506 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002
Fiber-Based Optical Parametric Amplifiers
and Their Applications
Jonas Hansryd, Peter A. Andrekson, Member, IEEE, Mathias Westlund, Jie Li, and Per-Olof Hedekvist
Invited Paper
Abstract—An applications-oriented review of optical para-
metric amplifiers in fiber communications is presented. The
emphasis is on parametric amplifiers in general and single
pumped parametric amplifiers in particular. While a theoretical
framework based on highly efficient four-photon mixing is
provided, the focus is on the intriguing applications enabled by
the parametric gain, such as all-optical signal sampling, time-de-
multiplexing, pulse generation, and wavelength conversion.
As these amplifiers offer high gain and low noise at arbitrary
wavelengths with proper fiber design and pump wavelength
allocation, they are also candidate enablers to increase overall
wavelength-division-multiplexing system capacities similar to the
more well-known Raman amplifiers. Similarities and distinctions
between Raman and parametric amplifiers will also be addressed.
Since the first fiber-based parametric amplifier experiments
providing net continuous-wave gain in the for the optical fiber
communication applications interesting 1.5-
m region were
only conducted about two years ago, there is reason to believe
that substantial progress may be made in the future, perhaps
involving “holey fibers” to further enhance the nonlinearity and
thus the gain. This together with the emergence of practical and
inexpensive high-power pump lasers may in many cases prove
fiber-based parametric amplifiers to be a desired implementation
in optical communication systems.
Index Terms—Nonlinear optics, fiber-optic amplifiers and oscil-
lators, O-TDM, multiplexing, demultiplexing, optical sampling.
I. INTRODUCTION
P
ARAMETRIC amplification is a well-known phenomenon
in materials providing
nonlinearity [1]. However,
parametric amplification can also be obtained in optical fibers
exploiting the
nonlinearity. New high-power light sources
and optical fibers with a nonlinear parameter 5–10 times higher
than for conventional fibers [2], [3], as well as the need of am-
Manuscript received February 12, 2002; revised March 27, 2002. This work
was supported in part by the Swedish Strategic Research Foundation (SSF), the
Swedish Research Council (VR), and Chalmers Center for High-Speed Tech-
nology (CHACH).
J. Hansryd was with Chalmers University of Technology, Photonics
Laboratory, Department of Microelectronics MC2, SE-412 96 Göteborg,
Sweden. He is now with CENiX Inc., Allentown, PA USA 18106 (e-mail:
jhansryd@cenix.com).
P. A. Andreksonis with Chalmers Universityof Technology, SE-412 96 Göte-
borg, Sweden and also with CENiX Inc., Allentown, PA 18106 USA.
M. Westlund and P.-O. Hedekvist are with Chalmers University of Tech-
nology, SE-412 96 Göteborg, Sweden.
J. Li is with Chalmers University of Technology, SE-412 96 Göteborg,
Sweden and also with Ericsson Telecom AB, Stockholm, Sweden.
Publisher Item Identifier S 1077-260X(02)05481-3.
plification outside the conventional Erbium band has increased
the interest in such optical parametric amplifiers (OPA). The
fiber-based OPA is a well-known technique offering discrete or
“lumped” gain using only a few hundred meters of fiber [4], [5].
It offers a wide gain bandwidth and may in similarity with the
Raman amplifier [6] be tailored to operate at any wavelength
[7]–[11]. Although continuous-wave (CW) pumped fiber OPAs
have been experimentally investigated since the late 1980s
[12], it was not until about two years ago that net “black-box
gain” was achieved in the 1.5-
m region. An OPA is pumped
with one or several intense pump waves providing gain over
two wavelength bands surrounding the single pump wave, or in
the latter case, the wavelength bands surrounding each of the
pumps. As the parametric gain process do not rely on energy
transitions between energy states it enable a wideband and
flat gain profile contrary to the Raman and the Erbium-doped
fiber amplifier (EDFA). The underlying process is based on
highly efficient four-photon mixing (FPM)
1
relying on the
relative phase between four interacting photons [13]–[16]. Due
to the phase matching condition, the OPA does not only offer
phase-insensitive amplification, but also the important feature
of phase-sensitive parametric amplification. The phase-sensi-
tive amplifier only amplifies components of the same phase
as the signal, while attenuating components with the opposite
phase [9], [17], [18]. This property has many potential uses,
e.g., pulse reshaping [19], [20], as well as dispersive wave,
soliton-soliton interaction, and quantum noise suppression
[21]–[23]. Another very important application is the possibility
of in-line amplification with an ideal noise figure of 0 dB [18],
[24], [25]. This should be compared to the quantum limited
noise figure of 3 dB for standard phase-insensitive amplifiers.
