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Radio emission of an optical soliton propagating in a ring fiber
Optics Communications 100 (1993) 437-442 North-Holland
OPTICS COMMUNICATIONS
Radio emission of an optical soliton propagating in a ring fiber V.V. K l i m o v P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospekt, Moscow, Russian Federation
and V.S. L e t o k h o v Institute of Spectroscopy, Russian Academy of Sciences, 142092 Troitsk, Moscow Region, Russian Federation Received 11 March 1993
Considered are emission processes occurring in the course of propagation of an optical soliton along a ring fiber. The spatial and temporal characteristics of radiation are calculated. It is demonstrated that conditions are possible in which radiation becomes observable.
1. Introduction The generation of solitons in optical fibers has recently become the object of many theoretical and experimental studies [ 1-4]. It has been shown theoretically [ 1 ] that in the approximation of a smooth pulse change over the length of the wave, its propagation is described by a nonlinear Schr~Sdinger equation allowing for soliton pulses, i.e., such pulses as propagate without changing shape. In refs. [2,3], such solitons have been experimentally demonstrated to exist, and in ref. [ 5 ], the generation of solitons 4 periods long at a frequency of 1.32 Ixm has even been reported. The considerable interest in such pulses is due to the expectations that high-throughput communication systems will be created on their basis. This prospect in mind, theoretical and experimental research has mainly been directed towards the stabilization of the form of solitons propagating over great distances (see, for example, ref. [6] ). Within the framework of this approach, no consideration has been given to the problem of emission of energy by solitons into the surroundings. But the similarities between solitons and elementary particles allow the problem to be raised of the emission of radiation by a propagating soliton. Naturally, no uniform rectilinear motion here will cause any no-
ticeable emission in a direction across the fiber: all the energy of the soliton is transferred along the fiber, as is the case with the uniform rectilinear motion of a charged particle. Extending this analogy further, there may take place the emission of radiation by a soliton propagating along a ring fiber (an analog of synchrotron radiation). It is the study of exactly this effect and the possibility of its observation that the present work has been devoted to.
2. Qualitative approach Let us first consider the problem of emission of radiation by a soliton in qualitative terms. It should primarily be noted here that the character of radiation emitted by a soliton will materially depend on the direction of its electric field. To illustrate, if in the course of propagation of the soliton the dielectric polarization vector of the fiber lies in the plane of the fiber ring, the system as a whole features an electric dipole moment directed along the radius-vector of the soliton, and since the electric dipole moment rotates together with the soliton, then a¢¢ 0, and so there takes place the emission of radiation whose frequency is governed by the rotational frequency of the soliton. But if in the course of propagation of the so-
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
437
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liton the dielectric polarization vector of the fiber is directed along the symmetry axis of the ring, the radiation will predominantly be of a magnetic-dipole character, for in that case t h ¢ 0. Other polarization directions give rise to mixed types of radiation. The next important point is the modulation of the soliton in its propagation direction. If we assume that emission occurs at radio frequencies and the size of the soliton is negligibly small compared to the optical wavelength )~opt, it is evident that in the absence of modulation the contributions to the field from the radiation emitted by the different portions of the soliton are added together in the same phase, whereas any modulation gives rise to a destructive interference, and so it should be expected that the soliton radiation will be a maximum at a minimum modulation. By the way, this is also clear from the fact that the "static" electric field outside the fiber is maximal when there are no soliton elements characterized by opposite polarization signs.
3. Theoretical model
15 July 1993
angular velocity of the rotating pulse; Po is the radius of the ring, and v, is the propagation velocity (group) of the pulse. The polarization directions and pulse shapes selected enable one to solve the problem in analytical form. Other types of polarization are being studied in a similar fashion, and the results will be published in an individual paper. As to the fact that the form of the soliton remains unchanged in the course of propagation, we believe that its radiation loss is small and that its energy store is being constantly replenished. The effect of radiation on the dynamics of the soliton will be considered elsewhere. The form of polarization specified, the problem reduces to the solution of the Maxwell equations which in our case are reduced to the wave equation for the electric field
[ --V2-I-C-202/OI 2] E=4~[WC-2O2/Ot 2] P ,
which uses ordinary notation. Using the Green function (delayed potential) method, the solution of eq. (2) may we written in the form
E= [VV--c-2 02/Ot2]F , Consider the electric polarization wave P(r, t) propagating along a ring fiber without suffering any change in its form (fig. 1 )
P=Pof(P, z) g(q~-t2ot) ,
z ....
