Electrostrictive fluid pressure from a laser beam
EElectrostrictive fluid pressure from a laser beam
Simen ˚A. Ellingsen and Iver Brevik Fluids Engineering Division, Department of Energy and Process Engineering,Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Recent times have seen surge of research activity on systems combining fluid mechanics and electromagneticfields. In radiation optics, whenever information about the distribution of pressure in a dielectric fluid isrequired, the contribution from electrostriction becomes important. In the present paper we calculate howthe local pressure varies with position and time when a laser beam is imposed in a uniform fluid. A Gaussianintensity profile of arbitrary time dependence is assumed for the beam, and general results are derived inthis case. For demonstration we analyze two different cases: first, that the beam is imposed suddenly(mathematically in the form of a step function); secondly, that the beam is switched on in a soft way. In bothcases, simple analytical expressions for the pressure distribution are found.PACS numbers: 47.65.-d, 42.65.Sf, 47.35.Rs, 47.85.Np
I. INTRODUCTION
Over the last decade, considerable experimental re-search has been conducted into the interplay of optics andfluid mechanics. The field of optofluidics, the applicationof microfluidic flows for optical purposes, is a topic in itsinfancy and is seen by many as holding very considerabletechnological promise . Appreciable progress in opticalmanipulation of microflows has also been achieved; someexamples are found in Refs. 2–7. These impressive exper-imental developments create the need also for improvedtheoretical understanding of liquid-laser interactions. Asa step towards this end we have investigated herein thetime-dependent electrostrictive pressure due to a laserbeam propagating through a liquid.The electrostriction force density is a local force den-sity acting within all dielectric fluids and solids subjectedto an electric field, proportional to gradient of the fieldintensity. The force acts towards higher field strength,thus causing a local pressure increase in the center of alaser beam propagating through the medium. From ageneral viewpoint, when including also solid elastic me-dia, the electrostrictive susceptibility is strongly relatedto the elasto-optic susceptibility as they both representthe interaction of electric fields and an elastic deforma-tion in the medium. A very general account of media ofthis kind can be found in Nelson’s book , and a compre-hensive statistical mechanical theory is provided by Rasa-iah, Stell and co-workers . When restricting ourselvesto electrostrictive phenomena in fluids we may mention,for instance, the early paper of Brueckner and Jorna in which it is discussed how the electrostrictive couplingbetween laser light and the medium can be importantfor rapidly growing instabilities, and hence may lead tolarge density fluctuations. Another work dealing withelectrostriction in fluids is the series of papers by Zan-chini and Baretta ; here a discussion is given of therole of electrostriction for the attractive force between ca-pacitor plates completely immersed in a dielectric fluid.A work of particular interest in our context is the mi-crogravity experiment of Zimmerli et al. on electrostric-tion in a fluid under extraterrestrial conditions . This work is actually a generalization of the classic elec-trostriction experiment of Hakim and Higham (TheHakim-Higham experiment is discussed more closely inRef. 18, section 2.3.). The last-mentioned authors mea-sured the pressure between closely spaced electrodes im-mersed in a nonpolar liquid (carbon tetrachloride and n-hexane). The essential new element in the experiment ofZimmerli et al. was to examine electrostriction in a liquid,sulfur hexafluoride (SF ) near the liquid-vapor criticalpoint, T c = 319 K, where the compressibility was sevenorders of magnitude higher than in the liquids studied byHakim and Higham. Because of this compressibility, evena moderate electric field produced an easily measurabledensity change in the nonpolar fluid. To avoid distur-bances from stratification gravity effects on Earth, theexperiment was carried out in outer space, in the SpaceShuttle Columbia in 1994. For the application of electro-dynamic theory to fluids, the experiment is important.However, generally speaking the occurrence of an elec-trostrictive pressure term in a dielectric fluid has for themost part received only moderate attention in practi-cal situations. The reason for this is quite evident: theelectrostrictive force density is generally written as a gra-dient, and the total electrostrictive force on a dielectricspecimen, which may be written as an exterior surfaceintegral, gives no contribution to the force on the spec-imen as a whole if the surroundings are non-dielectric.Under usual physical conditions, the electrostrictive ef-fect therefore only becomes of importance in considera-tions of local pressure distributions in matter, accessibleexperimentally usually via optical methods.Although deformations and flows generated by elec-tromagnetic fields can generally be described without in-clusion of electrostriction, this may not be so for their stability . This point was emphasized by one of us al-ready some time ago and we return to it in the follow-ing few examples. The deformations of fluids by lasers,under static conditions as well as under time-dependentconditions, are therefore adequately described in terms ofMaxwell’s stress tensor, without electrostriction includedat all. See, for instance, some of the Bordeaux papers inRefs. 