course notes of non linear optics

7/23/2019 Course notes of non linear optics

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Applied Nonlinear Optics

David C. Hutchings

Dept. of Electronics and Electrical Engineering

University of Glasgow


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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 Mid-

winter [2] and Yariv [3]. Like the bulk of the literature in nonlinear optics, these books do notalways 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


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iv


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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,

E ds=

q

0=

1

0

dV , (1.1)

where is the charge density. The divergence theorem can be used to change this surfaceintegral to a volume integral,

E ds= E dV , (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= 0

. (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 fieldwill 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 non-

uniform field. A polarisation fieldPcan be defined equal to the dipole moment per unit volume.

If there are Ndipoles per unit volume consisting of charges+q andq separated by r then

1


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2 CHAPTER 1. BIREFRINGENCE

P= N qr. If we assume for a moment that the induced charge separation r E,1 then thepolarisation field is proportional and parallel to the electric field. Conventionally this is written

in SI units as P= 0E, where the dimensionless constant of proportionality is called theoptical susceptibility.

It can be shown that the effective charge density due to the polarisation of a medium is given

by,p= P. Inserting this into the differential form of Gauss theorem gives,

E= 10

(+p) = 1

0( P) , (1.4)

where denotes the free (not polarisation) charge density. This can be rearranged to give (0E + P) =. For convenience a new field is introduced at this point equal to the quantityinside the bracket, D =0E + P, conventionally known as the electric displacement. Note that

unlike Eand P the electric displacement is not immediately related to a physical quantity, butallows 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= and is one of theset of differential relations known as Maxwells equations. Using the susceptibility to substitute

for the polarisation field provides,

D=0 (1 +) E=0rE , (1.5)

where r=1 + is known as the dielectric constant or relative permittivity. Note that thevacuum values of these dimensionless quantities are=0 andr= 1.

The divergence theorem and Stokes theorem are employed in deriving the remainder ofMaxwells 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 = , B = 0 , E = B

t,

H = j +Dt

.

The current is related to the electric field by the conductivity, j=E. In this course the usualcase will be of zero current j=0 and non-magnetic material B=0H.

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

1This 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.


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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. HenceEq. (1.5) cannot be written with the relative permittivity as a scaler quantity. To generalise Eq.

(1.5) to anisotropic media (such thatD andE can have different directions), the scaler dielectric

constantris replaced with a second rank tensor,

1

0

DxDy

Dz

=

xx xy xzyx yy yz

zx zy zz

ExEy

Ez

. (1.6)

In a more compact notation,1

0D

i=

j=x,y,z

i jE

j . (1.7)

We can prove that the tensori jis symmetric using energy conservation. The energy densityof an electric field in a dielectric is,

We=1

2E D= 1

2i,j

Eii jEj . (1.8)

Differentiating gives the power flow into a unit volume,

Wet

=1

2i j

i jEit

Ej+EiEjt . (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 Maxwells equations and assuming no currentsj = 0gives a power flow per unit volume,

E Dt

+ H Bt

. (1.11)

The first of these terms relates to the rate of change of energy of the electric field,

Wet

=E Dt

=i j

i jEiEjt

. (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 ifji= i j and the dielectric tensor is symmetric having a totalof 6 different elements.

Note that we have not yet specified the axes for our cartesian co-ordinate system {x,y,z}. Itis always possible to choose a set such that the (symmetric) dielectric tensor is diagonal. The


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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,

1

0

DxDy

Dz

=

x 0 00 y 0

0 0 z

ExEy

Ez

, (1.13)

wherex=xx etc. The set of axes in this case is known as the principal dielectric axes, whichmay 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.

No birefringence Isotropic

Cubic

0 00 0

0 0

Uniaxial birefringence

(1 optic axis)

Hexagonal

Tetragonal

Trigonal

x 0 00 x 00 0 z

Biaxial birefringence

(2 optic axes)

Orthorhombic

Monoclinic

Triclinic

x 0 00 y 0

0 0 z

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,

E=1

2

E0e

i(tkr) + E0ei(tkr)

, (1.14)

H=1

2

H0e

i(tkr) + H0ei(tkr)

. (1.15)

The wavevectork=ns/cwheresis a unit vector in the direction of propagation of the wave.The phase velocity is given by vp = cs/n. Inserting these into Maxwells equation E=


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1.3. EM WAVE PROPAGATION 5

B/twith a nonmagnetic medium (B=0H) gives,

H0= 10

k E0= n0c

s E0 . (1.16)

Thus the direction of the magnetic field amplitude is perpendicular to both the direction of

propagations and electric field E0. Similarly we can use the Maxwell equation H= j +D/twithj=0 to obtain,

D0= 1

k H0= nc

s H0 . (1.17)

Hence the electric displacement is perpendicular to the direction of propagationsand the mag-

netic fieldH0. However, as we have seen for anisotropic media, E0 and D0 are not necessarilyparallel. We also note that the power flow is given by the Poynting vectorS= E H whichis perpendicular to E0 andH0. 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 directions and the Poynting vector are not necessarily parallel

and hence the phase and group velocities may also not be parallel.

E D

S=ExH

k

v

v

g

p

H

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

0c2s s E0= n20 [E0 s (s E0)] . (1.18)

Consider just one component of this expression and introduce the dielectric tensor,

D0i = n2

D0i

i 0si (s E0)

,

D0i = 0si (s E0)

1

i 1

n2

1. (1.19)


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SinceD0andsare perpendicular,D0 s=0. This provides the relation,

i

s2i

n2

i 1

1=0 , (1.20)

which is known as Fresnels equation. This is quadratic in the square of the refractive index (n2)

and therefore provides two possible solutions given a propagation directions. 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 po-

larisations. Let the two roots be n and n with electric displacement amplitudes D0 and D0

respectively. Using Eq. (1.19), the scaler product of these amplitudes can be written,

D0 D0 = 0nns E02i

s

2

i(n2/i 1) (n2/i 1) ,

= (0n

ns E0)2n2 n2

i

n2

n2/i 1 n2

n2/i 1

s2i , (1.21)

where partial fractions have been employed to obtain the final form. Now Fresnels equation

(1.20) states that the summation over each of the terms is zero. Hence D0 D0= 0 and therforeD0and D

0 are mutually perpendicular.

