Abstract

A method for designing fiber Bragg gratings with desired complex transmission coefficients is proposed. The transmission coefficient of a fiber grating satisfies the minimum-phase condition when the linear phase from the pure propagation is ignored. Therefore only a finite bandwidth is considered for the synthesis. The algorithm is based on a result of Krein and Nudel’man [Prob. Peredachi Inf. 11, 37 (1975)]. A numerical algorithm is developed, and by numerical examples it is demonstrated that it is possible to realize gratings with specified complex transmission responses inside the considered bandwidth. The method is also applicable for thin-film filters.

© 2001 Optical Society of America

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References

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  1. K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
    [CrossRef]
  2. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  3. R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [CrossRef]
  4. K. Hinton, “Dispersion compensation using apodized Bragg fiber gratings in transmission,” J. Lightwave Technol. 16, 2336–2346 (1998).
    [CrossRef]
  5. E. Brinkmeyer, “Simple algorithm for reconstructing fiber grating from reflectometric data,” Opt. Lett. 20, 810–812 (1995).
    [CrossRef] [PubMed]
  6. L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
    [CrossRef]
  7. G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
    [CrossRef]
  8. F. Ouellette, “Limits of chirped pulse-compression with an unchirped Bragg grating filter,” Appl. Opt. 29, 4826–4829 (1990).
    [CrossRef] [PubMed]
  9. B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
    [CrossRef]
  10. N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
    [CrossRef]
  11. J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, Trondheim, Norway, 2000), available online at http://www.fysel.ntnu.no/Department/Avhandlinger/dring/index.html#2000 .
  12. M. G. Krein, P. Ya. Nudel’man, “On some new problems for Hardy class functions and continuous families of functions with double orthogonality,” Dokl. Akad. Nauk SSSR 206, 537–540 (1973).
  13. M. G. Krein, P. Ya. Nudel’man, “Approximation of functions by minimum-energy transfer functions of linear systems,” Probl. Peredachi Inf. 11, 37–60 (1975).
  14. V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
    [CrossRef]
  15. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).
  16. A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962), Chap. 10.
  17. L. Aizenberg, Carleman’s Formulas in Complex Analysis (Kluwer Academic, Dordrecht, The Netherlands, 1993).
  18. A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
    [CrossRef] [PubMed]
  19. A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
    [CrossRef]
  20. N. Young, An Introduction to Hilbert Space (Cambridge U. Press, Cambridge, UK, 1988), Chap. 7.
  21. G. H. Song, “Theory of symmetry in optical filter responses,” J. Opt. Soc. Am. A 11, 2027–2037 (1994).
    [CrossRef]
  22. N. M. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1997).
    [CrossRef]

1999 (1)

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

1998 (2)

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

K. Hinton, “Dispersion compensation using apodized Bragg fiber gratings in transmission,” J. Lightwave Technol. 16, 2336–2346 (1998).
[CrossRef]

1997 (5)

N. M. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1997).
[CrossRef]

L. Poladian, “Group-delay reconstruction for fiber Bragg gratings in reflection and transmission,” Opt. Lett. 22, 1571–1573 (1997).
[CrossRef]

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

1996 (1)

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

1995 (2)

V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

E. Brinkmeyer, “Simple algorithm for reconstructing fiber grating from reflectometric data,” Opt. Lett. 20, 810–812 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1990 (1)

1985 (1)

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

1975 (1)

M. G. Krein, P. Ya. Nudel’man, “Approximation of functions by minimum-energy transfer functions of linear systems,” Probl. Peredachi Inf. 11, 37–60 (1975).

1973 (1)

M. G. Krein, P. Ya. Nudel’man, “On some new problems for Hardy class functions and continuous families of functions with double orthogonality,” Dokl. Akad. Nauk SSSR 206, 537–540 (1973).

Agrawal, G. P.

Aizenberg, L.

L. Aizenberg, Carleman’s Formulas in Complex Analysis (Kluwer Academic, Dordrecht, The Netherlands, 1993).

Brinkmeyer, E.

Brodzeli, Z.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Dhosi, G.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1997).
[CrossRef]

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Feced, R.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Giles, C. R.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Hill, K. O.

