Experimental and theoretical investigation of third-harmonic generation in

Experimental and theoretical investigation of third-harmonic generation in

phase-matched dye solutions A. PENZKOFER, W. LEUPACHER

Naturwissenschaftliche Fakultät II - Physik, Universität Regensburg, D-8400 Regensburg, FRG

Received 24 August 1987

The p h a s e - m a t c h e d t h i r d - h a r m o n i c light generation in dye s o l u t i o n s is studied experi- mentally a n d theoretically. In t h e experiments p i c o s e c o n d light pulses of a passive m o d e - l o c k e d N d - g l a s s laser are c o n v e r t e d t o t h e t h i r d - h a r m o n i c frequency. A t h i r d - harmonic c o n v e r s i o n efficiency of u p t o 4 x 1 0 ~⁴ w a s achieved for o n e of t h e dyes investigated (1,3,3,1 ' ^ ' ^ ' - h e x a m e t h y l i n d o c a r b o c y a n i n e iodide in h e x a f l u o r o i s o p r o - p a n o l ) . T h e theoretical calculations determine t h e influence of various d y e a n d solvent parameters o n t h e c o n v e r s i o n efficiency. T h e c o n v e r s i o n efficiency is f o u n d t o be limited by excited-state a b s o r p t i o n of p u m p laser light a n d t h i r d - h a r m o n i c light f r o m t h e ST -state t o higher singlet states. T h e ST -state is mainly p o p u l a t e d by t w o - p h o t o n a b s o r p t i o n . A m p l i f i e d s p o n t a n e o u s emission may reduce t h e l i m i t i n g effects of e x c i t e d - state a b s o r p t i o n . Phase changes caused by t h e n o n - l i n e a r refractive index a n d t h e refractive index dispersion w i t h i n t h e spectral b a n d w i d t h of t h e laser pulses reduce t h e c o n v e r s i o n efficiency. Under ideal c o n d i t i o n s c o n v e r s i o n efficiencies u p t o 1 0 % may be achieved.

1. I n t r o d u c t i o n

C o l l i n e a r phase-matched a n d resonantly enhanced t h i r d - h a r m o n i c light generation is o b t a i n a b l e f o r some dye s o l u t i o n s w i t h the S⁰—S^t a b s o r p t i o n peak between the f u n d a m e n t a l a n d t h i r d - h a r m o n i c frequency [1-8]. T h e p h a s e - m a t c h i n g at a c e r t a i n dye c o n c e n t r a t i o n is achieved b y the a n o m a l o u s d i s p e r s i o n o f the refractive i n d e x o f the dye i n the S^x - a b s o r p t i o n b a n d . T h e resonant enhancement is due to the S0- S , t w o - p h o t o n resonance.

F o r n a n o s e c o n d light pulses o f a N d - Y A G laser t h i r d - h a r m o n i c c o n v e r s i o n efficiencies u p to 2 x 10~⁹ have been o b t a i n e d i n the dye 1, 3, 3, 1', 3', 3^/- h e x a m e t h y l i n d o c a r b o c y a n i n e i o d i d e ( H M I C I ) d i s s o l v e d i n h e x a f l u o r o i s o p r o p a n o l ( H F I P ) [5]. I n a recent p a p e r [8] the efficient g e n e r a t i o n o f t h i r d - h a r m o n i c light w i t h p i c o s e c o n d p u m p pulses o f a m o d e - l o c k e d N d - g l a s s laser i n the dye 1, 3 ,r^ ' - t e t r a m e t h y l - l ^ ' - d i o x o p y r i m i d o - ö j ö ' - c a r b o c y a n i n e h y d r o g e n sulphate ( P Y C ) d i s s o l v e d i n H F I P has been studied. A c o n v e r s i o n efficiency u p to 2 x 10~⁴ was achieved. T h e l i m i t a t i o n s o f t h i r d - h a r m o n i c generation at h i g h p u m p pulse intensities were a n a l y s e d b y a n a l y t i c a l estimates.

I n this p a p e r we present e x p e r i m e n t a l results o n the t h i r d - h a r m o n i c generation w i t h p i c o s e c o n d N d - g l a s s laser p u m p pulses for the dyes H M I C I , P Y C a n d safranine T .

0306-8919/88 $03.00 + .12 © 1988 Chapman and Hall Ltd. 2 2 7

A c o n v e r s i o n efficiency u p to r\ « 14 x 1 0 ~4 has been o b t a i n e d for H M I C I i n H F I P . T h e l i m i t a t i o n s o f the energy c o n v e r s i o n at h i g h p u m p pulse intensities for the dyes H M I C I , safranine T a n d P Y C are discussed i n d e t a i l . T h e m e a s u r e d energy c o n v e r s i o n curves are fitted b y c o m p u t e r s i m u l a t i o n s .

T h e l i m i t a t i o n s o f t h i r d - h a r m o n i c g e n e r a t i o n are a n a l y s e d b y n u m e r i c a l c a l c u l a t i o n . T h e effects o f l i n e a r a b s o r p t i o n at the t h i r d - h a r m o n i c frequency, o f t w o - p h o t o n a b s o r p t i o n , o f excited-state a b s o r p t i o n at the f u n d a m e n t a l a n d t h i r d - h a r m o n i c frequency, o f S J - S Q a m p l i - fied s p o n t a n e o u s e m i s s i o n , a n d o f p h a s e - m i s m a t c h due to the n o n - l i n e a r refractive i n d e x a n d the refractive i n d e x d i s p e r s i o n w i t h i n the spectral pulse b a n d w i d t h are i n c l u d e d . T h e analysis indicates that u n d e r f a v o u r a b l e c o n d i t i o n s (extremely w e a k a b s o r p t i o n at t h i r d - h a r m o n i c frequency) t h i r d - h a r m o n i c energy c o n v e r s i o n efficiencies u p to 1 0 % s h o u l d be achievable for p i c o s e c o n d p u m p pulses.

F o r t h i r d - h a r m o n i c generation i n gases a t h o r o u g h d i s c u s s i o n o f the l i m i t i n g processes is g i v e n i n [9, 10].

2. E x p e r i m e n t a l results

T h e t h i r d - h a r m o n i c generation experiments we c a r r i e d o u t w i t h a passively m o d e - l o c k e d N d - g l a s s laser [11]. T h e e x p e r i m e n t a l a r r a n g e m e n t is s h o w n i n F i g . 1. T h e N d - p h o s p h a t e glass laser generates pulses o f d u r a t i o n 5 to 6 ps at a w a v e l e n g t h o f AL = 1.054 / m i (laser glass H o y a L H G 5 , saturable a b s o r b e r K o d a k dye N o . 9860 i n 1,2-dichloroethane). A single pulse is selected f r o m the pulse t r a i n generated, a n d is a m p l i f i e d b y d o u b l e passage t h r o u g h a n N d - p h o s p h a t e glass a m p l i f i e r (glass: H o y a L H G 7 ) . T h e intensity o f the p i c o s e c o n d pulses is d e t e r m i n e d b y t r a n s m i s s i o n measurements t h r o u g h a saturable a b s o r b e r [12]

( p h o t o d e t e c t o r s P D 1 a n d P D 2 ) . T h e t w o - p h o t o n a b s o r p t i o n i n the dye s o l u t i o n s (sample S) is m o n i t o r e d b y t r a n s m i s s i o n measurements w i t h p h o t o d e t e c t o r s P D 1 a n d P D 3 . T h e t h i r d - h a r m o n i c s i g n a l is detected b y the p h o t o m u l t i p l i e r P M . T h e t h i r d - h a r m o n i c energy c o n v e r s i o n efficiency is d e t e r m i n e d b y c a l i b r a t i n g the p h o t o m u l t i p l i e r P M relative to the p h o t o d e t e c t o r P D 1 .