1.4 The Index EllipsoidUsing Fresnels equation based on the propagation direction is rather cumbersome when analys-

ing 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,

We=1

2E D= 1

2

D2xx

+D2y

y+

D2y

y

. (1.22)

Substituting r= n2 for the various directions and settingDx/

2We=x, etc. gives the following,

x2

n2x+

y2

n2y+

z2

n2z=1 . (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 nxfor thex-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 meansthat the index ellipsoid will have cylindrical symmetry round thez-axis (optic axis). There are

two distinct cases of uniaxial birefringence;nz> nxknown as positive unixial birefringence and


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1.4. THE INDEX ELLIPSOID 7

nz< nx known as negative unixial birefringence. These are illustrated in Fig. 1.2. The use ofthe index ellipsoid is as follows. The direction of propagationkis drawn from the origin and

makes an angle of to the optic axis. The intersection of the plane normal to the direction ofpropagation 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 semi-

axis in the case of a negative uniaxial crystal) always lies in the xy-plane and therefore has a

lengtha=nx=noindependent of the angle; a ray with this polarisation is called the ordinaryray. The length of the other semi-axis is dependent on the angle , b= ne(); a ray withthis 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(= 0) = nz= ne andne(=/2) = nx= no. For the general case, wedecompose b= ne() into components parallel and perpendicular to the optic axis, x

2 +y2 =

(ne() cos)2

andz =ne() sin, and insert into the equation for the index ellipsoid to give foruniaxial crystals,

1

n2e()=

cos2

n2o+

sin2

n2e. (1.24)

z

x

k

b

a

(a) (b)z

x

zz xn >n n


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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 twoimages 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 sta-

tionary 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 polarisa-

tion. What happens in the general case where the light is some combination of these po-

larisation states? Suppose we initially have a linear polarisation Din0 = D0(o cos +e sin)where o and e are unit vectors parallel to the ordinary and extraordinary polarisation direc-

tions respectively. Now suppose the crystal thicknessd is such that the difference in optical

path length between the ordinary and extraordinary rays is an odd integer of half-wavelengths

(ne() no)d= (2N+ 1)/2. A crystal of this thickness is termed a half-wave plate. On exitthe ordinary and extraordinary rays will be out of phase, Dout0 =D0(o cos e sin).2 It canbe seen that this is equivalent to the transform and corresponds to flipping the polar-isation angle around the optic axis. In particular for =45, the output (linear) polarisation isperpendicular 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() no)d= (2N+ 1)/4 known as a quarter waveplate. Now the ordinary and extraordinary rays will be /2 out of phase with each other onoutput from the crystal. Again if we consider the particular case of an input linear polarisation

at 45 to the optic axisDin0 =D0(o e)/

2 then the output polarisation will beDout0 =D0(o ie)/

2. The positive and negative cases correspond to the separate cases of N+ /4 and

N+ 3/4 optical path length differences. In this case we have generated circularly polarisedlight 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).

2For simplicity the common phase factor has been omitted since the polarisation state only depends on the

relative phase of the polarisation components


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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 Pockels 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

xx+

y2

yy+

z2

zz+

2yz

yz+

2xz

xz+

2xy

xy=1 , (2.1)

or in shorthand notation, i,j=x,y,zi ji j

=1. Now conventionally the electro-optic coefficient r

relates the change in 1/ (i.e. 1/n2) to the electric field E. Generalising this to anisotropic

crystals,

1

i j

=

1

i j

E

1

i j

E=0

=k

ri jkEk . (2.2)

The electro-optic coefficient ri jkis a third rank tensor as it relates a second rank tensor (1/i j)to a vector (E). It has 27 elements but only 18 are independent sinceji = i j and thereforerjik= ri jk. This symmetry is employed to write the electro-optic tensor in a contracted notationwhere the i j subscripts are replaced by: xx= 1, yy= 2, zz= 3, yz= zy = 4, xz= zx = 5 andxy=yx=6 and theksubscripts by: x=1,y=2 andz=3. This allows the electro-optic tensor

9


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10 CHAPTER 2. THE ELECTRO-OPTIC EFFECT

to be written as a 6 3 matrix:

(1/1)(1/2)(1/3)(1/4)(1/5)(1/6)

=

r11 r12 r13r21 r22 r23r31 r32 r33r41 r42 r43r51 r52 r53r61 r62 r63

E1E2

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 i jandri jkare unchanged. However, under inversion Ek Ek. Hencekri jkEk= kri jkEkfor any specifiedEwhich requires that all the electro-optical coefficientsare zero,ri jk= 0. For other symmetry classes, group theory can be employed to investigate theform of the electro-optic tensor, as shown in table 2.1.

2.2 Examples

2.2.1 KDP

Potassium dihydrogen phosphate KH2PO4(known as KDP) belongs to the symmetry class 42m

and hence exhibits uniaxial birefringence with the principal dielectric axes being the same asthe 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

n2o+

y2

n2o+

z2

n2e+ 2r41Exyz + 2r41Eyxz + 2r63Ezxy=1 . (2.4)

Now let us consider the case where the electric field is parallel to z such thatEx=0 andEy=0so the 4th and 5th terms can be ignored. Now as we stated in the previous chapter on birefrin-

gence, 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

by 45 about the z-axis. Thus we have x (x y)/2, y (x +y)/2 and z z. Oninserting these in Eq. (2.4) we obtain for the new optical indicatrix,

1

n2o+ r63Ez

x2 +

1

n2o r63Ez

y2 +

z2

n2e=1 . (2.5)

This is the same form as the optical indicatrix for a biaxial crystal, x2/n2x +y2/n2y +z2/n2z = 1where,

nx = no 1 + r63Ezn2o

1/2 no r63n3oEz

2 ,


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2.2. EXAMPLES 11

ny = no 1 r63Ezn2o

1/2 no+r63n3oEz

2 ,

nz = 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 x andy directions andthese will have different phase velocities.

y

x x

y

xyn

n

n

no

o

x

y

E z

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(ny nx)d= /2, wheredis the crystal thickness. Thus we requirethe application of an electric field E

()z where, r63n

3oE

()z d= /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 voltageV() =/(2n3or63). For KDP at a visible wavelength of=550 nm,n3or63 34 pmV1 and a voltage ofV() 8 kV is required.

