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Focal properties of liquid films deformed by heating with a Gaussian laser beam
Volume 73, number 1
OPTICS C O M M U N I C A T I O N S
1 September 1989
FOCAL P R O P E R T I E S OF L I Q U I D F I L M S D E F O R M E D BY H E A T I N G W I T H A GAUSSIAN LASER BEAM G. DA COSTA ~ and R. ESCALONA Laboratoire d'Optique P.M. Duffieux, Facultb des Sciences et des Techniques, Route de Gray. 25030 Besan¢on Cedex. France Received 13 February 1989; revised manuscript received 11 May 1989
The focal properties of liquid films of oils deformed by heating with a gaussian laser beam are numerically studied. A new method to measure the refractive index of oils is developed. The method is based on far-field properties of the caustics of the laser beams reflected and transmitted by the liquid film.
1. Introduction Temperature dependence of density and surface tension is known to be responsible of surface deformation in inhomogeneously heated liquid film [ 1,2 ]. The particular case corresponding to heating with a gaussian laser beam was studied in several papers [ 3,4 ]. Interferometric records [ 5 ] show that the free liquid surface undergoes dilatation in the early heating stage, followed by contraction for longer heating time. The liquid surface profile as a function of time is given by [ 6 ] y=yo[w3/4-C(w
- 1 ) ]1/2 ,
( 1)
where
w=l/[l+Df(fl, 7)] ,4
f(/~, y) =
] ,Sz/( i + y)
(2)
film surface, which results in turn in a modification of the structure of the reflected and transmitted beams. This is an example of self-interaction of a laser beam at low power level. A geometrical approach to the study of light intensity distribution in the reflected and transmitted beams is presented in the next section.
2. Properties of light rays reflected and transmitted by the laser-heated liquid film The equations of the reflected and transmitted rays (r, t) (fig. I ) corresponding to the incident ray (i) are respectively -1 - Yr -- 2y' (x)
(4)
(xr-x),
du
ue""
(3)
In these equations, Yo is the initial film thickness,
fl=x/a, and 7= t/to are respectively the dimensionless horizontal coordinate and heating time, x and a are defined in fig. 1 and to is a time constant defined in ref. [6]. Finally, C and D are dimensionless parameters containing the thermohydrodynamic properties of the liquid sample [6]. The incident laser beam (fig. 1 ) produces a deformation of the liquid On Sabbatical Year leave from: Departamento de Fisica, Universidad Simon Bolivar, A.P. 89000, Caracas, Venezuela.
y, =
-1
(n-1)y'(x)
(x,-x).
(5)
In these equations the refractive index n is assumed to be a real number, (x, y(x)) are the coordinates of surface point S and (xr, Yr), (x,, y, ) are the coordinates of current points of rays (r, t) respectively. As the equation y(x) of the surface profile is given at any time instant by eqs. ( 1 ) - ( 3 ), eqs. (4), (5) represent the families of reflected and transmitted rays with parameter x. In the geometric optics approximation light intensity is a maximum in the neighborhood of the corresponding envelopes,
0 030-4018/89/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 73, number I
OPTICS COMMUNICATIONS
Y
I
L \ \
O
\
i • . '.."
\ .°.
.
.\".
! i:. i x e
O
\ Fig. 1. A liquid sample lying over a transparent support of width e is heated by a gaussian laser beam (diameter 2a at the inflexion points ofthe power distribution ). Oxy is a coordinate system, ( i, r, t ) are light rays respectively incident, reflected and transmitted at the surface point S. (Er, E,) are the envelopes of the families of reflected and transmitted rays. (R, T) are the contact points of rays ( r, t ) with the envelopes (E, E, ).
