 Elana 1 . 515 420.86 438 .47 17 . 61 As in figure 52 PA-6 1 . 555 273 . 73 -228 .83 502 . 56 As in figure 123 PET 5000 1 . 615 105 . 06 -79 .03 184.09 As in figure 124 PET 1 . 615 53 .43 -102 .42 155 . 85 As in figure 125 PPT 1 . 615 6 . 76 -214 .65 221 . 41 As in figure 12

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Figure 12 . Microscopical images in polarized light (a) and (c) and optical Fourier transforms(b) and (d) of a cylindrical birefringent textile fibre (see fibre sample No . 2 in table 1)immersed in a liquid medium whose refractive index n' is near to the refractive indicesn 1l and nl of the fibre .

thus the pattern shown in figure 12(d) is approximately described by a squaredsincus function, i .e . (sincx) 2 .-(sin x/x) 2 . Very narrow bright fringes seen at themarginal zones of the fibre (figures 12 (a) and (c)) contribute only slightly to the lightdistributions (figures 12 (b) and (d)) in the Fourier plane .

From conventional polarizing-light microscopy it results that figure 12 (a) showsthe optical path difference b, at the fibre centre, defined by equation (11) . On theother hand, figure 12(c) corresponds with equation (12) if the fibre centre ismaximally bright. When the light wavelength 2 is continuously varied, the fibreimages shown in figures 12 (a) and (c) and their respective optical Fourier transforms(figures 12 (b) and (d)) repeat . The overall number of these repetitions within thevisible spectrum is directly proportional to the fibre birefringence . Consequently, wehave also in this instance the situation described by equations (13) and (14) for aseries of particular wavelengths A1, 22, 23We can therefore use the sameprocedure as before (see §6) for determining the spectral dispersion of fibrebirefringence, though spectral dispersion n'(2) of the immersion liquid may beunknown. The assessment of the maximum darkness and maximum brightness ofthe fibre centre (figures 12(a) and (c)) can be improved if the optical Fouriertransforms (figures 12(b) and (d)) are observed or recorded . This procedure issometimes preferable to conventional double-refracting micro-interferometry [5] .

It is self-evident that between two specific patterns shown in figures 12 (a) and (b)and 12 (c) and (d) there are many other configurations of bright and dark fringes forlight wavelengths other than 2 , 2 2, 2 3, . . . However, all these configurations onlyslightly differ from each other ; the most characteristic are those shown in figure 12 .

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Optical FTs of polymeric textile fibres

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8. Optical Fourier transforms with elliptical shapeTable 4 presents data for the same polymeric textile fibres as tables 1 and 3, but

the fibres are now immersed in a liquid (e .g . water) whose refractive index n' is muchsmaller than n 11 and n1 . The focal lengthsf 11 andfl (see figure 1) are actually relativelylong, but the difference df between f1 and f I , is equal to several pm (except for fibreNo 1) . The optical Fourier transforms that arise in this situation are shown infigure 13 .

The fringes are not straight-line or annular, but rather elliptical interferencefringes occur . However, these behave as the annular fringes shown in figures 7 and 9do when the light wavelength is varied . The longer axis of the ellipse is parallel andthe shorter is perpendicular to the condenser slit . For particular wavelengthsAt, 2 3, As, . . . one of the elliptical dark fringes becomes reduced to the central darkpatch of oval shape . This situation can be described . by equation (11), while for theother particular wavelengths AZ, 21, 2 6 , . . . the centre of the interference patternbecomes maximally bright and equation (12) applies . We can therefore determinethe fibre birefringence in the same way as described earlier in § 6 . The ellipticalfringes are of course irregular if the cylindrical fibre suffers from local optical and/orgeometrical defects .

