Spin-Orbit Interactions of Light Fundamentals and Emergent Applications Graciana Puentes12 1-Departamento de Fsica Facultad de Ciencias Exactas y Naturales

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Spin-Orbit Interactions of Light: Fundamentals and Emergent Applications

Graciana Puentes1,2

1-Departamento de Fsica, Facultad de Ciencias Exactas y Naturales,

Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires, Argentina

2-CONICET-Universidad de Buenos Aires, Instituto de Fsica

de Buenos Aires (IFIBA), Ciudad Universitaria, Buenos Aires,

Argentina.

(Dated: 7 October 2022)

We present a comprehensive review of recent developments in Spin Orbit Interactions

(SOIs) of light in photonic materials. In particular, we highlight progress on detection

of Spin Hall Eﬀect (SHE) of light in hyperbolic metamaterials and metasurfaces.

Moreover, we outline some fascinating future directions for emergent applications of

SOIs of light in photonic devices of the upcoming generation.

arXiv:2210.02921v1 [physics.optics] 6 Oct 2022

I. INTRODUCTION

Light’s polarization degrees of freedom, also known as Spin Angular Momentum (SAM)

and orbital degrees of freedom, also known as Orbital Angular Momentum (OAM), can be

coupled to produce a wide variety of phenomena, known as Spin-Orbit Interactions (SOIs) of

light. Due to their fundamental origin and diverse character, SOIs of light have become cru-

cial to a variety of active ﬁelds, such as singular optics, photonics, nano-optics and quantum

optics, when dealing with SOIs at the single-photon level. Among a variety of fascinating

exotic phenomena, SOIs show the remarkable spin-dependent transverse shift in light inten-

sity, also known as the Spin Hall Eﬀect (SHE) and the Spin Orbit Conversion (SOC) of light.

The various plane-wave components of the beam that travel in slightly diﬀerent direc-

tions and acquire slightly diﬀerent complex reﬂection or transmission coeﬃcients are what

cause the regular SHE of light at a planar interface to form [1-4]. A handy quantum-like

framework with generalized wavevector-dependent Jones-matrix operators at he interface,

and expectation values of the position and momentum of light, provides a theoretical expla-

nation of the photonic SHE [3-6]. Such a description also uniﬁes the longitudinal (in-plane)

beam shifts connected to the Goos—Hanchen (GH) eﬀect [3-6] and transverse SHE shifts,

also known as the Imbert—Fedorov (IF) shifts in the case of the Fresnel reﬂection/refraction

[7-10].

2D metamaterials, commonly referred to as metasurfaces, are a an interdisciplinary area

that encourages the use of substitute methods for light engineering based on spatially or-

dered meta-atoms and subwavelength-thick metasurfaces of diﬀerent compositions. They

display exceptional qualities in light manipulation in a 2D interphase. Metasurfaces can

attain their 3D counterpart functions, such as invisibility cloaking and negative refractive

index. In addition, they can eliminate some of the 3D metamaterial current restrictions,

such as high resistivity or dielectric loss, for example. Additionally, the creation of meta-

surfaces via conventional methods for nanofabrication, such as electron beam lithography

techniques, are readily available in the semiconductor industry.

In this review, we provide a summary of recent ﬁndings and future potentials for applica-

tions of SHE of light in photonic materials. As an optical equivalent of the solid-state spin

Hall eﬀect, SHE of light warrants promissing opportunities for examination of innovative

photonic materials and nanostructures physical characteristics, such as in ﬁguring out the

magnetic and metallic thin ﬁlms’ material characteristics, or the optical characteristics of

two-dimensional atomically thin metamaterials, with unmatched spatial and angular pre-

cision resolution, a trait that SHE and other combined technologies can provide, utilizing

quantum weak measurements and quantum weak ampliﬁcation methods. Additionally, we

provide a summary of recent developments in primary 2D metamaterials and metasurfaces

applications for producing and manoeuvering Spin Angular Momentum (SAM) and Orbital

Angular Momentum (OAM) of light, for applications in multicasting and multiplexing,

spin-based metrology or quantum networks.

II. SPIN HALL EFFECT OF LIGHT (SHEL)

Using the terminology of the closely related work, we begin with the theoretical descrip-

tion of the problem. Namely, the SHE of light in a tilted photonic material. Figure 1 (a), (b)

and (c) depict the problem’s geometry. The normalized Jones vector describes the incident

z-propagating paraxial beam’s polarization |ψi= (Ex, Ey)T(Tstands for the transposition

operator), hψ|ψi=|Ex|2+|Ey|2= 1. In the (x, z) plane, the photonic material is tilted

so that its axis forms a θangle (labeled ϑfor ease of notation in the following sections)

with the z-axis and transmits mostly the y-polarization. The dichroic action of the photonic

material in this geometry, and in the zero-order approximation of the incident plane-wave

ﬁeld, can be characterized by the Jones matrix, of the form:

M0=



Tx(θ) 0

0Ty(θ)



,(1)

the Jones vector of the transmitted wave is |ψ0i=ˆ

M0|ψi.Tx,y, which can depend on θ,

are the amplitude transmission coeﬃcients for the x- and y-polarized waves. While Tx= 0

and Ty= 1 for an ideal polarizer, we can assume that |Tx/Ty|  1 for real dichroic plates.

