Ultrafast Optics and X-Rays Division - IMPRS: …Attosecond Soft X-ray Pulses 7 Corkum, 1993 Time...

IMPRS: Ultrafast Source Technologies Lecture III: April 16, 2013: Ultrafast Optical Sources

Franz X. Kärtner

1

1

Markus Drescher

Institut für Experimentalphysik

markus.drescher@desy.de

Optische Ultrakurzzeitphysik

Ultrafast Optical Physics

An introductory course to

quickly evolving physical dynamics,

studied with ultrashort optical pulses

1

Ultrafast Optical Physics: 1. Introduction 2

Eadweard Muybridge

Animal Locomotion (1887)

Illumination: Sun

Shutter: mechanical

Time-resolution 1/60 sec.

Ultrafast Optical Physics: 1. Introduction 3

History: the roots of capturing fast moving objects120 years later...

100 nm

Ultrafast Optical Physics: 1. Introduction 4

an electron escapes from an atom...

Time resolution: 0.000 000 000 000 000 100 seconds

http://www.eadweardmuybridge.co.uk/

Is there a time during galopping, when all feet are of the ground? (1872) Leland Stanford

Eadweard Muybridge (* 9. April 1830 in Kingston upon Thames; † 8. Mai 1904, britischer Fotograf & Pionier der Fototechnik.

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

2

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:

*)

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

3

5

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!

X-ray Imaging

Chapman, et al. Nature 470, 73, 2011

Optical Pump

X-ray Probe

(Time Resolved)

Imaging before destruction: Femtosecond Serial X-ray crystalography 6

Attosecond Soft X-ray Pulses

7

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?!

High Energy Laser Systems

8

§  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!

3. Basics of Optical Pulses

9

( n - 1 ) T R n T R ( n + 1 ) T R

τF W H M

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

10

Typical Lab Pulse:

3.1 Electromagnetic Waves

Transverse electromagnetic wave (TEM) (Teich, 1991)

11

See Chapter: 2.1.2 Plane-Wave Solutions (TEM-Waves)

12

3.2 Optical Pulses ( propagating along z-axis)

: Wave amplitude and phase

: Wave number

0( )( )ccn

Ω =Ω

: Phase velocity of wave

13

At z=0

Spectrum of an optical wave packet 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

14

Carrier and Envelope

Carrier Frequency

Envelope:

Electric field and envelope of an optical pulse

15

Pulse width: Full Width at Half Maximum of |A(t)|2

Spectral width : Full Width at Half Maximum of |A(ω)|2 ~ _

16

Pulse width and spectral width: FWHM

Often Used Pulses

Fourier transforms to pulse shapes listed in table 2.2 [16]

17

Fourier transforms to pulse shapes listed in table 2.2, continued [16]

18

19

3.3 Linear Pulse Propagation

Envelope + Carrier Wave

Electric field and pulse envelope in time domain

20

Taylor expansion of dispersion relation at the center frequency of the wave packet

21

In the frequency domain:

3.4 Dispersion

Taylor expansion of dispersion relation:

Equation of motion in frequency domain:

Equation of motion in time domain:

22

i) Keep only linear term:

Time domain:

Group velocity:

Compare with phase velocity:

23

Retarded time:

Or start from (2.63)

Substitute:

24

ii) Keep up to second order term:

Decomposition of a pulse into wave packets with different center frequency. In a medium with dispersion the wave packets move at different relative group velocity

Ã( )ω

Disp

ersi

on R

elat

ion

ω0 ω1−ω1

k”ω212

Spec

trum

1 2 3

∼Δvg1 ∼Δvg3∼Δvg2

25

Gaussian Pulse:

Substitute:

Gaussian Integral:

Apply

26

Pulse width

Propagation:

Initial pulse width:

Exponent Real and Imaginary Part:

FWHM Pulse width:

determines pulse width chirp

z-dependent phase shift

27

After propagation over a distance z=L:

For large distances:

28

Magnitude of the complex envelope of a Gaussian pulse, |A(z, t’ )|, in a dispersive medium

29

Chirp:

(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

30

31

3.5 Sellmeier Equations and Kramers-Kroenig Relations

( ) 0, for 0t tχ = <Causality of medium impulse response:

Leads to relationship between real and imaginary part of susceptibility

Approximation for absorption spectrum in a medium:

( )rχ Ω

32

Example: Sellmeier Coefficients for Fused Quartz and Sapphire

Contribution of absorption lines to index changes

33

Typical distribution of absorption lines in medium transparent in the visible.

