1

The Generation of UltrashortLaser Pulses

The importance of bandwidth

More than just a light bulb

Two, three, and four levels – rate equations

Gain and saturation

2

But first: the progress has been amazing!

YEAR

Nd:glass

S-P DyeDye

CW Dye

Nd:YAG

Diode

Nd:YLF

Cr:YAG

Cr:LiS(C)AFEr:fiber

Cr:forsterite

Ti:sapphire

CPM

w/Compression

ColorCenter

1965 1970 1975 1980 1985 1990 1995

Dye

2000

SHO

RTE

ST P

ULS

E D

UR

ATI

ON

10ps

1ps

100fs

10fs

2005

Nd:fiber

The shortest pulse vs. year (for different media)

3

Continuous vs. ultrashort pulses of light

Continuous beam:

Ultrashort pulse:

Irradiance vs. time Spectrum

time

time

frequency

frequency

A constant and a delta-function are a Fourier-Transform pair.

4

Long vs. short pulses of light

Long pulse

Short pulse

Irradiance vs. time Spectrum

time

time

frequency

frequency

The uncertainty principle says that the product of the temporal and spectral pulse widths is greater than ~1.

5

But a light bulb is also broadband.

What exactly is required to make an ultrashort pulse?

Answer: A Mode-locked Laser

Okay, what’s a laser, what are modes, and what does it mean to lock them?

Light bulbs, lasers, and ultrashort pulses

6

There are 3 conditions for steady-state laser operation.

Amplitude conditionthresholdslope efficiency

Phase conditionaxial modeshomogeneous vs. inhomogeneous gain media

Transverse modesHermite Gaussiansthe “donut” mode

x y

|E(x,y)|2

7

Rate equations for a two-level system

Rate equations for the population densities of the two states:

21 2 2( )dN BI N N AN

dt

12 1 2( )dN BI N N AN

dt

2 2 2

d N BI N ANdt

AbsorptionStimulated emission Spontaneous

emission

1 2N N N 1 2N N N

If the total number of molecules is N:

2 1 2 1 22 ( ) ( )N N N N NN N

2

d N BI N AN A Ndt

2

1

N2

N1

LaserPump

Pump intensity

8

How does the population difference depend on pump intensity?

0 2BI N AN A N In steady-state:

2d N BI N AN A Ndt

/( 2 ) 21

NN AN A BI B IA

1 2 /

sat

NNI I

/satI A Bwhere:Isat is called the saturation intensity.

2

1

N2

N1

LaserPump

( 2 ) A BI N ANSolve for N:

9

Why inversion is impossible in a two-level system

2

1

N2

N1

LaserPump

It’s impossible to achieve a steady-state inversion in a two-level system!

Why? Because absorption and stimulated emission are equally likely.

1 2 /

sat

NNI I

Population difference

Recall that 1 2N N N

For population inversion, we require 0 N

N is always positive, no matter how hard we pump on the system!

0 2Isat

0

0.5 N

N

Pump intensity

popu

latio

n di

ffere

nce

N

4Isat 6Isat

a plot of this function

Even for an infinite pump intensity, the best we can do is N1 = N2 (i.e., N = 0)

10

Rate equations for a three-level system

So, if we can’t make a laser using two levels, what if we try it with three?

Assume we pump to a state 3 that rapidly decays to level 2.

21 2

dN BIN ANdt

11 2

dN BIN ANdt

1 22 2d N BIN ANdt

Fast decay

Laser Transition

Pump Transition

1

23

1 2N N N 1 2N N N

The total number of molecules is N:

Level 3 decays fast and so N3 = 0.

22N N N

12N N N

11

Why inversion is possible in a three-level system

Now if I > Isat, N is negative!

1 /1 /

sat

sat

I IN NI I

/satI A Bwhere, as before:

Isat is the saturation intensity.

d N BIN BI N AN A Ndt

In steady-state: 0 BIN BI N AN A N

Fast decay

Laser Transition

Pump Transition

1

23

11

B A IA BIN N NA BI B A I

Solve for N:

12

Rate equations for a four-level system

Now assume the lower laser level 1 also rapidly decays to a ground level 0.

22 2( )dN BI N N AN

dt

Laser Transition

Pump Transition

Fast decay

Fast decay

1

2

3

020 2

dN BIN ANdt

As before:

d N BIN BI N A Ndt

At steady state: 0 BIN BI N A N

0 2N N N

The total number of molecules is N :

0 2N N N 1 0,N 2N N Because

13

Why inversion is easy in a four-level system

0 BIN BI N A N

Now, N is negative—for any non-zero value of I!