We found above that the electro-optic coefficient appears in the factorn3r. 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 n3rwhich 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 propaga-

tion. With the change in polarisation, the transmission of the output polariser is given by


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T=sin2V/(2V()) and is shown in Fig. 2.2(b). We note that for small voltages, the trans-mission 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/2 point. Since this bias pointis equivalent to a quarter wave plate, a more elegant solution is to incorporate an additional

conventional quarter wave plate.

V

InputPolariser

OutputPolariser

x

y

z

T

V

/20

(a) (b)

1

0.5

0

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 m). 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 lengthL and thickness d, the required voltage to achieve a phase differencebetween the two polarisation components isV() =d/(2Ln3r41). Note that we have gained a

factor ofd/Lin comparison to the longitudinal modulator and thus the voltage can be reducedby increasing the length or decreasing the thickness. For GaAs at 10m,n=3.3,r14=1.6 pmV

1and takingL=5 cm andd=0.5 cm, we obtainV() 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 thex ory 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= n3or63V/2. A possible geometry for a GaAs phase


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2.2. EXAMPLES 13

Input

Polariser

Output

PolariserV

[110]

[001]

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=n3r14V 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

0is modulated with a sinusoidal voltage of frequencym, then the frequency spectrum of thetransmitted light develops side bands at 0 m,0 2m, etc. What we have accomplishedhere is mixing of two frequencies, one optical and one electrical. This brings us neatly to the

subject of the next chapter: optical frequency mixing.


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Triclinic

1

r11

r12

r13

r21

r22

r23

r31

r32

r33

r41

r42

r43

r51

r52

r53

r61

r62

r63

Monoclin

ic

2

0

r12

0

0

r22

0

0

r32

0

r41

0

r43

0

r52

0

r61

0

r63

m

r11

0

r13

r21

0

r23

r31

0

r33

0

r42

0

r51

0

r53

0

r62

0

Orthorhombic

222

0

0

0

0

0

0

0

0

0

r41

0

0

0

r52

0

0

0

r63

mm2

0

0

r13

0

0

r23

0

0

r33

0

r42

0

r51

0

0

0

0

0

Tetrago

nal

4

0

0

r13

0

0

r13

0

0

r33

r41

r51

0

r51

r41

0

0

0

0

4

0

0

r13

0

0

r13

0

0

0

r41

r51

0

r51

r41

0

0

0

0

422

0

0

0

0

0

0

0

0

0

r41

0

0

0

r41

0

0

0

0

4mm

0

0

r13

0

0

r13

0

0

r33

0

r51

0

r51

0

0

0

0

0

42m

0

0

0

0

0

0

0

0

0

r41

0

0

0

r41

0

0

0

r63

Cubic

43m

23

0

0

0

0

0

0

0

0

0

r41

0

0

0

r41

0

0

0

r41

432

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Trigonal

3

r11

r22

r13

r11

r22

r13

0

0

r33

r41

r51

0

r51

r41

0

r22

r11

0

32

r11

0

0

r11

0

0

0

0

0

r41

0

0

0

r41

0

0

r11

0

3m

0

r22

r13

0

r22

r13

0

0

r33

0

r51

0

r51

0

0

r22

0

0

Hexagonal

6

0

0

r13

0

0

r13

0

0

r33

r41

r51

0

r51

r41

0

0

0

0

6mm

0

0

r13

0

0

r13

0

0

r33

0

r51

0

r51

0

0

0

0

0

622

0

0

0

0

0

0

0

0

0

r41

0

0

0

r41

0

0

0

0

6

r11

r22

0

r11

r22

0

0

0

0

0

0

0

0

0

0

r22

r11

0

6m2

0

r22

0

0

r22

0

0

0

0

0

0

0

0

0

0

r22

0

0

Table

2.1:Theformoftheelectro-optictensorforthecrystalsymm

etryclasseswithnoinversionsymmetry.Abaroveranentry

indicatesthenegative.


19/43

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 os-

cillator. This can be pictured as electrons attached to their nuclei by springs with resonant

frequency and damping . An optical wave provides a forcing term through the dipole in-teraction with the electron and hence the motion of the electron around its equilibrium position

can be described by the linear differential equation,

d2r(t)

dt2 + 2

dr(t)

dt+2r(t) =

e

m0

E(t) . (3.1)

One way to solve this differential equation is to Fourier transform it: r(t) =

r()eitd

and hence the solution of Eq. 3.1 is straightforward,

r() = em0

E()

2 2 2i . (3.2)

Now the polarisation is given P =Ner and the optical susceptibility is defined P() =0()E(), so the optical susceptibility can be written,

() =

Ne2

0m0

1

2 2 2i . (3.3)

Neglecting absorption, the refractive index is given by n() =

1 + Re() and is shown inFig. 3.1 for frequencies around resonance . This is the characteristic shape of the refractiveindex 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()/dis positive which is termed normal dispersion.

Now suppose the potential for the electron is cannot strictly be described as that of a har-

monic 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


20/43

16 CHAPTER 3. OPTICAL FREQUENCY MIXING

-4 -2 0 2 40.6

0.7

0.8

0.9

1

1.1

1.2

1.3

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(0)/.

higher order. The differential equation describing the electrons motion then has additional

anharmonic terms which we will consider tor2,

d2r(t)

dt2 + 2

dr(t)

dt+2r(t) r2 = e

m0E(t) . (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 beperformed 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 harmonicLorentz equation (3.1). The next highest order gives,

d2r1(t)

dt2 + 2

dr1(t)

dt+2r1(t) =r

20(t) , (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) =

[E0ei0t+E0 e

i0t]/2. The Fourier transform of this isE() = [E0(

0)+E

0(+0)]/2.