34

Transparency range of some materials according to Saleh and Teich, Photonics p. 175.

35

Group Velocity and Group Delay Dispersion

Group Delay:

36

3.6 Nonlinear Pulse Propagation

37

3.6.1 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

Spectrum of a Gaussian pulse subject to self-phase modulation

3.6.2 Self-Phase Modulation (SPM)

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

3.6.3 Nonlinear Schroedinger Equation (NSE)

3.6.3.1 Solitons of the Nonlinear Schroedinger Equation

Propagation of a fundamental soliton

3.6.3.2 The Fundamental Soliton

Area Theorem

Nonlinear phase shift soliton aquires during propagation:

Balance between dispersion and nonlinearity:

Soliton Energy:

Pulse width:

Important Relations

General Fundamental Soliton Solution

Change of center frequency!

A soliton with high carrier frequency collides with a soliton of lower carrier frequency.

3.6.3.3 Higher Order Soliton (Soliton Collision)

Amplitude of higher order soliton composed of two fundamental solitons with the same carrier freuqency

3.6.3.3 Higher Order Soliton (Breather Soliton)

Spectrum of higher order soliton composed of two fundamental solitons with the same carrier freuqency

3.6.3.3 Higher Order Soliton (Breather Soliton)

Solution of the NSE for a rectangular shaped initial pulse

Rectangular Shaped Initial Pulse and Continuum Generation

Phase matching of soliton and continuum

Avoid resonance catastrophy for:

Kelly Sidebands

47

Pulse compression

3.7 Pulse Compression

Dispersion negligible, only SPM

Optimium Dispersion and nonlinearity

3.7.1 General Pulse Compression Scheme

48

Fiber-grating pulse compressor to generate femtosecond pulses

3.7.2 Spectral Broadening with Guided Modes

49

3.7.3 Dispersion Compensation Techniques

Pulse Compression:

Variable dispersion by grating and prism pairs

50

Figure 3:14: Amplitude of the envelope of a Gaussian pulse in a dispersive medium

Grating Pair

Phase difference between scattered beam and reference beam”

Disadvantage of grating pair: Losses ~ 25%

51

Figure 3.16: Prism pair

Prism Pair

Combination of grating and prims pairs can eliminate 3rd order dispersion 52

(a) Standard mirror, (b) simple chirped mirror, (c) double-chirped mirror

3.7.4 Dispersion Compensating Mirrors

High reflecitvity bandwidth of Bragg mirror:

53

Comparison for different chirping of the high-index layer

54

Proposed structures that avoid GTI-effects

55

Double-chirped mirror pair

56

Grating pair and LCM pulse shaper

Dispersion Compensation with 4f-Pulse Shaper

57

Acousto-Optic Programable Dispersive Filter (AOPDF)

Dispersion Compensation with Acousto-Optic Programable Filter (DAZZLER)

58

Hollow fiber compression technique

3.7.5 Hollow Fiber Compression Technique

59

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)

4.1 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?

60

4 Laser Dynamics

γ10 à ∞

w = N2

Rp à ∞

N0=0 w = N1

Figure 4.6: Four-level laser.

3

0

1

2

N

N

N

N

3

2

1

0

γ32

21

10

Rp γ

γ

61

w = N2

γ10 à ∞

γ32 à ∞

62

Rate Equations and Cross Sections

e.g. semiconductors: T2 ~ 50fs

Can be used for time dependent intensity varying much slower than T2:

Interaction cross section:

63

Lorentzian line shape:

Dipole matrix element of transition

Intensity:

Steady state inversion:

Saturation intensity:

64

Saturation Energy:

T1 = τL For laser media:

Rate equations for a laser with two-level atoms and a resonator.

