HAL Id: jpa-00223176
https://hal.archives-ouvertes.fr/jpa-00223176
Submitted on 1 Jan 1983
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
GENERALIZED SATURATION CRITERIA FOR PHOTOTHERMAL MEASUREMENTS
L. Aamodt, J. Murphy
To cite this version:
L. Aamodt, J. Murphy. GENERALIZED SATURATION CRITERIA FOR PHOTOTHER- MAL MEASUREMENTS. Journal de Physique Colloques, 1983, 44 (C6), pp.C6-115-C6-119.
�10.1051/jphyscol:1983617�. �jpa-00223176�
JOURNAL DE PHYSIQUE
ColIoque C 6 , suppl6ment au nO1O, Tome 44, octobre 1983 page C 6 - 115
G E N E R A L I Z E D SATURATION CRITERIA FOR PHOTOTHERMAL MEASUREMENTS
L.C. Aamodt and J . C . Murphy
Johns Hopkins U n i v e r s i t y , Applied Physics Laboratory, Johns Hopkins Road, Laurel, Maryland 20707, U.S.A.
, ,
Resume- 03 g & n & r a l i s e 1$ concept de s a t u r a t i o n photothermique l a
d e t e c t i o n p a r d e f l e x i o n photothermique e t
2
1 ' e x c i t a ; i o n 1ocalis;e.On t r a i t e l a s a t u r a t i o n photoacoustique comme cas s p e c i a l .
Abstract- The concept of photothermal s a t u r a t i o n i s g e n e r a l i z e d t o i n c l u d e photothermal d e f l e c t i o n d e t e c t i o n and l o c a l i z e d e x c i t a t i o n . P h o t o a c o u s t i c s a t u r a t i o n i s t r e a t e d a s a s p e c i a l case.
The p h o t o a c o u s t i c s i g n a l becomes i n c r e a s i n g l y i n s e n s i t i v e t o changes i n t h e o p t i c a l a b s o r p t i o n c o e f f i c i e n t a s o p t i c a l a b s o r p t i o n i n c r e a s e s . This e f f e c t
--
c a l l e d photoacoustic s a t u r a t i o n o r , a t t i m e s , o p t i c a l s a t u r a t i o n -- a l s o depends upon t h e thermal p r o p e r t i e s of t h e sample. I n a thermally heterogeneous m a t e r i a l , i t i s p o s s i b l e f o r t h e degree of o p t i c a l s a t u r a t i o n t o vary from point t o p o i n t on t h e sample s u r f a c e because of p o s i t i o n a l changes i n t h e thermal d i f f u s i o n l e n g t h even when o p t i c a l a b s o r p t i o n remains i n v a r i a n t . For example, i n p h o t o a c o u s t i c imaging, i f t h e e x c i t a t i o n beam i s scanned o v e r a region where t h e o p t i c a l a b s o r p t i o n c o e f f i c i e n t i s e s s e n t i a l l y c o n s t a n t ( b u t does vary somewhat), s e n s i t i v i t y t o t h e s e small v a r i a t i o n s i n a b s o r p t i o n can change p o i n t by p o i n t because of s p a t i a l thermal v a r i a t i o n s .
