Enhanced backscattering due to total internal reflection at a dielectric-air interface

Transcription

1 JOURNAL OF MODERN OPTICS, 1995, VOL. 42, NO. 2, Enhanced backscattering due to total internal reflection at a dielectric-air interface R. E. LUNAt, E. R. MENDEZ Centro de Investigacion Cientifica y de Educacion Superior de Ensenada, Apdo. Postal 2732, Ensenada, B. C , Mexico JUN Q. LU and ZU-HAN GU Surface Optics Corporation, P.O. Box , San Diego, CA 92126, USA (Received 20 November 1993 ; revision received 4 April 1994) Abstract. We report the observation of enhanced backscattering in the scattering of light from a photoresist film with a one-dimensional randomly rough interface deposited on a flat parallel glass plate. The random interface is illuminated from the photoresist side, entering the sample through the glass plate. Angular scattering measurements for this sample are presented and compared with results obtained numerically using two approximate models to describe the interaction between the light and the sample. The observed backscattering enhancement is believed to be due to multiple scattering processes at the photoresist-air interface, where total internal reflection can take place. 1. Introduction Recently, several observations of enhanced backscattering and other related effects in the angular distribution of light scattered from characterized randomly rough surfaces have been reported in the literature [1-5]. In parallel, numerical techniques to deal with surfaces with one-dimensional random profiles have been developed and employed to study these problems [6-9]. Most of the attention has been devoted to scattering from metallic surfaces although, more recently, rough dielectric surfaces have also been considered [9-13]. It has been suggested [1, 2] that for optically rough metallic surfaces, the enhancement is due to multiple scattering of waves at the random interface. By now, this idea seems to be well accepted and supported by the available evidence (see e.g. the numerical work carried out in [9] and [13]). Also in agreement with this model, numerical calculations for p-polarized illumination show that an enhanced backscattering peak can be observed for light scattered from a random rough air-dielectric interface when the surface is reflective enough [9]. Similarly, Nieto-Vesperinas and Sanchez-Gil [12] have shown through numerical work that enhanced backscattering can be observed in the light scattered from a random dielectric-air interface, when the light is incident from the dielectric side. They have linked this observation with the phenomenon of total internal reflection. t On leave from : Escuela de Ciencias Fisico-Matematicas, Universidad Autonoma de Sinaloa, Mexico /95 $ Taylor & Francis Ltd.

2 258 R. E. Luna et al. 4 x 3 Figure 1. Schematic diagram showing the sample geometry. The light is incident from the side of the flat glass substrate at an angle 0 0. With more complex structures, on the other hand, such as rough dielectric films deposited on flat reflecting substrates, sharp backscattering enhancement peaks have been observed experimentally and in numerical simulations [14,15]. The mechanism responsible for the enhancement is, in this case, the constructive interference between waves that follow reciprocal scattering paths when traversing (twice) the randomly rough air-dielectric interface. The situation is similar to that of the double passage configuration in which a mirror is illuminated through a thin phase screen [16]. Preliminary experimental results with purely dielectric samples, consisting of a photoresist film with an interface that has two-dimensional roughness, deposited on a glass plate, have shown the presence of a small backscattering enhancement peak when the sample is illuminated from the side of the flat glass plate [17]. Here, we report angular light scattering measurements from a sample that consists of a photoresist film with a one-dimensional randomly rough interface deposited on a flat parallel glass plate (see figure 1). The light is incident on the sample from the side of the parallel glass plate, and we measure the angular distribution of the reflected light. Our results show well-defined and strong backscattering enhancement peaks. These experimental results are compared with numerical simulations of the problem using two approximate models to describe the sample geometry. It is argued that the most important reflection is that arising from the photoresist-air interface, where total internal reflection can take place, and we use this assumption to devise our models. In section 2 we describe, briefly, the technique employed for the fabrication of the samples, and the instrument employed in the measurements. The approximate models and the numerical simulation are described in section 3, and the experimental and numerical results are presented and discussed in section 4. Finally, a summary of the results, together with our conclusions, are presented in section 5.

