Abstract

An analysis is given for the temporal compression of linearly chirped pulses propagating in strongly dispersive media. The wave vector k(ω) of the dispersive medium is expanded in a Taylor series through terms cubic in frequency. Analytic expressions are developed for the shape of the compressed pulse, and numerical examples given to illustrate the influence of the cubic term in k(ω). This term is shown to give asymmetric broadening of the compressed pulse envelope, even when it is very small in magnitude compared to the lower-order terms in the Taylor series.

© 1977 Optical Society of America

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References

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  1. J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
    [Crossref]
  2. J. R. Klauder, Bell System Tech. J. 39, 809 (1960).
    [Crossref]
  3. C. E. Cook, Proc. IRE 48, 310 (1960).
    [Crossref]
  4. M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), p. 493.
  5. A. Laubereau and D. von der Linde, Z. Naturforsh. A 25, 1626 (1976).
  6. J. D. McMullen, J. Appl. Phys. (to be published).
  7. D. G. Anderson and J. I. H. Askne, Proc. IEEE 62, 1518 (1974).
    [Crossref]
  8. D. Anderson, J. Askne, and M. Lisak, Proc. IEEE 63, 715 (1975).
    [Crossref]
  9. D. Anderson, J. Askne, and M. Lisak, Phys. Rev. A 12, 1546 (1975).
    [Crossref]
  10. D. Rader, Am. J. Phys. 41, 420 (1973).
    [Crossref]
  11. R. A. Fisher, Am. J. Phys. 44, 1002 (1976).
    [Crossref]
  12. D. Rader, Am. J. Phys. 44, 1005 (1976).
    [Crossref]
  13. G. I. Terina, Radio Engr. and Electron. Phys. 17, 475 (1972).
  14. handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. GPO, Washington, D. C., 1965), pp. 446–478.

1976 (3)

A. Laubereau and D. von der Linde, Z. Naturforsh. A 25, 1626 (1976).

R. A. Fisher, Am. J. Phys. 44, 1002 (1976).
[Crossref]

D. Rader, Am. J. Phys. 44, 1005 (1976).
[Crossref]

1975 (2)

D. Anderson, J. Askne, and M. Lisak, Proc. IEEE 63, 715 (1975).
[Crossref]

D. Anderson, J. Askne, and M. Lisak, Phys. Rev. A 12, 1546 (1975).
[Crossref]

1974 (1)

D. G. Anderson and J. I. H. Askne, Proc. IEEE 62, 1518 (1974).
[Crossref]

1973 (1)

D. Rader, Am. J. Phys. 41, 420 (1973).
[Crossref]

1972 (1)

G. I. Terina, Radio Engr. and Electron. Phys. 17, 475 (1972).

1960 (3)

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
[Crossref]

J. R. Klauder, Bell System Tech. J. 39, 809 (1960).
[Crossref]

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

Albersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
[Crossref]

Anderson, D.

D. Anderson, J. Askne, and M. Lisak, Phys. Rev. A 12, 1546 (1975).
[Crossref]

D. Anderson, J. Askne, and M. Lisak, Proc. IEEE 63, 715 (1975).
[Crossref]

Anderson, D. G.

D. G. Anderson and J. I. H. Askne, Proc. IEEE 62, 1518 (1974).
[Crossref]

Askne, J.

D. Anderson, J. Askne, and M. Lisak, Phys. Rev. A 12, 1546 (1975).
[Crossref]

D. Anderson, J. Askne, and M. Lisak, Proc. IEEE 63, 715 (1975).
[Crossref]

Askne, J. I. H.

D. G. Anderson and J. I. H. Askne, Proc. IEEE 62, 1518 (1974).
[Crossref]

Cook, C. E.

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

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
[Crossref]

Fisher, R. A.

R. A. Fisher, Am. J. Phys. 44, 1002 (1976).
[Crossref]

Klauder, J. R.

J. R. Klauder, Bell System Tech. J. 39, 809 (1960).
[Crossref]

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
[Crossref]

Laubereau, A.

A. Laubereau and D. von der Linde, Z. Naturforsh. A 25, 1626 (1976).

Lisak, M.

D. Anderson, J. Askne, and M. Lisak, Phys. Rev. A 12, 1546 (1975).
[Crossref]

D. Anderson, J. Askne, and M. Lisak, Proc. IEEE 63, 715 (1975).
[Crossref]

McMullen, J. D.

J. D. McMullen, J. Appl. Phys. (to be published).

Price, A. C.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
[Crossref]

Rader, D.

D. Rader, Am. J. Phys. 44, 1005 (1976).
[Crossref]

D. Rader, Am. J. Phys. 41, 420 (1973).
[Crossref]

Skolnik, M. I.

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), p. 493.

Terina, G. I.

G. I. Terina, Radio Engr. and Electron. Phys. 17, 475 (1972).

von der Linde, D.

