nonlinear optics: ws 2005/06

Institute of Physis and Physical Technologies, Clausthal University of Technology

Nonlinear Optics: WS 2005/06

Vorlesung NLO-e Kap.3 Vers. 1.1.doc 1 of 18 30.10.06

3 Wave propagation in nonlinear media

Starting point for the nonlinear wave equation are Maxwell's equations and the nonlinear polarisation P = Plin + PNL. To derive the nonlinear wave equation we investigate the simultaneous propagation of three electric fields E1, E2, E3 which finally results in a system of coupled differential equations describing effects like second harmonic generation and parametric processes. The Maxwell equations for non-magnetic media are

Ht

EDt

jHvvvvvvv

∂∂

−=×∇∂∂

+=×∇ 0, μ

with current density Ej

vvσ= and conductivity σ. For the dielectric displacement we

have NLPEEPED

vvvvvv++=+= χεεε 000

with χε +=1 and component PNL, i of the nonlinear polarisation ∑=

kjkjijkiNL EEdP

,, .

Insertion into Maxwell's equation results in

NLPt

Et

EH∂∂

+∂∂

+=×∇vvvv

0εεσ

Taking the curl operator rot on both sides of the second Maxwell equation leads to the nonlinear wave equation

LNPt

Et

Et

Evvvv

2

2

02

2

0002

∂∂

+∂∂

+∂∂

=∇ μεεμσμ

We assume an isotropic medium with LNPE

vv|| and go over to a scalar notation.

Let us consider the propagation of three plane waves E1, E2 and E3 with frequencies ω1, ω2 and ω3 along the z-direction of the medium:

( )..)(21),( )()(

10)(

11111 ccezEtzE zkti += −ωωω

( )..)(21),( )()(

20)(


( )..)(21),( )()(

30)(





The total instantaneous light field is )(

3)(

2)(

1321 ωωω EEEE ++=

and has to fulfil the nonlinear wave equation. Now we have a look at the different frequencies that may contribute to the nonlinear polarization PNL and which are of the form dijk Ej Ek , for example [ ]( )zkkti

NL eEEdP )()(2010

)()( 21212121 Re +−+++ = ωωωωωω or [ ]( )zkkti

NL eEEdP )()(*2030

)()( 23232323 Re −−−−− = ωωωωωω In general all different sums and differences of the frequencies ω1, 2, 3 may contribute to PNL. However, only those terms that are synchronous with either ω1, ω2 or ω3 can drive an oscillation at these frequencies, while all others being nonsynchronous. An example of the last statement is the case 213 ωωω += where the term )( 21 ωω +

NLP oscillates at frequency ω3 and thus forms a source of an oscillating wave with frequency ω3. In physical terms, we have a power flow from the fields at ω1 and ω2 into that of ω3, or vice versa. In what follows we assume the case 213 ωωω += to be fulfilled. The nonlinear wave equation has to be valid for all times t, i.e. we can treat the terms oscillating at different frequencies independently. For this term oscillating at frequency 1ω we get:

)(12010

)(110

)(1

2 111 ωωω εεμσμ Et

Et

E∂∂

+∂∂

=∇

[ ]⎥⎦⎤

⎢⎣⎡ +

∂∂

+ −−− ..21 )()()(

20)(

302

2

02323

*23 cceEE

td zkkti ωωωωμ

Because we have assumed a propagation along the z-direction only, the Laplace operator 2∇ reduces to the second derivative along z

[ ]..21 )()(

12

2)(

12 1111 cceE

zE zkti

o +∂∂

=∇ −ωωω

..221 )()(

102

1)(

101)(

102

211111 cceEkE

zkiE

zzkti +⎥

⎦

⎤⎢⎣

⎡−

∂∂

−∂∂

= −ωωωω




Next we assume slowly varying amplitudes of the propagating waves )(

0i

iE ω , i.e. we use the approximation

)(101

)(102

211 ωω E

zkE

z ∂∂

<<∂∂

This leads to

..221 )()(

101)(

102

1)(

12 11111 cceE

zkiEkE zkti +⎥

⎦

⎤⎢⎣

⎡∂∂

+−=∇ −ωωωω

All components oscillate with tie 1ω , and with 1/ ωit →∂∂ it follows

..221 111 )(

101)(

102

1 cceEz

kiEk zki +⎥⎦

⎤⎢⎣

⎡∂∂

+− −ωω

⎥⎦⎤

⎢⎣⎡ +−= − ..

