Table Of ContentSpin and orbital Hall effects for diffracting optical beams in gradient-index media
Konstantin Y. Bliokh1,2 and Anton S. Desyatnikov1
1Nonlinear Physics Centre, Research School of Physics and Engineering,
The Australian National University, Canberra ACT 0200, Australia
2Institute of Radio Astronomy, 4 Krasnoznamyonnaya Street, Kharkov 61002, Ukraine
Weexaminetheevolutionofparaxial beamscarryingintrinsicspinandorbitalangularmomenta
(AM) in gradient-index media. A parabolic-type equation is derived which describes the beam
diffraction in curvilinear coordinates accompanying the central ray. The center of gravity of the
beam experiences transverse AM-dependent deflections – the spin and orbital Hall effects. The
spin Hall effect generates a transverse translation of the beam as a whole, in precise agreement
9 with recent geometrical optics predictions. At the same time, the orbital Hall effect is significantly
0 affected by the diffraction in the inhomogeneous medium and is accompanied by changes in the
0 intrinsic orbital AM and deformations of thebeam.
2
PACSnumbers: 42.25.Bs,42.15.-i,42.25.Ja,42.50.Tx
n
a
J
I. INTRODUCTION II. MAXWELL EQUATIONS IN THE
7
RAY-ACCOMPANYING COORDINATE FRAME
] Spin-dependenttransversetransport–thespin-Hallef-
s
c fect (SHE) – of classical waves and quantum particles Maxwellequationsforthemonochromaticelectricfield
ti is currently attracting growing attention in condensed- E in a gradient-index dielectric medium read
p
matter [1], high-energy [2], and optical [3, 4, 5, 6, 7, 8]
o k−2∇2+ε E k−2∇(∇ E)=0, (1)
. physics. This effect appears under bending of the wave 0 − 0 ·
s trajectory in an external potential and is closely related (cid:0) (cid:1)
c where k = ω/c (ω is the wave frequency) and ε = ε(r)
i tosuchfundamentalphenomenaastheBerryphase,con- 0
s is the dielectric constant of the medium. Classical geo-
servation of the total angular momentum (AM) of the
y metricaloptics (GO) showsthat inthe short-wavelength
h wave, and spin-orbit interaction.
limit the wave propagates as a classical point particle
p The optical SHE deals with evolution of Gaussian- moving along the ray trajectory r =r (s) given by [13]
[ c c
type wave beams bearing intrinsic spin AM. A general-
2 ization of this effect for higher-orderLaguerre-Gaussian- r˙ =t , t˙ = ∇⊥εc. (2)
v type beams with phase singularities (vortices) has been c 2ε
3 put forward recently [9, 10, 11]. As such beams carry c
1
well-defined intrinsic orbital AM [12], an orbital-Hall ef- Here the overdot stands for the derivative with respect
1
fect (OHE) appears under the bending of their trajecto- to the parameter s, which is the trajectory arc length,
0
. ries. t is the unit vector tangent to the trajectory, ∇⊥ =
2 Extensive studies over the past several years mostly ∇ t(t ∇) is the gradient in the plane orthogonal to
1 − ·
8 considered the semiclassical trajectory equations tracing the ray, and the subscript “c” means that the function
0 evolution of the center of gravity of a wave beam rather is taken on the trajectory, i.e., ∇⊥εc ≡ (∇⊥ε)|r=rc, etc.
v: than the propagation of real extended beams. At the Equations (2) define the centralreference ray and realis-
i sametime,thetypicaltransverseshiftofthebeam’scen- tic beams evolve in the vicinity of it.
X ter of gravity due to the SHE or OHE is proportional The wave-beam propagationis described using parax-
r to the wavelength and is rather small as compared to ialapproximationinthe vicinityofthe GOray(2). This
a
the characteristic scale of the beam deformations in an implies the smallness of the two parameters:
inhomogeneous medium. Therefore, it is important to
λ w
give a picture of the evolution of realistic beams in a µ = 1 and µ = 1, (3)
1 2
w ≪ L ≪
gradient-index medium, which includes SHE, OHE, and
the diffraction processes. where λ is the wavelength, w is the characteristic beam
Below we provide such a description of paraxial opti- width,andL ∇ε/ε−1 isthecharacteristicscaleofthe
∼| |
cal beams carrying spin and orbital AM and evolving in mediuminhomogeneity. Weaimtodescribetheevolution
a smooth gradient-index medium. Although we consider of a paraxialbeam keeping the terms up to the µ3 order
Maxwell equations, our analysis can readily be extended withrespecttothecombinedparameterµ=max(µ ,µ )
1 2
toquantumwaveequationsdescribingevolutionofquan- inMaxwellequations. Previously,this problemhas been
tum particles in external potentials. In particular, the solved with an accuracy of µ2 [14], i.e., in the lowest-
OHE arises from the Laplace operator in curvilinear co- order approximation that describes diffraction but does
ordinates and is universal for any beams with vortices. not account for the Hall effects.