I n photoacoustic measurements, t h e degree of o p t i c a l s a t u r a t i o n i s determined by t h e r a t i o 6 / ~ where 6 i s t h e thermal d i f f u s i o n l e n g t h and p i s t h e o p t i c a l a b s o r p t i o n l e n g t h . When 6 / ~ > > 1 t h e s i g n a l s a t u r a t e s , otherwise it i s e i t h e r p a r t i a l - l y s a t u r a t e d 6 = 1
,
o r t o t a l l y u n s a t u r a t e d ( 6 / p < < 1 ) . When a t h e r m a l l y homogeneous sample i s used, and one-dimensional a n a l y s i s i s permitted by sample geometry, t h i s simple c r i t e r i o n i s adequate.This dependence of s a t u r a t i o n on j u s t two parameters i s a consequence of t h e thermo-acoustic coupling mechanism i n p h o t o a c o u s t i c c e l l s where t h e a c o u s t i c wavelength i n t h e c e l l f l u i d i s much l a r g e r than a f l u i d thermal d i f f u s i o n l e n g t h a t t y p i c a l photoacoustic modulation f r e q u e n c i e s . Under t h e s e circumstances, t h e photoacoustic s i g n a l i s p r o p o r t i o n a l t o t h e sample s u r f a c e temperature i n t e g r a t e d o v e r a d i s t a n c e comparable t o a n a c o u s t i c wavelength. Thus photoacoustic d e t e c t i o n cannot observe s p a t i a l v a r i a t i o n s i n t h e s u r f a c e temperature s i n c e t y p i c a l a c o u s t i c wavelengths a r e much l a r g e r t h a n t h e sample dimensions. From the viewpoint of s a t u r a t i o n , t h i s means t h a t t h e p h o t o a c o u s t i c s i g n a l i s i n s e n s i t i v e t o t h e shape of t h e temperature p r o f i l e on t h e sample s u r f a c e . Any t r a n s v e r s e h e a t flow i n t h e sample which merely r e d i s t r i b u t e s t h e s u r f a c e temperature l e a v i n g t h e average s u r f a c e temperature c o n s t a n t has no e f f e c t on t h e p h o t o a c o u s t i c s i g n a l . A s a consequence, t h e p h o t o a c o u s t i c s i g n a l depends only upon how deep i n t h e sample h e a t i s generated ( a s measured by p ) , and how f a r h e a t can t r a v e l i n a h a l f cycle of t h e modulation frequency ( a s measured by 6). When h e a t generated i n t h e sample can reach t h e s u r f a c e i n a h a l f c y c l e , t h e sample i s s a t u r a t e d .
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983617
JOURNAL DE PHYSIQUE
The p h o t o a c o u s t i c s i g n a l ' s i n s e n s i t i v i t y t o t h e s u r f a c e temperature p r o f i l e i s d i r e c t l y r e l a t e d t o i t s i n s e n s i t i v i t y t o heat flow p a t t e r n s i n t h e sample a s long a s t h i s flow does not a l t e r the average temperature on t h e sample s u r f a c e . When l o c a l i z e d d e t e c t i o n i s used, t h e photothermal s i g n a l monitors t h e s u r f a c e temperature only a t c e r t a i n p o i n t s , l i n e s , o r a r e a s on t h e sample s u r f a c e and t h u s i s markedly a f f e c t e d by any h e a t flow t h a t moves h e a t i n t o o r away from t h e s e p o s t i o n s . I n OBD d e t e c t i o n , f o r example, t h e s i g n a l i s p r o p o r t i o n a l t o t h e i n t e g r a t e d sample temperature along t h e p r o j e c t i o n of t h e probe beam on t h e sample s u r f a c e . Thus OBD d e t e c t i o n i s s e n s i t i v e t o any h e a t flow p a t t e r n t h a t moves h e a t toward o r away from t h i s p r o j e c t i o n .
When t h e concept of o p t i c a l s a t u r a t i o n i s a p p l i e d t o o t h e r photothermal measurements, i t must be g e n e r a l i z e d , and t h e c r i t e r i o n f o r s a t u r a t i o n must be modified. When l o c a l i z e d d e t e c t i o n i s used, o p t i c a l s a t u r a t i o n no l o n g e r depends only on t h e two l e n g t h s , and 6, but i t a l s o depends upon t h e r a d i u s of t h e e x c i t a t i o n beam ( R ) , and t h e two r a t i o s , 6/p, and 6/R, must be c o n s i d e r e d .
Figure 1. ( a ) Photothermal s a t u r a t i o n curve f o r PA d e t e c t i o n . The t a n g e n t s U , P , and S show t h e slope under u n s a t u r a t e d , p a r t i a l l y s a t u r a t e d , and s a t u r a t e d c o n d i t i o n s , r e s p e c t i v e l y . ( b ) o and 1-0 p l o t t e d vs l o g 6.
