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Azimuthal modulation of probe absorption and transfer of optical
vortices.
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Page 1 of 5
Azimuthal modulation of probe absorption and transfer of optical vortices.
Aamen Shujaat1,2 , Urgunoon Saleem1,2 , Muqaddar Abbas1 ,∗ and Rahmatullah1†
1
Quantum Optics Lab. Department of Physics, COMSATS University, Islamabad, Pakistan and
2
Both authors contributed equally to this work.
(Dated: July 2, 2020)
I.
INTRODUCTION
an
In recent era, the coherent control of optical properties
of a medium has received substantial interest. Coherent
interaction of light with multilevel atoms can dramatically modify the systems response to an optical field due
to quantum interference between the excitation amplitudes of optical transitions. This leads to an interesting phenomenon known as electromagnetically induced
transparency (EIT) in which the absorption of a weak
probe field is suppressed via a strong coupling field [1, 2].
EIT has important applications in quantum information
processing [3], non-linear optics [4, 5], slow light [6], optical switching and storage of the photons [7–10].
around the propagation axis with an undefined phase at
the center known as an optical vortex or phase singularity [14–18]. Recently, the tight focusing properties of a
hollow Gaussian beam (HGB) [19], a linearly polarized
circular Airy Gaussian vortex beam (CAiGVB) [20] and
a radially polarized symmetric Airy beam (RPSAB) [21]
have been investigated. Optical beams with OAM also
called structured light have widely been explored in optical communications [22, 23], optical trapping [24, 25],
optical data storage and quantum technologies [26, 27],
quantum information [28] and optical tweezers [29].
us
PACS numbers:
cri
pt
We analyze azimuthal modulation of probe field absorption profile in a closed four-level inverted
Y-type atomic system. We employ a Laguerre-Gaussian beam and a microwave field to realize
spatially varying atomic susceptibility, which is responsible for the generation of structured light.
An extra nonvortex control beam is used to ensure electromagnetically induced transparency at the
core of Laguerre-Gaussian beam. Our proposed model exhibits both absorption and gain at different
azimuthal angles, and orbital angular momentum of light can be identified via absorption profile of
the probe field. We further discuss the transfer of optical vortices from Laguerre-Gaussian beam to
the transmitted probe field.
ce
pte
dM
Light is an electromagnetic wave which can carry linear and angular momentum. Linear momentum of light
is always in the direction of propagation and on the other
hand, angular momentum of light consists of both spin
angular momentum (SAM) and orbital angular momentum (OAM). The SAM is responsible for the circular polarization of light while OAM gives light a helical shape
corresponding to the phase lΦ, where Φ is the azimuthal
angle with respect to the beam axis and l represents
the winding number or the azimuthal quantum number.
The most common example of light carrying OAM is
Laguerre-Gaussian (LG) beams [11]. When LG-beams
interfere, different types of shapes can be appeared in
the interfered plane of a vortex beam which depends on
winding numbers of both interfering beams, their relative
angle, coinciding or not coincident their centers. These
patterns can be classified into classes fork-shape, starlike shape, spiral zone plate and also like electric field
lines of two separated charges. These patterns are similar
to moiré patterns of superimposing two gratings consisting of azimuthal dependent terms in their transmission
phase functions [12, 13]. The amplitude is distributed
∗ Electronic
address: muqaddarabbas@comsats.edu.pk
address: rahmatktk@comsats.edu.pk
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AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-110782.R1
† Electronic
It is possible to identify the OAM of light in an azimuthally modulating atomic susceptibility where the information about OAM is obtained by mapping the spatially modulating probe absorption profile [30, 31]. Recently, Radwell et al. experimentally demonstrated spatially dependent EIT in cold rubidium atoms driven by
the vortex probe light beam [32]. This scheme is further theoretically analyzed using density matrix formalism [33]. As the intensity of a vortex beam is zero at the
center of the vortex, therefore, EIT vanishes at core center and an extra control field having no vortex is needed
to maintain EIT [34]. Here we are going to mention that
several systems have been reported where the vortices
present in the control field have been transferred to probe
light beam. Such transferred of optical vortices have been
observed in atomic systems [35–40], semiconductor quantum dots [41] and semiconductor quantum wells [42, 43].