2
Equation(8) can be projected onto the plane (ξ ,ξ ) or-
1 2
thogonal to the ray. In so doing, we notice that [16]
∂E = ∂E⊥ +E t˙, E t˙ =i(∇ E )∇⊥εc,(9)
(cid:18)∂s(cid:19) (cid:18) ∂s (cid:19) k k ⊥· ⊥ 2ε k
⊥ ⊥ c c
whereweusedEqs.(2),(6)and(7). Thisyieldsthewave
equation for the transverse electric field E :
⊥
FIG. 1: (Color online.) Geometry of the wave propagation
along a curved geometrical-optics ray. Depicted are: a ray 1 ∂ 1∂E 1 ∂ ∂E
accompanying coordinate frame attached to the unit vectors k−2 ⊥ + h ⊥ +εE
(e1,e2,t), direction of the inhomogeneity bending the ray, 0 (cid:20)h∂s(cid:18)h ∂s (cid:19) h∂ξi (cid:18) ∂ξi (cid:19)(cid:21)⊥ ⊥
∇to⊥tεacnkdt˙t˙,(atnhdedlairregcetidoonusbolef athrreowH)a.ll effects orthogonal both +k0−2(cid:18)∇ε⊥εc ×∇⊥(cid:19)×E⊥ =0. (10)
c
In the vicinity of the GO ray, we introduce a ray-
As we will see, the last term in Eq. (10), which is of
accompanyingcoordinateframe(ξ ,ξ ,s)attachedtothe
1 2 the order of µ3, describes the SHE of light. This term
unit vectors (e ,e ,t). These vectors evolve along the
1 2 originatesfromthe combinationof the polarizationterm
ray: t = t(s), ei = ei(s), i = 1,2, see Fig. 1. To make ∇ (∇ E) and the Coriolis term 2ik E t˙ [16].
the (ξ ,ξ ,s) frame orthogonal, (e ,e ) must obey the ⊥ · c k
1 2 1 2
parallel transport equation along the ray [14, 15]:
e˙ = e t˙ t, (4)
i i
− ·
(cid:0) (cid:1)
III. PARABOLIC-TYPE EQUATION
The Lame coefficients of the coordinates (ξ ,ξ ,s) are
1 2
equal to [14, 15]
Full wave Eq. (10) can further be simplified to a
∇ ε
h =h =1 , h h=1 ⊥ c ξ, (5) parabolic-type equation via the WKB ansatz:
1 2 s
≡ − 2ε ·
c
where ξ is the radius-vector in the (ξ1,ξ2) plane. E⊥(s,ξ)=e(s)εc−1/4(s)W(s,ξ)eiΦ(s), (11)
Theelectromagneticwaveisnear-transverseintheray
coordinates:
s
where Φ(s)=k ε (s′)ds′ is the GO phase along the
E=E +E t=E e +E t , E µ E . (6) 0 c
⊥ k i i k (cid:12) k(cid:12)∼ | | ray,e(s)=αe1(Rs0)p+βe2(s) is the unit polarizationvec-
Fromthe equation∇ (εE)=0,stem(cid:12)mi(cid:12)ng fromEq.(1), tor,e∗ e=1,andW (s,ξ)istheunknownslowlyvarying
· ·
it follows that in the lowest-order approximation in µ, envelopeofthewave. Thecomponentsofthepolarization
vector is unchanged along the ray, α,β = const, which
∇ ε
∇ E ⊥ c E and E ik−1∇ E , (7) signifies the parallel-transport law for the wave electric
· ≃− ε · ⊥ k ≃ c ⊥· ⊥
c field, related to the Berry phase [14, 17]. Owing to
Eq. (4), derivatives e˙ t do not contribute to the equa-
where kc =k0√εc is the central wave number. tion(10)forE . NotiekthatonlythelastterminEq.(10)
UsingEq.(5)andthefirstofEqs.(7),MaxwellEq.(1) ⊥
involvesthewavepolarization,andthistermisdiagonal-
in the (ξ ,ξ ,s) coordinates takes the form
1 2 ized in the basis of circular polarizations: e=e +iσe ,
1 2
1 ∂ 1∂E 1 ∂ ∂E whereσ = 1isthewavehelicityduetothespin. Substi-
k−2 + h +εE ±
0 (cid:20)h∂s(cid:18)h ∂s(cid:19) h∂ξ (cid:18) ∂ξ (cid:19)(cid:21) tuting Eq.(11) with a circularpolarizationinto Eq. (10)
