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Ultrashort Laser Pulses I Description of pulses Intensity and phase The instantaneous frequency and group delay Zero th and first-order phase The linearly chirped Gaussian pulse Prof. Rick Trebino Georgia Tech www.frog.gatech.edu

Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

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Page 1: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Ultrashort Laser Pulses I

Description of pulses

Intensity and phase

The instantaneous frequency and group delay

Zeroth and first-order phase

The linearly chirped Gaussian pulse

Prof. Rick TrebinoGeorgia Techwww.frog.gatech.edu

Page 2: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

An ultrashort laser pulse has an intensity and phase vs. time.

102

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

Neglecting the spatial dependence for now, the pulse electric field is given by:

Intensity PhaseCarrierfrequency

A sharply peaked function for the intensity yields an ultrashort pulse.The phase tells us the color evolution of the pulse in time.

( )I t

Ele

ctr

ic fie

ld X

(t)

Time [fs]

Page 3: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The real and complex pulse amplitudes

( ) exp{ }( )( )E I tt i tφ∝ −

Removing the 1/2, the c.c., and the exponential factor with the carrier frequency yields the complex amplitude, E(t), of the pulse:

This removes the rapidly varying part of the pulse electric field and yields a complex quantity, which is actually easier to calculate with.

( )I t is often called the real amplitude, A(t), of the pulse.

( )I t

Ele

ctr

ic fie

ld X

(t)

Time [fs]

Page 4: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The Gaussian pulse

where τHW1/e is the field half-width-half-maximum, and τFWHM is the intensity full-width-half-maximum.

The intensity is:

2

0 1/

2

0

2

0

( ) exp ( / )

exp 2ln 2( / )

exp 1.38( / )

HW e

FWHM

FWHM

E t E t

E t

E t

τ

τ

τ

= −

= −

= −

For almost all calculations, a good first approximation for any ultrashort pulse is the Gaussian pulse (with zero phase).

2 2

0

2 2

0

( ) exp 4ln 2( / )

exp 2.76( / )

FWHM

FWHM

I t E t

E t

τ

τ

∝ −

∝ −

Page 5: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Intensity vs. amplitude

The intensity of a Gaussian pulse is √2 shorter than its real amplitude. This factor varies from pulse shape to pulse shape.

The phase

of this pulse

is constant, φ(t) = 0,

and is not

plotted

here.

Page 6: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

It’s easy to go back and forth between the electric field and the intensity and phase:

The intensity:

Calculating the intensity and the phase

Im[ ( )]arcta( ) n

Re[ ( )]

E tt

E tφ

= −

φ(t) = − Im{ln[E(t)]}

The phase:

Equivalently,

−φ(ti)

Re

ImE(ti)

I(t) = |E(t)|2

Also, we’ll stop writing “proportional to” in these expressions and take E, X, I, and S to be the field, intensity,

and spectrum dimensionless

shapes vs. time.

Page 7: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The Fourier Transform

To think about ultrashort laser pulses, the Fourier Transform is essential.

( ) ( ) exp( )t i t dtω ω∞

−∞= −∫%X X

1( ) ( ) exp( )

2t i t dω ω ω

π∞

−∞= ∫ %X X

We always perform Fourier transforms on the real or complex pulse electric field, and not the intensity, unless otherwise specified.

Page 8: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

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

ω

ϕ

ω ϕ

ω ωω ω

ω ω

= − +

− − + − −

− −%X

The frequency-domain electric field has positive- and negative-frequency components.

Note that φ and ϕ are different!

Note that these two terms

are not complex

conjugates of each other

because the FT integral is

the same for each!

Page 9: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The complex frequency-domain pulse field

Since the negative-frequency component contains the same infor-mation as the positive-frequency component, we usually neglect it.

We also center the pulse on its actual frequency, not zero.

So the most commonly used complex frequency-domain pulse field is:

Thus, the frequency-domain electric field also has an intensity and phase.

S is the spectrum, and ϕ is the spectral phase.

( ) exp{ ( )( ) }S iω ωω ϕ≡ −%X

Page 10: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The spectrum with and without the carrier frequency

Fourier transforming X (t) and E(t) yields different functions.

We usually use just this component.

( )E ω% ( )ω%X

Page 11: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The spectrum and spectral phase

The spectrum and spectral phase are obtained from the frequency-domain field the same way the intensity and phaseare from the time-domain electric field.

Im[ ( )]arctan

Re[)

( )](

ωω

ϕ ω = −

%

%

X

X

{ }Im ln[( ) ( )]ϕ ω ω= − %X

or

2

( () )S ωω = %X

Page 12: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Intensity and phase of a Gaussian

The Gaussian is real, so its phase is zero.

