Incident polarization angle and temperature dependence ofpolarization and spectral response characteristics inoptical fiber couplers

Yoshinori Namihira, Toshio Kawazawa, and Hiroharu Wakabayashi

The incident polarization angle and temperature dependence of the polarization and spectral responsecharacteristics of three different types of fiber coupler are presented. The couplers are (1) the biconical-fused-twisted-taper single-mode fiber (coupler A), (2) the asymmetric-etched-fused-taper wavelength divi-sion multiplex (coupler B), and (3) the biconical-polished polarization maintaining fiber (coupler C), respec-tively. It is confirmed experimentally that the polarization characteristics of couplers A and B vary greatlywith temperature, but those of coupler C are independent of temperature. Also, the wavelength dependencecharacteristics of the power splitting ratio of couplers B and C have almost no change with temperature.However, the wavelength dependence of coupler A is greatly changed with temperature. Comparing couplersA and B, it is postulated that the sinusoidal variations of the polarization state vs the incident polarizationangle are due to the stress birefringence caused by the fiber twisting when the fused fiber coupler is fabricatedand packaged.

1. IntroductionSingle-mode optical fiber directional couplers are

useful for optical fiber communication systems, opticalfiber sensor applications, etc. because of their compati-bility with single-mode fiber (SMF) systems comparedto bulk-optic devices. The optical fiber directionalcouplers with nominally isotropic SMF have been fab-ricated by several techniques. The two-fiber four-port couplers can be classified into three types accord-ing to their fabrication techniques: the etched,'polished,2 and fused3-5 couplers etc. These couplershave been used as fiber optic devices, e.g., a fiber sensorusing a variable coupling ratio, a wavelength filter,7 8

and a polarizing beam splitter.9 Also, the asymmetricfused-taper SMF couplers can be fabricated and have awavelength flattened response. The asymmetry canbe introduced by pretapering,' 0 preetching,"1, 2 andprepolishing13 one fiber prior to the coupler fabrica-tion or even by using two dissimilar optical fibers.

When this work was done all authors were with KDD R&D Labo-ratories, 2-1-15, Ohara, Kamifukuoka, Saitama 356, Japan; T.Kawazawa is now with KDD Chikura Submarine Cable LandingStation, Japan.

Received 14 May 1990.0003-6935/91/091062-08$05.00/0.© 1991 Optical Socioty of America.

In some applications, especially in coherent opticalcomunication systems, the single-mode fiber couplerswill be used to monitor the state of polarization (SOP)on the heterodyne/homodyne receiver for the compen-sation' 4 of polarization mismatching in the polariza-tion control schemes' 5 and in dual-detector hetero-dyne/homodyne receivers' 6 etc. To achieve thesefunctions, the polarization characteristics of the twooutput ports of the couplers have to be as similar aspossible. Therefore, the investigation of polarizationcharacteristics' 7-2 ' in the fiber couplers is very impor-tant.

Also, in wavelength division multiplex (WDM)22,23

systems for coherent optical communications, thespectral response24' 25 as well as the polarization char-acteristics of the single-mode optical fiber couplers isimportant.

In this paper, the incident polarization angle and thetemperature dependence of the polarization charac-teristics and the spectral response of the power split-ting ratio in three types of optical fiber coupler, such asthe biconical-fused-twisted-taper SMF (coupler A,York),3-5 asymmetric-etched-fused-taper WDM (cou-pler B, Gould)," and biconical-polished polarization-maintaining fiber (PMF) [polarization-maintainingand absorption reducing (PANDA)] (coupler C, Fuji-kura),26 are investigated experimentally.

To achieve accurate polarization measurements, thedynamic range must be more than 30 dB. In theconventional spectral response measurement, it is very

1062 APPLIED OPTICS / Vol. 30, No. 9 / 20 March 1991

0; v

Cooled

Fiber Analyzer Ge-PD

difficult to achieve an accurate polarization spectralresponse measurement because of the weak power ofthe optical source (halogen lamp etc.) and low detectorsensitivity etc.

