Download Applied Nonlinear Optics

Applied Nonlinear Optics
David C. Hutchings
Dept. of Electronics and Electrical Engineering
University of Glasgow
ii
Preface
This course will cover aspects of birefringence, the electro-optic effect and optical frequency
conversion. It will be necessary to have a good grounding in electro-magnetic theory and optics
prior to this course. Recommended textbooks for this material are Shen [1], Zernike and Midwinter [2] and Yariv [3]. Like the bulk of the literature in nonlinear optics, these books do not
always employ a consistent SI notation. I would recommend Butcher and Cotter [4], although
not as applied as the above references, for its consistent notation (SI units) and formalism.
iii
iv
Chapter 1
Birefringence
1.1 Dielectrics (revision)
In a vacuum, Gauss’ theorem states that the closed surface integral of the normal component of
the electric field to the surface is proportional to the total charge enclosed by the surface,
Z
E · ds = ∑
q
1
=
ε0 ε0
Z
ρ dV
,
(1.1)
where ρ is the charge density. The divergence theorem can be used to change this surface
integral to a volume integral,
Z
Z
∇ · E dV
E · ds =
,
(1.2)
which allows us to relate two volume integrals. Now since these are over the same region which
we can arbitrarily choose, the integrands must be equal, i.e.,
∇·E =
ρ
ε0
.
(1.3)
Hence we have used the divergence theorem to transform the original integral equation [Eq.
(1.1)] for the electric field into a differential one.
Let us now address the issue of an electric field in a dielectric medium. The electric field
will induce dipoles in the medium, for example by distortion of the electron clouds or aligning
polar molecules preferentially along the direction of the field. In an extended medium and
a uniform field the average charge density due to these induced dipoles is zero except at the
surfaces. This is because the charge on one end of an induced dipole will be neutralised by
the opposite charge on the end of an adjacent dipole. At the surface there is no adjacent dipole
to cancel the charge. Thus additional polarisation charges have been generated at the surface
which must be accounted for. Similar polarisation charges are generated in the case of a nonuniform field. A polarisation field P can be defined equal to the dipole moment per unit volume.
If there are N dipoles per unit volume consisting of charges +q and −q separated by r then
1
CHAPTER 1. BIREFRINGENCE
2
P = Nqr. If we assume for a moment that the induced charge separation r ∝ E,1 then the
polarisation field is proportional and parallel to the electric field. Conventionally this is written
in SI units as P = ε0 χE, where the dimensionless constant of proportionality χ is called the
optical susceptibility.
It can be shown that the effective charge density due to the polarisation of a medium is given
by, ρ p = −∇ · P. Inserting this into the differential form of Gauss’ theorem gives,
∇·E =
1
1
(ρ + ρ p ) = (ρ − ∇ · P)
ε0
ε0
,
(1.4)
where ρ denotes the free (not polarisation) charge density. This can be rearranged to give
∇ · (ε0 E + P) = ρ. For convenience a new field is introduced at this point equal to the quantity
inside the bracket, D = ε0 E + P, conventionally known as the electric displacement. Note that
unlike E and P the electric displacement is not immediately related to a physical quantity, but
allows various electro-magnetic relations to be written in a far simpler form. However, D can
be thought of as a vacuum field, consisting of the contribution to the electric field with the effect
of the dielectric medium subtracted. The simpler form of Eq. (1.4) is ∇ · D = ρ and is one of the
set of differential relations known as Maxwell’s equations. Using the susceptibility to substitute
for the polarisation field provides,
D = ε0 (1 + χ) E = ε0 εr E ,
(1.5)
where εr = 1 + χ is known as the dielectric constant or relative permittivity. Note that the
vacuum values of these dimensionless quantities are χ = 0 and εr = 1.
The divergence theorem and Stokes theorem are employed in deriving the remainder of
Maxwell’s equations. We shall state these without proof here as we will be making frequent use
of them throughout this course. In SI units they are given by:
∇·D = ρ ,
∇·B = 0 ,
∂B
∇×E = −
,
∂t
∂D
∇×H = j+
.
∂t
The current is related to the electric field by the conductivity, j = σE. In this course the usual
case will be of zero current j = 0 and non-magnetic material B = µ0 H.
1.2 The Dielectric Tensor
In an anisotropic media the dipoles may be constrained so their direction is different to that of
the electric field. One can think of numerous mechanical analogies where the motion of a mass
1 This
corresponds to the simplest case of a medium which is linear and isotropic. Most of the course considers
the extension of this to cases where the two vectors are (i) not parallel or (ii) not simply proportional.
1.2. THE DIELECTRIC TENSOR
3
is constrained by e.g. ramps or string so that acceleration is not parallel to the force. In such
systems, the constraint reduces the symmetry. If the dipoles are not parallel to the field, then
the polarisation P and the electric displacement D are also not parallel to the field E. Hence
Eq. (1.5) cannot be written with the relative permittivity as a scaler quantity. To generalise Eq.
(1.5) to anisotropic media (such that D and E can have different directions), the scaler dielectric
constant εr is replaced with a second rank tensor,

 


Dx
εxx εxy εxz
Ex
1
Dy  =  εyx εyy εyz   Ey  .
(1.6)
ε0
Dz
εzx εzy εzz
Ez
In a more compact notation,
1
Di =
ε0
∑
εi j E j
.
(1.7)
j=x,y,z
We can prove that the tensor εi j is symmetric using energy conservation. The energy density
of an electric field in a dielectric is,
1
1
We = E · D = ∑ Ei εi j E j
2
2 i, j
.
(1.8)
Differentiating gives the power flow into a unit volume,
¶
µ
∂E j
∂We 1
∂Ei
= ∑ εi j
E j + Ei
∂t
2 ij
∂t
∂t
.
(1.9)
We can also calculate power flow using the Poynting vector which is the power flow across a
unit area, S = E × H. Using the divergence theorem gives the power flow per unit volume,
∇ · S = ∇ · (E × H) = E · ∇ × H − H · ∇ × E
.
(1.10)
We can substitute for the vector curls using Maxwell’s equations and assuming no currents j = 0
gives a power flow per unit volume,
E·
∂D
∂B
+H·
∂t
∂t
.
(1.11)
The first of these terms relates to the rate of change of energy of the electric field,
∂E j
∂D
∂We
= E·
= ∑ εi j Ei
∂t
∂t
∂t
ij
.
(1.12)
Now these two forms of power flow must be the same and hence Eqs. (1.9) and (1.12) must be
equal. This can only be the case if ε ji = εi j and the dielectric tensor is symmetric having a total
of 6 different elements.
Note that we have not yet specified the axes for our cartesian co-ordinate system {x, y, z}. It
is always possible to choose a set such that the (symmetric) dielectric tensor is diagonal. The
CHAPTER 1. BIREFRINGENCE
4
basis of this is that a symmetric matrix is specific case of a Hermitian matrix and has pure real
eigenvalues and eigenvectors. Thus if we change to a new co-ordinate system where the axes
are parallel to these eigenvectors, the dielectric tensor becomes diagonal,

 


Dx
εx 0 0
Ex
1
Dy  =  0 εy 0   Ey  ,
(1.13)
ε0
Dz
0 0 εz
Ez
where εx = εxx etc. The set of axes in this case is known as the principal dielectric axes, which
may be different from the usual crystal axes.
The dielectric tensor may be further simplified by considering the crystal symmetry. There
are three distinctive cases summarised in table 1.1. It can be seen that generally the higher
the degree of symmetry, the lower the degree of birefringence. In most of the above crystal
systems the principal dielectric axes correspond to the usual cartesian crystalline axes. The two
exceptions to this are the monoclinic and triclinic crystal systems.


ε 0 0
Isotropic
 0 ε 0 
No birefringence
Cubic
0 0 ε
Uniaxial birefringence
(1 optic axis)
Hexagonal
Tetragonal
Trigonal
Biaxial birefringence
(2 optic axes)
Orthorhombic
Monoclinic
Triclinic