Motivated by the above studies, we present a scheme
for the generation and detection of structured light in
a closed four-level inverted Y-type atomic system. We
employ LG beam to realize azimuthal phase-dependent
variation of the probe field absorption profile. We also
consider a nonvortex beam which maintains the EIT at
core of the vortex beam, and this feature is missing
in Refs [30, 32, 33]. We observe both absorption and
gain at different azimuthal angles and also analyze spatially structured transparency. Moreover, our proposal
can be implemented to distinguish different modes of LG
beam. Due to closed-loop structure, optical vortices can
be transferred from control to probe beam.
AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-110782.R1
Page 2 of 5
2
(a)
(b)
0.15
ρ̇cc = (γa − γc ) ρcc + iΩ∗c1 ρac + iΩB eiϕ ρbc
−iΩ∗B e−iϕ ρcb + iΩc1 ρca ,
(5)
ρ̇dd = iΩc2 ρad − iΩ∗c2 ρda − γd ρdd ,
(6)
0.10
y/w
0.05
ρ̇ab = (i∆p − γab ) ρab + iΩp (ρbb − ρaa ) + iΩc1 ρcb
cri
pt
0
+iΩ∗c2 ρdb − iΩB eiϕ ρac ,
x/w
FIG. 1: (a) Schematic diagram of a closed four-level inverted
Y-type atomic system. Here, Ωp , Ωc1 , Ωc2 and ΩB are the
Rabi frequencies of probe, LG, control and microwave fields
respectively. (b) Intensity profile of LG beam for l = 1.
+iΩ∗c2 ρdc − Ω∗B e−iϕ ρab ,
ρ̇da = (i∆c2 − γda ) ρad − iΩ∗p ρdb − iΩ∗c1 ρdc
+iΩc2 (ρaa − ρdd ),
THEORY AND DISCUSSION
A.
Model and Equations
(7)
ρ̇ac = (i∆c1 − γac ) ρac + iΩc1 (ρcc − ρaa ) + iΩp ρbc
(8)
(9)
ρ̇cb = (i(∆p − ∆c1 ) − γcb ) ρbc − iΩp ρca + iΩ∗c1 ρab
+iΩB eiϕ (ρbb − ρcc ) ,
(10)
ρ̇dc = (i(∆c1 + ∆c2 ) − γdc ) ρcd − iΩc1 ρda − iΩ∗B e−iϕ ρdb
+iΩc2 ρac ,
(11)
ρ̇db = (i(∆p + ∆c2 ) − γdb ) ρdb − iΩp ρda − iΩB eiϕ ρdc
+iΩc2 ρab
(12)
dM
an
We consider a four-level inverted-Y type atomic system
consisting of two excited states |ai and |di and two closely
lower lying level |bi and |ci as shown in Fig. 1 (a). A
weak probe field Ep drives the atomic transition |bi to
|ai and two control fields Ec1 and Ec2 are coupled to
atomic transitions |ci ↔ |ai and |ai ↔ |di respectively.
A microwave beam EB acts along |bi ↔ |ci transition.
We consider Ep , Ec2 and EB are non-vortex beams and
Ec1 is a LG beam. Fig. 1 (b) shows the intensity profile
of LG beam.
The corresponding Hamiltonian for the proposed system under the rotating-wave and dipole approximation
is written as;
us
II.
H = −h̄[∆p |ai ha| + (∆p − ∆c1 ) |ci hc|
+(∆p + ∆c2 ) |di hd|] − h̄[Ωp |ai hb|
+Ωc1 |ai hc| + Ωc2 |di ha|
+ΩB eiϕ |ci hb| + H.c.],
(1)
pte
where ϕ = ϕB + ϕc1 − ϕp is the relative phase and
∆p = ωab − υp , ∆c1 = ωac − υc1 and ∆c2 = ωda − υc2
are the detunings of probe (Ep ) and control (Ec1 , Ec2 )
fields. The corresponding Rabi frequencies are defined
℘ E
as Ωp = abh̄ p , Ωc1 = ℘ach̄Ec1 , Ωc2 = ℘adh̄Ec2 and
℘bc EB
. Here, ℘ab , ℘ac , ℘ad and ℘bc are transiΩB =
h̄
tion dipole elements. We utilize the Liouville equation
to describe the dynamics of the proposed system
ce
i
1
ρ̇ = − [H, ρ] − {Γ, ρ} .
