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PT -symmetry breaking in dual-core1
phosphate-glass photonic crystal fibers2
MATT IA LONGOBUCCO,1,2,3 LEXUA N THE TAI,4VIET HUNG3
NGUYEN,5JARO S Ł AW CIMEK,1BARTOSZ PAŁUBA ,2RY S Z A R D4
BUCZY ´
NSKI 1,2 AND MAREK TRIPPENBACH2
5
1
Department of Glass, Łukasiewicz Research Network - Institute of Microelectronics and Photonics, Aleja
6
Lotników 32/46, 02-668 Warsaw, Poland7
2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland8
3
School of Electrical and Electronics Engineering, Nanyang Technological University, 50 Nanyang Avenue,
9
639798 Singapore10
4Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland11
5International Training Institute for Materials Science (ITIMS), Hanoi University of Science and12
Technology (HUST), No 1 - Dai Co Viet Str., Hanoi, Vietnam13
*mattia.longobucco@imif.lukasiewicz.gov.pl, mlongobucco@fuw.edu.pl14
Abstract: We investigate the properties of a soft glass dual-core photonic crystal fiber for
15
application in multicore waveguiding with balanced gain and loss. Its base material is a phosphate
16
glass in a P
2
O
5
-Al
2
O
3
-Yb
2
O
3
-BaO-ZnO-MgO-Na
2
O oxide system. The separated gain and loss
17
channels are realized with two cores with ytterbium and copper doping of the base phosphate
18
glass. The ytterbium-doped core supports a laser (gain) activity under excitation with a pump at
19
1000 nm wavelength, while the copper-doped is responsible for strong attenuation at the same
20
wavelength. We establish conditions for an exact balance between gain and loss and investigate21
pulse propagation by solving a system of coupled generalized nonlinear Schrödinger equations.
22
We predict two states of light under excitation with hyperbolic secant pulses centered at 1000
23
nm: 1) linear oscillation of the pulse energy between gain and loss channel (
PT
-symmetry
24
state), with strong power attenuation; 2) retention of the pulse in the excited gain channel (broken
25
P T -symmetry), with very modest attenuation. The optimal pulse energy levels were identified26
to be 100 pJ (first state) and 430 pJ (second state).27
1. Introduction28
PT
-symmetry is a theoretical concept elaborated by Bender et al. in 1998 [1]. It is based on
29
the possibility of treating a Hermitian quantum system as a combination of two non-Hermitian
30
non-isolated subsystems [2]. Both of them are characterized by a nonzero flux of probability,
31
positive and negative respectively, across their boundaries; however, the non-isolated combined
32
system has no net flux of probability, i.e. it could exhibit real spectra or equivalently real
33
eigenvalues [3]. It was shown that the condition for
PT
-symmetry is that the complex potential
34
𝑉
involved satisfies the requirement
𝑉(𝑥)=𝑉∗(−𝑥)
. This concept finds application in several
35
fields of science, such as atomic systems [4], mechanics [5] and electronics [6].36
In the field of optics and the paraxial propagation regime, the condition on complex potential
37
translates into the one on the real and imaginary part of the refractive index, which should be
38
symmetric and anti-symmetric, respectively [7], i.e.39
𝑛(𝑥, 𝑦)=𝑛∗(−𝑥, 𝑦)(1)
This means that the two refractive index profiles should be symmetric with respect to the
40
central symmetry point and should have the same absolute values [8,9] If this condition is not
41
satisfied, the eigenvalues of the system cease to be real and the parity-time symmetry breaks down
42
(it is referred as
PT
-symmetry breaking), leading to complex spectra [10]. Theoretical works
43
predicted such scenario in several optical systems [7,11, 12], and also experimental verifications
44
were achieved in periodic structures [13], photonics lattices [14], semiconductor-based dual
45
microring laser resonators [15], plasmonic systems [16] and - recently - in high power large-area
46
lasers [17].47
One of the simplest realizations of
PT
-symmetric optical system is a coupled waveguide, with
48
one subjected to gain (active waveguide) and the other one to loss (dissipative waveguide) [18,19]
49
Moreover, it is possible to benefit the optical properties of gain/loss waveguides even in the
50
nonlinear regime, i.e. studying
PT
-symmetry in nonlinear directional couplers [20]. In these
51
systems, the non-Hermitian eigenvectors formally maintain the same structural form of the
52
corresponding linear one [8, 21,22] It has been demonstrated that such systems are beneficial
53
for all-optical switching in the nonlinear regime because of the possibility to lower the required
54
switching power [23], achieve faster transition [24] and support stable switching states due to the
55
possibility to support solitons [25, 26] The exact analytical formalism describing the switching
56
dynamics in nonlinear
PT
-symmetric couplers has been presented in [27]. In the last decades,
57
dual-core [28,29] and multicore optical fibers [30], which are one of the possible implementations
58
of the nonlinear directional coupler, have attracted significant interest in the implementation of
59
nonlinear
PT
-symmetric systems due to their possible application in all-optical signal processing.
