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A A The electric field vector in the second core is proportional to (a2 ocJb2) \|/2 and

In document Optical fibre couplers (Page 181-191)

BIREFRINGENT COUPLERS

A A The electric field vector in the second core is proportional to (a2 ocJb2) \|/2 and

is polarized at a corresponding angle to the a2 axis. This vector is simply a reflection through the x or y symmetry axes of the electric field vector in core 1. Between the two cores, the direction and amplitude of the electric field vector, which depends on the position (x,y), can be deduced from (5-12) when the unperturbed field amplitude functions, \jj?x and \|/2 , are known. However, knowledge of the electric field between the cores is not crucial to a physical understanding of birefringent couplers, as most power propagates in the core regions.

Fig. 5.3 shows schematically the electric field vector directions in each core for an arbitrary non-zero tilt angle 9, for all four modes. Three distinct situations are covered : Q « 1 (nearly isotropic coupler), Q = 1 (intermediate case), and Q » 1 (highly birefringent coupler). The angle between the vector and the a axis in each core comes from (5-21) and (5-17). In the limiting cases, Q = 0 and Q = *=, the modal fields would be polarized exactly parallel to the (x,y) axes or the (am,bm) axes (m = 1, 2),respectively, and when Q = 1, the field directions lie exactly midway between these limits. (This last fact can be shown by putting Q = 1 into (5-17) to find = Arctan (9/4) , with similar expressions for (j)j(j=2,3,4).) Except in the limiting cases, the modal electric fields are not all polarized parallel or at right angles to each other, generally, leading to interesting consequences as we show below.

1 7 0

Fig. 5.4 : Propagation constants, ßj , for each normal mode (j = 1,...,4) of the birefringent coupler, plotted against Q = ( ßa - ßb )/2C, for couplers having tilt angles 0 between the optical axes of (a) 0°, (b) 45° , (c) 90° . For simplicity, the graph is plotted around <ß> = (ßa + ßb )/2 on the horizontal axis.

The corresponding modal propagation constants are plotted against Q in Fig. 5.4, for three situations: 9 = 0°, (parallel core optical axes), 9 = 45° and 9 = 90° (perpendicular core optical axes). For plotting convenience, we let <ß> = (ßa + ßb )/2 remain constant as Q varies. The numbering (j=l,...,4) for each mode is based on the propagation constants, so that ßi > ß2 > ß3 > ß4-

Clearly, in the limit Q = 0, the propagation constants reduce to the well-known isotropic coupler values, given in (2-44). For large Q, ßj tend asymptotically towards ßa ± C cos9 or ßb ± C cos9 ,which is consistent with modes polarized parallel to the optical axes in each core. The modes of the highly birefringent coupler are analogous to the isotropic coupler modes, therefore, being odd and even modes polarized parallel to am or bm , but the coupling constant C in (2-44) is effectively replaced by C cos9 in this limit.

In Table 5.1, we have listed, for each mode, the general expressions for the electric field ej(x,y) from (5-12), aj(Q,9) from (5-17) and <|>j (Q,9) from (5-18). In Tables 5.II and 5.IV, we give the asymptotic expressions for ßj( 9 ) otj( 9 ) and <{>j( 9 ), respectively, (j=l,...,4), for the limits Q = 0, Q = 1, and Q °° . The left hand columns of Tables 5.II - 5.IV apply to those couplers which are intended to be isotropic, but have had some anisotropy introduced to their cores during fabrication. The right hand columns apply to couplers designed to hold polarization in coherent systems, but are not anisotropic enough to cause any polarization selectivity or splitting at the output ports. The middle columns give representative results for the intermediate couplers, in which the most pronounced depolarization effects can occur if the optical axes are not parallel.