4, 6, and 7 and also some related theoretical pa- a r X i v : . [ phy s i c s . f l u - dyn ] A ug pers in Refs. 20 and 21. For full stability considerations,however, electrostriction could play an important role.Electrostriction can moreover be made useful, for ex-ample in laser-induced thermal acoustics (LITA) whichhas been made a tool for measuring a range of quantitiesincluding the speed of sound in a medium . The elec-trostrictive force, being able to effectuate local changes indensity, can also locally alter a medium’s refractive index,giving rise to non-linear optical response even at moder-ate field strength , resulting in such effects as anomalousStokes gain , and optical self-focusing .A few examples may serve to illustrate the typical roleof electrostriction. Firstly, it is noteworthy that the elec-trostrictive force is of the same order of magnitude as theusual electromagnetic force density − ( ε / E ∇ ε actingin regions where the permittivity ε varies with position,typically in dielectric boundary layers. (We write theconstitutive relations as D = ε ε E , B = µ µ H , so that ε and µ become nondimensional.) Consider for definitenessthe usual textbook situation in which two metal platesare partially immersed in a dielectric liquid (the situa-tion is discussed, e.g. in section 2.2 of Ref. 18). Whenthere is an electric field between the plates, the liquid isknown to rise to a height h that can be easily calculated.The main physical principle here is the balance betweenthe lifting surface force at the free surface and the grav-itational force. A nontrivial point is, however, the effectof the electrostrictive force: its presence is necessary tosecure that the excess electric pressure in the liquid isstrong enough to make it possible for the liquid to riseas a coherent whole . The electrostrictive effect thusserves to stabilize the system - it is intimately related tothermodynamic stability - but it does not play a role forthe magnitude of the height h .The observability of electrostrictive phenomena de-pends on one single physical variable in the system,namely the velocity of sound. In the classic experimentof Ashkin and Dziedzic , our second example, a narrowbeam of light was sent through a dielectric liquid and anelevation of the free surface (in the order of 1 µ m) wasobserved. The mean time between molecular collisions insuch a liquid is of order 1 ps, and sound needs only a timeof about 7 ns to traverse the cross section of the beam(waist radius w = 4 . µ m). This means that the elasticpressure has had sufficient time to build itself up long be-fore experimental effects become visible. As shown froma detailed calculation in Fig. 9 in Ref. 18, the maximumdynamic electrostrictive pressure occurred at times t ∼ t = 0 and diedout after 8-10 ns. By contrast, the hydrodynamical mo-tion of the free surface needed hundreds of nanosecondsto develop. For practical purposes it is thus legitimateto introduce the concept of a hydrodynamical pressure p , and to assume that the electromagnetic forces start toact simultaneously throughout the liquid.A third example of essentially the same kind is fur-nished by the radiation pressure experiment of Zhang andChang from 1988 : the authors observed the radiation- induced deformation of the surface of a micrometer-droplet illuminated by a laser beam. Again, the elec-trostrictive pressure in the interior of the droplet is coun-terbalanced by an elastic pressure having had sufficienttime to build itself up. The deformation of the dropletcan thus be calculated as though there were no elec-trostrictive effects at all. The theory of this experimenthas been worked out in Refs. 28 and 29.Under special circumstances it becomes possible tomake the electrostrictive effect discernible in force mea-surements even without relying upon optical meth-ods. The Goetz-Zahn experiment belongs to thiscategory . The main principle of this experiment wasto apply a high frequency electric field E = E cos ωt be-tween the two plates of a condenser filled with a liquid(polar or nonpolar), and to measure the appropriate har-monic component of the force between the plates. Strongfields were necessary for this