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IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 12, DECEMBER 2003
Raman Amplifier Model in Single-Mode
Optical Fiber
Idan Mandelbaum and Maxim Bolshtyansky
Abstract—Equations to describe the process of Raman amplification in single-mode optical fiber and Stokes and anti-Stokes
spontaneous emission generation with conservation of photon
number are derived from first principles. The numerical simulation of Stokes and anti-Stokes spontaneous emission is in good
agreement with experimental results. The importance of the
sometimes-omitted anti-Stokes spontaneous emission terms in
wavelength-division-multiplexed situations for optical signal-tonoise ratio is demonstrated by the simulation of light propagation
through transmission fiber with and without Raman amplification.
Index Terms—Optical fiber amplifiers, optical fiber communication, Raman scattering.
figuration and counterconfiguration, which includes effects due
to group velocity, Rayleigh backscattering fiber loss, as well
as Stokes and anti-Stokes spontaneous emission. The equation
conserves photon number in all modes and, thus, corrects the
problem in [1] and [4], that gives negative power of ASEs in
our simulations. In addition, the model is compared to experimental results to verify the validity of this approach.
A derivation of the model is given in Section II; the experimental results are given in Section III. Finally, we conclude in
Section IV.
I. INTRODUCTION
II. MODELING
ISTRIBUTED Raman amplification can be found in both
long-haul and ultralong-haul optical networks and is becoming an enabling technology for ultrawide-band communication, with over 100-nm flat bandwidth already reported [1].
In addition, it can be used for increasing the bandwidth of Erbium-doped fiber amplifiers in hybrid systems [2]. Raman amplifiers are being studied for use in metro applications as a flexible and simple way to supply gain anywhere in the communication spectrum. Thus, the accurate modeling of Raman amplification becomes extremely important for modern communication
system design.
The problem of Raman amplifier modeling has been discussed in scientific literature for quite some time. Stolen
et al. [3] describe modeling of Stokes amplified spontaneous
emission (ASE) and multiple Stokes shifts in single-mode fiber.
Kidorf et al. [1] give detailed equations for a Raman amplifier
model applied to single-mode optical fiber, in which an attempt
has been made to conserve photon number. Achtenhagen
et al. [4] verify the gain predicted by this model and add
Rayleigh backscattering and wavelength scaling approximation
to account for effective area. Perlin et al. [5] add anti-Stokes
spontaneous emissions. Berntson et al. [6] give a similar model
to [1], but gives correct the equation for spontaneous absorption
of signal photons into the fundamental mode.
This letter, to the best of our knowledge and for the first time,
explicitly gives the Raman propagation equation in both cocon-
Consider the Raman interaction between two arbitrary bands
and with single-mode polarwith infinitesimal bandwidth
ized light in both bands. The longer wavelength band will be
referred to as the signal and the shorter wavelength band as
the pump. Two processes describe the Raman interaction between the two wavelengths [7]. The first process is responsible
for Raman gain and Stokes spontaneous emission generation as
well as pump depletion and is referred to here as the Raman
emission process. The second process is responsible for antiStokes spontaneous emission generation and is referred to here
as Raman absorption. Raman emission is the process whereby
a pump photon generates a signal photon and a phonon at a
frequency that is the difference between the pump and signal
frequencies. This process has a rate of
,
where and are the photon densities at the pump and signal
, respectively,
is the
frequencies within the bandwidth
number of phonons at the frequency that is the difference between the pump and signal frequencies, is the annihilation
is the creation operator with subscripts deoperator, and
is a polarization as
scribing pump, signal, and phonons.
well as frequency-dependent emission probability coefficient,
that will be related later to the Raman gain coefficient. Raman
absorption is the process where a signal photon and a phonon
combine to produce a photon at the pump frequency, with a rate
of
, where
is some frequency and polarization-dependent absorption probability coefficient that will be related later to the Raman gain coefficient. The rate equation for
the pump and signal photons can be written as
D
Manuscript received May 30, 2003; revised July 18, 2003.
I. Mandelbaum was with the JDS Uniphase Corporation, Trenton, NJ 08638
USA. He is now with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail: im218@columbia.edu).
M. Bolshtyansky is with the JDS Uniphase Corporation, Trenton, NJ 08638
USA.
Digital Object Identifier 10.1109/LPT.2003.819760
1041-1135/03$17.00 © 2003 IEEE
(1)
MANDELBAUM AND BOLSHTYANSKY: RAMAN AMPLIFIER MODEL IN SINGLE-MODE OPTICAL FIBER
Using the values for
and
as propagation equation
, this equation can be written
(2)
where is the group velocity of pump or signal, denoted by the
subscript. Converting (2) from photon number to power density
equations for the depolarized pump and signal
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As presented in (3) and (4), these terms represent only interaction between the two wavelength bands under consideration,
thus, these equations cannot be used directly and should be integrated over all other wavelengths. The third term in (3) and (4)
is responsible for the addition of spontaneously emitted photons
at the signal and pump wavelength into the fundamental mode.
