Abstract

We consider linear propagation through shallow, nonuniform gratings, such as those written in the core of photosensitive optical fibers. Though, of course, the coupled-mode equations for such gratings are well known, they are often derived heuristically. Here we present a rigorous derivation and include effects that are second order in the grating parameters. While the resulting coupled-mode equations can easily be solved numerically, such a calculation often does not give direct insight into the qualitative nature of the response. Here we present a new way of looking at nonuniform gratings that immediately does yield such insight and, as well, provides a convenient starting point for approximate treatments such as WKB analysis. Our approach, which is completely within the context of coupled-mode theory, makes use of an effective-medium description, in which one replaces the (in general, nonuniform) grating by a medium with a frequency-dependent refractive index distribution but without a grating.

© 1994 Optical Society of America

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  1. H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 15– 81.
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  3. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  4. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [Crossref] [PubMed]
  5. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. (to be published).
  6. C. Elachi, “Waves in active and passive periodic structures,” Proc. IEEE 64, 1666–1698 (1976).
    [Crossref]
  7. C. M. Ragdale, D. Reid, I. Bennion, “Fiber grating devices,” in Fiber Laser Sources and Amplifiers, C. M. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171, 148–156 (1990).
    [Crossref]
  8. K. Hill, “Aperiodic distributed-parameter waveguide for integrated optics,” Appl. Opt. 13, 1853–1856 (1974).
    [Crossref] [PubMed]
  9. G. I. Stegeman, D. G. Hall, “Modulated index structures,” J. Opt. Soc. Am. A 7, 1387–1398 (1990).
    [Crossref]
  10. F. Ouelette, “Dispersion cancellation using linearly chirped Bragg gratings in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
    [Crossref]
  11. D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
    [Crossref]
  12. See, e.g.,D. Z. Anderson, V. Mizrahi, T. Erdogan, A. E. White, “Production of in-fibre gratings using a diffractive optical element,” Electron Lett.29, 566–568 (1993).
    [Crossref]
  13. See, e.g.,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982), Chap. 8.
  14. H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).
  15. C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
    [Crossref] [PubMed]
  16. See, e.g., T. K. Gaylord, W. E. Beard, M. G. Moharan, “Zero-reflectivity high spatial-frequency rectangular groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
    [Crossref] [PubMed]
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  18. L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), Chap. 7.
  19. N. Fröman, P. O. Fröman, JWKB Approximation (North-Holland, Amsterdam, 1976).
  20. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
    [Crossref]

1993 (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[Crossref]

1992 (1)

C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[Crossref] [PubMed]

1990 (2)

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

G. I. Stegeman, D. G. Hall, “Modulated index structures,” J. Opt. Soc. Am. A 7, 1387–1398 (1990).
[Crossref]

1989 (1)

1987 (1)

1986 (1)

1976 (1)

C. Elachi, “Waves in active and passive periodic structures,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

1974 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Anderson, D. Z.

See, e.g.,D. Z. Anderson, V. Mizrahi, T. Erdogan, A. E. White, “Production of in-fibre gratings using a diffractive optical element,” Electron Lett.29, 566–568 (1993).
[Crossref]

Beard, W. E.

Bennion, I.

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

C. M. Ragdale, D. Reid, I. Bennion, “Fiber grating devices,” in Fiber Laser Sources and Amplifiers, C. M. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171, 148–156 (1990).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Buus, J.

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

de Sterke, C. M.

C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[Crossref] [PubMed]

Dodd, R. K.

See, e.g.,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982), Chap. 8.

Eilbeck, J. C.

See, e.g.,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982), Chap. 8.

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

Erdogan, T.

See, e.g.,D. Z. Anderson, V. Mizrahi, T. Erdogan, A. E. White, “Production of in-fibre gratings using a diffractive optical element,” Electron Lett.29, 566–568 (1993).
[Crossref]

Fröman, N.

N. Fröman, P. O. Fröman, JWKB Approximation (North-Holland, Amsterdam, 1976).

Fröman, P. O.

