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The polarisation behaviour of second-order modes in a circular core fibre due to Kerr nonlinearity
Optics Communications North-Holland
OPTICS COMMUNICATIONS
94 ( 1992) 373-378
The polarisation behaviour of second-order modes in a circular core fibre due to Kerr nonlinearity W.
Samir and S.J. Garth
Department of Mathematics, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT 2606, Australia Received
29 April 1992
We consider the effect of Kerr nonlinearity on the polarisation behaviour of a non-birefiingent, circular core optical tibre supporting the fundamental mode and the proper second-order modes. When the fundamental mode is absent, previous work has revealed the interesting phenomena of polarisation rotation modulation of an initially linearly polarised (LP) secondorder “mode”. This LP mode is actually a linear combination of the proper second-order modes in circular core fibres. The presence of the fundamental mode, however, gives rise to a more complex behaviour and destroys the presence of a steady state condition in which the relative phase between the modes remains constant at a critical power level. Since it is more practical to have some power excited in the fundamental mode, it is important to discuss its effect on the polarisation properties of circular core nonlinear bi-modal tibres, which may have device applications as polarisation switches.
1. Introduction Few mode tibres are finding increasing application as nonlinear optical libre devices. In particular, there has recently been a considerable interest in polarisation or intensity switching in bi-modal fibres [ 1- 111. The aim of this paper is to explore polarisation modulation on nonlinear circular core libres that support the second order mode set. There are two categories of bi-model fibres. The first refers to single mode birefringent fibres, which actually support two orthogonally polarized fundamental modes. The second category refers to few mode fibres that operate in the region that supports the fundamental mode set and the second order mode set. These modes have different properties depending on whether the libre has a noncircular or circular core. In a noncircular bi-modal fibre there are actually six modes present: the fundamental mode and the linearly polarised (LP) odd and even second order modes, and each mode is non-degenerate with respect to x and y polarisation states. In the circular core Iibre there are live modes: the fundamental mode and four proper second order modes, desig0030-4018/92/$05.00
0 1992 Elsevier
Science Publishers
nated by even and odd HE*,, TEol and TMo,. In this case the second order modes have slightly different propagation constants, and as the modes propagate they beat together resulting in a rotation of the initial polarisation state [ 12,13 1. This paper is concerned with nonlinear effects in the latter case. Polarisation rotation in nonlinear birefringent fibres has been extensively analysed [ 2-5 1. At low powers the natural polarisation state is elliptical, and at high powers the nonlinear interaction induces ellipse rotation. The intensity dependent refractive index gives rise to instabilities in the polarisation state
[4,51. Elliptical core fibres that support the second order LPll modes have been investigated as possible interferometers based on nonlinear phase modulation [ 6-8 1. The phase modulation is observed as a change in intensity of the output radiation pattern as the first and second order modes interfere. For sufficient power, the odd and even LP,, modes can couple power between themselves [ 9 1. There have only been limited investigations into the distinctive waveguide effect of polarisation rotation of the second-order modes on circular core nonlinear bi-modal libres [ 10,111. When nonlinear
B.V. All rights reserved.
373
Volume 94, number 5
OPTICS COMMUNICATIONS
effects become important the polarisation rotation is modulated, and the possibility of new polarisation switching devices based on the proper second-order modes arises. In ref. [ lo] the modulation was achieved by injecting an intense fundamental mode pump wave into a circular core bi-modal fibre operating at a signal wavelength different to that of the pump. Reference [ 111 considered the self-phase modulation case in which the pump beam was launched into the secondorder modes, where it was shown that for intense enough light the polarisation rotation could be suppressed and a truly linearly polarised mode would propagate. Furthermore, there was a critical or transition point at which the second-order modes evolved into the proper TEol mode and remained in that state. This fascinating polarisation behaviour was, however, investigated analytically by ignoring the effect of the fundamental mode. In almost any real situation some power will be excited in the fundamental mode and then nonlinear interactions between this mode and the second-order modes must be considered. The aim of this paper is to look at the more general case where the fundamental (HE, ,) mode and the proper second-order modes (odd HEzi and TEol) are excited. We stress that birefringence is not involved in this problem; the polarisation rotation is due solely to the intricate waveguide effect of launching proper second-order modes on perfectly circular core libres. Since the difference in propagation constants between the fundamental and second-order modes is large compared to the propagation difference between the second-order mode set, power exchange between the two sets will not occur to any large degree. However, the fundamental mode will still modulate the phase difference between the second-order mode set. This is a general result and is discussed in ref. [9]. In sect. 2 we derive the coupled power equations that describe the nonlinear interaction between the fundamental and proper second-order modes. Section 3 gives numerical results, where we show that the fundamental mode changes the critical point behaviour as well as the polarisation rotation behaviour discussed in ref. [ 111. Section 4 is a discussion 374
1 December 1992
of our results and sect 5 forms a conclusion.
