Spontaneous emission of a two-level atom with an arbitrarily polarized electric dipole in front of a flat dielectric surface

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by  Fam Le Kien

Spontaneous emission of a two-level atom with an arbitrarily polarized electric dipole
in front of a flat dielectric surface
Fam Le Kien
Wolfgang Pauli Institute, Oskar Morgensternplatz 1, 1090 Vienna, Austria
A. Rauschenbeutel
Vienna Center for Quantum Science and Technology, Institute of Atomic and Subatomic Physics,
Vienna University of Technology, Stadionallee 2, 1020 Vienna, Austria
(Dated: February 8, 2016)
We investigate spontaneous emission of a two-level atom with an arbitrarily polarized electric
6 dipoleinfrontofaflatdielectricsurface. Wetreat thegeneralcasewheretheatomicdipolematrix
1 elementisacomplexvector,thatis,theatomicdipolecanrotatewithtimeinspace. Wecalculatethe
0 ratesof spontaneous emission intoevanescent and radiation modesand studytheangular densities
2 of the rates in the space of wave vectors for the field modes. We show that, when the ellipticity of
theatomicdipoleisnotzero,theangulardensityofthespontaneousemission rateoftheatommay
b
havedifferentvaluesformodeswithoppositein-planewavevectors. Wefindthatthisasymmetryof
e
theangulardensityofthespontaneousemissionrateundercentralinversioninthespaceofin-plane
F
wave vectors is a result of spin-orbit coupling of light and occurs when the ellipticity vector of the

4 atomicdipolepolarization overlapswiththeellipticityvectorofthefieldmodepolarization. Dueto
thefast decay of thefield in theevanescentmodes, thedifferencebetween theratesof spontaneous

] emission into evanescent modes with opposite in-plane wave vectors decreases monotonically with
h
increasingdistancefromtheatomtotheinterface. Duetotheoscillatorybehavioroftheinterference
p
between the emitted and reflected fields, the difference between the rates of spontaneous emission
-
t intoradiationmodeswithoppositein-planewavevectorsoscillateswithincreasingdistancefromthe
n
atom to theinterface. This differencecan be positive or negative, dependingon theatom-interface
a
distance, and is zero for the zero distance.
u
q
[
I. INTRODUCTION of the atom is a real vector oriented along a given direc-

2 tion is space. In this condition, the rate of spontaneous
v
emissionintoevanescentmodesisalwayssymmetricwith
7 The study of individual neutral atoms in the vicini-
respect to central inversion in the plane of the interface.
4 ties of material surfaces has a long history [1–5] and has
5 attracted a lot of interest over decades [6–23]. The pos- Ina realisticquantumemitter, the dipole canbe ellip-
1 sibility to control and manipulate individual atoms near ticallypolarized,thatis,thedipolematrixelementvector
0
surfaces can find applications for quantum information canbeacomplexvector. Forexample,inanalkaliatom,
.
1 [24–26] andatom chips [27, 28]. Cold atoms canbe used thedipolematrixelementvectordM′M forthetransition
0 as a probe that is very sensitive to surface-induced per- between the Zeeman levels with the magnetic quantum
6 turbations[29]. Manyapplicationsrequireadeepunder- numbers M′ and M is a real vector, aligned along the
1
standingandaneffectivecontrolofspontaneousemission quantizationaxisz,fortheπtransitions,whereM′ =M,
:
v of atoms near to material objects. butisacomplexvector,lyinginthexy plane,fortheσ±
i transitions,where M′ =M 1. When the dipole matrix
X It is well known that the spontaneous emission rate ±
element vector is a circularly polarized complex vector,
r of an atom is modified by the presence of an interface
a thedipoleoftheemitterisnotalignedalongafixeddirec-
[8–19]. Such a modification has been demonstrated ex-
tion but rotates with time in space. It has recently been
perimentally [8]. A semiclassical approach to the prob-
shownthatspontaneousemissionandscatteringfroman
lemofsurface-modifiedradiativepropertieshasbeenpre-
atomwithacirculardipoleinfrontofananofibercanbe
sented [9]. A quantum-mechanical linear-response for-
asymmetric with respect to the opposite axial propaga-
malism has been developed for anatom close to anarbi-
tion directions [30–35]. These directional effects are the
traryinterface[10–12]. Analternativeapproachbasedon
signatures of spin-orbit coupling of light [36, 37]. They
mode expansion has been used for an atom near a per-
are due to the existence of a nonzero longitudinal com-
fect conductor [13]. The Green function approach has
ponent that is in phase quadrature with respect to the
been applied to a multilayered dielectric [14]. A quan-
radialtransversecomponentofthenanofiberguidedfield.
tum treatment for the internal dynamics of a multilevel
Thepossibilityofdirectionalemissionfromanatominto
atomnearamultilayereddielectricmediumhasbeenper-
propagatingradiationmodes ofa nanofiberand the pos-
formed[15]. Spontaneousradiativedecayoftranslational
sibility of generation of a lateral force on the atom have
levels of an atom in front of a semi-infinite dielectric has
been pointed out [34].
been studied [16]. In the previous treatments [9–18], it
was assumed that the induced dipole of the atom is lin- Spontaneousemissionofanatomissimilartotheemis-
earlypolarized,thatis, the dipole matrixelement vector sion of a dipole-like particle. Spontaneous emission of
2
a two-level atom and radiation of a classical oscillating field interaction. In Sec. III we calculate the rates
dipolehaveidenticalradiationpatterns,identicalrateen- of spontaneous emission into evanescent and radiation
hancement factors, and very similar decay rates [18]. A modes, and study the angular densities of the rates in
radiating dipole can, in general, oscillate in all three di- the space of wave vectors. In Sec. IV we present the re-
mensionswithrelativephases. Recently,emissionofpar- sultsofnumericalcalculations. Ourconclusionsaregiven
ticles with circularly polarized dipoles began to attract in Sec. V.
muchattention[38–46]. Ithasbeenshownthatthenear-
field interference of a circularly polarized dipole coupled
to a dielectric or metal leads to unidirectional excitation II. MODEL DESCRIPTION
ofguidedmodesorsurfaceplasmonpolaritonmodes[38–
44]. Thiseffecthasbeenexperimentallydemonstratedby We consider a space with one interface [see Fig. 1(a)].
shining circularly polarized light onto a nanoslit [38, 39] WeuseaCartesiancoordinatesystem x,y,z . Thehalf-
orcloselyspacedsubwavelengthapertures[40]inametal spacex<0 isoccupiedby anondisper{siveno}nabsorbing
film and by exciting a nanoparticle on a dielectric inter- dielectric medium (medium 1). The half-space x > 0 is
facewithatightlyfocusedvectorlightbeam[43,44]. The occupied by vacuum (medium 2). We examine an atom,
generationoflateralforcesbyspin-orbitcouplingoflight with an upper energy level e and a lower energy level g,
scattered off a particle at an interface between two di- located at a fixed point on the x axis in the half-space
electric media has been demonstrated [45, 46]. In order x>0. The energies of the levels e and g are denoted by
to enhance the selective coupling of light to plasmonic ¯hω and ¯hω , respectively.
