Propagation characteristics of fast light in an
erbium-doped fiber amplifier
Senfar Wen1,2,*and Sien Chi3,4 1
Institute of Electrical Engineering, Chung Hua University, 30 Tung Shiang, Hsinchu, Taiwan 300, China 2
Institute of Engineering Science, Chung Hua University, 30 Tung Shiang, Hsinchu, Taiwan 300, China 3
Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, China
4
Department of Electrical Engineering, Yuan Ze University, 135 Yuandong Road, Chungli, Taiwan 320, China
*Corresponding author: [email protected]
Received October 25, 2007; revised April 9, 2008; accepted April 16, 2008; posted April 18, 2008 (Doc. ID 88998); published May 29, 2008
A perturbation method is used to study the interactions among the signal power, pump power, and metastable population density for fast light in an erbium-doped fiber amplifier. The impact of temporal pump depletion (TPD) on fast light is investigated in which TPD is the response of the pump power to the temporal variation of the metastable population density. It is found that the gain coefficient and the absolute value of the negative group velocity are overestimated without considering the TPD. The effects of high-order dispersions on fast light are also shown. © 2008 Optical Society of America
OCIS codes: 260.2030, 060.2320.
1. INTRODUCTION
The propagation of an optical pulse in the highly disper-sive medium for slow or fast light was intendisper-sively inves-tigated [1–10]. It was reported that slow and fast light can be observed by using the effect of coherent population oscillation (CPO) in an erbium-doped fiber amplifier (EDFA) [9,10]. Due to the interaction between the fields and the erbium ions in an EDFA, there is the spectral gain dip of a narrow bandwidth. According to the Kramers–Kronig relations, the EDFA becomes a highly dispersive medium for the pulse in the presence of a gain dip. The spectral gain dip provides the dispersion for fast light. The case of a 9 m erbium-doped fiber strongly pumped by a 980 nm semiconductor laser diode was shown in [10]. The 1550 nm input pulse of 0.5 ms in width and 0.5 mW in power superimposed on a strong continu-ous wave (cw) control beam of the same wavelength was launched into the EDFA. The pulse backward propaga-tion owing to a negative group velocity was reported to be experimentally observed. The group index of the pulse is estimated to be about −4000.
Because the interested pulse width is much longer than the polarization dephasing time, the interaction of the pulse and the population of the doped erbium ions in an EDFA can be described by the coupled equations of power evolution equation and rate equation. A perturbation method to derive the time delay and gain (loss) coefficient of a sinusoidally modulated wave was used in [9]. This method linearizes the coupled equations by assuming that the power of the sinusoidally modulated wave is much less than that of the control beam. Under this as-sumption, the temporal variation of the population inver-sion induced by the sinusoidally modulated wave can also be assumed to be much less than the steady-state popu-lation inversion induced by the control beam. From the
linearized coupled equations, the gain coefficient and group velocity of the sinusoidally modulated wave can be derived. However, in this paper, we show that this pertur-bation method is not accurate in an EDFA even for the case that the assumption of perturbation is valid. The nu-merical results solved from the complete coupled equa-tions without linearization show that the gain coefficient and the absolute value of the negative group velocity are overestimated.
We find that the inaccuracy is due to the temporal pump depletion (TPD) that is not included in the above perturbation method. The pump power depleted by the control beam is not time varying. An optical pulse de-pletes the metastable population density. The pump power is absorbed more when the metastable population density is depleted. The TPD is the pump power temporal variation responding to the temporal variation of the metastable population density absorbed by the optical pulse. In this paper, we develop the perturbation method including the TPD effect. It is shown that our method is accurate compared with the results directly solved from the complete coupled equations. The impact of the TPD on the gain coefficient and group velocity is shown. In addi-tion, the pulse delay time and pulse shape distortion resulting from high-order dispersions induced by CPO in an EDFA are also studied.
