1

The Generation of Ultrashort Laser Pulses II

The phase condition

Trains of pulses – the Shah function

Laser modes and mode locking

2

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

3

( ) 1/exp)(

)(=ω−= cLi

Eangle

Eanglert

before

afterq

L

c

rt

q ⋅π

=ω2

(q = an integer)

An integer number of wavelengths must fit in the cavity.

Steady-state condition #2:

Phase is invariant after each round trip

length Lm

collection of 4-level systems

Round-trip length Lrt

Phase condition

4

“axial” or “longitudinal” cavity modes

Mode

spacing:

∆ν = c/Lrt

ω0

laser gain

profile

qL

c

rt

q ⋅π

=ω2

(q = an integer)

Longitudinal modes

fits

does not fit

But how does this translate to the case of short pulses?

5

Femtosecond lasers emit trains of

identical pulses.

where I(t) represents a single pulse intensity vs. time and T is the time

between pulses.

Every time the laser pulse hits the output mirror, some of it emerges.

The output of a

typical ultrafast laser

is a train of identical

very short pulses:

R = 100% R < 100%

Output

mirror

Back

mirror

Laser medium

t-3T -2T 0 T 2T 3T-T

Intensity

vs. time

I(t) I(t-2T)

6

The Shah Function

The Shah function, III(t), is an infinitely long train

of equally spaced delta-functions.

The symbol III is pronounced shah after the Cyrillic character III, which is

said to have been modeled on the Hebrew letter (shin) which, in turn,

may derive from the Egyptian , a hieroglyph depicting papyrus plants

along the Nile.

III( ) ( )

m

t t mδ∞

=−∞

≡ −∑

t-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

� �

7

If ω = 2nπ, where n is an integer, the sum diverges; otherwise, cancellation occurs.

{III( )} III(t ω π∝ / 2 )Y

) exp( )m

t m i t dtδ ω∞ ∞

=−∞−∞

= ( − −∑∫

III(t)

So:

exp( )m

i mω∞

=−∞

= −∑

) exp( )m

t m i t dtδ ω∞∞

=−∞ −∞

= ( − −∑ ∫

t-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

… …

F{III(t)}

2

ωπ-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

… …

{III( )}t =Y

The Fourier Transform of the Shah Function

8

The Shah Function

and a Pulse Traint

-3T -2T 0 T 2T 3T-T

E(t)

t-3T -2T 0 T 2T 3T-T

E(t)

t-3T -2T 0 T 2T 3T-T

E(t)f(t) f(t-2T)f(t) f(t-2T)

where f(t) is the shape of each pulse and T is the time between pulses.

III( /( () ))t TE t f t= ∗But E(t) can also can be written:

(( ))m

f t mE t T∞

=−∞

−= ∑An infinite train of

identical pulses can

be written:

( )m

f t mT∞

=−∞

−= ∑To do the integral, set:

t’/T = m or t’ = mT

( ) (III( / ) ( / ) )m

t T t T mf t f t t dtδ∞∞

=−∞ −∞

′−′ ′∗ = −∑ ∫Proof:

convolution

9

An infinite train of

identical pulses can

be written:

The Fourier Transform of an Infinite Train of Pulses

E(t) = III(t/T) * f(t)t

-3T -2T 0 T 2T 3T-T

E(t)

t-3T -2T 0 T 2T 3T-T

E(t)

t-3T -2T 0 T 2T 3T-T

E(t)f(t) f(t-2T)f(t) f(t-2T)

II ) () / 2( I(E T Fω πω ω)∝%

The spacing between frequencies (modes) is then δω = 2π/T or δνδνδνδν = 1= 1= 1= 1////T.

The Convolution Theorem says that the Fourier Transform of a

convolution is the product of

the Fourier Transforms. So:

ωΤ / 2π-4 -2 0 2 4

F(ωωωω) ( )E ω%

10

The Fourier Transform of a Finite Pulse Train

A finite train of identical pulses can be written:

[III(( ) ( () )/ ) ]t T g tE t f t= ∗

where g(t) is a finite-width envelope over the pulse train.

g(t)

t-3T -2T 0 T 2T 3T-T

E(t)

f(t)

[III( / 2 ) ( )( ) ]( )TE FG ωωω ω π∝ ∗%

F(ωωωω)

-6π/T -4π/Tω

0 2π/T 4π/T 6π/T-2π/T

( )E ω%

Use the fact

that the Fourier

transform of a

product is a

convolution<

G(ωωωω)

11

A laser’s frequencies are often called longitudinal modes.

They’re separated by 1/T = c/2L, where L is the length of the laser.

Which modes lase depends on the gain and loss profiles.

Frequency

Inte

nsity

Here,

additional

narrowband

filtering has

yielded a

single

mode.

