A pulse forming mechanism is needed otherwise lasers run “continuous wave” (CW)

Three types of pulsed operation

Pulsed Lasers

1. Gain switched (micro or millisecond pulses typically) turn gain on and off (flash lamps, modulate pump)

2. Q-switched (nanosecond pulses) modulate cavity loss on times scales > round trip time

3. Modelocked (picosecond to femtosecond pulses) modulate cavity loss periodically at roundtrip time

ADVANCES IN SHORT PULSE GENERATION

The “Mode” in “Mode-locking” is longitude Models

0

Log(

I out/I i

n)

Gain

Absorption

No absorptionBelow energy gap

Gain region

Loss region

Many longitudal modes can co-exists in the same cavity

Mode-Lock

The relative phases of the simultaneously lasing cavity modes must be “locked” at proper values.

Roughly:

How to achieve Mode-lock

Laser with many modes but without mode-locking

• Relative phases of the modes are random

• Output looks completely random but repeats itself in time equal to 2L/c

• Slow photodetectors see only the average power

Q: which of the above are still true under mode-lock condition?

How to achieve Mode-lock

Design a cavity so that the round-trip loss of the light is BIG, unless the modes are “lock” to generate short pulses.

Two Types of techniques:

1. Saturated absorbers due to limited number of absorbing particles (e.g., dye molecules)

2. Nonlinear effect: Kerr lens

Optical Kerr Effect

A change in the refractive index of a material in response to an electric field.

The change is proportional to the square of the electric field

All materials show Kerr effect

Optical Kerr Effect is the special case where the electric field is from the light itself

Kerr Lens Mode-locking

• Kerr Lens & Aperture gives increased transmission at high Intensity

• Increased transmission at high intensity

• Short, intense pulse preferred in laser

• Kerr effect instantaneous

Dispersion in OpticsThe dependence of the refractive index on wavelength has two effects on a pulse, one in space and the other in time.

Dispersion disperses a pulse in space (angle):

Dispersion also disperses a pulse in time:

“Chirp”d2n/d2

“Angular dispersion”dn/d

Both of these effects play major roles in ultrafast optics.

In most of the region, n increases as frequency of the light increase

angular frequency.

Group Velocity:

The velocity with which the envelope of the wave propagate through space.

1v /g dk d

: angular frequency.

k: angular wave number k = 2 /

is the same in or out of the medium; but k = k0n, where k0 is k in vacuum, and n is what depends on the medium.

v = / k

A Simple Derivation of Group Velocity:

v v / 1g phase

dn

n d

Group Velocity in terms of Optical dispersion

Vphase = c0 / n

Conclusion: group velocity has nothing to do with the true velocity of the light

The group velocity is less than the phase velocity in non-absorbing regions.

vg = c0 / (n + dn/d)

Except in regions of anomalous dispersion (which are absorbing), dn/d is negative, that is, near a resonance. So vg > vphase for these frequencies!

Group Velocity vs. WavelengthWe more often think of the refractive index in terms of wavelength,so let's write the group velocity in terms of the vacuum wavelength 0.

0 0

0

v / 1g

c dn

n n d

Consider the simplest case: dn/d0 = = constant

0/)/1(

0

0

0

c

ddn

d

vd g

This means vg is independent of wavelength

The effect of group velocity dispersion

GVD means that the group velocity will be different for different wavelengths in the pulse.

vgr(yellow) < vgr(red)

early times

late times

is the “group velocity dispersion.”1

( )vg

dk

d

Calculation of the GVD (in terms of wavelength)

Recall that:2

0 0

02

d

d c

20 0

0 0 02

dd d d

d d d c d

00

v /g

dnc n

d

20

00 0 0

1 1

v 2g

d d dnn

d c d c d

20

020 0 02

d dnn

c d d

2 20

02 20 0 0 02

dn d n dn

c d d d

and

3 20

0 2 20 0

( )2

d nGVD k

c d

Okay, the GVD is:

Simplifying:Units:s2/m or(s/m)/Hz or s/Hz/m

GVD yields group delay dispersion (GDD).

