Abstract. In this paper we present the development of a new direction in so-called optofluidics,
namely the research of photonic crystal fibers (PCF) infiltrated with liquids. In particular we concentrate on the flagship application of PCF, the process of Supercontinuum Generation (SG), in
which injected monochromatic pulse may be dramatically broadened (spectrally), which creates a
coherent beam generation of high brightness comparable to that of monochromatic lasers. The supercontinuum is formed when a collection of nonlinear processes act together upon a pump beam
in order to cause severe spectral broadening of the original pump beam. Explanation of this process is based on numerical simulations for Generalized Nonlinear Schrodinger equation (GNLSE) ¨
which describes the rich nonlinear dynamics of pulse propagation in nonlinear dispersive media.
All nonlinear phenomena involved in SG will be analyzed. We present specially activity of the
Polish-Vietnamese Group from the beginning in 2016 to recent time in this domain. Some recent
scientific projects concerning fiber physics of our Group in the near future, especially applications
in Biology and Medicine will be mentioned.

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Communications in Physics, Vol. 31, No. 1 (2021), pp. 1-22
DOI:10.15625/0868-3166/15547
REVIEW PAPER
SUPERCONTINUUM GENERATION IN PHOTONIC CRYSTAL FIBERS
INFILTRATEDWITH LIQUIDS
CAO LONG VAN†
Institute of Physics, University of Zielona Go´ra, Prof. Szafrana 4a, 65-516 Zielona Go´ra, Poland
†E-mail: caolongvanuz@gmail.com
Received 28 September 2020
Accepted for publication 14 October 2020
Published 5 January 2021
Abstract. In this paper we present the development of a new direction in so-called optofluidics,
namely the research of photonic crystal fibers (PCF) infiltrated with liquids. In particular we con-
centrate on the flagship application of PCF, the process of Supercontinuum Generation (SG), in
which injected monochromatic pulse may be dramatically broadened (spectrally), which creates a
coherent beam generation of high brightness comparable to that of monochromatic lasers. The su-
percontinuum is formed when a collection of nonlinear processes act together upon a pump beam
in order to cause severe spectral broadening of the original pump beam. Explanation of this pro-
cess is based on numerical simulations for Generalized Nonlinear Schro¨dinger equation (GNLSE)
which describes the rich nonlinear dynamics of pulse propagation in nonlinear dispersive media.
All nonlinear phenomena involved in SG will be analyzed. We present specially activity of the
Polish-Vietnamese Group from the beginning in 2016 to recent time in this domain. Some recent
scientific projects concerning fiber physics of our Group in the near future, especially applications
in Biology and Medicine will be mentioned.
Keywords: Photonic Crystal Fiber, Supercontinuum Generation, Generalized Nonlinear Schro¨dinger
equation.
Classification numbers: 42.55.Tv; 88.60.np; 42.25.-p.
©2021 Vietnam Academy of Science and Technology
2 SUPERCONTINUUM GENERATION IN PHOTONIC CRYSTAL FIBERS INFILTRATED WITH LIQUIDS
I. INTRODUCTION
Supercontinuum is a phenomenon based on a set of effects resulted from combined inter-
action of linear and nonlinear properties of optical medium involved in photonic crystal fibers
(PCFs). The input monochromatic pulse of light is broadened spectrally to form a continuous
spectrum in a beam of high brightness, even comparable to that of a monochromatic laser. The
output spectrum becomes much wider than the input (Fig. 1).
Fig. 1. Supercontinuum generation.
Discovery of the lasers realized experimentally by Maiman [1] for the first time in 1960
gives as a light source with very high intensity. This created a very important domain of physics,
namely nonlinear optics [2]. To this time the majority of optical effects are considered only in
the case of a linear material response of a medium to an incoming light field. However when the
intense light field is high, nonlinear optical effects appear which dramatically modify properties of
the incoming light. Experimental nonlinear optics started from the experiment on second harmonic
generation performed in 1961 by Franken et al. [3]. In this experiment, two photons having the
same frequency made up a photon of the doubled frequency by the nonlinear material responses
of the medium. In this phenomenon, a light spectrum of an enormous broaden bandwidth was
generated together with new frequency components, so one can say that a phenomenon, later
called as Supercontinuum Generation (SG) has been observed there for the first time.