2. COUPLED EQUATIONS
The energy levels of an EDFA can be approximated as a three-level system. However the decay rate from the up-per level to the metastable level is much faster than the decay rate from the metastable level to the ground level. Because the population density of the upper level is
ligible, the signal and pump powers in an EDFA can be described by the following equations [11]:
Ps z + ngs c Ps t = −共␣s+␣ls兲Ps+共␣gs+␣ls兲N2Ps, 共1兲 Pp z + ngp c Pp t = −共␣p+␣lp兲Pp+共␣gp+␣lp兲N2Pp, 共2兲
where Psis the signal power including the pulse and
con-trol beam powers; Ppis the forward pump power; ngsand ngp are the group indexes of the signal and pump in the
absence of doped erbium ions, respectively; c is the light velocity in vacuum;␣sand ␣pare the intrinsic fiber loss
coefficients at the signal and pump wavelengths, respec-tively; ␣ls and ␣lp are the absorption coefficients at the
signal and pump wavelengths, respectively, which are due to doped erbium ions when population is completely in the ground level;␣gsand␣gpare the gain coefficients at the
signal and pump wavelengths, respectively, which are due to doped erbium ions when population is completely in the metastable level. In Eqs.(1)and(2), N2= n2/ ntis the
nor-malized metastable population density in which n2and nt
are the population density of the metastable level and the doping density, respectively.
The normalized metastable population density can be described by the rate equation [11],
dN2 dt = 1
冉
Ps Psth + Pp Ppth冊
− 1 冉
1 + Ps Psis + Pp Ppis冊
N2, 共3兲where is the lifetime of the metastable level and
Pkth=Aehvknt ␣lk , 共4a兲 Pkis= Aehvknt 共␣gk+␣lk兲 , 共4b兲
where k = s and p; hvsand hvpare the photon energies of
the signal and pump, respectively; and Aeis the effective
doping area.
Equations (1)–(3) are the coupled equations that describe the interaction of the optical fields and doped erbium ions. The coupled equations can be numerically solved with the initial conditions
Ps共z = 0,t兲 = Pc0+ Pa0共t兲, 共5兲
Pp共z = 0,t兲 = Pp0, 共6兲
where Pc0is the input control beam power, Pa0共t兲 is the
input pulse power envelope; and Pp0is the input forward
pump power. In this paper, we will consider the Gaussian input pulse
Pa0共t兲 = Ps0exp关− 共t/T0兲2兴, 共7兲 where Ps0is the pulse peak power and the FWHM pulse
width Tw= 2关ln共2兲兴1/2T0.
For the considered EDFA, =10.5 ms, Ae= 3.14m2, nt= 1⫻1025m−3, the absorption cross section at 980 nm in wavelength is 2.1⫻10−25m, the absorption and emission
cross sections at 1550 nm in wavelength are 3.2 ⫻10−25m and 3.78⫻10−25m, respectively, and the EDFA length L = 10 m. At 980 nm in wavelength, ␣p
= 1.7 dB/ km, ␣lp= 4.52 dB/ m, and ␣gp= 0. At 1550 nm in
wavelength, ␣s= 0.4 dB/ km, ␣ls= 3.11 dB/ m, and ␣gs
= 3.67 dB/ m. Since the delay time contributed from the group index ng= 1.5 in an EDFA is Lng/ c = 50 ns, which is
much less than the interested millisecond pulse width, the terms with ngs and ngp in Eqs. (1) and (2),
respec-tively, are negligible. We take the pulse width Tw
= 0.5 ms and the pump power Pp0= 180 mW to show the
numerical results in Section 4, where the control beam power Pc0and the peak power Ps0are varied.
