Abstract

A first-order perturbation theory similar to the one widely used in quantum mechanics is developed for transverse-electric and transverse-magnetic photonic resonance modes in a dielectric microsphere. General formulas for the resonance frequency shifts in response to a small change in the exterior refractive index and its radial profile are derived. The formulas are applied to three sensor applications of the microsphere to probe the medium in which the sphere is immersed: a refractive-index detector, an adsorption sensor, and a refractive-index profile sensor. When they are applied to a uniform change in the refractive index in the surrounding medium, the formulas give the same results that one would obtain from the exact resonance equations for the two modes. In the application to adsorption of a thin layer onto the sphere surface, the results are identical to the first-order terms in the exact formulas obtained for the adsorption layer. In the last-named example, a scheme is proposed for instantaneous measurement of the refractive-index profile near the sphere’s surface.

© 2003 Optical Society of America

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References

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  1. A. Serpengüzel, S. Arnold, and G. Griffel, “Enhanced coupling to microsphere resonances with optical fibers,” Opt. Lett. 20, 654–656 (1995).
  2. A. T. Rosenberger, J. P. Rezac, “Evanescent-wave sensor using microsphere whispering-gallery modes,” in Laser Resonators III, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 3930, 186–192 (2000).
    [CrossRef]
  3. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
    [CrossRef]
  4. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
    [CrossRef] [PubMed]
  5. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of opticalwhispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
    [CrossRef]
  6. C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of reso-nances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [CrossRef]
  7. G. Griffel, S. Arnold, D. Taskent, A. Serpenguezel, J. Connolly, and D. G. Morris, “Morphology-dependent resonances of a microsphere-optical fiber system,” Opt. Lett. 21, 695–697 (1996).
    [CrossRef] [PubMed]
  8. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
    [CrossRef] [PubMed]
  9. V. S. Ilchenko, X. S. Yao, and L. Maleki, “Pigtailing the high-Q microsphere cavity: a simple fiber coupler for optical whispering-gallery modes,” Opt. Lett. 24, 723–725 (1999).
    [CrossRef]
  10. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
    [CrossRef]
  11. D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
    [CrossRef]
  12. 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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
    [CrossRef] [PubMed]
  13. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]
  14. A typographical error in Ref. 13 [Eq. (5b)] was corrected.
  15. L. M. Folan, “Characterization of the accretion of material by microparticles using resonant ellipsometry,” Appl. Opt. 31, 2066–2071 (1992).
    [CrossRef] [PubMed]

2003 (1)

2002 (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[CrossRef]

2000 (1)

A. T. Rosenberger, J. P. Rezac, “Evanescent-wave sensor using microsphere whispering-gallery modes,” in Laser Resonators III, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 3930, 186–192 (2000).
[CrossRef]

1999 (1)

1996 (2)

1995 (1)

1993 (1)

1992 (2)

1991 (1)

1990 (1)

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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

1989 (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of opticalwhispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

1962 (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Arnold, S.

Barber, P. W.

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
[CrossRef]

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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Braginsky, V. B.

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of opticalwhispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[CrossRef]

Chowdhury, D. Q.

Connolly, J.

Folan, L. M.

Gorodetsky, M. L.

M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. 21, 453–455 (1996).
[CrossRef] [PubMed]

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of opticalwhispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Griffel, G.

Hill, S. C.

D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
[CrossRef]

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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Holler, S.

Ilchenko, V. S.

Johnson, B. R.

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Lam, C. C.

Leung, P. T.

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of reso-nances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
[CrossRef]

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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[CrossRef]

Maleki, L.

Morris, D. G.

Rezac, J. P.

A. T. Rosenberger, J. P. Rezac, “Evanescent-wave sensor using microsphere whispering-gallery modes,” in Laser Resonators III, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 3930, 186–192 (2000).
[CrossRef]

Rosenberger, A. T.

A. T. Rosenberger, J. P. Rezac, “Evanescent-wave sensor using microsphere whispering-gallery modes,” in Laser Resonators III, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 3930, 186–192 (2000).
[CrossRef]

Savchenkov, A. A.

Serpenguezel, A.

Serpengüzel, A.

