Ultrashort Laser Pulses II

Relative importance of spectrum and spectral phase

More second-order phase

Higher-order spectral phase distortions

Pulse and spectral widths

Time-bandwidth productProf. Rick TrebinoGeorgia Techwww.frog.gatech.edu

The frequency-domain electric field

The frequency-domain equivalents of the intensity and phaseare the spectrum and spectral phase.

Fourier-transforming the pulse electric field:

102

( ) exp{ [ ]}( ) .() .t i t c ctI t φω= − +X

yields:

12

1

0

0 02

0( ) exp{ [ ]}

( ) exp{ [ ( )]

( ( )

}

) i

S i

Sω

ω

ϕ

ω ϕ

ω ωω ω

ω ω

= − +

− − + − −

− −%X

Note that φ and ϕ are different!

The spectral phase is the key quantity in determining the shape and duration of a short pulse.

The relative importance of intensity and phase

Photographs of Rick and Linda:

Composite photograph

made using the

spectral intensity of

Linda’s photo and the

spectral phase of

Rick’s (and inverse-

Fourier-transforming)

Composite photograph

made using the

spectral intensity of

Rick’s photo and the

spectral phase of

Linda’s (and inverse-

Fourier-transforming)

The spectral phase is more important for determining the intensity!

Frequency-domain phase expansion

( )2

000 1 2( ) ...

1! 2!

ω ωω ωϕ ω ϕ ϕ ϕ−−= + + +

Recall the Taylor series for ϕ(ω):

As in the time domain, only the first few terms are typically required to

describe well-behaved pulses. Of course, we’ll consider badly behaved

pulses, which have higher-order terms in ϕ(ω).

0

1

d

d ω ω

ϕϕω =

=where

is the group delay tgr.

0

2

2 2

d

d ω ω

ϕϕω =

= is called the “group-delay dispersion.”

The Fourier transformof a chirped pulse

Writing a linearly chirped Gaussian pulse as:

or:A Gaussian with

a complex width!

( )2 2

0 0( ) exp exp .t E t i t t c cα ω β ∝ − + + X

( ) ( )2

0 0( ) exp exp . .t E i t i t c cα β ω ∝ − − + X

where 21/ tα ∝ ∆

A chirped Gaussian pulseFourier-Transforms to

another Gaussian.

( )2

0 0

1/ 4( ) expE E

iω ω ω

α β ∝ − − −

%

( ) ( )2 2

0 0 02 2 2 2

/ 4 / 4( ) exp expE E i

α βω ω ω ω ωα β α β

∝ − − − − + + %

neglecting the negative-frequency

term due to the c.c.Fourier-Transforming yields:

Rationalizing the denominator and separating the real and imaginary parts:

The group delay vs. ω for a chirped pulse

The group delay of a wave is the derivative of the spectral phase:

( ) /gr

t d dω ϕ ω≡

( )2

02 2

/ 4( )

βϕ ω ω ωα β

= −+

For a linearly chirped Gaussian pulse, the spectral phase is:

( )02 2

/ 2 grt

β ω ωα β

= −+

So: The delay vs.

frequency is linear.

But when the pulse is long (α 0):

which is the same result as ωinst(t).

( )0

1

2grt ω ω

β= −

0( ) 2inst t tω ω β= +This is not the inverse of the instantaneous frequency, which is:

2nd-order phase: positive linear chirp

Numerical example: Gaussian-intensity pulse w/ positive linear chirp, ϕ2 = 14.5 rad fs2.

Here the quadratic phase has stretched what would have been a 3-fs pulse (given the spectrum) to a 13.9-fs one.

2nd-order phase: negative linear chirp

Numerical example: Gaussian-intensity pulse w/ negative linear chirp, ϕ2 = –14.5 rad fs2.

As with positive chirp, the quadratic phase has stretched what would have been a 3-fs pulse (given the spectrum) to a 13.9-fs one.

The frequency of a light wave can also vary nonlinearly with time.

This is the electric field of aGaussian pulse whose fre-quency varies quadraticallywith time:

This light wave has the expression:

Arbitrarily complex frequency-vs.-time behavior is possible.

But we usually describe phase distortions in the frequency domain.

Nonlinearly chirped pulses

( ) ( )2 3

0 0( ) Re exp / expGE t E t i t tτ ω γ = − +

2

0( ) 3inst t tω ω γ= +

E-field vs. time

X (t)

3rd-order spectral phase: quadratic chirp

Longer and shorter wavelengths coincide in time and interfere (beat).

Trailing satellite pulses in time indicate positive spectral cubic phase, and leading ones indicate negative spectral cubic phase.