e g
and dielectric waveguides on the nanoscale, a variety of WeusetheformalismofRef. [55]todescribethequan-
complex nanoantenna designs have been proposed and tum radiation field in the space with one interface. We
experimentally demonstrated [47–54]. Despite recent in- label the modes of the field by the index α = (ωKqj),
terestinspin-orbitcouplingoflightscatteredoffparticles where ω is the mode frequency, K = (0,K ,K ) is the
y z
[39, 41–46], a systematic study of the radiation pattern projection of the wave vector onto the dielectric surface
of a circularly polarized dipole in front of an interface is yz plane, q = s,p is the mode polarization index, and
absent. We note thatthe theoryofRef. [18]is validonly j =1,2 stands for the medium ofthe input of the mode.
forlinearlypolarizeddipoles andmustbe modifiedto be Foreachmodeα=(ωKqj),theconditionK kn must
j
used for circularly polarized dipoles [43, 44]. be satisfied. Here, k = ω/c is the wave num≤ber in free
Spontaneous emission of a two-level atom and radia- space,n >1istherefractiveindexofthedielectric,and
1
tion of a classical oscillating dipole are similar but dif- n =1 is the refractiveindex of the vacuum. We neglect
2
ferent phenomena. A two-level atom is a quantum sys- the dependence of the dielectric refractive index n on
1
tem. The dipole moment of the atom is coupled to the the frequency and the wave number.
field parametrically. Meanwhile, the dipole moment of a The mode functions are given, for x<0, by [55]
classicaloscillatingdipole iscoupleddirectlytothe field.
A quantum atom does not obey the classical equations UωKs1(x)= eiβ1x+e−iβ1xr1s2 s,
of motion when the atomic state is far from the ground UωKp1(x)=e(cid:0)iβ1xp1++e−iβ1xr(cid:1)1p2p1−,
state. Theinitialconditionsforspontaneousemissionare
that the field is in the vacuum state and the atom is in UωKs2(x)=e−iβ1xts21s,
the excited state. The spontaneous emission is initiated UωKp2(x)=e−iβ1xtp21p1−, (1)
by the vacuum field fluctuations. The expression for the
and, for x>0, by
damping rate of a classical oscillating dipole is different
fromthatforthespontaneousemissionrateofatwo-level UωKs1(x)=eiβ2xts12s,
atom. Inordertogetafullunderstandingofspontaneous
emission, the quantum model must be used. UωKp1(x)=eiβ2xtp12p2+,
Inviewoftherecentresultsandinsights,itisnecessary UωKs2(x)= e−iβ2x+eiβ2xr2s1 s,
to develop a systematic theory for spontaneous emission UωKp2(x)=e(cid:0)−iβ2xp2−+eiβ2xr(cid:1)2p1p2+. (2)
of a two-level atom with an arbitrarily polarized dipole
in front of a flat dielectric surface. We construct such In Eqs. (1) and (2), the quantity β = (k2n2 K2)1/2,
i i −
a theory in the present paper. We calculate the rates with Reβ 0 and Imβ 0, is the magnitude of
i i
≥ ≥
of spontaneous emission into evanescent and radiation the x component of the wave vector in medium i =
modes, and study the angular densities of the rates in 1,2. The quantities risi′ = (βi − βi′)/(βi + βi′) and
the space of wave vectors for the field modes. We focus tsii′ = 2βi/(βi + βi′) are the reflection and transmis-
on the case where the ellipticity of the atomic dipole is sion Fresnel coefficients for a TE mode, while the quan-
nroottatzeesrow,itthhattimise, tinhespcaascee.where the dipole of the atom t2intiiensi′βripii/′(β=in(2i′β+in2iβ′i′−n2iβ)i′anr2ie)/th(βeirne2ifl′e+ctβioin′na2i)ndantrdantspiim′ i=s-
The paper is organized as follows. In Sec. II we de- sion Fresnel coefficients for a TM mode. The vector
scribe the model and present the expressions for the s=[Kˆ xˆ]isthepolarizationvectorfortheelectricfield
×
modes of the field and for the Hamiltonian of the atom- in a TE mode, while the vectors p =(Kxˆ β Kˆ)/kn
i+ i i
−
3
(a) complete and orthogonalbasis for the field.
Note that a light beam propagating from the dielec-
medium 2: vacuum tric to the interface may be totally reflected because
n > n = 1. This phenomenon occurs for the modes
1 2
z interface α = (ωKqj) with j = 1 and k < K kn1. For such
≤
a mode, the magnitude of the x component of the wave
K
vector in medium 2 is β = i√K2 k2, an imaginary
2
y number. This mode does not propa−gate in the x direc-
tioninthe vacuumside ofthe interfacebutdecaysexpo-
φ atom
x nentially. Such a mode is an evanescent mode. We note
that, in the case of the p evanescent mode, that is the
mode α = (ωKp1) with k < K kn , the vector p
1 2+
medium 1: dielectric ≤
for the polarizationof the field in the half-space x>0 is
a complex vector. The modes with 0 K k are called
≤ ≤
radiation modes. For convenience, we use the indices µ
and ν to label the evanescent and radiation modes, re-
spectively,thatis,weusethenotationsµ=(ωKq1)with
(b) Single-input modes
k <K kn and ν =(ωKqj) with 0 K k.
1
≤ ≤ ≤
evan q1 rad q1 rad q2 The total quantized electric field is given by [55]
K > k K < k K < k
k ¯h
E(r,t)=i eiK·RU (x)a e−iωt+H.c.,
α α
4πsπǫ0βj
α
X
(3)
where a is the photon annihilation operator for the
α
mode α, R = (0,y,z) is the projection of the posi-
tion vector r = (x,y,z) onto the interface plane, and
= ∞dω knj KdK 2πdφ is the generalized
(c) Single-output modes α qj 0 0 0
summationoverthe modes. Here, φ is the azimuthalan-
evan q1 rad q1 rad q2 gPle of thPe veRctor KRwith respeRct to the y axis in the yz
K > k K < k K < k plane. Thecommutationruleforthephotonoperatorsis
[aα,a†α′]=δ(ω−ω′)δ(Ky−Ky′)δ(Kz−Kz′)δqq′δjj′. When
dispersionintheregionaroundthefrequenciesofinterest
is negligible, the mode functions U satisfy the relation
α
−∞∞dxn2(x)U∗ωKqj(x)·Uω′Kq′j′(x)=2πc2(βj/ω)δ(ω−
Rω′)δqq′δjj′. Here, n(x) = n1 for x < 0, and n(x) = n2
for x > 0. Hence, we can show that the energy of the
field is ǫ drn2(x)E(r)2 = ¯hω(a†a + a a†)/2.
