1

2. Nonlinear Optics in Waveguides

2.1. An Evolution Equation for Waveguide Modes

In the last chapter we have learned about the nonlinear response of the material. Together

with Maxwell’s equations we can now determine any kind of field propagation in the

presence of a nonlinear response. Here we concentrate on the field evolution in a straight

waveguide, which extends in z-direction. Such a waveguide is defined by a translationally

invariant distribution of the linear dielectric constant , which we for simplicity assume to be

isotropic and real valued. In case that absorption plays a role the imaginary part of can be

incorporated as a perturbation later.

In what follows we operate at a fixed carrier frequency . Hence all interacting fields are

decomposed like

1 11

,2

i t i tF r t F r e F r e

(1)

Unfortunately the complete set of equations including all nonlinear interaction is much too

difficult to be solved in a self-consistent way. A quite common and always justified approach

is to assume all nonlinear response to be small compared with the linear one. In what follows

we assume that all nonlinearly induced index changes (or induced phase changes) are orders

of magnitude smaller than the linear index of the material (or the linear phase evolution).

Hence we solve the field equations for the linear or unperturbed system first and discuss

nonlinearly induced changes in the sense of a perturbation theory afterwards.

1. unperturbed system

For fixed frequencies and in the absence of any nonlinear polarization Maxwell’s equations

transform into

1 0 1 1 0 1E i H H i E

, (2)

where , , ,r x y

contains the structure of the waveguide. Here we are only

interested in guided modes, which propagate along the waveguide and have a fixed field

structure in transverse direction as

1 0 0, , , , , expE x y z E x y i z

1 0 0, , , , , expH x y z H x y i z

(3)

2

Field structure 0E

and 0H

and propagation constant 0 are eigenvectors and eigenvalues

of a non-Hermitian matrix defined by Eq.(2). Guided modes have real valued 0 and their

field structure decays exponentially outside the waveguide.

2. perturbed system (index s)

0 0s s s s sE i H H i E P

, (4)

where sP

contains all perturbations including the action of the nonlinearity, but also those

which have been neglected in Eq.(2) (as e.g. the imaginary part of ).

We are now going to compare the perturbed and unperturbed fields by evaluating the

expression 1 1= s sE H E H

, which has a similar structure as the Poynting vector known

from the energy balance of electromagnetic fields. Using Eqs.(2) and (4) we obtain

1 1 1div =div s s sE H E H i E P

. (5)

Next we integrate above expression with respect to the transverse coordinates x and y making

use of the exponential decay of

with respect to x and y approaching infinity, which results

in the identity

div x y z zdx dy dy dx dx dy dx dy dx dyx y z z

Thus we obtain

1 1 1s s sz

dx dy E H E H i dx dy E Pz

(6)

The z-dependence of the unperturbed field is that of a guided modes and therefore quite trivial

(see Eq.(3)). To further evaluate Eq.(6) we assume the perturbation to affect the amplitude,

but not the transverse field structure of a certain guided mode (index m), which need not

coincide with that of the unperturbed field (index 0). Under this assumption the perturbed

fields look quite similar to Eq.(3) as

, , , , ,s m mE x y z u z E x y

, , , , ,s m mH x y z u z H x y

, (7)

but exp mi z has been replaced by mu z .

Inserting Eqs.(3) and (7) into Eq.(6) yields

3

0 00 0 0

i z i zm m s

zdx dy E H E H u z e i e dx dy E P

z

(8)

We first shortly discuss the case of vanishing polarization 0sP

, where mi zu z e holds.

In that case Eq.(8) can be simplified to

0

0 0 0 0mi zm m m

zdx dy E H E H i e

. (9)

If both modes are different ( 0m ) the integral must vanish. Hence, we have found an

orthogonality relation for guided modes as

0 0 0 04m m mz

dx dy E H E H

(10)

where is the Kronecker symbol and 0 the guided power of the mode defined as

0 0 0 0 01

4 zdx dy E H E H

. (11)

Evaluating Eq.(8) for nonvanishing polarization 0sP

corresponds to an expansion of the

perturbation into the orthogonal set of guided modes. Consequently one obtains an evolution

equation for the amplitudes of guided modes, which are driven by the polarization sP

0 004 s

iu z i u z dx dy E P

z

.

According to the current setting u(z) has no unit. It would be more convenient to link this

amplitude to the total guided power. That we do by rescaling the amplitude as

0U z u z . (12)

Now 2U z corresponds to the total guided power in mode 0 and follows the evolution

equation

0 004

si

U z i U z dx dy E Pz

. (13)

Note that according to the definition of 0 (see Eq.(11)) Eq.(13) does not depend on the actual

scaling of the mode fields.