2. Theory If a single mode libre is operated above its cutoff value ( VC> 2.4), the second order modes propagate. Because of the circular nature of the core-cladding boundary, these modes are the proper circular modes TMol, HE,, (odd and even) and TEe, modes. If initially x-polarised light moved off-axis in the y-direction or at angle to the y-axis is launched into the fibre, then three modes will be excited: the (fundamental) HE,,, the odd HEzl and the TE,,, modes (see ref. [ 111 for a discussion on initial launching conditions). We designate these modes by E,,, E, and E2 respectively, and they are given by [ 12,13 1. E. = $4,-,(z)tyo(r)
exp(&z)
6 =Ql(z)vI(r)
exp(iPlz)
Xexp( -iot) & = ~&(z)wz(Y) xexp(
-iwt)
(sin BP+cos
exp( -iot)
f ,
133) ,
ew(&z) (sin of-cos
0j)
,
(1)
where I,v~(r) and vi (r) = I,&(r) are the transverse (radial) fields of the fundamental and second-order modes, respectively; and A,(z) and PJ o’= 0, 1,2 ) are the modal amplitudes and propagation constants of the fundamental, odd HE*, and TE,,, modes, respectively. We define the total field E as: E= t
(E,+c.c.)
,
j=O
where C.C. indicates complex conjugate. Substituting into Maxwell’s equations and using the standard slowly varying approximation we obtain (3)
,~08,~w,eXP[i(P,z-wwt)]=-i14r~,
where PNL is the third-order quency w PNL=~o~[2(E-E*)E+
polarisation
(E*E)E*]
for fre-
(4)
and x, to and p. are the third-order susceptibility and free space electric and magnetic permeabilities. We assume no power exchange between the fun-
Volume 94, number
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5
damental mode and the the large difference in Following the procedure the normalised coupled d W/dZ=O
second-order modes due to propagation constants [ 91. in ref. [ 111 we thus obtain power and phase equations:
,
1992
Note that as R=O, this gives W=O, i.e. there is no power in the fundamental mode, and eqs. (5) and ( 11) reduce to the case in ref. [ Ill. The normalised power in the x- and y-polarised parts of the field are given by: P,/P=~+~W+Jm0s(~),
dV/dZ=-Q[2UVsin(2@)+2W@sin(@)],
PJP=
dU/dZ=Q[2UVsin(2@)+2Wmsin(@)], do/dZ=
1 December
COMMUNICATIONS
f - 4 w-JizVcos(@)
.
(13)
1- ;Q( U- V)
x{[1+2cos(2~)]+(2W/JUv)cos~},
(5) 3. Numerical results
where Z=Bz,
Q=P/B
.
The normalised W=P,/P,
(6)
powers are given by
V=P,/P,
U=P,/P,
(7)
where m P,=(Cton,)2]Aj]2
s
VjdS
(8)
0
is the power in mode j, and P= PO+ PI + P2 is the total power. The integration is over the infinite crosssection of the fibre. B is the propagation difference between the odd HE2i and TEol modes B=l& -PI
(9)
and @ is the relative @=Bz+q&,,
.