The linear solutionr0will be driven at the same frequency and hence we obtain,

r0(t) = e2m0

E0e

i0t

2 20 2i0+

E0 ei0t

2 20 2i0

. (3.6)

When this expression is squared, the terms have time dependencies ofe2i0t or are constant.Thus there exists driving terms for r1(t) is at frequencies20. Hence the anharmonic termhas 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


21/43

3.1. LORENTZ MODEL 17

frequency mixing. Since we have only one optical frequency present the only combinations are

at0

0, i.e. 20and 0.

The frequency mixing aspect can be seen more clearly if we take an optical input field that isbichromatic,E(t) = [E1e

i1t +E2ei2t + cc]/2 where the cc denotes the complex conjugate.The linear solutionr0(t)will contain terms oscillating at the same frequencies,

r0(t) = e2m0

E1e

i1t

2 21 2i1+

E2ei2t

2 22+ 2i2+ cc

. (3.7)

On squaring this will produce a term oscillating at 1+2,

r20(t)1+2 = e2

2m2

0

E1E2ei(1+2)t

2 21 2i11

2 22 2i2

1. (3.8)

On inserting this into Eq. (3.5) this provides for r1(t) (e.g. by the Fourier transform methoddescribed previously),

r1(t)|1+2 = e2

2m0E1E2e

i(1+2)t2 (1+2)2 2i(1+2)12 21 2i11 2 22 2i21 . (3.9)

Hence since the polarisation is given by P= Ner, there is also a polarisation componentoscillating at the sum frequency,

P()|1+2 = Ne3

2m0E1E2(1 2)

2 (1+2)2 2i(1+2)

12 21 2i11 2 22 2i21 . (3.10)

We can extend the definition of the optical susceptibility to include these higher order effects:

P()|1+2 =20(2)(1,2)E1E2(1 2) , (3.11)

where(2) is referred to as the second-order optical susceptibility. The factor of 2 is includedsince we must allow for both orderingsE

1E

2andE

2E

1. Eq. 3.11 is specific to the interaction of

two monochromatic sources. It can be generalised by summing over all possible frequencies,

P(2)() =0

da

db(2)(a,b)E(a)E(b)(a b) . (3.12)

The delta function ensures that the output polarisation oscillates at summations of the frequen-

cies. Although this form looks a bit different from the linear definition of the optical susceptib-

ility, we can write,

P(1)() =0

da(1)(a)E(a)(a) =0(1)()E() , (3.13)


22/43


which shows the same form. The total contribution to the polarisation is then P() =P(1)() +P(2)().

By comparing Eqs. (3.10) and (3.11) we obtain,

(2)(1,2) = Ne3

40m0

2 (1+2)2 2i(1+2)

12 21 2i11 2 22 2i21 ,

= 20m

20

4N2e3(1)(1+2)

(1)(1)(1)(2) , (3.14)

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 Millers rule which

states(2)(1,2) = (1)(1+2)

(1)(1)(1)(2), 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

Pi() = P(1)i () + P

(2)i () + P

(3)i () + . . . and the nonlinear polarisation contributions can be

written,

P(1)i = 0

da(1)i j (a)Ej(a)(a) ,

P(2)i = 0

da

db(2)i jk(a,b)Ej(a)Ek(b)(a b) ,

P(3)i = 0

da

db

dc(2)ijkl (a,b,c)Ej(a)Ek(b)El(c)

(a b c) ...

. =

..

. (3.15)Summation over the repeated indices j,kandl is implicit in the above.

3.2.1 1st order

(1)i 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 polarisa-

tion and the electric field, the interference gives rise to optical absorption or refraction; Re(1)

corresponds to refraction and Im(1) to absorption.


23/43

3.2. THE NONLINEAR SUSCEPTIBILITY TENSOR 19

3.2.2 2nd order

(2)

i jkis a third rank tensor (27 elements). If we use a monochromatic inputE(t) = (E0ei0t

+E0e

i0t)/2 then evaluating the integrals gives for the second-order polarisation,

P(2)i () =

04

(2)i jk(0,0)E0jE0k+

(2)i jk(0,0)E0jE0k

( 0)

+(2)i jk(0,0)E0jE0k( 20)

+(2)i jk(0,0)E0jE0k(+ 20)

. (3.16)

As we indicated in the Lorentz anharmonic model, for a monochromatic input of frequency

0, the second-order polarisation has frequency components at

20 and 0. These give rise

to second harmonic generation and optical rectification respectively. Note that there are no

components at the original frequency0. If we have a combination of frequencies present (1and2 say) we can produce the sum (=1+2) and difference (=1 2). This evenapplies if2=0 i.e. DC and gives an alternative description of the electro-optic effect.

3.2.3 3rd order

(3)ijkl is a fourth rank tensor (81 elements). For a monochromatic input of frequency 0, evalu-

ation of the integrals gives frequency components of the third-order polarisation at = 30(which describes third harmonic generation) and at =

0. 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)

ik j (2,1) =

(2)

i jk(1,2) . (3.17)This property where the direction indices and frequency arguments can be permutated

[e.g.(j,1) (k,2)] is called intrinsic permutation symmetry.

Reality condition The field E(t) and the polarsiation P(t) are physical quantities and hencemust be real. This then requires of the susceptibility, for example,

(2)i jk(1,2) =

(2)i jk (1,2) , (3.18)

that is the conjugate of the susceptibility is equivalent to negating the frequencies.


24/43


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 sym-

metry, overall permutation symmetry also applies where the combination(i,1 2)isincluded in the sets which can be permutated leaving the nonlinear susceptibility invari-

ant. So, for example,

(2)jik(1 2,2)

(2)i jk(1,2) . (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 nonlinearsusceptibility is invariant under permutations of the direction indices. This gives, for

example,

(2)ik j (1,2)

(2)i jk(1,2) . (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 polar-

isation. The linear susceptibility was defined P() =0()E() which is a product.

