Detection of laser generated ultrasonic waves in ablation regime by plane and confocal Fabry- Perot interferometers: Simulation

Transcription

1 INDT6 Proceedings of the 3 rd Iranian International NDT onference Feb -, 6, Olympic Hotel, Tehran, Iran INDT 6-T3 More Info at Open Access Database Detection of laser generated ultrasonic waves in ablation regime by plane and confocal Fabry- Perot interferometers: imulation Vahid Haghighi, edighe Malekmohamadi Department of Physics, University of Isfahan, Isfahan, Iran address: vahidreal@gmail.com Department of Physics, University of Isfahan, Isfahan, Iran address: sm-364@gmail.com Abstract Generation and detection of ultrasonic waves by laser in ablation regime have been simulated. Ultrasonic wave has been simulated by the ablation regime. By high laser power density, the ultrasonic pulse shape is of Heaviside step function shape. Interferometer is one of the famous methods to detect ultrasonic wave. We have simulated received waves by both plane and confocal Fabry-Perot interferometers. In this case the confocal Fabry-Perot interferometer would have comparable sensitivity and a better frequency response. This configuration would be more compact and more practical. Keywords: aser ultrasonic, Ablation, Pulsed laser, Fabry-Perot interferometer Introduction aser generation of ultrasound has been utilized to evaluate and characterize materials nondestructively in a broad range of engineering applications [].Ultrasound generation by means of laser irradiation has been studied since White discovered the phenomenon in 963[]. The combination of laser generation with optical detection provides a completely remote ultrasonic system for material inspection []. The optical detection presents many advantages: the vibrating surface can be investigated at a distance without disturbing the acoustic field, the measurement is local and can be calibrated from optical wavelength, and it has a wide pass band. Furthermore, as there is no mechanical contact, a large area can be scanned by moving either the object or the light beam [3]. In this paper, we have simulated generation and detection of ultrasonic wave by laser in ablation regime. In addition, influence of various parameters in acoustic waveform has been investigated in both time and frequency domains. Then we have simulated received waves by both plane and confocal Fabry-Perot interferometers. Finally properties of these methods in various conditions have been discussed. Theory When a low power laser pulse strikes onto a solid surface, one part of the incident energy is absorbed and heat converted. In that case the ultrasonic waves are generated in specimen under thermo-elastic and ablation regimes. In thermo-elastic regime, the fast temperature rise, localized near the surface produces a thermal expansion which in turn creates a transient elastic stress field. While in ablation regime, by increasing the optical power densities, the solid surface is melted and evaporated. Momentum transfer produced by removal material gives rise to a force normal to the surface [4, 5]. Figure : The laser beam of energy Q and time dependence δt is incident on the surface at z=. The ultrasonic waveform can be obtained by solving the wave propagation equations. We have assumed that the laser impulse is incident on the free plane surface at z = of an elastic material as shown in Fig.. The elastic and thermal constants of the material are as followsμ and ame constants, ρ density, k thermal diffusivity, K thermal conductivity, specific heat capacity, and α coefficient of thermal expansion. The laser pulse arriving at point has an energy Q and varies with time as a function (where is a Dirac impulse) [5]. eady [6] has studied the effects of the absorption of laser radiation having a power density W H(t) on an opaque surface, where H(t)is a Heaviside unit step. et us consider a laser pulse having the following power density[5,6]:

2 W t) W H( t) H( t ) ( where K AW 4 is the laser pulse duration. The material will vaporize for absorbed power densities such that [5]: v ( T v Ti ) / () INDT6-T3 () At low incident power density, the material drops rapidly to below its vaporization temperature after the end of each laser pulse and as a result the normal stress is pulsed. However, at high power density, vaporization continues for a long time after the end of each laser pulse. The variation in the normal stress as a function of time lies between two limiting cases: the same shape of each laser pulse and a pulse shape of the Heaviside unit step type [5,8]. In order to model vaporization process, the force equivalent to the vaporization will be assumed to have amplitude to act normally to the surface, and to vary as a pulse or as a step H (t). Under ablation conditions, displacement along the axis of symmetry for a normal force a is given by[5, 7]: fa g g uz(,, t) t t (3) and are longitudinal and shear waves velocity respectively and g, g Where is the sample thickness, given as follows: ( )( ) / [( ) 4 ( ) ( g / ) ] H t g 4 ( )( ) s H t / / [( ) 4 ( ) ( ) ] t t, For a laser pulse having finite shape, the displacement is obtained by convolution of Equations 3 with [5]. Directivity patterns of longitudinal and shear waves are employed to determine the position of detector. These patterns are measured at the plane which is normal to the sample surface and contains the laser incident point. We assumed that thermal conductivity and source expansion is negligible. Therefore ultrasound source can be assumed as point expander. In this case, the directivity patterns of the ultrasound waves are obtained from the following equations [4, 9, and ]: for directivity of longitudinal wave: k' cos ( k' sin ) u ra ( ) / / ( k' sin ) 4 sin ( sin ) ( k' sin ) (4) and for directivity of shear waves: / sin ( k' sin ) u A ( ) / / k'( sin ) 4sin ( sin ) ( k' sin ) (5) Where k' and is the angle of propagation direction with respect to the normal to the surface. Ultrasonic waveform In our simulation an aluminum sample with 5mm thickness has been considered. The power density required for ablation threshold were calculated for the steel and aluminum by using the equation and the results are summarized in Table. Table : Ablation threshold calculated for Aluminum and teel. sample Ablation threshold aser pulse aser beam diameter aser pulse energy (mj) (MW/cm) duration(ns) (mm) Aluminum teel fa are First, we assume one laser pulse with time dependence as a delta function. In this case we can directly use the equation 3 to obtain ultrasound wave format the epicenter and on the other side of irradiation point as shown in Fig. (a). The ongitudinal wave arrives on the other side of sample at time t. Time dependence of these waves has the same

3 t INDT6-T3 shape of laser pulse. The shear waves are detected at time. The amplitude of shear waves in ultrasonic waveform is weaker than longitudinal waves as shown in Fig. (a).ultrasonic displacement for high power density of laser pulse as a stepped pulse shape H(t)as simulated in Fig. (b). In this case, the amplitude of the shear wave is more than longitudinal wave. (a) x -9 t ts (b) t time (s) Time(s) ts x -6 x -.5 (c) t Time (s) ts x -6 Figure : Ultrasonic displacement at the epicenter under ablation conditions. a) Delta function force. b) tep force. c) Gaussian force. Now, we consider more practical case that is i.e. considering the laser pulse with Gaussian shape time dependence. In this case the ultrasound displacement for a Gaussian laser pulse of duration nsis simulated in Fig. (c). Temporal changes of longitudinal wave are similar to the laser pulse time dependence and its amplitude is greater than shear waves as shown in this figure. We have shown influence of laser pulse energy in ultrasonic waveform in both time and frequency domain in Fig. 3.

4 INDT6-T3 As it can be seen in this figure, ultrasonic waveform will be invariant and only the amplitude of wave will be change for various laser pulse energy when laser power density remains constant. Moreover we see that frequency band width is within MHz range. x F=N F=N time (s) Time(s) x -6 5 x -3 Amplitude (a.u.) frequency (MHz) Figure 3: The ultrasonic signal for two different forces in both time and frequency domains.. Optical detection of ultrasonic signals As we mentioned already the optical detection methods present many advantages. Most of these methods are based on interferometer procedures such as Fabry-Perot interferometers. The Fabry-Perot interferometer is based on small frequency changes in the scattered light from the sample surface. Thus, the output of this detection method is proportional to the "speed" of the surface [4]. As it is well known the Fabry Perot interferometer consist of two high reflective parallel mirrors with separation of h[].we have simulated receiving of the ultrasonic signals with both plane and confocal Fabry Perot interferometer methods as explained below. Plane Fabry-Peort interferometer (PFPI) PFPI is containing of two plane parallel mirrors with distance of h. The output of interferometer is inclusive of some frequency peaks. The transmission modes of a PFPI with respect to frequency are given in Fig. 4.The operation point of interferometer is chosen at half height of one of these frequency peaks. The high finesse and resolution are obtained with higher reflectivity of mirrors []. Figure 4: schematic of one laser ultrasonic inspection by optical detecting method. The change in frequency of light reflected or scattered normally from a moving surface is given by the usual Doppler shift formula [4]: δ D = λ t Where u is the normal component of the surface velocity and λ is the optical wavelength. Thus the change in the output of a Fabry-Perot interferometer is proportional to surface velocity. Assuming no light losses, the change in output is given by [4]: = + 4 / (7) (6)