3. PERTURBATIVE SOLUTION
The signal power along the EDFA can be written as
Ps共z,t兲=Pc共z兲+Pa共z,t兲. For the CPO effect, the pulse
power Pa共z,t兲 is much less than the control beam power Pc共z兲 [12], i.e., Pc共z兲 !Pa共z,t兲兩. For simplicity the z
de-pendence of all variables will not be shown in the follow-ing, unless they are specified. The normalized metastable population density can be written as N2共t兲=Nc+ Na共t兲,
where Ncand Na共t兲 are the normalized metastable
popu-lation densities corresponding to Pc and Pa共t兲,
respec-tively, hence兩Nc兩 兩Na共t兲兩. The signal power depletes the
pump power through the metastable population density. The corresponding pump power can be written as Pp共t兲
= Ppc+ Ppa共t兲, where Ppc and Ppa共t兲 are the pump powers
corresponding to Nc and Na共t兲, respectively, hence Ppc兩Ppa共t兲兩. The powers and normalized metastable
population density can be written as
Ps共t兲 = Pc+
冕
P˜a共⍀兲exp共− i⍀t兲d⍀, 共8兲 Pp共t兲 = Ppc+冕
P˜pa共⍀兲exp共− i⍀t兲d⍀, 共9兲 N2共t兲 = Nc+冕
N˜a共⍀兲exp共− i⍀t兲d⍀, 共10兲where P˜a共⍀兲, P˜pa共⍀兲, and N˜a共⍀兲 are the Fourier
trans-forms of Pa共t兲, Ppa共t兲, and Na共t兲, respectively. Note that P
˜a共⍀兲 and P˜pa共⍀兲 are the spectra of the power envelopes.
Because Pa共t兲, Ppa共t兲, and Na共t兲 are real, we have the
relations P˜a共⍀兲=P˜a* 共−⍀兲, P˜pa共⍀兲=P˜pa* 共−⍀兲, and N˜a共⍀兲
= N˜a* 共−⍀兲.
Substituting Eqs.(8)–(10)into Eq.(3)and equating the terms of the same order of magnitude, we have
Nc=
冉
Pc Psth +Ppc Ppth冊
共c兲−1, 共11兲 N˜a共⍀兲 =冋
冉
P˜a Psth +P ˜ pa Ppth冊
−冉
P ˜ a Psis +P ˜ pa Ppis冊
Nc册
共c − i⍀兲−1, 共12兲 where c is the resonant angular frequency definedc=
冉
1 + Pc Psis+ Ppc Ppis冊
−1. 共13兲Substituting Eqs.(8)–(12)into Eqs.(1)and(2)and equat-ing the terms of the same order of magnitude, we have the coupled equations dPc dz = −共␣s+␣ls兲Pc+共␣gs+␣ls兲NcPc, 共14兲 dPpc dz = −共␣p+␣lp兲Ppc+共␣gp+␣lp兲NcPpc, 共15兲 dP˜a共⍀兲 dz = cssP ˜ a共⍀兲 + cspP˜pa共⍀兲, 共16兲 dP˜pa共⍀兲 dz = cppP ˜ pa共⍀兲 + cpsP˜a共⍀兲, 共17兲
where the coefficients
css= i ngs c ⍀ − 共␣s+␣ls兲 + 共␣gs+␣ls兲
冋
Nc+冉
Pc Psth − Pc Psis Nc冊
共c − i⍀兲−1册
, 共18兲 csp=共␣gs+␣ls兲冉
Pc Ppth− Pc PpisNc冊
共c − i⍀兲 −1, 共19兲 cpp= i ngp c ⍀ − 共␣p+␣lp兲 + 共␣gp+␣lp兲冋
Nc+冉
Ppc Ppth− Ppc PpisNc冊
⫻共c − i⍀兲−1册
, 共20兲 cps=共␣gp+␣lp兲冉
Ppc Psth −Ppc Psis Nc冊
共c − i⍀兲−1. 共21兲For the case without TPD, the gain coefficient and propagation constant of P˜a共⍀兲 are the real and imaginary
parts of css, respectively. “Propagation constant” usually
refers to the electric field, but for the present analysis it refers to the power envelope. The group index of Pa共t兲 can
be obtained from the derivative of the propagation con-stant with respect to⍀ at ⍀=0. However, TPD always ex-ists. Thus, the gain coefficient and propagation constant of P˜a共⍀兲 must be solved from the coupled Eqs.(14)–(17).
The following shows the numerical solving procedures. Step 1. The cw powers Pcand Ppcalong the EDFA are
solved from Eqs.(14)and (15)with the boundary condi-tions Pc共z=0兲=Pc0and Ppc共z=0兲=Pp0in which Ncis given
by Eq.(11). Note that Eqs.(14)and(15)are independent of Eqs.(16)and(17).
