Multi-mode Waveguides with Tailored Dispersion - a Way for Coherent and Dispersion-Free Propagation of Classical and Quantum Optical Signals

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by  C. A. Valagiannopoulos

Multi-mode Waveguides with Tailored Dispersion - a Way for Coherent and
Dispersion-Free Propagation of Classical and Quantum Optical Signals
C. A. Valagiannopoulos,1 A. Mandilara,1 S. A. Moiseev,2,3 and V. M. Akulin3,4
1Department of Physics, School of Science and Technology, Nazarbayev University,
53, Kabanbay batyr Av., Astana, 010000, Republic of Kazakhstan.
2Quantum Center, Kazan National Research Technical University, 10 K. Marx, Kazan, 420111, Russia.
3Laboratoire Aim´e-Cotton CNRS UMR 9188, L’Universit´e Paris-Sud et L’E´cole Normale Sup´erieure de Cachan,
Baˆt. 505, Campus d’Orsay, 91405 Orsay Cedex, France
4Institute for Information Transmission Problems of the Russian Academy of Science,
Bolshoy Karetny per. 19, Moscow, 127994, Russia.
Weshowhowbyatailoredbreakingofthetranslationalsymmetryinwaveguidesandopticalfibers
one can control chromatic dispersions of the individual modes at any order and thereby overcome
7 the problem of coherent classical and quantum signal transmission at long distances. The method
1 is based on previously developed quantum control techniques and provides analytic solution for a
0 general class of models. In practice, this requires mode couplers and the possibility of tailoring
2 the dispersion laws around the operating frequency. We develop the method for spatial modes
of a waveguide and we further apply the technique to design dispersion-less propagation for the
n
polarization modes of a uniaxial material.
a
J
PACSnumbers: 03.67.HkQuantumcommunication,42.50.-pQuantumoptics,42.79.SzOpticalcommunica-
2
tionsystems,multiplexers,anddemultiplexers,42.81.-iFiberoptics
1

]
h Transmission of information at long distances via ple example of two spatial mode waveguide of a rectan-
p multi-mode optical fibers/waveguides is limited by the gular profile. The exact analytical expressions for the
- necessitytocompensateforthespreadinpackagearrival mode dispersions permit to suggest an explicit spatial
t
n times caused by the discrepancy in the group velocities distribution of ideal scatterers coupling the modes and
a of different waveguide modes. At the same time the dis- which act as perturbations backing the translational in-
u
persion effects need to be compensated for each of the variance. More specifically, we derive 4k non-linear alge-
q
modes. Is there a way to overcome these problems and braicequationsinanalyticformthatdeterminetheposi-
[
construct a multi-mode waveguide with identical group tions {L ,...,L } of M = 4k scatterers, such that sig-
1 M
1 velocitiesofdifferentmodesandnoquadraticdispersion? nalsonthetwomodespropagateinphase,synchronously
v
Hereweshowthatbybreakingthetranslationinvariance andwithoutdispersionuptok-thorderofTaylorexpan-
1
in a specific way one can accomplish such a task. sion. At the same time, we emphasize that, by analogy,
5
2 There is a vast amount of work [1]-[2] related the dis- the same problem can be solved for any finite number N
3 persion problem and the method developed here has un- of modes by finding numerically a proper distribution of
0 avoidableoverlapwithseveralofthem. Themethodthat M =N2k generic [11] ideal scatterers along the pertur-
1. wesuggestprovidesdispersionalongthefiber/waveguide bation period. In Section II we apply the method for
0 byperturbingthetranslationsymmetrysimilarlytofiber designing dispersion-less propagation in an actual struc-
7 Bragg gratings [3] and the tailoring of photonic band ture with imperfect scattering carried out by flakes of
1 structures techniques [4]. This also assumes availabil- graphene in a uniaxial background. With this example,
v: ity of mode couplers which have been extensively used we explicitly show how the method can be efficiently ad-
i in the design of dispersion compensation filters for two justed to given realistic conditions and materials.
X
[7] and higher number of modes [8]. Furthermore, for
r
a the development of the method we borrow previously
developed ideas for implementing quantum control [5]
over compact semigroups [6]. The main contribution of I. PROPOSED METHOD WITH IDEALIZED
our method is that the discrepancy in the mode disper- SCATTERERS
sion laws can be compensated up to any predetermined
order k of the Taylor expansion around a given oper-
Consider a rectangular a × b waveguide with con-
ational signal frequency, in complete analogy with the
ducting walls, shown in Fig. 1 a). If we consider a
regime of Quantum Error Protection [5],[9] and similar
frequency-dependent refraction index n(ω), then the
to the regime of Dynamic Decoupling [10]. Moreover in
mode wavevectors A , and z-components of the vector
l,j
thecasewhereadvancedtailoringofthedispersions’laws
potential of the electromagnetic field mode functions, as
is possible, the method can be extended to compensate
these defined up to normalization constants α ,
l,j
for higher than quadratic order dispersions.