Here, d0r = ∞ dx| ∞|dy ∞ αdz is tαheαintegrαalαover
FIG. 1. (Color online) (a) Atom in front of the flat surface R −∞ −∞ −P∞
of a semi-infinite dielectric medium. The half-space x< 0 is the whole space.
R R R R
occupied by a dielectric (medium 1). The half-space x>0 is We now present the Hamiltonian for the atom–field
occupiedbyvacuum(medium2). Theatomliesonthexaxis interaction. In the dipole and rotating-wave approxima-
inthehalf-space x>0. Theaxesy andz lie intheinterface. tionsandintheinteractionpicture,theatom–fieldinter-
Thein-planewavevectorKliesintheinterfaceyz plane. (b) action Hamiltonian is
Representation of single-input modes. (c) Representation of
single-output modes. In (b) and (c), the input and output H = i¯h G σ†a e−i(ω−ω0)t+H.c., (4)
int α α
parts of the modes are shown by the solid red and dashed −
α
bluearrows, respectively. X
where σ† = e g describes the atomic transition from
| ih |
the lowerlevelg tothe upper levele, ω =ω ω is the
0 e g
−
angular frequency of the transition, and
andp =(Kxˆ+β Kˆ)/kn arerespectivelythepolariza-
i− i i
tionvectorsforthe right-andleft-movingcomponentsof k
the electric field in a TM mode in medium i. Here, the Gα = 4π πǫ ¯hβ eiK·R(UωKqj ·deg) (5)
notation Vˆ = V/V stands for the unit vector of an ar- 0 j
bitraryvectorV, with V V = V 2+ V 2+ V 2 is the coefficient opf coupling between the atom and the
x y z
≡| | | | | | | |
beingthelengthofthevectorV. ItisclearfromEqs.(1) mode α = (ωKqj). In expression (5), d = eDg is
eg
p h | | i
and (2) that each mode α = (ωKqj) has a single input the matrix element of the dipole moment operator D of
in medium j [see Fig. 1(b)]. The set of the modes α is a the atom. In general, d can be a complex vector.
eg
4
The time reverse of the mode α = (ωKqj) is also Inserting Eq. (8) into Eq. (6) yields the Heisenberg–
a mode of the field. We introduce the label α˜ = Langevin equation
(ω, K,q,˜j) for the time reverse of the mode α =
γ
(ωK−qj). The mode function of the mode α˜ is given by ˙ = ([σ†, ]σ+σ†[ ,σ])+ξO. (9)
O 2 O O
U =U∗. Itisclearthatthemodeα˜ hasasingleoutput
α˜ α
coming from the interface into medium j [see Fig. 1(c)]. Here,
Like the set of the modes α, the set of the modes α˜ is
a complete and orthogonal basis for the field. We can γ =2π Gα 2δ(ω ω0) (10)
| | −
use the basis formed by the modes α˜ instead of the ba- α
X
sis formed by the modes α. We note that an evanescent
is the rate of spontaneous emission and ξ is the noise
modeα=(ωKqj)withj =1andk <K kn hasasin- O
≤ 1 operator. We emphasize that Eq. (9) can be applied to
gle input and a single output in the dielectric. Thus, we
anyatomicoperators. Duetothepresenceofthefunction
have(ωKqj)=(ωKq˜j)whenj =1andk <K kn . In
other words, there is no difference between sin≤gle-i1nput δ(ω−ω0), all the parameters needed for the calculation
of the decay rate are to be estimated at the frequency
evanescent modes and single-output evanescent modes
ω =ω . We will adopt this convention in what follows.
[see the left panels of Figs. 1(b) and 1(c)]. 0
Inthehalf-spacex>0,wheretheatomisrestrictedto,
the rate of spontaneous emission γ can be decomposed
as
III. SPONTANEOUS EMISSION RATE
γ =γ +γ , (11)
evan rad
We use the mode expansion approach and the
where
Weisskopf–Wigner formalism [56] to derive the micro-
scopicdynamicalequationsforspontaneousradiativede- k0n1 2π
caarbyitorfatrhyeaattoommic. Wopeerfiartsotrstu.dyTthheeHtiemiseenebvoelrugtieoqnuaotfioann γevan =2π KdK |Gω0Kq1|2dφ (12)
q=s,p Z Z
for this operator is O X k0 0
istherateofspontaneousemissionintoevanescentmodes
˙ = (G [σ†, ]a e−i(ω−ω0)t+G∗a†[ ,σ]ei(ω−ω0)t). and
O α O α α α O
α
X (6) k0 2π
Meanwhile, the Heisenberg equation for the photon an- γrad =2π KdK |Gω0Kqj|2dφ (13)
nihilation operatoraα is a˙α =G∗ασei(ω−ω0)t. Integrating qX=s,pjX=1,2Z0 Z0
this equation, we find
istherateofspontaneousemissionintoradiationmodes.
t In the particular case where the atom is in free space,
aα(t)=aα(t0)+G∗α dt′σ(t′)ei(ω−ω0)t′. (7) that is, where n1 = n2 = 1, we have γevan = 0 and
Zt0 γ =γrad =γ0. Here,
Here, t0 is the initial time. For convenience, we take ω3d2
t0 =0. γ0 = 3π0ǫ ¯hegc3 (14)
0
We considerthe situation wherethe field is initially in
the vacuum state. We assume that the evolution time is the natural linewidth of the two-level atom [56].
t t0 and the characteristic atomic lifetime τ are large Intheremainingpartofthispaper,weanalyzethecon-
−
ascomparedtothecharacteristicopticalperiodT. Since sequences of expressions (10)–(13). We note that these
the continuum of the field modes is broadband, the cor- expressions,apartfromanormalizationconstantequalto
relation time of the field bath is short as compared to γ , can be obtained by using the model of an arbitrarily
0
the atomic lifetime τ. Hence, the Markov approxima- polarized classical oscillating dipole. Consequently, the
tion σ(t′) = σ(t) can be applied to describe the back results of the remaining part of this paper can be used
action of the second term in Eq. (7) on the atom [56]. not only for spontaneous emission of a two-level atom
Under the condition t t0 T, we calculate the inte- with an arbitrarily polarized dipole but also for the rate
− ≫
gral with respect to t′ in the limit t t0 . We set enhancementfactorandtheradiationpatternofanarbi-
− → ∞
aside the imaginary part of the integral, which describes trarily polarized classical oscillating dipole. We empha-
the frequency shift. Such a shift is usually small. We size that expression (14) cannot be derived by using the
can effectively account for it by incorporatingit into the classicalformalism. Inaddition,Eq.(9)standsforatwo-
atomic frequency and the surface–atom potential. With level atom but not for a classicaloscillating dipole. This
the above approximations, we obtain equationdescribesnotonlythe decayofthe atomic level
population inversion but also the decay of the atomic
a (t)=a (t )+πG∗σ(t)δ(ω ω ). (8) coherence.