However, using the cooled Ge-PIN photodiode to 77K (-1960C) with liquid nitrogen and the lock-in am-plification technique, automatic polarization spectralresponse measurement systems with a measuring dy-namic range of -36 dB have been developed.25

II. ExperimentFigure 1 shows the schematic diagram of the experi-

mental setup for measuring the polarization spectralresponse characteristics of the optical fiber couplers.25

In the study, three different types of SMF coupler wereemployed: the biconical-fused-twisted-taper SMF(coupler A, York),3-5 the asymmetric-etched-fused-taper WDM (coupler B, Gould),1" and the biconical-polished PMF (PANDA) (coupler C, Fujikura).26

Each of the couplers' parameters [power splitting ra-tio, excess loss, core/cladding diameter, and numericalaperture (N.A.)] is given in Table I.

The measuring wavelength range was X = -1.20-1.70 Am. A halogen lamp with 100-W optical powerwas used as an optical source, and a cooled Ge-PINphotodiode was used as an optical detector. The de-tector parameters are shown in Table I1.25 The outputpower of the single-mode fiber is about -54 dBm; theminimum received power is more than -90 dBm (Ta-ble II). Thus the measuring dynamic range is morethan 36 dB.25

The measurements were carried out as follows: Lin-ear polarized light was launched into the input port ofthe optical fiber coupler using a polarizer, and theoptical intensity I(0) of the output polarization ellipsewas measured at four inclination angles: 0 = 0, 45, 90,and -45° (135°) through the rotating analyzer.27 Theincident polarization angle dependence was measuredby changing the incident polarization angle Oi of theinput port of the coupler.25 The degree of linear polar-ization P, and the output polarization angle T can be

Fig. 1. Polarization and spectral response charac-teristics of measuring systems for optical fiber cou-

plers.

Table I. Optical Fiber Coupler Parameters

Coupler A Coupler 13 Coupler C

Blconlcul-Fused Asymmetric-Etched Bicolcial-P.lishedTwite d.Tper SMF Fused-Taper WM PMF (PANDA Fiber)

Coupler (York) Coupler (Gould) Coupler (Fujikura)

splitting ~- 45 - 44 -40 % (X.Pol)Splitting - 45% (@),13=I ,1p,1.55pm) _ 50 % (Y.Pol)

Ratio (@lS1.7Pn-1.63srn) - 50 % X=131g

(@Xy=1.36gf,156pm) ( @X_ 1.31 urn

-0.2 dB (X-Pol)Excess - 0.10 dB - 0.003 dB ( Pol)

Loss (@ _ 1.55 m) (@X = 1.55m) (@ 1.31 lim)

Core/cladding - 9/125 m - 9/125 im 7/124 jim

Diameter

NA - 0.117 - 0.120

Table II. Cooled Ge-PIN Photodlode Parameters

Noise Short Detector Incident Detector Minimum

Temp. Current Current Surface Total Sensitivity Received

Area Power Power

T (K) IN Its A Pi S Pm

77K 0.2pA 18.1,uA 19.6mm2 0.104mW 174mA/W -9OdBm

(-196-C) (1 pW)

obtained simultaneously by the computer25 using thefollowing equations 27:

-= 1(00) -(90) 2+1 (45)-I(-450)12 (1)Ps I(0°) + I(900) )

(2)= 1 ta-' I(45) - I(-450)12 l I(0°) I(900) -

Also, using the channel selector, the polarizationspectral response characteristics of the two outputports of the optical fiber couplers can also be measuredautomatically. 2 5

20 March 1991 / Vol. 30, No. 9 / APPLIED OPTICS 1063

Halogen Monochromator

Lamp Chopper Polarizer

100w 1

0

9oo600300

00

300-60°-90

0

Oa

.J .°0

a

300 600 900 1200 150° 1800 2100Incident Polarization Angle Oi

(A)

1.0

0.5

0

300 600 900 120° 1500 1800Incident Polarization Angle Oi

M .0.

'o-

o m

.LI

vC I. c;

6a0

0

.I

00.

(B)Fig. 2. (A) Polarization straight-through (1-2) characteristics ofthe biconical-twisted-fused-taper WDM (coupler A) vs incident po-larization angle 0i: (a) degree of linear polarization P,; (b) outputpolarization angle A. (B) Polarization splitting (1-3) characteris-tics of coupler A against incident polarization angle Oi: (a) degree of

linear polarization P,; (b) output polarization angle .