The structure of this paper is the following. In Sec- (cid:16) x(cid:17) (cid:16) y(cid:17)
tion I we illustrate and develop the method at a sim- Al,j(x,y,z)=αl,jsin πla sin πjb ei(κl,jz−ωt) (1)
2
a)
a The logarithm of U(cid:98)L is proportional to the sum of the
Pauli matrix σ with a frequency dependent pre-factor
(cid:98)z
b
κ (ω)−κ (ω)
F (ω)= 1,1 2,1 (4)
z 2
1
2
+ -
and the identity matrix I(cid:98)also multiplied by a frequency
dependent pre-factor
2
1 κ (ω)+κ (ω)
F (ω)= 1,1 2,1 . (5)
b) I 2
For ω ∼= ω the coefficient F (ω) does not contain the
0 I
n((cid:2))=1+15/(16-(a(cid:2)(cid:4)c)²) quadratic dispersion term (ω−ω )2 in its Taylor expan-
0
(cid:3)a sion as presented in Fig. 1 caption.
Now we assume that at some point L along the prop-
z
agationlinelocatesascattererwhichflipstheamplitudes
ofthemodes. Thiscanbeachieved,forinstance,byanti-
symmetric Gaussian perturbation of the refraction index
shown in Fig. 1a). By choosing the size δn of the refrac-
(cid:2)a/c tion index perturbation
−(x−x0)2−(y−y0)2−(z−Lz)2
δn(x,y,z)=δn(e α2 α2 β2
FIG.1: a)Rectangulara×bwaveguidewithconductingwalls.
Co-factorsofthemodefunctionsgivenbyEq.(1)fortwofirst −(x−a+x0)2−(y−y0)2−(z−Lz)2
−e α2 α2 β2 ),
modesareshownbydashedlines. Byellipsesweshowascat-
terer, whichisapairofsmoothpositive(+)andnegative(-)
the perturbation matrix elements
perturbationsoftherefractionindexthathavealengthlonger
thatthemodeswavelengthstillremainingshorterthanthein- (cid:90)
verse of the mode wavevectors difference. b) Dispersion law V = δn(x,y,z)A (x,y,z)A (x,y,z)dxdydz =0
l,l l,1 l,1
for the two first modes for the case a = 2b and in the pres-
(cid:90)
ence of the frequency dependent refraction index n(ω) has
V = δn(x,y,z)A (x,y,z)A (x,y,z)dxdydz (cid:54)=0
a domain near ω = 3c/a (shown by a rectangle) where the 1,2 1,1 2,1
secondderivatives∂2κ/∂ω2 ofthefirstandthesecondwaveg- (6)
uide modes have compensate each for the other. As a result,
Eq.(5)forn(ω)=1+15/(16−(aω/c)2)yieldsTaylorexpan-
canbesetinsuchawaythatthemodescatteringmatrix
sionF (ω)=3.36984+5.23733(aω/c−2.98307)+O((aω/c−
I
2.98307)3)
S(cid:98)=exp(is(cid:98)) (7)
with the scattering action matrix [12]
follow different dispersion laws
s=F (ω)σ (8)
(cid:98) x (cid:98)x
(cid:115)
(cid:16)ω(cid:17)2 (cid:18)lπ(cid:19)2 (cid:18)jπ(cid:19)2 is proportional to the Pauli matrix σ(cid:98)x with a coefficient
κ (ω)= n2(ω) − − . (2)
l,j c a b
π F(cid:48)(cid:48)(ω )
F (ω)= +F(cid:48)(ω )(ω−ω )+ x 0 (ω−ω )2+...
x 2 x 0 0 2 0
For a = 2b, around the frequency ω0 ∼= 3c/a, (9)
there exist only two waveguide modes (j =1,l=1) and For a smooth enough perturbation (βκ(cid:29)1), the back
(j =1,l=2), as it is shown in Fig. 1b). Moreover, by a scattering is negligible. The assumption F (ω ) = π is
x 0 2
properchoiceofthedependencen(ω)showninFig.1b), not crucial and as we show in Section II one can even
one can find a regime where the dispersion ∂2κ/∂ω2 at use this angle as an additional free parameter in the
ω0 for the first mode is opposite to that of the second problem. We also note here that frequency dependence
one. Propagation of a two-mode signal at ω ∼= ω0 along mode-couplers, as fiber gratings [13], can fit to this gen-
a waveguide of length L results in accumulation of the eral model of scatterers.
