Abstract

A variational method was applied to find the wave function of a transverse-electric whispering gallery mode (WGM) in a spheroidal resonator of a uniform refractive index (RI). It was found that the electric field is tangential to the resonator surface as in the sphere, up to the linear order of the ellipticity. Using the wave function, the resonance shift due to adsorption of a thin, uniform dielectric layer onto the surface and the shift by a uniform RI change in the surroundings were evaluated in the perturbation theory. The shift by the RI change is not affected by the ellipticity, but the shift by the layer adsorption now depends on the meridional order. However, the correction is not large unless the ellipticity is large and the meridional order is away from the one for the equatorial mode of WGM.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on the silica microsphere surface using transverse electric/transverse magnetic whispering gallery modes,” Biophys. J. 92, 4466–4472 (2007).
    [CrossRef]
  2. X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
    [CrossRef]
  3. I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003).
    [CrossRef]
  4. I. Teraoka and S. Arnold, “Theory on resonance shifts in TE and TM whispering gallery modes by non-radial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381–1389 (2006).
    [CrossRef]
  5. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  6. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
    [CrossRef]
  7. I. Teraoka and S. Arnold, “Variational principle in whispering gallery mode sensor responses,” J. Opt. Soc. Am. B 25, 1038–1045 (2008).
    [CrossRef]
  8. V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
    [CrossRef]
  9. L. M. Folan, “Characterization of the accretion of material by microparticles using resonant ellipsometry,” Appl. Opt. 31, 2066–2071 (1992).
    [CrossRef]

2011

V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
[CrossRef]

2008

I. Teraoka and S. Arnold, “Variational principle in whispering gallery mode sensor responses,” J. Opt. Soc. Am. B 25, 1038–1045 (2008).
[CrossRef]

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

2007

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on the silica microsphere surface using transverse electric/transverse magnetic whispering gallery modes,” Biophys. J. 92, 4466–4472 (2007).
[CrossRef]

2006

2003

1993

1992

1990

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Arnold, S.

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Bhandarkar, U. V.

V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
[CrossRef]

Fan, X.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Folan, L. M.

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Jain, V.

V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
[CrossRef]

Johnson, B. R.

Joshi, S. C.

V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
[CrossRef]

Keng, D.

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on the silica microsphere surface using transverse electric/transverse magnetic whispering gallery modes,” Biophys. J. 92, 4466–4472 (2007).
[CrossRef]

Krishnagopal, S.

V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
[CrossRef]

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Noto, M.

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on the silica microsphere surface using transverse electric/transverse magnetic whispering gallery modes,” Biophys. J. 92, 4466–4472 (2007).
[CrossRef]

Shopova, S. I.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Sun, Y.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Suter, J. D.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Teraoka, I.

Vollmer, F.

White, I. M.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Young, K.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Zhu, H.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Anal. Chim. Acta

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef]

Appl. Opt.

Biophys. J.

M. Noto, D. Keng, I. Teraoka, and S. Arnold, “Detection of protein orientation on the silica microsphere surface using transverse electric/transverse magnetic whispering gallery modes,” Biophys. J. 92, 4466–4472 (2007).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Rev. A

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with applications to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef]

Phys. Rev. Spec. Top. Accel. Beams

V. Jain, U. V. Bhandarkar, S. C. Joshi, and S. Krishnagopal, “Analytical study of higher order modes of elliptical cavities using oblate spheroidal eigenvalue solution,” Phys. Rev. Spec. Top. Accel. Beams 14, 042002 (2011).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1.
Fig. 1.

(a) Spheroid with a cut out to show the radial distance A(θ). (b) Cross section of the spheroid by a plane containing the axis of rotation (z). Surface normal e^n is not parallel to the radial vector to e^r, and they form an angle of ξ. Likewise, the tangential vector within the cross section, e^t, is not parallel to the unit vector e^θ in the direction of polar angle θ.

Fig. 2.
Fig. 2.

Adsorption of a dielectric layer of thickness t. The distance across the layer in the radial direction is t/cosξ.