N. Fröman, P. O. Fröman, JWKB Approximation (North-Holland, Amsterdam, 1976).

Gaylord, T. K.

Gibbon, J. D.

See, e.g.,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982), Chap. 8.

Glenn, W. H.

Hall, D. G.

Hill, K.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 15– 81.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), Chap. 7.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), Chap. 7.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Meltz, G.

Mizrahi, V.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. (to be published).

See, e.g.,D. Z. Anderson, V. Mizrahi, T. Erdogan, A. E. White, “Production of in-fibre gratings using a diffractive optical element,” Electron Lett.29, 566–568 (1993).
[Crossref]

Moharan, M. G.

Morey, W. W.

Morris, H. C.

See, e.g.,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982), Chap. 8.

Ouelette, F.

Poladian, L.

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[Crossref]

Ragdale, C. M.

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

C. M. Ragdale, D. Reid, I. Bennion, “Fiber grating devices,” in Fiber Laser Sources and Amplifiers, C. M. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171, 148–156 (1990).
[Crossref]

Reid, D.

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

C. M. Ragdale, D. Reid, I. Bennion, “Fiber grating devices,” in Fiber Laser Sources and Amplifiers, C. M. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171, 148–156 (1990).
[Crossref]

Robbins, D. J.

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

Sipe, J. E.

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. (to be published).

Stegeman, G. I.

Stewart, W. J.

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

White, A. E.

See, e.g.,D. Z. Anderson, V. Mizrahi, T. Erdogan, A. E. White, “Production of in-fibre gratings using a diffractive optical element,” Electron Lett.29, 566–568 (1993).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Electron. Lett. (1)

D. Reid, C. M. Ragdale, I. Bennion, D. J. Robbins, J. Buus, W. J. Stewart, “Phase-shifted Moiré grating fibre resonators,” Electron. Lett. 26, 10–12 (1990).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Phys. Rev. A (1)

C. M. de Sterke, “Simulations of gap soliton generation,” Phys. Rev. A 45, 2012–2018 (1992).
[Crossref] [PubMed]

Phys. Rev. E (1)

L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E 48, 4758–4767 (1993).
[Crossref]

Proc. IEEE (1)

C. Elachi, “Waves in active and passive periodic structures,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

Other (10)

C. M. Ragdale, D. Reid, I. Bennion, “Fiber grating devices,” in Fiber Laser Sources and Amplifiers, C. M. Digonnet, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1171, 148–156 (1990).
[Crossref]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. (to be published).

H. Kogelnik, in Integrated Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1975), pp. 15– 81.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

See, e.g.,D. Z. Anderson, V. Mizrahi, T. Erdogan, A. E. White, “Production of in-fibre gratings using a diffractive optical element,” Electron Lett.29, 566–568 (1993).
[Crossref]

See, e.g.,R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, H. C. Morris, Solitons and Nonlinear Wave Equations (Academic, London, 1982), Chap. 8.

H. A. Macleod, Thin-Film Optical Filters (Hilger, London, 1969).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), Chap. 7.

N. Fröman, P. O. Fröman, JWKB Approximation (North-Holland, Amsterdam, 1976).

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Figures (9)

Fig. 1
Fig. 1

Parameters (a) σ ¯ and (b) κ ¯ as function of the grating depth κ for a cosinusoidal structure with σ = 0 and constant ϕ. The dotted lines indicate lowest-order results, in which κ ¯ = κ and σ ¯ = 0, while the dashed line in (a) indicates our second-order results. The solid curves indicate exact results obtained by integration of the wave equation directly.

Fig. 2
Fig. 2

Reflectivity as a function of frequency for a Fabry–Perot filter. The refractive-index contrast has been taken to be 5.

Fig. 3
Fig. 3

Location in the complex frequency plane of the poles (filled circles) and the zeros (open circles) of the amplitude reflection coefficient for a Fabry–Perot filter with refractive-index contrast equal to 5.

