Polarization properties of fiber lasers with twist ... - OSA Publishing

Polarization properties of fiber lasers with twist-induced circular birefringence Ho Young Kim, El Hang Lee, and Byoung Yoon Kim

We have experimentally observed and theoretically analyzed the polarization properties of fiber lasers with twist-induced birefringence. Twisting a fiber induces the circular birefringence of a fiber laser cavity, and this birefringence reduces the effects of intrinsic linear birefringence on the polarization properties of fiber lasers. The frequencies of their polarization eigenmodes coincide with each other gradually as the twist rate increases, and the directions of polarization eigenmodes deviate from the birefringence axis at a much larger twist rate than the magnitude of intrinsic linear birefringence. We describe the successful experimental results for Nd and Er fiber lasers. © 1997 Optical Society of America

1. Introduction

Fiber lasers have been extensively investigated for their applications to fiber laser sensors1–3 and short pulse laser sources.4,5 In particular, short pulse laser sources, such as soliton sources, have been developed for long distance light communication systems and for high speed optical switching systems.6 The performance of these systems strongly depends on the polarization properties of laser sources.4,6 The development of polarimetric fiber laser sensors opened up new possibilities of active fiber sensors with frequency readout by eliminating complicated electronic signal processing and using the polarization characteristics of fiber lasers.1–3 These new types of sensor have been studied for widespread use, one of which is expected to be developed in the near future for the performance management systems of high speed lightwave communication networks. Since the polarization characteristics of fiber lasers play key roles in the performance of such systems, a thorough understanding of them is necessary for further development of fiber lasers and their applications.

H. Y. Kim and E. H. Lee are with the Electronics and Telecommunications Research Institute, Yusong P.O. Box 106, Taejon 305606, Korea. B. Y. Kim is with the Department of Physics, Korean Advanced Institute of Science and Technology, 373-1 Kusongdong, Yusong-gu, Taejon 305-701, Korea. Received 7 May 1996; revised manuscript received 7 March 1997. 0003-6935y97y276764-06$10.00y0 © 1997 Optical Society of America 6764

APPLIED OPTICS y Vol. 36, No. 27 y 20 September 1997

Fiber lasers have two polarization eigenmodes. Their polarization characteristics, such as direction, frequency, and state of polarization, are determined uniquely by the total birefringence in a laser cavity. The external perturbations induce other birefringence, resulting in the change of polarization characteristics of the two polarization eigenmodes. For example, the lateral stress induces a linear birefringence and makes the polarization mode beat ~PMB! frequencies change linearly.1,2 The change in PMB frequency is monitored to measure external perturbations, providing a direct frequency readout without a complicated signal processor. The directions of their polarization eigenmodes are aligned with the birefringence axes of the fiber.1–3 These characteristics are useful for development of simple angular position sensors. However, in fabrication of the fiber lasers, we frequently encountered another important external perturbation: the twist of the cavity fiber. Twisting a fiber induces circular birefringence, and, if the fiber has an intrinsic linear birefringence as is usually the case, the total birefringence in this fiber results in elliptical birefringence.7 Because the properties of the total birefringence including twist-induced birefringence and linear birefringence can be changed by variation of the twist rate, the polarization characteristics of laser output evolve in a much more complicated way as a function of twist rate than the well-known case of polarization evolution in a twisted passive optical fiber.7 The basic difference between the polarization evolution of fiber lasers and light that passes through a birefringent fiber results because a fiber laser with a linear cavity formed by two mirrors must satisfy an additional condition that the state of polarization ~SOP! must return

the fiber induces circular birefringence, ~2 2 g! t, where g is a constant ~;0.16! for the fiber material.7 The resultant elliptical birefringence in the fiber is the sum of linear and circular birefringences: v0 5 bi 1 ~ g 2 2!t,

(1)

and the phase retardation ~d! between the polarization modes through the fiber is Fig. 1. Schematic of a fiber laser cavity.

