*Lectures on Nonlinear Optics - Lecture 9*

# Two case studies in nonlinear optics

**lect9.pdf**
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Lecture 9 in Portable Document Format.

### Contents

- General process for solving problems in nonlinear optics
- Formulation of the two case studies in this lecture
- Second harmonic generation
- Optical Kerr-effect - Field corrected refractive index
- References

In this lecture, we will focus on examples of electromagnetic wave propagation in nonlinear optical media, by applying the forms of Maxwell's equations as obtained in the eighth lecture to a set of particular nonlinear interactions as described by the previously formulated nonlinear susceptibility formalism.

The outline for this lecture is:

- General process for solving problems in nonlinear optics
- Second harmonic generation (SHG)
- Optical Kerr-effect

## 1. General process for solving problems in nonlinear optics

The typical steps in the process of solving a theoretical problem in nonlinear optics typically involve:

## 2. Formulation of the two case studies in this lecture

In order to illustrate the scheme as previously outlined, the following exercises serve as to give the connection between the susceptibilities, as extensively analysed from a quantum-mechanical basis in earlier lectures of this course, and the wave equation, derived from Maxwell's equations of motion for electromagnetic fields.

### 2.1. Case study I: Second harmonic generation in negative uniaxial media

Consider a continuous pump wave at angular frequency ω, initially
polarized in the *y*-direction and propagating in the positive
*x*-direction of a negative uniaxial crystal of crystallographic
point symmetry group 3*m*. (Examples of common crystals belonging
to this class: β-BaB_{2}O_{4}
(beta barium
borate, or BBO), LiNbO_{3}
(lithium niobate).)

**Task I (a):**
*Formulate the polarization density of the medium for
the pump and second harmonic wave.*

**Task I (b):**
*Formulate the system of equations of motion for the
electromagnetic fields.*

**Task I (c):**
*Assuming no second harmonic signal present at the input,
solve the equations of motion for the second harmonic field, using the
non-depleted pump approximation, and derive an expression for the conversion
efficiency of the second harmonic generation.*

### 2.2. Case study II: Optical Kerr-effect - continuous wave case

In this setup, a monochromatic optical wave is propagating in the positive
*z*-direction of an isotropic optical Kerr-medium.
We here consider the case of a continuous optical wave, that is to say a
non-pulsed beam. As will be shown in the
Lecture 9, sufficiently intense
optical pulses will in media possessing optical Kerr-effect experience
different changes of the experienced refractive index at leading and
trailing edges of the pulses, leading to the interesting phenomenon of
optical solitons. However, *basics first*.

**Task II (a):**
*Formulate the polarization density of the medium
for a wave polarized in the xy-plane.*

**Task II (b):**
*Formulate the polarization density of the medium
for a wave polarized in the x-direction.*

**Task II (c):**
*Formulate the wave equation for continuous wave propagation
in optical Kerr-media. The continuous wave is x-polarized and propagates
in the positive z-direction.*

**Task II (d):**
*For lossless media, solve the wave equation and give
an expression for the nonlinear, intensity-dependent refractive index
n = n_{0}+n_{2}|E_{ω}|^{2}.
*

## 3. Second harmonic generation

### 3.1. The optical interaction

In the case of second harmonic generation (SHG), two photons at angular frequency ω combine to a photon at twice the angular frequency,

This interaction is for the second harmonic wave (at angular frequency ω)described by the second order susceptibility

where, in the usual
convention of Butcher and Cotter (as throughout applied
this series of lectures), ω_{σ} = 2ω is
the generated second harmonic frequency of the light.

