Abstract

The diffraction efficiency of a tilted in-core fiber grating is analyzed with the scattering formalism and the first Born approximation. Without any prior physical assumptions about the shape or direction of the scattered wave, it is shown that diffraction occurs when the so-called Bragg conditions are nearly satisfied and the interaction process can be described by a pair of coupled first-order differential equations that are exactly the same as those obtained through the coupled-mode analysis.

© 1997 Optical Society of America

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References

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  1. K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
    [CrossRef]
  2. 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]
  3. G. Meltz, W. W. Morey, J. R. Dunphy, “Fiber Bragg grating chemical sensor,” in Chemical, Biochemical, and Environmental Fiber Sensors III, M. T. Wlodarczyk, ed., Proc. SPIE1587, 350–361 (1991).
  4. S. M. Melle, K. Liu, R. M. Measures, “Practical fiber-optic Bragg grating strain gauge system,” Appl. Opt. 32, 3601–3609 (1993).
    [CrossRef] [PubMed]
  5. G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).
  6. A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
    [CrossRef]
  7. A. Bouzid, M. A. G. Abushagur, “Thin-film approximate modeling of in-core fiber gratings,” Opt. Eng. 35, 2793–2797 (1996).
    [CrossRef]
  8. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  9. W. H. Carter, “Three-dimensional wave theory of optical image formation from scattered sound,” J. Opt. Soc. Am. 60, 1366–1374 (1970).
  10. W. H. Carter, “On some diffraction efficiency equations for a thick grating hologram,” Opt. Commun. 103, 1–7 (1993).
    [CrossRef]
  11. J. G. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), 48–54.

1996 (1)

A. Bouzid, M. A. G. Abushagur, “Thin-film approximate modeling of in-core fiber gratings,” Opt. Eng. 35, 2793–2797 (1996).
[CrossRef]

1995 (1)

A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
[CrossRef]

1993 (2)

S. M. Melle, K. Liu, R. M. Measures, “Practical fiber-optic Bragg grating strain gauge system,” Appl. Opt. 32, 3601–3609 (1993).
[CrossRef] [PubMed]

W. H. Carter, “On some diffraction efficiency equations for a thick grating hologram,” Opt. Commun. 103, 1–7 (1993).
[CrossRef]

1989 (1)

1978 (1)

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

1970 (1)

Abushagur, M. A. G.

A. Bouzid, M. A. G. Abushagur, “Thin-film approximate modeling of in-core fiber gratings,” Opt. Eng. 35, 2793–2797 (1996).
[CrossRef]

A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
[CrossRef]

Azzam, R. M. A.

A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
[CrossRef]

Bouzid, A.

A. Bouzid, M. A. G. Abushagur, “Thin-film approximate modeling of in-core fiber gratings,” Opt. Eng. 35, 2793–2797 (1996).
[CrossRef]

A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
[CrossRef]

Carter, W. H.

W. H. Carter, “On some diffraction efficiency equations for a thick grating hologram,” Opt. Commun. 103, 1–7 (1993).
[CrossRef]

W. H. Carter, “Three-dimensional wave theory of optical image formation from scattered sound,” J. Opt. Soc. Am. 60, 1366–1374 (1970).

Dunphy, J. R.

G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).

G. Meltz, W. W. Morey, J. R. Dunphy, “Fiber Bragg grating chemical sensor,” in Chemical, Biochemical, and Environmental Fiber Sensors III, M. T. Wlodarczyk, ed., Proc. SPIE1587, 350–361 (1991).

El-Sabae, A.

A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
[CrossRef]

Farina, J. D.

G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).

Fujii, Y.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Glenn, W. H.

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]

G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).

Goodman, J. G.

J. G. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), 48–54.

Hill, K. O.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Johnson, D. C.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Kawasaki, B. S.

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

Leonberger, F. J.

G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).

Liu, K.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Measures, R. M.

Melle, S. M.

Meltz, G.

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]

G. Meltz, W. W. Morey, J. R. Dunphy, “Fiber Bragg grating chemical sensor,” in Chemical, Biochemical, and Environmental Fiber Sensors III, M. T. Wlodarczyk, ed., Proc. SPIE1587, 350–361 (1991).

G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).

Morey, W. W.

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]

G. Meltz, W. W. Morey, J. R. Dunphy, “Fiber Bragg grating chemical sensor,” in Chemical, Biochemical, and Environmental Fiber Sensors III, M. T. Wlodarczyk, ed., Proc. SPIE1587, 350–361 (1991).

