Abstract

A numerical model based on a scalar beam propagation method is applied to study light transmission in photonic bandgap (PBG) waveguides. The similarity between a cylindrical waveguide with concentric layers of different indices and an analogous planar waveguide is demonstrated by comparing their transmission spectra that are numerically shown to have coinciding wavelengths for their respective transmission maxima and minima. Furthermore, the numerical model indicates the existence of two regimes of light propagation depending on the wavelength. Bragg scattering off the multiple high-index/low-index layers of the cladding determines the transmission spectrum for long wavelengths. As the wavelength decreases, the spectral features are found to be almost independent of the pitch of the multi-layer Bragg mirror stack. An analytical model based on an antiresonant reflecting guidance mechanism is developed to accurately predict the location of the transmission minima and maxima observed in the simulations when the wavelength of the launched light is short. Mode computations also show that the optical field is concentrated mostly in the core and the surrounding first high-index layers in the short-wavelength regime while the field extends well into the outermost layers of the Bragg structure for longer wavelengths. A simple physical model of the reflectivity at the core/high-index layer interface is used to intuitively understand some aspects of the numerical results as the transmission spectrum transitions from the short- to the long-wavelength regime.

© 2002 Optical Society of America

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References

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  1. P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).
  2. B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698.
    [Crossref] [PubMed]
  3. P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
    [Crossref]
  4. C. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–122 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113
    [Crossref] [PubMed]
  5. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
    [Crossref] [PubMed]
  6. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.
  7. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999).
    [Crossref]
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    [Crossref]
  9. T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660–1662 (2001).
    [Crossref]
  10. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000).
    [Crossref]
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    [Crossref]
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  13. M. Deopura, C. K. Ullal, B. Temelkuran, and Y. Fink, “Dielectric omnidirectional visible reflector,” Opt. Lett. 26, 1197–1199 (2001).
    [Crossref]
  14. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748.
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  17. BeamPROP software, version 4.0, Rsoft, Inc.
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    [Crossref]

2002 (1)

2001 (4)

2000 (4)

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. 18, 50–56 (2000).
[Crossref]

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

C. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–122 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113
[Crossref] [PubMed]

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[Crossref]

1999 (2)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999).
[Crossref]

1998 (1)

1980 (1)

M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” App. Opt. 19, 2240–2246 (1980).
[Crossref]

1978 (1)

1974 (1)

P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).

Abeeluck, A. K.

Allan, D. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Andres, P.

Astle, H. W.

P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).

Bennett, P. J.

Birks, T. A.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[Crossref]

Bise, R. T.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.

Botten, L. C.

Broderick, N. G. R.

Burdge, G. L.

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

Cregan, R. F.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

de Sterke, C. M.

Deopura, M.

Eggleton, B. J.

N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002).
[Crossref]

B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698.
[Crossref] [PubMed]

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

C. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–122 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113
[Crossref] [PubMed]

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.

Engeness, T. D.

Feit, M. D.

M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” App. Opt. 19, 2240–2246 (1980).
[Crossref]

Ferrando, A.

Fink, Y.

Fleck, J. A.

M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” App. Opt. 19, 2240–2246 (1980).
[Crossref]

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[Crossref]

Hale, A.

B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698.
[Crossref] [PubMed]

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

Headley, C.

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[Crossref]

Ibanescu, M.

Jacobs, S. A.

Joannopoulos, J. D.

Johnson, S. G.

Kaiser, P. V.

P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).

Kerbage, C.

Knight, J. C.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Kranz, K. S.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.

Litchinitser, N. M.

Mangan, B. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Marom, E.

McPhedran, R. C.

Miret, J. J.

Mogilevtsev, D.

Monro, T. M.

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[Crossref]

Richardson, D. J.

Roberts, P. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

Russell, P. St. J.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[Crossref]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[Crossref]

Silvestre, E.

Skorobogatiy, M.

Soljacic, M.

Steel, M. J.

Strasser, T. A.

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

Temelkuran, B.

Trevor, D. J.

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.

Ullal, C. K.

Weisberg, O.

Westbrook, P. S.

White, T. P.

Windeler, R. S.

B. J. Eggleton, C. Kerbage, P. S. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698.
[Crossref] [PubMed]

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

C. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–122 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113
[Crossref] [PubMed]

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.

Yariv, A.

P. Yeh, A. Yariv, and E. Marom,”Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
[Crossref]

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, Inc., New York, 1984).

Yeh, P.

P. Yeh, A. Yariv, and E. Marom,”Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
[Crossref]

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, Inc., New York, 1984).

App. Opt. (1)

M. D. Feit and J. A. Fleck, “Computation of mode eigenfunctions in graded-index optical fibers by the propagating beam method,” App. Opt. 19, 2240–2246 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (1)

P. S. Westbrook, B. J. Eggleton, R. S. Windeler, A. Hale, T. A. Strasser, and G. L. Burdge, “Cladding-Mode Resonances in Hybrid Polymer-Silica Microstructured Optical Fiber Gratings,” IEEE Photon. Technol. Lett. 12, 495–497 (2000).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

Opt. Express (3)

Opt. Lett. (5)

Science (1)

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999).
[Crossref] [PubMed]

The Bell System Technical Journal (1)

P. V. Kaiser and H. W. Astle, “Low-loss single-material fibers made from pure fused silica,” The Bell System Technical Journal 53, 1021–1039 (1974).

