Intracavity cw difference frequency generation by mixing three photons and using Gaussian laser beams

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Intracavity cw difference frequency generation by mixing three photons and using Gaussian laser beams

Tran-Ba-Chu, M. Broyer

To cite this version:

Tran-Ba-Chu, M. Broyer. Intracavity cw difference frequency generation by mixing three photons and using Gaussian laser beams. Journal de Physique, 1985, 46 (4), pp.523-533.

�10.1051/jphys:01985004604052300�. �jpa-00209992�

Intracavity

^cw

difference frequency generation by mixing ^three photons

and using ^Gaussian laser beams

Tran-Ba-Chu

(*)

^{and M.} Broyer

Laboratoire de Spectrométrie Ionique ^et Moléculaire (**),

Universite Lyon ^{I, Bât.} 205, Campus ^{de La} Doua, ⁶⁹⁶²² Villeurbanne Cedex, ^France (Reçu le 1 er octobre 1984, accepté le 7 decembre 1984)

Résumé. 2014 La théorie de la génération intracavité d’onde par différence de fréquence ^en régime ^{continu en effec-}

tuant le mélange ^{de trois} photons ^{est étudiée en} supposant que l’onde de pompe ^{est un} faisceau focalisé circulaire

ou elliptique ^et que le cristal ^non linéaire est placé dans la cavité du laser (onde signal). ^{On montre} que la puis-

sance de l’onde générée ^{par le} système intracavité est considérablement plus élevée que celle d’un système ^extra-

cavité opérant dans les mêmes conditions. L’expression ^du rapport ^{de ces deux} puissances appelée ^{« facteur} d’aug- mentation » ~ ^{contient des} paramètres représentant l’effet du champ laser dans la cavité (onde signal) ^{et celui de} l’amplification paramétrique. Ce facteur devient très grand lorsque ^la puissance de l’onde de pompe atteint la valeur de résonance ou lorsque le laser de l’onde signal opère ^{à un} régime proche du seuil. Des résultats numériques

ont été obtenus en se basant sur des systèmes ^de génération intracavité d’onde par différence de fréquence ^utilisant

un laser continu à colorant en anneau ou un laser YAG continu avec soit un cristal LiNbO3 (synchronisation ^de phase ^non critique) ^{soit un cristal} LiIO3 (synchronisation ^de phase critique). L’analyse ^a démontré que la généra-

tion intracavité d’onde par différence de fréquence ^en régime ^{continu est une méthode} favorable pour créer ^une

source convenable d’infrarouge pouvant être utilisée en spectroscopie ^à haute résolution.

Abstract. ²⁰¹⁴ The theory ^of Intracavity ^cw Difference Frequency Generation (I.D.F.G.) by ^a three-photon mixing using Gaussian laser beams has been studied assuming that the pump ^wave is either a Gaussian circular or elliptical focusing ^laser beam, ^with the non-linear crystal placed ^{in the} signal ^laser resonator. It is shown that the difference

frequency generated power of the intracavity system is many times larger ^than that obtained by extracavity ^diffe-

rence frequency generation operating ^{under the same} conditions. The general expression for the enhancement factor indicates that this increased power is due ^to the high ^field intensity ^{of the} signal ^wave within the cavity ^{and to}

the parametric amplification effect. This factor becomes _very large ^when either the pump power reaches ^{a resonant} value or when the signal ^laser operates ^{close to} the threshold. Numerical results have been obtained for I.D.F.G.

based on either cw-ring dye ^or cw-YAG:Nd3+ laser using ^{a 90°} phase matching LiNbO3-crystal ^{and a critical} phase matching LiIO3-crystal respectively. It is shown that cw-I.D.F.G. is a favourable method for producing ^an

efficient source of I.R. radiation suitable for uses in high resolution spectroscopy.

Classification Physics ^Abstracts 42.65201342.65C

1. Introduction.

High

resolution spectroscopy of atomic and molecular lines in the infrared

region

^{of the} spectrum

requires

tunable radiation sources.

Using

different semi- conductor materials, ^{it is in}

principle possible

^{to tune}

diode lasers over a wide

spectral

range (0.6-32

um),

but a

single

^diode

gives

^a very small tunable band -

approximately

^{100 A}

[1].