α α 0 α − 0
5
A. Spontaneous emission into evanescent modes 2n η/(η + in2ξ), and η n2 κ2 = n2 1 ξ2.
1 1 ≡ 1− 1− −
The explicit expressions for T and T in terms of ξ are
s p
p p
The rate of spontaneous emission from the atom at a given as
position x>0 into evanescent modes is
2ξ n2 1 ξ2
T = 1− − ,
γevan =γesvan+γepvan, (15) s pn21−1 (21)
where the notation T = 2n21 ξ n21−1−ξ2.
p n2 1 (n2+1)ξ2+1
k0n1 2π 1− p1
γeqvan =2π KdK |Gω0Kq1|2dφ (16) In the half-space x>0, the wave vector of an evanes-
kZ0 Z0 cent mode is (β2,Ky,Kz), where β2 =ik0ξ. The param-
eters ξ and κ = 1+ξ2 and the angle φ characterize
with q =s,p stands for the rate ofspontaneous emission the components ofthe complex wavevector(β ,K ,K )
2 y z
into the q-type evanescent modes. of an evanescentpmode in the half-space x > 0 via
Weintroducethenotationκ=K/k0,wherek0 =ω0/c, the relations β2/k0 = iξ, Ky/k0 = κy = κcosφ, and
for the normalized magnitude of the in-plane compo- K /k =κ =κsinφ.
z 0 z
nent K of the wave vector. In addition, we introduce The functions Fs and Fp are respectively the an-
evan evan
the notation ξ = 1 κ2 for the normalized mag- gulardensitiesofthespontaneousemissionratesintothe
| − |
nitude of the out-opf-plane component β2xˆ of the wave TE evanescent modes µ=(ω0Ks1) and the TM evanes-
vector in the half-space x > 0. In the case of evanes- cent modes µ = (ω Kp1), with k < K k n , in the
0 0 0 1
cent modes, we have β2 = ik0ξ, 1 ≤ κ ≤ n1, and wavevectorspace. ThefunctionFevan isth≤eangularden-
0≤ξ =√κ2−1≤ n21−1. Inthiscase,theparameter sity of the spontaneous emission rate into both s and p
ξ determines the penetration length Λ = 1/k0ξ of the types of evanescent modes. In the limit κ 1, that is,
evanescent mode inpthe half-space x>0. We change the K k , we have →
0
integration variable of the first integral in Eq. (16) from →
3 1
K to ξ. Then, we obtain limF = [n2 u 2+ u 2sin2φ
κ→1 evan 2π n2 1 1| x| | y|
√n21−1 2π +|puz|21c−os2φ−Re(u∗yuz)sin2φ]. (22)
γ =γ ξdξ F (ξ,φ)dφ,
evan 0 evan
In the limit κ κ =n , that is, K K =k n ,
Z0 Z0 → max 1 → max 0 1
(17) theratedensityF forevanescentmodestendstozero,
evan
√n21−1 2π that is, we have limκ→κmaxFevan =0.
γq =γ ξdξ Fq (ξ,φ)dφ, In the half-space x<0, the wave vector of an evanes-
evan 0 evan
cent mode is (β ,K ,K ), where β = k η. Let θ
Z Z 1 y z 1 0
0 0
be the angle between the axis x and the wave vec-
where tor (β1,Ky,Kz) of the evanescent mode in the dielec-
tric medium. This angle is determined by the formu-
Fevan =Fesvan+Fepvan, (18) las n1sinθ = κ = 1+ξ2 and n1cosθ = η for
−
θ [π/2,π arcsin(1/n )]. We find F (ξ,φ)ξdξdφ =
with ∈ − p1 evan
P (θ,φ)sinθdθdφ, where
evan
3 −
Fs = T e−2ξk0x[u 2sin2φ+ u 2cos2φ
evan 4πξ s | y| | z| Pevan =n1ηFevan =−n21cosθFevan (23)
Re(u∗u )sin2φ] (19)
− y z is the angular distribution of spontaneous emission into
and evanescent modes with respect to the spherical angles θ
andφ. TheexplicitexpressionforP canbeeasilyob-
3 evan
Fp = T e−2ξk0x[u 2(1+ξ2)+ u 2ξ2cos2φ tained by substituting Eq. (18) together with Eqs. (19)
evan 4πξ p | x| | y|
and (20) into Eq. (23). In the particular case where the
+|uz|2ξ2sin2φ+Re(u∗yuz)ξ2sin2φ dipole polarization vector u is real, this expression re-
+2ξ 1+ξ2Im(u∗u cosφ+u∗u sinφ)]. (20) duces to the result for the far-field limit of the radiation
x y x z pattern in the forbidden zone of the dielectric [18].
Here,u ,u ,apndu aretheCartesian-coordinatecompo- As already pointed out in the previous section, an
x y z
nents of the unit vector u=d /d for the polarization evanescent mode µ = (ω Kq1) with k < K k n
eg eg 0 0 0 1
≤
of the dipole matrix element d . In Eqs. (19) and (20), has a single input and a single output in the dielectric.
eg
wehaveintroducedtheparametersT (ξ/2η)ts 2 and Consequently,there is nodifference betweensingle-input
T (ξ/2η)tp 2, which are proporsti≡onal to t|h1e2|trans- evanescent modes and single-output evanescent modes.
p ≡ | 12|
mittivity of light coming from medium 1 to medium 2. The propagation direction of the evanescent mode in
Here,wehaveusedthenotationsts =2η/(η+iξ),tp = the interface plane yz is characterized by the vector
12 12
6
K = (0,K ,K ). The transformation K K is We can easily show that
y z
→ −
done by the transformation φ φ + π. We observe
→ 3
tthhaetcaolelffithceientetrsmssini2nφe,xcporse2ssφi,onan(1d9)sinar2eφa,swsohciicahteddownitoht ∆Fevan = 8π n2 1 ξ2[u∗×u]·[U∗ω0Kp1×Uω0Kp1].
1− −
vary with respect to the transformation φ φ + π. (25)
Thus, the rate density Fs has the same val→ue for the We note thatpthe vector i[u∗ u] is the ellipticity vec-
evan ×
s evanescentmodes with the opposite in-plane wavevec- tor of the atomic dipole polarization. Meanwhile, the
tors K and −K. Meanwhile, the terms in the last line vector −i[U∗ω0Kp1 × Uω0Kp1] is proportional to the el-
ofexpression(20)containthecoefficientscosφandsinφ, lipticity vector of the local electric polarization of the
which change their sign when we replace φ by φ + π. TM evanescent mode µ = (ω Kp1) with K > k at the
0 0
This means that the rate density Fp may take differ- position of the atom. Equation (25) indicates that the
evan
ent values for the p evanescent modes with the opposite difference∆F isaresultoftheoverlapbetweentheel-
evan
in-plane wave vectors K and K. This asymmetry in lipticity vector of the atomic dipole polarizationand the
spontaneous emission occurs w−hen either Im(u∗u ) or ellipticity vector of the local electric polarization of the
x y
Im(u∗u )isnotzero,thatis,whentheatomicdipolepo- TM evanescent mode µ = (ω Kp1) with K > k . The
x z 0 0
larizationvectoruisacomplexvectorandhasanonzero electric part of the other evanescent mode, that is, the
projection onto the axis x. The fact that u is a com- TEmodeµ=(ω Ks1)withK >k ,islinearlypolarized
0 0
plex vector means that the direction of the mean dipole in the half-space x > 0. This mode does not contribute
D(t) = d uσ†eiω0t+u∗σe−iω0t of the atom rotates to ∆F .