21)( 11)(

1001021101 cceEi zkiωεεμωσμω

⎥⎦⎤

⎢⎣⎡ +− −− ..

21 )()(

20)(

30021

2323 cceEEd zkki*ωωμω

For the wave number we find

010112 εεμωλπ

== nk

or 010

21

21 εεμω=k . Multiplication of the not complex conjugated part of the equation

before last with zkieki 11/ results in

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+−z

EkikiE

ki

)(10

11

)(10

1010

21

1

1 221

ωωεεμω

)(10

1010

21

1101

1

21 ωεεμωσμω E

ki

kii ⎟

⎟⎠

⎞⎜⎜⎝

⎛−=

zkkkieEEdki )()(

20)(

301

021

231*

23

21 −+−−− ωωμω




As a result we now have a system of three coupled differential equations that describe the evolution of the complex field amplitudes E0 i along the propagation direction z:

zkkkieEEdiEz

E )()(20

)(30

01

01)(10

01

01)(

10 123*

231

2

22−−−−−=

∂∂ ωωω

ω

εεμω

εεμσ

zkkkieEEdi

Ez

E )()(30

)(10

02

02)(20

02

02)(

20 2133*

12

2

22−−−−−=

∂∂ ωωω

ω

εεμω

εεμσ

zkkkieEEdiEz

E )()(20

)(10

03

03)(30

03

03)(

30 321213

3

22−+−−−=

∂∂ ωωω

ω

εεμω

εεμσ

The first term on the right hand side of each equation describes damping of the wave, while the second one accounts for the nonlinear coupling of the interacting waves („wave mixing“). This term also includes a phase term, which leads to an oscillation of the coupling strength along the propagation direction. 3.1 Frequency doubling The first experiment in nonlinear optics in 1961 by Franken et al. (Phys. Rev. Lett. 7, 118 (1961)) consisted of generating the second of a Ruby laser 694 nm (red) → 347 nm (UV) into the ultra violet spectral region.

The conversion efficiency achieved in this first experiment, i.e. the ratio of power of UV light compared to the red pump light power, was about 810/ −≈rotUV PP . Today conversion efficiencies up to 30 % in single-pass operation and up to 80 % inside an optical resonators can be obtained. Second harmonic generation is a special case of the differential equation system derived before for ωωωωω 2, 321 === and )(

2010ωEEE == , )2(

30ωEE = .

Furthermore we will neglect absorption in the following discussion, i.e. we assume 0321 === σσσ .




This results in

[ ] zkieEdizd

Ed Δ−=2)(

0

0)2(

ωω

εεμω

with )()2(13 22 ωω kkkkk −=−≡Δ

We now focus on frequency doubling in a crystal sample of length L: 0 ⎯ z → L Let us first assume that conversion efficiency is low, so the relation

.)0()( )()( constEzE =≈ ωω is fulfilled, which is the case of non-depleted pump approximation. Only wave )(ωE is used as input, so the boundary condition is

0)0()2( =ωE . Integration of the two above differential equations along the propagation direction z from 0 to L gives us the solution for the amplitude )()2( LE ω at the rear face of the crystal

[ ] dzeEdiLEL

ki∫ Δ−=0

z2)(

02

0)2( )( ω

ω

ω

εεμω

( )[ ]ki

eEdiLki

Δ−

−=Δ 12

02

0 ω

ω εεμω

The intensity of the frequency doubled wave is proportional to

( ) ( ) ( ) ( ) ( )( )

242

20

02222

)(11||)()(~

*

keeEdLELEI

LkiLki

Δ−−

−=Δ−Δ

ω

ω

ωωω

εεμω

( ) ( ) ( )2

242220

02

)2/(411||1Lk

eeLEdn

LkiLki

Δ−−

=Δ−Δ

ω

ωεμω

( ) ( ))2/(2/sin2

24

22

22

0

0

LkLkLE

nd

ΔΔ

= ω

ω

ωεμ




Intensity is defined as power per area

FPI

FPI ω

ωω

ω =≡ and22

with the effective beam area F. This results in

( ) 20

2ωω

ωωε EncFIFP ==

( )

FP

ncE ω

ω

ω

ε0

2 2=

The efficiency of second harmonic generation is given by the ratio

( )

( ) 2

222

ω

ω

ω

ωηE

EPP

=≡ ,

( )

( )( )

2

22

22

222

0

02

22

)2/()2/(sin

LkLkE

nLd

E

EΔ

Δ== ω

ωω

ωω

εμη

( )( ) F

PLk

Lkn

Ldnc

ω

ωω

ωεμ

ε 2

2

22

222

0

0

0 2/2/sin2

ΔΔ

=

( )( ) F

PLk

LknnLd ω

ωω

ωεμ

2

2

22

22223

0

0

2/2/sin2

ΔΔ

⎟⎟⎠

⎞⎜⎜⎝

⎛= .

