Nonlinear Optics
Homework 4
due Wednesday, 8 March 2017

Pulse transform code

constants and unit definitions: internally, all numbers in SI, except pressure in atm
angles in Mathematica are internally in radians, my variables are in degrees

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));W=1; V=1;
c=2.9979 10^8;                          (*speed of light,in m/s*)
eps0=8.8542 (10^(-12));              (*permittivity of free space*)
re=2.82 (10^(-15));	              (*m classical electron radius*)
e=1.6 (10^(-19));	                   (*Coul electron charge*)
Natm=760 3.3 (10^16)/(cm^3); (*no.particles in 1 atm*)
Torr=1/760;
eV=1.6 (10^(-19));
hbar=(10^(-34));

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));

taylor expanded phase

0th and 1st order terms are not included
1/n! terms should be included in coeff

Clear[kk2,kk3,kk4,kk5,Δω];
Δω[λ_]:=2 π c(-1/lambda0+1/λ);
ϕTaylor[λ_,kk2_,kk3_,kk4_,kk5_,kk6_]:=(1/(2!))kk2 ((Δω[λ])^2)+(1/(3!))kk3 ((Δω[λ])^3)+(1/(4!))kk4 ((Δω[λ])^4)+(1/(5!))kk5 ((Δω[λ])^5)+(1/(6!))kk6 ((Δω[λ])^6);
ϕ1Taylor[λ_,kk2_,kk3_,kk4_,kk5_,kk6_]=-((λ^2)/(2 π c)) D[ϕTaylor[λ,kk2,kk3,kk4,kk5,kk6],λ];
ϕ2Taylor[λ_,kk2_,kk3_,kk4_,kk5_,kk6_]=-((λ^2)/(2 π c)) D[ϕ1Taylor[λ,kk2,kk3,kk4,kk5,kk6],λ];
ϕ3Taylor[λ_,kk2_,kk3_,kk4_,kk5_,kk6_]=-((λ^2)/(2 π c)) D[ϕ2Taylor[λ,kk2,kk3,kk4,kk5,kk6],λ];
ϕ4Taylor[λ_,kk2_,kk3_,kk4_,kk5_,kk6_]=-((λ^2)/(2 π c)) D[ϕ3Taylor[λ,kk2,kk3,kk4,kk5,kk6],λ];

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];

plot the output pulse shape in the time domain

set up grids

frequency variable: ω

ntpts=(2^12);
dt = 5.fs;
λ0=800.nm;
ω0=2π c/λ0;

tSpan=ntpts dt;
tLst=Table[i dt,{i,-ntpts/2,ntpts/2-1}];
dω=2π/(ntpts dt);
ωLst=Table[ω0+iω dω,{iω,-ntpts/2,ntpts/2-1}];
ωSpan=ntpts dω; 

λLst=2π c/ωLst;
dλ=λ0 dω/ω0;
Print["tSpan: ",tSpan/fs," fs, dt: ",dt/fs," fs"];
Print["ωSpan: ",ωSpan fs," rad/fs, dω: ",dω fs," /fs"];
Print["Grid range. λMin: ",λLst[[ntpts]]/nm," nm. λMax: ",λLst[[1]]/nm," nm, dλ: ",dλ/nm," nm"];

define pulse shape

τIn and Δω are 1/e2 half widths in intensity

τIn=40fs;
ΔωIn=2/τIn;

AtInLst=Exp[-(tLst^2)/((τIn)^2)];
AωInLst=FFT[AtInLst,ntpts,dt];

Print["time domain"];
tRange=6τIn;
ListPlot[Transpose[{tLst/fs,((Abs[AtInLst])^2)}],Joined->True,PlotRange->{{-tRange/fs,tRange/fs},All},AxesLabel->{"t (fs)","intensity"}]

Print["spectral domain"];
ωRange=3ΔωIn;
ListPlot[Transpose[{ωLst fs,((Abs[AωInLst])^2)}],Joined->True,PlotRange->{{(ω0-ωRange)fs,(ω0+ωRange)fs},All},AxesLabel->{"ω (rad/fs)","intensity"}]

test sample phase profiles

test with Taylor phase

lambda0=800. nm;

kk2=0;
kk3=100000 (fs^3);
kk4=0 1000000(fs^4);
kk5=kk6=0;
ϕsystem[λ_]:=ϕTaylor[λ,kk2,kk3,kk4,kk5,kk6];

phaseTstLst=ϕsystem[λLst]-(ϕsystem[λ0]);

AtLst=InverseFFT[AωInLst Exp[I phaseTstLst],ntpts,dt];
Print["time domain"];
ListPlot[Transpose[{tLst/fs,((Abs[AtLst])^2)}],Joined->True,PlotRange->{{-tRange/fs,tRange/fs},All}]

