Polarization-selective magneto-optical modulation

We study magneto-optical coupling in a ferrimagnetic sphere resonator made of Yttrium iron garnet. We find that the resonator can be operated in the telecom band as a polarization-selective optical modulator. Intermodulation gain can be employed in the nonlinear regime for amplification.


I. INTRODUCTION
Information is commonly transmitted by modulating a monochromatic carrier wave.The method of single sideband modulation (SSM) allows reducing both transmission power and bandwidth, in comparison with simpler methods such as amplitude, frequency and phase modulation [1].In the radio frequency band SSM can be implemented using electronic circuits, however, SSM implementation in the optical band is challenging, since it requires that different out of phase modulation methods are simultaneously applied [2,3].

II. EXPERIMENTAL SETUP
The experimental setup is schematically shown in Fig. 1.Optical components and fibers are red colored, whereas blue color is used to label microwave (MW) components and coaxial cables.A MW cavity made of a loop gap resonator (LGR) allows achieving a relatively large coupling between magnons and MW photons [29][30][31][32].The LGR is fabricated from a hollow concentric aluminium tube.A sapphire (S) strip of 260 µm thickness is inserted into the gap in order to increase its capacitance, which in turn reduces the frequency f c of the LGR fundamental mode.An FSR made of Yttrium iron garnet (YIG) having radius of R s = 125 µm is held by two ceramic ferrules (CF) inside the LGR.The two CFs, which are held by a concentric sleeve, provide transverse alignment for both input and output single mode optical fibers.Fiber longitudinal alignment is performed by maximizing optical transmission.(circulator), VNA (vector network analyzer), SA (spectrum analyzer) and SG (signal generator)] and coaxial cables are blue colored.TL2 together with two PBSs (labelled as PBS1 and PBS2) and two differential photo detectors (labelled as DPD1 and DPD2) operate as a polarization-selective optical spectrum analyzer (OSA) [28].A power amplifier is serially connected to the SG.The MWA is weakly coupled to the FSR-LGR system.
The angular frequency of the Kittel mode ω m is approximately given by ω m = µ 0 γ e H s , where H s is the static magnetic field, µ 0 is the free space permeability, and γ e /2π = 28 GHz T −1 is the gyromagnetic ratio [10].The applied static magnetic field H s is controlled by adjusting the relative position of a magnetized Neodymium using a motorized stage.The static magnetic field is normal to the light propagation direction k, and the magnetic field of MW drive is nearly parallel to k.The LGR-FSR coupled system is encapsulated inside a metallic rectangular shield made of aluminum.The LGR is weakly coupled to a microwave loop antenna (MWA).
The plot in Fig. 2 exhibits a vector network analyzer (VNA) reflectivity measurement of the LGR-FSR coupled system.The static applied magnetic field H s in this measurement is varied near the value corresponding to avoided-crossing between the FSR and LGR resonances.Both motorized polarization controllers (labelled as PC1 and PC2 in Fig. 1) have three optomechanical components (paddles), which act as either quarter or half wave plates.The paddles' angles of PC1 (PC2) are denoted by θ 1A , θ 1B and θ 1C (θ 2A , θ 2B and θ 2C ).The in-cident light state of polarization (SOP) can be manipulated using PC1.We observe that intensity of lower wavelength λ L − λ SB anti-Stokes sideband and higher wavelength λ L + λ SB Stokes side band depend on the input SOP.SSM in the transmission spectrum, with either single anti-Stokes side band, or with single Stokes side band, can be obtained by adjusting PC1.The plot shown in Fig. 4(a) exhibits the measured anti-Stokes side band intensity as a function of microwave driving frequency f p = ω p / (2π) and PC1 angle θ 1C near the avoidedcrossing region.The plot shown in Fig. 4(c) exhibits simultaneously measured Stokes side band intensity in the same region.We clearly observe appreciable anti-Stokes and Stokes intensity in Fig. 4 (a) and (c), respectively, when driving frequency ω p / (2π) becomes close to FSR resonance ω m / (2π).However, they are asymmetric.For a certain range of PC1 position, SSM is obtained, i.e.only one side band, either anti-Stokes or Stokes, is observed.Contrary to other experimental setups, in which the FSR is optically coupled by either a tapered optical fiber or a prism, for our setup, for which the measured optical transmission only weakly depends on the input wavelength λ L , SSM can be obtained in wide range of λ L .
A rotating lambda plate polarimeter is employed to measure the input SOP.The polarimeter measurements reveal that the input SOP for the two extreme cases (SSM of either anti-Stokes or Stokes peak) are orthogonal to each other (i.e.separated by a diameter on the Poincaré sphere).Our experimental setup (see Fig. 1) allows measuring the SOP of both sidebands.While the plots shown in Fig. 4 display the total optical intensity I T = I DPD1 + I DPD2 , the intensity I DPD1 (I DPD2 ) is separately displayed in the top (bottom) row plots shown in Fig. 5.These two intensities I DPD1 and I DPD2 represent two orthogonal SOP, which can be set by adjusting PC2 (see Fig. 1).The left (right) column plots in Fig. 5 display the measured intensity of the left anti-Stokes (right Stokes) sideband at wavelength λ L − λ SB (λ L + λ SB ), whereas the intensity at the central wavelength λ L is displayed by the central column plots in Fig. 5.For the measurements shown in Fig. 5, PC1 is set to a nearly SSM state.By varying the setting of PC2, we find that the central peak at wavelength λ L is maximized (minimized) in the same region where the sidebands at wavelength λ L ± λ SB are minimized (maximized).This observation implies that in the region of SSM, the SOP of the sidebands is nearly orthogonal to the SOP of the incident light.This orthogonality can be exploited at the receiver end of a data transmission system based on our proposed MO modulation, since it allows demodulation by polarization filtering-out of the carrier at wavelength λ L .