A closer examination of cxj (Q,9 ) shows that, for all values of Q and 9 ,

a ia 4 = a 2a 3 = (5-22)

implying that the field vectors of modes 1 and 4 are perpendicular to each other in the cores, and likewise for the field vectors of modes 2 and 3. Thus, we need only two independent angles to specify the core field directions of all four modes. Fig. 5.5 shows the relationship between the angles and and the electric field polarization vectors of the four modes in each core, in the special case: Q = 1 and 9 = 45°. In Fig. 5.6, we plot (j>x and ({>3 against Q for three different values of 9. W henQ = 0°, <t>i = <|>3 = 9/2, and as Q increases, -» 0° while <J>3 -» 90°. At Q = 1 exactly, the angles and <|>3 are 9/4 and 45° + 9 /4, respectively, midway between their limiting values at Q = 0 and Q = <».

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a I as 2 £ 3

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§ 1 2 1 S' S' O ' S' £ £ £ + 1 1 -f— + <D05 CD <D05 CD O O O O CJ CJ CJ 1 + 1 o i +* . * 1 CQ l a ? l a ? I CQ 11 CN1 ■ « 1 CC CQ CQ CQ 0 ? 0 ? i * <D o CD 05 cD 05 <D05 a o U CN CN :N CM II CJ CJ y CJ a X CQ XCQ XCQ ACQ II "c s II ^ « CQ CQ CQ CQ O o 3 o T— v ¥ O'1 CJ T + a CJ X CJ X y X y A CQ CQ CQ II % 11^ II CQ CQ CQ CQ

1 7 6

Fig. 5.5 : Relative orientations of the polarization vectors for each mode in each core with respect to the optical axes ^ ,a2 , , b2 . The orientations of all eight vectors may be specified by two angles fa (Q,0) and fa (Q, 0).

Fig. 5.6 : Modal polarization angles as functions of Q = ( ßa - ßb )/2C. Plotted are <)>, (dashed) and <(>3 (solid) against Q for the cases (a) 9 = 90”, (b) 0 = 45”, (c) 0 = 2” . Note that, at Q = 0, <j>i = d>3 = 0/2, and, as Q 4 - , 6, * 0” while <t>3 * jt /2.

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C

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5.4 PROPAGATING FIELDS AND POWER TRANSFER 5.4.0 Field Launched into the Coupler

W hen light is launched into one o f the fibres leading into a polished-type evanescent coupler, the field entering the coupling region itself will excite a combination of the coupler m odes, including higher order and radiation m odes. W hen the fibres com prising the coupler are single-m oded in isolation from each other, the higher order fields all propagate in the cladding (which is modeled as being infinite, here) and after any appreciable distance along the coupler, the only power propagating in the core is due to the presence of the four low est order coupler modes. We may therefore ignore higher-order modes here and write our plane-polarized input field as:

eQ(x,y) = [äj cos% + sin%] Y jfoy) (5-23)

where % is the angle between the input polarization vector and the zli axis.

Each normal modal field of the coupler can be expressed in terms o f the angles <j>j(Q,0) (j = 1...4), calculated from (5-17) and (5-21):

ej(x,y) = [äj cos<J)1 + sin^J ^ (x ,y ) + co s^ - &2 sin ^ ] y 2(x,y)

with similar expressions for e2 ,e3 , and e4 .

(5-24)

5.4.1 Excitation of the Four Normal Modes

G enerally, the input field excites all four normal modes of the coupler, and the resultant field at a distance z along the coupler is:

V"1 ;2 V '

E(x,y,z) = 2 , A. ej(x,y) e = ^ AE.(x,y,z) (5-25)

j=i j-i

the norm al mode angles <j)j (j= l,...,4). To determ ine the co-efficients, Aj ,we use the orthogonality relationship in (5-6) and we scale the functions \j?i(x,y) and \j72(x,y) s0 that:

J

y 1(x,y)dA =

J

\|/^(x,y) dA = 1 (5-26)

\ o Ao

Thus, from (5-25), (5-26) and (5-6):

J e . e;

In document Optical fibre couplers (Page 181-191)