The mechanism for continuum polarization in laser induced breakdown spectroscopy of Si(111)

Spectrochimica Acta Part B 74–75 (2012) 3–10

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The mechanism for continuum polarization in laser induced breakdown spectroscopy of Si(111) John S. Penczak Jr. a, Yaoming Liu a, Richard D. Schaller b, c, Daniel H. Rich d, Robert J. Gordon a,⁎ a

Department of Chemistry, University of Illinois at Chicago, Chicago, IL 60607‐7061, USA Argonne National Laboratory, Center for Nanoscale Materials, Argonne, IL 60439, USA Department of Chemistry, Northwestern University, Evanston, IL 60208, USA d Department of Physics and The Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev, P.O.B 653, Beer-Sheva 84105, Israel b c

a r t i c l e

i n f o

Article history: Received 22 November 2011 Accepted 19 June 2012 Available online 26 June 2012 Keywords: Polarization-resolved laser-induced breakdown spectroscopy PRLIBS Silicon ablation

a b s t r a c t Polarization of the plasma luminescence produced by both nanosecond and femtosecond laser ablation of Si(111) was analyzed under different conditions of fluence and detection geometry. It is shown that the luminescence is partially polarized and is directed in the plane of the crystal. The time evolution of the plasma emission signal was also investigated with the use of a streak camera. The mechanism for polarization is proposed to be preferential reflection of s-polarized light (i.e., light polarized normal to the plane of laser incidence) by the melted surface, in agreement with the Fresnel equations. Earlier reports of much stronger polarization are shown to be erroneous. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Laser-induced breakdown (LIB) spectra generally consist of a series of sharp peaks riding on top of a continuous background [1,2]. The peaks arise from fluorescent transitions of ions and neutral atoms and molecules produced in a plasma, whereas the continuum is usually attributed to ion-electron recombination and bremsstrahlung. Various methods have been implemented to suppress the continuum so as to increase the signal/background ratio. In particular, time-gating of the detected signal can reduce the continuum, which generally decays much more rapidly than the discrete fluorescence. Several years ago, Liu et al. [3] discovered that under certain conditions the continuum emission from Si(111) produced with femtosecond (fs) pulses from an 800nm Ti:Sapphire laser is strongly polarized, suggesting the possibility of suppressing the background by simply placing a polarizer before the detector. The polarization was found to be strongest at short wavelengths, increasing to over 90% below 350 nm. Later studies showed this effect to be a general phenomenon, occurring also for metals [4,5] and dielectrics [5]. Subsequently, Liu et al. [6] and Majd et al. [7] reported that a polarized continuum is generated also by nanosecond (ns) pulses, using, respectively, the 532nm second harmonic and the 1.064 μm fundamental frequency of a Nd: YAG laser. Most recently, Asgill et al. [8] attempted to repeat these findings, using 1.064μm ns pulses at a fluence of 1020 J/cm2 to ablate Cu, and found little or no polarization. In an effort to reconcile this discrepancy, Penczak et al. [9] extended the fluence range of their earlier ns ⁎ Corresponding author. E-mail address: [email protected] (R.J. Gordon). 0584-8547/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sab.2012.06.023

studies of Al and found that the magnitude of the polarization, P, decreases monotonically with fluence between 5.4 and 497 J/cm2. They also found that P is greater for an s-polarized laser than for p-polarization. With these differences in mind, they showed that the various ns studies are mostly consistent. Despite the seeming consistency between the findings of the various groups, the mechanism for producing strong continuum polarization remained a mystery. The observations of the Gordon group [3–6,9] have some of the characteristics of scattering of the incident laser beam by the surface. The emission appears in a cone around the specular direction, and the angle of the polarization vector is consistent with dipole emission from a surface (i.e., a p-polarized laser produced p-polarized emission, an s-polarized laser produced s-polarization, and intermediate polarizations produced the corresponding projections onto the observation plane). Nevertheless, the increasing polarization at shorter wavelengths seemed to rule out a simple reflection or scattering mechanism. Accordingly, they speculated that the polarization is produced either by some alignment process in the plasma or possibly by some frequency up-conversion of the laser beam in the plasma. While the findings of Majd et al. are in general agreement with of those of Gordon and co-workers, they found that the emission is s-polarized and is independent of the polarization state of the laser [7]. They nevertheless also favored a plasma alignment mechanism. Asgill et al. [8] attributed the very small s-polarization (P≤5%) that they observed in some experiments to selective reflection by the surface of s-polarized light generated in the plasma, in accordance with the Fresnel equations. The present study is a further attempt to elucidate the polarization mechanism and to reconcile the observations of the various groups. We reason that any polarization produced in the plasma itself, either

