Thermal spin photonics in the nearfield of nonreciprocal media
Abstract
The interplay of spin angular momentum and thermal radiation is a frontier area of interest to nanophotonics as well as topological physics. Here, we show that a thick planar slab of a nonreciprocal material, despite being at thermal equilibrium with its environment, can exhibit nonzero photon spin angular momentum and nonzero radiative heat flux in its vicinity. We identify them as the persistent thermal photon spin (PTPS) and the persistent planar heat current (PPHC) respectively. With a practical example system, we reveal that the fundamental origin of these phenomena is connected to spinmomentum locking of thermally excited evanescent waves. We also discover spin magnetic moment of surface polaritons in nonreciprocal photonics that further clarifies these features. We then propose a novel thermal photonic imaging experiment based on Brownian motion that allows one to witness these surprising features by directly looking at them using a lab microscope. We further demonstrate the universal behavior of these nearfield thermal radiation phenomena through a comprehensive analysis of gyroelectric, gyromagnetic and magnetoelectric nonreciprocal materials. Together, these results expose a surprisingly little explored research area of spinrelated thermal radiation phenomena or thermal spin photonics with prospects for new fundamental and technological avenues.
pacs:
I Introduction
Thermal spin photonics merges the fields of the thermal radiation and the spin angular momentum of light. Thermal radiation plays an important role in energy conversion and renewable technologies fan2017thermal (); tervo2018near () while the spin angular momentum property of light is fundamentally relevant in the context of spincontrolled nanophotonics le2015nanophotonic (); mitsch2014quantum (), chiral quantum optics lodahl2015interfacing () and spintronics vzutic2004spintronics (). Despite extensive work in the past few decades, there has been very little overlap between these two areas. Important developments in this field include spinpolarized (circularly polarized) farfield thermal radiation from chiral absorbers shitrit2013spin (); wu2014spectrally (); yin2013interpreting () and the definition of the degree of polarization in the thermal nearfield setala2002degree () for reciprocal media. In stark contrast, the primary aim of this work is to explore thermal spin photonics (spinrelated thermal radiation phenomena) in the nearfield of nonreciprocal media.
Our work utilizes fluctuational electrodynamics and is fundamentally beyond the regime of Kirchhoff’s laws which is valid only for farfield thermal emission from bodies at equilibrium. One striking example where spin angular momentum of thermal radiation is not captured by Kirchhoff’s laws, is circularly polarized thermal emission from coupled nonequilibrium antennas demonstrated in our recent work khandekar2019circular (). This approach of exploiting interacting nonequilibrium bodies is a paradigm shift from ‘all’ conventional approaches of achieving spin angular momentum of light by employing either polarization conversion or structural chirality. Practically, this nonequilibrium mechanism can enable temperaturebased reconfigurability of the spin state of emitted thermal radiation. Our current work deals with bodies at thermal equilibrium with their surroundings, and reveals surprising spin angular momentum features in their nearfield arising in presence of nonreciprocity.
To show the universal nature of these nonreciprocal thermal spin photonic effects, we develop a framework to analyze equilibrium thermal radiation properties of a planar slab of a generic bianisotropic material described by arbitrary permittivity (), permeability () and magnetoelectric susceptibilities (). Such a material is reciprocal if the material properties satisfy.
It is nonreciprocal if any one of these conditions is violated. With fluctuational electrodynamic analysis, we show that a nonreciprocal planar slab at thermal equilibrium with its environment exhibits nonzero spin angular momentum of thermal radiation in its nearfield. We identify it as the persistent thermal photon spin (PTPS) because it exists without any temperature difference analogous to wellknown persistent electronic charge current bleszynski2009persistent (); buttiker1983josephson () that exists without any voltage difference. The PTPS is also accompanied by locally nonzero radiative heat flux parallel to the surface which we call as the persistent planar heat current (PPHC).
We reveal that spinmomentum locking van2016universal (); bliokh2015quantum (); petersen2014chiral () of thermally excited evanescent waves, plays a fundamental role in facilitating both these phenomena with a practical example system. Our work thus provides the first generalization of the spinmomentum locking of light wellknown in topological photonics and atomic physics to thermally excited waves. We consider a doped Indium Antimonide (InSb) slab at room temperature with an arbitrarily directed magnetic field. Our calculations reveal that each thermally excited surface plasmon polariton supported by InSb slab has transverse spin locked to its momentum. We show for the first time that the spinmomentum locked polariton also carries spin magnetic moment which leads to polaritonic energy/frequency shift through Zeeman type interaction with the applied magnetic field. For InSb sample with doping concentration cm, the polaritonic spin magnetic moment is found to be around where is Bohr magneton. The polaritonic magnetic moment depends asymmetrically on the momentum for forward and backward propagating polaritons leading to asymmetric energy shifts. This clarifies the fundamental origin of PTPS and PPHC, resulting from asymmetric contributions of forward and backward propagating evanescent waves.