Fig. 4
Fig. 4

Reflectivity as a function of frequency of a uniform grating with κL = 5

Fig. 5
Fig. 5

Location in the complex frequency plane of the pole (filled circles) and the zeros (open circles) of the amplitude reflection for a uniform grating with κL = 5.

Fig. 6
Fig. 6

Band diagram for Gaussian profile grating showing as a function of position the frequencies for which light can propagate and the grating is transparent (clear region) and the frequencies for which light is evanescent and the grating acts as a distributed mirror (shaded region).

Fig. 7
Fig. 7

Location of the Fabry–Perot-like fringes in a Gaussian profile grating as a function of the effective grating strength (κL)eff. The solid curves indicate the results according to the WKB approximation, while the circles give exact results. As discussed in the text, the deviations, which are largest for small negative detunings, are due to Goos–Hänschen shifts, which cannot be correctly determined within the WKB approximation.

Fig. 8
Fig. 8

Reflectivity as a function of frequency for a Gaussian profile grating with (κL)eff = 5. The two sharp resonances for small negative detunings are due to resonance transmission.

Fig. 9
Fig. 9

Energy density as a function of position for the strongest transmission resonance of a Gaussian profile grating with (κLeff = 5 Δ/κ ≈ −0.7(top curve). This detuning is indicated by the dotted line in the band diagram. The dashed lines indicate the borders of the regions where the grating is reflective at this detuning. The units are chosen such that the incoming beam and thus also the transmitted beam have unit energy densities.

Equations (93)

