Abstract

The results of a theoretical study of the throughput efficiency limitations of passive picosecond pulse shapers are presented. One goal of the authors was to provide a point of reference with which various pulse shaper schemes can be compared. The maximum efficiency of any general pulse stacker is derived and the special cases of pseudosquare, linear ramps, and exponential pulse distributions treated in detail.

© 1978 Optical Society of America

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References

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  1. J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
    [Crossref]
  2. J. L. Emmett, J. Nuckolls, and L. Wood, Sci. Am. 230, 24 (1974).
    [Crossref]
  3. W. E. Martin and D. Milam, Appl. Opt. 15,(12), 3054 (1976).
    [Crossref] [PubMed]
  4. C. E. Thomas and L. D. Sieber, Appl. Opt. 15, 462 (1976).
    [Crossref] [PubMed]
  5. Papoulis, Systems and Transforms With Applications in Optics (McGraw-Hill, New York, 1968).

1976 (2)

1974 (1)

J. L. Emmett, J. Nuckolls, and L. Wood, Sci. Am. 230, 24 (1974).
[Crossref]

1972 (1)

J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
[Crossref]

Emmett, J. L.

J. L. Emmett, J. Nuckolls, and L. Wood, Sci. Am. 230, 24 (1974).
[Crossref]

Martin, W. E.

Milam, D.

Nuckolls, J.

J. L. Emmett, J. Nuckolls, and L. Wood, Sci. Am. 230, 24 (1974).
[Crossref]

J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
[Crossref]

Papoulis,

Papoulis, Systems and Transforms With Applications in Optics (McGraw-Hill, New York, 1968).

Sieber, L. D.

Thessen, A.

J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
[Crossref]

Thomas, C. E.

Wood, L.

J. L. Emmett, J. Nuckolls, and L. Wood, Sci. Am. 230, 24 (1974).
[Crossref]

J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
[Crossref]

Zimmerman, G.

J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
[Crossref]

Appl. Opt. (2)

Nature (1)

J. Nuckolls, L. Wood, A. Thessen, and G. Zimmerman, Nature 239, 139 (1972).
[Crossref]

Sci. Am. (1)

J. L. Emmett, J. Nuckolls, and L. Wood, Sci. Am. 230, 24 (1974).
[Crossref]

Other (1)

Papoulis, Systems and Transforms With Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (10)

FIG. 1
FIG. 1

Pulse shapes in the time domain of a square output pulse, an apodized square output pulse, and a Gaussian input pulse.

FIG. 2
FIG. 2

Pulse shape in the time domain of a pseudosquare pulse synthesized using three Gaussians.

FIG. 3
FIG. 3

Frequency domain spectral intensity of the square pulse synthesized using three Gaussians together with the spectrum of the Gaussian input pulse. The shaded region represents energy lost in the pulse shaping process.

FIG. 4
FIG. 4

Maximum efficiency for a passive pulse shaper generating a linear ramp distribution of 2, 3, 4, and 10 output pulses as a function of the spacing of the pulses.

FIG. 5
FIG. 5

Maximum efficiency for a passive pulse shaper generating a distribution of two pulses as a function of the ratio of the amplitudes for temporal spacings between 0.5σ and 10σ.

FIG. 6
FIG. 6

Maximum efficiency for a passive pulse shaper generating an exponential distribution of three pulses as a function of the amplitude ratio X for temporal spacings between 0.5σ and 10σ.

FIG. 7
FIG. 7

Maximum efficiency for a passive pulse shaper generating an exponential distribution of four pulses as a function of the amplitude ratio X for temporal spacings between 0.5σ and 5.0σ.

FIG. 8
FIG. 8

Comparison of the input and output spectrum for the case of synthesizing a pseudosquare pulse using two stacked Gaussians of equal amplitude and optimum spacing for producing a smooth pulse.

FIG. 9
FIG. 9

Contrast between the spectra of the input Gaussian and two output Gaussians spaced in time by 5σ.

FIG. 10
FIG. 10

Illustration of efficiency decrease as the output pulse spacing is increased for the case of a pulse stacker which produces smooth pseudosquare pulses for the optimum value of τ.