Detecting nearfield thermal radiation effects of nonreciprocal media is an open challenge. We note however, that our discovered effects PTPS and PPHC are significantly enhanced in the nearfield due to a large density of thermally excited evanescent states. In particular, we show the striking result that at a distance m from the slab surface, the magnitude of PTPS exceeds the spin angular momentum density contained in the laser light carrying typical power of mW. This immediately motivates experimental validation of our predicted effects by probing optical forces and torques on small absorptive particles in the thermal nearfield of an InSb slab. We also propose a new experimental approach for probing nonreciprocal thermal fluctuations phenomena through a novel imaging experiment that witnesses Brownian motion of small absorptive particles in the thermal nearfield of a nonreciprocal slab. We predict that Brownian movement will be sufficiently influenced by these additional thermal spin photonic forces and torques that it can be directly viewed using a lab microscope.
We also demonstrate the universal behavior of both these nearfield thermal radiation phenomena with a comprehensive analysis of the key classes of nonreciprocal media namely, gyroelectric (), gyromagnetic () and magnetoelectric () materials. The general analysis describes the origin and the nature of these features for any given material type and further reveals that a nonreciprocal material is necessary but not sufficient to observe PTPS and PPHC. While thermal photonics is so far limited to isotropic, anisotropic and gyroelectric materials, our general analysis paves the way for future inquiries using generic bianisotropic materials.
We note that recent interesting works have explored the persistent phenomena in many body system of finite gyroelectric spheres zhu2016persistent (), a single gyroelectric sphere in dipolar regime ott2018circular () and inside a gyroelectric waveguide silveirinha2017topological (). However, we emphasize that the fundamental connection with spinmomentum locking of evanescent waves van2016universal (); bliokh2015quantum (); petersen2014chiral () has not been explored till date. We note that the spinmomentum locking of thermally excited waves opens a new degree of freedom for directional radiative heat transport at the nanoscale. We also point out that spherical nanosystems zhu2016persistent (); ott2018circular () are challenging to fabricate and not easily amenable to experimental verification. Our work answers an open question whether a planar slab, canonical geometry suitable for nearfield explorations, can exhibit these phenomena. As mentioned above, we address experimental detection as well as provide a universal description for all nonreciprocal material types as opposed to only gyroelectric type zhu2016persistent (); ott2018circular (); silveirinha2017topological (). The discovery of spin magnetic moment of surface polaritons in nonrecirprocal photonics is also fundamentally new. It invites related studies of spindependent quantum light matter interactions and spinquantization. Finally, we remark that the theoretical framework employed here is amenable for studying numerous topics in the future related to not only fluctuational buhmann2012macroscopic (); bermel2010design (); zhu2014near (); zhu2018theory (); ben2016photon (); ekeroth2017thermal () but also quantum clegg1995fluorescence (); novotny2012principles (); fuchs2017casimir (); gangaraj2018optical (); latella2017giant () electrodynamic effects by using generic, bianisotropic materials rather than a restricted class of material types explored previously.
Ii Results
Theory. We consider a planar geometry shown in figure 1 comprising of a semiinfinite halfspace of generic homogeneous material, interfacing with semiinfinite vacuum halfspace at . We focus on the thermal radiation on the vacuum side of this geometry where the physical quantities such as energy density , Poynting flux and spin angular momentum density are welldefined joulain2003definition (); barnett2016optical () and measurable in suitable experiments kalhor2016universal (); nieto2004near ():
(1)  
(2)  
(3) 
where denotes the position vector and denotes the thermodynamic ensemble average. The spin angular momentum density (3) has so far been studied primarily for nonthermal light joulain2003definition (); barnett2016optical (), where it leads to proportionate optical torque on small, absorptive particles canaguier2013force (); nieto2010optical (). We have generalized it here and in our recent work khandekar2019circular () to thermally generated electromagnetic fields in vacuum. We calculate both electric and magnetic type thermal spin angular momentum density given by and respectively. Throughout the manuscript, all quantities are described in SI units and the dependence on frequency (such as ) is suppressed assuming time dependence in Maxwell’s equations. The above quantities are to be integrated over frequency to obtain the total densities/flux rates as . Keeping in mind the future explorations using generic, bianisotropic materials, we prefer to use vector potential in Landau gauge to obtain associated electromagnetic fields (, ). The electromagnetic field correlations required for calculation of densities and flux rates above are obtained from the vector potential correlations. These correlations evaluated at two spatial points are expressed in the matrix form as . Here, the vector quantities are written as column vectors such that where denotes the transpose and is complex conjugation. We focus on the thermal equilibrium properties of radiation where both vacuum and material halfspaces are at same thermodynamic temperature . The vector potential correlations are then obtained by making analogies with Kubo’s formalism which describes equilibrium correlations of fluctuating thermodynamic quantities. The correlations are (see supplementary material for derivation):
(4) 
Here is the average thermal energy of the harmonic oscillator of frequency at temperature . The Green’s tensor relates the vector potential to all the source currents such that . We derive the Green’s function given in the methods section for a planar slab of a generic, bianisotropic medium (see supplement for derivation).