Laser Transition

Pump Transition

Fast decay

Fast decay

1

2

3

0

/1 /

sat

sat

I IN NI I

/satI A Bwhere:Isat is the saturation intensity.

1

B A IN BIN A BI N

B A I

Solve for N:

14

Two-, three-, and four-level systems

Two-level system

Laser Transition

Pump Transition

At best, you get equal populations.

No lasing.

Four-level systems are best.

Four-level system

Lasing is easy!

Laser Transition

Pump Transition

Fast decay

Fast decay

Three-level system

If you hit it hard, you get lasing.

Laser TransitionPump

Transition

Fast decay

15

Population inverstion in two-, three-, and four-level systems

0 2Isat 4Isat 6Isat 8Isat 10Isat

0

N

-N

population inversion

2 levels

3 levels

4 levels

intensity of the pump

population difference

N

16

What is the saturation intensity?A is the excited-state relaxation rate: 1/

/satI A B B is the absorption cross-section, , divided by the energy per photon, ħ: / ħ

satI

The saturation intensity plays a key role in laser theory. It is the intensity which corresponds to one photon incident on each molecule, within its cross-section , per recovery time .

Both and depend on the molecule, and also the frequency of the light.

ħ ~10-19 J for visible/near IR light

~10-12 to 10-8 s for molecules

~10-20 to 10-16 cm2 for molecules (on resonance)

105 to 1013 W/cm2

For Ti:sapphire, Isat ~ 300 kW/cm2

17

gain

length Lm

collection of 4-level systems

Steady-state condition #1:

Amplitude is invariant after each round trip

Gain g must be large enough to compensate for reflection losses R1 and R2 at the mirrors.

Threshold condition: what about losses?

1 2 exp 2 1 afterm

before

Er r gL

E 1 2

1 1ln2

m

gL r r

2j jR rNote:

Note: this ignores other sources of loss in the laser.

18

A 5 mm Ti:sapphire crystal, in a cavity with one high reflector and one 5% output coupler

Example: 317-2·10 cmNth

This corresponds to ~0.5% of the Ti3+ ions in the crystal.

Threshold population inversion

length Lm

collection of 4-level systems

But we have seen that the gain at resonance is: 0 0

12

g N

Threshold inversion density: 0 1 2

1 1ln2

thresh

m

NL R R

1 2

1 1ln4

m

gL R R

19

0 2Isat 4Isat 6Isat 8Isat 10Isat

0

1

-1

population inversion

2 levels

3 levels

4 levels

intensity of the pump

population difference

Nlasing!

Nthreshold

A threshold value of pump intensity

minimum pump intensity required to turn on a 4-level laser, Ithreshold

20

Round-trip length Lrt

cavity gainper round trip

net cavity lossper round trip,

including mirrors

More generally:

Gain vs. loss in a laser cavity

0

2

1 2 m m cafter

before

ER R e e

E

Cavity lifetime: c rt cT

Cavity Q-factor: the fractional power loss per optical cycle 2 rt cQ L

length Lm

collection of 4-level systems

21

Threshold inversion condition becomes c = m

Output power is determined by gain saturation:

gainloss

sat

circ

mmm

IILggL

1

44 0occ 0

satoc

mocsatcircocout ILgIII

14

0

0

(valid only above threshold, Iout 0)

Laser output power

or02

cthresh

m

NL

22

Recall that the unsaturated gain is proportional to the pumping rate Rp:

pRNg

1021

102100

,

1pout sat oc

p th

RI I

R

Thus, output power can also be written:

“Slope efficiency” = output power

pump power

Slope efficiency

Output power is linear in the pump rate (above threshold but below saturation)

23

pump poweroutp

ut la

ser p

ower

threshold

The concept of slope efficiency only applies for Ith < Ipump < Isat.

The slope of this line is called the “slope efficiency” of the laser.

Above threshold (but not too far above), the output laser power is a linear function of the input pump power.

For example, a slope efficiency of 50% means that for every two additional pump photons we add, one additional laser photon is generated.

Slope efficiency

Essentially all lasers exhibit this behavior.

24

distributed feedback diode laserA. Jechow, et al., Opt. Lett (2007)

= 975 nm

Slope efficiency - examples

diode-pumped thulium lasersP. Černý and H. Jelínková, SPIE 2006

threshold: 33 mA

slope efficiency: 0.74 W/A

saturation regime