On orbital angular momentum conservation in Brillouin light scattering within a ferromagnetic sphere

Magnetostatic modes supported by a ferromagnetic sphere have been known as the Walker modes, each of which possesses an orbital angular momentum as well as a spin angular momentum along a static magnetic field. The Walker modes with non-zero orbital angular momenta exhibit topologically non-trivial spin textures, which we call \textit{magnetic quasi-vortices}. Photons in optical whispering gallery modes supported by a dielectric sphere possess orbital and spin angular momenta forming \textit{optical vortices}. Within a ferromagnetic, as well as dielectric, sphere, two forms of vortices interact in the process of Brillouin light scattering. We argue that in the scattering there is a selection rule that dictates the exchange of orbital angular momenta between the vortices. The selection rule is shown to be responsible for the experimentally observed nonreciprocal Brillouin light scattering.


I. INTRODUCTION
The coupling between electron spins in solids and light is in general very weak. This is because the coupling is inevitably mediated by the orbital degree of the electrons and is realized through spin-orbit interaction for orbits and spins and electric-dipole interaction for orbits and light, respectively 1 . Although it is possible to coherently (non-thermally) manipulate collective excitations of spins in spin-ordered materials by means of ultrafast optics, where the electric field density of an optical pulse is high both temporally and spatially [2][3][4][5] , an attempt to realize coherent optical manipulation of magnons in the quantum regime is hindered by the weakness of the spin-light coupling 6 . Given the encouraging development of circuit quantum magnonics, where microwave photons and magnons are strongly coupled, enabling a coherent energy exchange at the single-quantum level 7-9 , the similar energy exchange between optical photons and magnons has been anticipated.
To overcome the weakness of the spin-light interaction, cavity optomagnonics has been investigated [10][11][12][13][14][15][16] . In cavity optomagnonics, the density of states of optical modes are engineered with an optical cavity to enhance spinlight interaction. In particular, spheres of ferromagnetic insulators supporting whispering gallery modes (WGMs) for photons and a spatially uniform magnetostatic mode, called the Kittel mode, for magnons are used as a platform of the cavity optomagnonics. With spheres made of typical ferromagnetic insulator, yttrium iron garnet (YIG), the pronounced sideband asymmetry [11][12][13] , the nonreciprocity 11 , and the resonant enhancement 12,13 of magnon-induced Brillouin scattering have been demonstrated.
In this context, it is interesting to examine the behavior of magnetostatic modes beyond the simplest Kittel mode. The magnetostatic modes residing in a ferromagnetic sphere under a uniform static magnetic field are known as the Walker modes 17,18 . They exhibit, in general, topologically non-trivial spin textures about the axis along the applied magnetic field and might be called magnetic quasi-vortices. The magnetic quasi-vortices can be characterized by their orbital angular momenta along the symmetry axis 19,20 . Photons in optical whispering gallery modes possess not only spin angular momenta but also orbital angular momenta, too, which echoes the concept known as optical vortices 21 . Within the ferromagnetic sphere, the optical vortices can interact with the magnetic quasi-vortices in the course of the Brillouin light scattering. The total orbital angular momentum is then expected to be conserved as long as the symmetry axis of the WGMs coincides with that of the Walker modes, imposing a selection rule on the Brillouin scattering processes.
In this article, the Brillouin scattering hosted in a ferromagnetic sphere is theoretically investigated putting a special emphasis on the orbital angular momentum exchange between the optical vortices and the magnetic quasi-vortices. We establish a selection rule imposed by the orbital angular momentum conservation for the Brillouin scattering hosted in a ferromagnetic sphere. The experimentally observed Brillouin scattering by various Walker modes reported in the accompanying paper 22 , which reveals that the scattering is either nonreciprocal or reciprocal depending on the orbital angular momentum of the magnetic quasi-vortices, is then analyzed with the theory developed here and found to be explained well. The result would provide a new area for chiral quantum optics 23 and topological photonics 24,25 based on optical vortices and magnetic quasi-vortices.

