IMPRS: Ultrafast Source Technologies...1.1 Dispersive Pulse Propagation 1.2 Nonlinear Pulse...
Transcript of IMPRS: Ultrafast Source Technologies...1.1 Dispersive Pulse Propagation 1.2 Nonlinear Pulse...
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IMPRS: Ultrafast Source Technologies Lecture III: Feb. 24, 2015: Ultrafast Optical Sources
Franz X. Kärtner
http://www.eadweardmuybridge.co.uk/
Is there a time during galloping, when all feet are off the ground? (1872) Leland Stanford
Eadweard Muybridge, * 9. April 1830 in Kingston upon Thames; † 8. Mai 1904, Britisch pioneer of photography
Harold Edgerton, * 6. April 1903 in Fremont, Nebraska, USA; † 4. Januar 1990 in Cambridge, MA, american electrical engineer, inventor strobe photography.
What happens when a bullet rips through an apple?
http://web.mit.edu/edgerton/
ms µs
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Physics on femto- attosecond time scales?
A second: from the moon to the earth
A picosecond: a fraction of a millimeter, through a blade of a knife A femtosecond: the period of an optical wave, a wavelength An attosecond: the period of X-rays, a unit cell in a solid
*F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009)
X-ray EUV
Time [attoseconds] Light travels:
*)
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How short is a Femtosecond
1 s 250 Million years
10 s
dinosaurs 60 Million years
dinosaures extinct 10 s = 1µs -6
10 s = 1 fs -15
Strobe photography
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Pump - Probe Measurements
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Structure, Dynamics and Function of Atoms and Molecules Struture of Photosystem I
Chapman, et al. Nature 470, 73, 2011
Todays Frontiers in Space and Time
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X-ray Imaging
6 Chapman, et al. Nature 470, 73, 2011
Optical Pump
X-ray Probe
(Time Resolved)
Imaging before destruction: Femtosecond Serial X-ray crystalography
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Attosecond Soft X-ray Pulses
Corkum, 1993
Elec
tric
Fie
ld, P
ositi
on
Time
Ionization
Three-Step Model Trajectories
First Isolated Attosecond Pulses: M. Hentschel, et al., Nature 414, 509 (2001) Hollow-Fiber Compressor: M. Nisoli, et al., Appl. Phys. Lett. 68, 2793 (1996)
High - energy single-cycle laser pulses!
How do we generate them? 7
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Short Pulse Laser Systems
Laser Oscillators (nJ), cw, q-switched, modelocked: Semiconductor, Fiber, Solid-State Lasers
Laser Amplifiers: Solid-State or Fiber Lasers
Regenerative Amplifiers
Multipass Amplifiers
Chirped Pulse Amplification
Parametric Amplification and Nonlinear Frequency Conversion
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Content
1. Basics of Optical Pulses 1.1 Dispersive Pulse Propagation 1.2 Nonlinear Pulse Propagation 1.3 Pulse Compression
2. Continuous Wave Lasers 3. Q-switched Lasers 4. Modelocked Lasers 5. Laser Amplifiers 6. Parametric Amplifiers
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1. Basics of Optical Pulses
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Peak Electric Field:
TR : pulse repetition rate W : pulse energy Pave = W/TR : average power τFWHM : Full Width Half Maximum pulse width
Pp : peak power
Aeff : effective beam cross section ZFo : field impedance, ZFo = 377 Ω
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Typical Lab Pulse:
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Time Harmonic Electromagnetic Waves
Transverse electromagnetic wave (TEM) (Teich, 1991)
See previous class: Plane-Wave Solutions (TEM-Waves)
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Optical Pulses ( propagating along z-axis)
: Wave amplitude and phase
: Wave number
0( )( )cc
nΩ =
Ω: Phase velocity of wave
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At z=0
Spectrum of an optical pulse described in absolute and relative frequencies
For Example: Optical Communication; 10Gb/s Pulse length: 20 ps Center wavelength : λ=1550 nm. Spectral width: ~ 50 GHz, Center frequency: 200 THz,
Carrier Frequency
Absolute and Relative Frequency
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Electric field and envelope of an optical pulse
