Polarization independent nonlinear fiber sagnac inteferometer

15 September 2000

Optics Communications 183 Ž2000. 389–393 www.elsevier.comrlocateroptcom

Polarization independent nonlinear fiber sagnac inteferometer E.A. Kuzin a , N. Korneev a , J.W. Haus b

b,)

, B. Ibarra-Escamilla

b,a

a INAOE, A.P. 51 y 216, Puebla, Pue., Mexico Electro-Optics Program, The UniÕersity of Dayton, Dayton, OH 45469, USA

Received 12 April 2000; received in revised form 15 June 2000; accepted 10 July 2000

Abstract In general, the propagation of light in a twisted, birefringent fiber evolves with a period and phase change non-linearly dependent on the initial polarization state. We show both analytically and numerically that a critical value for the ratio of twist rate to birefringence can be chosen so that the period and nonlinear phase change are no longer sensitive to the input polarization state. This makes possible the design of a input-state independent Sagnac interferometer for mode-locked laser applications. q 2000 Published by Elsevier Science B.V. PACS: 42.81.Wg; 42.65.Wi; 42.55.Wd Keywords: Sagnac interferometer; Birefringent optical fibers; Nonlinear phase change

Sagnac interferometers, also called nonlinear optical loop mirrors ŽNOLMs., have been investigated in relation to such applications as optical switching, demultiplexing, phase conjugation, and fiber laser mode-locking w1,2x. A conventional NOLM uses the phase shift caused by self-phase modulation to produce an intensity-dependent transmission with an asymmetric coupler. The experimental operation of such a fiber NOLM is complicated by the presence of both birefringence and optical activity induced by twist. As a consequence their operation depends in a quite complicated way on input polarization w3x. We previously considered an unconventional Sagnac interferometer using birefringent fiber with a

) Corresponding author. Tel: q1-937-229-2394; fax: q1-937229-2097; e-mail: [email protected]

0.5r0.5 coupler w4–6x. The input polarizations are adjusted so that counter-propagating beams have different polarization rotation and the switching mechanism in this case occurs by means of nonlinear polarization rotation. The transmission variation can be as strong as the conventional self-phase modulation effect at least for fibers as long as to 10 beat lengths. In some cases the critical powers are close to that for a conventional NOLM and contrast is high. The fact that the nonlinear polarization rotation is important and must not be neglected even for relatively long fibers is a significant complication in the NOLM analysis. In this paper we apply the Poincare´ sphere method to analyze the impact of parameters and input conditions on the Sagnac interferometer’s operation, including nonlinear polarization rotation. The Poincare´ sphere method was used earlier for the visualization of nonlinear polarization evolution in a twisted fiber

0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 8 6 8 - 3

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in Ref. w7x; a detailed discussion of the visualization of the trajectory on the Poincare´ sphere can be found in Ref. w8x. However, this method was not used in an analysis of the NOLM. In particular, earlier papers did not present results for the nonlinear phase change required to describe the NOLM’s transmission characteristics. We show that analysis using the Poincare´ sphere makes it possible to predict the NOLM’s operation for arbitrary fiber twist, birefringence, and length. Consequently, we are able to eliminate the polarization sensitivity of the NOLM within certain limits. Recently input polarization sensitivity has been experimentally reduced with a high twist rate fiber w9x, in which the twist length is much shorter than the beat length. In contrast, we show for the fiber twist length of the same order as the birefringence length that by correlating fiber twist with fiber birefringence and applying an appropriately selected polarization controller a truly polarization insensitive NOLM can be created. To use the Poincare´ sphere for the NOLM it is necessary to analyze the nonlinear phase shifts for the trajectory’s period on the Poincare´ sphere and the dependence of the period on the initial conditions and the physical parameters of the NOLM. The period and phase shift follow from solutions of the equations describing the nonlinear propagation of an optical beam in a birefringent fiber with twist. It is convenient to use the complex amplitudes of rightX X and left-circular polarization fields, Cq and Cy in the rotating basis, for which the evolution equations are w10x 2

X X X X X Ez Cq s ig Cq q i kCy q i b Cq q 2 Cy

ž

2

X q;

/C

Ž 1. 2

X X X X X Ez Cy s yig Cy q i kCq q i b Cy q 2 Cq

ž

2

X y.