3 Backscattering at dielectric-air interface 259 aperture SIDE VIEW lens 7 cylindrical lens x l source Figure 2. Schematic diagram of the optical system employed for the fabrication of the samples. 2. Experimental details The sample was fabricated with a variation of the technique described by Gray [18] for the fabrication of two-dimensional randomly rough surfaces, and employed subsequently by several authors for the fabrication of two-dimensional and quasi one-dimensional surfaces [1-5]. Basically, the technique consists of exposing photoresist-coated (Shipley AZ 1650) plates to speckle patterns with the appropriate statistical properties. The procedure followed to prepare and develop the plates was similar to the one described in [2]. The optical system employed for the fabrication of the surface is shown schematically in figure 2. A Gaussian beam arising from a HeCd laser (wavelength µm) is focused and cleaned by the spatial filter. The expanding beam is converted into a converging Gaussian beam by an optical system consisting of one or more lenses. Subsequently, a cylindrical lens and a diffuser are introduced into the system. The cylindrical lens focuses the Gaussian beam onto a line on the plane where the diffuser is placed. This arrangement produces elongated `speckles' (uncorrelated areas in the radom intensity distribution) on the plane of the photoresist-coated plate ; the correlation length of the speckle field produced is several times larger in the x2 direction than in the x 1 one. In order to ensure far-field conditions for the diffraction pattern of the diffuser, the photoresist-coated plate is placed on what would be the plane of focus of the system in the absence of the cylindrical lens and the diffuser. A rectangular mask, approximately ten speckle correlation lengths wide (in the x2 direction), is placed just before the photoresistcoated plate, which is scanned in the x2 direction. Assuming a linear response of the photoresist, the resulting samples should have a surface profile with an approximately Gaussian probability density function of heights and a Gaussian correlation function. This profile should provide a good approximate to a realization of a Gaussian random process with a Gaussian correlation function. The fabrication of adequate samples was one of the most difficult aspects of the work reported here. The resulting surface profile of the sample was measured with a Dektak 3030

4 260 R. E. Luna et al. lens Figure 3. Schematic diagram of the bidirectional scatterometer employed in the measurements. stylus machine. The stylus radius is approximately gm, the sampling interval was 0. 1 gm, and the stylus loading was 1 mg. Statistical properties of the profile defining the air-dielectric interface were calculated from 10 measured profiles. A total of 2000 points were taken for each trace. The standard deviation of heights 6 was estimated to be about 1. 0 gm, and the value of the correlation length a, as the 1 /e value of the correlation function, was 3. 2 gm. Both the histogram of heights and the measured correlation length were approximately Gaussian, but deviations from the assumed model (Gaussian random process with a Gaussian correlation function) were found in the second derivative of the measured profile. We present angular light scattering measurements in the plane of incidence for the characterized sample when the air-photoresist interface is illuminated through the glass plate and the photoresist film. Scattering data were taken with an automated bidirectional scatterometer, shown schematically in figure 3. An expanded laser beam (wavelength X = gm) was sent through a periscope with a series of mirrors from which it exited horizontally toward the sample. For reflective measurements the last mirror of the periscope obstructs the detector in the vicinity of the backscattering direction and, in order to reduce this problem, the beam was focused on this last mirror ; the obstruction can then be made narrower, permitting measurements close to the backscattering direction. Thus, with this arrangement, the surface was illuminated by a divergent beam. To reduce the speckle noise produced in the far field of the surface it was necessary to average spatially over many `speckles'. This was achieved by illuminating an approximately 20 mm diameter area of the surface and by using a field lens at the detector that integrated over an angle of about In normal circumstances the angle subtended by the detector is smaller than the spatial variations in the structure of the mean intensity, and it is expected that this integration had only a small effect on the measurements. It was also verified that the detector viewed the entire illuminated area of the diffuser, irrespective of the angle of detection. The detector was mounted on a 73 cm long arm that rotated about the sample in the horizontal plane. A PC-compatible computer was used to control the position of the detector and store the data. Wide angle scans (from - 90 to 90 ) were taken with this instrument. 3. Description of the simulation The configuration of the sample is shown schematically in figure 1. It essentially consisted of a photoresist film with a one-dimensional randomly rough interface deposited on a flat parallel glass plate. In our experiments, the light illuminated the