Pulse width: Full Width at Half Maximum of |A(t)|2
Spectral width : Full Width at Half Maximum of |A(ω)|2 ~ _
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Pulse width and spectral width: FWHM
Often Used Pulses
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Fourier transforms to pulse shapes listed in table 2.2 [16]
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Electric field and pulse envelope in time domain
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Taylor expansion of dispersion relation at pulse center frequency
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In the frequency domain:
1.1 Dispersion
Taylor expansion of dispersion relation:
i) Keep only linear term:
Time domain:
Group velocity:
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Compare with phase velocity:
Retarded time:
ii) Keep up to second order term:
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Gaussian Pulse:
Pulse width
Initial pulse width:
FWHM Pulse width:
determines pulse width chirp
z-dependent phase shift
z = L
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Magnitude Gaussian pulse envelope, |A(z, t’ )|, in a dispersive medium
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Phase:
(a) Phase and (b) instantaneous frequency of a Gaussian pulse during propagation through a medium with positive or negative dispersion
Instantaneous Frequency:
k”>0: Postive Group Velocity Dispersion (GVD), low frequencies travel faster and are in front of the pulse
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Sellmeier Equations
( )rχ Ω
Example: Sellmeier Coefficients for Fused Quartz and Sapphire
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Typical distribution of absorption lines in medium transparent in the visible.
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Transparency range of some materials, Saleh and Teich, Photonics p. 175.
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Group Velocity and Group Delay Dispersion
Group Delay:
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1.2 Nonlinear Pulse Propagation
The Optical Kerr Effect
Without derivation, there is a nonlinear contribution to the refractive index:
Polarization dependent
Table 3.1: Nonlinear refractive index of some materials
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Spectrum of a Gaussian pulse subject to self-phase modulation
Self-Phase Modulation (SPM)
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Intensity
Front Back
Phase
Time t
Time t
Time t
Instantaneous Frequency
(a)
(b)
(c)
.
(a) Intensity, (b) phase and c) instantaneous frequency of a Gaussian pulse during propagation
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1.3 Pulse Compression
Dispersion negligible, only SPM
Optimium Dispersion and nonlinearity
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Fiber-grating pulse compressor to generate femtosecond pulses
Spectral Broadening with Guided Modes and Compression
Pulse Compression:
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Grating Pair
Phase difference between scattered beam and reference beam”
Disadvantage of grating pair: Losses ~ 25%
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Prism Pair
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3.7.4 Dispersion Compensating Mirrors
High reflecitvity bandwidth of Bragg mirror:
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Hollow fiber compression technique
3.7.5 Hollow Fiber Compression Technique
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Figure 4.5: Three-level laser medium
0
1
2
N
N
N2
1
0
γ
γ21
10
Rp
0
1
2
N
N
N2
1
0
γ
γ21
10
Rp
a) b)
2.1 Laser Rate Equations How is inversion achieved? What is T1, T2 and σ of the laser transition? What does this mean for the laser dyanmics, i.e.for the light that can be generated with these media?
2 Continuous Wave Lasers
γ10 ∞
w = N2
Rp ∞
N0=0 w = N1
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Four-level laser 3
0
1
2
N
N
N
N
3
2
1
0
γ32
21
10
Rp γ
γ
w = N2
γ10 ∞
γ32 ∞
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Rate equations for a laser with two-level atoms and a resonator.
