Chemical Physics 142 (1990) 123-131 North-Holland
A B S O R P T I O N A N D E M I S S I O N S P E C T R O S C O P I C I N V E S T I G A T I O N O F C Y A N O V I N Y L D I E T H Y L A N I L I N E D Y E V A P O R S
A . V . D E S H P A N D E l, A . B E I D O U N , A . P E N Z K O F E R
Naturwissenschafiliche Fakultat II-Physik, Universit&t Regensburg, D-8400 Regensburg, FRG
a n d
G . W A G E N B L A S T
BASFAktiengeseltschaft, D-6700 Ludwigshafen, FRG
Received 28 September 1989
The dyes 4-dicyanovinyl-N,N-diethylaniline ( D C V A ) and 4-tricyanovinyl-N,N-diethylaniline ( T C V A ) are investigated spec- troscopically in the vapor phase. The absorption and emission spectra are compared with solution spectra. The thermal stability of the dye powder is determined. The saturated vapor density versus temperature is deduced from absorption measurements. The vapor absorption peaks at 382 nm ( D C V A ) and 437 nm ( T C V A ) . The fluorescence quantum efficiency of both dyes in the vapor phase is approximately 1.5 x 10~4.
1. Introduction
T h e a n i l i n e ( s t y r y l ) dyes 4 - d i c y a n o v i n y l - N , N - d i e - t h y l a n i l i n e ( D C V A , 4 - d i e t h y l a m i n o - p , P - d i c y a n o - styrene [ 1 ], p - d i e t h y l a m i n o b e n z o l m a l o n i t r i l e [ 2 ] ) a n d 4 - t r i c y a n o v i n y l - N , N - d i e t h y l a n i l i n e ( T C V A , 4- diethyiamino-a,p,p-tricyanostyrene [ 1 ] ) belong to the d i p o l a r m e r o p o l y m e t h i n e group [ 3 ]. T h e i r struc- tural formulae are s h o w n i n figs. 4 a n d 5 below. T h e y are used for c o l o r i n g synthetic polymer fibers [ 1 ] a n d are a p p l i e d as s u b l i m a b l e dyes in heat-transfer re- c o r d i n g materials [ 4 , 5 ] a n d i n p h o t o c o n d u c t i v e re- c o r d i n g materials [ 6 ] .
In this paper the usability o f these dyes as gain me- d i a i n dye v a p o r lasers is investigated. F o r this pur- pose t h e r m o d y n a m i c a n d spectroscopic properties o f the dyes are s t u d i e d . A b s o r p t i o n a n d fluorescence spectra o f the dye vapors are d e t e r m i n e d a n d c o m - pared w i t h s o l u t i o n spectra. T h e fluorescence q u a n - t u m efficiency is f o u n d to be very s m a l l ( < 7 * 1 . 3 x l 0 -4 for D C V A , a n d < ? * 1 . 6 x i 0 -4 for
1 On leave from Department of Chemical Technology, Univer- sity of Bombay, Bombay 400 019, India.
0039-6028/90/$ 03.50 © Elsevier Science Publishers B.V.
(North-Holland)
T C V A ) . T h e thermal stability o f the dyes is tested [ 7 ] a n d the saturated v a p o r density versus tempera- ture is deduced from absorption measurements [ 7 - 9 J .
2. E x p e r i m e n t a l
T h e t h e r m a l stability o f the dye stuff is i n v e s t i - gated by heating the dye materials i n an evacuated stainless steel vessel a n d d e t e r m i n i n g the fraction o f undestroyed dye. T h e a b s o r p t i o n spectra o f solutions o f heated a n d unheated dye are c o m p a r e d . T h e tech- nique is described i n ref. (7 ].
T h e spectroscopic measurements are carried out by using a stainless steel v a p o r cell w i t h sapphire w i n - dows (cell length 5 c m ) . D e t a i l s o f the v a p o r cell are given i n ref. [ 8 ] . T h e transmission measurements arc performed i n a c o n v e n t i o n a l spectrophotometer ( B e c k m a n type A c t a M I V ) [ 7,8 ].