4

2.2. Field Evolution in Channel Waveguides under the Action of

Quadratic Nonlinearities

2.2.1. The Equation of Motion

We are now interested in the field evolution happening in a waveguide in the presence of a

quadratic nonlinearity. In general the field evolution in the presence of a nonlinearity cannot

be restricted to a limited number of discrete frequencies. However, in most cases only a few

frequencies are generated with considerable amplitude. Here we restrict to classical second

harmonic generation, where only two frequencies are involved and a fundamental harmonic

wave (FH) is frequency doubled to a second harmonic one (SH). Following the notation

introduced in chapter one the following types of polarization come into play:

3

(2) (2)0 2

, 1

| 2 , | 2 ,j ki ijk

j k

P K E E

3

(2) (2)0

, 1

2 2 | , 2 | ,j ki ijk

j k

P K E E

(14)

Note that | 2 , 1K and 2 | , 1 2K . Further we assume that only two

modes, one at FH and one at SH frequency interact as

(2)

(2)2

FH4

2SH 2 2

4

i U dx dy E Pz

i U dx dy E Pz

(15)

Expressing the optical fields by the mode fields defined in Eqs.(7) and (12), inserting this

expression in (14) and finally replacing the nonlinear polarizations in Eq.(15) yields.

(2)2FH

(2) 22 SH

FH 0

SH 2 0

i U U Uz

i U Uz

. (16)

The magnitude of the nonlinear interaction is contained in the effective nonlinear coefficients

as

3 2(2) (2)0FH

, , 1 2

3(2)0

22 , , 1

| 2 ,4

| 2 ,4

j k

i

i j k

ijki j k

ijki j k

E Edx dy E

dx dy E E E

5

3

(2) (2)02FH

2 , , 1

2 | ,4 i j kijk

i j k

dx dy E E E

Note that above integrals are taken transverse to the waveguide. The orientation of the

coordinates is defined by the waveguide geometry and not by the nonlinear crystal. Because

typical tensors of the nonlinear response are given with respect to the crystal axes an

additional coordinate transformation might be required.

Above integrals also take into account a potential space dependence of the nonlinear

coefficient and the mutual overlap between FH and SH fields.

If we follow the above made assumption that no further fields except the modes at FH and SH

frequencies are involved energy conservation (Manley-Row relation) must hold or the power

flux 2 2

2U U must be constant along the waveguide. Together with Eq.(16) we obtain:

2 22

22

(2) (2) 22 2 2FH SH

0

2

U Uz

U UiU i iU i cc

z z

iU U U U iU U U cc

Because and 2 are assumed to be real respective terms cancel each other with

their complex conjugate. Finally we find that energy conservation is equivalent to

(2) (2)22 FH SH0 Im U U

.

Above relation must hold on each point z of the waveguide and for all injected fields. Hence it

can only be fulfilled if

(2) (2) (2)FH SH eff

(17)

holds. Hence, there is only a single nonlinear coefficient with the unit 1 m W in the

system of evolution Eqs.(16).

A further transformation makes the system of evolution equations more applicable. As

presented in Eqs.(16) the field amplitudes still contain the fast, i.e. on the wavelength scale,

spatial oscillations of the phase. These fast oscillation do not contain new information, but

make a numerical solution difficult and cumberson. We therefore remove the fast oscillations

by the following transformation

2exp and exp 2U z a z i z U z b z i z , (18)

where a and b are the slowly varying envelopes of the FH and SH fields, respectively.

6

Eqs.(16) now transforms into

(2)eff

(2) 2eff

FH 0

SH 0

i a a bz

i b b az

(19)

where 2a z and 2

b z are the guided power in the FH and SH waves and

2 2 (20)

is the mismatch between the two propagation constants.

2.2.2. Normalization

The aim of this chapter is twofold. First we want to come closer to a solution of the system of

Eqs.(19) and second we now introduce a general scheme, which can be applied to estimate

relevant scales of nonlinear evolution equations without actually solving them. We thus

introduce dimensionless quantities A and B and relevant physical units as

0 0 0, and ,iz Z Z a P A b P B e (21)

where 0Z is a characteristic length, 0P a relevant power level and a constant phase. Eq.(19)

expressed in these quantities reads as

(2)0 0eff

(2) 20 0 0eff

FH 0

SH 0

i

i

i A Z P e A BZ

i B Z B Z P e AZ

(22)

We now chose the normalization such, that all constants in Eq.(22) become unity. Thus we

obtain the characteristic length

01

Z

(23)

for the two waves to run out of phase. The characteristic power to induce a noticeable

nonlinear action

2

0 2 (2)(2)eff0 eff

1P

Z

(24)

is reduced by a growing nonlinear coefficient, but grows with the squared mismatch. This

explains why second harmonic generation is such a rare phenomenon. Particular care has to

be taken to reduce the mismatch. Otherwise required power levels grow to infinity.