G
/
d3p Pof(P', z') g[9'-[2°(t-lr-r'l Ir - r ' l / c ) ]
(4)
In expression (4), use is made of a cylindrical coordinate system. Insofar as we are interested in the radiation field in the far zone (R>>p2/v~To), we then may make standard simplifications in calculating eq. (4), as a result of which we get the following expression for F:
×g{~o'-Oo[t-R/c+sin 0 c o s ( 9 - ~ ' ) p'/c Ro
+cosO z'/c]} ,
~ ~ " /~.
--%~o --
"
~_2a
Vg Fig. 1. Geometry of the problem under consideration. 438
F=
// ,/
,
~
where F is described by the integral
F = P o / R - ~ p' dp' dz' d~' f(p', z') .
'
(3)
( 1)
where Po is the constant vector directed along the axis of the fiber ring, OZ; f(p, z) = exp [ - (p2 + z 2 ) / 2 a 2 ] describes the field distribution over the cross section of the fiber; g(~o) = YgN exp(iN~) is the 2n-periodic function describing the pulse shape; 12o= vg/po is the
waveguide
(2)
(5)
where 0 and (p are angles in a spherical coordinate system (fig. 1 ). Substituting into eq. ( 5 ) the expression for g expanded in harmonics, we obtain
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15 July 1993
It is not very difficult to find that of the two solutions of this equation the one suitable for the application of the steepest descent method is the point
F = P o / R - ~', gNexp [ -iN£2o(t-R/c) ] × ~p'dp'dz'd~o'f(p',z')
~0= 3~z+i ln[(1 _f12)1/2/fl]
.
× exp{iN[ ~o'- g2o/c(sin 0 cos (~o- ~0') p' + c o s 0 z ' ) 1}.
(6)
The integrals with respect to p' and z' are of the gaussian type and are easy to calculate under the assumption that the radius of the fiber is much smaller than that of the fiber ring, which allows integration with respect to p' to be extended between - o o and
In the ultrarelativistic case, the two saddle points merge, and so eq. (8) must be modified accordingly. In the case of fiber transmission, this situation apparently cannot occur, and we do not consider it here. By using eq. (3), we can now readily find the expression for the electric field: (27ra)2po £22
E= [Po-n(Pono) ]
Woo:
R
c2
- ~ guexp{iN[~o-f2o(t-R/c) ]
F = 27ra2p°p° ~ gN R
+ (aN/po)2(1 -/72)/2}
×exp{iN[~o-g2o(t-R/c) l - [at2oN cos 0/c]2/2}
XN2(--i)NJN(Nfl) [1 -i(a/po)2Nflcos~o *]
.
2n
(9)
× j do' exp{iN[~0'-flcos~0'
0
+ ½i(a/po)2Nfl2cosZq) ' ] } X [1 -i(a/po)ZNflcos~o '] .
(7)
In this expression, fl= vg/c sin 0. By analogy with the synchrotron radiation (see, for example, ref. [ 7 ] ), it can be assumed that in this sum the main part is played by terms with large INI values, which makes it possible to employ asymptotic methods. The results obtained confirm the validity of this supposition. So, if INI >> 1, the integral with respect to ~0' in eq. (7) can be calculated by the steepest descent method. In that case, we restrict ourselves to the radiatively most interesting case of relativistic pulse group velocities, i.e., the case fl~ 1. If, in addition, it is assumed here that INI << (po/a) 2, which is also justified by the results obtained, one may then write the following expression instead of eq. (7)
In this expression, n is the unit vector in the observation direction, and only those terms are retained which are responsible for radiation, i.e., those inversely proportional to R. The elementary analysis of eq. (9) shows that there exists a harmonic whose amplitude is a maximum. To find the serial number N* of this harmonic, assume that 1 << N* <
Ju(Nfl) = exp{N[ ( 1 __f12) 1/2_ arth ( 1 __f12) 1/2]} [2nN( 1 _f12) 1/2] 1/2
(10) to find the serial number of the maximum-amplitude harmonic to be 3 N * = 2[arth(l_fl2)l/2
F = (2~za)2p°p° ~ gN R
(l_fl2)l/2] ,
(11)
× exp{iN[~o--g2o( t--R/c) ] + (aN/po)2( 1 -/72)/2}
or, considering the fact that (1 _fl2)1/2 is small in the directions important for radiation,
×(-i)UJN(Nfl)[1-i(a/po)2Nflcos~o*],
N * ~ 9 / 2 [ ( 1 - f 1 2 ) 1/2] 3
(8)
where JN is a Bessel function,/7= vg/c, and ~0" is the saddle point defined by the equation 1 + fl sin ~0"= 0.
