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Article Number: OPTC 00932
DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers 1
DIFFRACTIVE SYSTEMS
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Introduction
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Photonic crystal fibers (PCFs) are very similar to
normal optical fibers in that they consist of a core
surrounded by cladding, such that light is guided
within the core of the fiber. The primary difference
between PCF and standard optical fibers is that PCFs
feature an air-silica cross-section, whereas standard
optical fibers have an all-glass cross-section. An
electron micrograph of a typical PCF is shown in
Figure 1. The air holes extend along the axis of the
fiber for its entire length and the core of the fiber is
formed by a defect, or missing hole, in the periodic
structure. The core is formed of solid glass, whose
refractive index is that of pure silica (or whatever
other glass is chosen), and the cladding is formed by
the air-glass mixture, whose effective refractive index
depends on the ratio of air-to-glass, also known as the
air-fill fraction, that comprises the structure. The
resulting effective-index of the cladding will be lower
compared with that of the core and, as such, will
provide the refractive index variation necessary to
support total internal reflection at the core-cladding
boundary, and guide light in a manner similar to that
of standard optical fibers. The fiber design (i.e., size,
shape, and the air-fill fraction) dictates solutions to
Maxwell’s equations for light propagating within the
fiber. Valid solutions are referred to as ‘modes’ which
propagate along the fiber in a known manner, and
have a well-defined shape in the transverse direction
(i.e., they have a well-defined transverse mode
structure).
Nonlinear-optical effects in fibers result from the
interaction of optical fields with the glass via the xð3Þ ;
or Kerr nonlinearity. The phenomenon of nonlinear
refractive index is a manifestation of a light-material
interaction mediated by xð3Þ : The magnitudes of the
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J E Sharping and P Kumar, Department of Electrical and
Computer Engineering, Northwestern University, 2145 N
Sheridan Road, Evanston, Illinois 60208-3118, USA
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q 2004, Elsevier Ltd. All Rights Reserved.
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J E Sharping and P Kumar, Northwestern
University, Evanston, IL, USA
components of the third-order susceptibility tensor in
glass, xð3Þ ; are generally quite small compared with
the analogous second-order ðxð2Þ Þ terms for materials
exhibiting such nonlinearities (e.g., lithium niobate,
beta-barium borate (BBO), etc.). The relatively small
xð3Þ nonlinearity in optical fibers makes them ideal for
wavelength-division multiplexed optical communication where light propagation subject to a minimum
of nonlinear effects is critical. Nonlinearity does,
however, eventually become an issue in wavelengthdivision multiplexed systems as the launched optical
power increases and as the channel spacing decreases.
On the other hand, one can utilize nonlinear-optical
effects in soliton communication systems and to build
useful photonic devices. Despite the weak xð3Þ
nonlinearity, the net nonlinear-optical effect in fibers
can be large due to the ability to tightly confine
intense fields within the core of an optical fiber and
maintain the interaction over a long distance as the
guided fields propagate through the fiber.
The study of nonlinear-fiber optics has benefited P0015
from dramatic improvements in optical fiber and
fiber-optic device fabrication. The importance of
understanding nonlinear-fiber optics is driven by the
need to develop fiber-integrated devices, and also by
the need to understand and mitigate the problems
that these nonlinearities cause in optical communication systems.
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Nonlinear Optics in Photonic
Crystal Fibers
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Figure 1 An electron micrograph showing the periodic
microstructure of a typical PCF. The core is formed by the
‘missing hole’ in the center of the microstructure. (From Ranka,
et al. q2000 Optical Society of America), courtesy of OFS.
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Article Number: OPTC 00932
2 DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers
This section introduces the unique linear- and
nonlinear-optical properties of PCFs in order to
understand the reasons why nonlinear-optical effects
are often enhanced in such fibers. These discussions
pertain to PCFs which are ‘highly nonlinear’. It is
essential to clarify that ‘highly nonlinear’ in this
context does not mean that the xð3Þ is any larger than
that of standard telecommunication fibers, rather that
the effect of this nonlinearity is enhanced due to the
fiber’s very small core.
Photonic crystal fibers feature a variety of interesting
properties. From the standpoint of nonlinear-fiber
optics there are four very useful fundamental properties of PCFs:
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. a mechanically robust optical fiber can be fabri2
cated with an extremely small core (a few mm );
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mode over an extremely broad wavelength range
(370 nm –1600 nm);
. there are new degrees of freedom that allow one to
manipulate the fiber’s group-velocity dispersion
(GVD) properties; and
. many, but not all, PCFs are polarization maintaining as a result of form birefringence present in the
core.