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E ( r , t ) = x ̂ E ( z ) exp ( i ω t ) + c . c . , H ( r , t ) = ŷ H ( z ) exp ( i ω t ) + c . c . ,
d E d z = i ω μ 0 H ( z ) , d H d z = i ω ( z ) E ( z ) = i ω 0 n 2 ( z ) E ( z ) ,
d 2 E d z 2 + k 2 [ n ( z ) n 0 ] 2 E ( z ) = 0 ,
n ( z ) / n 0 = 1 + σ ( k 0 z ) + 2 κ ( k 0 z ) cos [ 2 k 0 z + ϕ ( k 0 z ) ] ,
Δ ( ω ω 0 ) / ω 0 ,
k 2 [ n ( z ) / n 0 ] 2 k 0 2 { 1 + 2 [ σ ( ξ ) + Δ ] + 4 κ ( ξ ) cos [ 2 ξ + ϕ ( ξ ) ] } ,
d 2 E d ξ 2 + { 1 + 2 [ σ ( ξ ) + Δ ] 2 κ ( ξ ) exp ( i γ ) + 2 κ ( ξ ) exp ( i γ ) } E = 0 ,
E ( ξ ) = a + ( ξ ) exp ( i ξ ) + a ( ξ ) exp ( i ξ ) ,
d 2 E d ξ 2 [ a + ( ξ ) + 2 i d a + d ξ ] exp ( + i ξ ) + [ a ( ξ ) 2 i d a d ξ ] exp ( i ξ ) ,
a + ( ξ ) = u ( ξ ) exp [ + i 2 ϕ ( ξ ) ] , a ( ξ ) = υ ( ξ ) exp [ i 2 ϕ ( ξ ) ] ,
d u ( ξ ) d ξ = + i [ σ ̂ ( ξ ) u ( ξ ) + κ ( ξ ) υ ( ξ ) ] , d υ ( ξ ) d ξ = i [ σ ̂ ( ξ ) υ ( ξ ) + κ ( ξ ) u ( ξ ) ] ,
σ ̂ ( ξ ) = σ ( ξ ) + Δ 1 2 ϕ ξ .
d d ξ [ | u ( ξ ) | 2 | υ ( ξ ) | 2 ] = 0 .
A + ( z ) = 1 2 [ n ( z ) n 0 ] 1 / 2 [ E ( z ) + Z 0 H ( z ) n ( z ) ] , A ( z ) = 1 2 [ n ( z ) n 0 ] 1 / 2 [ E ( z ) Z 0 H ( z ) n ( z ) ] ,
d A + d z = + i ω c n ( z ) A + ( z ) + 1 2 ( d { ln [ n ( z ) ] } d z ) A ( z ) , d A d z = i ω c n ( z ) A ( z ) + 1 2 ( d { ln [ n ( z ) ] } d z ) A + ( z ) .
d d z [ | A + ( z ) | 2 | A | 2 ] = 0
G ( γ ) = + g m exp ( i m γ ) ,
n ( z ) / n 0 = 1 + η σ ( η k 0 z ) + η κ ( η k 0 z ) G [ 2 k 0 z + ϕ ( η k 0 z ) ] .
( ω ω 0 ) / ω 0 = η Δ
ω n ( ξ ) / c k 0 = 1 + η [ σ ( η ξ ) + Δ + κ ( η ξ ) G ( γ ) ] + η 2 Δ [ σ ( η ξ ) + κ ( η ξ ) G ( γ ) ] ,
γ = 2 ξ + ϕ ( η ξ ) .
A + ( ξ ) = u ( ξ ) exp ( i ξ ) exp [ + i 2 ϕ ( ξ ) ] , A ( ξ ) = υ ( ξ ) exp ( i ξ ) exp [ i 2 ϕ ( ξ ) ] .
d u ( ξ ) d ξ = + i η [ σ ̂ ( ξ 1 ) + κ ( ξ 1 ) G ( γ ) ] u ( ξ ) + i η 2 Δ [ σ ( ξ 1 ) + κ ( ξ 1 ) G ( γ ) ] u ( ξ ) + 1 2 exp ( i γ ) × ( d d ξ { ln [ 1 + η σ ( ξ 1 ) + η κ ( ξ 1 ) G ( γ ) ] } ) υ ( ξ ) , d υ ( ξ ) d ξ = i η [ σ ̂ ( ξ 1 ) + κ ( ξ 1 ) G ( γ ) ] υ ( ξ ) i η 2 Δ [ σ ( ξ 1 ) + κ ( ξ 1 ) G ( γ ) ] υ ( ξ ) + 1 2 exp ( + i γ ) × ( d d ξ { ln [ 1 + η σ ( ξ 1 ) + η κ ( ξ 1 ) G ( γ ) ] } ) u ( ξ ) ,
σ ̂ ( ξ 1 ) = σ ( ξ 1 ) + Δ 1 2 ϕ ( ξ 1 ) ξ 1 .
