Abstract

We demonstrate an inductive method for computing exact derivatives of reflection phase for layered media by using the transfer-matrix formalism. The algorithm scales linearly with the number of layers. We show a physically realistic approximation that leads to an efficient procedure for accurately computing dispersion significantly faster than with standard finite-difference methods. We discuss the theory behind the approximation and show results for a dispersion-compensating chirped mirror from a Ti:sapphire laser.

© 2006 Optical Society of America

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References

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  1. A. Stingl, M. Menzner, C. Spielmann, F. Krausz, and R. Szipöcs, 'Sub-10-fs mirror-dispersion-controlled Ti:sapphire laser,' Opt. Lett. 20, 602-604 (1995).
    [CrossRef] [PubMed]
  2. I. D. Jung, F. X. Kärtner, N. Matuschek, D. H. Sutter, F. Morier-Genoud, U. Keller, V. Scheuer, M. Tisch, and T. Tschudi, 'Self-starting 6.5-fs pulses from a KLM Ti:sapphire laser,' Opt. Lett. 22, 1009-1011 (1997).
    [CrossRef] [PubMed]
  3. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, 'Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,' Opt. Lett. 24, 411-413 (1999).
    [CrossRef]
  4. T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
    [CrossRef]
  5. K. Atkinson, An Introduction to Numerical Analysis (Wiley, 1989).
  6. J. R. Birge, C. Jirauschek, and F. X. Kaertner, 'Efficient analytic computation of group delay dispersion from optical interference coatings,' in Optical Interference Coatings Topical Meeting (Optical Society of America, 2004), paper ThA6.
  7. J. Kong, Electromagnetic Theory (EMW, 2001).
  8. N. Matuschek, F. X. Kärtner, and U. Keller, 'Theory of double-chirped mirrors,' IEEE J. Sel. Top. Quantum Electron 4, 197-208 (1998).
    [CrossRef]
  9. F. X. Kärtner, N. Matuschek, T. Schibli, U. Keller, H. A. Haus, C. Heine, R. Morf, V. Scheuer, M. Tilsch, and T. Tschudi, 'Design and fabrication of double-chirped mirrors,' Opt. Lett. 22, 831-833 (1997).
    [CrossRef] [PubMed]
  10. F. X. Kaertner, U. Morgner, T. R. Schibli, E. P. Ippen, J. G. Fujimoto, V. Scheuer, G. Angelow, and T. Tschudi, 'Ultrabroadband double-chirped mirror pairs for generation of octave spectra,' J. Opt. Soc. Am. B 18, 882-885 (2001).
    [CrossRef]

2003 (1)

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
[CrossRef]

2001 (1)

1999 (1)

1998 (1)

N. Matuschek, F. X. Kärtner, and U. Keller, 'Theory of double-chirped mirrors,' IEEE J. Sel. Top. Quantum Electron 4, 197-208 (1998).
[CrossRef]

1997 (2)

1995 (1)

Angelow, G.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
[CrossRef]

F. X. Kaertner, U. Morgner, T. R. Schibli, E. P. Ippen, J. G. Fujimoto, V. Scheuer, G. Angelow, and T. Tschudi, 'Ultrabroadband double-chirped mirror pairs for generation of octave spectra,' J. Opt. Soc. Am. B 18, 882-885 (2001).
[CrossRef]

Atkinson, K.

K. Atkinson, An Introduction to Numerical Analysis (Wiley, 1989).

Birge, J. R.

J. R. Birge, C. Jirauschek, and F. X. Kaertner, 'Efficient analytic computation of group delay dispersion from optical interference coatings,' in Optical Interference Coatings Topical Meeting (Optical Society of America, 2004), paper ThA6.

Chen, Y.

Cho, S. H.

Fujimoto, J. G.

Haus, H. A.

Heine, C.

Ippen, E. P.

Jirauschek, C.

J. R. Birge, C. Jirauschek, and F. X. Kaertner, 'Efficient analytic computation of group delay dispersion from optical interference coatings,' in Optical Interference Coatings Topical Meeting (Optical Society of America, 2004), paper ThA6.

Jung, I. D.

Kaertner, F. X.

