Optimized absorption in a chaotic double-clad ber amplier Valérie Doya, Olivier Legrand, and Fabrice Mortessagne Laboratoire de Physique de la Matière Condensée, CNRS UMR 6622, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice, France [email protected]

Double-clad bers with a doped singlemode core and a noncylindrical multimode chaotic cladding are shown to provide optimal pump power absorption in power ampliers. Based on the chaotic dynamics of ray in such bers, we propose a quantitative theory for the pump absorption ratio and favorably compare its predictions to numerical results obtained via an c 2001 Optical Society of America adapted beam-propagation scheme. OCIS codes:

030.6140, 060.2320, 080.2740, 250.4480

Optical ber ampliers, and more specically erbium-doped ber ampliers (EDFA) are now commonly used to restore optical signals in long-haul optical links. In double-clad 1

ber (DCF) ampliers, a high saturation power requires a strong pump power. To lower the latter, the shape of the inner cladding needs to be optimized. Recent progress in design 1

optimization has clearly established that the standard circular geometry cannot achieve very high absorption eciency, as stressed by the generalized use of oset DCFs, but a 2

systematic tool of analysis is not yet at hand to predict absorption characteristics of nonstandard double-clad geometries. The inner cladding may be viewed as an external multimode ber and recently, for the rst time, non-standard multimode bers have been investigated,

3

both theoretically and experimentally, within the framework of Wave Chaos , which provides 4

performing novel tools for their analysis. This original approach has been quite successful in the understanding of semiclassical properties of waves in bounded systems, where the dynamics of rays is generically chaotic, like microwave cavities, elastic waves in thin plates 4

or blocks, or sound in rooms. Multimode optical bers provide ideal experimental systems 5

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to investigate the spatial properties of the diuse eld induced by chaotic ray dynamics.

3

A better understanding of phenomena involved in light propagation in optical multimode bers can be reached with the help of Wave Chaos techniques, and we believe that more ecient devices can thence be designed. In this Letter, we propose to optimize pump power absorption in double-clad EDFA's by resorting to the ergodicity of the two-dimensional (2-D) motion of rays when projected onto the transverse section of the ber. From the ensuing properties of uniformity and isotropy of the diuse eld, we deduce a theoretical prediction for the pump decrease, which is found to rapidly become uniform along the ber in the case of an ideally chaotic system. This prediction is then tested against the results of numerical simulations of the true wave character of light propagation, performed through an adapted Beam-Propagation scheme including absorption within the doped core via a complex nonlinear index. It is clearly shown that the results obtained for the completely chaotic geometry are in fair agreement with the prediction deduced from the ergodic ray 2

dynamics. In double-clad EDFAs, the signal launched into the single-mode doped core is amplied as a result of the absorption of the pump power injected into a multimode inner cladding. It 7

can be proved that, if the inner cladding transverse shape is a truncated disk, the dynamics of rays is made fully chaotic in the strongest sense (i.e. with exponential sensitivity to initial 8

rays' coordinates and directions). In this situation, a typical ray trajectory tends to cover the whole available phase space (i.e. position and direction) uniformly without building 9

caustics which are generally found in regularly shaped waveguides. In chaotic bers, it has been shown that almost all propagating modes are speckle-like,

3, 10

which implies that each

individual mode is statistically uniformly distributed over the section of the ber. Hence this permits to achieve maximal overlapping of the pump power with the doped core, thus yielding ecient pump absorption. To account for the decrease of the pump along the ber due to absorption at the doped core, we use the assumption that light is uniformly and isotropically distributed in the transverse section of the inner cladding. This assumption is well founded in highly multimode bers with non-standard shapes as recently exemplied in a ber such as the D-shaped one shown in Fig. 1. Fig. 1(a) shows a typical single ray trajectory projected onto the transverse section, which tends to visit any part of the section after a long enough propagation length. The speckle pattern on Fig. 1(b) is a typical near-eld intensity distribution (inverse video) actually measured at the output of an optical multimode ber specially designed and fabricated in our laboratory to study the statistical properties of the transmitted light. The 3