Finally, using the Green’s function and vector potential correlations above, we obtain the electromagnetic field correlations at two spatial points and in vacuum:
(5)  
(6)  
(7) 
where for is the differential curl operator. The densities and flux rates are then calculated using definitions (1),(2) and (3) using above correlations with and making use of the fluctuation dissipation relation given by Eq.(4).
Possibility of observing nonzero spin angular momentum and heat flux despite thermal equilibrium. We now show that a nonreciprocal medium can lead to nonzero spin angular momentum density (3) and nonzero Poynting flux (2) in its thermal nearfield. We make use of insightful expressions derived here and timereversal symmetry arguments, but in this section we do not refer to any specific material.
First, the timereversal symmetry considerations reveal the possibility of observing PTPS and PPHC using nonreciprocal materials. For the planar geometry, electromagnetic waves are characterized by their inplane conserved propagation wavevector . Poynting flux and spin angular momentum of each wave are and respectively. In then follows that under timereversal symmetry at thermal equilibrium, heat and angular momentum associated with thermally excited waves are negated by heat and angular momentum carried by waves (, ) resulting into zero flux rates at thermal equilibrium. Because of timereversal symmetry for reciprocal media, the persistent phenomena (nonzero flux rates at equilibrium) cannot be observed. For nonreciprocal media, this is no longer true and one can expect to see nonzero flux rates in the absence of cancellation. We identify the resulting nonzero spin angular momentum density as the persistent thermal photon spin (PTPS) and nonzero heat flux as the persistent planar heat current (PPHC). Although this rudimentary analysis proves that nonreciprocity is necessary to observe PTPS and PPHC, full fluctuational electrodynamic calculations below, confirm and reveal much more, including the result that nonreciprocity is not a sufficient condition.
We use the insightful semianalytic expressions for PTPS and PPHC perpendicular to the planar slab (see supplement for their derivation) for making following important general comments . The nearfield Poynting flux in direction,
(8) 
for any material at thermal equilibrium with vacuum halfspace. Similarly, the electric and magnetic parts of spin angular momentum density along direction are:
(9) 
It follows that the total perpendicular thermal spin is zero for any material at thermal equilibrium. Here are the Fresnel reflection coefficients for light incident on the planar slab having perpendicular wavevector (conserved) parallel wavevector and making an azimuthal angle with xaxis of the geometry. is the vacuum wavevector (see methods and supplement). For reciprocal media, li2000symmetries () and the electric and magnetic type persistent thermal photon spin (PTPS) are separately zero. On the other hand, for nonreciprocal media and interestingly, even though electric and magnetic type PTPS are separately nonzero, total PTPS perpendicular to slab is always zero. We note that the semianalytic expressions for PTPS and PPHC parallel to the slab are not conducive for mathematical simplification and general insights but full calculations of examples below reveal their existence and nature.
Before describing PTPS and PPHC with examples, it is important to point out that the intuition confounding presence of nonzero heat current at thermal equilibrium does not lead to thermodynamic contradictions. In particular, because of the nonzero heat flow parallel to the surface, it could be expected that one end will be hotter than the other end. However, given the infinite transverse extent of the system considered above, there is no end that can be heated or cooled ishimaru1962uni (). On the other hand, since the two distinct halfspaces are separated by a welldefined interface, no macroscopic flux rates can exist across the boundary by definition of thermal equilibrium between the halfspaces. The fluctuational electrodynamic theory produces a consistent result above that the Poynting flux () and the total spin angular momentum () perpendicular to the surface are identically zero for any material. In the following, we demonstrate PTPS and PPHC in a practical example system revealing their fundamental connection with spinmomentum locking of evanescent waves.
