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Attenuation and modal dispersion models for spatially multiplexed co-propagating helical optical channels in step index fibers

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Attenuation and modal dispersion models for spatially multiplexedco-propagating helical optical channels in step index ﬁbers
Syed H. Murshid
n
, Abhijit Chakravarty, Raka Biswas
Optronics Laboratory, Department of Electrical and Computer Engineering, Florida Institute of Technology, 150 W University Boulevard, Melbourne, FL 32901, USA
a r t i c l e i n f o
Article history:
Received 19 March 2010Received in revised form15 June 2010Accepted 17 June 2010Available online 6 August 2010
Keywords:
Optical ﬁber communicationOptical multiplexingSkew rays
a b s t r a c t
Spatial reuse of optical frequencies has been shown to be possible through a novel spatial domainmultiplexing (SDM) technique that uses spatial multiplexer at the input end to launch multiplechannels of the same wavelength inside a single strand of carrier ﬁber and then employs spatial ﬁlteringmethods to de-multiplex the different optical channels at the output end. The individual SDM channelsare conﬁned to dedicated spatial locations inside the ﬁber while traversing through it owing to helicalpropagation of light. This presents attenuation and dispersion models of such a system. Experimentallyobtained beam proﬁle and resultant crosstalk of two such spatially multiplexed co-propagating SDMchannels of the same wavelength over standard step index multimode optical ﬁbers are also presented.
&
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Fiber optic data transfer facilities have gradually become anintegral part of the infrastructure at data centers worldwide.Current infrastructure used at most data centers is generally adecade old and requires up-gradation as legacy multimode ﬁbersare typically used for low bandwidth applications and these ﬁberscan only support limited distances at data rates exceeding 1 Gb/s.However optical ﬁbers employed in building back bones aregenerally limited to 100–300 m in length and they are seldomreplaced as new ﬁbers are usually added to the same conduit asthe old ﬁbers. As a result many investigative endeavors have beenundertaken on various methods for extending the distance of multimode ﬁbers at higher data rates such as bandwidthenhancement in multimode ﬁber, electronic dispersion compen-sation, and wavelength tuning control loops [1]. Most of thesetechniques are incremental in nature and the goal of increasingtotal capacity in optical communications and networking requiresnew concepts for basic transmission media. SDM is a novelmultiplexing technique that employs helical propagation of lightand can signiﬁcantly enhance the data rates by allowingco-propagating channels of the same wavelength inside stepindex optical ﬁbers. Spatial reuse of optical frequencies has been acherished goal in optical ﬁber communications for a long time;however it proved elusive for all practical purposes until SDM wasreported as it was almost impossible to rule out interferenceowing to the small dimensions of core region of the ﬁbers. Manydifferent techniques such as mode group diversity [2,3] andmodal multiplexing [4] using slightly off-axis meridional rayshave been mentioned in the literature but these are mostlylimited to graded index ﬁbers and proved to be useful only forshort distances as predicted by Gloge’s power ﬂow equation [5].Gambling et al. [6] showed that the steady state solution toGloge’s power ﬂow equation does support donut shaped outputrings but these rings collapse to a Gaussian beam proﬁle afterpropagating short distances. Multiple papers [7,8] have presentedsolutions to Gloge’s power ﬂow equation using experimental,numerical and analytical techniques, ever since it was ﬁrstpublished in 1971. All these endeavors unanimously attest tothe fact that modal multiplexing is not feasible for any reasonabledistances as the length of the ﬁber greatly affects the outputpattern, and a sufﬁciently long ﬁber produces a circular Gaussianproﬁle. After a ﬁber length of 100–200 m donut shaped beamcollapses into a disk type circular output. As a result such modalmultiplexing techniques could not be used for any reasonabledistances and its application was limited to short distancesand they often require complex signal processing algorithmssuch as zero forcing algorithm [2], which renders them to be veryexpensive for most applications. On the contrary the SDMhas already been successfully tested for distances exceeding akilometer where multiple ﬁber patches, on different spools, wereintegrated together using mechanical and fusion splices. Thisapproach offers a practical solution for increasing the bandwidthdistance product of most legacy multimode ﬁbers as it supportsboth step index and graded index ﬁbers where helicallypropagating channels are launched using selective input anglesat the input end and simple spatial ﬁltering techniques are used todescramble the channels at the output end.
Contents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/optlastec
Optics & Laser Technology
0030-3992/$-see front matter
&
2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.optlastec.2010.06.004
n
Corresponding author.