An important result is the proportionality of η to the squares of both sample length L2 (i.e. large interaction lengths) and d2 (i.e. large nonlinear coefficients) and the intensity Iω of the pump wave.




Now we want to discuss second harmonic generation for the case of pump wave depletion, i.e. we now have ( ) ( ) )0()( ωω ELE < . For simplification we use the notation

3,2,1== lwEnA ll

ll ith

ω

For the intensity of each wave l = 1, 2, 3 we get

20

2 lll

l EncFPI ε

==

20

2 ll Ac εω

=

The three coupled differential equations now read

z* kieAAiAzdAd Δ−−−= 321

112

κα

z* kieAAiAzd

Ad Δ−−−= 31222

2κ

α

zkieAAiAzd

Ad Δ−−= 21333

2κ

α

with 321

321

0

02 nnnd ωωω

εμ

κ =

lll εεμσα 00 /= , l = 1, 2, 3 321 kkkk +−−=Δ For the case of second harmonic generation with 21 AA = for the two pump wave amplitudes and with ωωω == 21 , we neglect again the influence of damping ,

0=lα . As a result we find for the amplitude 3A of the frequency-doubled wave at frequency ωω 23 =

zkieAAizdAd Δ−−= *

131 κ

zkieAizdAd Δ−= 2

13 κ

We further assume phase matching to be fulfilled, i.e. we take 0=Δk .




Without loss of generality we chose a phase reference of the differential equation system by assuming A1 (0) to be a real quantity. As a direct consequence now also A1 (z) is a real number and we get the following system

131 ˆ AA

zdAd

κ−= ,

21

3ˆ

Azd

Adκ= ,

with 3333

ˆˆ AiAAiA −=+= or . This can be interpreted in a way that the wave with amplitude A3 is shifted by π/2 relative to that with amplitude A1, i.e. the amplitudes A1 and 3A are in phase. For the nonlinear interaction with boundary condition A3(0) = 0 we have conservation of energy in the form .)(ˆ)( 2

32

1 constzAzA =+ )0(2

1A=

Thus we may write

[ ])(ˆ)0(ˆ

23

21

21

3 zAAAzdAd

−== κκ

or dzzAA

Ad κ=− )(ˆ)0(

ˆ23

21

3

Integration and using the relation

⎟⎠⎞

⎜⎝⎛=

−∫ ax

axadx atanh1

22

gives the amplitude of the frequency-doubled wave ( )zAAzA )0(tanh)0()(ˆ

113 κ= . The efficiency of second harmonic generation is now

( )

( ) ( )zAA

zAP

P )0(tanh)0()(

12

21

23

2

κη ω

ω

=== .




3.2 Phase matching Prerequisite for efficient second harmonic generation is phase matching of the interacting waves 0=Δk , or in other words equal phase velocity of pump and frequency-doubled beam. Using the deBroglie relation p = ħ k this is identical with conservation of momentum.

i, j, k : polarization direction

with )(

1)(

1)(

3jik kkkk −−=Δ , ωωωωω === 213 ,2 and

3,2,1,2

=== lnc

nk l

l

l

ll

ωλπ

Phase matching 0=Δk is fulfilled if

( ) 021 )()()2( =−− ωωω ωωω jik nnnc

or )()()2(2 ωωωjik nnn +=

In the most simple case )()( ωω

ji nn = of a linearly polarized pump wave with i = j, this requires )()2( ωω nn = . For the case that phase matching is not exactly fulfilled, 0≠Δk , the frequency-doubled wave propagates either faster or slower then the pump wave. This leads to a phase mismatch between the two waves that grows during propagation along z. As a result the power conversion oscillates as well among pump wave and second harmonic and follows a sinc function 22 )2/(/)2/(sin xkxk ΔΔ .