The phase is chosen as

7

(2)effarg . (25)

It removes the phase of the nonlinear coefficient completely. Different from linear interaction

the phase of the coefficient of the quadratic nonlinearity has no physical relevance.

It has to be pointed out that normalization is not a unique procedure. For example in case of

vanishing mismatch the sample length is much smaller than the inverse mismatch therefore

becoming the relevant scale. Following Eq.(24) required power levels scale inverse to the

squared sample length in case of vanishing mismatch.

The final set of scaled equations now reads as:

2

FH 0

SH 0

i A A BZ

i B B AZ

, (26)

where can have the values -1, 0 and +1.

2.2.3. Conservation laws – the Hamiltonian

We have already shown that energy conservation holds. Hence, the fields in Eq.(26) must

obey

2 2const.Q A B (27)

There is another conserved quantity called Hamiltonian

2 2H B B A B A B . (28)

This can be easily checked with the help of Eq.(26) as

2

2 2 2

2 4 2 2 2 2

2 .

2 .

2 .

0

H B B A B AB A ccz z z z

i B B A i A B A i AB A B cc

i B i A i A B i B A i B A cc

To get an idea why this quantity is called “Hamiltonian” we remember quantum mechanics,

where we also have two conserved quantities, namely the norm of the wave function

2const.dV r

and the expectation value of the Hamiltonian

ˆ const.dV r r H

8

The first one corresponds to the total power introduced in Eq.(27) where the second one to the

above introduced Hamiltonian.

2.2.4. Solution

We follow here a classical scheme, which allows for the solution of nonlinear ordinary

differential equations as introduced in Eq.(26). We first express the complex variables as real

ones by separating amplitudes and phases into

( ) ( ) exp ( )AA Z A Z i Z and ( ) ( ) exp ( )BB Z B Z i Z . (29)

Obviously our system has four independent variables to be determined.

Inserting Eq.(29) into Eq.(26) and dividing by exp ( )Ai Z and exp ( )Bi Z yields:

2

FH ( ) ( ) ( ) ( ) ( ) exp ( ) 2 ( ) 0

SH ( ) ( ) ( ) ( ) ( ) exp 2 ( ) ( ) 0

A B A

B A B

i A Z A Z Z A Z B Z i Z i ZZ Z

i B Z B Z Z B Z A Z i Z i ZZ Z

(30)

Actually the system of equations (30) consists of four real valued equations. Evaluating the

imaginary part of Eq.(30) yields

2

FH ( ) ( ) ( ) sin ( ) 0

SH ( ) ( ) sin ( ) 0

A Z A Z B Z ZZ

B Z A Z ZZ

. (31)

Here we have introduced the phase difference ( ) ( ) 2 ( )B AZ Z Z , for which an

evolution equation is likewise derived from the real part of Eq.(30) as

2

( ) ( ) 2 ( )

( )2 ( ) cos ( )

( )

B AZ Z ZZ Z Z

A ZB Z Z

B Z

. (32)

Eqs.(31) and (32) demonstrate that the field evolution depends on three real valued variables

only. As in all coherent systems without a fixed time reference the absolute phase does not

matter and only a phase difference as ( )Z plays a role.

We can further reduce the number of free variables by employing Eq.(27) and removing one

of the amplitudes as

9

2

2

( ) ( ) sin ( )

( )( ) 2 ( ) cos ( )

( )B

B Z Q B Z ZZ

Q B ZZ B Z Z

Z B Z

(33)

A further reduction of free variables is obtained by using the second conserved quantity

namely the Hamiltonian. To this end we first express the Hamiltonian from Eq.(28) by the

remaining variables while employing Eq.(27) as

2 2

2 2

( ) 2 ( ) ( ) cos ( )

( ) 2 ( ) ( ) cos ( )

H B Z A Z B Z Z

B Z Q B Z B Z Z

(34)

Eq.(34) allows to remove ( )B Z from Eq.(33) as

2

22

2

2

sin

12

B Q BZ

H BQ B

B Q B

(35)

A further simplification can be obtained by multiplying Eq.(35) with B and introducing the

SH power as 2X B

2 24X X Q X H X

Z

. (36)

Given the SH power at the entrance of the waveguide at Z=0 as 00X Z X we can

determine the SH power at Z=L as

0

2 204

X L L

X

dXdZ

X Q X H X

(37)

A solution of Eq.(37) still requires solving the above integral and inverting the resulting

function of the variable X. This finally leads to so-called elliptic functions.