60
In particular, dual-core fibers (DCFs) consist of two parallel channels throughout their whole
61
length: one of the channels should provide gain for the guided light along propagation (gain
62
channel), while the other one should cause losses to the propagating light (loss channel). To
63
satisfy Eq.(1), the amount of provided gain and loss should be equal.64
In this manuscript we present a proof of concept, a possible implementation of a
PT
-symmetric
65
optical system in the form of a DC photonic crystal fiber (PCF). The crucial features of such
66
systems are two regions of dynamics (stable and unstable) separated by the so-called exceptional
67
point. It is the area around the exceptional point that seems to be the most interesting, and we can
68
tune it by changing the system parameters, e.g. the intensity of the incident pulse.69
The base material of the fibre is phosphate glass in a P
2
O
5
-Al
2
O
3
-Yb
2
O
3
-BaO-ZnO-MgO-
70
Na
2
O oxide system [31]. The gain and loss channels are implemented by ytterbium-based and
71
copper-based doping, respectively. The fibre is suitable for fabrication by the stack-and-draw
72
method [32]. We present the numerical studies of nonlinear phenomena in such
PT
-symmetric
73
optical systems, first evaluating the effective parameters and then showing the predictions of
74
pulse propagation in such systems using a simple model. These considerations can also be seen
75
as an interesting perspective for all-optical switching using fibre-based devices.76
2. Materials and design77
The implementation of
PT
-symmetric DC PCF requires first of all the gain channel. We propose
78
to take advantage of the Yb-doped phosphate glass photonic crystal fiber laser fabricated by our
79
group [33]. Fig.1a presents the SEM images of the cross-section of the fabricated fiber with
80
different magnifications. The core material is phosphate glass doped with 6% mol of Yb
2
O
3
81
(15.69
·
10
20
Yb
3+
cm
-3
). In the frame of this study, a laser generation is demonstrated at the
82
central wavelength of approximately 1
µ
m with more than 400 dB
·
m
-1
of pump absorption
83
and the highest generation power of 150 W
·
m
-1
. The pump is a laser diode with a wavelength
84
of 973.5 nm and 3 nm bandwidth. The threshold power for laser activity is 8.7 W, while the
85
maximum pump power could reach the value of 35 W. The maximum output laser power in the
86
CW regime is 9 W. The slope efficiency, which indicates the power conversion between the pump
87
and the laser beam – i.e. power pump/laser gain– could be as high as 36.2%. For our purpose,
88
we consider this value as the target value of gain.89
Next, we focus on the loss channel. In order to respect the balance between gain and loss, we
90
need to use in the loss channel a glass that realizes a 36.2% power attenuation – or equivalently
91
73.8% transmission – at 1
µ
m and the same length of 6 cm. In order to estimate the loss coefficient
92
for the loss channel
𝛼
, we use the standard Beer-Lambert law,
𝑃(𝑧)=𝑃0𝑒−𝑖 𝛼𝑧
, where
𝑃0
is
93
the input power and z is the propagation distance. Considering that
𝑃(𝑧=
6 cm)
/𝑃0
should be
94
≈
73.8%, the absorption coefficient resulted in the value of
𝛼=
7.49 m
-1
. Here we propose
95
CuO-doped glass for the loss channel. As the phosphate glass has a much higher attenuation than
96
silica (0.46 vs 0.001 m
-1
) [31], the estimated required percentage of copper doping should be
97
rather low, at the level of 0.015% CuO.98
Fig. 1. (a) Refractive index profiles of the undoped phosphate glass (blue curve)
and Yb-doped one (red curve) in the spectral range 430-1490 nm. (b) Difference of
the refractive indices between undoped and Yb-doped phosphate glasses in the same
spectral range.