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Hinton, K.

Kailath, T.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Klibanov, V.

V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

Krein, M. G.

M. G. Krein, P. Ya. Nudel’man, “Approximation of functions by minimum-energy transfer functions of linear systems,” Probl. Peredachi Inf. 11, 37–60 (1975).

M. G. Krein, P. Ya. Nudel’man, “On some new problems for Hardy class functions and continuous families of functions with double orthogonality,” Dokl. Akad. Nauk SSSR 206, 537–540 (1973).

Krug, P. A.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

Lenz, G.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Levy, B. C.

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Litchinitser, N. M.

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1997).
[CrossRef]

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Meltz, G.

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Muriel, M. A.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Nudel’man, P. Ya.

M. G. Krein, P. Ya. Nudel’man, “Approximation of functions by minimum-energy transfer functions of linear systems,” Probl. Peredachi Inf. 11, 37–60 (1975).

M. G. Krein, P. Ya. Nudel’man, “On some new problems for Hardy class functions and continuous families of functions with double orthogonality,” Dokl. Akad. Nauk SSSR 206, 537–540 (1973).

Nussenzveig, M.

M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

Ouellette, F.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

F. Ouellette, “Limits of chirped pulse-compression with an unchirped Bragg grating filter,” Appl. Opt. 29, 4826–4829 (1990).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962), Chap. 10.

Patterson, D. B.

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

Poladian, L.

Sacks, P. E.

V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

Skaar, J.

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, Trondheim, Norway, 2000), available online at http://www.fysel.ntnu.no/Department/Avhandlinger/dring/index.html#2000 .

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Song, G. H.

Stephens, T.

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

Tikhonravov, A. V.

V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

Young, N.

N. Young, An Introduction to Hilbert Space (Cambridge U. Press, Cambridge, UK, 1988), Chap. 7.

Zervas, M. N.

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

Appl. Opt. (2)

Dokl. Akad. Nauk SSSR (1)

M. G. Krein, P. Ya. Nudel’man, “On some new problems for Hardy class functions and continuous families of functions with double orthogonality,” Dokl. Akad. Nauk SSSR 206, 537–540 (1973).

Electron. Lett. (1)

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, F. Ouellette, “Dispersion compensation using a fibre grating in transmission,” Electron. Lett. 32, 1610–1611 (1996).
[CrossRef]

IEEE J. Quantum Electron. (2)

R. Feced, M. N. Zervas, M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, R. E. Slusher, “Dispersive properties of optical fibers for WDM systems,” IEEE J. Quantum Electron. 34, 1390–1402 (1998).
[CrossRef]

Inverse Probl. (1)

V. Klibanov, P. E. Sacks, A. V. Tikhonravov, “The phase retrieval problem,” Inverse Probl. 11, 1–28 (1995).
[CrossRef]

J. Lightwave Technol. (5)

K. O. Hill, G. Meltz, “Fiber Bragg grating technology: fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: theoretical model and design criteria for nearly ideal pulse recompression,” J. Lightwave Technol. 15, 1303–1313 (1997).
[CrossRef]

N. M. Litchinitser, B. J. Eggleton, G. P. Agrawal, “Dispersion of cascaded fiber gratings in WDM lightwave systems,” J. Lightwave Technol. 16, 1523–1529 (1997).
[CrossRef]

K. Hinton, “Dispersion compensation using apodized Bragg fiber gratings in transmission,” J. Lightwave Technol. 16, 2336–2346 (1998).
[CrossRef]

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

Opt. Lett. (2)

Probl. Peredachi Inf. (1)

M. G. Krein, P. Ya. Nudel’man, “Approximation of functions by minimum-energy transfer functions of linear systems,” Probl. Peredachi Inf. 11, 37–60 (1975).

SIAM J. Appl. Math. (1)

A. M. Bruckstein, B. C. Levy, T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Other (5)

N. Young, An Introduction to Hilbert Space (Cambridge U. Press, Cambridge, UK, 1988), Chap. 7.

M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).

