Femtosecond soliton propagation in an optical fiber
Chi-Feng Chen
1, Sien Chi
2, Boren Luo
21 Industrial Technology Research Institute, Hsin-Chu 300, Taiwan R. O. C.
2 Institute of Electro-Optical Engineering, National Chiao Tung University, Hsin-Chu 300, Taiwan R. O. C.
Abstract: An accurate wave equation beyond the slowly
varying envelope approximation for femtosecond soliton propagation in an optical fiber is derived by the iterative method. The derived equation contains higher nonlinear terms than the generalized nonlinear Schro¨dinger equation obtained previously. For a silica-based weakly guiding single mode fiber, it is found that those more higher-order non-linear terms, whose coefficients are proportional to the sec-ond-order dispersion parameter, are much smaller than the shock term. The 2.5-fs fundamental solitons is numerically simulated by using the generalized nonlinear Schro¨dinger equation and the full Maxwell’s equations. Comparing these two results, we have found that the generalized nonlinear Schro¨dinger equation well describes the propagation of the pulse even containing a single optical cycle.
Key words: Femtosecond soliton slowly varying envelope ap-proximation – nonlinear Schro¨dinger equation
1. Introduction
As the rapid development of the laser technology, ul-trashort optical pulses containing only a few optical cy-cles and with pulsewidth less than 10 fs have been gen-erated [1]. The propagation of an ultrashort pulse in a fiber is usually described by the generalized nonlinear Schro¨dinger equation based on the slowly varying en-velope approximation (SVEA) [2]. The validity of this equation becomes questionable when the pulse con-tains only a few optical cycles. To resolve this problem, several approaches, which do not make the SVEA, have been employed [3–7]. The full-vector nonlinear Maxwell’s equations have been solved by direct inte-gration [3, 4], but this is very time consuming. By using an operator method and assuming that the nonlinear-ity is small [5, 6] or by representing the electric field as the superposition of monochromatic waves [7], modi-fied wave equations can be derived. In addition, an iterative method has been used to derive a wave equa-tion for ultrashort pulses [8].
In this paper, we will derive a wave equation using the electric field expansion of [7], the iterative method
of [8] and the order of magnitude considerations of [9]. Through the first iteration, we obtain a wave equation, which has four more higher-order nonlinear terms than the equation obtained previously [5–8]. For a silica-based weakly guiding single mode fiber, we have found those more higher-order nonlinear terms, the coefficients of which are proportional to the second-order dispersion parameter, are much smaller than the shock term. We numerically investigate 2.5-fs funda-mental solitons by using the generalized nonlinear Schro¨dinger equation and the full Maxwell’s equations [3, 4]. After we compare these two results, we have found that the generalized nonlinear Schro¨dinger equation well describes the propagation of the pulse even containing a single optical cycle.
2. Derivation of the wave equation
We derive the wave equation for a femtosecond pulse propagating in a single-mode fiber with a third-order nonlinearity. The electric field Eðx; y; z; tÞ which propa-gates in the fiber along the z-direction can be ex-pressed by
Eðx; y; z; tÞ ¼ Fðx; yÞ fðz; tÞ ; ð1Þ where Fðx; yÞ is the normalized linear eigenfunction of the mode excited in the fiber and fðz; tÞ can be further represented as a superposition of monochromatic waves,
fðz; tÞ ¼ 1 2p
ð
jðz; wÞ exp fi½bðwÞz wtg dw ; ð2Þ where bðwÞ ¼ nðwÞw=c is the mode propagation con-stant at frequency w, c is the velocity of light in va-cuum, and nðwÞ is effective refractive index. It is cus-tomary to express fðz; tÞ by
fðz; tÞ ¼ Aðz; tÞ exp ½iðb0z w0tÞ ; ð3Þ
where Aðz; tÞis the field envelope, w0 is the angular
frequency of the carrier wave, and b0¼ bðw0Þ. From
the Maxwell’s equations, we obtain the wave equation r2E 1 c2 @2E @t2 ¼ m0 @2P L @t2 m0 @2P NL @t2 ; ð4Þ
International Journal for Light and Electron Optics
0030-4026/02/113/06-267 $ 15.00/0 Received 25 January 2002; accepted 28 May 2002.