T h e dyes H M I C I , P Y C a n d safranine T were investigated. T h e solvent H F I P ( C F³- C H O H- C F 3 ) was used. T h e a b s o r p t i o n cross-section a n d s t i m u l a t e d e m i s s i o n cross- section spectra o f the dyes at the p h a s e - m a t c h i n g c o n c e n t r a t i o n are p l o t t e d i n F i g . 2 [13, 14]. ( F o r the d e t e r m i n a t i o n o f the a b s o r p t i o n spectra see [15], a n d for the d e t e r m i n a t i o n o f the fluorescence spectra see [16].)

M.L.LASER SWITCH AMPLIFIER

PM S L

C=h+-^--B---^f

i

PD3

cfclDC

ü

PD2 PD1

Figure 1 Experimental set-up. PD1 to PD3, photodetectors; DC, saturable absorber cell for intensity detection;

L, lens; S, dye sample; F, filters; P M , photomultiplier.

u i i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i—i i i r = i

WAVELENGTH X [nm]

Figure 2 Absorption and emission cross-section spectra of dyes investigated. The spectra are for the phase- matching concentrations of the dyes. Solid curves, HMICI in HFIP; broken curves, PYC in HFIP; chain- broken curves, safranine T in HFIP.

3

0 0

CC t -34 UJ 10

ID a:

Q UJ M US

<

I

1 - m i -1—r -•••["" 1

o

- o

r -

/ \°

r

-

(a) (b)

(c)

- - -

o

— c

o / o /

-

1 i / i-.J ^{i i} 1 , 1 1 / 1 „i,_L ^{i l}

-

o / i 1 1 ^{I i} 0.04 0.08 0 0.04 0.08 0

CONCENTRATION C [mol/dm³]

0.2 0.4

Figure 3 Third-harmonic generation versus dye concentration. Experimental points are obtained for pump pulse peak intensities /^{0 L} < 4 * 1 0⁹W c m "². Curves are calculated for /^{0 L} = 4 x 10⁹Wcnrf² by use of Equations 35 and 28 and data from Table I. The wave-vector mismatch Ak (Equation 17) is determined by assuming nj - /?|, oc C (/ = L,3; n^{S /}refractive index of solvent; C concentration), (a) HMICI; (b) PYC; (c) safranine T. Solvent: HFIP.

F"

t 10 "h-

/

/ 7

/

>

o

>-

ID or

7

10 1 -

7 oc>—-co

I N P U T PEAK I N T E N S I T Y IQL [ W/ c m2]

Figure 4 Third-harmonic conversion effi- ciency versus pump pulse intensity for HMICI in HFIP. Circles and solid curves, sample length / = 1 mm; triangles and

broken curves, / = 0.1 mm. Curves 3,3' are calculated using the data of Fig. 2 and Table 1. The other curves use the same data except: 1, aft = (rg> = «ig* = 0; 2,2',

<4 = *ex,³ = 0; 4,4', <T^e = 0. Structural for- mula of HMICI is included.

T h e n o r m a l i z e d energy c o n v e r s i o n t]^E/Io^L versus dye c o n c e n t r a t i o n is s h o w n i n F i g . 3 for the three investigated dyes. t]E = W3(l)/WL(0) is the t h i r d - h a r m o n i c energy c o n v e r s i o n a n d 7^{0 L} is the peak intensity o f the i n p u t laser light. S a m p l e lengths o f / = 1 m m were used. T h e p u m p laser intensity was set to I^0L < 4 x 1 0⁹W c m "² (no s a t u r a t i o n effects). T h e c o n - centrations o f m a x i m u m energy c o n v e r s i o n are the p h a s e - m a t c h i n g dye c o n c e n t r a t i o n s C^{P M} ( H M I C I , C^PM = 0 . 0 8 m o l d m -³; P Y C , C^PM = 0 . 0 8 2 5 m o l d m "³; safranine T , C^PM = 0 . 3 3 m o l d m "3) .

T h e t h i r d - h a r m o n i c energy c o n v e r s i o n s versus i n p u t p u m p pulse intensity at the phase- m a t c h i n g dye c o n c e n t r a t i o n s are d e p i c t e d i n F i g s 4, 5 a n d 6 for the dyes H M I C I , P Y C a n d safranine T , respectively. D y e cell lengths o f / = 1 m m (circles) a n d / = 0 . 1 m m (triangles) were used. T h e energy c o n v e r s i o n rises q u a d r a t i c a l l y w i t h i n p u t laser peak intensity 7^{0 L} u p to I^0L « 2 x 1 0^{1 0} W e m^{- 2}. A t higher i n p u t intensities the energy c o n v e r s i o n rises m o r e s l o w l y a n d n o further increase o f energy c o n v e r s i o n is o b s e r v e d a b o v e I^0L « 4 x 1 0^HW c m -².

T h e t h e o r e t i c a l curves i n F i g s 4 to 6 are e x p l a i n e d i n S e c t i o n 4, after a theoretical d e s c r i p t i o n o f t h i r d - h a r m o n i c g e n e r a t i o n i n dye s o l u t i o n s i n S e c t i o n 3.

T h e t w o - p h o t o n a b s o r p t i o n cross-sections, a^l, f o r the s i m u l t a n e o u s a b s o r p t i o n s o f t w o p u m p laser p h o t o n s (frequency v^L) a n d the S^x -excited state a b s o r p t i o n cross-sections a\^x have been d e t e r m i n e d p r e v i o u s l y [14]. T h e d a t a are listed i n T a b l e 1.

Parameter C^{P M} (moldm ³)

T^F (PS)

#F

?FC (PS)

?ex (PS)

*v,6 (PS)

*v,7 (PS)

a^L (cm-¹)

a3 (cm²)

<r^eLm (cm²)

^^{X S E} (cm²)

^ex,3 (CHI2)

-^L (cm²) crft (cm4s) o § ( a n⁴s )

^S(cm⁴s) Hu

TxLx ( ~ w3; coL, coL, ö)L) (Cm4V- j y i L ( - ö3; ö> L . coL, «L) ( C m4v -4)

^ ( - c o3; c oL, a ;L, a ;L) ( m2V -2) (-<o³; % , G>^L> ^WL ) (m²V~²)

A,THG (mm)

)^L) ( C m⁴V -⁴)

HFIP HMICI P Y C Safranine T Comments

_ _{0.08 0.0825 0.33}

•1.273 1.292 1.296 1.297

1.284 1.292 1.296 1.297

- 92 9.5 120 [14]

- 2.7 x 10~² 2.4 x 10"³ 7.4 x 10~3 [14]

- 0.7 0.7 0.7 Assumed [31]

- 0.1 0.1 0.1 Assumed [32]

- 4 4 4 Assumed [33]