eg evan
h i h i
withtimeinspace. Theasymmetryofspontaneousemis- Consider a light field with the electric component
sioninto evanescentmodes with respectto centralinver- E = (Ee−iωt+c.c.)/2, where E = ǫ is the envelope of
E
sionintheinterfaceplaneisaconsequenceoftheinterfer- the positive-frequency component, with being the am-
ence between the emission from the out-of-plane dipole plitude and ǫ being the polarization vecEtor. It is known
component u and the emission fromthe in-plane dipole that the local electric spin density S of the light field is
x
components u and u where u has a phase lag with relatedtothe ellipticity vector i[ǫ∗ ǫ]=Im[ǫ∗ ǫ]of
y z x
− × ×
respectto u or u . When the dipole polarizationvector the local electric polarization via the formula
y z
u is a real vector, the rate density F for evanescent
evan ǫ ǫ
modes is symmetric with respect to central inversion in S= 0 Im E∗ E = 0 2Im ǫ∗ ǫ . (26)
4ω × 4ω|E| ×
the interfaceplane. Itisinterestingtonote that,accord-
(cid:2) (cid:3) (cid:2) (cid:3)
ingtoEq.(22),inthelimitκ 1,theratedensityFevan It follows from Eqs. (25) and (26) that
→
is symmetric with respect to central inversion in the in-
tue.rface plane for an arbitrary dipole polarization vector ∆Fevan = 2πǫ n32ω0 1 ξ2i[u∗×u]·Sω0Kp1, (27)
0 1− −
It is clear from Eqs. (19) and (20) that the differ-
ence ∆Fevan Fevan(ξ,φ) Fevan(ξ,φ+π) between the whereSω0Kp1 isthpelocalelectricspindensityofthefield
rate densities≡of spontaneo−us emission into the evanes- in the TM evanescent mode µ = (ω0Kp1) with the in-
cent modes with the opposite in-plane wave vectors K plane wave number K > k0 and the positive-frequency-
and K is componentenvelopeEω0Kp1 =Uω0Kp1. Inthehalf-space
− x>0,wheretheatomislocated,theelectricpolarization
3 vector of the TM evanescent mode µ = (ωKp1) with
∆F = 1+ξ2T e−2ξk0x
evan π p (24) K >k is
pIm(u∗u cosφ+u∗u sinφ).
× x y x z κxˆ iξKˆ
ǫωKp1 = − . (28)
κ2+ξ2
We note thatthe sign(plus orminus)ofthe ratedensity
difference ∆Fevan for evanescent modes depends on the The ellipticity vector of thepelectric polarization of the
dipole polarization vector u and the azimuthal angle φ
field is found to be
of the in-plane wave vector K in the yz plane. However,
dthisetasingcneoxf∆anFdevtahnedeoveasnneosctednetp-menoddoenptehneetartaotmio-ninptaerrfaamce- Im ǫ∗ωKp1×ǫωKp1 =2ξ1+1+2ξ2ξ2[Kˆ ×xˆ], (29)
eter ξ. When the dipole polarization vector u is a real p
(cid:2) (cid:3)
vector, the rate density difference for evanescent modes which leads to the local electric spin density
with opposite in-plane wave vectors is ∆F =0.
evan
ǫ 2n2 n2 1 ξ2
unTdehrecaesnytmraml eitnrvyerdsieognreienotfhteheintaenrgfauclaerpdlaennesitiys Fchevaarn- SωKp1 = ω0n2 11(n21+−1)ξ−2+1ξ 1+ξ2e−2ξkx[Kˆ×xˆ].
1− 1
acterized by the factor ζ = ∆F /Fsum, where p (30)
Fevan evan evan
Fsum F (ξ,φ)+F (ξ,φ+π). It is clear that the We note that [Kˆ xˆ]=yˆsinφ ˆzcosφ.
evan ≡ evan evan × −
asymmetry factor ζ depends on ξ and φ. However, It follows from Eqs. (28) and (29) that the ellipticity
Fevan
ζ does not depend on the distance x. of the local electric polarization of the TM evanescent
Fevan
7
mode µ = (ωKp1) with K > k arises as a consequence element vector d as well as the orientations and circu-
eg
of the fact that field in the TM evanescent mode has a lations of the field mode profile vector e(α). This com-
longitudinalcomponentthatisalignedalongthein-plane ponent is the spontaneous emission rate averaged over
wave vector K. The phase of this component is shifted the orientation of the dipole matrix element vector d
eg
by π/2 from the phase of the transversecomponent that in space.
is aligned along the axis x. According to Eq. (33b), the vector component γ(1) of
α
Equation (30) shows that the local electric spin den- thespontaneousemissionratedependsontheoverlapbe-
tsoitryKSˆωKofp1thiseainv-epcltaonrethwaatvedevpeecntdors oKn.thIendpiraerctticiounlavr,eca- tawreeepnrothpeorvteioctnoarlstio[dt∗ehge×edlleigp]taicnitdy−vie[ect(αo)r∗o×fet(hαe)],awtohmicihc
reverse of Kˆ leads to a reverse of the electric spin den- electric dipole polarization and the ellipticity vector of
sity vector SωKp1. This is a signature of the so-called the electric field polarization, respectively. The vector
spin-orbit interaction of light [36, 37]. Thus, the differ- i[d∗ d ] characterizes an effective magnetic dipole
eg × eg
ence between the rates of spontaneous emission into the produced by the rotation of the electric dipole, and is
evanescentmodeswiththeoppositein-planepropagation responsibleforthevectorpolarizabilityoftheatom. The
directions K and K is a consequence of spin-orbitcou- vector i[e(α)∗ e(α)]characterizesaneffectivemagnetic
− − ×
pling of light. field and is responsible for the local electric spin density
We observe from Eqs. (27) and (30) that the local of light. The vector component γ(1) of the rate can be
α
electric spin density Sω0Kp1 of the TM evanescent mode considered as a result of the interaction between the ef-
µ = (ω0Kp1) with K > k0 and, consequently, the rate fective magnetic dipole and the effective magnetic field.
difference∆Fevan forevanescentmodeswithoppositein- Due to spin-orbit coupling of light [36, 37], a reverse of
plane propagation directions reduce exponentially with the propagation direction leads to a reverse of the spin
increasingdistancexfromthe atomtothedielectricsur- density of light and, consequently, to a reverse of the
face. For x = 0, the magnitudes of Sω0Kp1 and ∆Fevan vector component γ(1) of the spontaneous emission rate.
