On Classical Electromagnetic Fields (cont.)
VII. Nonlinear
Optics  Classical Picture (pdf
copy)
An Extended Phenomenological Model of Polarization:
As an introduction to the subject of nonlinear optical phenomena,
we write, in the spirit of Equation
[ I4 ], the most general form of higher order terms in the phenomenological
electric field expansion of the polarization density (which may then be
inserted into Equations [ I3
]) as

[ VII1 ]

The wave vector and frequency dependent second
and third order susceptibilities are then defined as

[ VII2a ]

and

[ VII2b ]

Thus, we may write quite generally [1]

[ VII3 ]

A Simple Classical Model of Nonlinear Optical Resonce:
A simple LorentzDude model is often used in the literature
as a valuable guide to the understanding of the frequency behavior of the
nonlinear dielectric response.[2]
We assume that the potential energy of a onedimensional nonlinear (anharmonic)
oscillator may be written

[ VII4 ]

Thus, the equation of motion of a particle moving in that potential
becomes

[ VII5 ]

We can analyze the response of the oscillator by expanding
the displacement in powers of the electric field
 viz.

[ VII6 ]

where
is proportional to the nth power of the field .
Inserting this expression into Equation [ VII5 ] and equating like powers
of ,
we obtain the following hierarchy of equations:

[ VII7a ]


[ VII7b ]


[ VII7c ]

In the frequency domain we see that

[ VII8a ]

or

[ VII8b ]

where .
Thus, knowing ,
we may then consider
a driving term in Equation [ VII7b ]  viz.

[ VII9 ]

Therefore, in the frequency domain

[ VII10a ]

which, in view of Equation [ 118b ], becomes

[ VII10b ]

We may treat the third order terms in a similar manner. We
write Equation [ VII7c ] as

[ VII11 ]

In the frequency domain

[ VII12 ]

Using Equations [ VII8b ] and [ VII10b ], we obtain

[ VII13 ]

In particular, for an input

[ VII14 ]

a. Second harmonic generation
is due to the terms

[ VII15a ]

and

[ VII15b ]

b. Sum frequency generation
is due to the term

[ VII16a ]

c. Difference frequency generationis
due to the term

[ VII16b ]

d. Optical rectificationor
dc
generation is due to the terms

[ VII17a ]

and

[ VII17b ]

e. Third harmonic generation
is due to the terms

[ VII18a ]

and

[ VII18b ]

f. Intensity dependent propagation
is due to the terms

[ VII19a ]

and

[ VII19b ]

g. Raman generation
(inelastic scattering) involves terms like

[ VII20 ]

Note that, according to this simple anharmonic oscillator model,
Raman generation may be enhanced by a resonance at a frequency !
Also note that, for this model (see Equation [ VII13a ] above), the ratio

[ VII21 ]

is a constant independent of frequency! This observation is
consistent with the famous empirical Miller Rule
which declares that the ratio

[ VII22 ]

has only a weak dispersion and is almost a constant for a wide
range of materials!
Second Harmonic Generation  Perturbation Analysis:
We may write the nonlinear, macroscopic Maxwell equations in
the form

[ VII23 ]


[ VII24 ]

Suppose that we have an input driving or pump field

[ VII25 ]

Then the components of the polarization with frequency
are given by

[ VII26 ]

It is a convenience to resolve this
and the resultant second harmonic field into longitudinal and transverse
components  viz.

[ VII27a ]

and

[ VII27b ]

Therefore, we may write the
or second harmonic components of Equations [ VII23 ] and [ VII24 ]

[ VII28a ]

and

[ VII28b ]

To satisfy these equations two conditions must hold  viz.

[ VII29a ]

and

[ VII29b ]

where

[ VII30 ]

We now write
where
is a slowly varying function of z  viz.

[ VII31 ]

so that Equation [ VII29b ] may be written

[ VII32 ]

where
If we assume that the driving field stays constant,
we can directly integrate Equation [ VII32 ] to obtain the
exceedingly
famous and important equation for the spatial variation of the
second harmonic field  viz.

