Abstract

Tapered- and straight-core fiber microlenses of hyperbolic shape are studied with the segmented beam propagation method (Se-BPM). This new formulation extends to a large scale the finite-difference time-domain method for calculating propagation of the wave field in guiding systems. It is based on partitioning an entire computational domain into subdomains along the direction of propagation. The Helmholtz equation can be solved directly for each subdomain, and an iterative procedure is used to propagate the field from one subdomain to another. The Se-BPM is compared with other approaches that are commonly used to analyze straight-core fiber microlen devices in the paraxial approximation. We deal mainly with small-spot-size fiber microlenses where this approximation does not apply. We show that the emergent beam is not Gaussian in the far field. Instead of the usual far-field characterization we propose a near-field characterization of the fiber microlens. This is possible with the near-field scanning optical microscopy technique.

© 2005 Optical Society of America

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References

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  1. M. Kawachi, T. Edahiro, H. Toba, “Microlens formation on VAD single-mode fiber ends,” Electron. Lett. 18, 71–72 (1982).
    [CrossRef]
  2. G. Eisenstein, D. Vitello, “Chemically etched conical microlenses for coupling single-mode lasers into single-mode fibers,” Appl. Opt. 21, 3470–3474 (1982).
    [CrossRef] [PubMed]
  3. H. Ghafoorishiraz, T. Asano, “Microlens for coupling a semiconductor laser to a single-mode fiber,” Opt. Lett. 11, 537–539 (1986).
    [CrossRef]
  4. H. Kuwahara, M. Sasaki, N. Tokoyo, “Efficient coupling from a semiconductor laser into single-mode fibers with tapered hemispherical ends,” Appl. Opt. 19, 2578–2583 (1980).
    [CrossRef] [PubMed]
  5. H. M. Presby, C. A. Edwards, “Near 100-percent-efficient fiber microlenses,” Electron. Lett. 28, 582–584 (1992).
    [CrossRef]
  6. H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
    [CrossRef]
  7. C. W. Barnard, J. W. Y. Lit, “Mode transforming properties of tapered single-mode fiber microlenses,” Appl. Opt. 32, 2090–2094 (1993).
    [CrossRef] [PubMed]
  8. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 19.
  9. C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser-to-fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
    [CrossRef]
  10. S. Gangopadhyay, S. Sarkar, “ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for the laser diode to single-mode fiber coupling by way of a hyperbolic lens on the fiber tip,” Appl. Opt. 36, 8582–8586 (1997).
    [CrossRef]
  11. B. Hermansson, D. Yevick, J. Saijonmaa, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” J. Opt. Soc. Am. A 1, 663–671 (1984).
    [CrossRef]
  12. W. Y. Su, G. W. Chern, L. A. Wang, “Analysis of cladding-mode couplings for a lensed fiber integrated with a long-period fiber grating by use of the beam-propagation method,” Appl. Opt. 41, 6576–6584 (2002).
    [CrossRef] [PubMed]
  13. S. P. Le Blanc, “Characterization of lensed optical fibers for optimal coupling efficiency and assembly yield,” in Technical Proceedings of the National Fiber Optic Engineers Conference 2002, NFOEC 2002, 15–19 September 2002, Dallas, Tex. (IEEE, Piscataway, N.J., 2002), pp. 1084–1094.
  14. H. J. W. M. Hoekstra, “On beam propagation methods for modeling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
    [CrossRef]
  15. C. Vassallo, “Reformulation for the beam-propagation method,” J. Opt. Soc. Am. A 10, 2208–2216 (1993).
    [CrossRef]
  16. A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
    [CrossRef]
  17. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 81.
  18. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  19. M. D. Feit, J. A. Fleck, “Analysis of rib waveguides and couplers by the propagating beam method,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  20. D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
    [CrossRef]
  21. H. H. Lin, A. Korpel, “Heuristic scalar paraxial beam-propagation method taking into account continuous reflections,” J. Opt. Soc. Am. B 8, 849–857 (1991).
    [CrossRef]
  22. P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
    [CrossRef]
  23. D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
    [CrossRef]
  24. M. Scalora, M. E. Grenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
    [CrossRef]
  25. J. Yamauchi, K. Nishio, H. Nakano, “Hybrid numerical technique combining the finite-difference beam-propagation method and the finite-difference time-domain method,” Opt. Lett. 22, 259–261 (1997).
    [CrossRef] [PubMed]
  26. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 252.
  27. femlab Reference manual, www.femlab.com

2002

2000

H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
[CrossRef]

1997

1995

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

1994

M. Scalora, M. E. Grenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

1993

1992

H. M. Presby, C. A. Edwards, “Near 100-percent-efficient fiber microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

1991

H. H. Lin, A. Korpel, “Heuristic scalar paraxial beam-propagation method taking into account continuous reflections,” J. Opt. Soc. Am. B 8, 849–857 (1991).
[CrossRef]

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

1989

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

1988

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

1986

1984

1982

1981

1980

An, H. L.

H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
[CrossRef]

Asano, T.

Bardyszewski, W.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

Barnard, C. W.

Ben-Ami, N.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Ben-Ami, U.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Chern, G. W.

Dragone, C.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser-to-fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

Edahiro, T.

M. Kawachi, T. Edahiro, H. Toba, “Microlens formation on VAD single-mode fiber ends,” Electron. Lett. 18, 71–72 (1982).
[CrossRef]

Edwards, C. A.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser-to-fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

H. M. Presby, C. A. Edwards, “Near 100-percent-efficient fiber microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

Eisenstein, G.

Feit, M. D.

Fish, G.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Fleck, J. A.

Gangopadhyay, S.

Ghafoorishiraz, H.

Glasner, M.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

Grenshaw, M. E.

M. Scalora, M. E. Grenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

Hermansson, B.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

B. Hermansson, D. Yevick, J. Saijonmaa, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” J. Opt. Soc. Am. A 1, 663–671 (1984).
[CrossRef]

Hoekstra, H. J. W. M.

H. J. W. M. Hoekstra, “On beam propagation methods for modeling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
[CrossRef]

Kaczmarski, P.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

Kawachi, M.

M. Kawachi, T. Edahiro, H. Toba, “Microlens formation on VAD single-mode fiber ends,” Electron. Lett. 18, 71–72 (1982).
[CrossRef]

Khachatryan, E.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Korpel, A.

Kuwahara, H.

Lagasse, P. E.

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
[CrossRef]

Le Blanc, S. P.

S. P. Le Blanc, “Characterization of lensed optical fibers for optimal coupling efficiency and assembly yield,” in Technical Proceedings of the National Fiber Optic Engineers Conference 2002, NFOEC 2002, 15–19 September 2002, Dallas, Tex. (IEEE, Piscataway, N.J., 2002), pp. 1084–1094.

Lewis, A.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Lieberman, K.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Lin, H. H.

Lit, J. W. Y.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 19.

Nakano, H.

Nishio, K.

Presby, H. M.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser-to-fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

H. M. Presby, C. A. Edwards, “Near 100-percent-efficient fiber microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

Saijonmaa, J.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 252.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 81.

Sarkar, S.

Sasaki, M.

Scalora, M.

M. Scalora, M. E. Grenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

Shalom, S.

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 19.

Su, W. Y.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 81.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 252.

Toba, H.

M. Kawachi, T. Edahiro, H. Toba, “Microlens formation on VAD single-mode fiber ends,” Electron. Lett. 18, 71–72 (1982).
[CrossRef]

Tokoyo, N.

van der Donk, J.

Van Roey, J.

Vassallo, C.

Vitello, D.

Wang, L. A.

Yamauchi, J.

Yevick, D.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

B. Hermansson, D. Yevick, J. Saijonmaa, “Propagating-beam-method analysis of two-dimensional microlenses and three-dimensional taper structures,” J. Opt. Soc. Am. A 1, 663–671 (1984).
[CrossRef]

Appl. Opt.

Electron. Lett.

M. Kawachi, T. Edahiro, H. Toba, “Microlens formation on VAD single-mode fiber ends,” Electron. Lett. 18, 71–72 (1982).
[CrossRef]

H. M. Presby, C. A. Edwards, “Near 100-percent-efficient fiber microlenses,” Electron. Lett. 28, 582–584 (1992).
[CrossRef]

P. Kaczmarski, P. E. Lagasse, “Bidirectional beam propagation method,” Electron. Lett. 26, 675–676 (1988).
[CrossRef]

D. Yevick, M. Glasner, “Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures,” Electron. Lett. 25, 1611–1613 (1989).
[CrossRef]

IEEE Photon. Lett.

D. Yevick, W. Bardyszewski, B. Hermansson, M. Glasner, “Split-operator electric-field reflection techniques,” IEEE Photon. Lett. 3, 527–529 (1991).
[CrossRef]

J. Lightwave Technol.

C. A. Edwards, H. M. Presby, C. Dragone, “Ideal microlenses for laser-to-fiber coupling,” J. Lightwave Technol. 11, 252–257 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

M. Scalora, M. E. Grenshaw, “A beam propagation method that handles reflections,” Opt. Commun. 108, 191–196 (1994).
[CrossRef]

H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

H. J. W. M. Hoekstra, “On beam propagation methods for modeling in integrated optics,” Opt. Quantum Electron. 29, 157–171 (1997).
[CrossRef]

Ultramicroscopy

A. Lewis, K. Lieberman, N. Ben-Ami, G. Fish, E. Khachatryan, U. Ben-Ami, S. Shalom, “New design and imaging concepts in NSOM,” Ultramicroscopy 61, 215–220 (1995).
[CrossRef]

Other

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 81.

S. P. Le Blanc, “Characterization of lensed optical fibers for optimal coupling efficiency and assembly yield,” in Technical Proceedings of the National Fiber Optic Engineers Conference 2002, NFOEC 2002, 15–19 September 2002, Dallas, Tex. (IEEE, Piscataway, N.J., 2002), pp. 1084–1094.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 19.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), p. 252.

femlab Reference manual, www.femlab.com

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Figures (19)

Fig. 1
Fig. 1

Schematics of the Se-BPM.

Fig. 2
Fig. 2

Geometrical two-dimensional model for (a) the straight-core fiber lens and (b) the tapered-core fiber lens. For these models three spatial regions are considered: 1, core; 2, cladding; 3, air.

Fig. 3
Fig. 3

Intensity cross sections inside the tapered fiber lens starting from z = −32 μm when the propagating field is the single fiber mode until z = 20 μm. The fiber core width is 4.8 μm, and the numerical aperture (NA) of the fiber is 0.13. The fiber lens has a 1.68 μm radius of curvature at the height of the hyperbola, and the wedge angle θ = 93.3°. The width of the single fiber mode on the left boundary is 10.4 μm. A gradual transformation of the single fiber mode as the field propagates in the tapered core is shown.

Fig. 4
Fig. 4

Pointing vector Sz = Re[i(∂zΦ)Φ*) distribution in the zx plane of the output beam from a cylindrical hyperbolic tapered-core fiber lens. The waist radius at the focus is 1.0 μm for wavelength λ = 1.55 μm. The focal length is f = 4 μm. The negative values of Sz correspond to the backpropagating field.

Fig. 5
Fig. 5

Phase image of the emergent electromagnetic field from the fiber lens. The parameters of the fiber lens are the same as in Fig. 4.

Fig. 6
Fig. 6

Intensity cross sections near the focus of the fiber lens. The focus corresponds to z = 42 μm. The 1/e2 width (spot size) at the focus is 2 μm.

Fig. 7
Fig. 7

Laser to fiber coupling efficiency η against offset δz from the focus f = 4 μm between the lens with the R = 1.68−μm lens and the laser. The maximum coupling efficiency with a perfect antireflection coating, AC, is η = 0.914. The maximum coupling efficiency without the antireflection coating is η = 0.836.

Fig. 8
Fig. 8

(a) Waist radius ω0 against the radius of curvature R for straight-core and tapered-core cylindrical hyperbolic fiber lenses with various radii of curvature R within the range of 1–45 μm. The wedge angle of all the fiber lenses was θ = 93.3°. The Gaussian optics curve was calculated from the Eq. (3). (b) Dependence ω0 on R for small R in the range of 0.5–4.5 μm.

Fig. 9
Fig. 9

Dependence of the focal length f of the emergent beam from the fiber lens on the radius of curvature R. The data were obtained from 23 simulations with different radii of curvature R starting from 0.5 to 13 μm.

Fig. 10
Fig. 10

Dependence of focal length f on the beam-waist radius for straight-core and tapered-core cylindrical hyperbolic fiber lenses. The data points were obtained by NSOM measurements on the fields emerging from tapered-core circular hyperbolic fiber lenses (Nanonics Imaging, Ltd.).

Fig. 11
Fig. 11

Dependence of coupling efficiency on waist radius for straight-core and tapered-core cylindrical hyperbolic fiber lenses with a perfect antireflection coating.

Fig. 12
Fig. 12

Straight-core and tapered-core fiber lens transmittivity T with waist radius ω0.

Fig. 13
Fig. 13

Coupling efficiency η = ηx2 against waist radius ω0 for straight-core and tapered-core circular hyperbolic fiber lenses with a perfect antireflection coating.

Fig. 14
Fig. 14

Dependence of coupling efficiency ηT = Tηx2 on waist radius ω0 for straight-core and tapered-core circular hyperbolic fiber lenses without antireflection coating.

Fig. 15
Fig. 15

Schematics of the experimental setup for NSOM fiber-lens characterization.

Fig. 16
Fig. 16

A CCD image of a tapered-core fiber lens of small spot size. The bar is 25 μm.

Fig. 17
Fig. 17

Three-dimensional image of lens topography obtained from an AFM image at the very end of the fiber lens.

Fig. 18
Fig. 18

Topography cross section corresponding to the line in the inset AFM image. The determined from the image radius of curvature of the fiber lens is 2.8 μm.

Fig. 19
Fig. 19

Cross section corresponding to the blue line of the NSOM image (inset) collected at the focus of the fiber lens. The determined from the image spot size is 1.6 μm.

Equations (26)

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

θ a = 2 tan - 1 ( n 1 2 - n 2 2 ) 1 / 2 n 2 ,
ω 2 = ω 0 [ 1 + ( λ f π ω 0 2 ) 2 ] 1 / 2 ,             R 2 = ( π λ ) 2 ( ω 0 ω 2 ) 2 f ,
R 2 = n 2 n 1 - n 2 R .
ω 0 = ω f R 2 ( R 2 2 + π 2 ω f 2 λ 2 ) 1 / 2 ,
f = π ω 0 ( ω f 2 - ω 0 2 ) 1 / 2 λ .
R π ω f 2 λ n 1 - n 2 n 2 ,
ω 0 = λ π ω f n 1 - n 2 n 2 R ,
f = n 2 n 1 - n 2 R .
η = ψ v ψ f * d x d y 2 ψ v 2 d x d y ψ f 2 d x d y
η = η x η y ,
η ( d 0 ) = T ( d 0 ) η x ( d 0 ) η y ( d 0 ) ,
η i = 2 ω 3 i ω f [ ( ω f 2 + ω 3 i 2 ) 2 + ( k 3 2 ω f 4 ω 3 i 4 ) / ( 4 R 3 i 2 ) ] 1 / 2 ,
M i = ( A i B i C i D i ) = [ 1 0 ( 1 - n ) / ( n R i ) 1 / n ] ( 1 d 0 0 1 ) .
2 Φ + k 0 2 n 2 Φ = 0 ,
Ω = k = 1 , , n Ω k
Φ ( r ) = ϕ ( r ) exp ( i n kr ) ,
Φ ( r ) = i n k ϕ ( r ) exp ( i n kr ) + ϕ ( r ) exp ( i n kr ) .
η · Φ = i η k n Φ ,
k = 2 π λ Ψ ( r ) Ψ ( r ) ,
Φ ( r ) = Φ f ( r ) + Φ b ( r )
Φ f ( r ) = ϕ f ( r ) exp ( i n k f r ) , Φ b ( r ) = ϕ b ( r ) exp ( i n k b r ) ,
Φ f ( r ) = i n k f Φ f ( r ) ,             Φ b ( r ) = i n k b Φ b ( r ) .
η Φ - i η k b n Φ = - i η ( k b - k f ) n Φ f ,
k f = 2 π λ ψ f ( r ) ψ f ( r ) ,             k b = 2 π λ ψ b ( r ) ψ b ( r ) .
η Φ = i k 0 n Φ .
η Φ - i k 0 n Φ = - 2 i k 0 n Φ f .

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