α
achieve their maximum values, which depend on ξ. In
According to Eq. (33c), the tensor component γ(2)
the limit ξ →0,thatis, κ→1,we haveSω0Kp1 =0 and, of the spontaneous emission rate depends on the scalαar
hence, ∆F =0.
evan product of the irreducible tensors d∗ d and
Inordertogetdeepinsightintotheunderlyingphysics { eg ⊗ eg}2
ofasymmetrybetweenthe ratesofspontaneousemission e(α)∗ e(α) 2 for the atomic dipole and the field mode
{profile,⊗respec}tively. Thetensor d∗ d isresponsi-
intooppositein-planepropagationdirections,weperform { eg⊗ eg}2
the following general tensor analysis: It is clear that the ble for the tensor polarizability of the atom. In general,
rate γα of spontaneous emission into a mode α with the {e(α)∗⊗e(α)}2 and, hence γα(2) depend on the azimuthal
modeprofilefunctione(α) isproportionaltothequantity angle φ of the in-plane wave vector K in the yz plane.
d e(α) 2, that is, We canshowthat,for the evanescentmodes,inthe half-
eg
| · | space x > 0, the tensor e(α)∗ e(α) and, hence, the
2
{ ⊗ }
γ = d e(α) 2, (31) tensorcomponentγ(2) oftherateγ donotchangewhen
α α eg α α
N | · |
we reverse the direction of the in-plane wave vector K.
where α is a parameter that does not depend on the We now calculate the rates of spontaneous emission
relativeNorientationbetweendeg ande(α). Itfollowsfrom into evanescent modes propagating into separate sides
Eq. (A8) of Appendix that we can decompose the rate of a plane containing the axis x, on which the atom is
γα as located. Without loss of generality, we choose the plane
xy. The rates γ(+) and γ(−) of spontaneous emission
γ =γ(0)+γ(1)+γ(2), (32) evan evan
α α α α into evanescent modes propagating into the +z and z
−
sides, respectively, are given by
where
γ(0) = Nα d 2 e(α) 2, (33a) √n21−1 π
α 3 | eg| | | γ(+) =γ ξdξ F (ξ,φ)dφ,
evan 0 evan
γ(1) = Nα[d∗ d ] [e(α)∗ e(α)], (33b) Z0 Z0
α 2 eg × eg · × (34)
γ(2) = d∗ d e(α)∗ e(α) . (33c) √n21−1 2π
α Nα{ eg⊗ eg}2·{ ⊗ }2 γ(−) =γ ξdξ F (ξ,φ)dφ.
evan 0 evan
InEq.(33c),thenotation A∗ A standsforthetensor Z Z
2 0 π
productofrank2oftheco{mpl⊗exv}ectorsA∗ andA. The
quantitiesγ(0),γ(1),andγ(2)arecalledthescalar,vector, We find
α α α
anAdctceonrsdoirngcotmopEoqn.en(3ts3ao)f,tthheersactaelaγrα,coremsppeocnteivnetlyγ.α(0) of γe(v+a)n = γe2van + ∆e2van,
(35)
the spontaneous emission rate does not depend on the γ ∆
orientationsandcirculationsofthe atomic dipole matrix γe(v−a)n = e2van − e2van,
8
where [10, 11, 18]
√n21−1 √n21−1
3 3
γ = γ (1 u 2)T (ξ) γ =γk = γ T (ξ)e−2ξk0xdξ. (39)
evan 4 0 −| x| s (36) evan evan 4 0 k
Z Z
0 (cid:8) 0
+[u 2(2+ξ2)+ξ2]T (ξ) e−2ξk0xdξ
| x| p Here,wehaveintroducedtheparametersT⊥ =(1+ξ2)Tp
istherateofspontaneousemissioninto(cid:9)evanescentmodes and Tk =Ts+ξ2Tp, whose explicit expressions are
in all directions [10, 11, 18] and 2n2 n2 1 ξ2
T = 1 1− − ξ(1+ξ2),
⊥ n2 1 (n2+1)ξ2+1
∆evan = π6γ0Im(u∗xuz) √Zn21−1ξ 1+ξ2Tp(ξ)e−2ξk0xdξ Tk = n1212−−1(cid:18)p11+ (n21+n121ξ)ξ22+1(cid:19)ξqn21−1−ξ2. (40)
0 p
(37) In both cases, we have ∆evan =0.
is the difference between the rate components γ(+) and
evan
γ(−) fortheoppositesides+z and z,respectively. Itis
evan − B. Spontaneous emission into radiation modes
clearfromEq.(37)thattheratedifference∆ depends
evan
onthe imaginarypartofthe crosstermu∗u , thatis, on
x z The rate of spontaneous emission from the atom at a
theellipticityofthepolarizationoftheatomicdipolevec-
position x>0 into radiation modes is
tor in the xz plane. Meanwhile, Eq. (36) shows that the
rate γ for all evanescent modes does not depend on
evan γ = γqj , (41)
the ellipticity of the dipole polarization. We note that rad rad
the sign (plus or minus) of the rate difference ∆evan for qX=s,pjX=1,2
evanescentmodes is determinedby the signof Im(u∗u )
x z where the notation
and, hence, does not depend on the distance x. In the
limit x , we have γ 0 and ∆ 0. When k0 2π
evan evan
the dipo→le∞polarizationvector→u is a real vect→or,the rate γrqajd =2π KdK |Gω0Kqj|2dφ (42)
difference for evanescent modes propagatinginto the op- Z Z
0 0
posite sides +z and z is ∆ =0.
evan
−
The asymmetry between the rates γ(+) and γ(−) for with q = s,p and j = 1,2 stands for the rate of sponta-
evan evan
neous emission into the qj-type radiation modes.
the+z and z sides,respectively,ischaracterizedbythe
− We again use the notation κ = K/k for the normal-
factor ζ =∆ /γ . It is interesting to note that, 0
evan evan evan
izedmagnitudeofthein-planecomponentKofthewave
unlike the asymmetry factor ζ for the angular rate
Fevan vector and the notation ξ = 1 κ2 for the normal-
densities F (ξ,φ) and F (ξ,φ+π), the asymmetry
evan evan | − |
factor ζevan for the side rates γe(v+a)n and γe(v−a)n depends on iwzaedvemvaecgtnoirtuindethoefthhaelf-osupta-ocef-pxpla>ne0.coFmorproandeinattiβon2xˆmoofdtehse,
thedistancex. Thereasonisthat,accordingtoEqs.(37)
we have β =k ξ, 0 κ 1, and 0 ξ =√1 κ2 1.