(12)
The above expression for the serial number of the maximum-amplitude harmonic agrees fully with the 439
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15 July 1993
can be factored outside the integral sign. After differentiating, the expression for the electric field in the wave zone takes on the form 2ha 2190 -(22
E = - [Po-n(Pon) ] - -R ×
£
sin ~ +
c2 2n
f12+ 2fl2COS2(~
( l + f l s i n q~) 5
o
~ d~0g(~0) _ o
(15)
where g((o) is the function describing the modulation of the pulse. This expression can conveniently be presented in the form Fig. 2. Spectral composition of radiation as a function of the serial number o f the harmonic and observation angle ( v J c = 0.9 ).
assumptions made above when calculating integral (7) and with the use of approximation (10) to find the number itself. It can be seen from expressions ( 11 ) and ( 12 ) that iffl is not too close to unity, then N*•o<< O~opt, and so the radiation frequency falls within the radio frequency range. It is also clear from the above analysis that the results obtained hold true, provided that the pulse spectrum g~ varies but not too rapidly in the radio frequency band. In our case of femtosecond pulses, the analysis is absolutely correct.
The fact of radio emission proved, one can neglect in eqs. (9) and (5) the thickness of the fiber in comparison with the radio wavelength. As a result, we have the following expression for the electric field E=
- [Po-n(Pon)
2na2po 02 ] ¢2R Ot2 (13)
2n 0
dq/ g[~o-g2o(t-R/c)+~] . 1 + f l s i n q)(~,)
(14)
In the integral obtained, the second term has a sharp maximum at ( o - g 2 o ( t - R / c ) + ~ = 0 , while the first practically does not change in this region and hence 440
(16)
where the factor F allowing for the suppression of radiation as a result of modulation is given by
F= --1 f dtg(t) ,
(17)
I"o d
where for brevity use is made of the notation q)= cb[f2o(t-R/c)-(o ], and ro is the parameter characterizing the duration of the pulse. Expression (16) is the final result of our calculations and can serve as a starting point for further studies. In the text below, we will use eq. (16) to find the radiation power flow and power. The radiation power flow is defined by the standard expression
c [EH]R2 = c
4nn ]EI2R2'
(18)
and substituting it into eq. (16), we get
dI
To calculate the integral with respect to q¢, we change the variables q~'= ~(q/), ~u= ~0'- fl cos (o', and as a resuit, it assumes the form
]
002 flsin ~+flz+2fl2cos2qgF, X 2na2vgro - R c2 (l+flsin ~) 5
dI
0
f
[Po-n(Pon)
d ~ - 47r
2rt
× t dq~'g[~o-g2o(t-R/c)+~o'-flcos(o'].
E=-
¢ p2{UgZO~z(ZT~a~4ff6sin, 0 × (sin ~:~q-fl--I-2fl c 0 s 2 ~ ) 2 / . 2 . (1 + f l s i n ~)lo
(19)
Recall that/~= Vg/C, fl=/~sin 0, and q~= qb[f2o(t-R/ c)-~0]. The angular dependences of the radiation power flow are shown in fig. 3. To find the total radiative power, it is necessary to integrate expression (19) with respect to the angles. First we integrate with respect to the azimuth angle
Volume 100, number 5,6
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15 July 1993
or, recalling that t h a t / ~ 1,
l=cp2
(~0)2(~0a)4
10
3(1_/~2)6 F 2 .
(22)
4. D i s c u s s i o n of the results
Fig. 3. Soliton radiation directivity diagram ( t = R / c, vg/c = 0.9 ).
20
-
i
]
Ioc [p2(1-/~2) 21 - 1 ,
..jf-~
15 J4
/" ."