purpose, of the order of E = 10 V/m. The reason why the electrostrictive partof the force was detected, was that the liquid was in astate of mechanical non-equilibrium; the fields were os-cillating so quickly that the elastic counter pressure hadinsufficient time to build. To propagate a distance of 1cm, sound needs a time of about 10 µ s, which is morethan the period 2 π/ω ∼ µ s of the oscillations. Thiscase was discussed in detailed in Ref. 18, section 2.4.In the following we will calculate the distribution ofthe electrostrictive pressure in a liquid under typical ex-perimental conditions. The practical utility of such anundertaking lies in the increasing need to have detailedinformation about the stress distribution in soft matterin the presence of lasers. In view of the increasing in-terest in liquid-laser interactions at a microscopically de-tailed level, such a calculation is timely. We consider fordemonstration two different time evolutions of the laserbeam; the case in which the pulse is switched on and offabruptly, and the case of a softly peaked pulse with expo-nential tail, while our final result is valid for an arbitrarytime evolution and can be of use, for example, in ap-plications within laser-induced acoustics. In both casesrelatively simple expressions are obtained for the pres-sure distribution in the liquid as a function of distancefrom the beam axis and time [corrections due to beamdivergence are ignored]. A natural time and length scalefor the problem are provided by the beam width and thevelocity of sound. Our final results in the two cases areEqs. (33) and (51), in which the pressure distribution isgiven as function of time and radius rescaled with respectto these scales. A. Basic formalism
We regard a laser beam propagating along the z di-rection in a homogeneous liquid, with beam intensity de-scribed by a Gaussian so that the squared electric fieldin cylindrical coordinates r = ( ρ, θ, z ), E ( r , t ) = E ( r ) T ( t ) (1) ρzn w w xyzn FIG. 1. The situation considered: A Gaussian laser beamof width w (cid:29) λ propagates through a medium of relativepermittivity ε ( λ : wavelength). The laser intensity varies intime according to a function T ( t ). with (c.f. e.g. Ref. 18 and chapter 16 of Ref. 33) E ( r ) = 2 Pπε ncw ( z ) exp (cid:20) − ρ w ( z ) (cid:21) . (2)The situation is shown in figure 1. Here, P is the totalincident laser power, n is the liquid’s index of refraction,and because of inevitable beam divergence its radius is z dependent and of the form w ( z ) = w (cid:113) z /l R (3)where w is the waist radius and l R = πw /λ is theRayleigh length with λ the wavelength of the laser lightin the liquid, and c is the velocity of light in vacuum.In the following we shall assume the beam width w tobe much greater than the laser wavelength λ , so that w ( z ) ≈ w and remains approximately constant over apropagation distance of many beam diameters.The electrostrictive force due to the presence of thebeam is ∇ χ where χ ( ρ, t ) = ε E ( ρ, t ) (cid:37) m d n d (cid:37) m = χ (0) T ( t ) exp (cid:20) − ρ w ( z ) (cid:21) (4)where (cid:37) m is the fluid’s density and χ (0) is the initial andoriginal value of χ . The speed of light is much greaterthan that of sound, hence we can assume all changes inthe electromagnetic laser field to be instantaneous andsimultaneous throughout the liquid. We then clearly have χ ( ρ, t ) = χ ( ρ ) T ( t ). Assuming the liquid to obey the one-component Lorentz-Lorenz (Clausius-Mossotti) relation (cid:37) m αε = n − n + 2 (5)where α is molecular polarizability gives χ ( ρ ) = ε E ( ρ )( n − n + 2) . (6)Explicitly, χ (0) = ( n − n + 2) P πncw . (7) It is to be noted that the Clausius-Mossotti relation, ac-cording to which the permittivity depends on the fluiddensity only, refers to a non-polar fluid. The relation isconvenient to use in practice, as one does not have to dis-tinguish between the isothermal derivative ( ∂n /∂(cid:37) m ) T and the adiabatic derivative ( ∂n /∂(cid:37) m ) S . It is found toapply with good accuracy for gases, for many liquids,and even for some solids. For polar liquids, in order toget better accuracy, the Onsager relation is available with various modifications . In our theory the use ofanother relation for n ( (cid:37) m ) amounts only to a differentform of χ (0), and since we will work to linear order inthe density variations in the liquid, χ (0) will enter onlyas a prefactor throughout the below theory. Hence thechoice of relation n ( (cid:37) m ) does not affect the following cal-culations and results in any essential way.We ought to point out at this point that we ignoreother effects besides electrostriction, in particular, thequadratic electro-optic effect. It is known that in strongelectric fields a liquid can become optically anisotropic,the polarizability α (cid:107) parallel to the field being differentfrom the polarizability α ⊥ in the