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Signal and crosstalk analysis using
optical convolution of transmitted
optical signals
Ikechi A. Ukaegbu, Anel Poluektova, Elochukwu
Onyejegbu, Aresh Dadlani, Hyo-Hoon Park
Ikechi A. Ukaegbu, Anel Poluektova, Elochukwu Onyejegbu, Aresh Dadlani,
Hyo-Hoon Park, "Signal and crosstalk analysis using optical convolution of
transmitted optical signals," Proc. SPIE 10912, Physics and Simulation of
Optoelectronic Devices XXVII, 1091219 (26 February 2019); doi:
10.1117/12.2505099
Event: SPIE OPTO, 2019, San Francisco, California, United States
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Signal and Crosstalk Analysis Using Optical Convolution of
Transmitted Optical Signals
Ikechi A. Ukaegbu a, Anel Poluektovaa, Elochukwu Onyejegbua, Aresh Dadlania, Hyo-Hoon Parkb
aDepartment of Electrical & Computer Engineering, Nazarbayev University
53 Kabanbay Batyr Av., 010000 Astana, Kazakhstan; bDepartment of Electrical Engineering, Korea
Advance Institute of Science and Technology, 291 Daehak-ro Yuseong-gu, Daejeon, 34141, Korea
ABSTRACT
An optical system which consists of a transmitter array, a fiber array, and a receiver array, experience some signal loss
and crosstalk as the signals travel from the transmitter to the receiver. Signal loss and crosstalk occur at the interface
between the light source (Vertical Cavity Surface Emitting Laser, or the VCSEL) and the fiber array, and also at the
interface between the fiber array and the detector (photodetector). In order to obtain the real-time analysis of the
transmitted and crosstalk signals, optical convolution is employed in this work. Optical convolution of the radiated
signals (from the VCSEL) and the fiber array is performed to determine the signal intensity at the receiver end and also
the amount of crosstalk in the array system. Transmitted signal intensity and crosstalk are essential for defining signal
integrity and reliability during the packaging of optoelectronic transmitter and receiver modules in an optical system. A
theoretical analysis of transmitted and crosstalk signals is performed with various separation distances between the
transmitter modules and the fiber array and with a zero separation distance between the fiber array and the photodetector.
The analysis is also performed for a top-emitting VCSEL (for the planar transmitter module) and bottom-emitting
VCSEL (for the multi-chip transmitter module). The optical convolution allows us to obtain the real-time and the actual
transmitted and crosstalk signals at the receiver end of an optical array system. It also provides optical system
performance analysis.
Keywords: Optical convolution, Transmitted signals, Crosstalk
1. INTRODUCTION
Optical interconnection technology is seen as a promising technology to meet the high bandwidth and high-speed
requirements of the next generation computer systems. Optical interconnects exhibit several advantages over electrical
interconnects such as low power consumption, high bandwidth, and low inter-channel crosstalk. In order to achieve
larger throughput of high speed signals from one point to another in the computer systems, there is need for multi-
channel optoelectronic and optical link systems. Some of such links might be between chips or boards. However, most of
the optoelectronic and optical components used in optical interconnection technology require electrical interconnects for
power and signal transmission. These electrical interconnects affect the overall performance and bandwidth of the
optical link. As clock frequency increases, reaching the gigahertz/terahertz band and signal transmission rate reaching
the gigabit/terabit range, signal integrity issues between parallel signal paths, such as crosstalk, become serious issues in
multichannel/multi-conductor interconnects. Crosstalk is the phenomenon where energy is coupled from one signal path
onto another in parallel multichannel links. Crosstalk could be electrical, when electromagnetic fields from links and
structures surrounding them interact; optical, when optical power in one channel is coupled to an adjacent channel at the
interconnect points along the optical link; or both electrical and optical.
