Pulse propogation

Question 1

What are the two concpets which describe a pulse

Answer 1

Envelope and carrier frequency
also the pahses inside that for the wave

Question 2

Two equiv discription of waves

Answer 2

Temporal (time) and spatial(frequency)
related by FT

Question 3

what is wl and dw

Answer 3

Carriel firency
the mean frequency cnetre of pulse
deltaw is the frequency spectral width (bandwidth) typically the width is much smaller than th ecarrier feq

definesd such that E(Omega) is centred at the origin Omega = 0

Question 4

What is typically the DC component of a pulse’s E FIELD not I

Answer 4

0 since integral -inft to tinfty of E(t) dt = 0 sicne the pulse oscillated rapidly and the detector measures an avg E field (ofc the I will be non zero)

Question 5

What are the two concpets which describe a pulse

Answer 5

Envelope and carrier frequency
also the pahses inside that for the wave

Question 6

Length of an optical cycle

Answer 6

T = 2pi/wl
T approx 2 fs for VIS phtotos
slow varying envelope aopprox not always correct but we assume it is here

Question 7

Phase function phi(t)

Answer 7

Time dependent phase of the wave packet. wl is typically choisen to minimise the variation in phi(T)

Question 8

how do you get the time dependent carrier freq

Answer 8

You take the phase factor Gamma(t) from the E field:
E(t) = 1/2 Ep(t)exp(iGamma(t))
and you take its first derivative givving
w(t) = wl + d/dt(phi(t)) < this is basically an instatanous frequency
where phi(t) is the time dep phase factor of the wavepacket

Question 9

Explain the orignin of chirp

Answer 9

w(t) derived has a first deriv wrt time dep phase factor. If non zero d/dt then the carreir freq wl changes up or down over time as the pulse propogates.

dphi/dt = b if b!=0 then it represents a correction of the carrier freq

if b is a constant then thats fine no chirp sicne the wl stays wl+b over time, but if b = f(t), then the correction to wl changes with time and the corresp pulse is Frequency Modulated/Chirped

Question 10

How do you knwo if up or down chirped

Answer 10

If d2phi(t)/dt^2 + then the d/dt(phi(t)) + then
w(t) = wl + b >wl, so the carrier frequency increases -> UP chirp

Question 11

How does a doen chirped pulse look

Answer 11

Frequency of oscillations ver sweezed together at start of wavepacket and more spread out at the end fo the packet
the oscillation speed degreated with time since w(t) = wl + b where b is -

Question 12

How is Tp and delta wp def

Answer 12

The pulse duration (actual pulse duration)
FWHM of intensity profile |Ep(t)|^2

detla wp
FWHM of the spectral intensity |E(Omega)|^2

Question 13

Minimum duration bandwidth product

Answer 13

pulse duration Tp and bandwidth dwp cannot vary independently fo each other
dwp tp >= 2 pi cB
where CB is a const O(1) dependin gon actual pulse shape

Question 14

Unchirped pulses minimum duration bandwidth product

Answer 14

pulses not freq modulated are bandwidth limited or Foureir lim
these pulses excibit the shortest possible duration at a given spectral width and shape

Question 15

oulse duration for short pulses

Answer 15

In fs domain the exact pulse shape is difficult to determine
closest for single pulses experimentally is the autocorr function

Question 16

Why can the autocorr func not give you back the pulse phase

Answer 16

A(tau) is symmetric
therefore its Fourier transform is real so no info about the pulse shape can be extracted (E field is complex to repr oscillations)
no info regarding phase or coherence is contained in the autocorreleation

Question 17

Linear chirp

Answer 17

Is freq modulated
not bandwidth limited and so the pulse will NOT exhibit the shorted possible duration for its givne spectral width and pulse shape it will be longer than theoretically possible

Question 18

diff betwen tg and tp

Answer 18

Tp is the actual pulse shape
for fs very difficult to know, assuem gaussian but very rarely is that the case actually
tG is the gaussian FWHM pulse length and these two are related by
tp = sqrt{2ln2}Tg

so tp> tg

Question 19

How does the addition of chirp lead to a duration bandwidth product exceeding the Foureir lim

Answer 19

The occurrence of chirp (a != 0)results in additional spectral components which enlarge the spectral width by a factor sqrt{1+a^2}

note also that the spectral phase will be changed quadratically with dreq if the input pulse is linearly chirped

Question 20

How does the source term in maxwell eqn break up

Answer 20

P = Plin +Pnl
these are sources due to the reoisne of the material to the field

Question 21

how does X(t) relate to dispersion

Answer 21

For a non dip medium an infinite bandwidth where X(Omega) is contsant the reponse of the medium is instanoues and memory free.

More generally X(t) describes a finite reponse time of the medium. In the freq domin this mean s NONZERO dispersion

Question 22

Tay exp of dk around carrier freq wl

Answer 22

k(Omega) = k(wl) +dk
dk = dk/dOmega term (group veocity 1/Vg, kl’) and d2k/dOmega2 term (group velocity disp kl’’)

here again assuem delta k &laquo_space;kl

Question 23

Define Vg

Answer 23

Group velocity of the pulse
vg = (dk/dOmega)^-1 evaluated at wl
this is the new fram of ref we take on (retarded frame that moves with the pulse)

Question 24

Define GVD

Answer 24

Group velocity dispersion
kl” = deriv of 1/Vg wrt w i.e.
(d^2l/dOmega^2) eval at wl

it is a material property

Question 25

chirp rel to gvd

Answer 25

Chirp = material thickenss x GVD x bandwidth

Question 26

what happens with GVD = 0

Answer 26

kl’’ =0
so the pulse envelope does not change at all in the system of local coords
in the retarded ref frame that travels with vg, the pulse will always look idetical

Question 27

How do you solve prob with nonzero GVD

Answer 27

Either directly in time domain or in freq domain

Question 28

WHat do we see when we look at non zero GVD

Answer 28

The spectrum (in amplitude) of the pulse |E(omega,z|^2 remaoms constant
the spectral components resp for chirp must appear at the expense of the envelope shape which has to become broader

Question 29

For a guass pulse that is linearly chripsed and has GVD how can we get a zero y(Z) param

Answer 29

if chrip sign and GVD sign are opposite and balance withtheir factors in the y(Z) expression properly then we can et y(z) = 0

Question 30

How does chirp help you to get a shorter pulse

Answer 30

Chirp generally leads to broadeing, but + or - chirp can work to get a pulse shorter (PRISM THING?)

Question 31

what is phi(t)

Answer 31

The chirp param
it can start as zero and become nonzero (if its deriv or 2nd deriv is+ or -) and so a initially unchirped pulse can become come chirped

Question 32

How is a chirped gaussian pulse in time like a gaussian beam spatially

Answer 32

pulse duration -> beam waist (both incr with time/space)
slope of the chirp -> curvature of the G beam ( controls the thingie above)

A PULSE lienarly chirped is completely chracteriszed by its posisiton and min pulse length
just as
A SPATIALLY GAUSSIAN BEAM
is unquely defined by its position adn size of its waist