5 Backscattering at dielectric-air interface 261 rough interface after traversing the glass plate and the photoresist film. The thickness of the glass plate was about 5 mm and the average thickness of the photoresist film, d, was about 10 gm. By measuring various samples it was noticed that, at least over a small range of variations in thicknesses about these values, none of these two parameters seemed to be critical for the results. Rigorous numerical simulations of light scattering by this complex structure would be difficult to implement, mainly due to the comparatively large thickness of the glass plate. However, since it is argued here that the mechanism responsible for the enhancement is multiple scattering at the photoresist-air interface, approximate models for describing the interaction between the incident field and the sample should provide a reasonable description of its scattering properties. We have used two models for describing this interaction, which are explained in the following. In both models, the field incident on the sample was assumed to be a Gaussian beam making an angle of incidence 0o with respect to the normal to the sample. In the first model, the field incident on the rough photoresist-air interface, after traversing the two flat interfaces, was calculated according to the well known laws of refraction. The interaction between the beam and the rough photoresist-air interface was treated quite rigorously, employing the techniques described in [9] and [12]. This provided the angular spectrum of the scattered field inside the photoresist. To propagate the light back, through the lower layers of the sample and into air, the amplitude and propagating angles of the scattered field inside the photoresist were modified to take into account the refraction that occurs at these lower interfaces. The accuracy of this approximation depends on the relative importance of multiple scattering effects due to these flat interfaces. In the following, we shall refer to this model as the two-media model. In the second model, the field incident on the photoresist film, after traversing the glass plate, was calculated by considering the refraction due to the air-glass interface. The interaction between the field and the film was treated rigorously [17], and the resulting angular spectrum was then propagated through the glass plate and into air, taking into account the refraction at the lower interface. In the following we shall refer to this model as the film model. This model assumes that the lower, flat interface introduces multiple scattering effects that can be neglected. It is expected that both models give reasonable results for relatively small angles of incidence and scattering. For large scattering angles the reflectivities of the lower interfaces approach a value of one and this can produce significant amounts of multiple scattering. So, for the numerical simulation with the two-media model we assumed that the system consisted of a one-dimensional rough interface separating air (refractive index n = 1) from a semi-infinite photoresist medium. The interface is illuminated by a Gaussian beam incident from the photoresist side with an angle of incidence calculated using the values of the angle of incidence in air, and of the refractive indices of glass and photoresist. For the second model we assumed that the system consisted of a photoresist film with a one-dimensional rough interface, deposited on a semi-infinite glass substrate. The film is illuminated by a Gaussian beam incident from the glass side with an angle of incidence calculated using the values of the angle of incidence in air, and of the refractive index of glass. The average film thickness, d, was 10 gm. In all the calculation, the wavelength of the incident light (in air) was taken to be 2 = µm, and the indices of refraction of the photoresist film and the glass substrate at this wavelength were taken to be np = 1.64 and ng = 1. 51,

6 262 R. E. Luna et al. respectively. The width of the Gaussian beam was w = L cos 0 ;/4, where 0; is the angle of incidence inside the photoresist (two-media calculations) or inside the glass (film calculations). The statistical parameters of the surface employed in the simulations were those determined experimentally. That is, the standard deviation of heights 6 = 1 µm, and the correlation length a = 3.2 gm. The length of the rough surface was taken to be L = 40 gm, and the number of points along the surface used for the numerical implementation of the technique was N=400. The total number of surface profiles used in the averaging procedure was Np = 500 for all the results. 4. Results and discussion Far field experimental results for our sample are shown in figures 4-7. Also in these figures we show the results of the Monte Carlo type numerical simulations obtained with our approximate models. In figure 4 (a) we show angular light scattering measurements of the sample taken with the bidirectional scatterometer depicted in figure 3, when the sample is illuminated by a p-polarized beam incident at an angle of 10 with respect to the normal. The relatively strong specular component due to the air-glass interface was blocked and it is not shown in the figure ; a `hole' encompassing a few degrees about the specular direction is visible in the figure. The exact retro-reflected signal was also blocked by the illuminating mirror of the scatterometer, but this region is only a fraction of a degree wide. It is clearly seen in this figure that there is a fairly strong backscattering peak at It is also worth noting that the scattering pattern is wide, and that significant amounts of scattered light are found at large angles of scattering. Numerical results for the incoherent component of the differential reflection coefficient ((ar P /a05 ) for p polarization, or (ar s /a0 5) for s polarization), which represents the fraction of the incident energy scattered per unit angle [9] are shown in figure 4 (b). The results were obtained with the approximate models described above, assuming p-polarized light and an angle of incidence of Oo = 10 in air. The thicker curve represents the results obtained with the two-media model, and the thinner one the results obtained with the film model. It can be seen that despite the simplicity of the assumed models, the numerical results are qualitatively similar to those shown in figure 4 (a). As with the experimental results, there is a clear backscattering peak and the overall shape of the two curves is quite similar. Nevertheless, the peak obtained with the simulation is not as pronounced as the one in the experimental curve. Comparison of the two numerical curves in figure 4 (b) can also be illustrative, as their differences are due to the fact that in the film calculations we are including the photoresist-glass interface in a rigorous manner. One can see that the two curves agree quite well in the neighbourhood of the backscattering direction, and that the overall shape of the curves is quite similar. However, the film calculations predict a slightly lower differential reflection coefficient than the two media calculations. In figure 5 we show results obtained when the sample is illuminated by a p-polarized beam incident at an angle of 20. As before, for the experimental curve (fig. 5 (a)) the strong specular reflection due to the air-glass interface is not shown. Also, as in the previous case, there is a strong backscattering peak, this time at When compared with the 10 incidence case, we notice that the enhanced backscattering peak is smaller but slightly wider. This is the general behaviour found when the angle of incidence is increased. Figure 5 (b) shows the incoherent component of the mean differential reflection coefficient of the scattered light for a p-polarized beam incident at an angle of 20 obtained with our numerical

