Abstract

Chirped long-period fiber gratings are analyzed for management of dispersion in optical fiber communications systems. A ray model is used to derive simple analytic expressions that describe the transmission, chromatic delay, and dispersion properties of chirped long-period fiber gratings. A numerical model based on coupled-mode theory is used to verify the accuracy of the analytic expressions and explore design issues of the chirped long-period grating. With certain reasonable restrictions, chirped long-period gratings are found to be a viable and desirable alternative to existing dispersion compensation techniques.

© 2000 Optical Society of America

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  1. A. M. Vengsarkar, W. A. Reed, “Dispersion-compensating single-mode fibers: efficient designs for first- and second-order compensation,” Opt. Lett. 18, 924–926 (1993).
    [CrossRef] [PubMed]
  2. A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fibers based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
    [CrossRef]
  3. D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–128 (1991).
    [CrossRef]
  4. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters,” Opt. Lett. 12, 847–849 (1987).
    [CrossRef] [PubMed]
  5. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1293 (1997).
    [CrossRef]
  6. K. O. Hill, F. Bilodeau, T. Kitagawa, S. Therlault, D. C. Johonson, J. Albert, “Chirped in-fibre Bragg gratings for compensation of optical-fibre dispersion,” Opt. Lett. 19, 1314–1316 (1994).
    [CrossRef] [PubMed]
  7. T. Komukai, M. Nakazawa, “Fabrication of non-linearly chirped fiber Bragg gratings for higher-order dispersion compensation,” Opt. Commun. 154, 5–8 (1998).
    [CrossRef]
  8. L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
    [CrossRef]
  9. F. Ouellette, J. F. Cliché, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1737 (1994).
    [CrossRef]
  10. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
    [CrossRef]
  11. R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
    [CrossRef]
  12. H. Po, J. D. Cao, B. M. Laliberte, “High-power neodymium-doped single transverse-mode fiber laser,” Electron. Lett. 29, 1500–1501 (1993).
    [CrossRef]
  13. A. B. Buckman, Guided-Wave Photonics (Saunders College Publishing, Forth Worth, Texas, 1992), pp. 185–186.
  14. D. S. Staradubov, V. Grubsky, J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10, 1590–1592 (1998).
    [CrossRef]
  15. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
    [CrossRef]

1999

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

1998

D. S. Staradubov, V. Grubsky, J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10, 1590–1592 (1998).
[CrossRef]

T. Komukai, M. Nakazawa, “Fabrication of non-linearly chirped fiber Bragg gratings for higher-order dispersion compensation,” Opt. Commun. 154, 5–8 (1998).
[CrossRef]

1997

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1293 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

1996

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

1994

F. Ouellette, J. F. Cliché, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1737 (1994).
[CrossRef]

K. O. Hill, F. Bilodeau, T. Kitagawa, S. Therlault, D. C. Johonson, J. Albert, “Chirped in-fibre Bragg gratings for compensation of optical-fibre dispersion,” Opt. Lett. 19, 1314–1316 (1994).
[CrossRef] [PubMed]

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fibers based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

1993

A. M. Vengsarkar, W. A. Reed, “Dispersion-compensating single-mode fibers: efficient designs for first- and second-order compensation,” Opt. Lett. 18, 924–926 (1993).
[CrossRef] [PubMed]

H. Po, J. D. Cao, B. M. Laliberte, “High-power neodymium-doped single transverse-mode fiber laser,” Electron. Lett. 29, 1500–1501 (1993).
[CrossRef]

1991

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–128 (1991).
[CrossRef]

1987

Abramov, A. A.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

Albert, J.

Antos, A. J.

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fibers based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

Bayon, J. F.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Bhatia, V.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Bilodeau, F.

Buckman, A. B.

A. B. Buckman, Guided-Wave Photonics (Saunders College Publishing, Forth Worth, Texas, 1992), pp. 185–186.

Cao, J. D.

H. Po, J. D. Cao, B. M. Laliberte, “High-power neodymium-doped single transverse-mode fiber laser,” Electron. Lett. 29, 1500–1501 (1993).
[CrossRef]

Chraplyvy, A. R.

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–128 (1991).
[CrossRef]

Cliché, J. F.

F. Ouellette, J. F. Cliché, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1737 (1994).
[CrossRef]

Delevaque, E.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

DiGiovanni, D. J.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

Eggleton, B. J.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1293 (1997).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760–1773 (1997).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Espindola, R. P.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

Feinberg, J.

D. S. Staradubov, V. Grubsky, J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10, 1590–1592 (1998).
[CrossRef]

Gagnon, S.