T h e absolute absorption cross sections <r( A) o f the vapors are calculated by equating the total S0- S , ab- sorption cross-section integrals o f the vapors a n d the
dye solutions i n cyclohexanc, i.e. it is set i^%a(P)dv^i\b%aL(v)4P [7,9] (i>=A~1 is the wavenumber, aL(P) is the a b s o r p t i o n cross section o f the s o l u t i o n at w a v e n u m b e r P). T h e integrals extend over the S0- S , a b s o r p t i o n bands ( a b s ) . A good agree- ment between the S0~ S i a b s o r p t i o n cross-section i n - tegrals o f dye vapors a n d dye solutions was found i n previous studies on organic dyes [ 8 - 1 1 ] . S o m e rea- soning o f the e q u a l i t y o f the a b s o r p t i o n cross-section integrals is given in ref. [ 1 0 ] .
The saturated v a p o r densities Ns are d e r i v e d from the t r a n s m i s s i o n measurements by
A ^ = - l n [ 7 ( Aa > p) ] / a ( Aa.p) / , ( 0 where Aap is the wavelength o f m a x i m u m S0- S i ab- sorption and / is the v a p o r cell length. Surplus dye is inserted in the dye reservoir to achieve an e q u i l i b - r i u m between the condensed dye and the dye i n the vapor phase.
The fluorescence spectra are measured with a home- made spectrofluorometer [ 1 2 ] , T h e front-face c o l - lection technique is a p p l i e d . A m e r c u r y l a m p ( O s - ram H B O 200 W / 4 ) c o m b i n e d w i t h interference fil- ters serves as e x c i t a t i o n source. T h e fluorescence q u a n t u m efficiencies are obtained by c o m p a r i n g the fluorescence signals o f the vapors w i t h the fluores- cence signals o f standard solutions o f k n o w n q u a n - tum efficiency.
3. Results
3.1 Thermal stability
The thermal stability o f the dyes is d e t e r m i n e d by c o m p a r i n g a b s o r p t i o n spectra o f m e t h a n o l i c solu- tions o f heated a n d u n h e a l e d dye stuff. T h e ratios
• \n[TU))V/minJ l n[ rn h( A . ,p) ] J ^ / m ,n w < n h
( 2 )
are c o m p a r e d . In the d e n o m i n a t o r are the parame- ters o f the nonheated reference sample while i n the numerator are the parameters o f the heated sample.
m ,n w is the mass o f dye inweighted. V is the v o l u m e of the s o l u t i o n . / is the length o f the t r a n s m i s s i o n measurement cell.
Fig. 1 presents examples for D C V A . T h e solid curve is the n o r m a l i z e d a b s o r p t i o n spectrum o f untreated
300 400
WAVELENGTH X Inm)
Fig. 1. Spectral analysis of thermal stability of D C V A powder.
Solid curve, unheated dye. Dashed curve, temperature 2 3 0aC heating period 1.2 h. Dash-dotted curve, temperature 2 3 06C heating period 3.1 h.
dye ( a b s o r p t i o n peak at 435 n m ) . T h e dashed c u r v e shows the spectral changes o f a sample heated to 2 3 0 ° C for 1.2 h. T h e total a b s o r p t i o n integral de- creases, i n d i c a t i n g the d e c o m p o s i t i o n i n v o l a t i l e c o m p o n e n t s . A p r o d u c t a b s o r b i n g at longer wave- length is f o r m e d (peak a b s o r p t i o n a r o u n d 500 n m ) , H e a t i n g the dye to 2 3 0 ° C for 3.1 h (dash-dotted c u r v e ) causes a nearly c o m p l e t e destruction o f the dye.
T h e t h e r m a l d e c o m p o s i t i o n o f T C V A is illustrated i n fig. 2. T h e dashed curve belongs to a heating pe- r i o d o f 2.2 h at 230 ° C a n d the dash-dotted curve is measured for a heating p e r i o d o f 3.1 h at 2 3 0 ° C . A g a i n a c o m p o n e n t absorbing at longer wavelengths is f o r m e d ( a b s o r p t i o n peak a r o u n d 600 n m ) . O n e d e c o m p o s i t i o n c o m p o n e n t has its a b s o r p t i o n peak at 435 n m . T h i s a b s o r p t i o n peak c o i n c i d e s w i t h the ab-
1—'—i—>~
,1 I , I i 1,,, » 1 , t I WAVELENGTH X (nml
Fig. 2. Spectral analysis of thermal stability of T C V A powder.
Solid curve, unheated dye. Dashed curve, 2 3 0 ° C for 2.2 h. Dash- dotted curve, 2 3 0 ° C for 3.1 h.
s o r p t i o n peak o f D C V A , It m a y be that T C V A de- composes partially i n t o D C V A .