Fig. 1a reports the refractive index profile of the phosphate glass (blue curve) and Yb-doped
99
one (red curve) in the spectral range 430-1490 nm, while Fig. 1b shows the difference between
100
the two refractive indices in the same spectral range. The difference between the refractive
101
indices is rather low, reaching a maximum value of approximately 1.4
·
10
-3
at 1490 nm. At 1000
102
nm, the difference is 1.04
·
10
-3
. The nonlinear refractive index
𝑛2
of phosphate glass is 0.99
·103
10
-19
m
2·
W
-1
at 1064 nm [34], approximately 4 times higher than the fused silica one (0.246
·104
10-19 m2·W-1 [35]).105
3. Finding optical properties106
For the particular design of the fiber (including geometry and the material), we can find the
107
optical properties of the setup using numerical tools, for instant commercial Lumerical software.
108
We started with the design shown in Fig. 2a (schematic plot of the proposed fiber). This seemed
109
to be a natural choice. Notice that due to the small refractive index difference between Yb-doped
110
and undoped phosphate glass (see Fig. 1b), we need to introduce a photonic lattice of air holes
111
in the undoped phosphate glass in order to support the fundamental modes and improve the
112
coupling efficiency between the cores [36]. Still, the concentration of copper doping in the loss
113
channel is much lower than the ytterbium one in the gain channel (0.015% mol of CuO and 6%
114
mol of Yb
2
O
3
respectively) and further studies are required to get more experience with this kind
115
of glass. Therefore, we decided to postpone the study of this particular setup until we examine it
116
thoroughly, both theoretically and in real experiments.
Fig. 2. (a) Structure of the designed DC PCF laser: Yb-doped phosphate glass for the
gain channel, Cu-doped one for the loss channel, and undoped phosphate glass for the
cladding. An extra photonic lattice of air holes is introduced to support the coupling
between the cores. (b) Fiber structure used for the simulation phase: the material of
the loss channel has been substituted from Cu-doped phosphate glass to Yb-doped
phosphate glass.
117
Here we consider the structure in Fig. 2b, where we replace the Cu-doped core with another
118
Yb-doped one. The core diameter
𝑑
and the lattice pitch
Λ
(marked with yellow arrows in Fig.
119
2b) were the same as the optimized soft glass DCF presented in [37]. We used the same structure
120
with
𝑑=
1.85
µ
m and
Λ =
1.6
µ
m and added an extra photonic lattice of air holes with diameter
121
𝑑𝐴=
1.4
µ
m surrounding the two Yb-doped cores. The distance between centers of the cores is
122
then 2Λ = 3.2
µ
m, as in the fiber structure in [38].123
Subsequently, the new structure was characterized in the context of optical field propagation
124
in the linear regime. The commercial Mode Solution software from Lumerical was used to
125
calculate the spectral dependences of the field mode profile, the corresponding effective index,
126
and the waveguide losses for each fundamental mode. All the relevant quantities were acquired
127
in the spectral window between 500 and 2400 nm, which sufficiently covers the wavelength of
128
our interest (1000 nm).129
Fig. 3 shows the dispersion profiles of the fundamental supermodes, with horizontal polarization
130
direction (along X-axis) and dual-core symmetric state. The dispersion is normal (
𝐷 < 0
) in
131
the wavelength range of 800-1200 nm. We calculated another important linear parameter, the
132
coupling length
𝐿c
. From the theory of nonlinear directional couplers, we know that, in the
133
DCFs, the input radiation coupled in one of the two cores experiences (in the linear regime)
134
periodic oscillations between cores with a period equal to 𝐿c, defined as135
Fig. 3. Simulated dispersion curves of the fundamental Symmetric-X supermode of the
dual-core structure in Fig. 3b. The fiber shows normal dispersion in the whole range of
800-1200 nm.