A general conclusion from that chapter is the acknowledgement of the importance of

conserved quantities. Given a nonlinear system with N real valued variables or unknowns we

need to find N-1 conserved quantities or symmetries to come to an analytical solution.

Therefore the existence of a time independent Hamiltonian is so important. It not only adds a

further conserved quantity, but also excludes dissipative effects, which usually prohibit the

existence of conserved quantities at all.

10

2.2.5. Phase-Matched SHG

We now demonstrate a solution of Eq.(37) for a particular case, where it leads to well known

analytical functions. We study the case of complete phase-matching (=0) and assume that

there is no SH power at the input

200 0B Z X . (38)

We first evaluate the conserved quantities at the input. The total energy Q is given by the FH

field only and the Hamiltonian vanishes following Eq.(28)

0H (39)

Inserting Eqs.(38) and (39) into (37) yields

0 2

X LdX

LQ X X

, (40)

which can be transformed to

0

1ln

X LQ X

LQ Q X

(41)

and solved as

2tanhX L Q QL (42)

Hence, total conversion occurs for infinite propagation length L. Conversion efficiency

X Q depends on the product QL only. Hence, required power scales inverse with the

squared propagation length, a relation which we could already guess as a result of

normalization (see chapter 2.2.2).

Up-conversion in the phase-matched case

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SH power / input power

propagation length in (2)in eff1 P

11

2.2.6 Downconversion

A general property of Hamiltonian systems is the reversibility of their evolution. If SH

generation is possible, as discussed in the previous chapter, also the opposite process called

parametric downconversion should be observed.

We assume phasematching 0 and all power to be initially concentrated in the SH wave as

200B Z X Q and 2

0 0A Z . (43)

Therefore the Hamiltonian vanishes following Eq.(28)

0H . (44)

All the conserved quantities have the same values as for SHG, only the initial value of X

(X=Q instead of X=0) has changed. Hence, the whole derivation of a solution is performed in

a similar way as in the last chapter, only the boundaries have changed compared with Eq.(41)

as

1

ln

X L

Q

Q XL

Q Q X

(45)

However, the function on the left side of Eq.(45) becomes infinite on the lower boundary.

Hence, X(L) can only deviate from Q, if the propagation length L tends to infinity. This is

consistent with an inversion of the SH process discussed above. The latter one also needs an

infinite propagation length to obtain complete conversion. Therefore, a downconversion

seems to require an infinite propagation distance to happen. Looking back at Eq.(36)

2X Q X XZ

(46)

we in fact notice, that

X L Q (47)

is in fact a solution because it allows for 0XZ

for all Z. Hence, the system can stay in a

state with all the power in the SH field. It formally requires an infinite propagation length to

obtain noticeable down-conversion. However, any small perturbation will start the process

after a finite length. In real systems quantum noise triggers down-conversion. In the phase-

matched case complete conversion is finally obtained and SH generation starts as discussed in

the previous chapter and follows Eq.(42).

12

Down-conversion for the phase-matched case (The down-conversion process starts from

quantum fluctuations)

2.2.7 Visualization of the field dynamics using the Hamiltonian

Although Eq.(36) can be solved in principle a direct visualization of respective solutions

might be helpful. Here we demonstrate a way to obtain a more global understanding also for

the non-phasematched case. We make use of the Hamiltonian H given in Eq.(34)

2 2

3 2

( ) ( ) ( )2 1 cos ( )

B Z B Z B ZHZ

Q QQ QQ

, (48)

where Q is the total power, ( )B Z the amplitude of the SH field, ( )Z the phase difference

between SH and FH fields and the scaled mismatch. Above equation can also be expressed

in rescaled amplitudes as

2 2, , 2 1 cosH B B B B

(49)

where 3 2, andH H Q Q B B Q holds. The scaled SH amplitude is

restricted to 0 1B , where ( )Z is periodic.

The system can only move on lines of fixed Hamiltonian. Hence, expressing H as a contour

plot of B and already displays the dynamics of the system. For complete phase-

matching SH generation corresponds to a contour line of 0H , which starts in the origin

(zero SH power) and approaches the outer circumference of the displayed circle. For all other

initial conditions, where FH and SH fields are injected together the initial phase difference

plays a role. Usually a periodic motion along a contour line with finite H occurs and no

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SH power / input power

propagation length in (2)in eff1 P

13

complete conversion neither to the FH nor to the SH wave occurs. When starting at the

extrema of H (crosses) no conversion occurs at all and the solution is completely stationary.