F. X. Kaertner, U. Morgner, T. R. Schibli, E. P. Ippen, J. G. Fujimoto, V. Scheuer, G. Angelow, and T. Tschudi, 'Ultrabroadband double-chirped mirror pairs for generation of octave spectra,' J. Opt. Soc. Am. B 18, 882-885 (2001).
[CrossRef]

J. R. Birge, C. Jirauschek, and F. X. Kaertner, 'Efficient analytic computation of group delay dispersion from optical interference coatings,' in Optical Interference Coatings Topical Meeting (Optical Society of America, 2004), paper ThA6.

Kärtner, F. X.

Keller, U.

Kim, J.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
[CrossRef]

Kong, J.

J. Kong, Electromagnetic Theory (EMW, 2001).

Krausz, F.

Kuzucu, O.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
[CrossRef]

Matuschek, N.

Menzner, M.

Morf, R.

Morgner, U.

Morier-Genoud, F.

Scheuer, V.

Schibli, T.

Schibli, T. R.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
[CrossRef]

F. X. Kaertner, U. Morgner, T. R. Schibli, E. P. Ippen, J. G. Fujimoto, V. Scheuer, G. Angelow, and T. Tschudi, 'Ultrabroadband double-chirped mirror pairs for generation of octave spectra,' J. Opt. Soc. Am. B 18, 882-885 (2001).
[CrossRef]

Spielmann, C.

Stingl, A.

Sutter, D. H.

Szipöcs, R.

Tilsch, M.

Tisch, M.

Tschudi, T.

IEEE J. Sel. Top. Quantum Electron (2)

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kärtner, V. Scheuer, and G. Angelow, 'Towards single-cycle laser systems,' IEEE J. Sel. Top. Quantum Electron 9, 990-1001 (2003).
[CrossRef]

N. Matuschek, F. X. Kärtner, and U. Keller, 'Theory of double-chirped mirrors,' IEEE J. Sel. Top. Quantum Electron 4, 197-208 (1998).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (4)

Other (3)

K. Atkinson, An Introduction to Numerical Analysis (Wiley, 1989).

J. R. Birge, C. Jirauschek, and F. X. Kaertner, 'Efficient analytic computation of group delay dispersion from optical interference coatings,' in Optical Interference Coatings Topical Meeting (Optical Society of America, 2004), paper ThA6.

J. Kong, Electromagnetic Theory (EMW, 2001).

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Figures (2)

Fig. 1
Fig. 1

Diagram showing transfer-matrix notation.

Fig. 2
Fig. 2

(a) GD of chirped mirror computed under various approximations. (b) GDD of same mirror under various approximations.

Tables (1)

Tables Icon

Table 1 Relative Execution Times under Various Approximations

Equations (34)