𝐿c=𝜋
|𝛽S−𝛽A|,(2)
where
𝛽S
and
𝛽A
are the propagation constants of symmetric and antisymmetric supermodes of
136
the fiber, respectively [38,39].137
Fig. 4. Simulated coupling length
𝐿c
spectral characteristics for fundamental X- and
Y-polarized modes of the DC PCF in Fig. 3b. The values are calculated using Eq. (2).
Fig. 4 reports the coupling length characteristics of the two fundamental supermodes, with
138
horizontal and vertical polarization direction (X- and Y-polarization, respectively) in the same
139
wavelength range of Fig. 3 (800-1200 nm). At 1000 nm, the values of
𝐿c
are 4.7 and 7.3 cm,
140
respectively.141
We calculated other linear and nonlinear parameters at the wavelength of interest i.e. 1000 nm,
142
which were used for the numerical simulations. They include the effective refractive indices of
143
the cores
𝑛eff
, the propagation constants
𝛽0
,
𝛽1
,
𝛽2
, the coupling coefficient
𝜅
and the nonlinear
144
parameter
𝛾
. All the values except
𝜅
were calculated for a single core structure, that was obtained
145
by filling one of the cores with undoped phosphate glass with diameter
𝑑=
1.85
µ
m and including
146
one air hole with diameter
𝑑𝐴=
1.4
µ
m. We calculate the dispersion profile of the single-core
147
fibers (left and right core separately) using Mode Solution software from Lumerical, including
148
also the spectral dependences of the field mode profile, the corresponding effective index, and
149
the waveguide losses for each fundamental mode. We also calculated the coupling coefficient
𝜅150
between the two single-core modes based on the overlap integrals [39].151
4. Numerical simulations of pulse propagation: method and results152
In this section, we develop numerical methods to study
PT
-symmetric dynamics in our system.
153
The goal of this part of the investigation is twofold. First, we identify the most important
154
parameters (eliminating the others) and then we study the role of chromatic dispersion of the
155
crucial characteristics: pulse dispersion, inter-channel coupling, and gain/loss coefficient.156
4.1. Generalized Nonlinear Schrodinger Equation157
In order to have a complete view of the system dynamics, the coupled generalized nonlinear
158
Schrödinger equations (CGNSE) were solved numerically, including effects like coupling
159
coefficient dispersion, self-steepening nonlinearity, and its spectral dependence, stimulated
160
Raman contribution, cross-phase modulation, and waveguide losses. The resulting mathematical
161
model is a system of two equations expressed in the following set of equations (𝑟=1, 2)162
𝜕 𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑧 =(−1)𝑟+1−𝑖𝛿0𝐴(𝑟)(𝑧, 𝑡 ) − 𝛿1
𝜕 𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑡 +
𝑙
𝑖𝑙
𝑙!𝛼(𝑟)
𝑙
𝜕𝑙𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑡𝑙+
+1
2
𝑚
𝑖𝑚+1
𝑚!𝛽(𝑟)
𝑚
𝜕𝑚𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑡 𝑚+
𝑛
𝑖𝑛+1
𝑛!𝜅(𝑟)
𝑛
𝜕𝑛𝐴(3−𝑟)(𝑧, 𝑡)
𝜕𝑡 𝑛+
+𝑖𝛾 (𝑟)1+𝑖𝜏shk (𝑟)𝜕
𝜕𝑡 ∞
−∞
𝑅(𝜏)𝐴(𝑟)(𝑧, 𝑡 −𝜏|2𝑑𝜏 +𝜎(𝑟)|𝐴(3−𝑟)(𝑧, 𝑡) |2𝐴(𝑟)(𝑧, 𝑡 )
(3)
where
𝑟=
1, 2 denotes the number of the core (1 – gain channel, 2 – loss channel),
𝐴𝑟
is the
163
corresponding electric field amplitude and quantities
𝛿0=(𝛽(𝑟)
0−𝛽(3−𝑟)
0)
and
𝛿1=(𝛽(𝑟)
1−𝛽(3−𝑟)
1)164
represent the difference between the phase and group velocities respectively. Furthermore,
𝛼(𝑟)
𝑘
,
165
𝛽(𝑟)
𝑘
and
𝜅(𝑟)
𝑘
are the k-th order of Taylor expansion coefficients around the central frequency
166
of gain/loss coefficient, propagation constant (dispersion) and coupling coefficient, respectively.