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by some directed motion of electrons or by frequency up-conversion of the laser beam, would be short-lived, on a time scale comparable to that of the laser pulse. To test this hypothesis, we used an ultrafast streak camera to measure the time-dependence of the polarized emission produced by ablating a Si(111) sample with a fs laser. Observation of polarization for periods substantially greater than the laser pulse would argue strongly against a plasma alignment mechanism. 2. Methods Three sets of experiments were performed, each with its own apparatus. In all the experiments, Si(111) wafers were ablated in air with single laser shots, and the emitted light was collected and analyzed with a spectrograph. The first set of experiments was performed at the University of Illinois at Chicago (UIC), using the apparatus depicted in Fig. 1. Laser pulses of ~65 fs duration were generated by a Ti:Sapphire laser (Spectra Physics Tsunami oscillator and Spitfire amplifier). The output of the regenerative amplifier had a peak intensity at 805 nm and a bandwidth of 24 nm (full width at half maximum), falling to 1% at 772 and 834 nm. A half-wave plate and polarizer were used to reduce the pulse energy, E, to within the range of 10–100 μJ, with a shot-to-shot variation of less than 3%. The polarization state of the laser was varied using a second half-wave plate. The laser beam was focused onto the sample with a microscope objective (10×, NA=0.25). The radius of the focused beam, ω0, was measured with a scanning knife edge and was found to have a value of 1.8±0.2 μm. This quantity is the Gaussian radius of the electric field, corresponding to an irradiance of −2r 2 =ω0 2

Iðr Þ ¼ I 0 e

;

ð1Þ

and a peak fluence of 2

F ¼ 2E cosθin =πω0 ;

ð2Þ

where θin is the angle of incidence. The sample was mounted on an automated sub-μm precision xyz translation stage. The stage advanced 100 μm between laser shots to expose a fresh surface for each pulse, with an average of 20 shots taken for each data point. The angles of incidence and detection (θdet) were manually adjusted by moving the translation stage and laser beam optics. A notch filter (Chroma Technology, E690SPUV6) was used to block scattered laser

Fig. 1. Schematic drawing of the apparatus used at UIC, including half wave plates (λ/ 2), polarizers (P1 and P2), lenses (L1, L2, and L3), and an interference filter (F).

