Abstract

Eigenmodes of spherical dielectric cavities are investigated within the framework of geometrical optics. A model for the coupling of internal and external rays is presented. Approximate expressions for the expansion coefficients of Mie theory are derived. Formulas for the modulus and the phase are given. The systematic behavior of the phase angle is investigated. All expressions are explicit equations and may be used to calculate the expansion coefficients as a function of order l or size parameter x. All features of the presented results are understandable in terms of light rays. A plane interface analog is presented. The so-called resonance regime (Λ/n<x<Λ), where n is the relative refractive index and Λ=l+1/2, is considered especially. An implicit equation for these resonance positions is rederived, and explicit relations for their strengths and widths are given. All findings are in agreement with results derived from Mie theory.

© 1999 Optical Society of America

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References

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  1. F. R. Faxvog, D. M. Roessler, “Optical absorption in thin slabs and spherical particles,” Appl. Opt. 20, 729–731 (1981).
    [CrossRef] [PubMed]
  2. C. F. Bohren, “Scattering by a sphere and reflection by a slab: some notable similarities,” Appl. Opt. 27, 205–206 (1988).
    [CrossRef] [PubMed]
  3. P. Chýlek, J. Zhan, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
    [CrossRef]
  4. P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990). Chap. 1.
  5. J. A. Lock, “Interference enhancement of the internal fields at structural resonances of a coated sphere,” Appl. Opt. 29, 3180–3187 (1990).
    [CrossRef] [PubMed]
  6. J. A. Lock, L. Yang, “Interference between diffraction and transmission in the Mie extinction efficiency,” J. Opt. Soc. Am. A 8, 1132–1134 (1991).
    [CrossRef]
  7. E. Hecht, Optik (Addison-Wesley, Bonn, 1989), Chap. 9.
  8. G. Roll, T. Kaiser, S. Lange, G. Schweiger, “Ray interpretation of multipole fields in spherical dielectric cavities,” J. Opt. Soc. Am. A 15, 2879–2891 (1998).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).
  10. P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
    [CrossRef]
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  12. Note that, in contrast to this discussion, in the book of Bohren and Huffman11 an outgoing wave is represented by a Hankel function of the second kind, which is a consequence of the exp(+ωt) time dependence assumed by Bohren and Huffman in contrast to the exp(-ωt) time dependence assumed here. In the literature a discussion of the consequences of a reversed time dependence can be found.13
  13. K. S. Shifrin, I. G. Zolotov, “Remark about the notation used for calculating the electromagnetic field scattered by a spherical particle,” Appl. Opt. 32, 5397–5398 (1993).
    [CrossRef] [PubMed]
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  15. I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
    [CrossRef]
  16. P. Chýlek, “Large-sphere limits of the Mie-scattering functions,” J. Opt. Soc. Am. A 63, 699–706 (1973).
    [CrossRef]
  17. J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
    [CrossRef]
  18. S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
    [CrossRef]
  19. J. D. Love, A. W. Snyder, “Optical fiber eigenvalue equation: plane wave derivation,” Appl. Opt. 15, 2121–2125 (1976).
    [CrossRef] [PubMed]
  20. G. Roll, T. Kaiser, G. Schweiger, “Controlled modification of the expansion order as a tool in Mie computations,” Appl. Opt. 37, 2483–2492 (1998).
    [CrossRef]
  21. C. C. Lam, P. T. Leung, K. Young, “Explicit asymptotic formulas for the positions, widths, and strength of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [CrossRef]
  22. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [CrossRef]

1998 (2)

1996 (1)

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

1993 (2)

1992 (1)

1991 (1)

1990 (1)

1989 (1)

1988 (1)

1981 (2)

F. R. Faxvog, D. M. Roessler, “Optical absorption in thin slabs and spherical particles,” Appl. Opt. 20, 729–731 (1981).
[CrossRef] [PubMed]

P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
[CrossRef]

1976 (1)

1973 (1)

P. Chýlek, “Large-sphere limits of the Mie-scattering functions,” J. Opt. Soc. Am. A 63, 699–706 (1973).
[CrossRef]

1967 (1)

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

1960 (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Barber, P. W.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990). Chap. 1.