Laser Rate Equations

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

65

Intracavity power: P Round trip amplitude gain: g

66

Output power: Pout

small signal gain ~ στL - product

Laser Rate Equations:

Out

put P

ower

, Gai

n

S m a l l s i g n a l g a i n g 0g = = l0 gth

g = = lgth

P = 0

Output power versus small signal gain or pump power 67

Steady State: d/dt = 0 Pvac = 0

Case 1: Case 2:

gs = g0 P s = 0

4.2 Continuous Wave Operation

68

4.3 Stability and Relaxation Oscillations Perturbations:

69

Spectroscopic parameters of selected laser materials

4.4 Lasers and Its Spectroscopic Parameters

70

71

Quality factor of relaxation oscillations:

Relaxation Oscillations 72

(a)Losses

t

High losses, laser is below threshold

(b)Losses

t

Build-up of inversion by pumping

(c)Losses

t

In active Q-switching, the losses are reduced,after the laser medium is pumped for as long as theupper state lifetime. Then the loss is reduced rapidlyand laser oscillation starts.

(d)

Gain

Losses

t

Length of pump pulse

"Q-switched" Laserpuls

Laser emission stops after the energy stored in the gain medium is extracted.

Gain

Gain

Active Q-switching 73

4.5 Short pulse generation by Q-Switching 4.5.1 Active Q-Switching

Asymmetric actively Q-switched pulse.

74

Q-switched microchip laser 75

Passively Q-switched Laser

76

4.5.2 Passive Q-Switching

1 TR

~

77

5. Mode Locking

Pulse width time

f 0 f 1 = f 0 - Δ f

f 0 , f 1 , f 2

f 3 = f 0 -2 Δ f

f 4 = f 0 +2 Δ f f 2 = f 0 + Δ f

f 0 , f 1 , f 2 , f 3 , f 4

Spec

trum

78

Superposition of longitudinal modes:

Slowly varying complex pulse amplitude

79

N modes with equal amplitude E0 and phase φm =0

Scaling of pulse train with number of modes N:

Laser intensity for mode-locked operation and multimode lasing with random phase. 80

Actively modelocked laser.

81

5.1 Active Mode Locking

loss modulation

Parabolic approximation at position where pulse will form;

Master Equation:

82

Schematic representation of actively modelocked laser.

83

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:

84

Gaussian pulse with spectrum:

FWHM spectral width:

Time bandwidth product:

Pulse shaping in time and frequency domain.

85

Pulse width depends only weak on gain bandwidth. 10-100 ps pulses typical for active mode locking:

86

saturated gain = linear losses + losses in mdoulator + losses due to gain filtering

Additional gain for multimode operation:

Saturated gain is approximately:

Mode Locking

f

1-M

n0-1

MM

n0+1fn0f f

87

Active mode locking can be understood as injection seeding of neighboring modes by those already present.

Saturation characteristic of an ideal saturable absorber and linear approximation.

88

5.2 Passive Mode Locking

89

Saturable absorber provides gain for the pulse

There is a stationary solution:

Easy to check with:

90

Leads to:

Pulse energy, pulse width and saturated gain:

Schematic representation of gain and loss dynamics in passive mode locking.

91

Shortest pulse:

For shortest pulse half of absorption

depth can be used to overcome gain

filtering losses, only marginally stable!