Practical Example of InSb slab. We consider doped Indium Antimonide (InSb) slab which has been most widely studied in context of coupled magnetoplasmonsurface polaritons kushwaha2001plasmons (); hu2015surface () and whose material permittivity dispersion has been wellcharacterized experimentally chochol2017experimental (); hartstein1975investigation (); palik1976coupled (); chochol2016magneto (). The permittivity dispersion is derived from an extended Lorentz oscillator model which further sheds light on the microscopic origin of the magnetooptic effect arising from the cyclotron motion of bound electrons of the material. For the sake of completeness of our study, we consider the magnetic field to be pointing along any arbitrary direction in our geometry and derive the semianalytic form of permittivity given by where:
This is obtained from an extended Lorentz oscillator model in which bound/free electrons (charge , effective mass , position ) are described as mechanical oscillators, that further experience additional Lorentz force () in presence of applied magnetic field . Each for describes the cyclotron frequency in direction given by . We obtain the parameters from Ref. palik1976coupled (); chochol2016magneto () for InSb sample of doping density cm. , rad/s, rad/s, rad/s, rad/s, rad/s, where kg is electron mass. It is found that InSb slab described by above parameters supports surface plasmon polaritons (SPPs) at rad/s (m) localized close to the interface with vacuum. Below, we first analyze these polaritons.
Spin magnetic moment of InSb surface polaritons. As shown schematically in fig.2(a), we consider a magnetic field applied along direction (parallel to the surface). The surface plasmon polariton characterized by its conserved inplane momentum makes an angle with (applied field direction) such that . Each such polariton also carries a transverse spin locked to its momentum, depicted in the schematic (spin momentum locking van2016universal ()). Figure 2(b) displays the dispersion of polaritons for different angles , obtained numerically as described in the methods section. The dispersion in the absence of magnetic field or with magnetic field perpendicular to the surface is same as dispersion (green curve) for all angles. In presence of magnetic field along direction, the polaritons characterized by (with positive spin component along applied magnetic field) are redshifted while those characterized by (with negative spin component along applied magnetic field) are blueshifted. This is further demonstrated in fig. 2(c) where is obtained as a function of angle for a fixed , for two different values of magnetic field. We numerically find that the energy shift for each polariton increases linearly with magnetic field in the weak field regime (T) while the dependence is complicated for strong applied fields. All these results strongly indicate that the polaritons have a spin magnetic dipole moment () parallel to the transverse spin that interacts with the applied magnetic field. The energy of this Zeeman interaction is described by the Hamiltonian,
It follows that the magnetic field induced frequency shift for each polariton is:
(10) 
We find that the magnetic moment for each polariton depends not only on momentum but also on the angle . This is evident upon closer inspection of fig 2(c) where the energy shift as a function of angle is not sinusoidal but exhibits slight deviation (see maximum and minimum values). In fig. 2(d), the magnetic moment in units of (Bohr magneton) as a function of is displayed for two different angles showing that or . This asymmetry in polaritonic spin magnetic moment and magneticfield induced energy shift lies at the origin of the asymmetry in the spin angular momentum and heat flux carried by thermally excited polaritons, resulting into PTPS and PPHC parallel to the surface. In the following, we analyze PTPS and PPHC computed from full fluctuational electrodynamic calculations.
PTPS and PPHC in thermal nearfield of InSb slab. We compute the spin angular momentum density (Eq.3) and Poynting flux (Eq.2) in thermal nearfield of InSb slab in presence of magnetic field of strength T along direction. Both vacuum and material are assumed to be at thermodynamic equilibrium temperature of K. Based on the discussion of polaritons in the previous section, we calculate spinresolved quantities in the sense described schematically in fig.3(a). In particular, the contributions of electromagnetic waves characterized by (waves) and those characterized by (waves) are calculated separately.
Figure 3(b,c,d) demonstrate the frequency spectra of energy density , spin angular momentum density and poynting flux at a distance of m above the surface of InSb. All these figures depict the separate contributions of waves (red curves) and waves (blue curves) along with the sum total (green curves). As evident from fig.3(b), the collective energy density of waves is redshifted and that of waves is blueshifted similar to polaritons, leading to broadening of total energy density spectrum (green), compared to the spectrum in the absence of magnetic field (black dashed line). The asymmetric overall contributions of and waves result into nonzero spin angular momentum density and nonzero Poynting flux at thermal equilibrium i.e. PTPS and PPHC. Note that these persistent quantities contain contributions from not only surface localized polaritons but also other evanescent waves. For instance, another small peak apparent in 3(b) and clearly visible in 3(d) is not related to the polaritons studied in previous section. Nonetheless, the polaritons make a dominant contribution to PTPS and PPHC.