Now under Fourier transforming, a product becomes a convolution so we have in the timedomain,

P(t) = 02

()E(t ) d . (3.21)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 convolutionE(t)cannot influenceP(t)ift< t i.e. < 0.This then requires () = 0 for < 0. One way of expressing this is to set () = ()()where() is the Heaviside (step) function. If we Fourier transform this expression, theproduct becomes a convolution and we obtain the relationship,

() = 1i

P

() d , (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 solely in terms of its imaginary part and vice versa. Hence if only Im issupplied (across the entire spectral range), Re can be generated. This is one exampleof a dispersion relation of which the best known is the Kramers-Kronig relation relating

refractive index to absorption coefficient. For a wider discussion on dispersion relations

and their applicability to the nonlinear case see [5].


25/43

Chapter 4

Second-Order Optical Nonlinearities

4.1 Contracted tensor for 2nd-order nonlinearities

In determining the second-order polarisation P(2)(), summation takes place over all per-mutations, e.g. in the case of second harmonic generation, if the electric field has compon-

ents along the y and z axes, there will be a polarisation generated parallel to the x axis:

(2)xyzEyEz+

(2)xzyEzEy. Since the same combination of fields occurs in both terms then we can

contract these to a single term. If we useP(2)() = [P20( 20) + P20(+ 20)]/2andE() = [E0(0) + E0(+0)]/2 then we can re-write the second-order polar-isation as,

P

20x

P20

y

P20

z

=0

d11 d12 d13 d14 d15 d16d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

(E0x )2

(E0y )2

(E0z )2

2E0

y E0

z

2E0

x E0

z

2E0

x E0

y

, (4.1)

where the second subscript on the coefficient drelates 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 contractedd-tensor for SHG

is simply related to the conventional susceptibility tensor by di jk(,) =(2)i jk(,)/2. Note

that we have used intrinsic permutation symmetry to reduce the 27 elements to 18 independentones.

If Kleinmann symmetry can be applied then the contracted tensor notation can be extended

to nondegenerate interactions (1 =2),

P1

+2x

P1+2

y

P1+2

z

=20

d11 d12 d13 d14 d15 d16d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

E1

x E2

x

E1

y E2

y

E1

z E2

z

E1

y E2

z +E2

y E1

z

E1

z E2

x +E2

z E1

x

E1

x E2

y +E2

x E1

y

, (4.2)

21


26/43

22 CHAPTER 4. SECOND-ORDER OPTICAL NONLINEARITIES

Triclinic 1

d11 d12 d13 d14 d15 d16d21 d22 d23 d24 d25 d26d31 d32 d33 d34 d35 d36

Monoclinic 2

0 0 0 d14 0 d16d21 d22 d23 0 d25 0

0 0 0 d34 0 d36

m

d11 d12 d13 0 d15 00 0 0 d24 0 d26

d31 d32 d33 0 d35 0

Orthorhombic 222

0 0 0 d14 0 00 0 0 0 d25 0

0 0 0 0 0 d36

mm2

0 0 0 0 d15 00 0 0 d24 0 0

d31 d32 d33 0 0 0

Tetragonal 4

0 0 0 d14 d15 00 0 0 d15 d14 0

d31 d31 d33 0 0 0

4

0 0 0 d14 d15 00 0 0 d15 d14 0

d31 d31 0 0 0 d36

422

0 0 0 d14 0 00 0 0 0 d14 0

0 0 0 0 0 0

4mm

0 0 0 0 d15 00 0 0 d15 0 0

d31 d31 d33 0 0 0

42m 0 0 0 d14 0 00 0 0 0 d14 0

0 0 0 0 0 d36

Cubic 432

0 0 0 0 0 00 0 0 0 0 0

0 0 0 0 0 0

43m

23

0 0 0 d14 0 00 0 0 0 d14 0

0 0 0 0 0 d14

Trigonal 3

d11 d11 0 d14 d15 d22d22 d22 0 d15 d14 d11

d31 d31 d33 0 0 0

32

d11 d11 0 d14 0 00 0 0 0 d14 d11

0 0 0 0 0 0

3m

0 0 0 0 d15 d22d22 d22 0 d15 0 0

d31 d31 d33 0 0 0

Hexagonal 6 0 0 0 d14 d15 0

0 0 0 d15 d14 0d31 d31 d33 0 0 0

6

d11 d11 0 0 0 d22d22 d22 0 0 0 d11

0 0 0 0 0 0

622

0 0 0 d14 0 00 0 0 0 d14 0

0 0 0 0 0 0

6mm

0 0 0 0 d15 00 0 0 d15 0 0

d31 d31 d33 0 0 0

6m2

0 0 0 0 0 d22d22 d22 0 0 0 0

0 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.

wheredi jk(1,2) =(2)i jk(1,2)/2=

(2)i jk(2,1)/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 P2i =0di jkEj E

k and

then apply the inversion operator, for materials with inversion symmetry di jk is unaltered and,

P2i =0di jk(Ej)(Ek). This is only consistent ifdi jk= di jkand hencedi jk= 0. Grouptheory 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


27/43

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 classmm2

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 pmV1, d24= 3.92 pmV1,d32=4.74 pmV

1 andd33=18.5 pmV1. Kleinmann symmetry specifiesd15 d31andd24 d32which is only approximately true (within around 30%) in this case. Note that in this example

the on-diagonal elementd33 dzzz is several times larger than the off-diagonal elements. Thisbehaviour is quite common among materials which exhibit a second order nonlinearity.

di jk di jdxyy= dyxy d12=d26dxzz= dzxz d13=d35

dxyz=dyzx= dzxy d14=d25=d36dxzx=dzxx d15=d31dxxy=dyxx d16=d21dyzz=dzyz d23=d34dyyz=dzyy d24=d32

Table 4.2: Equalities among thed-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,