Ill. Results and Discussions

A. Polarization Characteristics vs Incident PolarizationAngle

Figures 2(A) and (B) give the temperature T depen-dence of the polarization straight-through (1-2) andsplitting (1-3) characteristics of the biconical-fused-twisted-taper SMF coupler A3-5 vs the incident polar-ization angle A, respectively. The degree of linearpolarization characteristics P and the output polar-ization angle I in the two output ports of coupler Achange sinusoidally with the incident polarization an-gle Oi (Refs. 20 and 25) depending on whether thetemperature T variations are the same as the PMFcoupler.28 The variations of P, were also found to bealmost linearly proportional to the temperaturechange T (-25oC). Also, it may be considered thatthe variations of P and 'I are thought to be due tooptical anisotropy occurring from the photoelastic (op-toelastic) effect 29' 30 on the basis of the shearing stresscaused by the fiber twisting when the fiber coupler wasfabricated and packaged.3 '

1.0

0.5

0

600 900 120° 1500 1800 2100Incident Polarization Angle Oi

(.0

0.5

> 60'. 300:

<-30'

-60°- ()

(A)

2'~~~~~~~~~~ T.40'C30'C--

10 \ --',-

1 =l.30 pm o'c---

300 600 90' 1200 1500 1800Incident Polarization Angle Oi

210O

(B)Fig. 3. (A) Polarization straight-through (1-a2) characteristics ofthe asymmetric-etched-fused-taper SMF (coupler B) vs incidentpolarization angle Oi: (a) degree of linear polarization P0; (b) outputpolarization angle I. (B) Polarization splitting (1-a3) characteris-tics of couplerB against incident polarization angle 0: (a) degree of

linear polarization P,; (b) output polarization angle I.

Figures 3(A) and (B) represent the temperature Tdependence of the polarization straight-through(1-2) and splitting (1-'>3) characteristics of the asym-metric-etched-fused-taper WDM coupler B as afunction of the incident polarization angle 0. Thetemperature T variations were from 0 to 400C. InFigs. 3(A) and (B), (a) and (b) indicate the degree oflinear polarization P, and the output polarization an-gle I, respectively. From Figs. 3(A) and (B), it wasfound that the variations of polarization characteris-tics of the splitting port (1--3) were greater than thoseof the straight-through port (1--2). It may be consid-ered that the fiber core is asymmetrical because of itswavelength-flattened response characteristics.10-' 3 Itwas confirmed that the linear polarization states P, inthe splitting port of coupler B changes sinusoidallywith the incident polarization angle variation in thesame manner as the PMF (PANDA) coupler with theintrinsic birefringent axes.28 It may be consideredthat the behavior is due to the stress birefringencecaused by the photoelastic effect 29'30 when the fibercoupler was fabricated and packaged.3

1064 APPLIED OPTICS / Vol. 30, No. 9 / 20 March 1991

}.0 1 . . . . . . . . . . . . . . . . . . I .0

.5 I

O(a) \ .C 7 O- .0m

M o0

00

.

"0

0

0.

=_ hc

0

0

. . . I I I

(b) 6

4. ~~~~~~~~~~~~~4

:, ., .. , . .

D~Upler A

. I . . I

'1 = 40'C 10 °----

0 - ---(.0 0Oc_ _

. . I I . I . .

X=1.30 gm -

A . . I . I . . I . .

.

0 N

. ._

1J0

0 O

r0

0

0._

0

0 >

Z c.

2.-`

- atI X I

.__ 4 _< 3Coupler B _ 7

I . . . . .

. . . . . .

.. . . . . . . . . I I .

2

. . . . . .

Al

. . . . .

. . I . .

do

__� _---a --� ��7 ---�:- �-;;

Cl

0

. . ,. . . . I :. . . .

300 600 900 1200 1500 1800 2100

Incident Polarization Angle Oi

(A)

1.0

11W.

