October 1, 2002 / Vol. 27, No. 19 / OPTICS LETTERS 1681

p-shifted Sagnac interferometer for characterizationof femtosecond first- and

second-order polarization mode dispersion

I. Golub and E. Simova

Institute for Microstructural Sciences, National Research Council, Building M-23A, Montreal Road, Ottawa, Ontario K1A 0R6, Canada

Received May 29, 2002

A novel configuration for measuring femtosecond polarization mode dispersion (PMD) with a p-shifted Sagnacinterferometer is studied. The advantages of this configuration include a response that is independent of theorientation of the device under test. This independence allows characterization of the first- and second-orderPMD by measurement of the wavelength dependence of the differential group delay and of the orientation ofthe device’s principal states of polarization. The method provides an extension of the dynamic range of theSagnac interferometer technique to few-femtosecond PMD measurement. © 2002 Optical Society of America

OCIS codes: 060.0060, 060.2310, 060.2270, 060.2300, 060.2330, 260.1440.

Polarization mode dispersion (PMD) in f iber andfiber devices is caused by the wavelength-depen-dent birefringence and wavelength dependence ofthe birefringence axes’ orientation, or the associ-ated principal states of polarization (PSP). Theuse of a Sagnac interferometer for characterizationof PMD was proposed by Olson et al.1 Comparedwith the fixed analyzer method,2 the Sagnac in-terferometer had the advantage of better contrastbetween minima and maxima in the transmissionspectrum. Since these methods rely on countingfringes to measure femtosecond PMD, they requireuse of a tunable laser or a broadband source of severalhundred nanometers while measuring wavelength-averaged PMD. Moreover, neither of these methodscan determine PSP or the second-order PMD.

In this Letter we present a novel p-shifted configu-ration of the Sagnac interferometer for PMD charac-terization. The main advantage of this configurationis that the half-wave plate (HWP) inserted into theloop makes the method insensitive to the orientationof the birefringence axes of the device under test(DUT). As a result, one can characterize PMD valuesthat correspond to a part of a fringe in both the f ixedanalyzer method and a conventional Sagnac interfer-ometer. This allows extension of the dynamic rangeto femtosecond PMD values and measurement of thewavelength dependence of PMD. Furthermore, theorientation of the birefringence axes of the DUT canbe determined and the related second-order PMDderived.

Usually, the PMD is described as the vector V�v� �Dtq̂ in Stokes space,3,4 where Dt is the differentialgroup delay (DGD) and q̂ is a unit Stokes vector inthe direction of the fast PSP. The second-order PMDis the derivative

dV�v�dv

�dDt

dvq̂ 1 Dt

dq̂dv

, (1)

where the first component is parallel to V�v� andcauses polarization-dependent chromatic dispersion

0146-9592/02/191681-03$15.00/0

and the second term, perpendicular to V�v�, isthe rate of rotation of the PSPs and is known tocause signal depolarization. It has been shown [seeEqs. (9)–(13) of Ref. 4] that there is a straightforwardrelationship between the frequency-dependent PMDvector, V�v�, and the DUT’s Jones matrix, JDUT �v�.Matrix JDUT �v� can be presented as

JDUT �v� � Q21�v�DDUT �v�Q�v� , (2)

where DDUT �v� is a diagonal matrix with diagonal ele-ments that are the eigenvalues of JDUT �v� and Q21�v�and Q�v� represent two complementary rotations withthe rows of Q�v� as the eigenvectors of JDUT �v�:

DDUT �v� �

8>><>>:

exp∑i

d�v�2

∏0

0 exp 2

∑i

d�v�2

∏9††=††;

,

Q�v� �

"cos a�v� sin a�v�

2sin a�v� cos a�v�

#, (3)

where d�v� is the frequency-dependent birefringence,defined as d�v� � DtDv 1 �1�2� �dDt�dv�Dv2, withDv as the deviation from the optical carrier fre-quency v; a�v� � a0 1 �da�v��dv�Dv, with a0 as thefrequency-independent part of the orientation angle,and da�v��dv � k, with k specifying the frequency-dependent rotation rate of the PSPs.