As the demand for high speed and high data-rate optoelectronic chips and components increases, chip integration
technology is experiencing a lot of improvements where new forms of interconnect and packaging techniques are being
witnessed [1]. Such techniques include flip-chip bonding in multichip modules (MCM). Though wire-bonding
technology is the most widely used technology for electrical interconnects between chips, flip-chip bonding technology
is attracting a lot interest in optoelectronic interconnects [2] due to reduced interconnect length and reduction in
electrical parasitics [3]. These packaging technologies are possible due to various structures of the optoelectronic
interconnect components such as the light source in the optical link, namely, the VCSELs (Vertical Cavity Surface
Emitting Lasers). In view of these structures, 4 VCSEL types have been utilized in an optical link for transmission and
Physics and Simulation of Optoelectronic Devices XXVII, edited by Bernd Witzigmann,
Marek Osiński, Yasuhiko Arakawa, Proc. of SPIE Vol. 10912, 1091219 · © 2019 SPIE
CCC code: 0277-786X/19/$18 · doi: 10.1117/12.2505099
Proc. of SPIE Vol. 10912 1091219-1
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crosstalk analysis. These structures include: top-emitting VCSEL and bottom-emitting VCSEL, where the short-
wavelength (850nm) and long-wavelength (1310nm) of each type are all considered.
In this work, we consider an optical system which consists of a transmitter array, a fiber array, and a receiver array. As
light signals travel from the transmitter to the receiver, some signal loss and crosstalk occur. Signal loss and crosstalk
occur at the interface between the light source (Vertical Cavity Surface Emitting Laser, or the VCSEL) and the fiber
array, and also at the interface between the fiber array and the detector (photodetector). Several works have been
dedicated to the performance and total crosstalk analysis of optoelectronic links [4], [5]. However, much attention have
not been paid to the optical crosstalk component of the total crosstalk. Also, it is important to obtain the real-time
analysis of the transmitted and crosstalk signals. In order to achieve this, optical convolution of the transmitted signal is
employed in this work. Optical convolution of the radiated signals (from the VCSEL) and the fiber array is performed to
determine the signal intensity at the receiver end and also the amount of crosstalk in the array system. A theoretical
analysis of transmitted and crosstalk signals is performed with various separation distances between the transmitter
modules and the fiber array and with a zero separation distance between the fiber array and the photodetector. The
analysis is also performed for a top-emitting VCSEL (for the short- and long- wavelength VCSEL of the planar
transmitter module) and bottom-emitting VCSEL (for the short- and long- wavelength VCSEL of the multi-chip
transmitter module).
2. THE OPTICAL LINK
The block diagram of an optoelectronic link is shown in Figure 1. It is made up of a transmitter, an optical source,
optical link/medium, a photodetector, and a receiver. An optoelectronic transmitter is made up of the transmitter and the
optical source while an optoelectronic receiver is made up of a photodetector and a receiver. At the transmitter side, the
transmitter converts the input signal from the electrical source into a large current used modulate the optical source. The
light output propagates through the optical link/medium, which is optical fiber, free space or waveguide. The optical
signal from the optical link/medium is collected by the photodetector, which generates an electric current. The optical
signal is converted into electrical signals by the optoelectronic transmitter and is amplified to the required signal level.
Driver IC Receiver IC
Photodiode
VCSEL
Optoelectronic
Transmitter Op toelectr onic
Rece iver
Optical Med ium
Input Signa l
Rece ived Si gnal
Planar Tx
MCM Tx
Fiber
Figure 1. Block diagram of an optoelectronic link
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In this work, we have designed two kinds of transmitter (Tx) module structures, namely, planar transmitter module and
multichip transmitter modules. Each of these Tx modules were designed using four types of 1x4 VCSEL array chips,
namely, short-wavelength (850nm) top-emitting VCSEL, long-wavelength (1310nm) top-emitting VCSEL, short-
wavelength (850nm) bottom-emitting VCSEL, and long-wavelength (1310nm) bottom-emitting VCSEL. The
characteristics of the VCSELs are shown in Table 1. In order to implement the transmitter modules, Teflon-based
evaluation boards are used. The planar transmitter modules are prepared by attaching the driver IC and the 1x4 VCSEL
array chips, separately on the evaluation board using conductive epoxy and wire-bonding technology. On the other hand,
the MCM modules are prepared by flip-chip bonding the 1x4 VCSEL array chips to the driver IC. Gold stud bumps are
used in the flip-chip bonding process. The bonded chips are then placed and bonded on the evaluation board using
conductive epoxy and wire-bonding technology is used to make the necessary connect between the driver IC and the
evaluation board. A photograph a planar (with top-emitting VCSEL) transmitter and MCM (with bottom-emitting
VCSEL) are shown in Figure 1 as incepts.