The further development of nonlinear optics provided to the appearance of conventional
optical fibers which made the revolution not only in nonlinear optics but also in higher technolo-
gies as in telecommunication for last few decades. This discovery is treated as one of the greatest
scientific advents in the XX century. A new field of physics appeared, namely nonlinear fiber
optics [4]. This field had developed intensively, so new nonlinear phenomena were discovered,
such as self-phase modulation in fibers [5] and parametric mixing processes [6, 7]. In particular
optical solitons [8] and their dynamics [9, 10] have been highly interested by researchers. These
CAO LONG VAN 3
nonlinear phenomena played an important role in explanation of SG mentioned above. Now we
have an ultra-broad spectrum spanning almost the complete visible region of light [11, 12].
Guiding of light in optical fibers based on the phenomenon of total internal reflection (TIR).
A break point in development of optical fibers has been made in the 1950s by covering of glass
fiber with a material with a lower refractive index [13,14]. This type of fiber structure including the
core and cladding is used as the basic solution until now. Since the first experimental realization
of low-loss optical fiber in 1970 [15], worldwide telecommunication experienced extraordinary
growth.
However the conventional fiber guiding the light by TIR has several limits related to op-
tical nonlinearities and attenuation. Several new guiding mechanisms minimizing these effects
have been proposed. Among these mechanisms, discovery of Photonic Crystal Fiber (PCF) is a
milestone in the development of fiber optics. This has been started by papers [16, 17] through
extending band-gap mechanism in solid state physics to photonics. In 1991 the first experimental
observation of the photonic band-gap effects has been demonstrated in Ref. [18]. Inspired by this
paper, Russel and his collaborators fabricated the first PCF it by the method of stacking silica
capillaries in a hexagonal pattern. The light guiding in the core is based the band-gap mechanism.
The typical structure of the first PCF consists of micro-structured fibers made from silica glass
with air-holes arranged in a hexagonal lattice with a solid core, named now solid-core photonic
crystal fiber [19]. The central region was replaced by a glass rod acting as the core (see Fig. 2(a)).
The advantage of solid-core PCF in comparison with conventional fiber is that by changing
its geometrical structure, namely the diameter and lattice pitch of air-holes in the cladding region,
one can freely manages its dispersion (D). From other side, one can use various materials to
moving the zero dispersion wavelength (ZDW) and also making modification of the dispersion’s
shape. A typical example that for the PCFs made of silica, the ZDW can be shifted into the visible
region, in consequence a range of various pulse sources would be effectively used for generating
broadband SC can cover both of the near-infrared and visible spectrum. It is available to achieve a
large negative dispersion and ultra-flat or normal dispersion, which is essential for SG process in
the required wavelength range.
In other paper [20] Russel and his coworkers tried to guide the light in the air-core (see
Fig. 2(b)). The larger air core is now in the central region. PCF with this structure can confine
more than 99% of the light in its core. It called later called hollow-core photonic crystal fiber
(HC-PCF). It has a second name as photonic band-gap (PBG) fiber because of its guiding mecha-
nism. Further studies showed that the exact structure of the cladding is not essential for obtaining
broadband guidance [22, 23], but the geometry of the first layer surrounding the core is crucial
for the transmission character of the fiber [24, 25]. Wang et al. experimentally demonstrated in
Ref. [25] that loss of this fiber depends crucially on the core shape. A breakthrough to guide the
light in the air-core fiber is achieved by Pryamikov et al. [26] when they fabricated a fiber with a
single cladding layer and the core formed by a single row of silica capillaries (see Fig. 2(c)). In
comparison with PBG-fibers, this fiber has relatively broader transmission windows in the mid-
infrared range (MIR) and lower loss. It is named negative-curvature core fiber. This fiber is also
called as anti-resonant (AR) fiber. The disadvantage of PBG fiber is that its attenuation is still not
low enough for communication. Therefore, HC-PCFs are not suitable for data transmission for a
long distance. However the large possibilities of modifying their structure, and thus their optical
properties gave us many various applications not achieved by conventional ones. One can mention
4 SUPERCONTINUUM GENERATION IN PHOTONIC CRYSTAL FIBERS INFILTRATED WITH LIQUIDS
Fig. 2. Scanning electron microscope (SEM) images of (a) solid-core PCF [19], (b) PBG-
fiber [20], (c) AR fiber [21].
