Pulse propogation Flashcards
What are the two concpets which describe a pulse
Envelope and carrier frequency
also the pahses inside that for the wave
Two equiv discription of waves
Temporal (time) and spatial(frequency)
related by FT
what is wl and dw
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
What is typically the DC component of a pulse’s E FIELD not I
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)
What are the two concpets which describe a pulse
Envelope and carrier frequency
also the pahses inside that for the wave
Length of an optical cycle
T = 2pi/wl
T approx 2 fs for VIS phtotos
slow varying envelope aopprox not always correct but we assume it is here
Phase function phi(t)
Time dependent phase of the wave packet. wl is typically choisen to minimise the variation in phi(T)
how do you get the time dependent carrier freq
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
Explain the orignin of chirp
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
How do you knwo if up or down chirped
If d2phi(t)/dt^2 + then the d/dt(phi(t)) + then
w(t) = wl + b >wl, so the carrier frequency increases -> UP chirp
How does a doen chirped pulse look
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 -
How is Tp and delta wp def
The pulse duration (actual pulse duration)
FWHM of intensity profile |Ep(t)|^2
detla wp
FWHM of the spectral intensity |E(Omega)|^2
Minimum duration bandwidth product
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
Unchirped pulses minimum duration bandwidth product
pulses not freq modulated are bandwidth limited or Foureir lim
these pulses excibit the shortest possible duration at a given spectral width and shape
oulse duration for short pulses
In fs domain the exact pulse shape is difficult to determine
closest for single pulses experimentally is the autocorr function
Why can the autocorr func not give you back the pulse phase
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
Linear chirp
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
diff betwen tg and tp
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
How does the addition of chirp lead to a duration bandwidth product exceeding the Foureir lim
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
How does the source term in maxwell eqn break up
P = Plin +Pnl
these are sources due to the reoisne of the material to the field
how does X(t) relate to dispersion
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
Tay exp of dk around carrier freq wl
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 «_space;kl
Define Vg
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)
Define GVD
Group velocity dispersion
kl” = deriv of 1/Vg wrt w i.e.
(d^2l/dOmega^2) eval at wl
it is a material property
chirp rel to gvd
Chirp = material thickenss x GVD x bandwidth
what happens with GVD = 0
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
How do you solve prob with nonzero GVD
Either directly in time domain or in freq domain
WHat do we see when we look at non zero GVD
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
For a guass pulse that is linearly chripsed and has GVD how can we get a zero y(Z) param
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
How does chirp help you to get a shorter pulse
Chirp generally leads to broadeing, but + or - chirp can work to get a pulse shorter (PRISM THING?)
what is phi(t)
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
How is a chirped gaussian pulse in time like a gaussian beam spatially
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
