O. Vázquez-Estrada and A. García-Valenzuela
Vol. 31, No. 4 / April 2014 / J. Opt. Soc. Am. A
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Optical reflectivity of a disordered monolayer of highly scattering particles: coherent scattering model versus experiment Omar Vázquez-Estrada1,2 and Augusto García-Valenzuela1,* 1
Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Apartado Postal 70-186, Distrito Federal 04510, Mexico 2 e-mail: [email protected] *Corresponding author: [email protected] Received December 19, 2013; accepted January 23, 2014; posted February 7, 2014 (Doc. ID 203481); published March 17, 2014
Recently a multiple-scattering model for the reflectivity of a disordered monolayer of scattering particles on a flat surface was put forth [J. Opt. Soc. Am. 29, 1161 (2012)]. The approximate theoretical model provides relatively simple formulas for the reflectivity, but it was developed for a monodisperse monolayer. Here we extend the model to the case of a polydisperse monolayer and derive the appropriate formulas to calculate the optical transmissivity of the monolayer supported by a flat interface. A second objective of this paper is to test the approximate theoretical model against experimental data with highly scattering particles. We prepared monolayers of three different surface coverage fractions of polydisperse alumina and titanium dioxide particles deposited randomly on a glass slide. We measured their optical reflectivity and transmissivity versus the angle of incidence using a laser with a wavelength of 670 nm. Using the nominal values for the particles’ most probable radius and refractive index, we fitted the theoretical model to the experimental curves and found that it reproduces very well the experimental curves. Interestingly, a dip in the reflectivity curves at large angles of incidence is present for the alumina monolayers but not in the titanium dioxide monolayers. The dip corresponds to a maximum in the scattering efficiency by the alumina monolayers. The theoretical model reproduces very well this behavior. © 2014 Optical Society of America OCIS codes: (290.5825) Scattering theory; (290.4210) Multiple scattering; (290.5850) Scattering, particles; (240.0240) Optics at surfaces; (120.5700) Reflection; (120.7000) Transmission. http://dx.doi.org/10.1364/JOSAA.31.000745
1. INTRODUCTION The optical reflectivity of disordered monolayers of particles has been of interest for many years now [1–17]. Particular attention was paid in the past to the case of monolayers of small metallic particles whose size is much smaller than the incident wavelength adsorbed on a flat surface (see, for instance, [2–11]). The problem of light reflection from monolayers of particles with size comparable to the wavelength has been addressed in more recent studies (see, for instance, [12–17]). However, comparatively fewer works have been devoted to this problem. In this case the particles may scatter the incident light efficiently and thus, in addition to a coherently reflected and/or transmitted optical beam, diffuse light propagating in all directions is also present. Nevertheless, the coherent reflectance, also referred to as the reflectivity, can be easy to measure when the particles are supported by a flat interface and the surface coverage fraction is small or moderate. The coherent transmittance or transmissivity of this type of monolayers can also be easily measured if the particles are supported by a transparent substrate. Although this problem has been studied in relatively few occasions it is not difficult to foresee some interesting applications. In particular, detecting and characterizing the optical properties of air or waterborne particles could be done by forcing their adsorption on a substrate (maybe selectively, according to their size 1084-7529/14/040745-10$15.00/0
or composition) and then analyzing the optical reflectivity and transmissivity of the partially covered surface. In order to eventually develop such type of applications it is necessary to devise theoretical models and verify they can actually describe real measurements. Such a model or models would be essential aids in pursuing applications of optical reflectivity measurements. Recently a multiple-scattering model for coherent reflectance of a sparse, random, and monodisperse monolayer of spherical particles was developed in [1]. Formulas were derived for the coherent reflectance of disordered monolayers only and no comparison with experimental measurements was attempted at the time. Thus, the objectives of this work are: (i) to extend the model to the case of polydisperse monolayers and derive also the formulas for the coherent tansmittance; (ii) to perform measurements of the coherent reflectance and transmittance of random monolayers of highly scattering particles supported on a transparent substrate; and (iii) to compare the predictions of the theoretical model with experimental data. We make no attempt to extend the model to nonspherical particles. Although in many experiments and potential applications particles are not spherical, when the particles’ shape is on the average spheroidal, we may expect they behave as equivalent spherical particles to some degree of approximation. © 2014 Optical Society of America