For Ti:sapphire

gain

main cavity auxilary cavity

bias phase : π

t

nonlinear fiber

loss

artificial fast saturable absorber

Principle mechanism of APM

5.3 Kerr-Lens and Additive Pulse Mode Locking 5.3.1 Additive Pulse Mode Locking

92

Noninear Mach-Zehnder Interferometer

Kerr

Kerr

a

1-r2

b1

b21-r2

93

NLMZ using nonlinear polarization rotation in a fiber

94

200 MHz Soliton Er-fiber Laser modelocked by APM

•  167 fs pulses

•  200 pJ intracavity pulse energy

•  200 pJ output pulse energy

ISO

PBS

λ/4

λ/2

λ/4

collimatorcollimator

WDM980 nm Pump

10 cmSMF

50 cmEr doped fiber 10 cm

SMF

10 cmSMF

10 cmSMF

ISO

PBS

λ/4

λ/2

λ/4

collimatorcollimator

WDM980 nm Pump

10 cmSMF

50 cmEr doped fiber 10 cm

SMF

10 cmSMF

10 cmSMF

J. Chen et al, Opt. Lett. 32, 1566 (2007). K. Tamura et al. Opt. Lett. 18, 1080 (1993).

95

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Laser beam

Intensity dependent refractive index: "Kerr-Lens"

Self-Focusing Aperture

Time

Inte

nsity

Time In

tens

ity

Time

Inte

nsity

5.3.2 Kerr Lens Modelocking

Lens Refractive index n >1

Semiconductor saturable absorber mirror (SESAM) or Semiconductor Bragg mirror (SBR) 97

5.4 Semiconductor Saturable Absorbers

Pulse width of different laser systems by year.

5.5 Oscillators: Historical Development

99

6. Short Pulse Amplification 6.1 Cavity Dumping 6.2 Laser Amplifiers 6.2.1 Frantz-Nodvick Equation 6.2.2 Regenerative Amplifiers 6.2.3 Multipass Amplifiers 6.3 Chirped Pulse Amplification 6.4 Stretchers and Compressors 6.5 Gain Narrowing 6.6 Pulse Contrast 6.7 Scaling to Large Average Power by Cryogenic Cooling 6.8 Parametric Amplifiers (Cerullo)

[1] Largely follows lecture on Ultrafast Amplifiers by Francois Salin, http://www.physics.gatech.edu/gcuo/lectures/index.html.

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

1-10 W average power

Pul

seen

ergy

(J)

103

Schemes for generating high energy laser pulses.

Pulse energies from different laser systems

100

6.1 Cavity Dumping

With Bragg cell

With Pockels cell 101

Laser oscillator

Amplifier medium

Pump pulse

Energy levels of amplifier medium

Seed pulse

Amplified pulse

6.2 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.

102

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

fin =Fin/Fsat

G=F

out/F

in G0=3

Gain and extraction efficiency for a small signal gain of G0 = 3. 103

6.2.1 Frantz-Nodvik Equation

a) Multi-pass amplifier

pump

input output

gain

Pockels cell

polarizer

gain

pump

input/output

b) Regenerative amplifier

Basic Amplifier Schemes

104

6.2.2 Regenerative Amplifier Geometries

Two regens. The design in (a) is often used for kHz-repetition-rate amplifiers and the lower (b) at a 10-20-Hz repetition rate. The lower design has a larger spot size in the Ti:sapphire rod. The Ti:sapphire rod is usually ~20-mm long and doped for 90% absorption.

thin-film polarizer

Pockels cell

Faraday rotator

105

6.2.3 A Multi-Pass Amplifier

A Pockels cell (PC) and a pair of polarizers are used to inject a single pulse into the amplifier

106

6.3 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

107

d

f 2f

f

d

grating grating

Okay, this looks just like a “zero-dispersion stretcher” used in pulse shaping. But when d ≠ f, it’s a dispersive stretcher and can stretch fs pulses by a factor of 10,000!

With the opposite sign of d-f, we can compress the pulse.

6.4 Stretchers and Compressors

108

Alexandrite

Ti:sapphire

Excimers

0,0001

0,001

0,01

0,1

1

10

100

1 10 100 1000 10 4 10

5 10 6

Nd:Glass

Dyes

Flue

nce

(J/c

m2 )

Pulse Duration (fs)

109

Achieveable fluences using chirped pulse amplification for various stretching ratios. Compression of the pulses enables femtosecond pulses.

Achieveable fluences

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

110

6.5 Gain Narrowing

Influence of gain narrowing in a Ti:sapphire amplifier on a 10 fs seed pulse

If a pulse of 1018 W/cm2 peak power has a “little” satellite pulse one millionth as strong, that’s still 1 TW/cm2! This can do some serious damage!

