A One-Dimensional Solution of the Photoacoustic Wave Equation and its Relationship with Optical Absorption

Transcription

1 Int J Thermophys (2013) 34: DOI /s A One-Dimensional Solution of the Photoaousti Wave Equation and its Relationship with Optial Absorption D. Cywiak M. D. Barreiro-Argüelles M. Cywiak A. anda-curiel C. Garia-Segundo G. Gutierrez-Juárez Reeived: 14 February 2012 / Aepted: 25 September 2013 / Published online: 13 Otober 2013 Springer Siene+Business Media New York 2013 Abstrat A study based on the general solution of the one-dimensional photoaousti (PA) wave equation for an aousti plane soure is presented. This study relies on obtaining the impulse response of the PA system onsidering a heating funtion proportional to the ambert Beer law and spatially bounded by a retangle funtion. The PA pressure is obtained by onvoluting the impulse response with the temporal profile of the exiting pulse of light. With the obtained solution, it is possible to omprise, on a single expression, the PA pressure profile for optially thin samples (whose dimensions are smaller than the optial absorption length) and optially opaque samples (whose dimensions are larger than the optial absorption length). In the limit of weak absorption, the solution is in good agreement with the one for a uniform distribution of energy. Also, a study of the dependene of the aousti pressure with the duration of the laser pulse is inluded. Keywords Green s funtion method ambert Beer law aser-indued ultrasound Plane soure Photoaousti effet D. Cywiak M. D. Barreiro-Argüelles A. anda-curiel G. Gutierrez-Juárez (B) División de Cienias e Ingenierías, Universidad de Guanajuato-Campus eón, oma del Bosque 103, omas del Campestre, eón, GTO, Méxio ggutj@fisia.ugto.mx M. Cywiak Centro de Investigaiones en Óptia, A. C., oma del Bosque 115, omas del Campestre, eón, GTO, Méxio C. Garia-Segundo Centro de Cienias Apliadas y Desarrollo Tenológio (CCADET-UNAM), Méxio, Méxio

2 1474 Int J Thermophys (2013) 34: Introdution Photoaousti (PA) spetrosopy is based on the PA effet, whih onsists of the generation of sound waves after the absorption of eletromagneti (EM) radiation. There are several mehanisms by means of whih the PA effet an be generated; a survey of these mehanisms an be found in [1,2]. One of them, in whih this work is foused, is the thermoelasti expansion mehanism, whih is of interest in the siene of materials, in biomedial instrumentation; and partiularly in the emerging PA-based imaging methods, mainly beause the involved phenomena are non-invasive for biologial tissue. Basially, the thermoelasti expansion mehanism onsists of the generation of a sound or stress wave after the absorbed EM energy has indued a loal inrease in temperature [3,4]. For short optial pulses, in the absene of visosity and inhomogeneities, it has been demonstrated [5,6] that if heat and stress are onfined within the illuminated region, then the propagation of the stress wave, in terms of the veloity potential is 2 ϕ( r, t) 1 2 ϕ( r, t) 2 t 2 = β H( r, t). (1) ρc p In Eq. 1 ϕ( r, t) is the veloity potential, is the sound speed in the propagating media, r is the position, t is the time, β is the oeffiient of thermal expansion, ρ is the density of the absorber in equilibrium, C p is the speifi heat apaity at onstant pressure, and H( r, t) is the energy density per unit time distributed within the heated region during radiation. One Eq. 1 is solved for ϕ( r, t), the pressure profile of the aousti stress wave is alulated as p( r, t) = ρ ϕ( r, t). (2) t Equation 2 is valid in the thermoelasti regime (first-order approximation), where p( r, t) represents the pressure of the aousti stress wave. In this report, we solve Eq. 1 by means of the Green s funtion method. Unlike what is ommonly disussed [7,8], we inlude the ontribution of a heat distribution within the absorbing region to be proportional to ambert Beer s law multiplied by a retangle funtion whih aounts for the thikness of the one-dimensional heated objet. By inluding this distribution, the analytial model beomes more realisti. Our analytial expression takes into aount the relationship between the spatial dimensions of the absorber objet and the optial absorption length. 2 Theory 2.1 Calulation of the Energy Density Distribution per Unit Time within the Absorber et Q( r) be the laser fluene as a funtion of position r. We will onsider the ase of an objet immersed in a homogeneous ideal fluid with an absorption oeffiient muh

