Polarization selection in external cavity birefringent ... - CiteSeerX

Polarization selection in external cavity birefringent Vertical-Cavity Surface Emitting Lasers A. Valle a , L. Pesqueraa , J. Dellundeb, and K. A. Shore c a Instituto de Fsica de Cantabria (CSIC-UC) Facultad de Ciencias,Avda.Los Castros s/n E-39005 Santander Spain. b Departament d'Estructura i Constituents de la Materia Facultat de Fsica, Universitat de Barcelona Diagonal 647, E-08028 Barcelona Spain c University of Wales,Bangor School of Electronic Engineering and Computer Systems Bangor ,LL 57 1 UT Wales UK

ABSTRACT

Calculations are reported of the eect of weak optical feedback on the polarization properties of birefringent Vertical-Cavity Surface Emitting Lasers (VCSELs) in an external cavity con guration. Attention is focussed on the competition between two orthogonal polarizations of the fundamental (LP ) transverse mode of the weakly index VCSEL. Ohmic heating eects on polarization selection and sensitivity of external cavity VCSELs are analysed. We show that control of the emission polarization can be exercised even for very small external re ectivities ( of the order 10? % ) by appropriate choice of optical feedback delays. The range of currents in which a particular polarization is selected narrows as thermal eects become important. A controlled displacement of the external re ector with respect to the laser can maintain the polarization over a wide current range in the presence of thermal eects. The polarization selectivity is shown to be dependent upon the strength of optical feedback. Polarization is also shown to be highly sensitive to small changes in optical feedback delay and to very small unwanted re ections. Keywords: Semiconductor lasers, vertical-cavity laser polarization, optical feedback 01

4

1. INTRODUCTION

As Vertical Cavity Surface Emitting Lasers (VCSELs) move from the research laboratory into commercial production increased attention is being given to a wide range of potential applications of these devices including optical data recording and optical switching. The performance of VCSELs in applications such as magneto{ optic discs and coherent detection require control to be exercised over the emission polarization. For that reason considerable eort has been given to characterising the polarization properties of stand{alone VCSELs. The VCSEL lasing power can be distributed between two orthogonal linear polarizations in such a way that they can be randomly oriented in the plane of the active layer. Switching between linearly polarized states with the same fundamental transverse mode pattern is found as the current injection is increased. Several mechanisms have been proposed to explain this switching. In practical situations, VCSELs may be subject to weak optical feedback eg. lasers used to read pit patterns in the surface of optical discs may suer uncontrolled light re ections. Also unwanted re ections from the near or far end of an optical ber can modify the performance of VCSELs On the other hand, optical feedback can be utilised in order to control the emission polarization of VCSELs. It is therefore pertinent to establish the changes in polarization which can arise in VCSELs subject to weak optical feedback. The present paper addresses these issues and provides the rst results obtained from a 1{11

1,3,4

5,12,13

14

15,16

Other author information: (Send correspondence to A.V.) A.V: Email: [email protected]; Telephone: 34-42-201465; Fax: 34-42-201459.

theoretical analysis of the weak optical feedback response of birefringent weak index-guided VCSELs . Particular emphasis is given to determining whether those eects are sensitive to the location of the feedback mirror. The work described here represents the convergence of three themes treated previously in relation to the in uence of spatial hole burning (SHB) on the transverse and polarization mode structure of stand-alone VCSELs and the eects of optical feedback on VCSEL transverse mode structure. The purpose of the present paper is then to identify how optical feedback may in uence polarization mode excitation in VCSELs. We consider two situations: feedback induced polarization mode selection and sensitivity to unwanted feedback. This analysis will be performed in two stages. In the rst one thermal eects are not taken into account: we will show this is an ideal situation for polarization control by using an external cavity. In the second one ohmic heating eects are incorporated in our model through a current dependent wavelength shift. Thermal eects are then found to be detrimental for polarization control. The paper is organised as follows. The device model and numerical procedure are presented in section 2. In section 3 the results obtained by neglecting heating eects are presented. In section 4 the in uence of ohmic heating in polarization control and sensitivity of external cavity VCSELs is studied. A summary of our results and conclusions is given in section 5. 13,17

18

2. DEVICE MODEL AND NUMERICAL PROCEDURE

The model utilised in the present work incorporates both spatial dependence of carrier and optical eld pro les and time delay eects associated with weak optical feedback. The cylindrical VCSEL structure considered in this work is illustrated schematically in Fig. 1. It is appropriate to utilize a cylindrical coordinate system, (r; ), to describe the spatial distribution of charge carriers and optical elds in the structure.