5 INDT6-T3 = (8) where is the light input to the interferometer, a is the amplitude of surface displacement and ν w = c/hf where is the interferometer finesse. Fig. 5 transmission =98.5 % avity length=mm Finesse=8 free spectral range=.3ghz resolution=7. MHz frequency x 4 Figure 5: The transmission modes of a plane Fabry-Perot interferometer with laser speckle radius of 3mm. We have simulated the ultrasound signal that receiving by plane Fabry-Perot interferometer as shown in Fig. 6. The ratio of output to input power of Fabry Perot is in order of. output of Fabry-Perot(Watt).4 x W= mw =98.5 % avity length= mm f= MHz time (s) x -6 Figure 6: The received ultrasound signal with plane FPI. onfocal Fabry-Perot interferometer (FPI) One important factor in the optical detection of ultrasound waves is light gathering power from extensive sources. High power response for a plane Fabry-Perot interferometers equivalent to low light gathering power [4].An alternative Fabry-Perot arrangement with a confocal configuration is preferred for measurements on rough surfaces because of its much greater etendue, i.e. ability to accept input beams appreciably inclined to the axis without loss of resolution [4, ]. A FPI consists of two coaxial partially transmitting mirrors of equal radii of curvature, separated by a distance []. Figure 7: Path of rays in a confocal FPI: (a) incident beam parallel to the FPI axis; (b) inclined incident beam []. Neglecting spherical aberration, all light rays entering the interferometer parallel to its axis would pass through the focal point F and would reach the entrance point P again after having passed the FPI four times. Because of spherical aberration, rays with different distances ρ from the axis will not all go through F but will intersect the axis at different positions F depending on ρ and θ. Also, each ray will not exactly reach the entrance point P after

6 INDT6-T3 four passages through the confocal FPI since it is slightly shifted at successive passages. However, it can be shown that for sufficiently small angles θ, all rays intersect at a distance ρ(ρ, θ) from the axis in the vicinity of the two points P and P' located in the central plane of the FPI (Fig. 7) []. Interference only occurs between beams arising from alternate transits around the interferometer cavity. There are thus two output beams whose amplitudes are governed by multiple interference between beams arising from odd and even number of transits respectively. When the number of transits is large, each of these evidently has half the amplitude calculated for the output beam of the plane Fabry-Perot interferometer, and hence a quarter of the intensity. These result in a general reduction in output by a factor of two compared with the plane Fabry-Perot interferometer [4]. Frequency shift through the ultrasonic wave propagation changes the output power of the FPI that can be obtained from the following equation [4]: = + 4 / (9) = = F () = c c F = () o the maximum value of is. We have obtained the transmission intensity of the FPI when the mirrors separation and their reflectivity are similar to what we have done in previous section. These results are given in Fig. 8. transmission =98.5 % onfocal avity length=mm Finesse=4 free spectral range=.75ghz resolution=7. MHz frequency(hz) x 4 Figure 8: The transmission modes of a confocal Fabry-Perot interferometer with laser beam radius of 3mm. output of Fabry-Perot(Watt) 7 x W= mw =98.5 % onfocal cavity length = mm f= MHz time (s) x -6 Figure 9: The received ultrasound signal with FPI. Moreover, we have simulated the ultrasonic signal detected by the FPI infig.9. Although, the ratio of output to input power of FPI is in the order of, but for the rough surfaces inspection the FPI is preferred because of its much greater light gathering power.