(2)
h̄
2
The second term represents various incoherent decay processes. Therefore, the equations of motion can be obtained as
ρ̇aa = (γd − γa ) ρaa + iΩp ρba − iΩ∗p ρab + iΩc1 ρca
−iΩ∗c1 ρac + iΩ∗c2 ρda − iΩc2 ρad ,
(3)
B.
Perturbative Analysis of Equations of Motion
In this subsection, we solve Eqs. (3-12) under weak
probe field approximation where the intensities of the
probe and microwave fields are low as compared to the
control fields, i.e., ΩB < Ωp < Ωc1 , Ωc2 . The density
matrix equations can be solved by expressing them in
terms of the perturbation expressions
(0)
(1)
(2)
ρij = ρij + λρij + λ2 ρij + ......(i, j = a, b, c, d), (13)
(0)
(1)
where λ represents the interaction order. Here ρij , ρij
(2)
and ρij are the zeroth, first and second order solutions
in Ωp respectively. Under the weak field approximation
(1)
only the first-order terms ρij are important. In order to
(1)
obtain ρij , we substitute Eq. (13) into Eqs. (3-12).
Our aim is here to study the absorption profile of
the probe field corresponding to the atomic transition
(1)
|ai ←→ |bi and hence we need ρab . We assume that
initially all the populations are in ground state |bi, i.e.,
(0)
(0)
ρbb = 1 and ρij = 0. After applying the perturbation,
Eqs. (3-12) are reduced to the following equations
(1)
(1)
(1)
(1)
ρ̇ab = (i∆p − γab )ρab + iΩp + iΩc1 ρcb + iΩ∗c2 ρdb , (14)
ρ̇bb = (γa + γc ) ρbb + iΩ∗p ρab − iΩp ρba
+iΩB eiϕ ρcb − iΩ∗B e−iϕ ρbc ,
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(4)
(1)
(1)
(1)
ρ̇cb = (i(∆p − ∆c1 ) − γcb )ρcb + iΩ∗c1 ρab + iΩB eiϕ ,(15)
Page 3 of 5
AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-110782.R1
3
(a)
(b)
(a)
-2
-2
(b)
x10
x10
1.0
1.0
1.5
1.5
0.5
0.5
1.0
1.0
0.0
0.0
0.5
-1.0
y/w
-0.5
0
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-1.0
0
F(radian)
1
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6
F(radian)
0.5
y/w
0
-0.5
-0.5
-1.0
-1.0
cri
pt
-0.5
-1.5
FIG. 2: Absorption profile of probe field as a function of the
azimuthal angle Φ for (a) l = 1 and (b) l = 2. The other
parameters are γab = γ = 1MHz, γcb = 10−3 γ, γdb = 10−2 γ,
Ωc1 = Ωc2 = γ, Ωp = 0.1γ, ΩB = 0.01γ, ϕ = 0 and ∆p =
∆c1 = ∆c2 = 0.
x/w
(1)
(1)
(16)
We solve the Eqs. (14-16) under steady state condition
(1)
ρ̇ij = 0 and get required expression for ρab as
iΓcb Γdb Ωp
(1)
ρab =
2
2
Γab Γcb Γdb + Γdb |Ωc1 | + Γcb |Ωc2 |
Γdb Ωc1 ΩB eiϕ
−
2
2 , (17)
Γab Γcb Γdb + Γdb |Ωc1 | + Γcb |Ωc2 |
(d)
-2
x10
1.5
1.5
1.0
1.0
0.5
y/w
0
0.5
0
-0.5
-0.5
-1.0
-1.0
-1.5
x/w
-1.5
x/w
FIG. 3: Absorption profile of the probe field as a function of
x/w and y/w for (a) l = 1, (b) l = 2, (c) l = 3 and (d) l = 4.
We choose ǫc1 = γ. The remaining parameters are the same
as those given in Fig. 2.
one gain and one absorption peak for l = 1 as shown in
Fig. 2 (a). The maxima of the gain and absorption peak
lies at Φ = π/2 and Φ = 3π/2 respectively. Fig. 2 (a)
also shows zero absorption (EIT) at different Φ, e.g., 0,
π and 2π. The absorption profile exhibits two gain and
two absorption peaks by increasing the OAM from l = 1
to l = 2, see Fig 2 (b).