Taskent, D.

Teraoka, I.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[CrossRef]

Vollmer, F.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[CrossRef] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[CrossRef]

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Yao, X. S.

Young, K.

C. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of reso-nances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
[CrossRef]

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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4049 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Lett. (5)

Phys. Lett. A (1)

V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of opticalwhispering-gallery modes,” Phys. Lett. A 137, 393–397 (1989).
[CrossRef]

Phys. Rev. (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Phys. Rev. A (1)

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 application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[CrossRef] [PubMed]

Proc. SPIE (1)

A. T. Rosenberger, J. P. Rezac, “Evanescent-wave sensor using microsphere whispering-gallery modes,” in Laser Resonators III, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE 3930, 186–192 (2000).
[CrossRef]

Other (1)

A typographical error in Ref. 13 [Eq. (5b)] was corrected.

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Figures (4)

Fig. 1
Fig. 1

Sensitivity fTE (solid curves) and fTM (dashed curves) of the resonance frequencies of TE and TM modes, respectively, in a microsphere to a uniform refractive-index change in the surrounding medium, plotted as a function of size parameter k0a. These curves are for the first three orders (n=1, 2, 3) for m1=1.47 and m2=1.33. Dashed–dotted curves represent approximations [formulas (54) and (55)]; the lower curve represents fTE.

Fig. 2
Fig. 2

Expected sensing limit Δm2,min of refractive-index change in the surrounding medium in microspheres of different values of k0a for the first three modes. Solid curves are from the intrinsic linewidth of resonance; dashed curves denote the sensing limit for the current distributed-feedback laser driver’s resolution, Δk/k0=10-8.

Fig. 3
Fig. 3

Refractive-index profile across the interface surrounding a microsphere.

Fig. 4
Fig. 4

Frequency shift as a result of a refractive-index change of Δm2(r)=Δm2(){1-exp[-Γ(r-a)]} reduced by the shift that is due to a constant Δm2, plotted as a function of 1/Γ.

Equations (117)