Step 2. P˜a共z,⍀兲 and P˜pa共z,⍀兲 are solved from Eqs.(16)
and (17) with the boundary conditions P˜a共z=0,⍀兲
= P˜a0共⍀兲 and P˜pa共z=0,⍀兲=0, where P˜a0共⍀兲 is the Fourier
transform of the input pulse envelope Pa0共t兲, and there is
no initial temporal pump power variation. Note that the coefficients given by Eqs. (18)–(21) along the EDFA require the cw powers Pc and Ppc solved in Step 1. The
approximate solutions of the pulse shape and temporal pump power variation are
Pa共z,t兲 =
冕
P˜a共z,⍀兲exp共− i⍀t兲d⍀, 共22兲 Ppa共z,t兲 =冕
P˜pa共z,⍀兲exp共− i⍀t兲d⍀, 共23兲respectively. The integration of Eqs.(22)and (23)can be numerically calculated with an inverse fast Fourier trans-form (FFT) routine.
Step 3. Because the coupled Eqs.(16)and(17)are lin-ear, we define the spectral transmittance of the pulse en-velope at the distance z as T共z,⍀兲=P˜a共z,⍀兲/P˜a0共⍀兲, which
can be written as T共z,⍀兲=兩T共z,⍀兲兩exp关i共z,⍀兲兴, and 共z,⍀兲 is the phase of T共z,⍀兲. The gain coefficient and propagation constant of the spectral component of the pulse envelope are ga共z,⍀兲=d ln共兩T共z,⍀兲兩兲/dz and
a共z,⍀兲=d共z,⍀兲/dz, respectively.
The group delay time along the EDFA evaluated at ⍀=0 is Td0共z兲=d共z,⍀兲/dz兩⍀=0. However, the actual pulse delay time should be evaluated from the pulse shape given by Eq. (22). We will show in Section 4 that the actual pulse delay time is significantly influenced by high-order dispersions.
At the output of the EDFA, the accumulated gain ␥a共⍀兲=ln共兩T共z=L,⍀兲兩兲 and the phase shift a共⍀兲
=共z=L,⍀兲. The gain coefficient and propagation constant are even and odd functions, respectively, for the CPO in a two-level system [12]. In Section 4, it is shown that␥a共⍀兲
and a共⍀兲 are also even and odd functions, respectively.
Thus we may expand them as
␥a共⍀兲 + ia共⍀兲 =
兺
k=0 k:even Q 1 k!␥ak⍀ k+ i兺
k=1 k:odd Q 1 k!ak⍀ k, 共24兲where Q is an integer; ␥ak and ak are the coefficients
obtained by numerically fitting ␥a共⍀兲and a共⍀兲 with
Eq.(24). For the cases considered in this paper, we take
Q = 29 so that␥a共⍀兲 anda共⍀兲 can be fitted well.␥akand
ak represent dispersion coefficients that are the
deriva-tives of␥a共⍀兲 anda共⍀兲 at ⍀=0, respectively. It is noticed
that a1= Td0共z=L兲. In Eq. (24), the even and odd order
terms can be called the gain dispersion and phase shift dispersion, respectively. For studying the effect of disper-sion induced by CPO on the pulse shape, we define the output pulse shape:
Pa共M兲共t兲 =
冕
P˜a0共⍀兲exp关␥a共M兲共⍀兲 + ia共M兲共⍀兲 − i⍀t兴d⍀,共25兲 where M is an integer not larger than Q; ␥a共M兲共⍀兲 and a共M兲共⍀兲 are the partial accumulated gain and phase shift
including the dispersions up to the Mth order, respec-tively, i.e., they are the accumulated gain and phase shift given in Eq. (24) except that the terms of order larger than M are dropped. Comparing the pulse shapes of Pa共M兲 ⫻共t兲 and Pa共M−1兲共t兲, one can clearly see the effect of the Mth
order dispersion on the pulse shape.
For the case without TPD, we may solve Eqs.(14)and
(16) with the coefficient csp= 0. From the solutions, the
output pulse shape, gain coefficient, and propagation constant can be calculated with similar methods as shown above.
4. NUMERICAL RESULTS
It is found that the approximate solutions solved from Eqs.(14)–(17)are nearly the same as the exact solutions solved from Eqs.(1)–(3)when the control beam power Pc0
is about 100 times larger than the peak pulse power Ps0.