εx 0 0
 0 εx 0 
0 0 εz


εx 0 0
 0 εy 0 
0 0 εz
Table 1.1: The form of the dielectric tensor for the various crystal symmetries indicated.
1.3 EM Wave propagation
Now consider the propagation of a monochromatic electro-magnetic plane wave through the
medium. Such a wave has electric and magnetic fields given by,
i
1h
−i(ωt−k·r)
∗ i(ωt−k·r)
,
E0 e
+ E0 e
E=
2
i
h
1
H=
.
H0 e−i(ωt−k·r) + H∗0 ei(ωt−k·r)
2
(1.14)
(1.15)
The wavevector k = nωŝ/c where ŝ is a unit vector in the direction of propagation of the wave.
The phase velocity is given by v p = cŝ/n. Inserting these into Maxwell’s equation ∇ × E =
1.3. EM WAVE PROPAGATION
5
−∂B/∂t with a nonmagnetic medium (B = µ0 H) gives,
H0 =
1
n
k × E0 =
ŝ × E0
ωµ0
µ0 c
.
(1.16)
Thus the direction of the magnetic field amplitude is perpendicular to both the direction of
propagation ŝ and electric field E0 . Similarly we can use the Maxwell equation ∇ × H = j +
∂D/∂t with j = 0 to obtain,
1
n
D0 = − k × H0 = − ŝ × H0
ω
c
.
(1.17)
Hence the electric displacement is perpendicular to the direction of propagation ŝ and the magnetic field H0 . However, as we have seen for anisotropic media, E0 and D0 are not necessarily
parallel. We also note that the power flow is given by the Poynting vector S = E × H which
is perpendicular to E0 and H0 . The Poynting vector also provides the direction for the group
velocity. Fig. 1.1 shows the relative geometry of these vectors in an anisotropic medium. This
illustrates that the propagation direction ŝ and the Poynting vector are not necessarily parallel
and hence the phase and group velocities may also not be parallel.
E
D
vg
vp
S=ExH
H
k
Figure 1.1: Relative geometry of the field vectors and the phase and group velocities for an
em-wave in an anisotropic medium. The magnetic field H is directed out of the page.
Combining Eqs. (1.16) and (1.17) gives,
D0 = −
n2
ŝ × ŝ × E0 = n2 ε0 [E0 − ŝ (ŝ · E0 )]
µ0 c2
.
Consider just one component of this expression and introduce the dielectric tensor,
·
¸
2 D0i
D0i = n
− ε0 si (ŝ · E0 )
,
εi
µ
¶
1
1 −1
D0i = ε0 si (ŝ · E0 )
−
.
εi n2
(1.18)
(1.19)
CHAPTER 1. BIREFRINGENCE
6
Since D0 and ŝ are perpendicular, D0 · ŝ = 0. This provides the relation,
∑
µ
s2i
i
¶−1
n2
−1
=0
εi
,
(1.20)
which is known as Fresnel’s equation. This is quadratic in the square of the refractive index (n2 )
and therefore provides two possible solutions given a propagation direction ŝ. Hence the origin
of the name of this phenomenon; birefringence which means literally double refraction.
It can be shown that these two roots for the refractive index correspond to orthogonal polarisations. Let the two roots be n0 and n00 with electric displacement amplitudes D00 and D000
respectively. Using Eq. (1.19), the scaler product of these amplitudes can be written,
D00 · D000
¡
¢2
= ε0 n0 n00 ŝ · E0 ∑
s2i
,
02
002
i (n /εi − 1) (n /εi − 1)
µ
¶
n02
n002
(ε0 n0 n00 ŝ · E0 )2
− 002
s2i
=
02 /ε − 1
n002 − n02 ∑
n
n
/ε
−
1
i
i
i
,
(1.21)
where partial fractions have been employed to obtain the final form. Now Fresnel’s equation
(1.20) states that the summation over each of the terms is zero. Hence D00 · D000 = 0 and therfore
D00 and D000 are mutually perpendicular.
1.4 The Index Ellipsoid
Using Fresnel’s equation based on the propagation direction is rather cumbersome when analysing birefringence. A much easier method is based on the direction of the electric displacement.
The energy density of an electric field in a birefringent dielectric is,
Ã
!
1 D2x D2y D2y
1
+
+
.
(1.22)
We = E · D =
2
2 εx
εy
εy
√
Substituting εr = n2 for the various directions and setting Dx / 2We = x, etc. gives the following,
x 2 y 2 z2
+ + =1
n2x n2y n2z
.
(1.23)
This equation represents an ellipsoid and is conventionally referred to as the index ellipsoid or
optical indicatrix. The intercepts of this surface with the cartesian axes are at ±nx for the x-axis,
etc. This ellipsoid can be used to find the two allowed directions for the polarisation and their
associated refractive indices.
The remaining discussion will focus on uniaxial birefringence where ny = nx . This means
that the index ellipsoid will have cylindrical symmetry round the z-axis (optic axis). There are
two distinct cases of uniaxial birefringence; nz > nx known as positive unixial birefringence and
1.4. THE INDEX ELLIPSOID
7
nz < nx known as negative unixial birefringence. These are illustrated in Fig. 1.2. The use of
the index ellipsoid is as follows. The direction of propagation k is drawn from the origin and
makes an angle of θ to the optic axis. The intersection of the plane normal to the direction of
propagation and the index ellipsoid generates an ellipse with semi-axes a and b. For a uniaxial
crystal, the semi-axis a (minor semi-axis in the case of a positive uniaxial crystal, major semiaxis in the case of a negative uniaxial crystal) always lies in the xy-plane and therefore has a
length a = nx = no independent of the angle θ; a ray with this polarisation is called the ordinary
ray. The length of the other semi-axis is dependent on the angle θ, b = ne (θ); a ray with
this polarisation is called the extraordinary ray. When the direction of propagation is parallel
or perpendicular to the optic axis, the refractive index of the extraordinary ray can be written
down immediately, ne (θ = 0) = nz = ne and ne (θ = π/2) = nx = no . For the general case, we
decompose b = ne (θ) into components parallel and perpendicular to the optic axis, x2 + y2 =
(ne (θ) cos θ)2 and z = ne (θ) sin θ, and insert into the equation for the index ellipsoid to give for
uniaxial crystals,
1
cos2 θ sin2 θ
+ 2
.
(1.24)
=
n2e (θ)
n2o
ne
(a)
z
(b)
k
z
b
θ
x
x
a
n z >n x
n z <n x
Figure 1.2: The index ellipsoid or optical indicatrix for (a) a positive uniaxial crystal and (b) a
negative uniaxial crystal. The y-axis is directed into the page.
If we re-examine the derivation of Eq. (1.19), it can be seen that if n2 = εi for i = x, y or z
then ŝ · E0 = 0 and the electric field amplitude E0 is perpendicular to the propagation direction
and hence parallel to the electric displacement amplitude D0 . For the ordinary ray in a uniaxial
crystal this is satisfied for any propagation direction since n20 = εx = εy . Hence the derivation
of the term ordinary since the phase and group velocities are parallel as in a non-birefringent
medium. For the extraordinary ray, this condition only occurs for propagation parallel or perpendicular to the optic axis. In general, the phase and group velocities are not parallel.
Calcite (crystalline calcium carbonate) is a common uniaxial birefringent crystal and is useful in demonstrating the properties of birefringence. Place a piece of calcite over a piece of
CHAPTER 1. BIREFRINGENCE
8
paper with text on it and you will see a double image. If you have a polariser, place it over the
calcite and rotate it. You should find at some point that only one image will be visible. Rotate
the polariser by 90◦ and only the other image will be visible. This demonstrates that the two
images correspond to orthogonal polarisations. Now remove the polariser and rotate the calcite.
You should find that one of the images remains stationary and the other rotates about it. The stationary image corresponds to the ordinary ray where the phase and group velocities are parallel
and correspondingly the rotating image corresponds to the extraordinary ray.
1.5 Wave Plates
So far we have only considered light to be exclusively of ordinary or extraordinary polarisation. What happens in the general case where the light is some combination of these polarisation states? Suppose we initially have a linear polarisation Din
0 = D0 (ô cos φ + ê sin φ)
where ô and ê are unit vectors parallel to the ordinary and extraordinary polarisation directions respectively. Now suppose the crystal thickness d is such that the difference in optical
path length between the ordinary and extraordinary rays is an odd integer of half-wavelengths
(ne (θ) − no )d = (2N + 1)λ/2. A crystal of this thickness is termed a half-wave plate. On exit
2
the ordinary and extraordinary rays will be out of phase, Dout
0 = D0 (ô cos φ − ê sin φ). It can
be seen that this is equivalent to the transform φ → −φ and corresponds to flipping the polarisation angle around the optic axis. In particular for φ = 45◦ , the output (linear) polarisation is
perpendicular to the input.
Now let us consider a crystal of thickness such that the optical path length difference is an
odd number of quarter wave-lengths (ne (θ) − no )d = (2N + 1)λ/4 known as a quarter wave
plate. Now the ordinary and extraordinary rays will be π/2 out of phase with each other on
output from the crystal. Again if we consider
polarisation
√ the particular case of an input linearout
◦
in
at 45√ to the optic axis D0 = D0 (ô − ê)/ 2 then the output polarisation will be D0 = D0 (ô ±
iê)/ 2. The positive and negative cases correspond to the separate cases of Nλ + λ/4 and
Nλ + 3λ/4 optical path length differences. In this case we have generated circularly polarised
light from a linearly polarised input. A quarter wave plate can also perform the reverse, changing
circularly polarised light to linear (since two consecutive quarter wave plates make a half wave
plate).
2 For
simplicity the common phase factor has been omitted since the polarisation state only depends on the
relative phase of the polarisation components
Chapter 2
The Electro-optic Effect
2.1 The Electro-optic tensor
The change in refractive index with a DC electric field is known as the electro-optic effect. In
this chapter we will consider the linear electro-optic effect or Pockel’s effect where ∆n ∝ E.
There also exists the quadratic electro-optic effect or Kerr effect which is associated with higher
order effects considered later in this course. The principal application of the electro-optic effect
is in optical modulators where we use an external influence (applied voltage) to change the
optical properties of a material.
The general form of the optical indicatrix if the axes do not necessarily correspond to the
principal dielectric axes is,
x2
y2
z2 2yz 2xz 2xy
+
+
+
+
+
=1
εxx εyy εzz εyz
εxz
εxy
,
(2.1)
or in shorthand notation, ∑i, j=x,y,z εiijj = 1. Now conventionally the electro-optic coefficient r
relates the change in 1/ε (i.e. 1/n2 ) to the electric field E. Generalising this to anisotropic
crystals,
µ ¶ µ ¶
µ ¶
1
1
1
(2.2)
∆
=
−
= ri jk Ek .
εi j
εi j E
εi j E=0 ∑
k
The electro-optic coefficient ri jk is a third rank tensor as it relates a second rank tensor (1/εi j )
to a vector (E). It has 27 elements but only 18 are independent since ε ji = εi j and therefore
r jik = ri jk . This symmetry is employed to write the electro-optic tensor in a contracted notation
where the i j subscripts are replaced by: xx = 1, yy = 2, zz = 3, yz = zy = 4, xz = zx = 5 and
xy = yx = 6 and the k subscripts by: x = 1, y = 2 and z = 3. This allows the electro-optic tensor
9
CHAPTER 2. THE ELECTRO-OPTIC EFFECT
10
to be written as a 6 × 3 matrix:

∆(1/ε1 )
 ∆(1/ε2 )

 ∆(1/ε3 )

 ∆(1/ε4 )

 ∆(1/ε5 )
∆(1/ε6 )


 
 
 
=
 
 
 