Equations (36)

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θ ( ω ) = H ( ω ) I ( ω ) .
E o = - f * ( t ) f ( t ) d t = - θ * ( ω ) θ ( ω ) d ω ,
θ ( ω ) = 1 2 π - f ( t ) e i ω t d t
f ( t ) = 1 2 π - θ ( ω ) e - i ω t d ω .
ɛ = ( - θ * N ( ω ) θ N ( ω ) d ω ) / ( - I * N ( ω ) I N ( ω ) d ω ) .
θ ( t ) = 1 ,             - b < t < b θ ( t ) = 0 ,             otherwise
θ ( ω ) = ( 2 / π ) 1 / 2 b ( sin ω b ) / ω b .
I ( t ) = e - a t 2
I ( ω ) = ( 1 / 2 a ) e - ω 2 / 4 a .
I N ( ω ) = e - ω 2 / 4 a ,
θ N ( ω ) = ( sin ω b ) / ω b .
θ A ( ω ) [ ( sin ω b ) / ω b ] e - ω 2 / 4 a .
ɛ = [ - ( sin ω b ω b e - ω 2 / 4 a ) 2 d ω ] / [ - ( e - ω 2 / 4 a ) 2 d ω ] .
ɛ = 2 b 2 a π 0 sin 2 x x 2 e - x 2 / 2 a b 2 d x .
I ( t ) = δ ( t ) ,
θ ( t ) = n = 1 k A n δ ( t - n τ )
H ( ω ) = n = 1 k A n e i ω n τ .
θ * ( ω ) θ ( ω ) = H * ( ω ) H ( ω ) I * ( ω ) I ( ω ) .
H * N ( ω ) H N ( ω ) = ( i = 1 n A i ) - 2 × ( i = 1 n A i 2 + 2 j = 1 n - 1 i = 1 n - j A i A i + j cos ( j ω τ ) ) .
I ( t ) = e - a t 2 .
I ( ω ) = ( 1 / 2 a ) e - ω 2 / 4 a .
ɛ ( τ ) = ( i = 1 n A i ) - 2 ( i = 1 n A i 2 + 2 j = 1 n - 1 i = 1 n - j A i A i + j e - j 2 a τ 2 / 2 ) .
F ( A 1 , A 2 A n , τ ) = - ( θ ( t ) - i = 1 n A i e - [ ( t - i τ ) / σ ] 2 ) d t .
θ ( t ) = 1 ,             τ / σ < t < 3 τ / σ θ ( t ) = 0 ,             otherwise .
F ( A 1 , A 2 , A 3 , τ ) = τ / σ 3 τ / σ ( 1 - A 1 e - [ ( t - τ ) / σ ] 2 + A 2 e - [ ( t - 2 τ ) / σ ] 2 + A 3 e - [ ( t - 3 τ ) / σ ] 2 ) d t .
A i = i b .
ɛ ( τ ) = ( i = 1 n i ) - 2 [ i = 1 n i 2 + 2 j = 1 n - 1 i = 1 n - j i ( i + j ) e - j 2 τ 2 / 2 ] .
k = 1 n k = n ( n + 1 ) / 2
k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) / 6.
ɛ ( τ ) = 4 n 2 ( n + 1 ) 2 ( n ( n + 1 ) ( 2 n + 1 ) 6 + j = 1 n - 1 [ ( n - j ) ( n - j + 1 ) ( 2 n - 2 j + 1 ) / 3 + j ( n - j ) ( n - j + 1 ) ] e - j 2 τ 2 / 2 ) .
ɛ = 2 ( 2 n + 1 ) / 3 ( n + 1 ) n
X A 2 / A 1 A 3 / A 2 = A N / A N - 1 ,
ɛ ( X , τ ) = ( i = 0 n X i ) - 2 [ i = 0 n X 2 i + 2 j = 1 n X j e - j 2 τ 2 / 2 i = 0 n - j X 2 i ] .
H n * ( ω ) H n ( ω ) = 1 [ H * ( ω ) H ( ω ) ] max × ( i = 1 n A i 2 + 2 j = 1 n - 1 i = 1 n - j A i A i + j cos ( j ω τ ) ) .
ɛ ( τ ) = 1 [ H * ( ω ) H ( ω ) ] max × ( i = 1 n A i 2 + 2 j = 1 n - 1 i = 1 n - j A i A i + j e - j 2 a τ 2 / 2 ) .
H n * ( ω ) H n ( ω ) = 1 [ H * ( ω ) H ( ω ) ] max × [ 4 - 2 cos ( ω τ ) + 2 cos ( 3 ω τ ) ]