The spatial coherence length lc is given by

)(42 )()2()()2( ωωωω

λππnnkkk

l c −=

−=

Δ=

At z = lc both waves are π out of phase, i.e. power starts to be transferred back to the pump wave. For high conversion efficiency exact phase matching is therefore inevitable. Numerical example: wavelength λ = 1 μm, dispersion 2)()2( 10~ −−− ωω nn :

25104

m102

6

=×

= −

−

cl μm

The crystal length is assumed to be L = 1 cm. This results in an increase of efficiency as a result of exact phase matching of )()2( ωω nn = (compared to the case without phase matching) of more then five orders of magnitude

5

2

opt 106.1~ ×=⎟⎟⎠

⎞⎜⎜⎝

⎛

clL

ηη

.

To allow for exact phase matching the birefringence of anisotropic crystals is of advantage. Nonlinear crystals that are used for frequency doubling lack a center of inversion symmetry, and therefore always have at least two different refractive indices. For the case of an optically uniaxial sample these are ordinary no and extraordinary refractive index ne, respectively. In this case and with a purely ordinary or extraordinarily polarized pump wave this process is called type I phase matching; if the pump wave has a mixed polarization the phase matching is of type II. Because with the usual situation of normal dispersion the refractive index increases with light frequency ω, i.e. for identical polarisation we have ( ) ( )ωω nn >2 . Using birefringence the two possible solutions are

( ) ( )ωω

oe nn =2 or ( ) ( )ωωeo nn =2 .

In the following table the refractive indices of the nonlinear crystal KDP (KH2PO4) are given. Exact phase matching is thus possible for a pump wavelength of about 2 μm. However, in a real system the exact wavelength of pump light is usually fixed, e.g. by typical pump lasers like Nd:YAG or Nd:YVO4 at 1.064 μm. Therefore an additional degree of freedom to allow for phase matching is necessary. This becomes possible for example by changing the propagation direction of wave inside the crystal.




As an example we assume an unaxial crystal with negative birefringence )( oe nn < and take into account that the extraordinary refractive index depends on the propagation direction of the wave, i.e. it is a function of the angle θ between Poynting vector and the optical axis of the crystal:

2

2

2

2

2)(sin)(cos

)(1

eoe nnnθθ

θ+= ,

with ( ) ( ) ( )ωωω θ 222 )( oee nnn << . As a result there may exist an angle (this depends actually on the magnitude of birefringence) mθ , so that the following relation is fulfilled: ( ) ( )ωω θ ome nn =)(2 . In this case we have an ordinarily polarized pump wave (i = j) and an extraordinarily polarized second harmonic )( ik ≠ propagating in the same direction. The phase matching angle mθ can be obtained by equating the above equation for the case of negative birefringence )( oe nn < and with the relation ( ) ( ) )(2

meo nn θωω = :

2)2(

2

2)2(

2

2)( )()(sin

)()(cos

)(1

ωωω

θθ

e

m

o

m

o nnn+= ,

2)2(2)2(

2)2(2)(2

)()()()(

)(sin −−

−−

−−

= ωω

ωω

θoe

oom nn

nn .

−>

−> ω −>

2ω −>




As an example we take frequency doubling of a Ruby laser in a KDP crystal with pump wavelength nm693)( =ωλ . With the numbers

( ) ( )

( ) ( ) 534.1506.1

487.1466.12

2

==

==ωω

ωω

oo

ee

nn

nn

we find the phase matching angle to be mθ = 50.4 °. an alternative way to obtain exact phase matching is to use a mixed polarization of the pump wave (type II phase matching). Here the pump wave is polarized at an angle of 45° to the optical axis )( ji ≠ , i.e. we now have

( ) ( ) ( )[ ])(21)(2

meome nnn θθ ωωω += .

Again we make use of the dependence of refractive index on propagation direction

2)2(

2

2)2(

2

2)2( )()(sin

)()(cos

)(1

ωωωθθ

e

m

o

m

e nnn+=

and 2)(

2

2)(

2

2)( )()(sin

)()(cos

)(1

ωωω

θθ

e

m

o

m

e nnn+=

From the phase matching condition it follows that

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡++=⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−−21

2)(

2

2)(

2)(

21

2)2(

2

2)2(

2

)()(sin

)()(cos

21

)()(sin

)()(cos

ωωω

ωωθθθθ

e

m

o

mo

e

m

o

m

nnn

nn .