transverse direction.The field dependence of the polarizability in turn influ-ences the refractive index through the Lorentz-Lorenz re-lation (the optical analogue of the Clausius-Mossotti re-lation). On may estimate the magnitude of the quadraticelectro-optic effect by considering the nonlinear quantumtheory of the refractive index in an isotropic medium.For instance, regarding an artificial medium consistingof two-level atoms (see p. 147 of Ref. 18 and further ref-erences therein), it follows that α (cid:107) = α / (1 + E /E ).Here α is the polarizability in the absence of the fieldand E is the constant E = ¯ hω s / ( √ ex s ), where ¯ hω s isthe energy difference between the two energy levels of theatom and x s the matrix element of the position opera-tor between the same levels. For simple non-polar liquids,typically ω s ∼ s − , so that if the matrix element isin the order of the Bohr radius, we obtain E ∼ to10 V/m. This is a very high value, implying that thequadratic electro-optic effect can be safely ignored underusual physical conditions.To study the pressure that builds due to the elec-trostrictive force, consider the linearized Euler equation (cid:37) m ∂ t v = −∇ p + ∇ χ (8)[ ∂ t ≡ ∂/∂t ]. It follows from (8) and the fact that thesame equation must hold before the arrival of the beam,that ∇× v = 0, and hence we may write the velocity fieldin the form of a velocity potential, v = ∇ Φ . (9)Thus, because of gauge invariance of potentials we getthe simple equation p ( ρ, t ) = − (cid:37) m ∂ t Φ( ρ, t ) + χ ( ρ, t ) . (10)Writing p ( ρ, t ) = p + p (cid:48) ( ρ, t ) and (cid:37) m ( ρ, t ) = (cid:37) m + (cid:37) (cid:48) m ( r, t ) where p , (cid:37) m are the constant, initial pressureand density, and p (cid:48) (cid:28) p , (cid:37) (cid:48) m (cid:28) (cid:37) m , we linearize w.r.t. p (cid:48) , (cid:37) (cid:48) m and go to the gauge frame, p (cid:48) ( ρ, t ) = − (cid:37) m ∂ t Φ( ρ, t ) + χ ( ρ, t ) . (11)The second term on the right hand side is due to thedirect electrostrictive force from the local electric fielddensity whereas the first term comes from propagationof pressure waves from the surrounding area. Sphericalpressure waves are emitted from all points within thebeam when the electric field varies with time. [Fieldvariations over an optical period is of no consequence;the material response time is much longer than this, andthese periodic field variations simply average to zero.]With the continuity equation ∂ t (cid:37) m + ∇ · ( (cid:37) m v ) = 0 (12)and noting that p (cid:48) = u (cid:37) (cid:48) m (13)where u = [( ∂p (cid:48) /∂(cid:37) m ) S ] / is the speed of sound, thegoverning equation for Φ( ρ, t ) is obtained( ∇ − u − ∂ t )Φ( ρ, t ) = − (cid:37) m u ∂ t χ ( ρ, t ) . (14) II. STEP-FUNCTION SWITCH-ON OF BEAM
Assume first that the beam is switched on suddenly at t = 0 and off again at t . T ( t ) = Θ( t ) − Θ( t − t ) (15)where Θ is the unit step function. Because the velocityof light is immensely greater than the velocity of sound,we may safely assume the onset of electrostrictive forcesto be simultaneous throughout the liquid. We have ∂ t χ ( ρ, t ) = χ ( ρ ) ∂ t T ( t ) = χ ( ρ )[ δ ( t ) − δ ( t − t )] . (16)We define the Green’s function of Eq. (14), which bydefinition satisfies( ∇ − u − ∂ t ) G ( r , r (cid:48) , t, t (cid:48) ) = δ ( r (cid:48) − r ) δ ( t (cid:48) − t ) (17)which has solution (e.g. Ref. 36 p. 244) G ( r (cid:48) , r , t, t (cid:48) ) = − π | r (cid:48) − r | δ (cid:18) t (cid:48) − t + | r (cid:48) − r | u (cid:19) . (18)This solution requires that scattering of sound waves offfluid surfaces may be neglected. In microscopic systemswhere such effects are of importance, a Green’s func-tion taking these boundaries into account is required –Green’s functions for the scalar wave equation have beenfound for a wealth of geometries. From (14) and (18) it follows that (noting δ ( ax ) = δ ( x ) / | a | )Φ( ρ, t )= − (cid:37) m u (cid:90) t −∞ d t (cid:48) (cid:90) d r (cid:48) G ( r (cid:48) , r , t, t (cid:48) ) ∂ t χ ( r (cid:48) , t (cid:48) ) (19)= χ (0) π(cid:37) m u (cid:90) π d θ (cid:48) (cid:90) ∞ d z (cid:48) (cid:90) ∞ ρ (cid:48) d ρ (cid:48) exp( − ρ (cid:48) /w ) R × [ δ ( tu − R ) − δ ( ut − ut − R )] (20)where we define R = | r (cid:48) − r | ; R = | R | . (21)Since the geometry is obviously symmetric under z → − z and θ → − θ we have halved the integration ranges of z (cid:48) and θ (cid:48) above. For the same reason we are free to choose r to lie on the x axis without loss of generality: z = 0, θ = 0. To integrate out the remaining delta function weintegrate with respect to R using R = ρ + ρ (cid:48) − ρρ (cid:48) cos θ (cid:48) + z (cid:48) (22)to substituted z (cid:48) R = d Rz (cid:48) = d R (cid:112) R − ρ − ρ (cid:48) + 2 ρρ (cid:48) cos θ (cid:48) . (23)The delta functions in Eq. (20) pick out two sphericalshells of values of ( r (cid:48) , θ (cid:48) ) centered at