Figure 4 describes PTPS spectrum [4(a,b,c)] and also demonstrates that the total frequencyintegrated PTPS [4(d)] can compete with the angular momentum density contained in laser light. For brevity, we focus only on PTPS. As shown in fig 4(a), the electrictype PTPS is evidently much larger than the magnetictype PTPS. We describe later [fig 6] that this holds universally for any gyroelectric type nonreciprocal material. Fig 4(b) demonstrates the change in the spectrum as a function of distance from the surface. At each frequency, the sign (direction) of PTPS stays the same while the magnitude decays exponentially as a function of distance from the surface. This also indicates (although not shown separately) that PTPS and PPHC arise from the waves that are evanescent on the vacuum side of the geometry. Fig 4(c) depicts the dependence of the spin angular momentum density on the applied magnetic field. First, the PTPS spectrum broadens as magnetic field is increased from (light green) to (green). Second, while perpendicular magnetic field by itself does not lead to PTPS, it does affect the spectrum observed in presence of parallel magnetic field. This is evident from green and dark green () curves. When magnetic field is at oblique angles to the surface such as this example (dark green), the cyclotron motion in yzplane due to is intercoupled with that in xyplane due to . Due to this intercoupled cyclotron motion of underlying charges, the perpendicular component of magnetic field (which does not lead to PTPS by itself) affects the PTPS spectrum obtained with magnetic field parallel to the surface.
Finally, we plot the total frequency integrated PTPS () in fig 4(d) as a function of distance () for increasing values of magnetic field. The total PTPS lies along direction (antiparallel to applied magnetic field). In order to get a qualitative estimate of the overall strength of PTPS, fig.4(d) also displays the total spin angular momentum density contained in monochromatic (polariton frequency), circularly polarized laser light of power mW focused to mm spot size. Evidently, PTPS which originates from the intrinsic fluctuations in the medium can compete with the total spin angular momentum density contained in the laser light. It can even surpass it at separations m. from the surface. This large enhancement of PTPS arises from a large density of thermally excited evanescent and surface waves in the nearfield, otherwise inaccessible in the farfield. The same figure also depicts the dependence on the temperature with solid lines (K) and dashed lines (K) which comes from the mean thermal energy given by . Since the mean thermal energy is approximately constant over the entire frequency range, PTPS increases/decreases proportionately with the temperature.
We note that the total frequencyintegrated Poynting flux for this practical example (not shown) is along direction. The direction of integrated PPHC is related to the underlying cyclotron motion of electrons induced by magnetic field silveirinha2017topological (). Inside the bulk of InSb, the cyclotron motions of electrons cancel each other but at the surface, this cancellation is incomplete. The direction of this incomplete cyclotron motion coincides with the direction of the PPHC.
We further note that the practical example considered here can also lead to unidirectional energy transport because of nonreciprocity wang2009observation (); rechtsman2013photonic (); hu2015surface (). However, unidirectional transport is not a cause of PTPS and PPHC and this is explained in the following. Figure 2(b) indicates that the unidirectional transport due to polaritons can occur along directions for which polaritonic momentum has a dominant component (maximally blueshifted polaritons). Although PPHC spectrum in fig.3(d) has a predominant contribution along the same direction, both PPHC and PTPS spectra shown in fig. 3(c,d) show that smaller frequencies with bidirectional polaritons also lead to nonzero persistent quantities. While recent works buddhiraju2018absence () have started to explore the role of nonlocality in context of nonreciprocity and unidirectional transport, we leave nonlocality aside for future work.
Experimental Proposal. Here we address the experimental detection of the persistent thermal photonic phenomena based on our calculations above and by proposing a new thermal photonic experiment that can provide a direct visual evidence. It is wellknown that light carrying momentum and spin/orbital angular momentum can exert optical forces and torques on small absorptive particles in its path. Many works have explored this lightmatter interaction in optical tweezers ukita2010optical (); angelsky2012circular (); canaguier2013force () by nonthermal means and in entirely passive systems where forces and torques originate from intrinsic quantum and thermal fluctuations chan2001quantum (); haslinger2018attractive (); bao2018inhomogeneity (). We are interested in the latter for nonintrusive (without disturbing thermal equilibrium) detection of the persistent phenomena. We therefore consider a passive system shown schematically in figure 5 where aqueous or fluidic environment covers the gyrotropic InSb thick slab and contains suspended small micrometer size absorptive and nonmagnetic particles. The particles perform Brownian motion about their positions at finite equilibrium temperature. Upon application of magnetic field, the particles experience additional optical force and torque associated with PPHC and PTPS respectively. While the average motional energies remain constant at thermal equilibrium, the additional forces and torques lead to preferential changes in their mean positions and angular orientations which can be detected under a microscope kirksey1988brownian (); kawata1992movement (). In the following, we estimate the additional accelerations using certain simplifying approximations. A rigorous description of dynamics volpe2011microswimmers (); mijalkov2013sorting () is beyond the scope of this work and at this point unnecessary since we would like to merely detect presence of the persistent thermal photonic phenomena.
