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Mode evolution in tapered couplers with slight systematic asymmetry
Optics Communications 96 ( 1993) 41-44 North-Holland
OPTICS COMMUNICATIONS
Mode evolution in tapered couplers with slight systematic asymmetry L. L e r n e r Optical Sciences Centre, Australian National University, Canberra ACT. 2601 Australia
Received 12 August 1992
We analysesinglecore excitation in a single-modetwin-coreweaklyguidingtapered couplerwith a slightsystematicasymmetry and show that although initially only one couplersupermodeis excited a second supermodeis subsequentlyexcitedover a certain range of core separations. We present a criterion which can be used to determine the degree of this excitation and consequently the mode evolution in the coupler.
1. Introduction In the standard analysis of weakly-guiding singlemode symmetric tapered couplers one assumes that single core excitation results in the generation of equal amounts of even and odd supermodes. However at the large core separations at which power is normally coupled into the core, the supermode of a coupler with even a very slight systematic asymmetry will be essentially that of an isolated core [ 1 ], and if that mode were to propagate adiabatically no interference would be observed in the coupler. Since an ideal symmetric coupler in which the second mode is assumed to be initially present has been observed to be a good approximation of experimental devices which might contain a systematic asymmetry, the question arises as to the conditions under which a second mode is generated in such devices and the parameters determining its generation. We show that couplers are not adiabatic over their entire length, so that power initially coupled into a supermode propagates in the tapered region until a non-adiabatic section of the coupler is reached and power is coupled out of the supermode. Unlike the case of the ideal symmetric coupler, tapering is essential to this mechanism [ 2 ], however for small core asymmetries and refractive index differences the transition is essentially the abrupt excitation problem of the symmetric coupler, where mode evolution
is not strongly dependent on the tapering angle, and differences from conventional coupled mode theory are largely unnoticed (although the influence of tapering angles has been observed in ref. [3] ). We demonstrate that these differences can become significant if the tapering angle of the fused coupler is sufficiently small over a particular range of core separations. The coupler becomes adiabatic (an insignificant amount of the second supermode is excited) for sufficiently large refractive index differences and core asymmetries and we provide an adiabaticity criterion which serves as a limit within which conventional coupled mode theory can be applied to the symmetric coupler.
2. Analysis The local supermodes of a coupler with a small asymmetry parameter (q = p ~ / P 2 - 1, where Pl and P2 are the respective core half-widths) can be approximated by [ 1 ] ~l +a~2
O + = ~ a=
a~l-~2
' 0-= x/l+a2,
fl~ -- fl2 +
1+
(1)
2C where ~ , fl, ~2, P2, are the normalized modes and
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. Allfights reserved.
41
Volume 96, number 1,2,3
OPTICS C O M M U N I C A T I O N S
propagation constants of the two isolated modes with fli > f12, ~+ and ~t_ are the even and odd modes of the coupler, and C is the coupling constant between the two isolated modes [ 1 ]. The approximation inherent in eq. ( l ) is a leading order asymptotic expression in the parameters r = f O l ~2 dA and q. We check the validity of this approximation for our analysis, which is most sensitive to the ratio of powers in (tl and ~2, by comparing the parameter a in eq. (1) with that obtained from the exact solution for slab cores. Figure l shows that the ratio of a for the exact even mode to that for the approximate even mode of eq. ( l ) is very close to unity over the entire range of core separations d and asymmetries q of interest here. We can thus confidently use the modes of eq. ( 1 ) in our analysis. Figure 2 shows the variation of the ~2 component of g)+ as a function of d for q = 0.0 I. This component undergoes a rapid change over a small range of d, consequently in this region approximately half of the power present in the initially excited core must
r 1.Ol
(a)
1.oo8
1 February 1993
Q 1
0.8
0.6
0.4
0.2
5
10
15
20
d/p Fig. 2. Variation of a in eq. ( 1 ) as a function of core for q=0.01 and V=0.8.