=~o~ ~ "~
5
]
.//
0
! / -5
//
-I0 0
20
40
e theta,
60
80
dgs
Fig. 4. Intensityofemissionintothefingbelt(~/c=0.9). ~0. The result of this integration is the power flow through a spherical belt and is expressed in terms of elementary functions as
dI
d(cos0)
_zfvgzo'~2{2na'~ 4
#4sin20
W5 × 2-~ (1155WS-2772W6+2170W 4 - 5803 W 2 + 27 )/-2,
(20)
where W = (l-fl2sinZO) -wz. The graph of W as a function of 0 is presented in fig. 4. The expression for the total radiated power is obtained after integrating eq. (20) with respect to 0
,=cP~(~)
2
4
( ~ o a) 2/~n(8/~4+15/~2+2)
15(1_~2)6
The total radiation power found in the preceding section depends materially on the pulse velocity, and if it is close enough to the velocity of light, efficient emission of radiation is possible. Note for comparison that in the case of synchrotron radiation [7], the radiation power is described by the formula
r2,
which gives a slower rise of the radiation power with the increasing pulse speed. Reducing the radius of the fiber ring will cause a rapid radiation power rise (as 1/p] ) and will displace the radiation spectrum maximum towards the high-frequency side (foc 1/Po). At Po acN*2 opt, our results are naturally inapplicable, for the soliton in that case emits all of its energy during a time equal to its life span, which means that in this region the notion of soliton has no sense. To reveal the effect of the pulse shape (modulation) on the efficiency of radiation, consider the following three types of femtosecond pulses (see fig. 5 ): (a) classical soliton of the envelope g l ( t ) =
cos( o9t + 6) /ch ( t/ro); (b) asymmetrical pulse gz(t) =cos(o9t+8) (t/Zo) exp(1 - t/r0); (c) Partially rectified pulse g 3 ( t ) = g l ( t ) [1 + eg, (t)]. In all these cases, o9 stands for the optical frequency and 6 is the phase shift which can apparently be controlled. The suppression factors for these pulses are calculated in analytical form n cos 6 cosh (nu/2) ' FE -- e[cos 6( 1 - u a) - 2 u sin 6] (1+u2)2
Fa-'-FI + E [ n / 2 + n u c o s ( 2 6 ) / s i n h ( n u ) ]
(21)
(23)
.
(24)
In expressions (24), u = ogro, and if the pulse width 441
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15 July 1993
b.
Fig. 5. Modulated pulse examples: (a) classical soliton, (b) asymmetrical pulse, and (c) partially rectified pulse. Table 1 Total radiation intensity, nW. Type of pulse Classical Asymmetric Rectified
Pulse duration ('Cp/Topt) 0.25
0.5
1
2
3
0.73 0.73 0.92
0.29 0.64 0.5
0.01 0.26 0.17
0.000001 0.08 0.41
0 0.03 0.92
zp is measured at the half-power level, ro is in all cases approximately equal to 0.6rp. Let us estimate the radiated power level for all the above three types of signals, assuming that cp2o/4n=I011 W / c m 2, ).opt= 1 ~tm, a = 0 . 5 2opt, po= 5 cm, and (vJc)2=0.9, and that the n o n l i n e a r i t y of the signal of the third type is governed by the coefficient e = 0.1. We will also take that the phase shift is selected optimal for the emission of radiation (fi~ = ~ 3 = 0 , tan fi2= 2O9Zo/[ (09%) 2 - 1 ] ). In that case, N * ~ 142 and the radiation m a x i m u m occurs at a frequency oc 100 GHz, i.e., falls within the millimeter wavelength range. The total radiation intensities for the above three types of signal are listed in table 1. It follows from the table that to achieve a noticeable emission of radiation, one can take one of the following two courses. First, one can try and generate very short, actually n o n m o d u l a t e d solitons whose practical realization will apparently require special experiment a n d theoretical studies which first of all should involve media with high polarization saturation levels, for saturation limits the life span of solitons [8]. 442
Secondly, one can try a n d detect solitons by letting them pass through a m e d i u m featuring second-order nonlinearities. In the latter case, the radiation efficiency will grow higher with the increasing pulse duration. The above analysis has thus shown that when solitons propagate in ring fibers, there can occur a sufficiently efficient radio emission qualitatively similar to the synchrotron radiation.
References
[ 1] A. Hasegawa and F. Tappert, At. Phys. Lett. 23 (1973) 142, 171. [2] L.F. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. [3]J.P. Gordon, L.F. Mollenauer, R.H. Stolen and W.J. Tomlinson, Optics Lett. 8 (1983) 289. [4] A. Hasegawa, Optical solitons in fibers, Springer Tracts of Modern Physics, Vol. 116 (Springer, Berlin, 1989). [5] A.S. Gouveia, A.S. Gomes and J.R. Taylor, J. Mod. Optics 35 (1988) 7. [6] L.F. Mollenauer and K. Smith, Optics Lett. 13 (1988) 675. [7] L.D. Landau and E.M. Lifschits, Field theory (Moscow, Nauka, 1973 ) (in russian ). [8] S. Gatz and J. Hermann, J. Opt. Soc. Am. B 8 ( 1991 ) 2296.