The fact that small-core PCFs can be fabricated is
clear from Figure 1 by taking note of the fact that the
center defect region which comprises the core is about
1.7 mm in diameter. Photonic crystal fibers with even
smaller cores have been fabricated.
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. a fiber can be made to guide in a single transverse
where L is the spacing between air holes, l is the
wavelength of light, and Vcutoff is the cutoff condition
for the PCF. A similar expression for the V parameter
is commonly used to understand the modal behavior
of standard fibers where larger the V is, the more
transverse modes are supported within the fiber. In
standard fibers the cutoff condition below which only
a single mode can propagate within the core of a fiber
is given by Vcutoff , 2:405: In the case of PCFs, a
numerical method should be used to determine Vcutoff :
Mechanically robust PCFs can be fabricated where
the dispersion in neff ðlÞ (i.e., the variation of neff with
l) offsets dispersion in nco ðlÞ and compensates for the
2pL=l coefficient in eqn [1]. Therefore, the light
within the fiber propagates in a single, Gaussian-like
mode because for all wavelengths Veff , Vcutoff :
A graph of the variation of Veff with L=l is shown
in Figure 2, where d represents the size of an air hole.
Conceptually, the effective index model can be P0060
understood by noting that at short wavelengths the
mode field is confined well within the all-silica core,
but as l increases the mode field extends further into
the air-glass cladding and Veff and neff ðlÞ both
decrease.
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Figure 2 Variation of Veff for different relative hole diameters
d =L: The calculation assumes a fiber with an air-glass crosssection where the refractive index of air and glass were taken to
be 1 and 1.45, respectively. The dashed line marks Veff ¼ 2:405;
the cutoff value for a step-index fiber. (From Birks, et al. q1997
Optical Society of America)
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PCF Properties
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Transverse Mode Structure
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A widely accepted model used to describe the
transverse modal behavior of PCFs is called the
effective-index model. The effective-index model can
be used to understand why some PCFs are ‘endlessly
single mode’, meaning that the fiber guides in a single
transverse mode over an exceptionally wide wavelength range (370 nm –1600 nm). In the effectiveindex model, the refractive index of the core, nco ðlÞ; is
that of glass, and the refractive index of the cladding,
neff ðlÞ; assumes a value in between that of glass and
air. In the context of PCFs, one makes a modification
to the standard expression describing single-mode
behavior in step-index fibers:
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Veff ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2pL
nco ðlÞ2 2 neff ðlÞ2 , Vcutoff
l
½1
Dispersion in PCF
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Other critical differences between PCFs and standard P0065
optical fibers lie in the dispersion properties. When
light propagates through a fiber its behavior depends
on the light’s optical frequency:
Eðt; zÞ ¼ Aðt; zÞei½vt2bðvÞz
½2
Equation [2] describes the mode as it propagates
through the fiber. It is decomposed into a slowly
varying envelope, Aðt; zÞ; and a rapidly varying
exponential component where v is the frequency of
Article Number: OPTC 00932
DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers 3
D [ps/nm km]
Total dispersion
0
–10
Waveguide dispersion
– 20
1.1
1.2
1.3
1.4
Wavelength [mm]
1.5
1.6
Figure 3 Plots of the theoretical dispersion coefficient, D; as a
function of wavelength for a standard optical communication fiber.
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the wavelength-dependent mode depends on the
properties of the waveguide (i.e., the core size and
refractive index contrast between the core and
cladding). Empirical models can be used to describe
the material, waveguide, and total GVD for standard
communication fibers. Such a set of curves are given
in Figure 3 where it can be seen that it is possible
to have positive, negative, or zero values for D:
For historical reasons, the regions where D is negative
(b2 is positive) exhibit ‘normal GVD’, while those
where D is positive (b2 is negative), exhibit ‘anomalous GVD’. The wavelength corresponding to D ¼ 0
is referred to as the zero-dispersion wavelength ðl0 Þ;
which for most silica glass fibers is about 1,300 nm.
In contrast with standard optical fibers, where the P0085
waveguide contribution to D is always less than zero,
small-core PCFs can be fabricated where the waveguide contribution to GVD is positive and quite large.
As such, in PCFs, l0 can be shifted to wavelengths
shorter than the intrinsic dispersion zero of glass.
