Nonlinear optics midterm questions

1. Phase matching in a plasma and partially-ionized gas

a. Assume frequency conversion is taking place in a plasma, with the refractive index function 
n(ω)=1-(1/2)(((Subscript[ω, p])^2)/(ω^2)), 
where ((Subscript[ω, p])^2)=(((Subscript[N, e])(e^2))/((Subscript[ϵ, 0])(Subscript[m, e]))) defines the plasma frequency and (Subscript[N, e]) is the number density of the electrons. 

Calculate an expression for the phase mismatch for (q^th) harmonic generation, Δk=q (Subscript[k, 1])-(Subscript[k, q]).

b. Suppose the medium is a thin slab of thickness L. What is the maximum thickness before the phase mismatch leads to back-conversion?

c. What are the conditions on Δk for a harmonic signal to build up in the tight focusing limit (L infinite)? Would you expect to see a harmonic signal for frequency conversion in a plasma?

d. Now assume the gas is partially ionized (ionization fraction η), so that there is a mix of neutral atoms, electrons and ions. Let (Subscript[N, 0]) be the total number density, then (Subscript[N, e])=η (Subscript[N, 0]) and the atomic density is (Subscript[N, a])=(1-η)(Subscript[N, 0]). For the fundamental frequency (assumed to be near the visible or near-IR range), the refractive index is slightly greater than 1 by an amount proportional to the gas density. Let (Subscript[δn, 1])=n((Subscript[ω, 1]))-1 for the un-ionized gas at atmospheric pressure, where (Subscript[N, a])≡(Subscript[N, atm]). Show that the refractive index for the fundamental can be written as:
n((Subscript[ω, 1]))=1+((Subscript[δn, 1])/(Subscript[N, atm]))(1-η)(Subscript[N, 0])-η (Subscript[N, 0])((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, 1])^2))

e. For a harmonic with large q, (Subscript[ℏω, q]) is sufficiently large that all electrons can be considered free. Find an expression for the ionization fraction η for which Δk = 0 in the plane wave limit (no focusing or beam divergence).

f. Extra credit: suppose the nonlinear medium is within the confocal parameter of the focused fundamental beam. By adding the effect of the Gouy phase (letting (Tan^(-1))[z/(Subscript[z, R])]~z/(Subscript[z, R])) calculate the relationship between (Subscript[N, 0]) and η to achieve phase matching.

The refractive index in a plasma is given by
n(ω)=1-(1/2)(((Subscript[ω, p])^2)/(ω^2))
compare refractive indices: 
n((Subscript[ω, 1]))=1-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))
n((Subscript[ω, 3]))=1-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 3])^2))=1-(1/18)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))
we have a higher refractive index at the harmonic, so the medium is normally-dispersive. 

suppose we want to perform third harmonic conversion in an isotropic plasma. Is it possible to get any signal in the tight-focusing geometry? If so, what are the constraints on the focusing conditions?  

Δk=3((Subscript[ω, 1])/c)n((Subscript[ω, 1]))-((Subscript[ω, 3])/c)n((Subscript[ω, 3]))=((Subscript[ω, 3])/c)(n((Subscript[ω, 1]))-n((Subscript[ω, 3])))
Δk=((Subscript[ω, 3])/c)(1-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))-(1-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 3])^2))))=((Subscript[ω, 3])/(2c))((Subscript[ω, p])^2)((1/((Subscript[ω, 3])^2))-(1/((Subscript[ω, 1])^2)))=((Subscript[ω, 3])/(2c))(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))((1/9)-1)=-(4/3)(((Subscript[ω, p])^2)/((Subscript[ω, 1])c))

Δk=((Subscript[ω, 3])/c)(n((Subscript[ω, 1]))Cos[(Subscript[θ, 1])]-n((Subscript[ω, 3]))Cos[(Subscript[θ, 3])])
Δk=((Subscript[ω, 3])/c)((1-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2)))Cos[(Subscript[θ, 1])]-(1-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 3])^2)))Cos[(Subscript[θ, 3])])=((Subscript[ω, 3])/c)(Cos[(Subscript[θ, 1])]-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))Cos[(Subscript[θ, 1])]-Cos[(Subscript[θ, 3])]+(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 3])^2))Cos[(Subscript[θ, 3])])=((Subscript[ω, 3])/c)(Cos[(Subscript[θ, 1])]-Cos[(Subscript[θ, 3])]-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))(Cos[(Subscript[θ, 1])]-(1/9)Cos[(Subscript[θ, 3])]))=
((Subscript[ω, 3])/c)(Cos[(Subscript[θ, 1])]-(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))Cos[(Subscript[θ, 1])]-Cos[(Subscript[θ, 3])]+(1/2)(((Subscript[ω, p])^2)/((Subscript[ω, 1])^2))(1/9)Cos[(Subscript[θ, 3])])

