Stochastic model for quantifying effect of surfaceroughness on light reflection by diffuse reflectancestandards

Peng Tian,a,b,c Xun Chen,d Jiahong Jin,a,b,c Jun Q. Lu,a,b Xiaohui Liang,e and Xin-Hua Hua,b,*aInstitute for Advanced Optics, Hunan Institute of Science and Technology, Yueyang, ChinabEast Carolina University, Department of Physics, Greenville, North Carolina, United StatescHunan Institute of Science and Technology, School of Physics and Electronics Science, Yueyang, ChinadClemson University, Department of Bioengineering and COMSET, CU-MUSCBioengineering Program, Charleston, South Carolina, United StateseAMMS, Experimental Instrument Plant, Beijing, China

Abstract. Diffuse reflectance standards of known hemispherical reflectance Rh are widely used in optical andimaging studies. We have developed a stochastic surface model to investigate light reflection and roughnessdependence. Through Monte Carlo simulations, the angle-resolved distributions of reflected light have beenmodeled as the results of local surface reflection with a constant reflectance Rs representing the overall abilityof a reflectance standard. The surface was modeled by an ensemble of random Gaussian surface profiles para-meterized by a mean surface height δ and transverse correlation length a. By decreasing δ∕a, the calculatedreflected light distributions were found to transit from Lambertian to specular reflection regime. Reflected lightdistributions were measured with three standards by nominal reflectance Rh valued at 10%, 80%, and 99%. Thecalculated results agree well with the measured data in their angular distributions at different incident angles bysetting Rs ¼ Rh and δ ¼ a ¼ 3.5 μm.© 2018 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.57.9.094104]

Keywords: reflectance standard; scattering; rough surface; Monte Carlo simulation.

Paper 181041 received Jul. 18, 2018; accepted for publication Aug. 28, 2018; published online Sep. 13, 2018.

1 IntroductionDiffuse reflectance standards are very useful devices forcalibrating instrument and sensors designed for light reflec-tion measurement in, for example, multispectral reflectanceanalysis of materials and remote sensing.1–6 The roughsurfaces of these devices are essential for hemisphericaldispersion of reflected light intensity. A key challenge isto understand quantitatively the dependence of reflectedlight distribution on the degree of surface roughness.Extensive literature is available on device construction, spec-tral calibration, and angular characterization of the reflectedlight distribution.7–12 In contrast, much remains to be learnedabout the effect of surface roughness and its modeling.13,14

Clear understanding of relations between surface roughnessand light distribution by diffuse reflectance standards issignificant for not only providing essential insight on butalso accurate simulation of objects in machine vision thatcarry ubiquitously rough surfaces.9,15–19 In this report, wepresent a stochastic rough surface model of diffuse reflec-tance implemented by Monte Carlo (MC) simulations toanalyze directional-hemispherical light reflection. Themodel employs an effective surface treatment of turbid sam-ples that is simplified from our previous models of biologicaltissues with rough surfaces to allow detailed study of sur-face-roughness dependence of light reflection with signifi-cantly reduced simulation times.20–23 The calculatedangular distributions of reflected light are compared to themeasured data obtained with diffuse reflectance standardsof known hemispherical reflectance values. Besides its

theoretical implications, the ability to understand and simu-late the angular distribution of the hemispherically reflectedlight provides a quantitative and touch-free means to revealsurface roughness of diffuse reflectance standards and otherdevices producing diffuse reflection.

2 Methods

2.1 Diffuse Reflectance Standards and LightMeasurement

Diffuse reflectance standards are made of sintered polytetra-fluoroethylene (PTFE) resin in powder form.7,24 With arough surface and porous structure, the PTFE layer of afew millimeters in thickness acts as a strong turbid target,which scatters incident light into the hemispherical spacediffusely.7 By doping with black pigments, the nominal val-ues of hemispherical reflectance Rh can be adjusted andremain as a constant approximately in a wide spectral regionfrom 350 to 1800 nm. To validate the effective surface modelfor this study, we have performed angle-resolved measure-ment of light reflected from each of three standards illumi-nated by a monochromatic beam of wavelength λ andincident angles (θi0;ϕi0). A photodiode (FDS100, Thorlabs)rotating on a semicircular orbit of radius rd was used to de-tect reflected light intensity IRðθr;ϕrÞ in the y–z plane withφr ¼ 90 deg or −90 deg. Both sets of angles, (θi0;ϕi0) and(θr;ϕr), are defined by the Cartesian coordinate systemshown in Fig. 1(a) that also illustrates the experimentalconfiguration of reflection measurement.

