Abstract
We present a formulation of electromagnetic spinorbit coupling in magnetooptic media, and propose an alternative source of spinorbit coupling to nonparaxial optics vortices. Our treatment puts forth a formulation of nonreciprocal transversespin angularmomentumdensity shifts for evanescent waves in magnetooptic waveguide media. It shows that magnetizationinduced electromagnetic spinorbit coupling is possible, and that it leads to unequal spin to orbital angular momentum conversion in magnetooptic media evanescent waves in opposite propagationdirections. Generation of freespace helicoidal beams based on this conversion is shown to be spinhelicity and magnetizationdependent. We show that transversespin to orbital angular momentum coupling into magnetooptic waveguide media engenders spinhelicitydependent unidirectional propagation. This unidirectional effect produces different orbital angular momenta in opposite directions upon excitationspinhelicity reversals.
Introduction
The spinorbit interaction (SOI) of light has been the subject of extensive studies in the last few years^{1,2,3,4,5,6,7,8,9,10,11,12}. Recent experiments have demonstrated strong directional coupling of circularly polarized light (optical spin) in nanophotonic waveguides, where the optical helicity determines the direction of optical flow of the incoupled light^{1}. That is, transversely polarized light acting on nanoparticles on the waveguide cladding can be made to excite beams (optical orbital angular momentum) propagating normal to the incoming wave. The propagation direction of the guided mode being determined by the helicity of the incoming light. Unidirectional surfaceplasmon excitation has also been observed in spatially symmetric structures, the surface wave direction switchable with the sense of circularly polarized optical excitation^{12}. Other studies have demonstrated optical helicoidal beams, where light in a whispering gallery or ring resonator is made to emit waves possessing orbital angular momentum in free space, as illustrated schematically in Fig. 1^{11}. Finally, spintoorbital angular momentum conversion in tightly focused nonparaxial optical fields in free space has also been demonstrated, where circularly polarized light without a vortex actually exhibits circulating orbital momentum^{8}.
This type of spinorbit interaction in optical wave propagation paves the way to spincontrolled photonics. The use of transverse spin angular momentum and the coupling of transversely propagating circularly polarized beams to waveguide and surface plasmon modes permits selective directional addressing of guided light and quantum states, and enriches the store of spindependent tools available to integrated and nanophotonics^{1,9,10,11,12}.
However, the above studies have not addressed the effects of magnetooptic nonreciprocity on optical spinorbit coupling, nor ways to induce such coupling by magnetic means. The analysis we present here discusses the generation of optical orbital angular momenta induced through magnetooptic spinorbit coupling. It analyzes the effect of nonreciprocity on spininduced transverse optical momenta, as well as magnetization tuning and magnetization reversal effects on unidirectionally spininduced orbital angular momenta normal to the optical spin. It is known that transverse elliptical polarization of a given helicity occurs in the evanescent tail of optical waveguides, that transversely magnetized magnetooptic waveguides evince a nonreciprocal phase shift, and that the Faraday Effect rotates the polarization of light. Yet the effect of magnetooptic media on the orbital angular momentum shifts in unidirectionallycoupled light upon transverse optical spin reversal, the effect of Faraday rotation upon spin angular momentum conversion and the nonreciprocal transfer to orbital momenta due to electromagnetic spinorbit coupling, and the magnetic tuning of spinorbit coupling and its effect on the induced orbital angular momenta have not been addressed. It is these phenomena that are explored in the present report.
Thus, we address the spindependent magnetization control of the propagation direction and induced orbital angular momenta. Circularlypolarized beams of a given helicity evanescentlycoupled to optical waveguides in the presence of a transverse magnetic field to the optical channel, can be made to switch phasevelocity, alter orbital angular momentum, or cancel unidirectional propagation upon magnetic field tuning, reversals or rotations.