We can define delays in terms of the velocities and the medium length L.

The phase delay:

00

0

( )v ( )

k

00

1( )

v ( )g

k

1( )

vg

dk

d

The group delay:

The group delay dispersion (GDD):

so: 0

0 0

( )

v ( )

k LLt

0 00

( ) ( )v ( )g

g

Lt k L

1( )

vg

dGDD L k L

d

so:

so:

Units: fs2 or fs/Hz

GDD = GVD L

Manipulating the phase of light

Recall that we expand the spectral phase of the pulse in a Taylor Series:

20 1 0 2 0( ) [ ] [ ] / 2! ...

So, to manipulate light, we must add or subtract spectral-phase terms.

20 1 0 2 0( ) [ ] [ ] / 2! ...H H H H

and we do the same for the spectral phase of the optical medium, H:

For example, to eliminate the linear chirp (second-order spectral phase), we must design an optical device whose second-order spectral phase cancels that of the pulse:

2 H 2 0d2d 2

0

d2 H

d 2

0

0i.e.,

group delay group delay dispersion (GDD)phase delay

So how can we generate negative GDD?

This is a big issue because pulses spread further and further as they propagate through materials.

We need a way of generating negative GDD to compensate.

Angular dispersion yields negative GDD.

Suppose that some optical element introduces angular dispersion.

Inputbeam

Opticalelement

L

If frequency 0 propagates a distance L to plane S’, then frequency sees a phase delay of

d2d2

0

0 L

cdd 0

2

The GDD due to angular dispersion is always negative!

P0

A prism pair has negative GDD

How can we use dispersion to introduce negative chirp conveniently?

This term assumesthat the beam grazes the tip of each prism

d2d 2

0

4Lsep

03

2c2

dn

d 0

2

Lprism

03

2c2

d2n

d2

0

This term allows the beam to pass through an additionallength, Lpriam, of prism material.

Assume Brewsterangle incidenceand exit angles.

Vary the second term to tune the GDD!

Lsep

Always negative!

Always positive (in visible and near-IR)

Adjusting the GDD maintains alignment.

Original prism position

New prism position

Original paththrough prism

Any prism in the compressor can be translated perpendicular to the beam path to add glass and reduce the magnitude of negative GDD.

Remarkably, this doesnot misalign the beam.

New beam paththrough prism

Original and newpath out of the prism

Pulse Compressor

This device has negative group-delay dispersion and hence can compensate for propagation through materials (i.e., for positive chirp).

The longer wavelengths have a longer path.

It’s routine to stretch and then compress ultrashort pulses by factors of >1000

Ti:Sapphire laser

• Optically active atoms TiO3 <1% by Weight• Host solid is sapphire (Al2O3)• Ti interacts with solid so the E2, E1 broadened significantly (inhomogeneous broadening). The atoms in the solid vibrate and interactwith the Ti atoms.• Gain bandwidth huge (100 THz)• Can use as tunable laser • Makes for ultra short pulses

Titanium Sapphire Ti3+: Al2O3

E 3/2 excited state

2T2 ground state

Spin forbidden: long emission lifetime 300s

Reasonable extinction coefficient

Huge Stokes shift, Broad emission spectrum compared to dyes

Revolutionized laser industry in ability to make tunable short pulses

Tuning Range and Power of Ti:Sapphire

100 femtosecond pulses10 nm FWHM bandwidth

Longer wavelengths Less damaging

900 is often goodCompromise betweenPower and viability

Excite essentially every dyeWith this wavelength range

1. Pumped by Argon ion or doubled Nd laser2. Birefringence filter used for color tuning

Output coupler

end

END

Introduction to Non-linear Optical effects

•NLO effects have been observed since 19thC •Pockels effect •Kerr effect

•High fields associated with laser became available in the 1960s and gave rise to many new NLO effects

•second harmonic generation (SHG) •third harmonic generation (THG) •stimulated Raman scattering •self-focussing

•In NLO we are concerned with the effects that the light itself induces as it propagates through the medium

•In linear optics the light is deflected or delayed but its frequency (wavelength) is unchanged

Pulse Compression Simulation

Using prism and grating pulse compressors vs. only a grating compressor

Resulting intensity vs. time with only a grating compressor:

Resulting intensity vs. time with a grating compressorand a prism compressor:

Note the cubic spectral phase!