Problem 1: SH interferometric autocorrelation and SH FROG traces

a. Signal in time domain

Since we want to be able to calculate the output signal for different pulse shapes that are synthesized by adding spectral phase, we will implement the time delay τ in the spectral domain (multiply by Exp[i ω τ]). Make a plot in the time domain of the intensities of both the unshifted pulse and a pulse delayed by τ = 50fs. Make a second plot of the intensity of the combined pulses, along with the intensity of the second harmonic (((Abs[(E^2)])^2) ).

b. Interferometric autocorrelation signal

Put the calculation above in a loop, and calculate the time integrated signal (you can use  Total[ ((Abs[(Subscript[E, 2])])^2) ] for a list of delays. Plot this interferometric autocorrelation signal for the transform-limited pulse and for a pulse with enough third-order phase to show an effect on the trace. Normalize the signal of the autocorrelation so that the peak at zero delay is = 8. You should find an 8:1 ratio between the peak signal and the signal at long delay.

c. FROG trace calculation

Calculate the Fourier transform of the SH signal at this delay from part (a), and plot the spectral intensity of the SH signal. Now define a 2D list for the spectral intensity vs delay. Make a loop to calculate the SH spectrum over a list of delays. After the loop, make a ListDensityPlot of the resulting FROG trace. Do this for a transform-limited pulse and a pulse with the same 3rd order dispersion used in part (b).

Problem 2: practice with Fourier transforms

a. optical shutter

A CW laser of frequency (Subscript[ω, 0]) is passed through a fast optical shutter, which passes the beam only during a time (Subscript[T, w]), centered at t = 0. Write an analytic expression for the wave in the time domain, then do the Fourier transform to write a representation in ω space.

b. acousto-optic modulator

A Gaussian laser pulse with central frequency (Subscript[ω, 0]) and 1/e field half width (Subscript[t, 0]) is passed through an acousto-optic modulator, which modulates the amplitude of the pulse with the function a(t)=(1+(Subscript[a, m])cos((Subscript[ω, m])t) ). Write an expression for the wave in the time domain, then do the Fourier transform to write a representation in ω space.

Problem 3: self-phase modulation SPM

a. temporal phase modulation

A pulse passes through a nonlinear medium that has an intensity-dependent refractive index: n(I)=(Subscript[n, 2])I . 
Write an expression for the time-dependent propagation phase (k L) for a Gaussian pulse that passes through a length L of the nonlinear medium. Calculate the instantaneous frequency vs time, and for a B-integral of 10, make a plot of a the instantaneous frequency vs time.

b. spectral broadening with SPM

For the conditions in part (a), apply the nonlinear temporal phase shift to the time-domain E-field, then calculate the spectrum of the output. Feel free to vary the B-integral to see how the spectrum evolves with input intensity.

Problem 4: Doubling and mixing of chirped pulses

a. When a transform-limited pulse is frequency doubled, the output pulse duration is ideally reduced by a factor (Sqrt[2]), since ((Subscript[A, 1])^2)(t)=((((Subscript[A, 0])exp[-(t^2)/((Subscript[t, 0])^2)]))^2). The envelope of a linearly-chirped Gaussian pulse in the time domain can be written as (Subscript[A, 1])(t)=(Subscript[A, 0])exp[-(t^2)/((τ(β))^2)-i β (t^2)]. Ideally (no phase mismatch or depletion), frequency doubling will produce a signal that is (Subscript[A, 2])(t)∝((Subscript[A, 1])^2)(t). After squaring the input field, describe how the pulse duration τ and chirp rate β are affected by the doubling process. The doubled pulse will have a new transform-limited pulse duration (Subscript[t, 0])' and second-order spectral phase (Subscript[φ, 2])'. Working with the expressions for τ and β, relate (Subscript[t, 0])' to (Subscript[t, 0]) and (Subscript[φ, 2])' to (Subscript[φ, 2]).

b. now consider sum-frequency mixing of two pulses ((Subscript[A, a])(t) and (Subscript[A, b])(t) ) with the same central frequency but opposite chirp rates ((Subscript[β, b])=-(Subscript[β, a])). What is the output pulse duration and chirp rate of the mixing process (Subscript[A, 2])(t)∝(Subscript[A, a])(t) (Subscript[A, b])(t)?

c. Now introduce a relative time delay T between the pulses (the math works out easier if you shift them in opposite directions by T/2). Describe the nature of the output pulse in this case as a function of the delay T: temporal shape and width, peak amplitude, instantaneous frequency characteristics.