Positive dispersion: 2 n ω 2 > 0, 2 n λ 2 > 0 Negative dispersion: 2 n ω < 0, 2 n 2 λ < 0 2
φ(z,ω) = k ( n ω )z E( z,t)= 1 2π E ( z = 0,ω )e iωt iφ z,ω e ( ) dω
φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0 ( ) dφω ( ) E( z,t)= 1 2π dω + ω0 E ( z = 0,ω )e iωt iφ z,ω e ( ) dω ( ) 2 d 2 φω ( ) 1 2! ω ω 0 dω 2 +... ω0 ω 0
φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0 ( ) dφω ( ) E( z,t)= 1 2π dω + ω0 E ( z = 0,ω )e iωt iφ z,ω e ( ) dω ( ) 2 d 2 φω ( ) 1 2! ω ω 0 dω 2 +... ω0 ω 0
φ(z,ω) = k ( n ω )z φ( ω )= φ 0 + ω ω 0 ( ) dφω ( ) E( z,t)= 1 2π dω + ω0 E ( z = 0,ω )e iωt iφ z,ω e ( ) dω ( ) 2 d 2 φω ( ) 1 2! ω ω 0 dω 2 +... ω0 ω 0
t p (0) t p (z) t t τ p ( z) τ p ( 0) = 1 + 4ln2 d 2 φ dω 2 τ 2 p ( 0) 2 d 2 φ dω >> τ 2 2 p( 0) τ p () z d 2 φ dω Δω 2 p
p τ τ 2 2 2 2 d 4ln2 d 1 p p ϕ ω τ τ τ = +
n 2 1 = B k λ 2 Positive dispersion: 2 n ω 2 > 0, 2 n λ 2 > 0 k λ 2 C k up-chirped dω dt > 0 B 1 =0.00048673, C 1 =(65 nm) 2, B 2 =0.000058583, and C 2 =(132 nm) 2
τ p () z d 2 φ dω Δω 2 p
n= n() λ
Dispersion of material n( ) Snellius Law: Refraction is frequency dependent d.h. = ( )
t t
J. A. Valdmanis et al., Opt. Lett. 10, 131, 1985
Pump-Laser Ti:Saphir Kristall (2.3 mm, 0.25 wt.%) M 2 M 3 M 4 SESAM M 1 M 7 M 5 M 6 Quartzprismen Auskoppel- Spiegel
θ 1 ( λ 0 )= θ 2 ( λ 0 )= α 2 tanθ B = n 2 = n, and θ B + θ B = 90 o n 1 α = 180 o 2θ B = π 2θ B
n n = sin αδ ( min )+ δ min 2 ( ) sin αδ min 2
θ 1 ( λ 0 )= θ 2 ( λ 0 )= α 2 tanθ B = n 2 = n, and θ B + θ B = 90 o n 1 θ 2 ( λ)= arcsin( nsin θ 2 )= arcsin n( λ)sin π 2θ B arcsin sinθ B n( λ) α = 180 o 2θ B = π 2θ B
C θ 2 G H A β L P = 2 CDE opt = 2 AB = 2Lcosβ β θ 2 D E F s h B φ p = kp ( λ)= 2π λ P ( λ) d 2 φ P dω 2 = λ 3 2π c 2 d 2 P dλ 2 d 2 P dλ 2 = P β 2 n β λ 2 n + 2 β n n 2 λ 2 + β n n 2 d 2 P λ dβ 2 β = 0 d 2 P dλ = 2 θ 2 2 n n λ 2 L 8 dn dλ 2 L < 0
EFG = BH CDE = AB = L cosβ E F G = BH= BH C D E = AB= L cos β dφ P dω = d dω kp(ω ) ( )= 2L c cosβ 2ω L c sinβ dβ dω, dβ dω < 0
400 2000 200 Fused Quartz 0 SF10 d 2 φ/dω 2 [fs 2 ] -200 0 d 2 φ/dω 2 [fs 2 ] -2000-4000 -400-6000 0.0 0.1 0.2 L [m] 0.3 0.4 0.0 0.1 0.2 L [m] 0.3 0.4
d 2 φ dω >> τ 2 2 p( 0) 400 200 Fused Quartz 2000 0 SF10 d 2 φ/dω 2 [fs 2 ] -200 0 d 2 φ/dω 2 [fs 2 ] -2000-4000 -400-6000 d 3 φ 200 0-200 -400 0.0 dω 3 >> τ p d 3 φ/dω 3 [fs 3 ] ( ) 3 0 0.1 0.2 L [m] 0.3 0.4 Fused Quartz d 3 φ/dω 3 [fs 3 ] 0-5 -10 0.0 0.1 0.2 L [m] 0.3 SF10 0.4-600 -15x10 3 0.0 0.1 0.2 L [m] 0.3 0.4 0.0 0.1 0.2 L [m] 0.3 0.4
2000 20 mm Fused Silica Prisms 35 cm apex separation 1000 adjustable negative GDD allows compensation of material dispersion strong higher-order contributions 0 600 4 mm 700 800 900 Wavelength (nm) 1000
λ < λ h λ > λ h Any wavelength < h will not be refracted by the second prism and will be lost during propagation through the prism pair. λ h
1.0 400 Ti:S-gain Prism + Ti:S-Crystal 0.8 GDD, fs 2 200 0 6 x Standard Mirror (measured) Total Dispersion 0.6 0.4 Spektrum -200 6 x Chirped Mirror 0.2 600 700 800 900 1000 0.0 Wavelength, nm Goals: ---> High Reflectivity from 650 nm - 950 nm ---> Constant Total GDD from 650 nm - 950 nm