*Address all Correspondence to: Xin-Hua Hu, E-mail: hux@ecu.edu 0091-3286/2018/$25.00 © 2018 SPIE

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2.2 Modeling of Light Reflection by a DiffuseReflectance Standard

Turbid materials like PTFE layers with rough surfaces can bemodeled conventionally according to the radiative transfertheory for treatment of absorption and scattering in thebulk and the Fresnel equations for accounting of specularreflections at the surface. The computational cost of suchan implementation, however, is very high using the statisticalapproach of MC simulations.21–23 In addition, the hemi-spherical reflectance Rh determined for each diffuse reflec-tance standard provides the ratio of integrated power ofreflected light to that of incident light. As a result, we intendto develop an effective surface model that is capable of char-acterizing surface roughness of turbid objects with known Rh

without the time-consuming photon-tracking in the bulk asshown in our previous studies.21–23 A local surface reflec-tance Rs is defined for this purpose to quantify the abilityof a turbid object, such as a PTFE layer for light reflectionat surface and by bulk scattering. Furthermore, local surfacereflections are treated as specular ones by a randomly roughsurface across the x-y transverse plane at z ¼ 0. The surfaceprofile is described by z ¼ ζðρÞ, where z represents the sur-face height at a transverse location given by ρ ¼ ðx; yÞ basedon a previous model of light interaction with rough surface.21

The simulation method is briefly summarized below in thecontext of diffuse reflectance standards.

Since the wavelength of incident light is much smallerthan the transverse dimensions of examined area on the sur-face of a reflectance standard, one can replace the electro-magnetic field of the incident beam at r ¼ ðρ; zÞ of arough surface by a field existing on the plane tangentialto the surface at r.20 This plane approximation allows usto model the rough surface of a diffuse reflectance standardas a mesh of triangular elements with a local reflectance Rs.As stated above, we further assume that the local surfacereflection is specular in nature with Rs, representing the over-all ability to reflect light by the standard in order to signifi-cantly reduce computational time. For comparison to themeasured data, the value of Rs is related to the calibratedreflectance value Rh of the diffuse reflectance standard pro-vided by the vendor. Under these assumptions, reflection ofincident light by a standard depends on the surface profile interms of local incidence angle and Rs characterizing thelight–matter interaction for the reflectance standard.

A rough surface of profile given by z ¼ ζðρÞ is numeri-cally generated as a stationary stochastic process character-ized by a Gaussian function of correlation between twolocations of ρ and ρ 0 and a zero mean. The random numbersused for z must satisfy

EQ-TARGET;temp:intralink-;e001;326;697hζðρÞζðρ 0Þi ¼ δ2 exp½−ðρ − ρ 0Þ2∕a2� (1)

and

EQ-TARGET;temp:intralink-;e002;326;655hζðρÞi ¼ 0; (2)

where δ is the rms value of the surface departure from themean surface of x–y plane describing the height fluctuationand a is the transverse correlation length of the surfaceroughness. An MC code has been developed to calculatereflected light distribution IRcðθr;φrÞ that can be comparedto the measured data of IRðθr;ϕrÞ. To fulfill the stochasticrequirement, MC simulations of light reflection by therough surface characterized with a parameter set of (a; δ)have been performed on an ensemble of 50 statisticallyindependent surface profiles z ¼ ζðρÞ satisfying bothEqs. (1) and (2). The reflected light distributions calculatedfrom the 50 profiles were summed to obtain ensemble aver-aged IRcðθr;ϕrÞ, as indicated by the angular brackets inEqs. (1) and (2).

All rough surface profiles were set to the same squareareas of 1 mm side length to reduce memory requirementand are divided into a grid of 501 × 501 basic units onthe x–y plane with grid points ρij and −250 ≤ i, j ≤ 250.The incident beam is represented by N0 photons injectedover the area of a profile scaled down from the measuredbeam profile on x–y plane. The transverse locations ρ ofinjected photons are determined by two random numbersthat were transformed from uniform distributions between0 and 1 into numerical distributions along x- and y-axisthat are consistent with the measured profile of the incidentbeam. Each photon injected on a surface profile hits the sur-face M times before it exits. The maximum value of M wastruncated to 20 to avoid long loops of tracking in the MCsimulations with negligible loss of accuracy. At each hittinglocation, a random number RN uniformly distributedbetween 0 and 1 is compared to the predetermined surfacespecular reflectance Rs. The tracked photon is reflected ifRN ≤ Rs or absorbed if RN > Rs. A reflected photon ispropagated along the specular reflection direction deter-mined by a triangle mesh representing one of the 50 profiles.