7 INDT6-T3 Figure : Photoelectric recording of the spectral light power transmitted of a scanning confocal FPI. To see the effect of this light gathering power we consider Fig..We see that the solid angle accepted by the detector behind the aperture with radius b is Ω = π r. The light power transmitted to the detector is proportional to the product of the solid angle Ω and area A in the central plane, which is imaged by the lens onto the aperture. o the étendue is given by[4, ]: = Ω = F = Etendue quantity of FPI is increased with increasing the curvature radius of the mirrors. This quantity for the plane FPI is obtained from following equation [4]: = hf = h where is the mirrors aperture radius. The resolving power of confocal and plane FPI can be obtained from the following equations []: Δ = F (FPI) (4) = Δ (3) () (PFPI) (5) (Where D is the mirror diameter of PFPI). We see that while the spectral resolving power is proportional to Et for the FPI, it is inversely proportional to Et for the PFPI. This is because the étendue increases with the mirror separation h for the FPI but decreases proportional to /h for the PFPI. We have shown the étendue and the spectral resolving power of few FPI configurations in Table as an example. It can be seen that for a given light-gathering power, the FPI can have a much higher spectral resolving power ( ) than the PFPI. Moreover, for the same étendue as a confocal interferometer of the same spacing and mirror reflectivity (and hence approximately the same line width), a mirror diameter about five times the separation of them would be required for PFPI. This is clearly impractical in most circumstances. There for FPI is more suitable for rough sample surfaces inspection. Table :The étendue and the spectral resolving power of few FPI configurations. Etendue (mm ) pectral resolving power Mirrors radius (mm) Mirrors curvature radius (mm) Mirrors separation (mm) Mirror reflectivity % onclusion Generation of ultrasound waves under ablation conditions is more considered in some applications that the normal propagation of ultrasonic waves is necessary. In this regime, we have shown that the waveform changes by increase of

8 INDT6-T3 laser power density. The ultrasonic pulses have the same shape of the laser pulse at low power density. Whereas for high laser power density, the ultrasonic pulse shape is of Heaviside step function shape. At the constant power density, with increasing the laser pulse energy the ultrasonic signal amplitude is increased while its waveform remains unchanged. Under ablation conditions, the longitudinal wave is generated normal to the surface and has a little interference with shear waves. Moreover, detection of ultrasound waves by plane and confocal Fabry-Perot Interferometers has been studied in this paper. For a mirror-like reflecting specimen surface, we may choose a plane Fabry-Perot interferometer to ultrasound detection since etendue is not a consideration. While for rough surfaces, detection of these waves by confocal FPI is more efficient because of its greater light gathering power. In this case the confocal Fabry-Perot interferometer would have comparable sensitivity and a better frequency response. This configuration would be more compact and more practical. Acknowledgments I highly appreciated to Dr. oltanalketabi since to all of his contribution. Also sincerely thanks to Optic lab in Isfahan University. Nomenclature µ ame constants ame constants, ρ density, α coefficient of thermal expansion. Function Dirac impulse λ F Ω k K Q W A D Angle of propagation direction Optical wavelength Interferometer finesse the solid angle thermal diffusivity, thermal conductivity, pecific heat capacity, energy power density Thickness, ongitudinal wave velocity hear waves velocity adii of curvature area in the central plane Étendue Mirrors aperture radius Mirror diameter of PFPI eferences. MiBao, I. harles Ume, Parametric tudies of aser Generated Ultrasonic ignals in Ablative egime: Time and Frequency Domains, J. Nondestruct. Eval. () M. White, Generation of Elastic Waves by Transient urface Heating, J. Appl. Phys. 34 (693) B. ulshaw, aser Ultrasound for the Non ontact haracterization of themechanical Properties of Materials, International ymposium on aser Ultrasonics, (8). 4..B. cruby, aser Ultrasonics Techniques and Applications, New York, J.D. Aussel, Generating acoustic waves by laser: theoretical and experimental study of the emission source, Ultrasonics, 6(988) J.F. eady, Effects of High-Power laser adiation, Academic Press, New York, 97, pp J.E.inclair, solutions for point multipole sources in an elastic half-space, J. Phys. D (979) J. Dewhurst, D.A. Hutchins,.B. Palmer, and.b. cruby, Quantitative measurements of lasergenerated acoustic waveforms, J. Appl. Phys. 53 (98) J.Daviest,.Edward, G.. Taylor,.B Palmer, aser-generated ultrasound: its properties, mechanisms and multifarious applications, J. Phys. D Appl. Phys. 6 (993) N. Hopko, I. harles Ume, aser Ultrasonics: imultaneous Generation by Means of Thermoelastic Expansion and Material Ablation, Journal of Nondestructive Evaluation, 8(999) 9-98.

9 INDT6-T3. Wolfgang Demtröder, aser pectroscopy Vol. : Experimental Techniques, pringer-verlag, Berlin Heidelberg, 8.