August 15, 1995 / Vol. 20, No. 16 / OPTICS LETTERS 1713

Response of fiber lasers to an axial magnetic field

H. Y. Kim

Electronics and Telecommunication Research Institute, P.O. Box 8, Taedok Science Town, Taejon 305-606, Korea

B. K. Kim, S. H. Yun, and Byoung Yoon Kim

Department of Physics, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong,Yusong-gu, Taejon 305-701, Korea

Received May 11, 1995

The effects of an axial magnetic f ield applied to a fiber laser cavity on the polarization characteristicsof the fiber laser output are investigated. The experimental and theoretical results show sensitive changes inthe polarization mode beat frequency as a function of the applied magnetic f ield. 1995 Optical Society ofAmerica

An axial magnetic field induces circular birefringencein optical fibers, resulting in the rotation of thepolarization angle for a linearly polarized input (Fara-day effect).1 The rotation angle is linearly propor-tional to the magnitude of the applied field. Accuratemeasurements of the polarization angle variation pro-vide a means for monitoring the magnetic field. TheFaraday effect in optical fibers has been successfullyutilized for the measurement of the electric currentthat generates the magnetic field.2 – 5 For someother optical fiber sensors, such as fiber gyroscopes,the Faraday effect should be suppressed becauseit introduces nonreciprocal phase shift between thecounterpropagating optical waves in the fiber. Thisnonreciprocity is the major feature that distinguishesthe Faraday effect from the effect of fiber twist, whichalso induces circular birefringence.2

The recent development of fiber laser sensors hasopened up new possibilities of active fiber sensors withfrequency readout, eliminating complicated electronicsignal processing.6 The polarization characteristicsof fiber lasers play key roles for the sensors, and athorough understanding of them is necessary for fur-ther development of such sensors. Previously we ana-lyzed relatively complicated polarization behavior froma fiber laser cavity having a uniform elliptical birefrin-gence induced by twisting a fiber cavity with intrinsiclinear birefringence.7 In this Letter we describe theeffects of another important parameter, nonreciprocalbirefringence induced by an external magnetic field,on the polarization properties of fiber lasers. Unlikefor the case of twist-induced circular birefringence, theFaraday effect makes the eigen states of polarization(SOP’s) elliptical polarizations at the location of themirrors in the existence of an intrinsic linear bire-fringence in the fiber. The polarization mode beat(PMB) frequencies vary as a function of the magneticfield strength and can be utilized for magnetic fieldor current sensing with frequency readout. A theo-retical analysis predicts the experimental observationwith reasonable agreement.

The SOP of the laser output has to satisfy a laserresonance condition in that it has to come back tothe same SOP after a complete round trip inside the

0146-9592/95/161713-03$6.00/0

laser cavity.6,8 If a Fabry–Perot fiber cavity has auniform linear birefringence, the two eigen SOP’s arelinear, with their directions coinciding with those of thebirefringence axes.6 The two polarization componentsgenerally have different frequencies that depend onthe magnitude of the birefringence of the fiber cavity.When the fiber cavity has a random birefringencealong its length, the output SOP’s are not so simple.An interesting property is that the eigen SOP’s at thelocation of mirrors, and therefore the laser outputs,are always linearly polarized in the absence of anonreciprocal element in the cavity.7

Let us consider a fiber laser formed with two pla-nar mirrors and an amplifying fiber (length l) withuniform intrinsic linear birefringence b sradymd andtwist-induced circular birefringence aT sradymd. Itis assumed that the laser cavity does not havepolarization-dependent loss. When uniform mag-netic field is applied along the fiber axis, additionalcircular birefringence is induced in the fiber withthe magnitude aH ­ 2VH sradymd,9 where V is theVerdet constant of 1.6 3 1026 sradyAd at 1.06 mmfor fused silica10 and H is the axial magnetic fieldsAymd. The twist-induced birefringence is reciprocalin that circularly polarized light (e.g., right-handedcircular polarization), defined in reference to itspropagation direction, will experience the samerefractive index regardless of the propagation directionalong the fiber.11 For the nonreciprocal circularbirefringence induced by the magnetic field, thecircularly polarized light will experience differentrefractive indices depending on whether the lightis propagating along or against the direction of theapplied magnetic field.2,9 This difference results in asignif icant consequence in polarization properties offiber lasers. Although it is customary to define theSOP with respect to the propagation direction of light,the following discussions are based on the laboratoryframework for the simplicity of calculation. In thelaboratory framework, for a given circular SOP (e.g.,electric-field vector rotating in a fixed direction in thelaboratory frame), aT changes sign under the reversalof propagation direction,7 whereas aH remains thesame for the two cases.

1995 Optical Society of America

1714 OPTICS LETTERS / Vol. 20, No. 16 / August 15, 1995

The propagation of light from one fiber end to theother end can be represented by a unitary Jonesmatrix Af .7,8 For the light traveling in the backwarddirection, the Jones matrix Ab is no longer the trans-pose of Af as in the case of a reciprocal situation with-out a magnetic field.7 The magnitudes of ellipticalbirefringence for the forward and backward propaga-tions are Vf ­ fsaT 1 aH d2 1 b2g1/2 and Vb ­ fs2aT 1

aH d2 1 b2g1/2. Thus, for the light that makes acomplete round trip in the laser cavity of length lstarting from the output mirror in Fig. 1, the Jonesmatrix describing the propagation of light becomesAf Ab. If we neglect common optical phase terms, itcan be represented as