In this section, we take the ratio Pc0/ Ps0= 10 [10], which
Fig. 1. With input control beam power Pc0= 0.5 mW, (a) input and output pulse shapes, (b) pump power temporal variation and nor-malized metastable population density at EDFA output end, (c) gain coefficient spectra at several distances, (d) propagation constant spectra at several distances, (e) peak-power delay time Tpeakand the group delay time Td0along the EDFA evaluated at⍀=0, and (f) output pulse shapes Pa共M兲共t兲 synthesized up to several M dispersion orders and the approximate solution with TPD calculated from Eq.(22)without dispersion expansion. (f) Corresponding values of M (arrows). The exact solution is solved from Eqs.(1)–(3). (a)–(e) Approximate solutions with and without TPD are shown for comparison.
will result in a slight discrepancy between the approxi-mate solution and the exact solution. The propagation characteristics of fast light with Pc0/ Ps0= 10 and 100 are
similar. The cases with Pc0= 0.5, 0.1, and 2.5 mW are
considered in Subsections 4.A–4.C.
A. Pc0= 0.5 mW
Figures 1(a)–1(f) show the numerical results with Pc0
= 0.5 mW. Figure1(a) shows the input and output pulse shapes in which the input pulse shape is enlarged 100 times so that it can be clearly shown. The approximate so-lutions with and without TPD are also shown in Fig.1(a). Figure1(a)shows that there is negative power at the tail of the output pulse. The pulse peak power is amplified from 0.05 to 2.42 mW, and the cw power is amplified from 0.5 to 151 mW. Thus the signal power Ps, which comprises
both the pulse power Paand the cw power Pc, at the pulse
tail is still amplified. The leading edge of the output pulse depletes the metastable population density. The gain pro-vided by the metastable population density at the pulse tail is lower than that at the pulse leading edge for the signal power. Because the amplified signal power at the pulse tail is less than that at the pulse leading edge, there is the negative power at the pulse tail when the cw power is subtracted from the signal power to derive the pulse shape. This result agrees with the output pulse shape measured in [10]. In Fig.1(a), one can see that, without TPD, the pulse gain is overestimated, and the absolute value of the pulse delay time is underestimated. The dis-crepancy between the exact solution and the approximate solution with TPD is due to the pulse peak power that is not small enough compared with the control beam power. Figure 1(b) shows the pump power temporal variation and normalized metastable population density at the EDFA output end in which the approximate solutions
Ppa共t兲 and Na共t兲 are also shown. Ppa共t兲 and Na共t兲are
calcu-lated from Eqs. (23)and (12), respectively. For the case without TPD, P˜pa= 0 in Eq.(12). One can clearly see that
the depletion of the metastable population density is un-derestimated for the case without considering TPD, which leads to the underestimation of the CPO effect. Figures
1(c) and 1(d) show the gain coefficient and propagation
constant spectra, respectively, at several distances. In Figs. 1(c) and 1(d), the approximate solutions with and without TPD are shown. At a 2.5 m distance, the gain co-efficient spectra and the propagation constant spectra for the cases with and without TPD are about the same be-cause TPD is not yet significantly built up. After about a 5 m distance, for the case without TPD, the gain coeffi-cient and the absolute value of the negative slope of the propagation constant at⍀=0 are overestimated and un-derestimated, respectively. Thus the pulse gain and the absolute value of the negative group velocity are overes-timated. Figures2(a)and2(b)show the accumulated gain and phase shift, respectively, for the approximate solu-tions with and without TPD. From the results, we study the effect of gain dispersion and phase shift dispersion on pulse propagation in the following.