The results change, if a finite mismatch occurs. Eq.(49) demonstrates that the dynamics

basically depends on the scaled mismatch. An increase of power is similar to a reduction of

the mismatch. Any finite mismatch breaks the symmetry in the H plot. Now there exist no

contour line connecting the origin with the circumference. Hence, power conversion is never

complete neither in up- nor in down-conversion. When starting with all the poer in the FH

wave the evolution is always periodic. Hence, up-conversion is followed by down-conversion.

0.4

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.4

-0.4

-0.4

-0.4

0

0

0

0

0

0

0.4

0.4

B Q

=0

0.4

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.4

-0.4

-0.4

-0.4

0

0

0

0

0

0

0.4

0.4

-0.4

-0.4

-0.4

-0.4

0

0

0

0

0

0

0.4

0.4

B Q

=0

-0.4

-0.4

0

0

00

0.4

0.4

0.4

0.4

0.4

0.8

0.8

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

B Q

0.5Q

-0.4

-0.4

0

0

00

0.4

0.4

0.4

0.4

0.4

0.8

0.8

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

B Q

0.5Q

-0.4

-0.4

0

0

00

0.4

0.4

0.4

0.4

0.4

0.8

0.8

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

B Q

-0.4

-0.4

0

0

00

0.4

0.4

0.4

0.4

0.4

0.8

0.8

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

B Q

B QB Q

0.5Q

Contour plots of the scaled Hamiltonian H for zero (left image) and non-vanishing

mismatch.

2.2.8. Low-Depletion-Limit

If not particular care is taken the phase mismatch is rather big and conversion efficiency is

small. To have a better access to the mismatch itself we return to the unscaled Eqs.(19), but

assume that the FH power stays roughly the same in the whole sample.

FH const.a z P (50)

Consequently, we can replace a in the equation of the SH field as

(2)FHeffSH 0i b b P

z

. (51)

Assuming

0 0b z (52)

we immediately end up with the solution

14

(2)eff

FH exp 1b z P i z

. (53)

The power in the SH wave

2

22 (2) 2SH FHeff

sin2

4z

P b z P

(54)

decays quickly with growing mismatch and oscillates with a period of

SH

FH SH

2L

n n

, (55)

where FHn and SHn are the effective indices of the modes at FH and SH frequencies and SH

is the vacuum wavelength of the SH wave.

2.3. Pulse evolution in fibers

2.3.1. Evolution equation

The system we have in mind is a standard optical fiber as it is used in telecommunication.

Here we assume a so-called single mode fiber made from fused silica. The only relevant

nonlinearity in this amorphous material is the cubic one. All optical fields should be centered

around a mean frequency 0. Hence, the only relevant nonlinearity is the so-called self-phase

modulation (see chapter 1.1.5.) represented by (3) | , ,ijkl .

Here we start from Eq.(13)

00

, , , , , , ,4

si

U z i U z dx dy E x y P x y zz

(56)

which describes all the amplitude of the waveguide mode in frequency space. Because we aim

on describing pulses a transformation to time dependent quantities via a Fourier

transformation is required. To avoid convolution integrals we make use of the narrow

spectrum of our optical excitation. Hence, in the relevant frequency domain a Taylor

expansion can still ensure reasonable accuracy. It is quite common to expand the propagation

constant of the guided mode as

20 0 01

2g

D

v (57)

15

where 0 0 , 0

1

gv

and 0

2

2D

yields. Usually vg is called

the group velocity and D the group velocity dispersion.

The normalized mode profile 0 0, ,E x y

as it appears in Eq.(56) is only weakly

frequency dependent as long as the fiber is operated far from cut-off. Hence we can assume

0 0 0 0 0, , , ,E x y E x y

(58)

Inserting Eqs.(57) and (58) into Eq.(56) yields

20 0 0

0 00 0

0

1, ,

2

, 04

g

s

Di U z U z

z v

dx dy E P z

(59)

Before we return to the temporal domain we shortly have a look at Eq.(1). It tells us how to

construct the total optical field from all its frequency components as

0

0

0 0

0

0 0

0

, ,1, , , ,

2

, ,1,

2

, ,1,

2

i t

i t

E x yE x y z t d U z e cc

E x yd U z e cc

E x yU t z cc

(60)

Hence we can introduce the temporal profile and the time dependent polarization as

, , i tU t z d U z e

and , , , , , , i ts sP t x y z d P x y z e

(61)

When applying the same transformation onto Eq.(59) each prefactor in Eq.(59), which

consists of 0

n transforms into 0

ni t . Consequently we obtain

20 0 0

0 00 0

0

1, ,

2

, ,, , , 0

4

g

s

Di U t z i t i t U t z

z v

E x yi t dx dy P t x y z

. (62)

Now we want to express the nonlinear polarization by the acting optical field while making

use of Eq.(60) and assuming Kleinmann symmetry (see chapter 1.1.5, Eq.(27))

16

(3)2

0 0 0 0 03 20

220 0 0 0

3, , , 2 , , , ,

4

, , , , , ,

yyzzsP t x y z E x y E x y

E x y E x y U t z U t z

(63)

Inserting expression (63) into the integral in Eq.(62) allows to perform the integration via the

transverse coordinates thus reducing Eq.(62) to a self-consistent evolution equation for just

the amplitudes ,U t z .

