Abstract

It is well known that a fiber taper—an optical fiber whose size varies along its length—must exceed a certain minimum length to be adiabatic and low-loss. In contrast, we show that an optical fiber with a logarithmic refractive index profile can be adiabatically tapered over any length, however short. Its mode field distribution is independent of the fiber’s size and so remains the same along a taper. We report an experimental fiber in which tapers shorter than 2 mm can have losses as low as 0.03 dB. The fiber is compatible with standard telecoms fiber but with low bend loss and should have applications where it is desirable to make tapered fiber components (such as fused couplers and photonic lanterns) as short as possible. The index profile is analogous to the logarithmic potential of quantum mechanics.

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References

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2016 (1)

2015 (2)

2014 (1)

2013 (1)

2007 (1)

2004 (2)

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

1997 (1)

1996 (1)

D. A. Morales, “Analytical formulas for the eigenvalues and eigenfunctions of a d-dimensional hydrogen atom with a potential defined by Gauss’ law,” Int. J. Quantum Chem. 57, 7–15 (1996).
[Crossref]

1992 (2)

T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[Crossref]

M. Ohashi, M. Tateda, K. Tajima, and K. Shiraki, “Fluorine concentration dependence of viscosity in F-doped silica glass,” Electron. Lett. 28, 1008–1010 (1992).
[Crossref]

1991 (1)

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

1989 (1)

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7, 1139–1152 (1989).
[Crossref]

1987 (1)

J. D. Love, “Spot size, adiabaticity and diffraction in tapered fibres,” Electron. Lett. 23, 993–994 (1987).
[Crossref]

1986 (2)

K. P. Jedrzejewski, F. Martinez, J. D. Minelly, C. D. Hussey, and F. P. Payne, “Tapered-beam expander for single-mode optical-fibre gap devices,” Electron. Lett. 22, 105–106 (1986).
[Crossref]

D. B. Mortimore and J. V. Wright, “Low-loss joints between dissimilar fibres by tapering fusion splices,” Electron. Lett. 22, 318–319 (1986).
[Crossref]

1985 (1)

1984 (1)

1980 (1)

B. R. Johnson, “On a connection between radial Schrödinger equations for different power-law potentials,” J. Math. Phys. 21, 2640–2647 (1980).
[Crossref]

1979 (1)

C. Quigg and J. L. Rosner, “Quantum mechanics with applications to quarkonium,” Phys. Rep. 56, 167–235 (1979).
[Crossref]

1978 (2)

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a Si Cl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Solid State Sci. Tech. 125, 1298–1302 (1978).
[Crossref]

F. Gesztesy and L. Pittner, “Electrons in logarithmic potentials I. Solution of the Schrödinger equation,” J. Phys. A Math. Gen. 11, 679–686 (1978).
[Crossref]

1976 (1)

Abebe, M.

Alvarado, J. C.

Amezcua-Correa, R.

Antonio-Lopez, J. E.

Artiglia, M.

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7, 1139–1152 (1989).
[Crossref]

Birks, T. A.

S. Yerolatsitis, I. Gris-Sánchez, and T. A. Birks, “Adiabatically-tapered fiber mode multiplexers,” Opt. Express 22, 608–617 (2014).
[Crossref]

J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett. 22, 1129–1131 (1997).
[Crossref]

T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[Crossref]

K. Harrington, S. Yerolatsitis, D. Van Ras, and T. A. Birks, “Endlessly adiabatic fibre,” in Optical Fiber Communication Conference Postdeadline Papers (Optical Society of America, 2017), paper Th5D.2.

Black, R. J.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

Bures, J.

Burns, W. K.

Canning, J.

Cheung, G.

Coppa, G.

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7, 1139–1152 (1989).
[Crossref]

Di Vita, P.

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7, 1139–1152 (1989).
[Crossref]

DiGiovanni, D. J.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Digweed-Lyytikäinen, K.

DiMarcello, F. V.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Fan, M. F.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Fini, J. M.

Fleming, J. W.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Fontaine, N. K.

Gesztesy, F.