separation
transfer to the second core for the guide to be adiabatic. If the characteristic width (Ad) of this region is too small compared to the coupling length (Ad/O << l / C ) , where 0 is the local tapering angle of the coupler (in radians), this region cannot be adiabatic. Since Ad~ l(Op/Od) we introduce the parameter k = IO(Op/Od)/CI so that for an adiabatic section of the coupler we must have k<< l, where the fractional power in ~2 is given approximately by p=a2/( l + a 2 ) . Using eq. ( 1 ) we can write the adiabaticity criterion as
1.oo6
k(d)=
O(d) g Og fl~_fl2 (l+g2)3/20d'
fl, --fie g=
2----~
(2)
1.oo4 1.oo2
5
I0
15
d/p 1.014
(b)
1.012 r 1.oo8 1. 006 1.oo4 1.oo2
~
0 .01
0.015
0.02
q
Fig. 1. The ratio r, of parameter a for the exact even mode to that for the approximate even mode eq. ( 1 ) of a slab two-mode weaklyguiding coupler (a) as a function of core separation d for q = 0.01, and (b) as a function of the asymmetry parameter q for d = 10 (in units where the core half-width p = 1 ).
42
The criterion k is sufficient to determine non-adiabaticity in a section of the coupler. Support for our result is provided by a detailed calculation of the power transfer between the two lowest supermodes of a twin-core coupler P+_ (d) [4], for which kZ(d) provides the asymptotic variation. The adiabaticity criterion in eq. (2) is however more general than the analysis in ref. [ 4 ] as it applies to power transfer between the supermode and all other modes. It should be noted that the above adiabaticity criterion applies to power transfer between the supermodes of a coupler, and is not related to insertion losses due to the non-adiabaticity of an individual core. Since the two phenomena are not related they can be investigated separately. We can obtain a more specific result if we specialize to slab geometry. In this case Og/Od= ( W/p)g, and since q is small we can write fll--fl2=qU2W/ [tip2( 1 + W) ] so that eq. (2) becomes
Volume 96, number 1,2,3
k= O(d) tip( 1+ W) q
U2
OPTICS COMMUNICATIONS
g2
(1 +g2)3/2 ,
(3)
where we approximate fl and W by the average values over the two cores. Figure 3 shows that the factor containing g in eq. (3) varies rapidly with d and is strongly peaked. If we assume that 0 varies slowly in comparison, we find that the adiabaticity criterion attains its maximum at g ~ x / ~ , i.e. p 2v/2W din= ~ l o g ~ +2p,
1 February 1993
q = 0.3% and A = 0.003 we have 0= << 2 × 10-4 and it is almost impossible to achieve adiabaticity in such a coupler, its length would have to be greater than 2×10ap. However, an asymmetry parameter of q= 1%, with`4= 0.02, arid V=0.8 gives 0= << 2 × 10 -3, with a minimum length of 2.5 × 103p. Such a tapered coupler will not operate as an ideal symmetric coupler. Although eq. (3) was derived for slab geometry a calculation in cylindrical geometry gives
(4)
0
k=C(V,q,d) ~
g2
(1+g2)3/2,
(7)
and the value of k at the maximum is
k= Om2V(I+W) .4= n2°-n2~ q 3x/~U z ' 2n2oo '
(5)
where 0m is the value of the tilt angle at the peak maximum eq. (4). The peak width, defined by the value o f x at which k drops to half its maximum value is given approximately by
a=2.23p/OmW.
where C( V, q, d) depends weakly on q and d, thus the qualitative behaviour of the adiabaticity characteristic is the same as in slab geometry.
