ENEE 791, Fall 2018 Problem Set 1
Issued: Tuesday August 28, 2018 Due: Thursday September 6, 2018
Problem 1.1
Many optical materials exhibit more than one resonant frequency. Imagine, for example that
some electrons in the material are bound more tightly than others, giving rise to a variety of
different spring constants. To model this case, the equation for the linear susceptibility is
often generalized to include a series of terms:
Ne 2 X fj
.1/ .!/ D (1)
m0 j .j2 i2 j ! ! 2/
where the weighting factors fj represents strength of the j -th resonance, which has a
resonant frequency of j and corresponding damping coefficient j . (Alternatively, you
could consider Nfj to be the sub-population of electrons that have a resonant frequency of
j .)
Show that, in the low-damping limit (where we assume j D 0), the refractive index can be
described by the Sellmeier equation:
B1 2 B2 2 B3 2
n2 D 1 C C C C ::: (2)
2 C1 2 C2 2 C3
and derive an expression for the Sellmeier coefficients Bj and Cj in terms of the resonant
frequencies and oscillator weights.
Problem 1.2
Derive an equation for n and in terms of the real and imaginary parts of .1/ R C i I .
Problem 1.3
To get an order-of-magnitude estimate of Miller’s constant that was introduced in the
text, assume that the linear and nonlinear terms in the restoring force become equal to one
another when the displacement x is equal to the interatomic spacing, or lattice constant
a N 1=3 . Estimate the value of when 0 D 2c= D 1 m and a D 2Å.
Problem 1.4
In a linear material, the polarization P .t/ is related to E.t/ by the convolution integral:
Z
P .t/ D 0 χ.1/ . /E.t /d
This equation can be converted to the spectral domain by using the Fourier transform:
Z
O
P.!/ D P .t/e Ci !t dt
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ENEE 791, Fall 2018 Problem Set 1
O
and similarly for E.!/. Use these definitions to prove that
Z
O
P.!/ D 0 .!/EO .!/; where .!/
.1/ .1/
χ.1/ .t/e i !t dt
Problem 1.5
In this problem we consider a greatly simplified version of the argument for intrinsic per-
mutation symmetry. Consider the second-order tensor relationship between vectors E and
P:
Pj D j kl Ek El
where the summation over k and l is implied. To simplify the discussion, we omit the
frequency or time arguments on all quantities1, and we have omitted the scale factor of 0 .
(a) First we will prove that the tensor j kl is not unique. Let us assume that there exists
another tensor j0 kl that describes the same relationship between E and P. This means
that
Pj D j kl Ek El D j0 kl Ek El
for all possible choices of the vector E. Now, if we define the difference tensor
Dj kl D j0 kl j kl
then one can easily see that Dj kl Ek El D 0 for any possible choice of the vector E.
Prove that Dj kl must be antisymmetric in its last two arguments:
Dj kl D Dj lk
Note: this implies that the diagonal elements Dj kk D 0 for all k.
(b) You have now shown that if there are two different tensors j kl and j0 kl that de-
scribe the same relationship between E and P, the difference Dj kl D j0 kl j kl is
antisymmetric in its last two arguments. Now prove that the converse is also true:
You can modify the tensor j kl by adding to it any antisymmetric2 tensor Dj kl
without changing the underlying relationship between E and P.
(c) Finally, suppose you have a tensor j kl that is not symmetric in its last two indices:
j kl ¤ j lk . Show that you can always construct a new tensor j0 kl D j kl C Dj kl
that is symmetric in the last two indices. What is the appropriate relationship between
j0 kl and j kl and what is the antisymmetric tensor Dj kl ?
1Recall that intrinsic permutation symmetry only applies if you permute the frequency or time arguments
and corresponding indices in the same way.
2By “antisymmetric” we mean in the last two indices .k; l/.
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ENEE 791, Fall 2018 Problem Set 1
(d) Suppose E and P are related by a cubic tensor product:
Pj D j klm Ek El Em
where j klm is a fourth-rank tensor with 81 components. Suppose you want to
construct a new tensor j0 klm that is symmetric with respect to the interchange of
any of its last three indices .klm/. By generalizing your result from (c), how would
you construct this new symmetrized tensor j0 klm ? You are not required to prove that
the new tensor yields the same relationship between E and P.
Hint: how many distinct permutations of .klm/ are there?
Problem 1.6
The nonlinear polarization for second-harmonic generation was described by
0
Pj .2!/ D j kl . 2!I !; !/Ek .!/El .!/
2
whereas the nonlinear polarization for sum-frequency generation is given by:
Pj .!3 / D 0 j kl . !3 I !1 ; !2 /Ek .!1 /El .!2 /
Many people are confused about why the factors in these two equations differ by a factor
of two. In this problem we examine this question more carefully.
To begin, lets assume that the electric fields at !1 and !2 are co-polarized3. In this case,
we can regard P , E and .2/ as scalar quantities. The input electric field is given by:
1 i !1 t 1 i !2 t
E.t/ D E.!1 /e C E.!2 /e C c.c.
2 2
According to the derivation presented in class and in the notes, the component of the non-
linear polarization at !3 D !1 C !2 is given by:
1 i !3 t
P .t/ D P .!3 /e C c.c.
2
where
P .!3 / D 0 .2/ . !3 I !1 ; !2 /E.!1 /E.!2 / : (3)
Let us now assume that the two interacting waves at !1 and !2 have equal amplitudes and
frequencies:
E.!1 / D E.!2 / A; !1 D !2 !
In this case, the input electric field can be written as:
1 i !t
E.t/ D E.!/e C c.c.; where E.!/ 2A
2
3This is referred to as a Type I interaction.
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ENEE 791, Fall 2018 Problem Set 1
If we then use the Eq. 3 in the limit that !1 D !2 to calculate P .!3 D 2!/ we obtain
0 .2/
P .2!/ D 0 .2/ . 2!I !; !/A2 D . 2!I !; !/E.!/2
4
Thus, we now have factor of 41 , which does not agree with the result given earlier for
second-harmonic generation.
(a) What is wrong with this derivation?
(b) Show that if you properly consider the limit that !1 D !2 you obtain the correct result
that
0
P .2!/ D .2/ . 2!I !; !/E.!/2
2
Also prove that this result applies even if the waves !1 and !2 have different ampli-
tudes.
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