V:= Aeff L Mode volume fL: laser frequency I: Intensity Vg: group velocity at laser frequency NL: number of photons in mode
σ: interaction cross section w: inversion
Rate Equations and Cross Section
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Intracavity power: P Round trip amplitude gain: g
Output power: Pout
small signal gain ~ στL - product
Laser Rate Equations:
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Output power versus small signal gain or pump power
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Steady State: d/dt = 0 Pvac = 0 Case 1: Case 2:
gs = g0 P s = 0
2.2 Continuous Wave Operation
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Spectroscopic parameters of selected laser materials
Lasers and Its Spectroscopic Parameters
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Here active Q-switching
3 Q-Switched Lasers
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4. Modelocked Lasers
Pulse width time
f 0
f 1 = f 0 - ∆ f
f 0 , f 1 , f 2
f 3 = f 0 -2 ∆ f
f 4 = f 0 +2 ∆ f f 2 = f 0 + ∆ f
f 0 , f 1 , f 2 , f 3 , f 4
Spec
trum
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Actively modelocked laser
4.1 Active Mode Locking
loss modulation
Parabolic approximation at position where pulse will form;
Master Equation:
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Compare with Schroedinger Equation for harmonic oscillator
with
Eigen value determines roundtrip gain of n=th pulse shape
Pulse shape with n=0, lowest order mode, has highest gain.
This pulse shape will saturate the gain and keep all other pulse shapes below threshold.
Pulse width:
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Pulse shaping in time and frequency domain.
Pulse width depends only weak on gain bandwidth. 10-100 ps pulses typical for active mode locking!
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f
1-M
n0-1
MM
n0+1fn0f f
Active mode locking can be understood as injection seeding of neighboring modes by those already present.
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Saturation characteristic of an ideal saturable absorber and linear approximation.
4.2 Passive Mode Locking
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Saturable absorber provides gain for the pulse There is a stationary solution:
Easy to check with: Shortest pulse:
For Ti:sapphire
Fast Saturable Absorber Modelocking
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Das Bild kann zurzeit nicht angezeigt werden.
Laser beam
Intensity dependent refractive index: "Kerr-Lens"
Self-Focusing Aperture
Time
Inte
nsity
Time In
tens
ity
Time
Inte
nsity
Kerr Lens Modelocking
Lens Refractive index n >1
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Semiconductor saturable absorber mirror (SESAM) or Semiconductor Bragg mirror (SBR)
Semiconductor Saturable Absorbers
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Pulse width of different laser systems by year.
Modelocking: Historical Development
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5. Laser Amplifiers
5.1 Cavity Dumping 5.2 Laser Amplifiers 5.2.1 Frantz-Nodvick Equation 5.2.2 Regenerative and Multipass Amplifiers 5.3 Chirped Pulse Amplification 5.4 Stretchers and Compressors 5.5 Gain Narrowing
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Repetition Rate, Pulses per Second 109 106 103 100 10-3
10-9
10-6
100
10-3
Oscillators
Cavity-dumped oscillators
Regenerative amplifiers
Regen + multipass amplifiers
1-100 W average power
Puls
eene
rgy
(J)
103
Pulse energies from different laser systems
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5.1 Cavity Dumping
With Bragg cell
With Pockels cell
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Laser oscillator
Amplifier medium
Pump pulse
Energy levels of amplifier medium
Seed pulse
Amplified pulse
5.2 Laser Amplifiers
Laser amplifier: Pump pulse should be shorter than upper state lifetime. Signal pulse arrives at medium after pumping and well within the upper state lifetime to extract the energy stored in the medium, before it is lost due to energy relaxation.
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Multi-pass gain and extraction
0 1
2 3
5.2.1 Franz-Nodvik Equations
1 1.2 1.4 1.6 1.8
2 2.2 2.4 2.6
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 F = Fin / Fsat
G =
Fou
t / F
in G0=3
η =
Fou
t / F
ext F: Fluence = Area
Energy
Amplifier medium Fin Fout
Fext
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a) Multi-pass amplifier
pump
input output
gain
Pockels cell
polarizer
gain
pump
input/output
b) Regenerative amplifier
5.2.2 Basic Amplifier Schemes
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Multipass amplifier pulse-growth/gain-extraction
Multipass amplifier: Theory and numerical analysis Lowdermilk and Murray, JAP 51, No. 5 (1980)
DIODE PUMP
Initial gain Go
Round-trip transmission T
0 4 8 12 16 20 240
0.1
0.2
J'0 n,
n
0 4 8 12 16 20 240
0.1
0.2
g'0 n, g'o⋅
n
PASS NUMBER
0 4 8 12 16 20 24
0 4 8 12 16 20 24
( Fi/Fsat )
( Gi )
Fluence
Gain
Pulse growth dynamics dictated by the system’s
GAIN and LOSS ratio
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5.3 Chirped-Pulse Amplification
Chirped-pulse amplification in- volves stretching the pulse before amplifying it, and then compressing it later.