.2

0P: 0.5

a

._Z

0.

a5 n

Wavelength X (m )

(A)

1.0

0.5

0

00 300 600 900 1200 1500 1800 2100

Incident Polarization Angle Oi

(B)

Fig. 4. (A) Polarization straight-through (1-2) characteristics of

the biconical-polished PANDA fiber (coupler C) vs incident polar-ization angle O: (a) degree of linear polarization P0; (b) outputpolarization angle A. (B) Polarization splitting (1-3) characteris-tics of coupler C against incident polarization angle O: (a) degree of

linear polarization PI; (b) output polarization angle A.

Figures 4(A) and (B) are the temperature T depen-dence of polarization straight-through (1- 2) andsplitting (1-'3) characteristics of the polished PMFcoupler C,26 respectively, against the incident polariza-tion angle 0i. From Figs. 4(A) and (B), it was observedthat P, and I experience almost no changes with tem-perature T variations. P, changes sinusoidally withthe incident polarization angle 0. Also, the behaviorof P, was found to be the same as that of the fibersqueezer.2 9 '3 0

B. Power Splitting Ratio vs Wavelength

Figures 5(A) and (B) represent the incident polar-ization angle As and the temperature T dependence ofthe spectral response characteristics of power splittingratio R in coupler A. From Fig. 5(A), it was confirmedthat the incident polarization angle Oi had almost noeffect on the spectral response characteristics of thepower splitting ratio R in coupler A. Also, from Fig.5(B), it was found that there was a moderate tempera-

Wavelength x m (jim

(B)

Fig. 5. (A) Incident polarization angle Oi dependence of spectralresponse characteristics of the power splitting ratio in coupler A; (B)temperature T dependence of the spectral response characteristics

of the power splitting ratio in coupler A.

ture change effect on the spectral response characteris-tics of the power splitting ratio R in coupler A.

Figures 6(A) and (B) show the incident polarizationangle Oi and the temperature T dependence of thespectral response characteristics of the power splittingratio R {= P3/(P 2 + P3); P2 and P3 are optical power ofoutput ports 2 and 3, respectively} in coupler B, respec-tively. From Figs. 6(A) and (B), it was confirmed thatthere was almost no effect from the incident polariza-tion angle As and the temperature change on the spec-tral response characteristics of the power splitting ra-tio R in coupler B.

Figures 7(A) and (B) indicate the incident polariza-tion angle Oi and the temperature T dependence, re-spectively, of the spectral response characteristics ofthe power splitting ratio in coupler C. From Fig. 7(A),it was confirmed that there was a large effect from theincident polarization angle Oi on the spectral responsecharacteristics of the power splitting ratio R in couplerC.

From Fig. 7(B), it was found that the power splittingratio R has a small temperature dependence for cou-pler C.

20 March 1991 / Vol. 30, No. 9 / APPLIED OPTICS 1065

31

-3'-6-9

0 (b) I I } 2

0 .----- '\ Coupler C r0I

'-'C I I U-

(a) V/ o'C----; V X = 1.30jlm

I I I .

a.

.

94

0.A.

.

F- W . . . . I . . . . I .

.A> 4IiK 3l Coupler C lI I

, % , , , ., ! . I| . .1.70

1.0 Z-

0.5 -

0

X 1.0

v +

if

r% 0.5

'aW.

c:~

L.

004 0

1.

..1.0

0+

C40

X 0.5C

A

L.4)

004. 01

I.:

.20 1.30 1.40 1.50 1.60 1.7

Wavelength (m )

(A)

1.30 1.40 1.50 1.60 1.70

Wavelength x (im )

(B)

Fig. 6. (A) Incident polarization angle Oi dependence of the spectralresponse characteristics of the power splitting ratio in coupler B (B)temperature T dependence of the spectral response characteristics

of the power splitting ratio in coupler B.

C. Stress Birefringence in the Fused (Coupling) RegionsFigure 8 shows the wavelength dependence of polar-

ization characteristics of coupler A in the case of Oi 45°. From Fig, 8, it was confirmed that a quarter beatlength LBI4 can be estimated from the wavelengthbetween 1.25 m and 2 _ 1.55 Atm. In the sameway, B, LB, and f in the case of the fused region of Lf 30 mm can be estimated to be B _ 5.56 X 10-5, LB 27.9 mm (at X - 1.55 ,im), and f _ 88.6 (g/mm), respec-tively.

