Abstract

Theories of a technique for compression of an ultrashort femtosecond laser pulse beyond its Fourier-transform limit based on the pulse shaping technique are proposed. The technique is called superresolution in time domain (STD). Global optimization theories for the design of a mask to modulate the spectrum of an input pulse to obtain a STD output pulse are proposed. Several design examples illustrate the feasibility of STD. Some fundamental limits of STD are also provided.

© 2006 Optical Society of America

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  1. K. Ohno, T. Tanabe, and F. Kannari, "Adaptive pulse shaping of phase and amplitude of an amplified femtosecond pulse laser by direct reference to frequency-resolved optical gating traces," J. Opt. Soc. Am. B 19, 2781-2790 (2002).
    [CrossRef]
  2. N. Karasawa, L. Li, A. Suguro, H. Shigekawa, R. Morita, and M. Yamashita, "Optical pulse compression to 5.0 fs by use of only a spatial light modulator for phase compensation," J. Opt. Soc. Am. B 18, 1742-1746 (2001).
    [CrossRef]
  3. E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, "Pulse compression by use of deformable mirrors," Opt. Lett. 24, 493-495 (1999).
    [CrossRef]
  4. F. Verluise, V. Laude, Z. Cheng, Ch. Spielmann, and P. Tournois, "Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping," Opt. Lett. 25, 575-577 (2000).
    [CrossRef]
  5. E. Zeek, R. Bartels, M. M. Murnane, H. C. Kapteyn, and S. Backus, G. Vdovin, "Adaptive pulse compression for transform-limited 15-fs high-energy pulse generation," Opt. Lett. 25, 587-589 (2000).
    [CrossRef]
  6. P. Wnuk and C. Radzewicz, "Bimorph piezo deformable mirror for femtosecond pulse shaping," Opt. Express 13, 4154-4159 (2005).
    [CrossRef] [PubMed]
  7. C. Radzewicz, P. Wasylczyk, W. Wasilewski, and J. S. Krasinski, "Piezo-driven deformable mirror for femtosecond pulse shaping," Opt. Lett. 29, 177-179 (2004).
    [CrossRef] [PubMed]
  8. J. Seres, A. Müller, E. Seres, K. O'Keeffe, M. Lenner, R. F. Herzog, D. Kaplan, C. Spielmann, and F. Krausz, "Sub-10-fs, terawatt-scale Ti:sapphire laser system," Opt. Lett. 28, 1832-1834 (2003).
    [CrossRef] [PubMed]
  9. A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-1960 (2000).
    [CrossRef]
  10. D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
    [CrossRef]
  11. C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
    [CrossRef]
  12. J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, 1989), Chap. 2.
    [CrossRef]
  13. C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998), p. 31.
  14. T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
    [CrossRef]
  15. L. E. Elsgolc, Calculus of Variations (Pergamon, 1961), Chap. I.

2005 (1)

2004 (1)

2003 (1)

2002 (1)

2001 (1)

2000 (3)

1999 (2)

E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, "Pulse compression by use of deformable mirrors," Opt. Lett. 24, 493-495 (1999).
[CrossRef]

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

1993 (1)

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

Backus, S.

Bartels, R.

Blyth, T. S.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

Cheng, Z.

Elsgolc, L. E.

L. E. Elsgolc, Calculus of Variations (Pergamon, 1961), Chap. I.

Herzog, R. F.

Iaconis, C.

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

Kane, D. J.

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

Kannari, F.

Kaplan, D.

Kapteyn, H.

Kapteyn, H. C.

Karasawa, N.

Krasinski, J. S.

Krausz, F.

Laude, V.

Lenner, M.

Li, L.

Maginnis, K.

Morita, R.

Mourou, G.

Müller, A.

Murnane, M.

Murnane, M. M.

Ohno, K.

O'Keeffe, K.

Radzewicz, C.

Robertson, E. F.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

Rulliere, C.

C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998), p. 31.

Russek, U.

Seres, E.

Seres, J.

Shigekawa, H.

Spielmann, C.

Spielmann, Ch.

Strayer, J. K.

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, 1989), Chap. 2.
[CrossRef]

Suguro, A.

Tanabe, T.

Tournois, P.

Trebino, R.

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

Vdovin, G.

Verluise, F.

Walmsley, I. A.

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

Wasilewski, W.

Wasylczyk, P.

Weiner, A. M.

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

Wnuk, P.

Yamashita, M.

Zeek, E.