A. Laubereau and D. von der Linde, Z. Naturforsh. A 25, 1626 (1976).

Am. J. Phys. (3)

D. Rader, Am. J. Phys. 41, 420 (1973).
[Crossref]

R. A. Fisher, Am. J. Phys. 44, 1002 (1976).
[Crossref]

D. Rader, Am. J. Phys. 44, 1005 (1976).
[Crossref]

Bell System Tech. J. (2)

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, Bell System Tech. J. 39, 745 (1960).
[Crossref]

J. R. Klauder, Bell System Tech. J. 39, 809 (1960).
[Crossref]

Phys. Rev. A (1)

D. Anderson, J. Askne, and M. Lisak, Phys. Rev. A 12, 1546 (1975).
[Crossref]

Proc. IEEE (2)

D. G. Anderson and J. I. H. Askne, Proc. IEEE 62, 1518 (1974).
[Crossref]

D. Anderson, J. Askne, and M. Lisak, Proc. IEEE 63, 715 (1975).
[Crossref]

Proc. IRE (1)

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

Radio Engr. and Electron. Phys. (1)

G. I. Terina, Radio Engr. and Electron. Phys. 17, 475 (1972).

Z. Naturforsh. A (1)

A. Laubereau and D. von der Linde, Z. Naturforsh. A 25, 1626 (1976).

Other (3)

J. D. McMullen, J. Appl. Phys. (to be published).

M. I. Skolnik, Introduction to Radar Systems (McGraw-Hill, New York, 1962), p. 493.

handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun (U. S. GPO, Washington, D. C., 1965), pp. 446–478.

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Figures (1)

FIG. 1
FIG. 1

Waveform of compressed, chirped-pulse intensity with T = 1 ns, δωm = −100δω0, and τ2 = 0.1 ns, for various values of group dispersion τ3 = 0 (——), τ3 = 3 ps (- - -), τ3 = 4 ps (-· -· -· -·), τ3= 5 ps (-· · -· · -· · -· ·), and τ3 = 10 ps (·····).

Equations (22)

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E ( 0 , t ) = E 0 e 2 t 2 / T 2 cos ( ω 0 t + δ ω m t 2 / 2 T )
( 0 , ω ) = ( 2 π δ ω 0 δ ω ) 1 / 2 E 0 { exp [ 2 ( ω ω 0 ) 2 δ ω 0 ( δ ω 0 + i δ ω m ) i 1 2 tan 1 ( δ ω m δ ω 0 ) ] + exp [ 2 ( ω + ω 0 ) 2 δ ω 0 ( δ ω 0 i δ ω m ) + i 1 2 tan 1 ( δ ω m δ ω 0 ) ] ,
δ ω [ ( δ ω 0 ) 2 + ( δ ω m ) 2 ] 1 / 2
E ( x , t ) = d ω [ ( + ) ( 0 , ω ) e i k ( ω ) x + ( ) ( 0 , ω ) e i k ( ω ) x ] e i ω t ,
k ( ω ) x = k 0 x + τ 1 ( ω ω 0 ) + 1 2 ( 1 2 τ 2 ) 2 ( ω ω 0 ) 2 + 1 3 τ 3 3 ( ω ω 0 ) 3 +
k ( ω ) x = k 0 x τ 1 ( ω + ω 0 ) + 1 2 ( 1 2 τ 2 ) 2 ( ω + ω 0 ) 2 1 3 τ 3 3 ( ω + ω 0 ) 3 +
τ 1 d k d ω | ω 0 x ,
τ 2 2 4 d 2 k d ω 2 | ω 0 x ,
τ 3 3 1 2 d 3 k d ω 3 | ω 0 x .
E ( x , t ) = Re ( 2 E 0 ( 2 π δ ω 0 δ ω ) 1 / 2 exp { i [ k 0 x ω 0 t 1 2 tan 1 ( δ ω m δ ω 0 ) ] } exp [ a τ 3 3 ( τ 1 t + 2 3 a 2 τ 3 3 ) ] Ai ( τ 1 t + a 2 / τ 3 3 τ 3 ) ) ,
a 2 / δ ω 0 ( δ ω 0 + i δ ω m ) i 1 8 τ 2 2 .
E ( x , t ) = Re { E 0 ( T T x ) 1 / 2 exp { i [ k 0 x ω 0 t + 1 2 tan 1 ( ( τ 2 / T ) 2 1 + ( δ ω m / δ ω 0 ) ( τ 2 / T ) 2 ) ] } × exp [ 2 ( τ 1 t ) 2 T x 2 ( 1 + i { δ ω m δ ω 0 [ 1 + δ ω m δ ω 0 ( τ 2 T ) 2 ] + ( τ 2 T ) 2 } ) ] S ( x , t ) } ,
T x = T { [ 1 + δ ω m δ ω 0 ( τ 2 T ) 2 ] 2 + ( τ 2 T ) 4 } 1 / 2
S ( x , t ) = exp [ 1 3 ( τ 3 ( τ 1 t ) 2 a ) 3 ] ( 1 + τ 3 3 ( τ 1 t ) ( 2 a ) 2 ) 1 .
δ ω m | opt = δ ω 0 ( T τ 2 ) 2 ,
I = d y exp ( i ( τ 1 t ) y a y 2 + i τ 3 3 3 y 3 ) ,
ξ = y + i a / τ 3 3 ,
I = exp [ a τ 3 3 ( τ 1 t + 2 3 a 2 τ 3 3 ) ] × d ξ exp [ i τ 3 3 3 ξ 3 + i ( τ 1 t + a 2 τ 3 3 ) ξ ] .
lim | z | Ai ( z ) ( 1 / 2 π 1 / 2 z 1 / 4 ) exp ( 2 3 z 3 / 2 ) ,
z ( a 2 / τ 3 4 ) [ 1 + ( τ 3 3 / a 2 ) ( τ 1 t ) ] .
z 3 / 2 a 3 τ 3 6 [ 1 + 3 2 τ 3 3 a 2 ( τ 1 t ) + 3 8 ( τ 3 3 a 2 ( τ 1 t ) ) 2 + ] ,
z 1 / 4 a 1 / 2 τ 3 [ 1 + 1 4 τ 3 3 a 2 ( τ 1 t ) + ] .