Abstract

We present a complete end-to-end characterization of Gaussian pulse propagation through optical fibers and waveguides with an arbitrary dispersion profile. Our model takes into account the possible chirping of the source and it also encompasses the influence of the source linewidth. We modeled the arbitrary dispersion by taking as much of the coefficients from the Taylor series representing the fiber and waveguide propagation constant as desired. The model is used to study the impact of higher-order dispersion terms in the propagated pulse shape and rms time width. Also outlined are applications to the calculation of capacity limits in optical communications systems limited by high-order dispersion terms and to other fields such as the calculation of aberrations in temporal imaging systems and spatial diffraction.

© 2003 Optical Society of America

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References

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  1. F. P. Kapron and D. B. Keck, “Pulse transmission through a dielectric optical waveguide,” Appl. Opt. 10, 1519–1523 (1971).
    [CrossRef] [PubMed]
  2. K. Jurgensen, “Transmission of Gaussian pulses through single-mode dielectric optical waveguides,” Appl. Opt. 16, 22–23 (1977).
    [CrossRef]
  3. K. Jurgensen, “Gaussian pulse transmission through single-mode fibers accounting for source linewidth,” Appl. Opt. 17, 2412–2415 (1978).
    [CrossRef]
  4. F. Kapron, “Maximum information capacity of fiber-optic waveguides,” Electron. Lett. 13, 96–97 (1977).
    [CrossRef]
  5. M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to second-order dispersion,” Appl. Opt. 18, 678–682 (1979).
    [CrossRef] [PubMed]
  6. M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to second-order dispersion: effect of the optical source bandwidth,” Appl. Opt. 18, 2237–2240 (1979).
    [CrossRef] [PubMed]
  7. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653–1660 (1980).
    [CrossRef] [PubMed]
  8. D. Marcuse, “Pulse distortion in single-mode fibers. Part 2,” Appl. Opt. 20, 2969–2974 (1981).
    [CrossRef] [PubMed]
  9. D. Marcuse, “Pulse distortion in single-mode fibers. 3: Chirped pulses,” Appl. Opt. 20, 3573–3579 (1981).
    [CrossRef] [PubMed]
  10. J. Capmany and M. A. Muriel, “Optical pulse sequence transmission through single-mode fibers: interference signal analysis,” J. Lightwave Technol. 9, 27–36 (1991).
    [CrossRef]
  11. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1998).
  12. H. Kogelnik, “High-capacity optical communications,” IEEE J. Sel. Top. Quantum Electron. 6, 1279 (2000).
    [CrossRef]
  13. T. Yamamoto and M. Nakazawa, “Third- and fourth-order active dispersion compensation with a phase modulator in a terabit-per-second optical time-division multiplexed transmission,” Opt. Lett. 26, 647–649 (2001).
    [CrossRef]
  14. M. D. Pelusi, Y. Matsui, and A. Suzuki, “Fourth-order dispersion suppression of ultrashort optical pulses by second-order dispersion and cosine phase modulation,” Opt. Lett. 25, 296–299 (2000).
    [CrossRef]
  15. M. Amemiya, “Pulse broadening due to higher order dispersion and its transmission limit,” J. Lightwave Technol. 20, 591–597 (2002).
    [CrossRef]
  16. J. Lgsgaard, A. Bjarklev, and S. E. Barkou-Libori, “Chromatic dispersion in photonic crystal fibers: fast and accurate scheme for calculation,” J. Opt. Soc. Am. B 20, 443–448 (2003).
    [CrossRef]
  17. C. Bennett and B. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
    [CrossRef]
  18. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
    [CrossRef]
  19. C. V. Bennett and B. H. Kolner, “Subpicosecond single-shot waveform measurement using temporal imaging,” in Lasers and Electro-Optics Society Annual Meeting 1999 12th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 1999), paper ThBB1.
  20. C. Bennett and B. Kolner, “Principles of parametric temporal imaging—Part I: Systems configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
    [CrossRef]
  21. J. W. Goodman, Introduction to Fourier Optics 2nd ed. (McGraw-Hill, New York, 1996).
  22. We omit in the description of this model the spatial profile e(x, y) of the fundamental mode because it is not relevant for the discussion.
  23. We use the commonly accepted terminology on dispersion by which β2 is responsible for the fiber first-order dispersion and β3 is responsible for the fiber second-order dispersion. Thus, in general, βk is responsible for the k th-order to first-order dispersion. Note that with this terminology the odd dispersion orders correspond to β2, β4, β6 ... and the even to β3, β5, β7 ... .
  24. M. D. Pelusi et al., “Fourth-order dispersion compensation for 250-fs pulse transmission over 139-km optical fiber,” IEEE Photon. Technol. Lett. 12, 795–797 (2000).
    [CrossRef]
  25. J. Capmany, D. Pastor, S. Sales, and B. Ortega, “Effects of fourth-order dispersion in very high-speed optical time-division multiplexed transmission,” Opt. Lett. 27, 960–962 (2002).
    [CrossRef]
  26. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).
  27. The reader is warned again about the fact that the end-to-end (input-to-output electrical signals) propagation system is not linear. Linearity is achieved only for electric fields within the fiber and waveguide that is closed by two nonlinear operations: The input electrical field is proportional to the square root of the input electrical signal, and the output electrical signal is proportional to the square of the output electrical field.
  28. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003).