Table 1: Electrical and optical characteristics of the short and long wavelength VCSELs
Parameter VCSEL Type Wavelength Min Typical Max Unit
Threshold current
Top 850 nm 1 2
mA
1310 nm 1 1.4 2
Bottom 850 nm 1 2
1310 nm 1 1.7 2
Output power
Top 850 nm ~1.0
mW
1310 nm 0.5 ~0.7
Bottom 850 nm ~1.0 2
1310 nm ~0.9 2
Wavelength
Top 850 nm 840 850 860
nm
1310 nm 1290 1310 1360
Bottom 850 nm 840 850 860
1310 nm 1290 1310 1360
Beam divergence
Top 850 nm 15 20 25
deg
1310 nm 7 9 11
Bottom 850 nm 15 20 25
1310 nm 7 9 11
Aperture diameter
Top 850 nm 11
µm
1310 nm 9
Bottom 850 nm 10
1310 nm 12
Numerical
aperture (NA)
Top (MM) 850 nm 0.13 0.2 0.26
1310 nm 0.13 0.2 0.25
Bottom (MM) 850 nm 0.13 0.2 0.26
1310 nm 0.13 0.2 0.25
3. OPTICAL CONVOLUTION
Convolution is one of the most important and fundamental concepts in signal analysis and processing. If we know the
impulse response of a system, then through convolution, it is possible to construct the output system of any arbitrary
input signal. If we have an impulse, f[n] and impulse response, h[n], then the output, g[n] is given as:
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[]=[]∗ℎ[]=∑[].ℎ[−]
, (1)
where * denotes convolution. In many signal processing problems, it is desirable to cross-correlate two signal. However,
in this work, we consider a signal and a system. In order words, optical convolution in this work is defined as an
operation between a signal and a system in which the input signal undergoes changes and the output is delivered. Our
optical system is made up of an optoelectronic transmitter, a fiber system (propagation medium), and a receiver as
shown in Figure 1 and described in section 2. Since the transmitter is made up of the driver IC and VCSEL chips, we
will consider the VCSEL chips and their structures in this work. Thus, to perform optical convolution, we consider the
radiated beams from the VCSELs and the nature of the surface from which the beams are radiated. The radiated beam
and its nature is considered as the signal, while the fiber (propagation medium) is considered as the system.
4. SIGNAL ANAYISIS USING OPTICAL CONVOLUTION
4.1 Optical signal and the Gaussian beam
In this work, the optical signal from the VCSEL is considered as a Gaussian beam. The Gaussian beam, like most laser
beams is collimated and almost monochromatic [6]. The distinguishing feature between Gaussian beams and other laser
beams lies in the former’s behavior along the path of propagation and in the transverse plane to this direction of
propagation. In the plane transverse to the propagation direction, the intensity of the beam decreases in a typical
Gaussian shape. Additionally, though the beam is collimated, it expands as it propagates [7]. To derive the mathematical
representation of a Gaussian beam in 3-D, we start with a plane wave, the simplest type of 3-D waves [8]. A plane wave
propagating long the z-direction is represented by the wave function:
Ψ(,,,)=() (2)
We then modify the constant amplitude of the plane wave in equation (2) to reflect a decrease in amplitude as the wave
moves away from the source, along the direction of propagation (z-axis). Hence we have:
=(,)=()
⁄, (3)
where the beam width, =1+
, ω0 is the beam waist, zR denotes the Rayleigh range, λ is the wavelength of the
beam. The beam width is also known as the beam spot. As stated earlier, Gaussian beam expands along the direction of
propagation (z-axis in this case). Since the wave is expanding, the wavefront must be spherical [8]. This is because a
wave always propagates in a direction perpendicular to its wavefront. The wavefront is defined as the surface that
contains all points of the wave that carry the same phase. This spherical wavefront is not accounted fully by the plane-
wave phase term eikz, so we must use a modified expression. The spatial part of a spherical wavefront has the form eikr,
where =++. Suppose that we are examining the wave far from the origin but close to the z-axis such that
x ≪ z and y ≪ z. Then we can approximate r using the binomial approximation, to obtain
=
+1
≅1+
(4)
Now with the adjustments to the plane wave equation in terms of amplitude, phase and wavefront, we can say that the
equation of the Gaussian beam is approximated by:
Ψ(,,,)≈()/)()()/) (5)
The first term contains the Gaussian profile, the second term has the unidirectional wave term, and the third term has the
correction to the previous term that accounts for the curvature of the wavefront.