For the case with TPD shown in Fig.2(a), the gain dip of narrow bandwidth will result in serious high-order dis-persions. The first-order dispersion accelerates fast light without pulse distortion. Higher-order dispersion not only distorts the pulse shape as shown in Fig. 1(a) but also delays the pulse and slows down fast light. Figure 1(e)
shows the peak-power delay time Tpeak, the group delay time Td0with TPD, and the group delay time Td0without
TPD. One can see that兩Td0兩 of the case with TPD is much
larger than that of the case without TPD. In Fig. 1(e),
Tpeakis only about one-half of Td0with TPD. The average
group index can be calculated as navg= cTd/ L in which Td
is the delay time. We have navg= −3443, −6176, and −1093 for Td= Tpeak, Td0 with TPD, and Td0 without TPD,
re-spectively. Figure 1(f) shows the output pulse shapes
Pa共M兲共t兲 with partial high-order dispersions, and the cases
with M = 0, 1, 2, 3, and 5 are shown. In Fig. 1(f), the approximate solution with TPD calculated from Eq.(22)
without dispersion expansion is also shown for compari-son. In Fig.1(f), one can see how the combined effect of high odd order dispersions slows down fast light. The peak-power delay time of Pa共1兲共t兲 is Td0. The third-order
dispersion increases the pulse delay time and slows down fast light. Thus the absolute value of the peak-power de-lay time is decreased. The fifth-order dispersion acceler-ates fast light, but it is not able to recover the slow down resulting from the third-order dispersion. The dispersions of order larger than the fifth order further slightly
crease the pulse delay time and slow down fast light. Therefore, the group velocity of fast light cannot be de-fined as the velocity derived from the slope of propagation constant at⍀=0. In Figs. 1(e) and 1(f), one can see the significant modification of group velocity by high-order dispersions.
It is interesting to note that, comparing Pa共2兲共t兲 with Pa共1兲共t兲 shown in Fig. 1(f), one can see that the second-order gain dispersion significantly narrows the pulse width. It can easily be derived that if there only ex-ists the second-order gain dispersion, the output FWHM pulse width of the Gaussian input pulse given by Eq.(7)is
Tw2= 2关ln共2兲兩T02− 2␥a2兩兴1/2. 共26兲
If␥a2⬍T02/ 2, the pulse width is narrowed; otherwise, it is broadened. For the case shown in Fig. 1(f), T0= 0.3 ms
共Tw= 0.5 ms兲 and ␥a2= 0.0569 ms2. We have Tw2
= 0.256 ms, and the pulse is significantly compressed. The compressed pulse width enhances the unsymmetric pulse shape distortion due to the third-order dispersion in which a3= 0.0257 ms3. The dispersions of order higher
than the third-order smooth out the oscillating tail of
Pa共3兲共t兲. The resulting FWHM pulse width is 0.42 ms. In
general the pulse width may be broadened or narrowed depending on system parameters, such as pulse width, control beam power, and pump power [13]. Under a small signal assumption, dispersion coefficients change with the control beam and pump powers.
B. Pc0= 0.1 mW
With a lower Pc0, the depletion of pump power by the
am-plified control beam power is less, and the recovery of the
metastable population density (gain) is better. This re-sults in less pulse shape distortion but a slowing down of fast light induced by CPO. Figures 3(a)–3(f) show the same numerical results as Figs.1(a)–1(f), respectively, ex-cept that Pc0= 0.1 mW and the input pulse shape is
en-larged 500 times in Fig.3(a). Comparing Fig. 3(a) with Fig.1(a), one can see that the output pulse shape is better maintained, and the absolute value of the pulse delay time is decreased as expected. Comparing Fig.3(b)with Fig. 1(b), one can see that the depletion of the pump power and metastable population density by the pulse are larger because of a higher pulse gain and output pulse power. In Fig.3(c), at 2.5 and 5 m distances, the gain co-efficients for the cases with and without TPD are about the same because the pulse power is still low and TPD is not yet significantly built up, as are the propagation con-stant spectra shown in Fig.3(d). At a 7.5 m distance, TPD is high enough so that the difference between the cases with and without TPD becomes apparent. In Fig.3(e), we have average group indexes navg= −1794, −2223, and −877 for the peak-power delay time Tpeak, Td0 with TPD, and Td0without TPD, respectively.
Figures2(a)and 2(b)also show the accumulated gain and phase shift, respectively, for the case with Pc0
= 0.1 mW. The wide bandwidth of the gain dip for this case decreases the high-order dispersions so that the pulse shape is better maintained. For this case, ␥a2
= 0.0105 ms2, we have Tw2= 0.438 ms from Eq. (26), and
the pulse compression owing to␥a2is slight. In Fig.3(f),
one can see that, including only up to the fifth-order dis-persion, Pa共5兲共t兲 is about the same as the pulse shape cal-culated from Eq.(22). The resulting FWHM pulse width is slightly narrowed and is 0.472 ms. Fast light slowed down due to the third-order dispersion is less significant than the case with Pc0= 0.5 mW.