OPTICS COMMUNICATIONS
Radio emission of an optical soliton propagating in a ring fiber V.V. K l i m o v P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninskii Prospekt, Moscow, Russian Federation
and V.S. L e t o k h o v Institute of Spectroscopy, Russian Academy of Sciences, 142092 Troitsk, Moscow Region, Russian Federation Received 11 March 1993
Considered are emission processes occurring in the course of propagation of an optical soliton along a ring fiber. The spatial and temporal characteristics of radiation are calculated. It is demonstrated that conditions are possible in which radiation becomes observable.
1. Introduction The generation of solitons in optical fibers has recently become the object of many theoretical and experimental studies [ 1-4]. It has been shown theoretically [ 1 ] that in the approximation of a smooth pulse change over the length of the wave, its propagation is described by a nonlinear Schr~Sdinger equation allowing for soliton pulses, i.e., such pulses as propagate without changing shape. In refs. [2,3], such solitons have been experimentally demonstrated to exist, and in ref. [ 5 ], the generation of solitons 4 periods long at a frequency of 1.32 Ixm has even been reported. The considerable interest in such pulses is due to the expectations that high-throughput communication systems will be created on their basis. This prospect in mind, theoretical and experimental research has mainly been directed towards the stabilization of the form of solitons propagating over great distances (see, for example, ref. [6] ). Within the framework of this approach, no consideration has been given to the problem of emission of energy by solitons into the surroundings. But the similarities between solitons and elementary particles allow the problem to be raised of the emission of radiation by a propagating soliton. Naturally, no uniform rectilinear motion here will cause any no-
ticeable emission in a direction across the fiber: all the energy of the soliton is transferred along the fiber, as is the case with the uniform rectilinear motion of a charged particle. Extending this analogy further, there may take place the emission of radiation by a soliton propagating along a ring fiber (an analog of synchrotron radiation). It is the study of exactly this effect and the possibility of its observation that the present work has been devoted to.
2. Qualitative approach Let us first consider the problem of emission of radiation by a soliton in qualitative terms. It should primarily be noted here that the character of radiation emitted by a soliton will materially depend on the direction of its electric field. To illustrate, if in the course of propagation of the soliton the dielectric polarization vector of the fiber lies in the plane of the fiber ring, the system as a whole features an electric dipole moment directed along the radius-vector of the soliton, and since the electric dipole moment rotates together with the soliton, then a¢¢ 0, and so there takes place the emission of radiation whose frequency is governed by the rotational frequency of the soliton. But if in the course of propagation of the so-
0030-4018/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
437
Volume 100, number 5,6
OPTICS COMMUNICATIONS
liton the dielectric polarization vector of the fiber is directed along the symmetry axis of the ring, the radiation will predominantly be of a magnetic-dipole character, for in that case t h ¢ 0. Other polarization directions give rise to mixed types of radiation. The next important point is the modulation of the soliton in its propagation direction. If we assume that emission occurs at radio frequencies and the size of the soliton is negligibly small compared to the optical wavelength )~opt, it is evident that in the absence of modulation the contributions to the field from the radiation emitted by the different portions of the soliton are added together in the same phase, whereas any modulation gives rise to a destructive interference, and so it should be expected that the soliton radiation will be a maximum at a minimum modulation. By the way, this is also clear from the fact that the "static" electric field outside the fiber is maximal when there are no soliton elements characterized by opposite polarization signs.
3. Theoretical model
15 July 1993
angular velocity of the rotating pulse; Po is the radius of the ring, and v, is the propagation velocity (group) of the pulse. The polarization directions and pulse shapes selected enable one to solve the problem in analytical form. Other types of polarization are being studied in a similar fashion, and the results will be published in an individual paper. As to the fact that the form of the soliton remains unchanged in the course of propagation, we believe that its radiation loss is small and that its energy store is being constantly replenished. The effect of radiation on the dynamics of the soliton will be considered elsewhere. The form of polarization specified, the problem reduces to the solution of the Maxwell equations which in our case are reduced to the wave equation for the electric field
[ --V2-I-C-202/OI 2] E=4~[WC-2O2/Ot 2] P ,
which uses ordinary notation. Using the Green function (delayed potential) method, the solution of eq. (2) may we written in the form
E= [VV--c-2 02/Ot2]F , Consider the electric polarization wave P(r, t) propagating along a ring fiber without suffering any change in its form (fig. 1 )
P=Pof(P, z) g(q~-t2ot) ,
z ....