A difficult but necessary requirement for phase-sensitive OPAs
is the need for a strict control of the phases of all involved light
waves. The most usual experimental implementation of such
an amplifier is thus through a nonlinear Kerr interferometer
where the phase of only one light source needs to be tracked
[17], [26].
For the phase-insensitive OPA, two photons at one or two
pump wavelengths with arbitrary phases will interact with a
signal photon. A fourth photon, the idler, will be formed with a
phase such that the phase difference between the pump photons
and the signal and idler photon satisfies a phase matching con-
dition (this is further discussed below). The phase-insensitive
1
In the literature, four-photon mixing is also referred to as four-wave mixing.
1077-260X/02$17.00 © 2002 IEEE
HANSRYD et al.: FIBER-BASED OPTICAL PARAMETRIC AMPLIFIERS AND THEIR APPLICATIONS 507
Fig. 1. Frequency components generated due to FPM for two pumps at frequencies
!
and
!
and a weak signal at
!
.
OPA lacks the ability of amplification with a subquantum-lim-
ited noise figure, while the requirements for its implementation
are substantially relaxed and it still offers the important prop-
erties of high differential gain, optional wavelength conversion
and operation at an arbitrary wavelength.
As the Kerr effect, similarly to the Raman process, relies on
nonlinear interactions in the fiber, the intrinsic gain response
time for an OPA is in the same order as for the Raman amplifier
(a few femtoseconds). This prevents in many, but not all cases,
the amplifier from operating in a saturated mode. In return, it
allows for ultrafast all-optical signal processing. Potential ap-
plications, further discussed below, include in-line amplifica-
tion [4], [5], return-to-zero (RZ)-pulse generation [27], optical
time-division demultiplexing (O-TDM) [28], [29], transparent
wavelength conversion [10], [11], all-optical limiters [28], [30],
and all-optical sampling [31].
The remaining part of this paper describes the theory and
the possible applications of phase-insensitive fiber based
parametric amplifiers to high-speed and long-haul transmission
systems. The focus is on degenerated parametric amplification
using one strong pump at a single wavelength, while the results
in general are possible to extend to two pumps at different
frequencies [8], [13], [32], [33], [53].
In Section II, we will describe the theory, the limitations and
advantages of fiber based OPAs, and also give a brief compar-
ison with the well known Raman amplifier. In Section III, we
will discuss and demonstrate some general applications of the
OPA. We will start with a CW pumped linear amplifier and a
transparent wavelength converter followed by a review of ex-
periments demonstrating high speed O-TDM applications. The
paper is concluded with a brief discussion on future develop-
ments.
II. T
HEORY
A. Four-Photon Mixing
To understand the parametric gain process, relying on highly
efficient FPM, we will describe it from three view angles. In
an intuitive approach, the nondegenerated process starts with
two waves at frequencies
and that copropagate together
through the fiber. As they propagate they will continuously beat
with each other. The intensity modulated beat note at frequency
will modulate the intensity dependent refractive index
of thefiber. When a third wave atfrequency is added,
it will become phase modulated (PM) with the frequency
, due to the modulated . From the PM, the wave at will
develop sidebands at the frequencies
. The am-
plitude of the sidebands will be proportional to the amplitude of
the signal at
. In the same way, will beat with and PM
. As a consequence the wave at will generate sidebands
at
, where will coincide with
the previously mentioned
. It should be noted
that from a FPM process including three incident waves and all
possible degenerated and partially degenerated processes, nine
new frequencies will be generated [34]. Fig. 1 shows all nine
frequencies. It also shows that some FPM-products will overlap
with the signal frequency, in this case
. These products will
result in a gain for the signal, i.e., provide parametric amplifica-
tion. In general, the remaining weaker frequencies are usually
neglected with the exception of the stronger frequency compo-
nent at
. The two over-
lapping components (
and )at are here referred to
as the generated idler. In the degenerated case with one pump,
and will coincide and light will only be transferred to the
signal and the idler frequency.