Stretching factors of up to 10,000 and recompression for 30fs pulses can be implemented.
G. Mourou and co-workers 1985
t
t
Short pulse
oscillator
Pulse compressor
Solid state amplifier(s)
Dispersive delay line
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Chirped Pulse Amplifier System
Oscillator – Stretcher – Multiple Amplifiers - Compressor Chain
Oscillator Stretcher Amplifier 1
Compressor
Amplifier 2
Amplifier 3
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5.4 Stretchers and Compressors
64
f 2f f-d
grating
negative dispersion
grating
Phase Mask
f 2f f+d
grating
grating
positive dispersion
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Ti:sapphire gain cross section
10-fs sech2 pulse in
0
0.2
0.4
0.6
0.8
1
0
0.5
1
1.5
2
2.5
3
650 700 750 800 850 900 950 1000
Nor
mal
ized
spe
ctra
l int
ensi
ty
Norm
alized Gain)
Wavelength (nm)
32-nm FWHM
longer pulse out
65-nm FWHM
Influence of gain narrowing in a Ti:sapphire amplifier on a 10 fs seed pulse
Rouyer et al., Opt. Lett. 18, 214 (1993).
In general when applying gain G with bandwidth to a pulse with input bandwidth the output bandwidth is
5.5 Gain Narrowing
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6. Optical Parametric Amplifiers +++= 3)3(
02)2(
0)1(
0 EEEP χεχεχε
ω1
ω2
ω1
ω2
ωpump
ωsignal
ωidler = ωp - ωs
Non-linear polarization effects
ωsignal
Optical Parametric Amplification (OPA)
sk ik
pk
+ = Momentum conservation (vectorial):
(also known as phase matching)
sk
pk
ik
α
2ω1 2ω2
ω1 − ω2
ω1 + ω2
χ(2)
+ = sω iω pωEnergy conservation:
⇒ Broadband gain medium!
Courtesy of Giulio Cerullo 66
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Ultrabroadband Optical Parametric Amplifier
Broadband seed pulses can be obtained by white light generation
Broadband amplification requires phase matching over a wide range of signal wavelengths
G. Cerullo and S. De Silvestri, Rev. Sci. Instrum. 74, 1 (2003).
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Phase matching bandwidth in an OPA If the signal frequency ωs increases to ωs+∆ω, by energy conservation the
idler frequency decreases to ωi-∆ω. The wave vector mismatch is
ω∆ω∆ω
ω∆ω
∆
−=
∂∂
+∂∂
−=gigs
is
vvkkk 11
( )
gigs vvL 11
12ln2 2/12/1
−
≅
γπ
ν∆
The phase matching bandwidth, corresponding to a 50% gain reduction, is
⇒ the achievement of broad gain bandwidths requires group velocity matching between signal and idler beams
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Broadband OPA configurations
vgi = vgs : Operation around degeneracy ωi = ωs = ωp/2
Type I, collinear configuration Signal and idler have same refractive index
vgi ≠ vgs : Non-collinear parametric amplifier (NOPA):
sk
pk
ik
α Pump
Signal
Pump and Signal at angle α
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Noncollinear phase matching: geometrical interpretation
kp
ks
ki Ωα
In a collinear geometry, signal and idler move with different velocities and get quickly separated
vgs
vgi
vgs
vgiΩ
In the non-collinear case, the two pulses stay temporally overlapped
Note: this requires vgi>vgs (not always true!)