^{Colour centre lasers}

[2]

cover only ^{a small} range in the near infrared spectrum (*) Permanent address : Institute of Applied Physics, Hanoi, ^Vietnam.

(* *) ^{Associe au CNRS no 171.}

and, ⁱⁿ addition, ^{there are} several technical

problems

in its

development

which have not yet been resolved.

Many

of the tunable I.R. sources can be obtained

by

Difference Frequency Generation

(D.F.G.).

^Pulsed

dye

^{lasers can be used to} generate coherent radiation which is tunable over a wide _range of I.R.

frequencies (see,

^for

example, [3],

^with

comprehensive references).

Because the D.F.G. _process is non-linear, ^these

high

power pulsed systems ^{were chosen to achieve}

high

visible to infrared conversion

efficiency.

^{However the} spectral control of the

pulsed dye

^{lasers was}

generally

minimal, ^{so the}

resulting

^I.R. linewidths were of the order of several cm-1.

The cw-source of radiation demonstrates better

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004604052300

frequency stability

and smaller linewidth [4]. However,

in the conventional case (i.e. with the non-linear

crystal

outside the laser

cavity)

^both theoretical and

experimental

results have

given

^a very low conversion

efficiency [4-5].

In the case of

cw-optical parametric

oscillator

[6-7]

and cw-second harmonic

generation [8-9],

it has been shown that a

high

conversion

efficiency

^{can be}

obtained

by inserting

the non-linear element into the

cavity

of the continuous wave laser. The

experimental

work of Lahmann et al. on the D.F.G. in cw-YAG and

cw-dye

laser has

given

^{the same result}

[10].

In this work, Lahmann et al. have used the non-colli-

near difference

frequency mixing

^{waves to eliminate}

the influence of the double refraction on the difference

frequency generation

conversion

efficiency.

In this paper ^we describe our theoretical

study

^of

Intracavity

Difference Generation

(I.D.F.G.) pumped by

^{either a} Gaussian circular or

elliptical focusing eigenmode

^{laser beam and} concentrate on the operat-

ing

condition for which the D.F.G. _power is large. ^The

non-linear

crystal

^is

placed

^{in the}

cavity

^{of a laser}

whose medium causes a

homogeneously

^broadened

gain

transition. The treatment was

performed

^{with a}

Gaussian

eigenmode

^{of this}

coupled

cw-laser. The

properties

^{of the}

dye

^laser used in this _paper are

described in

[ 11-12],

^{i.e. the}

dye

is considered to be

homogeneously

broadened. Since the narrow band- width is achieved with the

help

^{of a}

frequency

^selective

tuning

element, it is also assumed that the

dye

^laser

radiation is

essentially

monochromatic

whereupon

the

approximate

conditions for I.D.F.G. are satisfied.

Otherwise, ^{the hole}

burning,

^mode

pulling pushing

effects are

ignored

^{in this} paper since

they

^{are absent in}

the

cw-ring dye

^laser

operated

^{with a} undirectional device. To

simplify

^the

problem,

the saturation effect in the active

dye

medium will also be

ignored.

^In

this _case, a low concentration of the

dye

^{would be}

required

^{to avoid}

quenching

^the

gain

^{of the}

exciting

laser.

The method of derivation in this work is similar to that used in our

previous

paper

[7]

and in the _paper of Oshmann and Harris

[4]

^{for the}

optical parametric

oscillator. Our numerical results were

applied

^to

I.D.F.G. systems ^{based on a 900}

phase matching LiNbo3-crystal

^{and a critical}

phase matching LiI03- crystal. Figure

^{1 shows the structure of the}

intracavity

Fig. ^{1. -} Intracavity difference frequency generation ^sche-

matic configuration. M1 ^and M2 reflect 100 % ^{for the} signal laser and are transparent for the pump and idler lasers.

difference

frequency generation

^{in the case of the}

waves

mixing collinearly.

The non-linear

crystal

^of

length

^{I is}

placed

in the laser resonator of

length

^L.

The _pump beam passes

through

the mirror

M 2

^and

focuses in the

crystal.

^{The two mirrors} M 1,

M 2

^are

transparent ^for the difference

frequency generated

wave, however

they

reflect the

signal (laser)

^wave

100 %.