The constants are W(Z)=
phase difference
1111. When power is excited in the fundamental mode, as described here, the critical point is affected. The transition (critical) point is now given by
(10) of the motion that arise from eq. (5)
W(0) >
US V= 1 - W= C ,
(a constant)
,
4UV[1+2cos(2@)]+16W,/&os(~) +(4/Q)(U-V)=8C-X2,
(11)
where the initial conditions U( 0) = V( 0) = C/2 have been used to calculate the rhs of eq. ( 11)) and
CCL 1+R’
In the case where there is no power excited in the fundamental mode (i.e. W=O, R=O, C= 1 in our equations), there exists a critical power ( Qcrit= $ ) where the phase steadily increases (d@/dz> 0) for powers below the critical point and oscillates around zero with diminishing amplitude for powers above the critical point. At the critical power, however, the phase reaches a steady state and is locked at += 0.66 rad where U is locked at unity, Vat zero and P, and Py are locked at $P each. At this critical point, the steady state situation is achieved where the true mode of the system is the TEo, mode. The system of equations for this case has a complete analytical solution
R=z
U+V’
(12)
R defines the ratio of the power in the fundamental mode to the power in the second-order modes.
Qcti,=;+&.
(14)
Equation ( 14) is found by substituting V= 0 (and hence U= C) and eq. ( 12) in eq. ( 11). In this case there is no longer a steady state situation where the phase is locked to a constant value. This is due to the fact that U and V continue to exchange power, even when there is zero power in V. By contrast, when the fundamental mode is not present, when the power in V goes to zero it remains at zero (and U remains at unity). Power exchange no longer occurs and the phase is locked to a constant value [ 111. This is because all higher derivatives of V vanish when V= 0. When the fundamental mode is present, however, the higher derivatives of eq. (5) do not all vanish when V= 0, and mode U is able to couple energy back into 375
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OPTICS COMMUNICATIONS
mode V. The coupling process continues, and no steady state situation is obtained. The behaviour of the period at the critical point is altered by introducing power in the fundamental mode. Figure 1 gives a plot of period 2, versus Q( =P/B) for different ratios R. It is clear that the peak in 2, no longer approaches infinity (steady state) as R becomes non-zero. As R is increased, the peak is smoothed out and finally vanishes. A complete and clear picture of the behaviour of the modal powers and the polarisations can be seen from the phase-portraits. In figs. 2-4, we explore the phase-portrait of V( = P, /P) versus @and PJP versus @for three different levels of power excited in the fundamental mode. Figure 2 depicts the case where there is no power in the fundamental mode, i.e., R = 0 which is the case analysed in detail in ref. [ 111. From fig. 2a we see how at the critical point, QCtit= 2, the power is locked
R=O (no fundamental
mode1
1 December
at zero while the phase is locked at 0.66 radians. Below the critical point, Q< 2, the power oscillates as the phase is increasing monotonically. Above the critical point, Q> {, the power oscillates around 0.5 with diminishing amplitude while the phase oscillates around zero with diminishing amplitude also. Figure 2b shows the polarisation rotation behaviour. At the critical point, the polarised powers are locked at 0.5. Before the critical point, we see the complete switching of power to the other cross polarisation. In other words, light that is initially x-polarised has rotated by 90” to become completely y-polarised. At very large nonlinearity, however, the initial x-polarisation state becomes the steady state solution. Figure 3 depicts the situation when 0.1% of the total power is excited in the fundamental mode, i.e., R=O.OOl. It is clear that before and after the transition point not much change has occurred compared to the case where R = 0. However, at the transition point, the steady state is destroyed. Figure 4 gives the case where R is increased to 0.1, that is 10%
1
0
0
J
0.0
0.5
1.o
1.5
2.0
2.5
I
3.0
Q = PIB Fig. 1. Period Z, versus Q( = P/B), for different
1992
Fig. 3. (a) Normalised modal power V( = PI/P) versus relative phase (8 and (b) P,,,/P versus 4; for R = 0.00 1 and a range of Q
ratios R. 1.c
O!
1.c
/ P,/P
V
P,/P
0.t 0.1
i
2x
Fig. 2. (a) Normalised modal power V( = PI /P) versus relative phase @and (b ) P&P versus @; for R = 0 and a range of Q values.
376
Fig. 4. (a) Normalised modal power V( =P, /P) versus relative phase Q, and (b) P&P versus @; for R=O.l and a range of Q values.
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of the power is excited in the fundamental mode. In this case we see a more exaggerated picture of that depicted in fig. 3. It is clear though, that at $J= II the modal power V is no longer able to couple enough power back to regain its initial state, as in the previous cases.
