The Spectrum

We define the spectrum, S(), of a wave E(t) to be:

2( ) { ( )}S E t F

This is the measure of the frequencies present in a light wave.

7. Usando el teorema de Rayleigh, calcular:

sen2 t

t2 dt

f t

0 , t T

2

1 , T

2t

T

2

0 , T2 t

ˆ f () T

2

sen(T

2)

T

2

f t 0 , t T

2

1 , t T2

ˆ f () T

2

sen(T

2)

T

2

f t 0 , t T

22T

, t T2

ˆ f () sen(

T

2)

T

2

f t 0 , t 12

, t 1

ˆ f ( )2

d

f (t)2dt

Rayleigh

f (t)2dt

2

dt 1

1

sen2 t

t2 dt

The Pulse Width

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 dtf

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 “second-order moment.”

The Full-Width-Half-Maximum

“Full-width-half-maximum” is 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 () similarly in the frequency domain (t ).

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 () has a minimum.

(Different definitions of the widths and the Fourier Transform yield different constants.)

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

(0) (0) (0)

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

(0) (0) (0)

Ft f t dt f t i t dt

f f f

fF d F i d

F F F

(0) (0)2

(0) (0)

f Ft

F f 2t 1t

Combining results:

or:

Define the widths assuming f(t) and F() peak at 0:

1 1( ) ( )

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

f F

The Time-Bandwidth Product

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

Time-Bandwidth Product (TBP)

t TBP

A pulse's TBP will always be greater than the theoretical minimumgiven 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 isthe theoretical minimum is called "Fourier-Transform Limited."

The coherence time (c = 1/)indicates the smallest temporal structure of the pulse.

In terms of the coherence time:

TBP = t = t / c

= about how many spikes are in the pulse

A similar argument can be made in the frequency domain, where theTBP is the ratio of the spectral width to the width of the smallestspectral structure.

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

t

c

I(t)A complicated

pulse

time

Temporal and Spectral Shapes

Parseval’s Theorem Parseval’s Theorem says that the energy is the same, whether you integrate over time or frequency:

Proof:

2 21( ) ( )

2f t dt F d

2

( ) ( ) *( )

1 1( exp( ) *( exp( )

2 2

1 1( ) *( ') exp( [ '] ) '

2 2

1 1( ) *( ') [2 ')] '

2 2

f t dt f t f t dt

F i t d F i t d dt

F F i t dt d d

F F d d

21 1

( ) *( ) ( )2 2

F F d F d

Time domain Frequency domain

f(t)

|f(t)|2

F()

|F()|2t

t

Parseval's Theorem in action

The two shaded areas (i.e., measures of the light pulse energy) are the same.

The Pulse Width

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 dtf

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 “second-order moment.”

The Full-Width-Half-Maximum

Full-width-half-maximum is 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 () similarly in the frequency domain (t ).

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 () has a minimum.

(Different definitions of the widths and the Fourier Transform yield different constants.)

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 f 2t 1t

Combining results:

or:

Define the widths assuming f(t) and F() peak at 0:

1 1( ) ( )

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

f F

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

(0) (0) (0)

fF d F i d

F F F

The Uncertainty Principle

For the rms width, t ≥ ½

There’s an uncertainty relation for x and k: k x ≥ ½

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[ ( )]( ) arctan

Re[ ( )]

E tt

E t

(t) = -Im{ ln[E(t)] }

The phase:

Equivalently,

(ti)

Re

ImE(ti)

√I(t i)

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

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

The spectral phase of a time-shifted pulse

( ) exp( ) ( )f t a i a F FRecall the Shift Theorem:

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

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

What is the spectral phase anyway?The spectral phase is the abs phase of each frequency in the wave-form.

t0

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

() = 0

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

And it has cancellation everywhere else.

1

2

3

4

5

6

Now try a linear spectral phase: () = a.

t

(1) = 0

(2) = 0.2

(3) = 0.4

(4) = 0.6

(5) = 0.8

(6) =

1

2

3

4

5

6

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

And this is what occurs!

The spectral phase distinguishes a light bulb from an ultrashort pulse.

Complex Lorentzianand its Intensity and Phase

L( ) 1

a i

1

a ia ia i

a

a2 2 i

a2 2

2 2

1Intensity( )

a

Im[ ( )]Phase( ) arctan arctan( / )

Re[ ( )]

La

L

0

Imaginarycomponent

Realcomponent

0

a

Real part Imag part

Intensity and Phase of a decaying exponential and its Fourier transform

Time domain:

Frequency domain:

(solid)

Light has intensity and phase also.

A light wave has the time-domain electric field:

Intensity Phase

Equivalently, vs. frequency:

Spectral Phase

(neglecting thenegative-frequency

component)

Spectrum

Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the light wave.

0( ) Re exp ( )( ) ( )E t iI tt t

( )( ) exp ( )SE i

The minus signs are just conventions.

We usually extract out the carrier frequency.

Fourier Transform with respect to space

F {f(x)} = F(k)

( ) ( ) exp( )F k f x ikx dx

If f(x) is a function of position,

We refer to k as the “spatial frequency.”

Everything we’ve said about Fourier transforms between the t and domains also applies to the x and k domains.

x

k

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.

The Shah Function

The Shah function, III(t), is an infinitely long train of equally spaced delta-functions.

t

III( ) ( )m

t t m

The Fourier Transform of the Shah Function

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

{III( )} III(t F

) exp( )

)exp( )

exp( )

m

m

m

t m i t dt

t m i t dt

i m

t

III(t)

F {III(t)}

2

The Shah Function and a pulse train

( )m

f t mT

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

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

Set t’ /T = m or t’ = mT

An infinite train of identical pulses (from a laser!) can be written:

( / ) ( )m

t T m f t t dt

An infinite train of identical pulses can be written:

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

where f(t) represents a single pulse and T is the time between pulses. The Convolution Theorem states that the Fourier Transform of a convolution is the product of the Fourier Transforms. So:

The Fourier Transform of an Infinite Train of Pulses

( )

III( / ) (2

E

FT

If this train of pulses results from a single pulse bouncing back and forth inside a laser cavity of round-trip time T. The spacing between frequencies is then = /T or = 1/T.

The Fourier Transform of a Finite Pulse Train

A finite train of identical pulses can be written:

( ) {III( / ) ( )} ( )E t t T g t f t

( ) {III( / 2 ) ( )} ( )E T G F

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

A laser’s frequencies are often called “longitudinal modes.”They’re separated by 1/T = c/2L.

Which modes lase depends on the gain profile.

Frequency

Inte

nsity

Here, additional narrowband filtering has yielded a single mode.

Laser Modes

The 2D generalization of the Shah function: “The Bed of Nails” function

We won’t do anything with this function, but I thought you might like this colorful image… Can you guess what its Fourier transform is?