/C

Ž 2. Here g s Ž hrŽ2 n. y 1. q f y0.95q Žwhere q is the fiber twist rate in radians per unit length, h s 0.13–0.16 for silica fibers, and n is the refractive index. k s pd nrl describes the birefringence, and b s 4p n 2r3 l A eff is the nonlinearity Ž A eff being the effective-modal area, and n 2 the Kerr coefficient, and l the wavelength.. These equations use the steady-state approximation, which neglects fiber dispersion and will be valid when the pulse widths are

sufficiently long to yield a dispersion length longer than the fiber length. By using scaled variables z zkrp and C "s X X Ž0.< 2 q < Cy Ž0.< 2 is the CX" r P IN Žwhere P IN s < Cq input power. the equations

™

(

2

Ez Cqs i gCqq ip Cyq i PN Cq q 2 Cy

ž

2

/C

q;

Ž 3. Ez Cys yi gCyq ip Cq 2

q i PN Cy q 2 Cq

ž

2

/.C

y;

Ž 4.

are now expressed in terms of two parameters the normalized input power PN s bp P IN rk, and ratio of twist to birefringence g s gprk. 2 In terms of the Stokes parameters: A s Cq 2 ) ) ) y Cy , B s iŽ Cq Cyy Cq Cy ., and C s Cq Cyq ) Cq Cy , which satisfy A2 q B 2 q C 2 s 1 the polarization state is a point Ž A, B,C . on the surface of the Poincare´ sphere. Equations for the Stokes parameters have closed form solutions, which are periodic functions of z. So, it is sufficient to calculate them over only one period. However, the period depends on the intensity, input polarization, and the physical parameters of the Sagnac interferometer. Geometrical analysis of trajectories on the Poincare´ sphere is very useful in guiding numerical simulation of NOLMs for design purposes. Without nonlinearity the polarization evolution on the Poincare´ sphere is a planar, circular trajectory, as illustrated in Fig. 1. The propagation can be visualized as the rotation of a point on the sphere around the axis passing through fixed points 1 and 2, Žthe eigen-modes.. The location of these points is determined by an angle F 0 , tanŽF 0 . s krg s prg, in terms of the twist and birefringence. For zero twist the points are 1a and 2 a, which correspond to linear polarization parallel to the slow and fast principal axes, respectively. From the Poincare´ sphere it is easily seen that any trajectory may be selected by giving the Stokes parameters of the input polarization in the plane B s 0. This circle A2 q C 2 s 1 corresponds to elliptic polarization with axes of the ellipse parallel to the birefringent fiber’s principal axes. Angles F s 0, and F s p correspond to circularly polarized beams. A specific trajectory is selected by the angle F on

E.A. Kuzin et al.r Optics Communications 183 (2000) 389–393

Fig. 1. A projection of the Poincare´ sphere on the A – C plane. 1a and 2 a are eigen-modes for zero twist Žprincipal polarization axes.. The eigen-modes 1b and 2 b are rotated in the A – C plane due to the fiber twist; at low intensities the trajectories are circles on the Poincare´ sphere orthogonal to 1b –2 b axis.

the A–C projection corresponding to the initial conditions. When the nonlinear terms are included in the evolution equations the trajectories are no longer planar. For a relatively weak nonlinearity there are still two eigen-modes, but two additional modes appear for PN ) 2p . In this paper we are only concerned with powers that lie below this threshold. The transmission of the NOLM is controlled by the difference in phases of the counter-propagating fields in the loop. This phase shift has two distinct parts. One is periodic and the other non-periodic part is due to the nonlinear terms and accumulates around the loop. Both of these and the resulting relative phase shifts between counter-propagating beams depend on fiber twist, birefringence, fiber length and the input polarization w4,5x. To analyze the NOLM’s operation we need to calculate period on the Poincare´ sphere and the phase shift per period. Fig. 2 presents the trajectory period for several scaled powers when the fiber is not twisted. The power PN s 1 may be considered as moderate. It will be shown below that this power corresponds to the transmission switching power for a conventional NOLM with fiber length equal to 10–20 beat-lengths. Low birefringent fibers were recently measured w11,12x, and beat lengths were found in the range 2–20 m. So, a fiber’s length considered in our model vary in the range from tens of meters to several

391

hundred meters of a low birefringence telecommunication fiber; this is a reasonable length for most of the laser applications. For moderate powers the nonlinear dependence of the period was found to be relatively low for F f 0.2 and higher at F s "pr2. For high input power, PN ) 3, the F dependence increases drastically for angles close to ypr2. Even a small fiber twist eliminates the abrupt increase of the polarization-rotation period for points close to the eigen-mode. But generally the dependence of the period on input polarization persists for high input power as well as for low. The family of such dependencies enable us to select from numerous possibilities those which are more convenient for particular purposes. A particularly interesting case is a polarization insensitive NOLM. It is useful to apply a perturbation analysis for weak nonlinearity. Details of these calculations will be published elsewhere. For the period change it yields the result: DTrT0 s y

PN 4p

Ž sin2F 0 y 2cos 2F 0 .