and(36),thedifference∆evan betweenandthesumγevan Wechange2thei0ntegr≤ation≤variableof≤thefirsti−ntegra≤lin
of the side rates γ(+) and γ(−) are given by different
evan evan Eq. (42) from K to ξ. Then, we obtain
integrals over the variable ξ. The kernels of these inte-
gralsaredifferentfromeachother althoughthey contain 1 2π
a common exponential factor e−2ξk0x. Due to the in- γ =γ ξdξ F (ξ,φ)dφ,
rad 0 rad
tegration over ξ, the x dependence of ∆ is different
evan Z Z
0 0
from that of γ . Consequently, the asymmetry factor (43)
evan
1 2π
ζ = ∆ /γ for the side rates γ(+) and γ(−) is
evan evan evan evan evan γqj =γ ξdξ Fqj (ξ,φ)dφ,
a function of the distance x. rad 0 rad
Z Z
Intheparticularcasewherethe dipolematrixelement 0 0
vector d is perpendicular to the interface, we obtain
eg where
[10, 11, 18]
F = Fqj , (44)
rad rad
√n21−1 q=s,pj=1,2
X X
3
γ =γ⊥ = γ T (ξ)e−2ξk0xdξ, (38)
evan evan 2 0 ⊥ with
Z
0 3
Fs1 = (1 r2) u 2sin2φ+ u 2cos2φ
and, in the particular case where the dipole matrix el- rad 8πξ − s | y| | z|
ement vector d lies in the interface plane yz, we find Re(u∗u )(cid:2)sin2φ , (45)
eg − y z
(cid:3)
9
Frsa2d = 8π3ξ 1+rs2+2rscos(2ξk0x) |uy|2sin2φ FpWe=inFtpr1od+uFcep2th,ewhniocthatairoentsheFrasandgu=laFrrdsa1edn+sitFierssa2dofatnhde
rad rad rad
+ u(cid:2) 2cos2φ Re(u∗u )sin(cid:3)(cid:2)2φ , (46) spontaneous emission rates into the radiation modes of
| z| − y z
the s and p types, respectively. We find
(cid:3)
3 3
Frpa1d = 8πξ (1−rp2)[|ux|2(1−ξ2)+|uy|2ξ2cos2φ Frsad = 4πξ 1+rscos(2ξk0x) |uy|2sin2φ+|uz|2cos2φ
+ u(cid:8) 2ξ2sin2φ+Re(u∗u )ξ2sin2φ] R(cid:2)e(u∗u )sin2φ (cid:3)(cid:2) (51)
| z| y z − y z
−2(1−rp2)ξ 1−ξ2Re(u∗xuycosφ+u∗xuzsinφ) , and (cid:3)
(47)
p (cid:9) 3
Fp = u 2(1 ξ2)+ u 2ξ2cos2φ
and rad 4πξ | x| − | y|
+ u 2ξ2si(cid:2)n2φ+Re(u∗u )ξ2sin2φ
Fp2 = 3 (1+r2)[u 2(1 ξ2)+ u 2ξ2cos2φ | z| y z
rad 8πξ p | x| − | y| + 3 r cos(2ξk x) u 2(1 ξ2) (cid:3) u 2ξ2cos2φ
p 0 x y
+ u(cid:8) 2ξ2sin2φ+Re(u∗u )ξ2sin2φ] 4πξ | | − −| |
| z| y z u 2ξ2(cid:8)sin2φ Re(cid:2)(u∗u )ξ2sin2φ
+2(1 r2)ξ 1 ξ2Re(u∗u cosφ+u∗u sinφ) −| z| − y z
+ 3 −r pcops(2ξ−k x) u 2x(1y ξ2) ux2zξ2cos2(cid:9)φ +2ξ 1−ξ2sin(2ξk0x)Im(u∗xuycos(cid:3)φ+u∗xuzsinφ) .
p 0 x y (52)
4πξ | | − −| | p (cid:9)
−|uz|2ξ2(cid:8)sin2φ−Re(cid:2)(u∗yuz)ξ2sin2φ It is clear that Frad =Frsad+Frpad.
The mode function U , givenby Eqs.(1) and (2), de-
+2ξ 1−ξ2sin(2ξk0x) (cid:3) scribes the mode α = α(ωKqj), which has a single in-
Im(u∗u cosφ+u∗u sinφ) . (48)
× p x y x z put incident from medium j to the interface. The func-
tion U = U∗ describes the mode α˜ = (ω, K,q,˜j),
ηH)e/r(eξ,+weη)haavnedinrtrodurcped=th(enn2ξotatiηo)n(cid:9)/s(nr2sξ≡+rη2s)1 f=or(ξth−e whichhα˜as a sinαgle output comingfrom the inte−rfaceinto
reflection coefficiepn≡ts o2f1light c1om−ing from1 medium 2 to medium j. The density Fq˜j of the rate of spontaneous
rad
medium 1, where η n2 κ2 = n2 1+ξ2. The emission into a single-output mode (ωKq˜j) can be ob-
≡ 1− 1−
explicit expressions for the reflection coefficients r and tained from that for the single-input mode (ω, K,q,j)
p p s −
r are given in terms of ξ as by replacing the dipole polarization vector u with its
p
complex conjugate vector u∗, that is, by applying the
ξ n2 1+ξ2 transformation = (u u∗,φ φ+π) to Fqj . The
r = − 1− , T → → rad
s transformation does not change the rate density func-
ξ+pn21−1+ξ2 (49) tions Fsj [see ETqs. (45) and (46)], Fs [see Eq. (51)],
rad rad
r = n21ξp− n21−1+ξ2. Frpad [see Eq. (52)], and Frad. However, the transforma-
p n2ξ+ n2 1+ξ2 tion reverses the sign of the term in the last line of
1 p 1− Eq.(4T7)andtheterminthethirdlineofEq.(48)forFp1
rad
In the half-space x>0, tphe wave vector of a radiation andFp2,respectively. Thesetermscanceleachotherand
rad
mode is (β2,Ky,Kz), where β2 = k0ξ. The parameters therefore do not appear in the expressions for Frpad and
ξ and κ = 1 ξ2 and the angle φ characterize the F . Thus, the functions Fsj , Fs , Fp , and F de-
− rad rad rad rad rad
componentsofthewavevector(β2,Ky,Kz)ofaradiation scribethe distributionsofthe emissionratesnotonlyfor
p
modeinthehalf-spacex>0viatherelationsβ2/k0 =ξ, single-input modes but also for singe-output modes.
Ky/k0 =κy =κcosφ, and Kz/k0 =κz =κsinφ. We observe that all the terms in expression (51) are
The functions Fsj and Fpj are respectively the an- associatedwith the coefficients sin2φ, cos2φ, andsin2φ,
rad rad
gular densities of the spontaneous emission rates into which do not vary with respect to the transformation
the radiation modes ν = (ω Ksj) and (ω Kpj), with φ φ + π. Hence, the rate density Fs for the TE
0 0 → rad
0 K k ,inthe wavevectorspace. ThefunctionF radiation modes has the same value for the opposite in-
0 rad
≤ ≤
is the angular density of the spontaneous emission rate planepropagationdirectionsKand K. Meanwhile,the
−
into both s and p types of radiation modes. In the limit terms in the last line of expression (52) contain the co-
κ 1, that is, K k , we have efficients cosφ and sinφ, which change their sign when
0
→ → we replace φ by φ+π. Hence, the rate density Fp for
3 1 rad
lim F = [n2 u 2+ u 2sin2φ the TM radiation modes may take different values for
κ→1 rad 2π n2 1 1| x| | y| the opposite in-plane propagationdirectionsK and K.