f ( ξ ) = F ( ξ , η ξ , η 2 ξ , ) ,
ξ p = η p ξ ,
f ( ξ ) = F ( ξ 0 , ξ 1 , ξ 2 , )
d f ( ξ ) d ξ = F ξ 0 + η F ξ 1 + η 2 F ξ 2 + .
u ( ξ ) = u ( 0 ) ( ξ 1 , ξ 2 , ) + η u ( 1 ) ( ξ 0 , ξ 1 , ξ 2 , ) + η 2 u ( 2 ) ( ξ 0 , ξ 1 , ξ 2 , ) + ,
d u ( ξ ) d ξ = η [ u ( 0 ) ξ 1 + u ( 1 ) ξ 0 ] + η 2 [ u ( 0 ) ξ 2 + u ( 1 ) ξ 1 + u ( 2 ) ξ 0 ] + O ( η 3 ) .
u ( 0 ) ( ξ 1 , ξ 2 , ) ξ 1 + u ( 1 ) ( ξ 0 , ξ 1 , ) ξ 0 = i [ σ ̂ ( ξ 1 ) + κ ( ξ 1 ) G ( γ ) ] u ( 0 ) ( ξ 1 , ξ 2 , ) + κ ( ξ 1 ) G ( γ ) exp ( i γ ) υ ( 0 ) ( ξ 1 , ξ 2 , ) .
u ( 0 ) ( ξ 1 , ξ 2 ) ξ 1 + u ( 1 ) ( ξ 0 , ξ 1 , ) ξ 0 = i κ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) m 0 g m exp ( i m γ ) + i κ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) m 0 , 1 m g m exp [ i ( m 1 ) γ ] + i [ σ ̂ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) + κ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) ] ,
u ( 1 ) ( ξ 0 , ξ 1 , ) = κ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) m 0 g m 2 m exp ( i m γ ) + κ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) × m 0 , 1 m g m 2 ( m 1 ) exp [ i ( m 1 ) γ ] ,
u ( 0 ) ( ξ 1 , ξ 2 , ) ξ 1 = i [ σ ̂ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) + κ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) ] .
υ ( 1 ) ( ξ 0 , ξ 1 , ) = κ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) m 0 g m 2 m exp ( i m γ ) + κ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) × m 0 , 1 m g m 2 ( m + 1 ) exp [ i ( m + 1 ) γ ] ,
υ ( 0 ) ( ξ 1 , ξ 2 , ) ξ 1 = i [ σ ̂ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) + κ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) ] .
u ( 0 ) ( ξ 1 , ξ 2 , ) ξ 2 = i [ σ ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) + ν ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) ] ,
σ ( ξ 1 ) Δ σ ( ξ 1 ) κ 2 ( ξ 1 ) m 1 , 0 m 2 | g m | 2 2 ( m + 1 )
ν ( ξ 1 ) = [ σ ( ξ 1 ) + 1 2 ϕ ( ξ 1 ) ξ 1 ] κ ( ξ 1 ) 1 2 i κ ( ξ 1 ) ξ 1 + κ 2 ( ξ 1 ) m 0 , 1 2 m 3 2 m + 1 2 m ( m 1 ) g 1 m g m
υ ( 0 ) ( ξ 1 , ξ 2 , ) ξ 2 = i [ σ ( ξ 1 ) υ ( 0 ) ( ξ 1 , ξ 2 , ) + ν * ( ξ 1 ) u ( 0 ) ( ξ 1 , ξ 2 , ) ] .