As detailed in Ref. 21, rough surface heights zij ¼ ζðρijÞat grid points ρij are generated randomly for MC simulationswith distributions satisfying Eqs. (1) and (2). Thus, eachrough surface profile, as viewed in Fig. 1(b), is made of amesh of triangular elements that can be projected to thex–y plane as one pair of triangles dividing equally a squaregrid unit, as shown in Fig. 2(a), by the blue and red coloredzones. The mesh elements also serve as the tangential planesto the photon hitting location r ¼ ðρ; zÞ employed by theplane approximation discussed above. Figure 2(b) presentsthe projection of an element pair of the surface mesh tothe x–y plane surrounded by eight nearest-neighbor elementpairs with projections shown in faded colors. A photon hit-ting a surface element can be reflected to hit the otherelement supported by the same grid unit or one of the 16 ele-ments in the nearest-neighbor element pairs. To reduce the

Fig. 1 (a) Schematic of reflection measurement with (θi0;ϕi0) for inci-dent beam direction and a detector D located at r d for detectingIRðθr;ϕrÞ; (b) simulation configuration of light reflected at r ¼ ðρ; zÞwith im (m ¼ 1) as unit vector of incident direction, θim as local incidentangle, rm and rmþ1 as of reflection directions and Rs as local surfacespecular reflectance. The surface profile in (b) is shifted upward forclear presentation.

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complexity of modeling, we neglect secondary hits of sur-face elements beyond the nearest-neighbor pairs, whichwere verified to affect little the calculated results. By apply-ing restrictions on z coordinates of hitting locations anddirectional cosine along the z-axis, the number of possibletriangular elements hit by a reflected photon can be reducedto nine or less on average to reduce simulation time.

Once a photon hits a mesh element at r, the local surfacenormal vector nm is first calculated by

EQ-TARGET;temp:intralink-;e003;63;483nm ¼ −ζ 0xx − ζ 0

yyþ zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiζ 02x þ ζ 02

y þ 1q ; (3)

where m ¼ 1; : : : ;M is the sequence number of reflection,x; y, and z are the basis vectors, ζ 0

x and ζ 0y are partial deriv-

atives of surface profile function ζðρÞ. From Eq. (3), one canderive the incident angle θim given the unit vectors of im forthe incident direction of a tracked photon and rm for specularreflection direction. The interception of a reflected photonalong rm with a neighboring element is determined usingalgorithms widely used in computational geometry.25

Photon reflection and tracking among surface elements con-tinue with (im; rm) updated until it exits from the surface fromthe last reflection site at rm. A total of 17 detection bins wereused in the MC code to obtain IRcðθr;φrÞ or IRcðηÞ, whichare arranged on a semicircular orbit of rd from the centerof rough surface according to the experimental system.The bins are marked with a rotation angle η defined byθr sinφr of the photons exiting from a rough surface. Forcomparison to the measured data of IR, the center angularposition η of the bins ranges from −40 deg to 40 deg insteps of 5 deg. The detection bins have identical areas scaleddown from that of the photodiode sensor by the same factorsued for scaling the illuminated area in MC simulations fromthe measured one.

3 ResultsThree diffuse reflectance standards were used in this studythat are of 50.8 mm in diameter and Rh ¼ 99%, 80%, and10% in nominal reflectance (SRS-99-020, SRS-80-020,SRS-10-020, Labsphere Inc.). Distribution of light reflectedfrom a standard was measured as IRðθr;φrÞ with a photo-diode of 3.6 × 3.6 mm2 in area and rotating on a circularorbit of rd ¼ 70 mm, as shown in Fig. 1(a). A monochro-matic incident beam of λ in wavelength was obtainedfrom a xenon light source and a monochromator. The

beam was collimated by spherical mirrors with a divergenceangle of about 2.5 deg before incidence on a standard.Measurements of angle-resolved distributions of reflectedlight have been performed with λ ¼ 880 nm for the centerwavelength and 3 nm in bandwidth. The incident beam pro-file was measured in the plane transverse to the incidentdirection and displayed as the inset of Fig. 3. The profilewas found to be rectangular with half-maximum sizes ofabout 2.0 mm × 13 mm along, respectively, the x- and y0-axis in the transverse plane. The measured profile of incidentbeam has been used to obtain a numerically normalizeddistribution for injection of N0 photons in each of the MCsimulations performed on a rough surface profile describedin Eq. (1).