IV. MO COUPLING
The MO coupling giving rise to the optical sidebands originates from an interaction term in the system's Hamiltonian, which is denoted by V SB .This term V SB is commonly derived from the classical energy density associated with the interaction between magnetization and optical modes.For the case where only whispering gallery FSR optical modes participate in the interaction, the term V SB was derived in [11][12][13][14][15][16][17][18][19], whereas for our experimental configuration we consider the case where light propagates through the FSR bulk.
Consider an incident I (scattered S) optical field, having right and left handed circular polarization amplitudes E I+ and E I− (E S+ and E S− ), respectively.The timeaveraged energy density u m associated with MO coupling is given by u m = (1/4) Re U m , where and where The spherical symmetry of the FSR is partially broken by the two CFs that are employed for holding it (see Fig. 1).
In the semiclassical approximation V SB is derived from u m = (1/4) Re U m [see Eq. ( 1)].Consider a pair of optical modes having normalized scalar spatial waveforms, which in spherical coordinates are expressed as u n ′ (r, θ, ϕ) and u n ′′ (r, θ, ϕ), respectively.The contribution of this pair to the total interaction term V SB , which is denoted by V n ′ ,n ′′ , is expressed as where a n (b) is an annihilation operator for the n'th optical mode (magnon mode), and h.c.stands for Hermitian conjugate.The coupling coefficients g n ′ ,n ′′ ,± are given by (recall that in our experiment the static magnetic field is normal to the light propagation direction) where , and N s is the number of FSR spins (N s = 3.4 × 10 16 for the FSR under study).
The ratio of side band output optical power to the input optical power is denoted by η SB .The largest value of η SB is obtained at the triple resonance [11], for which the MW driving is tuned to the FSR resonance ω m , the laser frequency ω L matches the frequency of one optical mode, and the second one has a frequency detuned from ω L by ω m .For this case η SB ≃ (2n 0 R s g 0 /c) 2 N m [it is assumed that the overlap integral in Eq. ( 3) is of order unity], where N m is the averaged number of excited magnons in steady state.For the case where the MWA is nearly critically coupled to the FSR, at resonance N m ≃ P p / ( ω m κ m ), where κ m is the FSR damping rate.The values of P p = 20 dBm, ω m / (2π) = 3.8 GHz and κ m / (2π) = 1 MHz yield η SB ≃ 10 −5 .This rough estimate agrees with the experimentally observed value of η SB [see Fig. 5].

V. KERR NONLINEARITY
Magnetic anisotropy gives rise to Kerr nonlinearity in the FSR response [34].The nonlinearity can be exploited for modulation amplification [35].Modulation measurements in the nonlinear regime are shown in Fig. 6.The results indicate that the Kerr coefficient is negative (giving rise to softening).For the plots shown in the top (bottom) row of Fig. 6, the microwave driving frequency is swept upwards (downwards).The dependency on sweeping direction is attributed to nonlinearity-induced bistability, which, in turn, gives rise to hysteresis.

VI. SUMMARY
In summary, polarization-selective SSM in the telecom band is achieved using an FSR strongly coupled to an LGR.The modulator can be used in wide optical band, and it is compatible with ultra low temperatures.Future study will explore potential applications, including quantum state readout of superconducting circuits.
This work was supported by the Israeli science foundation, the Israeli ministry of science, and by the Technion security research foundation.