u ( 0 ) ξ = + i [ η σ ̂ ( ξ 1 ) + η 2 σ ( ξ 1 ) ] u ( 0 ) + i [ η κ ( ξ 1 ) + η ( 2 ) ν ( ξ 1 ) ] υ ( 0 ) , υ ( 0 ) ξ = i [ η σ ̂ ( ξ 1 ) + η 2 σ ( ξ 1 ) υ ( 0 ) ] i [ η κ ( ξ 1 ) + η 2 ν * ( ξ 1 ) ] u ( 0 ) .
σ ¯ ( ξ ) σ ̂ ( ξ ) + σ ( ξ ) = σ ( ξ ) + Δ 1 2 ϕ ( ξ ) ξ + Δ σ ( ξ ) κ 2 ( ξ ) m 1 , 0 m 2 | g m | 2 2 ( m + 1 ) ,
κ ¯ ( ξ ) exp [ i ϕ ( ξ ) ] κ ( ξ ) + ν ¯ ( ξ ) = κ ( ξ ) { 1 + κ [ σ ( ξ ) + 1 2 ϕ ( ξ ) ξ ] } i 2 κ ( ξ ) ξ + κ 2 ( ξ ) × m 0 , 1 2 m 3 2 m + 1 2 m ( m 1 ) g 1 m g m ,
u ( 0 ) ( ξ ) u ¯ ( ξ ) exp [ i 2 ϕ ( ξ ) ] , υ ( 0 ) ( ξ ) υ ¯ ( ξ ) exp [ i 2 ϕ ( ξ ) ] ,
d u ¯ ( ξ ) d ξ = + i [ σ ¯ ( ξ ) u ¯ ( ξ ) + κ ¯ ( ξ ) υ ¯ ( ξ ) ] , d υ ¯ ( ξ ) d ξ = i [ σ ¯ ( ξ ) υ ¯ ( ξ ) + κ ¯ ( ξ ) u ¯ ( ξ ) ] .
E ( z ) = [ n 0 / n ( z ) ] 1 / 2 [ A + ( z ) + A ( z ) ] , H ( z ) = [ n 0 n ( z ) ] 1 / 2 ( 1 / Z 0 ) [ A + ( z ) A ( z ) ] ,
A + ( z ) = { u ¯ ( ξ ) [ 1 + κ ( ξ ) m 0 g m 2 m exp ( i m γ ) ] + υ ¯ ( ξ ) exp [ ϕ ¯ ( ξ ) ] κ ( ξ ) × m 0 , 1 m g m 2 ( m 1 ) exp [ i ( m 1 ) γ ] } × exp ( + i ξ ) exp [ + i 2 ϕ ¯ ( ξ ) ] , A ( z ) = { υ ¯ ( ξ ) [ 1 κ ( ξ ) m 0 g m 2 m exp ( i m γ ) ] + u ¯ ( ξ ) exp [ i ϕ ¯ ( ξ ) ] κ ( ξ ) × m 0 , 1 m g m 2 ( m + 1 ) exp [ i ( m + 1 ) γ ] } × exp ( i ξ ) exp [ i 2 ϕ ¯ ( ξ ) ] ,
ϕ ¯ ( ξ ) ϕ ( ξ ) + ϕ ( ξ ) .
E eff = u ¯ ( ξ ) + υ ¯ ( ξ ) H eff = u ¯ ( ξ ) υ ¯ ( ξ ) ,
d E eff d ξ = i μ eff ( ξ ) H eff ( ξ ) , d H eff d ξ = i eff ( ξ ) E eff ( ξ ) ,
μ eff = σ ¯ ( ξ ) κ ¯ , eff = σ ¯ ( ξ ) + κ ¯ ( ξ ) ,
n eff ( ξ ) = ( eff μ eff ) 1 / 2 = [ σ ¯ 2 ( ξ ) κ ¯ 2 ( ξ ) ] 1 / 2 .
n eff ( ξ ) = { [ σ ( ξ ) + Δ 1 2 ϕ ξ ] 2 κ 2 ( ξ ) } 1 / 2 ,
n eff = ( Δ ̂ 2 κ 2 ) 1 / 2 ,
Δ ̂ σ + Δ 1 2 ϕ ξ
E eff ( ξ ) = exp ( ± in eff ξ ) , H eff ( ξ ) = ± Z 1 E eff ( ξ ) ,
Z ( μ eff eff ) = ( Δ ̂ κ Δ ̂ + κ ) 1 / 2 ,
κ < Δ ̂ < κ .
r = r 12 [ 1 exp ( 2 i k L ) ] 1 r 21 2 exp ( 2 i k L ) ,
k = ω / C ,
ω / C = p π / L ,
ω C = p π L i 2 L ln ( r 21 2 ) .
r = r 12 [ 1 exp ( 2 in eff l ) ] 1 r 21 2 exp ( 2 in eff l ) ,
r 12 = ( Z 1 ) / ( Z + 1 ) ,
r ( Δ ̂ = 0 ) = i tanh ( κ l ) .
Δ ̂ / κ = ± [ ( p π / κ l ) 2 + 1 ] 1 / 2
Δ ̂ κ ± [ ( p π / κ l ) 2 + 1 ] 1 / 2 i α p π / κ l [ ( p π / κ l ) 2 + 1 ] 1 / 2 ,
α = 1 2 κ l ln { [ ( p π / κ l ) 2 + 1 ) ] 1 / 2 + p π / κ l [ ( p π / κ l ) 2 + 1 ] 1 / 2 p π / κ l } .