2. Difference frequency mixing ⁱⁿ

cw-ring

dye ^laser cavity using ^circular focusing ^{of the laser beams.}

2 .1 BASIC EQUATIONS. ^-

Adopting

^{the same notation}

as in the

intracavity Single

Resonance Oscillator

(S.R.O.)

[7],

^we shall refer to the two incident waves as

pump and

signal

^waves

(the

pump ^wave

corresponding

to the

higher frequency),

while the wave

generated

at the difference

frequency

will be named idler wave.

The

frequencies

^{of these} waves are wp, ^{cvs and Wi}

satisfying

the relation wp ⁼ Wi ⁺ (os for _energy con-

servation. The

phase

^relation

corresponding

^{to maxi-}

mum energy conversion,

Op - Os

⁼

n/2 is applied

to the _process of difference

frequency generation [13].

The total electric field in the non-linear

crystal

may be

expressed

^as

where ei ^{is the}

polarization

^unit vector,

Uj(r)

^the

spatial

^{mode and}

Ej(t)

^the

^amplitude

of the electric field. In the I.D.F.G. device,

only

^the

signal

^{wave is}

at resonance, and if its TEMoo mode alone is at resonance, ^{we can} write

US(r)

^{as a sum of a}

right

^and

left

travelling

^Gaussian beam, ^i.e.

We shall consider the case in which the _pump laser beam is an

extraordinary

^wave with walk-off

angle

p, whilst the

signal

^{beam is an}

ordinary

^wave.

Boyd

^and

Kleinman

[14]

have shown that the reverse arrange- ment

produces

^{the same} result. If the centre of the non-linear

crystal

is located ^at the

origin

^{of a Cartesian}

coordinate system (X, Y, Z) which is centred in the laser

cavity

with the Z-axis

along

^the

longitudinal

axis, the one-way

travelling signal

^{wave takes the}

Gaussian form

[14]

Here

Wos

is the Gaussian beam waist of the

signal

and

bs( = kS W6s)

is the confocal parameter.

The Gaussian _pump beam is assumed to be focused in the centre of the

crystal

^{at Z = 0. The} one-way

travelling

pump ^wave has the form [14]

The

amplitude

of the difference

frequency generated (D.F.G.)

^{wave has not}

previously

^been

given,

^{it can be} determined,

using equations

(1), (3) ^and

(4)

^{from the}

following parabolic equation,

which describes the evolution of the

slowly varying

^wave

amplitude during

one passage

through

the non-linear

crystal [ 15], [7] :

The

coupling

^constant y; in

equation

(5) is of the form

The effective non-linear coefficient

deff

^is

given by

where _x is the non-linear

susceptibility.

The

phase-mismatching

^{Ak is}

given by

and the wave vectors of

interacting waves ki

^are

^given

by

The solution of

equation

(5) ^is

where

is the

degeneracy

parameter ^and

and

Equation

(10) ^{was obtained}

assuming bs

⁼

bp

⁼

kp w2 p ^{= kS W2}

^{Then 2}

^Z/bs

^{=T,. = tp = 2}

Z/bp

^{= I and}

T’ ⁼ 2

Z’Ibr

⁼

2 Z’/bp.

^The non-linear interaction is maximized when the confocal parameters of the pump and

signal

^{waves are identical}

(see [15]), ignoring

^the

depletion

^{of the} pump ^wave power and

using

^the

specified

field

approximation [ 16].

^Relation (10) defines the field of the D. F. G.

wave,

^its

spatial

distribution is

where I is the

crystal length.

Taking

^{into account}

polarizations

in the laser medium and in the non-linear

crystal, using

^{the treatment}

given

ⁱⁿ

[7],

together ^with

expressions

(1), (2), (3),

(4),

(10), (14) ^{and Lamb’s}

equations [ 17],

^{we have the}

following

rate

equations

^{in the case of small}

gain

^laser

where Lx,,

^{is the}

single-pass

power loss for the

signal

^{laser mode,} the time variable is defined as T 1 ⁼

ct/L (L-the dye

^laser

cavity length), us(Å.s)

is the stimulated cross-section,

Lo

^the

optical path length

^{in the}

dye jet, NS(i 1 )

the

population

inversion in the

single

^excited state,

P.