- 0.1 0.1 0.1 Assumed

0.07 0.181 0.07 0.263

0 2.6 x 10-^{1 8} 3.55 x 10"^{1 8} 2.1 x 10~¹⁸ Fig. 2

0 -1.5 x 10-^{1 9} - 7 x 10-^{2 0} - 2 x 10-^{1 9} [14]^a

0 3 x 10-^{1 7} 1 x 10-^{1 7} 1 x 10-^{1 7} Assumed

0 1.5 x 10"^{1 6} 2.5 x 10-'⁶ 2 x 10"^{1 6} Fitted

0 4 x 10-1 7 2 x 10~18 7 x 10-^{1 8} [4]

0 2 x 10-4 9 1.8 x 10-4 9 5 x 10~⁵⁰ [14]

0 1.8 x 10-4 8 1.62 x 10"4 8 4.5 x 10~49 Assumed*5

0 1.2 x 10-4 8 1.08 x 10~48 3 x 10-4 9 Assumedb

1 x 10-^{6 2} 1 x 10"^{6 2} 1 x 10"6 2 1 x 10~62 [8]^d •

0 2.0 x 10-^{5 9} 1.7 x 10-5 9 [8] 3.3 x 10-6 0 d

1.4 x 10-^{2 3} 1.4 x 10"^{2 3} 1.4 x 10-^{2 3} 1.4 x 10-^{2 3} [8]d

0 2.48 x 10~²² 2.0 x 10~²² [8] 1.7 x 1Ö"^{2 2} d

- 0.24 0.17 0.072 Equation 32

. ( - G >³; <w^L, -OJ^L, w³) = x%^x (

a < 7L ⁼ 0 used in all calculations.

bIt is assumed that XxLx (~ WL ' WL ' WL ) = x\

°n2,ij 0 =L, 3,y = L, 3) values are not known. n2ij = 0 is used in calculations except n2ij dNon-linear susceptibility of dyes for third-harmonic generation is assumed to be imaginary.

18) [7].

»³; co³, - c o³, 0)³).

values are stated.

i-c XXL,D(-^³;O)^L,OJ^L,(O^L) = - i x i³L ,^D( - < y³; ö ;^L, a ;^L, ö ;^L) (Equation

>-

o 10 CO

>

o z

>- ID or

~r~rT

1/ 2 >

/

/ 2

CH=CH-CH =< N-CH,

• • I

HSO '

10¹² INPUT PEAK INTENSITY I^{0 L} [W/cm²]

Figure 5 Third-harmonic generation in PYC dissolved in HFIP. Circles and solid curves, sample length / = 1 mm; triangles and broken curves, / = 0.1 mm. Curve 3,3' are calculated with the data of Fig. 2 and Table 1. Other curves belong to the same data except: 1, o<$ = ag> = <rg^} = 0; 2,2',

<4 = *ex,3 = 0; 4,4', <r^E = 0. Structural formula of PYC is included.

3. T h e o r y

In this section a realistic d e s c r i p t i o n o f t h i r d - h a r m o n i c g e n e r a t i o n i n dye s o l u t i o n s is pre- sented. T h e effects i n c l u d e the l i n e a r a b s o r p t i o n at the f u n d a m e n t a l a n d the t h i r d - h a r m o n i c frequency, the t w o - p h o t o n a b s o r p t i o n o f t w o p u m p laser p h o t o n s , the s i m u l t a n e o u s a b s o r p t i o n o f t w o t h i r d - h a r m o n i c p h o t o n s , the t w o - p h o t o n a b s o r p t i o n o f a p u m p laser p h o t o n a n d a t h i r d - h a r m o n i c p h o t o n , the excited-state a b s o r p t i o n at the f u n d a m e n t a l frequency, the excited-state a b s o r p t i o n at the t h i r d - h a r m o n i c frequency, the a m p l i f i e d s p o n t a n e o u s e m i s s i o n a n d the refractive i n d e x changes.

T h e energy-level system o f the dye m o l e c u l e s is presented i n F i g . 7. T h e l i g h t a b s o r p t i o n a n d e m i s s i o n processes are i n c l u d e d i n the figure. A s i m i l a r level system was used i n [14] to describe S⁰-S^x t w o - p h o t o n a b s o r p t i o n d y n a m i c s i n dye s o l u t i o n s . H e r e the level system is extended to i n c l u d e the t h i r d - h a r m o n i c generation, the t w o - p h o t o n a b s o r p t i o n s o f a p u m p laser a n d a t h i r d - h a r m o n i c p h o t o n (p$) a n d o f t w o t h i r d - h a r m o n i c p h o t o n s (erf]), the l i n e a r a b s o r p t i o n o f t h i r d - h a r m o n i c l i g h t (<r3) a n d the excited-state a b s o r p t i o n o f the t h i r d - h a r m o n i c l i g h t (<r^{ex 3}) .

T h e d y n a m i c s o f t h i r d - h a r m o n i c g e n e r a t i o n , t w o - p h o t o n a b s o r p t i o n , ground-state a b s o r p t i o n , excited-state a b s o r p t i o n , a m p l i f i e d s p o n t a n e o u s e m i s s i o n a n d p h a s e - m o d u - l a t i o n are described b y the f o l l o w i n g differential e q u a t i o n system (see also [14]). T h e equations are t r a n s f o r m e d to a m o v i n g frame b y t' = t — nz/c⁰ a n d z' — z , where / is the time, z the s p a t i a l p o s i t i o n i n the p r o p a g a t i o n d i r e c t i o n , n the refractive i n d e x a n d c0 the

- i—r - n r

1/ / 2 ' / 7 7

r

to

>

o z

>- ID CC

CHÄ

H N'

6

speed

dt'

Figure 6 Third-harmonic generation for safranineTin HFIP. Circles and solid curves, / = 1 mm; triangles and broken curves, / = 0.1 mm. Curves 3,3' are calculated with data of Fig. 2 and Table I. Same curves are obtained for a^E = 0. Curve 1 is for 4 L = ^L3 = 423 = 0, and curves 2,2' for _ CgX = aex3 = 0. Other data are the same as

1Ö10 1011 1012 for curves 3 and 3'. Structural formula of

INPUT PEAK INTENSITY I0 L [W/cm2] safranine T is included.

o f l i g h t in vacuo. T h e e q u a t i o n s are _L_L _I I I I

= -(N} - Ns)

hv^ II -

dN2

miff, - N^W)

/j

v

v

2(hvLf L

2 ASh j

°em,/YASE,/

2(hv,)2

N2 - N, N2

t

V

N2 + N,

ffexA.

+ (N, - J V⁷)

flVL TF

+

ASE r

°em,/iASE,i

Av

,/

/*VL

/*VL

2

(1)

(2) _ASE r

°ex iA S E

/zv

/zv

hvL

N3 N4 N5 T + TL + ASE + 3H

''F Lex ''ex Lex

dN4

dt'

lex,3 (N2 + N3

Tex,10 Tex,U

° t x 4 J V 4 )

(3)

(4)

Figure 7 Energy-level diagram of dye molecules including the processes of third-harmonic generation, two-photon absorption, excited-state absorption, stimulated emission and amplified spontaneous emission.