E-mail address:
murshid@ee.ﬁt.edu (S.H. Murshid).Optics & Laser Technology 43 (2011) 430–436
The organization and motivation of this manuscript areintroduced in Section 1. Fundamentals of SDM are introducedin Section 2. Section 3 reviews helical propagation of light inoptical ﬁbers whereas Section 4 establishes the effective length, inwhich photons traversing helical trajectories, must travel insidethe optical ﬁber. Experimentally measured attenuation of SDMchannels is compared to the attenuation predicted by the helicalpropagation model in Section 5. Section 6 of this manuscript usesthe effective length in conjunction with the effective areaoccupied by the individual channels and the effective
V
number(the normalized frequency parameter) to develop the dispersionmodel. Beam proﬁle of two co-propagating SDM channels is usedto experimentally determine the crosstalk in Section 7 and ﬁnallySection 8 offers the conclusions from this endeavor.
2. Fundamentals of SDM system
SDM [9] is a patented new multiplexing technique that allowssimultaneous transmission of multiple channels over a singleoptical ﬁber [10]. The individual SDM channels may operate at thesame wavelength or differing wavelengths [11]. Radially dis-tributed, dedicated spatial locations are assigned to every SDMchannel inside the carrier ﬁber as these channels traverse thelength of the carrier [12]. The location of each channel inside theﬁber is a function of the launch angle [8] and the orientation of the carrier ﬁber [13,14]. As a result the centermost channelappears either like a standard TEM
00
spot [15] or like a donutshaped circular ring, while all subsequent channels appear asconcentric circular rings [16]. The center spot and each ringrepresent a separate spatially modulated optical channel, therebyenhancing the bandwidth of existing and futuristic optical ﬁbersystems by multiple folds [17]. From the perspective of formalmodal description of propagation, each helical channel can beperceived as a combination of only a few modes. The angularlaunch conditions selectively excite those modes for each channel,which travel in tandem, to produce the unique output of theSDM channels [18]. We present experimental and analyticalresults of attenuation, dispersion and crosstalk for two spatiallymodulated optical channels operating at 635 nm over a kilometerlong standard 62.5/125
m
m step index multimode optical ﬁberusing the limited number of modes excited due to the inputlaunch conditions and the helical trajectory followed by the donutbeams [19].Fig. 1 shows the block diagram of a typical SDM system. Itconsists of the following components: (a) single-mode pigtailedlaser diodes, (b) spatial multiplexer or the beam combiner module(BCM) [19], (c) spatial de-multiplexer or the beam separatormodule (BSM) [19], (d) standard communication grade stepindex multimode ﬁber, and (e) optical photo-detectors. Bulkoptics versions of spatial multiplexer units have already beensuccessfully tested [11] whereas an integrated CAD model iscurrently being evaluated [20,21]. Spatial de-multiplexer unitshave also been successfully tested using both bulk optics model[19] and semiconductor based designs [15,16,22].
Multiple channels of same wavelength are spatially modulatedand demodulated using an optical ﬁber communication link asshown in Fig. 1. Single mode pigtailed laser sources operating at635 nm are used to launch the individual SDM channels into astandard 62.5/125 silica based step index multimode commu-nication channel using the BCM unit. The BCM spatially multi-plexes two or more laser inputs into a single optical ﬁber carrier.At the receiving end the BSM employs spatial ﬁltering techniquesto de-multiplex the individual channels and route them todedicated photo-detectors. The SDM technique adds a newdimension to existing multiplexing techniques and complementsTDM and WDM systems. The SDM channels follow independenthelical trajectories inside the ﬁber and do not interfere with eachother.
3. Helical propagation model
The beam from the SDM system travels in the helical paththrough the ﬁber due to angular dependency and relativeorientation of the launching beam. Each photon travels throughthe ﬁber in a helical path with a transmission angle
F
and an axialvelocity
v
z
. The transmitted angle depends on the launch angleof the beam. The helical ray is a special case of skew ray. Weconsider a helical ray traveling on a cylindrical surface of radius
r
,with a slant angle
F
(transmitted angle) as shown in Fig. 2. Therelation between the radius of the helical path
r
and the launchangle
F
[23] is given as follows:
n
ð
r
Þð
sin
2
F
Þ
=
r
¼
dn
ð
r
Þ
=
dr
ð
1
Þ
In a typical spatially multiplexed system, the incident anglevaries between 14
1
and 24
1
. The transmitted angle is given bysin
F
¼
sin
y
i
n
1
ð
2
Þ
where
y
i
is the incident angle or the launch angle.Fig. 3a and b attests to the fact that the bigger the input launchangle, the larger the radius of the output beam. The axial velocityof a photon traversing a helical path inside the ﬁber can begiven as
v
z
ð
r
Þ¼
cos
F
n
ð
r
Þ½
e
0
m
0
1
=
2
¼ ½
1
þð
r
=
n
Þ
dn
=
dr
1
=
2
n
½
e
0
m
0
1
=
2
ð
3
Þ
Fig. 1.