r11
r21
r31
r41
r51
r61
r12
r22
r32
r42
r52
r62
r13
r23
r33
r43
r53
r63



 E1

  E2 

 E3

.
(2.3)
As in the case of the dielectric tensor, symmetry considerations provide information as to
which electro-optic coefficient tensor elements are non-zero and independent. An important
property is that materials with inversion symmetry exhibit no electro-optic effect. Consider Eq.
(2.2) under the application of the inversion operator. If the material has inversion symmetry then
the material parameters εi j and ri jk are unchanged. However, under inversion Ek → −Ek . Hence
∑k ri jk Ek = − ∑k ri jk Ek for any specified E which requires that all the electro-optical coefficients
are zero, ri jk = 0. For other symmetry classes, group theory can be employed to investigate the
form of the electro-optic tensor, as shown in table 2.1.
2.2 Examples
2.2.1 KDP
Potassium dihydrogen phosphate KH2 PO4 (known as KDP) belongs to the symmetry class 42m
and hence exhibits uniaxial birefringence with the principal dielectric axes being the same as
the conventional crystal axes. The electro-optic tensor has three non-zero components, two of
which are equal. Including these in the modified optical indicatrix gives,
x2 y2 z2
+ + + 2r41 Ex yz + 2r41 Ey xz + 2r63 Ez xy = 1 .
n2o n2o n2e
(2.4)
Now let us consider the case where the electric field is parallel to z such that Ex = 0 and Ey = 0
so the 4th and 5th terms can be ignored. Now as we stated in the previous chapter on birefringence, by selecting a new co-ordinate set, this equation can be transformed so only the diagonal
components remain. In this case we shall employ a cartesian
co-ordinate system
that is rotated
√
√
◦
0
0
0
0
by 45 about the z-axis. Thus we have x → (x − y )/ 2, y → (x + y )/ 2 and z → z0 . On
inserting these in Eq. (2.4) we obtain for the new optical indicatrix,
¶
µ
¶
µ
1
z02
1
02
02
+
r
E
x
+
−
r
E
y
+
=1 .
(2.5)
z
z
63
63
n2o
n2o
n2e
02 02
02 02
This is the same form as the optical indicatrix for a biaxial crystal, x02 /n02
x + y /ny + z /nz = 1
where,
¡
¢−1/2
r63 n3o Ez
≈ no −
n0x = no 1 + r63 Ez n2o
2
,
2.2. EXAMPLES
11
¢−1/2
¡
r63 n3o Ez
n0y = no 1 − r63 Ez n2o
≈ no +
2
n0z = ne .
,
(2.6)
Here we have used the binomial expansion to first order only since the refractive index changes
induced by the electro-optic effect are generally orders of magnitude smaller than the refractive
index itself. If we consider light propagating parallel to the z-axis then the relevant cross-section
of the optical indicatrix is shown before and after the application of an electric field in Fig. 2.1.
A ray of general polarisation will be split into components along the x0 and y0 directions and
these will have different phase velocities.
y
y
y’
no
x’
n’y
n’x
x
Ez
x
no
Figure 2.1: Change in the cross-section of the optical indicatrix for KDP on application of an
electric field parallel to the optic axis.
The electro-optic effect in this example can be considered as an induced birefringence.
Hence this effect can be employed in such applications as half and quarter wave plates. For
a half wave plate we require (n0y − n0x )d = λ/2, where d is the crystal thickness. Thus we require
(π)
(π)
the application of an electric field Ez where, r63 n3o Ez d = λ/2. Since the DC field is being
applied along the z-axis which is the same as the direction of propagation, the applied field is
the applied voltage divided by the crystal length, Ez = V /d. Hence we require for an induced
half wave plate a voltage V (π) = λ/(2n3o r63 ). For KDP at a visible wavelength of λ = 550 nm,
n3o r63 ' 34 pmV−1 and a voltage of V (π) ' 8 kV is required.
We found above that the electro-optic coefficient appears in the factor n3 r. This is true in all
cases and not just the above example. Hence in selecting materials for the electro-optic effect
it should be the factor n3 r which is compared and not just the raw electro-optic coefficient.
Fortunately most tabulations of electro-optic coefficients include the refractive index.
With the inclusion of polarisers at the input and output, this KDP example can be used
to construct an electro-optic amplitude modulator as shown in Fig. 2.2(a). This geometry is
termed a longitudinal modulator as the electric field is applied along the direction of propagation. With the change in polarisation, the transmission of the output polariser is given by
CHAPTER 2. THE ELECTRO-OPTIC EFFECT
12
T = sin2 πV /(2V (π) ) and is shown in Fig. 2.2(b). We note that for small voltages, the transmission will depend quadratically on the applied voltage. In many cases a linear modulator is
required. This could be achieved by biasing the voltage to the V π/2 point. Since this bias point
is equivalent to a quarter wave plate, a more elegant solution is to incorporate an additional
conventional quarter wave plate.
(a)
x
(b)
T
1
0.5
z
Input
Polariser
y
V
0
Output
Polariser
0
π/2
π
V
Figure 2.2: Longitudinal geometry for an electro-optic modulator based on KDP.
2.2.2 GaAs
Gallium Arsenide and other semiconductors of zinc-blende (cubic) symmetry belong to the
class 43m and do not normally exhibit birefringence. There are three non-zero electro-optic
tensor elements all of which are equal. GaAs is not transparent at visible wavelengths but can
be employed as an electro-optic modulator in the infrared (e.g. 10 µm). The KDP modulator is
longitudinal which means that changing the crystal length does not change the voltage required
since it equally affects the electric field and the optical path length. This means that the large
required voltage quoted in the example has no prospect of being reduced. Furthermore the light
has to pass through the electrodes so these have the complication of being transparent or having
a hole in them. A much more attractive geometry is the transverse one where the electric field is
applied perpendicular to the propagation direction. An example transverse amplitude modulator
geometry for GaAs and other zinc-blende semiconductors is shown in Fig. 2.3. For this example
with a crystal of length L and thickness d, the required voltage to achieve a π phase difference
between the two polarisation components is V (π) = λd/(2Ln3 r41 ). Note that we have gained a
factor of d/L in comparison to the longitudinal modulator and thus the voltage can be reduced
by increasing the length or decreasing the thickness. For GaAs at 10 µm, n=3.3, r14 =1.6 pmV−1
and taking L=5 cm and d=0.5 cm, we obtain V (π) '9 kV.
The two modulators discussed so far are examples of an amplitude modulator. These are
basically using the induced birefringence to change the polarisation, which is then sent through
a polariser. Also of use is a phase modulator. This is conceptually easier to follow as it directly
uses the change in refractive index to change the optical path length and hence the optical
phase. For the KDP case, the input polarisation would be aligned with either the x0 or y0 axes
(so only the ordinary or extraordinary ray is input and the polarisation is maintained), and then
the change in optical path length is, ∆nd = n3o r63V /2. A possible geometry for a GaAs phase
2.2. EXAMPLES
13
[110]
Input
Polariser
[001]
V
Output
Polariser
Figure 2.3: Transverse geometry for an electro-optic modulator based on GaAs.
electro-optic modulator is to use a transverse field parallel to [001] to modulate an optical beam
with linear polarisation parallel to [110]: the change in optical path length is given by, ∆nL =
n3 r14V L/(2d).
If a time varying voltage is applied to a phase modulator, the time varying phase is equivalent
to altering the frequency of the optical beam. In particular, if an optical beam of single frequency
ω0 is modulated with a sinusoidal voltage of frequency ωm , then the frequency spectrum of the
transmitted light develops side bands at ω0 ± ωm , ω0 ± 2ωm , etc. What we have accomplished
here is mixing of two frequencies, one optical and one electrical. This brings us neatly to the
subject of the next chapter: optical frequency mixing.
CHAPTER 2. THE ELECTRO-OPTIC EFFECT
14
Triclinic
Orthorhombic
Tetragonal
Cubic
Trigonal
Hexagonal
1
222
4
43m
23
3
6










































r11
r21
r31
r41
r51
r61
0
0
0
r41
0
0
0
0
0
r41
r51
0
0
0
0
r41
0
0
r11
r11
0
r41
r51
r22
0
0
0
r41
r51
0
r12
r22
r32
r42
r52
r62
0
0
0
0
r52
0
0
0
0
r51
r41
0
0
0
0
0
r41
0
r22
r22
0
r51
r41
r11
0
0
0
r51
r41
0
r13
r23
r33
r43
r53
r63
0
0
0
0
0
r63
r13
r13
r33
0
0
0
0
0
0
0
0
r41
r13
r13
r33
0
0
0
r13
r13
r33
0
0
0










































mm2
4
432
32
6mm

0
 0

 0

 0
 r
51
0

0
 0

 0

 r41
 r
51
0

0
 0

 0

 0
 0
0

r11
 r
 11
 0

 r41
 0
0

0
 0

 0

 0

r
51
0
0
0
0
r42
0
0
0
0
0
r51
r41
0
0
0
0
0
0
0
0
0
0
0
r41
r11
0
0
0
r51
0
0

r13
r 
23

r33 

0 
0 
0

r
13
r13 

0 

0 
0 
0

0
0 

0 

0 
0 
0

0
0 

0 

0 
0 
0

r13
r13 

r33 

0 
0 
0







Monoclinic
422
3m
622
0
0
0
r41
0
0
0
0
0
0
r41
0
0
0
0
0
0
0

r22 r13
0
r22 r13
 0

0
r33
 0

r51
0
 0
 r
0
0
51
r22
0
0

0
0
0
0
0
 0

0
0
 0

0
0
 r41

0
r
0
41
0
0
0





















2
4mm
6





















r11
r11
0
0
0
r22
0
0
0
0
r51
0
0
0
0
r41
0
r61
r22
r22
0
0
0
r11
0
0
0
r51
0
0
r12
r22
r32
0
r52
0
0
0
0
0
0
0
r13
r13
r33
0
0
0
0
0
0
r43
0
r63








m














42m













6m2







0
0
0
0
0
r22
0
0
0
r41
0
0
r11
r21
r31
0
r51
0
r22
r22
0
0
0
0
0
0
0
0
r41
0
0
0
0
r42
0
r62
0
0
0
0
0
0
0
0
0
0
0
r63
r13
r23
r33
0
r53
0





