light. The filter transmitted >85% of the emitted light between 390 nm and 595 nm and rejected the laser light with transmittances of 1×10 −6, 2.3×10 −6, and 2.3×10 −5 at 772, 805, and 834 nm, respectively. The ablating pulse generated a plume, consisting of electrons, ground state and electronically excited atoms and ions, molecules, and particulates. Light emission from this plume was focused onto the 600 μm entrance slit of a spectrograph (Spectrapro 2300i, Princeton Instruments), in which the spectrum was dispersed by a 300 lines/ mm grating blazed at 500 nm, and was recorded with a non-gated, thermoelectrically cooled CCD (PIXIS 400, Princeton Instruments) camera. A Glan-Thompson polarizer (Red Optronics, GMP-6015) mounted on a motorized rotation stage was inserted in front of the entrance slit of the spectrograph to measure the polarization of the plasma emission. The second set of experiments was performed at the Center for Nanoscale Materials at Argonne National Laboratory (ANL). The primary difference between the apparatus at ANL and UIC is that the ANL setup used an ultrafast streak camera (Hamamatsu C5680) mounted on a spectrograph (Acton SP2150) to measure the emitted light as a function of time, with a resolution of 1.5% of the temporal gate (e.g. 750 ps for a time window of 50 ns, and a maximum resolution of 3 ps for a 140 ps window) and a slit width of 60 μm. As before, a Ti:Sapphire laser (Spectra-Physics MaiTai oscillator and Spitfire Pro amplifier, 35 fs pulse width) was used to ablate the sample. The laser was triggered at a rate of 18 Hz, while the sample was translated continuously at a speed great enough to ensure that the irradiated spots were separated by a minimum of 100 μm. The pulse energy was set between 50 and 100 μJ by means of a variable neutral density filter. The same notch filter and sample controller were used as in the UIC setup. The third set of experiments, performed at UIC, used the second harmonic of a Nd:YAG laser (Continuum Surelite, 532 nm, 4 ns pulse width) to ablate the sample. A variable neutral density filter was placed before the focusing optics to set the pulse energy of the beam between 9 μJ and 4 mJ. The laser was focused onto the sample by a 100 mm focal length convex lens. The focused laser beam had a radius of ω0 =22.6±1.4 μm, which was measured with a scanning knife edge. An interference filter (Thorlabs, FES0500) was mounted directly in front of the entrance slit to block scattered laser light. This filter has a transmission range of 380 to 500 nm and a transmittance of 1.3×10 −5 at 532 nm. The sample controller and spectrograph were the same as in the first apparatus, except that the slit width of the spectrograph was decreased to 200 μm. Special precautions were taken to ensure that the signal was not contaminated by artifacts such as scattered laser light or polarization contributed by the detection optics. The polarization preference of the spectrometer grating was calibrated using an unpolarized white light source, so that any effects caused by differences in diffraction efficiencies of s- and p-polarization could be accounted for [10]. As an additional precaution, an experiment was performed using a 2 m long, single-mode fiber (Ocean Optics, QP600-2-UV–VIS) placed in the optical path between the polarizer and spectrometer to act as a depolarizer. The degree of linear polarization of the laser source was 99.8%. We found that our particular fiber optics geometry resulted in the depolarization of the laser source by ~93%. The polarization of the emitted light was then measured with the fiber placed between the polarizer and the spectrograph. We obtained nearly identical results as those using the grating calibration without the fiber, thereby ruling out the possibility that the monochromator/detector affected the polarization measurements. We also carefully checked the polarization properties of the optics (lenses and filter) in the detection path to confirm that they did not have any polarization preference. The entrance slit of the spectrometer was widened to 600 μm so that any misalignment of the polarizer would not result in part of the signal being lost on the edge of the slit. We also verified the

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results using a photomultiplier tube (Hamamatsu R212), utilizing interference filters for different wavelength selections in the same range as the CCD measurements.

3. Results Fig. 2 shows typical LIB spectra of Si recorded at UIC, using either an s-polarized (upper panel) or p-polarized (lower panel) Ti-Sapphire laser to ablate Si(111) crystals. In both panels, the upper trace was obtained using the polarizer to transmit only s-polarized light, while the lower trace corresponds to p-polarized transmission. These data show that the emitted light is preferentially s-polarized. The most intense peaks are labeled in panel (a), and the corresponding transitions are listed in Table 1 [10]. To quantify the magnitude of the polarization, spectra were recorded with the polarizer angle, α, set at different values in 20° intervals. The intensities at each wavelength were fit to the Malus function, A+Bcos 2(α−α0), as illustrated in Fig. 3. The magnitude of the polarization, P, equals B/(2A+B), and the direction of the polarization is determined from the value of α0. Calibration of the polarizer with a linearly polarized laser gave a value of α0 =44° for s-polarized light. (Hereafter we will refer to the polarization angle as α0). The resulting P and α0 spectra are shown in Fig. 4 for s- and p‐laser polarizations. The top panel shows a typical polarization spectrum, in which P is seen to have a background value of 12–20% that increases gradually with wavelength. The slightly larger value of P for s-polarized excitation is not statistically significant. (In some runs p-polarized excitation gave slightly larger P.) Superposed on this background are structural

Fig. 2. Laser-induced breakdown spectra obtained with an (a) s-polarized and (b) p-polarized fs laser. The upper (red) and lower (black) curves correspond to spectra taken with the polarizer set to admit s-polarized or p-polarized light, respectively. The peaks were assigned to neutral and ionic transitions, using the NIST atomic line database.