r and having radii ut and, for t > t , u ( t − t ), respectively. Thus, consider-ing the term of radius ut [the second term is essentiallyidentical, as is clear from Eq. (20)] (cid:90) π d θ (cid:48) (cid:90) ∞ ρ (cid:48) d ρ (cid:48) f ( ρ (cid:48) ) (cid:90) ∞ d z (cid:48) R δ ( tu − R )= (cid:90) π d θ (cid:48) (cid:90) ∞ ρ (cid:48) d ρ (cid:48) f ( ρ (cid:48) ) (cid:90) ∞R d R δ ( R − tu ) √ R − R = (cid:90) π d θ (cid:48) (cid:90) ∞ ρ (cid:48) d ρ (cid:48) f ( ρ (cid:48) ) √ u t − R Θ( ut − R ) (24)where we define R ( t ) = ρ + ρ (cid:48) − ρρ (cid:48) cos θ (cid:48) . (25)In the above, the exponential in (20) depending only on ρ (cid:48) were gathered in a shorthand function f ( ρ (cid:48) ).The Θ function restricts the integration area to a cir-cular disc of radius ut centered at ( ρ, , R , ϕ ), ρ (cid:48) = ρ + R + 2 ρ R cos ϕ. (26)Now the Θ function is equivalent with the upper integra-tion limits, and can be removed. An equivalent expres-sion is obtained for the second term, but with t → t − t , φθ ρρ’ R y’ x’ Integration range < u t R FIG. 2. Coordinate shift: translating the z axis to the point( ρ, and only allowing contributions for t > t for reasons ofcausality.Using the relation (cid:90) π d ϕπ e − a cos ϕ = I ( a ) (27)where I ( x ) is the modified Bessel function of the firstkind of order zero, we thus obtain a general expressionfor Φ( ρ, t ):Φ( ρ, t ) = χ (0) (cid:37) m u Θ( t ) (cid:90) ut R d R I (cid:16) ρ R w (cid:17) exp (cid:16) − ρ + R ) w (cid:17) √ u t − R − Θ( t − t ) (cid:90) u ( t − t )0 R d R I (cid:16) ρ R w (cid:17) exp (cid:16) − ρ + R ) w (cid:17)(cid:112) u ( t − t ) − R (28)The gauge pressure can now be calculated Eq. (11), andwe shall do so in a moment. First, we will simplify ournotation considerably by introducing rescaled variablesfor time and length. A. Rescaled time and radius
For further analysis we introduce the typical time andlength scales ˜ ρ = w √ t = w √ u (29)satisfying u ˜ t = ˜ ρ . The corresponding dimensionless timeand space variables we define as ξ (cid:48) = ρ (cid:48) / ˜ ρ ; ξ = ρ/ ˜ ρ ; (30a) τ = t/ ˜ t ; τ = t / ˜ t ; (30b)and substituting for R → x = x = √ u t − R / ˜ ρ aswell as rescaling we get the much simpler looking general expressionsΦ( ρ, t ) = χ (0)˜ t(cid:37) m (cid:104) ˜Φ( ξ, τ ) − ˜Φ( ξ, τ − τ ) (cid:105) ; (31a)˜Φ( ξ, τ ) =Θ( τ )e − ξ (cid:90) τ d xI (2 ξ (cid:112) τ − x )e x − τ . (31b)The integral in Eq (31b) is simple to evaluate numeri-cally. With the re-scaled variables and the dimension-less ˜Φ function defined above, Eq. (11) becomes [there-scaled time envelope function is obviously T ( τ ) =Θ( τ ) − Θ( τ − τ )] p ( ξ, τ ) = − χ (0)[ ∂ τ ˜Φ( ξ, τ ) − ∂ τ ˜Φ( ξ, τ − τ ) − T ( τ )e − ξ ] . (32)Evaluating the derivatives we finally find the pressuredistribution as function of (rescaled) radius and time p ( ξ, τ ) = χ (0)[ P ( ξ, τ ) − P ( ξ, τ − τ )] (33a) P ( ξ, τ ) =2 τ e − ξ Θ( τ ) (cid:90) τ x d x e − x √ τ − x (cid:104) I (2 ξx ) − ξx I (2 ξx ) (cid:105) (33b)where in the last line we have substituted x → √ τ − x .Equations (33) are the final result of the pressure calcu-lation in the case of step-function switch-on and -off ofthe laser.The generalisation to a series of N pulses turned on attimes t i and off at times t i ( t i < t i < t i +1 ) is obvious: p ( ξ, τ ) = χ (0) N (cid:88) i =1 [ P ( ξ, τ i ) − P ( ξ, τ − τ i )] . (34)A plot of the functions ˜Φ( ξ, τ ) and P ( ξ, τ ) are shownin figure 3. One notes how the pressure on the axis risessteeply when the laser is switched on and then relaxesmonotonously towards a new equilibrium value which hasthe very simple form P ( ξ, τ ) τ (cid:29) −→ e − ξ . (35)We show this in the appendix. While it is not entirelytrivial to obtain the limit from the more general ex-pression, this result could have been obtained in a verystraightforward way by assuming a static laser beamfrom the beginning. After reaching the peak value, thepressure relaxes quickly and the stationary situation isreached already at around 3˜ t . The left panel of figure 3also shows how a pressure wave travels outwards from thecylinder axis after switch-on at velocity u , and similarly,a wave of negative gauge pressure at switch-off. This factallows closely controlled acoustic waves to be created bya laser beam, the principle behind laser-induced thermalacoustics, already used for measurement purposes . ξ τ Φ(ξ,τ)∼ P (ξ,τ)ξ τ FIG. 3. Left: the function ˜Φ( ξ, τ ) for τ = 5. Right: The reduced pressure p ( ξ, τ ) /χ (0) for τ = 5 as given in Eq. (33b). B. Pressure time variations on the axis