1 September 1989
ing to the same surface point S have the same abscissa (for XR(X)=XT(X)); ( b ) caustics (Er, E~) are identical except for a vertical scale change (for the ratio yR(x)/yv(x) is constant and equal to ( n - 1 ) / 2). The main difference between them is that real caustics o f reflected rays are virtual caustics o f transmitted rays and vice versa. (c) The asymptotic branches o f the caustics correspond to rays deflected at the neighborhood o f inflexion points o f the surface profile, where y" ( x ) = 0 . In that neighborhood the surface profile behaves as a linear mirror and light rays are deflected as a parallel beam. ( d ) Regression points of caustics correspond to light rays deflected at the neighborhood o f surface profile points where y" ( x ) is stationary, that is, where y" (x) = 0 . As the curvature radius is R~-I/y"(x), we conclude that the surface profile locally behaves as a circular mirror, thus focusing the deflected rays at a regression point. Considerations ( a - d ) are well known results of geometric optics [ 8 ]. Their validity is not limited to liquid films studied in the present paper. Taking profit o f items (a, b ) only the envelopes o f reflected rays are studied in what follows. The coordinates (XR, YR ) are numerically calculated for different values o f the p a r a m e t e r x and time t by substitution o f y(x) (eqs. ( 1 ) - ( 3 ) ) i n t o eqs. ( 6 ) , ( 7 ) . The resulting graphs are represented in figs. 2-5, where scales are not respected in order to emphasize the local pecul-
whose equations are calculated by s t a n d a r d m e t h o d s o f d i f f e r e n t i a i g e o m e t r y [7],
XR(X)=X--y'(x)/y"(X),
(6a)
yR(X)= i/2y"(X),
(6b)
Xr ( X ) = X - - y ' ( x ) / y " ( X ) ,
(7a)
)~-(X)= l / ( n - - I ) y " ( x ) .
(7b)
6~
These equations (as well as eqs. (4, 5 ) ) are valid with the hypothesis: lY' ( x ) l << 1 (small surface def o r m a t i o n ) and I I / y " ( x ) l >>e ( c u r v a t u r e radius o f the surface profile much greater than the support thickness). In eqs. ( 6 ) , ( 7 ) , (XR, YR) and (XT, Yr) are the coordinates o f the contact points (R, T ) o f each ray with its envelope. C o m p a r i s o n o f these equations shows that: ( a ) points (R, T ) correspond-
n
~1-
2
Fig. 2. Surface profile and its corresponding caustics in the early heating stage•
Volume 73, number 1
OPTICS COMMUNICATIONS
iarities of the surface profile and its caustics. Fig. 6 shows in detail the local structure of the central caustics of fig. 3. Each caustic branch is assigned with the same number as the corresponding region of the surface profile. Caustics branches are represented as the borders o f shadowed regions. In the early heating stage the surface profile is dilated as a result of diminution o f the liquid density. In fig. 2, points (1) correspond to light rays reflected far from the symmetry axis. Points (2, 6) are stationary points o f the curvature radius of the surface profile, and give rise to regression points (focal points) in the caustics. Points (3, 5) of the surface profile are inflexion points and give rise to asymptotic caustic branches. Also at the central point (4) the curvature radius is stationary. The real caustic branches are (1, 2, 3) and (5, 6, 1 ), while (3, 4, 5) is virtual. At a certain time instant tf (fig. 3) the topological structure of the virtual caustic abruptly changes. The cusp vertex (4) o f fig. 2 is split into three vertex (4, 5, 6). Vertex (5) corresponds to the central surface point, while vertex (4, 6) are two new focal points proceeding from surface points (4, 6) where the curvature radius is stationary. For heating times t > tf the cusp vertex (5) travels down to infinite. At a certain time instant t~ (fig. 4) the dilatation stage of the surface is finished. From now on the surface tension gradient provokes the
,,';i,~
1 September 1989
iiI & 2
.i~",'7
.
", i "
•
I,. L,'
:
•
i, ~--
2
8--
:.-_Q
-
4q
7/
'6l il....... 3 5
Fig. 4. At a later time instant t~>tc a pit is formed in the top of the dilated surface. At this precise time instant the cusp vertex (5) is at infinite.
_~\\\~illl 1
7 ~,~lill i|ii////~/5
"16//ff/~"9
V i I!.,,,~6 \,'il L
".I \i
I ~III~ I I I I i .
.
.
l
.
.
.
I''/l~eC'
!/" : IIIII~-III,~ ........
.
J
:L::..
',:Z.','~z~~',,
,,,'~':
5~ C
~2
"
I"
C'
43
8P 9
....
/ I
2--
I
'
----8 ....
4' I "6
Fig. 3. At a certain time instant tr the central vertex of fig. 2 is split into three vertex (4. 5, 6 in the above graphic ).
