Homework 5

useful constants and definitions

constants and unit definitions: internally, all numbers in SI

ps=(10^(-12));fs=(10^(-15));
mm=(10^(-3));cm=(10^(-2));cm3=(10^6);um=(10^(-6));nm=(10^(-9));angs=(10^(-10));pm=(10^(-12));
mJ=(10^(-3));uJ=(10^(-6));
c=2.9979 10^8;                          (*speed of light,in m/s*)
eps0=8.8542 (10^(-12));            (*permittivity of free space*)
e=1.6 (10^(-19));(*Coul electron charge*)

refractive index for BK7: nbk7

nbk7[lambda_]:=√(2.2718929-(1.0108077 (lambda/um)^2)/10^2+1.0592509/(10^2 (lambda/um)^2)+2.0816965/(10^4 (lambda/um)^4)-7.6472538/(10^6 (lambda/um)^6)+4.9240991/(10^7 (lambda/um)^8));

refractive index for fused silica: nSiO2

as1=.6961663;
as2=.4079426;
as3=.8974794;
bs1=.0684043^2;
bs2=.1162414^2;
bs3=9.896161^2;
nSiO2[lambda_]:=√(1+(as1 (lambda/um)^2)/((lambda/um)^2-bs1)+(as2 (lambda/um)^2)/((lambda/um)^2-bs2)+(as3 (lambda/um)^2)/((lambda/um)^2-bs3))

Define FFT functions for temporal domain

FFT[fLst_,npts_,dt_]:=(Sqrt[npts (dt^2)])RotateLeft[Fourier[RotateRight[fLst,npts/2]],npts/2];
InverseFFT[FLst_,npts_,dt_]:=(Sqrt[(1/(npts (dt^2)))])RotateLeft[InverseFourier[RotateRight[FLst,npts/2]],npts/2];

Self-phase modulation and spectral broadening

At high intensity, the bound electrons in The refractive index index has a nonlinear component which can be represented as 
n(I)=(Subscript[n, 0])+(Subscript[n, 2])I

The intensity of the beam must be high for the nonlinear component to make a difference in the refractive index. However, it doesn’t need to be so high that (Subscript[n, 2])I ≈ (Subscript[n, 0]). What matters is the nonlinear phase shift: (Subscript[ϕ, NL])= ((Subscript[ω, 0])/c)(Subscript[n, 2])I L, where L is the propagation length through the medium. This quantity is known as the B-integral, since in the general case where the intensity depends on z, 
B=(Subscript[ϕ, NL])= ((Subscript[ω, 0])/c)(Subscript[n, 2])(∫_0^L)I(z)dz .   
Because of dispersion, the duration of the pulse can also change during the propagation of the pulse. 

For a laser pulse, the intensity varies in time, and the pulse experiences a nonlinear, time-dependent phase shift: 
(Subscript[E, out])(t)=(Subscript[E, in])(t)exp[ i (Subscript[ϕ, NL])(t)] , where (Subscript[ϕ, NL])(t)= ((Subscript[ω, 0])/c)(Subscript[n, 2])I(t) L. 
This process is known as self-phase modulation (SPM).  The time-dependent phase leads to the generation of new frequency components and broadens the spectrum pulse. In turn, the broadened spectrum can be further compressed to get a shorter pulse duration. 

Below we will step through the design of an experiment where we can investigate this pulse compression process.

1. Experiment design: dispersion length

First we need to do some back-of-the-envelope calculations to determine what experimental conditions we would need to observe spectral broadening from SPM. 

The dispersion length is the propagation distance in the window that results in stretching of the pulse duration by approximately (Sqrt[2])(Subscript[τ, p]). From our calculation of the propagation of a Gaussian pulse through a dispersive material, we found that the dispersion length is 
(Subscript[L, D])=(((Subscript[τ, 0])^2)/(2 (Subscript[β, 2]))), where (Subscript[β, 2])(ω)=((∂_ω)^2) ((ω/c)n(ω)) 
(Note that in my scanned notes, I called this (Subscript[k, 2]). There are other definitions that are different from this by a factor of 2, but we’ll use this version.)

a. For fused silica and BK7 glass, plot (Subscript[β, 2])(λ) over the range 400nm to 1800nm, both curves on the same plot. 
Next, assuming that the laser pulse duration is (Subscript[τ, p])=50fs (1/(e^2) half-width).

b. Plot (Subscript[L, D])(λ) over the same wavelength range for both glasses on the same plot. Near the zero dispersion point, the dispersion length goes to infinity, and third-order dispersion becomes the next-most important order of dispersion.

c. What is the dispersion length for λ = 800nm in fused silica? For the following sections, we’ll pick an interaction length to be L=(Subscript[L, D])/2 so that dispersion is not a major issue.