F. Ouellette, J. F. Cliché, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1737 (1994).
[CrossRef]

Grubsky, V.

D. S. Staradubov, V. Grubsky, J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10, 1590–1592 (1998).
[CrossRef]

Guilloux, J. Y.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Hill, K. O.

Johonson, D. C.

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Kitagawa, T.

Komukai, T.

T. Komukai, M. Nakazawa, “Fabrication of non-linearly chirped fiber Bragg gratings for higher-order dispersion compensation,” Opt. Commun. 154, 5–8 (1998).
[CrossRef]

Laliberte, B. M.

H. Po, J. D. Cao, B. M. Laliberte, “High-power neodymium-doped single transverse-mode fiber laser,” Electron. Lett. 29, 1500–1501 (1993).
[CrossRef]

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Marcuse, D.

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–128 (1991).
[CrossRef]

Monerie, M.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Morvan, M.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Nakazawa, M.

T. Komukai, M. Nakazawa, “Fabrication of non-linearly chirped fiber Bragg gratings for higher-order dispersion compensation,” Opt. Commun. 154, 5–8 (1998).
[CrossRef]

Ouellette, F.

F. Ouellette, J. F. Cliché, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1737 (1994).
[CrossRef]

F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters,” Opt. Lett. 12, 847–849 (1987).
[CrossRef] [PubMed]

Po, H.

H. Po, J. D. Cao, B. M. Laliberte, “High-power neodymium-doped single transverse-mode fiber laser,” Electron. Lett. 29, 1500–1501 (1993).
[CrossRef]

Quetel, L.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Reed, W. A.

Rivoallan, L.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Smith, D. K.

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fibers based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

Staradubov, D. S.

D. S. Staradubov, V. Grubsky, J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10, 1590–1592 (1998).
[CrossRef]

Strasser, T. A.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

Therlault, S.

Tkach, R. W.

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–128 (1991).
[CrossRef]

Vengsarkar, A. M.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

A. M. Vengsarkar, W. A. Reed, “Dispersion-compensating single-mode fibers: efficient designs for first- and second-order compensation,” Opt. Lett. 18, 924–926 (1993).
[CrossRef] [PubMed]

Windeler, R. S.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

Electron. Lett.

R. P. Espindola, R. S. Windeler, A. A. Abramov, B. J. Eggleton, T. A. Strasser, D. J. DiGiovanni, “External refractive index insensitive air-clad long period fibre grating,” Electron. Lett. 35, 327–328 (1999).
[CrossRef]

H. Po, J. D. Cao, B. M. Laliberte, “High-power neodymium-doped single transverse-mode fiber laser,” Electron. Lett. 29, 1500–1501 (1993).
[CrossRef]

IEEE Photon. Technol. Lett.

D. S. Staradubov, V. Grubsky, J. Feinberg, “All-fiber bandpass filter with adjustable transmission using cladding-mode coupling,” IEEE Photon. Technol. Lett. 10, 1590–1592 (1998).
[CrossRef]

J. Lightwave Technol.

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fibers based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

D. Marcuse, A. R. Chraplyvy, R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Lightwave Technol. 9, 121–128 (1991).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1293 (1997).
[CrossRef]

F. Ouellette, J. F. Cliché, S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. 12, 1728–1737 (1994).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. Sipe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

T. Komukai, M. Nakazawa, “Fabrication of non-linearly chirped fiber Bragg gratings for higher-order dispersion compensation,” Opt. Commun. 154, 5–8 (1998).
[CrossRef]

Opt. Fiber Technol.: Mater., Devices Syst.

L. Quetel, L. Rivoallan, M. Morvan, M. Monerie, E. Delevaque, J. Y. Guilloux, J. F. Bayon, “Chromatic dispersion compensation by apodised Bragg gratings within controlled tapered fibers,” Opt. Fiber Technol.: Mater., Devices Syst. 3, 267–271 (1997).
[CrossRef]

Opt. Lett.

Other

A. B. Buckman, Guided-Wave Photonics (Saunders College Publishing, Forth Worth, Texas, 1992), pp. 185–186.

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Figures (12)

Fig. 1
Fig. 1

Chirped fiber gratings produce wavelength-dependent delay by (a) reflecting the core mode at a wavelength-dependent position in a Bragg grating thereby producing a wavelength-dependent propagation distance or by (b) coupling the core mode into a cladding mode, which has a larger group velocity, at a wavelength-dependent position.