167
Finally,
𝛾(𝑟)
is the nonlinear parameter,
𝜏shk (𝑟)
is the characteristic time of shock wave formation,
168
𝑅
is the Raman response function and
𝜎(𝑟)
is the overlap integral between the single core modes
169
defining for the cross-phase modulation effect in the r-th core. Both experimentally determined
170
instantaneous Kerr and delayed Raman response of the guiding PBG-08 glass are included in the
171
material nonlinear response function.172
Moreover, we introduced the gain and loss coefficient in the CGNLSE by modeling the function
173
of the loss coefficient
𝛼(𝑟)(𝜆)
. We modeled
𝛼(𝑟)(𝜆)
to have a Gaussian-like profile in the
174
wavelength domain as follows:175
𝛼(𝑟)(𝜆)=(−1)𝑟𝛼𝑝·exp (𝜆−𝜆0)2
2𝜎2
𝜆,(4)
where
𝜆0=
1000 nm, which corresponds to the frequency
𝜔0=2𝜋𝑐
𝜆0=1.8837 ·
10
15
rad
·
s
-1
,
176
𝜎𝜆=100
nm is the standard deviation of the Gaussian and
𝛼𝑝=7.49
m
-1
is the peak amplitude.
177
The CGNLSE in Eq. (3) was solved numerically by the Split-Step Fourier method with
178
160,000 steps [38]. After every 400 calculation steps, the field arrays were saved and then used
179
to plot the output propagation maps; this means that the whole propagation distance is divided
180
into 400 intervals. This approach represents a good compromise between the calculation time
181
and the resolution of the propagation distance (fiber length). We considered a fiber length of
182
30 cm, which is 10 times larger than the estimated coupling length at 1000 nm (see Fig. 4).
183
Using the Split-Step method, we considered the whole spectral behavior of loss and gain and
184
applied them always at the frequency step. The input pulse shape was approximated by the
sech2
185
function, which is a good approximation for ultrafast oscillators. The power envelope of the pulse
186
is expressed as:187
𝑃(𝑡)=0.88𝐸
𝑇FWHM sech21.763𝑡
𝑇FWHM (5)
At each of the 400 propagation steps, we integrated the pulse envelopes in each channel to
188
observe the trend of the energy transfer along propagation. Fig. 5 shows the propagation maps in
189
case of 1000 nm wavelength, 1 ps pulse width hyperbolic secant pulse excitation with energy 100
190
pJ (top row) and 430 pJ (bottom row). We checked that the
PT
-symmetry breaking takes place
191
at 430 pJ: an energy increase through the fiber length is predicted in the gain channel, with some
192
low-input features after 20 cm.193
Fig. 5. Time domain evolution of the field intensity in the excited (left) and non-excited
(right) core under excitation by 1000 nm central wavelength and 1 ps width pulses with
100 pJ (top) and 430 pJ (bottom) energies, in the case of gain channel (left) excitation
of a 30 cm length fiber with structure as in Fig. 2b.