G
/
d3p Pof(P', z') g[9'-[2°(t-lr-r'l Ir - r ' l / c ) ]
(4)
In expression (4), use is made of a cylindrical coordinate system. Insofar as we are interested in the radiation field in the far zone (R>>p2/v~To), we then may make standard simplifications in calculating eq. (4), as a result of which we get the following expression for F:
×g{~o'-Oo[t-R/c+sin 0 c o s ( 9 - ~ ' ) p'/c Ro
+cosO z'/c]} ,
~ ~ " /~.
--%~o --
"
~_2a
Vg Fig. 1. Geometry of the problem under consideration. 438
F=
// ,/
,
~
where F is described by the integral
F = P o / R - ~ p' dp' dz' d~' f(p', z') .
'
(3)
( 1)
where Po is the constant vector directed along the axis of the fiber ring, OZ; f(p, z) = exp [ - (p2 + z 2 ) / 2 a 2 ] describes the field distribution over the cross section of the fiber; g(~o) = YgN exp(iN~) is the 2n-periodic function describing the pulse shape; 12o= vg/po is the
waveguide
(2)
(5)
where 0 and (p are angles in a spherical coordinate system (fig. 1 ). Substituting into eq. ( 5 ) the expression for g expanded in harmonics, we obtain
Volume 100, number 5,6
OPTICS COMMUNICATIONS
15 July 1993
It is not very difficult to find that of the two solutions of this equation the one suitable for the application of the steepest descent method is the point
F = P o / R - ~', gNexp [ -iN£2o(t-R/c) ] × ~p'dp'dz'd~o'f(p',z')
~0= 3~z+i ln[(1 _f12)1/2/fl]
.
× exp{iN[ ~o'- g2o/c(sin 0 cos (~o- ~0') p' + c o s 0 z ' ) 1}.
(6)
The integrals with respect to p' and z' are of the gaussian type and are easy to calculate under the assumption that the radius of the fiber is much smaller than that of the fiber ring, which allows integration with respect to p' to be extended between - o o and
In the ultrarelativistic case, the two saddle points merge, and so eq. (8) must be modified accordingly. In the case of fiber transmission, this situation apparently cannot occur, and we do not consider it here. By using eq. (3), we can now readily find the expression for the electric field: (27ra)2po £22
E= [Po-n(Pono) ]
Woo:
R
c2
- ~ guexp{iN[~o-f2o(t-R/c) ]
F = 27ra2p°p° ~ gN R
+ (aN/po)2(1 -/72)/2}
×exp{iN[~o-g2o(t-R/c) l - [at2oN cos 0/c]2/2}
XN2(--i)NJN(Nfl) [1 -i(a/po)2Nflcos~o *]
.
2n
(9)
× j do' exp{iN[~0'-flcos~0'
0
+ ½i(a/po)2Nfl2cosZq) ' ] } X [1 -i(a/po)ZNflcos~o '] .
(7)
In this expression, fl= vg/c sin 0. By analogy with the synchrotron radiation (see, for example, ref. [ 7 ] ), it can be assumed that in this sum the main part is played by terms with large INI values, which makes it possible to employ asymptotic methods. The results obtained confirm the validity of this supposition. So, if INI >> 1, the integral with respect to ~0' in eq. (7) can be calculated by the steepest descent method. In that case, we restrict ourselves to the radiatively most interesting case of relativistic pulse group velocities, i.e., the case fl~ 1. If, in addition, it is assumed here that INI << (po/a) 2, which is also justified by the results obtained, one may then write the following expression instead of eq. (7)
In this expression, n is the unit vector in the observation direction, and only those terms are retained which are responsible for radiation, i.e., those inversely proportional to R. The elementary analysis of eq. (9) shows that there exists a harmonic whose amplitude is a maximum. To find the serial number N* of this harmonic, assume that 1 << N* <
Ju(Nfl) = exp{N[ ( 1 __f12) 1/2_ arth ( 1 __f12) 1/2]} [2nN( 1 _f12) 1/2] 1/2
(10) to find the serial number of the maximum-amplitude harmonic to be 3 N * = 2[arth(l_fl2)l/2
F = (2~za)2p°p° ~ gN R
(l_fl2)l/2] ,
(11)
× exp{iN[~o--g2o( t--R/c) ] + (aN/po)2( 1 -/72)/2}
or, considering the fact that (1 _fl2)1/2 is small in the directions important for radiation,
×(-i)UJN(Nfl)[1-i(a/po)2Nflcos~o*],
N * ~ 9 / 2 [ ( 1 - f 1 2 ) 1/2] 3
(8)
where JN is a Bessel function,/7= vg/c, and ~0" is the saddle point defined by the equation 1 + fl sin ~0"= 0.