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

nsp(r)={n1(r<a)n2(r>a),
E=exp(imφ)krSl(r)Xlm(θ).
Sl(r)={ψl(n1kr)(r<a)Blχl(n2kr)(r>a),
Xlm(θ)=e^θimsinθPlm(cosθ)e^ϕθPlm(cosθ),
A(θ)=a1γcos2θ,
nel(r,θ)={n1(r<A(θ))n2(r>A(θ)).
e^n·e^r=[1+(A/A)2]1/2,
ξ2γsinθcosθ.
Δλλ=Δ(n2)E0*·Epdr2[n(r)]2E0*·E0dr,
Δ(n2)={n12n22(a<r<A(θ))0(otherwise).
Δ(n2)E0*·Epdr=2π(n12n22)1k2×0p{m2sin2θ[Plm(cosθ)]2+[θPlm(cosθ)]2}sinθdθaA(θ)[Sl(r)]2dr.
Δ(n2)E0*·Epdr=2π(n12n22)aγk2[Sl(a)]2(l+m)!(lm)!l2m22l,
[n(r)]2E0*·E0dr=2π(n12n22)ak2[Sl(a)]2(l+m)!(lm)!l(l+1)2l+1.
Δλλ=γ2(1m2l2)
E=exp(imφ)krSl(r(1γcos2θ))[(e^θ+e^rqsinθcosθ)imsinθPlm(cosθ)e^ψθPlm(cosθ)],
ρr(1γcos2θ).
Er=exp(imφ)krSl(ρ)imqcosθPlm(cosθ),
Eθ=exp(imφ)krSl(ρ)imsinθPlm(cosθ),
Eϕ=exp(imφ)krSl(ρ)θPlm(cosθ).
WIEI=k2N0+N1q+N2q2D0+D2q2.
N0=l+ηγl[3(l+m)22l22m2l24]γl2(1m2l2),
N1=4(2η1)γm2l[1(l+m)24l2],
N2=2ηm2l{1(l+m)24l2γ[1(l+m)22l2+(l+m)416l4]},
D0=l+γl2(1m2l2),
D2=m2l{1(l+m)24l2+γ[1(l+m)22l2+(l+m)416l4]}.
N1D02(N0D2N2D0)qN1D2q2=0.
q=2γ
WIEI=k2(1γl2m2l2).
Δkk=γ2l2m2l2.
Δ(n2)={n32n22(A(θ)<r<A(θ)+t/cosξ)0(otherwise).
Δ(n2)E0*·Epdr=2π(n32n22)1k2×0π{m2(4γ2cos2θ+1sin2θ)[Plm(cosθ)]2+[θPlm(cosθ)]2}sinθdθA(θ)A(θ)+t/cosη[Sl(ρ)]2dr.
Δ(n2)E0*·Epdr=2π(n32n22)tk2[Sl(a)]2(l+m)!(lm)!l.
Δλλ=n32n22n12n22ta(1γ2l2m2l2).
Δ(n2)={Δ(n22)(r>A(θ))0(otherwise).
Δ(n2)E0*·Epdr=2πΔ(n22)1k2×0π{m2(4γ2cos2θ+1sin2θ)[Plm(cosθ)]2+[θPlm(cosθ)]2}sinθdθA(θ)[Sl(ρ)]2dr.
A(θ)[Sl(ρ)]2dr=11γcos2θa2k2[Sl(a)]2{l(l+1)(n2ka)21+1n2kaχlχl(χlχl)2},
χlχl=1n2ka{[l(l+1)(n2ka)2]1/212[1(n2ka)2l(l+1)]1}.
A(θ)[Sl(ρ)]2dr11γcos2θa2k2[Sl(a)]21ka(neff2n22)1/2.
Δ(n2)E0*·Epdr=2πΔ(n22)a2k2[Sl(a)]21ka(neff2n22)1/2×0πsinθdθ{m2sin2θ[Plm(cosθ)]2+[θPlm(cosθ)]2}(1γcos2θ),
Δλλ=Δ(n22)2(n12n22)1ka(neff2n22)1/2.
(lm)!(l+m)!11Plm(x)Pnm(x)dx=22l+1δln,
(lm)!(l+m)!11Plm(x)Pln(x)1x2dx=1mδmnunlessm=n=0,
(2l+1)xPlm(x)=(lm+1)Pl+1m(x)+(l+m)Pl1m(x),
(2l+1)(1x2)1/2Plm1(x)=Pl1m(x)Pl+1m(x),
(1x2)ddxPlm(x)=(l+m)Pl1m(x)lxPlm(x).
(lm)!(l+m)!11[Plm(x)]2dx=22l+1,