7 Backscattering at dielectric-air interface 263 Experimental Results (a) (b) Figure 4. Angular intensity distribution of the mean intensity in the far field for the case of p polarization and an angle of incidence of 10. (a) Measurements of the scattered intensity and, (b) numerical results for the incoherent component of the differential reflection coefficient calculated with our approximate models ; the bold line represents the two-media calculations, and the thin line the film calculations.

8 2 6 4 R. E. Luna et al. Experimental Results Scattering Angle (deg.) (a) (b) Figure 5. Angular intensity distribution of the mean intensity in the far field for the case of p polarization and an angle of incidence of 20. (a) Measurements of the scattered intensity and, (b) numerical results for the incoherent component of the differential reflection coefficient calculated with our approximate models ; the bold line represents the two-media calculations, and the thin line the film calculations.

9 Backscattering at dielectric-air interface Experimental Results '. ' Scattering Angle (deg.) (a) r c m 0.04 a m Scattering Angle (deg.) (b) Figure 6. Angular intensity distribution of the mean intensity in the far field for the case of s polarization and an angle of incidence of 10. (a) Measurements of the scattered intensity and, (b) numerical results for the incoherent component of the differential reflection coefficient calculated with our approximate models ; the bold line represents the two-media calculations, and the thin line the film calculations

10 2 6 6 R. E. Luna et al. Experimental Results Scattering Angle (deg.) (a) Scattering Angle (deg.) (b) Figure 7. Angular intensity distribution of the mean intensity in the far field for the case of s polarization and an angle of incidence of ( a) Measurements of the scattered intensity and, (b) numerical results for the incoherent component of the differential reflection coefficient calculated with our approximate models ; the bold line represents the two-media calculations, and the thin line the film calculations.

11 Backscattering at dielectric-air interface 267 simulations. A small, but well-defined peak is observed in the retro-reflection direction, namely As for the 10 incidence case, the numerical results are similar to those found experimentally, with some discrepancies pertaining mainly to the height and shape of the backscattering enhancement peak. As expected, the main differences between the experimental results and those obtained with our approximate models, occur at large scattering angles. Also, as in the previous case, there are some differences between the film and two media calculations. We find that, for most angles of scattering the two-media calculations predict larger scattering cross-sections than the film calculations, but that the two models coincide in the vicinity of the backscattering direction. The corresponding experimental and numerical results for s polarization are shown in figures 6 and 7. There are only slight differences with the case of p polarization and, given the similarity of the results, the same comments can be made for this case of s-polarized illumination. For the scattering from air-dielectric interfaces (light incident from air) with statistical parameters and optical properties similar to those of our sample, it has been shown numerically and experimentally that an enhanced backscattering peak is not present, or is weak [9,17] when p-polarized light is incident on the sample. For s-polarized light, on the other hand, a well defined peak can be observed in the backscattering direction. This difference may be attributed to the different reflectivities of the surface for these two kinds of polarization. With p-polarized light, it is only when the refractive index of the dielectric is increased, so that the surface becomes more reflective, that a backscattering peak appears. All this supports an explanation for the enhancement based on the multiple scattering of waves at the air-dielectric interface. On the other hand, for the analogous situation with a dielectric-air interface (light incident from the dielectric side), the reflectivity is much higher due to the possibility of having total internal reflection at the interface. Numerical studies of this problem have shown that backscattering enhancement occurs in this situation [12]. Given these circumstances, and the fact that the results obtained with our model are in relatively close agreement with the experimental ones, we believe that the main scattering processes occurring in our samples take place at the dielectric-air interface. In this case, the phenomenon of total internal reflection becomes the dominant mechanism responsible for the observed effects. It is to be noted that, in general, the experimental backscattering peaks obtained experimentally are wider than the peaks obtained with the numerical simulations, and that the numerical results show sharp oscillations around the backscattering peak. This discrepancy could be due, at least partly, to the smoothing of the experimental curves by the detecting aperture. The fact that the experimental backscattering peak is stronger than those predicted by our models could be due to the additional interaction and multiple scattering effects introduced by the lower air-glass interface. Note that although the rigorous inclusion of the glass-photoresist interface did not have a significant effect on the backscattering peak (compare e.g. the two numerical curves on figures 4 (b) to 7 (b)), the reflectivity of the glass-air interface is higher, and thus multiple scattering effects could have a non-negligible effect in this case, even for small angles of incidence and scattering. Another possibility for explaining these discrepancies could be the deviation, in the higher-order statistics, of the random profile of the sample from the assumed model of a Gaussian random process with a Gaussian correlation function.

