IMPRS: Ultrafast Source Technologies...1.1 Dispersive Pulse Propagation 1.2 Nonlinear Pulse...

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

1

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:

*)

2

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

Pump - Probe Measurements

4

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

5

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

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

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

8

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

9

1. Basics of Optical Pulses

10

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

Typical Lab Pulse:

11

Time Harmonic Electromagnetic Waves

Transverse electromagnetic wave (TEM) (Teich, 1991)

See previous class: Plane-Wave Solutions (TEM-Waves)

12

Optical Pulses ( propagating along z-axis)

: Wave amplitude and phase

: Wave number

0( )( )cc

nΩ =

Ω: Phase velocity of wave

13

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

14

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 ~ _

15

Pulse width and spectral width: FWHM

Often Used Pulses

16

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

17

Electric field and pulse envelope in time domain

18

Taylor expansion of dispersion relation at pulse center frequency

19

In the frequency domain:

1.1 Dispersion

Taylor expansion of dispersion relation:

i) Keep only linear term:

Time domain:

Group velocity:

20

Compare with phase velocity:

Retarded time:

ii) Keep up to second order term:

21

Gaussian Pulse:

Pulse width

Initial pulse width:

FWHM Pulse width:

determines pulse width chirp

z-dependent phase shift

z = L

22

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

23

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

24

Sellmeier Equations

( )rχ Ω

Example: Sellmeier Coefficients for Fused Quartz and Sapphire

25

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

26

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

27

Group Velocity and Group Delay Dispersion

Group Delay:

28

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

29

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

31

1.3 Pulse Compression

Dispersion negligible, only SPM

Optimium Dispersion and nonlinearity

32

Fiber-grating pulse compressor to generate femtosecond pulses

Spectral Broadening with Guided Modes and Compression

Pulse Compression:

33

Grating Pair

Phase difference between scattered beam and reference beam”

Disadvantage of grating pair: Losses ~ 25%

34

Prism Pair

35

3.7.4 Dispersion Compensating Mirrors

High reflecitvity bandwidth of Bragg mirror:

36

Hollow fiber compression technique

3.7.5 Hollow Fiber Compression Technique

37

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

38

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 ∞

39

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

40

Intracavity power: P Round trip amplitude gain: g

Output power: Pout

small signal gain ~ στL - product

Laser Rate Equations:

41

Output power versus small signal gain or pump power

42

Steady State: d/dt = 0 Pvac = 0 Case 1: Case 2:

gs = g0 P s = 0

2.2 Continuous Wave Operation

Spectroscopic parameters of selected laser materials

Lasers and Its Spectroscopic Parameters

43

Here active Q-switching

3 Q-Switched Lasers

44

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

45

Actively modelocked laser

4.1 Active Mode Locking

loss modulation

Parabolic approximation at position where pulse will form;

Master Equation:

46

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:

47

Pulse shaping in time and frequency domain.

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

48

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.

49

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

52

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

Semiconductor Saturable Absorbers

53

Pulse width of different laser systems by year.

Modelocking: Historical Development

54

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

55

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

56

5.1 Cavity Dumping

With Bragg cell

With Pockels cell

57

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.

58

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

60

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

61

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

62

Chirped Pulse Amplifier System

Oscillator – Stretcher – Multiple Amplifiers - Compressor Chain

Oscillator Stretcher Amplifier 1

Compressor

Amplifier 2

Amplifier 3

63

5.4 Stretchers and Compressors

64

f 2f f-d

grating

negative dispersion

grating

Phase Mask

f 2f f+d

grating

grating

positive dispersion

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

65

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

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).

67

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

68

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

69

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

70

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

71

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!

72

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

73

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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