^the power of the

dye

^{laser and h(’) the}

focusing

^function

determined

by [7]

where ç ⁼

l/b

^{is the}

focusing

parameter

a is the

phase-mismatching

parameter ( - ^{a =}

Akb/2)

^and

B = § (lkp)1/2

^{is the double} refraction parameter.

2

F(B, f4 ç) ^is determined

by

wherein

and the powers of the

interacting

^{waves are}

given by

Using

^the

approximation proposed by

0. Teschke et al.

[11]

^the

population

^rate

equation

^{of the}

dye

^{laser can be}

written as

where _6o is the

ground

^state

absorption

cross-section; TR the fluorescence

decay

^{time; S the area of the pump}

beam in the

dye jet;

h Planck’s constant; _vo ⁼

l//Ao, Po,

the wavenumber and the _power

respectively

^{of the}

second laser used to pump the

dye jet;

^{and N} is the number of unexcited molecules.

Equation (19)

is written for the fundamental Gaussian mode

making

^the

assumption

that the active

region

^is

confined within ^a distance which is less than the confocal parameter ^{of the} waist, i.e. the Gaussian beam can be considered to be at constant diameter.

Equations

(15),

(16),

(19) ^and (20) illustrate the

time-dependence

^{of the} power and the

phase

^{of the}

dye

laser (resonant wave) ^{in the case of a small}

gain

^laser.

Equations (10)

^and (18) determine the _power of the D.F.G.

wave

(idler

wave) when all characteristics of the

dye

laser beam

(signal

wave) ^and pump ^{wave are} known.

2.2 RESULTS AND DISCUSSION.

2.2.1 The _powers of ^the parametric ^{waves. -}

Equation (15)

^shows that, ⁱⁿ the non-linear interaction process within the

crystal,

^the

dye

laser beam

(signal wave)

^is

amplified by

the pump beam.

In the

steady

^state

regime

^{of a cw}

small-gain dye

^laser,

equations (15),

(16), (19) ^are

equal

^{to zero.}

Using equation (20)

ⁱⁿ

conjunction

^{with these}

equations

^{we obtain the}

following expressions

^{for the}

intracavity

power and

phase

^{of the}

dye

laser beam :

Using (21’)

^{and the}

phase

^relation

corresponding

^to maximum energy conversion, ^we

have §; = cPp -

^Const.

This means that the

spectral quality

of the D.F.G. wave is the ^{same as} that of the

cw-dye

^laser.

For the

general

^case,

using

Pike’s result

[ 12]

^and

(21 ),

the formula which determines the

intracavity cw-dye

laser _power becomes

- In the absence of the non-linear effect

(deff

⁼ 0),

equation

(22)

yields

where

P Oth

^{is the}

dye

laser threshold power, and aS is the total

single-pass

^{loss at the}

dye

^laser

wavelength.

as includes non-useful components ^{such as}

scattering, absorption by singlet

^and

triplet

^states, extraneous reflections, ^losses

at the non-linear

crystal

^{and so} forth. The losses introduced

by placing

^the

crystal

in the laser

cavity

^are

relatively

small, since the

crystal

^{faces are cut at Brewster}

angles,

^and any modification caused to the

path

of the laser beam

by

^the presence of the

crystal

^is

compensated

^either

by realigning

^the

cavity

^or

by using

^{a rhombic compen-}

sator

[ 18].

On the basis of

comparison

^{with the power formulae}

[12],

^we may make a

phenomenological

^correction

for the Gaussian beam effect

generated

^{in the}

dye jet [7]

^{to our}

expressions (23), (24),

^{which become}

where the parameter v is defined as the ratio of the areas of the second _pump laser and

dye

laser beam at the

jet

The

expressions

(25) ^and (26) ^{are similar to} Pike’s formulae [12J ^{in the case} where the mirrors of the laser reso- nator have reflection coefficients ⁼ 100

%.

- In the _presence of the D.F.G. effect

(deff =1=

0).

Using

^{the above} correction, ^we have the

following

formula for the

intracavity

power of the

dye

^laser

Formula (27) ^{shows that the} power of the

dye

^laser

P.