Table 2.1: The form of the electro-optic tensor for the crystal symmetry classes with no inversion symmetry. A bar over an entry
indicates the negative.
Chapter 3
Optical Frequency Mixing
3.1 Lorentz model
In the Lorentz model the motion of the electrons in the medium is treated as a harmonic oscillator. This can be pictured as electrons attached to their nuclei by springs with resonant
frequency Ω and damping γ. An optical wave provides a forcing term through the dipole interaction with the electron and hence the motion of the electron around its equilibrium position
can be described by the linear differential equation,
d2 r(t)
dr(t)
e
+ 2γ
+ Ω2 r(t) = − E(t) .
2
dt
dt
m0
One way to solve this differential equation is to Fourier transform it: r(t) =
and hence the solution of Eq. 3.1 is straightforward,
r(ω) = −
E(ω)
e
2
m0 Ω − ω2 − 2iγω
.
(3.1)
R∞
−∞ r(ω)e
−iωt dω
(3.2)
Now the polarisation is given P = −Ner and the optical susceptibility is defined P(ω) =
ε0 χ(ω)E(ω), so the optical susceptibility can be written,
χ(ω) =
1
Ne2
2
ε0 m0 Ω − ω2 − 2iγω
.
(3.3)
p
Neglecting absorption, the refractive index is given by n(ω) = 1 + Reχ(ω) and is shown in
Fig. 3.1 for frequencies around resonance Ω. This is the characteristic shape of the refractive
index around a resonance consisting of a maximum at a frequency just below the transition and
a minimum just above; apart from a small frequency regime around the resonance, dn(ω)/dω
is positive which is termed normal dispersion.
Now suppose the potential for the electron is cannot strictly be described as that of a harmonic oscillator. We aim to describe the situation that the motion of the electron is large enough
such that the Taylor series expansion of the restoring form has significant terms of quadratic or
15
CHAPTER 3. OPTICAL FREQUENCY MIXING
16
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
-4
-2
0
2
4
Figure 3.1: The form of the dispersion of the refractive index around a resonance in the Lorentz
model. The frequency (horizontal axis) has been scaled as (ω − ω0 )/γ.
higher order. The differential equation describing the electron’s motion then has additional
anharmonic terms which we will consider to r2 ,
d2 r(t)
dr(t)
e
+ 2γ
+ Ω2 r(t) − ξr2 = − E(t) .
2
dt
dt
m0
(3.4)
The differential equation is now nonlinear and is complicated to solve. However it is usual that
the anharmonic term is small compared with the other terms and a perturbation analysis can be
performed to gain some insight. The displacement from equilibrium position is expanded as
r = r0 + r1 + . . .. The differential equation taken to the lowest order (in r0 ) is just the harmonic
Lorentz equation (3.1). The next highest order gives,
d2 r1 (t)
dr1 (t)
+
2γ
+ Ω2 r1 (t) = ξr02 (t) ,
2
dt
dt
(3.5)
where Eq. (3.2) is used to provide the form for r0 (t).
Consider the case where the optical field is monochromatic and can be described E(t) =
[E0 e−iω0t +E0∗ eiω0t ]/2. The Fourier transform of this is E(ω) = [E0 δ(ω−ω0 )+E0∗ δ(ω+ω0 )]/2.
The linear solution r0 will be driven at the same frequency and hence we obtain,
¸
·
E0∗ eiω0t
E0 e−iω0t
e
+
.
(3.6)
r0 (t) = −
2m0 Ω2 − ω20 − 2iγω0 Ω2 − ω20 − 2iγω0
When this expression is squared, the terms have time dependencies of e±2iω0t or are constant.
Thus there exists driving terms for r1 (t) is at frequencies ±2ω0 . Hence the anharmonic term
has resulted in the motion of the electron having a frequency component at twice the optical
frequency and so will the polarisation. Similarly there will also be DC components produced
from the constant term. What has happened here is that the anharmonic term has resulted in
3.1. LORENTZ MODEL
17
frequency mixing. Since we have only one optical frequency present the only combinations are
at ω0 ± ω0 , i.e. 2ω0 and 0.
The frequency mixing aspect can be seen more clearly if we take an optical input field that is
bichromatic, E(t) = [E1 e−iω1t + E2 e−iω2t + cc]/2 where the cc denotes the complex conjugate.
The linear solution r0 (t) will contain terms oscillating at the same frequencies,
·
¸
e
E1 e−iω1t
E2 e−iω2t
r0 (t) = −
+
+ cc
.
(3.7)
2m0 Ω2 − ω21 − 2iγω1 Ω2 − ω22 + 2iγω2
On squaring this will produce a term oscillating at ω1 + ω2 ,
¯
r02 (t)¯ω
1 +ω2
=
£ 2
¤−1 £ 2
¤−1
e2
−i(ω1 +ω2 )t
2
2
E
E
e
Ω
−
ω
−
2iγω
Ω
−
ω
−
2iγω
1
2
1
2
1
2
2m20
.
(3.8)
On inserting this into Eq. (3.5) this provides for r1 (t) (e.g. by the Fourier transform method
described previously),
r1 (t)|ω1 +ω2 =
£
¤−1
ξe2
E1 E2 e−i(ω1 +ω2 )t Ω2 − (ω1 + ω2 )2 − 2iγ(ω1 + ω2 )
2m0
£
¤−1 £ 2
¤−1
× Ω2 − ω21 − 2iγω1
Ω − ω22 − 2iγω2
.
(3.9)
Hence since the polarisation is given by P = −Ner, there is also a polarisation component
oscillating at the sum frequency,
£
¤−1
Nξe3
E1 E2 δ(ω − ω1 − ω2 ) Ω2 − (ω1 + ω2 )2 − 2iγ(ω1 + ω2 )
2m0
£ 2
¤−1 £ 2
¤−1
× Ω − ω21 − 2iγω1
Ω − ω22 − 2iγω2
.
(3.10)
P(ω)|ω1 +ω2 = −
We can extend the definition of the optical susceptibility to include these higher order effects:
P(ω)|ω1 +ω2 = 2ε0 χ(2) (ω1 , ω2 )E1 E2 δ(ω − ω1 − ω2 ) ,
(3.11)
where χ(2) is referred to as the second-order optical susceptibility. The factor of 2 is included
since we must allow for both orderings E1 E2 and E2 E1 . Eq. 3.11 is specific to the interaction of
two monochromatic sources. It can be generalised by summing over all possible frequencies,
P(2) (ω) = ε0
Z ∞
−∞
Z ∞
dωa
−∞
dωb χ(2) (ωa , ωb )E(ωa )E(ωb )δ(ω − ωa − ωb ) .
(3.12)
The delta function ensures that the output polarisation oscillates at summations of the frequencies. Although this form looks a bit different from the linear definition of the optical susceptibility, we can write,
(1)
P (ω) = ε0
Z ∞
−∞
dωa χ(1) (ωa )E(ωa )δ(ω − ωa ) = ε0 χ(1) (ω)E(ω) ,
(3.13)
CHAPTER 3. OPTICAL FREQUENCY MIXING
18
which shows the same form. The total contribution to the polarisation is then P(ω) = P(1) (ω) +
P(2) (ω).
By comparing Eqs. (3.10) and (3.11) we obtain,
¤−1
Nξe3 £ 2
χ (ω1 , ω2 ) = −
Ω − (ω1 + ω2 )2 − 2iγ(ω1 + ω2 )
4ε0 m0
£
¤−1 £ 2
¤−1
× Ω2 − ω21 − 2iγω1
Ω − ω22 − 2iγω2
,
(2)
ξε20 m20 (1)
χ (ω1 + ω2 )χ(1) (ω1 )χ(1) (ω2 ) ,
(3.14)
4N 2 e3
where we have substituted the linear susceptibility in the final form. This shows that if we
have a resonance in the linear susceptibility either at one of the frequency components or their
sum, there is likely to be a resonance in the second-order susceptibility. Although this result is
specific to the anharmonic Lorentz model, there is an empirical rule called Miller’s rule which
states χ(2) (ω1 , ω2 ) = ∆χ(1) (ω1 + ω2 )χ(1) (ω1 )χ(1) (ω2 ), where the variation of ∆ with frequency
and material is much smaller than in the linear and nonlinear susceptibilities themselves.
= −
3.2 The Nonlinear Susceptibility Tensor
In the previous section, the vectorial nature of the electric field and polarisation was ignored. As
in the case of birefringence, this can be allowed for by using tensors for the linear and nonlinear
susceptibilities. We can also expand the nonlinear polarisation beyond second-order so that
(1)
(2)
(3)
Pi (ω) = Pi (ω) + Pi (ω) + Pi (ω) + . . . and the nonlinear polarisation contributions can be
written,
(1)
Pi
= ε0
(2)
Pi
= ε0
(3)
= ε0
Pi
Z ∞
Z−∞
∞
Z−∞
∞
−∞
(1)
dωa χi j (ωa )E j (ωa )δ(ω − ωa ) ,
Z ∞
dωa
dωa
Z−∞
∞
−∞
(2)
dωb χi jk (ωa , ωb )E j (ωa )Ek (ωb )δ(ω − ωa − ωb ) ,
Z ∞
dωb
−∞
(2)
dωc χi jkl (ωa , ωb , ωc )E j (ωa )Ek (ωb )El (ωc )
×δ(ω − ωa − ωb − ωc ) .
..
.
. = ..
(3.15)
Summation over the repeated indices j, k and l is implicit in the above.
3.2.1 1st order
(1)
χi j is a second rank tensor (9 elements) and is equal to the dielectric tensor less the identity
matrix (the tensor form allows birefringence to be described). The frequency of the polarisation
has to be the same as that of the electric field. Depending on the relative phase of the polarisation and the electric field, the interference gives rise to optical absorption or refraction; Reχ(1)
corresponds to refraction and Imχ(1) to absorption.
3.2. THE NONLINEAR SUSCEPTIBILITY TENSOR
3.2.2
19
2nd order
(2)
χi jk is a third rank tensor (27 elements). If we use a monochromatic input E(t) = (E0 e−iω0t +
E∗0 eiω0t )/2 then evaluating the integrals gives for the second-order polarisation,
(2)
Pi (ω) =
ε0
4
i
nh
(2)
(2)
∗
χi jk (ω0 , −ω0 )E0 j E0k
+ χi jk (−ω0 , ω0 )E0∗ j E0k δ(ω − 0)
(2)
+χi jk (ω0 , ω0 )E0 j E0k δ(ω − 2ω0 )
o
(2)
∗
+χi jk (−ω0 , −ω0 )E0∗ j E0k
δ(ω + 2ω0 )
.
(3.16)
As we indicated in the Lorentz anharmonic model, for a monochromatic input of frequency
ω0 , the second-order polarisation has frequency components at ±2ω0 and 0. These give rise
to second harmonic generation and optical rectification respectively. Note that there are no
components at the original frequency ω0 . If we have a combination of frequencies present (ω1
and ω2 say) we can produce the sum (ω = ω1 + ω2 ) and difference (ω = ω1 − ω2 ). This even
applies if ω2 = 0 i.e. DC and gives an alternative description of the electro-optic effect.
3.2.3 3rd order
(3)
χi jkl is a fourth rank tensor (81 elements). For a monochromatic input of frequency ω0 , evaluation of the integrals gives frequency components of the third-order polarisation at ω = ±3ω0
(which describes third harmonic generation) and at ω = ±ω0 . Since we have a component at
the same frequency this will act like an absorption or refraction, only in this case the effect will
be nonlinear because of the additional electric field terms.
3.2.4 Properties of the susceptibility tensor
Intrinsic permutation symmetry In Eq. (3.15) we are at liberty to exchange pairs of indices
j, k, etc. since these are just dummy indices which are summed over the directions x, y
and z. Since the electric field component product commutates, then we must have, for
example,
(2)
(2)
χik j (ω2 , ω1 ) = χi jk (ω1 , ω2 ) .
(3.17)
This property where the direction indices and frequency arguments can be permutated
[e.g. ( j, ω1 ) ⇔ (k, ω2 )] is called intrinsic permutation symmetry.
Reality condition The field E(t) and the polarsiation P(t) are physical quantities and hence
must be real. This then requires of the susceptibility, for example,
(2)
(2)∗
χi jk (−ω1 , −ω2 ) = χi jk (ω1 , ω2 ) ,
that is the conjugate of the susceptibility is equivalent to negating the frequencies.
(3.18)
CHAPTER 3. OPTICAL FREQUENCY MIXING
20
Overall permutation symmetry If all the optical frequencies and their combinations are well
removed from any material resonance then in addition to the intrinsic permutation symmetry, overall permutation symmetry also applies where the combination (i, −ω1 − ω2 ) is
included in the sets which can be permutated leaving the nonlinear susceptibility invariant. So, for example,
(2)
(2)
χ jik (−ω1 − ω2 , ω2 ) ' χi jk (ω1 , ω2 ) .
(3.19)
Note that unlike intrinsic permutation symmetry, overall permutation symmetry is only
an approximation, but one which is valid in most cases of interest. Now consider the
low frequency limit such that the dispersion in the susceptibility can be ignored. In this
situation, all frequencies could be replaced by zero which implies that all frequencies are
equivalent. Thus the susceptibility will be invariant if the frequency arguments alone are
permuted. Combining this with overall permutation symmetry means that the nonlinear
susceptibility is invariant under permutations of the direction indices. This gives, for
example,
(2)
(2)
χik j (ω1 , ω2 ) ' χi jk (ω1 , ω2 ) .
(3.20)
This property is called Kleinmann symmetry. Once again it is important to note that this
is just an approximation that applies in the low frequency limit, far from any material
resonances.
Causality So far we have been using the frequency domain for describing the optical polarisation. The linear susceptibility was defined P(ω) = ε0 χ(ω)E(ω) which is a product.
Now under Fourier transforming, a product becomes a convolution so we have in the time
domain,
Z
ε0 ∞
P(t) =
χ(τ)E(t − τ) dτ .
(3.21)
2π −∞
Now the principle of causality states that any feature in the input (electric field in this
case) cannot affect the output (polarisation) at earlier times. That is effect cannot precede
cause. Hence in the above convolution E(t − τ) cannot influence P(t) if t < t − τ i.e. τ < 0.
This then requires χ(τ) = 0 for τ < 0. One way of expressing this is to set χ(τ) = χ(τ)θ(τ)
where θ(τ) is the Heaviside (step) function. If we Fourier transform this expression, the
product becomes a convolution and we obtain the relationship,
1
χ(ω) = P
iπ
Z ∞
χ(Ω) dΩ
−∞
Ω−ω
,
(3.22)
where P is used to denote a principal parts integral. Because of the extra factor i in
this relation, by separating this equation into real and imaginary parts, one can relate the
real part of χ solely in terms of its imaginary part and vice versa. Hence if only Imχ is
supplied (across the entire spectral range), Reχ can be generated. This is one example
of a dispersion relation of which the best known is the Kramers-Krönig relation relating
refractive index to absorption coefficient. For a wider discussion on dispersion relations
and their applicability to the nonlinear case see [5].
Chapter 4
Second-Order Optical Nonlinearities
4.1 Contracted tensor for 2nd-order nonlinearities
In determining the second-order polarisation P(2) (ω), summation takes place over all permutations, e.g. in the case of second harmonic generation, if the electric field has components along the y and z axes, there will be a polarisation generated parallel to the x axis:
(2)
(2)
χxyz Ey Ez + χxzy Ez Ey . Since the same combination of fields occurs in both terms then we can
contract these to a single term. If we use P(2) (ω) = [P2ω0 δ(ω − 2ω0 ) + P−2ω0 δ(ω + 2ω0 )]/2
and E(ω) = [Eω0 δ(ω − ω0 ) + E−ω0 δ(ω + ω0 )]/2 then we can re-write the second-order polarisation as,