4. Discussion We now discuss the effect of the power in the fundamental mode on the power distribution in the different polarisations. By exciting a negligible amount of power in the fundamental mode, say R= 0.001, i.e. 0.1% of the total power, the phase and power behaviour do not change before and after the transition with respect to the R = 0 case; however, at the transition we no longer get the TEo, mode as the steady state mode. Also, as shown in tig. 3b, at the transition PJP and PJP are no longer locked at f , but have a flat region where they are equal to t and then continue to oscillate periodically. Hence, by choosing a proper tibre length over the flat region we can still get a 50-50 polarisation split. By exciting a moderate amount of power in the fundamental mode (RG 0.1)) we obviously no longer get a 100% polarisation splitting as shown in fig. 4b. We no longer get a 90” rotation (i.e. completely ypolarised light) at half the beat length Z=Bz= K, but instead we get elliptically polarised light. After a beat length we completely get the x-polarised light back as expected. As the power increases beyond the transition point, we get similar behaviour to the case where R = 0.At very large powers, the Iibre acts like a birefringent libre as in ref. [ 111.
5. Conclusions In this paper we have considered the effect of Kerr nonlinearity on the polarisation behaviour of a circular core optical fibre that supports the proper second-order modes. Previous work considered the case where the fundamental mode was absent [ 111. From a practical point of view, however, it is impossible to excite the higher order modes without exciting some fundamental mode. Hence, it should be of
1992
practical interest to see how the fundamental mode alters the polarisation rotation modulation discussed in the previous paper. By introducing the smallest percentage of fundamental mode, the critical point is destroyed in the sense that the relative phase @is no longer locked to a constant value as in the case when no fundamental mode was present. However, a transition point was found where the phase changes its state from a monotonically increasing to a periodically oscillating function. We have shown the behaviour of the period of the polarisation rotation as a function of the amount of the fundamental mode excitation; this clearly demonstrates how the critical point behaviour is suppressed. From a practical sense, when a polarisation switch is to be designed using bi-modal fibres, the amount of power excited in the fundamental mode becomes important and needs to be minimized. In this case, the experimentalist will need to decide on the angle of incidence which will excite as negligible amount of power in the fundamental mode as is practically possible. Although a mode converter [ 14,15 ] could be used to selectively excite the second-order modes, and a complete rotation of x-polarised light to y-polarised light may be observed, we have found that the smallest percentage of light in the fundamental mode will destroy the steady state TE,, mode behaviour described in ref. [ 111.
Acknowledgments The authors discussions.
thank Prof. C. Pask for many useful
References [ 1] R.J. Black, A. Henault, [2] [3] [4] [5] [6]
S. Lacroix and M. Cada, IEEE J. Quantum Electron. 26 ( 1990) 108 1. H.G. Winful,Appl. Phys. Lett. 47 (1985) 213. B. Crosignaniand P. Di Porto, Optica Acta 32 (1985) 1251. S. Trillo, S. Wabnitz, R.H. Stolen, G. Assanto, C.T. Seaton and G.I. Stegeman, Appl. Phys. L-ett. 49 ( 1986) 1224. H.G. Winful, Optics Lett. 11 (1986) 33. H.J. Park, C.C. Pohalski and B.Y. Rim, Optics Lett. 9 (1988) 776.
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[7] R.J. Black, A. Henault, L. Gagnon, F. Gauthier and S. Lacroix, Topical Meetings on Nonlinear guided wave phenomena; Physics and Applications, 1989 Technical Digest Series 2 (Optical Society of America, Washington DC.) pp. 66-69. [ 81 R.V. Plenty, I.H. White and A.R.L. Ravis, Electron Lett. 24 (1988) 1338. [9] S.J. Garth and C. Pask, J. Opt. Sot. Am. B 9 (1992) 243. [lo] S.J. Garth and C. Pask, Electron Lett. 25 (1989) 182.
378
1 December
1992
[ 111 S.J. Garth and C. Pask, Lightwave Technol. 8 (1990) 129. [ 121 A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, New York, 1983 ) [ 131 A.W. Snyderand W.R. Young, J. Opt. Sot. Am. B 68 (1978) 297. [ 141 R.C. Youngquist, J.L. Brooks and H.J. Shaw, Optics Lett. 9 (1984) 177. [ 151 W.V.Sorin,B.Y.KimandH.J.Shaw,OpticsLett. 11 (1986) 581.