^is

amplified

in the non-linear interaction _process when the pump power is smaller than ^{a resonant} power

ppre,

which is determined

by

Using

(16),

(10)

^and (18) ^and (25), ^one obtains the

expression

^for

intracavity

^D.F.G. output power

Piin’

where

hDF(B,

^{a, 14} ç) ^{is a}

focusing

^function

dependent

upon the double refraction parameter

B,

^the

spatial phase matching

parameter ^{a, the}

focusing parameter ç

^{and the}

degeneracy

parameter M. It is determined

by

and

FCB, 14

ç) ^is

given by ^(17’).

The function

hDF(B,

^{a, Jl,}

^ç)

^{allows us to calculate}

the maximum D.F.G. power under

optimized focusing

condition. In the case of weak

focusing

(j «

1)

^and

without

the

double refraction

(p = 0)

^{we have found}

that

hDF(B

⁼ 0, Jl,

Qopt, 03BE ¹⁾

^{~ ç. We see from}

expression

(29) that the D.F.G. output power

Pin’

^is

equal

^{to zero}

if the parametric

interaction is absent

(derr

^{= 0 or}

hDF(B,

^(1, u,

ç) = 0)

^or if the pump power

Pp

^is

^equal

^{to zero.} When the pump power

Pp reaches

a resonant value defined

by (28),

^{i.e. when the energy}

given by

the pump ^to

signal

^wave (laser wave) ^{in the}

parametric

interaction _process compensates ^{the loss}

the

^D.F.G. output power tends ^to

infinity.

^When

the pump power

Pp

^{is small (several tens of mWatts),}

since deff

^{is also small, the term}

Pp(16

^1[2

deff)2

^{2 12}

h(1)(B, a, Jl, ç)/cni np ns(W5p

⁺

W5s) çÂ.A Â.s ~

0, ^{i.e. the}

amplification

^{of the}

signal wave by

the pump ^wave is

ignored

^{and formula}

(29)

^becomes

Using (25)

^{we have}

where

Pos is

^the

intracavity dye

^laser power determined

by (25).

^{Since the}

signal

power inside the

dye

^laser

cavity

^is

higher

^{than that} outside, the I.D.F.G. tech-

nique permits

^an efficient conversion of the _pump power ^to D.F.G. _power.

In the

general

^case

(formula (29)),

^{a high D.F.G.}

power may be associated with high ^values

Po

^or

Pp.

However, in the first method,

absorption by

^excited

states and other saturation effects prevents ^{the use} of large laser power

Po

^{at the}

dye jet [19].

^{The second}

method is better since the D.F.G. output power

P int i

is a

hyperbolic

function of second order in

Pp,

^{so that}

the D.F.G. _power can increase

rapidly

^{with the pump}

power

Pp. ^Figure

2a shows the

dependence

^{of the}

D.F.G. _power

Pn’

^on the pump power

Pp

^as

^expressed

in

equation

(29). ^The

working

conditions are : an

intracavity

^D.F.G. system

using

⁹⁰⁰

phase matching LiNb03-crystal (p

^{= 0,}

aeff

^{= 1.5 x 10- 8 ues} [4] ;

ne = no = 2.24

[4] ;

^{1 = 5}

mm)

^and rhodamine 6G

dye

laser

(TR

^{= 5 x 10- 9 s;}

^No

^{= 1.28 x 1017}

^molecules/

cm 3; as (As

^{= 0.586}

um)

^{= 1.62 x} lO-16 cm2

[12];

co

(,1,0

^{= 0.5145}

pm)

^{= 1.6 x 10-16 cm2}

[12]).

^Two

argon ion laser

operating

ⁱⁿ

single

transversal mode

are used : one

operating

^{at 0.514} gm with a power of

Fig. ^{2. -} Dependence of the difference frequency generated

power ^on the pump power Pp ^{in the case of 900-phase} matching ^D.F.G. using ^a LiNb03-crystal ^{with weak} focusing.

a) Intracavity D.F.G. power

pint

^{as a} function of Pp. ^b) Extracavity D.F.G. power Piext’ ^{as a function of} Pp. ^The

values of the pump and signal wavelengths ^are

Ap

^{= 0.488 urn,} Ar, ⁼ 0.586 J.1m. The other parameter ^{values are} given ^{in the}

text.

3 W is used to pump the

dye jet giving

^{a resultant}

optical path length Lo

^{= 0.37 mm and a} pump ^area radius of 0.014 ^mm. The other has a pump

wavelength ).p

^{= 0.488 pm and}

^gives

^a beam waist of 0.068 mm at the centre of the

LiNbo3 crystal.