Brito Cruz, et al., Opt. Lett., 13, 123 (1988).

Pulse compressors achieve amazing results, but, if not aligned well, they can introduce spatio-temporal distortions, such as “spatial chirp.”

Propagation through a prism pair produces a beam with no angular dispersion, but the color varies spatially across the beam.

Care must be taken to cancel out this effect with the 3rd and 4th prisms.

Color varies across beam

0( ) exp[ ( ) ]E t i x t

Prism pairs are inside nearly every ultrafast laser, so we’re just asking for spatial chirp.

Spatial chirp is difficult to avoid.

Simply propagating through a tilted window causes spatial chirp!

Because ultrashort pulses are so broadband, this distortion is very noticeable—and problematic!

Color varies across beam

Different colors have different refractive indices and so have different refraction angles.

n()

Angular dispersion also causes pulse fronts to tilt.Phase fronts are perpendicular to the direction of propagation.

Because group velocity is usually less than phase velocity, pulse fronts tilt when light traverses a prism.

Pulse-front tilt and angular dispersion are manifestations of the same effect and their magnitudes are directly proportional to each other.

( ) ( )E t I t x

Angular dispersion causes pulse-front tilt even when group velocity is not involved.

Diffraction gratings also yield a pulse-front tilt.

Since gratings have about ten times the dispersion of prisms, they yield about ten times the tilt.

The path is simply shorter for rays that impinge on the near side of the grating. Of course, there’s angular dispersion, too.

Chirped mirror coatings offer an alternative to prisms and gratings for dispersion compensation.

Such mirrors avoid spatio-temporal effects, but they have limited GDD.

Longest wavelengths penetrate furthest.

Doesn’t work for < 600 nm

The required separation between prisms in a pulse compressor can be large.

The resulting negative GDD is proportional to the prism separation and the square of the dispersion.

It’s best to use highly dispersive glass, like SF10, or gratings.

Kafka and Baer, Opt. Lett., 12, 401 (1987)

Different prism materials

Compression of a 1-ps, 600-nm pulse with 10 nm of bandwidth (to about 50 fs).

Diffraction-grating pulse compressor

The grating pulse compressor also has negative second-order phase.

d2d 2

0

0

3

2c2d 2

Lsep

cos2 (' )

Lsep

where d = grating spacing(same for both gratings)

Grating #1

Grating #2

Note that, as in the prismpulse compressor, thelarger Lsep, the largerthe negative GDD.

Compensating 2nd and 3rd-order spectral phase

Use both a prism and a grating compressor. Since they have 3rd-orderterms with opposite signs, they can be used to achieve almost arbitrary amounts of both second- and third-order phase.

This design was used by Fork and Shank at Bell Labs in the mid 1980’sto achieve a 6-fs pulse, a record that stood for over a decade.

Grating compressorPrism compressor

input2 prism2 grating2 0

input3 prism3 grating3 0

Given the 2nd- and 3rd-order phases of the input pulse, input2 and input3,

solve simultaneous equations:

() = 2 L / = k() L

out( ) H () in ( )

We simply add spectral phases.

Spectral Phase and Optical Devices

H()˜ E in() ˜ E out()

Optical deviceor medium

The phase due to a medium is:k() L = k0 n() L

To account for dispersion, expand

k() = k() + 1[2 [

The Group-Velocity Dispersion (GVD)

210 0 0 0 02( ) ( ) ( ) ( ) ...k L k L k L k L

is the “group velocity dispersion.”

00

0

( )v ( )

k

00

1( )

v ( )g

k

1( )

vg

dk

d

1( )

vg

dk

d

The first few terms are all related to important quantities.The third one is new: the variation in group velocity with frequency: