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On multistability of nonlinear fiber interferometer with recirculating delay line
Volume 78, number 5,6
OPTICS COMMUNICATIONS
15 September 1990
On multistability of nonlinear fiber interferometer with recirculating delay line T.V. B a b k i n a , F . G . Bass, S.A. B u l g a k o v , V.V. G r i g o r ' y a n t s a n d V.V. K o n o t o p Institute for Radiophysics and Electronics, Ukr. SSR Academy of Sciences, Proscura str. 12, Kharkov, 310085, USSR and Institute of Radioengeneering and Electronics, Academy of Sciences of the USSR, Marxa Avenue 18, Moskow, 103907, USSR
Received 23 January 1990; revised manuscript received 10 May 1990
We investigate an optical fiber device based on the scheme proposed by Backman [A.B. Backman, J. Lightwave Technol. 7 (1989) 151 ] with a nonlinear fiber. The possibility of multistable regimes has been stated. Advantages and differences between such a scheme and a linear one studied earlier are discussed.
1. Introduction The m u l t i s t a b i l i t y o f interferometers with nonlinear elements is one o f the most p o p u l a r branches o f m o d e r n optics (a rich b i b l i o g r a p h y is presented by G i b b s [ 1 ], see also review [2] ). M o s t papers deal with the m u l t i s t a b i l i t y o f bulk interferometers such as F a b r y - P e r o t resonators a n d M a c h - Z e n d e r ( M Z ) interferometers including b o t h a cavity with a nonlinear m e d i u m a n d a feedback (see e.g. refs [3,4] ). At the same t i m e there is a c o m p r e h e n s i v e literature d e v o t e d to linear m o n o m o d e optical fiber sensors. It is stipulated by perspectives o f using such devices for precision m e a s u r e m e n t s [5 ]. So, in a recent p a p e r B a c k m a n [ 6 ] has p r o p o s e d a novel optical fiber interferometer with c o m m o n m o d e compensation. This interferometer consists o f an M Z sensor a n d a recirculating delay line ( R D L ) . The m a i n advantage o f B a c k m a n ' s sensor is a very high measuring sensitivity with zero e n v i r o n m e n t a l a n d wavelength sensitivities. It allows to believe that any analogous scheme containing a n o n l i n e a r element will be interesting for the multistability investigation. Moreover, by varying the p a r a m e t e r s o f B a c k m a n ' s sensor, one can obtain m o r e simple devices which have been studied in literature. N a m e l y , an R D L m a n u factured with a n o n l i n e a r fiber has been considered by Crossignani et al. [ 7 ]. An M Z interferometer with one n o n l i n e a r path has been used by I m o t o et al. [ 8 ] 398
for the n o n d e m o l i t o n i a l m e a s u r e m e n t o f a p h o t o n number. The object o f our work is investigation o f the B a c k m a n sensor with a nonlinear fiber in one o f the " M Z pathes". Firstly we present an analytical description o f the B a c k m a n sensor with a nonlinear fiber ( B S N F ) (section 2). Then we investigate the output versus input power by a numerical simulation (section 3).
2. Mathematical description of BSNF The structure o f B S N F is presented in fig. 1. It is an M Z interferometer, the two paths o f which are connected by a feedback fiber. In contrast to Backm a n ' s scheme [ 6 ], there is a fiber with an intensity d e p e n d i n g refractive index,
®
Fig. 1. The scheme of the Backman sensor with a nonlinear fiber.
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
Volume 78, number 5,6
OPTICS COMMUNICATIONS
n=no + n2I ,
(1)
15 September 1990
(
tN =exp~,--71LN + i ~n2lEsI 2 [ 1 - - e x p ( --271 L N ) ] ) . /
where I ~ [El 2 is the electric field intensity and n2 is the Kerr coefficient. Let us assume that a m o n o c h r o m a t i c wave
E ~ EI exp[i( t o t - 2~zz/2 ) ] ,
i k t / 2 ( 1 - 7 ) 1/2
ikl/2(1--7)1/2
/"
(I-k)1~2(1-7)×~2/
(2) Here k is the coupling coefficient (0 < k < l ) and describes the losses in the couplers. For the description of the M Z part of the device one can introduce the matrix 57/=diag[tsexp(iUs), tNexp(iUN) ] .