1.0 400 Ti:S-gain Prism + Ti:S-Crystal 0.8 GDD, fs 2 200 0 6 x Standard Mirror (measured) Total Dispersion 0.6 0.4 Spektrum -200 6 x Chirped Mirror 0.2 600 700 800 Wavelength, nm 900 1000 0.0 λ < λ h λ > λ h λ h
Λ ν x,m = m 1 Λ θ x,m m λ Λ t t t t
sinθ m = m λ Λ + sinθ i where m = 0, ± 1, ± 2,... x φ g ( x)= π m x Λ 2π
z L g x B θ i P' P Q b θ m A x sinθ m = m λ Λ + sinθ i where m = 0, ± 1, ± 2,... Beugungsgitter z.b. Blaze-Gitter φ = ω c L + φ g ( x), with x = L g tan( θ m ) φ g ( x)= π m 2π Λ x ( ) L = PABQ = PA+ b = b 1 + cos θ m + θ i = L g cosθ m ( ) 1+ cos θ m + θ i
sinθ m = m λ Λ + sinθ i L g d 2 φ dω = m2 λ 3 L g 2 2πc 2 Λ 1 m λ 2 Λ sinθ i 2 3/2 d 2 φ dω 2 L g angle of incidence angle of incidence 1200 lines/mm 300 lines/mm Λ 830 nm Λ 3.3 μm
L f stretcher something a Treacy grating pair cannot do! L f compressor like the Treacy grating pair
L f E out ( ω )= h ( ω ) E in ( ω )
A. M. Weiner, Rev. Sci. Instrum. 71, 1929 (2000) f = 300 mm knife-edge f = 300 mm SLM 640 pixels 300 l/mm grating 300 l/mm grating Possible bandwidth through Spatial Light Modulator: 400-1050 nm
R = 100% b AR-Coating R t R R GTI R GTI = 1 and R GTI = exp i2φ GTI ( ) d Glass substrate Air gap (non-absorbing spacer layer) r GTI exp( iφ GTI )= r 12 + r 23 e i2ϕ 1 + r 12 r 23 e = R t e i2ϕ i2ϕ 1 R t e i2ϕ t 0 t 0 = 2nd c ϕ = nkd = n ω c d ωt 0 2 dϕ dω t 0 2 tanφ GTI = Im r GTI Re r GTI ( ) ( ) = 1 R t ( )sin2ϕ 2 R t ( 1 + R t )cos2ϕ and d dω tanφ = ( 1 + tan2 φ) dφ dω ( ) R t sinωt 0 dt g dω = d 2 φ GTI = 2t 2 0 1 R t dω 2 1+ R t 2 R t cosωt 0 ( ) 2
R = 100% b R t AR-Coating R R GTI dt g dω d 2 d Glass substrate Air gap (non-absorbing spacer layer) Bandwidth of dt g dω 1 d Example: d = 2.25 μm, R t = 4% Example: d = 80 μm, R t = 4%
t p (0) t p (z) t t L 1 2 3 4 Substrate
Bragg-Mirror: SiO 2 - Substrate TiO / SiO 2 2 B - Layers 4 AIR 1.0 6 Intensity Reflexion 0.8 0.6 0.4 0.2 Phase 5 4 3 2 1 0.0 0.8 1.0 1.2 Wavelength (μm) 1.4 0 0.8 1.0 1.2 Wavelength (μm) 1.4
R λ/4 stack Substrate T Refr. index n H n S n I n L Physical distance r = ( n H n L ) ( n H + n L ) Δω ω B = 4 π arcsin( r) R 0 ( λ B )= 1 aqpm 1 1 + aqp m 1 2 p = n I n H q = n L n H a = n L n S
μ μ dt g dλ > 0 dt g dω < 0
λ Bragg Substrate 1 2 3 4 Chirp Bragg wavelength wavelength-dependent penetration depth engineerable dispersion compensation of arbitrary material dispersion increased high-reflection bandwidth Group Delay (fs) Szipöcs et al., Opt. Lett. 19, 201 (1994) 40 20 λ 4 λ 3 λ 2 GDD = 30 fs 2 (3.5 μm thick mirror stack) λ 1 0 600 800 1000 Wavelength (nm)
Substrate Gires-Tournois High Partial R = 5% Interferometer (GTI) Reflector Reflector Group Delay (fs) 60 40 20 0 700 800 900 Wavelength (nm) GTI desired
Substrate Substrate AR Anti-reflection coating matches first-layer impedance to air Kärtner et al., Opt. Lett. 22, 831 (1997) Matuschek et al., IEEE J. Sel. Top. Quantum Electron. 4, 197 (1998)