F. Gesztesy and L. Pittner, “Electrons in logarithmic potentials I. Solution of the Schrödinger equation,” J. Phys. A Math. Gen. 11, 679–686 (1978).
[Crossref]

Gibson, B. C.

Gonthier, F.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

Greborio, L.

Gris-Sánchez, I.

Guan, B.

Harrington, K.

K. Harrington, S. Yerolatsitis, D. Van Ras, and T. A. Birks, “Endlessly adiabatic fibre,” in Optical Fiber Communication Conference Postdeadline Papers (Optical Society of America, 2017), paper Th5D.2.

Henry, W. M.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

Hernández-Cordero, J.

Huang, B.

Huntington, S. T.

Hussey, C. D.

K. P. Jedrzejewski, F. Martinez, J. D. Minelly, C. D. Hussey, and F. P. Payne, “Tapered-beam expander for single-mode optical-fibre gap devices,” Electron. Lett. 22, 105–106 (1986).
[Crossref]

Jacques, F.

Jasapara, J.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Jedrzejewski, K. P.

K. P. Jedrzejewski, F. Martinez, J. D. Minelly, C. D. Hussey, and F. P. Payne, “Tapered-beam expander for single-mode optical-fibre gap devices,” Electron. Lett. 22, 105–106 (1986).
[Crossref]

Johnson, B. R.

B. R. Johnson, “On a connection between radial Schrödinger equations for different power-law potentials,” J. Math. Phys. 21, 2640–2647 (1980).
[Crossref]

Kippenberg, T. J.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

Knight, J. C.

Koenings, J.

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a Si Cl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Solid State Sci. Tech. 125, 1298–1302 (1978).
[Crossref]

Küppers, D.

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a Si Cl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Solid State Sci. Tech. 125, 1298–1302 (1978).
[Crossref]

Lacroix, S.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

J. Bures, S. Lacroix, C. Veilleux, and J. Lapierre, “Some particular properties of monomode fused fiber couplers,” Appl. Opt. 23, 968–969 (1984).
[Crossref]

Lapierre, J.

Leon-Saval, S. G.

Li, G.

Li, Y. W.

T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[Crossref]

Lines, M. E.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Lingle, R.

Lopez-Galmiche, G.

Love, J. D.

S. T. Huntington, B. C. Gibson, J. Canning, K. Digweed-Lyytikäinen, J. D. Love, and V. Steblina, “A fractal-based fiber for ultra-high throughput optical probes,” Opt. Express 15, 2468–2475 (2007).
[Crossref]

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

J. D. Love, “Spot size, adiabaticity and diffraction in tapered fibres,” Electron. Lett. 23, 993–994 (1987).
[Crossref]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Marcuse, D.

Martinez, F.

K. P. Jedrzejewski, F. Martinez, J. D. Minelly, C. D. Hussey, and F. P. Payne, “Tapered-beam expander for single-mode optical-fibre gap devices,” Electron. Lett. 22, 105–106 (1986).
[Crossref]

Minelly, J. D.

K. P. Jedrzejewski, F. Martinez, J. D. Minelly, C. D. Hussey, and F. P. Payne, “Tapered-beam expander for single-mode optical-fibre gap devices,” Electron. Lett. 22, 105–106 (1986).
[Crossref]

Monberg, E. M.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Morales, D. A.

D. A. Morales, “Analytical formulas for the eigenvalues and eigenfunctions of a d-dimensional hydrogen atom with a potential defined by Gauss’ law,” Int. J. Quantum Chem. 57, 7–15 (1996).
[Crossref]

Mortimore, D. B.

D. B. Mortimore and J. V. Wright, “Low-loss joints between dissimilar fibres by tapering fusion splices,” Electron. Lett. 22, 318–319 (1986).
[Crossref]

Nicholson, J. W.

Ohashi, M.

M. Ohashi, M. Tateda, K. Tajima, and K. Shiraki, “Fluorine concentration dependence of viscosity in F-doped silica glass,” Electron. Lett. 28, 1008–1010 (1992).
[Crossref]

Okonkwo, C. M.

Olivero, M.

Orta, R.

Payne, F. P.