(3)
This matrix is introduced analogously to the one describing the M Z sensor (see ref. [ 6 ] ). H o w e v e r in the case under consideration one of the elements, namely tN, is dependent on the light intensity (see below). Here we have written the "linear" phase shifts
Us=2zt(Ls/2)no,
The expression for E5 m a y be found by using matrix
g,
E5 -- ( 1 - k ) I/2( 1 - y ) l / E E
with frequency to and wavelength 2, where z is the coordinate along the fiber, is incident on the BSNF input, i.e. on port 1 (see fig. 1 ). The evanescent couplers C 1 and C2, which are identical, are described by the matrix [ 9 ]
R = ( ( 1- k ) l / Z ( 1 - 7 ) 1 / 2
(6)
UN=2n(LN/J.)no,
(4)
in the evident form, and introduced ts and tN for the transmission coefficient of the linear (with length Ls) and nonlinear (with length LN) device paths, respectively. As usual, ts = exp ( - 7oLs), where 7o is the linear fiber damping coefficient. For determination of tN we assume that light propagation in the nonlinear fiber is in the steady state. It is described by (in the slowly varying amplitude approximation):
d E / d z = - 71E + i( 21t/2 )n2 [El 2E,
(5)
where 71 is the nonlinear fiber d a m p i n g coefficient. The linear phase shift (see eqs. (3), ( 4 ) ) has been taken into account. By solving this equation and connecting the output (E7) and input (Es) fields of the nonlinear fiber, one obtains
1
+ i ( 1-7)l/2kl/2E2
.
(7) The matrices/~ and 57I allow to describe the MZ part of the device
(E4)..~.KMK(E2) E3
....
El
(8)
To close the system of equations, one has to take into account the relation between E 3 and E2, which follows from the linear feedback path description, E 2 = tfexp( - i U f ) E 3 .
(9)
As above tf= exp ( - 7oLr) is the feedback transmission coefficient, Lf is the feedback fiber length, and
Uf=2n(Lf/2)no.
(10)
By direct calculation from (2), (3), ( 6 ) - ( 9 ) finally obtains ~
one
t=E4/E1 = I ( 1 - 7 ) / A ] × {i [k( 1 - k) ]1/2 [ tNexp (iUN) + tsexp (iUs) ]
+(1--7)tftstN(exp[i(Uf+Us+UN)]},
(lla)
trec =E3/E~ = [ ( 1 - y ) /A] × [(1-k)tNexp(iUN)-ktsexp(iUs)],
(lib)
tN = e x p ( -- 71LN) ( - 27~ LN) ] x exp(i~t [ 1 -- exp :tyl [AI 2 I ( 1 - k ) 1/2
--ikl/2(1--7)tftNexp[i(Ur--UN)] 121EI2), (12) where a~ It is necessary to point out that in the linear limit n=0 our results differ from the same ones in ref. [ 6 ], where there is a statement "... Transmittance depends only on the phase difference between the two paths joining C1 to C2 ...". In our results there is an evident transmittance dependence on the both Ur and Us. 399
Volume 78, number 5,6
OPTICS COMMUNICATIONS
A = 1 - i ( 1 - y ) t r exp(iUf) [k( 1 - k )
]1/2
× [tsexp(iUs) +tNexp(iUN) ] .
(13)
2.5
~Z0
Parameters t and tr~c are the transmission coefficient for the whole scheme and RDL, respectively. Eqs. ( 11 ) - ( 13 ) are basic ones for multistability investigation. We solve them numerically in the next section.