3 Int J Thermophys (2013) 34: smaller than that of the objet, allowing the absorption in the region outside the objet to be negleted. If the optial absorption length aross the absorber is smaller than the width of the laser spot size, the shape of the sound soure is like a plane [9]. We will onsider this situation over an absorber of thikness. Also, we will assume that the laser beam hits the absorber at z = z 0 and that the energy distribution within the thikness of the absorber is given by ambert Beer s aw, then at any position inside the sample, the energy density per unit time absorbed by the sample, an be written as H(x, y, z; t) = αq(x, y, z 0 ) exp [ α (z 0 z)] f (t). (3) In Eq. 3, f (t) is the temporal profile of the laser beam, and α is the absorption oeffiient of the heated region. In order to inlude the thikness of the absorber objet, Eq. 3 an be multiplied by a retangle funtion, Π(z), entered at z = z 0 /2 and width, thus, [ ] z (z0 /2) H(x, y, z; t) = αq(x, y, z 0 ) exp [ α (z 0 z)] Π f (t). (4) Equation 4 stands for the energy distribution per unit time that has to be substituted into Eq. 1, whih is then solved for the pressure. 2.2 Solution to the Inhomogeneous PA Wave Equation for a Plane Absorber One-Dimensional Solution Assuming that the width of the laser beam is larger than the ross setion of the heated objet, a onstant fluene Q(x, y, z 0 ) = Q 0 an be assumed [9]. With this ondition, it is only neessary to solve Eq. 1 for ϕ(z, t). We will solve Eq. 1 by means of the Green s method; we will onsider perfet impedane oupling between the absorbing and propagating media. Under these onditions, the differential equation to be solved by using Green s method reads 2G(z, z 2 t, z, t ) 1 2 G(z, t, z, t ) 2 t 2 = δ(z z )δ(t t ). (5) In Eq. 5, G represents the Green s funtion. Now, the solution of Eq. 1 in one-dimension an be found in terms of G as ϕ(z, t) = β ρc p 0 G(z, t, z, t )H(z, t )dt dz. (6) To alulate the integral in Eq. 6, first one has to find the analytial expression for G. For this, we take the aplae transform in the temporal domain of Eq. 5 and the Fourier transform in the spatial domain. One both transforms are alulated, we proeed by applying the inverse transforms. Thus, the problem is redued to integrals over t and z,

4 1476 Int J Thermophys (2013) 34: where the integration over z depends on the hoie z > z or z < z. After integration one obtains ( )] G(z, t, z, t ) = G(z, t, z, t 2 [t Θ t + z z, z z > 0 ) = ( )] (7) G(z, t, z, t ) = 2 [t Θ t z z, z z < 0 where Θ(t) is the Heaviside funtion. Equation 7 is the well-known Green s funtion for an infinite medium. Substituting Eq. 7 into Eq. 6 and using the relation given in Eq. 2, it is possible to obtain the pressure profile as p(z, t) = β 2C p z H (z, t z ) z dz + β 2C p z H (z, t + z ) z dz. (8) The first term on the right-hand side of Eq. 8 is a PA plane traveling along the positive z axis, while the seond one represents a plane wave traveling along the negative diretion Impulse Response In this setion, we find the temporal impulse response of the PA pressure onsidering the spatial heat distribution given in Eq. 4. For this we hoose the soure of exitation as an impulse, f (t) = δ(t), where δ(t) is the Dira delta funtion. Using this ondition in Eq. 8, we find the impulse response for a plane sample to be { p δ (z, t) = p [ ( 0 2 Θ(t) exp α t z )] [( ) ] t z Π 2 [ ( + exp α t + z )] [( ) ]} t + z Π + 2, (9) where p 0 2 βαq 0 /C p and Π is the retangle funtion. For simpliity, we hose z 0 = 0. Equation 9 gives the impulse response of the PA pressure for a one-dimensional plane absorber with a heating funtion proportional to the ambert Beer law. As usual, taking the onvolution of the impulse response with the temporal profile gives the PA pressure for any finite temporal distribution Finite Pulse For a finite input signal, the pressure is alulated as the onvolution of the input signal with the impulse response. For a pulsed beam with a normalized Gaussian temporal profile with a 1/e width of t p, the pressure is given by