IEEE J. Quantum Electron. (2)

D. J. Kane and R. Trebino, "Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating," IEEE J. Quantum Electron. 29, 571-579 (1993).
[CrossRef]

C. Iaconis and I. A. Walmsley, "Self-referencing spectral interferometry for measuring ultrashort optical pulses," IEEE J. Quantum Electron. 35, 501-509 (1999).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Lett. (5)

Rev. Sci. Instrum. (1)

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-1960 (2000).
[CrossRef]

Other (4)

J. K. Strayer, Linear Programming and Its Applications (Springer-Verlag, 1989), Chap. 2.
[CrossRef]

C. Rulliere, Femtosecond Laser Pulses: Principles and Experiments (Springer, 1998), p. 31.

T. S. Blyth and E. F. Robertson, Further Linear Algebra (Springer, 2002), Chap. 1.
[CrossRef]

L. E. Elsgolc, Calculus of Variations (Pergamon, 1961), Chap. I.

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Figures (6)

Fig. 1
Fig. 1

Pulse shaping system to compress an optical pulse beyond its FTL.

Fig. 2
Fig. 2

Control of G and M for the design of the mask on the spectrum plane.

Fig. 3
Fig. 3

Linearization of nonlinear constraints (5b). Here X = 1 2 1 2 H ( μ ) A ( μ ) cos ( 2 π μ τ i ) d μ , Y = 1 2 1 2 H ( μ ) A ( μ ) sin ( 2 π μ τ i ) d μ , R = ϵ 1 2 1 2 H ( μ ) A ( μ ) d μ , κ ( p ) = ( p 1 ) π ( 2 P ) , p = 1 , 2 , , P , and P = 4 in this scheme.

Fig. 4
Fig. 4

(a) Gaussian spectrum amplitude. (b), (c) Intensity of the designed STD pulse of example 1 in Table 2 with the Gaussian spectrum amplitude shown in (a).

Fig. 5
Fig. 5

(a) Irregular spectrum amplitude of Eq. (8). (b), (c) Intensity of the designed STD pulse of example 4 in Table 2 with the irregular spectrum amplitude shown in (a).

Fig. 6
Fig. 6

S e u ( G ) for the Gaussian spectrum amplitude (solid curves) and for the irregular spectrum amplitude of Eq. (8) (open circles). The curves in the inset are of logarithmic vertical coordinate to magnify those of linear vertical coordinate.

Tables (2)

Tables Icon

Table 1 Locations of Definitions of All Variables in the Text

Tables Icon

Table 2 Design Examples with a Gaussian Spectrum Amplitude (Examples 1–3) and with the Irregular Spectrum Amplitude of Eq. (8) (Example 4)

Equations (61)