A simple analytical expression can be found for pointson the z axis, ξ = 0. Then we are left with˜Φ(0 , τ ) = Θ( τ ) (cid:90) τ d x exp( x − τ ) = Θ( τ ) F ( τ ) (36)where F ( x ) = e − x (cid:82) x d s e s is Dawson’s integral. Insert-ing this into Eq. (32) with F (cid:48) ( x ) = 1 − xF ( x ) we getsimply, in terms of the dimensionless units Eq. (30),˜Φ(0 , τ ) =Θ( τ ) F ( τ ) − Θ( τ − τ ) F ( τ − τ ); (37) p (0 , τ ) =2 χ (0)[Θ( τ ) τ F ( τ ) − Θ( τ − τ )( τ − τ ) F ( τ − τ )] . (38)The dimensionless pressure p (0 , τ ) /χ (0) is shown in figure4. The special case of the cylinder axis was consideredalso previously , in agreement with Eq. (pressureAxis). τ p ( , τ ) / χ ( ) FIG. 4. The pressure p (0 , τ ) on the axis, Eq. (38), as functionof dimensionless time τ , divided by the prefactor χ (0). Thepulse is turned on at τ = 0 and off at τ = τ = 5. III. LASER PULSE WITH GENERAL TIME EVOLUTION
Whereas we considered a sharp turn-on and turn-offof the beam in the previous section, here we will be con-cerned with a more typical pulse transient. Within thesame formalism as employed above, we are able to keepthe function T ( t ) general throughout, yielding results ofmore general validity. For numerical examples in the fol-lowing, we choose the following typical form for a softlaser pulse, T ( t ) = e τ τ e − τ/τ (39)so that t = ˜ tτ is again the duration of the pulse. Hereand forthwith we make use of the rescaled variables de-fined in Eq. 30. We have T (cid:48) ( τ ) = e τ (cid:18) ττ − τ τ (cid:19) e − τ/τ ; (40) T (cid:48)(cid:48) ( τ ) = e τ (cid:18) − ττ + τ τ (cid:19) e − τ/τ . (41)The velocity potential is again given by Eq. (19), where-with we get after rescaling˜Φ( ξ, τ ) = 14 π (cid:90) τ d τ (cid:48) (cid:90) d ξ (cid:48) e − ξ (cid:48) s δ ( τ (cid:48) − τ + s ) T (cid:48) ( τ (cid:48) )= 14 π (cid:90) s ≤ τ d ξ (cid:48) e − ξ (cid:48) s T (cid:48) ( τ − s ) (42)where we define s = ξ (cid:48) − ξ ; s = | s | . (43)As before, we are free to choose ξ = ( ξ, , , and notethat the integrand is invariant under rotation about the ξ (cid:48) x axis. The integral over ξ (cid:48) runs over all of space, sowe choose integration coordinates instead to be sphericalcoordinates with the origin at ξ in the ξ (cid:48) system and withthe ξ (cid:48) x axis as polar axis with polar angle θ s . In the newcoordinate system ( s, θ s , ϕ s ) the integral reads˜Φ( ξ, τ ) = 14 π (cid:90) τ s d sT (cid:48) ( τ − s ) (cid:90) π d ϕ s (cid:90) π sin θ s d θ s × exp[ − ξ − s (1 − sin θ s sin ϕ s ) − ξs cos θ s ] (44)where the exponent is − ξ (cid:48) ( ξ, s, θ s , φ s ), as can be seengeometrically from figure 5 when noting that the length z (cid:48) can be expressed as z (cid:48) = s sin θ s sin φ s and that | ξ (cid:48) | = ξ + s + 2 ξs cos θ s = ξ (cid:48) + z (cid:48) . sz’ θsφsξξ’ξ’ x’y’z’ ξ ’θ’ ( ξ’,θ’,z’ )=( s,θs,φs ) z’ | | FIG. 5. Coordinate transformation to spherical coordinateswith origin at ( ξ, ,
0) in the cylindrical ( ξ (cid:48) , θ (cid:48) , z (cid:48) ) system, andpolar axis x (cid:48) . We now make use of the relation (cid:90) π d φ s e s sin θ s sin φ s = 2 π e s sin θ s I ( s sin θ s )giving˜Φ( ξ, τ ) =e − ξ (cid:90) τ d sT (cid:48) ( τ − s )e − s × (cid:90) s d β e − β I [ ( s − β )] cosh 2 ξβ (45)where we substituted β = s cos θ s .As a consistency check we should be able to insertthe step-function time evolution derivative of Eq. (15), T (cid:48) ( τ ) = δ ( τ ) − δ ( τ − τ ) and get back the expressionEq. (31b). One quickly sees that this is so, provided (cid:90) τ d x e x I (2 ξ (cid:112) τ − x )= (cid:90) τ d β e
12 ( τ − β ) I [ ( τ − β )] cosh 2 ξβ (46)for arbitrary τ and ξ . We have verified the identity nu-merically in the positive-positive ( τ, ξ ) plane. The twodifferent derivations of the re-scaled velocity potential for the step-function pulse thus serve as derivation of thepotentially useful relation Eq. (46).As in the previous case, the expression can be simpli-fied on the axis. Let us first use Eq. (46) to write Eq. (45)as ˜Φ( ξ, τ ) =e − ξ (cid:90) τ d sT (cid:48) ( τ − s )e − s × (cid:90) s d x e x I (2 ξ (cid:112) s − x ) , (47)an expression which is still valid for an arbitrary T ( τ ).On the z axis ( ξ = 0) ˜Φ becomes simply˜Φ(0 , τ ) = (cid:90) τ d sT (cid:48) ( τ − s ) F ( s ) (48)where F ( x ) is again Dawson’s integral. The functions˜Φ(0 , τ ) and P (0 , τ ) are plotted in figure 6. P (0, τ ) τ Φ (0, τ ) ~ FIG. 6. The functions ˜Φ and P on the symmetry axis for τ = 5. The expression for the pressure is simple to write downfrom the generalization of Eq. (32), p (cid:48) ( ξ, τ ) = χ (0)[ T ( τ )e − ξ − ∂ τ ˜Φ( ξ, τ )] . (49)Analogous to and for comparison with Eq. (33a), we de-fine p ( ξ, τ ) = χ (0) P ( ξ, τ ) (50)now with P ( ξ, τ ) = e − ξ (cid:34) T ( τ ) − (cid:90) τ d sT (cid:48)(cid:48) ( τ − s )e − s (cid:90) s d x e x I (2 ξ (cid:112) s − x ) − T (cid:48) (0)e − τ (cid:90) τ d x e x I (2 ξ (cid:112) τ − x ) (cid:35) . (51)This the expression for the pressure given a general T ( τ ),although this form clearly requires the existence of T (cid:48) (0)which is not satisfied for the step function consideredin section II. For the soft pulse transient of Eq. (39), T (cid:48) (0) = 0 and the last term vanishes.We plot ˜Φ( ξ, τ ) and P ( ξ, τ ) using the