To study spatially modulating probe field absorption,
we consider control field of Rabi frequency Ωc1 as a LG
doughnut type beam
C.
dM
an
where Γab = γab − i∆p , Γcb = γcb − i(∆p − ∆c1 ) and
Γdb = γdb − i(∆p + ∆c2 ). The first term on the right
(1)
hand of ρab corresponds to standard EIT in a four level
inverted Y -type atomic system. The second term arises
due to microwave field and related to the gain process.
-2
x10
us
(1)
ρ̇db = (i(∆p + ∆c2 ) − γdb )ρdb + iΩc2 ρab .
-1.5
x/w
(c)
y/w
Azimuthal modulation of probe absorption
pte
The density matrix element ρab is a complex quantity consists of both real and imaginary parts. The real
part is related to the dispersion, and imaginary part corresponds to the absorption of the probe field. In this
section, we study the absorption profile of the probe field
corresponding to the atomic transition |ai ←→ |bi and
(1)
only analyze the imaginary part of ρab . First, we investigate the effect of the azimuthal angle on absorption
profile of probe field. In that case we consider control
field Ωc1 carries OAM l and other fields have no vortices.
The Rabi frequency Ωc1 is defined as
Ωc1 = |Ωc1 | eilΦ ,
(18)
ce
where Φ is the azimuthal angle.
In Fig. 2, we plot the probe field absorption profile
versus azimuthal angle Φ for two different values of OAM,
i.e., (a) l = 1 and (b) l = 2. The other parameters are
γab = γ = 1MHz, γcb = 10−3 γ, γdb = 10−2 γ, Ωc1 =
Ωc2 = γ, Ωp = 0.1γ, ΩB = 0.01γ, ϕ = 0 and ∆p = ∆c1 =
(1)
∆c2 = 0. We observe optical transparency (Im[ρab ] =
(1)
(1)
0), gain (Im[ρab ] < 0) and absorption (Im[ρab ] > 0) at
different values of azimuthal angle Φ. There are only
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r2
r
Ωc1 = ǫc1 ( )|l| e− w2 eilΦ ,
(19)
w
p
where Φ = tan−1 (y/x), r = x2 + y 2 , w is the beam
waist parameter and ǫc1 is the strength of the beam. We
consider the other fields are independent of azimuthal
angle Φ. As the intensity of the LG beam is zero as r → 0
(see Fig. 1(b)), therefore, in a three-level atomic system,
EIT vanishes at the center of the LG beam. In order
to maintain EIT, we consider an extra non-vortex beam
Ωc2 . In the absence of the first control field (Ωc1 = 0),
Eq. (17) reduces to
(1)
ρab =
iΓdb Ωp
Γab Γdb + |Ωc2 |
2,
(20)
which represents standard EIT in three-level ladder type
atomic system. Therefore, in our proposed model, EIT
still maintains even at the core of LG beam.
We further proceed to investigate the absorption profile of the probe field for different OAM numbers. Fig.
3 shows the spatially dependent probe absorption profile
Page 4 of 5
AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-110782.R1
4
Transfer of optical vortices
∂Ωp (z)
αp γab (1)
=i
ρ ,
∂z
2L ab
(22)
dM
iκΩc1 ΩB eiϕ 1 − e−βz ,
β
here Ωp (0) is the probe field at the entrance of the cell
and
αp γab
Γcb Γdb
,
2L Γab Γcb Γdb + Γdb |Ωc1 |2 + Γcb |Ωc2 |2
(23)
κ=
Γdb
αp γab
.
2L Γab Γcb Γdb + Γdb |Ωc1 |2 + Γcb |Ωc2 |2
(24)
pte
β=
y/w
-2
x10
-2
1.0
1.0
0.5
0.5
y/w
0
-0.5
-0.5
-1.0
-1.0
ce
0
y/w
-1
-1
-2
-2
x/w
-3
x/w
(c)
(d)
3
3
2
2
1
1
0
y/w
0
-1
-1
-2
-2
-3
-3
x/w
FIG. 5: Phase profile of the probe field as a function of x/w
and y/w for (a) l = 1, (b) l = 2, (c) l = 3 and l = 4. Here we
consider αp = 100. The remaining parameters are the same
as those given in Fig. 3.