~a. l0 0.5
0.0
3.O ~ 2.S "~ ~.0 "~t.5 C~ tO O.5 0.0
3. N u m e r i c a l s i m u l a t i o n
To examine the transcendental system ( 11 ) - ( 13 ) we use numerical methods, the results of which are presented in this section. Assume both couplers to be lossless ( y = 0 ) and equally dividing power (k = 0.5 ). The fiber losses are as large as 0.2 d B / k m . We also consider all path lengths to be o f the order of 2 km and equal to each other. For typical silica fibers the Kerr coefficient is 3 × 10 -2° m 2 / W at the wavelength 2 = 1.28 ~tm (the linear part o f the refractive index no is approximately equal to 1.45). The fiber diameter is 8 p,m. According to ref. [ 6 ] we present the linear phases in the form Us=~+2nM,
Uf=p+ 2nN ,
UN =r+ 2nK,
where M, N and K are large integers, and 0 < 0, P, r < 2. Without restrictions o f generality one can put r = 0. Then q~ is the relative difference between the MZ paths and p is a phase shift in the feedback path. Since multistability is an expected effect in the case of the system discussed, we introduce the designations t~ (with j = 1, 2 ..... m) for the different transmission coefficient values at the given input intensity. The integer m shows the number of system roots. At m > 1 we have several values o f output (or recirculating) intensities for a single input intensity. Such behaviour o f both recirculating (Prec ~ [trec [ 2 [E~ [ 2) and output (Po~,~ [t[2[El[ 2) powers versus input power (Pi,p~ JE~ [2) is illustrated in fig. 2a and b, respectively. As one may see from these figures, the first multistability zone is located in th e region near P~,p ~ 1 W. The next multistability zone appears with increasing input power. Thus, there is "periodicity" in the plot behaviour. A simple illustration of this 400
15 September 1990
•
*
o
I
•
2
°
t
3
q
,,~
Pinp (V¢') Fig. 2. The recirculating (Pr~) and output (Pout) powers versus input power (Pinp). 0 =0. fact may be given. To this end, we rewrite eq. (12 ) in the form
U=f(U) ,
(14)
where f ( U ) = { ( 1 _ y),/2[ ( 1 - k ) 1/2 - i k l / 2 ( 1 - 7 ) t f t s exp(i(p+~b) ) ]} × {1 --i( l --y)tf exp (ip) [k( 1 - k ) ],/2 × [tN exp(iUN) + ts exp(iO) ]} - l ,
(15)
tN=exp(--ylLN+inn2 1 - - e x p ( - - 2 y ' L N ) U) Y12 (16) The numerical solution o f eq. (14) at different input power values is given in fig. 3. The appearance o f new roots, which are the intersection points on the plot, with increasing input power is evident. The intersection zone location is directly related with the periodic character o f the function F = f ( U ) . It is useful to point out that one o f the most important device characteristics, i.e. transmittance T = It)2, has a behaviour similar to that plotted in fig. 2a. As one can see from fig. 2, the multistability zones for Prec and Pou, are symmetrical in a sense o f shape.
Volume 78, number 5,6
OPTICS COMMUNICATIONS
15 September 1990
20 ~
/
18
1
2.o
t6
1.6 £2 0.8
1.5
tq
~ t.0
iZ
0.5 0.0
0.9 0.0
~" ~0
6
3.0
z.5
~.0 1.5
t.5
2
0.5
0
d.o
t:o
~o
~o ~:o ~ 12 / Jr
61o
Fig. 3. The illustration of the numerical solution of the eq. (14) (see text). 0=0. It is naturally connected with the fact that the preserving flux remains low at small Yo, Y~ ( IE212 + IE412 is proportional to the total flux). Thus, the "upper" loops of the recirculating power correspond to the "lower" ones of the output power. There are variations of multistability loops for different values of 0 as a controlling parameter for operating the multistability process (see fig. 4). The phase shift p is another controlling parameter, commonly used for multistability operating. The existence of independent parameters is a potentially useful property for designing devices as sensor with both higher sensitivity and higher switching on (off) power. Thus, as pointed out above, Backman's scheme for some parameters is similar to an R D L sensor. However the fundamental difference of the devices studied here is the presence of multistability loops. These loops allow to maintain value being measured (Pinp in our case) in some interval having two limits. On (off) power switching events occur on the edges of the loop highest part for Pr~c (or the lowest for Pout). From our point of view the higher sensitivity of Backman's scheme, the common mode compensation property [6], the great possibilities in controlling the output parameters, the existence of multistability and its singularity, make this device not only
t.0
i.O
O.S
O.5
0.0
0.0 •
0
t
2
5 tl 5 INPlJ,'T
,
,
,
o
,
o I PO WER
2
3
q
5
(W)
Fig. 4. The shapes differenceof multistability loops for different values of O. very interesting for theoretical investigation but also very useful for practice (in the capacity of an interferometer, a controlled device with switching power and so on).