Af Ab ­

∑a 1 bi ic 1 dic 2 d a 2 bi

∏, (1)

where

a ­ cossdfy2dcossdby2d

2 coss2Bf 2 2Bbdsinsdfy2dsinsdby2d ,

b ­ cos 2Bf sinsdfy2d cossdby2d

1 cos 2Bb sinsdby2dcossdfy2d ,

c ­ 2 sins2Bf 2 2Bbdsinsdfy2dsinsdby2d ,

d ­ sin 2Bf sinsdby2dcossdby2d

2 sin 2Bb sinsdfy2dcossdfy2d ,

with Bf ­ harctanfsaT 1 aH dybgjy2, Bb ­harctanfs2aT 1 aH dybgjy2, and df ­ Vf l, db ­ Vbl.The laser output must satisfy the resonance conditionsuch that the optical wave has to come back to thesame phase and SOP after one round trip inside thelaser cavity.6 This requirement leads to an eigen-value equation whose solution provide two mutuallyorthogonal eigenpolarizations (eigenvectors) with theiroptical frequencies (eigenvalues).7 A straightforwardcalculation leads to elliptical eigenpolarization modeswith the PMB frequency s fpd given as

fp ­c

2pnlarccosfcossdfy2dsinsdfy2dsinsdby2dg

s0 # fp # cy2nld , (2)

where c is the speed of light in vacuum and nis the refractive index of fused silica. It follows thatthe change in the PMB frequency Dfp that is due to theapplied magnetic field is

Dfp ­c

2pnlharccosfcossdfy2dcossdby2d

2 coss2Bf 2 2Bbdsinsdfy2dsinsdby2dg

2 arccosscos d cos2 2B 1 sin2 2Bdj

s0 # Dfp # cy2nld , (3)

where d ­q

a2

T 1 b2 l and B ­ farctansaT ybdgy2.When b .. saH , aT d and aH ,, 1, Dfp becomes

negligible, which makes it diff icult to observe the

Faraday effect with a fiber laser cavity having alarge linear birefringence. When the fiber cavity ispredominantly circularly birefringent (i.e., aT .. b,aH d, Bf ø 2Bb ø B, and Eq. (3) can be written as

Dfp øc

2pnlfarccosscos d cos2 2B 1 cos dH sin2 2Bd

2 arccosscos d cos2 2B 1 sin2 2Bdg

s0 # Dfp # cy2nld , (4)

where dH ­ jaH jl. Substituting in Bf ø 2Bb ­ py4,one can get Dfp ø scy2pnlddH , showing a linear rela-tionship between Dfp and dH .

An easy method for inducing large circular birefrin-gence is to twist the fiber. When a fiber is twisted,the twist rate t induces reciprocal circular birefrin-gence, sg 2 2dt, in the reference frame rotating withthe fiber, where g is a constant of ,0.16.9 Figure 2(a)shows the behavior of the PMB frequency fp as afunction of the twist angle tl when b ­ 1.8 radymand aH ­ 0.137 radym, with a 5-cm-long untwisted(but rotated) section at the output end of the fibercavity. The periodic behavior is essentially the sameas that described in Ref. 7, with the same periodic-ity of ,196±. Figure 2(b) illustrates the change of thePMB frequency Dfp that is due to the introduction ofaH ­ 0.137 radym as a function of tl. It can be seenthat the Dfp increases with periodic peaks. The loca-tions of peaks coincide with the minima of fp, whered ­ 2Np (N an integer) in relation (4). We can alsocalculate, from Eq. (1), the directions of the major axesfor elliptical eigenpolarizations. For small values ofaH , the change in the output polarization direction

Fig. 1. Schematic of a fiber laser cavity with a magneticfield.

Fig. 2. Responses of (a) PMB frequency s fpd and (b) theshift of the PMB frequency that is due to the appliedmagnetic field sDfpd as functions of fiber twist angle saH ­0.137 radym, b ­ 1.8 radymd.

August 15, 1995 / Vol. 20, No. 16 / OPTICS LETTERS 1715

Fig. 3. Experimental (symbols) and theoretical (curves)results for the shift of PMB frequency as a function of theapplied current to the solenoid su ­ tld.

Fig. 4. Shift of PMB frequency as a function of fiber twistangle (amplitude of the applied ac current, 14 A).

turns out to be small and is not discussed in detail inthis Letter.

The fiber laser used for the experiment was con-structed with a 1.46-m-long, Nd-doped fiber (pro-vided by BT Laboratories) and two planar mirrors(99% and 90% ref lectance) glued to the ends of thefiber. The entire fiber cavity was kept straight, and a50-cm-long hollow solenoid (1530 turns) was used toproduce the axial magnetic field. An Ar31 laser at514.5-nm wavelength was used for pumping the fiberlaser through the high-ref lectance mirror. The opticalspectrum of the laser output did not change with the

application of a magnetic field. A rotatable polariza-tion analyzer and a rf spectrum analyzer were used tomeasure the PMB frequency. The longitudinal modebeat frequencies were observed at the harmonics of,70 MHz, as expected from the cavity length. ThePMB signals were observed between two adjacentlongitudinal mode beat signals.7 We measured thechange in the PMB frequency, Dfp, as a function of theapplied magnetic field (or current) by applying an accurrent at 60 Hz to the solenoid. Figure 3 shows theexperimental results (symbols) for the PMB frequencyshift as a function of the amplitude of ac current for dif-ferent values of twist angle su ­ tld, along with thecalculated results (curves). Figure 4 shows the de-pendence of Dfp on the twist angle of the fiber lasercavity at the applied current amplitude of 14 A to thesolenoid sH ­ 42, 840 Aymd, showing reasonable agree-ment with the theoretical predictions, except for thepeaks at higher twist angles.

In conclusion, we analyzed the response of PMBfrequency of the fiber laser output to an axial mag-netic field. A theoretical model describes the observedbehavior of the laser with reasonably good agreement.The results presented here may be used for fiber cur-rent sensors and polarization control of fiber lasers.

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