Transverse optical spin is a physically meaningful quantity that can be transferred to material particles^{1,4,5,6,7,8,9}. This has potentially appealing consequences for opticaltweezer particle manipulation, or to locate and track nanoparticles with a high degree of temporal and spatial resolution^{10}. Thus, developing means of control for the transverse optical spin is of practical interest.
We address the latter question for spin and orbital angular momenta, show that their magnitudes and sense of circulation can be accessed and controlled in a single structure, and propose a specific configuration to this end. Explicit expressions for these physical quantities and for the spinorbit coupling are presented. Moreover, we develop our treatment for nonreciprocal slab optical waveguides, resulting in a different response upon time reversals.
Consider the behavior of evanescent waves in a magnetic garnet cladding on silicononinsulator waveguides, as in Fig. 2. The treatment we present deals with transversemagnetic (TM) mode propagation. This allows us to obtain explicit expressions for nonreciprocal transverse spin momenta and angular momenta and to propose a means for magnetically controlling these objects, with potential application to integrated optical vortex beam emitters, optical tweezers and quantum computation^{9}. The conversion of transversespin to orbital angular momenta through spinorbit coupling relies on TM to transverseelectric (TE) mode conversion. We show that, in this case, mode conversion via Faraday rotation, channels electromagnetic spininducing linear momenta into orbital angular momenta that can then be converted into freespace helicoidal beams.
Results
MagneticGyrotropyDependent Evanescent Waves
The offdiagonal components ±ig of the magnetic garnet’s dielectric permittivity tensor, control the structure’s magnetooptic response. Please refer to the Supplementary Material for the wave equation and dispersion relation for a slab waveguide with magnetic garnet cover layer. The TM mode’s electricfield components in the top cladding are
where H_{y} is the optical magneticfield, β its propagation constant in the zdirection, , the decay constant in the (vertical) xdirection, k_{0} = 2π/λ, λ the freespace wavelength, and the coverlayer’s dielectricpermittivity constant. The other components, E_{y} = H_{x} = H_{z} = 0.
Notice that these two electric field components are π/2 out of phase, hence the polarization is elliptical in the cover layer, with optical spin transverse to the propagation direction. In addition, the polarization evinces opposite helicities for counterpropagating beams, as E_{z}/E_{x} changes sign upon propagation direction reversal.
This result already contains an important difference with reciprocal nongyrotropic formulations, where E_{z}/E_{x} = −iγ/β, and γ the decay constant in the top cladding. Equations 1 and 2 depend on the gyrotropy parameter g, both explicitly and implicitly through β. and are therefore magnetically tunable, as we shall see below.
We emphasize that the magnitude and sign of the propagation constant β change upon propagation direction reversal, and, separately, upon magnetization direction reversal. The difference between forward and backward propagation constants is also gyrotropy dependent. This nonreciprocal quality of magnetooptic waveguides is central to the proper functioning of certain onchip devices, such as MachZehnderbased optical isolators^{13,14}.
In a dielectric medium, the momentum density expression accounts for the electronic response to the optical wave. Minkowski’s and Abraham’s formulations describe the canonical and the kinetic electromagnetic momenta, respectively^{15}. Here we will focus on Minkowski’s version, , as it is intimately linked to the generation of translations in the host medium, and hence to optical phase shifts, of interest in nonreciprocal phenomena. is the displacement vector, and the magnetic flux density.
Dualsymmetric versions of electromagnetic field theory in free space have been considered by various authors^{2,8,9,15}. However, the interaction of light and matter at the local level often has an electric character. Dielectric probe particles will generally sense the electric part of the electromagnetic momentum and spin densities^{2,8,9,15}. Hence, we treat the standard (electricbiased) formulation of the electromagnetic spin and orbital angular momenta. In the presence of dielectric media, such as iron garnets in the nearinfrared range, the expression for Minkowski spin angular momentum becomes
The orbital momentum is
where , and is the relative dielectric permittivity of the medium^{3,8}. Please refer to the supplementary material for discussion about the origin of these expressions.