5 Int J Thermophys (2013) 34: p( r, t) = exp [ ( 2t ) ] 2 /t p p δ ( r, t t )dt / Introduing Eq. 9 into Eq. 10, wehavefortheregion < z <, exp [ ( 2t ) ] 2 /t p dt. (10) { [ p(z, t) = p ( 0 2 Θ(t) exp α t z ) ( ) ] 2 αtp + P (z, t) 4 [ ( + exp +α t + z ) ( ) ] } 2 αtp + P + (z, t). (11) 4 For brevity in Eq. 11, we have defined P ± (z, t) =± 1 2 { [ 2 erf (t ± z ) ± αt p t p 4 ± 2 ] [ 2 erf (t ± z ) ± αt ]} p. t p t p 4 We remark that Eq. 11 was obtained without taking any analytial approximation in the onvolution integral. It will be notied that the retangle funtion in Eq. 9 will maintain a bounded amplitude of the propagating pressure for all values of t and z. Similarly in Eq. 11, the term ontaining the subtration of the two error funtions behaves as a retangle funtion, maintaining a bounded amplitude of the resulting pressure distribution as well. Without these funtions, the result would diverge at infinity. The argument (αt p /4) 2 that appears in the exponential funtions of Eq. 11 an be written as (t p /4t a ) 2, where t a = 1/(α), is the time that the perturbation takes to ross the optial absorption length. This parameter is relevant as Eq. 11 ontains the two possible regimes: short-pulse regime when t p /t a < 1 and long-pulse regime otherwise. 3 Numerial Results In the following, we onsider typial values for water, = 1500 m s 1,β = K 1, and C p = J kg 1 K 1. Now, let the thikness of the sample be = m, and the laser fluene Q 0 = 200 J m 2 ; for simpliity let z 0 = 0. In Fig. 1a h, the PA signals are shown as a funtion of α. From Fig. 1a, one an appreiate that for optially thin samples (α 10 m 1 ), the temporal widths of the PA pulses remain proportional to the thikness of the absorber and the shape remains omparable to the shapes reported in [7,8] for homogeneous heat distribution in the short-pulse regime. Thus, learly, our solution oinides with previous reports. Now, for values α = 30 m 1 and greater, the shape of the PA pressure is mainly dominated by the exponential funtions that appear in Eqs. 9 and 11, respetively, giving the shape of these pressures. It an also be appreiated that the temporal widths are proportional to 1/α. This result is a onse-

6 1478 Int J Thermophys (2013) 34: Fig. 1 PA pressure generated by a plane aousti soure as a funtion of the optial absorption oeffiient, for the impulse response and a Gaussian temporal pulse. The distane from the soure to the observation point was taken to be z = 6mm<, and the duration of the Gaussian laser pulse t p = 10 ns quene of inluding the effet of an exponentially deaying absorption funtion (Beer s law) in the heat distribution equation. This analytial result has not been previously reported.