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Table 1 Assignment of peaks in the LIB spectra. Label λ (nm) a b c d

Assignment

385.6 Si I 3s3p2-3s24p 390.6 Si I 3s23p2-3s23p4s 400 Second harmonic of Ti: Sapphire laser 413.1 Si II 3s23d-3s24f

Label λ (nm)

Assignment

e f g

462.1 Si II 3s24d-3s27f 500.6 Si I 3s23p4s-3s23p6p 505.6 Si II 3s24p-3s24d

h

455

H′-X (5,0) band of the Si dimer (?)

Wavelengths for lines ‘a–d’ and ‘e–g’ are the values listed in the NIST database, ref. [10]. The assignment of feature ‘h’ is from ref. [14].

features, some of which are correlated with peaks in the LIB spectrum. Between 395 and 415nm there is a strong maximum in P produced by the s-polarized laser and a smaller minimum in P produced by the p-polarized laser, which lie in the vicinity of peak ‘c’ of the LIB spectrum (Fig. 2a). Features ‘a’, ‘b’, and ‘d’ may also be identified with peaks in the LIB spectrum, whereas the small dip at 448nm does not appear to have a resolvable counterpart in the LIB spectrum. Fig. 4b shows the polarization angle spectrum obtained from the fitted parameters in the Malus function. The value of α0 lies within a few degrees of the s-plane (i.e., normal to the plane of laser incidence) and displays a gradual falloff with increasing wavelength. Some of the structure seen in the P spectrum is evident here as well. Fig. 5 displays time-resolved measurements of the emission intensity, polarization, and polarization angle obtained with the streak camera at ANL. These data are slices of the full spectra taken at 480 nm. The top panel shows that the signal reaches a maximum at 2.9 ns and then falls to 1/e of that value within 1.0 ns. The middle panel shows that P is fairly constant (5–10%) during the first 10 ns and increases slowly to a value of 15–25% at 45 ns. The bottom panel shows that α0 remains within ±5° of the s-plane throughout the measured time range. Additional experiments were conducted using the ns (Nd:YAG) apparatus at UIC. Spectra produced with s- and p-polarized laser beams are shown in Fig. 6. Although the fs and ns spectra have features in common, there are significant differences between them. For example, peaks ‘a’ and ‘b’ in the fs spectrum differ in intensity by almost a factor of 2, whereas in the ns spectra their intensities are nearly equal. Also, peak ‘e’ is absent in the ns spectra, and peak ‘h’ is present only in the ns spectrum. (Peaks ‘f’ and ‘g’ are blocked by the 532 nm filter and therefore cannot be seen in the ns spectrum.)

Fig. 3. A typical plot of the emission intensity as a function of polarizer angle, using the UIC fs apparatus. The s-polarized laser has a fluence of 503.6J/cm2 and an angle of incidence of 30°, with the detector positioned perpendicular to the incident laser beam. The data are for a wavelength of 460nm. The curve is a least squares fit of the Malus function.

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Fig. 4. Polarization (panel (a)) and polarization angle spectra (panel (b)), showing the magnitude and angle of polarization measured at each wavelength, taken under the same conditions as Fig. 2. The vertical tic marks in panel (a) correspond to the locations of the assigned peaks in Fig. 2a. The horizontal line in the bottom panel corresponds to s-polarized emission. The experimental uncertainty in P is ±1.5%.

