Abstract

A fully analytic method for computing gradients of dispersion (to any order) for a dielectric multilayer coating is developed, and it is demonstrated how group delay gradients can be used to optimize the dispersion of such a filter. The algorithm complexity is linear with the number of layers and quadratic in dispersion order. To our knowledge, this is the first published algorithm for computing exact analytic gradients of dispersion. We show an approximation that speeds up the computation significantly, making it linear in dispersion order. MATLAB and C code implementing the algorithms are made available.

© 2007 Optical Society of America

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References

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    [CrossRef]
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2006 (1)

2005 (2)

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)

1997 (2)

1995 (1)

1994 (1)

1993 (1)

1978 (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).

Baumeister, P. W.

Birge, J. R.

Casperson, L. W.

Chen, Y.

Cho, S. H.

Dombi, P.

Ell, R.

Fuji, T.

Fujimoto, J. G.

Haus, H. A.

Heine, C.

Ippen, E. P.

Jirauschek, C.

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

Kaertner, F. X.

Kärtner, F. X.

Keller, U.

Kim, J.

O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, "Self-referenced 200 MHz octave-spanning Ti:saphhire laser with 50 attosecond carrier-envelope phase jitter," Opt. Express 13, 5163-5169 (2005).
[CrossRef] [PubMed]

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]

Lezius, M.

Matos, L.

Matuschek, N.

Menzner, M.

Morf, R.

Morgner, U.

Mücke, O. D.

O'Keeffe, K.

Popov, K. V.

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.

Szipöcs, R.

Tempea, G.

Tikhonravov, A. V.

Tilsch, M.

Tovar, A. A.

Tschudi, T.

Winter, A.

Yakovlev, V. S.

Appl. Opt. (4)

IEEE J. Sel. Top. Quantum Electron. (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]

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

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

Opt. Express (2)

Opt. Lett. (3)

Other (3)

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

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

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

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Figures (4)

Fig. 1
Fig. 1

Diagram showing transfer matrix notation.

Fig. 2
Fig. 2

(Color online) Spectral group delay of an example chirped mirror. A portion of the response past the high reflectivity region (wavelengths greater than approximately 1050   nm ) is shown to demonstrate that the approximation holds even when the group delay is rapidly varying.

Fig. 3
Fig. 3

(Color online) Spectral group delay dispersion of the chirped mirror shown in Fig. 2.

Fig. 4
Fig. 4

Exact and approximate gradient of the group delay of the filter shown in Fig. 2 at 800   nm . The gradient approaches zero for layers past the 40th as virtually all of the light is reflected by that point.

Equations (39)