Correspondence to: S. Chi E-mail: [email protected] http://www.urbanfischer.de/journals/optik
where m0is the permeability in vacuum, and the linear part PLand the nonlinear part PNLof the induced
po-larization are related to electric field Eðx; y; z; tÞthrough the following equations:
PLðx; y; z; tÞ ¼ "0 Ð 1 1 cð1Þðt t0Þ Eðx; y; z; tÞ dt0; ð5Þ PNLðx; y; z; tÞ ¼ "0Ð Ð 1 1 Ð cð3Þðt t1;t t2;t t3Þ
Eðx; y; z; t1Þ Eðx; y; z; t2Þ E*ðx; y; z; t3Þ dt1dt2dt3; ð6Þ
where "0 is the vacuum permittivity, cð1Þðt t0Þ is the
linear susceptibility response function, and cð3Þðt t
1;t t2;t t3Þ is the third-order nonlinear
sus-ceptibility response function. Substituting eqs. (1), (2), (5), and (6) into eq. (4), we have
@jðz; wÞ @z ¼ ikw2 2c2bðwÞ ð ð dw0dw00jðz; w0Þ jðz; w00Þ j*ðz; w0þ w00 wÞ cð3Þðw w0Þ exp ði Db zÞ þ i 2bðwÞ @2jðz; wÞ @z2 ; ð7Þ
where k¼Ð ÐjFðx; yÞj4dx dy=Ð Ð jFðx; yÞj2dx dy, cð3ÞðwÞ ¼Ð cð3ÞðtÞ exp ðiwtÞ dt is the third-order susceptibility, and Db¼ bðw0Þ þ bðw00Þ bðw0þ w00 wÞ bðwÞ. We
expand bðwÞ around w0 up to the fourth order, we
have bðwÞ ¼ b0þ b1Dwþb2 2 Dw 2þb3 6 Dw 3þb4 24Dw 4, where Dw¼ w w0, b0¼ bðw0Þ, bj¼ @jb @wjj w¼w0
for j¼ 1 to 4. b1 is the reciprocal group velocity. b2, b3, and b4 are the second-order, third-order, and fourth-order dispersion parameters, respectively. Sub-stituting jðz; wÞ ¼ ~AAðz; DwÞ exp fi½bðwÞ bðw0Þ zg
into eq. (7) and taking the inverse Fourier transform Aðz; tÞ ¼ 1
2p ð
~ A
Aðz; w w0Þ exp ½iðw w0Þ t dw, we have
@A @z ¼ H þ i 2b0 CbAh; ð8Þ where H¼ b1@A @t ib2 2 @2A @t2 þ b3 6 @3A @t3 þ ib4 24 @4A @t4 þ ig 1 þ i 2 w0 b1 b0 @ @t 1 w2 0 2b1 b0w0 þb 2 1 b20 b2 2b0 ! @2 @t2 # NA ; ð9Þ Cb¼ 1 ib1 @ @tþ b2 2b0 b21 b20 ! @2 @t2; ð10Þ Ah¼ @2A @2z þ 2b1 @ @tþ ib2 @2 @t2 b3 3 @3 @t3 @A @z ib1 @ @t b2 2 @2 @t2 ib3 6 @3 @t3 2 A ; ð11Þ where g¼n2w0 cAeff
, n2is the Kerr coefficient, Aeffis
effec-tive fiber cross section, Aeff¼
Ð Ð
jFðx; yÞj2dx dy k , and the higher order terms are neglected. The response Nðz; tÞ is described by [10] Nðz; tÞ ¼ ð1 aÞ jAðz; tÞj2þ a Ð t 1 dt0fðt t0Þ jAðz; t0Þj2: ð12Þ On the right-hand side of eq. (12), the first term repre-sents Kerr nonresonant virtual electronic transitions in the order of about 1 fs or less [4], the second term re-presents delayed Raman response, fðtÞ is the delayed response function, and a¼ 0:18 parameterizes the re-lative strengths of Kerr and Raman interactions. In this paper, fðtÞ models a single Lorentzian line centered on the optical phonon frequency 1=t1 and having a
band-width of 1=t2(the reciprocal phonon lifetime).
fðtÞ ¼t 2 1þ t22 t1 t22 expðt=t2Þ sinðt=t1Þ ; ð13Þ where t1¼ 12:2fs, t2¼ 32fs.