multimode core of this ber has a 120m diameter and is made of pure silica with index

n1 = 1:458 surrounded by an outer cladding composed of mixed silicon elastomers with index 3

n2 = 1:453, thus implying weak guidance. The width perpendicular to the at part is 90m. We now turn to the analysis of the pump absorption characteristics in a DCF with a Er3+ doped 5m diameter core of index n0 at the center of a D-shaped inner cladding with index n1 and the same dimensions as given above (see middle inset in Fig. 2). Guidance is obtained through step index proles with the linear index of the core n0 = 1:468, the index of the inner clad n1 = 1:458, and the index of the outer silicon clad n2 = 1:43. The main point of the following argument stems from the fact that the deterministic 2-D ray dynamics is strongly chaotic so that ergodization and mixing of rays is obtained as though a truly random dynamics were assumed. Therefore, provided that the fraction  of energy absorbed from a ray per passage through the doped core be small, it may be shown that the power decay length along the ber is given by the expression (the so-called Sabine's law) : 11

L = h`i tan  ;

(1)

where  is the mean angle of the rays with respect to the axis of the ber, and h`i is the mean transverse path between successive encounters of the core and is expressed as clad h`i = A ; P core

(2)

Aclad being the area of the inner cladding and Pcore the perimeter of the core. Then, denoting by  the inverse length of absorption along the core,  reads

 = `core tan 0 ; where n1 sin 

=

n0 sin 0 and `core

=

(3)

Acore=Pcore is the mean transverse path across the

core, given in terms of the area Acore of the core and of its perimeter Pcore. Of course, the 4

latter expression relies on the complete transverse isotropy of light resulting from the chaotic motion of rays. In practice, for values of  as high as 0:1 rad, and an index mismatch of the order of n0 ; n1 = 0:01, to an excellent approximation, tan 0 = (n1 =n0) tan . The inverse decay length thus reduces to

L;1 =  AAcore nn1 : clad

0

(4)

This simple expression is local, in the sense that it may depend on the coordinate along the ber, and can therefore allow for a nonlinear dependence of the  factor (as, for instance, in the case of a saturable absorption). It also has the rather intuitive interpretation of an inverse decay length being in proportion of the fraction of light intensity localized in the core, assuming a uniform distribution of intensity over the total cross section of the double-clad ber. One should also keep in mind that the above 2-D analysis assumes negligible bending, which is reasonable for a bend radius far larger that the cladding radius. To check the validity of the above mean eld theory, one models the true wave nature of light propagation in the double-clad ber with the help of the Beam Propagation Method (BPM) modied by allowing for the introduction of a saturable imaginary part of the index in the core. For the non-diractive part of the propagation scheme of the BPM, this amounts to writing an equation for the pump intensity within the core, of the type : dIp=dz = ;Ip where  = pa Nt =(1 + Iep) , pa being the absorption cross section, Nt the concentration of Er3+ per unit volume, and Iep the normalized pump intensity in units of threshold intensity Ip0 =

hp=(patsp). Obviously, the latter equations are valid in the case of a weak signal intensity. For all our simulations, the following parameters were used : pump wavelength p = 980 nm, erbium ions concentration Nt = 2  1026 m;3, absorption cross section pa = 2:5  10;25 m2, 5

spontaneous lifetime tsp = 10;2 s, and an initial pump power a little above the threshold. For all the numerical simulations presented here, the input is chosen to be a quasiplane wave with a mean angle  around 6 degrees. Fig. 2 displays the pump power decrease (in logarithmic scale) along 20 meters of double-clad ber for three dierent geometries as shown by the insets. For the standard circular ber (with the total area of the guiding cross section, including core and inner cladding, equal to 1:38  10;8 m2), the decrease saturates very rapidly due to the rapid exhaustion of the small fraction of modes of the inner cladding having non-negligible overlap with the core. For the D-shaped ber with parameters as indicated above (total guiding area equal to 1:19  10;8 m2), the decrease does not saturate and the visible curvature is essentially due to the nonlinear dependence of the absorption parameter . Another source of curvature is known to originate in the existence of the continuous family of special ray trajectories which pass through the center of the ber and do not hit the cut : these so-called marginally unstable trajectories may drastically slow down the exploration of phase space and therefore induce a long term nonexponential decrease. These are certainly the main responsible for the 10 % discrepancy 11

between the numerically obtained slope after 20 meters and the prediction given by Eq. 4, with  being evaluated at the corresponding intensity level. The slope deduced from the modied BPM scheme is, in this case, equal to 0:055 m;1 whereas the predicted value is