4.2. Simplified theoretical model194
To simplify the model in Eq. (3), we set
𝜎=
0 (no cross-phase modulation),
𝑡shk =
0 (no shock
195
wave formation), and
∞
−∞ 𝑅(𝜏)𝑑𝜏 =
1 (impulsive Raman response). Moreover, we limited the
196
dispersion terms to the second-order
𝛽(𝑟)
2
, and only linear coupling. Due to the sensitivity of
197
the system towards the change of gain and loss, we keep the gain/loss coefficient with the full
198
spectral dependence. We considered symmetric fiber structure, therefore
𝜅(1,2)
0=𝜅(2,1)
0=𝜅0
,
199
𝛽(1)
2=𝛽(2)
2=𝛽2and 𝛾(1)=𝛾(2)=𝛾. Set of Eqs. (3) takes the form as follows200
𝜕 𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑧 =−𝑖 𝛽2
2
𝜕2𝐴(𝑟)(𝑧, 𝑡)
𝑑𝑡2+𝑖𝜅0𝐴(3−𝑟)(𝑧, 𝑡 )+
𝛼(𝑟)∗𝐴(𝑟)(𝑧, 𝑡)+𝑖𝛾 𝐴(𝑟)(𝑧, 𝑡)2𝐴(𝑟)(𝑧, 𝑡)
(6)
where (
𝑟=1,2
). Since the function
𝛼(𝑟)
is defined in the frequency domain (
𝛼(𝑟)=𝛼(𝑟)(𝜔)
)
201
it is crucial to apply convolution according to the following property of Fourier transform:
202
(𝑓·𝑔)(𝑡)=(˜
𝑓∗˜𝑔)(𝑡). In our case, it is given by:
(𝛼(𝑟)·˜
𝐴(𝑟))(𝑧, 𝑡)=(
𝛼(𝑟)∗𝐴(𝑟))(𝑧, 𝑡).203
Fig. 6. Normalized integrated energies in the corresponding channels at the correspond-
ing energy levels of (a) 100 pJ and (b) 430 pJ for full (solid blue lines: gain channel,
solid red lines: loss channel) and simplified model (dashed blue lines: gain channel,
dashed red lines: loss channel).
For the simulation study with the simplified model, we used the optical parameters calculated
204
at 1000 nm central wavelength by Lumerical. All the parameters are the same for both cores.
205
The estimated value for the gain/loss coefficients (7.49 m
-1
) was included in the parameter
𝛼0
: it
206
has a positive sign for loss, indicating power attenuation, and negative one for gain, indicating
207
power increase. The other parameters were
𝛽2=4.21 ×10−25
m
·
s
-2
,
𝛾=0.3
W
-1 ·
m
-1
and
208
𝜅0=33.74 m−1.209
We generated, with our simplified model, 2D time-domain evolution plots with parameters
210
corresponding to those used in Fig. 5 and the difference was hard to notice. Therefore, in order
211
to trace subtle differences we looked at the pulse shapes in the time domain at 4 specific fiber
212
lengths for the two cases above. The result is shown in Fig. 6 for input energies 100 pJ and 430
213
pJ, respectively. Solid lines present the results of the simulation obtained using the full model
214
(red: unexcited, loss channel; blue: excited, gain channel), while dashed lines present the ones
215
obtained using the simplified model. In Fig. 7, we compare pulse shapes generated by both
216
methods; the four panels refer to lengths: (a) 8 cm, (b) 12 cm, (c) 21 cm, and (d) 30 cm. We
217
observed that there is a close correspondence between the two models in each reported case. A
218
small discrepancy between the models is observed only on the rising edge of the pulse in the loss
219
channel at 21 and 30 cm (i.e. between solid and dashed red lines of Fig. 7c and d for
𝑡
in the
220
range -1.0 to 1.0). In Fig. 8, which was calculated in the unstable regime, the four panels refer to
221
lengths: (a) 8 cm, (b) 10 cm, (c) 12 cm and (d) 14 cm. In this case, as observed in Fig. 8c and d,
222
the two models give significantly different results after 12 cm and 14 cm. The solid and dashed
223
curves differ much more from each other than in case of 8 cm and 10 cm of propagation distances.
224
It is not a surprise that in an unstable regime, propagation is sensitive to the fiber parameters used
225
in the extended and simplified models, which becomes more visible with increasing propagation
226
distance. However, the good news is, that the position of the exceptional point, the border between
227
Fig. 7. Snapshot of 100pJ pulses corresponding channels simulated in full model
(solid blue lines: gain channel, solid red lines: gain channel) and simplified model
(dashed blue lines: gain channel, dashed red lines: loss channel) at different propagation
distances: a) 8 cm, b) 12 cm, c) 21 cm and d) 30 cm.