In magnetooptic media, the dielectric permittivity is , depending on the helicity of the propagating transverse circular polarization. This is usually a small correction to , as g is two, or three, orders of magnitude smaller in iron garnets, in the near infrared range. For elliptical spins, where one helicity component dominates, we account for the admixture level of the minority component in through a weighted average.
Nonreciprocal Transverse MagnetoOptic SpinOrbit Coupling
In this section we present a formulation for the transversespin and orbital angular momentum densities, and nonreciprocal spinorbit coupling induced by evanescent fields in magnetooptic media. The magnitude and tuning range of these objects in terms of waveguide geometry and optical gyrotropy are expounded and discussed. We detail the differences in orbital angular momenta between transversely propagating beams induced by circularlypolarized light of opposite helicities. Their unequal response to given optical energy fluxes in opposite propagation directions and to changes in applied magnetic fields are analyzed. And we apply the recently proposed BliokhDresselNori electromagnetic spinorbit correction term to calculate the spinorbit interaction for evanescent waves in gyrotropic media^{8}.
Equations 1 to 4 yield the following expressions for the transverse Minkowski spin angular momentum and the orbital momentum densities in evanescent nonreciprocal electromagnetic waves,
These expressions depend on the magnetooptic gyrotropy parameter g and the dielectric permittivity of the waveguide core channel and of its cover layer under transverse magnetization. They yield different values under magnetic field tuning, magnetization and beam propagation direction reversals, and as a function of waveguide core thickness as discussed below. The propagation constant β is gyrotropy, propagationdirection, and waveguidecorethicknessdependent, and this behavior strongly impacts the electromagnetic spin and orbital momenta.
Consider now the electromagnetic spinorbit coupling induced by transversely propagating circularly polarized beams impinging on a gold nanoparticle on the surface of a silicononinsulator slab waveguide with Bi:YIG cover layer, as in Fig. 3(a). Alternatively, one may examine the response of a magnetic garnet waveguide on gadoliniumgallium garnet substrate and air cover as in Fig. 3(b). These configurations are similar to the chiral nanophotonic waveguide arrangement considered in ref. 1, except that we are now dealing with a magnetooptic nonreciprocal system. We assume (but do not prove), that the light emitted by the rotating dipole in the gold nanoparticle couples to the elliptically polarized evanescent tail of the same helicity as the rotating dipole, as was shown in refs 1 and 12. The sign of the helicity of the evanescent TM wave locksin the direction of propagation, resulting in unidirectional spinorbit coupling.
We now explore the difference in unidirectionallyexcited orbital momenta and coupling efficiency for opposite helicities in the magnetooptic system, based on Eq. 5 and Eq. 6. Figure 4(a) plots the shift in coupled orbital momentum per unit spin angular momentum, in a slab waveguide for opposite excitation helicities. This quantity is obtained from the difference in ratio of Eq. 6 to Eq. 5, for opposite propagation directions. The result is plotted as a function of magnetooptic gyrotropy. Plotted in the same figure, Fig. 4(b), we also have the coupling efficiency shift for unidirectional propagation for oppositehelicity circularly polarized excitations. The latter is obtained from the overlap of the circular polarization input to the evanescent tail elliptical polarization, obtained from Eq. 1 and Eq. 2. A derivation of this quantity can be found in the Supplementary Material. Notice that the nonreciprocal orbital momentum shift of the excited unidirectionallyoriented light is significant (0.1%) for typical magnetooptical gyrotropies (~0.001 to 0.01) found in the infrared regime in magnetic garnet materials. Even larger shifts (up to ~1%) obtain in the visible range. Larger shifts are possible for ferromagnetic metallic materials (plasmonic guiding) possessing significantly larger gyrotropies. These latter effects have yet to be explored both theoretically and experimentally.