Control over the GVD is essential for phase matching
certain nonlinear-optical interactions involving light
of different colors co-propagating within a fiber.
Indeed, several exciting applications of nonlinear
optics in PCF require a fiber with a l0 , 800 nm. The
GVD is also of great importance when working with
pulsed light in PCF, because GVD results in temporal
pulse broadening. It also governs pulse temporal
walkoff effects, limiting the effective interaction
length between pulses of different colors. This new
flexibility to manipulate the GVD curve, by varying
waveguide design parameters, is a key advantage
associated with using PCFs for nonlinear optics.
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where bi ¼ di b=dvi : The physical significance of the
various bi in eqn [3] are as follows: the phase-fronts
of the electric field move at a speed given by v=b0 ¼ vp ;
and the envelope, Aðt;zÞ; moves at a group velocity
given by 1=b1 : A GVD term, which governs temporal
spreading of the envelope, is given by b2 : Higherorder b terms are usually negligible for propagation
of pulses of $1 ps duration in optical fibers and are
lumped into the category of ‘higher-order chromatic
dispersion.’
The notation ‘b2 ’, as defined above, is often used in
the literature with dimensions of ps2/nm. However,
another expression is frequently used because of its
direct relationship to measured quantities. It is
straightforward to measure the relative delay, T;
between two pulses having different center wavelengths. Choosing a particular wavelength as a
reference, one can then measure relative delay as a
function of an injected pulse’s center wavelength. The
first derivative with respect to l of the relative delay
curve gives the GVD according to
!
1
›
vg
1 dT
2pc
D¼
¼
¼ 2 2 b2
½4
L dl
›l
l
Material dispersion
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½3
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bðvÞ ¼ b0 þ b1 ðv 2 v0 Þ þ b2 ðv 2 v0 Þ2 þ ···
2
20
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the mode, t is time, z is the position along the length
of the fiber, and bðvÞ is called the mode-propagation
constant. The general term used in describing the
frequency dependence of b is chromatic dispersion,
which includes contributions from the material as
well as the waveguide. Other types of dispersion
present in optical fibers include multi-modal (arising
from multiple guided transverse modes) and polarization-mode dispersion.
One way to understand the chromatic dispersion of
a mode propagating through an optical fiber is to
study the Taylor series expansion of the modepropagation constant, b; about the center frequency
of the field, v0 :
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where vg is the group velocity, and L is the length of
the fiber under test. The dimensions commonly used
for D ps/(nm km).
Chromatic dispersion in single-mode optical fibers
results from two different wavelength-dependent
fiber parameters. The medium itself, glass in this
case, has a wavelength-dependent refractive index.
This ‘material’ contribution has the same magnitude
regardless of the various parameters associated with
the waveguide. A second contribution has to do with
the design of the optical fiber. This ‘waveguide’
contribution to dispersion arises from the fact that
Birefringence in PCF
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The core of an optical fiber often exhibits some P0090
amount of anisotropy. The core may be elliptical in
form (shape) which leads to a phenomena referred to
as form birefringence. Since mode propagation
depends on the fiber structure, a fiber with an
Article Number: OPTC 00932
4 DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers
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. propagation losses are generally larger in PCFs
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. free-space coupling and splicing is difficult and can
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Nonlinear Phenomena
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The basic principles determining nonlinear effects in
PCF are the same as those for standard optical fibers
(00652). It is the new flexibility in PCFs to obtain
transverse-modal and GVD behavior different from
that of standard optical fibers that makes PCFs truly
interesting for nonlinear optics. The relevant nonlinear-optical effects are: self-phase modulation
(SPM); cross-phase modulation (CPM); third-harmonic generation (3HG); four-wave mixing (FWM);
Raman scattering; and Brillouin scattering.
Self-phase modulation (also known as the optical
Kerr effect) refers to the self-induced phase shift
experienced by an optical field as it propagates
through a fiber. It becomes particularly important
for the case of pulses of light propagating through
optical fibers. In small core PCFs, SPM is enhanced
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than for standard optical fibers; and
result in large coupling losses.
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due to the high intensity light propagating within the
core. Self-phase modulation can lead to substantial
spectral broadening of pulsed light propagating along
an optical fiber.