OPTICS COMMUNICATIONS
94 ( 1992) 373-378
The polarisation behaviour of second-order modes in a circular core fibre due to Kerr nonlinearity W.
Samir and S.J. Garth
Department of Mathematics, University College, University of New South Wales, Australian Defence Force Academy, Canberra ACT 2606, Australia Received
29 April 1992
We consider the effect of Kerr nonlinearity on the polarisation behaviour of a non-birefiingent, circular core optical tibre supporting the fundamental mode and the proper second-order modes. When the fundamental mode is absent, previous work has revealed the interesting phenomena of polarisation rotation modulation of an initially linearly polarised (LP) secondorder “mode”. This LP mode is actually a linear combination of the proper second-order modes in circular core fibres. The presence of the fundamental mode, however, gives rise to a more complex behaviour and destroys the presence of a steady state condition in which the relative phase between the modes remains constant at a critical power level. Since it is more practical to have some power excited in the fundamental mode, it is important to discuss its effect on the polarisation properties of circular core nonlinear bi-modal tibres, which may have device applications as polarisation switches.
1. Introduction Few mode tibres are finding increasing application as nonlinear optical libre devices. In particular, there has recently been a considerable interest in polarisation or intensity switching in bi-modal fibres [ 1- 111. The aim of this paper is to explore polarisation modulation on nonlinear circular core libres that support the second order mode set. There are two categories of bi-model fibres. The first refers to single mode birefringent fibres, which actually support two orthogonally polarized fundamental modes. The second category refers to few mode fibres that operate in the region that supports the fundamental mode set and the second order mode set. These modes have different properties depending on whether the libre has a noncircular or circular core. In a noncircular bi-modal fibre there are actually six modes present: the fundamental mode and the linearly polarised (LP) odd and even second order modes, and each mode is non-degenerate with respect to x and y polarisation states. In the circular core Iibre there are live modes: the fundamental mode and four proper second order modes, desig0030-4018/92/$05.00
0 1992 Elsevier
Science Publishers
nated by even and odd HE*,, TEol and TMo,. In this case the second order modes have slightly different propagation constants, and as the modes propagate they beat together resulting in a rotation of the initial polarisation state [ 12,13 1. This paper is concerned with nonlinear effects in the latter case. Polarisation rotation in nonlinear birefringent fibres has been extensively analysed [ 2-5 1. At low powers the natural polarisation state is elliptical, and at high powers the nonlinear interaction induces ellipse rotation. The intensity dependent refractive index gives rise to instabilities in the polarisation state
[4,51. Elliptical core fibres that support the second order LPll modes have been investigated as possible interferometers based on nonlinear phase modulation [ 6-8 1. The phase modulation is observed as a change in intensity of the output radiation pattern as the first and second order modes interfere. For sufficient power, the odd and even LP,, modes can couple power between themselves [ 9 1. There have only been limited investigations into the distinctive waveguide effect of polarisation rotation of the second-order modes on circular core nonlinear bi-modal libres [ 10,111. When nonlinear
B.V. All rights reserved.
373
Volume 94, number 5
OPTICS COMMUNICATIONS
effects become important the polarisation rotation is modulated, and the possibility of new polarisation switching devices based on the proper second-order modes arises. In ref. [ lo] the modulation was achieved by injecting an intense fundamental mode pump wave into a circular core bi-modal fibre operating at a signal wavelength different to that of the pump. Reference [ 111 considered the self-phase modulation case in which the pump beam was launched into the secondorder modes, where it was shown that for intense enough light the polarisation rotation could be suppressed and a truly linearly polarised mode would propagate. Furthermore, there was a critical or transition point at which the second-order modes evolved into the proper TEol mode and remained in that state. This fascinating polarisation behaviour was, however, investigated analytically by ignoring the effect of the fundamental mode. In almost any real situation some power will be excited in the fundamental mode and then nonlinear interactions between this mode and the second-order modes must be considered. The aim of this paper is to look at the more general case where the fundamental (HE, ,) mode and the proper second-order modes (odd HEzi and TEol) are excited. We stress that birefringence is not involved in this problem; the polarisation rotation is due solely to the intricate waveguide effect of launching proper second-order modes on perfectly circular core libres. Since the difference in propagation constants between the fundamental and second-order modes is large compared to the propagation difference between the second-order mode set, power exchange between the two sets will not occur to any large degree. However, the fundamental mode will still modulate the phase difference between the second-order mode set. This is a general result and is discussed in ref. [9]. In sect. 2 we derive the coupled power equations that describe the nonlinear interaction between the fundamental and proper second-order modes. Section 3 gives numerical results, where we show that the fundamental mode changes the critical point behaviour as well as the polarisation rotation behaviour discussed in ref. [ 111. Section 4 is a discussion 374
1 December 1992
of our results and sect 5 forms a conclusion.