2003

2002

2001

2000

C. Bennett and B. Kolner, “Principles of parametric temporal imaging—Part I: Systems configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

M. D. Pelusi et al., “Fourth-order dispersion compensation for 250-fs pulse transmission over 139-km optical fiber,” IEEE Photon. Technol. Lett. 12, 795–797 (2000).
[CrossRef]

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Fourth-order dispersion suppression of ultrashort optical pulses by second-order dispersion and cosine phase modulation,” Opt. Lett. 25, 296–299 (2000).
[CrossRef]

H. Kogelnik, “High-capacity optical communications,” IEEE J. Sel. Top. Quantum Electron. 6, 1279 (2000).
[CrossRef]

1994

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

1991

J. Capmany and M. A. Muriel, “Optical pulse sequence transmission through single-mode fibers: interference signal analysis,” J. Lightwave Technol. 9, 27–36 (1991).
[CrossRef]

1981

1980

1979

1978

1977

F. Kapron, “Maximum information capacity of fiber-optic waveguides,” Electron. Lett. 13, 96–97 (1977).
[CrossRef]

K. Jurgensen, “Transmission of Gaussian pulses through single-mode dielectric optical waveguides,” Appl. Opt. 16, 22–23 (1977).
[CrossRef]

1971

Amemiya, M.

Barkou-Libori, S. E.

Bennett, C.

C. Bennett and B. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

C. Bennett and B. Kolner, “Principles of parametric temporal imaging—Part I: Systems configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

Bjarklev, A.

Capmany, J.

J. Capmany, D. Pastor, S. Sales, and B. Ortega, “Effects of fourth-order dispersion in very high-speed optical time-division multiplexed transmission,” Opt. Lett. 27, 960–962 (2002).
[CrossRef]

J. Capmany and M. A. Muriel, “Optical pulse sequence transmission through single-mode fibers: interference signal analysis,” J. Lightwave Technol. 9, 27–36 (1991).
[CrossRef]

Jurgensen, K.

Kapron, F.

F. Kapron, “Maximum information capacity of fiber-optic waveguides,” Electron. Lett. 13, 96–97 (1977).
[CrossRef]

Kapron, F. P.

Keck, D. B.

Kogelnik, H.

H. Kogelnik, “High-capacity optical communications,” IEEE J. Sel. Top. Quantum Electron. 6, 1279 (2000).
[CrossRef]

Kolner, B.

C. Bennett and B. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

C. Bennett and B. Kolner, “Principles of parametric temporal imaging—Part I: Systems configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

Lgsgaard, J.

Marcuse, D.

Matsui, Y.

Miyagi, M.

Muriel, M. A.

J. Capmany and M. A. Muriel, “Optical pulse sequence transmission through single-mode fibers: interference signal analysis,” J. Lightwave Technol. 9, 27–36 (1991).
[CrossRef]

Nakazawa, M.

Nishida, S.

Ortega, B.

Pastor, D.

Pelusi, M. D.

M. D. Pelusi et al., “Fourth-order dispersion compensation for 250-fs pulse transmission over 139-km optical fiber,” IEEE Photon. Technol. Lett. 12, 795–797 (2000).
[CrossRef]

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Fourth-order dispersion suppression of ultrashort optical pulses by second-order dispersion and cosine phase modulation,” Opt. Lett. 25, 296–299 (2000).
[CrossRef]

Sales, S.

Suzuki, A.

Yamamoto, T.

Appl. Opt.

Electron. Lett.

F. Kapron, “Maximum information capacity of fiber-optic waveguides,” Electron. Lett. 13, 96–97 (1977).
[CrossRef]

IEEE J. Quantum Electron.

