Birefringence measurement of fiber-optic devices

Y. Yen and R. Ulrich

A spectrometer-polarimeter is described for the measurement of the wavelength-dependent general birefrin-gence of single-mode optical fibers and fiber-optic devices. The polarizer rotates, the analyzer is stepped,and a Fourier transform yields the principal axes and retardation. The Poincare representation of polariza-tion is used throughout.

1. Introduction

Birefringence in single-mode fibers results fromvarious internal causes (noncircular core,' internallateral stress components2 ), and external influences(lateral force,3 twisting, 4 bending, 5 etc.). Employingclever combinations of these effects, a number of usefulin-line fiber-optic devices have been proposed recently,for example, an adjustable polarization rotator6 orcompensator, spectral filters of the olc design,8 9 anoptical isolator,10 and various fiber-optic sensors.1 Totest the operation of such devices it is generally neces-sary to identify their eigenstates of polarization (orprincipal axes of birefringence) and their retardationand then determine how these quantities depend onsome parameter such as twist angle, wavelength, mag-netic field, or temperature.

For such measurements on single-mode fiber-opticdevices we designed a polarimeter as described in thispaper. The principle of operation is analyzed in Sec.II, using the Poincare representation of polarization. InSec. III the relevant technical details are given of theinstrument and the method of evaluation. Finally, inSec. IV the capability of the instrument is illustratedby showing some results obtained with a single-modefiber-optic bandpass filter.

II. Principle of Operation

The fiber-optic device under test is regarded as alossless general birefringent optical element. Used withmonochromatic light it transforms any given input stateof polarization (SOP) into a corresponding output state.

The authors are with Technische Universitdt Hamburg-Harburg,2100 Hamburg 90, Federal Republic of Germany.

Received 7 February 1981.0003-6935/81/152721-05$00.50/0.©) 1981 Optical Society of America.

This transformation can be represented on the Poincar6sphere12 by a general rotation vector U. For an arbi-trary input state Ci, then, the output state C2 is ob-tained by rotating C, on the surface of the sphere aboutthe direction of Q in a right-handed sense through theangle Q = I . Hence, the direction of Q defines theeigenstates of the fiber device, and the magnitude Qgives its retardation. The vector completely char-acterizes the birefringence of the device. It is the pur-pose of the polarimeter to determine . In formulatingthe problem this way, it is tacitly assumed that suitablereference axes have been defined for the definitions ofthe states of polarization.13

The arrangement of the polarimeter is shown in Fig.1. The light source is a Xe arc lamp. A gratingmonochromator M selects the wave number v of themeasurement. The light then passes through a com-bination of linear polarizer LPI and broadband quar-terwave retarder (Fresnel rhomb FR) to render it cir-cularly polarized. The linear polarizer LP2 is rotatedcounterclockwise (facing the light source) by a syn-chronous motor MI at a constant rate of fo = 21.4 Hz.Thus, the input light to the fiber-optic device FD hasconstant power and a linear polarization whose azimuthvaries periodically with 2fo. At the fiber output theanalyzer LP3 is set by a step motor M2 sequentially toa number of fixed azimuths am. The transmitted lightis measured by a PIN Si-photodiode D, whose signal isamplified in a narrowband preamplifier tuned to 2fo,and then processed in a vector lock-in amplifier (EG&G129A). The latter receives a reference pulse each timethe input polarizer LP2 passes through the state ofhorizontal polarization, i.e., two pulses per full revolu-tion. The unknown birefringence of the fiber undertest is then evaluated by a small desk-top computer (HP9810) from the digitized output signals of the lock-in atthe various azimuthal positions of LP3. The computeralso controls the wave number drive of the monochro-mator and a plotter.

1 August 1981 / Vol. 20, No. 15 / APPLIED OPTICS 2721

Fig. 1. Experimental setup of the spectrometer-polarimeter. Ex-planation of symbols in the text.