mode phases factors, that can be written in matrix form Wenowcometothekeyideaofourmethod. Thecom-
as multiplication of the two-component mode amplitude binedmodetransformationmatrixthatdescribesscatter-
vector by a matrix ing and then propagation for distance L, reads
U(cid:98)L =e−iL(Fz(ω)σ(cid:98)z+FI(ω)I(cid:98)). (3) U(cid:98)LS(cid:98)=e−iLFI(ω)I(cid:98)e−iLσ(cid:98)zFz(ω)eiσ(cid:98)xFx(ω). (10)
3
We repeat such a transformation 3 times with 3 differ- This makes the first of Eqs.(13) fulfilled as the identity,
entlengthparametersL , L , L , andrequirethatupto and thus the system becomes degenerate consisting of
1 2 3
the common phase factor e−i(L1+L2+L3)FI(ω) the result- just two equations
ing transformation matrix
0=cos[X −X −X ]+cos[X +X −X ] (16)
1 2 3 1 2 3
T(cid:98)=e−iL3σ(cid:98)zFz(ω)eiσ(cid:98)xFx(ω)e−iL2σ(cid:98)zFz(ω) +cos[X −X +X ],
1 2 3
eiσ(cid:98)xFx(ω)e−iL1σ(cid:98)zFz(ω)eiσ(cid:98)xFx(ω) (11) 0=(X −X −X )sin[X −X −X ]+
1 2 3 1 2 3
(X +X −X )sin[X +X −X ]+
equalstoanondegeneratesquarerootoftheidentityma- 1 2 3 1 2 3
trix [6] with the accuracy of the order of (ω−ω )3. For- (X1−X2+X3)sin[X1−X2+X3],
0
mally this requirement implies that DetT(cid:98) = −1 while
on three variables F (ω )L → X , F (ω )L →
the linear coefficient c (L ,L ,L ,ω) in the characteris- z 0 1 1 z 0 2
1 1 2 3 X , F (ω )L → X . The system Eq.(16) has a
tic polynomial 2 z 0 3 3
continuous family of solutions easy to find numer-
(cid:104) (cid:105) ically. One can get a countable number of solu-
Det T(cid:98)−λI(cid:98) =λ2+c1(L1,L2,L3,ω)λ−1 (12) tions if the system is complemented by the addi-
tional physical requirement (L +L +L )F (ω ) =
1 2 3 I 0
vanishestogetherwithitstwofirstfrequencyderivatives,
(X +X +X )F (ω )/F (ω ) = 2mπ. Then, for
1 2 3 I 0 z 0
that is
m = 10, and coefficients F (ω ) = 3.36984, F (ω ) =
I 0 z 0
0.549151 suggested by the Taylor expansions around the
c (L ,L ,L ,ω )=0,
1 1 2 3 0 frequency ω = 2.98307c/a, we numerically find one of
(cid:12) 0
∂∂∂2ωc1(L1,L2,L3,ω)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ω=ω0 =0, Xmm3aatn=ryixp7o.3s5si9b7l7e.soTluhtisiosnosl,uXtio1n=in2d.1e0ed26y9i,eXld2s =the0.t7r7a6n6s4fe2r,
∂ω2c1(L1,L2,L3,ω)(cid:12)(cid:12)ω=ω0 =0. (13) (cid:16)U(cid:98)L1S(cid:98)U(cid:98)L2S(cid:98)U(cid:98)L3S(cid:98)(cid:17)2 =ei195.3037ωc−/ωa0+O(cid:98) (17)
In such a situation the T(cid:98)2 becomes the identity matrix
over a period of length 2(L +L +L ) = 18.645a,
up to a cubic correction 1 2 3
(cid:16) (cid:17)3
T(cid:98)2 =eσ(cid:98)O[(ω−ω0)3], (14) bwihcerceorOr(cid:98)ec=tioin(1.85.1W1eI(cid:98)−no4t.e08hσ(cid:98)eyre+t4h.a4t8σ(cid:98)thx)e dωec−g/ωean0eracaycuin-
the system Eq.(13) is lifted when either the requirement
where σ is a 2 × 2 matrix of norm one. There-
(cid:98) F (ω )= π or the assumption Eq.(8) is relaxed.
fore both waveguide modes have identical phase x 0 2
To better ensure the identity of propagating signals in
shifts 2(L +L +L )F (ω), which does not contain
1 2 3 I different modes, the order k of the Taylor remainder can
quadratictermsintheirTaylorexpansionoverfrequency
be increased and, in principle, done as high as needed,–
thus resulting in the signal propagation with identical
the system Eq.(13) for such a case has to be extended to
group velocity in both modes and with no quadratic dis-
incorporate more variables L and require vanishing of
persion. Furthermore, in practice the DetT(cid:98)(cid:12)= −(cid:12)1 re- all higher-order derivatives ∂ki−1c1/∂ωk−1 at ω =ω0. In
quirementdoesnotneedtobeimposedsince(cid:12)(cid:12)DetT(cid:98)(cid:12)(cid:12)=1 contrast, in order to eliminate also the cubic and higher-
always holds for unitary matrices. In this case, an ad- order dispersion in the common phase of the modes, one
ditional frequency-independent global phase will appear has to properly tailor the frequency dependence of the
on T(cid:98)2 that we will ignore in the rest of the discussion refraction index n(ω), a task which can be possibly ad-
for two modes −this phase can be easily recompensed. dressed in the discipline of metamaterials. In the case
Now we let us turn to our particular example where where no explicit analytical derivatives of the mode dis-
wehaveanexplicitanalyticexpressionforthecoefficient persionareavailableallthederivativesenteringEqs.(13)
c (L ,L ,L ,ω ) have to be taken numerically.
1 1 2 3 0
c =2(cid:2)cosF (ω)−cos3F (ω)(cid:3) (15) Wenowbrieflyconsiderthegeneralformulationofthe
1 x x
probleminthecaseofN modeswithdifferentdispersion
{cos[F (ω)(L −L −L )]+
z 1 2 3 laws κ (ω). The aim is to achieve a tailored breaking of
l
cos[F (ω)(L +L −L )]+ translational symmetry such as all modes have identical
z 1 2 3
cos[F (ω)(L −L +L )]} dispersion laws up to the k-th order of Tailor expansion.
z 1 2 3
As earlier, we consider a sequence of mode scattering
−2cos3F (ω)cos[F (ω)(L +L +L )]
x z 1 2 3 transformations
while according to Eq.(4) cosF (ω) ∝ (ω−ω ), and
x 0 kN
therefore,tobeconsistentwithintheorderofapproxima- T(cid:98)({Ll},ω)=(cid:89)(cid:16)e−iLlφ(cid:98)(ω)eis(cid:98)(ω)(cid:17) (18)
tion, the term containing cos3F (ω) has to be ignored.