P1+2

=20d(1,2):E1

E2

. (4.3)

Furthermore, it is quite common to split the fields into magnitude and direction, i.e.E1 =e1E1

etc. wheree1 is a unit vector. A scaler quantitydeff is commonly used which hides the details

of the tensor multiplication: deff= ep [d(1,2): e1e2]. 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)/tand H(t) = j(t) +D(t)/t,assume a non-magnetic material B= 0Hand take j =E. We will split the polarisation intolinear and nonlinear components,D=0E + P=0rE + P

NL. Combining the two curl equa-

tions then gives the second order PDE,

E(t) = 0E(t)t

rc22E(t)

t2 0

2PNL(t)

t2 . (4.4)

Now the properties of the vector differential operators gives that E=( E) 2Eand if there are no free charges, we also have E= 0. The time derivatives can also be


28/43


simplified by Fourier transforming to the frequency domain to give,

2E() =i0 2r

c2

E() 02PNL() . (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() = E(,z)eikz where the value of the wavevector k=r/c is

obtained from solution of the wave equation [Eq. 4.5 without the nonlinear polarisation term].

On taking the space differential,

2E() d2E()

dz2 =

d2E(,z)

dz2 + 2ik

dE(,z)

dz k2E(,z)e

ikz . (4.6)

Note that the final term proportional to E 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) approxim-

ation. This allows us to use |d2E/dz2| |kdE/dz| and neglect the second-order derivative inthe envelope. Hence we reduce the second order differential equation to the first order one,

dE

(,z

)dz = 2 E(,z) +i

20

2kPNL()eikz , (4.7)

where we have substituted the absorption coefficient=0/k. Although this is termed theslowly 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 frequency1 and2, producing a polarisation at the sum frequency3= 1+2.Using the contracted d-tensor notation we have, P3 =20d(1,2): E1 E2 . Inserting thisinto Eq. (4.7) gives,

dE3

dz= 3

2E3 +

i232k3c2

d(1,2): E1E2 eikz ,

dE1

dz= 1

2E1 +

i212k1c2

d(3,1): E3E2 eikz ,dE2

dz= 2

2E2 +

i222k2c2

d(3,1): E3E1 eikz , (4.8)


29/43


where we have used k= k1+ k2 k3 andk3= k(3)etc. Also of relevance the interactionswhere the difference between 3 and1 produces 2 and the difference between 3 and2

produces1which are also listed. Note that E1 = (E1 ).

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 hence1,2,3= 0. We can also apply overall permutation symmetry whichstates d(3,2) = d(3,1) = d(1,2). For simplicity we use the substitution deff=d(1,2):e1e2. Hence Eqs. (4.8) can be rewritten as,

dE3

dz =

i232k3c2 d

effE1

E2

eikz

,

dE1

dz=

i212k1c2

deffE3 (E2 )eikz ,

dE2

dz=

i222k2c2

deffE3 (E1 )eikz . (4.9)

In SI units the irradiance is defined in terms of the electric field amplitude as, I =0cn()|E|2/2, wheren()is the refractive index. Differentiating this gives,

dI

dz=0cn0()

2 (E)dE

dz+E

dE

dz

. (4.10)Inserting the electric field derivatives and using k=n/cthen gives,

dI3

dz= 03

2

ideffE

1 E2

E3

eikz ideff

E1 E2 E3 eikz ,

dI1

dz= 01

2

ideff

E1 E2

E3 eikz ideffE1 E2 E3 eikz ,dI2

dz= 02

2

ideff

E1 E2

E3 eikz ideffE1 E2 E3 eikz . (4.11)We can see from the above that,

13

dI3

dz= 1

1dI

1

dz= 1

2dI

2

dz. (4.12)

Now irradiance is defined as optical energy flowing through a unit area per unit time. The

photon energy is hand hence the ratio of the irradiance to the optical frequency is proportionalto 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 3 is thenegative of the change in the number of photons at 1or2(which are equal): N3= N1=N2. These are known as the Manley-Rowe relations. These seem intuitively correct fromsimple energy conservation; we are far from material resonances so the only energy flow can be

between the waves of different frequency.


30/43


4.2.3 Sum frequency generation:1+2 3Let us assume that there is initially no light of the sum frequency3present. Let us also initiallyexamine the case of low conversion efficiency such that any depletion of the frequencies 1and2 can be neglected, i.e. dE

1/dz, dE2/dz 0. This simplification allows just one of thedifferential equations to be studied instead of the complete set. Let us also assume that the linear

loss can be neglected,3=0 and hence we have for the evolution of the light at frequency 3,

dE3

dz=

i3cn3

deffE1 E2 eikz , (4.13)

where we have used as shorthand for the refractive index n3= n(3). Since E1 and E2 are

treated as constant (low conversion efficiency), this can be easily integrated along the length of

the crystal to give,

E3 (L) = 3deff

cn3E1 E2

(eikL 1)k

= 3deffL

cn3E1 E2 eikL/2sinc

kL

2 , (4.14)

where sincx = (sinx)/x. Now writing this instead in terms of the irradiance, I =0cn()|E|2/2,

I3 (L) = 223

0c3

d2eff

n1n2n3L2I1I2 sinc2

kL

2 . (4.15)

There are several points to make about the generation of the sum frequency described by Eq.

(4.15): the irradiance of the generated light is (1) proportional to a material factor d2eff/n3 (we

will see this is the usual factor in second order processes) (2) proportional to both the irradi-

ance at1 and at2 (3) grows quadratically with distance (L2) and (4) depends on the factor

sinc2kL/2. This last factor is shown plotted in Fig. 4.1. It can be seen that this has a max-imum value of 1 at kL/2= 0 but falls off rapidly away from this. Hence it is important forefficient generation to operate atk= 0 i.e. k1+ k2= k3. This is termed phasematching sincewe are matching the phase velocity of the existing wave (k3) to that of the nonlinear polarisa-

tion (k1+ k2). Phase-matching can be thought of as photon momentum conservation (since the

photons momentum is given by hk) just as the Manley-Rowe relations describe photon energyconservation.

