# 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 [ I-4 ], 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 [ I-3 ]) as
 [ VII-1 ]
The wave vector and frequency dependent second and third order susceptibilities are then defined as
 [ VII-2a ]
and
 [ VII-2b ]
Thus, we may write quite generally [1]
 [ VII-3 ]

### A Simple Classical Model of Nonlinear Optical Resonce:

A simple Lorentz-Dude 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 one-dimensional nonlinear (anharmonic) oscillator may be written
 [ VII-4 ]

Thus, the equation of motion of a particle moving in that potential becomes
 [ VII-5 ]
We can analyze the response of the oscillator by expanding the displacement in powers of the electric field  -- viz.
 [ VII-6 ]
where  is proportional to the nth power of the field . Inserting this expression into Equation [ VII-5 ] and equating like powers of , we obtain the following hierarchy of equations:

 [ VII-7a ]
 [ VII-7b ]
 [ VII-7c ]

In the frequency domain we see that
 [ VII-8a ]
or
 [ VII-8b ]
where .   Thus, knowing , we may then consider  a driving term in Equation [ VII-7b ] -- viz.
 [ VII-9 ]
Therefore, in the frequency domain
 [ VII-10a ]
which, in view of Equation [ 11-8b ], becomes
 [ VII-10b ]
We may treat the third order terms in a similar manner. We write Equation [ VII-7c ] as
 [ VII-11 ]
In the frequency domain
 [ VII-12 ]
Using Equations [ VII-8b ] and [ VII-10b ], we obtain
 [ VII-13 ]
In particular, for an input
 [ VII-14 ]
a. Second harmonic generation is due to the terms
 [ VII-15a ]
and
 [ VII-15b ]
b. Sum frequency generation is due to the term
 [ VII-16a ]
c. Difference frequency generationis due to the term
 [ VII-16b ]
d. Optical rectificationor dc generation is due to the terms
 [ VII-17a ]
and
 [ VII-17b ]
e. Third harmonic generation is due to the terms
 [ VII-18a ]
and
 [ VII-18b ]
f. Intensity dependent propagation is due to the terms
 [ VII-19a ]
and
 [ VII-19b ]
g. Raman generation (inelastic scattering) involves terms like
 [ VII-20 ]
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 [ VII-13a ] above), the ratio
 [ VII-21 ]
is a constant independent of frequency! This observation is consistent with the famous empirical Miller Rule which declares that the ratio
 [ VII-22 ]
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
 [ VII-23 ]
 [ VII-24 ]
Suppose that we have an input driving or pump field
 [ VII-25 ]
Then the components of the polarization with frequency  are given by
 [ VII-26 ]
It is a convenience to resolve this  and the resultant second harmonic field into longitudinal and transverse components -- viz.
 [ VII-27a ]
and
 [ VII-27b ]
Therefore, we may write the  or second harmonic components of Equations [ VII-23 ] and [ VII-24 ]
 [ VII-28a ]
and
 [ VII-28b ]
To satisfy these equations two conditions must hold -- viz.
 [ VII-29a ]
and
 [ VII-29b ]
where
 [ VII-30 ]
We now write  where  is a slowly varying function of z -- viz.
 [ VII-31 ]
so that Equation [ VII-29b ] may be written
 [ VII-32 ]
where

If we assume that the driving field stays constant, we can directly integrate Equation [ VII-32 ] to obtain the exceedingly famous and important equation for the spatial variation of the second harmonic field -- viz.

 [ VII-33 ]

 Schematic diagram of the up-conversion 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 up-conversion 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 [ VII-32 ] must take into account the spatial variation of . To that end we assume perfect phase matching

and rewrite Equation [ VII-32 ] as

 [ VII-34 ]
Of course, as the second harmonic grows Equation [ VII-3 ] tells us that a nonlinear polarization at the pump frequency is generated -- viz.
 [ VII-35 ]
Repeating the arguments associated with Equations [ VII-28 ] through [ VII-31 ] we may write
 [ VII-36 ]
Combining these two equations, we find an equation governing the spatial evolution of  -- viz.
 [ VII-37 ]
Equations [ VII-34 ] and [ VII-37 ] 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.
 [ VII-38a ] [ VII-38b ]
To solve these equation we first write  and  with the boundary conditions  and .  Thus, the coupled equations reduce to the dimensionless form
 [ VII-39a ] [ VII-39b ]
where and .  We next separate  and  into phase and amplitude parts as
 [ VII-40 ]
and substitute into Equations [ VII-39 ] -- viz
 [ VII-41a ] [ VII-41b ]
Equating real and imaginary parts of these equations, we find
 [ VII-42a ] [ VII-42b ] [ VII-42c ] [ VII-42d ]
Since the phase enters only in the combination  these four equations reduce to three -- viz.
 [ VII-43a ] [ VII-43b] [ VII-43c]
Combining these three equations, we obtain
 [ VII-44a ]
which is, obviously, equivalent to
 [ VII-44b ]
or
 [ VII-44c ]
Since  as  the "const' must be zero and, hence,  must be  for all .  Thus, the original four coupled equations now reduce to two -- viz.
 [ VII-45a ]
and
 [ VII-45b ]
Combining these equations, we obtain
 [ VII-46a ]
or
 [ VII-46b ]
which is an assertion of the principle of energy conservation. Taking this equation together with Equation [ VII-45a ] we see that
 [ VII-47a ]
or
 [ VII-47b ]
and
 [ VII-47c ]
Finally, returning to the original variables, we see that
 [ VII-48a ]
and
 [ VII-48b ]

 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), Addison-Wesley (1992), ISBN 0-201-57868-9.

[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).

This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated February 21, 2000