In the region o f the S0- S , absorption b a n d o f the untreated dye the a b s o r p t i o n ratios o f the heated dye components are d e c o n v o l v e d roughly a n d the ratio o f the a b s o r p t i o n peak o f undestroyed dye to the ab- sorption peak o f untreated dye determines the frac- t i o n o f undestroyed dye. T h i s fraction is plotted i n fig. 3 versus heating time for different heating tem- peratures. B e l o w 2 0 0 ° C the dyes are very stable. T h e e x p e r i m e n t a l points indicate an over-exponential de- c o m p o s i t i o n process. T h e disintegration products seem to enhance the d e c o m p o s i t i o n rate.
J . 2. A bsorption spectra
T h e absorption spectra o f D C V A v a p o r a n d T C V A v a p o r are shown i n figs. 4 a n d 5, respectively. T h e c o r r e s p o n d i n g spectra o f the dye solutions i n cyclo- hexane are i n c l u d e d . T h e absolute absorption cross sections o f the vapors are d e t e r m i n e d by [ 8 - 1 1 ]
\n[T(9))f.b,ah(i>)dv
J^\n[T(P)]dP '
where the S0- S i absorption integrals o f the vapors are set equal to those o f the solutions ( /l b» ^ L . ( ^ ) Slllnm0L(v) d i > « 6 x l 0 -1 3 c m for D C V A i n c y c l o - hexane a n d Ja b, ( 7L( ? ) <W=.J * i 8 S S f fL( * ) d ? * 4 . 8 x
1 0 ~1 3 c m for T C V A i n cyclohexane; the integration ( 3 )
regions o f the vapors are 4 3 0 - 3 2 5 n m for D C V A a n d 4 9 0 - 3 4 0 n m for T C V A ) . T h e v a p o r spectra are blue shifted (A0.,p *= tfavp - 0«-,p * 2300 c m -1 for D C V A a n d « 2500 c m "1 for T C V A ) . T h e v a p o r spectra are s m o o t h e d c o m p a r e d to the cyclohexane spectra
( t h e r m a l s m o o t h i n g ) .
3.3. Saturated vapor density
T h e saturated v a p o r density, NSL versus v a p o r res- e r v o i r temperature is d i s p l a y e d i n fig. 6. T h e v a p o r cell temperature, d, is kept 1 5 ° C higher than the res- e r v o i r temperature, #f t, to a v o i d dye c o n d e n s a t i o n at the sapphire w i n d o w s . T h e values i n d i c a t e d by cir- cles are o b t a i n e d w h i l e heating u p a n d those i n d i - cated by triangles are measured w h i l e c o o l i n g d o w n . T h e y show the same temperature dependence.
T h e saturated vapor densities, ; VS, m a y be trans- ferred to saturated vapor pressures, pSt b y a p p l i c a - t i o n o f the ideal gas equation,
/?s = i Vs/ cB# , ( 4 )
where /cB is the B o l t z m a n n constant.
T h e saturated v a p o r pressure o r the saturated v a - por density m a y be determined theoretically from the thermal e q u i l i b r i u m c o n d i t i o n where the rate o f de- s o r p t i o n a n d the rate o f a d s o r p t i o n at the condensed dye surface are equal [ 8 , 9 ] . T h e a d s o r p t i o n rate is
[ 8 , 9 , 1 3 - 1 5 ] .
PsS
2 7 I W #R
1/2
( 5 )
where m is the m o l e c u l a r mass a n d S the d y e - d y e s t i c k i n g coefficient. T h e d e s o r p t i o n rate is [8,9,15]
idcs = > W d e s exp ( ( 6 )
where nwrf is the surface n u m b e r density o f dye m o l - ecules ( d i m e n s i o n c m "2) a n d J>def the attempt fre- quency o f desorption from the condensed dye. T h e e x p o n e n t i a l factor accounts for the escape p r o b a b i l - ity. Cde$ is the d y e - d y e d e s o r p t i o n energy.
T h e equality o f a d s o r p t i o n a n d d e s o r p t i o n rate,
" a d ^ d e s , results i n
Fig. 3. Fraction of undcstroyed dye versus heating time for different temperatures.
Fig. 4. Absorption cross-section spectra of D C V A in cyclohexane (solid curve) and D C V A vapor (dashed curve, 160*C). The struc- tural formula of D C V A is included.
E q . ( 7 ) is fitted to the experimental p o i n t s o f fig.