7
Fig. 5. In the final heating stage (t> to) a new real caustic branch is formed above the central region of the liquid surface. The cusp vertex (6) is a real focus which travels down from infinite for t> to. The asymptotic branches ( 5, 7) proceed from the inflexion points (5.7) of the surface profile. formation of a pit in the top of the central region. At this precise time instant ( t = t c ) the cusp vertex (5) is at infinite. The central pit works as an aspherical mirror. As a result, in the final heating stage (fig. 5) a new real caustic branch is formed in the central region above
Volume 73, number 1
OPTICS COMMUNICATIONS
the liquid surface. Cusp vertex ( 6 ) is a real focus which is placed at infinite at t=tc and travels down for t > to. The a s y m p t o t i c caustic branches (5, 7) proceed from the new inflexion points (5, 7) o f the surface profile. It is noted that, contrarily to intuition, surface points (C, C' ) where slope is null do not give rise to focal points in the caustics. Photographic and optoelectronic records o f the reflected beam in the far field are shown in figs. 7, 8. The reflected beam appears as a concentric fringe system due to interference between light rays coming mainly from the neighborhood o f focal points (2, 4, 6, 8, 10). As the p h e n o m e n o n is axisymmetric, in 3D space the lateral focal points (4, 8) and (2, 10) give rise to coherent ring-sources, while the central focal point works as a point source. The fringe system is limited by a bright external ring, which is due to light reflected at inflexion points (5, 7). Due to the higher light intensity in the central region o f the incident beam, only the central point source ( 6 ) and the first ring source (4, 8) play a significant role in practice. Fig. 9 shows the result o f a 2D numerical simulation o f interference between light rays coming from three point sources disposed as (4, 6, 8). The intensity distribution was calculated in the far-field, in an observation plane p e r p e n d i c u l a r to the s y m m e t r y axis. It is seen that the interference p h e n o m e n o n can be considered as the superposition o f a Newton ring system (due to interference between rays issued from
\, / \/
°it° 5 J__
Fig. 6. Detailed structure of the central caustic branch of fig. 3. Arrows show the direction followed by the contact point between the reflected ray and its envelope when the incident ray passes through points (4.5.6) of the surface profile. 4
I September 1989
!,
Fig. 7. Direct photograph of the reflected beam at the far field of the central region. The photograph corresponds to the final heating stage. The laser source is an argon laser at 100 mW output power. The samples are highly viscous oils from Venezuelan petroleum wells.
Fig. 8. Local intensity distribution along the radial coordinate of one fringe in the photograph of fig. 7. The record was done with an image acquisition system formed by a linear CCD camera and a microcomputer. As seen in fig. 7, the fringe system is formed by high-spatial frequency fringes modulated by a larger envelope.
points 4, 6 and 8, 6) and a Young fringe system with point sources (4, 8). In particular, the internal highfrequency m o d u l a t i o n o f the large fringes is clearly seen in figs. 8, 9. The width o f the internal (Young) fringes is nearly constant, while the width and separation o f the large fringes follow the square root law characteristic o f Newton rings.
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OPTICS COMMUNICATIONS
Fig. 9. Numerical simulation of an interference phenomenon between three point sources placed as the focal points (4, 6, 8) of fig. 5. The intensity distribution (vertical coordinate) is calculated in a horizontal plane in the far field above the liquid film. The horizontal coordinate is the distance from the symmetry axis. The above graph showsthe increasing width of the envelope, while the width of the internal narrow fringes remains constant.
