Abstract

Four-wave mixing in high refractive index materials, such as chalcogenide glass or semiconductors, is promising because of their large cubic nonlinearity. However, these materials tend to have normal dispersion at telecom wavelengths, preventing phase matched operation. Recent work has shown that the waveguide dispersion in strongly confining guided-wave structures can lead to anomalous dispersion, but the resulting four-wave mixing has limited bandwidth because of negative quartic dispersion. Here we first show that the negative quartic dispersion is an inevitable consequence of this dispersion engineering procedure. However, we also demonstrate that a slight change in the procedure leads to positive quartic dispersion, resulting in a superior bandwidth. We give an example in which the four-wave mixing bandwidth is doubled in this way.

© 2008 Optical Society of America

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  1. Q. Lin, J. D. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
    [CrossRef] [PubMed]
  2. A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
    [CrossRef] [PubMed]
  3. J. Meier, W. S. Mohammed, A. Jugessur, L. Qian, M. Mojahedi, and J. S. Aitchison, "Group velocity inversion in AlGaAs nanowires," Opt. Express 15, 12755-12762 (2007).
    [CrossRef] [PubMed]
  4. E. C. Mägi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr nonlinearity in sub-wavelength diameter As2Se3 chalcogenide fiber tapers," Opt. Express 15, 10324-10329 (2007).
    [CrossRef] [PubMed]
  5. C. M. B. Cordeiro, W. J. Wadsworth, T. A. Birks, and P. S. J. Russell, "Engineering the dispersion of tapered fibers for supercontinuum generation with a 1064 nm pump laser," Opt. Lett. 30, 1980-1982 (2005).
    [CrossRef] [PubMed]
  6. D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. B. Fu, and B. J. Eggleton, "Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires," Opt. Lett. 33, 660-662 (2008).
    [CrossRef] [PubMed]
  7. L. H. Yin, Q. Lin, and G. P. Agrawal, "Soliton fission and supercontinuum generation in silicon waveguides," Opt. Lett. 32, 391-393 (2007).
    [CrossRef] [PubMed]
  8. M. R. E. Lamont, C. M. de Sterke, and B. J. Eggleton, "Dispersion engineering of highly nonlinear As2S3 waveguides for parametric gain and wavelength conversion," Opt. Express 15, 9458-9463 (2007).
    [CrossRef] [PubMed]
  9. R. Zhang, J. Teipel, X. P. Zhang, D. Nau, and H. Giessen, "Group velocity dispersion of tapered fibers immersed in different liquids," Opt. Express 12, 1700-1707 (2004).
    [CrossRef] [PubMed]
  10. M. E. Marhic, N. Kagi, T. K. Chiang, and L. G. Kazovsky, "Broadband fiber optical parametric amplifiers," Opt. Lett. 21, 573-575 (1996).
    [CrossRef] [PubMed]
  11. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
    [CrossRef]
  12. M. E. Marhic, Y. Park, F. S. Yang, and L. G. Kazovsky, "Broadband fiber-optical parametric amplifiers and wavelength converters with low-ripple Chebyshev gain spectra," Opt. Lett. 21, 1354-1356 (1996).
    [CrossRef] [PubMed]
  13. M. Yu, C. J. McKinstrie, and G. P. Agrawal, "Modulational instabilities in dispersion-flattened fibers," Phys. Rev. E 52, 1072 (1995).
    [CrossRef]
  14. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, "Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber," Opt. Lett. 28, 2225-2227 (2003).
    [CrossRef] [PubMed]
  15. M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, "Selective FWM-based wavelength conversion realized by highly nonlinear fiber," in European Conference on Optical Communication (Cannes, France, 2006), paper Th1.3.5.
  16. M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, "Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance," in European Conference on Optical Communication (Glasgow, Scotland, 2005), paper Th4.4.4.
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, California, 2001).
  18. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
    [CrossRef] [PubMed]
  19. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, San Diego, California, 1991).
  20. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
    [CrossRef]
  21. V. G. Ta'eed, M. R. E. Lamont, D. J. Moss, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, "All optical wavelength conversion via cross phase modulation in chalcogenide glass rib waveguides," Opt. Express 14, 11242-11247 (2006).
    [CrossRef] [PubMed]
  22. K. P. Hansen, "Dispersion flattened hybrid-core nonlinear photonic crystal fiber," Opt. Express 11, 1503-1509 (2003).
    [CrossRef] [PubMed]
  23. W. H. Reeves, J. C. Knight, P. S. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
    [PubMed]

2008 (1)

2007 (4)

2006 (4)

2005 (1)

2004 (1)

2003 (2)

2002 (3)

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

W. H. Reeves, J. C. Knight, P. S. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
[PubMed]

1996 (2)

1995 (1)

M. Yu, C. J. McKinstrie, and G. P. Agrawal, "Modulational instabilities in dispersion-flattened fibers," Phys. Rev. E 52, 1072 (1995).
[CrossRef]

Agrawal, G. P.

Aitchison, J. S.

Birks, T. A.