Photoacoustic s a t u r a t i o n can be i l l u s t r a t e d by p l o t t i n g t h e p h o t o a c o u s t i c s i g n a l , S, vs. 5, ( t h e o p t i c a l a b s o r p t i o n c o e f f i c i e n t ) ( f i g u r e 1 ) . We use t h i s r e l a t i o n s h i p t o g e n e r a l i z e and q u a n t i f y t h e concept of s a t u r a t i o n . I n f i g u r e 1 , t h e s l o p e equals 1 when t h e sample i s u n s a t u r a t e d (S p r o p o r t i o n a l t o 5 ) , and e q u a l s z e r o i n t h e region where t h e s i g n a l i s s a t u r a t e d ( S independent of
$1-
We d e f i n e a s a t u r a t i o n c o e f f i c i e n t ,which i s , p h y s i c a l l y , one minus t h e s l o p e of t h e s a t u r a t i o n curve p l o t t e d using a log-log s c a l e . Thus, a = 1 i n d i c a t e s t o t a l s a t u r a t i o n w h i l e o = 0 i n d i c a t e s no s a t u r a t i o n . Although derived s p e c i f i c a l l y f o r photoacoustic d e t e c t i o n , e q u a t i o n 1 can be adopted a s t h e g e n e r a l d e f i n i t i o n of s a t u r a t i o n f o r a l l modes of photothermal d e t e c t i o n .
I n t h i s paper, we apply t h i s d e f i n i t i o n s p e c i f i c a l l y t o s a t u r a t i o n e f f e c t s i n a t h e r m a l l y t h i c k sample u s i n g PAS and OBD modes of d e t e c t i o n . The r e s u l t s obtained a r e a l s o r e l a t e d t o t h e thermal dependence of t h e s e modes of d e t e c t i o n . [We c o n s i d e r only t h e t r a n s v e r s e o p t i c a l beam d e f l e c t i o n component, and s p e c i f i c a l l y c o n s i d e r t h e s l o p e of t h i s component with r e s p e c t t o t r a n s v e r s e o f f s e t when t h e axes of t h e s e two beams a r e c o i n c i d e n t . This v a l u e i s t h e o r e t i c a l l y convenient t o use and i s e a s i l y measured e x p e r i m e n t a l l y . ]
I f we c o n s i d e r a thermally t h i c k sample i l l u m i n a t e d by an e x c i t a t i o n source having a g a u s s i a n energy d i s t r i b u t i o n , t h e p h o t o a c o u s t i c and o p t i c a l beam d e f l e c t i o n s i g n a l s can be w r i t t e n ,
where To i s t h e t o t a l i n t e n s i t y of t h e e x c i t a t i o n s o u r c e , w i s t h e a n g u l a r modulation frequency, K i s t h e sample thermal c o n d u c t i v i t y , C i s t h e sample thermal c a p a c i t y per u n i t volume, and v i s t h e s p a t i a l frequency. Here M and M' a r e m u l t i p l i c a t i v e c o n s t a n t s t h a t a r e independent of o p t i c a l and thermal p r o p e r t i e s of t h e sample and modulation frequency. t i s t h e photothermal response of t h e photoacoustic and photothermal systems.
By applying equation 1 t o e q u a t i o n 2 and 3 and i n t e g r a t i n g numerically, t h e v a l u e of a can be o b t a i n e d a s a f u n c t i o n of sample o p t i c a l and thermal parameters. Figure 2 shows a t o p o g r a p h i c a l p l o t t i n g of constant-a l i n e s i n
6 / R , BR(=R/p) space. A comparison of f i g u r e 2a and f i g u r e 2b shows t h a t f o r 6 / R < 1 , t h e o p t i c a l beam d e f l e c t i o n and p h o t o a c o u s t i c s i g n a l s have t h e same s a t u r a t i o n c h a r a c t e r i s t i c s , but a s t h e sample thermal d i f f u s i o n l e n g t h i n c r e a s e s r e l a t i v e t o t h e beam r a d i u s , t h e s a t u r a t i o n c h a r a c t e r i s t i c s d i f f e r . I n PAS d e t e c t i o n ( o r OBD d e t e c t i o n when 6 / R i s s m a l l ) , t h e sample can be s a t u r a t e d by e i t h e r i n c r e a s i n g B ( w i t h 6 c o n s t a n t ) o r by i n c r e a s i n g 6 ( w i t h B c o n s t a n t ) . (This i s e q u i v a l e n t t o moving along v e r t i c a l and h o r i z o n t a l l i n e s , r e s p e c t i v e l y , i n f i g u r e 2.) This i s not t h e case f o r OBD d e t e c t i o n when 6 / R i s l a r g e
.