When a pulse of light experiences normal GVD
(i.e., D , 0) as it propagates, the longer-wavelength
components travel faster than the shorter-wavelength
components. Anomalous GVD (i.e., D . 0) leads to
the opposite, short-wavelength components traveling
faster than the long-wavelength components. Groupvelocity dispersion generally leads to temporal
broadening of pulses as they propagate along a
fiber. Under ideal conditions, however, SPM in
combination with anomalous GVD, leads to pulses
which propagate without any temporal or spectral
broadening. These self-sustaining pulses are called
‘optical solitons’.
When waves of light having different wavelengths
co-propagate along a fiber, CPM can occur. It can be
understood as a phase shift induced on one wave,
due to the presence of the other wave. Cross-phase
modulation also leads to spectral broadening and
solitonic pulse propagation.
In 3HG and FWM, one or more photons are
destroyed and others are created. In 3HG, three
‘fundamental’ photons are destroyed to create one
with three times the energy of the fundamental
photons. In FWM, two fundamental photons are
destroyed while two others are created. While it is
straightforward to conserve energy in 3HG and
FWM, these interactions must be ‘phase-matched’,
meaning that the interacting waves must be made to
propagate in-phase over a meaningful length. Such
phase-matching conditions need to be carefully
considered when studying 3HG and FWM. Nevertheless, 3HG and FWM can be used to obtain
frequency shifts and all-optical amplification. In
comparison with other types of optical fibers, PCFs
are particularly useful for 3HG and FWM applications. The small core of the PCF allows interactions
to occur at much lower input power, and the new
flexibility associated with the GVD properties permits
phase matching in cases which are not possible using
standard optical fibers. Finally, the fiber’s endlessly
single-mode behavior permits very good transverse
mode overlap between interacting waves having
widely different center wavelengths (e.g., the fundamental wave and third-harmonic wave in 3HG).
Self-phase modulation, CPM, 3HG, and FWM are
photon –photon interactions wherein no energy is
exchanged with the medium itself. In contrast,
Raman scattering and Brillouin scattering result
from photon– phonon interactions. The differences
between Raman and Brillouin scatterings lie in the
energy of the phonons involved and the direction in
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elliptical core will exhibit mode propagation that
depends on the electric field’s polarization with
respect to the axes of the elliptical core. As a result
of birefringence, the polarization of the mode varies
as it propagates through the fiber (unless care is taken
to align the polarization of the injected light with
respect to the principal axes of the birefringence).
Polarization-maintaining (PM) fibers are designed
to include birefringence in a particular axis of the
fiber. By including a well-defined birefringence
throughout the length of a fiber that is larger than
that induced by external perturbations, fast and slow
axes of the optical fiber are created for all guided
wavelengths, giving two orthogonal ‘polarization
modes’. If light is injected into one of the polarization
modes (i.e., with its linear polarization along one of
the axes) it remains linearly polarized along that axis
as it propagates along the fiber. The two polarization
modes generally have different group velocities, so
pulsed light in each mode will take a different amount
of time to propagate through a given segment of fiber.
Most PCFs exhibit strong birefringence due to a
slightly elliptical core combined with a large corecladding index difference, and so they behave
similarly to PM fibers. Special care must be taken
when working with PCF to be sure that the
polarization of the light launched into the fiber is
aligned with one of the birefringent axes.
In practice, there are a few other features of PCF
that are of importance when discussing nonlinearoptic interactions:
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Article Number: OPTC 00932
DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers 5
Figure 4 In supercontinuum generation one observes a broad
continuum generated after short pulses of light from a Ti:Sapphire
laser propagate through a 75 cm-section of PCF. The spectrum
of the input pulse is shown as a dashed curve, while the
output is a solid curve. (From Ranka, et al. q2000 Optical Society
of America).
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from an input pulse whose spectral width is only
about 10 nm.
It is widely accepted that supercontinuum gener- P0160
ation results from a combination of solitonic effects
which conspire to generate the broad spectrum.
As the pulse propagates through the PCF, high-order
solitons are formed which then undergo a ‘fission’
or ‘Cherenkov radiation’ process that blue-shifts
energy into the normal dispersion regime. This
explanation is consistent with the fact that for
efficient supercontinuum generation the injected
pulse’s wavelength should be chosen in the anomalous dispersion regime but close to l0 : Under these
conditions, solitons are formed and there is a wide
phase matching bandwidth for the soliton fission
process.
Spectral broadening, due to SPM, is common in P0165
standard optical fibers, but what makes this particular experiment interesting is the remarkable width of
spectrum generated with a short piece of fiber (,3 m)
and with comparatively small optical powers.