In optical fibers the mode is only weakly guided within the silica and predominantly linearly

polarized. Therefore we assume the nonlinear coefficient to be spatially uniform within the

fiber core and the mode profile to be real.

0 00

0

4 2(3)00 0 02

0

, ,, , ,

4

9, , , ,

16

s

yyzz

E x ydx dy P t x y z

dx dy E x y U t z U t z

(64)

This expression can be further simplified by replacing the nonlinear coefficient in Eq.(64) by

the (3)

2 20

9

4yyzzncn

introduced in chapter 1.1.5. In addition the guided power introduced in

Eq.(11) can be completely expressed by the electric field provided that longitudinal

components ( 0zE ) completely vanish as

20 0

02dx dy E

(65)

combining Eqs.(64) and (65) and inserting n2 yields:

2

20 0 0

0 220

,, ,, , , ,

4s

effeff

U t zE x y ndx dy P t x y z n U t z

c An

(66)

where he have introduced the effective index effn c and the effective area of the mode

2

2 4

0 0 0 0, , , ,effA dx dy E x y dx dy E x y

.

Because effn almost coincides with the refractive index of the silica core n the factor 2 2effn n

can be put to 1. Under these conditions we can reformulate Eq.(62) as

17

2

0 0 0

22

00 22

0

1, ,

2

,11 , 0

g

effeff

Di U t z i i U t z

z v t t

U t zni n U t z

t c An

(67)

Eq.(67) is still not in a handy form. It contains the fast phase variations of the guided mode

like 0 0exp i z i t . Hence ,U t z should be decomposed into the fast varying phase and

into a slowly varying envelope as

0 0, , expU t z a t z i z i t (68)

Doing so removes the constant terms from Eq.(67) as

22

20 222

,, , ,

2g eff eff

a t z i ni Di n a t z a t z a t z

z v t c A cA tt

(69)

In Eq.(69) just the natural coordinates z and t were chosen. But, the zero of the time is not a

priori defined. It is useful to count time only from the moment where the pulse arrives. Hence,

we introduce a co-moving frame of reference, where time is counted like

gt z v . (70)

This has a profound impact on the derivatives in Eq.(69) like

, , ,g g g

i i ii a t z i a z i a z

z v t z v v z

(71)

Hence, this last transformation removes the first derivative with respect to time in Eq.(69) and

we end up with the evolution equation for an optical pulse propagating in a fiber

22

20 222

,, , ,

2 eff eff

a z i nDi n a z a z a z

z c A cA

(72)

The last transformation also tells us that motion with a certain velocity is always represented

by a first derivative with respect to time.

2.3.2. Discussion of terms

We are now dealing with Eq.(72) and discuss its different terms separately.

a) iz

This term is responsible for the evolution along the fiber. This is quite surprising, because

evolution is usually connected with time. All solutions of Eq.(72) have to be understood in the

18

following way: For a fixed propagation distance z we look for the temporal shape of the pulse

represented by the variable . For different fiber length z this pulse shape will differ. Hence,

the pulse shape evolves along the fiber.

b) 2

22

D

combined with the term responsible for evolution yields

2

2, 0

2

Di a z

z

. (73)

A similar equation, is well known in quantum mechanics as Schrödinger equation, which

reads in the one-dimensional case and in the absence of external potentials for a particle with

mass m as

2 2

2, 0

2i u t x

t m x

. (74)

This gives us a nice interpretation of the terms in Eq.(73). z and play the role of time t and

length x in quantummechanics, where D is quite similar to the inverse of a mass. In the same

way as wave packets spread in the quantum world also pulses tend to broaden in fibers. For an

initially Gaussian pulse 20

0, 0 Ta z A e

a complete analytical solution of Eq.(73) is

known as

2 20

2 22 200 0 0 0 0

( , ) exp exp1 1 1

A za z i

zi z z T z z T z z

pulse broadening phase variation

(75)

with the dispersion length

20

0 2

Tz

D . (76)

An initially Gaussian pulse broadens as seen in the first term of Eq.(74), but also its phase

evolves. Keeping in mind that phase is connected with frequency as

we conclude

that the quadratic phase variation in the pulse displayed in Eq.(75) corresponds to a time

dependent frequency shift. Hence, different frequency components arrive at different times .