vgs=vgi cosΩ
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Broadband OPA configurations
Pump wavelength
NOPA Degenerate OPA
400 nm (SH Ti:sapphire)
500-750 nm 700-1000 nm
800 nm (Ti:sapphire)
1-1.6 µm 1.2-2 µm
OPAs should allow to cover nearly continuously the wavelength range from 500 to 2000 nm (two octaves!) with few-optical-cycle pulses
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Tunable few-optical-cycle pulse generation
D Brida et al., J. Opt. 12, 013001 (2010). Can we tune our pulses even more to the mid-IR? Yes, using the
idler!
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Broadband pulses in the mid-IR
Simulations confirm the generation of broadband idler pulses, with 20-fs duration (≈2 optical cycles) at 3 µm
0,0
0,5
1,00,9 1 1,1 1,2 1,3
Gai
n [
104 ]
(a)2 3 4 5 6
LiIO3 KNbO3 PPSLT
(b)
0,9 1 1,1 1,2 1,30,0
0,5
1,0
Inte
nsity
(arb
. un.
)
Signal Wavelength (µm)
(c) TL duration:19 fs
2 3 4 5 6Idler Wavelength (µm)
(d) TL duration: 22.3 fs
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References B.E.A. Saleh and M.C. Teich, "Fundamentals of Photonics," John Wiley and Sons, Inc., 1991. P. G. Drazin, and R. S. Johnson, ”Solitons: An Introduction,”Cambridge University Press, New York (1990). R.L. Fork, O.E. Martinez, J.P. Gordon: Negative dispersion using pairs of prisms, Opt. Lett. 9 150-152 (1984). M. Nisoli, S. Stagira, S. De Silvestri, O. Svelto, S. Sartania, Z. Cheng, M. Lenzner, Ch. Spielmann, F. Krausz: A novel high energy pulse compression system: generation of multigigawatt sub-5-fs pulses, Appl. Phys. B 65 189-196 (1997). J. Kuizenga, A. E. Siegman: "FM und AM mode locking of the homoge- neous laser - Part II: Experimental results, IEEE J. Quantum Electron. 6, 709-715 (1970). H.A. Haus: Theory of mode locking with a slow saturable absorber, IEEE J. Quantum Electron. QE 11, 736-746 (1975). K. J. Blow and D. Wood: "Modelocked Lasers with nonlinear external cavity," J. Opt. Soc. Am. B 5, 629-632 (1988). J. Mark, L.Y. Liu, K.L. Hall, H.A. Haus, E.P. Ippen: Femtosecond pulse generation in a laser with a nonlinear external resonator, Opt. Lett. 14, 48-50 (1989)· E.P. Ippen, H.A. Haus, L.Y. Liu: Additive pulse modelocking, J. Opt. Soc. Am. B 6, 1736-1745 (1989). D.E. Spence, P.N. Kean, W. Sibbett: 60-fsec pulse generation from a self-mode-locked Ti:Sapphire laser, Opt. Lett. 16, 42-44 (1991). H. A. Haus, J. G. Fujimoto and E. P. Ippen, ”Structures of Additive Pulse Mode Locking,” J. Opt. Soc. Am. 8, pp. 2068 — 2076 (1991). U. Keller, ”Semiconductor nonlinearities for solid-state laser modelock- ing and Q-switching,” in Semiconductors and Semimetals, Vol. 59A, edited by A. Kost and E. Garmire, Academic Press, Boston 1999. Lecture on Ultrafast Amplifiers by Francois Salin, http://www.physics.gatech.edu/gcuo/lectures/index.html. D. Strickland and G. Morou: "Compression of amplified chirped optical pulses," Opt. Comm. 56,219-221,(1985). G. Cerullo and S. De Silvestri, "Ultrafast Optical Parametric Amplifiers," Review of Scientific Instr. 74, pp. 1-17 (2003).
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