Figure 9 represents the wavelength dependence ofthe polarization characteristics of coupler B in the caseof O = 45°. From Fig. 9, it was found that a quarterbeat length LB/4 (see Appendix) corresponds to thewavelength between X i 1.25 gim and 2 _ 1.60 Am.Here, if the fused region length L is assumed to be 30mm, the modal birefringence B [see Eq. (A4)], the beatlength LB [see Eq. (A6)], and the equivalent lateralforce f [see Eq. (A7)] can be estimated to be B 4.76 X10-5, LB 32.6 mm (at X 1.55 Mm), and f _ 75.9 (g/mm), respectively.

Figure 10 indicates the wavelength dependence ofthe polarization characteristics of coupler C in the case

0.1.

Ii

X 0.

A

O:

.

I-

C v

.

co0

, 1

0. I

A

04II

0

. 0.

0.UI

4)

o

51

Wavelength X (m )

(A)

Wavelength x (m )

(B)Fig. 7. (A) Incidentpolarization angle Oi dependence of the spectralresponse characteristics of the power splitting ratio in coupler C; (B)temperature T dependence of the spectral response characteristics

of the power splitting ratio in coupler C.

of Oi = 45°. From Fig. 10, it was found that the 5.5times of the beat length (5.5LB) was obtained from thewavelength range of X, 1.32 m to 2 1.33 gim. Inthe same way, the B, LB, and f in the PMF coupler C (Lf_ 2330 mm; total PMF length) can be estimated to beB 4.14 X 10-4 , LB _ 3.7 mm (at X 1.55 Am) [or LB _3.2 mm, (at X _ 1.33 m)], and f _ 659.8 (g/mm),respectively.

The comparisons of couplers A, B, and C are summa-rized in Table III. From Table III, it was confirmedthat the asymmetric-etched-fused-taper (wavelengthflattened type) WDM coupler B is superior to the otherbiconical-fused-taper SMF coupler A in polarizationas well as in spectral response characteristics.

IV. ConclusionsThe incident polarization angle and temperature

dependence of the polarization and the spectral re-sponse characteristics of the three types of fiber cou-pler, the biconical-fused-twisted-taper SMF couplerB, the asymmetric-etched-fused-taper WDM couplerB, and the biconical-polished PMF (PANDA) couplerC, wore first discussed experimentally.

1066 APPLIED OPTICS / Vol. 30, No. 9 / 20 March 1991

Without Analyzer

* Oi= O 1- -260"°----- }

120.---------4 3Coupler B

I . I . I . I

Without Analyzer

T = 40C 1- -220 'C ------

. I°C ............. 4 - 3Coupler B

I I . I * I

5'

I

.20

1.20 1.30 1.40 1.50 1.60 1.70

Wavelength (Am )

Fig. 8. Polarization spectral response characteristics of coupler A

in the case of Oi = 450: (a) degree of linear polarization P5; (b) outputpolarization angle IF.

1.0 . . . .

(a) - --0.5 -j4 -- \ p05 o- i 450 a m 2

,/ Coupler B f, I I I I I L

90, I . .600 (b)30° \

1 -- …/ - -= -- =---- -

3Q0

60°M9 Fl . . IA I I

1.20 1.30 1.4 1.-IU

Wavelength X (im )

1.01U t."U

Fig. 9. Polarization spectral response characteristics of coupler B

in the case of 0i = 450: (a) degree of linear polarization P0; (b) outputpolarization angle 'P.

From the results of the measurements for the polar-ization as well as the spectral response characteristicsof the three different types of couplerfollowing conclusions were obtained:

A, B, and C, the

I

0

50

a

To

.2

0 0

0

.e

to

I.

()

90 F600E30E

o0

-900:

1.3 1.31 1.32 1.33

Wavelength (m )1.34 1.35

Fig. 10. Polarization spectral response characteristics of coupler Cin the case of 0i = 450: (a) degree of linear polarization P5; (b) output

polarization angle T.

A. Fused-Twisted Type SMF Coupler A

(a) The changes in P8 and T were almost linearlyproportional to temperature.

(b) The 6i had little effect on the spectral responsecharacteristics of R.

B. Etched-Fused Type WDM Coupler B

(a) The variations of the polarization characteris-tics of the splitting port were greater than those of thestraight-through port.

(b) The linear polarization state P, changes sinusoi-dally with the incident polarization angle Oi as tem-perature T.

(c) There was almost no effect from Oi as well astemperature T on the spectral response characteristicsof the power splitting ratio R.

(d) The modal birefringence B, beat length LB, andequivalent lateral force f [the fused (coupling) region

length Lf _ 30 mm] were estimated as follows: B _4.76 X 10-5; LB _ 32.6 mm (at X _ 1.55 ,gm); f _ 75.9 (g/mm).

(e) The birefringent characteristics in the fused(coupling) region (fused length Lf 30 mm) were

Table III. Comparison of Three Optical Fiber Coupler Characteristics

Coupler A Coupler B Coupler C

Biconical-Fused Asymmetric-Etched Biconical-PolishedTwisted-Taper SMF Fused-Taper WDM PMF (PANDA Fiber)

Coupler (York) Coupler (Gould) Coupler (Fujikura)

I)egree of Linear i-i Large Medium Nothing

Polarization Ps

vs. Temperature 1-2' Large Large Nothing

Splitting Ratio Nothing Nothing Large

vs. 0,Splitting Ratio Medium Nothing Small

vs. Temperature

Birefringence B 5.56x|O-4 4.76x10-

5 4. 14x 1 0-4

Beat Length 27.9 32.6 3.7(@xl.55lim)

LB (mm) (@AX2l.55gm) (@Xa=1.55tm) 3.2(@X2El.33gm

Stress f (gmm) 88.6 75.9 659.8

20 March 1991 / Vol. 30, No. 9 / APPLIED OPTICS 1067

1. 1.0

-0 0.5._4

Oa 0, 0in 0

_

-

A.

5

C0 -4

9060300

-30-60-909

I I,_ __ . . . . . . .

I (b) X I ------ - - -- - _ E

0 a\

,0

: O = 45 t- -2

.5 (a) Coupler C

CI

1.00 1.10

(b)

I I I I I * *

0

_

0

0

.

0

1.10

A%

(HE § | | | | | g | w

It,

30

-n

estimated to be B 5.56 X 10-5, LB 27.9 mm (at X1.55 Am), f = 88.6 (g/mm).

C. Polished Type PMF (PANDA Fiber) Coupler C

(a) P changes sinusoidally with Oi.(b) P and T had no change with temperature T.(c) The Oi had a great effect on the spectral response

characteristics of R.(d) The birefringent characteristics in the PMF (Lf2330 mm, total PANDA fiber length) were estimated

as follows: B 4.14 X 10-4; LB _ 3.7 mm (at X 1.55Aim) [LB _ 3.2 mm (at X 1.33 Am)];f _ 659.8 (g/mm).

D. In General

From the above results, it was confirmed that thepolarization characteristics of coupler A as well ascoupler B are greatly changed with temperature.However, those of coupler C were independent of tem-perature change. The wavelength dependence char-acteristics of the power splitting ratio R of couplers Band C had almost no change with temperature. How-ever, those of coupler A had large changes with tem-perature. In coupler A, it may be considered that thestress birefringence was generated caused by the fibertwisting when the coupler was fabricated and pack-aged.

Accordingly, if the biconical-fused-twisted-taperSMF coupler, e.g., coupler A, is used for polarizationsensitive devices or for such WDM coherent opticalcommunication systems using the Er+3-doped fiberamplifiers etc., the polarization and temperature com-pensation of the coupler are required.

In addition, it may be considered that suitable fibercouplers for coherent optical fiber communication sys-tems and dual-detector heterodyne/homodyne receiv-ers are the single-mode optical fiber couplers with lessadditional birefringence due to external stress, e.g., theetched-fused type, coupler B. Also, it is very impor-tant to clarify the relationship between the stress bire-fringence and the polarization characteristics in thefused (coupling) regions of the fused SMF couplers.Therefore, further study of these problems is needed.

We would like to thank K. Ono and Y. Kushiro ofKDD R&D Laboratories for encouragement and F.Kondoh of Nihon University for experimental sup-port. We also thank C. D. Hussey of Limerick Univer-sity, Ireland, for fruitful discussions.

Appendix

The phase difference V(X) between the two orthogo-nal propagation modes, the HE,, x-mode and the HE,,y-mode, can be generally expressed as29' 30

p(X) = (L i - fly)Lf _ k(n - n)Lf _, (2r/X)BLf, (Al)

where 5f3(_ f - fly; f3x and fly are the phase constants inthe HE11 x-mode and HE,, y-mode, respectively) is thephase constant difference. Lf is the length of theoptical fiber. The k(=27r/X) is the wavenumber in freespace. B(_nm - n; n and n indicate the effective

Q 1.0I

e A' 0.5 Xi X

me a

0

.

0 0.'

, 4'E

Zo

9045.

0.-45.

-901.'40 1.45 1.50 1.55 1.60

Wavelength X (m)Fig. 11. Wavelength dependence ofthe polarization characteristicsof the PMF with 10% elliptical cladding in the case of Oi = 450: (a)

degree of linear polarization P8; (b) output polarization angle T.

refractive indices in the birefringent x-axis and y-axis,respectively) is the modal birefringence.

Here linear polarization is launched at Oi = 450 to theintrinsic birefringent axes, as in the PMF; the relation-ship between the degree of the linear polarization P8and the phase difference so is given by29' 30

P(X) = cos(,(X) = cos[(27r/X)BLf1]. (A2)

Here P8(X) is the periodical function of the phasedifference s(X). An example of the wavelength depen-dence of the PMF with 10% elliptical cladding (Hita-chi)32 vs the incident polarization angle Oi is shown inFig. 11. In Fig. 11, the interval bo(X) between the peakP8(X1) and peak P(X 2) of the degree of linear polariza-tion at wavelengths Xi and \ 2 is given by

6(X) = (X - 2(X2 ) = cos'P s (X - cos'1P8 (X2)

= (27r/IX)BLf - (27r/X 2)BLf

= 2rBLf(X2 - X1)/X1X2- (A3)

The phase difference 6so(X) [the wavelength intervalbetween P(X 1) and P(X 2)] corresponds to r. There-fore, from Eq. (A3) and 6 = r, the modal birefringenceB can be obtained as

B = X1X2/1[2Lf(X 2 - d]- (A4)

Meanwhile, the well known definition of the beatlength LB between the two orthogonal modes with thephase difference 53 is given by

LB 2r/60 = X/B fat XI. (A5)

From Eqs. (A4) and (A5), the beat length LB isexpressed as

LB = 2L{I1 - (X1 /X2)1 (at AX}. (A6)

According to Refs. 29 and 30, the relationship be-tween the equivalent lateral force f and birefringence Bis given by

f =_rbB/A4CX)l, (A7)

where f (kg/mm) is the lateral force in unit lengths ofthe optical fiber, b is the optical fiber cladding radius,

1068 APPLIED OPTICS / Vol. 30, No. 9 / 20 March 1991

(b) PMF (Ea10%) Lr43.6mm

I I I

and C(X) ( 3.08 X 10-5 mm2/kg at X 1.50 Mtm) repre-sents the photoelastic constant,29 30 respectively.

Accordingly, from Fig. 11 and Eqs. (A4), (A6), and(A7), it can be estimated that the modal birefringenceB, beat length LB, and equivalent lateral force f in thePMF (Hitachi) with the elliptical cladding (10% ellip-ticity)32 are as follows:

B _ 6.55 X 10-5; (A8)

LB _ 22.5 mm (at X _ 1.472 im); (A9)

f _ 104.4 (g/mm), (A10)

where X _ 1.435 pim, X2 _ 1.472 gm, and Lf _ 43.6 mm,respectively.

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20 March 1991 / Vol. 30, No. 9 / APPLIED OPTICS 1069