Ionization occurs at 1011 W/cm2 : so at 1021 W/cm2 we need a 1010 contrast ratio!

111

6.6 Contrast Ratio

Major sources of poor contrast Nanosecond scale:

pre-pulses from oscillator pre-pulses from amplifier ASE from amplifier

Picosecond scale: reflections in the amplifier spectral phase or amplitude distortions

0 -1 -2 -3 -4 -5 -6 -7 -8 -9

-10

Front Back

time

Spectral phase aberrations

Pre-pulses

ASE

0 ps 10 ns ns

FWHM

Amplified pulses often have poor contrast. Lo

g(E

nerg

y)

Pre-pulses do the most damage, messing up a medium beforehand. 112

Calculations for kHz systems Cryogenic cooling results in almost no focal power

In sapphire, conductivity

increases and dn/dt

decreases with T

Murnane, Kapteyn, and coworkers 113

6.7 Large Average Power: Cryogenic Cooling

10

15

20

25

30

35

40

45

50

0

1

2

3

4

5

6

7

8

100 150 200 250 300

THER

MA

L C

ON

DU

CTI

VIT

Y (W

/m K

)C

TE(ppm/K

), dn/dT (ppm/K

)

TEMPERATURE (K)

UNDOPED YAG

Ther

mal

Con

duct

ivity

(W/m

K) C

TE (ppm/K

), dn/dT (ppm/K

)

Temperature (K)

114

Thermal Properties of YAG

T. Y. Fan, and coworkers at Lincoln Laboratory

287-W Picosecond Amplifier

Yb-fiber oscillator (78 MHz)

30-mW Yb-fiber preamplifier

10-W Yb-fiber amplifier

LN2 Dewar Yb:YAG crystals

Fiber-coupled pump laser

DM

287 W, 5.5 ps, 3.7 µJ@78 MHz

Chirped volume Bragg grating

(CVBG) stretcher

λ/4

PBS

λ/2

λ/4

CVBG compress

or

Telescope

Isolator

PBS

>100 ps

λ/2

λ/4

TFP

L3

CM

DM L1 L2 L2 L1

PBS

FR

(a) (b)

Yb:YAG amplifier requires >watt-level seed power for efficient amplification

PBS: pol. beam splitter TFP: thin film polarizer Ls: lens, DM: dichr. mirror CM: curved mirror, λ/4, λ/2: waveplates FR: Faraday rotator

115

116

6. 8 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!

Ultrabroadband Optical Parametric Amplifier

n  Broadband seed pulses can be obtained by white light generation

n  Broadband amplification requires phase matching over a wide range of signal wavelengths

G. Cerullo and S. De Silvestri, Rev. Sci. Instrum. 74, 1 (2003). "117

118"

Phase matching bandwidth in an OPA If the signal frequency ωs increases to ωs+Δω, by energy conservation the

idler frequency decreases to ωi-Δω. The wave vector mismatch is

ωΔωΔω

ωΔω

Δ ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

∂

∂+

∂

∂−=

gigs

is

vvkkk 11

( )

gigs vvL 11

12ln2 2/12/1

−

⎟⎠

⎞⎜⎝

⎛≅γ

πνΔ

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

119!

Broadband OPA configurations

vgi = vgs : Operation around degeneracy ωi = ωs = ωp/2

ü  Type I, collinear configuration ü  Signal and idler have same refractive index

vgi ≠ vgs : Non-collinear parametric amplifier (NOPA):

sk

pk

ik

α Pump

Signal

ü  Pump and Signal at angle α

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

120

121"

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

122!

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!

123!

Broadband pulses in the mid-IR

l  Simulations confirm the generation of broadband idler pulses, with 20-fs duration (≈2 optical cycles) at 3 µm

0,0

0,5

1,0

0,9 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

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). R. Boyd, Nonlinear Optics ~Academic, New York, 1992. G. Cerullo and S. De Silvestri, "Ultrafast Optical Parametric Amplifiers," Review of Scientific Instruments 74, pp. 1-17 (2003).