Consider now the excited nonreciprocal spin angular momentum shifts per energy flow ,
An expression for the energy flow can be found in the Supplementary Material. Figure 2 plots the nonreciprocal transverse spinangularmomentumdensity shift per unit energy flux, as a function of silicon slab thickness in an SOI slab waveguide with Ce_{1}Y_{2}Fe_{5}O_{12} garnet top cladding. Calculations are performed for the same electromagnetic energy flux in opposite propagation directions, at a wavelength of 1550 nm, g = −0.0086. The nonreciprocal shift is normalized to the average spin angular momentum. The energy shift evinces a relatively stable value, close to 0.7% above 0.3 μm thickness. Its thickness dependence is a function of the ellipticity of the transverse polarization in the xz plane. Above 0.3 μm, the ellipticity ranges from 31.4° to 36.9°, where 45° corresponds to circular polarization. In other words, the ellipticity stays fairly constants, with a moderately small admixture of the minority circularly polarized component, ranging from 25% to 14%. Below 0.3 μm, the minority component admixture increases precipitously, reaching 87% at 0.13 μm. Magnetization reversals produce the same effect, for the corresponding transverse spinangularmomentumdensity shift. Refer to the Supplementary Material for full expression.
MagnetoOptic Gyrotropy Control of SpinOrbit Effects
The magnetooptic gyrotropy of an iron garnet can be controlled through an applied magnetic field. These ferrimagnetic materials evince a hysteretic response, such as the one displayed in Fig. 5 (inset) for 532 nm wavelength in a sputterdeposited film. The target composition is Bi_{1.5}Y_{1.5}Fe_{5.0}O_{12}. Shown here are actual experimental data extracted from Faraday rotation measurements. Below saturation, the magnetooptic response exhibits an effective gyrotropy value that can be tuned through the application of a magnetic field. These measurements correspond to a 0.5 μmthick film on a (100)oriented gadolinium gallium garnet (GGG) substrate. The optical beam is incident normal to the surface, and the hysteresis loop probes the degree of magnetization normal to the surface as a function of applied magnetic field. These data show that the electromagnetic spin angular momentum can be tuned below saturation and between opposite magnetization directions.
Figure 5 also reveals an interesting feature about the magnetooptic gyrotropy. The normalized nonreciprocal transverse spinangularmomentumdensity shift per unit energy flux linearly tracks the gyrotropy, and is of the same order of magnitude as g, although thicknessdependent. Yet, as pointed out before, this thickness dependence reflects the admixture of the minor helicity component in the spin ellipticity. At 0.4 μm, for example, when g = −0.0086. However, the major polarization helicity component contribution to is 84.4% at this thickness, translating into 0.00853 at 100%. At 0.25 μm, , and the major polarization helicity component contribution is 76.2%, translating into 0.0086 at 100%. We, thus, reinterpret the magnetooptical gyrotropy as the normalized spinangularmomentum density shift per unit energy flux.
Figure 1 illustrates schematically the induction of freespace vortex beams. The difference in freespace vortex beams orbital angular momenta are therefore apparent from Fig. 4, a consequence of the difference induced by coupling light with positive or negative spin helicities into the waveguide. The fact that the magnetooptic gyrotropy can be tuned, as discussed above, means that the magnitude orbital momenta and the phase of the coupled light (and hence the resonance of the coupled light in the resonator), can be tuned to resonate for either positive or negative helicity excitation beams, while at the same time suppressing the excitation of one or the other freespace helicoidal beams. These are novel proposed effects that translate into magnetic control of freespace helicoidal beams for opposite chiralities.
MagnetizationInduced Electromagnetic SpinOrbit Coupling
Bliokh and coauthors have studied the electromagnetic spinorbit coupling in nonparaxial optical beams^{8}. They find that there is a spin dependent term in the orbital angular momentum expression that leads to spintoorbit angular momentum conversion. This phenomenon occurs under tight focusing or the scattering of light^{8}. Here we consider an alternative source of electromagnetic spinorbit coupling, magnetizationinduced coupling in evanescent waves.