The P and α0 spectra shown in Fig. 7 were determined in the same manner as in the fs experiments and have similar properties. The continuum polarization ranges from 12 to 15% between 380 and 495 nm, increasing slightly with wavelength. The polarization angle of the continuum lies near the s-plane and varies weakly with wavelength. As before, the difference between s- and p-polarized excitation is not significant. Both P and α0 have structure correlated with peaks in the LIB spectrum. Closer examination of this structure reveals asymmetric profiles in both P and α0. There is a hint of such asymmetry in the fs data as well, although the poorer signal/noise ratio makes this effect more difficult to discern. In all the experiments described so far, the laser was incident at an oblique angle 30° from the normal, and the emission was detected perpendicular to the laser beam. To further characterize the polarized emission, additional measurements were performed with ns excitation in which the detection angle was varied, with θin fixed at 40°. These results are plotted in Fig. 8 for 488 nm emission. The polarization was found to have a maximum value near θdet =80°, falling off to near zero at θdet =0. The polarization direction of the emitted light was always in the s-plane. In a final set of experiments, we measured the fluence dependence of P. Using the ns apparatus, we varied the fluence from 1 J/cm 2 to over 400 J/cm 2. We found that although the general appearance of the polarization spectrum is for the most part independent of fluence, the value of P is greatest at low fluences. Fig. 9 shows a slice of the spectra at 460 nm as a function of fluence. This plot shows that P remains relatively constant at low fluence and then drops monotonically as the fluence is increased past approx. 40 J/cm 2. The angular study shown in Fig. 8 was performed in the region of the low fluence plateau.

Fig. 5. Time-resolved polarization data taken with the ANL fs laser at a fluence of 1700 J/cm2 and an angle of incidence of 30°. The data are for a wavelength of 480 nm. The panels show (a) the signal intensity, (b) the magnitude of the polarization, and (c) the polarization angle. The inset in (b) is a higher resolution scan of the first 2 ns of the polarization. The horizontal line in the bottom panel corresponds to s-polarized emission.

4. Discussion We begin our discussion by comparing the fs and ns LIB spectra shown in Figs. 2 and 6. Most of the features may be explained semi-quantitatively on the basis of the line intensities and Einstein A-coefficients listed on the NIST Atomic Spectra Database. More care is needed, however, to identify peaks ‘c’, ‘e, and ‘h’. The first of these is prominent only in the s-polarized emission produced by the s-polarized fs laser (Fig. 2a). More extensive studies performed with single and double pulses at varying angles of incidence [11] provide strong evidence that this peak is the second harmonic (SH) of the 800 nm Ti:Sapphire laser. The most likely mechanism for producing this emission is surface SH generation (SSHG [12]), although some contribution from the plasma-generated SH is also possible [13]. (The latter is more commonly produced by a p-polarized laser, which couples

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Fig. 6. Laser-induced breakdown spectra obtained with an s-polarized (panel (a)) and a p-polarized (panel (b)) ns laser at a fluence of 27.0 J/cm2 and an angle of incidence of 30°, with the detector positioned perpendicular to the incident laser beam. The upper (red) and lower (black) curves correspond to spectra taken with the polarizer set to admit s-polarized or p-polarized light, respectively.

efficiently to the plasma.) The observation of SHG far from the specular angle is most likely caused by a ripple produced by a Rayleigh-Taylor instability in the laser-generated plasma [11]. The shoulder at 400 nm seen in the ns spectrum may be assigned to several weak Si+ lines that coincidentally lie in this region [10]. Feature ‘e’, seen only in the fs spectrum, is assigned to fluorescence from a highly excited state of Si +, lying at 122,655 cm −1. Population of this state requires ten 800 nm photons and seven 532 nm photons. The observation of this feature only with the fs laser may be explained by the six orders of magnitude greater irradiance of the short pulse. Feature ‘h’, which appears only in the ns spectrum, requires a different explanation. Although there is a line in the NIST database at 455.3 nm having a large Einstein coefficient, the very high energy of the upper level (175,336 cm −1) makes it difficult to rationalize its population with the ns and not the fs laser. A more attractive assignment is to fluorescence from the neutral Si dimer. Recently Ojha and Gopal [14] discovered several new electronic states of Si2 populated by 532 nm ablation of Si. The most intense feature in their spectrum recorded in air matches the 455 nm peak in Fig. 6. Although we cannot explain its absence in the fs spectrum, it is well known that clustering in an ablation plume is sensitive to the temporal properties of the pulse [15,16]. We turn now to the possible sources of the observed polarization. One possibility is that it is derived from the polarization of the laser beam itself, either by scattering from the surface or by some non-linear frequency up-conversion process in the plasma. A second possibility is that the polarization is caused by an anisotropic velocity distribution of electrons in the plasma, which, upon recombining

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Fig. 7. Polarization (panel (a)) and polarization angle (panel (b)) spectra recorded under the same conditions as Fig. 6. The vertical tic marks in panel (a) correspond to the locations of the assigned peaks in Fig. 6a. The horizontal line in the bottom panel corresponds to s-polarized emission.

