DE modes

47

for𝜖 ≫ {𝑓𝑀 , 𝑔𝑀 , 𝑔 𝑀 }. Finally, 𝐺^± = 𝑔𝑀 ± 𝑓

2𝜖 𝑀

√𝑠𝑉𝑖𝜔 𝛿 _, 𝛿_{, ∓} 𝛿 _{, ∓} , (2.58) which can be written in terms of the MO constants defined in Eqs. (1.14) - (1.15) and lead to Eq. (2.27). The orthonormality of the SHs reflects the conservation of angular momentum and the orthonormality of WGMs in the radial direction leads to the radial selection rule.

2.7.2. DE modes

We calculate scattering of a WGM with 𝑃 ≡ {𝜈, 𝑙, 𝑚, 𝑇𝐸} into one with index 𝑄 ≡ {𝜈 , 𝑙 , 𝑚 , 𝑇𝑀} by a particular DE magnon given by 𝐴 ≡ {0, 𝑙 , 𝑙 } . We take the case of𝑚 > 0 which implies 𝑚 < 0 as discussed in the main text, Sec. 2.3.2. Here, we assume𝑙, 𝑚 ≫ 1, |𝑙 − 𝑚|, and similarly 𝑙 , |𝑚 | ≫ 1, |𝑙 + 𝑚 |. The coupling constants

ℏ𝐺^± = 𝑖𝒢_±ℰ ℰ 𝑀 𝑎

4 ℛΘ_±, (2.59)

where we divided the integrals into the angular (Θ) and the radial (ℛ) parts. The angular integral is

Θ_±= ∫ 𝑑Ω𝑌 [sin 𝜃𝑒^± (𝑌 )^∗] (sin 𝜃𝑒^± ) , (2.60) where 𝑑Ω = sin 𝜃𝑑𝜃𝑑𝜙 is the angular differential. As 𝑌 ∼ 𝑒 , we get thatΘ_± is non-zero only if𝑚 − 𝑚 ± 𝑙 = 0. As discussed in the text, 𝑚 ≈ −𝑚 , so Θ = 0.

Θ can be evaluated by using (𝑌 )^∗= (−1) 𝑌 ,

𝑌 ≈ 𝐿 ^/

√2 𝜋 ^/ sin 𝜃𝑒 , (2.61)

and the identity,

∫ 𝑑Ω 𝑌 𝑌 (𝑌 )^∗≈ √𝐿 𝐿

2𝜋𝐿 ⟨𝐿 , 𝑀 ; 𝐿 , 𝑀 |𝐿 , 𝑀 ⟩ ⟨𝐿 , 0; 𝐿 , 0|𝐿 , 0⟩ , (2.62) where the approximations holds for𝐿 ≫ 1 for 𝑖 ∈ {1, 2, 3}. We get

Θ = 𝜋 ^/ 𝑙 ^/

√𝑙𝑙

𝜋 ⟨𝑙, 𝑚; 𝑙 , |𝑚 || 𝑙 , 𝑚 ⟩ ⟨𝑙, 0; 𝑙 , 0| 𝑙 , 0⟩ . (2.63) The radial integral is

ℛ = ∫ 𝑗 (𝑘 𝑎𝜌) [𝑗 (𝑘 𝑎𝜌) − 𝑗 (𝑘 𝑎𝜌)] 𝜌 𝑑𝜌, (2.64)

2

where 𝜌 = 𝑟/𝑎. It quantifies the overlap between the DE modes and WGMs in the radial direction. It can be estimated by realizing that𝜌 ≈ exp(−𝑙 (1 − 𝜌)) for 𝑙 ≫ 1. Therefore, the magnetization of the DE magnon decays rapidly in a reduced length scale of 1/𝑙 (or in a length scale of 𝑎/𝑙 ). In such a small length, we can approximate WGMs by their value at the surface (𝜌 = 1) giving

ℛ ≈ 𝑗 (𝑘 𝑎)

𝑙 [𝑗 (𝑘 𝑎) − 𝑗 (𝑘 𝑎)] . (2.65)

We use the asymptotic form of the Bessel’s function, Eq. (1.24), along with

𝑘 𝑎 = 𝑙 + (𝛽 − 2 ^/ 𝑃 𝑙 ^/ ) (2

𝑙)

/

, (2.66)

and the Taylor expansion of the Airy’s function around its zeroes for large 𝑙,

Ai (−𝛽 + 2 ^/ 𝑃

𝑙 ^/ ) ≈ Ai (−𝛽 )2 ^/ 𝑃 𝑙 ^/ . We can find a similar function for𝑗 (𝑘 𝑎). We simplify

ℛ ≈𝜋 4(4

𝑙𝑙 )

/

Ai (−𝛽 )Ai (−𝛽 )𝑃 (1 + 𝑃 ). (2.67) Putting all the constants in Eq. (2.59), we arrive at the result mentioned in Eq.

(2.32).

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3

Selection rules for cavity-enhanced Brillouin light scattering from magnetostatic modes

This chapter has been published as J. A. Haigh, N. J. Lambert, S. Sharma, Y. M. Blanter, G. E. W. Bauer, and A. J. Ramsay Phys. Rev. B 97, 214423 (2018) [1].

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