Summarizing the conditions for phase matching Type I: both pump waves ( ) ( )ωω

ji EE , have the same polarization, i = j

Type II: pump waves ( ) ( )ωωji EE , have different polarization, ji ≠

positive birefringent

oe nn > negative birefringent

oe nn < type I

( ) ( ) )(2

meo nn θωω =

( ) ( )ωω θ ome nn =)(2

type II

( ) ( ) ( )( ))(

21)(2

meomo nnn θθ ωωω +=

( ) ( ) ( )( )ωωω θθ omeme nnn += )(

21)(2

Here it is important to mention that not all crystals can be phase-matched by changing the propagation direction for a given pump wavelength. For example, the mixed crystal SBN75 (Sr0.75Ba0.25Nb2O6, strontium-barium niobate), which has rather large SHG coefficients d, but at the same time has a small birefringence

012.0≤− eo nn but a larger dispersion ( ) ( ) 2.02 ≈− ωωee nn , is not phase-matchable in the

wavelength range 0.5 μm < λ < 3 μm. Already for smaller deviations of the exact phase matching condition the efficiency of second harmonic generation is considerable reduced:

2

2

)(

)2(

)2/()2/(sin~

LkLk

PP

ΔΔ

= ω

ω

η .

Let δ be the deviation of the exact phase matching angle mθθδ −= . For small values of δ we have the approximation ( ) ( ) δβθθ ωω 22)()( 2 ≈−=Δ oe kkk ,

with ( ) ( ) ( ) .),,( 22 constnnn eoo == ωωωββ . With the definition Lδβψ =: we can rewrite the efficiency in the form

( )

( ) 2

22 sin~ψ

ψη ω

ω

PP

= .

An experimental proof of this relation is given in the following diagram for the case of a KDP crystal.




Another option to achieve phase matching is to use the temperature dependence of refractive indices, where one can take advantage of the fact that in most case the refractive indices no and ne have different thermooptic coefficients. For optical uniaxial samples one finds

Tn

Tn oe

∂∂

>∂∂ .

As an example we take the uniaxial crystal LiNbO3 with oe nn < . Lithium niobate is a ferroelectric material with a Curie temperature Tc ≈ 1210° C. At temperatures above the Curie temperature, T > Tc, the crystal is in the paraelectric phase and shows inversion symmetry. On the other hand, at temperatures below Tc the crystal is ferroelectric and thus possesses a spontaneous polarization Ps (where the direction of polarization is related to the sign of the tensor elements dijk) as well as different refractive indices oe nn ≠ . Starting from room temperature an increase in T usually results in a decrease of birefringence and thus leads to ne → no for T → Tc. Here one has to take into account that the thermooptic coefficient dn/dT is a function of both temperature and wavelength and may show a more complicated behaviour. Typical numbers for LiNbO3 are

μm5.0=λ : , 5103 −×≈Tdnd o

μm1=λ : 5104 −×≈Tdnd e ,

As an approximation we may use

Tdnd

Tdnnd eoe ≈

− )( .

For the case of type I phase matching for the material KDP (see table in this section) frequency doubling of a Nd:YAG laser at a wavelength λ = 1.064 μm is possible by tuning the crystal temperature to the phase matching temperature Tpm. An example is given in the next diagram.

410−≈Tdnd e

0≈Tdnd o




3.3 Quasi phase matching In many cases materials that may be used for frequency doubling, for example the ferroelectric crystals LiNbO3 and LiTaO3, shown rather large diagonal elements of the d Tensors (e.g. d333), which cannot be used by type I or type II phase matching. Therefore, as has been shown in the proceeding section, the pump and frequency-doubled wave have orthogonal polarizations, thus using for example the coefficient d311 = d31 (or d312 = d36). In the case of LiNbO3 or LiTaO3 the ratio of the involved coefficients is 3/ 311333 ≈dd . Consequently, using the larger coefficient d333 would result in an increase of SHG efficiency of about one order of magnitude. An alternative technique for achieving phase matching is referred to as quasi phase-matching (QPM), where the nonlinear coefficient is periodically modulated along the propagation direction )(zdd = . The periodic modulation of the nonlinear coefficient d can be expanded in a Fourier series with dbulk being the nonlinearity of the homogeneously polarized crystal, the period length Λ and the Fourier coefficients am of order m. An example that is close to application is a purely binary modulation )(zdd = with amplitudes bulkd± .

∑ ⎟⎠⎞

⎜⎝⎛

Λ−=

mmbulk zmiaidzd π2exp)(




Insertion of the Fourier expansion of )(zdd = into the differential equation system (for the case )(

20)(

1021 ωω EE = ) results in

( )

∑ ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++−

Λ−−−=

mmbulk zkkkmiaiEEdiE

zd

Ed113

)(*10

)(30

10

01)(10

10

0110 2exp22

1311 π

εεμω

εεμσ ωωω

ω

For QPM the phase matching condition is m = 1, 2, 3, ....