x l=1
4
where s(ω) is a generic [11] frequency dependent N × the scattering angles (which now express linear polariza-
(cid:98)
N Hermitian scattering action matrix, and φ(cid:98)(ω) is tion tilts) are considered small to avoid significant back-
a real traceless diagonal N × N matrix with the el- scattering. It has been found [15], that a magnetically-
ements φ (ω) = κ (ω) − 1 (cid:80)N κ (ω), which de- biased graphene flake exhibits that property of polariza-
ll l N j=1 j tion rotation and therefore it is utilized in our consider-
scribes the mode phase difference accumulation per unit
ation. For the operational frequencies that this effect is
waveguide distance. The average mode wavevector
κ (ω) = 1 (cid:80)N κ (ω) contributes with a global phase recorded, the tilt is not changing substantially and thus
0 N l=1 l is assumed to be frequency-independent. Under these
κ (ω)(cid:80)N L , which is factored out from the product
0 l=1 l assumptions and as we show below, the first equation of
Eq.(18). Let us assume that by tuning the material re-
Eqs.(13)dictatesthatthesumofallthescatteringangles
fraction index one gets the desired analytic properties
of the half-scheme (see Fig. 2) should be equal to π/2.
κ (ω) at ω = ω . One thus have to find length intervals
0 0
{Ll}suchthatT(cid:98)wouldbeanondegenerateN-throotof n
(cid:16) (cid:17) +π/2 +π/2
the identity matrix up to the terms O (ω−ω )k .
0
To this end we consider the characteristic polynomial
χ -χ χ -χ
π/(2Ν) π/(2Ν)
N−1 ... ...
Det(cid:104)T(cid:98)−λI(cid:98)(cid:105)=λN + (cid:88) cn({Ll},ω)λn+eiϕ({Ll},ω) N N z
n=1
(19)
and require, that at ω = ω0 all coefficients cn vanish Lm/2 Lm/2 Lm/2 Lm/2
togetherwithalltheirderivativestill(k−1)-thorder. In Lg/Ν Lg/Ν Lg/ΝLg/Ν Lg/Ν Lg/Ν Lg/ΝLg/Ν
Lg Lg
addition, we require that ϕ({L },ω ) = 2πq with q <
l 0
N an integer number, and also all derivatives to vanish Lm+Lg Lm+Lg
including (k−1)-th order. All together we impose kN
conditionsonkN variables{Ll}, –asystemofnonlinear FIG.2: Thephysicalconfigurationoftheproposedstructure.
algebraic equations to be solved numerically for given The overall system is comprised of two identical components
φ(cid:98)(ω) and s(cid:98)(ω). Once a solution is found, the repetition each of which contains (N +2) graphene flakes suitably dis-
ofthescatteringsequenceN-times, correspondingtothe tributedalongalength(Lm+Lg)ofthez axisintotheback-
matrixT(cid:98)N({Ll},ω)=eO(cid:98)[(ω−ω0)k]resultsintherequired ground medium of refractive index n.
propagation with a common phase κ (ω)(cid:80)N L , which
0 l=1 l After extensive parametric checks, we have concluded
canbeinprinciplecontrolledbyapropertailoringofthe
that the minimal configuration which can make a total
refraction index n(ω). (cid:112)
transfer matrix Tˆ2 ∼= Iˆ⇔ Tˆ ∼= Iˆ, is that depicted in
Fig. 2. As the light arrives from the left side, it prop-
II. MATERIALIZATION WITH ACTUAL agates into our background medium for a total length
MEDIA Lm/2 before meeting a graphene sheet of scattering an-
gle χ. To this end, it is inserted to a stack of N identical
A. Presentation of the Configuration graphene flakes with distance Lg/N between two neigh-
boring ones; the graphenes are suitably polarized and
dopedtoachieveanoverallFaradayrotationofπ/2. The
In the aforementioned analysis, we have presented the
outcomingsignalpropagatesforanotherlengthL /2be-
method by considering, as a model example, the spatial m
forebeingsubjectedtoanoppositetiltof(−χ). Asindi-
modes of a waveguide. In this context, it is difficult to
cated above, to avoid transmissivity issues, the polariza-
think of structures conforming the strict conditions of
tion tilt χ is taken low and for simplicity equal to that
perfect scattering (7)-(8), even though the possibility of
of each graphene sheet of the stack: χ = π/(2N). The
tailoring the dispersion relations of the waveguide is re-
same half-structure repeats itself to form a planar stack
alistic. In this Section, we provide a second example of
of multiple layers which provides dispersion-free propa-
applicability of the method, addressing different issues
gation.
thatmayariseinpracticeandconsideringrealisticstruc-
tures in the mode conversion of the input signal.
Tosimplifytheapproach,weconsiderpropagationinto
B. Uniaxial Medium with Lorentz Dispersions
unboundedmedia(insteadofawaveguide);thus,therole
of two modes is played by the two polarizations of the
incoming field which are vertical to each other. The An electric field E(cid:126)(z,t) with a vector lying on the xy
background medium treats differently each direction of plane which propagates along the z axis into a uniax-
electric field which means that is uniaxial [14], namely ial medium, characterized by refractive indexes (nx,ny)
is characterized by a diagonal permittivity matrix. Fur- along the corresponding axes, is written as an inverse
thermore, since we are referring to real-world systems, Fourier transform: E(cid:126)(z,t) = (cid:82)+∞E(cid:126)(z,ω)A(ω)e+iωtdω.