Block diagram of SDM system.
Fig. 2.
Helical ray trajectory.
S.H. Murshid et al. / Optics & Laser Technology 43 (2011) 430–436
431
where
e
0
is the permittivity of free space,
m
0
is the permeability of free space, and
n
is the refractive index of the core of the ﬁber.Since we are using step index ﬁbers, the radial dependency of theaxial velocity can be ignored within the core region, and Eq. (3)can be expressed as a constant velocity [23–25]
v
z core
¼
1
=
n
½
e
0
m
0
1
=
2
ð
4
Þ
The length of the helix or the arc length is given by
½
r
2
þ
a
2
1
=
2
2
p
where ‘
r
’ is the radius of helix. The maximum valuefor an SDM channel is bound by the radius of the ﬁber and ‘
a
’ is aconstant; therefore 2
pa
is also a constant giving the radialseparation of the helical loops. The parametric equation of helix isgiven as
x
ð
t
Þ¼
r
cos
t
,
y
ð
t
Þ¼
r
sin
t
,
z
ð
t
Þ¼
a
t
,
t
A
½
0
,
2
p
Þ ð
5
Þ
Hence the time taken for a photon to travel one single helicalarc can be given as follows [23]:
½
r
2
þ
a
2
1
=
2
2
p
v
z core
¼ ½
r
2
þ
a
2
1
=
2
2
p
1
=
n
½
e
0
m
0
1
=
2
ð
6
Þ
Eq. (4) can be used to determine the number of helical loopsformed in a certain length of ﬁber, which can be used to calculatethe extra distance the photon will travel in the ﬁber due to helicalpropagation. The sum of the actual length of the ﬁber and theextra distance is called the effective length (
L
effective
) and knowl-edge of this length is important to properly determine thedispersion and attenuation of an SDM system.
4. Effective length for helically propagating channels
As discussed earlier photons traversing the length of a ﬁber ina helical path physically travel longer distances as compared tophotons traveling strictly along the srcin. Hence it is importantto determine the effective length that helically propagating SDMchannels encounter to cover a given length of ﬁber as attenuationand dispersion encountered by the optical energy traversing agiven length of ﬁber cable will depend on it. Fig. 2 depicts the‘‘path’’ of a moving photon as a curve inside a ﬁber [26–29]. Thephoton when launched at a particular angle travels in a helicalpath through the ﬁber as seen in Fig. 2. It is interesting to knowthe position of the photon as a function of time [30,31], as timecan be used as a parameter to determine the coordinates of thephoton. The position of the particle can be given by the vector [32]
2-dimensions
r
ð
t
Þ¼
x
ð
t
Þ
i
þ
y
ð
t
Þ
j
,
3-dimensions
r
ð
t
Þ¼
x
ð
t
Þ
i
þ
y
ð
t
Þ
j
þ
z
ð
t
Þ
k
The length of the helix or the arc length can be expressed as[29,30]
L
effective
¼
Z
2
p
0
½
r
2
sin
2
t
þ
r
2
cos
2
t
þ
a
2
1
=
2
dt
¼½
r
2
þ
a
2
1
=
2
2
p
ð
7a
Þ
and
L
¼
a
2
p
)
a
¼
L
2
p
ð
7b
Þ
where
L
is the distance of formation of one single helix from thepoint where the beam is launched:
L
effective
¼
2
p
½
r
2
þ
a
2
1
=
2
¼
2
p
r
2
þ
L
2
p
2
" #
1
=
2
¼½ð
2
p
r
Þ
2
þ
L
2
1
=
2
ð
7c
Þ
Applying simple trigonometry to Fig. 4 we gettan
¼
r L
hence
L
¼
r
tan
¼
4
r
tan
½
sin
1
ð
sin
y
i
=
n
1
Þð
8
Þ
where
F
is the transmitted angle, given in Eq. (2),
y
i
¼
incidentangle and
n
1
¼
refractive index of core
¼
1.47. Using Eq. 7(c) theeffective length that a photon in a single helix travels is given by
L
effective
¼ ð
2
p
r
Þ
2
þ
16
r
2
tan
2
½
sin
1
ð
sin
y
i
=
n
1
Þ
" #
1
=
2
ð
9
Þ
For
y
i
¼
18
1
¼
0.314 rad,
n
1
¼
1.47, radius of helix
r
¼
62.5
F
/2
F
max
,
F
max
¼
maximum transmitted ray angle for incidentangle
¼
y
i max
¼
maximum acceptance angle.So,
r
¼
(62.5
12.13)/(2
16.05
¼
23.6115)
m
m. From Eq. (9)we get effective length
L
effective
¼ ð
2
p
23
:
6115
Þ
2
þ
16
23
:
6115
2
tan
2
½
sin
1
ð
sin18
3
=
1
:
47
Þ
" #
1
=
2
¼
463
:
59
m
m
ð
9a
Þ
Fig. 3.