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δω=δE=½Re(δpE*),
××E-k2m2(r)E=0.
Mlμ(r, θ, φ)=exp(iμφ)kr Sl(r, k)Xlμ(θ).
Nlμ(r, θ, φ)=exp(iμφ)k2m2(r)1rr Tl(r, k)Ylμ(θ)+Tl(r, k)r2Zlμ(θ).
Xlμ(θ)=iπlμ(θ)e^θ-τlμ(θ)e^φ,
Ylμ(θ)=τlμ(θ)e^θ+iπlμ(θ)e^φ,
Zlμ(θ)=l(l+1)Plμ(cos θ)e^r,
πlμ(θ)=μsin θ Plμ(cos θ),
τlμ(θ)=θ Plμ(cos θ).
-2r2+VTE(r, k)+l(l+1)r2sl(r, k)=k2Sl(r, k).
VTE(r, k)=k2[1-m2(r)].
-2r2+VTM(r, k)+l(l+1)r2Tl(r, k)=k2Tl(r, k),
VTM(r, k)=VTE(r, k)+2m(r)dmdrddr.
VTE(r, k)=k2(1-m12)(r<a)k2(1-m22)(r>a),
ψl(z)=zjl(z),χl(z)=znl(z),
Sl(r, k)=ψl(m1kr)Bl(k)ϕl(m2kr),
ϕl(m2kr)=χl(m2kr)+βl(k)ψl(m2kr),
ψl(m1ka)=Bl(k)ϕl(m2ka),
m1ψl(m1ka)=Bl(k)m2ϕl(m2ka),
m1ψl(m1k0a)ψl(m1k0a)=m2χl(m2k0a)χl(m2k0a).
Tl(r, k)=ψl(m1kr)Al(k)ϕl(m2kr),
ϕl(m2kr)=χl(m2kr)+αl(k)ψl(m2kr),
ψl(m1ka)=Al(k)ϕl(m2ka),
ψl(m1ka)/m1=Al(k)ϕl(m2ka)/m2.
1m1ψl(m1k0a)ψl(m1k0a)=1m2χl(m2k0a)χ1(m2k0a).
0Sl(r, k)Sl(r, ka)m2(r)dr
=m120aψl(m1kr)ψl(m1kar)dr
+Bl(k)Bl(ka)m22a ϕl(m2kr)ϕl(m2kar)dr
0Sl(r, k)Sl(r, ka)m2(r)dr
=m1k2-ka2 [-kψl(m1ka)ψl(m1kaa)
+kaψl(m1ka)ψl(m1kaa)]-m2k2-ka2 Bl(k)Bl(ka)×[-kϕl(m2ka)ϕl(m2kaa)+kaϕl(m2ka)ϕl(m2kaa)].
0 Sl(r, k)Sl(r, ka)m2(r)dr=0.
Sl(r, k)|m2(r)|Sl(r, ka)=0,
f |p|g f(r)p(r)g(r)dr.
k|m2|ka=0.
0 Tl(r, k)Tl(r, ka)dr
=0a ψl(m1kr)ψl(m1kar)dr+Al(k)Al(ka)
×0 ϕl(m2kr)ϕl(m2kar)dr
Tl(r, k)|Tl(r, ka)=k|ka=0,
f |g f(r)g(r)dr.
k0|m2|k0=ψl2a2 (m12-m22),
k0|k0=a2 ψl2m12m22-1χlχl2+l(l+1)(m1ka)2.
HTE,0(k0)=-2r2+VTE(r, k0)+l(l+1)r2.
HTE,0(k0)S0=k02S0,
δVTE=2k0δk(1-m2)-2k02mδm.
[HTE,0(k0)+δVTE](S0+δS)=(k0+δk)2(S0+δS).
HTE,0(k0)δS+δVTES0=k02δS+2k0δkS0.
δS=k ckSl(r, k),
k ckk2Sl(r, k)+k ck[HTE,0(k0)-HTE,0(k)]Sl(r, k)
+δVTESl(r, k0)
=k02k ckSl(r, k)+2k0δkSl(r, k0),
HTE,0(k0)-HTE,0(k)=(k02-k2)[1-m2(r)].
-ckb(k02-kb2)kb|m2|kb+kb|δVTE|k0
=2k0δkkb|k0.
k0|δVTE|k0=2k0δkk0|k0.
δkk0TE=-k0|mδm|k0k0|m2|k0.
δVTM=δVTE+2 d(δm/m)drddr.
δkk0TM=-k0|m-1δm|k0k0|k0+k0|1k02m2d(δm/m)drddr|k0k0|k0,
δkk0TM=-1k02 k0|k0-1δmm3×dT0dr2+l(l+1)r2 T02dr.
δkk0=-(energyperturbation)2×(electricenergy).
Δkk0TE=-m2Δm2m12-m22×l(l+1)(m2k0a)2-1+1m2k0aχlχl-χlχl2=-m2Δm2m12-m22χl+1χl-1χl2-1,
Δkk0TM=-m2Δm2m12-m22×l(l+1)(m2k0a)2-1-1m2k0aχlχl-χlχl2l(l+1)(m1k0a)2+χlχl2.