4.2.4 Second harmonic generation:+ 2Second 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


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4 2 0 2 4

0.2

0.4

0.6

0.8

1.0

Figure 4.1: Plot of sinc2x, indicating the sensitivity of phase-matching.

sinc2kL/2 phase=matching dependence. Here we shall include the effects of pump depletionso we require both differential equations,

dE2

dz=

i

cn2deff(,)

E2

eikz ,

dE

dz=

i

cndeff(,)E

2

E

eikz , (4.16)

wherek= 2k k2. We can combine these two by differentiating the second equation andsubstituting for dE2/dzfrom the first to give,

d2E

dz2 + ik

dE

dz+ 2d2effc2n2n2

n E2 n2 E22 E =0 . (4.17)

Now if we initially have no second harmonic, E2(z=0) = 0, then we can use Manley-Roweto getI2(z) =I(z=0)I(z). In terms of electric fields this gives,n2| E2|2 =n(| E(z=0)|2 | E|2). Substituting gives a differential equation in E alone,

d2E

dz2 + ikdE

dz +

2d2effc2nn2

2

E2 E2(z=0)2 E =0 . (4.18)

Now let us just consider the case of perfect phase-matchingk=0. In that case the solution toEq. (4.18) is given by E(z) = E(z= 0)sech Gzwhere G2 =2d2eff| E(z= 0)|2/(c2nn2).In terms of irradiances we have,

I2(z) = I(0)tanh2 Gz ,

I(z) = I(0)sech2 Gz ,

G2 = 22

0c3d2eff

n2n2I(0) . (4.19)


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Note once again the material factord2eff/n3 appears. The functional forms of these are shown in

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 harmonicbecomes an appreciable fraction of that in the fundamental.

0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1

Figure 4.2: Plot of the growth of second harmonic and pump depletion with distance (normal-

ised to 1/G).

4.2.5 Parametric upconversion:1+2 3This is yet another extension of sum frequency generation. In this case we will consider inputs

at frequencies1 and2 generating the sum frequency at 3 for which initially is not present.The beam at 2 is assumed to be the more intense and hence its depletion can be ignored,dE2/dz 0. However, we will include the depletion of the beam at1. It is conventionalto call the intense (2) beam the pump and the less intense beam (1) the signal. In fact thedescription upconversion arises because we will be raising the frequency of the signal from 1to3. As we can ignore the evolution of the pump we need only consider the pair of coupleddifferential equations (making the usual assumption of remote from material resonances),

dE3 (z)dz

= i32cn3

deffE1 (z)E2 eikz ,

dE1 (z)

dz=

i12cn1

deffE3 (z)

E2

eikz . (4.20)

For perfect phase-matching, k= 0, Eqs. (4.20) can be combined to eliminate E3 which resultsin the undamped harmonic oscillator differential equation. The solution is E1 (z) = E1 (z=0) cos GzwhereG2 =13d

2eff| E2|2/(4c2n1n3). In terms of irradiances we have,

I1 (z) = I1 (0) cos2 Gz ,


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I3 (z) = 31

I1 (0) sin2 Gz ,

G2 = 1320c3

d2effn1n2n3

I2 . (4.21)

This oscillating behaviour is demonstrated in Fig. 4.3 which shows the relative power levels

at the frequencies1 and3. Complete conversion is obtained after a propagating a distancel=/(2G).

/2 3/2 20

0.2

0.4

0.6

0.8

1.0

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 to3I3/[1I1 (0)] (solid) andI1/I1 (0)(dashed).

4.2.6 Parametric downconversion:3 2 1As before the pump at 2will 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 fre-

quency3 and subsequently generate the lower (difference) frequency at 1. It turns out that

Eqs. (4.20) apply in this case also with a different initial condition. The result is the same asshown in Fig. 4.3 except that the origin is shifted to the point Gz=/2.

4.2.7 Parametric amplification:3 1+2In the previous examples the intense (pump) beam has been at one of the lower optical fre-

quencies. We will now examine the case where the pump beam is at the higher frequency 3.The other input we will take to be at frequency 1 termed the signal as we will concentrateon changes in this beam. The remaining at beam at frequency2 generated by the nonlinearinteraction is conventionally called the idler. Ignoring depletion of the pump and assuming we


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are remote from resonances, we have to consider the following pair of differential equations,

dE1 (z)

dz = i1

2cn1deffE3

E2 (z)

eikz ,dE2 (z)

dz=

i22cn2

deffE3

E1 (z)

eikz . (4.22)

Taking perfect phase-matching (k= 0), and combining Eqs. (4.22) to eliminate one of the elec-tric fields results in the undamped harmonic oscillator equation with a sign change. Assuming

that there is initially no light of frequency2we obtain the solution E1 (z) = E1 (z = 0) cosh Gz

whereG2 =12d2eff| E3|2/(4c2n1n2). In terms of irradiances we have,

I1 (z) = I1 (0) cosh2 Gz ,

I

2

(z) =

2

1I

1

(0) sinh

2

Gz ,

G2 = 12

20c3d2eff

n1n2n3I3 . (4.23)

For large amplifications,Gz 1, both signal and idler will be dominated by the factor e2Gz, i.e.it will appear we have a gain coefficient 2Gwhich 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 cav-

ity 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, oscil-

lation will preferentially occur at the maximum gain which automatically specifies operation at

perfect phase-matchingk1+ 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 2Gmust exceed

some critical value. SinceG

I3 , 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, 1 (say). The doubly resonant form hascavities for both generated frequencies, 1 and2. The doubly resonant OPO has the lower

threshold but is more complex to set up (since we have to satisfy that both these frequenciesare cavity modes and satisfy1+2=3) and as a consequence is less stable. A triply reson-ant 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


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1

2

3

2

1

2

3

2

3

1

3

1

2

3

1

2

(a) (b) (c)

(d) (e) (f)

