Download Nonlinear Optics (PHYC/ECE 568) Fall 2015

Nonlinear Optics (PHYC/ECE 568)
Fall 2015
Instructor: M. Sheik-Bahae
Midterm Exam, Open Text-Book, Closed Notes, Time: 2 hours
5:30pm-7:30pm
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Happy Thanksgiving!
1. Self-Phase Modulation (SPM) (20 points)
An ultrashort laser pulse with an electric field E (t )  E0 e t / 2t cos( 0 t ) is incident on a
nonlinear medium of length L with a cumulative nonlinear refraction (i.e. noninstantaneous with long decay time) described by
2
n(t )  
t
 I (t ')dt '
,

Where I(t) in the instantaneous intensity,
and ( cm2/J ) is the nonlinear coefficient.
Show (derive) how the spectrum of the laser
pulse changes (i.e. gets chirped) upon
exiting the material. Qualitatively draw and
compare the incident and transmitted E(t)
and |E()| assuming <0. (20 points)
𝜔
𝑡
𝜔
ΔΦ(𝑡) = 0 Δ𝑛(𝑡)𝐿 = 0 γ𝐿 ∫−∞ 𝐼(𝑡 ′ )𝑑𝑡′
𝑐
𝑐
∂ΔΦ(𝑡)
𝜔0
δω = −
=−
γ𝐿𝐼(𝑡)
∂t
𝑐
<0, therefore >0 for all t.
5
4
out
E2( t)+
3
2
E( t)
in
0
 1.5
2
3
0
Time
t
2
3
E(t)
2
p
2. Optical Rectification (25 points)
An ultrashort laser pulse with instantaneous electric
field E (t )  E0 e t / 2t cos( 0 t ) is incident on a 2nd order
NLO material with known (2).
2
E (t )
2
p
(2)
(a) Write down, and plot the second-order nonlinear
polarization term 𝑃̃(2) (𝑡) for optical rectification.
𝑃̃(2) (𝑡) =
𝑡2
− 2
𝜖0 𝜒 (2) 𝐸02 𝑒 𝑡𝑝
𝜕2 𝑃̃ (2)
(b) As you well know, the driving term in the Maxwell’s equation involves 𝜕𝑡 2 .
Qualitatively plot this term, and describe the generated waveform (temporal and spectral)
assuming tp=100 femtoseconds (femto=10-15 )
The generated wave form will look like the above (i.e. a single-cycle wave form)
Spectrally it has the contents of the incident pulse but at zero freq. Note there is no DC
term as the area under the curve is zero. The width of its spectrum is nearly 1/tp=10 THz.
(c) Show the photon-picture (i.e. the energy conservation arrows) in describing the new
frequencies that are generated.
3.
Spontaneous Parametric Down Conversion (30 points):
In a quantum optics experiment, photon pairs (signal and idler) at s and i (<s) are
generated in a (2) material by a pump beam at p polarized along the c-axis as shown
below. Using appropriate polarizers, we only detect photons generated in a type-I phase
matching. The normal dispersion no() and ne() are known.
(a) Draw the wave-vector matching diagram. What is the polarization of generated
photon pairs (show graphically)? (8 pts)
(b) What type of birefringence (positive, negative, none or both) will work in this
geometry? Explain. (7 pts)
As shown in (a), we must have |kp|<|ks|+|ki|. Since p=s+i, then ns, ni> np or no>ne
(negative birefringence only).
(c) Write down the equation(s) that will give s and i. (15 pts)
From the above triangle, we get the following two equations:
no[icos(i)+ scos(s)]=nep , and isin(i) - ssin(s)=0
(d) (Bonus: 10 pts) Repeat (c) but assume type-II phase matching where idler (signal) is
extra ordinary
In this case, assuming signal wave is e-ray, making an angle of /2+s with the c-axis.
The new equations are like in part c, but ne(s,s) is obtained from index ellipsoid
equation:
1
𝑛𝑒2 (𝜃𝑠 )
=
𝜋
2
𝑛𝑒2
sin2 ( +𝜃𝑠 )
+
𝜋
2
2
𝑛𝑜
cos2 ( +𝜃𝑠 )
noicos(i)+ s ne(s)cos(s)=nep
, thus
and noisin(i)- s ne(s)sin(s)=0
4. NLO susceptibilities: resonances and selection rules (25 points)
A fictional molecule has the following 3 energy levels. Draw the spectrum (for
0< <10 eV) for the (a) linear absorption coefficient α(), (b) two-photon absorption
(TPA) coefficient (), (c) SHG: |(2)(2;,)| and (d) THG: |(3)(3;,,)|. (25
points)
Be quantitative in your x-axis. Assume a finite broadening in your drawings. Point out
the resonances (diagrammatically) on your graph for each case and show the relative
strengths if obvious. Note: no calculations needed for this problem.
E (eV)
8
|3> s & p-type (mixed parity hybrid)
6
|2> p-type
0
|1> p-type (ground state)