to its original SOP after a complete round trip inside the cavity.8 Another interesting property is that the SOP’s of polarization eigenmodes at mirror locations, or of laser output, are always linearly polarized in the absence of a nonreciprocal element in the cavity. Therefore, the effects of twist-induced circular birefringence on the polarization properties of fiber lasers must be thoroughly analyzed to gain better knowledge of fiber laser output characteristics and optimal design of stable polarimetric fiber laser sensors. Here we report on our experimental and theoretical investigations of twisting effects on fiber laser cavities in terms of polarization properties of laser output, such as PMB frequencies and the directions of polarization eigenmodes. In Section 2 we describe the general theory of polarization properties of fiber lasers using Jones vector calculus. Based on the laser condition, the SOPs, frequencies, and directions of polarization eigenmodes are obtained for twistinduced birefringence in the case of uniform twist. Modified models for nonuniform twist cases are also presented. We found that the existence of short nontwist fiber parts at the ends of a fiber laser cavity causes significant changes in the polarization characteristics of a fiber laser and in the theory for polarization properties of uniformly induced birefringent fibers. In Section 3 we explain the experimental setup and results followed by a theoretical analysis. Some applications of the results in this paper are suggested. Section 4 contains the conclusion. 2. Theory

Let us consider a fiber laser formed with two planar mirrors and an amplifying fiber ~fiber length, l ! with uniform linear birefringence bi ~radym! as shown in Fig. 1. Lasing occurs when the pump laser is properly focused on one end of the fiber through a rear mirror. Here we consider only the polarization characteristics of the fiber laser output. At first we assume that the twist ~twist rate, t! is enforced uniformly through the fiber and other externally induced birefringence does not exist in the fiber cavity. At zero twist, since there is only intrinsic linear birefringence bi , the direction of polarization is aligned with this linear birefringence.1–3 We use the coordinate system that is aligned to the linear birefringence, fixed to the fiber, and revolves about the fiber axis at the twist rate. It is more convenient to use this coordinate system since the principal axes of linear birefringence are stationary.7 The twist in

d 5 uv0ul 5 $bi 2 1 @~ g 2 2!t#2l%1y2.

(2)

Evolution of the SOP through such a fiber is complicated and can be conveniently analyzed by using a Jones matrix O: 25

F

G

cos dy2 1 i cos 2B sin dy2 sin 2B sin dy2 , 2sin 2B sin dy2 cos dy2 2 i cos 2B sin dy2 (3)

where B is the birefringence parameter related to the ratio of circular to linear birefringence: B 5 $tan21@ g 2 2!tybi #%y2

(4)

for an optical wave that travels in the forward direction from rear mirror to front mirror.9,10 In the backward direction, birefringence differs from that of the forward direction because the direction of twist is opposite. However, when the difference exists only in the direction of circular birefringence in the same coordinate system as in the forward direction, backward birefringence has the same magnitude of phase retardation as the forward direction, but the transposed matrix of O, denoted as OT, correctly describes the behavior,11 assuming no nonreciprocal elements in the fiber cavity. For the light that makes a complete round trip in the laser cavity starting from the output mirror, the Jones matrix that describes the propagation of light becomes 22T, represented as 22T 5

S

a 1 bi ic

D

ic , a 2 bi

(5)

where a 5 cos d 1 ~sin22B!~1 2 cos d!, b 5 ~cos 2B!sin d, and c 5 ~cos 2B!~1 2 cos d!sin 2B. The laser output must satisfy the resonance condition in such a way that the optical wave has to return to the same phase and SOP after one round trip inside the laser cavity.8 The eigenvectors of the corresponding Jones matrix, 22T, satisfy the requirements obtained from the following eigenvalue equation: 22Tu6 5 l6u6 5 exp~6ih!u6,