(12)
The above expression for the serial number of the maximum-amplitude harmonic agrees fully with the 439
Volume 100, number 5,6
OPTICS COMMUNICATIONS
15 July 1993
can be factored outside the integral sign. After differentiating, the expression for the electric field in the wave zone takes on the form 2ha 2190 -(22
E = - [Po-n(Pon) ] - -R ×
£
sin ~ +
c2 2n
f12+ 2fl2COS2(~
( l + f l s i n q~) 5
o
~ d~0g(~0) _ o
(15)
where g((o) is the function describing the modulation of the pulse. This expression can conveniently be presented in the form Fig. 2. Spectral composition of radiation as a function of the serial number o f the harmonic and observation angle ( v J c = 0.9 ).
assumptions made above when calculating integral (7) and with the use of approximation (10) to find the number itself. It can be seen from expressions ( 11 ) and ( 12 ) that iffl is not too close to unity, then N*•o<< O~opt, and so the radiation frequency falls within the radio frequency range. It is also clear from the above analysis that the results obtained hold true, provided that the pulse spectrum g~ varies but not too rapidly in the radio frequency band. In our case of femtosecond pulses, the analysis is absolutely correct.
The fact of radio emission proved, one can neglect in eqs. (9) and (5) the thickness of the fiber in comparison with the radio wavelength. As a result, we have the following expression for the electric field E=
- [Po-n(Pon)
2na2po 02 ] ¢2R Ot2 (13)
2n 0
dq/ g[~o-g2o(t-R/c)+~] . 1 + f l s i n q)(~,)
(14)
In the integral obtained, the second term has a sharp maximum at ( o - g 2 o ( t - R / c ) + ~ = 0 , while the first practically does not change in this region and hence 440
(16)
where the factor F allowing for the suppression of radiation as a result of modulation is given by
F= --1 f dtg(t) ,
(17)
I"o d
where for brevity use is made of the notation q)= cb[f2o(t-R/c)-(o ], and ro is the parameter characterizing the duration of the pulse. Expression (16) is the final result of our calculations and can serve as a starting point for further studies. In the text below, we will use eq. (16) to find the radiation power flow and power. The radiation power flow is defined by the standard expression
c [EH]R2 = c
4nn ]EI2R2'
(18)
and substituting it into eq. (16), we get
dI
To calculate the integral with respect to q¢, we change the variables q~'= ~(q/), ~u= ~0'- fl cos (o', and as a resuit, it assumes the form
]
002 flsin ~+flz+2fl2cos2qgF, X 2na2vgro - R c2 (l+flsin ~) 5
dI
0
f
[Po-n(Pon)
d ~ - 47r
2rt
× t dq~'g[~o-g2o(t-R/c)+~o'-flcos(o'].
E=-
¢ p2{UgZO~z(ZT~a~4ff6sin, 0 × (sin ~:~q-fl--I-2fl c 0 s 2 ~ ) 2 / . 2 . (1 + f l s i n ~)lo
(19)
Recall that/~= Vg/C, fl=/~sin 0, and q~= qb[f2o(t-R/ c)-~0]. The angular dependences of the radiation power flow are shown in fig. 3. To find the total radiative power, it is necessary to integrate expression (19) with respect to the angles. First we integrate with respect to the azimuth angle
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OPTICS COMMUNICATIONS
15 July 1993
or, recalling that t h a t / ~ 1,
l=cp2
(~0)2(~0a)4
10
3(1_/~2)6 F 2 .
(22)
4. D i s c u s s i o n of the results
Fig. 3. Soliton radiation directivity diagram ( t = R / c, vg/c = 0.9 ).
20
-
i
]
Ioc [p2(1-/~2) 21 - 1 ,
..jf-~
15 J4
/" ."