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4.2 The optical fiber link and fiber transfer function
Optical fibers are usually categorized into single mode and multimode optical fibers. For single mode fibers, when the
optical field is observed, only one spot is seen and it is single mode and the basic mode. However, for the multimode
fiber, which has more than one mode, more than one spot is noticed in the optical field and the modes are of higher order
than the basic mode. A step index fiber is made of a core and clad, where the single mode fiber has a smaller core when
compared to a multimode fiber. In optical fibers, the V-number is usually used to describe the mode number of the
optical fiber. When V < 2.405, the fiber can support only one mode and is said to be a single traverse mode fiber. But
when V > 2.405, more than one mode is supported and the modes are represented with the mode number, M. The V-
number is given by:
=
−
=
, (6)
Where nCo and nCl are the refractive indices of the fiber core and cladding, respectively; a denotes the core radius; and
NA is the numerical aperture of the fiber. The relationship between the V-number and the mode number, M is given as:
≈
(7)
In this work, multimode fiber is used, where core/clad diameter is 100µm/110µm. Multimode fiber is used for easy
coupling of light from the VCSEL to the fiber. Based on (6) and (7), M gives a large number of modes, which is of no
consequence in our fiber system as it is only a few centimeters long. All the modes of the multimode fiber divert after
they exit the fiber and overlap each other.
In the linear domain and for single mode fiber cables, the transfer function is given by the Fresnel sine and cosine
integrals obtained using a recursive method called the Split-Step Fourier Transforms (SSFT) [9]. On the other hand, the
Volterra Series Transfer Function (VSTF) or the Modified Volterra Series Transfer Function (MVSTF) is used to
determine the transfer functions for both single mode and multimode fiber cables in the non-linear domain [9, 10]. These
non-linear effects include the Self-Phase-Modulation Effects (SPM), Cross-Phase Modulation Effects (CPM), Stimulated
Raman Scattering (SRS, Stimulated Brillouin Scattering (SBS) as well as the Four-Wave Mixing. It is important to note
that non-linear effects in optical systems increase with increase in input power level as well as increases in the fiber
length [9]. The Volterra Series Transfer Function (VSTF) itself is a mathematical tool than finds application in fields
outside optics [11]. It is generally used to approximate the non-linear effects in a system.
Although the SSFT method incorporates the non-linear effects in the fiber cable in the time domain into the calculation
of linear effects in the frequency domain, its recursive processes are considered a weakness, as it might introduce errors
into the calculation [9]. We would however use both methods in our analysis of the effects on single mode optical fibers
on light, then compare the results with empirical values to determine which of the methods gives the closest
approximation experimental values. This is despite the fact that the input power/intensity of the input signal and the
length of the fiber array used in this experiment suggests a linearity in the output characteristics fiber cable/array.
The generalized Non-Linear Schrödinger Equation (NLSE), shown below, is used to model all the known linear and
non-linear effects of optical fibers.
