Abstract

In optical communication, pulses have approximately a rectangular shape with smooth boundaries. In order to understand the dynamics of these pulses: (A) the generic dynamics of singular quantum mechanics’ wave functions was applied to sharp-boundaries pulses propagation in short dispersive medium and (B) an analytical expression for the propagation of a smooth rectangular pulse in dispersive medium was derived. This analytical expression consists of a couple of complex error functions and can be applied in good approximation to most rectangular pulses propagations in dispersive medium. (C) An analytical approximation was derived for the propagation of any pulse with sharp boundaries. This approximation, despite being analytical, can be applied to any sharp-boundaries pulse with any given shape. These exact expressions and approximations can be used in other systems where the Schrödinger dynamics hold, such as the paraxial approximation.

© 2012 Optical Society of America

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References

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  1. G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).
  2. G. Keiser, Optical Fiber Communications4th ed. (McGraw-Hill, 2010).
  3. G. Goldfarb and L. Guifang, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. 19, 969–971 (2007).
    [CrossRef]
  4. M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lank, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27, 3614–3622 (2009).
    [CrossRef]
  5. P. S. Henry, “Lightwave primer,” IEEE J. Quantum Electron. 21, 1862–1879 (1985).
    [CrossRef]
  6. G. P. Agrawal, P. J. Anthony, and T. M. Shen, “Dispersion penalty for 1.3 μm lightwave systems with multimode semiconductor lasers,” J. Lightwave Technol. 6, 620–625 (1988).
    [CrossRef]
  7. E. Granot and A. Marchewka, “Generic short-time propagation of sharp-boundaries wave packets,” Europhys. Lett. 72, 341–347 (2005).
    [CrossRef]
  8. A. Marchewka, E. Granot, and Z. Schuss, “Short time propagation of a singular wave function: some surprising results,” Optics Spectrosc. 103, 330–335 (2007).
    [CrossRef]
  9. E. Granot and A. Marchewka, “Universal potential-barrier penetration by initially confined wave packets,” Phys. Rev. A 76, 012708 (2007).
    [CrossRef]
  10. R. P. Feynman, “The space–time formulation of nonrelativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
    [CrossRef]
  11. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).
  12. M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
    [CrossRef]
  13. M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover, 1965).
  14. A. del Campo, G. Garcia-Calderon, and J. G. Muga, “Quantum transients,” Phys. Rep. 476, 1–50 (2009).
    [CrossRef]
  15. E. Granot and A. Marchewka, “Emergence of currents as a transient quantum effect in nonequilibrium systems,” Phys. Rev. A 84, 032110 (2011).
    [CrossRef]

2011 (1)

E. Granot and A. Marchewka, “Emergence of currents as a transient quantum effect in nonequilibrium systems,” Phys. Rev. A 84, 032110 (2011).
[CrossRef]

2009 (2)

2007 (3)

G. Goldfarb and L. Guifang, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. 19, 969–971 (2007).
[CrossRef]

A. Marchewka, E. Granot, and Z. Schuss, “Short time propagation of a singular wave function: some surprising results,” Optics Spectrosc. 103, 330–335 (2007).
[CrossRef]

E. Granot and A. Marchewka, “Universal potential-barrier penetration by initially confined wave packets,” Phys. Rev. A 76, 012708 (2007).
[CrossRef]

2005 (1)

E. Granot and A. Marchewka, “Generic short-time propagation of sharp-boundaries wave packets,” Europhys. Lett. 72, 341–347 (2005).
[CrossRef]

1988 (1)

G. P. Agrawal, P. J. Anthony, and T. M. Shen, “Dispersion penalty for 1.3 μm lightwave systems with multimode semiconductor lasers,” J. Lightwave Technol. 6, 620–625 (1988).
[CrossRef]

1985 (1)

P. S. Henry, “Lightwave primer,” IEEE J. Quantum Electron. 21, 1862–1879 (1985).
[CrossRef]

1952 (1)

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[CrossRef]

1948 (1)

R. P. Feynman, “The space–time formulation of nonrelativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

Abramowitz, M.