For a discussion of the procedure of evaluation, weconsider the time-dependent state C1(t) at the fiberinput [see Fig. 2(a)]. It circulates at constant angularvelocity = 4rfo in the positive (eastern) directionalong the equator of the sphere. Hence, the geo-graphical longitude of the input state is'2"14

24,0(t) = cot. (1)

As the unknown birefringence causes a rotation of allstates on the sphere, the output state C2(t) must alsocirculate along a great circle. We call that circle G andcharacterize it by a unit vector 0, normal to the planeof G, and so directed that C and the rotation of C2(t)form a right-handed screw. The geographical latitudeof C will be denoted' 2"14 by 2 G and its geographicallongitude by 20G. Particularly important is the nodalpoint N at which G intersects the equator going fromthe south to the north. This point has the longitude2ON = 2 G + 7r/2. The absolute position of C2(t) on thegreat circle G is described14 by an angle 2y(t), measuredalong G from the nodal point N. As Cl(t) and C2(t)must move at equal angular velocities, it follows fromEq. (1) that

2'y(t) = cot + 2 o. (2)

We have assumed that the origin of time (t = 0) is thatmoment when the input polarizer LP2 passes throughthe reference state H of horizontal linear polarization.In that moment, C2 has the distance 2 o from the nodalpoint N.

The actual analysis now requires two steps: (1) usingthe amplitude and phase of the detector, signal mea-sured at various azimuths of the analyzer LP3 for a de-termination of the parameters 2G, 2 0G, and 2 yo of thegreat circle G; (2) calculating from those parameters thebirefringence vector , i.e., the principal axes of bire-fringence and the retardation of the fiber device.

For the first step, it is noted that the detector signalis proportional to the momentary power transmittanceT(t) of the linear analyzer LP3 at azimuth a. Hence,as there is constant optical power in the fiber, constantfactors may be suppressed here and T(t) used directlyas the detector signal. The analyzer is characterizedby point A on the Poincar6 sphere, lying on the equatorat a longitude'4 of 2a. According to the Poincare for-malism, then, the detector signal varies as

TL(t) = cos2 A(t) = 1/2 + /2 cos2A(t). (3)

Here, 2z\(t) = AC2(t) denotes the time-varying angulardistance' 4 on the sphere between the circulating output

state C2(t) and the analyzer point A. This distanceoscillates symmetrically about its mean value 7r/2 re-gardless of the position of A. Therefore, the term 1/2cos2A(t) in Eq. (3) represents the ac part of the detectorsignal. We recognize that this ac part is largest whenA is adjusted to lie on the great circle G, either at N orat the opposite point.

In the general case we find the characteristics of thedetector signal are straightforward from the propertiesof the spherical triangle ANC2 [see Fig. 2(a)].

Ta(t) = 1/2 + P'd cos(wt + X), (4)

P2 = 1/4 -1/8 coS2 G[(a 4~~~G~ - '41w-(R)], (5)tan(2yo - X) = sin2 ,bG tan2(a - '1N)- (6)

Here, T is the (non-negative) ac amplitude of the de-tector signal and X its phase. Both quantities are di-rectly available at the output of the lock-in amplifier.From Eq. (5) those azimuths are obtained at which theac amplitude Ta becomes maximum or minimum.

(a)

(b)

Fig. 2. (a) Representation of the trajectories of the input state ofpolarization (equator) and output state (circle G) on the Poincar6sphere. (b) Determination of the birefringence vector Q from the

circle G.

2722 APPLIED OPTICS / Vol. 20, No. 15 / 1 August 1981

Tamax = 1/2, for a = ON or ON + 7r/2,

Tamin = /2 sinJ21G|, for a = ON ± r/4. (7b)

Therefore, we can determine the desired parameter OG= N - w/4 by searching for those azimuths a at whichthe ac detector signal assumes its extreme values. Fromthose values we can also find the second desired pa-rameter 2 1 G:

sinl2iG = Tmin/Tmax- (8)

Moreover, at the azimuthal position (a = N) of themaximum we have X = 2 o according to Eq. (6). Atthat azimuth, therefore, the third desired parameter 2'yomay be read directly from the vector lock-in amplifier.Equation (6) may also be used to determine the sign of24IG which remained uncertain in Eq. (8). By differ-entiating Eq. (6) at a = N, we find

sin2 4'G = -dX/daJ1=0N (9)

This provides an alternative possibility of measuring2kG, including its sign, because the right-hand side ofEq. (9) can be obtained directly from the experiment.