Figure 4.4: Summary of the six optical frequency conversion configurations discussed in this

section taking 1 +2= 3: (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-Kronig 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,| E(z)| | E(z=0)|, Eq. (4.18) becomes a linear homogeneous differential equation,

d2E

dz2 + ik

dE

dz + G2 E

=0 . (4.24)

Now let us take the approximation|k| G which is usually the case for low conversionefficiency. Then the second-order derivative can be neglected and the solution of the differen-

tial can be approximated as E(z) = E(z= 0)eiG2z/k. This shows we obtain a phase shift

= G2z/kwhich is proportional to the irradiance and distance propagated. Note that it isinversely proportional tokand hence can be made large for small phase mis-match (althoughthe 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

we identify an effective nonlinear refractive coefficient neff2 such that= 2neff2 Iz/then we


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obtain,

n

eff

2 =

4

0c

1

k

d2eff

n2n2 . (4.25)

For small phase mis-matches we cannot simplify Eq. (4.18) in this fashion. Analytic solu-

tions 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.

/2 3/2 2

Phase Shift

0

/4

/2

3/4

/2 3/2 2

Throughput

0

0.2

0.4

0.6

0.8

1.0

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. 2,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,

dE

dz=

i

cndeff(0,)E

DC E . (4.26)

The solution of this is E(z) = E(z=0)eideffEDCz/(nc), i.e. the optical beam develops a phase

shift=deffEDCz/(nc).

Now from the electro-optic effect we had (1/n2) = rEDC and hence for small changes inrefractive index produces a phase change = n3rEDCz/(2c). Comparing these two forms


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4.3. PHASE MATCHING 33

0 4 8 12 16

Optical Power (scaled)

0

1

2

3

Non

linearPhaseShift

0.0

0.2

0.4

0.6

0.8

1.0

FundamentalThroughput

kL=

0 1 2 3 4

Optical Power (scaled)

0

1

2

3

Non

linearPhaseShift

0.0

0.2

0.4

0.6

0.8

1.0

FundamentalThroughput

kL=2

Figure 4.6: Fundamental phase shift and throughput as a function of scaled irradiance for two

fixed values ofkL.

gives us that the electro-optic coefficient is simply proportional to the deff(0,) coefficient,r= 2deff/n4. Note that since the refractive index is dimensionless, the electro-optic coefficientand the dcoefficient are in the same units usually given in pmV1. If we take more care toinclude the tensor nature of these coefficients we would find that in the contracted notation it is

the transposes that are related,ri j= 2dji/n4. In fact we can see that by comparing Tables 4.1and 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 1 +2= 3(energy conservation). For efficientconversion phase-matching is also required, k1+ k2 = k3 (momentum conservation). Sincek= n/c, phase-matching is satisfied ifn1=n2=n3. The problem is that materials are usually

dispersive and the refractive index varies with frequency. Normal dispersion has the refractiveindices increasing with frequency, n3> n1, n2. This obviously makes it difficult to achievephase-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 ex-

traordinary refractive index in a uniaxial crystal as a function of angle between the propagation


38/43


direction and the optic axis is given by, ne() = (sin2/n2e+ cos

2/n2o)1/2. Fig. 4.7 demon-

strates how phase-matching could be achieved for second harmonic generation for a particular

propagation direction by selecting the propagation direction such thatno(m) =n2e (m)

z

x

m

on2

no

ne

2

ne

k

Figure 4.7: Birefringent phase-matching for second harmonic generation in a negative uniaxial

crystal. For propagation at a particular angle mto the optic axis,no =n

2e .

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.

x

z

k k k k k ko

1

e

1

o

2

e

2

o

3

e

3

Figure 4.8: Birefringent phase-matching for three wave interaction in a negative uniaxial crystal.

For propagation at a particular angle mto the optic axis,ko1+ k

o2= k

e3.

If we expand the phase mis-match around the anglemwhich corresponds to perfect phase-matching we have k+ O(2)where = m. Since the sinc2 efficiency dependenceof phase mis-match is very tight, this means that the propagation direction must, in general, be


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Type I: low frequency components negative uniaxial ko1+ k

o2= k

e3

positive uniaxial ke1+ ke2= ko3Type II: low frequency components negative uniaxial ko1+ k

e2=k

e3

positive uniaxial ko1+ ke2=k

o3

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 m,

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 varyas 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 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.

x

z

k ke

2

o

3

k

k

k1

2

3

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, = 0 and 90.Obviously the = 0 does not apply as there is no variation in refractive index between theorthogonal polarisations. The m= 90

can be exploited though to eliminate walk-off. This also


40/43


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 atm=90. Sometimes it is necessary to temperature tune the crystal to achieve this, but the advantagesof this geometry outweighes the disadvantage of the necessity of a uniform temperature stage.

To date the most attractive materials for second-order processes are LiB3O5 and -BaB2O4because 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-birefringentmaterials (e.g. for GaAs in the infrared, d14 200 pmV1) or for the on-diagonal elementswhere all the polarisation components are parallel (e.g. for LiNbO3at 1.3m,d33=32 pmV

1compared to d31 = 5.5 pmV

1 and for KTP at 0.88 , d33 = 18.5 pmV1 compared tod15 = 2.0 pmV

1). Quasi-phase-matching offers the possibility of using these coefficients,particularly in guided wave geometries.

0 Lc

2Lc

3Lc

4Lc

5Lc

distance

0.0

10.0

20.0

30.0

40.0

2

Irradianc

e

k=0

DR

DD

k=/ 0

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=0) and the quasi-phase-matching casesof (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 finitephase mis-match (k=0). Let us suppose that the refractive indices are such that this phasemis-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 in-

terference remains constructive is called the coherence length Lc=/|k|. Now in the case


41/43


of domain reversal we process the crystal so that when the second harmonic and the nonlin-

ear 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 seenthat 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 4/2

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 ofLc with a cor-

responding 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 non-

linearity 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 satisfies

k=2/. This grating couples the wavevector of the optical wave to k 2/although theoverall efficiency of this process is low.


42/43



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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).

course notes of non linear optics

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