pulse transient(39) in figure 7. Notice how much longer it takes forthe gauge pressure to relax to zero again compared tothe step function turn-off shown in figure 3. The be-haviour is otherwise comparable to that of figure 3. Thegenerality of Eq. (51) implies for example that one canderive the exact acoustic pressure wave field throughoutthe medium which is set up by a train of laser pulses withknown time dependence [reflection/absorption at mate-rial boundaries not included, of course – this would re-quire a different, geometry specific Green’s function inEq (19)].Also in this case the expression takes a simpler formon the symmetry axis, where we are left with P (0 , τ ) = T ( τ ) − T (cid:48) (0) F ( τ ) − (cid:90) τ d sT (cid:48)(cid:48) ( τ − s ) F ( s ) (52)where F ( s ) is again the Dawson integral. IV. CONCLUSIONS
We have calculated the local gauge pressure distribu-tion due to electrostrictive forces as a function of timeand position, in the presence of a Gaussian laser beampropagating through a bulk fluid. The resulting expres-sions are simple and may form a natural part of an ana-lytical toolkit for experimental interpretation. A typicalscale for length (˜ ρ ) and time (˜ t ) emerge, given by thebeam width and the velocity of sound. Our final resultsare valid for a general time evolusion of the propagatinglaser beam, and for demonstratino purposes we considertwo typical cases: a sudden switch-on/switch-off of thebeam power, as well as a soft laser pulse. Reasonablysimple expressions are obtained for both cases, as given,respectively, by our Eqs. (33) and (51) in terms of there-scaled distance from the beam axis ( ξ = ρ/ ˜ ρ ), andtime ( τ = t/ ˜ t ). On the beam axis, these expressions maybe simplified further. The pressure relaxation time, theduration before the gauge pressure has returned to zeroas the laser pulse is turned off, is surprisingly different inthe two cases for the same pulse duration.Provided a stationary laser is switched on from zeroto full intensity in a time shorter than the typical timescale ˜ t = w / √ u ( w : beam waist, u : speed of sound),our result for a step function switch on are valid. After atransient period in which the pressure rises sharply andthen relaxes to a stationary value, equilibrium is reachedafter a time of 3 − t , roughly the time for a sound waveto propagate the beam diameter. Under most circum-stances fluid motion is much slower than this, and elec-trostrictive pressure can be considered to be set up in-stantaneously. A cylindrical acoustic pressure wave trav-els outwards from the beam on switch-on, and similarlya wave of lower pressure is emitted at switch-off, allow-ing for highly controlled laser-induced acoustics in the medium, a principle already used in measurement set-ups . Our result (51) provides in a simple way theacoustic pressure field which results from an arbitrarytrain of laser pulses anywhere in the medium at any time.Optofluidics and electrohydrodynamics are fields of re-search seeing impressive advances and promises many ap-plications in optics, sensing and measurement, microflu-idics and microchemistry, see e.g. Refs. 1 and 2 and ref-erences therein. So far this field of research has beendriven by impressive experimental advances with theo-rists working to explain the observed phenomena. In thispaper we provide another theoretical building block to-wards this end. Electrostriction, while not typically re-quired to describe electromagnetically induced fluid de-formations and flows themselves, is often of importanceto their stability. This is examplified by perhaps the sim-plest example of electrohydrodynamics, that of a liquidrising between condensator plates due to the electric field.While the elevation can be calculated without referenceto electrostriction, the latter plays the tacit but crucialrole of enabling the liquid column to rise as a coherentwhole . The methods and results presented hereincould have many applications in analyses of laser drivenflow and fluid manipulation, and for the stability of these. Appendix A: Equilibrium value after step-function switch-on
We show here Eq. (asymptotic), the asymptotic equi-librium value at times τ (cid:29) τ (cid:29)