We now use order of magnitude considerations to simplify the calculation. The dispersion and the non-linear terms in the nonnon-linear Schro¨dinger equation are of the same order of magnitude for the fundamental soliton, hence, we have
b2 @ 2A @t2 bT2A2 0 gjAj 2 A N2 p ; ð14Þ where the parameter Np¼ ½gP0T02=jb2j
1=2
is the order of the soliton and Np¼ 1 for the fundamental soliton,
T0¼ Tw=1:763, Tw is the pulse full width at half
max-imum, and P0is peak power of the incident pulse. In a
silica-based weakly guiding single mode fiber, b0 n0w0 c , b1 n0 c, b 2 1 b0b2, and b31 b0b3. Defin-ing s¼ 1 w0T0
, we obtain from eq. (14) for the funda-mental soliton
jb2j T2
0
gjAj2 w20jb2j s2: ð15Þ By using the iterative technique and the order of mag-nitude considerations, we first neglect the second term on the right-hand side of eq. (8) and obtain in the zer-oth order approximation
@A
@z ¼ H : ð16Þ
From eq. (16), the first order approximation of Ahis
Ah¼ g2N2A 2gb2A ð1 aÞ @Aðz; tÞ @t 2 " þ a ðt 1 dt0fðt t0Þ @Aðz; t 0Þ @t0 23 5 2gb2@N @t @A @t : ð17Þ
By substituting eq. (16) into eq. (8), the first order ap-proximation of the wave equation is
@A @z ¼ b1 @A @t ib2 2 @2A @t2 þ b3 6 @3A @t3 þ ib4 24 @4A @t4 þ ig NA þ ia1 @ @t NA iga2 @2NA @t2 igb2A b0 ð1 aÞ@Aðz; tÞ @t 2 þ a ðt 1 dt0fðt t0Þ@Aðz; t0Þ @t0 2 2 4 3 5 igb2 b0 @N @t @A @t i 2b0 g 2N2A ; ð18Þ where a1¼ 2 w0 b1 b0 1 w0 , a2¼ 1 w2 0 2b1 b0w0 þb 2 1 b20 b2 2b0 b2
2b0, and retaining all terms to the order of s
4. If
we make a second iteration, we find that the equation does not change up to the order of s4. On the right-hand side of eq. (18), the term with coefficient a1 is of
order s3, and the last five terms representing nonlinear
high-order terms of order s4 which are newly derived terms. Comparing the s4 term with coefficient a2 with
the s3term with coefficient a1, we have
r¼ a2 @2NA @t2 max a1 @NA @t max jb2j b0T2 0 1 w0T0 jb2j c n0T0 : ð19Þ
We consider a single cycle pulse. At the wavelength l¼ 1:55 mm, b2¼ 20 fs2/mm, pulsewidth Tw 5.17 fs
and we have r¼ 1:4 103. At l¼ 0:8 mm,
b2¼ 38:5 fs2/mm, pulsewidth Tw 2.67 fs and we have
r¼ 5:1 103 [6]. Similarly we can show that other s4
terms are much smaller than the s3 term. Therefore,
all s4 terms can be neglected for the single cycle pulse
in the low loss window of the fiber.