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which the interactions occur. Raman scattering is an
interaction between a photon and an optical –
phonon mode of the molecules making up the
material. In the case of Raman scattering in glass,
the energy shift, associated with molecular
vibrational (Raman) modes, corresponds to frequencies of 1 –12 THz. Raman scattering occurs in the
forward and backward directions. With pulsed light,
stimulated Raman scattering can occur when the
lower-frequency spectrum of the pulse overlaps with
the spectrum of the Raman resonances excited by the
higher-frequency spectrum. When this happens,
energy can be efficiently shifted in spectrum towards
the peak of the Raman resonance. In the forward
direction, this ‘Raman self-frequency shift’ builds up.
Additionally, if the Raman self-frequency shift
occurs in the presence of anomalous dispersion, a
‘Raman soliton self-frequency shift’ can result. As
the injected power is increased, the spectral shift
between the injected pulse and the resulting Raman
soliton increases. The principal advantage associated
with PCF is the ability to generate Raman solitons
for a broader range of wavelengths than was
previously possible.
For Brillouin scattering, interaction with the
acoustical phonons results in frequency shifts of
about 10 GHz and the interaction only occurs in the
backward direction. Brillouin scattering is generally a
nuisance in fiber-based devices, leading to intensity
noise and other problems. The interesting feature of
Brillouin scattering in PCFs is that the threshold
intensity where problems begin to occur is higher for
PCFs than for standard optical fibers. The higher
threshold permits further optimization of fiber-based
devices wherein Brillouin scattering limits the
performance.
Experiment Examples
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In the following subsections, a selection of experiments, demonstrating a few of the relevant nonlinearoptical phenomena, are briefly described.
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Supercontinuum Generation
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One of the most exciting demonstrations of nonlinear optics in PCF is that of supercontinuum
generation. In a typical experiment, 100 femtosecond pulses from a mode-locked Ti:Sapphire laser
operating at a wavelength of 800 nm were injected
into the PCF. As the injected power was increased,
a broad continuum of spectrum was generated from
wavelengths of 400 nm up to 1,600 nm. Typical
data showing the input and output optical spectra
are given in Figure 4, where it can be seen that well
over an octave of frequency spectrum is generated
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Optical Switching in PCF
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Cross-phase modulation can be used to build an all- P0170
optical switch. Such a switch can be implemented as a
three-port device where the output port for a given
optical bit (the signal pulse), is determined by the
presence of an optical control (the pump pulse).
Switching can then be achieved by dividing the signal
pulse equally on two arms of an interferometer and
injecting the strong pump pulse only on one arm.
Because the pump pulse co-propagates with only one
of the two signal pulses, there exists an CPM-induced
phase difference fNL ¼ 2gPp L at the output of the
interferometer, where Pp is the peak power of the
pump pulse and L is the interaction length. By varying
the intensity of the pump pulses one can vary the
Article Number: OPTC 00932
6 DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers
Signal
(1546 nm, 2.6 ps)
Pump
(1537nm,4.9ps)
FPC2
FPC3
FPC1
50/50
EDFA
TBPF
BWDM
APC
POL
Detector
PBS
Synchronous pump
and signal input
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Sagnac
loop
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magnitude of this phase difference. If a p-phase shift
is achieved, one can switch the interference from
destructive to constructive, or vice-versa, thus realizing an all-optical switch.
Figure 5 shows an experimental setup used to
observe switching near 1,550 nm (a similar apparatus
using bulk-optic rather than fiber-optic components
can be used to conduct experiments near 780 nm).
The pump and signal are synchronous few-psduration pulses with a tunable wavelength separation
of 5 – 15 nm. The switching characteristics for this
implementation are shown in Figure 6. The apparatus
has the advantage of requiring short fiber lengths, low
switching powers, and allows switching of weak
pulses. It demonstrates the feasibility of using nonlinear optics in PCF to perform essential functions in
high-speed all-optical processing.
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Figure 5 Experimental setup used to demonstrate all-optical
switching near 1,550 nm. (EDFA, erbium-doped fiber amplifier;
BWDM, bandpass wavelength-division multiplexor; PBS, polarization beamsplitter; FPC, fiber polarization controller) (From
Sharping, et al. q2002 IEEE).
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(5.8m)
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Figure 6 Switching curves (open boxes and filled circles),
showing the relative power measured in each port of the switch
vs. the pump peak power, for experiments conducted near
(a) 1,550 nm and (b) 780 nm. The curves accompanying the data
are generated from numerical solutions of coupled wave
equations for XPM. (From Sharping, et al. q2002 IEEE).