2. Theory If a single mode libre is operated above its cutoff value ( VC> 2.4), the second order modes propagate. Because of the circular nature of the core-cladding boundary, these modes are the proper circular modes TMol, HE,, (odd and even) and TEe, modes. If initially x-polarised light moved off-axis in the y-direction or at angle to the y-axis is launched into the fibre, then three modes will be excited: the (fundamental) HE,,, the odd HEzl and the TE,,, modes (see ref. [ 111 for a discussion on initial launching conditions). We designate these modes by E,,, E, and E2 respectively, and they are given by [ 12,13 1. E. = $4,-,(z)tyo(r)
exp(&z)
6 =Ql(z)vI(r)
exp(iPlz)
Xexp( -iot) & = ~&(z)wz(Y) xexp(
-iwt)
(sin BP+cos
exp( -iot)
f ,
133) ,
ew(&z) (sin of-cos
0j)
,
(1)
where I,v~(r) and vi (r) = I,&(r) are the transverse (radial) fields of the fundamental and second-order modes, respectively; and A,(z) and PJ o’= 0, 1,2 ) are the modal amplitudes and propagation constants of the fundamental, odd HE*, and TE,,, modes, respectively. We define the total field E as: E= t
(E,+c.c.)
,
j=O
where C.C. indicates complex conjugate. Substituting into Maxwell’s equations and using the standard slowly varying approximation we obtain (3)
,~08,~w,eXP[i(P,z-wwt)]=-i14r~,
where PNL is the third-order quency w PNL=~o~[2(E-E*)E+
polarisation
(E*E)E*]
for fre-
(4)
and x, to and p. are the third-order susceptibility and free space electric and magnetic permeabilities. We assume no power exchange between the fun-
Volume 94, number
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5
damental mode and the the large difference in Following the procedure the normalised coupled d W/dZ=O
second-order modes due to propagation constants [ 91. in ref. [ 111 we thus obtain power and phase equations:
,
1992
Note that as R=O, this gives W=O, i.e. there is no power in the fundamental mode, and eqs. (5) and ( 11) reduce to the case in ref. [ Ill. The normalised power in the x- and y-polarised parts of the field are given by: P,/P=~+~W+Jm0s(~),
dV/dZ=-Q[2UVsin(2@)+2W@sin(@)],
PJP=
dU/dZ=Q[2UVsin(2@)+2Wmsin(@)], do/dZ=
1 December
COMMUNICATIONS
f - 4 w-JizVcos(@)
.
(13)
1- ;Q( U- V)
x{[1+2cos(2~)]+(2W/JUv)cos~},
(5) 3. Numerical results
where Z=Bz,
Q=P/B
.
The normalised W=P,/P,
(6)
powers are given by
V=P,/P,
U=P,/P,
(7)
where m P,=(Cton,)2]Aj]2
s
VjdS
(8)
0
is the power in mode j, and P= PO+ PI + P2 is the total power. The integration is over the infinite crosssection of the fibre. B is the propagation difference between the odd HE2i and TEol modes B=l& -PI
(9)
and @ is the relative @=Bz+q&,,
.