(6)

where l6 are the eigenvalues for the eigenvectors and h is the phase of the eigenvalue. These eigenvectors satisfy the resonance condition for the polarization. These eigenvectors are the polarization eigenmodes for the total birefringence experienced during one round trip. Equation ~6! explains that the eigen20 September 1997 y Vol. 36, No. 27 y APPLIED OPTICS

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' ~2 2 g!utul. Afterward, whenever the increment of twist angle reaches ;196° and d becomes 2Np, h in Eq. ~7! vanishes. The PMB frequency shows quasiperiodic behavior with a period of ;196° at large twist rates as shown in Fig. 2. These features can be used for the measurement of constant g and the magnitude of intrinsic linear birefringence of fiber material. Inserting the eigenvalues into Eq. ~6!, we obtain the matrix components of eigenmodes a1 and a2, and their ratios can be easily calculated as

S

D

~a2ya1!6 5 2b 6 Î1 2 a 2 yc 5 tan u6. Fig. 2. Theoretical PMB frequency of twisted fiber lasers ~ g 5 0.16!.

modes have the phase difference 2h, which can be expressed as h 5 cos21 a 5 cos21~cos d cos2 2B 1 sin2 2B!

(7)

during one round trip. One can obtain the frequency difference between eigenmodes, or the PMB frequency fp, by dividing this phase difference by one round trip through the laser cavity: fp 5

hc , p2nl

(8)

where n is the mean refractive index of the fiber and c is the speed of light in vacuum. As shown in Eq. ~7!, the PMB frequency is determined by d and B of the resultant birefringence as given by Eq. ~2! and ~4!, respectively. At a small twist rate, the PMB frequency variation with twist rate strongly depends on the magnitude of intrinsic linear birefringence ubi u. The simulated results of the PMB frequencies as functions of twist angle ~5 tl ! are shown in Fig. 2. The dotted curve is for bi 5 1.8 radym ~of the Nd-doped fiber used in the experiments! and the solid curve is for bi 5 104.5 radym ~of the Er-doped fiber!. For low intrinsic birefringence ~dashed curve in Fig. 2! the PMB frequency decreases rapidly as the twist rate increases and approaches zero with a damped oscillation at moderately large twist rates. Because of angle ambiguity, the PMB frequency vanishes whenever d 5 2Np. The fact that the PMB frequency approaches zero means that the two polarization eigenmodes become almost degenerate. It is reasonable to assume that any linearly polarized optical wave propagating in a purely circular birefringent medium experiences the same phase accumulation. On the other hand, with a high intrinsic birefringence ~solid curve in Fig. 2!, the PMB frequency does not decrease until the twist rate increases. At a small twist rate, because twist-induced circular birefringence is negligible and B ' 0.0, h ' d and the PMB frequency increases parabolically as shown in Eq. ~2!. When the twist rate increases to the extent that it is much larger than bi , the PMB frequency decreases in a pattern similar to that of low birefringent fiber and d 6766


(9)

If the ratio of these components has a real value, the SOP’s of polarization eigenmodes are linear. When the ratio is complex, the SOP’s are elliptical or circular. From Eq. ~9!, the SOP’s of the polarization eigenmodes in twisted fiber lasers are linearly independent of the twist rate and the intrinsic linear birefringence. As the inner product of two eigenmodes vanishes,

S DS D S a1 a2

1

a1 a2

2

5

DS

c c 2b 1 Î1 2 a 2 2b 2 Î1 2 a 2

D

(10)

5 a2 1 b2 1 c2 2 1 5 0, the directions of two eigenmodes are orthogonal to each other. The angle directions in Eq. ~9! are simplified to u1 5 $tan21 @tan~dy2!sin 2B#%y2, u2 5 u1 1 py2.