3. Application to refractive index measurement As shown in the preceding section, caustics o f reflected and transmitted beams differ only by a vertical scale factor equal to ( n - 1 ) / 2 . From eqs. (4), (5) it is seen that this is also the ratio between the slopes o f rays (r, t) reflected and transmitted at the same (though arbitrary) surface point S. This is valid in particular for inflexion points (5, 7) (fig. 5), which generate the external bright rings in both beams. Thus a simultaneous measurement o f the divergence o f the reflected and transmitted beams at any arbitrary time instant t > tc allows one to deduce the refractive index n. Typical simultaneous movie records of both beams are shown in ref. [9]. It may be noted that this method is valid only if the refractive index n is a real number, for this is the condition of validity ofeqs. ( 5 ), ( 7 ). This condition is approximately fulfilled if the light-absorption coefficient o f the liquid sample is low enough. On the other hand, light absorption must be high enough to allow appreciable surface deformation for the given laser power and the given values o f the parameters C, D (section l ) governing the liquid thermohydrodynamic behavior. These conditions are approximately valid for petroleum samples used in our experiments. The optical absorption coefficient at 6328 •~ is about l c m - ', so that the imaginary part of the refractive index is about l 0 - 2. Slopes of about 1/20 are usually observed in low viscosity oils irradiated
1 September 1989
at laser power as low as 5 roW. As shown in the photographs of ref. [ 9 ], both the reflected and the transmitted ring systems are clearly visible. The corresponding value o f the refractive index obtained by the present method is about 1.6, which fits values furnished by Maraven Petroleum Co. (Venezuela). Whether these conditions may be fulfilled by other liquids is a subject of current research. In our preliminary experiments using other liquids (water, alcohol, vegetal oils) thermocapillary surface deformation is not appreciable at these low power levels.
4. Additional remarks and conclusions Self-interactions of low power, CW laser beams are obtained by heating liquid films o f oils with strongly temperature-dependent properties. In the far-field o f the liquid film and for heating times t>tc the reflected and transmitted beams are formed by a concentric interference fringe system surrounded by an outer bright ring. Simultaneous measurement of the diameters of both outer rings at given observation planes above and below the liquid film allows one to deduce the value of the liquid refractive index. Experiments with low-viscosity oils are conducted at power levels o f about 5 mW, the laser beam being focused on the surface of the sample. The resulting critical time tc is then about 0.1 s. Appreciable surface deformation of high-viscosity oils requires utilisation o f an argon laser at 100 mW, the critical time tc being about 1 s. Preliminary experiments using other liquids (water, alcohol, vegetal oils) do not show appreciable surface deformation at these low power levels. One advantage o f the method suggested in the present paper is that the samples are easily prepared by deposing the liquid over a fiat glass plate. In other methods [ 10 ] the sample must have a prismatic or spherical shape. Besides, measurement can be performed in real time by reading the diameters of the bright rings with C C D cameras followed by a dataacquisition system. An apparent disadvantage is its low precision, which is mainly due to uncertainty in the measurement o f the rings diameters. This is a subject o f current research, whose aim is to develop methods for rapid optical identification of petroleum samples.
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OPTICS COMMUNICATIONS
Acknowledgements T h e authors gratefully t h a n k their colleagues o f the L a b o r a t o i r e d ' O p t i q u e P.M. D u f f i e u x for t h e i r help. O n e o f t h e m (G. D a C o s t a ) a c k n o w l e d g e s f i n a n c i a l s u p p o r t o f U n i v . S i m o n B o l i v a r ( V e n e z u e l a ) and o f C . N . R . S . ( F r a n c e ) d u r i n g his S a b b a t i c a l Year.
References [ 1 ] L. Landau and E. Lifschitz, Mechanics of fluids (MIR, Moscow, 1971 ) pp. 296-297.
1 September 1989
[2 ] V.G. Levich, Physicochemical hydrodynamics (Prentice Hall, Englewood Cliffs, N.J., 1962 ) pp. 384-390. [3] G. Da Costa and J. Calatroni, Appl. Optics 17 (1978) 2381. [4] G. Da Costa and J. Calatroni, Appl. Optics 18 (1979) 235. [ 5 ] J. Calatroni and G. Da Costa, Optics Comm. 42 ( 1982 ) 5. [ 6 ] G. Da Costa, J. Physique 43 ( 1982 ) 1503. [7] J.W. Bruce and P.J. Giblin, Curves and singularities (Cambridge University Press, Cambridge, London, New York, Melbourne, Sydney, 1984). [ 8 ] M. Born and E. Wolf, Principles of optics ( Pergamon Press, London, New York, Paris, Los Angeles, 1959 ) pp. 120-126. [9]G. Da Costa, F. Bcntolila and E. Ruiz, Phys. Lctt. 95A (1983) 313. [ 10] Advanced optical techniques, ed. A.C.S. Van Heel (NorthHolland Publishing Co., Amsterdam, 1967 ) pp. 251-254.