−∞
5
κ(ω)=Re[n(ω)]ω/c ω√ε /c free propagation into our background medium; this is
∞
possible only into an isotropic environment. However,
our material is uniaxial; thus, we demand n = n only
x y
at ω = ω . In Fig. 3, we schematically show the vari-
x-direction 0
ation of the quantities κ (ω) = Re[n (ω) ω/c] with
x,y x,y
respect to the frequency ω. Each curve begins from the
origin(ω =0)andtakesallthepositivevaluestobecome
ω/c infinite at ω = ωrx,y. For ω (cid:38) ωrx,y the represented
quantity becomes imaginary since the value under the
root in (21),(22) turns to very negative from very posi-
y-direction
tive. However, if ω passes a larger threshold (ω > ω
s rx
for the x direction), the refractive index gets again pos-
ωrx ωs ωry ω itive and remains so (no matter how high the frequency
O ω ω
0A 0B is – for ω → ∞ behaves like its linear asymptote). As
also indicated in Fig. 3, when one has ε > 1 (as we
∞
FIG. 3: Schematic graphs of the (real parts of) propagation assumed), the two refractive indexes can be equal only
constantsalongthetwoaxes(x,y)oftheuniaxialmediumas
when ω >ω and such an equality is achieved for op-
function of frequency ω. ry rx
erational frequencies within the interval ω < ω < ω .
s ry
Atthefrequency ω2 =ω2 +ω2/ε , therefractiveindex
s rx p ∞
of x axis gets nullified; for ω > ω its curve n (ω) has
The complex phasor of the electric field is given by [16]: s x
two common points with the graph of n (ω) at the fre-
y
E(cid:126)(z,ω)=xˆSxe−iωcnx(ω)z+yˆSye−iωcny(ω)z, (20) quencies ω0A and ω0B whose explicit expressions are not
shown here for brevity.
while A(ω) is the Fourier transform of the time- Therefore, we have two alternative central frequencies
dependent profile of the input signal which is assumed ω = ω at which the isotropy demand is well-served.
0A,B
to be concentrated around a central operating frequency Nevertheless, an additional constraint is related to the
ω0. The amplitudes (Sx,Sy) determine the sort (linear sign of the second derivative of propagation constants
or elliptical) and the tilt angle of the polarization [17]. κ(cid:48)(cid:48) . The whole dispersion-neutralization concept de-
x,y
It is clear thateach electric component“sees” adiffer- scribed in the Section I is based on the assumption that
ent permittivity and gets affected independently by each thequantitiesκ(cid:48)(cid:48)(ω)andκ(cid:48)(cid:48)(ω)areoppositeinthevicin-
x y
of the refractive indexes nx(ω),ny(ω) due to the uniax- ity of ω = ω0. Accordingly, we would like the func-
ial nature of the background medium. For this reason, tions κ (ω) and κ (ω) to have opposite second deriva-
x y
each transverse component of the electric field defines tives at their common points ω = ω . A necessary
0A,B
a mode with different dispersion relation nx,y(ω). We but not sufficient condition for it, is to have the same
assume that the frequency-dependent refractive indexes first derivatives; in particular, for κ(cid:48)(cid:48)(ω ) = −κ(cid:48)(cid:48)(ω ),
x 0 y 0
follow simple Lorentz models [18] and are written as fol- the two curves should be mirror-symmetric with re-
lows: spect to their common tangent. In other words, the
(cid:115) equation κ (ω) = κ (ω) should have a double root at
ω2 x y
nx(ω)= ε∞+ ωr2x−p ω2, (21) ωωp2==ω(0εA∞=−1ω)0(Bω.r2yS−ucωhr2xa)/c4onandditiloenadbsetcoomopeesrfaetaiosinballeffroer-
(cid:115) quency ω2 = (ω2 +ω2 )/2. Thus, one can search for
ω2 0 ry rx
n (ω)= 1+ p . (22) the desired regime κ(cid:48)(cid:48)(ω ) = −κ(cid:48)(cid:48)(ω ) in the parametric
y ω2 −ω2 x 0 y 0
ry vicinity of these values.