some of them: a large mode area (LMA) HC-PCF used for high power delivery [27], HC-PCF
with high birefringence [28] used to measure physical properties, while liquids and gases can be
detected by the sensor using suspended core PCF [29]. With the flexible capability of modifying
dispersion and mode area, HC-PCF is a most suitable for studying the nonlinear effects involved
in SG [11].
Furthermore the use of different materials for making PCFs, for example heavy metal oxide
glasses and tellurite which have much higher nonlinear coefficients than silica glasses, gives us
more broadband transmission into the mid-infrared region [30]. This creates new possibilities for
the fiber design. Reassuming, PCFs are powerful flexible tool for considering all optical effects.
Furthermore they have many effective applications.
On other hand, one can infiltrate liquid into the core of HC-PCF. By varying the liquids’
refractive index properly we can modify the guiding effect of the fiber from basing on modified
total internal reflection to basing on the photonic band gap effects. In consequence, the optical
characteristics of the fiber would highly depend on the temperature and concentration of liquids.
Therefore by using liquids, the dispersion properties of the fiber can be further modified without
changing its geometrical parameters by varying the temperatures and concentrations. One can
say that insertion of liquids in PCFs introduces new degrees of freedom for observing and con-
trolling nonlinear effects. In Sec. II, we will present in detail dispersion and its engineering in
PCF infiltrated with liquids. Due to the unique properties of the liquid, e.g. high nonlinearity,
high transparency, possibility of modifying the refractive index by changing ambient temperature,
liquid core PCFs have numerous applications. We can mention some examples: the liquid core
PCFs applied in e.g. sensing, laser, and SG; The AR fiber infiltrated with low-index liquid used
for optofluidic laser [31] and as temperature sensors [32]. . .
CAO LONG VAN 5
My paper is organized as follows. In Sec. II, I will briefly present an important concept,
namely the dispersion an how can modify it for different purposes, in other words how can make
dispersion engineering. For this purpose Mode Solutions (MODE) simulation program provided
by Lumerical to modelling PCF infiltrated with liquids for different applications, in particular for
SG is used. I will not demonstrate this in detail here, concentrating only on analyzing SG in
Sec. III. The last section contains conclusions.
II. DISPERSION AND ITS ENGINEERING IN PCF INFILTRATEDWITH LIQUIDS
In this section, we describe briefly theory of propagating light in optical fibers. At first we
start from linear propagation in standard step-index fibers for fixing some essential concepts, in
particular the dispersion. Next, we consider the propagation in PCFs. Finally we extend this to
nonlinear propagation which leads to Generalized Nonlinear Schro¨dinger Equations (GNLSE).
In which, the characterized parameters and quantities of light propagating process are clearly
mentioned.
A standard derivation of the NLSE in arbitrary dispersive media has been given in [33,34].
A more detailed derivation for the case of optical fibers can be found in an excellent book [4], in
which all nonlinear phenomena involved in the problem have been clearly described. If a source
less medium is assumed, one that does not generate current, but can still exhibit attenuation and
gain, from the system of Maxwell’s equation one can arrive at the wave equation (see also Eq. (1)
in Ref. [33]):
∇×∇×E= 1
c2
∂ 2E
∂ t2
−µ0 ∂
2P
∂ t2
= 0, (1)
where c is the light’s speed in vacuum and c2 = 1µ0ε
Generally, one can use a quantum mechanical to check the polarization P. But it is well
known that when the frequency is near a medium resonance, the quantum treatment is necessary.