Snyder, A. W.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. O. Hill, Y. Fujii, D. C. Johnson, B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647–649 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

W. H. Carter, “On some diffraction efficiency equations for a thick grating hologram,” Opt. Commun. 103, 1–7 (1993).
[CrossRef]

A. Bouzid, M. A. G. Abushagur, A. El-Sabae, R. M. A. Azzam, “Fiber-optic four-detector polarimeter,” Opt. Commun. 118, 329–334 (1995).
[CrossRef]

Opt. Eng. (1)

A. Bouzid, M. A. G. Abushagur, “Thin-film approximate modeling of in-core fiber gratings,” Opt. Eng. 35, 2793–2797 (1996).
[CrossRef]

Opt. Lett. (1)

Other (4)

G. Meltz, W. W. Morey, J. R. Dunphy, “Fiber Bragg grating chemical sensor,” in Chemical, Biochemical, and Environmental Fiber Sensors III, M. T. Wlodarczyk, ed., Proc. SPIE1587, 350–361 (1991).

G. Meltz, J. R. Dunphy, W. H. Glenn, J. D. Farina, F. J. Leonberger, “Fiber optic temperature and strain sensors,” in Fiber Optic Sensors II, A. V. Scheggi, ed., Proc. SPIE798, 104–114 (1987).

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

J. G. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), 48–54.

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Figures (5)

Fig. 1
Fig. 1

Fiber grating model for the scattering analysis.

Fig. 2
Fig. 2

FBA compared with CM and extended FBA; L = 0.1 mm, Δ n = 2 × 10- 4, and ϕ = 45°.

Fig. 3
Fig. 3

FBA compared with CM and extended FBA; L = 0.5 mm, Δ n = 2 × 10- 4, and ϕ = 45°.

Fig. 4
Fig. 4

FBA compared with CM and extended FBA; L = 0.5 mm, Δ n = 6 × 10- 4, and ϕ = 45°.

Fig. 5
Fig. 5

Slicing model for the derivation of the scattering coupled differential equations.

Equations (35)

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nr=n¯1+Δn sinK·r+Θ,r  Vn,r V
K=K0, 0, 12πΛ0, 0, 1,
Eir=xˆΨrt·exp-jβ·r,
β=β0, sin ϕ, cos ϕ=n¯ 2πλ00, sin ϕ, cos ϕ,
sas=-jβ2πws   FrEirexpjβas·rd3r,
ws=1 -ast2,ast2us2+vs2 1jast2-1otherwise.
Fr=β24π1-nrn¯20,otherwise-β2Δn2πsinK·r+Θ, r  V
sas=s+as+s-as,
s±as=γ0±exp-jβcos ϕ  Kβz×Ψcrt·exp-jβsin ϕy×expjβas·rd3r,
Ψcrt=Ψrt,rtx, y  D0, outside,
γ0±±exp±jΘn¯Δnβ24πλ0ws,
s±as=γ0±Lz sincβLz2ws-cos ϕ±Kβ.×λ0n¯2Ψcast*δusδvs -sin ϕ,
Ψcast Ψcrtexpjβast·rtd2rt.
ws=1 -us2+vs2=1-sin2ϕ=± cos ϕ.
K=±2β cos ϕ.
ws=-cos ϕK=2β cos ϕ,-π/2 < ϕ < π/2.
us0+Δus,vssin ϕ+Δvs,wscos ϕ-Kβ+Δws.
sas=γΨcus -Δus, vs-Δvs -sin ϕ,
γ-expjΘπn¯ΔnLzλ0 cos ϕsincβLz2 Δws.
Esr= sasexp-jβas·rd2ast.
Esr=γΨcx, yexp-jβps·r,
psΔus, sin ϕ+Δvs, cos ϕ-K/β+Δws.
rsx, yEsrEirz=0=γΨcx, yΨx, yexp-jβΔusx+Δvsy.
rs=γϑ,
ϑDΨx, y2 exp-jβΔusx+ΔvsydxdyΨx, y2dxdy
Esr=rszΨcx, y·exp-jβps·rrszzκ0*cos ϕ z,
κ0exp-jΘ πn¯Δnλ0 ϑ.
Eisr=rzΨcx, y·exp-jβp·rrz=r*z,
Eir=xˆAzΨx, y·exp-jβz cos ϕ+y sin ϕEsr=xˆAszexp-jβΔwszΨx,y·exp-jβ-z cos ϕ+y sin ϕ,
Az=rzAsz·exp-jβΔwszAszexp-jβΔwsz=rszAz,
Az=-zκAsz·exp-jβΔwszAsz=-zκ*AzexpjβΔwsz,
κκ0cos ϕ.
Az+dz=Az-zκAsz·exp-j2δβzAsz=Asz+dz-zκ*Az·expj2δβz,
dAzdz=κAsz·exp-j2δβzdAszdz=κ*Az·expj2δβz.
RAsz=0Az=02=κ2δβ2+α2 coth2αL, α2κ2-δβ2  0κ2δβ2+α2 cot2αL, α2=-α2  0.

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