Clear[ω1,ωp,θ1,θ3];
Δk=3((ω1)/c)((1-(1/2)(((ωp)^2)/((ω1)^2)))Cos[θ1]-(1-(1/18)(((ωp)^2)/((ω1)^2)))Cos[θ3]);
ωp=1;c=1;
ω1=30;
θ1=0;
Plot[Δk,{θ3,0,6Degree}]

;lk
n((Subscript[ω, 1]))=1+((Subscript[δn, 1])/(Subscript[N, atm]))(1-η)(Subscript[N, 0])-η (Subscript[N, 0])((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, 1])^2))
n((Subscript[ω, q]))=1-(Subscript[N, 0])((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, q])^2))
Δk=q ((Subscript[ω, 1])/c)(n((Subscript[ω, 1]))-n((Subscript[ω, q])))=q ((Subscript[ω, 1])/c)(1+((Subscript[δn, 1])/(Subscript[N, atm]))(1-η)(Subscript[N, 0])-η (Subscript[N, 0])((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, 1])^2))-1+(Subscript[N, 0])((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, q])^2)))
Δk=q ((Subscript[ω, 1])/c)(Subscript[N, 0])(((Subscript[δn, 1])/(Subscript[N, atm]))(1-η)-η ((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, 1])^2))+((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((q^2) ((Subscript[ω, 1])^2))))
Δk=q ((Subscript[ω, 1])/c)(Subscript[N, 0])(((Subscript[δn, 1])/(Subscript[N, atm]))(1-η)-((e^2)/(2(Subscript[ϵ, 0])(Subscript[m, e])))(1/((Subscript[ω, 1])^2))(η-(1/(q^2))))

2. 4-wave mixing for pulse characterization

A technique to characterize a pulse involves putting a mask on a beam with 3 holes arranged as shown below:

Graphics[{Circle[{-1,0},.25],Circle[{1,0},.25],Circle[{0,1},.25],Text[1,{-1,0}],Text[2,{1,0}],Text[3,{0,1}]},ImageSize->150]

The numbers correspond to the different fields (Subscript[E, 1]), (Subscript[E, 2]) and (Subscript[E, 3]), each of which is linearly polarized in the x direction. When those parallel beams are focused by a lens into a thin piece of glass, a fourth wave (Subscript[E, 4]) is generated. The beams are arranged on the output side as shown:

Graphics[{Circle[{-1,0},.25],Circle[{1,0},.25],Circle[{0,-1},.25],{GrayLevel[0.5],Disk[{0,1},.25]},Text[2,{-1,0}],Text[1,{1,0}],Text[3,{0,-1}],Text[4,{0,1}]},ImageSize->150]

Note that by passing through the lens focus the beam locations are inverted. The nonlinear interaction takes place through a (χ^((3))) four-wave mixing process, so that the nonlinear polarization is proportional to ((Subscript[E, T])^2)(SuperStar[(Subscript[E, T])]), where the total field is (Subscript[E, T])=(Subscript[E, 1])+(Subscript[E, 2])+(Subscript[E, 3]).

a. Write an expression for the nonlinear polarization (P^NL) (as a function of the individual fields (Subscript[E, 1]),(Subscript[E, 2]),(Subscript[E, 3])) that corresponds to the generation of (Subscript[E, 4]) in the direction shown above. Make it clear which component(s) of the ((Subscript[χ, ijkl])^((3))) tensor are involved and explain your reasoning.

we need an output that is in the -(Subscript[k, 3]) direction, so the third wave is conjugated.  so (Subscript[k, 1]) and (Subscript[k, 2]) have to cancel each other. So we have (Subscript[E, 1])(Subscript[E, 2])(SuperStar[(Subscript[E, 3])]) 
P=6(Subscript[ϵ, 0])(Subscript[χ, 1122])(Subscript[E, 1])(Subscript[E, 2])

b. Write a vector expression for the phase mismatch Δk in terms of the individual k-vectors ((Subscript[k, 1]) etc). By breaking the vectors into x, y and z components, show that Δk=0 for the geometry shown.

c. Each of the beams contains a pulse, so for example, (Subscript[E, 1])(r,t)=(Subscript[A, 1])(t)(e^(i((Subscript[k, 1])·r-ω t))). Experimentally, we can reflect the three beams off of separate input mirrors. One of these can be displaced to give one pulse a delay τ relative to the others. Write an expression for the nonlinear polarization that includes the delay for the three cases where the delay is applied to each of the three input beams. This is the nonlinearity that is used to perform an intensity autocorrelation to measure the pulse width. The non-delayed beams combine to form a “gate” for the other delayed beam.