Magneto-optical coupling in whispering-gallery-mode resonators

J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baumberg, A. J. Ramsay, A. Nunnenkamp, and A. J. Ferguson

Phys. Rev. A 92, 063845 – Published 29 December 2015

Abstract

We demonstrate that yttrium iron garnet microspheres support optical whispering-gallery modes similar to those in nonmagnetic dielectric materials. The direction of the ferromagnetic moment tunes both the resonant frequency via the Voigt effect as well as the degree of polarization rotation via the Faraday effect. An understanding of the magneto-optical coupling in whispering-gallery modes, where the propagation direction rotates with respect to the magnetization, is fundamental to the emerging field of cavity optomagnonics.

Received 22 October 2015

DOI:https://doi.org/10.1103/PhysRevA.92.063845

Physics Subject Headings (PhySH)

Cavity quantum electrodynamics Nonlinear waveguides Optomechanics

Waveguides

Atomic, Molecular & Optical

Authors & Affiliations

J. A. Haigh^1,*, S. Langenfeld², N. J. Lambert², J. J. Baumberg², A. J. Ramsay¹, A. Nunnenkamp², and A. J. Ferguson²

¹Hitachi Cambridge Laboratory, Cambridge, CB3 0HE, United Kingdom
²Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United Kingdom

^*jh877@cam.ac.uk

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
(a) Schematic of experimental setup. The light from the fiber-coupled laser is collimated, linearly polarized vertically (perpendicular to the plane), and focused onto the internal surface of the coupling prism. Two photodiodes on the output of the prism monitor the reflection of the input beam with vertical polarization (I) and the emission of any horizontally linear polarized light (II) from the YIG sphere. To account for fluctuations in laser intensity, all measurements are normalized to the input power monitored with a beam splitter and photodiode (III) on the input path. The YIG sphere is mounted on a $x y z$ positioning stage to locate it at the coupling point. (b) The optical axis of the birefringent rutile prism is out-of-plane, which separates the output angles of the two linear polarizations for analysis via the two photodiodes. (c) Definition of the polar $(ϕ_{0})$ and azimuthal $(θ_{0})$ angles of the magnetization $\vec{M}$ .
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Figure 2
Reflected intensity (PD I) as a function of input laser free-space wavelength $λ$ for varying radius YIG spheres: (a) $500 μ m$ , (b) $250 μ m$ , and (c) $125 μ m$ . Red lines are separate fits to each peak, which give (d) the peak separation (free spectral range) and (e) $Q$ factors. The red line in (d) is the expected free spectral range for the WGMs, $Δ λ$ , given by Eq. (1), and the red line in (e) is the expected $Q$ factor.
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Figure 3
Reflected intensity (PD I) on a single resonance as a function of (a) in-plane angle $ϕ_{0}$ $(θ_{0} = 90^{\circ})$ and (b) out-of-plane angle $θ_{0}$ $(ϕ_{0} = 45^{\circ})$ of the magnetization with respect to the WGM plane. The fitted gray lines are used to extract (c), (d) the resonant frequency as a function of the two angles. The gray lines in (c) and (d) are the expected shifts considering only the Voigt effect given by Eq. (9).
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Figure 4
Emitted intensity from WGM with horizontal linear polarization (PD II), opposite to the input beam. Emitted intensity as a function of laser wavelength for (a) in-plane angle $ϕ_{0}$ and (b) out-of-plane angle $θ_{0}$ of the magnetization. Gray dotted lines are double peak fit to data. (c) Peak intensity of the two peaks as a function of out-of-plane angle $θ_{0}$ [from fitting in (b)]. (d) Wavelength shift of the two peaks as a function of out-of-plane angle $θ_{0}$ . The gray lines are the expected dependence based on Voigt effect, same as Fig. 3, for the two modes with mainly vertical (solid) and horizontal (dotted) linear polarizations.
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Figure 5
Calculated transmission spectra $(| S_{22} |^{2})$ for rotation of the magnetization (a) in-plane $(ϕ_{0} = 0 \to 360^{\circ}, θ_{0} = 0)$ and (b) out-of-plane $(θ_{0} = 0 \to 360^{\circ}, ϕ_{0} = 45^{\circ})$ . These correspond to the measured data in Figs. 2, 2. Calculated emission spectra $(| S_{21} |^{2})$ for rotation of the magnetization (c) in-plane $(ϕ_{0} = 0 \to 360^{\circ}, θ_{0} = 0)$ and (d) out-of-plane $(θ_{0} = 0 \to 360^{\circ}, ϕ_{0} = 45^{\circ})$ . These correspond to the measured data in Figs. 3, 3.
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