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Agrawal, G. P.

G. P. Agrawal, P. J. Anthony, and T. M. Shen, “Dispersion penalty for 1.3 μm lightwave systems with multimode semiconductor lasers,” J. Lightwave Technol. 6, 620–625 (1988).
[CrossRef]

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).

Alfiad, M. S.

Anthony, P. J.

G. P. Agrawal, P. J. Anthony, and T. M. Shen, “Dispersion penalty for 1.3 μm lightwave systems with multimode semiconductor lasers,” J. Lightwave Technol. 6, 620–625 (1988).
[CrossRef]

del Campo, A.

A. del Campo, G. Garcia-Calderon, and J. G. Muga, “Quantum transients,” Phys. Rep. 476, 1–50 (2009).
[CrossRef]

Feynman, R. P.

R. P. Feynman, “The space–time formulation of nonrelativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

Garcia-Calderon, G.

A. del Campo, G. Garcia-Calderon, and J. G. Muga, “Quantum transients,” Phys. Rep. 476, 1–50 (2009).
[CrossRef]

Goldfarb, G.

G. Goldfarb and L. Guifang, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. 19, 969–971 (2007).
[CrossRef]

Granot, E.

E. Granot and A. Marchewka, “Emergence of currents as a transient quantum effect in nonequilibrium systems,” Phys. Rev. A 84, 032110 (2011).
[CrossRef]

E. Granot and A. Marchewka, “Universal potential-barrier penetration by initially confined wave packets,” Phys. Rev. A 76, 012708 (2007).
[CrossRef]

A. Marchewka, E. Granot, and Z. Schuss, “Short time propagation of a singular wave function: some surprising results,” Optics Spectrosc. 103, 330–335 (2007).
[CrossRef]

E. Granot and A. Marchewka, “Generic short-time propagation of sharp-boundaries wave packets,” Europhys. Lett. 72, 341–347 (2005).
[CrossRef]

Guifang, L.

G. Goldfarb and L. Guifang, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. 19, 969–971 (2007).
[CrossRef]

Hauske, F. N.

Henry, P. S.

P. S. Henry, “Lightwave primer,” IEEE J. Quantum Electron. 21, 1862–1879 (1985).
[CrossRef]

Hibbs, A. R.

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

Keiser, G.

G. Keiser, Optical Fiber Communications4th ed. (McGraw-Hill, 2010).

Kuschnerov, M.

Lank, B.

Marchewka, A.

E. Granot and A. Marchewka, “Emergence of currents as a transient quantum effect in nonequilibrium systems,” Phys. Rev. A 84, 032110 (2011).
[CrossRef]

E. Granot and A. Marchewka, “Universal potential-barrier penetration by initially confined wave packets,” Phys. Rev. A 76, 012708 (2007).
[CrossRef]

A. Marchewka, E. Granot, and Z. Schuss, “Short time propagation of a singular wave function: some surprising results,” Optics Spectrosc. 103, 330–335 (2007).
[CrossRef]

E. Granot and A. Marchewka, “Generic short-time propagation of sharp-boundaries wave packets,” Europhys. Lett. 72, 341–347 (2005).
[CrossRef]

Moshinsky, M.

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[CrossRef]

Muga, J. G.

A. del Campo, G. Garcia-Calderon, and J. G. Muga, “Quantum transients,” Phys. Rep. 476, 1–50 (2009).
[CrossRef]

Napoli, A.

Piyawanno, K.

Schuss, Z.

A. Marchewka, E. Granot, and Z. Schuss, “Short time propagation of a singular wave function: some surprising results,” Optics Spectrosc. 103, 330–335 (2007).
[CrossRef]

Shen, T. M.