(11)

When the twist rate is much smaller than the intrinsic linear birefringence, B ' 0.0 and u1 ' 0. Thus, the directions of polarization eigenmodes coincide with the birefringence axes ~twist axes! for the negligible twist rate in the fiber laser cavity and rotate at that twist rate. When t 5 0.0, we can adjust the orientation of the optical axes to the orientation of the intrinsic linear birefringence axes. When the twist rate is much greater than bi , B ' 2py4, and u1 ' 2dy4 ' ~ g 2 2!ty4 5 20.46t. The directions of polarization eigenmodes deviate from the birefringence axes. With laboratory experiments, the polarization direction ~5 u1 1 tl ! changes at 0.54t, which is slower than the twist rate. The simulated results of the directions of polarization eigenmodes for the twist angle ~5 tl ! are shown in Fig. 3 with the same conditions as those for PMB frequency ~dashed curve for Nd fiber and solid curve for Er fiber!. For low birefringent fiber such as that represented by a dashed curve, the directions of polarization eigenmodes deviate rapidly from the twist axes at a small twist rate and rotate at about half of the twist rate. But, for high birefringent fiber ~see the solid curve in Fig. 3!, the directions of polarization eigenmodes coincide with the twist rate at a small twist rate and deviate gradually from the twist axes as the twist rate increases. At a very large twist rate, they rotate at 0.54 of the twist rate.

Fig. 3. Theoretical directions of polarization eigenmodes of twisted fiber lasers ~ g 5 0.16!.

Fig. 5. Variation of PMB frequency as a function of twist angle ~bi 5 1.8 radym!.

Next we describe a fiber laser with a nontwisted part. As shown in Fig. 4, the twist is not applied to the portion adjacent to the near mirrors in the real experimental setup. Therefore, the resultant birefringence does not remain uniform throughout the fiber. The main twisted part has elliptical birefringence as described above and the nontwisted part has only intrinsic linear birefringence. The Jones matrix that describes the propagation of light through one round trip has to be modified from the matrix in Eq. ~5! by the matrix for the unperturbed section and is represented as

even when the twist rate is sufficiently high because the phase retardation of polarization modes through nontwisted parts is unaffected by twist. In Fig. 6 the behavior of the direction of polarization for the nonuniformly twisted fiber laser @curve ~3!# is much different from that of a uniformly twisted fiber laser @the curve ~1!#. At a small twist rate, the directions of polarization with a nontwisted part follow those with a uniformly twisted laser @curve ~1!# but abruptly deviate from them and increase to a twist angle for which d 5 2Np. They more or less rotate at a twist angle similar to curve ~2! in Fig. 6, which is another expression of the same solution for uniformly twisted lasers as obtained by a 90° jump of polarization angle with 195° periodicity of twist angle. The variation of directions such as that represented by curve ~3! in Fig. 6 demonstrates that the unaffected and only linear birefringence in the nontwisted part can be used to determine the polarization properties and make the direction of polarization revert to a rotation angle ~twist angle! wherever the resultant birefringence for the twisted part has no effect ~i.e., d 5 2Np!. The small nontwisted part generates significant changes in polarization characteristics of fiber lasers.

2t2tT 5

F

G

~a 1 bi!exp~idv! ic , ic ~a 2 bi!exp~2idv!

(12)

where dv ~5 bi ld! is the phase retardation in the nontwisted part of length ld. Therefore, Eqs. ~7! and ~11! were modified to h* 5 cos21~a cos dv 2 b sin dv!,

(13)

21

u6* 5 tan

H

3

J

c , a sin dv 1 b cos dv 6 @1 2 ~a cos dv 2 b sin dv!2#1y2 (14)

respectively. As shown in Fig. 5, the curve of PMB frequencies for a twist angle shift toward the origin because the length of twisted fiber was shortened and their twist rate increased comparatively. The lower bound of this curve is dv ~the phase retardation through the nontwisted part!, and does not vanish

Fig. 4. Model of a nonuniformly twisted fiber laser cavity.

Fig. 6. Dependence of polarization direction on twist angle: curve ~1!, uniformly twisted fiber laser; curve ~2!, another expression ~periodic 90° jump for a uniformly twisted fiber laser; curve ~3!, a nonuniformly twisted fiber laser, bi 5 1.8 radym. 20 September 1997 y Vol. 36, No. 27 y APPLIED OPTICS

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Fig. 7. Schematic diagram of the experimental setup.