1− −
+|puz|2cos2φ−Re(u∗yuz)sin2φ]. (50) TthheirsIamsy(mu∗muet)ryorinImsp(oun∗tuan)eiosunsoetmziesrsoio,nthoactcuisr,swwhheenntehie-
x y x z
Comparison between Eqs. (50) and (22) confirms that atomic dipole polarization vector u is a complex vector
lim F =lim F . in a plane containing the axis x. As already mentioned,
κ→1 rad κ→1 evan
10
the fact that u is a complex vector means that the di- where Sω0Kp2 is the local electric spin density of the
rection of the dipole of the atom rotates with time in field in the TM radiation mode ν = (ω Kp2) with
0
space. The asymmetry of spontaneous emission into ra- the positive-frequency-component envelope Eω0Kp2 =
diation modes with respect to central inversion in the Uω0Kp2. In the half-space x > 0, where the atom is
interface plane appears as a consequence of the interfer- located, the electric polarization vector of the TM radi-
ence between the emission from the out-of-plane dipole ation mode ν =(ωKp2) is
component u and the emission fromthe in-plane dipole
x
componentsuy anduz whereux hasa phaselagwithre- ǫωKp2 = 1 [κxˆ+ξKˆ +rpe2iξkx(κxˆ ξKˆ)], (56)
specttouy oruz. Whenthe dipolepolarizationvectoru √Z −
isarealvector,the ratedensityF forradiationmodes
rad
is symmetric with respect to central inversion in the in- whereZ =1+rp2+2rp(1−2ξ2)cos(2ξkx). Theellipticity
terface plane. We note that, according to Eq. (50), in vectoroftheelectricpolarizationofthemodeisfoundto
the limit κ 1, the rate density F is symmetric with be
rad
→
respect to central inversion in the interface plane for an
4
arbitrary dipole polarization vector u. Im ǫ∗ωKp2×ǫωKp2 = Zξ 1−ξ2rpsin(2ξkx)[Kˆ ×xˆ],
It is clear from Eqs. (51) and (52) that the difference
(cid:2) (cid:3) p (57)
∆F F (ξ,φ) F (ξ,φ+π)betweentheratedensi-
rad ≡ rad − rad which leads to the local electric spin density
tiesF (ξ,φ)andF (ξ,φ+π)ofspontaneousemission
rad rad
intotheradiationmodeswiththeoppositein-planewave
ǫ n2ξ n2 1+ξ2
vectors K and K is SωKp2 = 0ξ 1 ξ2 1 − 1− sin(2ξkx)
− ω − n2ξ+ n2 1+ξ2
3 p 1 p 1−
∆Frad = π 1−ξ2rpsin(2ξk0x) (53) [Kˆ xˆ]. p
× ×
pIm(u∗u cosφ+u∗u sinφ). (58)
× x y x z
We note that the sign (plus or minus) of the rate den- We note that [Kˆ xˆ]=yˆsinφ ˆzcosφ.
sitydifference∆F forradiationmodesdependsonnot × −
rad ItfollowsfromEqs.(56)and(57)thattheellipticityof
onlythedipolepolarizationvectoruandtheemissionaz-
the local electric polarization of the TM radiation mode
imuthal angle φ but also on the atom-interface distance ν =(ωKp2) arises as a consequence of the change of the
x and the out-of-plane wave-vector-component parame- polarizationvectorfromκxˆ+ξKˆ toκxˆ ξKˆ duetothere-
ter ξ. When the dipole polarization vector u is a real −
flection, the additional phase 2ξkx of the reflected beam
vector, the rate density difference for radiation modes
duetoaroundtripbetweenthepointxandtheinterface,
with opposite in-plane wave vectors is ∆F =0.
rad and the interference between the incident and reflected
The asymmetrydegreeofthe angulardensity F un-
rad beams. We note that the reflection leads to a change
der central inversion in the interface plane is character-
of the electric polarization vector in the case where the
ized by the factor ζ = ∆F /Fsum, where Fsum
Frad rad rad rad ≡ electriccomponentofthefieldliesintheincidenceplane,
F (ξ,φ)+F (ξ,φ+π). Itisclearthattheasymmetry
rad rad that is, the case of p modes.
factor ζFrad depends on not only ξ and φ but also x. Equation (58) shows that a reverse of Kˆ leads to a
We can easily show that
reverse of the spin density vector SωKp2. This is a sig-
3 nature of spin-orbit coupling of light [36, 37]. The dif-
∆Frad = 8πξ[u∗×u]·[U∗ω0Kp2×Uω0Kp2]. (54) ference between the rates of spontaneous emission into
the radiationmodes with the opposite in-plane propaga-
Asalreadymentioned,the vectori[u∗ u]is theelliptic-
× tion directions K and K is a consequence of spin-orbit
ity vector of the atomic dipole polarization. Meanwhile, −
coupling of light [36, 37], like in the case of evanescent
the vector −i[U∗ω0Kp2 ×Uω0Kp2] is proportional to the modes.
ellipticity vector of the electric polarization and, conse-
We observe from Eqs. (58) and (55) that the local
quently, to the electric spin density vectorof the TM ra-
diation mode ν = (ω Kp2) at the position of the atom. electric spin density Sω0Kp2 of the TM radiation mode
0 ν = (ω Kp2) and, consequently, the rate difference
Equation(54)indicatesthatthedifference∆F isare- 0
rad ∆F for radiation modes with opposite in-plane prop-
sult of the overlap between the ellipticity vector of the rad
agationdirections oscillate as sin(2ξk x) with increasing
atomic dipole polarization and the ellipticity vector of 0
distance x from the atom to the dielectric surface. For
the local electric polarization of the TM radiation mode
ν = (ω Kp2). The electric parts of the other radiation x=0, we have Sω0Kp2 =0 and, hence, ∆Frad =0. This
0 result is in contrast to the result for the case of evanes-
modes, that is, the modes ν = (ω Ks1), (ω Ks2), and
0 0 cent modes, where the magnitudes of the spin density
(ω Kp1),withK k ,arelinearlypolarizedinthe half-
spa0ce x>0. Thes≤e m0odes do not contribute to ∆F . Sω0Kp1 for the TM evanescent mode µ = (ω0Kp1) with
rad K > k and, hence, the rate difference ∆F achieve
Comparison between Eqs. (54) and (26) shows that 0 evan
their maximum values at the interface. The explanation
3ω for the fact that ∆F = 0 at x = 0 is simple. Indeed,
∆Frad = 2πǫ0ξi[u∗×u]·Sω0Kp2, (55) at the interface, therraedlative phase between the incident
0