OPTICS C O M M U N I C A T I O N S
1 September 1989
FOCAL P R O P E R T I E S OF L I Q U I D F I L M S D E F O R M E D BY H E A T I N G W I T H A GAUSSIAN LASER BEAM G. DA COSTA ~ and R. ESCALONA Laboratoire d'Optique P.M. Duffieux, Facultb des Sciences et des Techniques, Route de Gray. 25030 Besan¢on Cedex. France Received 13 February 1989; revised manuscript received 11 May 1989
The focal properties of liquid films of oils deformed by heating with a gaussian laser beam are numerically studied. A new method to measure the refractive index of oils is developed. The method is based on far-field properties of the caustics of the laser beams reflected and transmitted by the liquid film.
1. Introduction Temperature dependence of density and surface tension is known to be responsible of surface deformation in inhomogeneously heated liquid film [ 1,2 ]. The particular case corresponding to heating with a gaussian laser beam was studied in several papers [ 3,4 ]. Interferometric records [ 5 ] show that the free liquid surface undergoes dilatation in the early heating stage, followed by contraction for longer heating time. The liquid surface profile as a function of time is given by [ 6 ] y=yo[w3/4-C(w
- 1 ) ]1/2 ,
( 1)
where
w=l/[l+Df(fl, 7)] ,4
f(/~, y) =
] ,Sz/( i + y)
(2)
film surface, which results in turn in a modification of the structure of the reflected and transmitted beams. This is an example of self-interaction of a laser beam at low power level. A geometrical approach to the study of light intensity distribution in the reflected and transmitted beams is presented in the next section.
2. Properties of light rays reflected and transmitted by the laser-heated liquid film The equations of the reflected and transmitted rays (r, t) (fig. I ) corresponding to the incident ray (i) are respectively -1 - Yr -- 2y' (x)
(4)
(xr-x),
du
ue""
(3)
In these equations, Yo is the initial film thickness,
fl=x/a, and 7= t/to are respectively the dimensionless horizontal coordinate and heating time, x and a are defined in fig. 1 and to is a time constant defined in ref. [6]. Finally, C and D are dimensionless parameters containing the thermohydrodynamic properties of the liquid sample [6]. The incident laser beam (fig. 1 ) produces a deformation of the liquid On Sabbatical Year leave from: Departamento de Fisica, Universidad Simon Bolivar, A.P. 89000, Caracas, Venezuela.
y, =
-1
(n-1)y'(x)
(x,-x).
(5)
In these equations the refractive index n is assumed to be a real number, (x, y(x)) are the coordinates of surface point S and (xr, Yr), (x,, y, ) are the coordinates of current points of rays (r, t) respectively. As the equation y(x) of the surface profile is given at any time instant by eqs. ( 1 ) - ( 3 ), eqs. (4), (5) represent the families of reflected and transmitted rays with parameter x. In the geometric optics approximation light intensity is a maximum in the neighborhood of the corresponding envelopes,
0 030-4018/89/$03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
Volume 73, number I
OPTICS COMMUNICATIONS
Y
I
L \ \
O
\
i • . '.."
\ .°.
.
.\".
! i:. i x e
O
\ Fig. 1. A liquid sample lying over a transparent support of width e is heated by a gaussian laser beam (diameter 2a at the inflexion points ofthe power distribution ). Oxy is a coordinate system, ( i, r, t ) are light rays respectively incident, reflected and transmitted at the surface point S. (Er, E,) are the envelopes of the families of reflected and transmitted rays. (R, T) are the contact points of rays ( r, t ) with the envelopes (E, E, ).