3. Experiments and Applications

The fiber lasers for our experiments were constructed using a 0.95-m-length Nd-doped fiber and a 0.55-mlength Er-doped fiber ~provided by BT Laboratories! and two planar mirrors with 99% and 90% reflectance at each wavelength ~1.06 and 1.55 mm!, glued to the ends of the fiber as shown in Fig. 7. An Ar laser at 514.5-nm wavelength was used for pumping the fiber laser through the high reflectance mirror for which the threshold pump powers were 30 and 20 mW. At an incident pump power level of 60 mW, we obtained approximately 30 mW of signal power at 1.057-mm wavelength and approximately 40 mW at 1.55-mm wavelength. A rotatable polarization analyzer and a rf spectrum analyzer were used to measure the polarization direction and the PMB frequency as functions of the twist rate. To twist the fiber, the whole fiber was positioned in a straight line from end to end. A short portion ~approximately 5 cm long! of the output end of the fiber laser was glued to a holder that was mounted on a rotation stage. A much shorter portion of the other end of the laser cavity was also glued to a fixture. These constitute the nontwisted parts. With a rf spectrum analyzer, the longitudinal mode beat signals of laser output were observed at harmonics of approximately 110 and 180 MHz as estimated from the cavity length. When the laser output was measured with a polarization analyzer located in front of the photodetector, pairs of PMB signals between adjacent longitudinal mode beat signals were also observed. The lowest frequency of PMB signals is measured to be a PMB frequency. Since PMB frequency is symmetrical around a zero twist angle, to change a twist angle whose sign is set to negative

when a twist is made in a counterclockwise direction to the fiber axis, we can easily determine the location of a truly zero twist including the intrinsic twist in fiber. The experimental data for PMB frequency as a function of twist angle are plotted in Figs. 8 and Fig. 9 along with theoretical results. By measuring the polarization rotation angle of single-pass polarized light through the cavity fiber, the value of g was independently obtained to yield ;0.16, which was used for the simulation. For a low birefringent fiber laser, the PMB frequency decreased with ;196° periodicity as the twist angle increased. For a high birefringent fiber, the PMB frequency did not decrease and oscillated until the twist rate increased. The minima of PMB frequency of high birefringent fiber lasers apparently showed the effect of a nontwisted part. The PMB signals disappeared every 90° when the angle of polarization analyzer was rotated. This means that the laser output consists of two orthogonal linear polarization eigenmodes. By locating the angles that do not produce PMB signals, we can measure the direction of polarization. At zero twist, the orientation of intrinsic birefringence to optical axis could be measured, as mentioned above. Figures 10 and 11 show the experimental results of the output polarization angle as a function of twist angle of fiber cavity for Nd and Er fiber lasers, respectively. The dashed curves represent theoretical results assuming

Fig. 8. Variation of PMB frequency as a function of twist angle.

Fig. 10. Dependence of polarization direction on twist angle.

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Fig. 9. Variation of PMB frequency as a function of twist angle.

Fig. 11. Dependence of polarization direction on twist angle.

