Nonlinear Optics
Homework 5
due Thursday, 21 Feb 2013

Problem 1:

In third-harmonic microscopy, harmonic light is generated when there is an interface in the focus. As we discussed in class, the signal is not really generated at the interface, but it is localized near there because of the role of phase matching and the dependence of the signal on the cube of the intensity. In this problem, we will consider the longitudinal spatial resolution of this kind of microscopy. Here are the parameters: 
- We focus a Gaussian beam with a vacuum wavelength of 1 micron at f/1 (the beam diameter to 1/(e^2) is equal to the focal length).
- We are looking at a fused silica slide in water. Use the Sellmeier equations for fused silica shown below, and assume that in water n= 1.33 and we neglect dispersion in the water. You may assume that the thickness of the fused silica slide is much greater than the Rayleigh range of the focus. Be careful in how to introduce the refractive index of the slide into the equations for the focal spot size and the Rayleigh range: the wavelength that appears in these equations is its value in the medium i.e. λ = (Subscript[λ, vac])/n.
    
Calculate the third-harmonic signal vs. z-position as the focal spot is moved in the z-direction across the interface. The FWHM of this peak (full-width at half-maximum) is effectively the axial resolution of the imaging system in the longitudinal direction. We are not interested in the absolute strength of the signal, just how it varies with z. Use the absolute value of the expression 2.10.11b, paying attention to the limits of the integration.

dispersion for fused silica

as1=.6961663;
as2=.4079426;
as3=.8974794;
bs1=.0684043^2;
bs2=.1162414^2;
bs3=9.896161^2;
ns[λ_]:=√(1+((as1 (λ^2))/((λ^2)-bs1))+((as2 (λ^2))/((λ^2)-bs2))+((as3 (λ^2))/((λ^2)-bs3)));

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 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: pulse propagation

a. dispersive temporal focusing

We want to deliver a short pulse to a plane that is 5cm deep into fused silica. Calculate the second-order phase coefficient of fused silica, (Subscript[ϕ, 2]) (in (fs^2)/mm), then determine how much negative pre-chirp (in (fs^2)) is necessary for the pulse to be short at that depth. Make a plot of the pulse duration vs distance, and the intensity vs distance (assuming constant beam size).

b. 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 sketch or plot a representative profile.

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.