But in our case we are far from medium resonances, so we can use Taylor’s series for P as follows
(Eq. (4) in [33]):
P˜= ε0
(
χ˜(1)⊕ E˜+ χ˜(2)⊕ E˜E˜+ χ˜(3)⊕ E˜E˜E˜+ . . .+ χ˜(i)⊕ E˜i
)
. (2)
here χ i (i = 1,2,3, . . .) is i-th order of the material’s electric susceptibility. In the further we
will consider the fiber made by symmetric molecules SiO2, the second order susceptibility χ(2)
vanishes because of P(−E) = −P(E). If we take only the third order nonlinear effects related to
χ(3), the polarization induction consists of linear and nonlinear part:
P(r, t) = PL(r, t)+PNL(r, t), (3)
which are expressed correspondingly as:
PL(r, t) = ε0
t∫
−∞
χ(1)(t− t ′)E(r, t ′)dt ′, (4)
PNL(r, t) = ε0
∫ ∫ +∞∫
−∞
χ(3)(t− t1, t− t2, t− t3)
...E(r, t1)E(r, t2)E(r, t3)dt1dt2dt3, (5)
6 SUPERCONTINUUM GENERATION IN PHOTONIC CRYSTAL FIBERS INFILTRATED WITH LIQUIDS
Now we assume for simplicity that nonlinear polarization PNL in Eq. (3) is only a small perturba-
tion to the total polarization. This is reasonable because for silica fiber the nonlinear effects are
relatively weak. Thus in the first step, we put PNL = 0 in equation (1). Then it becomes linear in
E. Furthermore in the case of optical fibers we have propagation in an arbitrary direction, say Oz.
Thus in the linear regime, the linear wave equation has the form [4]:
∂ 2E˜
∂ρ2
+
1
ρ
∂ 2E˜
∂ρ2
+
1
ρ2
∂ 2E˜
∂ϕ2
+
∂ 2E˜
∂ z2
+n2k20E˜= 0 . (6)
where k0 = ω/c = 2pi/λ and E˜ is the Fourier transform of the electric field E(r, t). The general
solution can be presented as superposition of waves propagating in the z-direction:
E˜z(r,ω) = A(ω)F(ρ)exp(±imϕ)exp(iβ z) (7)
where A is normalization constant depending only on ω , m is an integer number and β is the
propagation constant defined as:
β (ω) =
[
n2(ω)k2o− k2(ω)
]1/2
. (8)
The propagation constant β is one of the key parameters to describe the light propagation
in the optical fiber. The values of β are different for individual modes and depends on the fre-
quency (wavelength). They are determined from the following equation for F (frequently called
as eigenvalue equation)
d2F
dρ2
+
1
ρ
dF
dρ
+
(
n2k20−β 2−
m2
ρ2
)
F = 0. (9)
This equation has been considered in many excellent books [35–37] for determining an
important parameter characterizing arbitrary fibers, namely apart from the relative core index dif-
ferences
∆=
n1−n2
n1
(10)
the normalized frequency V is defined by
V = k0a(n21−n22)1/2, (11)
where a is radius of the core and λ is wavelength. Thus the number of modes of a particular fiber
per specific wavelength depends on its design parameters and is determined by so called cut-off
condition: All other modes are beyond cut-off if the parameter V < VC, where VC is the smallest
solution of J0VC = 0 or VC ≈ 2.405. In practice fibers are designed to satisfy that V is close to
VC. The cut-off wavelength λC for single-mode fibers might be obtained with k0 = 2pi/λC and
V = 2.405 in Eq. (11), core radius should be below 2 µm for fiber to support a single mode in the
visible region. Single mode fiber is commonly used to transmit over longer distances [38].
It is well-known from Newton’s time that in light propagation through a medium, as in
an optical fiber, different modes need different time to pass through a certain distance. The total
dispersion of optical fiber defined as:
Dt =
√
τ20 − τ2i (12)
CAO LONG VAN 7
where τ0,τi are the width of the input pulse and the output pulse, the unit is seconds [s]. Usually
we are interested in pulse expansion per kilometer. Dispersion increases the width of the pulse, it
is calculated in units of [ps /km].