The transmission matrix of a two-port network (each
Assume that the two permittivities are characterized by
port corresponds to one polarization mode), which is
thesameplasmafrequencyω forbrevity,whilethehigh-
p
comprised only by a length L of a medium, as this de-
frequency limits of the dielectric constants are different
scribed above, is obviously written as:
(ε for the x direction and 1 for the y direction). The
∞
rεwe∞islol>nbaen1co,cemwfeereonqbeuveeidonuctsioelsaastωserurxmi,nyetaωhreeryaan>lsaolωydrsxiiffs.eforNrenortteeaatsnhodna,tssftiohnracaet UˆL =(cid:20)e−iκ0x(ω)L e−iκ0y(ω)L (cid:21). (23)
lossless medium (which is our case because we study the
propagation into it), the quantities n of (21) and (22)
x,y
are either positive or purely imaginary; we are obviously C. Graphene Flakes as Imperfect Scatterers
interestedforfrequencieswherebotharepositive: n >
x,y
0. As it is well-known [15], an one-atom-thick graphene
Since the polarization direction (angle) of our electric flake,beingdopedwithelectricchargecarriers(electrons)
field expresses the weights of each of the two superim- at graphene conduction band with energy µ (chemi-
c
posed modes, it should be retained as linear during the cal potential), exhibits specific surface conductivity γ
6
[19]. If, additionally, it gets biased normally to its sur-
101 8
face by a DC magnetic field B , the graphene sheet be-
0
comes anisotropic (and non-reciprocal) because its con- 7
ductivity matrix γˆ acquires non-zero off-diagonal ele- 6
ments (γ = −γ = γ (ω)) apart from the diagonal
xy yx O 5
oonbeesy(tγhxexK=uγbyoym=oγdDel(ω[1)9)]. aTnhdeacroerrneosptornepdrinogdufocermduhlearse, B (T)0100 4
for brevity. 3
Let us consider a single graphene flake into the afore-
2
mentioned uniaxial background medium with refractive
1
indexes n (ω), which is excited by the electric field of
x,y
(20). If one imposes the necessary boundary conditions 10−1 0
0 0.02 0.04 0.06 0.08 0.1
[15] for continuous tangential electric fields (cid:126)n × (E(cid:126)2 − μc (eV)
E(cid:126) ) = (cid:126)0 and discontinuous tangential magnetic fields (a)
1
(cid:126)n×(H(cid:126) −H(cid:126) )=γˆ·E(cid:126) ,oneobtainsthefollowingtransfer
2 1 2
matrix 101 1
2 0.95
Sˆ=
(2n +η γ )(2n +η γ )+η2γ2 0.9
x 0 D y 0 D 0 O
(cid:18) (cid:19)
n (2n +η γ ) −n η γ 0.85
x nxyη0γO0 D ny(2nxy+0ηO0γD) , (24) B (T)0100 0.8
0.75
where η is the wave impedance of free space and (cid:126)n the
0
unitary vector normal to the surface pointing from side 0.7
1 to side 2. This matrix connects the two polarization 0.65
components of the electric field from both sides of the
10−1 0.6
graphene boundary. Note that for an opposite mag- 0 0.02 0.04 0.06 0.08 0.1
μ (eV)
netic bias (B → −B ), the conductivity matrix of the c
0 0
flake gets transposed (γˆ →γˆT because γ changes sign); (b)
O
therefore, the same happens to the transfer matrix (24)
of the graphene sheet (Sˆ→SˆT) [20]. FIG.4: The: (a)tiltangleofthelinearpolarizationindegrees
and(b)normalizedsquaredamplitude,thatcharacterizesthe
transmittedwaveasfunctionsonthechemicalpotentialµ re-
c
quired for the doping of graphene and the magnetic bias B .
D. Calculation of the Overall Transmissivity 0
Thewhitelineshowsthecombinationsof(µ ,B )whichlead
c 0
to polarization tilt χ=π/(2N) for N =100. The conductiv-
Bytakingintoaccountthetransfermatrices(23),(24), ity of graphene has been computed with use of Kubo model
one can write down the total transfer matrix Pˆ of the [19] with vF ∼= 106 m/sec Fermi group velocity of the elec-
overall configuration shown in Fig. 2 as follows: trons in the hexagonal lattice, T ∼=300 K room temperature
and τ ∼=0.2 psec transport scattering time of the electrons.
(cid:18) (cid:16) (cid:17)N (cid:19)2
Pˆ = SˆTUˆ Uˆ Sˆ SˆUˆ . (25)
Lm Lg Lm
2 N 2
perfect transmissivity from the graphene flake, (ii) the
The symbol Sˆ is used for the scattering matrix of a sin- number N of graphene flakes is large enough so that the
galteωgr=apωhen,ebyflaaknegtlehaχt r=otπat/e(s2Nth)e.pIoflaorniezaetxiocnluddiersecfrtioomn Trotter’s formula is applicable (cid:16)e−ih(ω)LNgσˆz−i2πNσˆy(cid:17)N ∼=
0
(25)thecommonphaseforbothpolarizationcomponents e−ih(ω)Lgσˆz−iπ2σˆy, (iii) the signal is concentrated around
e−i(κx(ω)+κy(ω))(Lm+Lg)Iˆ,whichforω ∼=ω0 doesnotcon- ω =ω0,(iv)thetiltinthepolarizationangleχ=π/(2N)
tainthequadraticdispersionterm, oneobtainstheanal- of the linearly polarized field inflicted by the graphene
ogous to Eq.(11) relation: is practically independent from frequency ω across the
operational bandwidth around ω [21].
0
Tˆ ∼=ei2πNσˆye−ih(ω)L2mσˆze−ih(ω)Lgσˆz−iπ2σˆy We repeat the method described in Section I,
e−i2πNσˆye−ih(ω)L2mσˆz, (26) nam(cid:104)ely, we (cid:105)demand that the characteristic polynomial
Det T(cid:98)−λI(cid:98) does not possess, in the vicinity of ω =ω0,
where for ω ∼= ω , the Taylor expansion is used: h(ω) =
0 a first-order term with respect to λ. Accordingly, we
(κ(cid:48)(ω )−κ(cid:48)(ω ))(ω−ω )/2+κ(cid:48)(cid:48)(ω )(ω−ω )2.