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When particles in a flat monolayer are very small compared to the incident wavelength, the beam is reflected, or transmitted, only in the specular direction. However, as already said, when the particles are not small, the beam will not only be reflected and transmitted partially in the specular direction, but also, it will be scattered in multiple directions. The light scattering efficiency of the particles is determined not only by their size, but also by their refractive index contrast. When particles are not small compared to the wavelength of light, the field reflected and transmitted by a monolayer of these particles, can be divided into two components, a coherent component and a diffuse component. When the incident light comes in a well-defined optical beam and impinges on a disordered monolayer on a flat substrate, the coherently reflected beam travels in the specular direction whereas the coherently transmitted beam refracts into the substrate following Snell’s law. The coherent fields correspond to the average of the reflected or transmitted fields over all the permitted configurations of the particles in the monolayer, that is to the so-called configurational average of the electromagnetic fields. Fluctuations of the electromagnetic fields about their average give rise to what we call diffuse light. The diffuse reflected and transmitted electromagnetic fields have a null average but they do transport energy, since the average of the intensity associated with these fields is not zero. If the incident beam is well collimated, the energy carried by the coherent reflected and transmitted beams is also well collimated, and thus their energy fluxes are concentrated within small solid angles. On the other hand, diffuse light is scattered in all directions, so the energy flux is spread over large solid angles and in many cases contributes very little within the solid angles subtended by the coherently reflected and/or transmitted beams. We illustrate in Fig. 1 the coherent and diffuse light components when measuring the reflectivity of a disordered monolayer of particles. In practice, the reflectivity or specular reflectance of a monolayer will directly correspond to the coherent reflectance of the system, if the incident light beam illuminates a large area of the surface with a very large number of particles and if the diffuse light contribution within the solid angle
O. Vázquez-Estrada and A. García-Valenzuela
subtended by the coherently reflected beam is negligible. When the latter condition is not entirely satisfied we may still measure the coherent reflectance by measuring the power captured by the detector at angles near the direction of the coherent reflected beam just outside the solid angle subtended by the coherent beam and subtract this value from the specular measurement done with the same detector. In doing this we would assume the diffuse light is distributed over solid angles much broader than the coherent beam, and thus it could be considered constant within and around the solid angle of the coherent beam. In general, the relative amount of average power carried by the coherent and the diffuse components depends on the angle of incidence as well as on the size and refractive index of the particles. This work is organized as follows. First, in Section 2 we present an outline of the derivation of the coherent reflectance of a free-standing, polydisperse monolayer of spherical particles referring to the mathematical steps presented in detail in [1] for a monodisperse monolayer. Then we derive the formulas for the coherent reflection and transmission coefficients of an interface partially covered by a random monolayer. We include additional terms to take into account the contribution of a second interface behind the one supporting the monolayer, to model the case when the monolayer is deposited on the surface of a (optically) thick slab. In Section 3 we briefly discuss our experimental methodology to deposit monolayers of alumina and titanium dioxide particles on the surface of a glass slide and describe our measurements of the coherent reflectance and transmittance of the samples with a laser beam. In Section 4 we present a selection of experimental curves of the coherent reflectance and transmittance versus the angle of incidence along with theoretical curves fitted to the experimental curves by adjusting the surface coverage fraction. In Section 5 we discuss the results and compare the surface coverage fraction obtained from adjusting the model to the experimental curves with rough estimates obtained from a relatively simple image analysis of optical micrographs of the monolayers. Finally, in Section 6 we give our conclusions.
2. MULTIPLE-SCATTERING MODEL
Fig. 1. Coherent and diffuse components for a light beam reflected and transmitted by a polydisperse monolayer of particles.