4. C o n c l u s i o n
We have discussed only the stationary problem and found the multistability phenomenon. Naturally, the question arises, how does the BSNF work in the timedependent dynamics? The discussion of this problem will be published elsewhere. But here we may make some conclusion, based on the comparison of the BSNF with previously studied schemes. So, in the absence of the signal path (i.e. at t s = 0 ) , the above discussed device is a fiber analog of the bulk scheme, studied by Ikeda et al. [ 10 ]. Such a scheme is a ring cavity containing a nonlinear dielectric medium. Chaotic behaviour of the transmitted light has been observed. It allows one to expect the chaotic property of the BSNF in the dynamic regime. At the same time the corresponding dynamic systems are not identical, and some differences may occur (especially in the parameter regions interesting from the practical point of view). Note that chaos possibility 401
Volume 78, number 5,6
OPTICS COMMUNICATIONS
has n o t b e e n m e n t i o n e d in ref. [ 7 ] in w h i c h the n o n l i n e a r R D L has b e e n studied.
References [ 1 ] H.M. Gibbs, Optical bistability: controlling light with light. (Academic Press, New York, 1985). [2} E. Bernabeu, P.M. Mejias and R. Martinez-Herrero, Phys. Scr. 36 (1987) 312. [ 3 ] G.S. Agarwal and S.D. Gupta, Phys. Rev. A30 (1984) 2764.
402
15 September 1990
[4] P.M. Mejias, R. Martinez-Herrero and E. Bernabeu, Phys. Rev. A33 (1986) 1836. [5] D.A. Jackson, J. Phys. E Sci. lnstrum. 18 (1985) 981. [6] A.B. Backman, J. Lightwave Technol. 7 (1989) 151. [ 7 ] B. Crosignani, B. Diano, P. Diporto and S. Wabnitz, Optics Comm. 59 (1986) 309. [8]N. Imoto, S. Watkins and Y. Sasaki, Optics Comm. 61 (1987) 159. [9 ] P. Urquhart, Fiber laser resonators, in: The physics and technology of laser resonators (Institute of Physics Publishing, 1989). [ 10] K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709.
OPTICS COMMUNICATIONS
15 September 1990
On multistability of nonlinear fiber interferometer with recirculating delay line T.V. B a b k i n a , F . G . Bass, S.A. B u l g a k o v , V.V. G r i g o r ' y a n t s a n d V.V. K o n o t o p Institute for Radiophysics and Electronics, Ukr. SSR Academy of Sciences, Proscura str. 12, Kharkov, 310085, USSR and Institute of Radioengeneering and Electronics, Academy of Sciences of the USSR, Marxa Avenue 18, Moskow, 103907, USSR
Received 23 January 1990; revised manuscript received 10 May 1990
We investigate an optical fiber device based on the scheme proposed by Backman [A.B. Backman, J. Lightwave Technol. 7 (1989) 151 ] with a nonlinear fiber. The possibility of multistable regimes has been stated. Advantages and differences between such a scheme and a linear one studied earlier are discussed.
1. Introduction The m u l t i s t a b i l i t y o f interferometers with nonlinear elements is one o f the most p o p u l a r branches o f m o d e r n optics (a rich b i b l i o g r a p h y is presented by G i b b s [ 1 ], see also review [2] ). M o s t papers deal with the m u l t i s t a b i l i t y o f bulk interferometers such as F a b r y - P e r o t resonators a n d M a c h - Z e n d e r ( M Z ) interferometers including b o t h a cavity with a nonlinear m e d i u m a n d a feedback (see e.g. refs [3,4] ). At the same t i m e there is a c o m p r e h e n s i v e literature d e v o t e d to linear m o n o m o d e optical fiber sensors. It is stipulated by perspectives o f using such devices for precision m e a s u r e m e n t s [5 ]. So, in a recent p a p e r B a c k m a n [ 6 ] has p r o p o s e d a novel optical fiber interferometer with c o m m o n m o d e compensation. This interferometer consists o f an M Z sensor a n d a recirculating delay line ( R D L ) . The m a i n advantage o f B a c k m a n ' s sensor is a very high measuring sensitivity with zero e n v i r o n m e n t a l a n d wavelength sensitivities. It allows to believe that any analogous scheme containing a n o n l i n e a r element will be interesting for the multistability investigation. Moreover, by varying the p a r a m e t e r s o f B a c k m a n ' s sensor, one can obtain m o r e simple devices which have been studied in literature. N a m e l y , an R D L m a n u factured with a n o n l i n e a r fiber has been considered by Crossignani et al. [ 7 ]. An M Z interferometer with one n o n l i n e a r path has been used by I m o t o et al. [ 8 ] 398
for the n o n d e m o l i t o n i a l m e a s u r e m e n t o f a p h o t o n number. The object o f our work is investigation o f the B a c k m a n sensor with a nonlinear fiber in one o f the " M Z pathes". Firstly we present an analytical description o f the B a c k m a n sensor with a nonlinear fiber ( B S N F ) (section 2). Then we investigate the output versus input power by a numerical simulation (section 3).