x 0 y 0 0 x 0 0 obtain from the third condition in (13), namely the de-
Thederivationof(26)reliesontheassumptions: (i)the (cid:12)
mand for: ∂2 c (L ,L ,L ,ω)(cid:12) = 0, the following
tilt angle χ = π/(2N) is small so that we have almost ∂ω2 1 1 2 3 (cid:12)
ω=ω0
7
relation connecting the lengths L ,L :
m g 0.8 0.25
30 T
B =30 T
LLmg = 2cscπ(2χ) vities0.7 B00=10 T seitivis 0.2
·(cid:16)2cos2χ+(cid:112)cosχ(3cosχ+cos(3χ)+2πsinχ)(cid:17), missi0.6 sims0.15
wttwohtoearlcesocnχadt=itteiroπinn/s(g2iaNnn)g(.1le3f)Ooarnretehceiadhneaneltfai-cssiaclhlyleyvmesearitifisysπfite/hd2atawsthhietenis(fi2tfrho7sert) diagonal trans00..45 B0=1 T nart lanogaid-ffo00.0.15 10 T
1 T
our case. This degeneracy stems from the conditions of:
0.3 0
(i) uniaxial medium that dictates that κ (ω ) = κ (ω ) 0.8 0.9 1 1.1 1.2 0.8 0.9 1 1.1 1.2
x 0 y 0 / /
and (ii) frequency independence tilt angles χ of the 0 0
graphene flakes. In this way, for a fixed (large) num- (a)
ber N of graphene flakes equally distributed within a
lengthLg,weobta0.i8nviaEq.(27)thenecessarysizeofthe 0.25
30 T
backgroundmediumL sothattheschBem=e3i0ll uTstratedin
m 0
s
Fig. 2 leads tovitiesdis0p.7ersion-less and coheBre0n=t1p0r Topagation. eitivis 0.2
boueTernhoerpeepcrooarltdaioreindzEaadiagonal transmissilm.tiofa000rniN...e456nqlutyuimeltanetcrtyihtchraboelueSlmgoihnmicgruasolwawgtariiBvatoehp0ns=ihsne1[1n tT5eh]eflaaninkdteetrhhvuaassl simsnart lanogaid-ffo000..10.155 10 T
10 GHz < ω/(2π) < 60 GHz. The chemical potential is 1 T
0.3 0
keptsmall(below0.1eV0.)8since0w.9earein1terest1e.d1 forsm1.2all 0.8 0.9 1 1.1 1.2
tiltsχ=π/(2N),whilethemagneticb/iasispermittedto /
0 0
take large (>1 T) or even huge and unrealistic (>10 T)
(b)
valuessinceithelpsusdemonstratingourconcept. When
itcomestothebackgroundmedium,wehavenoticedthat
FIG. 5: The: (a) diagonal (ideally 1 in the vicinity of ω =
better final outcome is obtained when the resonant fre-
ω ) and (b) off-diagonal (ideally 0 in the vicinity of ω =ω )
quencies are not very close each other. Therefore, we 0 0
transmissivities of the overall structure in Fig. 2 as function
can make a pre-selection of ω = 10(2π) Grad/sec and
rx of the normalized frequency ω/ω0. Several magnetic bias are
ωry =60(2π)Grad/sec. Asfarasthehigh-frequencyper- consideredcombinedwithsuitablechemicalpotentialdictated
mittivityalongthexaxisisconcerned,itcantakeanor- by the white line of Figs 4 (µc ∼= 0.0229 eV for B0 = 1 T,
dinaryvalueaboveunity: ε∞ =2.5[22]. Finally,asmen- µc ∼= 0.0953 eV for B0 = 10 T and µc ∼= 0.1671 eV for B0 =
tioned above, the number of graphene layers is selected 30T).Therestofgrapheneparametersareidenticaltothose
quite high: N = 100. By carefully evaluating the dou- of Figs 4.
blederivativesofκ(cid:48)(cid:48) (ω),weconcludethatthecondition
x,y
κ(cid:48)(cid:48)(ω )=−κ(cid:48)(cid:48)(ω ) is achieved for a common plasma fre-
x 0 y 0
quency of Lorentz models ω ∼=36.19(2π) Grad/sec with It is clearly recorded that the transmission coefficient is
p
central operational frequency ω ∼=42.07(2π) Grad/sec. veryhigh; thisfindingisapositiveindicatorthatthede-
0
InFigs4, weexaminethebehaviorofagrapheneflake scribed concept for minimal dispersion may work well in
into the aforementioned uniaxial medium, at the oscil- the described real-world configuration.
lating frequency ω . In Fig. 4(a), we show the tilt angle However, this high transmissivity indicated by
0
(in degrees) that a linearly polarized incident field ex- Fig. 4(b) does not automatically give an overall power
periences by passing through a graphene sheet of chem- output close to 100%. This is referred to a single
ical potential µ and magnetic bias B . It is clear that graphene flake alone into our background medium; how-
c 0
the polarization rotation is a growing function of µ and ever, the device of Fig. 2, we have clusters of (N) of
c
gets maximized for a specific B . The white line cor- these sheets. Therefore, we should check the transmis-
0
responds to combinations of (µ ,B ) which give a tilt sionthroughtheentirestructure. Ifweassumethethick-
c 0
equal to χ = π/(2N) = π/200 ∼= 0.9o and thus are suit- ness of the graphene stack equal to Lg =500 nm (lattice
able for our specific application. One observes that for period L /N = 5 nm, for a tilt χ = π/200 (correspond-
g
every single bias (except for some very low B ), there is ingtoN =100),weobtainfrom(27)thatthelengthL
0 m
always a doping degree µ which constitutes a working is found equal to: L ∼=40.8 µm.