### 3.2. Symmetries of the medium

In this example we consider second harmonic generation in trigonal media
of crystallographic point symmetry group 3*m*.
(Example: LiNbO_{3}, lithium niobate)

For this point symmetry group, the nonzero tensor elements of the first
order susceptibility are (for example according to Table A3.1 in *The
Elements of Nonlinear Optics*)

which gives the ordinary refractive indices

for waves components polarized in the $x$- or $y$-directions, and the extraordinary refractive index

for the wave component polarized in the $z$-direction. Since we here are
considering a *negatively uniaxial crystal* (see Butcher and Cotter,
page 214), these refractive indices satisfy the inequality

The nonzero tensor elements of the second order susceptibility are (for example according to Table A3.2 in {\sl The Elements of Nonlinear Optics})

### 3.3. Additional symmetries

Intrinsic permutation symmetry for the case of second harmonic generation gives

which reduces the second order susceptibility in Eq. (1) to a set of 11 tensor elements, of which only 4 are independent. (We recall that the intrinsic permutation symmetry is always applicable, as being a consequence of the symmetrization described in lectures two and five.) Whenever Kleinman symmetry holds, the susceptibility is in addition symmetric under any permutation of the indices, which hence gives the additional relation

that is to say, reducing the second order susceptibility to a set of 11 tensor elements, of which only 3 are independent.

To summarize, the set of nonzero tensor elements describing second harmonic generation under Kleinman symmetry is

For the pump field at angular frequency ω, the relevant susceptibility describing the interaction with the second harmonic wave is [1]

For an arbitrary frequency argument, this is the proper form of the susceptibility to use for the fundamental field, and this form generally differ from that of the susceptibilities for the second harmonic field. However, whenever Kleinman symmetry holds, the susceptibility for the fundamental field can be cast into the same parameters as for the second harmonic field, since

Hence the second order interaction is described by the same set of tensor elements for the fundamental as well as the second harmonic optical wave whenever Kleinman symmetry applies.

### 3.4. The polarization density

Following the convention of Butcher and Cotter [2], the degeneracy factor for the second harmonic signal at 2ω is

where

that is to say, for the present case of second harmonic generation

For the fundamental optical field at ω, one might be mislead to assume
that since the second order interaction for this field is described by an
identical set of tensor elements as for the second harmonic wave, the
degeneracy factor must also be identical to the previously derived one.
This is, however, *a very wrong assumption*, and one can easily verify
that the proper degeneracy factor for the fundamental field instead is
given as

where

that is to say,

The general second harmonic polarization density of the medium is hence given as

while the general polarization density at the angular frequency of the pump field becomes [3]

For a pump wave polarized in the *yz*-plane of the crystal frame, the
polarization density of the medium hence becomes

and

### 3.5. The wave equation

Strictly speaking, the previously formulated polarization density
gives a coupled system between the polarization states of both the
fundamental and second harmonic waves, since both the *y*- and
*z*-components
of the polarization densities at ω and 2ω contain components
of all other field components.
However, for simplicity we will here restrict the continued analysis to the
case of a *y*-polarized input pump wave, which through the
χ^{(2)}_{zyy} = χ^{(2)}_{zxx}
elements give rise to a *z*-polarized second harmonic frequency component
at 2ω.

The electric fields of the fundamental and second harmonic optical waves
are for the forward propagating configuration expressed in their
*spatial envelopes* **A**_{ω} and
**A**_{2ω} as

Using the above separation of the natural, spatial oscillation of the light, in the infinite plane wave approximation and by using the slowly varying envelope approximation, the wave equation for the envelope of the second harmonic optical field becomes (see Eq. (6) in lecture eight)

while for the fundamental wave,

These equations can hence be summarized by the coupled system

where

is the *phase mismatch* between the pump and second harmonic wave.
In the case of perfect phase-matching, Δ*k* = 0.

### 3.6. Boundary conditions

Here the boundary conditions are simply that no second harmonic signal is present at the input,

together with a known input field at the fundamental frequency,

### 3.7. Solving the wave equation

Considering a nonzero Δ*k*, the conversion efficiency is regularly
quite small, and one may approximately take the spatial distribution of
the pump wave to be constant,
*A*^{y}_{ω}(*x*) ≈ *A*^{y}_{ω}(0)$.
Using this approximation [4],
and by applying the initial condition
*A*^{z}_{2ω}(0) = 0
of the second harmonic signal, one finds

In terms if the light intensities of the waves, one after a propagation
distance *x* = *L* hence has the second harmonic signal
with intensity *I*_{2ω}(*L*) expressed in terms of
the input intensity *I*_{ω}(*L*) expressed in terms
of as

that is to say, with the *conversion efficiency*

## 4. Optical Kerr-effect - Field corrected refractive index

As a start, we assume a monochromatic optical wave (containing forward
and/or backward propagating components) polarized in the *xy*-plane,

with all spatial variation of the field contained in

### 4.1. The optical interaction

Optical Kerr-effect is in isotropic media described by the third order susceptibility [5]

### 4.2. Symmetries of the medium

The general set of nonzero components of
χ^{(3)}_{μαβγ}
for isotropic media are from Appendix A3.3 of Butcher and Cotters book
given as

with

that is to say, a general set of 21 elements, of which only 3 are independent.

