The Generation of Ultrashort Laser Pulses

1

The Generation of Ultrashort Laser Pulses

The importance of bandwidth More than just a light bulb

Two, three, and four levels – rate equations Gain and saturation

First point: the progress has been amazing!

YEAR Nd:glass

S-P Dye Dye

CW Dye

Nd:YAG Diode

Nd:YLF

Cr:YAG Cr:LiS(C)AF

Er:fiber

Cr:forsterite

Ti:sapphire CPM

w/Compression

Color Center

1965 1970 1975 1980 1985 1990 1995 Dye

2000

SHORTEST PULSE DURATION

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.

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 or equal to ~1.

We’ll get to this part in a later lecture.

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

There are 3 conditions for steady-state laser operation.

Amplitude condition threshold

slope efficiency Phase condition

axial modes

homogeneous vs. inhomogeneous gain media Transverse modes

Hermite Gaussians the “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:

2

(

)

dN BI N N AN

dt   

1

(

)

dN BI N N AN

dt   

2 2

 d N     

BI N AN dt

Absorption Stimulated emission Spontaneous emission

1 2

N N N

  

1 2

N  N  N

If the total number of molecules is N:

2 1 2 1 2

2 N ( N N ) ( N N ) N N

   

    2

 d N       BI N AN A N dt

2

1

N₂

N₁ Laser Pump

Pump intensity

How does the population difference depend on pump intensity?

0   2BI N AN A N    

In steady-state:

d N 2

BI N AN A N dt

      

/( 2 )

1 2

    

 N AN A BI N

B I A 1 2 /

  

N N

I I

 /

I

A B

where:



2

1

N₂

N₁ Laser Pump

( A  2 BI N )   AN

Solve for N:

9

Why inversion is impossible in a two-level system

2

1

N₂

N₁ Laser Pump

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 /

  

N N

Population difference

I I

Recall that

  N N

 N

For population inversion, we require

  N 0

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

0 2I_sat

0 0.5 N N

Pump intensity

population difference N

4I_sat 6I_sat

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.

2

1 2

dN BIN AN dt  

1

1 2

dN BIN AN dt   

1 2

2 2

d N BIN AN dt

   

Fast decay

Laser Transition Pump

Transition 1 2 3

1 2

N N N

  

1 2

N  N  N

The total number of molecules is N:

Level 3 decays fast and so N₃ = 0.

2 N

   N N

2 N

   N N

11

Why inversion is possible in a three-level system

Now if I > I_sat, N is negative!

1 / 1 /

sat sat

N N I I

I I

  



sat

/

I  A B

where, as before:



d N BIN BI N AN A N dt

       

In steady-state:

0   BIN BI N AN A N     

Fast decay

Laser Transition Pump

Transition 1 2 3

 

 

1 1

 

  

 

B A I N N A BI N

A 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.

2

2 2

( )

dN BI N N AN dt   

Laser Transition Pump

Transition

Fast decay Fast decay

1 2 3

0

2

0 2

dN BIN AN dt  

As before:

d N BIN BI N A N dt

      

At steady state:

0 BIN BI N A N     

0 2

N  N  N

The total number of molecules is N :

0 2

N   N N

1

0,

N     N 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

N N I I

   I I



sat

/

I  A B

where:

I_sat is the saturation intensity.



   

 

1

      



B A I N BIN A BI N

B A I

Solve for N:

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 Transition Pump

Transition

Fast decay

15

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

0 2I_sat 4I_sat 6I_sat 8I_sat 10I_sat 0

N

-N

population inversion

2 levels

3 levels

4 levels

intensity of the pump population

difference

N

What is the saturation intensity?

A is the excited-state relaxation rate: 1/

sat

/

I  A B

  I





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, I_sat ~ 300 kW/cm²

17

gain

length L

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

  

after

m before

E r r gL

E

1 1

2 ln

 



    

g L r r

 2

j j

R r

Note:

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

mirror with reflection coefficient r₁

mirror with reflection coefficient r₂

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

Example: 17 3

-2ꞏ10



 N

cm

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

Threshold population inversion

length L

collection of 4-level systems

But we have seen that the

gain at resonance is:

 

1

   2

g  N 

Threshold inversion density:

0 1 2

1 1

2 ln

 

   

 

thresh

m

N L  R R

1 2

1 1

4 ln

 

  

 

m

g L R R

19

0 2I_sat 4I_sat 6I_sat 8I_sat 10I_sat 0

1

-1

population inversion

2 levels

3 levels 4 levels

intensity of the pump population

difference

N

lasing!

N_threshold

A threshold value of pump intensity

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

20

Round-trip length L_rt

cavity gain per round trip

net cavity loss per round trip, including mirrors More generally:

Gain vs. loss in a laser cavity

0

2

1 2

 





after before

E R R e e

E

   

Cavity lifetime:



 T



Cavity Q-factor: the fractional power loss

per optical cycle

 

 2

Q  L 

length L

collection of 4-level systems

21

Threshold inversion condition becomes 

= 

Output power is determined by gain saturation:

gain loss (separated into the output coupler plus all other losses)

sat circ m m

m

I I

L gL g









1

4 4

 

 

 

sat oc

m oc

sat circ

oc

out

g L I

I I

I  



 



 





 





 4 1

0 0

(valid only above threshold, I_out  0)

Laser output power

or

2





m

N L





2

m c

after before

E e

E

 



22

One consequence: the dependence on the output coupler

0

2 1

out sat oc

oc

P P  G

 

 

      

We’ve just shown this:

What output coupler should we use to maximize the output power?

 

2 2

1 0

out

sat OC sat

OC OC other OC other

P G G

P   P

      

       

OC 2G other other

    

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

P out

There is an optimum value for the output coupler, which depends on

the gain and loss factors. ^{g = 0.2}

_other = 0.01 P_sat= 6.8 Watts

23

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

R

N

g  



 













 





10 21

10 0 21

0

, p

1

out sat oc

p th

I I R

 ^ R ^

       

Thus, output power can also be written:

“Slope efficiency” = output power pump power

Slope efficiency

Output power varies linearly with the pump rate (above

threshold but below saturation)

pump power

output laser power

threshold

The concept of slope efficiency only applies for I_th < I_pump < I_sat. 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.

25

distributed feedback diode laser

A. Jechow, et al., Opt. Lett (2007)

 = 975 nm

Slope efficiency – some randomly chosen examples

diode-pumped thulium lasers

P. Černý and H. Jelínková, SPIE 2006

threshold: 33 mA

slope efficiency: 0.74 W/A

saturation regime