1 September 1989
ing to the same surface point S have the same abscissa (for XR(X)=XT(X)); ( b ) caustics (Er, E~) are identical except for a vertical scale change (for the ratio yR(x)/yv(x) is constant and equal to ( n - 1 ) / 2). The main difference between them is that real caustics o f reflected rays are virtual caustics o f transmitted rays and vice versa. (c) The asymptotic branches o f the caustics correspond to rays deflected at the neighborhood o f inflexion points o f the surface profile, where y" ( x ) = 0 . In that neighborhood the surface profile behaves as a linear mirror and light rays are deflected as a parallel beam. ( d ) Regression points of caustics correspond to light rays deflected at the neighborhood o f surface profile points where y" ( x ) is stationary, that is, where y" (x) = 0 . As the curvature radius is R~-I/y"(x), we conclude that the surface profile locally behaves as a circular mirror, thus focusing the deflected rays at a regression point. Considerations ( a - d ) are well known results of geometric optics [ 8 ]. Their validity is not limited to liquid films studied in the present paper. Taking profit o f items (a, b ) only the envelopes o f reflected rays are studied in what follows. The coordinates (XR, YR ) are numerically calculated for different values o f the p a r a m e t e r x and time t by substitution o f y(x) (eqs. ( 1 ) - ( 3 ) ) i n t o eqs. ( 6 ) , ( 7 ) . The resulting graphs are represented in figs. 2-5, where scales are not respected in order to emphasize the local pecul-
whose equations are calculated by s t a n d a r d m e t h o d s o f d i f f e r e n t i a i g e o m e t r y [7],
XR(X)=X--y'(x)/y"(X),
(6a)
yR(X)= i/2y"(X),
(6b)
Xr ( X ) = X - - y ' ( x ) / y " ( X ) ,
(7a)
)~-(X)= l / ( n - - I ) y " ( x ) .
(7b)
6~
These equations (as well as eqs. (4, 5 ) ) are valid with the hypothesis: lY' ( x ) l << 1 (small surface def o r m a t i o n ) and I I / y " ( x ) l >>e ( c u r v a t u r e radius o f the surface profile much greater than the support thickness). In eqs. ( 6 ) , ( 7 ) , (XR, YR) and (XT, Yr) are the coordinates o f the contact points (R, T ) o f each ray with its envelope. C o m p a r i s o n o f these equations shows that: ( a ) points (R, T ) correspond-
n
~1-
2
Fig. 2. Surface profile and its corresponding caustics in the early heating stage•
Volume 73, number 1
OPTICS COMMUNICATIONS
iarities of the surface profile and its caustics. Fig. 6 shows in detail the local structure of the central caustics of fig. 3. Each caustic branch is assigned with the same number as the corresponding region of the surface profile. Caustics branches are represented as the borders o f shadowed regions. In the early heating stage the surface profile is dilated as a result of diminution o f the liquid density. In fig. 2, points (1) correspond to light rays reflected far from the symmetry axis. Points (2, 6) are stationary points o f the curvature radius of the surface profile, and give rise to regression points (focal points) in the caustics. Points (3, 5) of the surface profile are inflexion points and give rise to asymptotic caustic branches. Also at the central point (4) the curvature radius is stationary. The real caustic branches are (1, 2, 3) and (5, 6, 1 ), while (3, 4, 5) is virtual. At a certain time instant tf (fig. 3) the topological structure of the virtual caustic abruptly changes. The cusp vertex (4) o f fig. 2 is split into three vertex (4, 5, 6). Vertex (5) corresponds to the central surface point, while vertex (4, 6) are two new focal points proceeding from surface points (4, 6) where the curvature radius is stationary. For heating times t > tf the cusp vertex (5) travels down to infinite. At a certain time instant t~ (fig. 4) the dilatation stage of the surface is finished. From now on the surface tension gradient provokes the
,,';i,~
1 September 1989
iiI & 2
.i~",'7
.
", i "
•
I,. L,'
:
•
i, ~--
2
8--
:.-_Q
-
4q
7/
'6l il....... 3 5
Fig. 4. At a later time instant t~>tc a pit is formed in the top of the dilated surface. At this precise time instant the cusp vertex (5) is at infinite.
_~\\\~illl 1
7 ~,~lill i|ii////~/5
"16//ff/~"9
V i I!.,,,~6 \,'il L
".I \i
I ~III~ I I I I i .
.
.
l
.
.
.
I''/l~eC'
!/" : IIIII~-III,~ ........
.
J
:L::..
',:Z.','~z~~',,
,,,'~':
5~ C
~2
"
I"
C'
43
8P 9
....
/ I
2--
I
'
----8 ....
4' I "6
Fig. 3. At a certain time instant tr the central vertex of fig. 2 is split into three vertex (4. 5, 6 in the above graphic ).