The constants are W(Z)=
phase difference
1111. When power is excited in the fundamental mode, as described here, the critical point is affected. The transition (critical) point is now given by
(10) of the motion that arise from eq. (5)
W(0) >
US V= 1 - W= C ,
(a constant)
,
4UV[1+2cos(2@)]+16W,/&os(~) +(4/Q)(U-V)=8C-X2,
(11)
where the initial conditions U( 0) = V( 0) = C/2 have been used to calculate the rhs of eq. ( 11)) and
CCL 1+R’
In the case where there is no power excited in the fundamental mode (i.e. W=O, R=O, C= 1 in our equations), there exists a critical power ( Qcrit= $ ) where the phase steadily increases (d@/dz> 0) for powers below the critical point and oscillates around zero with diminishing amplitude for powers above the critical point. At the critical power, however, the phase reaches a steady state and is locked at += 0.66 rad where U is locked at unity, Vat zero and P, and Py are locked at $P each. At this critical point, the steady state situation is achieved where the true mode of the system is the TEo, mode. The system of equations for this case has a complete analytical solution
R=z
U+V’
(12)
R defines the ratio of the power in the fundamental mode to the power in the second-order modes.
Qcti,=;+&.
(14)
Equation ( 14) is found by substituting V= 0 (and hence U= C) and eq. ( 12) in eq. ( 11). In this case there is no longer a steady state situation where the phase is locked to a constant value. This is due to the fact that U and V continue to exchange power, even when there is zero power in V. By contrast, when the fundamental mode is not present, when the power in V goes to zero it remains at zero (and U remains at unity). Power exchange no longer occurs and the phase is locked to a constant value [ 111. This is because all higher derivatives of V vanish when V= 0. When the fundamental mode is present, however, the higher derivatives of eq. (5) do not all vanish when V= 0, and mode U is able to couple energy back into 375
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mode V. The coupling process continues, and no steady state situation is obtained. The behaviour of the period at the critical point is altered by introducing power in the fundamental mode. Figure 1 gives a plot of period 2, versus Q( =P/B) for different ratios R. It is clear that the peak in 2, no longer approaches infinity (steady state) as R becomes non-zero. As R is increased, the peak is smoothed out and finally vanishes. A complete and clear picture of the behaviour of the modal powers and the polarisations can be seen from the phase-portraits. In figs. 2-4, we explore the phase-portrait of V( = P, /P) versus @and PJP versus @for three different levels of power excited in the fundamental mode. Figure 2 depicts the case where there is no power in the fundamental mode, i.e., R = 0 which is the case analysed in detail in ref. [ 111. From fig. 2a we see how at the critical point, QCtit= 2, the power is locked
R=O (no fundamental
mode1
1 December
at zero while the phase is locked at 0.66 radians. Below the critical point, Q< 2, the power oscillates as the phase is increasing monotonically. Above the critical point, Q> {, the power oscillates around 0.5 with diminishing amplitude while the phase oscillates around zero with diminishing amplitude also. Figure 2b shows the polarisation rotation behaviour. At the critical point, the polarised powers are locked at 0.5. Before the critical point, we see the complete switching of power to the other cross polarisation. In other words, light that is initially x-polarised has rotated by 90” to become completely y-polarised. At very large nonlinearity, however, the initial x-polarisation state becomes the steady state solution. Figure 3 depicts the situation when 0.1% of the total power is excited in the fundamental mode, i.e., R=O.OOl. It is clear that before and after the transition point not much change has occurred compared to the case where R = 0. However, at the transition point, the steady state is destroyed. Figure 4 gives the case where R is increased to 0.1, that is 10%
1
0
0
J
0.0
0.5
1.o
1.5
2.0
2.5
I
3.0
Q = PIB Fig. 1. Period Z, versus Q( = P/B), for different
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Fig. 3. (a) Normalised modal power V( = PI/P) versus relative phase (8 and (b) P,,,/P versus 4; for R = 0.00 1 and a range of Q
ratios R. 1.c
O!
1.c
/ P,/P
V
P,/P
0.t 0.1
i
2x
Fig. 2. (a) Normalised modal power V( = PI /P) versus relative phase @and (b ) P&P versus @; for R = 0 and a range of Q values.
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Fig. 4. (a) Normalised modal power V( =P, /P) versus relative phase Q, and (b) P&P versus @; for R=O.l and a range of Q values.
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of the power is excited in the fundamental mode. In this case we see a more exaggerated picture of that depicted in fig. 3. It is clear though, that at $J= II the modal power V is no longer able to couple enough power back to regain its initial state, as in the previous cases.