Simulated output for two incident angles (a) 18
1
and (b) 24
1
.
Fig. 4.
Single helix cone angle.
S.H. Murshid et al. / Optics & Laser Technology 43 (2011) 430–436
432
So if we launch the beam at 18
1
the radius of the beam ‘
r
’ canbe calculated and the beam will travel 463.59
m
m in order tocomplete one helix. For every
L
¼
439.2
m
m there is one loop of helix. So for 1000 m of ﬁber there will be 1000
10
6
/439.2
¼
2.277
10
6
helix loops. For 2.277
10
6
helix loops time differ-ence of arrival at the end of 1000 m ﬁber is
D
t
¼
0.27
10
6
s. Soin a 1000 m long ﬁber the beam will travel approximately anadditional length of 55 m. Hence SDM channels launched at 18
1
will travel an additional length of approximately 5.5% to traversea given length of ﬁber.
5. Attenuation of SDM channels
The SDM channels follow helical path inside the ﬁber. Aphoton launched inside the optical ﬁber at a particular angle canfollow a helical path as it traverses the length of the ﬁber.Therefore the effective length (
L
effective
) that a photon travelsinside the ﬁber in SDM systems due to helical propagation of lightis slightly longer than the distance covered by a photon thatstrictly follows the srcin of the ﬁber. The attenuation model forSDM system uses standard attenuation equations in conjunctionwith the
L
effective
to predict attenuation. Fig. 5 can be used todetermine the effective length of the SDM channels for 2
1
and 18
1
launch angle.Fig. 6 presents the block diagram of an SDM system supportingtwo spatially modulated channels of same wavelength over astandard 62.5/125
m
m multimode ﬁber. The BCM serves as spatialmultiplexer by launching light from two sources (L1 and L2) intothe carrier ﬁber. Physical construction of the BCM is also shown inFig. 6. After traversing through the ﬁber, the two channels areseparated with the help of the BSM unit, which also serves as thespatial de-multiplexer. In the experimental setup, attenuationmeasurement is divided into two parts. Initially the lossesbetween the source and the BCM unit are quantiﬁed and thenthe attenuation between the BCM unit and the BSM unit ismeasured.This is shown in Fig. 6 where initially loss is ﬁrst measuredfrom A1 to A2 and then from A2 to A3 for the center channel.Similarly the attenuation for the outer channel is determined byﬁrst measuring attenuation between B1 and B2 and thenmeasuring it between B2 and A3. Losses for three lengths of ﬁbers were measured using this process. Initially the total lengthof the carrier ﬁber between the BCM and A3 was 200 m, whichwas then increased 600 m and ﬁnally attenuation for a 1 km longSDM system was measured using commercially available opticalpower meters.Fig. 7 compares the experimentally measured values of attenuation for an outer SDM channel launched at 18
1
with thevalues predicted by the helical path model developed in Section 4using
L
effective
and standard attenuation equations. It can be notedfrom the graph that the two sets of data match closely. There wasa maximum difference of 2.6% between the two sets of values,which can be attributed to connector and splice losses included inthe experiment.
6. Modal dispersion model of SDM channels
The modal dispersion model of SDM channels utilizes standardoptical ﬁber dispersion equations where the
V
number primarilydepends upon the radius of the core for a given wavelength andthe total numbers of modes supported by a multimode ﬁber relieson the
V
number. The greater the radius/area of the ﬁber core, thegreater the
V
number leading to a signiﬁcantly larger number of modes. However the individual SDM channels do not occupy theentire core region of the ﬁber and each channel occupies only asmall portion of the available core area. Hence in order to developthe SDM modal dispersion model, we consider two parametersunique to the SDM system: (i) the effective
V
number (
V
effective
)encountered by the individual channels, which depends on the
Fig. 5.
(a) Incident angle vs. transmitted (slant) angle and (b) simulated cone angle vs. effective length for 200 m of ﬁbers (for 2
1
and 18
1
).
Fig. 6.
Block diagram showing measurement of losses at different stages.