Equations (3) and (4) are extended for the multiwavelength
case and combined to form one general equation, in which
multiple channels with optical bandwidths
are included in
the codirection and counterdirection. Here, the channels should
cover all the bandwidth under investigation, which includes the
bands where only ASE is generated and no signals are present
(3)
(4)
where is Planck’s constant, is the Boltzmann constant, is
the temperature, and are the pump and signal frequencies,
are the Raman gain and the fiber
respectively, and and
effective areas, respectively, which are defined in [7]. In the notation of this letter, is always taken to be the highest of the two
frequencies. The summation over in (3) and (4) represents the
spontaneous emission into all modes of the fiber, including radiation modes or a continuous spectrum that is not supported by
the fiber, for which case the summation becomes an integration.
Each mode is considered to have two polarization states and
denotes optical power within bandwidth
in both states, with
unit of watts per hertz in SI. In the derivation of (3) and (4), it
is assumed that 1) the phonons obey the Bose–Einstein distribution and are local, because at optical frequencies they cannot
efficiently propagate through glass; 2) spontaneous events occur
in two polarization states, thus additional factors of two for the
. The
second and third term in (3) and (4); and 3)
, the Raman
last assumption is justified because if
gain becomes temperature-dependent. In [9], it is shown that the
Raman gain is independent of temperature between 77 K and
300 K, which justifies the assumption.
Some of the terms in (3) and (4) are the result of photon
number conservation. These are the second terms in (3) and
(4). The contribution due to conservation of photon number can
be seen as a temperature and wavelength-dependent loss, but
it is not power-dependent. This contribution is the loss present
in the fiber due to the conversion of signal and pump photons
into spontaneously emitted Raman frequency shifted photons.
(5)
where the subscript represents the th wavelength, the superscripts of and represent the codirection and counterdirection, respectively, and and are the wavelength-dependent
loss and Rayleigh backscattering coefficients, respectively [10].
In (5), includes the loss due to Raman emissions into all
modes, as is usually measured, which means the corresponding
terms in (4) are included in . If needed, the sums over the
frequencies can be converted into integrals for a continuous
spectrum.
III. RESULTS
In order to check the validity of (5), the fiber loss and Raman
gain, as well as the Rayleigh backscattering coefficient are
measured for 13-km Lucent TrueWave RS transmission fiber
at room temperature. These parameters are directly used in the
simulation, and have not been adjusted to best fit experimental
data. The fiber is then pumped with 13 mW at 1560 nm. The
comparison of the simulated and experimental results is shown
in Fig. 1. Extremely good agreement between experiment and
simulation can be seen. The model, which does not account for
anti-Stokes spontaneous emission, would have no spontaneous
emission for wavelength shorter than 1560 nm and, thereby,
not agree with the experimental results. The values of the group
velocity used were obtained from group index and dispersion
data supplied with the fiber.
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Fig. 1. Simulation versus measurement for ASE generated by 1560-nm signal
at 300 K.
IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 15, NO. 12, DECEMBER 2003
Fig. 3. Comparison of OSNR between models with anti-Stokes and without
anti-Stokes spontaneous emissions in a system with 51-channel C -band in
100-km SMF-28 Fiber at 300 K.
1529.55 nm. It is interesting to note that, according to our
equations, when the temperature is 0 K, there is no anti-Stokes
spontaneous emissions generated, and therefore, the model
with and without anti-Stokes give the same results.
IV. CONCLUSION
Fig. 2. Comparison of OSNR between models with anti-Stokes and without
anti-Stokes spontaneous emissions in a system with two pumps at 1430 and 1454
copropagating with 51 channels. C -band in 100-km SMF-28 Fiber at 300 K. The
inset shows the ON–OFF gain of the system.
We have presented a model for Raman amplifiers, which
includes Raman emission and absorption processes, effects due
to group velocity, and that preserves photon number. In addition, fiber loss and Rayleigh backscattering have been added
and the model has been extended to support multiple pump
and signal wavelengths at both copropagating and counterpropagating directions. We also experimentally demonstrated
good agreement of the model and the experiment in describing
spontaneous emissions at anti-Stokes frequency shift. The
importance of the correct accounting of the anti-Stokes ASE
for OSNR calculation is also demonstrated.
REFERENCES
In order to demonstrate the importance of anti-Stokes
spontaneous emission in system applications, the optical
signal-to-noise ratio (OSNR) is calculated in pumped and
un-pumped configuration. Using (5), and turning ON and
OFF the anti-Stokes spontaneous emission term, OSNR for
a 100-km span of Corning SMF28 with 51 channels in the
-band (1529.55–1569.59 nm) is compared in pumped and
un-pumped configurations. The total input signal power is
20 dBm with flat channel distribution. Input OSNR of 100 dB
is used, with a detector bandwidth of 100 GHz. For the pumped
case, a pair of copropagating pumps at 1430 and 1454 nm with
powers 300 and 262.5 mW, respectively, is used in order to
achieve a reasonably flat spectrum at the output. The ON–OFF
gain, for this configuration, is shown in the inset of Fig. 2.
Comparing the simulated results for the two different OSNRs
at room temperature, which correspond to the different models
in the pumped configuration, is shown in Fig. 2. The figure
shows 2.17-dB discrepancy in the OSNR for the lowest signal
wavelength used, 1529.55 nm. The difference is much more
significant for the case where there is no pump and only the
signals are present, as shown in Fig. 3. In that case, a 20-dB
difference between the two noise figures can be observed at
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