k0|mδm|k0=m2Δm2[Bl(k0)]2a[χl(m2k0r)]2dr=[Bl(k0)]2a2 m2Δm2×-χl2+l(l+1)(m2k0a)2-1χl2+χlχlm2k0a,
k0|m-1δm|k0
=[Al(k0)]2a2Δm2m2×-χl2+l(l+1)(m2k0a)2-1χl2+χlχlm2k0a,
k0|1k02m2d(δm/m)drddr|k0
=Δm2k02m201m2 T0dT0dr δ(r-a)dr=Δm2k0m22 [Al(k0)]2χlχl.
Δkk0TE-m2Δm2m12-m22l+122-(m2k0a)2-1/2,
Δkk0TM-m2Δm2m12-m222(l+½)2/(m2k0a)2-1[(l+½)2-(m2k0a)2]1/2(l+½)2(m1k0a)2+(l+½)2(m2k0a)2-1.
Δkk0TE-m2Δm2(m12-m22)3/21k0a,
Δkk0TM-m2Δm2(m12-m22)3/22-m22m121k0a.
w=2m2m12-m221[χl(m2k0a)]2
Δm2,min=2/(k0aχl2fTE)=2/[k0a(χl+1χl-1-χl2)].
Δm2,min=2k0aχl21(l+½)2/(m1k0a)2+(χl/χl)2.
δm2=αexσ2ε0m2 δ(r-a+).
k0|mδm|k0=αexσ2ε0 [Bl(k)]2ϕl2
Δkk0TE=-αexσε0a1m12-m22.
δmm3dT0dr2+l(l+1)2 T02dr
=αexσ2ε0m22 k02[Al(k0)]2χl2l(l+1)(m2k0a)2+χlχl2.
Δkk0TM=Δkk0TEl(l+1)/(m2k0a)2+(χl/χl)2l(l+1)/(m1k0a)2+(χl/χl)2.
Δkk0TM/Δkk0TE2(l/m2k0a)2-1(1+m22/m12)(l/m2k0a)2-1.
Δkk0TM/Δkk0TE2-m2m12.
Δkk0TE=-2m2Δm2a(m12-m22) t,
Δkk0TM=Δkk0TEl(l+1)/(m1k0a)2+(ψl/ψl)2l(l+1)/(m2k0a)2+(ψl/ψl)2,
Δkk0=Δkk0inf-Δkk0dep,
Δkk0TEdep=-k0|m2[Δm2()-Δm2(r)]|k0k0|m22|k0=-m2a[χl(m2k0r)]2[Δm2()-Δm2(r)]dr(a/2)[χl(m2k0a)]2(m12-m22).
Δm2(r)/Δm2()=1-exp[-Γ(r-a)]
ψl(m1kaa)=ψl(m1ka)+m1aΔkψl(m1ka),
ψl(m1kaa)=ψl(m1ka)+m1aΔkl(l+1)(m1ka)2-1ψl(m1ka),
limΔk0m11(k2-ka2) [-kψl(m1ka)ψl(m1kaa)
+kaψl(m1ka)ψl(m1kaa)]
=-12m1k-m1akψl2+l(l+1)m1ka-m1ka×ψl2+ψlψl,
ϕl(m2kaa)=ϕl(m2ka)+m2aΔkϕl(m2ka)+Δkβlψl(m2ka),
ϕl(m2kaa)=ϕl(m2ka)+m2aΔk×l(l+1)(m2ka)2-1ϕl(m2ka)+Δkβlψl(m2ka).
limΔk01m2(k2-ka2) [-kϕl(m2ka)ϕl(m2kaa)
+kaϕl(m2ka)ϕl(m2kaa)]
=-12m2k-m2kaϕl2+l(l+1)m2ka-m2ka×ϕl2+ϕlϕl+βl2m2,
limΔk01m2 (k2-ka2)limr[kϕl+1(m2kr)ϕl(m2kar)
-kaϕl(m2kr)ϕl+1(m2kar)]=βl/2m2,
limΔk00Sl(r, k)Sl(r, k+Δk)[m(r)]2dr
=ψl2(a/2)(m12-m22).
Sl|m2|Sl=ψl2a2 (m12-m22).
Tl|Tl=ψl2a2m12m22-1l(l+1)(m1ka)2+χlχl2.
Nν(ν sech s)-exp[ν(s-tanh s)][(π/2)ν tan s]1/2,
 nl(z)-l+12-1×expl+12(s-tanh s)cosh ssinh1/2 s,
z=l+12sech s.
nlnl-1zl+122-z21/2+12-12l+122/z2-1-1,
χlχl-1zl+122-z21/2-121-z2/l+122-1.
χl+1(z)χl(z)=nl+1(z)nl(z)l+½z+l+½z2-11/2.
χl-1χl+1χl2-1l+122-(m2k0a)2-1/2,
ψl(z)=l+1z ψl(z)-ψl+1(z),
ψl-1(z)+ψl+1(z)=2l+1z ψl(z),
χl(z)=l+1z χl(z)-χl+1(z),
χl-1(z)+χl+1(z)=2l+1z χl(z).
ψl(z)χl(z)-ψl(z)χl(z)=1.
l(l+1)z2-1+1zχlχl-χlχl2
=χl-1χl+1χl2-1=1zχl-1χl+ddzχl-1χl.

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