Fig. 8. Polarization of 488 nm emission as a function of detection angle measured with the ns apparatus, using either an s- (black symbols) or p-polarized (red symbols) laser. Data were recorded at a fluence of 21.6 J/cm2 and an incident angle of 40°. The detection angle is measured with respect to the sample surface normal. Vertical error bars are a single standard deviation of the polarization measurement, and the horizontal error bars are an estimate of the experimental uncertainty in the detection angle. The curves Rs and Rp are the reflectance of liquid Si calculated from the Fresnel equations for s- and p-polarized light, respectively, assuming a refractive index of 2.89+4.98i. The curve labeled P is the polarization that would be produced by an unpolarized beam reflected from the liquid Si surface, whereas curve P′ is the calculated result if half of the light reaches the detector directly and half is reflected by the surface.

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light from the plume strikes the surface at all angles and is reflected at the specular angle. For a transparent material, the Fresnel equations predict that p-polarized light is totally transmitted at the Brewster angle, so that the reflected light is 100% s-polarized. For an absorbing medium, the reflectance of p-polarized radiation falls to a non-zero minimum at the Brewster angle, and the reflected beam has maximum s-polarization at that angle. For a complex index of refraction, n=η+iκ, the Fresnel equations give for the reflectances of s- and p-polarized light
Rs ¼

 n1 cosθ−n2 cosϕ 2 n1 cosθ þ n2 cosϕ

ð3Þ

and
Rp ¼ Fig. 9. Fluence dependence of the polarization obtained using either an s- (black squares) or p- (red circles) polarized ns laser. The data were recorded for an angle of incidence of 30°, with the detector positioned perpendicular to the incident laser beam. Error bars are a single standard deviation.

with the ions, convert their directed motion into polarization of the emitted light [17]. A third possibility is that reflection by the target surface of unpolarized emission from the plasma polarizes the outgoing radiation in accordance with the Fresnel equations. Lastly, it is possible that photoluminescence from an intact layer within the crystal is polarized. This effect is well-known for semiconductors having local asymmetry caused, for example, by strain induced by lattice mismatch between a quantum dot and the substrate on which it is grown [18,19] or by the cylindrical axis of a nanowire [20]. We rule out the first mechanism immediately because we found that the polarization lasts much longer than the pulse duration. We may rule out the second mechanism on the same grounds inasmuch as it is difficult to imagine a plasma alignment persisting for several ns without some external means of sustaining it. The observation that the polarization vector lies in the s-plane (i.e., normal to the plane of laser incidence) regardless of the laser polarization is further evidence against these two mechanisms. The last mechanism is also unlikely because crystalline Si has a diamond structure, for which the optical selection rules predict isotropic photoluminescence [20]. (Recently Hu et al. [21] reported that a properly shaped pulse train may induce symmetry-breaking in GaAs that results in polarized photoluminescence. These specialized conditions are absent in the experiments reported here.) We are left with the third mechanism, namely selective reflection by the surface of unpolarized light, resulting in an s-polarized outgoing wave. Previously, we ruled out this mechanism because the emission was not always observed to be s-polarized. The present study shows, however, that the data in our earlier experiments were obscured by scattered laser light, which appeared as an artifact at shorter wavelengths. Using notch filters to reject laser scatter and carefully correcting for polarization induced by optical components, we now find that the detected light is always within a few degrees of the s-plane. We conclude therefore that the most likely mechanism for the polarized signal is reflection of unpolarized plasma emission by the sample surface. Numerous pump and probe measurements of the time-resolved reflectance of semiconductors in the melting and ablative regimes have been reported [22–26]. Time-resolved microscopy revealed the formation of a highly reflective liquid phase that persists for nanoseconds [27]. We posit that in our experiment a surface layer of liquid Si selectively reflects the s-component of unpolarized emission from the plume. To test this hypothesis, we compared the polarization as a function of detection angle with the predictions of the Fresnel equations. In our model, the incident angle of the laser does not play a role in the reflection mechanism. Rather, we assume that

n1 cosϕ−n2 cosθ n1 cosϕ þ n2 cosθ

2

;