Time domain:

Frequency domain:

So the spectral phase

is zero, too.

A Gaussian

transforms

to a Gaussian

Intensity and Phase

Spectrum and Spectral Phase

Page 13: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The spectral phase of a time-shifted pulse

{ }( ) exp( ) ( )f t a i a Fω ω− = −YRecall the Shift Theorem:

So a time-shift simply adds some linear spectral phase to the pulse!

Time-shifted Gaussian pulse (with a flat phase):

Intensity and Phase

Spectrum and Spectral Phase

Page 14: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

What is the spectral phase?

The spectral phase is the phase of each frequency in the wave-form.

t0

All of these frequencies have zero phase. So this pulse has:

ϕ(ω) = 0

Note that this wave-form sees constructive interference, and hence peaks, at t = 0.

And it has cancellation everywhere else.

ω1

ω2

ω3

ω4

ω5

ω6

Page 15: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Now try a linear spectral phase: ϕ(ω) = aω.

By the Shift Theorem, a linear spectral phase is just a delay in time.

And this is what occurs!

t

ϕ(ω1) = 0

ϕ(ω2) = 0.2 π

ϕ(ω3) = 0.4 π

ϕ(ω4) = 0.6 π

ϕ(ω5) = 0.8 π

ϕ(ω6) = π

Page 16: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

To transform the spectrum, note that the energy is the same, whether we integrate the spectrum over frequency or wavelength:

Transforming between wavelength and frequency

The spectrum and spectral phase vs. frequency differ from the spectrum and spectral phase vs. wavelength.

ϕλ(λ) = ϕω (2π c / λ)

( ) ( )S d S dλ ωλ λ ω ω∞ ∞

−∞ −∞=∫ ∫

Changing variables:

= Sω (2π c / λ)2π c

λ2 dλ−∞

Sλ (λ) = Sω (2π c / λ)2πc

λ2

The spectral phase is easily transformed:

2

2(2 / )

cS c dω

ππ λ λλ

−∞

−= ∫

2

2d c

d

ω πλ λ

−=

2 cπωλ

=

Page 17: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The spectrum and spectral phase vs. wavelength and frequency

Example: A Gaussian spectrum with a linear spectral phase vs. frequency

vs. Frequency vs. Wavelength

Note the different shapes of the spectrum and spectral phasewhen plotted vs. wavelength and frequency.

Page 18: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Bandwidth in various units

(1/ )cδν δ λ⇒ =

In frequency, by the Uncertainty Principle, a 1-ps pulse has bandwidth:

δν = ~1/2 THz

cνλ

= (1/ ) / cδ λ δν⇒ =

So δ(1/λ) = (0.5 × 1012 /s) / (3 × 1010 cm/s) or: δ(1/λ) = 17 cm-1

In wavelength:

4 1(800 nm)(.8 10 cm)(17 cm )δλ − −×=Assuming an 800-nm wavelength:

using δν δt ∼ ½

2

1(1/ )δ λ δλ

λ−= 2 (1/ )δλ λ δ λ⇒ =

or: δλ = 1 nm

In wave numbers (cm-1), we can write:

Page 19: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The temporal phase, φ(t), contains frequency-vs.-time information.

The pulse instantaneous angular frequency, ωinst(t), is defined as:

The Instantaneous frequency

0( )inst

dt

dt

φω ω≡ −

This is easy to see. At some time, t, consider the total phase of the

wave. Call this quantity φ0:

Exactly one period, T, later, the total phase will (by definition) increase

to φ0 + 2π:

where φ(t+T) is the slowly varying phase at the time, t+T. Subtracting these two equations:

0 0 ( )t tφ ω φ= −

0 02 [ ] ( )t T t Tφ π ω φ+ = ⋅ + − +

02 [ ( ) ( )]T t T tπ ω φ φ= − + −

Page 20: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Dividing by T and recognizing that 2π/T is a frequency, call it ωinst(t):

ωinst(t) = 2π/T = ω0 – [φ(t+T) – φ(t)] / T

But T is small, so [φ(t+T) – φ(t)] /T is the derivative, dφ /dt.

So we’re done!

Usually, however, we’ll think in terms of the instantaneous

frequency, νinst(t), so we’ll need to divide by 2π:

νinst(t) = ν0 – (dφ /dt) / 2π

While the instantaneous frequency isn’t always a rigorous quantity,

it’s fine for ultrashort pulses, which have broad bandwidths.

Instantaneous frequency (cont’d)

Page 21: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Group delay

While the temporal phase contains frequency-vs.-time information,the spectral phase contains time-vs.-frequency information.