Brar, K.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Centanni, J. C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Chiang, T. K.

Choi, D. Y.

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

Coen, S.

Cordeiro, C. M. B.

de Sterke, C. M.

Eggleton, B. J.

Fauchet, P. M.

Foster, M. A.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

Fu, L. B.

Gaeta, A. L.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

Giessen, H.

Hansen, K. P.

Harvey, J. D.

Headley, C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Jorgensen, C. G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Jugessur, A.

Kagi, N.

Kazovsky, L. G.

Knight, J. C.

Lamont, M. R. E.

Leonhardt, R.

Lin, Q.

Lipson, M.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

Luther-Davies, B.

Madden, S.

Mägi, E. C.

Manolatou, C.

Marhic, M. E.

McKinstrie, C. J.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, "Modulational instabilities in dispersion-flattened fibers," Phys. Rev. E 52, 1072 (1995).
[CrossRef]

Meier, J.

Mohammed, W. S.

Mojahedi, M.

Moss, D. J.

Nau, D.

Nguyen, H. C.

Park, Y.

Qian, L.

Radic, S.

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Raybon, G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Reeves, W. H.

Roberts, P. J.

Roelens, M. A. F.

Russell, P. S. J.

Schmidt, B. S.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

Sharping, J. E.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

Ta'eed, V. G.

Teipel, J.

Turner, A. C.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

Wadsworth, W. J.

Wong, G. K. L.

Yang, F. S.

Yeom, D. I.

Yin, L. H.

Yu, M.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, "Modulational instabilities in dispersion-flattened fibers," Phys. Rev. E 52, 1072 (1995).
[CrossRef]

Zhang, J. D.

Zhang, R.

Zhang, X. P.

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

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, "Parametric amplifiers driven by two pump waves," IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, "Continuous-wave parametric gain synthesis using nondegenerate pump four-wave mixing," IEEE Photon. Technol. Lett. 14, 1406-1408 (2002).
[CrossRef]

Nature (1)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

Opt. Express (9)

W. H. Reeves, J. C. Knight, P. S. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002).
[PubMed]

K. P. Hansen, "Dispersion flattened hybrid-core nonlinear photonic crystal fiber," Opt. Express 11, 1503-1509 (2003).
[CrossRef] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

Q. Lin, J. D. Zhang, P. M. Fauchet, and G. P. Agrawal, "Ultrabroadband parametric generation and wavelength conversion in silicon waveguides," Opt. Express 14, 4786-4799 (2006).
[CrossRef] [PubMed]

V. G. Ta'eed, M. R. E. Lamont, D. J. Moss, B. J. Eggleton, D. Y. Choi, S. Madden, and B. Luther-Davies, "All optical wavelength conversion via cross phase modulation in chalcogenide glass rib waveguides," Opt. Express 14, 11242-11247 (2006).
[CrossRef] [PubMed]

M. R. E. Lamont, C. M. de Sterke, and B. J. Eggleton, "Dispersion engineering of highly nonlinear As2S3 waveguides for parametric gain and wavelength conversion," Opt. Express 15, 9458-9463 (2007).
[CrossRef] [PubMed]

E. C. Mägi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr nonlinearity in sub-wavelength diameter As2Se3 chalcogenide fiber tapers," Opt. Express 15, 10324-10329 (2007).
[CrossRef] [PubMed]

J. Meier, W. S. Mohammed, A. Jugessur, L. Qian, M. Mojahedi, and J. S. Aitchison, "Group velocity inversion in AlGaAs nanowires," Opt. Express 15, 12755-12762 (2007).
[CrossRef] [PubMed]

R. Zhang, J. Teipel, X. P. Zhang, D. Nau, and H. Giessen, "Group velocity dispersion of tapered fibers immersed in different liquids," Opt. Express 12, 1700-1707 (2004).
[CrossRef] [PubMed]

Opt. Lett. (6)

Phys. Rev. E (1)

M. Yu, C. J. McKinstrie, and G. P. Agrawal, "Modulational instabilities in dispersion-flattened fibers," Phys. Rev. E 52, 1072 (1995).
[CrossRef]

Other (4)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, San Diego, California, 1991).

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, "Selective FWM-based wavelength conversion realized by highly nonlinear fiber," in European Conference on Optical Communication (Cannes, France, 2006), paper Th1.3.5.

M. Hirano, T. Nakanishi, T. Okuno, and M. Onishi, "Broadband wavelength conversion over 193-nm by HNL-DSF improving higher-order dispersion performance," in European Conference on Optical Communication (Glasgow, Scotland, 2005), paper Th4.4.4.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, California, 2001).

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Figures (8)

Fig. 1.
Fig. 1.

(a) The dispersive phase-mismatch, Δβ, for the case when Δβ becomes less than -2γP. The dashed line has β 2<0 and β 4=0; the solid line has β2 <0 and β 4>0, which results in a broader bandwidth. (b) Δβ when β2 <0 and β 4>0, optimized to have zero-slope at -γP, giving it a broad and flat gain spectrum. The grey-shaded areas indicate the gain region. The real part of the exponential gain coefficient, g, is calculated from Eq. (3) and shown in (c) and (d) for Δβ curves shown in (a) and (b), respectively.