Fig. 2
Fig. 2

(a) Core-mode power transmission, (b) wavelength-dependent delay, and (c) dispersion for a uniform (solid) and Gaussian apodized (dotted) LPG with length (or FWHM) = 80 cm, Δneff=0.15, and chirp=-2.5×10-3[nm/cm].

Fig. 3
Fig. 3

(a) Core-mode power transmission, (b) wavelength-dependent delay, and (c) dispersion for a chirped Gaussian apodized LPG with FWHM=0.5×Leff (dotted line), 1×Leff (solid curve), 3×Leff (dashed–dotted curve), and 5×Leff (dashed curve), where Leff is the effective length of the grating defined in the text.

Fig. 4
Fig. 4

Comparison of results from the analytic expressions describing the behavior of chirped LPG’s from Section 2 (curves) and from the numerical model with use of coupled-mode theory (points).

Fig. 5
Fig. 5

The relationship between the effective length of a chirped LPG and the coupling constant that provides optimal coupling exhibits an inverse proportionality (κLeffconstant) for sufficiently long grating lengths (FWHM/Leff).

Fig. 6
Fig. 6

Effective indexes of the LP01 core mode and HE1m and EH1m cladding modes where the radial order m of the cladding modes increases for a decreasing effective index.

Fig. 7
Fig. 7

Coupling coefficient values for coupling between the LP01 core mode and both HE1m and EH1m cladding modes for the case of air surrounding the cladding and a recoating material with an index n=1.33 surrounding the cladding.

Fig. 8
Fig. 8

(a) The chirped LPG dispersion compensator is a fusion of two similarly chirped LPG’s. The device consists of two identical LPG’s that have opposite chirp of equal magnitude; the device couples the light out of the core, into the cladding, and back into the core. (b) Interference fringes at the band edges of the device are eliminated by use of either a core mode block (above) or a fiber Bragg grating (below) at the center of the device.

Fig. 9
Fig. 9

(a) Core-mode transmission and (b) wavelength-dependent delay and dispersion of a chirped LPG dispersion compensator with the core mode blocked (solid and dashed–dotted curves) or transmitted (dotted curve) at the center of the device.

Fig. 10
Fig. 10

Power carried by the HE1m (squares) and EH1m (circles) within the core region of the fiber, which is useful for estimating the cladding-mode losses induced by placement of a core-mode block at the center of the dispersion compensator.

Fig. 11
Fig. 11

Keeping the FWHM of an apodized grating fixed while varying the grating length with values of 1×Leff (dashed–dotted curve), 2×Leff (dotted curve), 3×Leff (solid curve), and 4×Leff (dashed curve) FWHM affects the truncation of the tails of the Gaussian profile as well as the separation between the peaks of the apodization.

Fig. 12
Fig. 12

(a) Core-mode transmission, (b) wavelength-dependent delay, and (c) dispersion for the case of the grating length equal to 1×Leff (dashed–dotted curve), 2×Leff (dotted curve), 3×Leff (solid curve), and 4×Leff (dashed curve) FWHM of the grating, where the FWHM is fixed.

Equations (21)

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

dg=dτdλ=τ2-τ1λ2-λ1=LGcneff,2-neff,1λ2-λ1=LGΔneffcΔλ,
chirp=dλdz=ΔλLGnmcm,
dg=Δneffc×chirp×1010 psnm.
dRdz=iσˆR(z)+iκS(z),
dSdz=-iσˆS(z)+iκ*R(z),
σˆ=δ+σ11-σ222-12dϕdz,
δ=πΔneffλ-πΛ0,
K(z)=σ+2κ cos2πΛ0z+ϕ(z).
Kg(z)=2πΛ(z)=ddzArg[K(z)]=2πΛ0+dϕdz.
λDλD0+dλDdzz,
Kg(z)=2πΔneffλD(z)2πΔneffλD01-zλD0dλDdz.
-12dϕdz=πΔneffzλD02dλDdz.
Tx=S(Lg)R(0)2=κ2κ2+σˆ2sin2(κ2+σˆ2Lg).
σˆLg=±3π/2=δ-12dϕdzLeff,
Leff=3×1e72Δneffchirpnmcm1/2λD0[m],
ddevice=2Δneffc×chirp×1010psnm,
LossΓcore=PcorePcore+Pclad+Psurround,
Pcore=12Re 02πdϕ0rcorerdr(ErclHϕcl*-Hrcl*Eϕcl),
Pclad=12Re 02πdϕrcorercladrdr(ErclHϕcl*-Hrcl*Eϕcl),
Psurround=12Re 02πdϕrcladrdr(ErclHϕcl*-Hrcl*Eϕcl),
Δλλvδneffneff,

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