The center directions of incident beam and reflected lightacquired by the photodiode are given by (θi0;φi0) and(θr;φr), respectively, with φi0 ¼ −90 deg and φr ¼ 90 degor −90 deg. To increase signal-to-noise ratio, the incidentbeam was modulated by a mechanical chopper atf0 ¼ 20 Hz and the acquired photodiode signals were ampli-fied and digitized by a 16-bit A/D convertor. Fourier trans-form was performed to extract the signal component at f0 forIRðηÞ. For each standard, we performed three measurementsof IR for calculation of mean values and standard deviations.Figure 3 presents the normalized intensity of IRðηÞmeasuredwith the standard of Rh ¼ 99% by symbols and error barsand examples of normalized calculated curves of IRcðηÞwith Rs ¼ Rh and different combinations of a and δ for θi0 ¼30 deg and φi0 ¼ −90 deg. The examples of the calculatedcurves IRcðηÞ are in two groups: one with a ¼ 5.0 μm bydash lines and one with a ¼ δ by solid lines. One canobserve from the group of dash lines that IRcðηÞ appearssymmetric when δ∕a is close to 1 and showing a peak towardthe specular reflection direction of a flat surface when δ∕adecreases from 1. These results demonstrate that the stochas-tic surface model presented in this report can yield reflectionregimes from Lambertian for significantly rough surfacesto specular for smoother surfaces by varying δ∕a with a con-stant Rs. For a ¼ δ, the IRcðηÞ curves in the solid-line groupexhibit symmetric and approximate Lambertian shapes,

Fig. 2 Schematic representation of (a) a pair of triangular elements ofa rough surface profile mesh projected to a square grid unit of the x–yplane and (b) an element pair projected to a grid unit of the x–y plane(z ¼ 0) in bright colors and other eight nearest-neighbor element pairsprojected in faded colors.

Fig. 3 Normalized intensity of reflected light versus rotation angleη ¼ θr sin ϕr with symbols and error bars for IRðηÞ measured atλ ¼ 880 nm from a diffuse reflectance standard of Rh ¼ 99% andsolid lines for IRcðηÞ calculated with Rs ¼ 99% and ensemblesrough surfaces generated using different combinations of (a; δ) inμm. Inset: incident beam profile measured in the transverse x–y0plane with color bars representing photodiode signals in μV.

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which is comparable to the measured data of IRðηÞ. Hence,we varied values of a under the condition of a ¼ δ to fitIRcðηÞ to IRðηÞ, as shown in Fig. 3, by the solid lines.Good agreements between IRcðηÞ to IRðηÞ can be achievedwhen a was set to between 3.0 and 3.5 μm.

Measurements and MC simulations of reflected light dis-tribution have been carried out with two more diffuse reflec-tance standards of Rh ¼ 80% and 10% at multiple incidentangles of θi0. Figure 4 plots the normalized data of IRðηÞacquired from the three standards at two values of θi0 forcomparison and only the normalized calculated curves ofIRcðηÞ∕IRcð0Þ with Rs ¼ Rh and surface parameters set toa ¼ δ ¼ 3.5 μm. Again, the calculated curves fit well tothe measured data on their angular dependences, as shownin Figs. 4(a) and 4(b), and the dependences of intensity atη ¼ 0 on Rs or Rh, as shown in the inset. Larger fluctuationscan be observed in IRcðηÞ calculated with Rs ¼ 10%, sinceincident photons have much higher probability to beabsorbed or terminated before exit the surface and hit oneof detection bins.

4 DiscussionThe diffuse reflectance standards considered here can beregarded as turbid media for their porous structures in thesense of irregular orientations of surface elements and granu-lar constituents of bulk leading to heterogeneity in refractiveindex distribution. Without doping, the pressed PTFE pow-ders exhibit little absorption and strong scattering due to thelarge refractive index mismatch between PTFE (∼1.35 in

visible and near-infrared regions)26 and air. As a result,the steady-state distribution of light reflected from suchan object is a consequence of specular reflection at surfaceand multiple scattering in bulk. Various analytical andnumerical methods have been developed to consider suchcases of light–matter interaction as random media consistingsimple shaped inclusions and/or 1-D randomly rough surfa-ces by effective medium theories or extended variants.27,28

These existing methods of field theory, however, are verydifficult to be applied here due to complex and unknowndetails on surface profile and bulk composition. The multiplescattering in turbid materials can also be modeled using theradiative transfer theory and implemented in MC simulationfor treatment of rough surfaces.21–23 While feasible, suchendeavors often require multiple optical parameters, suchas absorption and scattering coefficients that are wavelengthdependent and unavailable for many types of turbid samples.In addition, high cost of code development and computa-tional resources is needed to perform such numerical studies.