Bohren, C. F.

C. F. Bohren, “Scattering by a sphere and reflection by a slab: some notable similarities,” Appl. Opt. 27, 205–206 (1988).
[CrossRef] [PubMed]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Chýlek, P.

P. Chýlek, J. Zhan, “Interference structure of the Mie extinction cross section,” J. Opt. Soc. Am. A 6, 1846–1851 (1989).
[CrossRef]

P. Chýlek, “Large-sphere limits of the Mie-scattering functions,” J. Opt. Soc. Am. A 63, 699–706 (1973).
[CrossRef]

Everitt, J.

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

Faxvog, F. R.

Felsen, L. B.

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Hecht, E.

E. Hecht, Optik (Addison-Wesley, Bonn, 1989), Chap. 9.

Hill, S. C.

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990). Chap. 1.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Johnson, B. R.

Kaiser, T.

Keller, J. B.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Lam, C. C.

Lange, S.

Leung, P. T.

Lock, J. A.

Love, J. D.

Ludlow, I. K.

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

Maurer, S. J.

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Moser, P. J.

P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
[CrossRef]

Murphy, J. D.

P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
[CrossRef]

Nagl, A.

P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
[CrossRef]

Roessler, D. M.

Roll, G.

Rubinow, S. I.

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Schweiger, G.

Shifrin, K. S.

Snyder, A. W.

Überall, H.

P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

Yang, L.

Young, K.

Zhan, J.

Zolotov, I. G.

Ann. Phys. (N.Y.) (1)

J. B. Keller, S. I. Rubinow, “Asymptotic solution of eigenvalue problems,” Ann. Phys. (N.Y.) 9, 24–75 (1960).
[CrossRef]

Appl. Opt. (6)

J. Opt. Soc. Am. A (5)

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

Phys. Rev. E (1)

I. K. Ludlow, J. Everitt, “Systematic behavior of the Mie scattering coefficients of spheres as a function of order,” Phys. Rev. E 53, 2909–2924 (1996).
[CrossRef]

Proc. IEEE (1)

S. J. Maurer, L. B. Felsen, “Ray-optical techniques for guided waves,” Proc. IEEE 55, 1718–1729 (1967).
[CrossRef]

Wave Motion (1)

P. J. Moser, J. D. Murphy, A. Nagl, H. Überall, “Creeping-wave excitation of the eigenvibrations of dielectric resonators,” Wave Motion 3, 283–295 (1981).
[CrossRef]

Other (6)

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Note that, in contrast to this discussion, in the book of Bohren and Huffman11 an outgoing wave is represented by a Hankel function of the second kind, which is a consequence of the exp(+ωt) time dependence assumed by Bohren and Huffman in contrast to the exp(-ωt) time dependence assumed here. In the literature a discussion of the consequences of a reversed time dependence can be found.13

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).

P. W. Barber, S. C. Hill, Light Scattering by Particles: Computational Methods (World Scientific, Singapore, 1990). Chap. 1.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1993).

E. Hecht, Optik (Addison-Wesley, Bonn, 1989), Chap. 9.

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Figures (11)

Fig. 1
Fig. 1

Plane reference configuration. A plane wave is incident on a dielectric slab. One interface of the slab has a reflective coating. The ray R through point P is phase retarded with respect to ray J1+ in homogeneous space.

Fig. 2
Fig. 2

Plots of sin δ (upper row) and δ (lower row) as a function of k0d for different angles of incidence ϑe. We plotted δ¯ +10δ˜ instead of δ=δ¯+δ˜ to enhance the visibility of the oscillations that are due to δ˜. These oscillations also show as humps on the trace of sin δ as compared with that of sin δ¯.

Fig. 3
Fig. 3

Amplitude ratio τ (upper row) and sin δ and cos δ (lower row) as a function of the dimensionless size parameter k0d for ni=1.5 and the two angles of incidence ϑe=0° (on the left) and ϑe=60° (on the right).