The particles are absorptive, nonmagnetic (not influenced by presence or absence of magnetic field) and much smaller in size (m) and hence dipolar in nature at surfacepolariton wavelengths (m) of InSb. The entire system is at thermal equilibrium room temperature. Magnetic field of T is applied along direction resulting into persistent planar heat current and persistent thermal photon spin . Using expressions from references canaguier2013force (); nieto2010optical (); nieto2015optical (), timeaveraged optical force and optical torque parallel to the surface are calculated where is the refractive index of the aqueous medium. Here, is dipolar vacuum polarizability defined such that induced dipole moment is . We assume (water) and spherical particles of diameter m, mass density kgm and small (nonresonant) imaginary part of dispersionless polarizability where is the volume of the nanoparticle. For these parameters, the particles experience linear acceleration of m/s and angular acceleration of rad/s at distance of m from the interface. Note that we have ignored the influence of dipolar particles on the surface polaritons assuming sparse distribution in the aqueous medium and nonresonant polarization response at polariton wavelength. The accelerations can be further enhanced by choosing lossy particles or by reducing the particle size. They are enhanced by orders of magnitude by using resonant particles such as doped silicon basu2009review () whose response can be tailored by tuning the doping concentration. Overall, these calculations indicate that the persistent phenomena can be noticed based on statistical and time averaged translational and rotational diffusion of suspended particles along and about transverse directions. For clearly observing the rotational motion, one can use anisotropic particles in the shape of microrods as depicted in fig. 5. Since the additional changes in preferential directions are not possible in absence of magnetic field or with other homogeneous reciprocal media, their mere presence will be a clear indicator of the persistent thermal photonic phenomena. They can be readily perceived upon seeing through the microscope or by methodically tracking the particle movements.
Universal behavior of PTPS and PPHC. Here we describe the universal behavior of PTPS and PPHC with generic biansiotropic material types. A bianisotropic medium is often considered in the literature kriegler2010bianisotropic (); asadchy2018bianisotropic () to represent a superset of all types of media, more commonly described with following constitutive relations assuming local material response (in the frequency domain):
(11) 
are dimensionless permittivity and permeability tensors and are magnetoelectric coupling tensors. Based on the existing literature we categorize bianisotropic materials into following five wellstudied material types:

Isotropic materials: Most naturally existing dielectric or metallic materials with scalar and .

Uniaxial/biaxial anisotropic materials such as birefringent crystals that have diagonal and with unequal diagonal entries and .

Gyroelectric (Magnetooptic) materials such as semiconductors in external magnetic fields ishimaru2017electromagnetic () for which has nonzero offdiagonal components, is a scalar and .

Gyromagnetic materials such as ferromagnets and ferrites rodrigue1988generation () for which has nonzero offdiagonal components, is a scalar and .

Magnetoelectric materials such as chromite CrO, multiferroics CuCrO pyatakov2012magnetoelectric (); albaalbaky2017magnetoelectric () and topological insulators such as BiSe laforge2010optical () for which are diagonal and are nonzero tensors.
PTPS and PPHC in thermal nearfield of biansiotropic materials  

No.  Material Type  Example  
Uniaxial Anisotropic  , ,  
Gyroelectric (Magnetooptic)  , , , Gyrotropy axis along (parallel to surface)  
Gyromagnetic  , , , Gyrotropy axis along (parallel to surface)  
Gyroelectric (Magnetooptic)  , , , Gyrotropy axis along (perpendicular to surface)  
MagnetoElectric (Nonreciprocal)  , , , Isotropic magnetoelectric coupling  
MagnetoElectric (Reciprocal)  , ,  
MagnetoElectric (Nonreciprocal)  , , , Magnetoelectric cross coupling between fields 
Apart from the naturally existing examples given above, there exists a huge range of artificially designed metamaterials with/without bias fields/currents that can effectively provide any combination of these material types kriegler2010bianisotropic (); asadchy2018bianisotropic (). It is wellknown that such a bianisotropic material is reciprocal caloz2018electromagnetic () when,
(12) 
The material is nonreciprocal if atleast one of these conditions is violated.
