2.1 Ultrafast solid-state lasers - ETH - the Keller Group

Ref. p. 134] 2.1 Ultrafast solid-state lasers 97
Table 2.1.9. Dispersion compensation, its defining equations and figures. c: light velocity in vacuum, λ:
wavelength in vacuum, λ 0: center wavelength of pulse spectrum, ω: frequency in radians/second.
Quantity
Gires–Tournois
Interferometer (GTI)
(Fig. 2.1.16a)
Dispersion: 2nd order
Defining equation
d: thickness of Fabry–Perot
n: refractive index of material inside Fabry–Perot (airspaced n =1)
(Note: Material dispersion is neglected.)
t 0 = 2nd : round-trip time of the Fabry–Perot
c
R t: intensity reflectivity of top reflector of Fabry–Perot
(Bottom reflector is assumed to have a 100%-reflectivity.)
d 2 φ
dω 2 = −2t2 0 (1 − R t) √ R t sin ωt 0
(
1+Rt − 2 √ R t cos ωt 0
) 2
Four-grating compressor
(Fig. 2.1.16b)
Dispersion: 2nd order
L g: grating pair spacing
Λ: grating period
θ i: angle of incidence at grating
[ ( ) ]
d 2 2 −3/2
φ
dω = − λ3 L g λ
1 −
2 π c 2 Λ 2 Λ − sin θi
Dispersion: 3rd order
Four-prism compressor
(Fig. 2.1.16c)
d 3 φ
dω = − d2 φ 6 π λ
3 dω 2 c
1+ λ Λ sin θi − sin2 θ i
( ) 2
λ
1 −
Λ − sin θi
n: refractive index of prisms
θ B: angle of incidence of prism is at Brewster angle
θ B = arctan [n (λ 0)]
α = π − 2θ B : apex angle of prism
θ 2 (λ) = arcsin
[
n (λ)sin
(
π − 2θ B − arcsin
L: apex-to-apex prism distance
h: beam insertion into second prism
)]
sin θB
n (λ)
sin β = h L
cos θ 2
cos (α/2)
Dispersion: 2nd order
Dispersion: 3rd order
d 2 φ
dω =
λ3 d 2 P
2 2 π c 2 dλ 2
[ (
d 2 P
dλ =2 ∂ 2 n ∂θ2
2 ∂λ 2 ∂n
≈ 4
) ( )( ∂ 2 θ 2 ∂n
+
∂n 2 ∂λ
) ] 2
[ (
∂ 2 n
∂λ + 2n − 1 )( ∂n
2 n 3 ∂λ
(
d 3 φ
dω = −λ4 3 d2 P
3 4 π 2 c 3 dλ 2
d 3 P
dλ 3
+ λ d3 P
dλ 3 )
≈ 4
d3 n
dn
L sin β − 24
dλ3 dλ
d 2 n
dλ L cos β
2
) 2
]
( ) 2 ∂θ2 ∂n
L sin β − 2
L cos β
∂n ∂λ
( ) 2 ∂n
L cos β
∂λ
L sin β − 8
Landolt-Börnstein
New Series VIII/1B1