2. Experiment design: nonlinear length

Now we need to estimate the spot size and pulse energy to get sufficient intensity to see a nonlinear effect. We want the beam to be approximately collimated throughout the length of the sample.

a. Calculate the minimum focal spot size ((Subscript[w, 0]),intensity 1/(e^2) radius), so that the confocal parameter is equal to the window thickness. (The confocal parameter is 2(Subscript[z, R]), where (Subscript[z, R])=((π (Subscript[n, 0])((Subscript[w, 0])^2))/λ) is the Rayleigh range).

b. Given this minimum spot radius, we can estimate the intensity as I=(E/(τ π ((Subscript[w, 0])^2))). For future use, write a function that calculates corresponding intensity as a function of the pulse duration and energy.

c. Now calculate the approximate pulse energy such that the B-integral is equal to 1.  (B = ((Subscript[ω, 0])/c)(Subscript[n, 2])I L). In fused silica, the nonlinear coefficient has been measured to be (Subscript[n, 2])=2.5(x10^(-16))(cm^2)/W. Also then calculate the intensity for the focused beam under these conditions in W/(cm^2).

For this spot radius, this will be the minimum pulse energy we need to see an effect. For comparison, an ultrafast oscillator is typically 200mW average power at a repetition rate of 76MHz, so the energy per pulse would be about 2.5nJ. A typical ultrafast amplifier with a repetition rate of 1kHz has a pulse energy of 1mJ.

3. instantaneous frequency shift

a. The nonlinear phase shift is (Subscript[ϕ, NL])(t)=((Subscript[ω, 0])/c)(Subscript[n, 2])I(t)L, where I(t)=(Subscript[I, 0])exp[-2(t^2)/((Subscript[τ, 0])^2)]. 

Write a function for the nonlinear phase shift as a function of time, pulse duration and pulse energy. (We will hold the spot size and sample length constant. ) Plot ϕNL[t] for the energy calculated above. If your pulse energy is correct, the peak value should be 1.

b. Now calculate the instantaneous frequency shift: 
(Subscript[Δω, inst])(t)=-(∂_t) (Subscript[ϕ, NL])(t)
Calculate an expression for (Subscript[Δω, inst])(t), then plot the relative frequency shift: (((Subscript[Δω, inst])(t))/(Δω)), where Δω=2/(Subscript[τ, 0]) is the bandwidth of the incident pulse. Again, if your pulse energy is correct, the peak value should be of the order 1.

4. Numerical calculation of SPM spectrum

To see what the self-phase modulated spectrum looks like, we need to calculate it numerically. Set up time and frequency grids, sample the self-phase modulated pulse in the time domain, then calculate the pulse spectral intensity. Feel free to use the FFT functions and code for setting up the grids from my examples.  

To calculate a list for the nonlinear phase at a given pulse energy, you can use the function you defined above with the list of time points ϕNL[tLst, τ, E ]. The peak B-integral is just the maximum value of this list. 
 
Make plots of the spectrum vs frequency for peak B-integrals of 1, 5, and 20.  On each plot, also show the original spectrum for comparison. Normalize the plots by dividing the array you are plotting by its maximum value.

5. simple pulse compression

a. The self-phase modulated pulse is not any different in duration than the input pulse, but the if the right spectral phase is applied, the broader bandwidth can be compressed to get a shorter pulse. 

The best compression would be a pulse with spectral phase that is constant.  To see what the Fourier transform limit is for the broadened spectrum, calculate the inverse FFT of the absolute value of the spectrum for the case where the B-integral = 20.

b. Next calculate a list Exp[i(1/2)(((ωLst-ω0))^2)ϕ2Tst] and use this to tryto compress the pulse. Calculate the inverse FFT to get the pulse in the time domain. Plot the intensity in the time domain for the compressed pulse along with the original pulse shape.  Vary the value of the (Subscript[ϕ, 2]) used for compression so that you get a shorter pulse than the initial value. 

Normalize the plots to better compare the original with the compressed pulse.