8N5

~Bt⁷ = (N,

hv rA S E

(5)

dN^6J df

= *L

Trad

i SAT AT \ ^enU^ASE,/ ^6,i ~ Qd,i^\

^ASE,/ "T Mv3 Iy6,i) ,

"^VASE,/ ^Tv,6

(6)

dN¹

~d~F = (N3

v ) ^ e m 4 ^ 7 - 07^1

hv^L T^{v > 7} (7)

8N,

dt' = (N, (8)

dN⁹

= (N^{3 A r} ^ ö"ex,3^3 ^9

iy/9) j

hv³ T^{e x},³ (9)

SN]0 = o®(Ni - 7VI 0) Nl0

dt' 2(/*v3) (11)

" = e^{A S E (} ^ A v ^{A S E} ^ + (AT, - N^IASE., - (N³ - N⁵)a^I^ASEJ (12)

dz ' r^{r a d} 4?r

d z ' 2 \ hv^L A v³

+ (TV, - N7)alm) Ew + \x%GEl0E3t exp ( - i M z ' ) + «Ln2.L 3| £3 0|2£L 0

/ 2 n^Lc⁰

+ »L«2.LLI^LOP£LO] <^{1 3})

+ [ Ä ^ L O exp (iAfezO + n^Ln²,^{L 3}| £^{L O}|²£ 3 o + ^ . « l ^ o f ^ o ] (14) 2 «³c⁰

Ak = k³ - 3k^L = — (n, - n^L) = (n³ - «^L) = 6TIV^l(«³ - w^L) (17) The initial conditions for the number densities o f the level populations are N^x(t' = — oo,

r, z) = N⁰, N²(-oo) = N³(-oo) = N⁴(-oo) - 7 V⁵( - o o ) = J V⁸( - o o ) = #⁹( - o o ) ' =

^ i o ( — ° o ) = Nn( - o o ) = 0, i V6 > /( — o o ) = Q6jN0 a n d 7 V7( - o o ) = £7A V N0 is the t o t a l n u m b e r density o f dye molecules. T h e t e r m i n a l level o f amplified spontaneous e m i s s i o n (6) is d i v i d e d i n t o sublevels (6, /) to a c c o u n t for the v a r i a t i o n s o f the t h e r m a l level p o p u - l a t i o n (g⁶ⁱ) a n d o f the s t i m u l a t e d e m i s s i o n cross-section (a^j) w i t h frequency. T h e t h e r m a l level p o p u l a t i o n factors g⁶ⁱ a n d Q⁷ are a p p r o x i m a t e l y given b y g⁶ⁱ « ÖA^ASE,/)/*⁷^ ^{A N (}* Qi ~ ^A^L)/^ - Ö'ACVASE,/)an d Ö"A(VL)are the a p p a r e n t a b s o r p t i o n cross-sections at vA S E /

a n d vL, respectively [17, 18].

T h e i n i t i a l l i g h t intensities are I^L(t\ r, z = 0) = I^0L exp (—t'/tl) exp ( — r²/rl), 4S E, / ( *V, ^ = 0) = 0 ( / = 1, . . . , m ) a n d /³( f ' , r, z = 0) = 0 .1^{0 L} is the i n p u t p u m p pulse peak intensity. t⁰ = A f^L/ 2 [ l n ( 2 ) ]^{1 / 2} is h a l f the (\/e)-pxx\se w i d t h ( A /^L full w i d t h at h a l f m a x i m u m ) a n d r0 is the ( l / e ) - b e a m r a d i u s .

E q u a t i o n 1 describes the p o p u l a t i o n changes o f the S0- b a n d . N{ comprises the p o p u - lations o f levels 6 a n d 7. T h e first term is due to l i n e a r a b s o r p t i o n o f the t h i r d - h a r m o n i c light. T h e second t e r m is responsible for t w o - p h o t o n a b s o r p t i o n o f t w o p u m p laser p h o t o n s (frequency v^L, c i r c u l a r frequency CD^L = 2nv^L). T h e t h i r d t e r m gives the t w o - p h o t o n absorp- t i o n o f t w o t h i r d - h a r m o n i c p h o t o n s (frequency v³, co³ = 2nv³) a n d the f o u r t h t e r m gives the simultaneous a b s o r p t i o n o f a p u m p laser p h o t o n a n d a t h i r d - h a r m o n i c p h o t o n . T h e

fifth t e r m handles the a m p l i f i e d spontaneous e m i s s i o n , a n d the s i x t h t e r m gives the s t i m u - lated e m i s s i o n at the laser frequency. T h e last t e r m gives the S , - S⁰ r e l a x a t i o n .

T h e second e q u a t i o n includes the t w o - p h o t o n a b s o r p t i o n , the excited-state a b s o r p t i o n , the r e l a x a t i o n w i t h i n the Srb a n d a n d the r e l a x a t i o n t o the S0- b a n d . T h e t h i r d e q u a t i o n handles the S, -state d y n a m i c s . T h e v a r i o u s terms describe the level p o p u l a t i o n b y F r a n c k - C o n d o n r e l a x a t i o n , the excited-state a b s o r p t i o n s at frequencies v^L, v^{A S E} a n d v³, the a m p l i - fied spontaneous e m i s s i o n , the s t i m u l a t e d e m i s s i o n at the p u m p laser frequency v^L a n d v a r i o u s r e l a x a t i o n processes.

E q u a t i o n s 4, 5 a n d 9 describe excited-state a b s o r p t i o n s . E q u a t i o n 6 handles the t e r m i n a l level p o p u l a t i o n s o f the a m p l i f i e d spontaneous e m i s s i o n process. T h e first t e r m gives the spontaneous emission c o n t r i b u t i o n to the frequency interval Av⁶ⁱ. e^ASEi = E(v^ASEi)Av⁶ⁱ/q^F is the f r a c t i o n o f fluorescence l i g h t w i t h i n the frequency i n t e r v a l A v^{6 > /} a r o u n d v^{A S E}E ( v^{A S E i}) is the fluorescence q u a n t u m d i s t r i b u t i o n (j"e m E(v) dv = qF, qF fluorescence q u a n t u m effi- ciency, i n t e g r a t i o n over the S , - S0 fluorescence b a n d [16]). T h e second t e r m o f E q u a t i o n 6 gives the fluorescence l i g h t a m p l i f i c a t i o n , a n d the last t e r m causes t h e r m a l i z a t i o n w i t h i n the S⁰- b a n d w i t h a time constant T^{V 6}.

E q u a t i o n 7 describes the s t i m u l a t e d e m i s s i o n at the p u m p laser frequency. T h e first t e r m gives the s t i m u l a t e d e m i s s i o n a n d the second term causes p o p u l a t i o n t h e r m a l i z a t i o n w i t h i n the S⁰- b a n d .

E q u a t i o n 8 takes care o f the p o p u l a t i o n o f level 8 b y ground-state a b s o r p t i o n o f the generated t h i r d - h a r m o n i c light. E q u a t i o n s 10 a n d 11 h a n d l e the p o p u l a t i o n s o f levels 10 a n d 11 b y t w o - p h o t o n a b s o r p t i o n o f p h o t o n s o f frequency v^L a n d v³ a n d o f t w o p h o t o n s o f frequency v³, respectively, a n d the r e l a x a t i o n to level 3.

E q u a t i o n 12 describes the a m p l i f i c a t i o n o f fluorescence l i g h t . T h e first t e r m gives the seeding spontaneous e m i s s i o n i n a frequency i n t e r v a l Av⁶ⁱ a r o u n d v^{A S E /}. A Q is the s o l i d angle o f efficient a m p l i f i e d spontaneous e m i s s i o n . T h e second t e r m is responsible for a m p l i f i c a t i o n o f fluorescence l i g h t a n d the t h i r d t e r m takes care o f excited-state a b s o r p t i o n .