We consider control field Ωc1 carries the OAM and corresponding Rabi frequency Ωc1 is defined by Eq. (19).
The Eq. (22) shows that OAM l of the control field Ωc1
is transferred to the transmitted probe field. Initially at
the entrance of the atomic medium, the incoming probe
field does not carry any vortex. After traveling some distances, vortex of the control beam shifts to the probe
beam due to the closed-loop structure of the atomic system. The presence of the nonvortex beam Ωc2 avoid the
losses of the probe field. In Fig. 5, we plot phase profile
of the transmitted probe field at z = L for l = 1, 2, 3, 4
and αp = 100. The singularity at the center shows transfer of the optical vortex, and corresponding phases shift
from 0 to 2πl.
III.
CONCLUSION
x10
1.5
-1.5
-1.5
x/w
(b)
1.5
0
1
an
(21)
where αp is the optical depth of probe field and L is
the length of the atomic vapor cell. With Eq. (17), the
transmitted probe field can be expressed as
(a)
1
x/w
In this section, we demonstrate the transfer of optical
vortices from control to probe field. The time independent propagation equation of the probe field under slowly
varying envelope approximation can be written as
Ωp (z) = Ωp (0)e−βz −
2
-3
y/w
3
2
0
y/w
(b)
3
us
D.
(a)
cri
pt
for (a) l = 1, (b) l = 2, (c) l = 3 and (d) l = 4. We obtain pleasing petal-like structures proportional to 2l. The
bright petals indicate to absorption and the dark petals
reveal the gain and orange area corresponds to the optical
transparency. The above results imply that the behavior
of the absorption profile of the probe field depends on
the OAM l. The absorption profile displays 2l-symmetry
and unknown vorticity of a LG beam can be identified
by counting the number of petal structures. Further, we
investigate the effect of the relative phase on the probe
field absorption spectrum. The petal-like structures rotate clockwise as we change the relative phase, see Fig. 4.
The patterns rotate through an angle of 45◦ for ϕ = π/2
as shown in Fig. 4(a). By changing the relative phase
from ϕ = π/2 to ϕ = π, absorption in Fig. 3 (b) converts
into gain and vice versa as shown in Fig. 4(b).
x/w
FIG. 4: Absorption profile of the probe field as a function
of x/w and y/w for (a) ϕ = π/2 and (b) ϕ = π. Here we
consider l = 2. The remaining parameters are same as those
given in Fig. 3.
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In conclusion, we have studied spatially structured
transparency in a four-level inverted Y type atomic configuration illuminated by a pair of control fields, a probe
and a microwave field. A LG beam is used to periodically modulates the absorption profile in the azimuthal
plane. Such a periodic oscillation is the key factor behind
the generation of structured light. The other nonvortex
control beam maintains the condition for EIT at the vortex core and circumvents the losses in the probe field.
The absorption profile has shown 2l fold symmetry, and
one can identify the OAM associated with vortex beam.
The OAM can also be transferred from the control vortex
beam to transmitted probe beam.
Page 5 of 5
AUTHOR SUBMITTED MANUSCRIPT - PHYSSCR-110782.R1
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[23] Bozinovic N, Yue Y, Ren Y X, Tur M, Kristensen P,
Huang H, Willner A E and Ramachandran S 2013 Science
340 1545
[24] Friese M E J, Enger J, Rubinsztein-Dunlop H and Heckenberg N R 1996 Phys. Rev. A 54 1593
[25] Garcès-Chàvez V, McGloin D, Padgett M J, Dultz W,
Schmitzer H and Dholakia K 2003 Phys. Rev. Lett. 91
093602
[26] Kasap S O 1999 Optoelectronics (Prentice Hall, NJ, USA)
[27] Yao A M and Padgett M J 2011 Advances in Optics and
Photonics 3(2) 161
[28] Dada A C, Leach J, Buller G S, Padgett M J and Andersson E 2011 Nat. Phys. 7 677
[29] Padgett M J and Bowman R 2011 Nat. Photonics 5 343
[30] Han L, Cao M, Liu R, Liu H, Guo W, Wei D, Gao S,
Zhang P, Gao H and Li F 2012 Europhys. Lett. 99 34003
[31] Hamedi H R, Kudriašov V, Ruseckas J and Juzeliūunas
G 2018 Optics Express 26 28249
[32] Radwell N, Clark T W, Piccirillo B, Barnett S M and
Franke-Arnold S, 2015 Phys. Rev. Lett. 114 123603
[33] Sharma S and Dey T N 2017 Phys. Rev. A 96 033811
[34] Ruseckas J, Mekys A and Juzeliūnas G 2011 Phys. Rev.