For the rest of this chapter, we will focus on the degenerated
case including one pump at
, one signal at and one idler
lightwave at
. From the above discussion it follows that a re-
quirement for the FPM process to be “resonant” is that both a
phase-matching conditionbetween the waves is maintained, and
that the frequencies of the three waves are symmetrically posi-
tioned relatively to each other,
(1)
(2)
Here,
is the low power propagation mismatch, is the speed
of light in vacuum and
is the propa-
gation constant of each lightwave.
Parametric amplification can be viewed from a quantum me-
chanical picture. Here, the degenerated parametric amplifica-
tion is manifested as the conversion of two pump photons at fre-
quency
to a signal and an idler photon at frequencies and
. The conversion needs to satisfy the energy conservation re-
lation as in (2) and the quantum-mechanical photon momentum
conservation relation as in (1).
From an electromagnetic point of view we may consider
the interaction of three stationary copolarized waves at an-
508 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002
gular frequencies and , characterized by the slowly
varying electric fields with complex amplitudes
and , respectively. The total transverse field
propagating along the single-mode fiber may be written as [16]
(3)
where c.c is the complex conjugate which is usually omitted in
the calculations and
is the common transverse modal
profile which is assumed to be identical for all three waves
along the fiber. Using the basic propagation equation [35], it is
straightforward to derive three coupled equations for the com-
plex field amplitude of the three waves
[13], [14], [16],
[36]
(4)
(5)
(6)
Here, fiber loss have been neglected and
is the
nonlinearity coefficient where
is the fiber nonlinear param-
eter and
is the effective modal area of the fiber. Further-
more, the frequencies are assumed to be similar such that
are
equal for the three light waves and
is assumed approximately
real such that any Raman gain is negligible. The first two terms
on the right-hand side (RHS) of (4)–(6) are responsible for the
nonlinear phase shift due to self-phase modulation (SPM) and
cross-phase modulation (XPM),respectively. Thelast term isre-
sponsible for the energy transfer between the interacting waves.
If required, fiber loss may be included by adding the loss term
to the RHS of each equation, respectively.
B. Phase-Sensitive or Phase-Insensitive Parametric
Amplification
By rewriting (4)-(6) in terms of powers and phases of the
wavesfurther insight can be gained. Let
and , where for
[14], [36], [37].
(7)
(8)
(9)
(10)
Here,
describes the relative phase difference between the
four involved light waves
(11)
where
includes both the initial phase at and the
acquired nonlinear phase shift during propagation.
The first term of
on the RHS of (10) describes the linear
phase shift, while the second and third term describe the non-
linear phase shift. The attentive reader may note that a phase
term representing the time dependent phase difference,
is missing in (11). However, this term
will remain zero when , resulting in the
phase-matching condition previously stated in (2).
As can be observed from (7)–(10), by controlling the phase
relation
, we havethe opportunity to control the direction of the
power flow from the pump to the signal and the idler (
,
parametric amplification) or from the signal and the idler to the
pump (
, parametric attenuation). In other words, by
having signal, idler, and pump photons present at the fiber input
and adjusting the relative phase between them, we are able to
decide if the signal will be amplified or attenuated. This gives
us the possibility to create a phase-sensitive amplifier. As previ-
ously discussed in the introductory part of this paper, the major
obstacle for implementing such device is the difficulties of con-
trolling and maintaining the relative phase of the interacting
photons.