C. Bennett and B. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

C. Bennett and B. Kolner, “Principles of parametric temporal imaging—Part I: Systems configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

H. Kogelnik, “High-capacity optical communications,” IEEE J. Sel. Top. Quantum Electron. 6, 1279 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

M. D. Pelusi et al., “Fourth-order dispersion compensation for 250-fs pulse transmission over 139-km optical fiber,” IEEE Photon. Technol. Lett. 12, 795–797 (2000).
[CrossRef]

J. Lightwave Technol.

J. Capmany and M. A. Muriel, “Optical pulse sequence transmission through single-mode fibers: interference signal analysis,” J. Lightwave Technol. 9, 27–36 (1991).
[CrossRef]

M. Amemiya, “Pulse broadening due to higher order dispersion and its transmission limit,” J. Lightwave Technol. 20, 591–597 (2002).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1995).

The reader is warned again about the fact that the end-to-end (input-to-output electrical signals) propagation system is not linear. Linearity is achieved only for electric fields within the fiber and waveguide that is closed by two nonlinear operations: The input electrical field is proportional to the square root of the input electrical signal, and the output electrical signal is proportional to the square of the output electrical field.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, Calif., 2003).

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1998).

C. V. Bennett and B. H. Kolner, “Subpicosecond single-shot waveform measurement using temporal imaging,” in Lasers and Electro-Optics Society Annual Meeting 1999 12th Annual Conference (Institute of Electrical and Electronics Engineers, New York, 1999), paper ThBB1.

J. W. Goodman, Introduction to Fourier Optics 2nd ed. (McGraw-Hill, New York, 1996).

We omit in the description of this model the spatial profile e(x, y) of the fundamental mode because it is not relevant for the discussion.

We use the commonly accepted terminology on dispersion by which β2 is responsible for the fiber first-order dispersion and β3 is responsible for the fiber second-order dispersion. Thus, in general, βk is responsible for the k th-order to first-order dispersion. Note that with this terminology the odd dispersion orders correspond to β2, β4, β6 ... and the even to β3, β5, β7 ... .

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Figures (11)

Fig. 1
Fig. 1

Block diagram for the model describing the pulse propagation through optical fibers and waveguides with arbitrary dispersion. The shaded ovals include the end-to-end electric magnitudes.

Fig. 2
Fig. 2

Results of the effect of stand-alone dispersive terms from k=2 (first-order dispersion) to k=7 (sixth-order dispersion) for monochromatic and chirpless sources (V=0, C=0) computed with Eqs. (12) and (13) (solid curves) and Eqs. (14) (dashed curves). (a) First-order dispersion, (b) second-order dispersion, (c) third-order dispersion, (d) fourth-order dispersion, (e) fifth-order dispersion, (f) sixth-order dispersion. In each case the value of the relevant normalized dispersion parameter is shown in the graphs.

Fig. 3
Fig. 3

Output pulse for a system characterized by V=0, C=0, D2=0, D3=5/2, D4=5/3, D5=-5/6, D6=5/9, D7=-1/5 illustrating the influence of the interfering combination of several dispersion terms. Results computed with Eqs. (12) and (13) are shown by the solid curve and with Eqs. (14) by the dashed curve.

Fig. 4
Fig. 4

Output pulse for a system with only even dispersion orders (i.e., all the odd dispersion orders are zero). In this case, C=V=0, D2=0, D3=1, D4=0, D5=1, D6=0, D7=1. Results computed with Eqs. (12) and (13) are shown by the solid curve and with Eqs. (14) by the dashed curve.

Fig. 5
Fig. 5

Output pulse for a system with only odd dispersion orders. In this case C=V=0, D2=1, D3=0, D4=1, D5=0, D6=1, D7=0. Results computed with Eqs. (12) and (13) are shown by the solid curve and with Eqs. (14) by the dashed curve.

Fig. 6
Fig. 6

Effect of the chirp parameter on the compression and broadening of a pulse subject to the effect of fifth-order dispersion (D6=2). The reader can observe that a chirp parameter of C=-0.5 results in a pulse compression as compared with the no-chirp case (C=0) and positive chirp (C=0.5).

Fig. 7
Fig. 7

Normalized broadening factor given by Eqs. (17) for a system with only fifth-order dispersion D6 against the value of D6 and for several values of the chirp parameter (C=0, C=-0.5, and C=0.5). The results obtained from Eq. (16) and the numerical solution of Eqs. (14) are identical.