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e ( t ) = ν 0 Δ ν 2 ν 0 + Δ ν 2 h ( ν ) u ( ν ) exp ( i 2 π ν t ) d ν ,
E ( τ ) = exp ( i 2 π ν 0 t ) 1 2 1 2 H ( μ ) U ( μ ) exp ( i 2 π μ τ ) d μ ,
I ( τ STD + ) I ( 0 ) = I ( τ STD ) I ( 0 ) = I L ( τ FTL + ) I L ( 0 ) = I L ( τ FTL ) I L ( 0 ) = 0.5
max U ( μ ) I ( 0 ) , subject to I ( τ i ) I ( 0 ) ϵ , I ( τ i ) I ( 0 ) ϵ ,
i = 1 , 2 , , N , U ( μ ) 1 ,
max { A ( μ ) , B ( μ ) } { [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B ( μ ) d μ ] 2 } ,
{ 1 2 1 2 H ( μ ) [ A ( μ ) cos ( 2 π μ τ i ) + B ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 + { 1 2 1 2 H ( μ ) [ B ( μ ) cos ( 2 π μ τ i ) A ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 ϵ { [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B ( μ ) d μ ] 2 } ,
{ 1 2 1 2 H ( μ ) [ A ( μ ) cos ( 2 π μ τ i ) B ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 + { 1 2 1 2 H ( μ ) [ B ( μ ) cos ( 2 π μ τ i ) + A ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 ϵ { [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B ( μ ) d μ ] 2 } ,
A ( μ ) 2 + B ( μ ) 2 1 .
max A ( μ ) 1 2 1 2 H ( μ ) A ( μ ) d μ ,
[ 1 2 1 2 H ( μ ) A ( μ ) cos ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) A ( μ ) sin ( 2 π μ τ i ) d μ ] 2 ϵ [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 ,
1 2 A ( μ ) 1 2 ,
( 1 ) m [ 1 2 1 2 H ( μ ) A ( μ ) cos ( 2 π μ τ i ) d μ ] cos γ ( p ) + ( 1 ) n [ 1 2 1 2 H ( μ ) A ( μ ) sin ( 2 π μ τ i ) d μ ] sin γ ( p ) ϵ [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] cos π 4 P ,
m , n { 0 , 1 } , p = 1 , 2 , , P ,
max { A k } k = 1 K A k μ k 1 μ k H ( μ ) d μ ,
( 1 ) m [ k = 1 K A k μ k 1 μ k H ( μ ) cos ( 2 π μ τ i ) d μ ] cos γ ( p ) + ( 1 ) n [ k = 1 K A k μ k 1 μ k H ( μ ) sin ( 2 π μ τ i ) d μ ] sin γ ( p ) ϵ [ k = 1 K A k μ k 1 μ k H ( μ ) d μ ] cos π 4 P ,
m , n { 0 , 1 } , p = 1 , 2 , , P ,
1 2 A k 1 2 , k = 1 , 2 , , K .
H ( μ ) = [ 0.6 + 0.5 sin 2 ( 2 π μ + π 4 ) + 0.3 sin 2 ( 4 π μ π 4 ) ] exp [ ( 2 μ ) 4 ln w 0 ]
μ j , δ b = μ j b + δ sin ( j π 2 ) , j = 1 , 2 , , N b ,
4 π Δ τ RMS Δ μ RMS 1 ,
Δ τ RMS = [ I ( τ ) τ 2 d τ I ( τ ) d τ ] 1 2 ,
Δ μ RMS = [ 1 2 1 2 H ( μ ) U ( μ ) 2 μ 2 d μ 1 2 1 2 H ( μ ) U ( μ ) 2 d μ ] 1 2 = [ 1 2 1 2 H ( μ ) 2 μ 2 d μ 1 2 1 2 H ( μ ) 2 d μ ] 1 2 .
max U ( μ ) I ( 0 ) , subject to I ( τ 0.5 G ) I ( 0 ) = 0.5 ,
I ( τ 0.5 G ) I ( 0 ) = 0.5 , U ( μ ) 1 ,
A ( μ ) + i B ( μ ) = [ A 1 ( μ ) + i B 1 ( μ ) ] exp ( i ϕ 1 )
1 2 1 2 H ( μ ) A ( μ ) d μ = 1 2 1 2 H ( μ ) B ( μ ) d μ ,
ϕ 1 = n 1 π + arctan 1 2 1 2 H ( μ ) [ A 1 ( μ ) B 1 ( μ ) ] d μ 1 2 1 2 H ( μ ) [ A 1 ( μ ) + B 1 ( μ ) ] d μ ,
max { A ( μ ) , B ( μ ) } 1 2 1 2 H ( μ ) A ( μ ) d μ ,
{ 1 2 1 2 H ( μ ) [ A ( μ ) cos ( 2 π μ τ i ) + B ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 + { 1 2 1 2 H ( μ ) [ B ( μ ) cos ( 2 π μ τ i ) A ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 2 ϵ [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 ,
{ 1 2 1 2 H ( μ ) [ A ( μ ) cos ( 2 π μ τ i ) B ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 + { 1 2 1 2 H ( μ ) [ B ( μ ) cos ( 2 π μ τ i ) + A ( μ ) sin ( 2 π μ τ i ) ] d μ } 2 2 ϵ [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 ,
1 2 1 2 H ( μ ) A ( μ ) d μ = 1 2 1 2 H ( μ ) B ( μ ) d μ ,
A ( μ ) 2 + B ( μ ) 2 1 .
A ( μ ) = A 0 ( μ ) , B ( μ ) = B 0 ( μ )
A ( μ ) = B ( μ ) = [ A 0 ( μ ) + B 0 ( μ ) ] 2 .
1 2 1 2 H ( μ ) A 0 ( μ ) + B 0 ( μ ) 2 d μ = 1 2 1 2 H ( μ ) A 0 ( μ ) d μ = F max ,
[ 1 2 1 2 H ( μ ) A 0 ( μ ) cos ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B 0 ( μ ) sin ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B 0 ( μ ) cos ( 2 π μ τ i ) d μ ] 2
+ [ 1 2 1 2 H ( μ ) A 0 ( μ ) sin ( 2 π μ τ i ) d μ ] 2 2 ϵ [ 1 2 1 2 H ( μ ) A 0 ( μ ) d μ ] 2 .
{ 1 2 1 2 H ( μ ) [ A 0 ( μ ) + B 0 ( μ ) 2 cos ( 2 π μ τ i ) + A 0 ( μ ) + B 0 ( μ ) 2 sin ( 2 π μ τ i ) ] d μ } 2 + { 1 2 1 2 H ( μ ) [ A 0 ( μ ) + B 0 ( μ ) 2 cos ( 2 π μ τ i ) A 0 ( μ ) + B 0 ( μ ) 2 sin ( 2 π μ τ i ) ] d μ } 2 = 1 2 [ 1 × 1 2 1 2 H ( μ ) A 0 ( μ ) cos ( 2 π μ τ i ) d μ + 1 × 1 2 1 2 H ( μ ) B 0 ( μ ) cos ( 2 π μ τ i ) d μ ] 2 + 1 2 [ 1 × 1 2 1 2 H ( μ ) A 0 ( μ ) sin ( 2 π μ τ i ) d μ + 1 × 1 2 1 2 H ( μ ) B 0 ( μ ) sin ( 2 π μ τ i ) d μ ] 2 1 2 ( 1 2 + 1 2 ) { [ 1 2 1 2 H ( μ ) A 0 ( μ ) cos ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B 0 ( μ ) cos ( 2 π μ τ i ) d μ ] 2 } + 1 2 ( 1 2 + 1 2 ) { [ 1 2 1 2 H ( μ ) A 0 ( μ ) sin ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) B 0 ( μ ) sin ( 2 π μ τ i ) d μ ] 2 } 2 ϵ [ 1 2 1 2 H ( μ ) A 0 ( μ ) d μ ] 2 = 2 ϵ [ 1 2 1 2 H ( μ ) A 0 ( μ ) + B 0 ( μ ) 2 d μ ] 2 ,
{ [ A 0 ( μ ) + B 0 ( μ ) ] 2 } 2 + { [ A 0 ( μ ) + B 0 ( μ ) ] 2 } 2 = [ 1 × A 0 ( μ ) + 1 × B 0 ( μ ) ] 2 2 ( 1 2 + 1 2 ) [ A 0 ( μ ) 2 + B 0 ( μ ) 2 ] 2 1 ,
[ 1 2 1 2 H ( μ ) A ( μ ) cos ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) A ( μ ) sin ( 2 π μ τ i ) d μ ] 2 ϵ [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 + r i 2 = 0 ,
2 A ( μ ) 2 1 + C ( μ ) 2 = 0 ,
F [ A ( μ ) , r i , C ( μ ) , ω i , λ ( μ ) ] = 1 2 1 2 H ( μ ) A ( μ ) d μ + i = 1 N ω i { [ 1 2 1 2 H ( μ ) A ( μ ) cos ( 2 π μ τ i ) d μ ] 2 + [ 1 2 1 2 H ( μ ) A ( μ ) sin ( 2 π μ τ i ) d μ ] 2 ϵ [ 1 2 1 2 H ( μ ) A ( μ ) d μ ] 2 + r i 2 } + 1 2 1 2 λ ( μ ) [ 2 A ( μ ) 2 1 + C ( μ ) 2 ] d μ ,
δ A ( μ ) F [ A ( μ ) , r i , C ( μ ) , ω i , λ ( μ ) ] = 1 2 1 2 { H ( μ ) + 2 H ( μ ) i = 1 N ω i [ c i cos ( 2 π μ τ i ) + s i sin ( 2 π μ τ i ) ϵ c 0 ] + 4 λ ( μ ) A ( μ ) } δ A ( μ ) d μ = 0 ,
F [ A ( μ ) , r i , C ( μ ) , ω i , λ ( μ ) ] r i = 2 ω i r i = 0 , i = 1 , 2 , , N ,
δ C ( μ ) F [ A ( μ ) , r i , C ( μ ) , ω i , λ ( μ ) ] = 1 2 1 2 2 λ ( μ ) C ( μ ) δ C ( μ ) d μ = 0 ,
c 0 = 1 2 1 2 H ( μ ) A ( μ ) d μ ,
c 1 = 1 2 1 2 H ( μ ) A ( μ ) cos ( 2 π μ τ i ) d μ ,
s i = 1 2 1 2 H ( μ ) A ( μ ) sin ( 2 π μ τ i ) d μ .
H ( μ ) + 2 H ( μ ) i = 1 N ω i [ c i cos ( 2 π μ τ i ) + s i sin ( 2 π μ τ i ) ϵ c 0 ] + 4 λ ( μ ) A ( μ ) = 0 ,
2 λ ( μ ) C ( μ ) = 0 .
C ( μ ) 0 , μ [ 1 2 , 1 2 ] ,
C ( μ ) 0 , μ [ a , b ] , [ a , b ] [ 1 2 , 1 2 ] ,
( 1 2 i = 1 N ω i ϵ c 0 ) + i = 1 N 2 ω i c i cos ( 2 π μ τ i ) + i = 1 N 2 ω i s i sin ( 2 π μ τ i ) = 0 ,
μ [ a , b ] , [ a , b ] [ 1 2 , 1 2 ] ,
1 2 i = 1 N ω i ϵ c 0 = 0 ,
2 ω i c i = 0 , 2 ω i s i = 0 , i = 1 , 2 , , N .
c i 2 + s i 2 ϵ c 0 2 + r i 2 = 0 , i = 1 , 2 , , N .
ω i ϵ c 0 = 0 , i = 1 , 2 , , N .
C ( μ ) = 0 , μ [ 1 2 , 1 2 ] .
A ( μ ) { 1 2 , 1 2 } .

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