12 268 R. E. Luna et al. 5. Summary and conclusions We have presented measurements of the angular distribution of the light scattered by a rough one-dimensional dielectric film deposited on a flat glass plate when the incident beam impinges on the sample from the side of the glass plate. We have observed strong backscattering peaks both with p- and s-polarized illumination. Numerical simulations of the problem employing two approximate models for the sample and the measured statistical parameters of the rough interface, were presented and compared with the experimental results. Some differences have been found between the numerical calculations and the experimental results, although the overall shape of the curves agree quite well. We have found that the measured backscattering enhancement peaks are stronger than those predicted by the numerical simulations. This could be due to the fact that our models neglect the multiple scattering introduced by the flat air-glass interface. There are also some differences concerning the width and shape of the peak. The simulations show the presence of secondary maxima on the sides of the backscattering enhancement peak, a feature that was not observed experimentally. These oscillations are quite sharp and are particularly evident for angles of incidence of 10 or smaller. The facts that the secondary maxima were not found experimentally and that the experimental peak is wider could be partly attributed to the smoothing of the experimental scattering curve by the detector aperture. We believe that the reflection at the rough photoresist-air interface is responsible for most of the features observed ; the multiple scattering occurring at this rough interface, aided by total internal reflection effects, provides the basic mechanism for producing the backscattering enhancement. Our belief is supported by the similarity between the experimental and numerical results. Acknowledgments R. E. Luna wishes to express his gratitude for the support of the Consejo Nacional de Ciencia y Tecnologia and the Universidad Autonoma de Sinaloa. This work has been supported by the Consejo Nacional de Ciencia y Tecnologia through grant F , and by the US Army Research Office through grant DAAH04-93-C References [1] M$NDEZ, E. R., and O'DONNELL, K. A., 1987, Optics Commun., 61, 91. [2] O'DONNELL, K. A., and MtNDEZ, E. R., 1987, J. opt. Soc. Am. A, 4, [3] SANT, A. J., DAINTY, J. C., and KIM, M. J., 1989, Optics Lett., 14, [4] KIM, M. J., DAINTY, J. C., FRIBERG, A. T., and SANT, A. J., 1990, J. opt. Soc. Am. A, 7, 569. [5] O'DONNELL, K. A., and KNOVrs, M. E., 1992, J. opt. Soc. Am A, 9, 585. [6] NIETO-VESPERINAS, M., and SOTO-CRESPO, J. M., 1987, Optics. Lett., 12, 979. [7] SOTO-CRESPO, J. M., and NIETO-VESPERINAS, M., 1989, J. opt. Soc. Am. A, 6, 367. [8] MARADUDIN, A. A, MENDEZ, E. R., and MICHEL, T., 1989, Optics Lett., 14,151. [9] MARADUDIN, A. A., MICHEL, T., McGURN, A. R., and MENDEZ, E. R., 1990, Ann. Phys., 203, 255. [10] NIETO-VESPERINAS, M., SANCHEZ-GIL, J. A., SANT, A. J., and DAINTY, J. C., 1990, Optics Lett., 15, [11] SANCHEZ-GIL, J. A., and NIETO-VESPERINAS, M., 1991, J. opt. Soc. Am. A, 8, [12] NIETO-VESPERINAS, M., and SANCHEZ-GIL, J. A., 1992, J. opt. Soc. Am. A, 9, 424. [13] BRUCE, N., and DAINTY, J. C., 1993, J. mod. Optics, 38, 1471.

13 Backscattering at dielectric-air interface 269 [14] Lu, J. Q., MARADUDIN, A. A., and MICHEL, T., 1991, J. opt. Soc. Am. B, 8, 311. [15] MARADUDIN, A. A., Lu, J. Q., TRAN, P., WALLIS, R. F., CELLI, V., Gu, Z.-H., MCGURN, A. R., MENDEZ, E. R., MICHEL, T., NIETO-VESPERINAS, M., DAINTY, J. C., and SANT, A. J., 1992, Rev. Mexicana Fis., 38, 343. [16] JAKEMAN, E., 1988, J. opt. Soc. Am. A, 5, [17] Gu, Z.-H., Lu, J. Q., and MARADUDIN, A. A., 1993, J. opt. Soc. Am. A, 10, [18] GRAY, P. F., 1978, Optica Acta, 25, 765.