:

0 060

m;1. In the last case, the geometry of the inner cladding is designed with two cuts

(total guiding area equal to 0:91  10;8 m2), thus suppressing the above mentioned special trajectories and therefore leading to an excellent agreement between the numerical result (0:083 m;1) and the theoretical prediction (0:086 m;1). Here it should be remarked that even the short distance decrease (at about one meter) is observed to fairly agree with the 6

prediction, thereby indicating that ergodicity is established over shorter distances (typically a few centimeters are sucient to randomize almost any initial input).

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In conclusion, by noticing that multimode bers with non-standard transverse shapes generically induce chaotic dynamics of rays, we have proposed to study chaotic DCF's as a problem of Wave Chaos. We have shown that ergodicity of the ray dynamics may result in a strongly ecient improvement of the pump decrease, due to a maximal overlap of the pump intensity with the doped absorbing core. We have provided a theoretical argument to estimate the inverse decay length, and, through a ne control of the relevant parameters of the ber (notably by suppressing non chaotic ray trajectories), have been able to optimize the absorption characteristics of a double-clad EDFA. We have then tested our prediction against numerical results obtained via a modied BPM scheme. The agreement is excellent and illustrates the potential important role of Wave Chaos in designing double-clad EDFA's. The authors acknowledge fruitful discussions with Ph. Leproux, D. Pagnoux and Ph. Roy from the group Guided and Integrated Optics of IRCOM Limoges and are grateful to G. Monnom from LPMC Nice for his collaboration in the fabrication of the chaotic ber.

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References

1. E. Desurvire, Erbium-doped ber ampliers, Wiley Interscience (1994). 2. A. Liu, and K. Ueda, Optics Comm., 132, 511-518 (1996). 3. V. Doya, Du speckle aux scars : une expérience de chaos ondulatoire dans une bre optique, Thèse de doctorat, (2000). 4. H.-J. Stöckmann, Quantum Chaos an introduction, Cambridge University Press (1999). 5. R. L. Weaver, and O. I. Lobkis, Phys. Rev. Lett. 84, 4942-4945 (2000). 6. F. Mortessagne, O. Legrand, and D. Sornette, Chaos 3, 529541, (1993). 7. A. B. Grudinin, J. Nilsson, and P. W. Turner, New generation of cladding pump lasers and ampliers, Proc. Conference on Lasers and Electro-optics, Nice, France, Invited Paper CWA3 (2000). 8. S. Ree, and L. E. Reichl, Phys. Rev. E 60, 1607-1615 (1999). 9. M. V. Berry, Eur. J. Phys. 2, 91-102 (1981). 10. M. V. Berry, J. Phys. A : Math. Gen. 10, 2083-2091 (1977). 11. F. Mortessagne, O. Legrand, and D. Sornette, J. Acoust. Soc. Am. 94, 154-161 (1993).

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List of gures

Fig. 1. (a) a typical chaotic ray trajectory in the transverse section of a D-shaped ber; (b) a typical speckle-like near eld intensity actually measured at the output of an optical multimode ber.

Fig. 2. Pump power decreases in logarithmic scale along 20 meters of a double-clad EDFA for three dierent geometries indicated as insets. The two chaotic geometries lead to non-saturating absorption.

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(a)

(b)

Figure 1, V. Doya et. al. 10

1

ln(Pp (z )=Pp (0))

1

0.2 0 0

0.2

0.4

0.6

10 z (m)

0.8

1

Figure 2, V. Doya et. al. 11

1.2

1.4

1.6

1.8

20 2