(Exω0 )2



  (Eyω0 )2 

Px2ω0
d11 d12 d13 d14 d15 d16 
 (Ezω0 )2 
 2ω0 


(4.1)
 Py  = ε0  d21 d22 d23 d24 d25 d26  
ω0 ω0  ,
2E
E
y
z
2ω0


d31 d32 d33 d34 d35 d36 
Pz
2Exω0 Ezω0 
2Exω0 Eyω0
where the second subscript on the coefficient d relates to the conventional axes by, 1:xx, 2:yy,
3:zz, 4:yz or zy, 5:zx or xz and 6:xy or yx. It can be seen that the contracted d-tensor for SHG
(2)
is simply related to the conventional susceptibility tensor by di jk (ω, ω) = χi jk (ω, ω)/2. Note
that we have used intrinsic permutation symmetry to reduce the 27 elements to 18 independent
ones.
If Kleinmann symmetry can be applied then the contracted tensor notation can be extended
to nondegenerate interactions (ω1 6= ω2 ),


Exω1 Exω2



 ω1 +ω2 
Eyω1 Eyω2


Px
d11 d12 d13 d14 d15 d16 
ω1 ω2

E
E
z
z
ω
+ω
1
2
 = 2ε0  d21 d22 d23 d24 d25 d26   ω1 ω2
 Py
, (4.2)
ω2 ω1 

Ey Ez + Ey Ez 
ω1 +ω2


d31 d32 d33 d34 d35 d36  ω1 ω2
Pz
Ez Ex + Ezω2 Exω1 
Exω1 Eyω2 + Exω2 Eyω1
21
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
22

Triclinic
1
d12
d22
d32
d13
d23
d33
d14
d24
d34
d15
d25
d35

d16
d26 
d36
0
d22
0
0
d23
0
d14
0
d34
0
d25
0

d16
0 
d36
0
0
0
0
0
0
d14
0
0
0
d25
0

0
0 
d36
0
0
0
0
0
0
0
0
d31
0
0
0
0
0
0
0
0
d33
d14
0
0
d14
0
0
d14
d15
0
0
d14
0
0
d14
0

d15 0
d14 0 
0 0
0
0 
0 
0
0 
d36
0
0
0
0
0
0
d11
 d21
d31

Monoclinic
2
0
 d21
0

Orthorhombic
222
0
 0
0

Tetragonal
4


422


42m

0
0
d31
0
0
0
0
0
0

Cubic
432
0
 0
0

Trigonal
3
3m
d11
 d22
 d31
0
 d22
d31

Hexagonal
6


622


6m2

d11
d22
d31
0
d22
d31
0
0
0
0
d31 d31
0 0 0
0 0 0
0 0 0
0
0
d22 d22
0
0
0
0
0
0
0
0

0
0 
0

m
d11
0
d31
d12
0
d32
d13
0
d33
0
d24
0
d15
0
d35

0
d26 
0
0
 0
d31
0
0
d32
0
0
d33
0
d24
0
d15
0
0

0
0 
0
0
0
d31
0
0
d31

0 d14 d15
0
0 d15 d14
0 
0
0
0
d36 
0
0
d15 0
d15
0
0 
0
d33
0
0
0


mm2

4
4mm
0
0
 d31
0
 0
d31


43m
23

d22
d11 
0 
d22
0 
0
0
0
d33
0
0
d33
d14
d15
0
0
d15
0
d15
d14
0
d15
0
0
0
0
d33
d14
0
0
0 0
0 0
0 0
d14
d15
0
0
d14
0
0
0
0

d15 0
d14 0 
0 0
0
0 
0 
d22
0 
0
0
 0
0

32
d11
 0
0

6
6mm
d11
 d22
 0
0
 0
d31
0
0
0
0
0
0
d14
0
0
0
d14
0
d11
0
0
0
0
0
d14
0
0
d11
d22
0
0
0
d31
0 0 0
0 0 0
0 0 0
0
0
0
d15
d33
0