Figure 1.

Schematic diagram of the external{cavity VCSEL structure.

The active region of the device is taken to be of thickness d and radius a. The laser cavity is de ned by two highly re ecting mirrors separated by a distance, L, along the longitudinal axis. We consider that the injected current is distributed uniformly over the active region. The directions of the orthogonal polarizations are indicated by the vectors x1 and x2 . Subscripts i = 1; 2 will be used to denote the appropiate polarization. It has been noted that birefringence can arise in the structure of nominally isotropic VCSELs due to strain induced during the fabrication process. Birefringence eects are taken into account by assuming that the core refractive index (ncore ) in direction 1 is greater than in direction 2 (ncore ), while the cladding refractive indices (ncladd; ncladd) are the same for both directions. The appropriate modes for the assumed weakly index guided 8

1

1

2

2

structure are conventionally denoted as the LPmn modes. Attention is focused on the polarization states of the lowest order transverse mode LP whose pro les are characterised in detail in Ref. 13. Several mechanisms of selection of the fundamental transverse mode can be found in the literature. The transverse variation of the fundamental transverse mode power for the i-polarization direction is given by: 01

19

) pi (r) = J J(u(iur=a ) 2 0

2 0

i

) pi (r) = KK(w(iwr=a )

r  a;

if

2 0

otherwise;

i

2 0

(1)

= cladd = Here ui = a((ncore i i ) ? ) ; wi = a( ? (ni i ) ) ; and Jm and Km are Bessel functions of the rst and second kind. In the present model, a cavity resonance condition is imposed in the form L = q where q is an integer (here, q = 8). The wavevector i is obtained from the eigenvalue equation: 2

2 1 2 01

2 01

2 1 2

01

ui J (ui ) ? wi K (wi ) = 0 J (ui ) K (wi ) 1

(2)

1

0

0

Normalized intensity pro les are then de ned as i (r) = pi (r)=

1

Z 0

pi (r)rdr:

(3)

In this work we have taken ncore = 3:5002, ncore = 3:5, and an index step,
n = ncore ? ncladd = 0:01, obtaining in this way an spectral splitting between orthogonal polarizations of 0.5 Amstrongs which is of the order of typical experimental values. Observed splittings are independent of the injected current and temperature. Pro les of dierent polarizations for a given transverse mode are very similar. However, this small dierence is able to induce polarization selectivity and switching behaviour. In order to determine the temporal evolution of the modal gain and the power of the identi ed polarization states it is necesary to calculate the time development of the spatial distribution of the carrier density N (r; t) in the active region of the laser. In general, the dynamics of carriers is prescribed using a continuity equation applied to the active region: 1

2

2

2

6

9,20

13

17

13

@N = Dh 1 @ r @N i ? BN ? X a g jE j (r) + j (4) i i i i @t r @r @r ed i ; where Ei is the eld amplitude of the i-polarized LP mode, and ai = vg ?(2d)? . The numerical value and 0

2

2

=1 2

1

01

meaning of the parameters used in this work can be found in Table 1. The evolution of the carrier density is coupled to the corresponding rate equations: s

dEi = 1 ? i v ?g ? 1 E (t) +  exp(i!  )E (t ?  ) +  d a N (r; t)dr  (t) g i  i i i i dt 2 n pi where the modal gain of the i-polarization mode is given by R

0

gi =

Z 0

1

A(N (r; t) ? Nt ) i (r)rdr;

(5)

(6)

!i are the emission frequencies of the dierent polarizations in the solitary laser, is the spontaneous emission factor, and i (t) is a Gaussian white noise.

The present model incorporates the eect of optical feedback on the evolution of the complex mode amplitudes of the optical eld by using the Lang-Kobayashi equations in the form given in Ref. 18 . Re ected light is fed back into the cavity after an external roundtrip time  . The feedback strength is conventionally characterised by a parameter p (7)  = 1 (1p? R) Rext  L

R

Table 1.