G. P. Agrawal, P. J. Anthony, and T. M. Shen, “Dispersion penalty for 1.3 μm lightwave systems with multimode semiconductor lasers,” J. Lightwave Technol. 6, 620–625 (1988).
[CrossRef]

Spinnler, B.

Stegun, A.

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Europhys. Lett. (1)

E. Granot and A. Marchewka, “Generic short-time propagation of sharp-boundaries wave packets,” Europhys. Lett. 72, 341–347 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. S. Henry, “Lightwave primer,” IEEE J. Quantum Electron. 21, 1862–1879 (1985).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. Goldfarb and L. Guifang, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. 19, 969–971 (2007).
[CrossRef]

J. Lightwave Technol. (2)

M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lank, “DSP for coherent single-carrier receivers,” J. Lightwave Technol. 27, 3614–3622 (2009).
[CrossRef]

G. P. Agrawal, P. J. Anthony, and T. M. Shen, “Dispersion penalty for 1.3 μm lightwave systems with multimode semiconductor lasers,” J. Lightwave Technol. 6, 620–625 (1988).
[CrossRef]

Optics Spectrosc. (1)

A. Marchewka, E. Granot, and Z. Schuss, “Short time propagation of a singular wave function: some surprising results,” Optics Spectrosc. 103, 330–335 (2007).
[CrossRef]

Phys. Rep. (1)

A. del Campo, G. Garcia-Calderon, and J. G. Muga, “Quantum transients,” Phys. Rep. 476, 1–50 (2009).
[CrossRef]

Phys. Rev. (1)

M. Moshinsky, “Diffraction in time,” Phys. Rev. 88, 625–631 (1952).
[CrossRef]

Phys. Rev. A (2)

E. Granot and A. Marchewka, “Emergence of currents as a transient quantum effect in nonequilibrium systems,” Phys. Rev. A 84, 032110 (2011).
[CrossRef]

E. Granot and A. Marchewka, “Universal potential-barrier penetration by initially confined wave packets,” Phys. Rev. A 76, 012708 (2007).
[CrossRef]

Rev. Mod. Phys. (1)

R. P. Feynman, “The space–time formulation of nonrelativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
[CrossRef]

Other (4)

R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, 2002).

G. Keiser, Optical Fiber Communications4th ed. (McGraw-Hill, 2010).

M. Abramowitz and A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

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Figures (5)

Fig. 1.
Fig. 1.

A pulse (its real part) with a discontinuity at t=0.

Fig. 2.
Fig. 2.

The real part of the discontinuous pulse (16) for different values of a: 0,0.2, 0.5, 1.5 and 0.7ps1 (from the upper one to the lower one, respectively). The dispersive medium is a 1m of smf28 (i.e., β2=20ps2/km).

Fig. 3.
Fig. 3.

A comparison between the approximate solution (dashed curve) and the exact theory [solid curve—Eq. (16)] for the initial pulse generated by Eq. (13) for a=0.5. The vertical dashed line corresponds to τ1 (left one) and τ1, respectively.

Fig. 4.
Fig. 4.

Comparison between the exact solution (middle panel) Eq. (36), and the approximation [based on Eq. (44)] in the lower panel in the different regions of the approximation (marked by vertical dashed lines). The upper panel stands for the initial pulse. The vertical dashed line corresponds to (from left to right) T/2τ2, T/2τ1, T+2+τ1, T/2τ1, T+2+τ1, and T/2+τ2, respectively.

Fig. 5.
Fig. 5.

Numerical comparisons between the exact pulse propagation (solid black curve in the lower panels) and the approximation (dashed red curve) for three different initial pulses (a ,b, and c in the upper panels). In all the three plots, the transition was taken to be θ=2ps.