0
0 
d14
0
d14
0

0
d11 
0

d22
d11 
0

d15 0
0
0 
0
0
Table 4.1: The form of the d-tensor for the crystal symmetry classes which do not possess
inversion symmetry. A bar over an entry indicates the negative.
(2)
(2)
where di jk (ω1 , ω2 ) = χi jk (ω1 , ω2 )/2 = χi jk (ω2 , ω1 )/2.
As in the case of the electro-optic coefficient, the number of independent elements can be
further reduced with symmetry considerations. First of all if the material exhibits inversion
symmetry then all the d tensor elements are zero. If we start from Pi2ω = ε0 di jk E ωj Ekω and
then apply the inversion operator, for materials with inversion symmetry di jk is unaltered and,
−Pi2ω = ε0 di jk (−E ωj )(−Ekω ). This is only consistent if di jk = −di jk and hence di jk = 0. Group
theory can be applied to investigate other symmetry properties. Table 4.1 shows the form of the
d-tensor for the 18 crystal classes that do not exhibit inversion symmetry.
If Kleinmann symmetry can be applied (low frequency, well removed from resonances) the
4.2. EM PROPAGATION WITH A SECOND-ORDER NONLINEARITY
23
18 tensor elements are further reduced to 10. The relevant equalities are summarised in Table
4.2. As an example consider KTP (KTiOPO4 ) which has orthorhombic symmetry of class mm2
and hence by Table 4.1 has 5 independent, non-zero components. At a wavelength of 880 nm,
these have been measured as [6]: d15 = 2.04 pmV−1 , d31 = 2.76 pmV−1 , d24 = 3.92 pmV−1 ,
d32 = 4.74 pmV−1 and d33 = 18.5 pmV−1 . Kleinmann symmetry specifies d15 ' d31 and d24 '
d32 which is only approximately true (within around 30%) in this case. Note that in this example
the on-diagonal element d33 ≡ dzzz is several times larger than the off-diagonal elements. This
behaviour is quite common among materials which exhibit a second order nonlinearity.
di jk
dxyy = dyxy
dxzz = dzxz
dxyz = dyzx = dzxy
dxzx = dzxx
dxxy = dyxx
dyzz = dzyz
dyyz = dzyy
di j
d12 = d26
d13 = d35
d14 = d25 = d36
d15 = d31
d16 = d21
d23 = d34
d24 = d32
Table 4.2: Equalities among the d-tensor elements under application of Kleinmann symmetry.
Rather than completely write out Eqs. (4.1) and (4.2) every time, we will denote this by the
shorthand tensor multiplication,
Pω1 +ω2 = 2ε0 d(ω1 , ω2 ) : Eω1 Eω2
.
(4.3)
Furthermore, it is quite common to split the fields into magnitude and direction, i.e. Eω1 = ê1 E ω1
etc. where ê1 is a unit vector. A scaler quantity deff is commonly used which hides the details
of the tensor multiplication: deff = ê p · [d(ω1 , ω2 ) : ê1 ê2 ]. This allows the vector equation (4.3)
to be written in scaler form.
4.2 EM Propagation with a Second-Order Nonlinearity
4.2.1 Slowly Varying Envelope Approximation
Start from the Maxwell curl equations ∇ × E(t) = −∂B(t)/∂t and ∇ × H(t) = j(t) + ∂D(t)/∂t,
assume a non-magnetic material B = µ0 H and take j = σE. We will split the polarisation into
linear and nonlinear components, D = ε0 E + P = ε0 εr E + PNL . Combining the two curl equations then gives the second order PDE,
∂2 PNL (t)
∂E(t) εr ∂2 E(t)
− 2
− µ0
∇ × ∇ × E(t) = −µ0 σ
∂t
c ∂t 2
∂t 2
.
(4.4)
Now the properties of the vector differential operators gives that ∇ × ∇ × E = ∇(∇ · E) − ∇2 E
and if there are no free charges, we also have ∇ · E = 0. The time derivatives can also be
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
24
simplified by Fourier transforming to the frequency domain to give,
µ
¶
ω2 εr
2
∇ E(ω) = −iωµ0 σ − 2
E(ω) − µ0 ω2 PNL (ω) .
c
(4.5)
Now let us insert a plane wave propagating in one direction only, which we will take to be
forward (taken as parallel to the z-axis here). We will allow the amplitude of this wave to vary
√
on propagation and take E(ω) = Ê(ω, z)eikz where the value of the wavevector k = εr ω/c is
obtained from solution of the wave equation [Eq. 4.5 without the nonlinear polarisation term].
On taking the space differential,
· 2
¸
d2 E(ω)
d Ê(ω, z)
dÊ(ω, z)
2
2
∇ E(ω) →
=
+ 2ik
− k Ê(ω, z) eikz .
(4.6)
dz2
dz2
dz
Note that the final term proportional to Ê will exactly cancel a term in Eq. 4.5. Now let us
assume that the nonlinear term is small (which is the case except in extreme circumstances
which will not be dealt with here) and hence the modification from the linear case will be small.
This tells us that the envelope will depart only slightly from its linear (constant) value. It will
certainly vary much more slowly than the oscillation of the underlying wave — hence this
approximation is conventionally termed the slowly varying envelope (or amplitude) approximation. This allows us to use |d2 Ê/dz2 | ¿ |kdÊ/dz| and neglect the second-order derivative in
the envelope. Hence we reduce the second order differential equation to the first order one,
dÊ(ω, z)
α
iω2 µ0 NL
= − Ê(ω, z) +
P (ω)e−ikz
dz
2
2k
,
(4.7)
where we have substituted the absorption coefficient α = ωµ0 σ/k. Although this is termed the
slowly varying envelope approximation, the key approximation is the unidirectional nature of
the wave propagation. If propagation both in the forward and backward directions occurs, then
coupled terms arise which prevent this simplification. On the right-hand side of Eq. 4.7 the first
term describes linear loss and the second term describes a nonlinear polarisation source for the
wave.
Eq. (4.7) is relevant for any order of nonlinearity. In this chapter we are concentrating on
second order nonlinearities for frequency mixing. Let us consider the interaction between two
waves of frequency ω1 and ω2 , producing a polarisation at the sum frequency ω3 = ω1 + ω2 .
Using the contracted d-tensor notation we have, Pω3 = 2ε0 d(ω1 , ω2 ) : Eω1 Eω2 . Inserting this
into Eq. (4.7) gives,
dÊω3
dz
dÊω1
dz
dÊω2
dz
iω23
α3 ω3
Ê +
d(ω1 , ω2 ) : Êω1 Êω2 ei∆kz ,
2
2
2k3 c
iω21
α1
= − Êω1 +
d(ω3 , −ω1 ) : Êω3 Ê−ω2 e−i∆kz
2
2
2k1 c
iω22
α2
= − Êω2 +
d(ω3 , −ω1 ) : Êω3 Ê−ω1 e−i∆kz
2
2k2 c2
= −
,
,
(4.8)
4.2. EM PROPAGATION WITH A SECOND-ORDER NONLINEARITY
25
where we have used ∆k = k1 + k2 − k3 and k3 = k(ω3 ) etc. Also of relevance the interactions
where the difference between ω3 and ω1 produces ω2 and the difference between ω3 and ω2
produces ω1 which are also listed. Note that Ê−ω1 = (Êω1 )∗ .
4.2.2
Manley-Rowe Relations
Now consider the case that we are considering frequencies well removed from any material
resonances. This is equivalent to stating we are concerned with a frequency region that is
transparent and hence α1 , α2 , α3 = 0. We can also apply overall permutation symmetry which
states d(ω3 , −ω2 ) = d(ω3 , −ω1 ) = d(ω1 , ω2 ). For simplicity we use the substitution deff =
d(ω1 , ω2 ) : ê1 ê2 . Hence Eqs. (4.8) can be rewritten as,
dÊ ω3
dz
dÊ ω1
dz
dÊ ω2
dz
iω23
deff Ê ω1 Ê ω2 ei∆kz ,
2k3 c2
iω21
=
deff Ê ω3 (Ê ω2 )∗ e−i∆kz
2k1 c2
iω22
=
deff Ê ω3 (Ê ω1 )∗ e−i∆kz
2
2k2 c
=
,
.
(4.9)
In SI units the irradiance is defined in terms of the electric field amplitude as, I ω =
ε0 cn(ω)|E ω |2 /2, where n(ω) is the refractive index. Differentiating this gives,
·
µ ω ¶∗ ¸
ω
dI ω ε0 cn0 (ω)
ω ∗ dE
ω dE
=
(E )
+E
.
(4.10)
dz
2
dz
dz
Inserting the electric field derivatives and using k = nω/c then gives,
i
¢∗
¡
¢∗
¡
dI ω3
ε 0 ω3 h
ideff Ê ω1 Ê ω2 Ê ω3 ei∆kz − ideff Ê ω1 Ê ω2 Ê ω3 e−i∆kz
=
dz
2
ω
1
¡ ω ω ¢∗ ω −i∆kz
¡ ω ¢∗ i∆kz i
dI
ε 0 ω1 h
ω1 ω2
1
2
3
=
ideff Ê Ê
Ê e
− ideff Ê Ê
Ê 3 e
dz
2
i
¢∗
¡
¢∗
¡
dI ω2
ε 0 ω2 h
=
ideff Ê ω1 Ê ω2 Ê ω3 e−i∆kz − ideff Ê ω1 Ê ω2 Ê ω3 ei∆kz
dz
2
We can see from the above that,
1 dI ω3
1 dI ω1
1 dI ω2
=−
=−
ω3 dz
ω1 dz
ω2 dz
.
,
,
.
(4.11)
(4.12)
Now irradiance is defined as optical energy flowing through a unit area per unit time. The
photon energy is h̄ω and hence the ratio of the irradiance to the optical frequency is proportional
to the number of photons passing through a unit area per unit time or in other words the photon
flux. Hence Eq. (4.12) can be restated as the change in the number of photons at ω3 is the
negative of the change in the number of photons at ω1 or ω2 (which are equal): ∆N3 = −∆N1 =
−∆N2 . These are known as the Manley-Rowe relations. These seem intuitively correct from
simple energy conservation; we are far from material resonances so the only energy flow can be
between the waves of different frequency.
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
26
4.2.3 Sum frequency generation: ω1 + ω2 → ω3
Let us assume that there is initially no light of the sum frequency ω3 present. Let us also initially
examine the case of low conversion efficiency such that any depletion of the frequencies ω1
and ω2 can be neglected, i.e. dE ω1 /dz, dE ω2 /dz ≈ 0. This simplification allows just one of the
differential equations to be studied instead of the complete set. Let us also assume that the linear
loss can be neglected, α3 = 0 and hence we have for the evolution of the light at frequency ω3 ,
dÊ ω3
iω3
=
deff Ê ω1 Ê ω2 ei∆kz
dz
cn3
,
(4.13)
where we have used as shorthand for the refractive index n3 = n(ω3 ). Since Ê ω1 and Ê ω2 are
treated as constant (low conversion efficiency), this can be easily integrated along the length of
the crystal to give,
ω3 deff ω1 ω2 (ei∆kL − 1)
Ê Ê
cn3
∆k
∆kL
ω3 deff L ω1 ω2 i∆kL/2
Ê Ê e
sinc
=
cn3
2
Ê ω3 (L) =
,
(4.14)
Now writing this instead in terms of the irradiance, I ω =
where sinc x = (sin x)/x.
ε0 cn(ω)|E ω |2 /2,
2
2ω23 deff
∆kL
I (L) =
L2 I ω1 I ω2 sinc2
3
ε0 c n1 n2 n3
2
ω3
.
(4.15)
There are several points to make about the generation of the sum frequency described by Eq.
2 /n3 (we
(4.15): the irradiance of the generated light is (1) proportional to a material factor deff
will see this is the usual factor in second order processes) (2) proportional to both the irradiance at ω1 and at ω2 (3) grows quadratically with distance (L2 ) and (4) depends on the factor
sinc2 ∆kL/2. This last factor is shown plotted in Fig. 4.1. It can be seen that this has a maximum value of 1 at ∆kL/2 = 0 but falls off rapidly away from this. Hence it is important for
efficient generation to operate at ∆k = 0 i.e. k1 + k2 = k3 . This is termed phasematching since
we are matching the phase velocity of the existing wave (k3 ) to that of the nonlinear polarisation (k1 + k2 ). Phase-matching can be thought of as photon momentum conservation (since the
photon’s momentum is given by h̄k) just as the Manley-Rowe relations describe photon energy
conservation.
4.2.4 Second harmonic generation: ω + ω → 2ω
Second harmonic generation is just a special case of sum frequency generation where an optical
wave interacts with itself to generate the sum frequency. Instead of three coupled differential
equations to consider, in this case we require just two. Let us assume that we are removed from
resonances so that linear loss can be neglected and overall permutation symmetry applied. In
the low conversion efficiency approximation we get a result similar to the previous case with a
4.2. EM PROPAGATION WITH A SECOND-ORDER NONLINEARITY
27
1.0
0.8
0.6
0.4
0.2
−4π
−2π
0
2π
4π
Figure 4.1: Plot of sinc2 x, indicating the sensitivity of phase-matching.
