Abstract

An asymptotic method for synthesis and analysis of optical pulses using Hermite–Gaussian functions is outlined. This enables distortion of arbitrary pulse waveforms in monomode fibers to be studied analytically, even with chirping of the light source. This pulse-modeling technique is proposed to be particularly suitable for the synthesis of analytically tractable representations of realistic pulse shapes, providing a flexible and effective platform for analytic system optimization.

© 1989 Optical Society of America

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References

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  1. D. Marcuse, Appl. Opt. 20, 3573 (1981).
    [CrossRef] [PubMed]
  2. F. Koyama, Y. Suematsu, IEEE J. Quantum Electron. QE-21, 292 (1985).
    [CrossRef]
  3. G. F. Agrawal, M. J. Potasek, Opt. Lett. 11, 318 (1986).
    [CrossRef] [PubMed]
  4. K. Pettermann, U. Krüger, Arch. Elektron. Übertra-gungstech. 40, 283 (1986).
  5. H. J. A. da Silva, J. J. O’Reilly, Proc. IEEE 2, 588 (1988).
  6. L. Franks, Signal Theory (Prentice-Hall, Englewood Cliffs, N.J., 1969), Chap. 3.
  7. M. Abramowitz, A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 691.
  8. C. H. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
    [CrossRef]

1988 (1)

H. J. A. da Silva, J. J. O’Reilly, Proc. IEEE 2, 588 (1988).

1986 (2)

G. F. Agrawal, M. J. Potasek, Opt. Lett. 11, 318 (1986).
[CrossRef] [PubMed]

K. Pettermann, U. Krüger, Arch. Elektron. Übertra-gungstech. 40, 283 (1986).

1985 (1)

F. Koyama, Y. Suematsu, IEEE J. Quantum Electron. QE-21, 292 (1985).
[CrossRef]

1982 (1)

C. H. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
[CrossRef]

1981 (1)

Agrawal, G. F.

da Silva, H. J. A.

H. J. A. da Silva, J. J. O’Reilly, Proc. IEEE 2, 588 (1988).

Franks, L.

L. Franks, Signal Theory (Prentice-Hall, Englewood Cliffs, N.J., 1969), Chap. 3.

Henry, C. H.

C. H. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
[CrossRef]

Koyama, F.

F. Koyama, Y. Suematsu, IEEE J. Quantum Electron. QE-21, 292 (1985).
[CrossRef]

Krüger, U.

K. Pettermann, U. Krüger, Arch. Elektron. Übertra-gungstech. 40, 283 (1986).

Marcuse, D.

O’Reilly, J. J.

H. J. A. da Silva, J. J. O’Reilly, Proc. IEEE 2, 588 (1988).

Pettermann, K.

K. Pettermann, U. Krüger, Arch. Elektron. Übertra-gungstech. 40, 283 (1986).

Potasek, M. J.

Suematsu, Y.

F. Koyama, Y. Suematsu, IEEE J. Quantum Electron. QE-21, 292 (1985).
[CrossRef]

Appl. Opt. (1)

Arch. Elektron. Übertra-gungstech. (1)

K. Pettermann, U. Krüger, Arch. Elektron. Übertra-gungstech. 40, 283 (1986).

IEEE J. Quantum Electron. (2)

F. Koyama, Y. Suematsu, IEEE J. Quantum Electron. QE-21, 292 (1985).
[CrossRef]

C. H. Henry, IEEE J. Quantum Electron. QE-18, 259 (1982).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

H. J. A. da Silva, J. J. O’Reilly, Proc. IEEE 2, 588 (1988).

Other (2)

L. Franks, Signal Theory (Prentice-Hall, Englewood Cliffs, N.J., 1969), Chap. 3.

M. Abramowitz, A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), p. 691.

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Figures (4)

Fig. 1
Fig. 1

Approximation of a super-Gaussian pulse3 with Hermite–Gaussian functions. With n = 16 the approximation error is negligible.

Fig. 2
Fig. 2

Approximation of a simulated laser pulse4 with Hermite–Gaussian functions. The error is negligible for n = 32.

Fig. 3
Fig. 3

Approximation (n = 32) of a chirped (α = 2) simulated laser pulse with Hermite–Gaussian functions.

Fig. 4
Fig. 4

Synthesized pulse obtained by considering only three terms (c0 = 1.0, c5 = −0.4, and c9 = −0.2) to represent the in-phase field component R(t).

Equations (12)

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p ( t ) = i = 0 c i Ψ i ( t ) ,
c i = ( p , Ψ i ) = - p ( t ) Ψ i ( t ) d t
Ψ i ( t ) = ( 2 i × i ! × π ) - 1 / 2 H i ( t ) exp ( - t 2 / 2 ) ,
Ψ i ( t ) = ( - 1 ) i ( 2 i × i ! × π ) - 1 / 2 × i ! × k = 0 [ 1 / 2 ] { 2 i - 2 k k ! × ( i - 2 k ) ! d ( i - 2 k ) d t ( i - 2 k ) [ exp ( - t 2 / 2 ) ] } ,
E in ( t ) = P in ( t ) exp [ j ϕ ( t ) ] = R ( t ) + j Q ( t ) ,
P in ( t ) = [ i = 0 r i Ψ i ( t ) ] 2 + [ i = 0 q i Ψ i ( t ) ] 2 ,
H i p ( Ω ) = exp [ - j β ( 2 ) × L × Ω 2 / 2 ] ,
χ ( t ) = ( 2 / T 2 ) i / 4 exp [ - ( t / T ) 2 ] × exp ( j { ( t / T ) 2 × β ( 2 ) × L - tan - 1 [ β ( 2 ) L ] 2 } ) ,
T 2 = 2 { i + [ β ( 2 ) L ] 2 } .
Ψ i ( t ) = χ ( t ) ( - 1 ) i ( 2 i × i ! × π ) - 1 / 2 × i ! × k = 0 [ i / 2 ] [ ( - 2 × u ) 1 - 2 k k ! × ( 1 - 2 k ) ! × H i - 2 k ( v ) ] ,
v = u × t = ( 1 - j β ( 2 ) × L 2 { 1 + [ β ( 2 ) L ] 2 } ) 1 / 2 × t .
P out ( t ) = | i = 0 r i × Φ i ( t ) | 2 + | i = 0 q i × Φ i ( t ) | 2 ,

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