Fig. 4
Fig. 4

Pictorial representation of ray paths and wave fronts of fields whose radial dependence is given by Hankel functions of the first and the second kind, respectively. All rays are tangent to a caustic circle of radius rc, which is connected to the order l by krc=l+1/2=Λ. The Hankel function of the second kind represents an incoming wave, whereas the Hankel function of the first kind represents an outgoing wave. At large distances the field resembles a spherical wave.

Fig. 5
Fig. 5

Ray paths (a) without and (b) with a spherical scatterer. Caustics are drawn as dashed circles. The scattered ray S through point P is phase retarded with respect to the ray J1+ in homogeneous space.

Fig. 6
Fig. 6

Plot of the r dependence of the external electric field for different values of δ. In the radiating region (kr>Λ =100.5), a change in δ shifts the phase pattern radially, whereas it causes dramatic amplitude changes in the evanescent region (kr<Λ).

Fig. 7
Fig. 7

Pictorial representation of ray paths associated with the three cases, considered separately in Section 3. Arrows indicate the direction of propagation of rays in radiating fields, whereas curly lines represent evanescent fields. The caustics are represented by dashed circles. See the text for details.

Fig. 8
Fig. 8

Absolute value (top plot), real part (middle plot), and imaginary part (bottom plot) of the Mie expansion coefficient of the internal field c60 plotted as a function of the size parameter x in the interval 60.6x100. The exact values according to Mie theory are drawn as solid curves, and the approximate values are shown as dashed curves. The top plot also shows the envelopes of |c|, i.e., τres and τoff, as dotted–dashed curves. For xΛ there are small deviations between the exact and approximate values, but for larger size parameters the agreement is excellent.

Fig. 9
Fig. 9

Spectrum of the absolute value of the Mie expansion coefficient |bl| as a function of order l for a sphere of size parameter x=100 (upper plot) and trace of sin δ (lower plot). The continuous envelope in the upper plot is |sin δ|, which was calculated continuously. The vertical lines represent the Mie theory values, calculated for integer values of order l.

Fig. 10
Fig. 10

Absolute value (top plot), real part (middle plot), and imaginary part (bottom plot) of the Mie expansion coefficient of the internal field c60 plotted as a function of the size parameter x in the interval 40.3x60.4. Exact values are shown as solid curves, and approximate values are plotted as dashed curves. |c60| is plotted on a logarithmic scale; the real and imaginary parts of b60 are normalized by the absolute value.

Fig. 11
Fig. 11

Plot of |c60| on a logarithmic scale in a size parameter interval, which includes all considered cases. Exact values are plotted as solid curves, and approximate values are plotted as dashed curves. Significant deviations can be found only in the vicinity of the caustics (vertical lines).

Equations (109)