Actual Laser Modes

12

Mode-locked vs. non-mode-locked light

Mode-locked pulse train:

A train of

short pulses

( ) ( 2 / )m

F m Tω δ ω π∞

=−∞

= −∑

Non-mode-locked pulse train:

exp( )( ) ( ) ( 2 / )m

m

iE F m Tω ω ϕ δ ω π∞

=−∞

= −∑%

Random phase for each mode

A mess<( ) () /exp( 2 )m

m

F m Tiω δ ω πϕ∞

=−∞

= −∑

( ) ( III( / 2 )E F Tω ω ω π= )%

13

Generating short pulses = Mode-locking

Locking vs. not locking the phases of the laser modes (frequencies)

Random

phases

Light bulb

Intensity vs. time

Ultrashort

pulse!

Locked

phases

Time

Time

Intensity vs. time

14

Q: How many different modes can oscillate simultaneously in

a 1.5 meter Ti:sapphire laser?

A: Gain bandwidth ∆λ = 200 nm ⇒ ∆ν = (c/λ2) ∆λ ~ 1014 Hz

∆νbandwidth/∆νmode = 106 modes

That seems like a lot. Can this really happen?

Therefore ∆ν = c/Lrt ≈ 100 MHz

200 nm

bandwidth!

Wavelength (nm)

Ti:sapphire: how many modes lock?

Ti: sapphire crystal

pump

Typical cavity layout:

length of path from

M1 to M4 = 1.5 m

15

independent of ω

We have seen that the gain is given by:

2

00

1/12

1)(

ζ+σ

⋅+∆

⋅=ωsatII

Ng

( )γ

ω−ω=ζ 02

where

ωq

laser

gain

profileωq+1

ωq−1

losses Q: Suppose the gain is increased

further. Can it be increased so that the

mode at ωq+1 oscillates in steady state?

A: In an ideal laser, NO!

Homogeneous gain media

Suppose the gain is increased to a point where it

equals the loss at a particular frequency, ωq

which is one of the cavity mode frequencies.

16

Suppose that the collection of 4-level systems do not all share the same ω0

Consider a collection of sites, with fractional number between ω0 and ω0 + dω0:

000 )()( ωω=ω dNgdN

( ) ( ) )(; 000 ωωωχω=ωχ ∫ gd h

The modified susceptibility is:

homogeneous “packet”

inhomogeneous

line

“packets” are mutually independent -

they can saturate independently!

Inhomogeneous gain media

17

Lorentian

homogeneous

line shape

Gaussian

inhomogeneous

distribution of

width ∆ω( ) ( )2

00)2ln(4

00 ;

ω∆ω−ω

−

⋅ωωχω=ωχ ∫ ed h

No closed-form solution

For strong inhomogeneous broadening (∆ω >> γ):

χ"(ω) = Gaussian, with width = ∆ω (NOT γ)χ'(ω): no simple form, but it resembles χh'(ω)

Examples:

Nd:YAG - weak inhomogeneity

Nd:glass - strong inhomogeneity

Ti:sapphire - absurdly strong inhomogeneity

R6G - intermediate case (?)

Inhomogeneous broadening

18

Q: Suppose the gain is increased above threshold

in an inhomogeneously broadened laser. Can it be

increased so that the mode at ωq+1 oscillates cw?

A: Yes! Each homogeneous packet saturates independently

“spectral hole burning”

multiple oscillating cavity modes

a priori, these modes

need not have any

particular phase

relationship to one

another

ωq

laser gain

profile

ωq+1ωq−1

losses

ωq−2 ωq+2 ωq+3

Hole burning

19

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

20propagation kernel

Steady-state condition #3:

Transverse profile reproduces on each round trip

How would we determine these modes?

"transverse modes": those which reproduce themselves on

each round trip, except for overall amplitude and phase factors

An eigenvalue problem:

( ) ( ) ( )nm nm 0 0 nm 0 0 0 0E x, y K x, y;x , y E x , y dx dyβ ⋅ = ⋅∫∫

Condition on the transverse profile

21

( ) ( ) ( ), = ⋅nm n mE x y u x u y

Solutions are the product of two functions, one for each transverse dimension:

"TEM" = transverse electric and magnetic

"nm" = number of nodes along two principal axes

Notation:

( )2

2

2exp

= ⋅ −

n n

x xu x H

w w

where the un’s are Hermite Gaussians:

w = beam waist

parameter

Transverse modes

22

TEM00 TEM01 TEM02 TEM13

http://www.physics.adelaide.edu.au/optics/

x y

|E(x,y)|2

12 mode

x y

|E(x,y)|2

A superposition of the 10 and

01 modes: the “donut mode”

Transverse modes - examples