sinc2 ∆kL/2 phase=matching dependence. Here we shall include the effects of pump depletion
so we require both differential equations,
¡ ¢2
dÊ 2ω
iω
=
deff (ω, ω) Ê ω ei∆kz ,
dz
cn2ω
ω
¡ ¢∗
dÊ
iω
=
deff (ω, ω)Ê 2ω Ê ω e−i∆kz
dz
cnω
,
(4.16)
where ∆k = 2kω − k2ω . We can combine these two by differentiating the second equation and
substituting for dÊ 2ω /dz from the first to give,
2 h
¯ ω ¯2
¯ 2ω ¯2 i ω
ω2 deff
d2 Ê ω
dÊ ω
¯
¯
+ i∆k
+ 2 2
nω Ê
− n2ω ¯Ê ¯ Ê = 0
dz2
dz
c nω n2ω
.
(4.17)
Now if we initially have no second harmonic, Ê 2ω (z = 0) = 0, then we can use Manley-Rowe
to get I 2ω (z) = I ω (z = 0) − I ω (z). In terms of electric fields this gives, n2ω |Ê 2ω |2 = nω (|Ê ω (z =
0)|2 − |Ê ω |2 ). Substituting gives a differential equation in Ê ω alone,
2 h ¯
¯ ¯
¯2 i ω
ω2 deff
dÊ ω
d2 Ê ω
ω ¯2 ¯ 2ω
¯
+ i∆k
+ 2
2 Ê
− Ê (z = 0)¯ Ê = 0
dz2
dz
c nω n2ω
.
(4.18)
Now let us just consider the case of perfect phase-matching ∆k = 0. In that case the solution to
2 |Ê ω (z = 0)|2 /(c2 n n ).
Eq. (4.18) is given by Ê ω (z) = Ê ω (z = 0)sech Gz where G2 = ω2 deff
ω 2ω
In terms of irradiances we have,
I 2ω (z) = I ω (0)tanh2 Gz ,
I ω (z) = I ω (0)sech2 Gz ,
2
2ω2 deff
2
G =
I ω (0) .
3
2
ε0 c nω n2ω
(4.19)
28
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
2 /n3 appears. The functional forms of these are shown in
Note once again the material factor deff
Fig. 4.2. Note that for small distances the growth of SHG is proportional to z2 as we discovered
previously. However, it can be seen that this saturates when the power in the second harmonic
becomes an appreciable fraction of that in the fundamental.
1
0.8
0.6
0.4
0.2
0.5
1
1.5
2
2.5
3
Figure 4.2: Plot of the growth of second harmonic and pump depletion with distance (normalised to 1/G).
4.2.5 Parametric upconversion: ω1 + ω2 → ω3
This is yet another extension of sum frequency generation. In this case we will consider inputs
at frequencies ω1 and ω2 generating the sum frequency at ω3 for which initially is not present.
The beam at ω2 is assumed to be the more intense and hence its depletion can be ignored,
dÊ ω2 /dz ≈ 0. However, we will include the depletion of the beam at ω1 . It is conventional
to call the intense (ω2 ) beam the pump and the less intense beam (ω1 ) the signal. In fact the
description upconversion arises because we will be raising the frequency of the signal from ω1
to ω3 . As we can ignore the evolution of the pump we need only consider the pair of coupled
differential equations (making the usual assumption of remote from material resonances),
dÊ ω3 (z)
iω3
=
deff Ê ω1 (z)Ê ω2 ei∆kz ,
dz
2cn3
¢∗
¡
dÊ ω1 (z)
iω1
=
deff Ê ω3 (z) Ê ω2 e−i∆kz
dz
2cn1
.
(4.20)
For perfect phase-matching, ∆k = 0, Eqs. (4.20) can be combined to eliminate Ê ω3 which results
in the undamped harmonic oscillator differential equation. The solution is Ê ω1 (z) = Ê ω1 (z =
2 |Ê ω2 |2 /(4c2 n n ). In terms of irradiances we have,
0) cos Gz where G2 = ω1 ω3 deff
1 3
I ω1 (z) = I ω1 (0) cos2 Gz
,
4.2. EM PROPAGATION WITH A SECOND-ORDER NONLINEARITY
ω3 ω 1
I (0) sin2 Gz ,
ω1
2
ω1 ω3 deff
I ω2 .
=
3
2ε0 c n1 n2 n3
29
I ω3 (z) =
G2
(4.21)
This oscillating behaviour is demonstrated in Fig. 4.3 which shows the relative power levels
at the frequencies ω1 and ω3 . Complete conversion is obtained after a propagating a distance
l = π/(2G).
1.0
0.8
0.6
0.4
0.2
0
π/2
π
3π/2
2π
Figure 4.3: Plot of the evolution of the signal and the upconverted power levels with distance
(normalised to 1/G). The irradiances are normalised to ω3 I ω3 /[ω1 I ω1 (0)] (solid) and I ω1 /I ω1 (0)
(dashed).
4.2.6 Parametric downconversion: ω3 − ω2 → ω1
As before the pump at ω2 will be assumed to be more intense than the other frequencies present.
This case is similar to the upconversion case except that we initially start with the higher frequency ω3 and subsequently generate the lower (difference) frequency at ω1 . It turns out that
Eqs. (4.20) apply in this case also with a different initial condition. The result is the same as
shown in Fig. 4.3 except that the origin is shifted to the point Gz = π/2.
4.2.7 Parametric amplification: ω3 → ω1 + ω2
In the previous examples the intense (pump) beam has been at one of the lower optical frequencies. We will now examine the case where the pump beam is at the higher frequency ω3 .
The other input we will take to be at frequency ω1 termed the signal as we will concentrate
on changes in this beam. The remaining at beam at frequency ω2 generated by the nonlinear
interaction is conventionally called the idler. Ignoring depletion of the pump and assuming we
30
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
are remote from resonances, we have to consider the following pair of differential equations,
¢∗
¡
iω1
dÊ ω1 (z)
=
deff Ê ω3 Ê ω2 (z) e−i∆kz ,
dz
2cn1
ω
2
¡
¢∗
dÊ (z)
iω2
=
deff Ê ω3 Ê ω1 (z) e−i∆kz .
(4.22)
dz
2cn2
Taking perfect phase-matching (∆k = 0), and combining Eqs. (4.22) to eliminate one of the electric fields results in the undamped harmonic oscillator equation with a sign change. Assuming
that there is initially no light of frequency ω2 we obtain the solution Ê ω1 (z) = Ê ω1 (z = 0) cosh Gz
2 |Ê ω3 |2 /(4c2 n n ). In terms of irradiances we have,
where G2 = ω1 ω2 deff
1 2
I ω1 (z) = I ω1 (0) cosh2 Gz ,
ω2 ω 1
I ω2 (z) =
I (0) sinh2 Gz ,
ω1
2
ω1 ω2 deff
2
G =
I ω3 .
3
2ε0 c n1 n2 n3
(4.23)
For large amplifications, Gz À 1, both signal and idler will be dominated by the factor e2Gz , i.e.
it will appear we have a gain coefficient 2G which is proportional to the square root of the pump
irradiance.
One application of the optical parametric amplifier is to directly amplify an imput signal.
Alternatively, it can be used in a similar fashion to a laser in that the amplifier is placed in a cavity and the oscillating optical power is initially amplified from noise (e.g. quantum fluctuations
or black-body). This device is called an optical parametric oscillator (OPO). Like a laser, oscillation will preferentially occur at the maximum gain which automatically specifies operation at
perfect phase-matching k1 + k2 = k3 . If there is some means of controlling the phase-matching,
then this provides a means of tuning the OPO. Again like a laser, for oscillation to occur optical
gain must exceed optical losses (transmission at mirrors, absorption, scattering, etc.) and the
feedback is positive. For a laser, this gives rise to a threshold in whatever pumping mechanism is
used. For an OPO the same form
√ of threshold exists in that the gain coefficient 2G must exceed
some critical value. Since G ∝ I ω3 , this provides a threshold in the optical pump power.
There are several different forms of OPO. In the simplest, singly resonant form, a cavity
is formed for only one of the generated frequencies, ω1 (say). The doubly resonant form has
cavities for both generated frequencies, ω1 and ω2 . The doubly resonant OPO has the lower
threshold but is more complex to set up (since we have to satisfy that both these frequencies
are cavity modes and satisfy ω1 + ω2 = ω3 ) and as a consequence is less stable. A triply resonant OPO has also been sometimes used where an additional cavity for the pump increases the
circulating optical power in the nonlinear element compared to the input power level.
A summary of the six configurations discussed is shown in Fig. 4.4.
4.2.8 Phase shift in fundamental (cascaded nonlinearity)
The principle of causality states that output cannot precede an input. One of the consequences
of this is if some frequency component is attenuated then the remaining frequency components
4.2. EM PROPAGATION WITH A SECOND-ORDER NONLINEARITY
ω1
ω2
ω
ω2
2ω
ω3
(a)
ω3
ω3
ω1
ω3
(c)
ω3
ω1
(d)
ω1
(b)
ω2
31
ω1
ω2
ω2
(e)
(f)
Figure 4.4: Summary of the six optical frequency conversion configurations discussed in this
section taking ω1 + ω2 = ω3 : (a) sum frequency generation, (b) second harmonic generation, (c)
parametric upconversion, (d) parametric downconversion, (e) parametric amplification and (e)
optical parametric oscillation. In each case the larger arrow indicates the more intense pump.
must have a phase shift to maintain causality. Mathematically this is known as a dispersion
relation of which the most well known example is the Kramers-Krönig relation relating the
refractive index to the absorption coefficient spectrum. Now in the case of frequency conversion,
the power is being attenuated in at least one of the input beams. Now of course the power
is being transferred to another frequency instead of a material excitation, but causality still
applies and there will be an associated phase shift. The mechanism by which this occurs is
that away from perfect phase-matching, conversion to the new frequency will be followed by
back-conversion when the nonlinear polarisation source and the electric field drift out of phase.
This interference can produce a wave with a different phase to the original. Of course since
the frequency conversion is a nonlinear process and depends on irradiance, the associated phase
shift will also be irradiance dependent.
The nonlinear phase shift will be present in all the configurations studied but we use second
harmonic generation as an example. If we assume a low conversion efficiency, |Ê ω (z)| ≈
|Ê ω (z = 0)|, Eq. (4.18) becomes a linear homogeneous differential equation,
d2 Ê ω
dÊ ω
+
i∆k
+ G2 Ê ω = 0
dz2
dz
.
(4.24)
Now let us take the approximation |∆k| À G which is usually the case for low conversion
efficiency. Then the second-order derivative can be neglected and the solution of the differen2
tial can be approximated as Ê ω (z) = Ê ω (z = 0)eiG z/∆k . This shows we obtain a phase shift
∆φ = G2 z/∆k which is proportional to the irradiance and distance propagated. Note that it is
inversely proportional to ∆k and hence can be made large for small phase mis-match (although
the approximation in dering this breaks down close to perfect phase matching) and that the sign
of the phase shift is the same as the phase mis-match which can, in principle, be changed. If
eff
we identify an effective nonlinear refractive coefficient neff
2 such that ∆φ = 2πn2 Iz/λ then we
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
32
obtain,
neff
2 =
2
4π 1 deff
ε0 c ∆kλ n2ω n2ω
.
(4.25)
For small phase mis-matches we cannot simplify Eq. (4.18) in this fashion. Analytic solutions are possible using Jacobi elliptic functions [7] but in many cases numerical solutions are
more convenient. For example in Fig. 4.5 we show the fundamental phase shift and throughput
as a function of distance for three different values of initial fundamental irradiance (this was
calculated using the built-in numerical differential equation solver in Mathematica). Note that
for the small phase mis-matches the phase shift is not simply linearly dependent on distance as
indicated by the low conversion efficiency approximation. We can also plot the phase shift and