The timeaveraged spin, , and orbital, , angular momenta conservation laws put forth in ref. 8 each contain a term, responsible for spinorbit coupling, in the form
We have modified the original expressions to include a dielectric permittivity factor to account for the material response of the medium. The indices i = x, y, z. Expression 8 appears with opposite signs in the spin and orbital conservation laws, signaling a transfer of angular momentum from spin to orbital motion. As it stands, so far in our treatment, this term equals zero, since the spin points in the ydirection and the electricfield components of the TM wave point in the x, and zdirections. A way to overcome this null coupling, and enable the angular momentum transfer, is to rotate the applied magnetic field about the xaxis away from the ydirection, as in Fig. 6. This action induces a Faraday rotation about the zaxis, generating a spinorbit coupling term in the angular momentum conservation laws. An inplane rotation of the magnetization M to the zaxis will induce TM to TE mode conversion and electromagnetic spinorbit interaction in the magnetooptic medium^{16,17}. Hence, nonzero electromagnetic field components and , and spinorbit coupling, are induced in the propagating wave. The spatial, nonintrinsic, component, characteristic of orbital motion, emerges in the form of a zdependence in the angular momentum, embodied in the partial or total evanescence of the major circularlypolarized transversespin component as the wave propagates along the guide.
In what sense is there an angular momentum transfer from spin to orbital, in this case? As the polarization rotates in the xy plane due to the Faraday Effect, there will be a spatiallydependent reduction in the circulating electric field spincomponent of the electromagnetic wave along the propagationdirection. This can be seen as a negative increase in circular polarization with z, i.e., an orbital angular momentum in the opposite direction to the electromagnetic spin.
More specifically, the TE mode, with an electricfield component only in the ydirection, carries no transverse spin angular momentum, as per Eq. 3. Where does this angular momentum go? It goes into orbital angular momentum, according to Eq. 8. This electromagnetic orbital angular momentum (OAM) in the TE mode may be converted into freespace OAM via a helicoidal beam emitter as proposed in ref. 11. These authors demonstrate an integrated compact vortex beam emitter through the use of a circular microring or microdisk optical resonator, as in Fig. 1, furnished with an embedded angular grating. The grating partially converts the whispering gallery mode in the microring into freespace radiation. The device is configured to emit vortex beams from quasiTE light fed into the microring via a straight waveguide bus coupled to it. At issue here is not the Faraday rotation per se, but the conversion of transverse spin angular momentum into orbital angular momentum.
Finally, we derive an explicit expression for the spinorbit coupling term. We assume that Faraday rotation induces the E_{y}, H_{z} terms via TM to TE mode conversion, where , and . is the electric field amplitude corresponding to full TM to TE conversion, is the specific Faraday rotation angle, and are the coverlayer decay constant and the propagation constant for the TE mode, respectively. For simplicity, we assume no linear birefringence in the waveguide, so . The spin to orbital angularmomentum coupling term is then
Hence full angular momentum conversion from transverselycoupled forward (or backward) propagating positive (negative) helicity light in the zdirection upon Faraday rotation generates a spinto orbital momentum transfer in the cover layer.
Discussion
There are two key results put forth in this article. A formulation of the interaction between nonreciprocal transverse electromagnetic spin and orbital angular momenta in evanescent waves in magnetooptic media. And a treatment of magnetically induced spinorbit coupling in electromagnetic waves. Our analysis examines the effect of magnetooptic nonreciprocity on the orbital angular momenta of unidirectional light in nanophotonic waveguide interfaces generated by electromagnetic spinorbit interaction. We explore the role of optical chirality on the induced orbital momenta on transversely propagating light and quantify its response to oppositespin optical excitations. The dependence of induced orbital momenta on magnetooptical gyrotropy is detailed.