stable and unstable regimes, in both models is very similar. Hence, the simplified model can be
228
used for finding critical intensity in the nonlinear regime.229
4.3. The role of dispersion230
In this section, we investigate the role of the dispersion of crucial parameters on the stability
231
of pulse propagation in a waveguide with two coupled channels: intra-channel coupling
𝜅
and
232
gain/loss coefficient
𝛼
. We restrict the analysis to the linear regime and study the dynamics
233
described by the equation234
𝜕 𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑧 =−𝑖 𝛽2
2
𝜕2𝐴(𝑟)(𝑧, 𝑡)
𝜕𝑡2+𝑖𝜅∗𝐴(3−𝑟)(𝑧, 𝑡) + 𝛼∗𝐴(𝑟)(𝑧, 𝑡).(7)
Note that we have two terms on the right hand side of the equation (7) in the form of a convolution.
235
Each of the functions
𝜅
and
𝛼
depend on the frequency
𝜔
. For the current study, we have
236
chosen Gaussian functions for all dispersion profiles and each of them is characterized by three237
parameters: width
𝜎
, central frequency
𝜔0
and maximums
𝜅0
and
𝛼0
respectively. For example,
238
the inter-core coupling function will be defined as239
𝜅(𝜔)=𝜅0exp (𝜔−𝜔0,𝜅 )2
𝜎2
𝜅.(8)
The dispersion in the gain/loss coefficient
𝛼(𝜔)
is introduced by analogy. We have checked that
240
our conclusions are the same if we use Lorenzian functions instead of Gaussians in the frequency
241
domain. In this formulation, the constant coefficient corresponds to the Gaussian function, which
242
is a very broad function of frequency (formally for
𝜎→ ∞
). In each of the studies reported
243
below, we looked for the exceptional point that lies on the boundary between stable and unstable
244
propagation regions. Unstable propagation is characterized by the exponential growth of the
245
signal in the gain channel, while in the lossy channel the pulse decays rapidly to zero.246
Fig. 8. Snapshot of 430 pJ pulses corresponding channels simulated in full model (solid
blue lines-gain channel, solid red lines-gain channel) and simplified model (dashed blue
lines: gain channel, dashed red lines: loss channel) at different propagation distances:
a) 8 cm, b) 10 cm, c) 12 cm and d) 14 cm.
In the case of linear propagation considered here, it is sufficient to examine the maximal real
247
part of the eigenvalue
ℜ(𝜆)max
of the characteristic equation derived from Eq. (7). It is worth
248
noting, that this result remains consistent regardless of the shape of intensity of the input pulse.
249
In the nonlinear scenario, however, our approach becomes ineffective. Then, in order to predict
250
the dynamics of the system and to identify the specific exceptional points, we must resort to
251
direct simulations. The ability to find these exceptional points depends on various parameters
252
associated with the channels, and this requires a different methodology compared to the linear
253
case.254
First, we considered the case where all coefficients are constant. In this case, the system
255
becomes unstable when the magnitude of
𝛼
is greater than the inter-channel coupling
𝜅
, regardless
256
of the value of the dispersion
𝛽2
, which is a well-known result. Then, we introduced the dispersion
257
to
𝜅
and
𝛼
in the manner described above. A summary of this study is shown in Fig. 9. We clearly
258
observed that, as long as the width of the coupling 𝜎𝜅is greater than the gain/loss profile given259
by
𝜎𝛼
, we have a region of stable dynamics (while the maximums
𝜅0> 𝛼0
). In the opposite case,
260
when the width of the gain/loss function exceeds the width of the coupling function,
ℜ(𝜆)max
is
261
always positive and tends to linear growth (with increasing value of
𝛼0
) as the gain/loss profiles
262
become more and more narrow. This gives us a clue about the possible components of our system.