k B,min = 2π 600 nm k B,max = 2π 900 nm
0 prism pair: tunability, better DCM designs -100-200 100 (GDD= T g / ) 0 20 Double-Chirped Mirrors: trade off: bandwidth vs. GDD oscillations non-normal incidence: smoother average GDD (average of several DCMs) 99 600 700 800 900 1000 1100 ) 2x 20 Wavelength (nm)
(a) Bragg Mirror: SiO 2 Substrate TiO / SiO 2 2 λ /4-Layers B Air (b) Simple-Chirped Mirror: Bragg Wavelength λ Chirped SiO 2 Substrate B λ 1 λ 2 Negative Dispersion: λ λ 2 > 1 GTI (c) Double-Chirped Mirror: Bragg Wavelength and Coupling Chirped SiO 2 Substrate AR Coating Air { Impedance Matching (n d < h h < n d ) l l
10 0 10-1 10-2 10-3 10-4 10-5 Δλ = 230 nm 600 800 Wavelength (nm) 1000 80 60 40 20 0 600 With AR Without AR R AR < 10-4 700 800 900 Wavelength (nm) 1000 cannot make arbitrarily low reflectivity and arbitrarily broad bandwidth at the same time J.A. Dobrowolski et al., Appl. Opt. 35, 644 (1996)
Substrate Perfect impedance matching with double-chirp and n substrate = n first layer Front-reflection from substrate is not interfering with wave from mirror stack (by geometry) AR coating required only for loss reduction
20 0-20 -40-60 -80 BASIC DCM: Target Design 20 0-20 -40-60 -80 conventional DCM: Target Design 100.0 600 800 1000 Wavelength (nm) 1200 600 700 800 900 Wavelength (nm) 1000 99.8 99.6 99.4 99.2 99.0 600 BASIC conventional 800 1000 1200 Wavelength (nm) 1400 R > 99.7% (600-1240 nm) T g variations = 0.3 fs rms GDD var. = 3.8 fs 2 rms 260 THz bandwidth
50 0-50 Error simulations Measured GDD Design Layer deposition error: 0.2 nm rms 600 700 800 900 Wavelength (nm) 1000 future 50 0-50 Error simulations Measured GDD Design Layer deposition error: 0.1 nm rms 600 700 800 900 Wavelength (nm) 1000
N. Matuschek: Theory and design of double-chirped mirrors, Ph.D. Thesis, ETH Zurich (1999) Hartung-Gorre Verlag, ISBN 3-89649-501-1
n o,k o = kn o, and ω,ω a,ω b [ ω 0 ± Δω ] k A,ω A n e, k e = kn e ω o + ω A = ω e, because ω A << ω ω o ω e k o + k A = k e k( n e n o )= k A = ω A υ A
+ S ( Δt)= R E S * ()E t R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, 393-395 (1990).
S ( ω )= E S ( ω ) E * R ( ω ) + S( Δt)= R E S ()E t * R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, 393-395 (1990).
S ( ω )= E S ( ω ) E * R ( ω ) ϕω ( ) + S ( Δt)= R E S * ()E t R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, 393-395 (1990). GDD = d 2 ϕω ( ) dω 2 ϕω ( )
S ( ω )= E S ( ω ) E * R ( ω ) + S ( Δt)= R E S * ()E t R ( t Δt)dt K. Naganuma, K. Mogi, H. Yamada, Opt. Lett. 15, 393-395 (1990). GDD = d 2 ϕω ( ) dω 2 ϕω ( )
Phase Velocity υ p ω k n c n Group Velocity υ g dω dk n Group Delay T g Tg = z = dφ υ g dω, φ k n z c n nz c 1 1 dn λ dλ n dn λ 1 dλ n Dispersion 1. Order Dispersion 2. Order Dispersion 3. Order dφ dω d 2 φ dω 2 d 3 φ dω 3 nz c dn λ 1 dλ n λ 3 z d 2 n 2πc 2 dλ 2 λ 4 z 4π 2 c 3 3 d 2 n dλ + λ d 3 n 2 dλ 3