Also take note of the fact that Tx,y = exp(∓iΦ/2) relates to the birefringent waveplate issue

discussed in [11].

Considering that in the paraxial approximation the beam consists of a superposition

of plane waves with their wavevector directions labelled by small angles Θ= (Θx,Θy)'

(kx/k, ky/k) [see Fig. 1(a)], the Jones matrix incorporates Θ-dependent corrections and can

be wrtten as [11]:

M(Θ) = 



Tx(1 + ΘxXx)TxΘyYx

−TyΘyYyTy(1 + ΘxXy)



,(2)

here

Xx,y =dlnTx,y

dθ Yx,y = (1 −Ty,x

Tx,y

)cot(θ),(3)

are the well-known Goos-Hanschen (GH) and Spin Hall (SHEL) terms [11], the latter

being the main focus of this review.

III. QUANTUM WEAK AMPLIFICATION TECHNIQUES

A detailed explanation of the entire theory governing transverse shifts of light in tilted

uniaxial crystals was ellaborated in [11,12,13]. The anisotropic transverse shift, or SHE of

light, is represented by the expectation value of the position operator hˆ

Yi, for an input state

deﬁned by a Jones vector |ψi, and a transmitted state described by a Jones vector |ψ0i:

hˆ

Yi=hψ0|ˆ

Y|ψi=cot(ϑ)

k[−σ(1 −cos(φ0)) + χsin(φ0)],(4)

where kis the wave-vector, φ0refers to the diﬀerence in phase between ordinary and extraor-

dinary waves as they propagate through a birefringent material, ϑrepresents the tilt angle of

the photonic medium, and (σ, χ, τ) represent the Stokes parameters for the light beam. The

SHE can be measured directly, using the sub-wavelength shift of the beam centroid [14-18]

[Eq. (3)]. Various alternative techniques, such as quantum weak measurements [19-27] are

also used. The latter technique enables substantial ampliﬁcation, when employing almost

crossed polarizers for pre-selection and post-selection of the polarization input and output

states, as they relate to the system.

The output polarizer corresponds to a post-selected polarization state |ψi= (α0, β0)T

whereas the input polarizer corresponds to a pre-selected state |ψi= (α, β)T, here (α(α0), β(β0))

represent the input(output) polarization components of the input(output) beam (Ex(E0

x), Ey(E0

y)),

and Tstands for transposition operation. As opposed to traditional expectation values, weak

values hˆ

Yiweak might display a quantum weak ampliﬁcation eﬀect and lie beyond the opera-

tor’s spectrum, moreover weak values can take imaginary values. We examine the quantum

weak ampliﬁcation of the SHE shift using an initial beam with epolarization |ψi= (1,0)T,

and a nearly orthogonal polarization state |φi= (, 1)T,||<< 1, for the post-selection

polarizer. In this conﬁguration, the SHE weak value results in:

hˆ

Yiweak =1

k sin(φ0) cot(ϑ) + z

k (1 −cot(φ0)) cot(ϑ),(5)

here zRstands for the Rayleigh length. The second angular term, becomes dominant in the

far ﬁeld zone, and presents weak ampliﬁcation due to two reasons: First, due to the fact

Fig. 1 (a) General 3D geometry of the problem displaying the angle ϑbetween the anisotropy axis

of the plate and the beam axis z. (b) The wave vectors in-plane deﬂections (Θx) cause the well-

known birefringence shift hˆ

Xiwhich is comparable to the GH shift, by altering the angle between k

and the anisotropy axis. (c) View along the anisotropy axis of the crystal is shown. The transverse

Θydeﬂections of the wave vectors rotate the corresponding planes of the wave propagation with

respect to the anisotropy axis by the angle ϑ≈Θy/sin(Θy). This causes a new helicity-dependent

transverse shift hˆ

Yi, i.e., a spin-Hall eﬀect similar to the IF shift. Further details are in the text.

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Spin-OrbitInteractionsofLight:FundamentalsandEmergentApplicationsGracianaPuentes1;21-DepartamentodeFsica,FacultaddeCienciasExactasyNaturales,UniversidaddeBuenosAires,CiudadUniversitaria,1428BuenosAires,Argentina2-CONICET-UniversidaddeBuenosAires,InstitutodeFsicadeBuenosAires(IFIBA),CiudadUniversitar...

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