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Mie scattering analysis of spherical Bragg “onion” resonators

Wei Liang, Yong Xu, Yanyi Huang, Amnon Yariv, J. G. Fleming, and Shawn-Yu Lin

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Abstract

Combining the Mie scattering theory and a transfer matrix method, we investigate in detail the scattering of light by spherical Bragg “onion” resonators. We classify the resonator modes into two classes, the core modes that are confined by Bragg reflection, and the cladding modes that are confined by total internal reflection. We demonstrate that these two types of modes lead to significantly different scattering behaviors.

©2004 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. A SEM image of the onion resonator.

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Fig. 2.
Fig. 2. Light scattering by a spherical Bragg “onion” resonator.

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Fig. 3.
Fig. 3. Radial dependence of the electrical field of TE mode and the magnetic field of TM mode.

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Fig. 4.
Fig. 4. Spectrum of the onion resonator modes (calculated with Nclad =4).

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Fig. 5.
Fig. 5. Spectrum of total scattering efficiency. The number of Bragg pairs is Nclad =4.

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Fig. 6.
Fig. 6. Comparison of scattering resonances numerically obtained using the Mie scattering method developed in section 2, and those given by a Lorentzian approximation. The number of Bragg pair is Nclad =4.

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Fig. 7.
Fig. 7. Scattering resonances of the TE10 and TE24 modes (with Nclad =4 and 5). The dashed lines represent the resonant scattering coefficient -Re(αl) (whose values are shown on the left y-axis), and the solid lines represent the total scattering efficiency Qtotscatt (whose values are shown on the right y-axis). The dashed vertical lines give the position of modal wavelength.

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Fig. 8.
Fig. 8. Comparison of the scattering resonance width Γ res and the modal quality factor as a function of the Bragg pair number. ΓresL and Γrestot represent, respectively, the resonance width of the scattering coefficient — Re(α) and the total scattering efficiency Qtotscatt (see Fig. 7(a)).

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Fig. 9.
Fig. 9. Effect of absorption on scattering resonances. The number of Bragg pairs is Nclad =4.

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Fig. 10.
Fig. 10. Comparison between plane and Gaussian incident wave. In both (a) and (b), the upper line is under the assumption the incident wave is a plane wave; the lower one is under the assumption the incident wave is a Gaussian beam with the waist width of 5µm with the sphere located at the center of the beam. In (a), the values of the upper and lower lines are shown on the left and right y-axis respectively. The number of Bragg pair is Nclad =4.

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Equations (28)