3. Conclusion
(6)
Thus the peak width is independent of the coupler asymmetry. For taper geometries where we cannot assume 0' (d) << g' (d) in the vicinity of the peak, eqs. ( 4 ) (6) depend on the specific form of the taper. A coupler parameter of V=0.8 gives k=0.78 Om/qv/-~, and since the tapering angle has to satisfy this only over the range given by eq. (5), the minimum length of a coupler satisfying k = 1 is of order 3.6p/qv/J. For a slightly asymmetric coupler with 1 E ~0.8
We have shown that the excitation of a single core in a sin#e-mode weakly guiding coupler with a slight systematic core asymmetry, will result in the propagation of one supermode in the tapered region until over a certain range o f core separations, centred on a value given by eq. (4) power is transfered to the second supermode. If coupler asymmetry is only slight, this region simulates the abrupt excitation of the ideal symmetric coupler. A criterion used to determine the adiabaticity of a coupler section shows that the tapering angle required to maintain an adiabatic regime is inversely proportional to the asymmetry parameter of the coupler (eq. (5)), thus in couplers with substantial asymmetry one can attain small amounts of power transfer to other modes and consequently little interference.
0.6 0.4
Acknowledgement
0.2 i
i
i
5
10
a/p
15
20
Fig. 3. Variation of the adiabaticity criterion (sealed to the peak maximum) with core separation d for q = 0.01 assuming 0 varies slowly in comparison.
I wish to thank J.D. Love. This work was carried out under a Generic Industrial Research and Development Grant, No. 17009, from the Department of Industry, Technology and Commerce. Permission to publish is acknowledged. 43
Volume 96, number 1,2,3
OPTICS COMMUNICATIONS
References [! ] A.W. Snyder and J.D. Love, Optical waveguide thoery (Chapman & Hall, London, 1983) p. 390. [2] J.D. Love and L. Lerner, Proc. 16th Australian Conf. on Optical Fibre Technology, Adelaide, 1991, p. 56.
44
1 February 1993
[ 3 ] T.A. Birks and C.D. Hussey, Optics Lett. 13 ( 1988 ) 681. [ 4 ] L. Lemer and J.D. Love, Coupling between the guiding modes of a single-mode tapered coupler with slight systematic asymmetry, J. Lightwave Technol., submitted.
OPTICS COMMUNICATIONS
Mode evolution in tapered couplers with slight systematic asymmetry L. L e r n e r Optical Sciences Centre, Australian National University, Canberra ACT. 2601 Australia
Received 12 August 1992
We analysesinglecore excitation in a single-modetwin-coreweaklyguidingtapered couplerwith a slightsystematicasymmetry and show that although initially only one couplersupermodeis excited a second supermodeis subsequentlyexcitedover a certain range of core separations. We present a criterion which can be used to determine the degree of this excitation and consequently the mode evolution in the coupler.
1. Introduction In the standard analysis of weakly-guiding singlemode symmetric tapered couplers one assumes that single core excitation results in the generation of equal amounts of even and odd supermodes. However at the large core separations at which power is normally coupled into the core, the supermode of a coupler with even a very slight systematic asymmetry will be essentially that of an isolated core [ 1 ], and if that mode were to propagate adiabatically no interference would be observed in the coupler. Since an ideal symmetric coupler in which the second mode is assumed to be initially present has been observed to be a good approximation of experimental devices which might contain a systematic asymmetry, the question arises as to the conditions under which a second mode is generated in such devices and the parameters determining its generation. We show that couplers are not adiabatic over their entire length, so that power initially coupled into a supermode propagates in the tapered region until a non-adiabatic section of the coupler is reached and power is coupled out of the supermode. Unlike the case of the ideal symmetric coupler, tapering is essential to this mechanism [ 2 ], however for small core asymmetries and refractive index differences the transition is essentially the abrupt excitation problem of the symmetric coupler, where mode evolution
is not strongly dependent on the tapering angle, and differences from conventional coupled mode theory are largely unnoticed (although the influence of tapering angles has been observed in ref. [3] ). We demonstrate that these differences can become significant if the tapering angle of the fused coupler is sufficiently small over a particular range of core separations. The coupler becomes adiabatic (an insignificant amount of the second supermode is excited) for sufficiently large refractive index differences and core asymmetries and we provide an adiabaticity criterion which serves as a limit within which conventional coupled mode theory can be applied to the symmetric coupler.