^{At the}

wavelength A,S

^{= 0.586} pm, the

dye

laser makes a

signal

^beam

waist of 0.070 ^mm in the

crystal

^{and one of 0.014 mm}

at the dyejet(Ws

^{= 0.014 mm i.e. v =} 1). The LD.F.G.

system operates ^{in the weak}

focusing

^condition,

whereupon

^the

focusing

functions take the

following

values hDF

(B =

0, ç 1, 14

Uopt) = ç

^{and h(1)}

(B =

0, j « 1, Jl,

(JoPt) ~ ç

^[7]. The threshold of the

dye

^laser

is 2 W and the total

single-pass

loss is 0.32

(see

^for-

mula

(26)).

This loss is

acceptable

since, ^{from its}

definition in 2.1), ai is the total

single-pass loss]

including

non-useful components.

In figure

^{2 we see the} forceful increase of the D.F.G.

output power caused

by

^the

parametric amplification

of the

signal

^wave

(dye

laser). To compare

Pn’

^with

the D.F.G. _power in the case of the non-linear

crystal being placed

outside the

dye

^laser

cavity,

^{we can}

calculate the _power in the latter case.

Using

(3), (4), (6), (10) ^and (18), ^{we have the} following

expression

for this D.F.G. _{power :}

where hDF is determined

by

(30) ^and

p:xt

^{is the} power of the

signal

(laser) beam outside of the

cavity.

Formula (32) ^is

general

^for

extracavity

^D.F.G.

using

Gaussian laser beams. In the

special

^case

of weak focusing

and without double refraction

h(B

⁼ 0, j « 1, 6op,,

u) --- 03BE, equation (32)

^becomes

The

expression

(33) ^{is in} agreement ^with

Boyd’s

^and

Ashkin’s formulae under the same conditions

[5].

Using

the above

dye

laser with the transmission coefficient of the output ^mirror

Topt, equation (33)

becomes

Taking Topt

^{= 15}

^%

^{into account, the}

dependence

^of

the D.F.G. _power on

Pp

^as

^expressed

^{in (34) is shown}

in

figure

^{2b. The}

working

conditions

being

^{the same}

as those

quoted by

the above I.D.F.G. (i.e. ^same

focusing

condition, ^same

dye

^{laser and same} crystal, etc...) ^with

Pp

^{= 50 mW,}

^Pose Topt

^{= 50 mW, equa-}

tion (34)

yields Pfxt

^{= 7} uW. This result is in agreement with the

experimental

result of A. S. Pine

[4],

^whereas

formula (29)

gives

^a power of 51

J.1W

for the I.D.F.G.

system operated ^{in the same condition which means}

that it

gives

^an enhancement factor of about 7.

2.2.2 The enhancement

factor of

^D.F.G. power. ^- To deduce the

requirements

^{to be met in}

practice,

let us

study

the enhancement of D.F.G. power. We

write the enhancement factor as

Using equations

(29) ^and

(32)

^{we have the}

general

formula for the enhancement

factor q

where

T.pt

is the transmission coefficient for the mirror which results in the maximal _power output ^{from the} laser in the absence of the

D.F.G.-crystal.

It is obvious from (35) that the enhancement factor is determined

by

two terms : the

signal

^wave

high

field effect inside the laser

cavity expressed by

^the

factor

1 /T opt

^{and the}

parametric amplification

^effect

responsible

^{for the} appearance of the

remaining

^factor.

When

Pp

^{is small,} the enhancement factor becomes n~

1 / Top,,

i. e. the first effect is overcome. The _para- metric

amplification

effect becomes remarkable

only

when the

following experimental

conditions are met :

a) ^The pump power ^must be

relatively high

^but

smaller than the resonant value determined

by

(28).

fl)

^{It is} necessary ^{to use} the

optimal focusing

method to obtain the maximal value of the function

h(l)(B,

^a, M, ç)

[ 15].

The

input

laser beams must be

chopped

^{to reduce}

the thermal effect which is associated with

high input

laser powers. A

slight

energy

absorption

^{in the}

crystal

causes non-uniform

heating

which cancels the 900

phase matching by

temperature

tuning [19].

In

figure

^3, the enhancement

factor q

^is

plotted

versus pump power

Pp.