SYMBOL

a L D B Nt A d ?

pi n    R 1 2

Device and material parameters

VALUE 3 m 1 m 5 cm s? 10? cm s? 1:33  10 cm? 3  10? cm 0:03 m 0.06 1:5 ps 2 ns 3 0:87473m 0:87468m 0.99425 2

10

1

3

1

18

3

16

2

MEANING OF THE SYMBOL radius of the core length of the cavity diusion constant bimolecular recombination coecient carrier density at transparency gain coecient thickness of the active layer longitudinal con nement factor photon lifetime carrier lifetime linewidth-enhacement factor wavelength of polarization 1 wavelength of polarization 2 VCSEL mirror re ectivity

dependent on the laser cavity roundtrip time, L (L = 2L=vg ; where vg is the group velocity ), the laser mirror re ectivity R, the external re ectivity Rext and the coupling eciencies . Here the same value of  has been taken for both polarizations. Integration of the eqs. (4)-(5) is performed by using N (r; 0) corresponding to the stationary solution in the below threshold state. The boundary condition for the carrier density is taken as N (1; 0) = 0. After the initial time the laser is switched on with a current above threshold over time scales of the order of 50 ns by which the device has reached a steady state. The numerical procedures followed are as detailed in Refs. 17 and 22. 21

3. POLARIZATION CONTROL AND SENSITIVITY WITHOUT HEATING EFFECTS

The underlying physical mechanism for polarization selection in stand-alone VCSELs has been discussed by a number of authors and the phenomenon has been attributed to temperature dependence of the gain, polarization sensitive dependence of the carrier dynamics , anisotropic losses, and SHB. The relative contributions of these mechanisms is dependent upon both basic material parameters and device structure. Here we focus attention on eects arising due to SHB. The impact of optical feedback on polarization characteristics of the external cavity VCSEL is studied in this section without heating eects. This will help to better understand in the following section the role played by thermal eects. As we saw in previous section modal gain is found from an overlap integral between the carrier density pro le and the transverse mode intensity pro le. Now, it has been con rmed in earlier work on stand-alone VCSELs that at low currents SHB is not signi cant so that carriers accumulate near the centre of the device and hence the modal gain of the better con ned polarization (mode 1) is greater. On the other hand, when SHB becomes signi cant at higher injection currents the situation is reversed and the less well-con ned polarization mode (mode 2) has the higher modal gain. In this way polarization switching behaviour due to SHB (from the longer to the shorter wavelength) can occur as shown in Fig. 2 where the Light-Current and modal gain characteristics have been plotted. Now attention is given to the changes in mode competition which arise due to optical feedback eects. On the basis of previous work on tranverse mode selection in VCSELs it is expected that optical feedback will, in particular, change threshold currents, Jth;i , of the respective polarization modes. Jth;i is de ned as the lower injected current density at which i{polarized power exceeds 0.01 mW. Depending upon the phase of the returned light with respect to that emitted from the laser the eect of optical feedback can be either to increase or decrease the laser threshold. The modi cation of Jth;i is determined by conditions of constructive or destructive 5

12

10

13

13

18

interference between the emitted and re ected light and thus changes of the external cavity dimension on optical wavelength scales are expected to give rise to alternating increase and decrease of the threshold gain and hence threshold currents of the respective polarizations. The phase changes in the external cavity are dependent upon the oscillating frequencies of the polarizations under consideration being essentially determined by a term of the form cos(!i ). Optical feedback eects become important when the dierence of feedback induced losses is of the order or greater than the dierence of modal gains between both polarizations. This modal gain dierence is very small since both polarization intensity pro les are very similar. The previous value is around 0:005 ps? when the laser is near threshold. This value can be taken as an estimation of the order of magnitude of the modal gain dierence between polarizations. The dierence of losses is given by vg (gth; ? gth; ) = 2(cos !  ? cos !  )=?. This dierence is maximum when (! ?! ) = (2n+1). At this external roundtrip time, feedback eects become important when  > 7:5  10? ps? , that corresponds to a very small external re ectivity, Rext = 10? %. This estimation is con rmed with the results presented in the following paragraph. We focus our attention on short external cavities. This can correspond to the case when a VCSEL is butt{ coupled to an optical ber, but there is a small air gap between the ber and the VCSEL. When considering polarization mode selection it is appreciated that only very small frequency dierences - of order 20 GHz- exist between two adjacent modes. Then, for small external cavity round trip times  , it is found that the modes can be enhanced or suppressed essentially in unison since cos(!  )  cos(!  ). For larger values of  the dierences in these terms become signi cant. Speci cally in the case where (! ? ! )  (2n +1) where n is an integer it is possible to enhance one mode and suppress the other by arranging, for example, that cos(!  ) = 1 and cos(!  ) = ?1. This eect is illustrated in the upper part of Figure 3 where the threshold gain of both polarizations is shown for  = 0:00025 ps? , that corresponds to an external re ectivity of Rext = 10? %. In the lower part of Figure 3 the corresponding values of Jth;i are shown. 1