K. P. Jedrzejewski, F. Martinez, J. D. Minelly, C. D. Hussey, and F. P. Payne, “Tapered-beam expander for single-mode optical-fibre gap devices,” Electron. Lett. 22, 105–106 (1986).
[Crossref]

Pellegrino, P.

Perrone, G.

Pittner, L.

F. Gesztesy and L. Pittner, “Electrons in logarithmic potentials I. Solution of the Schrödinger equation,” J. Phys. A Math. Gen. 11, 679–686 (1978).
[Crossref]

Potenza, M.

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7, 1139–1152 (1989).
[Crossref]

Quigg, C.

C. Quigg and J. L. Rosner, “Quantum mechanics with applications to quarkonium,” Phys. Rep. 56, 167–235 (1979).
[Crossref]

Reed, W. A.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Regio, P.

Rosner, J. L.

C. Quigg and J. L. Rosner, “Quantum mechanics with applications to quarkonium,” Phys. Rep. 56, 167–235 (1979).
[Crossref]

Ryf, R.

Sanchez-Mondragon, J.

Sharma, A.

M. Artiglia, G. Coppa, P. Di Vita, M. Potenza, and A. Sharma, “Mode field diameter measurements in single-mode optical fibers,” J. Lightwave Technol. 7, 1139–1152 (1989).
[Crossref]

Shiraki, K.

M. Ohashi, M. Tateda, K. Tajima, and K. Shiraki, “Fluorine concentration dependence of viscosity in F-doped silica glass,” Electron. Lett. 28, 1008–1010 (1992).
[Crossref]

Shubochkin, R.

Sillard, P.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Spillane, S. M.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

Steblina, V.

Stewart, W. J.

J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and F. Gonthier, “Tapered single-mode fibres and devices part 1: adiabaticity criteria,” IEE Proc. J. Optoelectron. 138, 343–354 (1991).
[Crossref]

Sun, Y.

Tajima, K.

M. Ohashi, M. Tateda, K. Tajima, and K. Shiraki, “Fluorine concentration dependence of viscosity in F-doped silica glass,” Electron. Lett. 28, 1008–1010 (1992).
[Crossref]

Tateda, M.

M. Ohashi, M. Tateda, K. Tajima, and K. Shiraki, “Fluorine concentration dependence of viscosity in F-doped silica glass,” Electron. Lett. 28, 1008–1010 (1992).
[Crossref]

Vahala, K. J.

T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004).
[Crossref]

Van Ras, D.

K. Harrington, S. Yerolatsitis, D. Van Ras, and T. A. Birks, “Endlessly adiabatic fibre,” in Optical Fiber Communication Conference Postdeadline Papers (Optical Society of America, 2017), paper Th5D.2.

Veilleux, C.

Velazquez-Benitez, A. M.

Villarruel, C. A.

Wilson, H.

D. Küppers, J. Koenings, and H. Wilson, “Deposition of fluorine-doped silica layers from a Si Cl4/SiF4/O2 gas mixture by the plasma-CVD method,” J. Solid State Sci. Tech. 125, 1298–1302 (1978).
[Crossref]

Wisk, P.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Wright, J. V.

D. B. Mortimore and J. V. Wright, “Low-loss joints between dissimilar fibres by tapering fusion splices,” Electron. Lett. 22, 318–319 (1986).
[Crossref]

Yablon, A. D.

A. D. Yablon, M. F. Fan, P. Wisk, F. V. DiMarcello, J. W. Fleming, W. A. Reed, E. M. Monberg, D. J. DiGiovanni, J. Jasapara, and M. E. Lines, “Refractive index perturbations in optical fibers resulting from frozen-in viscoelasticity,” Appl. Phys. Lett. 84, 19–21 (2004).
[Crossref]

Yerolatsitis, S.

S. Yerolatsitis, I. Gris-Sánchez, and T. A. Birks, “Adiabatically-tapered fiber mode multiplexers,” Opt. Express 22, 608–617 (2014).
[Crossref]

K. Harrington, S. Yerolatsitis, D. Van Ras, and T. A. Birks, “Endlessly adiabatic fibre,” in Optical Fiber Communication Conference Postdeadline Papers (Optical Society of America, 2017), paper Th5D.2.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

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A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Corning SMF-28e+ Product Information, https://www.corning.com/media/worldwide/coc/documents/PI1463_07-14_English.pdf .