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E → inc = e ̂ x e ikz = ∑ l = 1 ∞ i l 4 π ( 2 l + 1 ) 2 [ j l ( kr ) ( X → l , + 1 + X → l , − 1 ) + 1 k ∇ → × j l ( kr ) ( X → l , + 1 − X → l , − 1 ) ]
E → sc x = ∑ l = 1 ∞ i l 4 π ( 2 l + 1 ) 2 [ α l 2 h l 1 ( kr ) ( X → l , + 1 + X → l , − 1 ) + β l 2 ∇ → k × h l 1 ( kr ) ( X → l , + 1 − X → l , − 1 ) ]
E → sc x ≈ ie ikr kr ( cos ⁡ ϕ · S 2 ( θ ) e ̂ θ − sin ⁡ ϕ · S 1 ( θ ) e ̂ ϕ )
S 2 ( θ ) = ∑ l = 1 ∞ ( 2 l + 1 ) l ( l + 1 ) ( α l 2 P l 1 sin ⁡ θ + β l 2 dP l 1 d θ )
S 1 ( θ ) = ∑ l = 1 ∞ ( 2 l + 1 ) l ( l + 1 ) ( α l 2 dp l 1 d ⁡ θ + β l 2 P l 1 sin θ )
d σ sc d Ω = dP sc I inc d Ω = 1 2 · c ε 0 ∣ E → sc x ∣ 2 · r 2 1 2 · c ε 0 ∣ E → inc ∣ 2
= 1 k 2 ( ∣ S 2 ∣ 2 cos 2 ⁡ ϕ + ∣ S 1 ∣ 2 sin 2 ⁡ ϕ )
σ total = − π k 2 ∑ l = 1 ∞ ( 2 l + 1 ) Re [ α l + β l ]
[ H → E → ] = [ j l ( k n r ) X → l , m h l 1 ( k n r ) X → l , m Z n i k n ∇ → × j l ( k n r ) X → l , m Z n i k n ∇ → × h l 1 ( k n r ) X → l , m ] · [ A n B n ] ( TM )
[ E → H → ] = [ j l ( k n r ) X → l , m h l 1 ( k n r ) X → l , m − i Z n k n ∇ → × j l ( k n r ) X → l , m − i Z n k n ∇ → × h l 1 ( k n r ) X → l , m ] · [ C n D n ] ( TE )
[ j l ( k n r n ) h l 1 ( k n r n ) Z n k n ∂ ∂ r [ rj l ( k n r ) ] ∣ r n Z n k n ∂ ∂ r [ rh l 1 ( k n r ) ] ∣ r n ] ( A n B n ) =
[ j l ( k n + 1 r n ) h l 1 ( k n + 1 r n ) Z n + 1 k n + 1 ∂ ∂ r [ rj l ( k n + 1 r ) ] ∣ r n Z n + 1 k n + 1 ∂ ∂ r [ rh l 1 ( k n + 1 r ) ] ∣ r n ] ( A n + 1 B n + 1 ) ( TM )
[ j l ( k n r n ) h l 1 ( k n r n ) 1 Z n k n ∂ ∂ r [ rj l ( k n r ) ] ∣ r n ∂ Z n k n ∂ r [ r h l 1 ( k n r ) ] ∣ r n ] ( C n D n ) =
[ j l ( k n + 1 r n ) h l 1 ( k n + 1 r n ) ∂ Z n + 1 k n + 1 ∂r [ rj l ( k n + 1 r ) ] ∣ r n ∂ Z n + 1 k n + 1 ∂r [ r h l 1 ( k n + 1 r ) ] ∣ r n ] ( C n + 1 D n + 1 ) ( TE )
[ A co B co ] = M l TM · [ A out B out ]
[ C co D co ] = M l TE · [ C out D out ]
E → tot = ∑ l = 1 ∞ i l 4 π ( 2 l + 1 ) 2 [ ( j l + α l 2 h l 1 ) ( X → l , + 1 + X → l , − 1 ) + ∇ → k × ( j l + β l 2 h l 1 ) ( X → l , + 1 − X → l , − 1 ) ]
E → tot l , m ∝ ∇ ⇀ × ⌊ ( A out j l ( kr ) + B out h l 1 ( kr ) ) X → l , m ⌋ ( TM )
E → tot l , m ∝ ( C out j l ( kr ) + D out h l 1 ( kr ) ) X → l , m ( TE )
A out / B out = 2 / β l , C out / D out = 2 / α l
B co = ( M l TM ) 2 , 1 A out + ( M l TM ) 2 , 2 B out = 0
D co = ( M l TE ) 2 , 1 C out + ( M l TE ) 2 , 2 D out = 0
β l / 2 = − ( M l TM ) 2,1 / ( M l TM ) 2,2
α l / 2 = − ( M l TE ) 2,1 / ( M l TE ) 2,2
g l = exp { − [ ( l + 1 / 2 ) λ 2 π W 0 ] 2 }
Q tot scatt = σ total π R 2 = − 1 k 2 R 2 ∑ l = 1 ∞ ( 2 l + 1 ) Re [ α l + β l ]
− Re ( α or β ) = 2 1 + ( Q · 2 Δ λ / λ 0 ) 2
Q tot scatt = Q bg scatt + 2 l + 1 k 2 R 2 2 1 + ( Q · 2 Δ λ / λ 0 ) 2
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