Here we derive a multiple-scattering model to compute the coherent reflection and transmission coefficients of an electromagnetic plane wave incident on a monolayer of polydisperse particles sitting on a flat interface between media of refractive indices n1 and n2 . We use vector electrodynamics theory and take into account in an approximate way multiplescattering among the particles and the substrate. The model also takes into account approximately the exclusion volume of the particles, and the reflection and transmission coefficients for either TE (s) or TM (p) polarized light are given in terms of the elements of the scattering amplitude matrix of an isolated particle. The approximations incurred in deriving the model limit its validity to low surface coverage fractions of the monolayer but it can be used for particles of sizes either small or comparable to the wavelength of radiation. In [1], a detailed derivation of the model for the reflection of light from a monodisperse monolayer of spherical particles sitting on the interface is given. The corresponding details are omitted in this paper.
O. Vázquez-Estrada and A. García-Valenzuela
Vol. 31, No. 4 / April 2014 / J. Opt. Soc. Am. A
A. Free-Standing Monolayer First we consider a free-standing disordered monolayer of polydisperse particles in which an incident electric field, given by Ei r; t E ∘ expiki · r − iωtˆei , excites the particles. By a free-standing monolayer of particles we mean an ensemble of particles embedded in a homogeneous medium of refractive index n1 , randomly placed on an imaginary plane. Our analysis is restricted to spherical particles and we assume that all particles have the same refractive index, np . Additionally, we assume that all the particles touch a specific plane surface, but since the particles have different radii, their centers are no longer aligned on a plane as it was the case for a monodisperse monolayer. We will put origin of the coordinate system to the tangent plane that supports all the particles, see Fig. 2. The field scattered by any of the particles, say the nth particle, can be written as [18,19] Z E r s
3 0 3 00
↔
00
0
00
d r d r Gr; r · Ta r − rn ; r − rn ·
00 Eexc n r ;
(1)
where rn is the position vector of the nth particle with respect to its center, Eexc is the field that excites the nth particle, n
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where ρa is the density of probability of a particle to have a radius of a, pn rn is the density of probability of finding the nth particle of radius an at rn , and hEexc n in is the configurational average of the field exciting the nth particle while keeping its position fixed at rn . Remember, we will assume all particles sit on the plane z 0. Then the radius of a particle of radius a is located at z −a, see Fig. 2. Thus, the probability density pn rn depends on the particle radius. We assume it is uniform over the area of xy-plane covered by the monolayer and depends on zn as δzn − an . As in the case for monodisperse monolayers we can start with the so-called quasi-crystalline approximation (QCA) to set up an integral equation for the field exciting any of the particles. The difference here is that we must also average over the particle size distribution. We can write this equation as Z Z∞ Z ↔ i r 3r Eexc d3 r 0 Gr; r0 r; r E daρa d j n j 0 Z ↔ 00 · d3 r 00 prn jrj Ta r0 − rn ; r00 − rn · Eexc n r ; rn ; (4)
↔
Gr; r00 is the dyadic Green’s function of the vector wave ↔
equation in the background medium [20], Ta r0 − rn ; r00 − rn is the transition operator (T matrix) used in scattering theory for a particle of radius a [18,19]. For a collection of N particles we simply add up the field scattered by all the particles: Er Ei r
N Z X
↔
d3 r 0 d3 r 00 Gr; r0
n1 ↔
00 · Ta r0 − rn ; r00 − rn · Eexc n r :
(2)
The coherent reflected and transmitted fields correspond to the configurational average of the field scattered by the particles on either side of the monolayer. Since we are assuming the position of the particles is random and the particles have a size distribution, we must take the average over the position of the particle and over their size. We can write Z∞ Z Z ↔ hEri Ei r daρa d3 r n pn rn d3 r 0 Gr; r0 0 Z ↔ 3 00 00 (3) · d r Ta r0 − rn ; r00 − rn · hEexc n in r ; rn ;
where daρa is the number density of particles with radius between a and a da, prn jrj is the conditional density of probability of finding the nth particle around rn given that the jth particle is around rj . prn jrj depends on the radii of the jth particle and the nth particles. Clearly, it must be zero when jrn − rj j < l, where l is the sum of radii of the nth particle and jth particles and should approach a constant when the particles are far away from each other. To obtain an approximate solution to the integral Eq. (4), we replace prn jrj by Ujrn − rj j − lprn where U is the step function and we assume prn is given by A−1 δzn − an , where A is the area extended by the monolayer on the xy plane and an is the radius of the nth particle. Moreover, we assume the exciting field is in the form of two effective plane waves, one traveling in the incidence direction and the other in the specular direction [1]. We assume that all particles, regardless of their size, see the same effective field. This approximation may be regarded as equivalent to the effective field approximation used in the coherent reflection of light from a half space of a random distribution of particles [18,19]. It should be a good approximation for low surface coverage fractions. Then we write i Eexc ei E 2 expikr · rˆer ; p r; rp E 1 expik · rˆ
Fig. 2. Illustration of the monolayer supported by a flat interface. The particles are sitting on the interface and embedded in medium 1, with refractive index n1 .