1, Eq. (33b) becomes P ( ξ, τ ) → − ξ (cid:90) τ d xx e − x (cid:104) I (2 ξx ) − ξx I (2 ξx ) (cid:105) (A1)Using the following formulae from Ref. 37: (cid:90) ∞ d xx ν +1 e − αx J ν ( βx ) = β (2 α ) ν +1 e − β / α (cid:90) ∞ d xx ν − e − αx J ν ( βx ) =2 ν − β − ν γ ( ν, β / α )where γ ( a, x ) = (cid:82) x d t e − t t a − is the incomplete Gammafunction, using J n (i z ) = i n I n ( z ) and letting β = 2i ξ wefind (cid:90) ∞ d xx e − x I (2 ξx ) = 12 e ξ , (A2) (cid:90) ∞ d x e − x I (2 ξx ) = − ξ (1 − e ξ ) . (A3)Inserted into (A1) this immediately gives Eq. (35). C. Monat, P. Domachuk, and B.J. Eggleton, “Integrated optoflu-idics: A new river of light”, Nature Photonics , 106 (2007). N. Garnier, R. O. Grigoriev, and M. F. Schatz, “Optical Manip-ulation of Microscale Fluid Flow”, Phys. Rev. Lett. , 054501(2003). ξ τΦ(ξ,τ)∼ ξ τ P (ξ,τ) FIG. 7. Left: the function ˜Φ( ξ, τ ) from Eq. (45) or (47) for τ = 5. Right: The reduced pressure p ( ξ, τ ) /χ (0) for τ = 5 asgiven in Eq. (51). C. N. Baroud, M. R. de Sait Vincent, and J.-P. Delville, “Anoptical toolbox for total control of droplet microfluidics”, Lab ona Chip , 1029 (2007). E. Brasselet, R. Wunenburger, and J.-P. Delville, “Liquid Opti-cal Fibers with a Multistable Core Actuated by Light RadiationPressure”, Phys. Rev. Lett. , 014501 (2008). E. Verneuil, M. L. Cordero, F. Gallaire, and C. N. Baroud,“Laser-Induced Force on a Microfluidic Drop: Origin and Mag-nitude”, Langmuir , 5127 (2009). J-P. Delville, M. R. de Sait Vincent, R. D. Schroll, H. Chra¨ıbi,B. Issenmann, R. Wunenburger, D. Lasseux, W. W. Zhang, andE. Brasselet, “Laser microfluidics: fluid actuation by light”, J.Opt. A , 034015 (2009). R. Wunenburger, B. Issenmann, E. Brasselet, C. Loussert,V. Hourtane, and J.-P. Delville, “Fluid flows driven by light scat-tering”, J. Fluid. Mech. , 273 (2011). D. F. Nelson,
Electric, Optic, and Acoustic Interaction in Di-electrics (Wiley, New York, 1979). J. C. Rasaiah, D. J. Isbister, and G. Stell, “Nonlinear effects inpolar fluids: A molecular theory of electrostriction”, J. Chem.Phys. , 4707 (1981). S. L. Carnie and G. Stell, “Electrostriction and dielectric satu-ration in a polar fluid”, J. Chem. Phys. , 1017 (1982). J. C. Rasaiah, “Electrostriction and the dielectric constant of asimple polar fluid”, J. Chem. Phys. , 5710 (1982). K. A. Brueckner and S. Jorna, “Linear instability theory of laserpropagation in fluids”, Phys. Rev. Lett. , 78 (1966). E. Zanchini and A. Barletta, “Force between the plates of aplane capacitor in a fluid: A rigorous thermodynamic analysis”,Il Nuovo Cimento D , 481 (1994). A. Barletta and E. Zanchini, “Thermodynamic equilibrium indielectric fluids”, Il Nuovo Cimento D , 189 (1994). A. Barletta and E. Zanchini, “Can the definition of mechanicalstress tensor be applied to a dielectric fluid in an electrostatic ormagnetostatic field?”, Il Nuovo Cimento D , 177 (1994). G. A. Zimmerli, R. A. Wilkinson, R. A. Ferrell, and M. R.Moldover, “Electrostriction of near-critical fluid in micrograv-ity”, Phys. Rev. E , 5862 (1999). S. S. Hakim and J. B. Higham, “Experimental determinationof excess pressure produced in a liquid dielectric by an electricfield”, Proc. Phys. Soc. London , 190 (1962). I. Brevik, “Experiments in phenomenological electrodynamicsand the electromagnetic energy-momentum tensor”, Phys. Rep. , 133 (1979). I. Brevik, “Fluids in electric and magnetic fields: Pressure vari-ation and stability”, Can. J. Phys. , 449 (1982). A. Hallanger, I. Brevik, S. Haaland, and R. Sollie, “Nonlinear de-formations of liquid-liquid interfaces induced by electromagneticradiation”, Phys. Rev. E , 056601 (2005). O. J. Birkeland and I. Brevik, “Nonlinear laser-induced deforma-tions of liquid-liquid interfaces: An optical fiber model”, Phys.Rev. E , 066314 (2008). E. B. Cummings, “Laser-induced thermal acoustics: simple ac-curate gas measurements”, Opt. Letters , 1361 (1994). E. B. Cummings, I. A. Leyva, and H. G. Hornung, “Laser-induced thermal acoustics (LITA) signals from finite beams”,Appl. Opt. , 3290 (1995). J. P. Huang, “New nonlinear dielectric materials: Linear elec-trorheological fluids under the influence of electrostriction”,Phys. Rev. E , 042501 (2004). e.g. R. W. Boyd, S. G. Lukishova, and Y. R. Shen (eds.), Self-focusing: Past and present , Volume 114 of Topics in AppliedPhysics (Springer, Berlin, 2009) and references therein. A. Ashkin and J. M. Dziedzic, “Radiation Pressure on a FreeLiquid Surface”, Phys. Rev. Lett. , 139 (1979). J. Z. Zhang and R. K. Chang, “Shape distortion of a single wa-ter droplet by laser-induced electrostriction”, Opt. Lett. , 916(1988). H. M. Lai, P. T. Leung, K. L. Poon, and K. Young, “Electrostric-tive distortion of a micrometer-sized droplet by a laser pulse”, J.Opt. Soc. Am. B , 2430 (1989). I. Brevik and R. Kluge, “Oscillations of a water droplet illumi-nated by a linearly polarized laser pulse”, J. Opt. Soc. Am. B , 976 (1999). H. Goetz, “Die Elektrostriktion in Fl¨ussigkeiten und die Erzeu-gung von Ultraschall auf elektrostriktivem Wege”, Zeitscr. Physik , 277 (1955). H. Goetz and W. Zahn, “Die Elektrostriktion dipolloserFl¨ussigkeiten”, Zeitscr. Physik , 202 (1958). W. Zahn, “Die Elektrostriktion in Dipolfl¨ussigkeiten”, Zeitscr.Physik , 275 (1962). A. E. Siegman,
Lasers (University Science Books, Sausalito CA,1986). L. Onsager, “Electric Moments of Molecules in Liquids”, J. Am.Chem. Soc. , 1486 (1936). C. J. F. B¨ottcher,
Theory of Electric Polarization , Vol. 1, 2nded. (Elsevier, Amsterdam, 1973). J. D. Jackson,
Classical Electrodynamics I. S. Gradshteyn and I. M. Ryzhik,
Table of Integrals, Series andProducts §§