(lm)!(l+m)!11(1x2)[Plm(x)]2dx=2(lm)(lm1)(2l+1)2(2l1)+2(l+m+2)(l+m+1)(2l+3)(2l+1)2,
(lm)!(l+m)!11(1x2)2[Plm(x)]2dx=2(lm)(lm1)(lm2)(lm3)(2l+1)2(2l1)2(2l3)+8(l+m+2)(l+m+1)(lm)(lm1)(2l+3)2(2l+1)(2l+1)2+2(l+m+4)(l+m+3)(l+m+2)(l+m+1)(2l+5)(2l+3)2(2l+1)2,
(lm)!(l+m)!1111x2[Plm(x)]2dx=1m,
(lm)!(l+m)!11(1x2)[ddxPlm(x)]2dx=2l(l+1)2l+1m,
(lm)!(l+m)!11(1x2)2[ddxPlm(x)]2dx=2(2l+1)2[l2(l+m+1)(lm+1)2l+3+(l+1)2(l+m)(lm)2l1].
(××E)r=exp(imϕ)krimr{Sl(ρ)2rγcosθPlm(cosθ)+Sl(ρ)2γcosθ[(1+3qsin2θ2qcos2θ)Plm(cosθ)2qsinθcosθθPlm(cosθ)]+qrSl(ρ)[2sinθθPlm(cosθ)+(l2+l+2)cosθPlm(cosθ)]},
(××E)θ=exp(imϕ)krim{Sl(ρ)2γrcosθθPlm(cosθ)+Sl(ρ)l(l+1)r2sinθPlm(cosθ)Sl(ρ)(12γcos2θsinθ2qγcos2θsinθ)Plm(cosθ)+qr[1rSl(ρ)Sl(ρ)(1γcos2θ)][sinθPlm(cosθ)cosθθPlm(cosθ)]},
(××E)ϕ=exp(imφ)kr1r{Sl(ρ)r[qm2Plm(cosθ)tanθl(l+1)θPlm(cosθ)]+[rSl(ρ)(12γcos2θ)Sl(ρ)2γsin2θ]θPlm(cosθ)+Sl(ρ)[m22γq(1γcos2θ)tanθ4γl(l+1)sinθcosθ]Plm(cosθ)}.
k2r2E·××E=1r2[Sl(ρ)]2m2[q2(l2+l+2)cos2θ+q+l(l+1)sin2θ][Plm(cosθ)]2+1r2[Sl(ρ)]2{2m2q(qsinθ1sinθ)cosθPlm(cosθ)θPlm(cosθ)+l(l+1)[θPlm(cosθ)]2}+m2qrSl(ρ)Sl(ρ){2γ[1+q(5sin2θ2)]cos2θ(1γcos2θ)}[Plm(cosθ)]2+m22rSl(ρ)Sl(ρ)[2γq2cos2θsinθ+q(1γcos2θ)2γsinθ]cosθPlm(cosθ)θPlm(cosθ)+2γrSl(ρ)Sl(ρ){2l(l+1)sinθcosθPlm(cosθ)θPlm(cosθ)+sin2θ[θPlm(cosθ)]2}+Sl(ρ)Sl(ρ){m2(4qγcos2θ12γcos2θsin2θ)[Plm(cosθ)]2(12γcos2θ)[θPlm(cosθ)]2}.
01r2[Sl(ρ)]2dr=(1γcos2θ)2Ir,
01rSl(ρ)Sl(ρ)dr=Ir,
0Sl(ρ)Sl(ρ)dr=I0+2l(l+1)Ir1γcos2θ,
I0k20n2[Sl(r)]2dr=k2a2(n12n22)[Sl(a)]2,
Ir1201r2[Sl(r)]2dr,
ηl(l+1)Ir/I0.
1a20Sl2dr<0Sl2r2dr<a0Sl2r3dr,
UL=1l(l+1)(ka)4l(l+1)aΓ(aΓ)2n12n22,
12πk2WI2Ir11{m2[γ+4ql2γx2+q2l2(x2γx4)]+γl2(13x2)}[Plm(x)]2dxm2I011(4qγx21γx21x2)[Plm(x)]2dx+I011(1γx2)(1x2)[ddxPlm(x)]2dx,
(lm)!(l+m)!WIk22πI0=l+ηγl[3(l+m)22l22m2l24]γl2(1m2l2)+4(2η1)qγm2l[1(l+m)24l2]+2ηq2m2l{1(l+m)24l2γ[1(l+m)22l2+(l+m)416l4]},
E*·E=1kr[Sl(ρ)]2{m2(q2cos2θ+1sin2θ)[Plm(cosθ)]2+[θPlm(cosθ)]2}.
(lm)!(l+m)!EIk42πI0l+γl2(1m2l2)+q2m2l{1(l+m)24l2+γ[1(l+m)22l2+(l+m)416l4]}.

Metrics