So we can define the group delay vs. frequency, tgr(ω), given by:

tgr(ω) = dϕ / dω

A similar derivation to that for the instantaneous frequency can show that this definition is reasonable.

Also, we’ll typically use this result, which is a real time (the rad’s cancel out), and never dϕ/dν, which isn’t.

Always remember that tgr(ω) is not the inverse of ωinst(t).

Page 22: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Phase wrapping and unwrapping

Technically, the phase ranges from –π to π. But it often helps to “unwrap” it. This involves adding or subtracting 2π whenever there’s a 2π phase jump.

Example: a pulse with quadratic phase

Wrapped phase Unwrapped phase

The main reason for unwrapping the phase is aesthetics.

Note the scale!

Page 23: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Phase-blanking

When the intensity is zero, the phase is meaningless.

When the intensity is nearly zero, the phase is nearly meaningless.

Phase-blanking involves simply not plotting the phase when the intensity is close to zero.

The only problem with phase-blanking is that you have to decide the intensity level below which the phase is meaningless.

−φ(ωi)

Re

ImE(ωi)

Without phase blanking

Time or Frequency

With phase blanking

Time or Frequency

Page 24: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Phase Taylor Series expansions

We can write a Taylor series for the phase, φ(t), about the time t = 0:

where

where 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(t).

Expanding the phase in time is not common because it’s hard to measure the intensity vs. time, so we’d have to expand it, too.

2

0 1 2( ) ...1! 2!

t ttφ φ φ φ= + + +

1

0t

d

dt

φφ=

= is related to the instantaneous frequency.

Page 25: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Frequency-domain phase expansion

( )2

000 1 2( ) ...

1! 2!

ω ωω ωϕ ω ϕ ϕ ϕ−−= + + +

It’s more common to write a 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!

0

2

2 2

d

d ω ω

ϕϕω =

= is called the group-delay dispersion.

Page 26: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Zeroth-order phase: the absolute phase

The absolute phase is the same in both the time and frequency domains.

An absolute phase of π/2 will turn a cosine carrier wave into a sine.

It’s usually irrelevant, unless the pulse is only a cycle or so long.

Different absolute phases

for a single-cycle pulse

Notice that the two four-cycle pulses look alike, but the three single-cycle pulses are all quite different.

f (t)exp(iφ0 ) ⊃ F(ω )exp(iφ0 )

Different absolute phases

for a four-cycle pulse

Page 27: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

First-order phase in frequency: a shift in time

By the Fourier-transform Shift Theorem, f (t − ϕ1) ⊃ F (ω)exp(iωϕ1)

Time domain Frequency domain

ϕ1 = 0

ϕ1 = − 20 fs

Note that ϕ1 does not affect the instantaneous frequency, but the group delay = ϕ1.

Page 28: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

First-order phase in time: a frequency shift

By the Inverse-Fourier-transform Shift Theorem,

1 1( ) ( ) exp( )F f t i tω φ φ− ⊂ −

Time domain Frequency domain

1 0 /φ = fs

φ1 = − .07 / fs

Note that φ1 does not affect the group delay, but it does affect the

instantaneous frequency = –φ1.

Page 29: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

Second-order phase: the linearly chirped pulse

A pulse can have a frequency that varies in time.

This pulse increases its frequency linearly in time (from red to blue).

In analogy to bird sounds, this pulse is called a chirped pulse.

Page 30: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

( )2 2

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

The Linearly Chirped Gaussian Pulse

We can write a linearly chirped Gaussian pulse mathematically as:

ChirpGaussian

amplitude

Carrierwave

Note that for β > 0, when t < 0, the two terms partially cancel, so the phase changes slowly with time (so the frequency is low).And when t > 0, the terms add, and the phase changes more rapidly

(so the frequency is larger).

Page 31: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The instantaneous frequencyvs. time for a chirped pulse

A chirped pulse has:

where:

The instantaneous frequency is:

which is:

So the frequency increases linearly with time.

( )0( ) exp ( )E t i t tω φ∝ −

2( )t tφ β= −

0( ) /inst t d dtω ω φ≡ −

0( ) 2inst t tω ω β= +

Page 32: Ultrashort Laser Pulses I - Brown University...Phase Taylor Series expansions We can write a Taylor series for the phase, φ(t), about the time t = 0: where where only the first few

The Negatively Chirped Pulse

We have been considering a pulse whose frequency increaseslinearly with time: a positively chirped pulse.

One can also have a negativelychirped (Gaussian) pulse, whose instantaneous frequency decreases with time.

We simply allow β to be negativein the expression for the pulse:

And the instantaneous frequency will decrease with time:

( ) ( )2 2

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

0 0( ) 2 2inst t t tω ω β ω β= + = −