Fig. 2.
Fig. 2.

(a) Schematic of the effective index model of the mode-propagation constant and its consequences on the various orders of dispersion. (b) RSoft FemSIM results of a 1×1 µm waveguide with ncore =2 and nclad =1, which match the predicted features in (a). Although ω is the independent variable, similar results are obtained if ω is kept constant and the waveguide size varied; a low frequency is equivalent to a small size, high frequency to a large size.

Fig. 3.
Fig. 3.

Illustration of conventional β 2-engineered for a high-index waveguide. Total β 2 and β 4 are shown as a function of the transverse dimension of the waveguide. Typical normal material dispersion (β 2,mat ) is indicated by the horizontal dotted line. Markers show the two ZDPs and their corresponding β 4 values. β 4,mat is typically insignificant compared to β 4,wg for high-index waveguides and would be indistinguishable from the horizontal origin line.

Fig. 4.
Fig. 4.

Illustration of high-order dispersion engineering applied to β 2 and β 4 for a high-index waveguide. Total β 2 and β 4 are shown as a function of the transverse dimension of the waveguide. The magnitude of β 2,wg is now only slightly larger than β 2,mat . Markers show the targeted ZDP and its corresponding β 4 value. β 4,mat is typically small compared to β 4,wg for high-index waveguides and has not been included.

Fig. 5.
Fig. 5.

Simulated β 2 and β 4 at 1550 nm of an As2Se3 fibre taper, n=2.76, (a) suspended in air, and (b) immersed in a liquid of n=1.65. The vertical dotted line indicates the taper width at which the ZDP is at the target wavelength of 1550 nm.

Fig. 6.
Fig. 6.

(a) D-FWM dispersive phase-mismatch for taper A, 1.19 µm As2Se3 taper in air (blue, dashed line), and taper B, 0.86 µm immersed in a high-index fluid with n=1.65 (red, solid line). The left-hand side of Table 1 contains the D-FWM parameters for both tapers. (b) Signal gain for the same tapers with a taper length of 5 cm. The vertical lines indicate the position of the pumps, and the grey area denotes the gain region.

Fig. 7.
Fig. 7.

(a) ND-FWM dispersive phase-mismatch for taper A, 1.19 µm As2Se3 taper in air (blue, dashed line), and taper B, 0.84 µm immersed in a high-index fluid with n=1.66 (red, solid line). The right-hand side of Table 1 contains the ND-FWM parameters for both tapers. (b) Signal gain for the same tapers with a taper length of 5 cm. The vertical lines indicate the position of the pumps, and the grey area denotes the gain region.

Fig. 8.
Fig. 8.

Illustrations of the estimates used (solid green lines), to the generic waveguide dispersion curves (dashed red lines). (a) β 2 as a function of V-number. (b) Estimating the β-curve as three lines. (c) β 1 resulting from the estimate of β, indicating β 1,max . (d) Estimate of the β 1, falling from its maximum value to its asymptotic value over a range of ΔV.

Tables (2)

Tables Icon

Table 1. Degenerate and non-degenerate FWM parameters

Tables Icon

Table 2. Comparison of estimated values to simulation at 1550 nm

Equations (19)

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

Δ β = 1 2 ( β s + β i 2 β p )
= 1 2 β 2 ( Δ ω ) 2 + 1 24 β 4 ( Δ ω ) 4 ,
G s = 1 + ( γ P g ) 2 sinh 2 ( gz ) ,
g = ( γ P ) 2 κ 2 = ( γ P ) 2 ( Δ β + γ P ) 2 .
Δ ω c = 4 γ P β 2 + 1 12 β 4 ( Δ ω c ) 2 .
Δ ω c = 12 β 2 β 4 .
β = n eff ω c ; β i = d i d ω i β .
min β 2 , wg β 2 , mat .
min β 2 , wg 2 c Δ n ω 0 ,
β 2 , wg > ˜ β 2 , mat ,
min β 2 , wg 2 c ω 0 Δ n Δ V .
d d ω Δ β | ω = ω m = [ β 2 + 1 6 β 4 ( ω m ω p ) 2 ] ( ω m ω p ) .
β 4 = 6 β 2 ( ω m ω p ) 2 .
V = ω D c ( n core 2 n clad 2 ) 1 2 ,
V 1 = 1 , ω 1 = d ω d V V 1 , β ( ω 1 ) = n clad c ω 1 = n clad c d ω d V V 1
V 2 = 2 , ω 2 = d ω d V V 2 , β ( ω 2 ) = n core c ω 2 = n core c d ω d V V 2 .
β l , max β ( ω 2 ) β ( ω 1 ) ω 2 ω 1 = 2 n core n clad c .
β 2 , min Δ β 1 Δ V dV d ω = D c 2 ( Δ n ) 3 2 ( n core + n clad ) 1 2 .
β 2 , min 2 c Δ n ω 0 .

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