[ VII33 ]

Maker Fringes [4]


SHG Autocorrelation


Time Microscope System
Schematic diagram of the upconversion time microscope system.
The input waveform I(t) was a 4 bit 100 Gb/s word.
Input and output dispersion was obtained by two sets of diffraction
grating pairs. The upconversion time lens was achieved by sum frequency
generation of the input waveform and a linearly chirped pump E(t). The
measured temporal image of the 100 Gb/s 1101 input test pattern was a 1011
waveform at 8.5 Gb/s. (Source)

.
Second Harmonic Generation  Coupled Wave Analysis:
When the process of second harmonic generation takes place
under conditions of perfect phase match
i.e.
 the perturbation result breaks down if the path is sufficiently long.
Under these circumstances the pump beam will be depleted as the second
harmonic grows and a solution of Equation [ VII32 ] must take into account
the spatial variation of .
To that end we assume perfect phase matching
Phase Matching in Anisotropic
Media


and rewrite Equation [ VII32 ] as

[ VII34 ]

Of course, as the second harmonic grows Equation [ VII3 ]
tells us that a nonlinear polarization at the pump frequency is generated
 viz.

[ VII35 ]

Repeating the arguments associated with Equations [ VII28
] through [ VII31 ] we may write

[ VII36 ]

Combining these two equations, we find an equation governing
the spatial evolution of
 viz.

[ VII37 ]

Equations [ VII34 ] and [ VII37 ] are then the coupled differential
equation which describe the coupling of the first and second harmonic fields.
Handling all the "vectorness" in these two equation would obscure important
issues. Thus, we consider a pair of somewhat less complicated equations
which incorporate the essence of the problem  viz.

[ VII38a ]


[ VII38b ] 
To solve these equation we first write
and
with the boundary conditions
and .
Thus, the coupled equations reduce to the dimensionless form

[ VII39a ]


[ VII39b ] 
where
and .
We next separate
and
into phase and amplitude parts as

[ VII40 ]

and substitute into Equations [ VII39 ]  viz

[ VII41a ]


[ VII41b ] 
Equating real and imaginary parts of these equations, we find

[ VII42a ]


[ VII42b ] 

[ VII42c ] 

[ VII42d ] 
Since the phase enters only in the combination
these four equations reduce to three  viz.

[ VII43a ]


[ VII43b] 

[ VII43c] 
Combining these three equations, we obtain

[ VII44a ]

which is, obviously, equivalent to

[ VII44b ]

or

[ VII44c ]

Since
as
the "const' must be zero and, hence,
must be
for all .
Thus, the original four coupled equations now reduce to two  viz.

[ VII45a ]

and

[ VII45b ]

Combining these equations, we obtain

[ VII46a ]

or

[ VII46b ]

which is an assertion of the principle of energy
conservation. Taking this equation together with Equation [
VII45a ] we see that

[ VII47a ]

or

[ VII47b ]

and

[ VII47c ]

Finally, returning to the original variables, we see that

[ VII48a ]

and

[ VII48b ]

where

[1]
must vanish for any material that is invariant under inversion, since both
and are vectors, and
are thus odd under inversion symmetry. Note also that
for a given material has the same transformation properties the elastic
constants of that material. The nonzero elements of
and
for various crystal symmetries are compiled in Y.R. Shen, The Principles
of Nonlinear Optics (Wiley, New York, 1984).
[2] See, for example, N. Bloembergen, Nonlinear
Optics (The Advance Book Program), AddisonWesley (1992), ISBN 0201578689.
[3] Note that near resonance
where is the so called
complex Lorentzian.
[4] P.D. Maker, R.W. Terhune, N. Nisenoff, and C.M.
Savage, Phys. Rev. Lett., 8, 21 (1965).
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by R. Victor Jones, jones@deas.harvard.edu
Last updated February 21, 2000