II. ORBITAL ANGULAR MOMENTA
The schematics of the cavity optomagnonic system we investigate is shown in Fig. 1, where the Walker mode Here the distribution of the transverse magnetization of the (4, 0, 1) Walker mode on the equatorial plane is shown as an example. The Walker modes and the WGMs are assumed to share the symmetry axis (z-axis) along a static magnetic field H. and the WGMs share the symmetry axis (z-axis) along a static magnetic field H. The Walker modes and the WGMs generally exhibit nonzero orbital angular momenta. In this section we analyze the orbital angular momenta of these modes.

A. Orbital angular momenta of Walker modes
The orbital angular momentum density l (mmag) z of a magnon along the static magnetic field H ( z-axis) can be deduced from the dependence of the transverse magnetizations, M x (t) and M y (t), on the azimuthal angle φ as 19,20 where M ⊥ (t) = M 2 x (t) + M 2 y (t). As for the Walker mode with the index (n, m mag , r) 17,18 the orbital angular momentum L (mmag) z can be given by the volume integral z = 0, and L (0) z = 1, corresponding to the winding numbers of the respective spin textures of the Walker modes. Note that the magnetic field is applied parallel to z axis. of l (mmag) z over the entire sphere and depends on the index m mag , that is, While the Kittel mode [(1, 1, 0) mode] has no orbital angular momentum, L (1) z = 0, (4, 0, 1) and (3,1, 1) modes, for instance, have L (0) z ≈ 1 and L (−1) z ≈ 2, respectively. The approximation in the last line of Eq. (2) is due to the dipolar interaction with broken axial symmetry. As the applied static magnetic field H approaches infinity, the Zeeman energy becomes dominant over the dipole interaction energy, and thus "≈" becomes "=" in Eq. (2). Note also that for the Walker modes with n = m mag and n = m mag + 1, Eq. (2) is exact. We call the Walker modes with non-zero L (mmag) z as magnetic quasi-vortices. The prefix "quasi-" emphasizes the fact that the orbital angular momentum we defined in Eq. (2) is the approximated one and the fact that magnons are quasi-particle with finite lifetime. Figure 2 shows the spatial distributions of the transverse magnetizations for the representative Walker modes (1, 1, 0), (3,1, 1), (3, 1, 1), and (4, 0, 1). The modes having non-zero L z [e.g., (3,1, 1) and (4, 0, 1) in Fig. 2] exhibit the topologically non-trivial spin textures. Note that the orbital angular momentum L z here plays a similar role as the winding number or the skyrmion number in other literature 26 .

B. Orbital angular momentum of WGMs
The electric field of the WGM in an axially symmetric dielectric material has been extensively studied 27 . Now, for simplicity, we focus on the azimuthal mode index m which characterizes the azimuthal profile of the electric field of the fundamental WGM. In the spherical ba- the electric field of the WGMs of the counterclockwise (CCW) orbit can be written as where E (TE) and E (TM) correspond to the transverse electric (TE) and the transverse magnetic (TM) WGMs, respectively, and φ is the azimuthal angle. Note that the time-dependent electric field as a whole is written as For the clockwise (CW) orbit, the electric fieldsĒ (TE) andĒ (TM) can be written as E i (E o ) in Eqs. (4) and (7) shall be called the inner (outer ) component of the TM mode. To see this, Fig. 3(a) shows the radial intensity distributions of two components |E i | 2 and |E o | 2 (magenta and green dotted lines) along with the intensity profiles of the transverse for the TM electric field of a WGM. We can see that |E i | 2 has its maximum in the inner part of the resonator compared to |E o | 2 . The shift of the "centers of gravity" of the two components, |E i | 2 and |E o | 2 is a manifestation of the spin-Hall effect of light 28,29 , which originates from the spin-orbit coupling of light 30 .
From the dependence of the electric field on φ, the orbital angular momentum L z of the WGM, that is, the optical vortex 21 , can be straightforwardly deduced. First, let us consider the CCW orbit. As for the TE mode with the azimuthal mode index m = m TE , since there is no spin angular momentum, the orbital angular momentum is given by As for the TM mode with m = m TM , however, the spinorbit coupling of light has to be taken into account 30 . For the CW orbit, the similar argument leads us to the following: and the total angular momentum J (CW,TM,mTM) = −m TM is again well-defined. Note that for the CW orbit the outer (inner) component of TM mode is associated with σ + (σ − ), that is opposite to that for the CCW orbit. The orbital angular momenta of the WGMs can be visualized by sketching the trajectory of the head of the polarization vector of the electric fields [Fig 3(b)]. When the mode index is 10, the orbital angular momentum reads 10, 9 and 11 for the TE, inner TM, and outer TM components, respectively.