2. Analysis The local supermodes of a coupler with a small asymmetry parameter (q = p ~ / P 2 - 1, where Pl and P2 are the respective core half-widths) can be approximated by [ 1 ] ~l +a~2
O + = ~ a=
a~l-~2
' 0-= x/l+a2,
fl~ -- fl2 +
1+
(1)
2C where ~ , fl, ~2, P2, are the normalized modes and
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. Allfights reserved.
41
Volume 96, number 1,2,3
OPTICS C O M M U N I C A T I O N S
propagation constants of the two isolated modes with fli > f12, ~+ and ~t_ are the even and odd modes of the coupler, and C is the coupling constant between the two isolated modes [ 1 ]. The approximation inherent in eq. ( l ) is a leading order asymptotic expression in the parameters r = f O l ~2 dA and q. We check the validity of this approximation for our analysis, which is most sensitive to the ratio of powers in (tl and ~2, by comparing the parameter a in eq. (1) with that obtained from the exact solution for slab cores. Figure l shows that the ratio of a for the exact even mode to that for the approximate even mode of eq. ( l ) is very close to unity over the entire range of core separations d and asymmetries q of interest here. We can thus confidently use the modes of eq. ( 1 ) in our analysis. Figure 2 shows the variation of the ~2 component of g)+ as a function of d for q = 0.0 I. This component undergoes a rapid change over a small range of d, consequently in this region approximately half of the power present in the initially excited core must
r 1.Ol
(a)
1.oo8
1 February 1993
Q 1
0.8
0.6
0.4
0.2
5
10
15
20
d/p Fig. 2. Variation of a in eq. ( 1 ) as a function of core for q=0.01 and V=0.8.
separation
transfer to the second core for the guide to be adiabatic. If the characteristic width (Ad) of this region is too small compared to the coupling length (Ad/O << l / C ) , where 0 is the local tapering angle of the coupler (in radians), this region cannot be adiabatic. Since Ad~ l(Op/Od) we introduce the parameter k = IO(Op/Od)/CI so that for an adiabatic section of the coupler we must have k<< l, where the fractional power in ~2 is given approximately by p=a2/( l + a 2 ) . Using eq. ( 1 ) we can write the adiabaticity criterion as
1.oo6
k(d)=
O(d) g Og fl~_fl2 (l+g2)3/20d'
fl, --fie g=
2----~
(2)
1.oo4 1.oo2
5
I0
15
d/p 1.014
(b)
1.012 r 1.oo8 1. 006 1.oo4 1.oo2
~
0 .01
0.015
0.02
q
Fig. 1. The ratio r, of parameter a for the exact even mode to that for the approximate even mode eq. ( 1 ) of a slab two-mode weaklyguiding coupler (a) as a function of core separation d for q = 0.01, and (b) as a function of the asymmetry parameter q for d = 10 (in units where the core half-width p = 1 ).