2. Mathematical description of BSNF The structure o f B S N F is presented in fig. 1. It is an M Z interferometer, the two paths o f which are connected by a feedback fiber. In contrast to Backm a n ' s scheme [ 6 ], there is a fiber with an intensity d e p e n d i n g refractive index,
®
Fig. 1. The scheme of the Backman sensor with a nonlinear fiber.
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
Volume 78, number 5,6
OPTICS COMMUNICATIONS
n=no + n2I ,
(1)
15 September 1990
(
tN =exp~,--71LN + i ~n2lEsI 2 [ 1 - - e x p ( --271 L N ) ] ) . /
where I ~ [El 2 is the electric field intensity and n2 is the Kerr coefficient. Let us assume that a m o n o c h r o m a t i c wave
E ~ EI exp[i( t o t - 2~zz/2 ) ] ,
i k t / 2 ( 1 - 7 ) 1/2
ikl/2(1--7)1/2
/"
(I-k)1~2(1-7)×~2/
(2) Here k is the coupling coefficient (0 < k < l ) and describes the losses in the couplers. For the description of the M Z part of the device one can introduce the matrix 57/=diag[tsexp(iUs), tNexp(iUN) ] .
(3)
This matrix is introduced analogously to the one describing the M Z sensor (see ref. [ 6 ] ). H o w e v e r in the case under consideration one of the elements, namely tN, is dependent on the light intensity (see below). Here we have written the "linear" phase shifts
Us=2zt(Ls/2)no,
The expression for E5 m a y be found by using matrix
g,
E5 -- ( 1 - k ) I/2( 1 - y ) l / E E
with frequency to and wavelength 2, where z is the coordinate along the fiber, is incident on the BSNF input, i.e. on port 1 (see fig. 1 ). The evanescent couplers C 1 and C2, which are identical, are described by the matrix [ 9 ]
R = ( ( 1- k ) l / Z ( 1 - 7 ) 1 / 2
(6)
UN=2n(LN/J.)no,
(4)
in the evident form, and introduced ts and tN for the transmission coefficient of the linear (with length Ls) and nonlinear (with length LN) device paths, respectively. As usual, ts = exp ( - 7oLs), where 7o is the linear fiber damping coefficient. For determination of tN we assume that light propagation in the nonlinear fiber is in the steady state. It is described by (in the slowly varying amplitude approximation):
d E / d z = - 71E + i( 21t/2 )n2 [El 2E,
(5)
where 71 is the nonlinear fiber d a m p i n g coefficient. The linear phase shift (see eqs. (3), ( 4 ) ) has been taken into account. By solving this equation and connecting the output (E7) and input (Es) fields of the nonlinear fiber, one obtains
1
+ i ( 1-7)l/2kl/2E2
.
(7) The matrices/~ and 57I allow to describe the MZ part of the device
(E4)..~.KMK(E2) E3
....
El
(8)
To close the system of equations, one has to take into account the relation between E 3 and E2, which follows from the linear feedback path description, E 2 = tfexp( - i U f ) E 3 .
(9)
As above tf= exp ( - 7oLr) is the feedback transmission coefficient, Lf is the feedback fiber length, and
Uf=2n(Lf/2)no.