i i
andretainingonlytermsupto theµ3 order,wearriveat
∇ ε
+k−2∇ ⊥ c E =0. (8) the parabolic-type equation for W:
0 (cid:18) ε · ⊥(cid:19)
c
∂W 1 3 ∇ ε ∇ ε
2ik +∆ W +k2 (ξ ∇ )2ε (ξ ∇ ε )2 W =iσ ⊥ c ∇ W t+ ⊥ c ∇ W
c ∂s ⊥ 0(cid:20)2 · ⊥ c− 4ε · ⊥ c (cid:21) (cid:18) ε × ⊥ (cid:19)· ε · ⊥
c c c
∇ ε ∂W ∂ ∇ ε 1
2ik ξ ⊥ c ik ξ ⊥ c W k2 (ξ ∇ )3ε W. (12)
− c(cid:18) · εc (cid:19) ∂s − c(cid:18) · ∂s 2εc (cid:19) − 6 0h · ⊥ ci
3
Here ∇ =∂/∂ξ, ∆ = ∂2 + ∂2 , and in derivation of SHE and OHE. (Note that the second term originates
Eq. (12)⊥we used the⊥Tayl∂oξr12exp∂aξn22sion for ε(r): from the ∂∂hξ∂∂ξ term in the Laplace operator and has a
universal form for waves of any nature [18].) These two
ε ε +(ξ ∇ )ε + 1(ξ ∇ )2ε + 1(ξ ∇ )3ε . terms can be eliminated by a simple transformation:
c ⊥ c ⊥ c ⊥ c
≃ · 2 · 6 ·
W (s,ξ)=W˜ (s,ξ δ(s)). (13)
The first two terms of Eq. (12) represent usual −
parabolic equation, whereas the next terms in square SubstitutingEq.(13)intoEq.(12),wenoticethat ∂W =
brackets describe the influence of the medium inhomo- ∂W˜ δ˙ ∇ W˜ and the above-mentionedterms are∂csan-
geneity on diffraction. All the terms in the left-hand ∂s − · ⊥
celed when
side of Eq. (12) are of the order of µ2; they have been
orebptraeinseendticnorRreecft.io[1n4s].ofTthheeµr3igohrtd-hear.nd side of Eq. (12) δ˙ = kσ (cid:18)∇2⊥εεc ×t(cid:19)+ ki ∇2⊥εεc. (14)
c c c c
Aswewillsee,thefirsttwotermsintheright-handside
ofEq.(12),whichareproportionalto∇ W,describethe As a result, the parabolic equation acquires the form
⊥
∂W˜ 1 3
2ik +∆ W˜ +k2 (ξ ∇ )2ε (ξ ∇ ε )2 W˜ =
c ∂s ⊥ 0(cid:20)2 · ⊥ c− 4ε · ⊥ c (cid:21)
c
∇ ε ∂W˜ ∂ ∇ ε 1
2ik ξ ⊥ c ik ξ ⊥ c W˜ k2 (ξ ∇ )3ε W˜. (15)
− c(cid:18) · εc (cid:19) ∂s − c(cid:18) · ∂s 2εc (cid:19) − 6 0h · ⊥ ci
IV. SPIN AND ORBITAL HALL EFFECTS shifted along ξ on the distance signlχ, while the center
2
of gravity of the vortex is shifted along the ξ axis on
2
Equations (13)–(15) are the central results of the pa- distance lχ, i.e., in the opposite direction, see Fig. 2
−
per. While Eqs. (13) and (14) describe the transverse and cf. Ref. [11]. Taking this into account, one can
deformations of the beam due to the SHE and OHE, derive from Eq. (14) the differential equation describing
Eq. (15) describe all other deformations caused by the the shift of the center of gravity of the vortex (16):
diffraction in the medium. The σ-dependent term in
Eq.(14) is responsible for SHE; it takes the form as pre- δr˙{σ,l} = σ+l t˙ t , (17)
c k ×
dicted by the modified GO theories [3, 4, 5, 6, 8]. Equa- c (cid:0) (cid:1)
tions(13)and(15)(whichisindependentofpolarization) where we substituted ∇⊥εc = t˙ from Eq. (2). Equa-
imply that the SHE produces a perfect translation of the 2εc
tion (17) represents correction to the ray Eqs. (2) de-
wholebeam inthetransversedirection,withoutanyother
termining motion of the center of gravity of a wave
polarization-dependent distortions, see Fig. 2.
