Abstract

In mode-locked laser studies and other applications, it is desirable to derive relationships giving the time-domain length and the properties of the waveform under the envelope for pulses presenting narrow spectral widths, without performing inverse-Fourier-transform calculations. The method presented is based on an approximate expression for the transform of a sinusoid with variable frequency, and leads to simple rules for pulse-length evaluation from the frequency-domain phase function. If the phase function has no inflection points, the length is approximately equal to the range covered by the slope of the phase function, and the actual waveform sweeps monotonically in frequency. When inflection points occur, the pulse comprises several superimposed envelopes, each modulating a monotonically swept sinusoid. The effect of dispersive filters is also analyzed. Illustrative examples are discussed numerically.

© 1972 Optical Society of America

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References

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  1. C. E. Cook, Proc. IRE 48, 310, (1960).
    [Crossref]
  2. R. R. Cubeddu and O. Svelto, IEEE J. QE-5, 495 (1969).
    [Crossref]
  3. O. Svelto, Phys. Letters 17, 83 (1970).
  4. E. B. Treacy, Phys. Letters 28A, 34 (1968).
  5. E. B. Treacy, IEEE J. QE-5, 454 (1969).
    [Crossref]

1970 (1)

O. Svelto, Phys. Letters 17, 83 (1970).

1969 (2)

R. R. Cubeddu and O. Svelto, IEEE J. QE-5, 495 (1969).
[Crossref]

E. B. Treacy, IEEE J. QE-5, 454 (1969).
[Crossref]

1968 (1)

E. B. Treacy, Phys. Letters 28A, 34 (1968).

1960 (1)

C. E. Cook, Proc. IRE 48, 310, (1960).
[Crossref]

Cook, C. E.

C. E. Cook, Proc. IRE 48, 310, (1960).
[Crossref]

Cubeddu, R. R.

R. R. Cubeddu and O. Svelto, IEEE J. QE-5, 495 (1969).
[Crossref]

Svelto, O.

O. Svelto, Phys. Letters 17, 83 (1970).

R. R. Cubeddu and O. Svelto, IEEE J. QE-5, 495 (1969).
[Crossref]

Treacy, E. B.

E. B. Treacy, IEEE J. QE-5, 454 (1969).
[Crossref]

E. B. Treacy, Phys. Letters 28A, 34 (1968).

IEEE J. (2)

R. R. Cubeddu and O. Svelto, IEEE J. QE-5, 495 (1969).
[Crossref]

E. B. Treacy, IEEE J. QE-5, 454 (1969).
[Crossref]

Phys. Letters (2)

O. Svelto, Phys. Letters 17, 83 (1970).

E. B. Treacy, Phys. Letters 28A, 34 (1968).

Proc. IRE (1)

C. E. Cook, Proc. IRE 48, 310, (1960).
[Crossref]

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Figures (3)

Fig. 1
Fig. 1

Simple polygonal approximation for an angle function ζ(t) of monotonically increasing frequency.

Fig. 2
Fig. 2

Plot of the function - x 2 + 5 6 x 4 showing the location of the end points and the inflection points, and the three frequency ranges.

Fig. 3
Fig. 3

Time-domain location of the three pulse components for the example of Fig. 2.

Equations (33)

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and             f ( t ) = cos ζ ( t )             for             0 t T , f ( t ) = 0             elsewhere .
ω ( t ) 2 π ν ( t ) = d ζ ( t ) d t .
d ω ( t ) d t > 0
d ω ( t ) d t < 0.
ω ( t ) = ω 0 + μ ( t ) ,
Δ μ μ ( T ) ω 0 ,
A ( ω ) e i ϕ ( ω ) = 0 T cos ζ ( t ) e - i ω t d t = 1 2 0 T e i [ ζ ( t ) - ω t ] d t + 1 2 0 T e i [ - ζ ( t ) - ω t ] d t .
e i [ ζ ( t ) - ω t ] d t
A ( ω ) Δ t ,
ϕ ( ω ) ζ ( t ) - ω t .
T 2 π / Δ μ .
A ( ω ) d t d ω ,
ϕ ( ω ) ζ ( t ) - ω t ,
d ϕ ( ω ) d ω d ζ ( t ) d t d t d ω - t - ω d t d ω = - t ,
d ϕ ( ω ) d ω | ω = ω 0 0 ,
d ϕ ( ω ) d ω | ω = ω 0 + Δ μ T .
d t d ω = - d 2 ϕ ( ω ) d ω 2
μ ( t ) = Δ μ T t ,             i . e . , t ( μ ) = T Δ μ μ .
ϕ ( μ ) = - 0 μ t ( μ ) d μ = - 1 2 T Δ μ μ 2 .
μ ( t ) = Δ μ ( t / T ) p ,             i . e . ,     t ( μ ) = T ( μ / Δ μ ) 1 / p .
ϕ ( μ ) = - 0 μ T ( μ / Δ μ ) 1 / p d μ = T ( Δ μ ) 1 / p ( p p + 1 ) μ ( p + 1 ) / p .
ϕ ( μ ) = const × μ q
T = const × q ( Δ μ ) q - 1
μ ( t ) = cons t × t 1 / ( q - 1 ) .
ϕ ( μ ) = γ [ - ( μ / Δ μ ) 2 + 5 6 ( μ / Δ μ ) 4 ] ,
- Δ μ / 2 μ + Δ μ / 2.
μ Δ μ / 2 = x ,
ϕ ( x ) / γ = - x 2 + 5 6 x 4 ,
1 γ d ϕ ( x ) d x | x = ± 1 = ± 4 3 , 1 γ d ϕ ( x ) d x | x = ± ( 1 / 5 ) 1 2 = ± 4 3 ( 1 5 ) 1 2 .
t 1 = - 4 3 ( 1 5 ) 1 2 γ ( 2 / Δ μ )
t 4 = - 4 3 γ ( 2 / Δ μ )
t 3 - t 4 = ( 16 / 3 ) ( γ / Δ μ ) .
ϕ ( ω ) = ϕ ( ω ) + A Δ μ / 2 ( ω - ω 0 ) 2 = γ ( - x 2 + 5 6 x 4 ) + A x 2 ,