ð4Þ

where θ is the angle of incidence of the plasma luminescence, and ϕ is the angle of refraction [28]. Subscripts 1 and 2 refer to the medium of the incident wave and the absorbing/reflecting surface, respectively. As shown in Appendix A, Snell's law can be used to eliminate ϕ, yielding reflectances that depend solely on the angle of incidence, the real refractive index of medium 1, and the complex refraction index of medium 2 [29]. Assuming that the vapor pressure of Si above the reflecting surface is well below atmospheric during most of the period that light is gathered by the detector, we may to good approximation take n1 =1, while for n2 we take the literature values of η2 and κ2 for liquid Si [30]. If the light emitted by the plasma is unpolarized, the reflected light has polarization P¼

Rs −Rp : Rs þ Rp

ð5Þ

The problem is complicated by the fact that some of the light travels directly to the detector and some of it is reflected (and absorbed) by the surface, as depicted schematically in Fig. 10. If we assume that a constant fraction, f, is detected directly, the relative signals detected when the polarizer is set for maximum and minimum transmission are f+(1−f)Rs and f+(1−f)Rp, respectively, so that the observed polarization is given by   ð1−f Þ Rs −Rp  : P ¼ 2f þ ð1−f Þ Rs þ Rp ′

ð6Þ

Fig. 10. Schematic drawing of light emitted from the plasma plume and detected at angle θdet. Rays R1 and R2 represent, respectively, light emitted directly from the plasma without reflection and light emitted from the plasma and subsequently reflected by the thin liquid Si surface layer. Electric field orientation vectors are shown for rays R1 and R2 to illustrate the (i) unpolarized and (ii) s-polarized character of the respective rays, the latter caused by the Fresnel effect upon reflection. The positions of lenses L1 and L2 correspond to the same optical element positions illustrated in Fig. 1.

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In general, f is spatially dependent. Assuming arbitrarily a value of f=0.5, we obtain the results for Rs, Rp, P, and P′ plotted in Fig. 8. In adding the intensities of the direct and reflected rays, we have assumed that coherent interference averages to zero [31]. This assumption is justified because of spatial averaging of the emission originating from adjacent regions of the plume. Fig. 11 shows the calculated variation of P′ with wavelength for a fixed value of θdet =80°. The Fresnel mechanism predicts a gradual increase in polarization with wavelength, in qualitative agreement with the data. The quantitative agreement between P′ and the measured polarization in Fig. 8 is most likely fortuitous, considering the arbitrary choice of f and the assumption that n2 does not vary with time. Nevertheless, agreement between the calculated and measured values of the maximum angle (~80°) and the near-zero value of P at θdet = 0 strongly support the Fresnel mechanism. Structure in the polarization spectra (Figs. 4a and 7a) associated with discrete transitions may be explained by the spatial dependence of f. It is likely that atomic fluorescence and continuum emission are generated at different locations in the plasma, so that the relative contributions of reflected and directly transmitted waves are different, leading to maxima or minima in P′ at the discrete lines. The random structure in Fig. 11 is caused by the much poorer signal/noise ratio at an 80° incidence angle. The strong maximum in P at 400 nm (peak ‘c’ in Fig. 4a) produced by the s-polarized fs laser may be explained by addition of the s-polarization of the SH signal to the s-polarized reflection of the background continuum at that wavelength. The dip in P produced by the p-polarized laser is similarly explained by partial cancelation of p-polarized SHG by the s-polarized Fresnel reflection. This effect is naturally absent in the ns spectrum (Fig. 7a). The structure in the polarization angle spectra (Figs. 4b and 7b) is more difficult to explain. The small but systematic drift in the baseline of α0 in Fig. 4b suggests the existence of a weak birefringence of the plume. If real, this unexpected effect would imply the formation of some optically active molecules or clusters that rotate the polarization of continuum radiation passing through the plume, with the effective column length depending on where in the plume different free–free transitions occur. Such an effect would not be inconsistent with the assumption of n1 ≈1 near the surface. The asymmetric profiles of α0 are as of yet unexplained, although we note that a shift from s- to p-polarization has been observed in the blue wing of the SH signal produced in a plasma [13]. Direct comparison of our results with those of Majd et al. [7] and Hahn et al. [8] is not possible because they worked with Cu and our present study was performed with Si. Nevertheless, the discrepancies between the various groups are mostly resolved, at least qualitatively. In particular, the falloff in P with laser fluence (Fig. 9) is consistent