The equation (18) is used to describe the propagation. In dimensionless soliton units, it can be rewritten as
@ @xu¼ i 2 @2u @t2þ b @3u @t3þ ib4 24jb2j T2 0 @4u @t4þ i NNu 1 w0T0 @ @t NNu ib2 b0T2 0 1 2 @2NNu @t2 þ ð1 aÞ @u @t 2 " ( þa ðt 1 dt0 f ðt t0Þ @u @t0 23 5þ@ NN @t @u @tþ 1 2 N N2u 9 = ;; (20) where x¼ z LD , t¼t b1z T0 , u¼NPffiffiffiffiffiffiA P0 p , b b3 6jb2j T0 , LD¼ T2 0
jb2j is dispersion length, and NN¼ N2
PN
P0
. When the terms of order s4are neglected, eq. (20) reduces to
@ @x u¼ i 2 @2u @t2þ b @3u @t3þ ib4 24jb2j T2 0 @4u @t4 þ i NNu 1 w0T0 @ @t NNu : ð21Þ
3. Full Maxwell’s equations model
The Maxwell’s equations for the optical pulse linearly polarized in x-direction propagation in z-direction are written as @Hy @t ¼ 1 m0 @Ex @z ; ð22aÞ @Dx @t ¼ @Hy @t ; ð22bÞ Dx¼ "0"rExþ Px: ð22cÞ
Here m0 and "0 are the permeability and permittivity
coefficients in free space, "r is the relative permittivity,
Dx is the electric filed displacement and Pxis the
elec-tric polaritation.
Px consists of linear part PLx and nonlinear part PNLx ,
Px¼ PLx þ P NL
x . The linear polarization P L
z is given by
convolution of Ezðx; tÞ and first-order susceptibility
function cð1ÞðtÞ. PL zðx; tÞ ¼ "0 Ð 1 1 cð1Þðt ttÞ E zðx; tÞ dtt; ð23Þ where cð1ÞðtÞ ¼w 2 rð"s "rÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 r d2 4 s expðdt=2Þ sin ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 r d2 4 s t 0 @ 1 A:
wris dipole resonant frequency d is damping constant.
PNL
z is given by convolution of Ezðx; tÞ and third-order
susceptibility cð3ÞðtÞ PNLz ðx; tÞ ¼ "0 Ð 1 1 Ð 1 1 Ð 1 1 cð3Þðt tt 1;t tt2;t tt3Þ Ezðx; tt1Þ Ezðx; tt2Þ Ezðx; tt3Þ dtt1dtt2dtt3:ð24Þ
We consider the nonlinear polarization with single time convolution
PNLz ðx; tÞ ¼ "0cð3ÞEzðx; tÞ Ð 1 1
gðt ttÞ E2zðx; ttÞ dtt; ð25Þ
where cð3Þis the nonlinear coefficient. The response is
given by phonon interaction fðtÞ and nonresonant elec-tric effects dðtÞ.
gðtÞ ¼ ð1 aÞ dðtÞ þ a f ðtÞ ; ð26Þ where fðtÞ is given in eq. (13).
4. Numerical results
In an attempt to verify the validity of the eq. (21) for describing the propagating of ultrashort pulse, we
si-mulate the 2.5-fs fundamental soliton propagation by eqs. (20), (21) and the full Maxwell’s equations. The fiber parameters are: soliton wavelength l¼ 1:55 mm, wr¼ 8 1013rad/s, "1¼ 2:25, "s¼ 5:25, d¼ 1:0
109s1, and the nonlinearity g¼ 2
106W1mm1. From the permittivity function cð1ÞðtÞ, we obtain b1¼ 5:01 103fs/mm, b2¼ 24:56 fs2/mm,
b3¼ 61:97 fs3/mm, and b
4¼ 209:64 fs3/mm. eqs. (20)
and (21) are solved by the split-step Fourier method. eq. (22) is directly and iteratively computed by follow-ing the algorithm of the FD-TD method [4]. Figs. 1a and 1b are the pulse shapes of the 2.5-fs fundamental soliton in 5LD simulated by using the generalized
non-linear Schro¨dinger equation, eq. (21), and the full Max-well’s equations, eq. (22), respectively. Using the mov-ing frame relation t¼t b1z
T0
, we transform Fig. 1b
into temporal distribution. After the transformation, one can see that Fig. 1a and Fig. 1b are almost the same. From Fig. 1, the soliton phenomenon is induced by third-order dispersion [11, 12], self-steepening effect [13] and delayed Raman response [14, 15]. The angular frequency of the pulse is 200 THz, which is 10-times to the spectrum of Raman gain spectrum. It can be seen from Figs. 1a and b that the pulse is with oscillation structure and dispersive wave in tailing edge. The dominant effect by third-order dispersion is shown. We can find the pulse shape in Figs. 1a is consistent with that in Fig. 1b. The same propagation also simulated by using eq. (20), and it is found that two numerical re-sults by using eq. (20) and (21) differ less than 0.5%. It is demonstrated that eq. (21) could well describe the propagation of the 2.5-fs fundamental soliton in fiber. On the other hand, the coefficient of s4order is found
to be negligible.