E q u a t i o n 13 describes the changes o f the electric field strength o f the p u m p laser. T h e first term gives l i n e a r losses, the second a n d t h i r d terms give t w o - p h o t o n a b s o r p t i o n losses (2co^L a n d o^L + co³), the f o u r t h t e r m is d u e t o excited-state a b s o r p t i o n a n d the fifth t e r m handles the s t i m u l a t e d e m i s s i o n . T h e s i x t h t e r m [/THG = Xxxxxi — ^ l ^ L) ] gives the p u m p pulse r e d u c t i o n b y t h i r d - h a r m o n i c generation. T h e seventh t e r m ( « 2 , 0 ) takes care o f t h i r d - h a r m o n i c field-induced phase changes a n d the last t e r m («²,LL) gives the p u m p field-induced p h a s e - m o d u l a t i o n . n^{l x l} a n d n^{1 X L} are n o n - l i n e a r refractive indices. T h e wave- vector m i s m a t c h Ak is g i v e n b y E q u a t i o n 17.

T h e b u i l d - u p o f the t h i r d - h a r m o n i c l i g h t field is described b y E q u a t i o n 14. T h e first t e r m is d u e t o ground-state a b s o r p t i o n a n d the second t e r m is d u e t o excited-state a b s o r p t i o n . T h e t h i r d a n d f o u r t h terms are d u e t o t w o - p h o t o n a b s o r p t i o n losses (coL + co3 a n d 2a>3).

T h e fifth t e r m gives the t h i r d - h a r m o n i c generation. T h e s i x t h a n d seventh terms (n^2yL3 a n d

^2,33) are d u e to phase changes caused b y the p u m p pulse a n d the t h i r d - h a r m o n i c l i g h t , respectively. n^2X3 a n d «2,33 are n o n - l i n e a r refractive indices.

T h e r e l a t i o n between field strength a n d light intensity is presented b y E q u a t i o n s 15 a n d 16. A d e r i v a t i o n o f E q u a t i o n s 13 a n d 14 is g i v e n i n the A p p e n d i x , where relations between the p h a s e - c h a n g i n g susceptibilities a n d the n o n - l i n e a r refractive indices are also d e r i v e d . T h e third-order non-linear susceptibility, #THG> responsible for t h i r d - h a r m o n i c generation is g i v e n b y XTHG = XxL(-o)3; CDL, COL, CDL) = * £ L ( - a )3; G>L, G>L, COl) - iXxL(-a>3;

(DL, a>L, coL). It is c o m p o s e d o f solvent a n d dye c o n t r i b u t i o n s , i.e. [19]

XTHG = ZTHG,S + XTHG,D 0 8 )

XTHG,S = ZTHG,SO^S/^SO is the solvent c o n t r i b u t i o n . N^S is the n u m b e r density o f s o l - vent molecules at dye n u m b e r density N0. NSO is the n u m b e r density o f the neat solvent (N0 = 0). XTHG,S is generally real (no resonant c o n t r i b u t i o n s ) . XTHG,D is the apparent t h i r d - h a r m o n i c susceptibility o f the dye. It includes i n t r i n s i c dye m o l e c u l e c o n t r i b u t i o n s a n d d y e - s o l v e n t i n t e r a c t i o n c o n t r i b u t i o n s . I f dye aggregation [13, 16, 20] occurs, the d y e - d y e i n t e r a c t i o n contributes to ^^G, D [21]. ZTHG,D m a y be expressed i n terms o f a p p a r e n t h y p e r p o l a r i z a b i l i t i e s y^lcnj b y [19, 22] (see A p p e n d i x )

TO) / (3)

XTHG.D = ^ W ^ G . D I + « A G, D 3 ) * - Z ^ N A ^D (19)

7THG,DI is the a p p a r e n t h y p e r p o l a r i z a b i l i t y o f dye molecules i n the g r o u n d state (level 1) a n d 7THG,D3 is the apparent h y p e r p o l a r i z a b i l i t y o f dye molecules i n the S^x -state (level 3).

^THG = (ⁿl + 2 ) ( « [ + 2)³/81 is the L o r e n t z local-field c o r r e c t i o n factor. Since g r o u n d - state d e p l e t i o n is weak i n the t h i r d - h a r m o n i c generation process (see F i g . 8), the second

e q u a l i t y o f E q u a t i o n 19 is reasonably accurate [7THG,D ~ 7THG,DI = yolxxxi-^l G>L, coL)]. T h e m a g n i t u d e o f XJHG — Z( 3 ) ~ i %( 3 )" determines the efficiency o f t h i r d - h a r m o n i c

generation. T h e phase o f XTHG p r a c t i c a l l y does not influence the t h i r d - h a r m o n i c c o n v e r s i o n efficiency.

T h e relations between the t w o - p h o t o n a b s o r p t i o n cross-sections off a n d the i m a g - i n a r y parts o f the susceptibilities Xolxxxxi — ^il ty, — o)j9 o r the h y p e r p o l a r i z a b i l i t i e s 7D]XXXX( — <üj, —u>j, (D^T) are g i v e n i n the A p p e n d i x ( E q u a t i o n s A 2 3 to A 2 6 ) .

Figure 8 Normalized length-integrated ST -state populations of dyes investigated. Solid curves, HMICI in HFIP; broken curves, PYC in HFIP;

chain-broken curves, safranine T in HFIP. Curves are calculated with data of Table I. Pulse duration AtL = 5ps. (1) sample length / = 0.1 mm; (2) INPUT PEAK INTENSITY I0 L [W/cm2] / = 1 mm. Dye parameters are listed in Table I.

T h e relations between the n o n - l i n e a r refractive indices n^2ij a n d the real parts o f the n o n - l i n e a r susceptibilities XxL^x(~ <*>i\ °>j > <x>,) are d e r i v e d i n the A p p e n d i x ( E q u a t i o n s A 2 7 t o A 3 0 ) .

T h e refractive indices, nⁱ⁹ a n d the wave-vector m i s m a t c h , Ak, change w i t h excited-state level p o p u l a t i o n a c c o r d i n g to E q u a t i o n s A 7 a n d A 1 8 . T h e change o f Ak at h i g h p u m p pulse intensities d u e t o the p o p u l a t i o n - d e p e n d e n t refractive indices is n o t e x p l i c i t l y a n a l y s e d b e l o w . Its c o n t r i b u t i o n to p h a s e - m i s m a t c h is f o r m a l l y a d d e d to the n o n - l i n e a r refractive indices n^2ij (see F i g s 19 a n d 20, b e l o w ) .

T h e t h i r d - h a r m o n i c intensity c o n v e r s i o n is fj, = 73 ( 0 / 4( 0 ) . T h e time-integrated t h i r d - h a r m o n i c c o n v e r s i o n is

rj^Tl = f^{0 0} I³(t\ / ) d/ 7 f°° IL(t', 0) dt' (20) J — oo J — oo

F i n a l l y , the t h i r d - h a r m o n i c energy c o n v e r s i o n is

ffE = W3(l)/W^L(0) = 73(f', r , / ) drj \* r 7L( r ' , r, 0) dt'dr (21) A t l o w p u m p pulse intensities the p o p u l a t i o n changes o f the v a r i o u s levels are negligible.