Undert h e s e c o n d i t i o n s , s a t u r a t i o n becomes independent of 6 , a s i l l u s t r a t e d by t h e h o r i z o n t a l s l o p e of t h e constant-o l i n e s i n t h i s region. For OBD d e t e c t i o n t h e PAS c r i t e r i o n f o r s a t u r a t i o n i s s t i l l v a l i d i f t h e thermal d i f f u s i o n l e n g t h s i s small compared with t h e e x c i t a t i o n beam r a d i u s , but when t h e r a t i o , 6 / R i s l a r g e , s a t u r a t i o n become independent of thermal p r o p e r t i e s and t h e c r i t e r i o n f o r s a t u r a t i o n becomes R / p = BR > > I .
The t r a n s i t i o n between s a t u r a t i o n c r i t e r i a i s d i r e c t l y r e l a t e d t o h e a t flow p a t t e r n s w i t h i n t h e sample. When 6 i s l a r g e compared w i t h R , h e a t i s e a s i l y c a r r i e d away from t h e beam c e n t e r
--
t h e region monitored by t h e probe beam. The a b i l i t y t o r e a d i l y t r a n s f e r h e a t from t h i s region depends not only upon t h e thermal c o n d u c t i v i t y K , i n t h e sample, but a l s o upon how deeply i n t h e sample h e a t i s generated ( i . e . , upon 11). Competition between t r a n s v e r s e and normal h e a t flow i n t h e sample i s evident i n f i g u r e 2.C6-118 JOURNAL DE PHYSIQUE
O p t i c a l s a t u r a t i o n i s c l o s e l y r e l a t e d t o t h e s e n s i t i v i t y of t h e photoacoustic and photothermal s i g n a l s t o small changes i n thermal parameters. We d e f i n e t h e
"thermal c h a r a c t e r " (TC) of t h e sample a s
TC monitors t h e s e n s i t i v i t y of S t o small changes i n thermal parameters a s a does t o changes i n o p t i c a l parameters. While t h i s term appears t o be r e l a t e d only t o K , i t a l s o provides a measure of t h e s e n s i t i v i t y of S t o C , 6 , and t o t h e thermal admittance [defined a s Y = J ( u K c / ~ ) ] .
-
Unsaturated
PAS Thermal character
6 / R S I R
Saturated 95%
Unsaturated
FIG. 2 Topographical map of a . Constant-0 contour l i n e s s e p a r a t e s a t u r a t e d from u n s a t u r a t e d regions.
1
= 0.9
OBD Thermal character
I
FIG. 3 Topographical map of TC.
Constant-TC contour l i n e s s e p a r a t e a r e a s where S i s e s s e n t i a l l y c o n t r o l l e d by a s i n g l e thermal v a r i a b l e .
The r e s u l t obtained by applying e q u a t i o n 4 t o e q u a t i o n s 2 and 3 , is shown i n f i g u r e 3 . For PAS d e t e c t i o n , l i n e s of c o n s t a n t thermal c h a r a c t e r d i v i d e t h e space i n t o two d i s t i n c t r e g i o n s , one which depends , e x c l u s i v e l y on thermal c a p a c i t a n c e , and t h e o t h e r on thermal admittance ( i . e . , on t h e product, KC). For no combination of thermal and o p t i c a l parameters i s t h e r e an e x c l u s i v e dependance on
thermal c o n d u c t i v i t y . By c o n t r a s t , f o r OBD d e t e c t i o n , c o n s t a n t thermal c h a r a c t e r l i n e s s e p a r a t e t h e space i n t o t h r e e s e p a r a t e r e g i o n s , one which i s e x e l u s i v e l y dependent on C , one on Y, and one on K . A comparison of f i g u r e s 2 and 3 show t h a t f o r PAS d e t e c t i o n , an u n s a t u r a t e d s i g n a l corresponds t o a thermal dLpendence on C alone while f o r a s a t u r a t e d sample t h e dependence i s on t h e product, KC.