A. Magnetic Quasi-Vortices-Optical Vortices Interaction
Let us now see that the total orbital angular momentum is conserved in the Brillouin scattering process. The thorough treatment of the Brillouin scattering by magnons in WGMs can be found in Ref. [16]. In the following, we emphasize the role of orbital angular momenta in the Brillouin scattering process. The interaction Hamiltonian representing the Brillouin scattering is where the integrand E is the energy flux density and the integral runs over infinity in time t and the volume V of the WGM, E 1 (t) = E 1 e −iω1t and E * 2 (t) = E * 2 e iω2t are the input and scattered electric fields of WGMs, respectively. Here, the permittivity tensorǫ can be written in the Cartesian basis as 31 The interaction between the magnetic quasi-vortices and optical vortices in the course of the Brillouin scattering process can be understood best in the spherical basis. In this basis the permittivity tensor can be written as Here the term ǫ 0 ǫ r M 0 in Eq. (15) has been neglected. Henceforth, the term f M s M z in Eq. (16) is also ignored for it is independent of time and give no contribution to the Brillouin scattering. Here ω m /2π is the resonant frequency of the concerned Walker mode with the azimuthal mode index of m mag .
In the spherical basis the time-dependent transverse magnetization is given by Here note that the creation (annihilation) of a magnon decreases (increases) the spin angular momentum. As we shall show, the Brillouin scattering stems from the term with M + (M − ) representing the Stokes-scattering (anti-Stokes-scattering) associated with the creation (annihilation) of a magnon. Since the TE-to-TM or TM-to-TE transition process changes the spin angular momentum of magnon, these transitions give nonzero contributions to the Brillouin scattering given the conservation of the spin angular momentum. We shall see this more clearly in Sec. III B.

B. Selection rules
Since the interaction depends on the direction of the input field and its polarization, let us first suppose that the input field is the CW TE mode with mode index of m TE , that is, E 1 (t) = E (mTE) e −imTEφ e −iω1tê * 0 . In this case the Brillouin scattering results in producing photons in the CW TM mode as seen in the following. We can straightforwardly extend the argument to other cases, e.g., the TM mode input or the input to the CCW orbit.
With the CW TE mode as the input field, the energy flux density E in Eq. (14) reads where The first (second) term in the right-hand side of Eq. (18) represents the Stokes (anti-Stokes) scattering. The possibility of the scattered light being the CCW WGM is denied given the fact that we are concerned only with cases where L for the Stokes scattering and for the anti-Stokes scattering. Since the optical densities of states are modified in the presence of the WGMs, the probabilities of the scattering processes are affected by them, too. Furthermore, because of the axial symmetry of the system, the conservation of the total angular momentum is expected. The designated WGM of the Brillouin scattering can then be specified by the selection rule obtained by the conservation of the orbital angular momentum. To see this, we integrate E in Eq. (14) over the azimuthal angle φ as a part of the volume integral. From the first Stokes term in Eq. (18) As for the second anti-Stokes term in Eq. (18), the selection rule is and with Eqs. (2), (11), and (13), we have Next, let us briefly describe the results when the laser light is injected into the CCW-TE mode. The Stokes (anti-Stokes) scattering process gives the same conditions of the energy conservation, Eq.
that is, the same selection rule as Eq. (24). As for the Stokes scattering, represents the orbital angular momentum conservation, yielding that is, the same selection rule as Eq. (26). These selection rules regarding the orbital angular momentum are the main result of this paper. With the geometric birefringence 32-34 and densities of states of WGMs, these selection rules dictate the Brillouin light scattering by Walker-mode magnons hosted in a ferromagnetic sphere as shown below. In the next section we employ the selection rules to explain the experiment reported in the accompanying paper 22 , which reveals that the Walker-mode-induced Brillouin light scattering is either nonreciprocal or reciprocal depending on the orbital angular momentum of the magnon in the relevant Walker mode, that is, the magnetic quasi-vortex.