A 83 023812
[35] Ruseckas J, Kudriašov V, Yu I A and Juzeliūnas G 2013
Phys. Rev. A 87 053840
[36] Hamedi H R, Ruseckas J and Juzeliūnas G 2018 Phys.
Rev. A 98 013840
[37] Sabegh Z A, Maleki M A and Mahmoudi M 2019 Sci.
Rep. 9 3519
[38] Sabegh Z A, Mohammadi M, Maleki M A and Mahmoudi
M 2019 J. Opt. Soc. Am. B 36 2757
[39] Hamedi H R, Ruseckas J, Paspalakis E and Juzeliūnas G
2019 Phys. Rev. A 99 033812
[40] Hong Y, Wang Z, Ding D and Yu B 2019 Optics Express
27 29863
[41] Rahmatullah, Abbas M, Ziauddin and Qamar S 2020
Phys. Rev. A 101 023821
[42] Zhang Y, Wang Z, Qiu J, Hong Y and Yu B 2019 Appl.
Phys. Lett. 115 171905
[43] Qiu J, Wang Z, Ding D, Li W and Yu B 2020 Optics
Express 28 2975
ce
pte
dM
an
[1] Fleischhauer M, Imamoglu A, and Marangos J P 2005
Rev. Mod. Phys. 77 633
[2] Harris S E 1997 Phys. Today 50(7) 36
[3] Lukin M D and Imamoglu A 2001 Nature 413 273
[4] Lukin M D and Imamoglu A 2000 Phys. Rev. Lett. 84
1419
[5] Yan Y, Rickey E G and Zhu Y 2001 Opt. Lett. 26 548
[6] Xiao M, Li Y -Q, Jin S Z and Banacloche J G 1995 Phys.
Rev. Lett. 74 666
[7] Liu C, Dutton Z, Behroozi C H and Hau L V 2001 Nature
409 490
[8] Turukhin A V, Sudarshanam V S, Shahriar M S, Musser
J A, Ham B S and Hemmer P R 2001 Phys. Rev. Lett.
88 023602
[9] Hau L V, Harris S E, Dutton Z and Behroozi C H Nature
1999 397 594
[10] Phillips D F, Fleischhauer A, Mair A, Walsworth R L
and Lukin M D 2001 Phys. Rev. Lett. 86, 783
[11] Allen L, Beijersbergen M W, Spreeuw R J C and Woerdman J P 1992 Phys. Rev. A 45 8185
[12] Rasouli S and Yeganeh M 2015 Journal of Optics 17
105604
[13] Rasouli S and Yeganeh M 2016 Journal of the Optical
Society of America A 33 416
[14] Duttonand Z and Ruostekoski J 2004 Phys. Rev. Lett. 93
193602
[15] Pugatch R, Shuker R M, Firstenberg O, Ron A and
Davidson N 2007 Phys. Rev. Lett. 98 203601
[16] Wang T, Zhao L, Jiang L and Yelin S F 2008 Phys. Rev.
A 77
[17] Xie X -T, Li W -B and Yang X 2006 J. Opt. Soc. Am. B
23 1609
[18] Moretti D, Felinto D and Tabosa J W R 2009 Phys. Rev.
A 79 023825
[19] Ye F, Zou J, and Deng D 2020 Ann. Phys 532 1900548
[20] Zhuang J, Zhang L, and Deng D 2020 Optics Letters 45
296
[21] Xu C, Hu H, Liu Y, and Deng D 2020 Optics Letters 45
1451
[22] Wang J, Yang J Y, Fazal I M, Ahmed N, Yan Y, Huang
H, Ren Y X, Yue Y, Dolinar S, Tur M and Willner A E
2012 Nat. Photon. 6 488
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