5-cm-long nontwisted parts. The theoretical calculation predicts the experimental results with reasonably good agreement. For low birefringent fiber lasers, such as a Nd fiber laser, the directions of polarization eigenmodes deviated from the birefringence axis immediately after a zero twist rate and rotated at about half of the twist rate. However, the change of polarization direction jumped periodically with the continuous change of twist angle because of the effects of nontwisted parts, as shown in Fig. 10. For a high birefringent fiber laser, the orientation of linear birefringence was not set to coincide with the optical axis. As the twist rate of the cavity fiber was increased, the polarization directions of Er fiber laser rotated at the same twist rate until the twist rate was increased as shown in Fig. 11. The variation of the PMB frequency and the polarization angle for twist angle had a periodic behavior of ;196° at large twist rates and agreed well with the predicted values. The polarization properties of twisted fiber lasers that we have analyzed can be used to design a stable fiber laser. Because the PMB frequency vanishes at a large twist angle, a fiber laser insensitive to polarization variation can be formed by using a sufficiently twisted fiber in the cavity without any other polarization elements. If fibers with highly linear birefringence, such as a polarization-maintaining fiber, are used to form a laser cavity, the fiber laser will be stable under a small twist because their polarization properties remain nearly unchanged. This type of fiber laser is more useful for polarimetric sensors because of the PMB frequency of fiber lasers. Based on the fact that polarization directions change linearly for twist angle changes in fiber lasers, we can easily adjust the polarization direction of fiber lasers under twist and also design a simple twist sensor. 4. Conclusion

We have investigated the twist dependence of polarization characteristics, such as the SOP’s of polarization eigenmodes, the polarization direction, and the

PMB frequency of fiber laser output. The theoretical model that we derived suggests that the twistinduced circular birefringence creates elliptical birefringence in the laser cavity and reduces the effects of intrinsic linear birefringence. Thus, the frequencies of the two polarization eigenmodes coincide with each other, and the PMB frequency decreases with oscillation as the twist rate increases depending on the intrinsic linear birefringence. However, the directions of polarization eigenmodes deviate from the birefringence axis at a much larger twist rate than intrinsic linear birefringence. On the other hand, the states of polarization are linear independent of the twist rate and the magnitude of intrinsic linear birefringence. The theoretical model provides a good description of the observed behavior both from a low linearly birefringent fiber laser ~Nd fiber laser! and from a high linearly birefringent fiber laser ~Er fiber laser!. The results presented here can be applied to polarization control of fiber lasers and their application systems. The authors acknowledge K. H. Kim, K. J. Kim, and D. S. Lim of the research department at the Electronics and Telecommunications Research Institute for technical assistance. References 1. H. K. Kim, S. K. Kim, H. G. Park, and B. Y. Kim, “Polarimetric fiber laser sensors,” Opt. Lett. 18, 817– 819 ~1993!. 2. H. K. Kim, S. K. Kim, and B. Y. Kim, “Polarization control of polarimetric fiber laser sensors,” Opt. Lett. 18, 1465–1467 ~1993!. 3. G. A. Ball, G. Meltz, and W. W. Morey, “Polarimetric heterodyning Bragg-grating fiber-laser sensors,” Opt. Lett. 18, 1976 – 1978 ~1993!. 4. D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Pulse repetition rates in passive, selfstarting, femtosecond soliton fiber lasers,” Electron. Lett. 27, 1451–1452 ~1991!. 5. K. H. Kim, M. Y. Jeon, S. Y. Park, H. K. Lee, and E. H. Lee, “Gain dependent optimum pulse generation rates of a hybridtype actively and passively mode-locked fiber laser,” Electron. Telecommun. Res. Inst. J. 18, 1–14 ~1996!. 6. A. J. Stentz and R. W. Boyd, “Polarization effects and nonlinear switching in fiber figure-eight lasers,” Opt. Lett. 19, 1462– 1464 ~1994!. 7. R. Ulrich and A. Simon, “Polarization optics of twisted singlemode fibers,” Appl. Opt. 18, 2241–2251 ~1979!. 8. G. Stephan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733–746 ~1977!. 9. H. Y. Kim, S. K. Kim, H. J. Jeong, H. K. Kim, and B. Y. Kim, “Polarization properties of a twisted fiber laser,” Opt. Lett. 20, 386 –387 ~1995!. 10. B. Lamouroux, B. Prade, and A. Orszag, “Polarization effect in optical-fiber ring resonators,” Opt. Lett. 7, 391–393 ~1982!. 11. R. Ulrich, Fiber-Optic Rotation Sensors and Related Technologies ~Springer-Verlag, New York, 1982!, p. 54.

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