1

2 1

5

2

2

1

1

5

14

1

2

2

1

1

2

1

4

1.0

Power (mW)

0.8

0.6

0.4

0.2

0.0 0.0

2.0

4.0

6.0

8.0

10.0

8.0

10.0

−1

Modal Gain (ps )

11.12

11.11

11.10

11.09 0.0

2.0

4.0 6.0 2 Current Density (kA/cm )

Polarization resolved Light - Current characteristics (upper part) and corresponding modal gains (lower part) for the stand-alone VCSEL (polarization 1: solid line and polarization 2: dashed line ) and for the VCSEL in an external cavity (1: dotted line and 2: dashed-dotted line).

Figure 2.

vggth,1 vggth,2

−1

Threshold Gain (ps )

11.125 11.120 11.115 11.110 11.105 11.100 26.0000

26.0040

Jth,1 Jth,2

2

Threshold Current (kA/cm )

9.0000

26.0020

7.0000

5.0000

3.0000

1.0000 26.000

26.002 26.004 External Roundtrip Time (ps)

(Upper part) Threshold gains of 1 (squares) and 2 (circles) polarization modes as a function of external cavity roundtrip time; (Lower part) Threshold currents of 1 (squares) and (circles) polarization modes as a function of external cavity roundtrip time. The values of the threshold currents of the two modes when feedback is not present are also plotted with dotted and solid line for the 1 and 2 polarization modes respectively.

Figure 3.

It can be seen that selection of either mode can be eected by appropriate choice of  . When the feedback favours polarization 1 (see eg.  = 26:0028 ps)Jth; increases thus providing a wider range of currents at which polarization 1 dominates. This is also seen from Fig 2 where the L-I characteristic for both polarizations in the presence of optical feedback has been plotted ( = 0:00025 ps? ;and  = 26:0028 ps). Figure 3 shows that Jth; can be doubled for the very small value of external re ectivity (Rext = 0:0001%) used. In this case increasing the current leads to a deeper hole in the carrier density near the centre and to an accumulation of carriers near the cladding region. Then the overlap of carriers with the worse con ned polarization ( mode 2 ) is enhanced and the modal gain of that polarization increases - eventually reaching the threshold gain as is seen from Fig. 2. The previous behaviour is maintained in a wide range of values of  . We have checked that Jth; can be increased by 50 % with respect to its value without feedback when (! ? ! )  =4 - indicating in this way that any value of  such that (2n + 1=4) < (! ? ! ) < (2n + 7=4) is appropriate to implement the polarization feedback control mechanism. On the other hand, when the feedback favours polarization 2 (26:0008 ps <  < 26:0018 ps) the switching disappears and polarization 1 does not get the opportunity to lase. Now, when the current is increased, the accumulation of carriers near the cladding region leads to a decrease of the modal gain of the better con ned polarization. The polarization 1 is not able to reach its threshold gain and the selection of polarization 2 is achieved over all the current range and hence the situation is optimal for polarization selection. This behaviour is maintained for a range of (! ? ! ) similar to the previously considered. This is illustrated in Fig. 4 where the external cavity length range over which polarization 2 is dominating for all current values,
Lext , has been plotted as a function of the external cavity length Lext . This quantity gives an idea of the maximum length over which the external re ector can be displaced without changing the Light{Current curve. The maximum observed in this gure corresponds to the optimum situation in which the feedback induced loss dierence is maximum, (! ? ! )  , that is equivalent to a 4 mm external cavity length, (Lext = c=2). 2

1

2

2

2

2

1

2

2

1

1

1

0.20

∆Lext (µm)