In this study, we developed a different and simplified sur-face model to examine the effect of surface roughness on thereflected light distributions by diffuse reflectance standardswith known hemispherical reflectance Rh. By using a localsurface reflectance Rs representing the overall ability of lightreflection, the simplified model allows rapid calculation ofreflected light distribution of a standard, which is representedas a rough surface profiled as a mesh of stochastically imple-mented triangular elements, as illustrated in Fig. 2. As shownin Figs. 3 and 4, the calculated results of reflected light dis-tribution agree well with the measured data of three diffusereflectance standards if Rs is set as a constant and Rs ¼ Rh.The modeling of rough surface of a standard as stochasticmesh of local flats is consistent with the tangential planeapproximation of the field theory that requires local surfacesizes to be much larger than the light wavelength λ,20 whichis satisfied by the sizes of the surface profile parameters aand δ around 3 μm derived by curve fitting.

By assuming Rs as a constant, the calculated light reflec-tion can approach approximate Lambertian regime onlyfor the cases of a ∼ δ. We note from Eq. (1) that a and δrepresent statistically and, respectively, the characteristiclengths of transverse locations and vertical variations of,for example, peak-to-peak changes in a surface profile ζðρÞ.Consequently, a∕δ can be regarded as the average slopeangle of the profile and a ∼ δ indicates the condition ofslope angle around 45 deg for the ensemble of surface pro-files. This outcome suggests that objects exhibiting diffusereflection appearances that are of approximate Lambertiantype should likely possess the same surface profile featureof a ∼ δ, as revealed in this report. It is also interesting tonote from the calculated curves of IRc in Fig. 3 that underthe condition of a ¼ δ, increasing the value of a or thesize of surface roughness leads to steeper decrease of lightintensity as η moves away from 0. These results could beused to quantify surface roughness indirectly by straight-forward measurement of IRðηÞ in the cases of diffuse reflec-tance standards or samples with Lambertian or similar typesof diffuse reflection.

AcknowledgmentsP.T. acknowledges a visiting scholarship from ChinaScholarship Council (Grant No. 201508430225) and a

Fig. 4 Normalized intensity of reflected light versus rotation angle ηwith symbols and error bars for IRðηÞ measured at λ ¼ 880 nm fromthree diffuse reflectance standards and solid lines for IRcðηÞ calculatedwith a ¼ δ ¼ 3.5 μm and incident angle of (a) θi0 ¼ 30 deg;(b) θi0 ¼ 50 deg. Inset: IRcð0Þ versus Rh (symbols) and IRcð0Þ versusRs (lines). The angular dependence of Lambertian reflectance isrepresented by blue dash lines.

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research grant from the Education Department of HunanProvince, China (Grant No. 14A062). The authors thankMr. Jue Ding for performing a part of data acquisition.

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Peng Tian received his MS and PhD degrees in optical engineeringfrom Huazhong University of Science and Technology in China andhis major research interests are on the optical properties of biologicaltissues and turbid materials.

Xun Chen has anMS degree in biomedical engineering received fromTianjin University in China and is currently a PhD student in the studyof fluorescence imaging of biological cells.

Jiahong Jin received his MS degree in optics from Sun Yat-SenUniversity in China and is pursuing a PhD degree in biomedical engi-neering in the direction of multispectral imaging of biological tissues.

Jun Q. Lu received her MS degree from Nankai University in Chinaand PhD degree from the University of California at Irvine, both inphysics. Her main interests of research are in the theoretical andnumerical modeling of light-tissue and light–cell interactions.

Xiaohui Liang received her MS degree in biomedical engineeringfrom Tianjin University in China and her research interests are inthe field of precision instrumentation.

Xin-Hua Hu received his MS degree in physics from NankaiUniversity in China and Indiana University at Bloomington and PhDdegree in physics from University of California at Irvine. His researchinterests are on light interaction with biological tissues and cells.

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