5. Conclusion
In conclusion, we have used iterative method to derive a wave equation for femtosecond soliton propagation in an optical fiber. The derived equation contains high-er nonlinear thigh-erms than the equation obtained pre-viously. It is found that those more higher-order non-linear terms, the coefficients of which are proportional to the second-order dispersion parameter, are much smaller than the shock term in a silica-based weakly guiding single mode fiber. The propagations of 2.5-fs fundamental soliton by using the generalized nonlinear Schro¨dinger equation and the full Maxwell’s equations are numerically simulated. Comparing these two re-sults, we found that the generalized nonlinear Schro¨-dinger equation well describes the propagation of the pulse even containing a single optical cycle.
Acknowledgement. This work was supported in part by the National Science Council, Taiwan, R.O.C. under Contract NSC 89-2215-E-009-112, the Academic excellence program of R.O.C. Ministry of Education under Contract 90-E-FA06-1-4-90X023 and National Center for High-Performance Comput-ing.
References
[1] Fork RL, Brito Cruz CH, Becker PC, Shank CV: Com-pression of optical pulse to six femtosecond by using cu-bic phase compensation. Opt. Lett. 12 (1987) 483–488 [2] Agrawal GP: Nonlinear Fiber Optics 2nd, pp. 28–59,
Academic Press, New York, 1995
[3] Joseph RM, Hagness SC, Taflove A: Direct time integra-tion of Maxwell’s equaintegra-tions in linear dispersion media with absorption for scattering and propagation of femto-second electromagnetic pulses. Opt. Lett. 16 (1991) 1412– 1414
[4] Goorjian PM, Taflove A, Joseph RM, Hagness SC: Com-putational modeling of femtosecond optical solitons from Maxwell’s equation. IEEE J. Quantum Electron. 28 (1992) 2416–2422
[5] Blow KJ, Wood D: Theoretical description of transient stimulated Raman scatering in optical fiber. IEEE J. Quantum Electron. 25 (1989) 2665–2673
Fig. 1. Pulse shapes of the 2.5-fs fundamental soliton in 5 LD simulated by using a) the generalized nonlinear Schro¨dinger equation and b) the full Maxwell’s equations.
[6] Brabec T, Krausz F: Nonlinear optical pulse propagation in the single-cycle regime. Phys. Rev. Lett. 17 (1997) 3282–3285
[7] Mamyshev V, Chernikov SV: Ultrashort-pulse propaga-tion in optical fibers. Opt. Lett. 15 (1990) 1076–1078 [8] Chi S, Wen S: Derivation of a wave equation for pulse
propagation beyond a slowly varying envelope approxi-mation. Opt. Quantum Electron. 28 (1996)1351–1357 [9] Chi S Guo Q: Vector theory of self-focusing of an optical
beam in kerr media. Opt. Lett. 20 (1995) 1598–1600 [10] Stolen RH, Tomlinson WJ: Raman response function of
silica-core fibers. J. Opt. Am. B 6 (1989)1159–1166 [11] Wai PKA, Menyuk CR, Lee YC, Chen HH: Nonlinear
pulse propagation in the neighborhood of the zero-disper-sion wavelength of monomode optical fibers. Opt. Lett. 11 (1986) 464–466
[12] Wai PKA, Chen HH, Lee YC: Radiations by solitons at the zero group-dispersion wavelength of single-mode opti-cal fibers. Phys. Rev. A 41 (1990) 426–439
[13] Ohkuma K, Ichikawa YH, Abe Y: Soliton propagation along optical fibers. Opt. Lett. 12 (1987) 516–518 [14] Hodel W, Weber HP: Decay of femtosecond higher-order
solitons in an optical fiber induced by Raman self-pump-ing. Opt. Lett. 12 (1987) 924–926
[15] Tai K, Hasegawa A, Bekki N: Fission of optical solitons induced by stimulated Raman effect. Opt. Lett. 13 (1988) 392–394