0.15

0.10

0.05

0.00 0.0

2.0

4.0 Lext (mm)

6.0

8.0

External cavity length range in which we obtain a L-I curve dominated by polarization 2 vs. the external cavity length. Apart from  , the second principal parameter which characterises the behaviour of external cavity lasers is the strength of optical feedback, . This parameter would, in general, be relatively easy to adjust in practical device operation and hence the modi cations of the polarization behaviour due to variation in  may be of particular interest. Figure 5 shows the eect of varying  on Jth;i when the polarization 1 is favoured. It is seen that depending upon  , Jth; can exhibit dierent behaviours. Small changes of the external cavity length near the maximum of Jth; in Fig. 3 do not lead to big changes in Jth; as seen in Fig. 5. However, when  is far from that maximum small changes in the external cavity length cause a drastic change in Jth; . For instance in Fig. 5 a variation of Lext of 0.045 m (from  = 26:0022 ps to  = 26:0025 ps) can double Jth; for  = 0:0003 ps? , that corresponds to Rext = 1:5  10? %. Then, if Lext is not carefully controlled the polarization behaviour of the device can change drastically. As a consequence unwanted feedback can modify the emitted polarization.

Figure 4.

2

2

2

2

2

1

4

2

Threshold Current (kA/cm )

9.0

7.0

τ=26.0022 ps τ=26.0025 ps τ=26.0028 ps

J th,2

5.0

3.0 J

1.0 0.00001

th,1

0.00010 −1 Feedback strength (ps )

Eect of optical feedback strength on threshold currents of polarization 1 (lower symbols) and polarization 2 (upper symbols) for dierent values of external cavity round trip times  .

Figure 5.

4. POLARIZATION CONTROL AND SENSITIVITY. INFLUENCE OF HEATING EFFECTS

In this section we include ohmic heating eects by considering a current dependent wavelength shift,
=
I , of 0.3 nm/mA. The current induced external cavity phase change causes that an alternated domain of both modes appears as seen from Fig. 6 (d). This phase change is also made clear in Fig. 6 (a) where threshold gains 9

for both polarizations as a function of injected current are shown. When considering ohmic heating threshold gain becomes: h h  a
 (j ? j )ii; 1 ? (8) gth;i = v 1? 1 ? 2 cos 2c th i i
I g pi where jth is the solitary VCSEL threshold current density (1.9 kA=cm ). The oscillating behaviour of gth;i as a function of j due to the nonvanishing value of
=
I becomes apparent in Fig. 6 (a). Feedback induced losses determine polarization for low values of j . At high j values (bigger than 8kA=cm ) the modal gain 2

0

2

0

0

2

0

mechanism becomes dominant due to SHB (see Fig. 6 (b)) and polarization 2 dominates with the corresponding disappearance of the alternated polarization behaviour. The previous behaviour can be better understood by looking at Fig. 6 (c) where the dierence between modal gain and threshold gain for each polarization has been plotted. There is a clear correspondence between this graphic and the alternated L-I characteristics. This behaviour has been recently observed. The alternated domain seen in L-I curve turns Fig. 3 into a band structured diagram. This is seen in Fig. 7 where the switching current, Jsw , has been plotted as a function of  . Jsw is de ned as the current at which both polarizations have equal power and hence a particular polarization (indicated by its index) is dominating within each band appearing in Fig. 7. Thermal eects are clearly detrimental for polarization control since current must be in one of those bands if the polarization has to be xed. In this way a narrower current range in which a polarization is maintained is obtained when ohmic heating is considered. Width and slope of these bands are determined by modal gains and wavelength shift. This is illustrated with Fig. 7 where Jsw for dierent wavelength shifts are shown. The number and width of bands decreases when
=
I decreases since the number of oscillations of the threshold gain in the considered current range decreases. Results with no thermal eects are recovered when
=
I decreases as seen from Fig. 7: slope and width of bands increase in such a way that the situation depicted in Fig. 3 is recovered when
=
I = 0. A controlled dependence of the external cavity length on the injected current can be implemented to make the laser to recover the Ligh-Current characteristics without thermal eects ( Fig. 2 ). This dependence is shown in Fig. 8. The linear character of this control mechanism is due to the linear dependence of the wavelength shift on the injection current. With this control mechanism the posibility of polarization control for a wider current range is then recovered. Finally, we analyse in this section the eect of varying  on the switching current when thermal eects are present. Figure 9 shows this eect when the external cavity length is chosen to favour a particular polarization. 16

−1

Threshold Gain, vggth,i (ps )