C. Pc0= 2.5 mW
From the results shown above, it seems that we may en-hance the average negative group index and increase the absolute value of the pulse delay time by increasing Pc0.
However the increase of the input control beam power not only enhances the first-order dispersion coefficient a1
but also higher-order dispersion coefficients ak共k⬎1兲.
The enhanced higher-order dispersion coefficients may re-sult in a serious pulse shape distortion and slowing down of fast light. For example, Figs. 4(a) and 4(b) show the same case as Figs.1(a)and1(f), respectively, except that
Pc0= 2.5 mW, and the input pulse shape is enlarged 20
times in Fig.4(a). One can see thata1= −0.424 ms, which
is about an 85% pulse width, but the combined effect of higher-order dispersions decreases the peak-power delay time to be −0.144 ms (−4320 average group index) and se-riously distorts the pulse shape. In Fig.4(b), one can see that Pa共2兲共t兲 is broadened instead of narrowed. For this case,␥a2= 0.187 ms2, which is large enough to broaden the
pulse width. From Eq. (26), Tw2= 0.888 ms. Careful
sys-tem parameter optimization is able to improve the abso-lute value of the peak-power delay time under a certain constraint of the pulse shape distortion [13]. However, as the first-order dispersion is enhanced, higher-order dispersions are usually enhanced accordingly. The optimization should compromise between the first-order dispersion and higher-order dispersions.
5. CONCLUSIONS
Fast light can be realized by utilizing the CPO effect in an EDFA in which a pulse superimposed on a strong cw con-trol beam is launched into the EDFA. The pulse depletes the metastable population density. The pump power is ab-sorbed more when the metastable population density is depleted. In literature, the perturbation method analyz-ing fast light in an EDFA did not consider this pump power depletion. Thus the CPO effect is underestimated, and the derived gain coefficient and propagation constant are inaccurate. We have developed the perturbation method for solving the time varying parts of the signal power, pump power, and metastable population density. The coupled equations of the spectral components of the
Fig. 4. With input control beam power Pc0= 2.5 mW, (a) input and output pulse shapes and (b) output pulse shapes Pa共M兲共t兲 synthesized up to several M dispersion orders and the approximate solution with TPD calculated from Eq.(22)without dispersion expansion. (b) Corresponding values of M (arrows). The exact solution is solved from Eqs.(1)–(3). (a) Approximate solutions with and without TPD are shown for comparison.
signal power, pump power, and metastable population density are derived. From the coupled equations, we can accurately solve the gain coefficient and propagation con-stant of fast light in an EDFA. It is found that the pulse gain and the absolute value of the negative group velocity are over estimated if TPD is not considered. From the solved gain coefficient and propagation constant, we also study the pulse delay time and shape distortion resulting from high-order dispersions induced by CPO. The gain dispersion resulting from accumulated gain is shown. Ac-cumulated gain is the integration of the gain coefficient along an EDFA, which is an even function of frequency. The second-order gain dispersion may symmetrically broaden or compress the pulse depending on the value of its coefficient. The changes of the pulse shape by higher even order gain dispersions are complicated because of the combined effect with high-/odd-order phase shift dis-persions. The phase shift dispersion results from the ac-cumulated phase shift, which is the integration of the propagation constant along an EDFA and is an odd func-tion of frequency. The first-order phase shift dispersion undistortedly leads to a negative pulse delay time. Higher-/odd-order phase shift dispersions unsymmetri-cally distort the pulse shape and change the pulse delay time. For the shown examples, the third- and fifth-order dispersions result in slowing down and accelerating fast light, respectively. Thus the group velocity of fast light cannot be simply defined as the velocity derived from the first derivative of the propagation constant. The presented perturbation method can also be applied to analyze fast light in other resonant mediums with optical pumping.
ACKNOWLEDGMENT
This work was supported in part by Chung Hua Univer-sity, China, under contract CHU-95-TR-01.
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