For OBD d e t e c t i o n , t h e thermal dependence d i f f e r s depending upon t h e r a t i o , 6 / R . For small thermal d i f f u s i o n l e n g t h s ( r e l a t i v e t o beam r a d i u s ) , OBD d e t e c t i o n has t h e same thermal dependence a s PAS d e t e c t i o n . This i s understandable s i n c e h e a t flow i s s t i l l r e s t r i c t e d t o a region c l o s e t o t h e e x c i t a t i o n source which produces a s i t u a t i o n somewhat comparable t o t h e one-dimensional c a s e which governs PAS d e t e c t i o n ( i . e . , t r a n s v e r s e h e a t flow i s not s i g n i f i c a n t l y competing with normal h e a t flow). As 6 becomes l a r g e r and h e a t r e a d i l y flows away from t h e p o i n t of g e n e r a t i o n , t h e s i g n a l becomes independent of thermal parameters, s a t u r a t i o n a l s o changes, and t h e photothermal s i g n a l i s dominated by thermal c o n d u c t i v i t y .
The r e l a t i o n s h i p between thermal c h a r a c t e r and o p t i c a l s a t u r a t i o n can be made more s p e c i f i c by considering incremental changes i n S a s a l o c a l i z e d e x c i t a t i o n s o u r c e i s scanned over a heterogeneous sample whose sample o p t i c a l and thermal parameters vary s u f f i c i e n t l y slowly, s p a t i a l l y , s o t h a t t h e sample can be considered t o be l o c a l l y homogeneous. I f dx i s an incremental d i s t a n c e along t h e scan p a t h ,
where X i s t h e o p t i c a l wavelength of t h e e x c i t a t i o n source. I f we d e f i n e ,
where E i s any thermal, o p t i c a l , o r experimental v a r i a b l e a f f e c t i n g S ,
The PE a r e r e l a t e d through e q u a t i o n 2 and 3 , and a r e a l s o r e l a t e d t o both o and TC. For PAS d e t e c t i o n , o = 2TC = 3
+
2 PW, while f o r OBD d e t e c t i o n ,o = 2 T C - P - 1
R = 2 P
+
1-
P (Note t h a t o and TC a r e p r o p o r t i o n a l i n photoacoustic d e t e c t i o n , b u t W not i n %BD d e t e c t i o n . )Since P can be determined e x p e r i m e n t a l l y ( i f i n s t r u m e n t a l and sample dependent W frequency e f f e c t s can be s e p a r a t e d ) , t h e v a l u e of TC and o a r e e x p e r i m e n t a l l y determinable p o i n t by p o i n t on t h e sample s u r f a c e when PAS d e t e c t i o n i s used. Since P i s not e a s i l y determined e x p e r i m e n t a l l y R
--
changingof t h e beam r a d i u s while r e t a i n i n g a g a u s s i a n energy d i s t r i b u t i o n i n t h e beam i s d i f f i c u l t
--
t h e r e l a t i o n s h i p between a and TC i s not a s e a s i l y obtained f o r OBD d e t e c t i o n .I n summary, photothermal o p t i c a l s a t u r a t i o n i s detection-mode dependent, and does n o t , i n g e n e r a l , have a simple dependence on t h e r a t i o , 6 / p . In l o c a l i z e d d e t e c t i o n , t h e e x c i t a t i o n beam s i z e a l s o a f f e c t s t h e degree of o p t i c a l s a t u r a t i o n , and heat flow p a t t e r n s i n t h e sample a r e important. O p t i c a l s a t u r a t i o n and t h e s e n s i t i v i t y of t h e photothermal s i g n a l t o changes i n thermal parameters a r e a l s o r e l a t e d .