Equations (44)

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A(z,T)z+iβ222A(z,t)t2=0,
A(z,t)=K(tt,z)A(0,t)dt,
K(tt,z)=(2πiβ2z)1/2exp(i(tt)22β2z).
A(z=0,t)=exp(iωt)u(t),
A(z>0,t)=M(t,ω,z),
M(t,ω,z)12exp[i(ω2z2β2ωt)]erfc[i12β2z(ωtβ2z)],
A(z=0,t)=f(t)u(t),
A(z>0,t)=F(ω)M(t,ω,z)dω,
A(z>0,t)=M(t,iτ,z)f(τ)|τ=0.
A(z>0,t)=iπexp(it22β2z){η2iη3(1+tτ)+}f(τ)|τ=0.
A(z>0,t>0)1tizβ22πexp(it22β2z)f(0),
A(z>0,t<0)f(t)1tizβ22πexp(it22β2z)f(0).
A(z=0,t)=Aexp(at)u(t).
A(z=0,t=0)/t=aA.
A(z>0,t)=AM(t,ia,z),
A(z>0,t)=A2exp[ia2z2β2+at]erfc[1i2β2z(ia+tβ2z)].
A(z=0,t)=f(t)u(t)+f+(t)u(t).
A(z>0,t<0)f(t)1tizβ22πexp(it22β2z)[f(0)f+(0)],
A(z>0,t>0)f+(t)1tizβ22πexp(it22β2z)[f+(0)f(0)].
A(z>0,t)=(izβ2)3/2t212πexp(it22β2z)f(τ)τ|τ=0.
A(z=0,t)=t[f(t)u(t)+f+u(t)],
A(z>0,t<0)f(t)+(izβ2)3/2t212πexp(it22β2z)[f(0)f+(0)],
A(z>0,t>0)f+(t)+(izβ2)3/2t212πexp(it22β2z)[f+(0)f(0)]
A(z>0,t)A(z>0,t=0)1ttaizβ22πexp(i(tta)22β2z)ΔAa1ttbizβ22πexp(i(ttb)22β2z)ΔAb,
A(z,|t|<T/2)A[1+1t+T/2izβ22πexp(i(t+T/2)22β2z)1tT/2izβ22πexp(i(tT/2)22β2z)],
I(z,|t|<T/2)I[1+1t+T/22zβ2πcos((t+T/2)22β2z+π4)1tT/22zβ2πcos((tT/2)22β2z+π4)]
A(z,|t|>T/2)Aizβ22π[1t+T/2exp(itT2β2z)1tT/2exp(itT2β2z)]exp(it2+T2/42β2z).
A(z,|t|>T/2)2izβ2π2Atsin(tT2β2z)exp(it2+T2/42β2z).
I(z,|t|>T/2)I8zβ2πt2sin2(tT2β2z).
θt2β2zt2,
A(z=0,t)=A2erfc(tθ).
A(z,t)=Ag(z,t),
g(z,t)=12erfc(ti2β2z+θ2).
G(z,t,T)g(z,tT/2)g(z,t+T/2).
A(z=0,t)=AG(0,t,T)=A2[erfc(tT/2θ)erfc(t+T/2θ)].
A(z,t)=AG(z,t,T)=A2[erfc(tT/2i2β2z+θ2)erfc(t+T/2i2β2z+θ2)].
A(z=0,t)=An=x(n)G(0,tn/B,T),
A(z,t)=An=x(n)G(z,tn/B,T).
A(z,t)A2πi2β2z+θ2texp(t2(i2β2z+θ2)(2β2z)2+θ4).
A(z,t)=A2πi2β2ztexp(it22β2z(tθ2β2z)2).
A(z,t0)A2[12πti2β2z+θ2].
τ1[(2β2z)2+θ4]1/4,
τ22β2z/θ.
B(z>0,t){A(0,t)fort<τ2A(0,t)+A(0,0)tizβ22πexp(it22β2z)forτ2<t<τ1A(0,0)2[12πti2β2z+θ2]for|t|<τ1A(0,0)tizβ22πexp(it22β2z)forτ1<t<τ20forτ2<t.

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