^The

^working

conditions of the I.D.F.G. system ^{are similar to those}

given

ⁱⁿ

section 2.2.1. Figure ^3a illustrates the case of weak

focusing.

^{In the case of}

optimal focusing ’oPt

^{= 1.6}

(Wos

^{= 40}

J.1m), using

the result in

[7],

^{the value of the}

focusing

^function

h(1)(B

⁼ 0,03BE, ⁼ 1.6, ,u ^~

2/3; (JoPt)

is

equal hopt

^{= 0.6. The}

dependence

of the enhance- ment

factor q

^on

Pp

^{in this case} is shown in

figure

^3b.

We see in

figure

3 that the enhancement factor is _very

large

^{when the} pump power

Pp

takes the value close to resonant power value

ppe

determined

by (28).

It is obvious from

equation

(35) ^that yy increases as

the _pump power

Po

^{at the}

dye jet

decreases, ^{and the}

enhancement factor becomes _very

large

^when

PO -> Poth.

Fig. ^{3. -} Dependence of the enhancement factor n ⁼ pint/pext ^on the pump power

Pp

^{in the case of 90°-} phase matching ^D.F.G. using LiNb03-crystal. a) ^Weak focusing. b) Focusing parameter ç ^{= 1.6. The} other para- meter values are given ^{in the text.}

When the

dye

^laser operates ^{close to the} threshold,

it is better to use the

intracavity

^{method to obtain}

a

large

difference

frequency generated

^power.

3.

Intracavity

difference

frequency-mixing

^{in critical} phase

matching crystal

using

elliptical focusing

^{of the}

pump laser beam.

In their

experimental

^work

[10],

Lahmann et al. have shown that, ⁱⁿ

intracavity

difference generation, ^the thermal effect which cancels the

phase matching

condition can be eliminated

by using

^{a critical}

phase

matching crystal

in which the refractive index is almost

independent

^{of the} temperature. However, ⁱⁿ

the case of critical

phase matching,

the effective interaction

length,

^{and hence the}

efficiency

^{of conver-} sion, ^{are both}

significantly

^reduced

by

double refrac- tion processes

occurring during

^collinear

frequence mixing.

This reduction of the effective interaction

length

^is also encountered in

optical parametric

oscillators, ^{where it can be shown}

[20-21]

that para- metric

generation

conversion

efficiency

^{under condi-}

tions of

optimum elliptical focusing

^is

higher

^than

that obtained with

optimum

^circular

focusing.

In this section we

study

the I.D.F.G. with

elliptical focusing

of the pump laser beam. We consider the

parametric

interaction of an

extraordinary

pump ^wave with

ordinary signal

^and difference

frequency

^waves.

In a uniaxial

crystal,

double refraction occurs

only

in the

principal plane (i.e.

^{in the}

plane

of the direction of

propagation

^{and the}

optic axis).

Therefore,

by elliptical focusing,

^{it is}

possible

^{to focus more}

tightly

in the non walk-off

plane

^{to make}

overlap

^between

the

signal

^{and pump waves,} which results in a

large parametric

conversion

efficiency.

The treatment used in this section is similar to that

given

in section 2 and in

[7].

The Gaussian pump beam is assumed to be

elliptically

focused, ^{whereas the}

signal

^{beam is}

circularly

determined

by (3).

^Both

signal

^and pump beams are focused on the entrance of the

crystal

^{at Z = 0}

⁽ⁱⁿ

the presence of double refrac- tion

(B >

3), the result is the same as one would find if the

focusing positions

^{were at the centre of the}

crystal [22]).

Describing

^the

ellipticity

of the pump beam

by

^the parameter

the _one-way

travelling

pump beam is of the form

[21]

where ç

⁼

l/bp

^and

bp

⁼

kp Wpx W p Y = (bpx bpy)1/2, bp

^{is an} effective confocal parameter ^{of the} pump beam

[21 ]

and

Wp

^{is now defined}

^by bp

⁼

kp Wp. ^Using

the results in the section 2 and

equations

(3) ^and

(38)

^{we have the}

following expression

for the enhancement factor :

where

hel(ff

^{6, Jl, ç,}

^a)

^{is a}

focusing

function defined

by [21].

where the

following

abbreviations were used