Another uncertainty lies in the choice of the azi-muthal maximum of Ta,. According to Eq. (7a) twosuch maxima exist at a = N and at a = N + wr/2.Without additional means, then, the azimuths of thefast and slow axes of birefringence remain uncertain to7r/2. If desired this uncertainty could be removed byone extra measurement in which a X/4 plate is inserted.This is necessary, however, at one wave number only.At neighboring wave numbers a decision is possible fromthe fact that the dispersion of the axes must be a smoothanalytical function.

In the second step of the analysis the birefringencevector Q can be determined from G by a simple geo-metric argument [See Fig. 2(b)]. From any pair (C1 C2 )of corresponding input/output states a locus is obtainedfor the direction Q, because that direction must be onthe great circle that bisects the arc C1C2. From Eqs. (1)and (2) we find a first pair of such corresponding statesfor t = -2y o/, [Fig. 2(b)]. At that time C1 has theazimuth 2'01 = -2yo and C2 (t1) = N. Hence, the lon-gitude of Q must be that of point M, lying halfway be-tween C1 (t1 ) and C2 (t1 ):

20 = ON - 7y, (10)

tan24' = sin(ON + -yo) tan(i'G + 7/4). (11)

The last equation, expressing the latitude of Q, followsfrom a consideration of the spherical triangle QNM.The side of QN of this triangle must bisect at N theangle (24 + r/2) formed by G and the equator, ac-cording to the mentioned geometrical argument appliedat cot = 2. From the same triangle the retardationangle I I l can be obtained through the relation

cosQ/2 = cos(O + yo) sin(t'G + 7r/4). (12)

Thus we have completely determined the birefringencevector Q from the parameters of the circle G.

Ill. Experimental Details

The optical power that can be coupled from an inco-herent source into a single-mode fiber is essentiallydetermined by the spectral brightness of the source andby the optical bandwidth. The latter was selected inthis experiment by a grating monochromator of f =600-mm focal length and entrance/exit slit widths oftypically 0.7/0.5 mm. The corresponding resolution of12 cm-l gave an adequate detector signal (>5 X 10-11A) over the usable single-mode spectral range(11500-16500 cm-') of the fibers under test. The lightemerging from the exit slit was collimated with an f =60-mm lens.

All linear polarizers were of calcite (Glan-Thompsontype). The first, LP1, is set to pass maximum power ofthe partially polarized output light of the monochro-mator. In combination with the Fresnel rhomb, LP1produces circularly polarized light equivalent in thepresent application to unpolarized light. The Fresnelrhomb is optimized for the 0.7-1.5-jim wavelength range(BK7 glass, 53.5° angle). The rotating linear polarizerLP2 is selected and adjusted for minimum beam de-flection. Such a deflection causes a small modulation(typically 5%) of the power in the fiber, as can be mea-sured by removing LP3. This modulation is at therotation frequency fo, however, and is ignored thereforeby the lock-in amplifier operating at 2fo. Coupling intoand out of the single-mode fiber of -5-Aim core diameteris done by achromatic f = 10-mm lenses.

To locate the maximum of Ta and determine Tmaxand Tmin, the analyzer LP3 is rotated sequentially tosixteen azimuthal positions spaced by Aa = 7r/8. Ateach position the amplitude T. and phase X are read byan analog-to-digital converter and stored in the calcu-lator. All sixteen readings of Ta are used then toevaluate ON. According to Eq. (5), this azimuth ObN isrelated to the fourth harmonic of T2. Therefore, asixteen-point Fourier analysis of Ta is performed:

T2 = A + A4 cos4a + B4 sin4a, (13)

where

8A4 = cos24'G sin4N,

16= L T,(mAa) cos4mAa,

m=1

8B4 = cos24'G cos4 kN,

16= EZ P'(mAa) sin4mAa,

2 ON = 1/2 arctan(B 4 /A4 ),

2OG = 2 arccos[(A + B)1/ 2].