6. F o r b o t h D C V A a n d T C V A the surface n u m b e r density is assumed to be niurf=2x 1.014 c m2 ( m o l e c - ular s i z e w 0 . 5 n m2) . * >d e t/ S a n d Qdet are fitted. T w o sets o f parameters are needed for fitting the experi-
i I i t i—i—|—i—i—i—i—|—i—n—r - T—r i i i — | — i — i i r {1 r r
WAVELENGTH X Inm)
Fig, 5. Absorption cross-section spectra of T C V A in cyclohexane (solid curve) and T G V A vapor (dashed curve, 1 7 0 ° C ) . The struc- tural formula of T C V A is included.
Fig. 6. Saturated vapor densities of D C V A and T C V A versus temperature. For #R < #m, sublimation. For flR > #m, evaporation. The fitted parameters of the curves are given in table 1.
mental points i n the l o w temperature a n d the high temperature region. A t l o w temperatures the dye sub- limates ( e q u i l i b r i u m between s o l i d a n d v a p o r ) w h i l e at high temperatures the dye evaporates ( e q u i l i b -
rium between l i q u i d a n d v a p o r ) . T h e intersection p o i n t o f the curves determines the m e l t i n g point, r3m, o f the dye. T h e d e t e r m i n a t i o n o f the m e l t i n g temper- ature is not very accurate because the curves cross
Table 1
Thermodynamic properties of D C V A and T C V A
Parameter D C V A T C V A
sublimation
(2. (J) 1.82xl0~1 9 2.1 X l 0 ~1 9
1.3 X l O1 2 2.2 X 1 0, J
evaporation
e . ( J ) 1.44X10-1 9 1.72X10"1 9
"S»/5e is'1) 1.4 X l O9 3.1 X l O1 0 melting
C « ( J ) 3.8 X l O "2 0 3.8 X I O "2 0
0 « r C ) 135 ± 1 0 145 ± 1 0
under a s m a l l angle. F o r D C V A a m e l t i n g tempera- ture o f dm = 134.5 ± 0.5 ° C is reported [ 1 ] , A t l o w temperatures <2dC8 is equal to the s u b l i m a t i o n energy Q% (A%xNAQ% is the m o l a r latent heat o f s u b l i m a t i o n
[ 9 ] , NA is the A v o g a d r o c o n s t a n t ) , a n d at high t e m - peratures Qd e i is equal to the e v a p o r a t i o n energy Qe
(At*NAQc is the m o l a r latent heat o f e v a p o r a t i o n ) . T h e m e l t i n g energy is Qm = Qs - & (Am =At-Ac is m o l a r latent heat o f m e l t i n g ) .
T h e fitting parameters are listed i n table 1. T h e surface b i n d i n g energy o f T C V A is slightly higher than that o f D C V A (higher desorption energy, higher ef- fective attempt frequency o f escape yd e s/ 5 , and higher m e l t i n g t e m p e r a t u r e ) . T h e effective s u b l i m a t i o n es- cape frequencies, fd e s/ 5s, are considerably higher than the effective e v a p o r a t i o n escape frequencies,
yd c s/ 5e. In the case o f s u b l i m a t i o n , the i m p i n g i n g
molecules lose energy by inelastic c o l l i s i o n . O n l y i n the case o f total loss o f k i n e t i c energy they stick to the surface. Therefore the s t i c k i n g coefficient is 5S« : 1.
T h e attempt frequency o f escape is the surface v i b r a - t i o n frequency o f a single molecule. In contrast, i n the case o f e v a p o r a t i o n , the i m p i n g i n g molecules penetrate the surface a n d the s t i c k i n g coefficient ap- proaches toward u n i t y ( 5e; S 1). A m o l e c u l e t r y i n g to escape the l i q u i d drags m a n y molecules and there- fore the attempt frequency o f escape becomes the surface v i b r a t i o n frequency o f many molecules w h i c h lowers i>d c $ c o m p a r e d to pd c s.
3.4. Fluorescence analysis
T h e fluorescence q u a n t u m d i s t r i b u t i o n s , £ (A ) , o f D C V A a n d T C V A v a p o r are d e t e r m i n e d by calibrat-
ing the spectral fluorescence d i s t r i b u t i o n s , 5 ( A ) , to the total fluorescence signals, SEMSK(A) dA, o f refer- ence dye solutions o f k n o w n q u a n t u m efficiency 0R
[ 12,13,16 ] ( 5 ( A ) a n d 5R (A) arc given i n photons per wavelength i n t e r v a l ) . T h e reference dye solutions were 9,10-diphenylanthracene i n cyclohexane
( < ?R= : 0 . 9 0 ± 0 . Q 2 [ 1 7 ] , excitation wavelength
Ac x c= 3 6 5 n m ) for D C V A vapor, a n d c o u m a r i n 314
T i n ethanol (</R = 0 . 8 7 ± 0 . 0 7 [ 1 8 ] , Ae R C = 405 n m ) for T C V A vapor.