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Parametric (Mixing) Processes
P0180
The first set of experiments with controlled FWM in a
PCF achieved nondegenerate parametric gains over a
30 nm range of pump wavelengths near the l0 of the
PCF. The experiments also confirmed the wavelength
dependence of the GVD coefficient of the PCF near
l0 : Since the dispersion characteristics of these fibers
can be adjusted during the fabrication process, the
experiments demonstrate the potential for the use of
PCFs in broadband parametric amplifiers, wavelength shifters, and other optical communication
devices.
The experimental setup used to demonstrate
phase-matched FWM in PCF is shown in Figure 7.
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Figure 7 A schematic of the experimental setup used to
investigate FWM in PCFs. (From Sharping, et al. q2001 Optical
Society of America).
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Article Number: OPTC 00932
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DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers 7
Figure 8 A typical FWM spectrum observed at the output of the microstructure fiber. The inset shows a spectrum where higher-order
cascaded mixing is evident. (From Sharping, et al. q2001 Optical Society of America).
. PCFs can support a single transverse mode over
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an extremely broad wavelength range (370 nm –
1600 nm);
. PCF design parameters allow one to manipulate
the fiber’s GVD properties; and
. nonlinear interactions are enhanced due to the
polarization maintaining properties of PCFs which
result from form birefringence present in the core.
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The pump and the input signal are two synchronous
pulsed beams having 3 –5 nm wavelength separation
with the center wavelength tunable over a 720 –
850 nm range. The maximum peak power of the
pump pulses is .12 W: The two synchronous beams
are then combined and injected into the PCF. The
pump and signal’s optical paths are adjusted to
obtain temporal overlap in the PCF and their
polarizations are aligned by fiber polarization
controllers.
Figure 8 shows a typical FWM optical spectrum at
the output of a 6.1 m long PCF. Here the strong pump
beam and weak signal beam have wavelengths of
753 nm and 758 nm, respectively. The spectrum
shows the undepleted pump, the amplified signal,
and the generated idler at 747 nm. The spectra in
Figure 8 show that large gain is achievable for a
pump-to-signal spacing of 5 nm. Gain values of more
than 20 (13 dB) were obtained.
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These four properties combine to allow efficient
interactions to occur in PCFs which are either
inefficient or not possible at all in standard optical
fibers.
Conclusion
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In summary, the advantages of using photonic crystal
fibers for demonstrating nonlinear-fiber optical effects
arise from four novel properties:
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. the nonlinear coefficient is enhanced in small-core
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PCFs (core area of a few mm2);
See also
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Raman, Brillouin and Parametric Amplifiers: (00663). P0225
Nonlinear Sources: Raman Scattering (00862). Diffractive Systems: EM Theory of Photonic Crystals, 1D, 2D,
and 3D Band Structure Calculations (00926).
Further Reading
Agrawal GP (2000) Nonlinear Fiber Optics, 3rd edn.
Academic Press.
Birks TA, Knight JC and Russell PStJ (1997) Endlessly
single-mode photonic crystal fiber. Optical Letters 22:
961 – 963.
Hansryo J, Andrekson PK, Westlund M, Li J and
Hedekvist P (2002) Fiber-based optical parametric
Q2
Article Number: OPTC 00932
8 DIFFRACTIVE SYSTEMS / Nonlinear Optics in Photonic Crystal Fibers
Russell PStJ (2003) Photonic crystal fibers. Science 299:
358 – 362.
Sharping JE, Fiorentino M, Coken A, Kumar P and
Windele RS (2001) Four-wave mixing in microstructure
fiber. Optical Letters 26: 1048– 1050.
Shapring JE, Fiorentino M, Kumar P and Windelen RS
(2002) All-optical switching based on cross-phase
modulation in microstructure fiber. IEEE Photonics
Technology Letters 14: 77 – 79.
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amplifiers and their applications. IEEE Journal of
Selected Topics in Quantum Electronics 8: 506 – 520.
Monre TM, Richardson DJ, Broderick NGR and Bennett PJ
(2000) Modeling large air fraction holey optical fibers.
Journal of Lightwave Technology 18: 50 –56.
Ranka JK, Windeler RS and Stentz AJ (2000) Visible
continuum generation in air-silica microstructure optical
fibers with anomalous dispersion at 800 nm. Optical
Letters 25: 25– 27.