7
Fig. 5. In the final heating stage (t> to) a new real caustic branch is formed above the central region of the liquid surface. The cusp vertex (6) is a real focus which travels down from infinite for t> to. The asymptotic branches ( 5, 7) proceed from the inflexion points (5.7) of the surface profile. formation of a pit in the top of the central region. At this precise time instant ( t = t c ) the cusp vertex (5) is at infinite. The central pit works as an aspherical mirror. As a result, in the final heating stage (fig. 5) a new real caustic branch is formed in the central region above
Volume 73, number 1
OPTICS COMMUNICATIONS
the liquid surface. Cusp vertex ( 6 ) is a real focus which is placed at infinite at t=tc and travels down for t > to. The a s y m p t o t i c caustic branches (5, 7) proceed from the new inflexion points (5, 7) o f the surface profile. It is noted that, contrarily to intuition, surface points (C, C' ) where slope is null do not give rise to focal points in the caustics. Photographic and optoelectronic records o f the reflected beam in the far field are shown in figs. 7, 8. The reflected beam appears as a concentric fringe system due to interference between light rays coming mainly from the neighborhood o f focal points (2, 4, 6, 8, 10). As the p h e n o m e n o n is axisymmetric, in 3D space the lateral focal points (4, 8) and (2, 10) give rise to coherent ring-sources, while the central focal point works as a point source. The fringe system is limited by a bright external ring, which is due to light reflected at inflexion points (5, 7). Due to the higher light intensity in the central region o f the incident beam, only the central point source ( 6 ) and the first ring source (4, 8) play a significant role in practice. Fig. 9 shows the result o f a 2D numerical simulation o f interference between light rays coming from three point sources disposed as (4, 6, 8). The intensity distribution was calculated in the far-field, in an observation plane p e r p e n d i c u l a r to the s y m m e t r y axis. It is seen that the interference p h e n o m e n o n can be considered as the superposition o f a Newton ring system (due to interference between rays issued from
\, / \/
°it° 5 J__
Fig. 6. Detailed structure of the central caustic branch of fig. 3. Arrows show the direction followed by the contact point between the reflected ray and its envelope when the incident ray passes through points (4.5.6) of the surface profile. 4
I September 1989
!,
Fig. 7. Direct photograph of the reflected beam at the far field of the central region. The photograph corresponds to the final heating stage. The laser source is an argon laser at 100 mW output power. The samples are highly viscous oils from Venezuelan petroleum wells.
Fig. 8. Local intensity distribution along the radial coordinate of one fringe in the photograph of fig. 7. The record was done with an image acquisition system formed by a linear CCD camera and a microcomputer. As seen in fig. 7, the fringe system is formed by high-spatial frequency fringes modulated by a larger envelope.
points 4, 6 and 8, 6) and a Young fringe system with point sources (4, 8). In particular, the internal highfrequency m o d u l a t i o n o f the large fringes is clearly seen in figs. 8, 9. The width o f the internal (Young) fringes is nearly constant, while the width and separation o f the large fringes follow the square root law characteristic o f Newton rings.
Volume 73, number 1
OPTICS COMMUNICATIONS
Fig. 9. Numerical simulation of an interference phenomenon between three point sources placed as the focal points (4, 6, 8) of fig. 5. The intensity distribution (vertical coordinate) is calculated in a horizontal plane in the far field above the liquid film. The horizontal coordinate is the distance from the symmetry axis. The above graph showsthe increasing width of the envelope, while the width of the internal narrow fringes remains constant.