A p u m p pulse d e p l e t i o n does n o t o c c u r . T h e effects o f t w o - p h o t o n a b s o r p t i o n , excited- state a b s o r p t i o n a n d a m p l i f i e d s p o n t a n e o u s e m i s s i o n are d i m i n i s h i n g l y w e a k . T h e w h o l e e q u a t i o n system 1 to 14 reduces t o [23]:

—? = ~ ^LN0EL0 (22)

dz

^ = -±o3N0 + - ^ -Z? >G^ „ e x p ( - i A / c z ' ) (23)

oz 2n³ c⁰

F o r £"30(0) = 0 the s o l u t i o n s o f E q u a t i o n s 22 a n d 23 are g i v e n b y [23]:

Eu>(z') = £^Lo ( 0 ) e x p ( - ^^L7 V⁰z ' ) (24)

/ AT ^^{e X P}

£ a o ( * ' ) = ^ - z L 4 ( 0 ) e x p '

2n³c⁰ \ 2 J o³ - 3o-^L

( ^ 2 ^ i V o - i A ^ ) z ' ] - l

N⁰ - iAk

(25) T h e intensities at z' = / are ( E q u a t i o n s 15 a n d 16) [23]:

kd) = /L( 0 ) e x p ( - f fLt f0/ ) (26)

hil) = 'CTHGI^GI²4³(0) (27)

w i t h

col exp (-3a^LN⁰l) + exp (-<r³N⁰l) - 2 exp

( -

JV

/^J

(28) W i t h o u t a b s o r p t i o n (cr^L = 0, a³ = 0) E q u a t i o n 28 reduces to

col s i n² ( A W / 2 ) _ s i n²( A A : / / 2 )

— KTHG,0 , . , ,_,•> (2")

A t p h a s e - m a t c h i n g E q u a t i o n 29 is g i v e n b y

KTHG = ^ 4 2 = KTHG,(/ (3° )

I n the case o f Ak = 0, aL = 0 a n d Ö"3AT0/ ^> 1, E q u a t i o n 28 reduces to

^ (

Y /

Y

*THG < 3~T~2 TT = ^THCO — T T * (3 1) n3nlc40£2o \ <T³N⁰J V M ^ o /

T h e e q u a l i t y sign is v a l i d f o r a3N0l -> 00. C o m p a r i s o n o f E q u a t i o n s 31 a n d 30 indicates that the m a x i m u m c o n v e r s i o n efficiency i n case o f a b s o r p t i o n at frequency v3 (/ = 00) is e q u a l to the c o n v e r s i o n efficiency i n a cell o f e q u i v a l e n t l e n g t h /e q u = 2/(<73AT0) i f n o a b s o r p t i o n is present.

F o r n o n - l i n e a r m e d i a w i t h a b s o r p t i o n at the t h i r d - h a r m o n i c frequency one m a y define a r b i t r a r i l y a n effective i n t e r a c t i o n l e n g t h llTHG b y

3

h,THG ⁼ 77" (32)

<737V⁰

A t this length 6 0 . 3 5 % o f the m a x i m u m t h i r d - h a r m o n i c c o n v e r s i o n efficiency for / -> 00 are o b t a i n e d ( aL = 0).

T h e t h i r d - h a r m o n i c intensity c o n v e r s i o n efficiency is

1 , 1 = W) = k™gIA"g|2/l2(0) (33)

F o r G a u s s i a n - s h a p e d p u m p pulses the time-integrated c o n v e r s i o n efficiency is

iTi = ^ I ^ G P / O K O ) (34)

a n d the energy c o n v e r s i o n efficiency is

tin = ^ t ä U l K O ) (35) E q u a t i o n 35 is very useful to determine the t h i r d - o r d e r n o n - l i n e a r s u s c e p t i b i l i t y o f dye

s o l u t i o n s b y energy c o n v e r s i o n efficiency measurements. XTHG,D ( E q u a t i o n 18) is d e t e r m i n e d by m e a s u r i n g separately the t h i r d - h a r m o n i c g e n e r a t i o n o f the solvent. T h e real a n d i m a g i n - ary parts o f XTHG,D = ZTHG,D — 1XTHG,D are d e t e r m i n e d b y m e a s u r i n g the t h i r d - h a r m o n i c energy c o n v e r s i o n efficiency as a f u n c t i o n o f c o n c e n t r a t i o n [19].

T h e c o m p l e t e e q u a t i o n system 1 to 14 has to be s o l v e d to similate the t h i r d - h a r m o n i c g e n e r a t i o n at h i g h i n p u t p u m p pulse intensities a n d to analyse the l i m i t i n g effects to the t h i r d - h a r m o n i c g e n e r a t i o n .

4. S i m u l a t i o n o f e x p e r i m e n t a l results

T h e curves i n F i g s 4 ( H M I C I ) , 5 ( P Y C ) a n d 6 (safranine T ) are c a l c u l a t e d b y use o f the a b s o r p t i o n a n d e m i s s i o n curves o f F i g . 2, the spectroscopic d a t a o f T a b l e 1 a n d the parameters listed i n the figure c a p t i o n s .

T h e s o l i d curves are for / = 1 m m w h i l e the b r o k e n curves are for / = 0 . 1 m m . T h e curves 1 neglect t w o - p h o t o n a b s o r p t i o n a n d subsequent excited-state a b s o r p t i o n . E q u a t i o n s 35 a n d 28 are used i n the c a l c u l a t i o n s . T h e s e curves determine the t h i r d - h a r m o n i c n o n - l i n e a r susceptibilities |^THG I b y fitting to the e x p e r i m e n t a l p o i n t s at intensities

70 L < 2 x 1 01 0W c m ~2. T h e energy c o n v e r s i o n s are r e d u c e d b y the s t r o n g l i n e a r g r o u n d - state a b s o r p t i o n o f the generated t h i r d - h a r m o n i c l i g h t (<J³, short i n t e r a c t i o n l e n g t h /^{I > T H}G ) - T h e curves 3 , 3 ' represent the best-fitting curves (data listed i n T a b l e I). T h e effects i n c l u d e d are the t w o - p h o t o n a b s o r p t i o n processes co^L + co^L (o%l f r o m [14]), co^L + co³ ( ö f ] , here fitted) a n d a>³ + a>³ (afj, assumed, n o influence since I³ is s m a l l ) , the excited-state a b s o r p - t i o n o f p u m p l i g h t (<7g^X, f r o m [14]) a n d o f t h i r d - h a r m o n i c l i g h t ( c 7^{e x 3}, this w o r k ) a n d the a m p l i f i e d s p o n t a n e o u s e m i s s i o n (<r^^E f r o m F i g . 2 , <T£E is assumed f r o m the ground-state a b s o r p t i o n s p e c t r u m o f F i g . 2 ) . T h e effects o f refractive i n d e x changes are neglected since they are thought to be negligibly s m a l l for the achieved c o n v e r s i o n efficiences (see Section 5 ) .