−1

Modal Gain − Threshold Gain (ps )

9

11.13

(a)

11.12 11.11 11.10 11.09

11.13

−0.03 −0.05 (d)

0.8 Power (mW)

11.12 11.11 11.10

0.6 0.4 0.2

11.09 11.08

(c)

−0.01

(b)

−1

Modal Gain, vggi (ps )

11.08

0.01

2

4

6

8

10 2

Current Density (kA/cm )

0.0

2

4

6

8

10 2

Current Density (kA/cm )

(a) Threshold gains, (b) modal gains, and (c) dierence between them for 1 (solid) and 2 (dashed) polarization modes. (d) Polarization resolved Light{Current characteristics (1: solid line and 2: dashed line). Parameters in this gure are
=
I = 0:3 nm=mA;  = 0:00025 ps? ; and  = 26:0028 ps.

Figure 6.

1

2

Current Density (kA/cm )

8

2 1

6

2 1

4

2 1 2

1

2

2

Current Density (kA/cm )

2

8

1

2

1

2

6 4

1

2

1

2

1

2 2

2 26.000

26.001

1

2

26.002

26.003

External Roundtrip Time (ps)

Jsw vs  for two dierent wavelength shifts: (upper part)
=
I = 0:15 nm=mA and (lower part)
=
I = 0:3 nm=mA. The index indicates the polarization that dominates in each zone. When polarization 1 is favoured ( = 26:0028 ps) a band structured diagram appears due to ohmic heating eects. If feedback favours the other polarization( = 26:0013 ps) a band diagram again appears but now mode 2 (1) dominates where the mode 1 (2) was dominating at high feedback strengths. The diagram for low  does not appreciably change since optical feedback eects are negligible. Small changes in the external cavity length cause important changes in Jsw . For instance a variation of Lext of 0.06 m (from  = 26:0022 ps to  = 26:0026 ps) can change Jsw in 0.2 kA=cm for  = 0:0003 ps? . This change is of the order of the width of the bands in Fig. 9. Then, if Lext is not carefully controlled the polarization behaviour of the device can change drastically. As a consequence unwanted feedback can modify the emitted polarization. In fact, as seen from gure 9, very small unwanted feedback can modify the VCSEL polarization since the minimum feedback strength needed to change the polarization can be around  = 3:5  10? ps? , that corresponds to an external re ectivity of Rext = 2  10? %.

Figure 7.

2

1

5

1

External Cavity Length (mm)

6

Figure 8.

0:3 nm=mA.

3.904 3.903 3.902 3.901 3.900

2

4 6 8 2 Current Density (kA/cm )

10

Mechanism of control to avoid thermal detrimental eects in polarization selection.
=
I =

10

1

2

Current Density (kA/cm )

1 1 1

8

1

2

2

1

1

6

1 1 1 1

4 2

1

2

1

2

2

−4

10 −1 Feedback strength (ps )

−3

10

−4

10 −1 Feedback strength (ps )

−3

10

Eect of optical feedback strength on Jsw when  = 26:0028 ps (left), and when  = 26:0013 ps (right). The index indicates the polarization that dominates in each zone.
=
I = 0:3 nm=mA.

Figure 9.

5. SUMMARY AND CONCLUSIONS

In summary, the eect of weak optical feedback on the polarization properties of an external cavity VCSEL has been studied using numerical simulations. Ohmic heating eects on polarization selection of the fundamental transverse mode have been analysed. We have shown that optical feedback can be useful since selection of either polarization can be eected by the appropriate choice of the external cavity length. Selection of a particular polarization over wider current ranges is achieved as current dependent wavelength shift becomes smaller. When ohmic heating eects becomes important a controlled displacement of the external re ector with respect to the VCSEL can select the polarization over wider current ranges. The polarization selectivity has been shown to be dependent upon the strength of optical feedback and highly sensitive to small changes in optical feedback delay. This means that the external re ector position should remain xed to a tolerance level below a fraction of the optical wavelength if the objective is to avoid feedback induced polarization variations. Finally, we have shown that very small unwanted feedback can modify the VCSEL polarization.

ACKNOWLEDGEMENTS

The work of Dr. A. Valle and Dr. L. Pesquera was supported by CICYT project TIC95-0563-C05-01 and by Caja Cantabria. The work of K.A.Shore was partially supported by the UK EPSRC.

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