£ ( A ) is given by [12,13,16]
£ ( A ) =
, SR( A ) d A / H WRA f lR
5 ( A ) (\-RR-TK)Wexe,R
(l~R-T)WeKC Je m5R( A ) d A / iF.R QR ( 8 ) T h e absorbed excitation energies in the dye vapor a n d the reference dye s o l u t i o n are
K .b s = H ;xc( l - * - r ) a n d
w;b,,R = H/ e x c < R( i - . / eR^ rR) ,
r e s p e c t i v e l y . / ? is the reflectivity a n d 7* is the trans- m i s s i o n at Ae x c. T h e ratio o f the s o l i d angles o f fluo- rescence acceptance by the detector is
^ / A f l R = « F , R / " F * « F , R .
/ iF t R- is the average refractive index o f the reference
s o l u t i o n at the wavelength o f the fluorescence b a n d . nF« 1 is the refractive index o f the dye v a p o r i n the e m i s s i o n region. T h e spectral characteristics o f the elements i n the fluorescence path are taken i n t o c o n - s i d e r a t i o n i n the d e t e r m i n a t i o n o f the fluorescence signals 5 ( A ) a n d 5R( A ) ( c a l i b r a t i o n w i t h a halogen tungsten l a m p o f k n o w n c o l o u r temperature, for de- tails seercf. 1 1 6 ] ) .
T h e fluorescence q u a n t u m d i s t r i b u t i o n s £ ( A ) o f D C V A a n d T C V A v a p o r are d e p i c t e d i n fig. 7. T h e fluorescence signal is very weak reaching the detec- t i o n l i m i t o f o u r home-made spectrofluorimeter.
Therefore the shapes o f £ ( A ) are not very accurate.
T h e fluorescence q u a n t u m efficiencies q o f the v a - pors are o b t a i n e d by
g» | £ ( A ) d A . ( 9 )
T T—|—t r i i — | — i—T—i — i— | —r
WAVELENGTH X [nm]
Fig. 7. Fluorescence quantum distributions of D C V A vapor (solid curve, 1 5 0 ° C ) and T C V A vapor (dashed curve, d « 2 1 0 ° C ) .
T h e fluorescence q u a n t u m efficiencies are a p p r o x i - mately 1.3X 1 0 ~4 for D C V A v a p o r a n d
L 6 x l O -4f o r T C V A vapor.
T h e radiative lifetimes, rr a d, o f the vapors are cal- culated using the S t r i c k l e r - B e r g f o r m u l a [ 1 9 , 2 0 ] w h i c h exploits the fundamental relations between the t r a n s i t i o n p r o b a b i l i t i e s o f a b s o r p t i o n , s t i m u l a t e d e m i s s i o n , a n d spontaneous e m i s s i o n [ 2 1 ] . T h e S t r i c k l e r - B e r g f o r m u l a reads
Trad «A Jc mi i ( A ) / l3c U J A abs
where nF* I a n d A Ia« 1 are the refractive indices o f the vapors i n the S0- S i e m i s s i o n a n d absorption re- gion, respectively. c0 is the v a c u u m light velocity. T h e calculated r a d i a t i v e lifetimes are rr a d( D C V A ) « 7.2 ns a n d Tr a d( T C V A ) » 8 . 1 ns. T h e calculated fluores- cence lifetimes, TF= # Tr a d, o f the vapors are rF( D C V A ) * 0 . 9 ps a n d tF( T C V A ) « 1.3 ps.
T h e s t i m u l a t e d e m i s s i o n cross-section spectrum is o b t a i n e d from the E i n s t e i n A and B coefficients
[ 2 2 , 2 3 ] . A short d e r i v a t i o n is given i n the a p p e n d i x . T h e result is
<7emU)= Qnl*r^r a' <8 K «FC0Tr a d^ M>
In fig. 8 the a b s o r p t i o n cross-section spectra, cr(A), a n d the e m i s s i o n cross-section spectra aem(X) o f the D C V A a n d T C V A vapors are d i s p l a y e d . T h e atm spectra are very crude because E(X) is k n o w n o n l y very inaccurately.