ω = k C ( uniform medium ) , Δ ̂ = ( n eff ) 2 + κ 2 ( uniform grating ) .
E = p c , E 2 = p 2 c 2 + m 0 2 c 4 ,
I ( z ) = I 0 exp ( z 2 / w 2 ) [ 1 + cos ( 2 k 0 z ) ] ,
n ( z ) = n 0 + δ n exp ( z 2 / w 2 ) [ 1 + cos ( 2 k 0 z ) ] .
σ ( ξ ) = 2 κ 0 exp [ ξ 2 / ( k 0 w ) 2 ] , κ ( ξ ) = κ 0 exp [ ξ 2 / ( k 0 w ) 2 ] , G ( γ ) = cos ( γ ) , ϕ ( ξ ) = 0 ,
κ 0 = δ n / 2 n 0 .
d u ( ξ ) e ξ = i { 2 κ 0 exp [ ξ 2 / ( k 0 w ) 2 ] u ( ξ ) + Δ u ( ξ ) + k 0 exp [ ξ 2 / ( k 0 w ) 2 ] υ ( ξ ) } , d υ ( ξ ) d ξ = i { 2 κ 0 exp [ ξ 2 / ( k 0 w ) 2 ] υ ( ξ ) + Δ υ ( ξ ) + k 0 exp [ ξ 2 / ( k 0 w ) 2 ] u ( ξ ) } ,
r eff = 3 κ 0 exp [ ξ 2 / ( k 0 w ) 2 ] + Δ , μ r eff = κ 0 exp [ ξ 2 / ( k 0 w ) 2 ] + Δ ,
n eff = ( { 2 κ 0 exp [ ξ 2 / R 0 w ) 2 ] + Δ } 2 { κ 0 exp [ ξ 2 / ( R 0 w ) 2 ] } 2 ) 1 / 2 ,
3 κ 0 exp ( z 2 / w 2 ) < Δ < κ 0 exp ( z 2 / w 2 ) ,
Δ < 3 κ 0 , Δ > 0 ,
3 κ 0 < Δ < κ 0 ,
κ 0 < Δ < 0 .
| n ( ξ ) ξ | | n ( ξ ) | .
d 2 E eff ( ξ ) d ξ 2 d { ln [ μ eff ( ξ ) ] } d ξ d E eff ( ξ ) d ξ + μ eff ( ξ ) eff ( ξ ) E eff ( ξ ) = 0 , d 2 H eff ( ξ ) d ξ 2 d { ln [ eff ( ξ ) ] } d ξ d H eff ( ξ ) d ξ + μ eff ( ξ ) eff ( ξ ) H eff ( ξ ) = 0 ,
eff ( ξ ) = [ μ eff ( ξ ) ] 1 / 2 E eff ( ξ ) , H eff ( ξ ) = [ eff ( ξ ) ] 1 / 2 H eff ( ξ ) ,
d 2 eff ( ξ ) d ξ 2 + { [ n eff ( ξ ) ] 2 + 1 2 μ eff ( ξ ) d 2 μ eff ( ξ ) d ξ 2 3 4 [ μ eff ( ξ ) ] 2 [ d μ eff ( ξ ) d ξ ] 2 } eff ( ξ ) = 0 , d 2 H eff ( ξ ) d ξ 2 + { [ n eff ( ξ ) ] 2 + 1 2 eff ( ξ ) d 2 eff ( ξ ) d ξ 2 3 4 [ eff ( ξ ) ] 2 [ d eff ( ξ ) d ξ ] 2 } H eff ( ξ ) = 0 ,
d 2 eff ( ξ ) d ξ 2 + [ n eff ( ξ ) ] 2 eff ( ξ ) = 0 , d 2 H eff ( ξ ) d ξ 2 + [ n eff ( ξ ) ] 2 H eff ( ξ ) = 0 ,
E eff ( ξ ) [ μ eff ( ξ ) n eff ( ξ ) ] 1 / 2 exp [ ± i n eff ( ξ ) d ξ ] , H eff ( ξ ) ± [ z ( ξ ) ] 1 E eff ( ξ ) ,
R W K B = { 1 + exp [ 2 z 0 z 0 k 0 n eff ( z ) d z ] } 1 ,
z 1 z 1 k 0 n eff ( z ) d z = ( m + 1 2 ) π ,
N = 1 π k 0 n eff ( z ) d z + 1 2 = ( 3 4 π ) 1 / 2 δ n n 0 k 0 w + 1 2 ,
( κ L ) eff = κ ( ξ ) d ξ = π 2 δ n n 0 k 0 w ,
N = ( 3 / π ) ( κ L ) eff + 1 2 .
3 ( κ L ) eff ( 1 + Δ / κ 0 ) 5 / 4 ( m + 1 / 2 ) π .

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