c m
point. In Fig. 4(b), we show the same white line but on InFigs5,wepickthreecombinationsofgraphenedop-
the map of the transmissivity of a single graphene flake. ing and bias (µ ,B ) which lead to polarization tilt by
c 0
8
χ = π/(2N) (two of them are within the limits of the chemical potential µ ) and magnetic bias (B ).
c 0
mapsofFigs4)andrepresentthesquaredmagnitudesof
the four elements of matrix Pˆ from (25). We use light-
III. DISCUSSION
coloredthicklinesanddark-coloredthinlinesinorderto
represent each couple (diagonal or off-diagonal) of quan-
Chromatic dispersion is one of the main problems
tities. Ideally, the diagonal of them should be equal to 1
[23] in optical communications over long distances and
around ω = ω and the off-diagonal ones should vanish
0
many works have been devoted to this subject [1]-[2]. In
in the vicinity of ω = ω . Note that we expect neither
0
classical single-mode optical networks the most common
thediagonalelementstobeequaleachother(becausethe
methodseemstobeatthispointthedispersioncompen-
networkisnotsymmetric)northeoff-diagonalonestobe
sating fibers. In the multi-mode scenario the techniques
equal each other (because the network is not reciprocal).
are somehow different, and there is a number of elegant
In Fig. 5(a), the diagonal transmissivities are shown
suggestions, e.g. [24], [25], including ones [8] which are
with respect to the operational frequency ω normal-
based on the conversion of the signal between different
ized by the central frequency ω for various bias B =
0 0
modes, as the current work does, and on the tailoring of
1,10,30T. Inallthethreecasesweobserveasubstantial
the dispersion laws.
stability of the response in the neighborhood of ω =ω ,
0
which means that our dispersion-free structure can sup- Concerning dispersion compensation in single-mode
portwide-bandsignals. Asanticipated, thetransmission quantum signals, the direct extension of classical meth-
through the structure gets deteriorated far from ω = ω ods to quantum signals -containing low number of pho-
0
because the device has been designed to operate at this tons, is under question [26] since these may induce ad-
oscillation frequency. Notice that the two diagonal el- ditional source of photon losses. Quantum solitons [27]
ements of Pˆ have almost identical magnitude variation provideasolutiontothisproblembutontheotherhand
with respect to ω; this feature can be attributed to the these require elaborate means of preparation.
effective “averaging” performed by the multiple layers The method that we describe in this work provides
whichmitigatestheoverallasymmetry. Itisalsoremark- analytic expressions which can be employed to eliminate
able that despite the very high transmission revealed by quadratic dispersion at any Taylor order and even com-
Fig. 4(b), the total transmissivity is much weaker due pensate dispersion at higher orders. The results are ap-
to the presence of numerous graphene sheets (as implied plicable to both classical and quantum multi-mode net-
above). Most importantly, we see that larger magnetic works however it seems to be more of interest in the
fields B accomplish better results which means that latter case. The first interesting point of the method is
0
the transmission along the white line of Fig. 4(b) is not that under the condition of weak scattering, the method
equally high; even very small changes to the behavior of does not induce photon losses on the signal. The sec-
a single scatterer has a significant outcome to the total ond advantage is that the proposed method can produce
structure (comprised of (2N +2) scatterers). phase matching conditions along extended distances and
InFig.5(b),weshowthecorresponding(squaredmag- thereby induce nonlinear coupling between the quantum
nitudes of the) off-diagonal elements of the matrix Pˆ de- fields of different modes even for a typically weak non-
fined in (25) as functions of ω/ω . It is clear that all the linear permeability. This result that seems negative at
0
quantities vanish at ω = ω and acquire substantial val- first look, may pave new methods in multi-mode quan-
0
ues far from the central frequency for the obvious reason tum communication, where entanglement can be gener-
mentionedabove. Followingthetrendofthediagonalel- ated‘ontheway’andusedforerror-protectionofsignals
ements,thequantitiesinFig.5(b)are,onaverage,larger against photon losses. This idea is to be further devel-
as B increases; however, in all the three of the consid- oped in a future work.
0
eredcases,themagnitudesofoff-diagonaltermsaremuch
smaller than the respective diagonal ones (Fig. 5(a)). It
should be also stressed that, unlike the diagonal ones in
Acknowledgement
Fig.5(a),thetwooff-diagonalelementsdiffereachother;
this is the definition of of non-reciprocity [15] and is at-
tributed to the presence of external magnetic bias. AM acknowledges financial support from the Ministry
Therefore, we can conclude that in a realistic scheme of Education and Science of the Republic of Kazakhstan
likethatinFig.2,onecanemulatedispersion-freepropa- (Contract # 339/76 − 2015). AM is also thankful to
gation for a suitably modulated signal around frequency CCQCN at University of Crete for their hospitality and
ω = ω . Two electromagnetic entities are required: (i) partialfinancialsupportbytheEuropeanUnionSeventh
0
A properly designed uniaxial medium whose refractive Framework Programme (EU/FP7-REGPOT-2012-2013-
indexes n (ω),n (ω) follow common Lorentz dispersion 1) under Grant No. n316165. VA thanks G. A. Kaba-
x y
models. (ii) Graphene flakes with specific doping (with tiansky for attracting his attention to this problem.
9
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