A. Papoulis, The Fourier Integral and Its Applications (McGraw–Hill, New York, 1962), Chap. 10.

L. Aizenberg, Carleman’s Formulas in Complex Analysis (Kluwer Academic, Dordrecht, The Netherlands, 1993).

J. Skaar, “Synthesis and characterization of fiber Bragg gratings,” Ph.D. dissertation (Norwegian University of Science and Technology, Trondheim, Norway, 2000), available online at http://www.fysel.ntnu.no/Department/Avhandlinger/dring/index.html#2000 .

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Figures (9)

Fig. 1
Fig. 1

Original (solid curves) and reconstructed (dashed curves) Hilbert transform pairs for μ=1. For μ less than 0.1, the reconstructed and the original functions are virtually identical on this plot.

Fig. 2
Fig. 2

Rms error of the reconstructed transfer function G inside Ω for different values of μ.

Fig. 3
Fig. 3

Power transmission of the fiber grating in example B. The relevant wavelength interval corresponding to frequencies inside Ω is indicated with vertical dashed lines.

Fig. 4
Fig. 4

Transmission group delay of the designed fiber grating in example B. The relevant band Ω is indicated with vertical dashed lines. Note that the delay offset associated with the pure propagation has been removed because the input and output reference planes coincide.

Fig. 5
Fig. 5

Magnitude and phase of the complex coupling coefficient of the designed grating (example B).

Fig. 6
Fig. 6

Power transmission of the fiber grating in example C. The relevant wavelength interval corresponding to frequencies inside Ω is indicated with vertical dashed lines.

Fig. 7
Fig. 7

Transmission group delay of the designed fiber grating in example C. The relevant band Ω is indicated with vertical dashed lines.

Fig. 8
Fig. 8

Magnitude (top) and phase (bottom) of the complex coupling coefficient of the designed grating with nondispersive reflection response (example C).

Fig. 9
Fig. 9

Magnitude of coupling coefficient (top) and relative chirp of the designed grating (bottom) with dispersive reflection response (example C). The chirp is designed relative to the design Bragg wavelength 1550 nm.

Equations (30)

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

H(ω)=(Fh) (ω)=0h(t)exp(-iωt)dt,
h(t)=δ(t)+hL2(t).
H(ω)=HL2(ω)+1.
H(ω)1,ω.
H˜(ω)=ln[H(ω)].
arg H=H{ln|H|},
arg H(ω)=1π-ln|H(ω)|ω-ωdω,
ω1ω2|F(ω)-G(ω)|2dω2,
H(ω)=exp[G(ω)].
F(ω)=iddωln[Hdes(ω)],
H(ω)=exp[G1(ω)],
Re G1(ω)=ωIm G(ω)dω+Cforω-(ω1+ω2)2ωB0forω-(ω1+ω22>ωB,
Im G1(ω)=H{Re G1(ω)}.
-ddωIm G1(ω)-Re G(ω)
=-ddωHRe G1+HIm G=H-ddωRe G1+Im G)
=1π-ω-Im G(ω)dωω-ω+1πω+Im G(ω)dωω-ω
-G-π(ω--ω)+G+π(ω+-ω),
μ+12Ge(ω)-12πiΩˆGe(ω)dωω-ω=F(ω),ωΩ
G(ω)=12π0exp(-iωt)Ωˆexp(iωt)Ge(ω)dωdt
G(ω)=12Ge(ω)-12πiΩˆGe(ω)dωω-ω,ωΩ.
μ+12Ge-12iHGe=F,
-12πiΩGe(ω)dωω-ω=12π-exp(-iωt)u(t)-12×Ωexp(iωt)Ge(ω)dωdt,
(HGe)(m)=-iNn=1Nk=-(N-1)/2(N-1)/2sgn(k)Ge(n)×exp[i2πk(n-m)/N],
sgn(k)=-1for k<00for k=01for k>0.
(H)(m, n)=-iNk=-[(N-1)/2](N-1)/2sgn(k)exp[i2πk(n-m)/N]
form, n{1, 2 ,, NF},
H/(2μ+1)<1,μ>0.
u(ω)=(ω-ω0)/2ωwfor|ω-ω0|<ωw0for|ω-ω0| ωw,
F(ω)=a(ω-ω0)+i0,ωΩ,
F(ω)=b(ω-ω0)2+i0,ωΩ,

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