The nature of the elliptical optical Fourier transforms (figure 13) cannot be soeasily interpretated as that of interference patterns whose fringes are annular/circular (figures 7 and 9) or straight-line (figures 12 (b) and (d)) . At any rate, thisspecific interference pattern is produced by both the fibre focal lines and the fibre

Table 4 . Focal lengthsf 11 and f1 , and their difference df for the same textile fibres as in tables1 and 3, but the fibres are immersed in a liquid whose refractive index n' is much smallerthan n 1l and n1 .

Figure 13 . Optical Fourier transform of a cylindrical birefringent textile fibre (see fibresamples 2-5 in table 1) immersed in a liquid medium whose refractive index n' is muchsmaller than the refractive indices n il and n1 of the fibre .

Samplenumber

Name offibre n'

f il

(µm)f1

(µm)df

(µm)Type ofOFT

1 Elana 1 . 333 79.90 80-38 0 . 48 As in figure 52 PA-6

,1 . 333 33-48 41 . 71 8 . 23 As in figure 13

3 PET 5000 1-333 17.18 23-20 6 .02 As in figure 134 PET 1-333 20.32 33 .54 13-22 As in figure 135 PPT 1 . 333 5-80 14-61 8-81 As in figure 13

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Optical FTs of polymeric textile_ fibres

itself and can be considered as intermediate between the annular and straight-linefringe patterns .

9 . ConclusionsThe described optical Fourier transforms appear to be interesting from both

theoretical and practical points of view . They were produced by cylindricalpolymeric textile fibres . However, the procedure can also be applied to otherbirefringent fibres, including some optical (wave-guiding) fibres holding polariz-ation or birefringent properties .

The most useful optical Fourier transforms appear to be those presented in § 5 .They enable the fibre birefringence and its spectral dispersion to be measured in anew way described in §6 . This method does not require the mounting of highlybirefringent fibres in immersion media (liquids) as is usually necessary in conven-tional polarizing microscopy or microinterferometry of textile fibres . Moreover, themeasurements procedure does not require any special instrument, and a standardpolarizing microscope may be used, whose equipment should only slightly bemodified. In comparison to the known micro-interferometric techniques used forthe measurement of the fibre birefringence, the OFT method is more versatile andnot limited in its application to relatively-small optical path differences (b) due to thefibre birefringence . Conversely, it is more effective when b is not too small, saygreater than 5A .

Additionally, the optical Fourier transforms presented in § 5 permit us to testrapidly the cylindrical birefringent fibres, in particular, polymeric textile fibres,when their homogeneity and especially optical and geometrical micro-irregularitiesmust be assessed on relatively long fibre segments . The testing procedure is suitablefor fully automatic operation due to the invariance of the optical Fourier transformwhen the fibre under test translates within the field of view of the microscopeobjective .

However, it is important to note that the OFT procedure cannot be used for themeasurement of the refractive indices n ll and n1 ; only their mean difference(i .e . n il -n1) maybe determined. Consequently, this procedure should be treated as aspecial technique which supplements the micro-interferometric methods known todate .

On the other hand, the optical Fourier transform described in § 4 may beregarded as a useful tool for some optical purposes when extremely-narrow light slitsare necessary . For instance, a single focal line produced by a slightly birefringentfibre exhibits specific (and even astonishing!) properties, which permitted us todevelop a simple and versatile double-refracting interferometer with variabledirection of tilt of laterally sheared wavefronts for testing microscope objectives .

AcknowledgmentsI wish to thank Mrs B . Mirkowska for her help in making photocopies of the

illustrations .

References[1] PLUTA, M ., 1981, Einladung and Program zur Tagung 'Optik and Feinmechanik' der

DGaO, (Wetzlar: DGaO), p . 36 .[2] HANtzA, A. A ., and SIKORSKI, J ., 1978, J. Microsc ., 113, 15 .[3] PLUTA, M ., 1984, Optica Applicata, 25, 77 .[4] HANIZA, A. A., 1986, J. Microsc ., 142, 35 .[5] PLUTA, M ., 1972, J. Microsc ., 96, 309 .

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