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Ei-(x)=τ exp[-i(kixx-π/2)],
Ei+(x)=τ exp[+i(kixx-π/2)].
Ee-(x)=exp(+iδ)exp[-i(kexx-π/2)],
Ee+(x)=exp(-iδ)exp[+i(kexx-π/2)]
α[Ei-(d)+Ei+(d)]=Ee-(d)+Ee+(d),
1αddx[Ei-(x)+Ei+(x)]x=d=ddx[Ee-(x)+Ee+(x)]x=d,
ϕˆix(x)=nik0x cos ϑi-π/2,
ϕˆex(x)=nek0x cos ϑe-π/2
ατ cos[ϕˆix(d)]=cos[ϕˆex(d)-δ],
κατ sin[ϕˆix(d)]=sin[ϕˆex(d)-δ]
τ=1α11+F sin2 ϕˆix(d)1/2,
F=κ2α4-1.
τres=1α,
τoff=ακ.
δ=ϕˆex(d)-arctanκα2tan ϕˆix(d)
δ=δ¯+δ˜,
δ¯=ϕˆex(d)-ϕˆix(d),
δ˜=-arctan((κ/α2)-1)tan ϕ¯ix(d)1+(κ/α2)tan2 ϕˆix(d)=-arctanG sin ϕˆix(d)cos ϕˆix(d)1+G sin2 ϕˆix(d),
Ee±(x)=exp(iδ˜)exp{±i[ϕˆix(d)+kex(x-d)]}.
cos δ=τ{[cos ϕˆix(d)cos ϕˆex(d)]/τres+[sin ϕˆix(d)sin ϕˆex(d)]/τoff},
sin δ=τ{[cos ϕˆix(d)sin ϕˆex(d)]/τres-[sin ϕˆix(d)cos ϕˆex(d)]/τoff},
kη(r)=Λr,
kr(r)=k2r2-Λ2r,
Φr(r)=rcrkr(ρ)dρ.
ϕr(kr)=k2r2-Λ2-Λ arccosΛkr,
ψr=0ifkrΛ,
ψr(kr)=Λ2-k2r2-Λ arccoshΛkr,
ϕr=0ifkr<Λ.
hl(1)(kr)exp[+i(ϕr(kr)-π/4)](krk2r2-Λ2)1/2,krΛ12exp[ψr(kr)](krΛ2-k2r2)1/2-iexp[-ψr(kr)](krΛ2-k2r2)1/2,kr<Λ,
hl(2)(kr)exp[-i(ϕr(kr)-π/4)](krk2r2-Λ2)1/2,krΛ12exp[ψr(kr)](krΛ2-k2r2)1/2+iexp[-ψr(kr)](krΛ2-k2r2)1/2,kr<Λ.
jl(kr)cos[ϕr(kr)-π/4](krk2r2-Λ2)1/2krΛ12exp[ψr(kr)](krΛ2-k2r2)1/2,kr<Λ,
hl(1)(kr)=jl(kr)+iyl(kr),
hl(2)(kr)=jl(kr)-iyl(kr).
yl(kr)sin[ϕr(kr)-π/4](krk2r2-Λ2)1/2,krΛ-exp[-ψr(kr)](krΛ2-k2r2)1/2,kr<Λ.
Ee(r)=exp(+iδ)hl(2)(nek0r)+exp(-iδ)hl(1)(nek0r),
Ee(r)=2jl(nek0r)cos δ+2yl(nek0r)sin δ.
Ei(r)=τhl(2)(nik0r)+τhl(1)(nik0r)=2τjl(nik0r)
αEi(a)=Ee(a),
1αddr[Ei(r)]r=a=ddr[Ee(r)]r=a.
rci=Λ/(nik0),
rce=Λ/(nek0).
sin β=rcia=Λnx,
ddr[krzl(kr)]krdzl(kr)dr.
αcjl(nx)=jl(x)-bhl(1)(x),
1αcddr[jl(nk0)r]r=a=ddr[jl(nek0r)-bhl(1)(nek0r)]r=a,
b=(1/2)[1-exp(-2iβ)],
αc exp(iβ)[h(2)(nx)+h(1)(nx)]
=exp(+iβ)h(2)(x)+exp(-iβ)h(1)(x).
|b|=|sin δ|,
Re(b)=sin2 δ,
Im(b)=sin δ cos δ,
|c|=τ,
Re(c)=+τ cos δ,
Im(c)=-τ sin δ.
b=11-iu,
u=Im(b)Re(b)=cot δ.
u=Jl(nx)Yl(x)-α2Jl(nx)Yl(x)Jl(nx)Jl(x)-α2Jl(nx)Jl(x)
jl(nx)jl(nx)-α2yl(x)yl(x)=0