4. Discussion We now discuss the effect of the power in the fundamental mode on the power distribution in the different polarisations. By exciting a negligible amount of power in the fundamental mode, say R= 0.001, i.e. 0.1% of the total power, the phase and power behaviour do not change before and after the transition with respect to the R = 0 case; however, at the transition we no longer get the TEo, mode as the steady state mode. Also, as shown in tig. 3b, at the transition PJP and PJP are no longer locked at f , but have a flat region where they are equal to t and then continue to oscillate periodically. Hence, by choosing a proper tibre length over the flat region we can still get a 50-50 polarisation split. By exciting a moderate amount of power in the fundamental mode (RG 0.1)) we obviously no longer get a 100% polarisation splitting as shown in fig. 4b. We no longer get a 90” rotation (i.e. completely ypolarised light) at half the beat length Z=Bz= K, but instead we get elliptically polarised light. After a beat length we completely get the x-polarised light back as expected. As the power increases beyond the transition point, we get similar behaviour to the case where R = 0.At very large powers, the Iibre acts like a birefringent libre as in ref. [ 111.
5. Conclusions In this paper we have considered the effect of Kerr nonlinearity on the polarisation behaviour of a circular core optical fibre that supports the proper second-order modes. Previous work considered the case where the fundamental mode was absent [ 111. From a practical point of view, however, it is impossible to excite the higher order modes without exciting some fundamental mode. Hence, it should be of
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practical interest to see how the fundamental mode alters the polarisation rotation modulation discussed in the previous paper. By introducing the smallest percentage of fundamental mode, the critical point is destroyed in the sense that the relative phase @is no longer locked to a constant value as in the case when no fundamental mode was present. However, a transition point was found where the phase changes its state from a monotonically increasing to a periodically oscillating function. We have shown the behaviour of the period of the polarisation rotation as a function of the amount of the fundamental mode excitation; this clearly demonstrates how the critical point behaviour is suppressed. From a practical sense, when a polarisation switch is to be designed using bi-modal fibres, the amount of power excited in the fundamental mode becomes important and needs to be minimized. In this case, the experimentalist will need to decide on the angle of incidence which will excite as negligible amount of power in the fundamental mode as is practically possible. Although a mode converter [ 14,15 ] could be used to selectively excite the second-order modes, and a complete rotation of x-polarised light to y-polarised light may be observed, we have found that the smallest percentage of light in the fundamental mode will destroy the steady state TE,, mode behaviour described in ref. [ 111.
Acknowledgments The authors discussions.
thank Prof. C. Pask for many useful
References [ 1] R.J. Black, A. Henault, [2] [3] [4] [5] [6]
S. Lacroix and M. Cada, IEEE J. Quantum Electron. 26 ( 1990) 108 1. H.G. Winful,Appl. Phys. Lett. 47 (1985) 213. B. Crosignaniand P. Di Porto, Optica Acta 32 (1985) 1251. S. Trillo, S. Wabnitz, R.H. Stolen, G. Assanto, C.T. Seaton and G.I. Stegeman, Appl. Phys. L-ett. 49 ( 1986) 1224. H.G. Winful, Optics Lett. 11 (1986) 33. H.J. Park, C.C. Pohalski and B.Y. Rim, Optics Lett. 9 (1988) 776.
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[7] R.J. Black, A. Henault, L. Gagnon, F. Gauthier and S. Lacroix, Topical Meetings on Nonlinear guided wave phenomena; Physics and Applications, 1989 Technical Digest Series 2 (Optical Society of America, Washington DC.) pp. 66-69. [ 81 R.V. Plenty, I.H. White and A.R.L. Ravis, Electron Lett. 24 (1988) 1338. [9] S.J. Garth and C. Pask, J. Opt. Sot. Am. B 9 (1992) 243. [lo] S.J. Garth and C. Pask, Electron Lett. 25 (1989) 182.
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[ 111 S.J. Garth and C. Pask, Lightwave Technol. 8 (1990) 129. [ 121 A.W. Snyder and J.D. Love, Optical waveguide theory (Chapman and Hall, New York, 1983 ) [ 131 A.W. Snyderand W.R. Young, J. Opt. Sot. Am. B 68 (1978) 297. [ 141 R.C. Youngquist, J.L. Brooks and H.J. Shaw, Optics Lett. 9 (1984) 177. [ 151 W.V.Sorin,B.Y.KimandH.J.Shaw,OpticsLett. 11 (1986) 581.