+∝
A+
+j
+
=jγ|A|A−||
+
|A|+||
A+jA×(t−
)2|A(,z)|d
(8)
Where A = A (t, z), A slowly varying complex envelop of the input pulse; α = Linear attenuation coefficient of the
optical fiber; = 1/vg = Inverse of the optical carrier group velocity vg; β= Second-order dispersion parameter usually
called group velocity dispersion (GVD); β= Third-order dispersion parameter; γ= ω0 / cAeffn2 ; ω0 = assigned as the
central optical frequency of the carrier under consideration; Aeff is the effective cross section of the fiber and n2 is the
refractive index of the core section; a1=a0 / ω0, a2 and a3 are the nonlinear coefficients of the single mode fiber; c is the
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speed of light; QR =ω0 / cAeffGR, is the Raman constant with GR as the Raman gain coefficient factor; and sR(t) is the
time-domain representation of the Raman gain spectrum SR(ω). Equation (8) above cannot be solved analytically due to
its non-linear nature. Numerical methods like the SSFT or the Runga-Kutta method are used to reach a final solution.
Frequency Domain Transfer Function is given by:
H(ω)=() (9)
And Time Domain Impulse Response is given by:
h(t) =
(
) (10)
Based on [9], time domain step response in terms of Fresnel sine and cosine integrals is given as:
()=
=
14
⁄+14
⁄
, (11)
where ()=
; ()=
. As mentioned earlier, the weakness of most of the recursive
methods in solving the NLSE is that they do not provide much useful information to help the characterization of non-
linear effects. This is remedied by the VSTF which provides an elegant way of describing a system’s nonlinearities. The
Volterra series transfer function of a particular optical channel can be obtained in the frequency domain as a relationship
between the input spectrum X(ω) and the output spectrum Y(ω) [9]:
()=
∑…
(,…,,−−⋯−)×()
…()(−−⋯−)…
(12)
where Hn (ω1,…,ωn) is the nth-order frequency domain Volterra kernel including all signal frequencies of orders 1 to n.
Since the fiber length used in this work is a few centimeters, we consider only the first term of the equation above, which
is the solution for the first order transfer function. This is the linear transfer function of an optical fiber with the
dispersion factors as defined previously. A(ω) = A(ω,0), which is the amplitude envelope of the optical pulses at the
input of the fiber. Thus the first order transfer function is given as [9]:
(,)=()=(
) (13)
Where ()=−
++
−
; ω takes the values over the signal bandwidth and beyond in overlapping the
signal spectrum of other optically modulated carriers, while ω1 ... ω3 also take values over a similar range as that of ω.
4.3 Optical convolution of transmitted signals
In our optical system, optical convolution is performed between laser (VSCEL) beam and medium in which beam
propagates (fiber system). Basd on (1) and substituting (5) and (13), the convolution equation for our system is given as:
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g[n]=
(√
(())
(())[,(())
√
()((()))
+(√
(())
(())[,(())
√
]
()((())) (14)
5. MEASUREMENT AND SIMULATION RESULTS
As reported in previous works [4], [5], [12], our optical interconnection scheme is based on fiber-embedded optical PCB,
where the VCSEL based transmitter module is coupled onto a fiber input port. Based on the Gaussian beam equation of
the VCSEL, the beam profile is plotted for various VCSEL types used in this work at different separation distances as
shown in Table 1. Table 2 shows the bandwidth performance of the transmitted signals based on different transmitter
module type at different separation distances between the VCSEL and the fiber. In Table 3, the optical crosstalk
performance of the transmitted signals is show at various separation distances between the VCSEL and the fiber.
Table 1. Simulated Gaussian profile of the VCSEL beam at various separation distances from the fiber.