263
If we include a small nonlinearity, we expect essentially the same characteristics, with the shift
264
of the exceptional point.265
Finally, we investigated, whether the center frequency for the gain/loss and coupling profiles
266
need not be the same. To do this, we introduced the shift
Δ = |𝜔0, 𝛼 −𝜔0, 𝜅 |
between the two
267
profiles and set the width of the inter-channel coupling to be several times greater than the width
268
of the gain/loss parameter (
𝜎𝜅=6
and
𝜎𝛼=1
). As it can be seen in Fig. 10, as long as the
269
shift between the two profiles is smaller than the width of the coupling (note that we arbitrarily
270
choose the value of the gain/loss profile to be equal to one), an exceptional point appears. We
271
have carried out thorough studies, including various coupling widths and shifts, to establish
272
Fig. 9. Maximum value of the real part of the eigenvalues for different amplitudes of
the coupling coefficient. The width of the gain/loss coefficient is arbitrarily set equal to
one, and the widths of the coupling coefficients as a function of
𝜔
are given in the inset.
Note that in all cases where the width of the coupling is smaller than the width of the
gain/loss coefficient, we do not observe stable propagation.
that the features shown in Fig. 10 are representative over a wide range of parameters, including
273
proposed configuration. Although, our conclusion is purely qualitative, it serves the purpose of
274
our pilot study. The main insight from our simulations is that, for practical implementation, we
275
should look for setups with consistent coupling and carefully study how the relative shifts of the
276
resonances, in both 𝜅and 𝛼coefficients, vary with frequency.277
5. Conclusion278
We predicted
PT
-symmetry breaking in a DC PCF made of phosphate glasses synthesized
279
in-house. The fiber cores are made of phosphate glasses with 6% mol of Yb
2
O
3
(gain channel)
280
and 0.015% mol CuO doping (loss channels) and have the following structural parameters: core
281
diameters of 1.85
µ
m, lattice pitch of 1.6
µ
m, and extra photonic lattice of air holes with diameter
282
of 1.4
µ
m. The fiber exhibits normal dispersion of the fundamental supermodes in the range of
283
500-2000 nm and coupling length in the order of 5 cm at 1000 nm. We investigated the stability
284
of the
PT
-symmetric DC PCF system by simulating the propagation of hyperbolic secant pulses
285
with a width of 10 ps and 1 ps: the system is stable considering both temporal widths and
286
values of gain/loss and coupling coefficients ratio being
𝛼/𝜅≤
0.25. The designed fiber resulted
287
in a
𝛼/𝜅
value of 0.22. Then, we solved the system of CGNLSE with the Split-Step method,
288
considering excitation pulses with a wavelength of 1000 nm and width of 1 ps. We predicted two
289
regimes of light propagation through the designed fiber: 1) linear oscillations of the pulse energy
290
between the gain and loss channel (
PT
-symmetry state); 2) unstable dynamics with strong
291
enhancement in both channels (broken
PT
-symmetry). Initial input energies were 100 pJ and
292
430 pJ, respectively. The same scenarios were predicted considering pulses with the same input
293
energy and using a simplified theoretical model, which only includes second-order dispersion
294
term, linear coupling, first-order nonlinearity, and dispersive gain/loss coefficient. We carried out
295
an extensive investigation of the influence of dispersion on both the gain/loss and the coupling.
296
Our investigation led us to understand that stable dynamics prevails when the coupling width (
𝜎𝜅
)
297
exceeds the gain/loss profile width (
𝜎𝜅> 𝜎𝛼
), while a linear growth of intensity is predicted
298
Fig. 10. Maximal real part from the set of eigenvalues in the case when gain/loss and
interchannel coefficients are centered at frequencies shifted by
Δ
, as indicated in the
inset. Other parameters are: 𝛽2=4,𝜎𝛼=1,𝜎𝜅=6and 𝜅0=1.
in the opposite case. These predictions hold even when the relative frequency shift between
299
the gain/loss and coupling profiles is taken into account. The results presented here represent a
300
very promising prediction of
PT
-symmetric breaking using a manufacturable dual-core optical
301
fibre. This breakthrough has significant potential for several applications, including all-optical
302
switching and the development of robust high-power lasers.303
Funding. National Science Center, Poland (project No. 2019/33/N/ST7/03142, under the PRELUDIUM-
304
17 program) and Vietnam National Foundation for Science and Technology Development (NAFOSTED,
305
under grant number 103.01-2021.152).306
Disclosures. The authors declare no conflicts of interest.307
Data Availability. Data underlying the results presented in this paper are not publicly available at this
308
time but may be obtained from the authors upon reasonable request.309
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