For monochromatic light waves propagating along the waveguide in the z-direction (waveg-
uide axis), these constant phases move with phase velocity:
vp =
dz
dt
=
ω
β
. (13)
In practice, the light wave is not ideally mono-chromatic, there is a group of waves with
frequencies near each other spreading so that the final form has a waveform. This group of waves
propagates with group velocity:
vg =
∂ω
∂β
. (14)
We have the following relation for Group velocity:
vg =
c
ng
where
ng = n1−λ dn1dλ (15)
is called group index.
For short pulse propagation in a single mode fiber [33, 34], the wavelength dependence of
the propagation constant βω) is called chromatic dispersion, which causes the short light pulse to
be broadened. Expanding βω) into Taylor series around the central frequency ω0 we obtain:
β (ω) = β0 +β1(ω−ω0)+ 12β2(ω−ω0)
2 + ..., (16)
where:
βm =
(
dmβ (ω)
dωm
)
ω=ωo
(m = 0,1,2, . . .). (17)
The parameters β1 and β2 are related to the refractive index n = n1 and its derivatives
through the relations:
β1 =
1
vg
=
ng
c
(18)
β2 =
1
2
(
2
dn
dω
+ω
d2n
dω2
)
. (19)
Thus from physical point of view, the envelope of an optical pulse moves at the group ve-
locity, whereas the parameter β2 represents dispersion of the group velocity and is responsible
for pulse broadening. This phenomenon is known in literature as the group-velocity dispersion
(GVD), and β2 is the GVD parameter. The GVD is the change of the group velocity with fre-
quency. The wavelength range is said to be normal dispersion regime when β2 > 0. In this case,
group velocity decreasing with increasing optical frequency. The wavelength corresponding to
a value of β2 < 0 is called the anomalous dispersion regime (group velocity increasing with in-
creasing optical frequency). The wavelength where β2 = 0 is referred to as the zero-dispersion
wavelength (ZDW) (see Fig. 3).
8 SUPERCONTINUUM GENERATION IN PHOTONIC CRYSTAL FIBERS INFILTRATED WITH LIQUIDS
Fig. 3. Dispersion regimes in relation to the ZDW [39].
Another definition of the GVD is frequently used to describe the dispersion of fibers,
namely the parameter D:
D =
dβ1
dλ
=−2pic
λ 2
β2 ≈ λc
d2ne f f
dλ 2
. (20)
It is expressed in units of ps/km/nm. According with Fig. 3, the region with D < 0 is
normal dispersion, D > 0 (β2 < 0) corresponds to anomalous dispersion regime and where D = 0,
the wavelength is zero-dispersion wavelength (ZDW). The effects of higher-order dispersion will
be taken into account when the very short pulses launch into the fiber or the central wavelength
is very closed to the ZDW. The D of optical fiber depends on its geometrical structure and fiber
material, and it is approximate to the sum of waveguide dispersion DW and material dispersion
DM:
D = DW +DM (21)
DW (λ ) originates from the geometry of the fiber refractive index distribution that deter-
mines the D relation of the guided mode. DM originates from the dependence of the refractive
index of fiber material on the wavelength n2g = n2gλ , due to the difference in group velocities
of different spectral components in the fiber. Pulse expansion due to material dispersion can be
obtained by examining the group delay time in optical fiber [34]. The material dispersion DM and
the waveguide dispersion DW are given by the following formulas:
DM =−2piλ 2
dn2g
dω
=
1
c
dn2g
dλ
,DW =−2pi∆λ 2
[
n22g
n2(ω)
V d2(V b)
dV 2
+
dn2g
dω
d(V b)
dV
,
]
(22)
where n2g is the group index of the cladding material and V is given by Eq. (11), ∆ is given by the
Eq. (10) and b = β/k0−n2n1−n2 . The relation of (21) is presented on Fig. 4
CAO LONG VAN 9
Fig. 4. Total dispersion Dand relative contributions of material dispersion DM and waveg-
uide dispersion DW for a single mode fiber [4