For the general application of a phase-insensitive fiber-based
parametric amplifier as outlined in Fig. 2, we may consider an
intense pump at
and a weak signal at . Theidler is assumed
to be zero at
. For this special case at the fiber
input port. This can be understood as described by Inoue and
Mukai [37] by realizing that the idler will be generated after
an infinitesimal propagation distance in the fiber. Analyzing the
phases in (6):
, shows that the
phase of the initiated idler will be
; thus at the input port. Following (8)–(9), this
has the consequence that the signal and the idler will start to
grow immediately in the fiber.
C. Phase Matching Condition
Operating in a phase-matched condition
remains near
, the third term in (10) may be neglected and the following
approximation, first introduced by Stolen and Bjorkholm in[13]
is valid
(12)
Here, the phase mismatch parameter
is introduced and the
second approximation is valid when the amplifier is operating
in an undepleted mode
.
Expanding
in Taylor series to the fourth order around
the zero-dispersion frequency
the wavelength
dependent part,
of the phase mismatch parameter can be
rewritten as
(13)
HANSRYD et al.: FIBER-BASED OPTICAL PARAMETRIC AMPLIFIERS AND THEIR APPLICATIONS 509
Fig. 2. General scheme of phase-insensitive fiber-based optical parametric amplifier.
Here, and is the third and fourth derivativeof the propaga-
tion constant
at . When the pump frequency is chosen
to
the wavelength dependent part, of the phase
mismatch parameter
, and thus the OPA gain bandwidth may
be limited by the fourth-order dispersion [7]. As
is typically
in the
ps /km range, higher order dispersion becomes
an important and fundamental limiting factor as the operating
bandwidth
exceeds 100 nm.
By neglecting
, a convenient approximative transformation
of (13) may be done from the frequency domain to the more
generally used wavelength domain [28],
(14)
Here,
is the slopeof the dispersionat the zero-dispersion
wavelength and the approximation
has been made. This approximation is only valid for bandwidths
where
.
When
is positioned in the normal dispersion regime
, the accumulated phase mismatch will increase with in-
creasing signal wavelength
, thus decreasing the resulting ef-
ficiency of the process. By positioning the pump wavelength
in the anomalous dispersion regime
, it is possible
to compensate for the nonlinear phase mismatch
by the
linear phase mismatch
. For a fixed , the gain versus signal
wavelength
will thus be formed in two lobes on each side of
, each lobe having its peak gain for .
This process is identical to the phenomenon that is also referred
to as modulation instability [32], i.e., the parametric process es-
tablishes a balance between GVD and the nonlinear Kerr-effect.
As a sidenote, it can be mentioned that parametric amplifica-
tion with the pump positioned in the normal dispersion regime
may be achieved by using a birefringent fiber [16]. The linear
phase mismatch consist of material phase mismatch (due to
the properties of fused silica) and waveguide phase mismatch
(due to the design of the optical fiber). In the normal dispersion
regime, the nonlinear and the material phase mismatch contri-
bution will have the same sign. By placing the pump in the slow
propagation axis of the fiber, while the idler and signal is po-
sitioned in the fast axis, the sign of the waveguide mismatch
contribution will cancel the material and nonlinear mismatch
contributions and parametric amplification may occur.
Combining the expression for the maximum power flow i.e.,
with (14) shows that
(15)
Hence, the separation between the gain peaks for the signal
wavelength will increase with increasing
and with fixed.
Considering the special case when the pump becomes de-
pleted such that the condition
is no longer fulfilled.
Studying (12), we may note that the approximation performed
in the last equality is no longer correct. If we still assume that
we are operating in the phase matched regime
, the
nonlinear phase mismatch will decrease in order to keep the
total phase mismatch close to zero so that the optimum linear
phase mismatch will decrease compared to
predicted by
(15) with
. When the pump become so depleted that
, the power will start to oscillate between the pump
and the signal/idler as a consequence of
will start to oscil-
late between
and .