Fig. 7.
Experimental values of attenuation for SDM channel launched at 18
1
andattenuation values predicted by the helical propagation model.
S.H. Murshid et al. / Optics & Laser Technology 43 (2011) 430–436
433
area occupied by the individual channel and (ii) the effectivelength each channel traverses inside the ﬁber as predicted bythe helical propagation model. The
V
effective
in SDM tends tosigniﬁcantly reduce modal dispersion whereas the
L
effective
tendsto increase the dispersion by a small percentage. Reduceddispersion in SDM systems as compared to the dispersion instandard multimode ﬁber is expected and similar ﬁndings havebeen reported in the literature [33–40] by using techniques suchas off-axis launch and offset launch. For a helically propagatingSDM channel launched at an incident angle of 18
1
inside akilometer long ﬁber the
L
effective
encountered by the beam is1055.4 m as shown in Fig. 7. Hence modal dispersion can bedetermined modifying the standard modal dispersion equation[26]:
D
t
SDM modal
¼
L
effective
ð
n
1
n
2
Þ
c
1
p
V
effective
ð
10
Þ
where
n
1
(1.47) and
n
2
(1.456) are the refractive indices of thecore and cladding, respectively, ‘
c
’ is the speed of light. In SDMsystem the dispersion for the outer channel can be predicted bycalculating the number of modes that will propagate in a certainhelical channel for a given incident angle.Fig. 8 shows beam spot of a spatially multiplexed outer and thestandard intensity spot from multimode ﬁber. The dispersion forthe individual channels can be predicted by calculating thenumber of modes that will propagate inside the helical path for agiven incident angle. The dimensions of the two channels aredetermined by projecting them over a screen and applyingtrigonometric solutions. The effective
V
number for the ringwith outer radius
a
2
¼
15.6
m
m and inner radius of
a
1
¼
14.06
m
mis calculated by ﬁrst determining the total area occupied by thering and then determining the effective radius (
r
effective
) that isrequired to obtain a spot of similar area. Once the radii for thecenter and outer channels are known, they can be substituted todetermine the corresponding
V
effective
. The beam areas occupiedby standard ﬁber optic channels and the SDM channels arepresented for two different wavelengths in Table 1. It alsopresents the effective
r
effective
,
V
effective
, the total number of modes (no. of modes) and the modal dispersion for a kilometerlong standard multimode ﬁber optic channel and the center andouter SDM channels of the same length.Fig. 9 graphically represents the modal dispersion datapresented in Table 1 and highlights the advantages of the SDMsystem. The SDM system not only supports two simultaneouschannels of the same wavelength inside a single carrier ﬁber butalso increases the data carrying capacity of the ﬁber due tosigniﬁcant reduction in modal dispersion. We are currentlyreﬁning the design of our array concentric photodiodestructures [16,22] and plan to use those devices in futureendeavors to experimentally verify the results obtained fromour dispersion model and plan to report those ﬁndings in the nearfuture. These future endeavors will also allow us to determine thebit error rate for a two channel SDM system. These values formodal dispersion were obtained by modifying the standarddispersion equation and using effective length and effective
V
number instead of the standard length and standard
V
number.
7. Crosstalk analysis between partially multiplexed channels
Crosstalk is an important parameter in optical communica-tions [41–45] as it describes the isolation between co-propagatingchannels. Crosstalk between SDM channels is quantiﬁed bysimultaneously modulating all channels and determining theinﬂuence of one channel over the other channel at the output endof the carrier ﬁber. It has been experimentally veriﬁed that SDM
Fig. 8.
(a) Spatially multiplexed outer channel and (b) spatially multiplexed outerchannel superimposed on a standard output from multimode ﬁber.
Table 1
Modal dispersion and relevant parameters for two different operating wavelengths.
Channel
r
effective
Beam area (
l
m
2
)
V
effective
No. of modes Modal dispersion(ns/km)Improvement (%) Wavelength: 635 nm
Standard 31.25 3066.406 62 1922 44.23 –Center (SDM) 5.2 84.9056 10.3 53 32.433 26.67Outer (SDM) 6.75 143.42 13.15 86 37.48 15.26
Wavelength: 820 nm
Standard 31.25 3066.406 48.375 1170 43.59 –Center (SDM) 5.2 84.9056 8 32 28.32 35Outer (SDM) 6.75 143.42 10.45 55 34.42 21.03
Fig. 9.
Comparison between standard dispersion of multimode ﬁber and predicteddispersion due to helical propagation.
S.H. Murshid et al. / Optics & Laser Technology 43 (2011) 430–436
434

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