42
The criterion k is sufficient to determine non-adiabaticity in a section of the coupler. Support for our result is provided by a detailed calculation of the power transfer between the two lowest supermodes of a twin-core coupler P+_ (d) [4], for which kZ(d) provides the asymptotic variation. The adiabaticity criterion in eq. (2) is however more general than the analysis in ref. [ 4 ] as it applies to power transfer between the supermode and all other modes. It should be noted that the above adiabaticity criterion applies to power transfer between the supermodes of a coupler, and is not related to insertion losses due to the non-adiabaticity of an individual core. Since the two phenomena are not related they can be investigated separately. We can obtain a more specific result if we specialize to slab geometry. In this case Og/Od= ( W/p)g, and since q is small we can write fll--fl2=qU2W/ [tip2( 1 + W) ] so that eq. (2) becomes
Volume 96, number 1,2,3
k= O(d) tip( 1+ W) q
U2
OPTICS COMMUNICATIONS
g2
(1 +g2)3/2 ,
(3)
where we approximate fl and W by the average values over the two cores. Figure 3 shows that the factor containing g in eq. (3) varies rapidly with d and is strongly peaked. If we assume that 0 varies slowly in comparison, we find that the adiabaticity criterion attains its maximum at g ~ x / ~ , i.e. p 2v/2W din= ~ l o g ~ +2p,
1 February 1993
q = 0.3% and A = 0.003 we have 0= << 2 × 10-4 and it is almost impossible to achieve adiabaticity in such a coupler, its length would have to be greater than 2×10ap. However, an asymmetry parameter of q= 1%, with`4= 0.02, arid V=0.8 gives 0= << 2 × 10 -3, with a minimum length of 2.5 × 103p. Such a tapered coupler will not operate as an ideal symmetric coupler. Although eq. (3) was derived for slab geometry a calculation in cylindrical geometry gives
(4)
0
k=C(V,q,d) ~
g2
(1+g2)3/2,
(7)
and the value of k at the maximum is
k= Om2V(I+W) .4= n2°-n2~ q 3x/~U z ' 2n2oo '
(5)
where 0m is the value of the tilt angle at the peak maximum eq. (4). The peak width, defined by the value o f x at which k drops to half its maximum value is given approximately by
a=2.23p/OmW.
where C( V, q, d) depends weakly on q and d, thus the qualitative behaviour of the adiabaticity characteristic is the same as in slab geometry.
3. Conclusion
(6)
Thus the peak width is independent of the coupler asymmetry. For taper geometries where we cannot assume 0' (d) << g' (d) in the vicinity of the peak, eqs. ( 4 ) (6) depend on the specific form of the taper. A coupler parameter of V=0.8 gives k=0.78 Om/qv/-~, and since the tapering angle has to satisfy this only over the range given by eq. (5), the minimum length of a coupler satisfying k = 1 is of order 3.6p/qv/J. For a slightly asymmetric coupler with 1 E ~0.8
We have shown that the excitation of a single core in a sin#e-mode weakly guiding coupler with a slight systematic core asymmetry, will result in the propagation of one supermode in the tapered region until over a certain range o f core separations, centred on a value given by eq. (4) power is transfered to the second supermode. If coupler asymmetry is only slight, this region simulates the abrupt excitation of the ideal symmetric coupler. A criterion used to determine the adiabaticity of a coupler section shows that the tapering angle required to maintain an adiabatic regime is inversely proportional to the asymmetry parameter of the coupler (eq. (5)), thus in couplers with substantial asymmetry one can attain small amounts of power transfer to other modes and consequently little interference.
0.6 0.4
Acknowledgement
0.2 i
i
i
5
10
a/p
15
20
Fig. 3. Variation of the adiabaticity criterion (sealed to the peak maximum) with core separation d for q = 0.01 assuming 0 varies slowly in comparison.
I wish to thank J.D. Love. This work was carried out under a Generic Industrial Research and Development Grant, No. 17009, from the Department of Industry, Technology and Commerce. Permission to publish is acknowledged. 43
Volume 96, number 1,2,3
OPTICS COMMUNICATIONS
References [! ] A.W. Snyder and J.D. Love, Optical waveguide thoery (Chapman & Hall, London, 1983) p. 390. [2] J.D. Love and L. Lerner, Proc. 16th Australian Conf. on Optical Fibre Technology, Adelaide, 1991, p. 56.
44
1 February 1993
[ 3 ] T.A. Birks and C.D. Hussey, Optics Lett. 13 ( 1988 ) 681. [ 4 ] L. Lemer and J.D. Love, Coupling between the guiding modes of a single-mode tapered coupler with slight systematic asymmetry, J. Lightwave Technol., submitted.