T h e curves 4,4' d o n o t i n c l u d e the effect o f a m p l i f i e d s p o n t a n e o u s e m i s s i o n (the other parameters are the same as f o r curves 3 , 3 ' ) . T h e curves i n d i c a t e that a n o p t i m u m p u m p laser intensiy exists a b o v e w h i c h the t h i r d - h a r m o n i c c o n v e r s i o n efficiency reduces w i t h i n c r e a s i n g p u m p pulse energy because o f d o m i n a t i n g excited-state a b s o r p t i o n . I n case o f safranine T the curves 4,4' c o i n c i d e w i t h 3 , 3 ' , i n d i c a t i n g that the influence o f a m p l i f i e d spontaneous e m i s s i o n o n t h i r d - h a r m o n i c g e n e r a t i o n is d i m i n i s h i n g l y s m a l l . I n the case o f H M I C I slightly l o w e r t h i r d - h a r m o n i c energy c o n v e r s i o n efficiencies are c a l c u l a t e d at h i g h p u m p pulse intensities w i t h o u t a m p l i f i c a t i o n o f spontaneous e m i s s i o n . F o r the dye P Y C the a m p l i f i c a t i o n o f s p o n t a n e o u s e m i s s i o n is strongest [14]. It hinders a decrease o f t h i r d - h a r m o n i c c o n v e r s i o n efficiency at h i g h p u m p pulse intensities (S! -state level p o p u l a t i o n is l o w e r e d w i t h i n the p u m p pulse d u r a t i o n ) .

F o r the curves 2 , 2 ' the effects o f a m p l i f i e d s p o n t a n e o u s e m i s s i o n a n d excited-state a b s o r p t i o n at v^L a n d v³ are neglected. O n l y the effects o f t w o - p h o t o n a b s o r p t i o n o f t w o p u m p laser p h o t o n s ( a ^ , f r o m [14]) a n d o f a p u m p laser p h o t o n a n d a t h i r d - h a r m o n i c p h o t o n ( ö f ] , fitted here) are considered. T h e simultaneous a b s o r p t i o n o f t w o t h i r d - h a r m o n i c p h o t o n s is n e g l i g i b l y s m a l l because o f l o w t h i r d - h a r m o n i c c o n v e r s i o n efficiency (I³ s m a l l ) . T h e curves 2 , 2 ' i n d i c a t e that the t w o - p h o t o n a b s o r p t i o n sets a n u p p e r l i m i t o f the t h i r d - h a r m o n i c c o n v e r s i o n efficiency at h i g h p u m p pulse intensities. T h i s u p p e r l i m i t increases w i t h sample length, /, as l o n g as / < l^iTHG. F o r / > /^{I > T H}G ( E q u a t i o n 3 2 ) the u p p e r l i m i t o f c o n v e r s i o n efficiency decreases w i t h sample l e n g t h ( c o m p a r e curves 2 for / = 1 m m w i t h curves 2 ' f o r / = 0.1 m m ) .

I n F i g . 8 the c a l c u l a t e d S,-state p o p u l a t i o n s are presented for H M I C I ( s o l i d curves), P Y C ( b r o k e n curves) a n d safranine T ( c h a i n - b r o k e n curves). T h e length-integrated p o p u - l a t i o n (§⁰N³dz) t o w a r d s the e n d o f the p u m p pulse d u r a t i o n (/' = A /^L) is depicted. T h e curves 1 b e l o n g to / = 0.1 m m a n d the curves 2 to / = 1 m m (for details see [14]). T h e S! -state p o p u l a t i o n rises q u a d r a t i c a l l y w i t h p u m p pulse intensity at i n p u t intensities b e l o w 4 x 1 0^{1 0}W c m ~². A t h i g h i n p u t p u m p pulse intensities the S^{ -state p o p u l a t i o n levels off w i t h the t h i r d - h a r m o n i c l i m i t i n g effects.

5. I n f l u e n c e o f v a r i o u s d y e p a r a m e t e r s

T h e influence o f v a r i o u s dye parameters o n the t h i r d - h a r m o n i c c o n v e r s i o n efficiency is s t u d i e d i n the f o l l o w i n g . I n the n u m e r i c a l c a l c u l a t i o n s the parameters o f H M I C I ( T a b l e I a n d F i g . 2 ) are used except where otherwise stated.

I n F i g . 9 the influence o f the l i n e a r a b s o r p t i o n coefficient, a³ = N⁰a³, o n t h i r d - h a r m o n i c c o n v e r s i o n efficiency is investigated. Effects o f t w o - p h o t o n a b s o r p t i o n (<TLL =

°"L3 = ö i ? = 0), excited-state a b s o r p t i o n (o-gX = ö "^{e x 3} = a^^E '= 0), a m p l i f i e d s p o n t a n e o u s e m i s s i o n (p™^E = 0), l i n e a r p u m p laser a b s o r p t i o n (oc^L = GLN0 = 0), p h a s e - m i s m a t c h (Ait = 0) a n d n o n - l i n e a r refractive indices (n² LL — ⁿi 33 = ni u ⁼ 0 for d e f i n i t i o n see

ABSORPTION COEFFICIENT a3 [cm"1]

1Q-2 K T1 1 10_ 102 10J

>- in"¹

on a:

UJ >

o z

<

cc ID

Figure 9 Influence of linear absorption at third-harmonic frequency. HMICI data of Table I are used except off = off =

«L = 0' <4 = ^ex,3 = 0. The curves are for (1) /0 L = 1 07W c m "2 and / = 10cm; (2) /^{0 L} = 1 0⁸W c m "² and / = 10cm; (3) /0 L = 1 09W c m "2 and / = 10cm; (4) /^{0 L} = 1 0^{1 0}W c r T r² and / = 10cm; (4')/^{0 L} = 1 0^{1 0}W c r r T²a n d / = 1 cm;

(5) /⁰, 101 1 Wem /^{0 L} = 1 0^{1 2}W c m "² and /

2 and / = 1 cm; (6) 1 mm. Absorp- 10 ^{io-2 0 1(T}

ABSORPTION CROSS - SECTION °3 ^[cm2]

tion coefficients, a solutions of HMICI are indicated

3, of phase-matched , PYC and safranine T

E q u a t i o n s A 2 7 to A 2 9 ) are e x c l u d e d . T h e depicted curves s h o w a s t r o n g decrease o f t h i r d - h a r m o n i c c o n v e r s i o n efficiency for / > a³~¹. C u r v e s 4 (7^{0 L} = 1 0^{1 0}W c m ~², /, = 10 cm) a n d 4 ' ( 7^{0 L} = 1 0^{1 0} W c m ~², l² = 1 c m ) s h o w the influences o f sample length a n d a³, i n the case o f p u m p pulse d e p l e t i o n . I n case o f a^{3 _ 1} > l^x > l² the c o n v e r s i o n efficiency is higher for longer samples. I n a n intermediate r e g i o n l^x > a^{3 _ 1} > /², the shorter sample gives the higher c o n v e r s i o n , since i n the shorter sample n o t so m u c h p u m p l i g h t is transferred to the t h i r d - h a r m o n i c l i g h t w h i c h is a b s o r b e d . F o r large a b s o r p t i o n coefficients/, > l2 > a3 _ 1t h e c o n v e r s i o n efficiences become independent o f sample length, since the c o n v e r s i o n efficiency becomes very l o w a n d p u m p pulse d e p l e t i o n does n o t o c c u r .

T h e r e d u c t i o n o f t h i r d - h a r m o n i c l i g h t generation w i t h i n c r e a s i n g linear p u m p pulse a b s o r p t i o n is depicted i n F i g . 10 (/ = 1 c m ) . F o r / > OLE 1 the t h i r d - h a r m o n i c c o n v e r s i o n efficiency reduces q u a d r a t i c a l l y w i t h aL (n cc 70 2 L/e 2 f f, /e f f oc a f1) . T h e t h i r d - h a r m o n i c l i g h t is generated at the cell entrance before the p u m p pulse is reduced a n d it is transferred to the cell exit since n o a b s o r p t i o n o f t h i r d - h a r m o n i c l i g h t is assumed (a³ = 0, o^{2) = 0, = 0, Ak = 0, n² = 0).