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with our most recent results [6] and with the near-zero value of P reported by Asgill et al. [8] at 1020 J/cm 2. Possible reasons for the loss in polarization at higher fluences include plasma shielding of the surface, which could increase the value of f, and disruption of the laminar liquid layer at the Si surface, causing the light to be scattered non-specularly. Likewise, our observation of predominantly s-polarization is in agreement with the finding of Majd et al. [7], although their observation of larger P for s-polarized irradiation is unexplained. Within experimental error, we find that P is the same for both s- and p-polarized excitation. In summary, the Fresnel mechanism for the reflection of light at a surface provides a semi-quantitative explanation of the observed polarization of the continuum in laser-induced breakdown spectroscopy. In particular, it explains why the polarization is largely s-polarized. The results for fs and ns excitation are largely consistent. There remain, however, some aspects of our data associated with the discrete structure in P and α0 that are not fully understood and which are especially relevant for application to LIB spectroscopy. Acknowledgments This project was supported under Contract Number FA701407-C-0047, with the U.S. Air Force Surgeon General's Office (AF/SG) and administered by the Air Force District of Washington (AFDW). Use of the Center for Nanoscale Materials was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. Appendix A. Calculation of the polarization Eqs. (3) and (4) give the polarization of a light beam incident on a surface with incident angle θ and refraction angle ϕ. The latter angle may be calculated from a generalization of Snell's Law. We start by writing the wave number as a complex quantity → →

→

K¼k þi a

ðA1Þ

and the index of refraction as n ¼ η þ iκ:

ðA2Þ

The components of the wave vector in medium 2 are given by [20] k2 sinϕ ¼ k0 η1 sinθ

ðA3Þ

and h i 2 2 2 1=2 ; k2 cosϕ þ ia2 ¼ k0 η2 þ iκ 2 −η1 sin θ

ðA4Þ

where k0 is the wave number in a vacuum. Eq. (A4) may be decomposed into real and imaginary components. We write the argument of the square root as 2

2

2

2

iγ

V ¼ η2 −κ 2 −η1 sin θ þ 2iκ 2 η2 ¼ jV je :

ðA5Þ

The amplitude and phase of V are given by  2 2 2 2 2 2 2 2 jV j ¼ η2 −κ 2 −η1 sin θ þ 4κ 2 η2

ðA6Þ

and

Fig. 11. Polarization spectrum under the same laser conditions as Fig. 7a, measured at a detection angle (θdet) of 80°. The smooth curve is the calculated value of P′ with f=0.5.

tanγ ¼

2κ 2 η2 : η22 −κ 22 −μ 21 sin2 θ

ðA7Þ

10

J.S. Penczak Jr. et al. / Spectrochimica Acta Part B 74–75 (2012) 3–10

The real part of Eq. (A3) then becomes pffiffiffiffi k2 cosϕ ¼ k0 V cosðγ=2Þ:

ðA8Þ

Using Eq. (A3) to eliminate k2, we obtain the refraction angle, n sinθ tanϕ ¼ pffiffiffiffi 1 ; V cosðγ=2Þ

ðA9Þ

which may be inserted into Eqs. (3) and (4). In the limit of κ2 =0, we recover the ordinary form of Snell's Law: 2

2

2

V ¼ η2 −η1 sin θ

ðA10Þ

  η1 =η2 sinθ η1 sinθ ffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanϕ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   η22 −η21 sin2 θ 1− η1 =η2 2 sin2 θ

ðA11Þ

while Snell's Law is normally written as sinϕ ¼

η1 sinθ : η2

ðA12Þ

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