3. Application to refractive index measurement As shown in the preceding section, caustics o f reflected and transmitted beams differ only by a vertical scale factor equal to ( n - 1 ) / 2 . From eqs. (4), (5) it is seen that this is also the ratio between the slopes o f rays (r, t) reflected and transmitted at the same (though arbitrary) surface point S. This is valid in particular for inflexion points (5, 7) (fig. 5), which generate the external bright rings in both beams. Thus a simultaneous measurement o f the divergence o f the reflected and transmitted beams at any arbitrary time instant t > tc allows one to deduce the refractive index n. Typical simultaneous movie records of both beams are shown in ref. [9]. It may be noted that this method is valid only if the refractive index n is a real number, for this is the condition of validity ofeqs. ( 5 ), ( 7 ). This condition is approximately fulfilled if the light-absorption coefficient o f the liquid sample is low enough. On the other hand, light absorption must be high enough to allow appreciable surface deformation for the given laser power and the given values o f the parameters C, D (section l ) governing the liquid thermohydrodynamic behavior. These conditions are approximately valid for petroleum samples used in our experiments. The optical absorption coefficient at 6328 •~ is about l c m - ', so that the imaginary part of the refractive index is about l 0 - 2. Slopes of about 1/20 are usually observed in low viscosity oils irradiated
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at laser power as low as 5 roW. As shown in the photographs of ref. [ 9 ], both the reflected and the transmitted ring systems are clearly visible. The corresponding value o f the refractive index obtained by the present method is about 1.6, which fits values furnished by Maraven Petroleum Co. (Venezuela). Whether these conditions may be fulfilled by other liquids is a subject of current research. In our preliminary experiments using other liquids (water, alcohol, vegetal oils) thermocapillary surface deformation is not appreciable at these low power levels.
4. Additional remarks and conclusions Self-interactions of low power, CW laser beams are obtained by heating liquid films o f oils with strongly temperature-dependent properties. In the far-field o f the liquid film and for heating times t>tc the reflected and transmitted beams are formed by a concentric interference fringe system surrounded by an outer bright ring. Simultaneous measurement of the diameters of both outer rings at given observation planes above and below the liquid film allows one to deduce the value of the liquid refractive index. Experiments with low-viscosity oils are conducted at power levels o f about 5 mW, the laser beam being focused on the surface of the sample. The resulting critical time tc is then about 0.1 s. Appreciable surface deformation of high-viscosity oils requires utilisation o f an argon laser at 100 mW, the critical time tc being about 1 s. Preliminary experiments using other liquids (water, alcohol, vegetal oils) do not show appreciable surface deformation at these low power levels. One advantage o f the method suggested in the present paper is that the samples are easily prepared by deposing the liquid over a fiat glass plate. In other methods [ 10 ] the sample must have a prismatic or spherical shape. Besides, measurement can be performed in real time by reading the diameters of the bright rings with C C D cameras followed by a dataacquisition system. An apparent disadvantage is its low precision, which is mainly due to uncertainty in the measurement o f the rings diameters. This is a subject o f current research, whose aim is to develop methods for rapid optical identification of petroleum samples.
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OPTICS COMMUNICATIONS
Acknowledgements T h e authors gratefully t h a n k their colleagues o f the L a b o r a t o i r e d ' O p t i q u e P.M. D u f f i e u x for t h e i r help. O n e o f t h e m (G. D a C o s t a ) a c k n o w l e d g e s f i n a n c i a l s u p p o r t o f U n i v . S i m o n B o l i v a r ( V e n e z u e l a ) and o f C . N . R . S . ( F r a n c e ) d u r i n g his S a b b a t i c a l Year.
References [ 1 ] L. Landau and E. Lifschitz, Mechanics of fluids (MIR, Moscow, 1971 ) pp. 296-297.
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[2 ] V.G. Levich, Physicochemical hydrodynamics (Prentice Hall, Englewood Cliffs, N.J., 1962 ) pp. 384-390. [3] G. Da Costa and J. Calatroni, Appl. Optics 17 (1978) 2381. [4] G. Da Costa and J. Calatroni, Appl. Optics 18 (1979) 235. [ 5 ] J. Calatroni and G. Da Costa, Optics Comm. 42 ( 1982 ) 5. [ 6 ] G. Da Costa, J. Physique 43 ( 1982 ) 1503. [7] J.W. Bruce and P.J. Giblin, Curves and singularities (Cambridge University Press, Cambridge, London, New York, Melbourne, Sydney, 1984). [ 8 ] M. Born and E. Wolf, Principles of optics ( Pergamon Press, London, New York, Paris, Los Angeles, 1959 ) pp. 120-126. [9]G. Da Costa, F. Bcntolila and E. Ruiz, Phys. Lctt. 95A (1983) 313. [ 10] Advanced optical techniques, ed. A.C.S. Van Heel (NorthHolland Publishing Co., Amsterdam, 1967 ) pp. 251-254.