4. C o n c l u s i o n s
In this paper t h e r m o d y n a m i c a n d spectroscopic parameters o f the styryl dyes ( a n i l i n e dyes) D C V A and T C V A have been d e t e r m i n e d . T h e dyes are ther- mally stable up to 2 0 0 ° C . T h e saturated v a p o r den- sities at 200 ° C are rather high c o r r e s p o n d i n g to v a - p o r densities o f N$(DCVA)» 1 . 4 x 1 01 6 c m "3 a n d M s ( T C V A ) * 4 . 5 x 1 0 " c m " *3. T h e fluorescence q u a n t u m efficiency is very l o w ( < 7 ( D C V A ) « 1.3X
1 0 ~4, < ? ( T C V A ) » 1.6X 1 0 -4) . T h i s fact hinders the a p p l i c a t i o n o f the dyes as gain m e d i a i n dye vapor lasers. I f the excited-state absorption o f the dye v a - pors is weak for excitation wavelengths i n the S0- S i absorption region, the dye vapors m a y be a p p l i e d as fast saturable absorbers [24] for dye lasers i n the blue a n d violet spectral region ( 3 5 0 g A ^ 3 9 0 n m for D C V A vapor, a n d 400 <>X < 450 n m for T C V A v a - p o r ) . U l t r a s h o r t pulse generation by a m p l i f i e d spon-
WAVELENGTH \ (nml
Fig. 8. Absorption cross section, a, and emission cross section,
<reim spectra of D C V A vapor (solid) and T C V A vapor (dashed).
taneous e m i s s i o n [ 2 5 , 2 6 ] m a y b e possible by femto- second pulse e x c i t a t i o n [27 ] .
Acknowledgement
T h e authors thank D r , J . S c h m i d t for discussions and e x p e r i m e n t a l help. T h e y thank the D e u t s c h e Forschungsgemeinschaft for f i n a n c i a l support.
A p p e n d i x . R e l a t i o n between stimulated emission cross-section spectrum and radiative lifetime
T h e rate o f spontaneous e m i s s i o n x ~d is given b y the E i n s t e i n A coefficient a n d is related to the E i n - stein B coefficient by [ 2 1 , 2 2 , 2 8 ]
f3 Cg
( A . l )
C = C0/ A I F is the light velocity (c0 is the v a c u u m light v e l o c i t y , nF is the average refractive index i n the e m i s s i o n r e g i o n ) . T h e rate o f s t i m u l a t e d e m i s s i o n i n a m e d i u m w i t h spectral photon density nph(v) ( c m " "3
s "1) a n d excited-state level p o p u l a t i o n n u m b e r den- sity N ( c m "3) is
dnph(p)
d/ = Bu{v)NE'(v) ( A . 2 ) u (v) = rtph ( v) h v ( J c m 3 s ~1) is the spectral energy density o f the r a d i a t i o n field. E'(v) is the n o r m a l - i z e d spectral fluorescence q u a n t u m d i s t r i b u t i o n f u n c t i o n , i.e. E'{v)^E{v)/q a n d jemE'(v) di>= 1.
q-JtmE(v) dv is the fluorescence q u a n t u m effi- ciency. I n t r o d u c i n g the spectral light i n t e n s i t y I(v) =ZU(P)C=U(P)C0/nF^nph(i>)hpc0/nF,zq. ( A . 2 ) changes to
dl(p) _ hvc0
dt nFq BU(P)NE(P)
= ~BI(P)NE(P) . ( A . 3 )
T h e spatial change o f the spectral intensity I(v) ( W c m ~2 s ~1) is
dl(p) nFdI(p) hvnF
————- s= - — — ns
dz c0 dt c^q
= atm{p)I(p)N.
BE(P)1(P)N
( A . 4 ) E q . ( A . 4 ) defines the s t i m u l a t e d e m i s s i o n cross sec*
t i o n t o
, . hpnF n —, v
Coq cl
&np2nFqTrmii
E{v) ( A . 5 )
T h e last equality is o b t a i n e d by i n s e r t i o n o f eq. ( A . l ) i n t o eq. ( A . 5 ) . I n terms o f the v a c u u m wavelength, A = c0/ v , e q . ( A . 5 ) reads
c re m( / 0 = ae m(*>) =
inn2FcQqxrmd
E(X), ( A . 6 )
where the r e l a t i o n E(v)=E(X) dX/dP=E{X)X2/c0
has been used.
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