Haze of surface random systems: An approximate analytic approach

Approximate analytic expressions for haze (and gloss) of Gaussian randomly rough surfaces for various types of correlation functions are derived within phase-perturbation theory. The approximations depend on the angle of incidence, polarization of the incident light, the surface roughness, $\sigma$, and the average of the power spectrum taken over a small angular interval about the specular direction. In particular it is demonstrated that haze (gloss) increase (decrease) with $\sigma /\lambda$ as $\text{exp}\left[-A{\left(\sigma /\lambda \right)}^2\right]$ and decreases (increase) with $a/\lambda$, where $a$ is the correlation length of the surface roughness, in a way that depends on the specific form of the correlation function being considered. These approximations are compared to what can be obtained from a rigorous Monte Carlo simulation approach, and a good agreement is found over large regions of parameter space. Some experimental results for the angular distribution of the transmitted light through polymer films and their haze are presented and compared to the analytic approximations derived in this paper. A satisfactory agreement is found. In the literature haze of blown polyethylene films has been related to surface roughness. Few authors have quantified the roughness and others have pointed to the difficulty in finding the correct roughness measure.

Figure 1

The scattering geometry used in this study. The rough surface is defined by $z=\zeta \left(x\right)$. The region above the surface, $z>\zeta \left(x\right)$, is assumed to be vacuum $\left[{\epsilon }_0\left(\omega \right)=1\right]$, while the medium below is a dielectric characterized by a frequency-dependent dielectric function ${\epsilon }_1\left(\omega \right)$. Notice for which direction the angle of incident $\left({\theta }_0\right)$, scattering $\left({\theta }_s\right)$, and transmission $\left({\theta }_t\right)$ are defined as being positive. An angle of transmission is only well defined if the lower medium is transparent, i.e., if $\text{Re}{\epsilon }_1\left(\omega \right)>0$.

Figure 2

(Color online) Experimental height measurements obtained from atomic force microscopy (AFM) measurements on a LLDPE film. The material was of the narrow (molecular weight distribution) metallocene grade. (a) A contour plot of the experimental height function measured over a $50\times 50\mu {\text{m}}^2$ quadratic area. The color code is so that red corresponds to $0.015\mu \text{m}$ and blue to $-0.015\mu \text{m}$. (b) The height distribution function $P\left(\zeta \right)$ calculated from the AFM data (open circles) and a Gaussian fit (solid line) corresponding to $\sigma =0.04\mu \text{m}$. (c) The height-height correlation function $W\left(\left|x\right|\right)$ obtained from the AFM data (open circles) and fitted with an exponential correlation function $W\left(\left|x\right|\right)=\text{exp}\left(-\left|x\right|/a\right)$ (solid line) corresponding to a correlation length of $a=1.3\mu \text{m}$.

Figure 3

(Color online) The mean differential reflection coefficient, $⟨\partial R_p/\partial {\theta }_s⟩$, vs the scattering angle ${\theta }_s$ as obtained by rigorous Monte Carlo simulations. The incident $p$-polarized light of wavelength $\lambda =0.6328\mu \text{m}$ was incident at normal incident $\left({\theta }_0=0°\right)$ onto the rough dielectric surface, and for the incident wave a finite sized beam of half-width $g=6.4\mu \text{m}$ was used in order to reduce end effects. This surface of length $L=25.6\mu \text{m}=40.5\lambda$ separates vacuum, above the surface, from the dielectric medium of dielectric constant ${\epsilon }_1\left(\omega \right)=2.25$ below the surface. The statistical properties of the randomly rough surface was characterized by a Gaussian height distribution function of (rms) width $\sigma =0.037\mu \text{m}=0.058\lambda$ and an exponential correlation function of correlation length $a=1\mu \text{m}=1.58\lambda$. The surface was discretized at $N_{\zeta }=500$ equally distributed points, and the result was averaged over $N_{\zeta }=5000$ surface realizations. The vertical dash-dotted lines are at ${\theta }_{\pm }=\pm {2.5}^{\circ }$. Notice the specular (coherent) peak around ${\theta }_s={\theta }_0$. The haze in reflection for this surface is according to the numerical simulations $\mathcal{H}\left({\theta }_0\right)=0.33$, while the prediction of Eq. (27) is $\mathcal{H}\left({\theta }_0\right)=0.34$.

Figure 4

(Color online) Haze, $\mathcal{H}$, as a function of (a) the surface roughness $\sigma /\lambda$ for $a/\lambda =1.58$ at normal incidence, (b) the correlation length $a/\lambda$ for $\sigma /\lambda =0.058$ at normal incidence, and (c) the angle of incidence ${\theta }_0$ for $\sigma /\lambda =0.058$ and $a/\lambda =1.58$. For all figures the wavelength of the $p$-polarized incident light was $\lambda =0.6328\mu \text{m}$. The open symbols are results of rigorous Monte Carlo simulations, while the solid lines are the predictions of Eq. (27). The dashed-dotted line in the upper panel (reflection) of (c) corresponds to the position of the Brewster angle ${\theta }_0={\theta }_B$ determined by ${\text{tan}}^2{\theta }_B={\epsilon }_1/{\epsilon }_0$.

Figure 5

(Color online) Contour plots of [(a) and (c)] the rigorous Monte Carlo simulation results and [(b) and (d)] the analytic approximation (27) for haze obtained in [(a) and (b)] reflection and [(c) and (d)] transmission for light of wavelength $\lambda =0.6328\mu \text{m}$ incident normally $\left({\theta }_0=0°\right)$ onto the rough surface. The dielectric media that was separated from vacuum by a rough interface was characterized by the dielectric constant ${\epsilon }_1=2.25$. All results were averaged over at least $N_{\zeta }=500$ surface realizations. The random surfaces were all characterized by a Gaussian height distribution function of standard deviation $\sigma$ and an exponential height-height correlation function of correlation length $a$. Overall the agreement between the analytic and rigorous simulation results is satisfactory over large regions of parameter space.

Figure 6

(Color online) Experimental results for the angular distribution of the light being transmitted through a melt blown LLDPE film of the type described and partly characterized in Fig. 2. The (mean) thickness of the film was $d=50\mu \text{m}$, and the wavelength of the light being incident normally onto the top mean surface was $\lambda =0.6328\mu \text{m}$. It is the logarithm of the transmitted intensity in arbitrary units that is presented. The measurements were conducted with a spectrophotogoniometer built by SINTEF. The weak anisotropy seen in the transmitted intensity is caused by the polymers being preferentially oriented in the flow direction (vertical direction in the figure).

Figure 7

(Color online) The mean differential transmission coefficient $⟨\partial T_p/\partial {\theta }_t⟩$ vs angle of transmission ${\theta }_t$ for light of wavelength $\lambda =0.6328\mu \text{m}$ impinging normally $\left({\theta }_0=0°\right)$ onto a $d=40\text{−}\mu \text{m}$-thick LLDPE film $\left({\epsilon }_1=2.25\right)$. The experimental results (open circles) corresponds to a cut through Fig. 6. Due to the use of arbitrary units in the experiment, the amplitude of the measurements was adjusted to fit that of the simulation results. In obtaining the simulation results (solid line), the surface parameters represented by the solid lines in Figs. 2 were used, i.e., the parameters used were $\sigma =0.04\mu \text{m}$ and $a=1.3\mu \text{m}$ for the exponentially correlated rough surface. In order to replicate the unpolarized incident light used in the experiment, the simulation results were averaged over the $s$- and $p$-polarized results. The thickness of the film was also here $d=40\mu \text{m}$.