(14)

(15)

(16)

(17)

This method of evaluating 2 N and 2 G is preferredbecause it does not require a search procedure and be-cause it averages over errors caused by possible beamdeflection in LP3 .

In evaluating -yo, an additive error may exist frompossible electronic phase shifts in the preamplifier andlock-in. This problem is overcome by measuring thephase X for both directions of rotation of the polarizerLP2. In both measurements the electronic phase error

1 August 1981 / Vol. 20, No. 15 / APPLIED OPTICS 2723

Fig. 4. Trajectory of the birefringence vector (v) of the retarderof Fig. 3 shown on a world map. The parameter along the trajectory

is the wave number in units of 100 cm- 1 .

Z- V

0goE 01 (12000 14000 16000

FREQUENCY (cm-l)

Fig. 3. Measured dispersion of the birefringence of the retarder ofa three-stage fiber-optic Solc filter: (a) Azimuth of the birefringencevector, at modulo 1800. At the two positions marked byl the azimuthchanges very rapidly by 1800, because ( passes near the poles of thesphere. (b) Ellipticity of the eigenstates of polarization. (c) Phase

retardation. shown at modulo 1800.

is the same, but the optical phases yo have oppositesigns. Therefore the electronic phase shift can be de-termined once from those two measured X values andis accounted for in the further evaluations.

IV. Results

We used the method described to measure thewavelength dependent birefringence (Fig. 3) of the re-tarder of a three-element fiber-optic Solc filter.8 '9 Thethree identical birefringenct elements are coils of eightturns (12-mm diam) of an originally weak birefringencesingle-mode fiber (beat length -2 m). Due to thebending-induced birefringence, 5 each coil acts like awave plate of -5-jim absolute retardation (for example,a phase retardation of 147r at v = 14,100 cm-'). Thefast optical axes at the input/output ends of the coils[shown as dashed lines in the inset of Fig. 3(c)] are ar-ranged to form angles of 15, 45, and 750 with the hori-zontal reference plane, respectively.

In Fig. 3 is shown the measured dispersion of the totalbirefringence Q(v) as a function of wave number v. InFig. 4 the dispersion of the axes, given in Figs. 3(a) and(b), is represented again in the form of a trajectory ona world map of the Poincar6 sphere. This representa-tion shows with particular clarity the operation of this

1.0

z

za:

0.5

012000 14000

FREQUENCY (cm-1 )16 000

Fig. 5. Transmittance of the three-stage Solc filter with bandpasscharacteristic. --- measured; X, calculated from data of Fig. 3.

retarder. 8 The retarder becomes a 8olc filter if a linearpolarizer and analyzer are added at the input and out-put, respectively. Their optimum longitude and lati-tude can directly be read from Fig. 4. If they are bothset to the point marked P a filter with narrow pass-bands should result. The expected spectral transmit-tance characteristic has been calculated from themeasured birefringence 0(v) and is shown by the crossesin Fig. 5. This filter has then been realized by suitableadjustments of LP2 (now fixed) and LP3 to the point Pof Fig. 4. The light is modulated with a chopper nowto facilitate detection. The actually measured spectraltransmittance is shown in Fig. 5. For comparison, wehave also given the transmittance calculated from thebirefringence data of Fig. 3. The excellent agreementconfirms the validity and accuracy of the measured bi-refringence Q(v).

V. Conclusions

We have described a polarimeter employing a rotat-ing polarizer for the measurement of the wavelengthdependent birefringence of single-mode optical fibersand fiber-optic devices. The discussion and data re-duction have been based entirely on the Poincar6 rep-resentation of polarization, permitting a mathematically

2724 APPLIED OPTICS / Vol. 20, No. 15 / 1 August 1981

1o0

N- 90°

t900

N-

I-i

a-a

- yQ

o'/ 0\/ 0/) \.|r~~~~0

; \1 X IX X

x~~~~~~~~~~~~~

_ _ _ x\

I___ I_____________ 1 I ,.I- f ' ~

simple and intuitively appealing evaluation. Theusefulness of the method has been demonstrated bymeasuring a three-element fiber-optic golc filter as anexample. It is clear that the described method ofevaluation could be equally well employed for measur-ing the dependence of birefringence on other parame-ters than wavelength, e.g., temperature or pressure. Inany case we believe that the Poincare representation ofbirefringence as used here (e.g., in Fig. 4) will help toprovide a clearer understanding of general fiber-opticarrangements, and that it may even lead to the devel-opment of novel fiber-optic devices.