Table 1 summarizes PTPS and PPHC with generic bianisotropic media with suitable representative examples that describe the presence or absence of PTPS and PPHC with specific material type. Both isotropic and uniaxial/biaxial anisotropic materials being reciprocal in nature, do not lead to any persistent spin or heat current. The first example considers uniaxial anisotropic material with its anisotropy axis parallel to the surface (breaking the rotational symmetry) and the full calculations confirm the absence of persistent phenomena. The second and third examples in the table correspond to gyroelectric and gyromagnetic materials having antisymmetric permittivity and permeability tensors with nonzero offdiagonal components (hence nonreciprocal). For both examples, the gyrotropy axis is assumed to be along which leads to PTPS along and PPHC along direction. It is also found that PTPS of gyroelectric material is mostly electrictype while that of gyromagnetic material is mostly magnetictype. While the chosen parameters lead to plasmonic enhancement of the persistent phenomena, other parameters (dielectric and ) also show the same (zero or nonzero) features. In the fourth example with gyrotropy axis along (perpendicular to surface), no PPHC is observed. The components of electric and magnetic type PTPS are nonzero but cancel each other leading to zero total thermal spin. This proves that nonreciprocity is not sufficient to observe PTPS and PPHC.
The fifth and sixth examples consider isotropic, dielectric permittivity and permeability () and diagonal magnetoelectric susceptibilities (). For nonreciprocal susceptibilities (), it is found that both PPHC and PTPS are zero although electric and magnetic type contributions to PTPS are nonzero. This is qualitatively similar to gyrotropic media with gyrotropy axis perpendicular to the surface. When the offdiagonal components of are nonzero as considered in the seventh (last) example, PTPS and PPHC parallel to the surface are observed. Interestingly, when and components of fields are coupled, PTPS is along direction while PPHC is along direction. When and components of fields are coupled, PTPS and PPHC are parallel to the surface but they are not necessarily in a specific direction (not tabulated). For a magnetoelectric medium, both electric and magnetic contributions to PTPS are comparable to each other as opposed to gyrotropic media where one of them dominates. Figure 6 summarizes the important findings based on this general analysis.
We note that the material parameters are not entirely arbitrary but follow certain symmetry relations and are also constrained by conditions of causality and passivity silveirinha2010comment (); gustafsson2010sum (). The causality constraint leads to KramerKronig relations for frequency dependent parameters and it requires separate examination for different types of materials silveirinha2011examining (). The passivity requires that the material matrix is such that is positive definite silveirinha2010comment (). We do not discuss the frequency dependence of various parameters here since single frequency calculations are sufficient to describe the nature of PTPS and PPHC for a given material type. However, we make sure that all the parameters satisfy the constraint of passivity and note that without such constraints, the persistent phenomena can be incorrectly deduced for reciprocal systems that are actually nonpassive (nonequilibrium). The seven examples above and the analysis presented here are sufficient to predict the presence or absence of the PTPS and PPHC and their nature (directions, electric/magnetic type PTPS) for any given type of bianisotropic material.
Iii Conclusion
Modern thermal photonics utilizes fluctuational electrodynamic paradigm to explore new phenomena (nearfield radiative heat transfer song2015near ()) and new regimes (nonreciprocity zhu2018theory (), nonlinearities khandekar2017near () and nonequilibrium jin2016temperature ()) which are inaccessible to older paradigm of radiometry and Kirchhoff’s laws. And yet, thermal spin photonics is so far limited to inquiries based on Kirchhoff’s laws shitrit2013spin (); wu2014spectrally (); yin2013interpreting (). This work demonstrates new thermal spin photonic phenomena in the nearfield of nonreciprocal materials analyzed within fluctuational electrodynamic paradigm. It invites new inquiries and paves the way for new fundamental and technological avenues in thermal spin photonics. In particular, it will be useful in the near future for shaping spinangular momentum related radiative heat transport phenomena such as our recent work on circularly polarized thermal light sources khandekar2019circular ().
We revealed that spinmomentum locking of thermally excited evanescent waves plays a fundamental role in facilitating the surprising thermal equilibrium features of PTPS and PPHC. This opens a new degree of freedom for directional radiative heat transport at the nanoscale. We provided a new result that polaritons in nonreciprocal photonics can carry spin magnetic moment. To the best of our knowledge, it has not been calculated from rigorous numerical computations before. Spin magnetic moment of polaritons in nonreciprocal photonics invites further separate inquiries related to spindependent quantum light matter interactions and spinquantization.