throughput as a function of fundamental irradiance and this is shown in Fig. 4.6.
Phase Shift
3π/4
Throughput
1.0
0.8
π/2
0.6
0.4
π/4
0.2
0
π/2
π
3π/2
0
2π
π
π/2
3π/2
2π
Figure 4.5: Fundamental phase shift and throughput as a function of distance (scaled to 1/∆k).
The input irradiances have been scaled but are in the ratio 10(solid):5(dash):2(long dash).
4.2.9 Electro-optic effect revisited
The contracted d-tensor notation can also be used to describe the electro-optic effect. If we
take one of the fields as DC, e.g. ω2 , k2 → 0, then a “nonlinear” polarisation is obtained at the
same frequency as the optical input. Phase-matching will be automatic and the slowly varying
envelope approximation gives,
iω
dÊ ω
=
deff (0, ω)E DC Ê ω
dz
cnω
.
(4.26)
The solution of this is Ê ω (z) = Ê ω (z = 0)eiωdeff E z/(nc) , i.e. the optical beam develops a phase
shift ∆φ = ωdeff E DC z/(nc).
Now from the electro-optic effect we had ∆(1/n2 ) = rE DC and hence for small changes in
refractive index produces a phase change ∆φ = −ωn3 rE DC z/(2c). Comparing these two forms
DC
4.3. PHASE MATCHING
33
0.6
0.4
0.2
1.0
0.8
0.6
0.4
0.2
0.0
Nonlinear Phase Shift
0.0
3
2
1
0
∆kL=2π
Fundamental Throughput
0.8
Nonlinear Phase Shift
Fundamental Throughput
∆kL=π
1.0
0
4
8
12
16
Optical Power (scaled)
3
2
1
0
0
1
2
3
4
Optical Power (scaled)
Figure 4.6: Fundamental phase shift and throughput as a function of scaled irradiance for two
fixed values of ∆kL.
gives us that the electro-optic coefficient is simply proportional to the deff (0, ω) coefficient,
r = −2deff /n4 . Note that since the refractive index is dimensionless, the electro-optic coefficient
and the d coefficient are in the same units usually given in pmV−1 . If we take more care to
include the tensor nature of these coefficients we would find that in the contracted notation it is
the transposes that are related, ri j = −2d ji /n4 . In fact we can see that by comparing Tables 4.1
and 2.1 that the symmetry properties of the tensors reflect this transpose relation.
4.3 Phase matching
For second-order processes we have to satisfy ω1 + ω2 = ω3 (energy conservation). For efficient
conversion phase-matching is also required, k1 + k2 = k3 (momentum conservation). Since
k = ωn/c, phase-matching is satisfied if n1 = n2 = n3 . The problem is that materials are usually
dispersive and the refractive index varies with frequency. Normal dispersion has the refractive
indices increasing with frequency, n3 > n1 , n2 . This obviously makes it difficult to achieve
phase-matching.
4.3.1 Birefringent phase-matching
The most common technique to obtain phase-matching is to use birefringence to compensate
for dispersion. The idea is that at each frequency there is a pair of refractive indices (orthogonal
polarisations) and the difference between these is adjusted to compensate for dispersion. The extraordinary refractive index in a uniaxial crystal as a function of angle between the propagation
34
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
direction and the optic axis is given by, ne (θ) = (sin2 θ/n2e + cos2 θ/n2o )−1/2 . Fig. 4.7 demonstrates how phase-matching could be achieved for second harmonic generation for a particular
2ω
propagation direction by selecting the propagation direction such that nω
o (θm ) = ne (θm )
z
k
ω
ne
θm
x
ω
no
n 2ω
e
n 2ω
o
Figure 4.7: Birefringent phase-matching for second harmonic generation in a negative uniaxial
2ω
crystal. For propagation at a particular angle θm to the optic axis, nω
o = ne .
For a three wave interaction a possible configuration is shown in Fig. 4.8 for a negative
uniaxial crystal. Here there are a couple of possible phase-matching scenarios outlined in Table
4.3: in type I phase-matching the low frequency components have parallel polarisations, in type
II the low frequency components are orthogonal.
z
x
o
k1
e
k1
o
k2
e
k2
o
k3
e
k3
Figure 4.8: Birefringent phase-matching for three wave interaction in a negative uniaxial crystal.
For propagation at a particular angle θm to the optic axis, k1o + k2o = k3e .
If we expand the phase mis-match around the angle θm which corresponds to perfect phasematching we have ∆k ∝ ∆θ + O(∆θ2 ) where ∆θ = θ − θm . Since the sinc2 efficiency dependence
of phase mis-match is very tight, this means that the propagation direction must, in general, be
4.3. PHASE MATCHING
35
Type I: low frequency components k
negative uniaxial
k1o + k2o = k3e
positive uniaxial
k1e + k2e = k3o
Type II: low frequency components ⊥
negative uniaxial
k1o + k2e = k3e
positive uniaxial
k1o + k2e = k3o
Table 4.3: Birefringent phase-matching scenarios for three wave interactions.
set very accurately (critical phase-matching). For example for type II SHG in KTP at 1.064 µm,
in a crystal of length 1 cm, the tolerance on the propagation angle is 16 mrad. This is fine
in theory for plane waves but in practice, the optical beams have a finite spatial extent (e.g.
gaussian profile). Finite beams diverge which means that the propagation direction will vary
as we move across a wavefront. Therefore not all of the beam can be perfectly phase-matched
leading to a drop in conversion efficiency. A solution to this finite beam problem is to try and
arrange a geometry so that the proportionality constant in front of the linear term ∆θ is zero.
This is called tangential or non-critical phase-matching and is shown schematically in Fig. 4.9.
The wider acceptance angle in this configuration also means that it is experimentally easier to
set up. Note though from Fig. 4.9 that the wavevectors are not collinear. This means there will
be a limited distance of interaction where all 3 beams overlap, leading to a practical limitation
on the crystal length.
z
k1
k2
k3
x
e
k2
o
k3
Figure 4.9: Tangential phase-matching in a uniaxial crystal, k1 + k2 = k3 .
In birefringence for the extra-ordinary ray we showed that the phase and group velocities
were not parallel and hence walk-off occurs. This will limit the overlap region for optical
beams with different polarisations (as occurs in birefringent phase-matching). There are two
cases where the phase and group velocities are parallel for a uniaxial crystal, θ = 0◦ and 90◦ .
Obviously the θ = 0◦ does not apply as there is no variation in refractive index between the
orthogonal polarisations. The θm = 90◦ can be exploited though to eliminate walk-off. This also
36
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
has the advantage of automatically being tangential phase-matching with parallel wavevectors
thus also eliminating the other overlap problem discussed in the previous paragraph.
Hence the ideal situation for birefringent phase-matching is (non-critical) tangential at θm =
◦
90 . Sometimes it is necessary to temperature tune the crystal to achieve this, but the advantages
of this geometry outweighes the disadvantage of the necessity of a uniform temperature stage.
To date the most attractive materials for second-order processes are LiB3 O5 and β-BaB2 O4
because of the possibility of non-critical phase-matching and relatively low damage thresholds.
For a wider list of current second-order nonlinear materials consult [8] and references therein.
4.3.2 Quasi-phase-matching
Birefringent phase-matching can be seen to have some problems yet it is the most commonly
used technique. The largest nonlinear coefficients occur though for either non-birefringent
materials (e.g. for GaAs in the infrared, d14 ∼ 200 pmV−1 ) or for the on-diagonal elements
where all the polarisation components are parallel (e.g. for LiNbO3 at 1.3 µm, d33 = 32 pmV−1
compared to d31 = 5.5 pmV−1 and for KTP at 0.88 µ, d33 = 18.5 pmV−1 compared to
d15 = 2.0 pmV−1 ). Quasi-phase-matching offers the possibility of using these coefficients,
particularly in guided wave geometries.
40.0
∆k=0
2ω Irradiance
30.0
DR
DD
20.0
10.0
∆k=/ 0
0.0
0
Lc
2Lc
3Lc
distance
4Lc
5Lc
Figure 4.10: Growth of second harmonic (neglecting pump depletion) in the cases of (a) perfect
phase-matching (∆k = 0), (b) a phase mis-match (∆k 6= 0) and the quasi-phase-matching cases
of (c) domain reversal and (d) domain disordering.
The quasi-phase-matching technique is most easily explained with reference to Fig. 4.10.
On this plot the initial growth of SHG is shown for perfect phase-matching (∆k = 0) and a finite
phase mis-match (∆k 6= 0). Let us suppose that the refractive indices are such that this phase
mis-match is unavoidable. Now this oscillating behaviour is because the wave at the second
harmonic and the nonlinear polarisation have different phase velocities and so they oscillate
between constructive and destructive interference. The characteristic length for which the interference remains constructive is called the coherence length Lc = π/|∆k|. Now in the case
4.3. PHASE MATCHING
37
of domain reversal we process the crystal so that when the second harmonic and the nonlinear polarisation waves are about to be in anti-phase, the domain of the crystal is inverted (and
hence the nonlinear polarisation is inverted) such that the waves remain in phase. It can be seen
that this requires periodic domain inversion every coherence length (Fig. 4.10). With domain
reversal there is a drop in efficiency, which at the reversal planes corresponds to a factor 4/π2
for domain lengths equal to the coherence length.
There are other possibilities for quasi-phase-matching. First, the domain reversal is not
restricted to lengths of a single coherence length but can be any odd multiple of Lc with a corresponding drop in efficiency. Second, It is also possible to quasi-phase-match by periodically
disordering the material such that it gains inversion symmetry and hence the second order nonlinearity disappears for regions where destructive interference occurs. Third, it is possible to
quasi-phase-match with a refractive index grating such that the period of the grating Λ satisfies
∆k = 2π/Λ. This grating couples the wavevector of the optical wave to k ± 2π/Λ although the
overall efficiency of this process is low.
38
CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES
Bibliography
[1] Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
[2] F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973).
[3] A. Yariv, Quantum Electronics, (Wiley,New York, 1989).
[4] P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, edited by P. L. Knight
and W. J. Firth, Cambridge Studies in Modern Optics Vol. 9 (Cambridge University Press,
Cambridge, U.K. 1990).
[5] D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan and E. W. Van Stryland, Opt. and Quantum
Electron. 24, 1 (1992).
[6] H. Vanherzeele and J. D. Bierlein, Opt. Lett. 17, 982 (1992).
[7] C. N. Ironside, J. S. Aitchison and J. M. Arnold, IEEE J. Quantum Electron. 29, 2650
(1993).
[8] C. L. Tang, Chapter 38: Nonlinear Optics, Handbook of Optics, edited by M. Bass, E. W.
Van Stryland, D. R. Williams and W. L. Wolfe (OSA/McGraw-Hill, New York 1995).
39