(10)
By direct calculation from (2), (3), ( 6 ) - ( 9 ) finally obtains ~
one
t=E4/E1 = I ( 1 - 7 ) / A ] × {i [k( 1 - k) ]1/2 [ tNexp (iUN) + tsexp (iUs) ]
+(1--7)tftstN(exp[i(Uf+Us+UN)]},
(lla)
trec =E3/E~ = [ ( 1 - y ) /A] × [(1-k)tNexp(iUN)-ktsexp(iUs)],
(lib)
tN = e x p ( -- 71LN) ( - 27~ LN) ] x exp(i~t [ 1 -- exp :tyl [AI 2 I ( 1 - k ) 1/2
--ikl/2(1--7)tftNexp[i(Ur--UN)] 121EI2), (12) where a~ It is necessary to point out that in the linear limit n=0 our results differ from the same ones in ref. [ 6 ], where there is a statement "... Transmittance depends only on the phase difference between the two paths joining C1 to C2 ...". In our results there is an evident transmittance dependence on the both Ur and Us. 399
Volume 78, number 5,6
OPTICS COMMUNICATIONS
A = 1 - i ( 1 - y ) t r exp(iUf) [k( 1 - k )
]1/2
× [tsexp(iUs) +tNexp(iUN) ] .
(13)
2.5
~Z0
Parameters t and tr~c are the transmission coefficient for the whole scheme and RDL, respectively. Eqs. ( 11 ) - ( 13 ) are basic ones for multistability investigation. We solve them numerically in the next section.
~a. l0 0.5
0.0
3.O ~ 2.S "~ ~.0 "~t.5 C~ tO O.5 0.0
3. N u m e r i c a l s i m u l a t i o n
To examine the transcendental system ( 11 ) - ( 13 ) we use numerical methods, the results of which are presented in this section. Assume both couplers to be lossless ( y = 0 ) and equally dividing power (k = 0.5 ). The fiber losses are as large as 0.2 d B / k m . We also consider all path lengths to be o f the order of 2 km and equal to each other. For typical silica fibers the Kerr coefficient is 3 × 10 -2° m 2 / W at the wavelength 2 = 1.28 ~tm (the linear part o f the refractive index no is approximately equal to 1.45). The fiber diameter is 8 p,m. According to ref. [ 6 ] we present the linear phases in the form Us=~+2nM,
Uf=p+ 2nN ,
UN =r+ 2nK,
where M, N and K are large integers, and 0 < 0, P, r < 2. Without restrictions o f generality one can put r = 0. Then q~ is the relative difference between the MZ paths and p is a phase shift in the feedback path. Since multistability is an expected effect in the case of the system discussed, we introduce the designations t~ (with j = 1, 2 ..... m) for the different transmission coefficient values at the given input intensity. The integer m shows the number of system roots. At m > 1 we have several values o f output (or recirculating) intensities for a single input intensity. Such behaviour o f both recirculating (Prec ~ [trec [ 2 [E~ [ 2) and output (Po~,~ [t[2[El[ 2) powers versus input power (Pi,p~ JE~ [2) is illustrated in fig. 2a and b, respectively. As one may see from these figures, the first multistability zone is located in th e region near P~,p ~ 1 W. The next multistability zone appears with increasing input power. Thus, there is "periodicity" in the plot behaviour. A simple illustration of this 400
15 September 1990
•
*
o
I
•
2
°
t
3
q
,,~
Pinp (V¢') Fig. 2. The recirculating (Pr~) and output (Pout) powers versus input power (Pinp). 0 =0. fact may be given. To this end, we rewrite eq. (12 ) in the form
U=f(U) ,
(14)
where f ( U ) = { ( 1 _ y),/2[ ( 1 - k ) 1/2 - i k l / 2 ( 1 - 7 ) t f t s exp(i(p+~b) ) ]} × {1 --i( l --y)tf exp (ip) [k( 1 - k ) ],/2 × [tN exp(iUN) + ts exp(iO) ]} - l ,
(15)
tN=exp(--ylLN+inn2 1 - - e x p ( - - 2 y ' L N ) U) Y12 (16) The numerical solution o f eq. (14) at different input power values is given in fig. 3. The appearance o f new roots, which are the intersection points on the plot, with increasing input power is evident. The intersection zone location is directly related with the periodic character o f the function F = f ( U ) . It is useful to point out that one o f the most important device characteristics, i.e. transmittance T = It)2, has a behaviour similar to that plotted in fig. 2a. As one can see from fig. 2, the multistability zones for Prec and Pou, are symmetrical in a sense o f shape.