T h e influence o f p h a s e - m i s m a t c h is i l l u s t r a t e d i n F i g . 11. T h e s i t u a t i o n o f l o w p u m p pulse intensities is depicted where E q u a t i o n s 35 a n d 28 are a p p l i c a b l e . T h e r a t i o o f the energy c o n v e r s i o n envelope i n the case o f Ak ^ 0, rj^{E e n v}( A £ ) , to the c o r r e s p o n d i n g energy c o n v e r - s i o n i n the case o f Ak = 0, rj^E(0), is s h o w n . T h e r a t i o is given b y

ffcenvCM) . A 4

% ( 0 )

= m i n I 1,

( A W ) (36)

^ /^{E e n v}( A / c ) is o b t a i n e d f r o m E q u a t i o n 28 b y setting cos ( A W ) = 1. T h e w a v e v e c t o r m i s m a t c h reduces the effective i n t e r a c t i o n length to /^{e f f} « Ak~^x. I n some dye s o l u t i o n s w i t h the S⁰- S i a b s o r p t i o n b a n d i n the frequency r e g i o n between v^L a n d v³ p h a s e - m a t c h i n g at a c e r t a i n dye c o n c e n t r a t i o n , C^PM, is achievable due to the a n o m a l o u s refractive i n d e x d i s p e r s i o n i n the w a v e l e n g t h r e g i o n o f the a b s o r p t i o n b a n d .

I n the presence o f a b s o r p t i o n o f t h i r d - h a r m o n i c l i g h t , cr3, the energy c o n v e r s i o n envelope ( E q u a t i o n 28) becomes p r o p o r t i o n a l to (Ah² + G²³NllA)~\ T h e a l l o w e d p h a s e - m i s m a t c h , Ak, that reduces the energy c o n v e r s i o n a factor o f 2 is g i v e n by

. . a}N0 2.78 \

T h e second p a r t o f the m a x i m u m c o n d i t i o n is o b t a i n e d f r o m E q u a t i o n s 29 a n d 30, i.e.

s i n2( A / r / / 2 ) / ( A f c / 2 )2 = /2/ 2 . T h e a l l o w e d wave-vector m i s m a t c h versus a b s o r p t i o n coef- ficient a³ is d e p i c t e d i n F i g . 12.

F o r u l t r a s h o r t l i g h t pulses their spectral w i d t h s , A v^L (full w i d t h at h a l f m a x i m u m ) , are n o t negligible. I f the p h a s e - m a t c h i n g c o n c e n t r a t i o n , CPM, is adjusted to the c e n t r a l frequen- cies vL a n d v3, the spectral w i n g s o f the pulses are n o t phase-matched for t h i r d - h a r m o n i c generation because o f the refractive i n d e x d i s p e r s i o n D = dn³/dv — dn^L/dv. I n the case o f p h a s e - m a t c h i n g at the c e n t r a l frequency v^L the p h a s e - m i s m a t c h at frequency v^L = v^L + öv^L

for the process v^L + v^L + v^L v³ is g i v e n b y Ak(v^L) = 6nv^L(n³ — n^L) = 6növ^Ld[v^L(n³ —

nh)]/^L ⁼ 6nv^L(dn³/dv^L — dn^L/dv^L)öv^L = 6nv^LDöv^L. T h e effective p h a s e - m i s m a t c h Afc^{e f f} due to d i s p e r s i o n is f o u n d b y i n t e g r a t i n g o v e r the spectral intensity d i s t r i b u t i o n . F o r

ABSORPTION COEFFICIENT a3 [cm"1] ip~* urj i io lof ig!

1 1 TT] 1 1 TT| 1 1 IT] 1 1 IT] 1 I I I J

10 10 10 ABSORPTION CROSS

10 SECTION

10 [cm²

Figure 12 Allowed wave-vector mismatch as a function of sample length and linear absorption cross-section at third-harmonic frequency (Equation 37). The third-harmonic conversion efficiency reduces a factor of 2 compared with Ak = 0 for the curves shown.

G a u s s i a n pulses it is

Too exp Akef! = 6nv^LD

•41n ( 2 ) ( « 5 v^L/ A v^L)²] d (<5v^L)

| ^ e x p [ - 4 1 n ( 2 ) ( ^ v^L/ A v^L)²] d (3v^L) 6[ln ( 2 ) ] "2vL£ >

= 3 r c^{l / 2}[ l n (2)]^U2v^LDAv^L

A f . C n (38)

T h e last equality is f o u n d b y use o f the r e l a t i o n A v^LA /^L = c⁰Av^LAt^L = 21n (2)n~^] for F o u r i e r t r a n s f o r m - l i m i t e d G a u s i a n pulses [24]. E q u a t i o n 38 is v a l i d for strict t h i r d - h a r m o n i c generation v^L + v^L + v^L v³ w i t h A v³ = 3 A v^L. F o r b a n d w i d t h - l i m i t e d pulses frequency m i x i n g processes, v^{L 1} + v^{L 2} + v^{L 3} -> v³, w i t h i n the b a n d w i d t h A v^L take place a n d reduce the t h i r d - h a r m o n i c b a n d w i d t h to A v3 « A vL. T h e effective wave-vector m i s m a t c h reduces to

Ak\ BWL

eff tAfc eff (39)

Ak™^L versus d i s p e r s i o n D is p l o t t e d i n F i g . 13 for the p u m p pulse d u r a t i o n s At^L = l O p s (1), 1 ps (2), 100 fs (3) a n d 10 fs (4). F o r femtosecond pulses the effective p h a s e - m i s m a t c h becomes r e m a r k a b l e .

F o r c h i r p e d pulses (due to self-phase m o d u l a t i o n [25-29]) the p u m p pulse frequency changes w i t h i n the pulse d u r a t i o n a n d o n l y the strict t h i r d - h a r m o n i c generation is relevant.

In this case the effective wave-vector m i s m a t c h is given b y E q u a t i o n 38 (i.e. Ak*™ = Afc^{e f f}).

T h e r e d u c t i o n o f energy c o n v e r s i o n due to refractive i n d e x d i s p e r s i o n is i l l u s t r a t e d i n F i g . 14 for b a n d w i d t h - l i m i t e d p u m p pulses. rj^E(D)/rj^E(0) is depicted versus D for the case o f negligible a b s o r p t i o n . E q u a t i o n s 35, 29 a n d 30 are a p p l i e d , l e a d i n g to ( G a u s s i a n spectral

«TL I I I 1 1 1 I

R E F R A C T I V E I N D E X D I S P E R S I O N D

Figure 13 Effective wave-vector mismatch, Ak*«L, in third-harmonic generation of ultrashort light pulses phase-matched at centre frequency versus refractive index dispersion, D = dn³/dv - dn^L/dv (Equation 38). Bandwidth-limited Gaussian pulses are assumed. A^L = v[¹ = 1.054^m. Curves are for the pulse durations (1) Af^L ~ 10ps; (2) 1 ps; (3) 100fs; (4) 10fs. In the case of self-phase modu- lated (chirped) pulses the curves are for (1) Av^L = 0.5 c m^{- 1}; (2) 5 cm"¹; (3) 50 cm"¹; (4) 500 cm"¹. The refractive index dispersion of the solvent HFIP is indicated.