Data Repository, https://doi.org/10.15125/BATH-00444 .

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Figures (10)

Fig. 1.
Fig. 1. Simulated index profiles (upper) and λ=1550  nm fundamental-mode field distributions (lower) along (a) the step-profile fiber SMF-28 and (b) the realistic log-profile fiber, each tapered from 10 to 125 μm outer diameter.
Fig. 2.
Fig. 2. Calculated λ=1550  nm MFD of (blue) the step-profile fiber, (red) the realistic log-profile fiber, and (grey) a uniform silica rod, as functions of outer diameter. Diameters above 125 μm imagine an enlarged fiber.
Fig. 3.
Fig. 3. Logarithmic refractive index profile of Eq. (2), showing how it can be capped near the axis. The parameter NA gives the logarithmic decay rate of the profile. (Here and elsewhere we include negative values of r for visual effect; the absolute value of r is taken.)
Fig. 4.
Fig. 4. Calculated λ=1550  nm fundamental-mode field distributions Ψ(R) of (blue) the step-profile fiber and (red) the idealized log-profile fiber. The distributions are normalized to unit power and horizontally scaled to have the same MFD.
Fig. 5.
Fig. 5. Index profiles, relative to undoped silica, of (red) the capped log-profile design and (black) the measured experimental fiber drawn from the PCVD preform.
Fig. 6.
Fig. 6. (a) Optical micrographs of (left) the log-profile fiber and (right) the step-profile fiber SMF-28 (both 125 μm diameter) illuminated by transmitted white light. (b) Near-field patterns for λ=1550  nm at the output of the log-profile fiber when (left) butt-coupled to the step-profile fiber with various lateral offsets and (right) fusion-spliced to the step-profile fiber (not to the same scale).
Fig. 7.
Fig. 7. Loss distribution of tapers made from (black, all <1  dB) the log-profile fiber and (blue, all >1  dB) the step-profile fiber. The histogram bin width is 0.1 dB. Inset: Schematic of the programmed taper profile. The finite flame size made the short input taper 1.7–2.0 mm long in practice.
Fig. 8.
Fig. 8. (a) Representative optical micrographs of side views of nominally identical short input tapers (from 125 to 30 μm) made from the log-profile fiber and the step-profile fiber, labelled with their measured losses. The images have been cropped, but the aspect ratio is otherwise preserved. (b) The corresponding taper profiles as extracted from the images for (black) log- and (blue) step-profile fibers.
Fig. 9.
Fig. 9. Single-mode near-field λ=1550  nm images at the output of (top) the step-profile fiber and (bottom) the log-profile fiber, when tapered to the indicated diameters. The images are to the same scale and have been +40% brightness-enhanced to show fainter regions.
Fig. 10.
Fig. 10. Index and stress distributions for log-profile fibers drawn with (solid black lines) 40 g draw tension (this paper) and (broken lines) 120 g draw tension [13]. (a) Measured index profiles, along with the design profile (red line). (b) Measured stress profiles; positive values are tensile. (c) Index profiles of (a) with the effects of stress from (b) removed, as calculated with the formula in [29].

Equations (14)

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|2π(β1β2)dρdzΨ2Ψ1ρdA|1,
n2(r)=n02NA2ln(r/ρ)=n02+NA2ln(ρ)NA2ln(r),
Rd2ΨdR2+dΨdR+Rln(RER)Ψ=0,
R=rkNA
β2k2=n02NA2ln(REρkNA).
MFD=RIIλπNA,
|πkn0(β1β2)2dρdzn2ρΨ1Ψ2dA|1.
|2π(β1β2)dρdz1ρ|1.
Ψ(R)1+R24[ln(RRE)1].
Ω={Ψ1Ψ2dA}2{Ψ12dA}{Ψ22dA},
Rd2ΨldR2+(2l+1)dΨldR+Rln(RER)Ψl=0,
Ψ(R)=RlΨl(R)
n2(r)=BCrα.
n2(r)=n12(n12n22)(rαr1α)(r2αr1α).

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