Appendix A: Transverse permittivity tensor
The evolution of electromagnetic waves propagating inside a magnetized medium is governed by a 3 × 3 permittivity tensor [36][37][38].Consider a Cartesian coordinate system (x, y, z), for which the propagation direction is parallel to the z direction.In this system the static magnetic field (magnetization vector) is parallel to a unit vector denoted by ĥ ( m).The angle between ĥ = (h x , h y , h z ) = (sin θ cos ϕ, sin θ sin ϕ, cos θ) and m = (m x , m y , m z ) is assumed to be small.
From the 3 × 3 permittivity tensor, a 2 × 2 transverse permittivity tensor ǫ T can be derived.In a basis of circular SOP ǫ T is given by ǫ T = n 2 0 I + ǫ m , where n 0 is the medium refractive index, I is the 2 × 2 identity matrix, and the 2 × 2 matrix ǫ m (in a basis of circular SOPs) is given by [39] where m ± = (m x ± im y ) / √ 2. For YIG in the telecom band, the refractive index is n 0 = 2.19, and the dimensionless MO coupling coefficient is Q s ≃ 10 −4 [33].
The eigenvalues of ǫ m /n 2 0 (A1) are given by For the Faraday configuration, for which m x = m y = 0 and m z = 1, i.e. m is parallel to the propagation direction, the eigenvectors of ǫ m /n 2 0 represent circular SOPs, the corresponding eigenvalues are ±Q s , and MO coupling gives rise to circular birefringence, whereas for the Voigt (Cotton-Mouton) configuration, for which m z = 0 and m 2 x + m 2 y = 1, i.e. m is perpendicular to the propagation direction, the eigenvectors of ǫ m /n 2 0 represent colinear SOPs, the corresponding eigenvalues are x + m 2 y 2 /4], and MO coupling gives rise to colinear birefringence.Note that for the Faraday configuration, the SOP rotation angle that is accumulated over a traveling distance of a single optical wavelength is 2πQ s .
The assumption that the angle between the static magnetic field and the magnetization vector is small implies that m z ′ ≃ 1 and |m ± ′ | ≪ 1.To first order in |m ± ′ |, ǫ m can be expanded as ǫ m = ǫ m0 + ǫ m+ m + ′ + ǫ m− m − ′ , where ǫ m0 , which is given by [compare with Eq. (A1)] accounts for static magnetization, and where ǫ m± , which is given by 1 , (A3) accounts for magnetization precession.

Figure 1 :
Figure 1: Experimental setup.Optical fibers are installed on both sides of the FSR for transmission of light through the sphere.Optical components [TL (tunable laser), Att (optical attenuator), PC (polarization controller) and PBS (polarization beam splitter)] and fibers are red colored, and MW components [MWA (microwave loop antenna), S (splitter), C(circulator), VNA (vector network analyzer), SA (spectrum analyzer) and SG (signal generator)] and coaxial cables are blue colored.TL2 together with two PBSs (labelled as PBS1 and PBS2) and two differential photo detectors (labelled as DPD1 and DPD2) operate as a polarization-selective optical spectrum analyzer (OSA)[28].A power amplifier is serially connected to the SG.The MWA is weakly coupled to the FSR-LGR system.

Figure 2 :
Figure 2: VNA reflectivity in dB units as a function of magnetic field Hs at applied microwave power of −30 dBm.

Figure 3 :
Figure 3: The transmitted optical spectrum.For this measurement the TL1 is set at optical power of 31 mW and wavelength λL of 1538.887nm, and the driving microwave is set at frequency ωp/ (2π) of 3.79 GHz and power of Pp of 20 dBm.

Figure 4 :
Figure 4: Side bands in dBm units.(a) anti-Stokes intensity as a function of PC1 angle θ1C.(b) anti-Stokes intensity as a function of magnetic field Hs.(c) Stokes intensity as a function of θ1C.(d) Stokes intensity as a function of Hs.The magnetic field Hs in (a) and (c) is tuned near avoidedcrossing regime.TL1 is set at optical power of 31 mW and wavelength of λL of 1537.7 nm, and the driving microwave power is set at Pp = 20 dBm.In (a) and (c), θ1A = 170 • and θ1B = 85 • , and θ1C is varied from 0 • to 170 • , whereas in (b) and (d) (θ1A, θ1B, θ1C) = (170 • , 85 • , 60 • ) (for this setting both Stokes and anti-Stokes peaks are clearly visible near the FSR resonance).

Figure 5 :
Figure 5: Sideband SOP.The measured intensity IDPD1 (IDPD2) is shown (in dBm units) in the plots labeled by the letter 'a' ('b').The intensity at wavelengths λL − λSB, λL and λL + λSB is shown in the plots labelled by the numbers '1', '2' and '3', respectively.The TL1 is set at optical power of 31 mW and wavelength of λL of 1538.9 nm, the driving microwave is set at power Pp of 20 dBm.
a transverse permittivity tensor.The static part ǫ m0 is given by Eq. (A2) of appendix A. The diagonal elements of ǫ m0 give rise to the static Faraday effect, whereas the static Voigt (Cotton-Mouton) effect originates from the off-diagonal elements of ǫ m0 [see Eq. (A2)].The terms ǫ m+ m + ′ and ǫ m− m − ′ account for the effect of magnetization precession, where ǫ m± is given by Eq. (A3) of appendix A, and m ± ′ represent amplitudes of magnetization precession.Note that the matrix ǫ m± is proportional to e ±iϕ , where ϕ is the azimuthal angle [see Eq. (A3)].

Figure 6 :
Figure 6: Spectral peaks (in dBm units) in the nonlinear regime as a function of MW driving power Pp.The intensity of the left (right) sideband at wavelength λL − λSB (λL + λSB) is shown in the plots in the left (right) column, whereas the plots in the central column show the intensity of the central optical peak (at TL1 wavelength λL).For the plots shown in the top (bottom) row, the frequency fp is swept upwards (downwards).