Fig. 8
Fig. 8

Universal curves for capacity limits Cmaxβ1/p (bit/s) versus the optical fiber link length L parameterized by the order p of the most significant dispersion term. Results are shown for orders ranging from 2 to 10. The curves show a monotonic and asymptotic behavior toward a vertical line. This is a consequence of the Cmax1/L1/p behavior anticipated by Eq. (4), which in logarithmic units results in L-pCmax. These curves apply to any kind of fiber.

Fig. 9
Fig. 9

Example of denormalized curves for a dispersion-shifted fiber link. Capacity limits Cmax (bits/s) versus the optical fiber link length L parameterized by the order p of the most significant dispersion term for a standard dispersion-shifted fiber with |β2|1/2=1.47, |β3|1/3=0.52, |β4|1/4=0.137, |β5|1/5=0.068, and |β6|1/6=0.044.

Fig. 10
Fig. 10

Block diagram of a TIS.

Fig. 11
Fig. 11

Configuration of the one-dimensional Fresnel diffraction problem.

Equations (45)

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E(0, t)=f(t)ψ0(t),
ψ0(t)=A(t)expjw0t+Ct22T2.
e(0, w)=12π-f(t)ψ0(t)exp(-jwt)dt=-ϕ0(s)F(w-s)ds,
E(L, t)=-e(0, w)exp{j[wt-β(w)L]}dw.
P(L, t)=|E(L, t)|2=--E˜(0, u)E˜*(0, ν)×exp(j{(u-ν)t-[β(u)-β(ν)]L})dudν,
E˜(0, u)E˜*(0, ν)=--ϕ0(r)ϕ0*(s)×F(u-r)F*(ν-s)drds.
β(w)L=k=01k!dkβdwkw=w0(w-w0)kL=k=0βkLk! (w-w0)k=k=0Dk[T(w-w0)]k.
β(u)L-β(ν)L=β1L(u-ν)TT+n=2k=0n-1Dn×Tnnk(u-ν)n-k(ν-w0)k.
f(t)=Sexp-t22T2.
R(u)=A(t)A*(t+u)=P0exp-uW22,
ϕ0(r)ϕ0*(s)=T22πC RT2(r-s)CexpjT2(r-s)C×w0-12 (r+s),
F(u-r)=S2π T exp-T22 (u-r)2,
F(ν-s)=S2π T exp-T22 (ν-s)2.
x=(u-ν)T,
y=(ν-w0)T,
V=WT.
P(L, t)=-G(x, L)expjt-β1LTxdx.
G(x, L)=P0S2(1+V2+C2)1/2exp-jk=0NDkxk×exp-x2(2+V2+2jC)4(1+V2+C2)×-exp-u(u+x+jCx)1+V2+C2×exp-jr=2Nk=1r-1rkDrxr-kukdu.
fξ=k=2-(j)k-1βkTkk!kfτk=k=2-(j)k-1Dkkfτk,
τ=t-β1zT,
ξ=zL.
σ=-T2G(x, L)-1d2Gdx2x=0-jTG(x, L)-1dGdxx=021/2.
σ2σ02=1+4Ck=1EA1(k)D2kUk-1+2k=1El=1EA2(k, l)D2kD2lUk+l-1+2k=1Ol=1OA3(k, l)D2k+1D2l+1Uk+1,
U=1+V2+C2,
A1(k)=2k(2k)!4kk!,
A2(k, l)=4kl4k+l-1[2(k+l-1)]!(k+l-1)!,
A3(k, l)=(2k+1)(2l+1)4k+l[2(k+l)]!(k+l)!-(2k)!(2l)!k!l!.
(1+C2)E+2A1(E)D2EA2(E, E)C=0.
C4+2C2+C5D4+1=0,
Cmax|β2n|1/2n=144n4n-21/2(4n-2)A2(n, n)L(2n)!2n21/(4n) ifp=2n,
Cmax|β2n+1|1/2n+1=14(2n+1)2n1/24nA3(n, n)L(2n+1)!22n+1/221/(4n+2) ifp=2n+1.
E(0, t)=exp-t22τ2exp(jw0t),
T2=τ2+[Φ2(1)]2τ2,
C=Φ2(1)τ2-T2Φ2(f),
Dk=Φk(2)k!Tk.
P[L(1)+L(2), t]=exp-Φ2(1)τΦ2(2)2t2=exp-tτM2.
E(xi)=expxi22X2A(xi),
R(u)=A(x)A*(x+u)=P0exp-uWx22.
H(kx)=exp-jkd1-kxk21/2.
H(kx)expjd (kx2)2k,
H(kx)expjzn=1Bn(kx2n),
Bn=(2n-3)!!2nk2n-1n!.
D2n=Bnd/X2n,
D2n+1=0,
V=WxX.

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