7 Int J Thermophys (2013) 34: Fig. 2 PA pressure for the Gaussian temporal pulse for different values of the pulse duration. The distane from the soure to the observation point was taken to be z = 6mm. The absorption oeffiient is fixed at α = 100 m 1 It is interesting to notie (Fig. 1e h) that the maximum peak of the PA signal for the Gaussian temporal pulse is left shifted in time with respet to the maximum of the impulse response. This is beause the PA signals for the impulse response and the Gaussian pulse no longer oinide as one approahes the long-pulse regime. In this ase, the shape and width of the PA pulses for the Gaussian temporal pulse are proportional to the Gaussian distribution, whih is in good agreement with the longpulse regime. Figure 2a, b show the shape of the PA pressure as a funtion of the pulse duration, for a fixed value of α = 100 m 1. It an be seen that as the duration of the pulse inreases, the shape of the pressure tends to the Gaussian distribution as expeted, and the width is given by the duration of the laser pulse, whih is in good agreement with the long-pulse regime. For short pulses, the width of the pressure is given by the fator 1/α, whih is also in good agreement with the short-pulse regime. At this point it is important additionally to notie that by inluding Beer s law in our solution, it permits onfirmation of an expeted result, namely, if the energy of the laser is fixed, then the amplitude of the PA pressure diminishes as the duration of the laser pulse inreases. Before finishing this report, we want to remark that Eq. 11 inludes the two wellknown regimes of the pulsed PA effet for an aousti plane soure, short-pulse and long-pulse regimes. Moreover, it inludes the relationship between the thikness of the sample () and the absorption oeffiient of the absorber (α), i.e., if 1/α <,the width of the PA pressure is given by / for a short pulse and t p for a long pulse; if 1/α >, the width of the PA pressure is given by 1/α for a short pulse and t p for a long pulse. 4 Conlusion We solved the inhomogeneous PA wave equation that desribes the propagation of the stress generated by a plane aousti soure. In the wave equation, we have a heating funtion proportional to the ambert Beer law and spatially bounded by a retangle funtion to attain a more realisti model. Our solution showed that for high absorption

8 1480 Int J Thermophys (2013) 34: oeffiients the shape of the PA pressure is mainly dominated by an exponential funtion and the temporal width is proportional to the optial absorption length. For low absorption oeffiients, the temporal width of the PA pulse remains proportional to the thikness of the absorber and the shape remains similar to the shape of a homogeneous heat distribution. The advantage of our solution is that it inludes the two well-known regimes of the pulsed PA effet for an aousti plane soure, namely, short-pulse and long-pulse regimes. Also, we found two basi relationships between the thikness of the sample and the absorption oeffiient of the absorber: (i) if the optial absorption length is larger than the thikness of the absorber, the width of the PA pressure is proportional to the absorber thikness for a short pulse and to the laser temporal width for a long pulse and (ii) if the optial absorption length is smaller, the width of the PA pressure is proportional to the optial absorption length for a short pulse and to the laser temporal width for a long pulse. Aknowledgments This work was partially supported by the University de Guanajuato-Campus eón and by the Instituto de Cienia y Tenología del Distrito Federal, Méxio, through a grant ontrat with CCADET-UNAM. Referenes 1. W. Sigrist, K. Kneubühl, J. Aoust. So. Am. 64, 1652 (1978) 2. W. Sigrist, J. Appl. Phys. 60, 805 (1986) 3. M. Xu,.V. Wang, Rev. Si. Instrum. 77, (2006) 4. G. Paltauf, H. Shmidt-Kloiberpl, J. Appl. Phys. 82, 1525 (1997) 5. A.C. Tam, Rev. Mod. Phys. 58, 381 (1986) 6..V. Wang (ed.), Photoaousti Imaging and Spetrosopy, hap. 3 (CRC Press, Boa Raton, 2009). 7. G.J. Diebold, T. Sun, M.I. Khan, Phy. Rev. ett. 67, 3384 (1991) 8..V. Wang (ed.), Photoaousti Imaging and Spetrosopy, hap. 1 (CRC Press, Boa Raton, 2009) 9. Z. Zhao, Pulsed Photoaousti Tehniques and Gluose Determination in Human Blood and Tissue (Ph.D. Thesis Dissertation, University of Oulu, Oulu, 2002)