Separatio
n distance
850 Bottom-
emitting VCSEL
Beam
1310 Bottom-
emitting VCSEL
Beam
1310 Top-
emitting VCSEL
Beam
850 Top-
emitting
VCSEL Beam
10 um
50 um
250 um
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Table 2: Summary of measured bandwidth of transmitted signals at varying vertical distances between the VCSEL and
the fiber link showing the bandwidth performance of each transmitter type
Separation
Distance, µm
0 250 500
850nm Top-
emitting
-3-dB Bandwidth,
GHz
4.57 4.27 3.96
1310nm Top-
emitting
-3-dB Bandwidth,
GHz
2.16 2.0 1.96
850nm Bottom-
emitting
-3-dB Bandwidth,
GHz
3.0 2.46 2.36
1310nm Bottom-
emitting
-3-dB Bandwidth,
GHz
2.46 2.16 2.0
Table 3: Summary of measured crosstalk at varying vertical distances between the VCSEL and the waveguide to show
the effct of optical crosstalk contribution in the total crosstalk measurement
Gap distance, μm 0 250 500
850nm Top-
emitting
Crosstalk, @ < 1GHz dB -53 -49.91 -37.9
% optical crosstalk contribution 6.8% 28.5%
1310nm Top-
emitting
Crosstalk, @ < 1GHz dB -56.84 -55.0 -50.09
% optical crosstalk contribution 3.2% 10.5%
850nm Bottom-
emitting
Crosstalk, @ < 1GHz dB -59.97 -56.27 -50.27
% optical crosstalk contribution 6.2% 16.2%
1310nm Bottom-
emitting
Crosstalk, @ < 1GHz dB -62.56 -61.88 -58.57
% optical crosstalk contribution 1.1% 6.4%
500 um
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6. CONCLUSION
In this work, we are able to lay the groundwork for the application of optical convolution in optical systems for the main
purpose of transmitted signal information and the optical crosstalk performance of the system. Optical convolution of the
radiated signals (from the VCSEL) and the fiber array is performed to determine the signal intensity at the receiver end
and also the amount of crosstalk in the array system. A convolution equation was obtained for the radiated signal
through the fiber which could be applied for numerical signal analysis of signals at the receiver end. Measurement
results were shown for the received signals where transmitted signal and crosstalk performance values are obtained.
REFERENCES
[1] A. Dickinson and M. M. Downs, “An optical multi-chip module,” in Proc. of IEEE International Conference on
Computer Design on VLSI in Computer and Processors (ICCD), pp. 490-493, 1991.
[2] K. Takahashi, T. Ikeuchi, T. Tsuda, and T. Chuzenji, “High speed multi-chip modules using flip chip mount
technology for 10 Gbps transmission system,” in Proc. of Electronic Components and Technology (ECTC), 2001,
pp. 987-992, 2001
[3] D. Varoutas, A. Arapoyianni, and T. Sphicopoulos, “Modeling of electrical crosstalk in OIEC modules”, Fiber and
integrated optics, vol 24, pp91-111, 2005
[4] I. A. Ukaegbu, D.-W. Kim, M. H. Cho, T.-W. Lee, H.-H. Park, “Performance Analysis of Vertical and Horizontal
Transmitter Array Modules Using Short and Long Wavelength VCSELs for Optical Interconnects,” IEEE
Transactions on Components, Packaging and Manufacturing Technology, vol. 3, no. 5, pp. 740-748, May, 2013
[5] I. A. Ukaegbu, J. Sangirov, M. H. Cho, T.-W. Lee, H.-H. Park, “Analytical Model for Crosstalk Analysis of
Optoelectronic Transmitter Modules for Optical Interconnetcs,” Journal of Optical Engineering, Vol. 50, no. 7, pp.
075401(1)-(8), July, 2011
[6] Galvez, E. J. (2009). Gaussian Beams
[7] G. P. Agrawal, “Spectrum-induced changes in diffraction of pulsed optical beams,” Opt. Commun., vol. 157, no. 1–
6, pp. 52–56, 1998
[8] H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss
or Gain Variation,” Appl. Opt., vol. 4, pp. 1562–1569, 1965
[9] L. N. Binh, “Linear and nonlinear transfer functions of single mode fiber for optical transmission systems,” J. Opt.
Soc. Am. A, vol. 26, no. 7, p. 1564, 2009.
[10] B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photonics Technol. Lett.,
vol. 14, no. 1, pp. 47–49, 2002.
[11] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989)
[12] I. A. Ukaegbu, D.-W. Kim, M. H. Cho, T.-W. Lee, H.-H. Park, “2.5Gb/s/ch Long Wavelength Transmitter Modules
for Chip-to-chip Optical PCB Applications,” IEEE Photonics Technol. Lett., vol. 23, no. 19, pp. 1403–1405, 2011.
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