Y. Yen gratefully acknowledges many helpful dis-cussions with W. Eickhoff and financial support fromthe Alexander-von-Humboldt Stiftung. We thank H.P. Huber of AEG-Telefunken for providing the single-mode fiber.

Y. Yen is on leave from the Shangai Institute ofTechnical Physics, Academia Sinica, China.

References1. R. H. Stolen, V. Ramaswamy, P. Kaiser, and W. Pleibel, Appl.

Phys. Lett. 33, 699 (1978); J. D. Love, R. A. Sammut, and A. W.Snyder, Electron. Lett. 15, 615 (1979).

2. I. P. Kaminow and V. Ramaswamy, Appl. Phys. Lett. 34, 268(1979).

3. Y. Namihira, M. Kudo, and Y. Mushiako, Trans. Inst. Chem. Eng.60C, 391 (1977); M. Johnson, Appl. Opt. 18, 1288 (1979).

4. R. Ulrich and A. Simon, Appl. Opt. 13, 2241 (1979).5. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, Opt. Lett. 5, 273

(1980).6. R. Ulrich and M. Johnson, Appl. Opt. 11, 1857 (1979).7. H. C. Lefevre, Electron. Lett. 16, 778 (1980).8. M. Johnson, Opt. Lett. 5, 142 (1980).9. Y. Yen and R. Ulrich, Opt. Lett. 6, 278 (1981).

10. R. H. Stolen, in Digest of Topical Meeting on Integrated andGuided- Wave Optics (Optical Society of America, Washington,D.C., 1980), paper MB2.

11. S. C. Rashleigh, Opt. Lett. 5,392 (1980); W. Eickhoff, Opt. Lett.6, 204 (1981).

12. G. N. Ramachandra and S. Ramaseshan, in Handbuch derPhysik, Vol 25/1, S. Fluigge, Ed. (Springer, Berlin, 1962), p. 1; R.Ulrich, Opt. Lett. 1, 109 (1977).

13. At either end of the fiber a pair of fixed orthogonal reference axesis required normal to the direction of light propagation. One axisof either pair is arbitrarily assigned to mark horizontal H linearpolarization, the other axes vertical V. Circular polarizationstates are called left- (L) or right-(R) handed when the electricvector is seen (facing the source of light) to rotate in the mathe-matically positive (ccw) or negative (cw) sense, respectively. Allazimuths are counted then in the positive sense (ccw) from theH direction.

14. The factor 2 has been included here in the definition of the anglein accordance with the usual notation of the Poincar6 for-malism.

0

Meetings Calendar continued from page 2705

1982June

22-25 OSA XII Int. Quantum Electronics Conf., Munich Mtgs.Dept., OSA, 1816 Jefferson Place, N. W., Wash., D.C.20036

28-1 July Precision Electromagnetic Measurements Conf., BoulderD. Belsher, NBS, 1-4001, 325 Broadway, Boulder,Colo. 80303

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5-7 Applications of Laser-Doppler Anemometry to FluidMechanics Int. Symp., Lisbon F. Durst, Sonderfor-schungsbereich 80 An Der Universitat Karlsruhe, 7500Karlsruhe 1, Postfach 6380, FRG

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21-27 SPIE 26th Ann. Int. Tech. Symp. & Exhibit/15th Int.Cong. on High Speed Photography and Photonics, SanDiego SPIE, P.O. Box 10, Bellingham, Wash.98227

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? SPIE Industrial Applications of Infrared ThermographyConf., Milwaukee SPIE, P.O. Box 10, Bellingham,Wash. 98227

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? Coherence and Quantum Optics, 5th Rochester Mtg. L.Mandel, U. Rochester, Dept. Phys. & Astronomy,Rochester, N.Y. 14627

continued on page 2746

1 August 1981 / Vol. 20, No. 15 / APPLIED OPTICS 2725