Other (3)

R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, “Tunable photonic band gap fiber,” in OSA Trends in Optics and Photonics (TOPS) Vol. 70, Optical Fiber Communication Conference, Technical Digest, Postconference Edition (Optical Society of America, Washington DC, 2002), pp. 466–468.

BeamPROP software, version 4.0, Rsoft, Inc.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, Inc., New York, 1984).

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Figures (9)

Fig. 1.
Fig. 1.

(a) Cross-section of a Bragg fiber, where D is the core diameter, d is the thickness of each high-index layer of the Bragg mirror stack and Λ is the pitch. As discussed in the text, the material and physical parameters of the Bragg fiber were chosen with respect to an experimental PBG fiber, the cross-section of which is shown below the Bragg fiber. (b) Index profile of a planar waveguide (labeled W1) obtained by slicing the Bragg fiber in (a) along the dotted line; a centered Gaussian beam was launched along the low-index core of diameter D. Typical length L of the waveguide in our simulations was 5 cm.

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Fig. 2.
Fig. 2.

(a) Structure of a simple cylindrical waveguide and its planar analogue. (b) Computed transmission spectra of the cylindrical and planar waveguides. (c) Comparison between scalar BPM and vector BPM for a cylindrical waveguide with the same parameters as the cylindrical waveguide in (a) except that L = 50 μm and D = 6 μm.

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Fig. 3.
Fig. 3.

Normalized transmission spectrum of the 1-D PBG waveguide W1 as a function of the free-space wavelength λ0 [and the free-space wavevector k0] in the (a) [(b)] “long-wavelength” regime, and (c) [(d)] “short-wavelength” regime for four different values of the pitch. The thickness of the high-index layer in each Bragg mirror stack was fixed at d = 3.437 μm. The predicted positions of the minima from the ARROW model are shown with small arrows along the horizontal axis.

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Fig. 4.
Fig. 4.

Sensitivity of the transmission spectrum with respect to the thickness, d, of the high-index layer for a fixed pitch of 5.642 μm. The predicted positions of the minima from the ARROW model are indicated by small arrows along the horizontal axis for each plot.

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Fig. 5.
Fig. 5.

(a) Index profiles of 1-D PBG waveguides W1 and W2 with pitches 5.642 μm and 9.772 μm, respectively. (b) Comparison between the transmission spectra of 1-D waveguides W1 and W2. The predicted positions of the minima from the ARROW model are indicated by small arrows along the horizontal axis.

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Fig. 6.
Fig. 6.

Modes of the 1-D PBG waveguide W1 at an exciting wavelength of (a) 11.3 μm, (b) 5.07 μm, and (c) 0.632 μm. The lateral index profile of the waveguide is also shown. The mode spectra corresponding to the exciting wavelength of (d) 11.3 μm, (e) 5.07 μm, and (f) 0.632 μm are shown next to each mode shape plot as a function of the modal propagation constant β and the modal effective index neff. The modes are labeled as m1, m2,…, where m1 refers to the fundamental mode and m2,…are the higher-order modes. The results shown were obtained by launching at z = 0 a Gaussian beam of width 8 μm and centered off-axis at x = 4 μm. The length of the waveguide was set at L = 2 cm.

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Fig. 7.
Fig. 7.

Modes and index profile (a), and mode spectrum (b) of a 1-D waveguide consisting of a low-index core sandwiched between two high-index layers. The core size, the thickness of each high-index layer and the index contrast between the core and the cladding were identical to those of the waveguide shown in Fig. 6 with ten high-index/low-index layer pairs in each Bragg mirror on either side of the core. The modes are labeled as m1, m2…, where m1 is the fundamental mode and m2,…are the higher-order modes. The launch condition and the length of the waveguide were the same as in Fig. 6.

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Fig. 8.
Fig. 8.

(a) Simple model for deriving the reflection coefficient at the core/high-index layer boundary. The index of the core is nlow and the index of each cladding layer is nhigh. Ei, Er and Et are the incident, reflected and transmitted electric fields, respectively, at the boundary. θ is the half-angle of the diffracting Gaussian beam; θlow and θhigh are the angles of incidence and refraction, respectively, at the interface. (b) Variation of the magnitude of the reflection coefficient for TE and TM polarized beams as a function of the spot size (2w0) of the Gaussian beam and free-space wavelength (λ0).

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Fig. 9
Fig. 9

(a) Effect of varying the width of the launched Gaussian beam (centered at x = 0 at the input) on the transmission spectrum of the 1-D PBG waveguide W1 of length L = 5 cm. (b) Modes excited at λ0 = 0.632 μm in waveguide W1 by launching a Gaussian beam of width 4 μm and centered off-axis at x = 4 μm. The length of the waveguide was set at L = 2 cm. (c) Comparison between the mode spectra for beam widths of 8 μm and 4 μm with otherwise identical launch conditions.

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Equations (4)

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

T ( λ 0 , L ) = ∫∫ core Ψ ( λ 0 , x , y , L ) Ψ * ( λ 0 , x , y , L ) dxdy ∫∫ input Ψ ( λ 0 , x , y , 0 ) Ψ * ( λ 0 , x , y , 0 ) dxdy
λ m = 2 n low d m ( n high n low ) 2 − 1 , m = 1,2 , ……
λ ℓ = 4 n low d ( 2 ℓ + 1 ) ( n high n low ) 2 − 1 , ℓ = 0,1,2 , ……
λ > 2 d n high 2 − n low 2 .

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