Additionally, the results presented here provide a means for testing the BliokhDresselNori electromagnetic spinorbit coupling formulation in magnetooptic media^{8}. And, simultaneously, offer a way for magnetically inducing and controlling the electromagnetic spinorbit interaction. The approach may be used to generate OAM helicoidal beams through spin to orbital angular momentum conversion, and to magnetically tune the production of free space orbital angular momenta via ring resonators. The latter relies on experimental results already demonstrated by X. Cai and coworkers in ref. 11.
Our treatment of the nonreciprocal transverse electromagnetic spin is developed for slab waveguides with magnetic garnet cladding layers. This approach allows for an explicit analytical solution of the spin and orbital momenta and angular momenta that can be experimentally tested via prismcoupling in slab waveguides. As such, it allows for testing the controversial reality of electromagnetic spin momenta, in other words, the reality of the electromagnetic linear momenta that induce transverse spinangular momenta.
The article shows that transverse spin angular momentum in evanescent waves can be magnetically tuned, with possible applicability to nanoparticle optical manipulation, and that it evinces a precisely quantifiable nonreciprocal response in magnetooptic media. Moreover, we demonstrate that the shift in spin angular momentum per unit energy flow upon time or magnetization reversal corresponds to the magnetooptical gyrotropy. This finding provides a dynamic interpretation of the magnetooptic gyrotropy parameter, and gives a fresh perspective on the source of the nonreciprocal phase shift effect, often used in the design of integrated optical isolators.
The thickness dependence of the nonreciprocal transversespinshift upon time or magnetization reversals is found to reflect the admixture of minority circular polarization component in the elliptical spin configuration in evanescent waves. This admixture is very pronounced for thin slabs near cutoff, but wanes and saturates for thicker samples, where the majority circular polarization dominates.
Our treatment of electromagnetic spinorbit coupling also provides a means for magnetically inducing and modulating OAM and freespace helicoidal beams. These can be produced onchip, in magnetooptic waveguides with magnetooptic claddings, thus allowing for compact packaging of vortexbean sources. Nonreciprocal TE to TM mode conversion in semiconductor waveguides with magnetooptic upper claddings has already been demonstrated^{16,17}.
Numerous applications of optical angular momentum and vortex beams have been discussed in the literature. These include their use in optical tweezers^{18,19}, optical microscopy^{20}, and quantum and wireless communications^{21,22}.
Method
The theoretical tools to develop the formulation and angular momenta expressions presented here are simple differential equations and algebraic manipulation. C++ computer programming is used to evaluate waveguide propagation constants based on modal dispersion, and to compute spin and orbital angular momenta in evanescent waves in the garnet cladding. These calculations are used to analyze the nonreciprocal response of the spin and orbital angular momenta as a function of slab waveguide thickness. Radiofrequency magnetron sputterdeposition is used to fabricate bismuthsubstituted yttrium iron garnet films on gadolinium gallium garnet substrates, and a rotating polarizer setup is used to measure their Faraday rotation hysteresis.
Additional Information
How to cite this article: Levy, M. and Karki, D. Nonreciprocal Transverse Photonic Spin and MagnetizationInduced Electromagnetic SpinOrbit Coupling. Sci. Rep. 7, 39972; doi: 10.1038/srep39972 (2017).
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Acknowledgements
The authors thank Ramy ElGanainy for suggesting this problem and for useful discussions, and M.J. Steel for a critical reading of the manuscript.
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M.L. conceived the study and analyzed the role of nonreciprocity in transverse spin angular momentum in evanescent waves. He also conceived and analyzed the magnetic spinorbit coupling mechanism in magnetooptic waveguide media. D.K. wrote the programs to compute propagation constants, nonreciprocal phase shifts and spin and orbital angular momenta based on the expressions derived in this work. He also fabricated the bismuthsubstituted yttrium iron garnet films on GGG and measured their Faraday rotation hysteresis loop.
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Levy, M., Karki, D. Nonreciprocal Transverse Photonic Spin and MagnetizationInduced Electromagnetic SpinOrbit Coupling. Sci Rep 7, 39972 (2017). https://doi.org/10.1038/srep39972
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