This condition is assumed to be fulfilled for a certain m = n; for all other nm ≠ their contributions averages out to zero over distances that are large compared to the coherence length, i.e. for Λ>>L . The magnitude of the coefficients am can be obtained by applying the orthogonality of the different terms in the sum over m:

∑ ⎟⎠⎞

⎜⎝⎛

Λ−=

mm

bulk

zmiaid

zd π2exp)(

⎟⎠⎞

⎜⎝⎛

Λ−

Λ ∫Λ

zmiid

zd

bulk

π2exp)(1

0

∫ ∑Λ

⎟⎠⎞

⎜⎝⎛

Λ−⎟

⎠⎞

⎜⎝⎛

ΛΛ=

0

2exp2exp1 dzznizmiamm

ππnmma δ=

As an example we again treat the case of )(zdd = where the sign changes every halve grating period:

dzzmid

zdiabulk

m ⎟⎠⎞

⎜⎝⎛

Λ−

Λ= ∫

Λ π2exp)(

0

(with 0≠m )

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

Λ−−+⎟

⎠⎞

⎜⎝⎛

Λ−

Λ= ∫ ∫

Λ Λ

Λ

2/

0 2/

2exp2exp dzzmidzzmii ππ

[ ])(exp)2(exp1)(exp2

ππππ

mimimimi

i−+−−−−

Λ−Λ

=

[ ])2(exp)(exp21211 ππ

πmimi

m−+−−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+−+−−−−=

===43421434214434421

010

)2sin()2cos()sin(2)cos(212

1 πππππ

mmmimm

[ ]π

πππ m

mmm

)(cos1)(cos22211 −

=−= .

13 22 kkkm −=Δ=Λπ

dzzmid

zdiabulk

m ⎟⎠⎞

⎜⎝⎛

Λ−

Λ= ∫

Λ π2exp)(1

0




For the case m = 1 we obtain

bulkbulkmeff ddadπ2

== . Compared to the situation of true phase matching the efficiency of quasi phase-matching is reduced by a factor ( ) 4.0/2/ 2 ≈= πηηQPM . A numerical example of quasi phase-matching for SHG using LiNbO3 at a wavelength of 1 μm is given below:

( )16.225.2μm14)(42

)()2(

13 −=−

=−=Δπ

λπ ωω

ee nnkkk 16 m102.1 −×≈

1)()2( )(2

2 −−=Δ

=Λ ωωλπee nn

k μm5≈

The case Fall m = 1 requires a period length of only 5 μm or a coherence length of lc = 2.5 μm, respectively. Such small period lengths (the structure size is only half the period length) may be not easy to achieve in real applications. Therefore also quasi phase matching with m > 1 may be used. For the case m = 3 one requires less-demanding periods of 5 μm, however, at the same time the relative efficiency is further reduced to ( ) 05.03/2/ 2

3, ≈== πηη mQPM .

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For the fabrication of periodic structures for quasi phase matching the spontaneous polarization Ps of a ferroelectric crystal has to be reversed periodically. This can be achieved by applying an external voltage to grating-like ("finger") electrodes, which are formed by lithographic techniques on the crystal's surface. The dependence

)(EPs shows the typical hystersis behaviour of ferroelectric media.

The necessary coercive field Ec, at which the spontaneous polarization is reversed, is usually higher then the electrical break-down threshold of air (i.e., for LiNbO3 Ec = 22 kV/mm), thus a treatment in isolating atmosphere (e.g., silicone oil) has to be used.

The poling procedure cannot be understood as a simple switching of domains. Instead, this is a more complicated, time-dependent process, with the growing speed of the domains (which by itself strongly depend on the external poling field) as the most important parameter. A scheme of the poling process can be seen in the following figure. In a first step, needle-like start growing at the edges of the finger electrodes, because there the local electric field has a maximum value (a-c). Then the domains start to grow in lateral direction (d,e), which accidentally may result also in fusion of neighboured domains. Finally, if the poling procedure is stopped, a periodic modulation of the spontaneous polarization is achieved (f). The Figure on the right shows a photograph of the domain structure which has been made visible by etching of the crystal's surface.

nonlinear optics: ws 2005/06

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