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z gd i = 1 n k w i ( τ g ( k i ) τ g 0 ( k i ) + τ * ) 2 ,
z gd τ * = i 2 w i [ τ g ( k i ) τ g 0 ( k i ) + τ * ] = 0 .
τ * = 1 W i w i [ τ g ( k i ) τ g 0 ( k i ) ] ,
τ g ¯ τ g 0 ¯ ,
z GD i = 1 n k w i [ τ g ( k i ) τ g 0 ( k i ) + τ g ¯ τ g 0 ¯ ] 2 .
d z GD = 2 i w i [ τ g ( k i ) τ g 0 ( k i ) + τ g ¯ τ g 0 ¯ ] [ d τ g ( k i ) + d τ g ¯ ] .
T ( k ) = [ T 11 ( k ) T 12 ( k ) T 12 * ( k ) T 11 * ( k ) ] .
T l P l D l
= [ e i n ˜ l ( k ) d l k 0 0 e i n ˜ l ( k ) d l k ] × 1 2 [ 1 + p l ( k ) 1 p l ( k ) 1 p l ( k ) 1 + p l ( k ) ] ,
p l ( k ) { n ˜ l 1 ( k ) n ˜ l ( k ) TE   polarization, n ˜ l 1 ( k ) n l 2 ( k ) n ˜ l ( k ) n l 1 2 ( k ) TM   polarization .
Γ ( k ) = T 12 * ( k ) T 11 * ( k ) ,
T ( k ) = T 11 ( k ) | T 12 ( k ) | 2 T 11 * ( k ) ,
T ( l ,0 ) = T l T ( l 1 , 0 ) ,
T ( l , 0 ) k = T l k T ( l 1 , 0 ) + T l T ( l 1 , 0 ) k .
T l ( k ) P l D l
= [ i d l [ n ˜ l ( k ) + k n ˜ l ( k ) ] 0 0 i d l [ n ˜ l ( k ) + k n ˜ l ( k ) ] ] P l D l
i d l [ n ˜ l ( k ) + k n ˜ l ( k ) ] σ 3 T l ( k ) .
σ 3 [ + 1 0 0 1 ] ,
T = T ( n , l ) T l T ( l 1 , 0 ) .
T d l = T ( n , l ) T l d l T ( l 1 , 0 ) ,
= i k n ˜ l T ( n , l ) σ 3 T ( l ,0 ) ,
T ( k ) = T ( n , l ) T l T ( l 1 , 0 ) + T ( n , l ) T l T ( l 1 , 0 ) + T ( n , l ) T l T ( l 1 , 0 ) .
T ( k ) d l = i k n ˜ l T ( n , l ) σ 3 T ( l , 0 ) + T ( n , l ) T l d l T ( l 1 , 0 ) i k n ˜ l T ( n , l ) σ 3 T ( l , 0 ) ,
T ( k ) d l = i k n ˜ l [ T ( n , l ) σ 3 T ( l , 0 ) + T ( n , l ) σ 3 T ( l , 0 ) ] + 1 2 p T ( n , l ) [ 2 p ( d l k n ˜ + i ) ( n ˜ + k n ˜ ) i k n ˜ p i k n ˜ p e i 2 d l k n ˜ i k n ˜ p e i 2 d l k n ˜ 2 p ( d l k n ˜ i ) ( n ˜ + k n ˜ ) + i k n ˜ p ] T ( l , 0 ) ,
T ( k ) d l = i k n ˜ l [ T ( n , l ) σ 3 T ( l , 0 ) + T ( n , l ) σ 3 T ( l , 0 ) ] + i ( n ˜ + k n ˜ ) 2 p T ( n , l ) σ 3 T ( l , 0 ) + d l k n ˜ ( n ˜ + k n ˜ ) T ,
T ( k ) d l = i [ ( n + k n ) / 2 p n ] [ T ( n , l ) σ 3 T ( l , 0 ) + T ( n , l ) σ 3 T ( l , 0 ) ] 2 i k n [ T ( n , l ) σ 3 T ( l , 0 ) + T ( n , l ) σ 3 T ( l , 0 ) + T ( n , l ) σ 3 T ( l , 0 ) ] + i n 2 p T ( n , l ) σ 3 T ( l , 0 ) + ( d l n ˜ 2 + 2 d l n n k ) T + d l n k ( n ˜ + n ˜ k ) T .
T R = T | T | ,
T n l + 1 R = P l D l 1 | D l 1 | .
T ( n , l ) = ( T ( l , 0 ) R ) | T ( l , 0 ) R |
= [ D l 1 | D l 1 | ( P l 1 D l | D l | ) ( P n 1 D n | D n | ) ] | D l 1 | D l 1 | ( P l 1 D l | D l | ) ( P n 1 D n | D n | ) | .
T ( n , l ) = [ D l 1 ( P l 1 D l ) ( P n 1 D n ) ] | D l 1 | | D l | | D n | | D l 1 ( P l 1 D l ) ( P n 1 D n ) | ( | D l 1 | | D l | | D n | ) 2 .
T ( n , l ) = [ D l 1 ( P l 1 D l ) ( P n 1 D n ) ] .
Γ ( k ) = T 12 T 11 T 12 T 11 T 11 2 ,
ϕ ( k ) = [ Γ ] [ Γ ] [ Γ ] [ Γ ] | Γ | 2 ,
Γ ( k ) = 1 T 11 ( Γ T 11 + T 12 ) ,
| Γ ( k ) | 2 = 2 ( [ Γ ] [ Γ ] + [ Γ ] [ Γ ] ) ,
Γ ( k ) = 1 T 12 2 [ T 12 T 11 + T 11 T 12 + Γ ( 2 T 11 T 11 T 11 T 11 T 11 T 12 ) ] ,
ϕ ( k ) = [ Γ ] [ Γ ] [ Γ ] [ Γ ] | Γ | 2 ,
ϕ ( k ) = 1 | Γ | 2 [ [ Γ ] [ Γ ] [ Γ ] [ Γ ] ϕ ( [ Γ ] [ Γ ] [ Γ ] [ Γ ] ) + ϕ | Γ | 2 2 ] .

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