carrying well-defined spin and orbital AM. It is in
The OHE is more intricate and is described by the
agreementwith the geometrical-optics predictions of the
imaginary term in Eq. (14). To show this, let us first
SHE[3,4,5,6,8]andOHE[9],whicharedirectlyrelated
consider an unperturbed beam carrying a well-defined
totheconservationofthetotalAMintheproblem[5,9].
intrinsic orbital AM. Its field contains an optical vortex,
However,thel-dependentterminEq.(17)isvalidonly
which is described by the structure [12]
until one can neglect perturbations in the beam shape
W [ξ +isign(l)ξ ]|l| =ρ|l|exp(ilϕ), (16) causedbythe diffractioninagradient-indexmedium. In
l 1 2
∝ the lowest-order approximation, the diffraction-induced
wherel=0, 1, 2,...isthevorticity,characterizingthe deformations are described by Eq. (15) with the right-
± ±
value of intrinsic orbital AM per photon, whereas (ρ,ϕ) hand-side terms neglected [14]. Typically, the beam
are the polar coordinates in the ξ plane. It is easy to acquires elliptical deformations at distances comparable
see that small imaginary shift along, say, the ξ1 axis: with the characteristic inhomogeneity scale [19]. These
ξ1 ξ1 iχ, deforms the intensity distribution in the deformations do not affect the SHE, but they do af-
→ −
vortex along the orthogonalξ2 axis: fect the OHE, because elliptical deformations of a vor-
tex beam dramatically change the intrinsic orbital AM
|l| 2lχξ
|Wl|2 ∝hξ12+(ξ2−sign(l)χ)2i ≃ρ2|l|(cid:18)1− ρ22(cid:19). cgararrviietdy boyf tthheevboeratmex.[10In] acnodnttrhaeststhoiftspoifntAheMc,etnhteerino-f
Fromthisequationfollowsthatthenodal point W =0is trinsic orbital AM of the beam is not conserved upon
l
4
tion Eq. (15) in each particular problem.
This research was supported by the Linkage Interna-
tional Grant of the Australian Research Council.
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pendsontheabsolutevalueofthevortexcharge, l ,but, (e1,e2,t)producesadditionaltermswhichcanbeassoci-
in contrast to the OHE, Eq. (15) is independent|o|n the atedwiththeCorioliseffect: E˙ =E˙iei+E˙kt+Eie˙i+Ekt˙.
Indeed, the (e1,e2,t) frame rotates with the instant an-
sign of the vortex, sign(l). gular velocity Ω = t× t˙ with respect to s. The cor-
Toconclude,wehavederivedaparabolic-typeequation responding Coriolis term can be written as Ω × E =
which describes propagation and diffraction of paraxial −`E·t˙´t+Ekt˙ =Eie˙i+Ekt˙, where we used Eq.(4).
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[18] Similarly to [16], the term ∂h ∂ in the Laplace opera-
opposite helicities σ = 1 and vorticities l = l, and ∂ξ∂ξ
± ±| | tor can be associated with a Coriolis effect, because it
the diffraction effects described by Eq. (15). While the
is caused by derivatives of the frame parameters. Thus,
SHE turns out to be diffraction-independent, the OHE
both the SHE and OHE can be regarded as manifesta-
is crucially affected by the beam deformations upon the tionsoftheCoriolis effectunderthebendingofthewave
diffraction in a gradient medium. Due to this, calcula- trajectory.
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