S(ω)

tg(ω)ϕ(ω)

Spectrum and spectral phase

400 500 600 700

Because we’re plotting vs. wavelength (not frequency),

there’s a minus sign in the group delay, so the plot is correct.

3rd-order spectral phase: quadratic chirp

Numerical example: Gaussian spectrum and positive cubic spectral phase, with ϕ3 = 750 rad fs3

Trailing satellite pulses in time indicate positive spectral cubic phase.

Negative 3rd-order spectral phase

Another numerical example: Gaussian spectrum and negative cubic spectral phase, with ϕ3 = –750 rad fs3

Leading satellite pulses in time indicate negative spectral cubic phase.

4th-order spectral phase

Numerical example: Gaussian spectrum and positive quartic spectral phase, ϕ4 = 5000 rad fs4.

Leading and trailing wings in time indicate quartic phase. Higher-frequencies in the trailing wing mean positive quartic phase.

Negative 4th-order spectral phase

Numerical example: Gaussian spectrum and negative quartic spectral phase, ϕ4 = –5000 rad fs4.

Leading and trailing wings in time indicate quartic phase. Higher-frequencies in the leading wing mean negative quartic phase.

5th-order spectral phase

Numerical example: Gaussian spectrum and positive quintic spectral phase, ϕ5 = 4.4×104 rad fs5.

An oscillatory trailing wing in time indicates positive quintic phase.

Negative 5th-order spectral phase

Numerical example: Gaussian spectrum and negative quintic spectral phase, ϕ5 = –4.4×104 rad fs5.

An oscillatory leading wing in time indicates negative quintic phase.

The pulse length

There are many definitions of the width or length of a wave or pulse.

The effective width is the width of a rectangle whose height and area are the same as those of the pulse.

Effective width ≡ Area / height:

Advantage: It’s easy to understand.Disadvantages: The Abs value is inconvenient.

We must integrate to ± ∞.

1( )

(0)efft f t dt

f

∞

−∞

∆ ≡ ∫t

f(0)

0

∆teff

t

∆t

(Abs value is unnecessary for intensity.)

The rms pulse width

The root-mean-squared width or rms width:

Advantages: Integrals are often easy to do analytically.Disadvantages: It weights wings even more heavily,so it’s difficult to use for experiments, which can't scan to ± .

1/ 2

2 ( )

( )

rms

t f t dt

t

f t dt

∞

−∞∞

−∞

∆ ≡

∫

∫

∞

t

∆t

The rms width is the normalized second-order moment.

The Full-Width-Half-Maximum

Full-width-half-maximumis the distance between the half-maximum points.

Advantages: Experimentally easy.Disadvantages: It ignores satellite pulses with heights < 49.99% of the peak!

Also: we can define these widths in terms of f(t) or of its intensity, |f(t)|2.Define spectral widths (∆ω) similarly in the frequency domain (t → ω).

t

∆tFWHM

1

0.5

t

∆tFWHM

The Uncertainty Principle

The Uncertainty Principle says that the product of a function's widthsin the time domain (∆t) and the frequency domain (∆ω) has a minimum.

1 1 (0)( ) ( ) exp[ (0) ]

(0) (0) (0)

Ft f t dt f t i t dt

f f f

∞ ∞

−∞ −∞

∆ ≥ = − =∫ ∫

(0) (0)2

(0) (0)

f Ft

F fω π∆ ∆ ≥ 2tω π∆ ∆ ≥ 1tν∆ ∆ ≥

Combining results:

or:

Use effective widths assuming f(t) and

F(ω) peak at 0:

1 1( ) ( )

(0) (0)t f t dt F d

f Fω ω ω

∞ ∞

−∞ −∞

∆ ≡ ∆ ≡∫ ∫

1 1 2 (0)( ) ( ) exp (

(0) (0) (0)

fF d F i d

F F F

πω ω ω ω ω ω∞ ∞

−∞ −∞

∆ ≥ = [ 0)] =∫ ∫

1

Other width definitions yield

slightly different numbers.

For a given wave, the product of the time-domain width (∆t) and

the frequency-domain width (∆ν) is the

Time-Bandwidth Product (TBP)

∆ν ∆t ≡ TBP

A pulse's TBP will always be greater than the theoretical minimum

given by the Uncertainty Principle (for the appropriate width definition).

The TBP is a measure of how complex a wave or pulse is.

Even though every pulse's time-domain and frequency-domain

functions are related by the Fourier Transform, a wave whose TBP is

the theoretical minimum is called "Fourier-Transform Limited."

The Time-Bandwidth Product

The coherence time (τc = 1/∆ν)

indicates the smallest temporal

structure of the pulse.