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T ( n , 0 ) = [ a ( ω ) b ( ω ) b * ( ω ) a * ( ω ) ] [ | a ( ω ) | exp [ j ϕ a ( ω ) ] | b ( ω ) | exp [ j ϕ b ( ω ) ] | b ( ω ) | exp [ j ϕ b ( ω ) ] | a ( ω ) | exp [ j ϕ a ( ω ) ] ] .
Γ ( ω ) | Γ ( ω ) | exp [ j ϕ ( ω ) ] = b * ( ω ) a * ( ω ) .
ϕ ( ω ) = ϕ a ( ω ) ϕ b ( ω ) + π ,
ϕ ( k ) ( ω ) = ϕ a     ( k ) ( ω ) ϕ b     ( k ) ( ω ) .
k T ( , 0 ) ω k = k [ T T ( 1 , 0 ) ] ω k .
T ( , 0 ) = T T ( 1 , 0 ) ,
T ( , 0 ) ω = T ω T ( 1 , 0 ) + T T ( 1 , 0 ) ω ,
2 T ( , 0 ) ω 2 = 2 T ω 2 T ( 1 , 0 ) + 2 T ω T ( 1 , 0 ) ω + T 2 T ( 1 , 0 ) ω 2 ,
3 T ( , 0 ) ω 3 = 3 T ω 3 T ( 1 , 0 ) + 3 T ω 2 T ( 1 , 0 ) ω 2 + 3 2 T ω 2 T ( 1 , 0 ) ω + T 3 T ( 1 , 0 ) ω 3 ,
T ( 0 , 0 ) = I , k T ( 0 , 0 ) ω k = 0 , 0 < k m .
a r ( ω ) + j a i ( ω ) | a ( ω ) | exp [ j ϕ a ( ω ) ] .
ϕ a ( ω ) = 1 | a | ( a i cos ϕ a a i , sin ϕ a ) ,
a ( ω ) = a r , cos ϕ a + a i sin ϕ a ,
ϕ″ a ( ω ) = 1 | a | ( a″ i cos ϕ a a″ r    sin ϕ a 2 a ϕ a ) ,
a ( ω ) = α″ r cos ϕ a + α″ i sin ϕ a + | a | ( ϕ a ) 2 ,
ϕ a ( ω ) = 1 | a | [ a i cos ϕ a a‴ r sin ϕ a + | a | ( ϕ a ) 3 3″ a ϕ a 3 a ϕ″ a ] .
ϕ a ( ω ) = 1 | a | 2 ( a i a r a r a i ) ,
ϕ a ( ω ) = 1 | a | 2 [ ( a i 2 a r ϕ a ) a r ( a r 2 a i ϕ a ) a i ] .
T ( ω ) = { [ 1 + p ( ω ) ] exp [ i n ˜ ( ω ) d ω / c ] [ 1 p ( ω ) ] exp [ i n ˜ ( ω ) d ω / c ] [ 1 p ( ω ) ] exp [ i n ˜ ( ω ) d ω / c ] [ 1 + p ( ω ) ] exp [ i n ˜ ( ω ) d ω / c ] } ,
T ( ω ) = ( { i d / c [ 1 + p ( ω ) ] [ n ˜ ( ω ) + ω n ˜ ( ω ) ] + p ( ω ) } exp [ i n ˜ ( ω ) d ω / c ... ] { i d / c [ 1 p ( ω ) ] [ n ˜ ( ω ) + ω n ˜ ( ω ) ] p ( ω ) } exp [ i n ˜ ( ω ) d ω / c ... ] ) ,
p ( ω ) { n ˜ 1 ( ω ) n ˜ ( ω ) ,   TE  polarization, n ˜ 1 ( ω ) n 2 ( ω ) n ˜ ( ω ) n - 1 2 ( ω ) ,   TM  polarization .
T ( k ) ( ω ) D ^     ( k ) T ( D ( k ) 0 0 D ( k ) * ) T ,
T ( , 0 ) ω = D ^ ( 1 ) T ( 1 , 0 ) + T T ( - 1 , 0 ) ,
2 T ( , 0 ) ω 2 = D ^ ( 2 ) T ( 1 , 0 ) + 2 D ^ ( 1 ) T T ( 1 , 0 ) + T T ( 1,0 ) ,
3 T ( , 0 ) ω 3 = D ^ ( 3 ) T ( 1 , 0 ) + 3 D ^ ( 2 ) T T ( 1 , 0 ) + 3 D ^ ( 1 ) T T ( 1 , 0 ) + T T ( 1 , 0 ) ,
D ( 1 ) = i d c [ n ˜ ( ω ) + ω n ˜ ( ω ) ] ,
D (2) = d c { 2 n ˜ ( ω ) + d c [ n ˜ ( ω ) + ω n ˜ ( ω ) ] 2 } ,
D (3) = i d 2 c 2 [ n ˜ ( ω ) + ω n ˜ ( ω ) ] { 6 i n ˜ ( ω ) + d c [ n ˜ ( ω ) + ω n ˜ ( ω ) ] 2 } .
κ ( ω ) 2 r ( ω ) = 2 1 p ( ω ) 1 + p ( ω ) ,
τ g ( ω ) = 2 0 m t ( ω ) ω δ 2 ( m ) κ 2 ( m )  d m ,
τ g ( 0 ) ( ω ) = 2 0 m t ( ω ) ω | δ ( m ) | d m ,
= 2 π 0 m t ( ω ) 1 k B  d m .
k B ( m ) = k 0 + k 1 m ,
τ g ( 0 ) ( ω ) = 2 π c k 1 { ln [ 1 κ ( ω ) π ] + ln ( c k 0 ω ) } .

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