Equations (4)–(6) are general in the sense that they include a
depleted pump, higher order dispersion and a nonlinear phase
shift, they may also easily be solved numerically by using a
standard computer math package. An improved understanding
can be obtained by considering a strong pump and a weak signal
incident at thefiber input suchthat the pumpremains undepleted
during the parametric gain process. We may then set
and an analytical solution may be derived for the remaining
coupled equations as [13]
(16)
(17)
Here,
is the fiber interaction length and the parametric gain
coefficient
is given by
(18)
510 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 8, NO. 3, MAY/JUNE 2002
If the fiber is long or the attenuation is high, the interaction
length will be limited by the effectivefiber length
expressed
as [38]
(19)
where
is the loss coefficient of the fiber. In most applications
involving low-loss HNLF
such that . In the
remaining part of this paper that condition will be assumed. The
unsaturated single pass gain
and the unsaturated wavelength
conversion efficiency
may be written [7], [33]
(20)
(21)
In (20), the last equality stems from the Taylor expansion of
.
From (20), it may be noted that for signal wavelengths close
to
and . Inthe special case of perfect
phase matching
and , (20) may be rewritten
as
(22)
D. Discussion on Amplifier Gain and Bandwidth
The above expression shows that for the perfect
phase-matching case, the parametric gain is approximately
exponentially proportional to the applied pump power. A very
simple expression for the OPA peak gain may be obtained if
(22) is rewritten in decibel units as
(23)
where
is introduced as the
parametric gain slope in [dB/W/km]. Fig. 3 shows calculated
gain for a parametric amplifier with 1.4-W pump power and
500-m HNLF with
W km . The region for perfect
phase matching (exponential parametric gain) and the region
where
(quadratic parametric gain) is marked in the
figure. Fig. 4 shows the measured gain slope for the same
experimentalfiberparameters [5]. Theamplifierbandwidth may
be defined as the width ofeach gain lobe surrounding
[7], [8],
[13]. From (14), (18), (20) it may be observed that the amplifier
bandwidth for a fixed
will increase with decreasing as
the reduction in withrespect to will be “accelerated” by the
longer fiber length. On the other hand, since
increases as
decreases, the peak gain wavelength will bepushed further away
from . This is an important observation since the available
Fig. 3. Calculated gain for a fiber optical parametric amplifier with
P
:1
:
4
W,
L
:500
m,
:11
W
1
km ,
: 1559
nm,
:1560
:
7
nm,
dD=d
=0
:
03
ps/nm km. Arrows indicate regions with exponential gain and the region with
quadratic gain proportional to the applied pump power.
Fig. 4. Measured parametric gain slope
S
using 500-m HNLF with
=
11
W
1
km ,
= 1561
:
5
nm, and
dD=d
=0
:
03
ps/nm km. The
gain slopes were measured for the two peak wavelengths 1547 and 1579 nm,
respectively.
power was usually the limiting factor for conventional optical
fibers. By using a short HNLF it is possible to decrease
and
increase such that the maximum gain is fixed while the ampli-
fier bandwidth is increased. The benefit of using such a fiber is
demonstrated in Fig. 5. Here, the single pass gain is calculated
from (16) for different fiber lengths. The product
is con-
stant, resulting in a fixed maximum gain but an increased band-
width as the fiber length is decreased. The condition
correspond e.g., to a pump power of 1 W for a HNLF with
m and W km . Decreasing the fiber
length to 50 m would increase the bandwidth
20 times and require for instance W and
W km . Such a high could be achieved in novel types
of HNLF such as air-silica microstructured fibers (ASMF, also
called “holey fibers”) [39]–[41].
As discussed earlier, in the context of (13), we saw that in the
linear phase-matching regime that the limiting factor for wide
operating bandwidth is the fourth-order propagation constant
. However, as the impact of the nonlinear phase mismatch
increases, can be advantageously utilized
to increase and flatten the operational bandwidth by optimizing
[7], [42].
A second factor to take into account for the OPA gain band-
width is thefact that
in a realfiberis slightly distributed along
the fiber length [9], [43], [44]. This will broaden the resulting