This brings us back to the physical origin of the term displayed in Eq.(73). Following Eq.(57)

the group velocity dispersion D is defined as the second derivative of the propagation constant

and has a non-vanishing value as soon as the group velocity is dispersive or if different

frequency components travel with different speed. This now marks an important difference

19

between the Schrödinger equation (74) and Eq.(73). In contrast to mass m, which is a priori

positive group velocity dispersion can be positive or negative dependent on the type of fiber

used and on the frequency of operation. A fiber piece with negative D can compensate for the

dispersive broadening accumulated during propagation under the influence of positive group

velocity dispersion. Hence, Eq.(73) can also allow for pulse compression for certain z values.

Dependent on the sign of D z0 is positive or negative and the frequency chirp in Eq.(75) has

positive or negative sign. In case of normal dispersion ( 0D ) red light travels faster than

blue one and therefore arrives at earlier times and vice versa.

Phase and frequency evolution in a pulse and an example for a resulting field

c) 2

02

,

eff

a zn

c A

combined with the term responsible for evolution yields

2

02

,, 0

eff

a zi n a z

z c A

. (76)

It is easy to show that based on Eq.(76) for real valued n2 as it is the case in silica

4 4

2 0 02 2, 0

eff eff

a aa z ia i a cc i n i n

z z c A c A

(77)

holds and that therefore the power in the pulse remains the same at each time step . Based on

that result Eq.(76) can be integrated immediately resulting in a pure phase evolution of the

pulse like

2

02

, 0, , 0 exp

eff

a za z a z i n z

c A

. (78)

Hence the phase follows the power instantaneously resulting in a time-dependent frequency

shift. To a certain extend this can be balanced by the frequency shift, which is induced by the

D > 0

D < 0

D > 0 D < 0 E

D > 0

red

blue

20

action of group velocity dispersion thus resulting in a stable propagation of a so-called

temporal soliton.

Nonlinearly induced frequency shift in a pulse

d) 22 , ,eff

i na z a z

cA

combined with the term responsible for evolution yields

22, , , 0eff

i ni a z a z a z

z cA

. (79)

Obviously the i in Eq.(79) can be removed resulting in a purely real valued equation. Hence,

if we assume a to be real valued it will stay real and we can simplify Eq.(79) in the following

way:

223, , 0

eff

na z a z

z cA

(80)

We already know that a first derivative represents a motion of the pulse. In Eq.(69) this was

represented by the group velocity vg. In contrast to our earlier derivation this new quasi

velocity depends on the amplitude a itself in a nonlinear way. Hence, motion of the pulse will

depend on its velocity. In case of positive n2 parts of the pulse having a high amplitude will

arrive at later times , a process known as self-steepening. Eq.(80) can be solved. Respective

solutions are well known for water waves. They develop a singularity with infinite inclination

after a finite propagation length. Of course the effect of the self-steepening term in the

complete Eq.(73) is more complicated. However, the principal physics of its action remains

the same. It tends to force the pulse towards a catastrophic steepening, a process which is

counteracted by group velocity dispersion.

We are now going to estimate the magnitude of the self-steepening term. Assuming a rather

smooth pulse the time derivative in the self-steepening term can be approximated by the pulse

duration T0 as

pulse

nonlinearly induced frequency shift

21

2 22 2

0

1, , , ,

eff eff

i n na z a z a z a z

cA T cA

(81)

We can neglect the self-steepening term in Eq.(72) as long as the other nonlinear term

representing the self-phase modulation is much bigger yielding

2 2 22 2 20

0

1, , , , , ,

eff eff eff

i n n na z a z a z a z a z a z

cA T cA cA

Hence, self-steepening will be negligible as long as 0 01 T yields. Hence, the slowly

varying envelope of the pulse must cover many oscillations of the electric field or the spectral

width of the pulse must be much smaller than its carrier frequency. These conditions are still

well fulfilled for pulses as short as 100fs. Pulses used in telecommunication are still two

orders of magnitude longer. Hence, it is well justify to neglect the self-steepening term in the

following considerations.

2.3.3. The Nonlinear Schrödinger Equation

The equation, which we obtain after neglecting the self-steepening term as

22

022

,, 0

2 eff

a zDi n a z

z c A

, (82)

is quite general. A similar version describes diffraction and nonlinear propagation in film

waveguides. Without a formal derivation we give

22

022

0

,1, 0

2 eff

a x zi n a x z

z c dx

, (83)

where ,a x z is the envelope of a cw-beam (fixed frequency 0) propagating along z and

diffracting in x-direction in a film waveguide. 2,a x z carries now the unit W/m and deff is

the effective thickness of the film, which is defined in a similar way as the effective area in

Eq.(82).