ατcos[ϕˆr(nx)](nxn2x2-Λ2)1/2=cos[ϕˆr(x)-δ](xx2-Λ2)1/2,
κατsin[ϕˆr(nx)](nxn2x2-Λ2)1/2=sin[ϕˆr(x)-δ](xx2-Λ2)1/2,
κ=n2x2-Λ2x2-Λ21/2,
ϕˆr(kr)=ϕr(kr)-π/4.
τα2nκ1/2 cos[ϕˆr(nx)]=cos[ϕˆr(x)-δ],
τκnα21/2 sin[ϕˆr(nx)]=sin[ϕˆr(x)-δ],
τ=τres11+F sin2 ϕˆr(nx)1/2
F=κ2α4-1.
τres=nκα21/2,
τoff=nα2κ1/2.
δ¯=ϕˆr(x)-ϕˆr(nx),
δ˜=-arctanG sin ϕˆr(nx)cos ϕˆr(nx)1+G sin2 ϕˆr(nx).
cos δ=τ{[cos ϕˆr(nx)cos ϕˆr(x)]/τres+[sin ϕr(nx)sin ϕr(x)]/τoff},
sin δ=τ{[cos ϕˆr(nx)sin ϕˆr(x)]/τres-[sin ϕr(nx)cos ϕr(x)]/τoff}.
limx/Λ ϕˆr(x)=x-(l+1)π/2,
limx/Λ κ=n,
limx/Λ Re(b)=(α2 sin nx cos x-n cos nx sin x)2α4 sin2 nx+n2 cos2 nxforevenl(α2 cos nx sin x-n sin nx cos x)2α4 cos2 nx+n2 sin2 nxforoddl,
(τ/n)tan ξrescos ϕˆr(nx)
=(1/2)exp[+ψr(x)]cos δ-exp[-ψr(x)]sin δ,
(-τ/n)cot ξressin ϕˆr(nx)
=(1/2)exp[+ψr(x)]cos δ+exp[-ψr(x)]sin δ,
tan ξres=tanδB,res2=α2Λ2-x2n2x2-Λ21/2
cos δ=(-τ/n)exp[-ψr(x)][cot ξressin ϕˆr(nx)-tan ξrescos ϕˆr(nx)],
sin δ=(-1/2)(τ/n)exp[+ψr(x)][cot ξressin ϕˆr(nx)+tan ξrescos ϕˆr(nx)].
[cot ξressin ϕˆr(nx)±tan ξrescos ϕˆr(nx)]2
(tan ξres+cot ξres)sin2[ϕˆr(nx)±ξres],
sin2[ϕˆr(nx)+ξres]
sin2[ϕˆr(nx)-ξres]+sin[2ϕˆr(nx)]sin(2ξres),
τ=τres1Γ+F sin2[ϕˆr(nx)-ξres]1/2,
F=(tan ξres+cot ξres)2{exp[-4ψ(x)]+1/4},
τres=[n(tan ξres+cot ξres)]1/2 exp[-ψr(x)],
τoff=(1/2)[n(tan ξres+cot ξres)]1/2 exp[+ψr(x)],
Γ=sin[2ϕˆr(nx)]sin(2ξres).
2rciakr(ρ)dρ-π2-δB,res=(ν-1)2π,
τ=τres11+F sin2[ϕˆr(nx)-ξ]1/2,
sin2[ϕˆr(nx)-ξ]=sin2[ϕˆr(nx)-ξres]+Γ-1F.
Re(b)=sin2 δ=11+F sin2[ϕˆr(nx)-ξres].
F[ϕˆr(nx)-ξres]
Fϕˆr(nx0)+dϕˆr(nx)dxx0dx-ξres,
Fdϕˆr(nx)dxx0Δx2=1
dϕˆr(nx)dxx0=n2x02-Λ2x0,
Δx=2Λ2-x02x0(n2-1)pexp[2ψr(x0)],
p=1forTEmodesΛx02+Λnx02-1forTMmodes.
Δx=2(n2-1)px02yl2(x0),
τα2nκ1/2 exp[+ψr(nx)]
=exp[+ψr(x)]cos δ-2 exp[-ψr(x)]sin δ,
τκnα21/2 exp[+ψr(nx)]
=exp[+ψr(x)]cos δ+2 exp[-ψr(x)]sin δ,
κ=Λ2-n2x2Λ2-x21/2.
τ=2 exp[ψr(x)-ψr(nx)]α2/(nκ)+κ/(nα2)cos δ,
tan δ=12α2/(nκ)-κ/(nα2)α2/(nκ)+κ/(nα2)exp[2ψr(x)],

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