We showed that there are no thermodynamic contradictions with respect the existence of nonzero thermal photon spin (PTPS) and heat flux (PPHC) despite thermal equilibrium. However, it remains to be explored whether nonreciprocity and thermal equilibrium can always coexist, particularly for more complicated systems involving inhomogeneous magnetic field or several heat channels such as conduction. Pursuit of such fundamental inquiries in the future is important since paradoxical situations could be encountered that may invite revision or revalidation of fundamental theories kuhn2012structure ().
We described the universal behavior of the thermal spin photonic phenomena with a comprehensive analysis of key classes of nonreciprocal materials namely, gyroelectric, gyromagnetic and magnetoelectric media. This general analysis motivates future studies of thermal radiation li2018nanophotonic (), radiative heat transfer song2015near (), Casimir forces/torques capasso2007casimir () from generic bianisotropic materials. So far, thermal photonics exploring these effects is limited to a restricted class of isotropic, uni/biaxial anisotropic and in some recent works, gyroelectric materials. The theoretical framework and the Green’s function produced here can also be used to study environment assisted quantum nanophotonic phenomena such as Forster resonance energy transfer clegg1995fluorescence (), atomic transition shifts novotny2012principles () with general, bianisotropic materials. We leave all these promising directions of research aside for future work.
Iv Acknowledgments
This work was supported by the U.S. Department of Energy, Office of Basic Energy Science under award number DESC0017717 and the Lillian Gilbreth Postdoctoral Fellowship program at Purdue University (C.K.).
V Methods
Derivation of Green’s function. The Green’s function relating vector potential at to source current at in vacuum where denotes planar coordinates, is:
(13) 
is the total wavevector consisting of conserved parallel component and perpendicular component in vacuum . The () and () signs denote waves going away from and towards the interface respectively. It follows from Maxwell’s equations that they satisfy the dispersion relation where is real and can be real () or complex valued (). For simplicity, we write where is the angle subtended by with axis. Assuming , the integrand tensor is written using the polarization vectors ():
(14) 
The polarization vectors for with denoting waves going along directions are:
(15) 
The Fresnel reflection coefficient for describes the amplitude of polarized reflected light due to unit amplitude polarized incident light. The Green’s function above consists of two parts corresponding to the trajectories of electromagnetic waves generated at the source position and arriving at either directly () or upon reflection from the interface (). The Green’s function in Eq. 14 is derived for . For , only the vacuum part is modified to . The Fresnel reflection coefficients can be obtained experimentally or theoretically.
Fresnel reflection coefficients and polaritonic dispersion. We develop a tool to compute Fresnel reflection coefficients for a generic, homogeneous medium that can be described using the following constitutive relations assuming local response (in the frequency domain):
(16) 
are dimensionless permittivity and permeability tensors and are magnetoelectric coupling tensors. For isotropic materials, and are scalars. For gyroelectric (magnetooptic) and gyromagnetic media, the tensors and respectively have offdiagonal components and . The tensors are nonzero for magnetoelectric media. By writing electromagnetic fields inside the material as and using above constitutive relations in Maxwell’s equations, we obtain the following dimensionless dispersion equation for waves inside the material:
(17) 
Here, material tensor describes the constitutive relations and corresponds to the curl operator acting on plane waves. Because of the generality of this problem, we obtain numerically by solving for given . Depending on the nature of the material, there can be two (for isotropic media) or four (for anisotropic media) solutions of corresponding to propagation of electromagnetic waves. Overall, there are four eigensolutions spanning the nullspace of the matrix , out of which two solutions correspond to waves propagating in direction (transmitted waves in our geometry). The four Fresnel reflection coefficients are then obtained by matching the tangential components at the interface () of incident and reflected fields with the transmitted fields. Here, the transmitted fields are written in the basis of former two nullspace solutions while the incident and reflected fields are written in the basis of polarizations (Eq. 15). This procedure is also extended in this work to compute the polaritonic dispersion () of surface polaritons that decay on both sides of the interface. While that calculation does not involve Fresnel coefficients, the boundary conditions again lead to a homogeneous, linear problem of the form where contains the coefficients describing the decomposition of polaritonic fields into four eigenstates (polarizations in vacuum and two propagating solutions inside the medium) at the interface. By numerically solving , the polaritonic dispersion is obtained. The associated nullspace describes the polaritonic fields. Note that since is assumed to be realvalued and nondecaying, is complexvalued with the imaginary part describing the finite lifetime (quality factor) of the polaritons.
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