In terms of the coherence time:

TBP = ∆ν ∆t = ∆t / τc

= about how many spikes are in the pulse

A similar argument can be made in the frequency domain, where the

TBP is the ratio of the spectral width to the width of the smallest

spectral structure.

The Time-Bandwidth Product is a measure of the pulse complexity.

∆t

τc

I(t)A complicated

pulse

time

Temporal and spectral shapes and TBPs of simple ultrashort pulses

Diels and Rudolph,

Femtosecond

Phenomena

Time-Bandwidth Product

For the angular frequency and different definitions of the widths:

TBPrms = 0.5 TBPeff = 3.14

TBPHW1/e = 1 TBPFWHM = 2.76

Divide by 2π for the cyclical frequency ∆trms ∆νrms, etc.

Numerical example: A transform-limited pulse: A Gaussian-intensity pulse with constant phase and minimal TBP.

Notice that this definition yields

an uncertainty product of π, not 2π; this is because we’ve used the intensity and spectrum here, not the fields.

Time-Bandwidth Product

For the angular frequency and different definitions of the widths:

TBPrms = 6.09 TBPeff = 4.02TBPHW1/e = 0.82 TBPFWHM = 2.57

Divide by 2π for ∆trms ∆νrms, etc.

Numerical example: A variable-phase, variable-intensity pulse with a fairly small TBP.

Time-Bandwidth Product

For the angular frequency and different definitions of the widths:

TBPrms = 32.9 TBPeff = 10.7

TBPHW1/e = 35.2 TBPFWHM = 116

Divide by 2π for ∆trms ∆νrms, etc.

Numerical example: A variable-phase, variable-intensity pulse with a larger TBP.

A linearly chirped pulse with no structure can

also have a large time-bandwidth product.

For the angular frequency and different definitions of the widths:

TBPrms = 5.65 TBPeff = 35.5

TBPHW1/e = 11.3 TBPFWHM = 31.3

Divide by 2π for ∆trms ∆νrms, etc.

Numerical example: A highly chirped, relatively long Gaussian-intensity pulse with a large TBP.

The shortest pulse for a given spectrum has a constant spectral phase.

We can write the pulse width in a way that illustrates the relative contributions to it by the spectrum and spectral phase.

If B(ω) = √S(ω), then the temporal width, ∆trms, is given by:

2 2 2 2( ) ( ) ( )rmst B d B dω ω ω ϕ ω ω

∞ ∞

−∞ −∞′ ′∆ = +∫ ∫

Notice that variations in the spectral phase can only increase the pulse width.

Contribution due to

variations in the

spectrum

Contribution due to

variations in the

spectral phase

Note: this result

assumes that

the mean group

delay has been

subtracted from

ϕ’. That is, the

pulse is cen-tered at tgr = 0.

We can write the pulse width in a way that illustrates the relative contributions to it by the spectrum and spectral phase.

If B(ω) = √S(ω), then the temporal width, ∆trms, is given by:

The narrowest spectrum for a given intensity has a constant temporal phase.

2 2 2 2( ) ( ) ( )rms

A t dt A t t dtω φ∞ ∞

−∞ −∞′ ′∆ = +∫ ∫

Notice that variations in the phase can only increase the spectral width.

Contribution due to

variations in the

intensity

Contribution due to

variations in the

phase

We can also write the spectral width in a way that illustrates the relative contributions to it by the intensity and phase.

If A(t) = √I(t), then the spectral width, ∆ωrms, is given by:

Pulse propagation

What happens to a pulse as it propagates through a medium?

Always model (linear) propagation in the frequency domain. Of course,

you must know the entire field (i.e., the intensity and phase) to do so.

( ) ( ) exp[ ( ) / 2] exp[ ( ) ]out inE E L i n k Lω ω α ω ω= − −% %

In the time domain, propagation is a convolution—much harder.

( )inE ω% ( )outE ω%( )

( )n

α ωω

ω ω

( ) ( ) exp[ ( ) ]out inS S Lω ω α ω= −

( ) ( ) ( )out in n Lc

ωϕ ω ϕ ω ω= +

Pulse propagation(continued)

( ) ( ) exp[ ( ) / 2] exp[ ( ) ]out inE E L i n k Lω ω α ω ω= − −% %

( ) exp[ ( ) / 2] exp[ ( ) ]inE L i n Lc

ωω α ω ω= − −%using k = ω /c:

Separating out the spectrum and spectral phase:

Rewriting this expression:

( )inE ω% ( )outE ω%

Next lecture: what does pulse propagation in a medium do to short pulses?