The Sagnac interferometer has been studied theo-retically with the Jones calculus.5 – 7 We include aHWP in the loop, as shown in Fig. 1 (a polarizationcontroller, PC, is used for canceling the loop’s residualbirefringence). Effectively, here the HWP introducesa p-shift between the clockwise and counterclockwisepropagating beams, which now have orthogonal polar-izations upon entry into the DUT. The polarizationproperties of the loop and the DUT can be described bythe electrical f ield vectors into and out of the fiber cou-pler, Ein and Eout, respectively, and the corresponding

© 2002 Optical Society of America

1682 OPTICS LETTERS / Vol. 27, No. 19 / October 1, 2002

Fig. 1. Configuration of the p-shifted Sagnac interferom-eter.

Jones matrix, J�v�. At the exit of the coupler, forarm 1 (ref lection arm) and arm 2 (transmission arm),the electric f ield vectors’ components, Eout

1n and Eout2n ,

respectively, are

Eout1n � K1�2

n �1 2 Kn�1�2Ja�v�E in1n

1 K1�2n �1 2 Kn�1�2Jc�v�E in

1n ,

Eout2n � KnJa�v�E in

1n 1 �1 2 Kn�Jc�v�E in1n , (4)

where the coeff icient Kn represents the splitting ratioof the coupler, with the index n referring to the x ory component; Jc�v� and Ja�v� are the clockwise andcounterclockwise Jones matrices, respectively, for thepropagation of the light through the loop.

As can be seen from Eq. (4), an uneven splittingratio, Kn fi 0.5, of the wavelength-dependent f iber-optic coupler results in a limited interference contrastwhen the intensity distribution is recorded in arm 2,whereas for arm 1 the contrast is not limited by thevalue of Kn. Indeed, the fact that arm 1 is morestable and gives better results was confirmed in theexperiment, as will be shown below. The followingJones matrices for the HWP and the DUT [accordingto Eqs. (2) and (3)] were assumed in the calculations:

JcHWP �

24 i cos 2r i sin 2r

i sin 2r 2i cos 2r

35 , Ja

HWP �

24 i cos 2r 2i sin 2r

2i sin 2r 2i cos 2r

35 ,

JcDUT �

26664

expµi

d

2

∂cos2 a 1 exp

µ2i

d

2

∂sin2 a 2i sin 2a sin

d

2

2i sin 2a sind

2exp

µ2i

d

2

∂cos2 a 1 exp

µi

d

2

∂sin2 a

37775 ,

JaDUT �

26664

expµi

d

2

∂cos2 a 1 exp

µ2i

d

2

∂sin2 a 22i sin 2a sin

d

2

22i sin 2a sind

2exp

µ2i

d

2

∂cos2 a 1 exp

µi

d

2

∂sin2 a

37775 , (5)

where r and a are the orientation angles of the bire-fringence axes of the HWP and the DUT, respectively,relative to the coordinate system defined by the planeof the loop and the coupler, and d is the birefringenceof the DUT. Assuming Kn � 0.5, the transmission inarm 2, for example, is

T �l� � cos2d�l�2

sin2 2r . (6)

It is straightforward to show that the ref lection fromarm 1 and transmission in arms 1 and 2 are comple-mentary. Equation (6) shows that the transmissionresponse of the p-shifted Sagnac interferometer is in-dependent of the orientation angle, a, of the DUT’sbirefringence axes and the polarization of the inputlight. The response depends only on the DUT’s bire-fringence, provided that the HWP’s orientation angle,r, is optimized at 45± relative to the coordinate system.The result in Eq. (6) is significant, because it allows usto recover the wavelength-dependent phase and the as-sociated PMD through intensity measurement.

For comparison, a similar equation can be derivedfor the conventional Sagnac interferometer with onlythe DUT present:

T �l� � sin2 d�l�2

sin22a . (7)

The drawbacks of the conventional Sagnac interferom-eter, namely, the dependence on the DUT’s orientationangle, a, become apparent from Eq. (7). Since thefiber-optic DUT’s axes cannot be rotated, there are twounknowns in the equation. This problem is bypassedthrough the fringe-counting technique,1,2 but thenthe limiting factor in measuring femtosecond PMD isthe bandwidth of the source. Moreover, only thewavelength-averaged PMD is measured, andsecond-order PMD cannot be evaluated.

We can determine the birefringence axes of the DUTby first measuring the birefringence of the DUT withour scheme and Eq. (6) and then removing the HWP,measuring the corresponding T �l�, and calculatingangle a from Eq. (7). To the f irst order, PMD isthe DGD between the two orthogonal PSPs and is

calculated as DGD � dd�dv, where v is the opticalfrequency.

As can be seen from Eqs. (1), (6), and (7), theproposed technique allows complete characterizationof the second-order PMD caused by linear frequencydependence of the DGD (or polarization-dependent

October 1, 2002 / Vol. 27, No. 19 / OPTICS LETTERS 1683

Fig. 2. Calculated DGD for the f iber-optic coupler frommeasured intensity in arm 1 of the p-shifted Sagnac inter-ferometer.

Fig. 3. Calculated DGD for the f iber-optic coupler frommeasured intensity in arm 2 of the p-shifted Sagnac inter-ferometer.

Fig. 4. Calculated DGD for the f iber-optic coupler fromthe JME.

chromatic dispersion) as well as that caused by thefrequency dependence of PSPs through differentiationof the DGD and angle a�v� with respect to frequency.

Measurements were carried out with the setupshown in Fig. 1. A tunable laser, a broadband 50:50fiber-optic coupler, and a broadband HWP were usedto ensure operation of the interferometer for a widerange of wavelengths. To verify the possibility ofmeasuring femtosecond PMD, we tested another cou-pler as the DUT. The transmission, T �l�, in Eqs. (6)and (7) was normalized by careful calibration of themeasurement system to account for the losses of theloop, the HWP, and the DUT. The DGD calculatedfrom the intensity scan from arm 1 with Eq. (6) is pre-sented in Fig. 2 for a wavelength step of 0.3 nm. TheDGD calculated from the intensity scan from arm 2simultaneously with arm 1 and under the same condi-tions is presented in Fig. 3. It can be seen that the

Fig. 5. Orientation of the DUT’s birefringence axes cal-culated with the p-shifted Sagnac interferometer in con-junction with a conventional Sagnac interferometer. Darkcurve, calculated from measurements in arm 1, light curve,calculated from arm 2.

measurements in ref lection arm 1 contain less noise,although arm 1 includes an additional component,such as a circulator (see Fig. 1). For comparison, theDGD measured with a commercial instrument basedon Jones matrix eigenanalysis8 (JME) is shown inFig. 4. It can be seen that there is good agreementbetween the two methods. Figure 5 presents theangle of the DUT’s birefringence axes obtained fromsuccessive measurements with the p-shifted Sagnacinterferometer and with the conventional Sagnacinterferometer and calculated from Eqs. (6) and (7).

We have proposed and studied a novel p-shiftedconfiguration of the Sagnac interferometer for charac-terization of PMD. The p shift allows recovery of thephase as a function of wavelength from the intensityscan. Thus, the dynamic range is extended towardthe few-femtosecond range without the need for useof a broadband source while measuring wavelength-dependent DGD. A comparison with the more-complex JME method showed good agreement.Combining the proposed configuration with the con-ventional Sagnac interferometer allows determinationof the DUT’s birefringence axes and their wavelengthdependence. Consequently, the scheme allows com-plete characterization of first- and second-order PMD.

The authors are grateful to Xiaoyi Bao, DongfengLiu, and David Waddy from the Department of Physicsof the University of Ottawa, Ottawa, Canada, for theirhelp with the JME measurements. E. S. Simova’se-mail address is eli.simova@nrc.ca.

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