Equations of the type (82) or (83) can be regarded as a nonlinear continuation of the

Schrödinger Equation (74). They are therefore summarized under the pseudonym Nonlinear

Schrödinger Equation (NLS). The NLS is a quite general nonlinear equation describing all

types of waves, as water waves or plasma waves, as well as the evolution of Bose-Einstein-

Condensates (BEC). Of course, in all these equations relevant quantities appear in different

units. Hence, a respective normalization is required to highlight the underlying physics.

22

2.3.4.Normalization

We have already seen in chapter 2.2.2. that normalization of an evolution equation is a simple,

but effective way to gain insight into the relevant scales of a process and to identify relevant

quantities. We will do now the same for Eq.(82). We assume all relevant quantities as the

coordinates and the field to consist of a dimensionless variable denoted by a capital letter and

a “unit” marked with the index “0” as:

0 0 0, andz Z Z T T a P A . (84)

Inserting Eqs.(84) into Eq.(82) yields

2

20 0 20 02 2

0

1, , 0

2 eff

Z D ni Z P A T Z A T Z

Z cAT T

(85)

It is quite natural to request that all constants in Eq.(85) should become unity as

0 0 20 02

0

1 and 1eff

Z D nZ P

cAT

(86)

According to Eq.(84) we have 3 normalization quantities or “units” to choose, but have only

to requirements to fulfill. This hints at a hidden symmetry of the NLS, which we will employ

later. At the moment we just fix one quantity, say T0. We set it to the initial pulse width.

Consequently we get the relevant length

20

0

TZ

D (87)

and power scale

0 20 2 0 0 2 0

eff effcA cA DP

n Z n T

. (88)

According to Eq.(75) a low power pulse will broaden by a factor of 5 after a propagation

length of Z0. Nonlinear effects will start to play a role, if the peak power exceeds P0. Doubling

the initial pulse width results in a four times increase of the propagation length to see similar

effects at a quarter of the previously required power levels.

The scale equation only contains the signs of n2 and D as

2

222

1sign sign , , 0

2i D n A T Z A T Z

Z T

(89)

23

In fact only the product of the two signs is relevant as we now demonstrate by a little

transformation. We take the complex conjugate of Eq.(89) and multiply with -1. Finally we

obtain an equation for ,A T Z

2 2

22

1sign sign , , 0

2i D n A T Z A T Z

Z T

, (90)

in which the signs have been flipped. Hence, only the product 2sign D n has some physical

significance. In what follows we work with the equation

2

2

2

1, , 0

2i A T Z A T Z

Z T

, (91)

where 1 corresponds either to anomalous dispersion (D > 0) and a positive n2 or to normal

dispersion (D < 0) and a negative n2.

Finally we discuss the influence of the ambiguity of the scaling parameter. Let’s assume that

we already know a solution of Eq.(91) namely ,A T Z . This corresponds to the unscaled

solution

0 0 0, ,a z P A T T Z z Z (92)

We can normalize this solution again taking another set of normalization units, say

1 1 1, , andT Z P . Consequently we obtain a new normalized solution

20 1 11 1

1 0 01

1, , , ,

P T ZA T Z a t TT z Z Z A T T Z Z A T T Z Z

P T ZP

,

(93)

where 1 0T T can be chosen as an arbitrary number. Hence, we have found a way to

generate a whole set of solutions from just a single one. In the next chapters we will just try to

find those generic solutions.

2.3.5. Plane Wave Solutions

We first derive the simplest possible solution of Eq.(91), such with a constant amplitude 0A .

We make the ansatz

0 expA A i Z T , (94)

where is a shift of the normalized propagation constant and that of the frequency.

Inserting the ansatz (94) into Eq.(91) yields an equation

2202 0A ,

24

which defines a nonlinear dispersion relation

2202 A . (95)

it is of great practical importance to check whether a nonlinear solution is stable against tiny

fluctuations. Those infinitesimally small perturbations can growth exponentially and can

finally destroy the basic solution. This scenario is essentially determined by the initial steps,

where the perturbation is still extremely small but might start growing. We therefore extend

the ansatz (94) by adding a small perturbation like

0, , expA T Z A A T Z i Z T (96)

We insert the ansatz (96) into the evolution equation (91), but keep only those terms, which

are linear in ,A T Z as

2

2

2

1, , 0

2i A T Z A T Z

Z T