=cos Ž F y F 0 . sinF 0 .

Ž 5.

Note that the first-order intensity contribution vanishes when tan2F 0 s 2; this result corresponds to a critical value for the twist to beat length ratio near g c s 2.22, which also corresponds to that calculated earlier by Menyuk in a different context w13x. We analyzed the operation of the Sagnac interferometer

Fig. 2. The dependence of the trajectory period on the angle F Ždefined on the Poincare´ sphere in Fig. 1. for fiber without twist.

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E.A. Kuzin et al.r Optics Communications 183 (2000) 389–393

for this twist numerically both for low and high nonlinearity. Fig. 3 displays orbit periods on the Poincare´ sphere for the critical twist rate g c s 2.22 and for PN equal to 0 Žlow intensity., 1, and 6. The relatively low dependence of the periods on a trajectory’s initial polarization is found for low and moderate powers. Note that at the highest power the period depends on polarization, because of quadratic and higher-order intensity corrections to the result in Eq. Ž5.. The nonlinear phase shift including all contributions are presented for the right circularly polarized component Cq in Fig. 4 for several values of g. The left circularly polarized component Cy has a similar dependence. Without twist the nonlinear phase shift for Cq is the same as for Cy. For g c s 2.22 a low dependence of the phase shift on the initial polarization state is observed. For high twist rate a large variation of the phase shift with initial polarization state is observed in the phase shifts for Cq and Cy. F 0 s 0.158 at the twist rate g s 20, and is close to its infinite limit value F 0 s 0. So g s 20 may be considered as very large twist rate. The elimination of F-dependence of the period and the phase shift on input field polarization for the critical twist rate leads to an improved design for NOLM operation. It depends on the self-phase modulation effect, which is roughly the average of the phase shown in the Fig. 4 for g c .

Fig. 3. For g c s 2.22 the period of the Stokes parameters is nearly constant for moderate power at all input polarizations. At high power the period strongly depends on the input polarization.

Fig. 4. The left circular polarization phase shift for power PN s1. The phase variation versus input polarization is minimum at the value g c s 2.22.

We analyzed the NOLM operation with fiber twisted at the critical twist rate. A polarization controller is required in the loop to compensate for the relative orientations of the input polarization at each end. For instance, it determines the linear transmission, because it adjusts the projection of the polarization for two counter-propagating fields on one another. The angle u denotes the rotation of the linear polarization from the polarization controller. We chose to insert it in the fiber loop at one of the output coupler ports and adjusted it so that the counter-propagating fields have the same polarization orientation with respect to the principal axes at the fiber ends. This orientation minimizes the transmission at low intensities. To illustrate our results we calculate parameters of the NOLM for the 0.4r0.6 coupler, with two ports connected by a low-birefringence fiber that is 10 beat lengths long. For our scaling parameters a normalized power equal to unity corresponds to the switching intensity of a conventional NOLM with a 0.4r0.6 coupler and fiber length around 10 beat lengths, that is typical for fiber laser applications. So PN s 1 may be considered a moderate input power for NOLMs, as was mentioned above. With our design the linear input polarization orientation u was changed from 0 to pr2. In other words the initial values for the trajectories on the Poincare´ sphere lie in the A s 0 plane, see Fig. 1. The dependence of the transmission on input power

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on the input polarization is practically eliminated for light intensities comparable to those used in conventional NOLM experiments. This new condition for polarization insensitive operation can be applied to building more robust versions of mode-locked fiber lasers.

Acknowledgements

Fig. 5. The NOLM transmission for 2 input polarization states with the polarization controller adjusted to the optimum. One trajectory demonstrates the sensitivity to the polarization controller; the polarization angle is shifted from the optimal value by 1 rad.

is practically the same for all input polarization orientations. The results for two values of u are presented in Fig. 5 by dotted and dashed lines. The critical normalized power where the first transmission maximum was found, is equal to 1.2, close to the conventional NOLM results. The solid line presents dependence of the transmission on input power when the polarization controller was not optimally adjusted. A very high deviation of the input power dependence is observed. This shows that NOLM operation drastically depends on the polarization controller adjustment. Calculations have also been made for the NOLM without fiber twist. The deviation of the dependencies was found to be equal to 10%–15%. A small deviation of the transmission curves have been found for the relatively high twist rate g s 5. We have analyzed the operation of a Nonlinear Sagnac interferometer with birefringent fiber. By choosing a critical fiber twist and adjusting a polarization controller the dependence of the transmission

J.W.H. was supported by a grant from the National Science Foundation INT-9722619. E.K. was supported by a CONACyT project 28498A. We are grateful to Craig Spencer for a thorough and critical reading of the manuscript.

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