### 4.3. Additional symmetries

By applying the intrinsic permutation symmetry in the middle indices for optical Kerr-effect, one generally has

which hence slightly reduces the set (3) to still 21
nonzero elements, but of which now only two are independent.
For a beam polarized in the *xy*-plane, the elements of interest are
only those which only contain *x* or *y* in the indices, that is
to say, the subset

with

that is to say, a set of eight elements, of which only two are independent.

### 4.4. The polarization density

The degeneracy factor *K*(-ω;ω,ω,-ω) is
calculated as

From the reduced set of nonzero susceptibilities for the beam polarized in
the *xy*-plane, and by using the calculated
value of the degeneracy factor in the
convention of
Butcher and Cotter,
we hence have the third order electric polarization density at
ω_{σ} =ω given as

with

For the optical field being linearly polarized, say in the *x*-direction,
the expression for the polarization density is significantly simplified,
to yield

that is to say, taking a form that can be interpreted as an intensity-dependent
(∼|E^{x}_{ω}|^{2}) contribution to the
refractive index (see Butcher and Cotter §6.3.1).

### 4.5. The wave equation - Time independent case

In this example, we consider continuous wave propagation (That is to say, a
time independent problem with the temporal envelope of the electrical field
being constant in time) in optical Kerr-media, using light polarized in the
*x*-direction and propagating along the positive direction of the
*z*-axis,

where, as previously,
*k* = ω*n*_{0}/*c*.
From material handed out during the third
lecture (notes on the
Butcher and Cotter
convention), the nonlinear
polarization density for *x*-polarized light is given as

with

and the time independent wave equation for the field envelope
**A**_{ω}, again using
Eq. (6)
of lecture eight, becomes

or, equivalently, in its scalar form

### 4.6. Boundary conditions - Time independent case

For this special case of unidirectional wave propagation, the boundary condition is simply a known optical field at the input,

### 4.7. Solving the wave equation - Time independent case

If the medium of interest now is analyzed at an angular frequency far
from any resonance, we may look for solutions to this equation with
|**A**_{ω}(*z*)| being constant (for a lossless medium).
For such a case it is straightforward to integrate the final wave
equation to yield the general solution

or, again equivalently, in the scalar form

which hence gives the solution for the real-valued electric field
**E**(**r**,*t*) as

From this solution, one immediately finds that the wave propagates with an effective propagation constant

that is to say, experiencing the intensity dependent refractive index

with

## 10. References

[1] Keep in mind that in the convention of Butcher and Cotter, the frequency arguments to the right of the semicolon may be writen in arbitrary order, hence we may in an equal description instead use

for the description of the second order interaction between light and matter.

[2]
See course material on the Butcher and Cotter convention, handed out during the
third lecture.
Notice that for the first order polarization density, one at
optical frequencies *always* has the trivial degeneracy factor

[3] Keep in mind that a negative frequency argument to the right of the semicolon in the susceptibility is to be associated with the complex conjugate of the respective electric field; see Butcher and Cotter, section 2.3.2.

[4]
For an outline of the method of
solving the coupled system exactly in terms of Jacobian elliptic
functions (thus allowing for a depleted pump as well), see one of the
pioneering works in nonlinear optics by J. A. Armstrong,
N. Bloembergen, J. Ducuing, and P. S. Pershan, *Interactions between Light
Waves in a Nonlinear Dielectric*
Phys. Rev.
**127**, 1918-1939, (1962).

DOI: 10.1103/PhysRev.127.1918

[5] Again, keep in mind that in the convention of Butcher and Cotter, the frequency arguments to the right of the semicolon may be writen in arbitrary order, hence we may in an equal description instead use

or

for this description of the third order interaction between light and matter.