=~o~ ~ "~
5
]
.//
0
! / -5
//
-I0 0
20
40
e theta,
60
80
dgs
Fig. 4. Intensityofemissionintothefingbelt(~/c=0.9). ~0. The result of this integration is the power flow through a spherical belt and is expressed in terms of elementary functions as
dI
d(cos0)
_zfvgzo'~2{2na'~ 4
#4sin20
W5 × 2-~ (1155WS-2772W6+2170W 4 - 5803 W 2 + 27 )/-2,
(20)
where W = (l-fl2sinZO) -wz. The graph of W as a function of 0 is presented in fig. 4. The expression for the total radiated power is obtained after integrating eq. (20) with respect to 0
,=cP~(~)
2
4
( ~ o a) 2/~n(8/~4+15/~2+2)
15(1_~2)6
The total radiation power found in the preceding section depends materially on the pulse velocity, and if it is close enough to the velocity of light, efficient emission of radiation is possible. Note for comparison that in the case of synchrotron radiation [7], the radiation power is described by the formula
r2,
which gives a slower rise of the radiation power with the increasing pulse speed. Reducing the radius of the fiber ring will cause a rapid radiation power rise (as 1/p] ) and will displace the radiation spectrum maximum towards the high-frequency side (foc 1/Po). At Po acN*2 opt, our results are naturally inapplicable, for the soliton in that case emits all of its energy during a time equal to its life span, which means that in this region the notion of soliton has no sense. To reveal the effect of the pulse shape (modulation) on the efficiency of radiation, consider the following three types of femtosecond pulses (see fig. 5 ): (a) classical soliton of the envelope g l ( t ) =
cos( o9t + 6) /ch ( t/ro); (b) asymmetrical pulse gz(t) =cos(o9t+8) (t/Zo) exp(1 - t/r0); (c) Partially rectified pulse g 3 ( t ) = g l ( t ) [1 + eg, (t)]. In all these cases, o9 stands for the optical frequency and 6 is the phase shift which can apparently be controlled. The suppression factors for these pulses are calculated in analytical form n cos 6 cosh (nu/2) ' FE -- e[cos 6( 1 - u a) - 2 u sin 6] (1+u2)2
Fa-'-FI + E [ n / 2 + n u c o s ( 2 6 ) / s i n h ( n u ) ]
(21)
(23)
.
(24)
In expressions (24), u = ogro, and if the pulse width 441
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OPTICS COMMUNICATIONS
15 July 1993
b.
Fig. 5. Modulated pulse examples: (a) classical soliton, (b) asymmetrical pulse, and (c) partially rectified pulse. Table 1 Total radiation intensity, nW. Type of pulse Classical Asymmetric Rectified
Pulse duration ('Cp/Topt) 0.25
0.5
1
2
3
0.73 0.73 0.92
0.29 0.64 0.5
0.01 0.26 0.17
0.000001 0.08 0.41
0 0.03 0.92
zp is measured at the half-power level, ro is in all cases approximately equal to 0.6rp. Let us estimate the radiated power level for all the above three types of signals, assuming that cp2o/4n=I011 W / c m 2, ).opt= 1 ~tm, a = 0 . 5 2opt, po= 5 cm, and (vJc)2=0.9, and that the n o n l i n e a r i t y of the signal of the third type is governed by the coefficient e = 0.1. We will also take that the phase shift is selected optimal for the emission of radiation (fi~ = ~ 3 = 0 , tan fi2= 2O9Zo/[ (09%) 2 - 1 ] ). In that case, N * ~ 142 and the radiation m a x i m u m occurs at a frequency oc 100 GHz, i.e., falls within the millimeter wavelength range. The total radiation intensities for the above three types of signal are listed in table 1. It follows from the table that to achieve a noticeable emission of radiation, one can take one of the following two courses. First, one can try and generate very short, actually n o n m o d u l a t e d solitons whose practical realization will apparently require special experiment a n d theoretical studies which first of all should involve media with high polarization saturation levels, for saturation limits the life span of solitons [8]. 442
Secondly, one can try a n d detect solitons by letting them pass through a m e d i u m featuring second-order nonlinearities. In the latter case, the radiation efficiency will grow higher with the increasing pulse duration. The above analysis has thus shown that when solitons propagate in ring fibers, there can occur a sufficiently efficient radio emission qualitatively similar to the synchrotron radiation.
References
[ 1] A. Hasegawa and F. Tappert, At. Phys. Lett. 23 (1973) 142, 171. [2] L.F. Mollenauer, R.H. Stolen and J.P. Gordon, Phys. Rev. Lett. 45 (1980) 1095. [3]J.P. Gordon, L.F. Mollenauer, R.H. Stolen and W.J. Tomlinson, Optics Lett. 8 (1983) 289. [4] A. Hasegawa, Optical solitons in fibers, Springer Tracts of Modern Physics, Vol. 116 (Springer, Berlin, 1989). [5] A.S. Gouveia, A.S. Gomes and J.R. Taylor, J. Mod. Optics 35 (1988) 7. [6] L.F. Mollenauer and K. Smith, Optics Lett. 13 (1988) 675. [7] L.D. Landau and E.M. Lifschits, Field theory (Moscow, Nauka, 1973 ) (in russian ). [8] S. Gatz and J. Hermann, J. Opt. Soc. Am. B 8 ( 1991 ) 2296.