Volume 78, number 5,6
OPTICS COMMUNICATIONS
15 September 1990
20 ~
/
18
1
2.o
t6
1.6 £2 0.8
1.5
tq
~ t.0
iZ
0.5 0.0
0.9 0.0
~" ~0
6
3.0
z.5
~.0 1.5
t.5
2
0.5
0
d.o
t:o
~o
~o ~:o ~ 12 / Jr
61o
Fig. 3. The illustration of the numerical solution of the eq. (14) (see text). 0=0. It is naturally connected with the fact that the preserving flux remains low at small Yo, Y~ ( IE212 + IE412 is proportional to the total flux). Thus, the "upper" loops of the recirculating power correspond to the "lower" ones of the output power. There are variations of multistability loops for different values of 0 as a controlling parameter for operating the multistability process (see fig. 4). The phase shift p is another controlling parameter, commonly used for multistability operating. The existence of independent parameters is a potentially useful property for designing devices as sensor with both higher sensitivity and higher switching on (off) power. Thus, as pointed out above, Backman's scheme for some parameters is similar to an R D L sensor. However the fundamental difference of the devices studied here is the presence of multistability loops. These loops allow to maintain value being measured (Pinp in our case) in some interval having two limits. On (off) power switching events occur on the edges of the loop highest part for Pr~c (or the lowest for Pout). From our point of view the higher sensitivity of Backman's scheme, the common mode compensation property [6], the great possibilities in controlling the output parameters, the existence of multistability and its singularity, make this device not only
t.0
i.O
O.S
O.5
0.0
0.0 •
0
t
2
5 tl 5 INPlJ,'T
,
,
,
o
,
o I PO WER
2
3
q
5
(W)
Fig. 4. The shapes differenceof multistability loops for different values of O. very interesting for theoretical investigation but also very useful for practice (in the capacity of an interferometer, a controlled device with switching power and so on).
4. C o n c l u s i o n
We have discussed only the stationary problem and found the multistability phenomenon. Naturally, the question arises, how does the BSNF work in the timedependent dynamics? The discussion of this problem will be published elsewhere. But here we may make some conclusion, based on the comparison of the BSNF with previously studied schemes. So, in the absence of the signal path (i.e. at t s = 0 ) , the above discussed device is a fiber analog of the bulk scheme, studied by Ikeda et al. [ 10 ]. Such a scheme is a ring cavity containing a nonlinear dielectric medium. Chaotic behaviour of the transmitted light has been observed. It allows one to expect the chaotic property of the BSNF in the dynamic regime. At the same time the corresponding dynamic systems are not identical, and some differences may occur (especially in the parameter regions interesting from the practical point of view). Note that chaos possibility 401
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has n o t b e e n m e n t i o n e d in ref. [ 7 ] in w h i c h the n o n l i n e a r R D L has b e e n studied.
References [ 1 ] H.M. Gibbs, Optical bistability: controlling light with light. (Academic Press, New York, 1985). [2} E. Bernabeu, P.M. Mejias and R. Martinez-Herrero, Phys. Scr. 36 (1987) 312. [ 3 ] G.S. Agarwal and S.D. Gupta, Phys. Rev. A30 (1984) 2764.
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[4] P.M. Mejias, R. Martinez-Herrero and E. Bernabeu, Phys. Rev. A33 (1986) 1836. [5] D.A. Jackson, J. Phys. E Sci. lnstrum. 18 (1985) 981. [6] A.B. Backman, J. Lightwave Technol. 7 (1989) 151. [ 7 ] B. Crosignani, B. Diano, P. Diporto and S. Wabnitz, Optics Comm. 59 (1986) 309. [8]N. Imoto, S. Watkins and Y. Sasaki, Optics Comm. 61 (1987) 159. [9 ] P. Urquhart, Fiber laser resonators, in: The physics and technology of laser resonators (Institute of Physics Publishing, 1989). [ 10] K. Ikeda, H. Daido and O. Akimoto, Phys. Rev. Lett. 45 (1980) 709.