(5)
where ki kix aˆ x kiy aˆ y kiz aˆ z and kr kix aˆ x kiy aˆ y − kiz aˆ z , eˆ i and eˆ r are unitary polarization vectors, and E 1 and E 2 are the amplitudes of the effective planes waves, which we must solve from a consistency requirement. Introducing Eq. (5) in the right-hand side of the QCA integral equation, and using the momentum representation of the transition operator, performing the integral over d3 r n on different portions of the monolayer with appropriate plane wave expansions for the dyadic Green’s function as it was done for the monodisperse case [1], and ignoring all integrals of only evanescent waves, yields
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η Eind j r; rj ≈ − E 1 2
Z 0 r
∞
O. Vázquez-Estrada and A. García-Valenzuela
where
daρaSa ki ; ki expki · r
Sa k ; ki exp−2ikiz a expikr · r Z∞ η − E2 daρaSa ki ; kr exp2ikiz a expki · r 2 0 Sa kr ; kr expikr · r;
(6)
where the phase factors exp−2ikiz a and exp2ikiz a take into account that the center of particles of radius a are at z −a (and not at the origin), and η
Sa q; p
2π ; k2m cos θi
(7)
↔ km ↔ ˆ · Ta q; p · eˆ p : I − qˆ q 4πi
(8)
We added the subscript a to the transition operator and to the vector S to indicate they are functions of the particle radius. If we demand the left-hand side of Eq. (6) to be equal to Eq. (5) we obtain two consistency equations from which we can solve for the amplitude of the effective exciting planes waves. We get Z∞ η E 1 eˆ i E i eˆ i − E1 daρaSa ki ; ki 2 0 Z∞ η daρaSa ki ; kr exp2ikiz a; − E2 2 0 Z∞ η daρaSa kr ; ki exp−2ikiz a E 2 eˆ r − E 1 2 0 Z∞ η daρaSa kr ; kr ; − E2 2 0
(9)
(10)
where Sa kr ; ki and Sa kr ; kr are given by Eq. (8) above. As mentioned in [1], these latter vectors are related to the elements of the amplitude scattering matrix of an isolated particle. In fact one can show that, Sa ki ; ki S a 0ˆei , Sa kr ; kr S a 0ˆer , Sa kr ;ki S j;a π −2θi ˆer , and Sa ki ; kr S j;a π − 2θi ˆei , where in the last two identities the subscript j takes the value 1 or 2 when the incident wave to the monolayer is TE (s) or TM (p) polarized, respectively. S 1;a θ and S 2;a θ are the elements of the amplitude scattering matrix of an isolated spherical particle of radius a as defined in the book by Bohren and Huffman [21]. S0 S 1;a 0 S 2;a 0 is the so-called forward scattering amplitude of the particle. Taking the scalar product on both sides of the expressions (9) and (10) with eˆ i and eˆ r , respectively, and solving the obtained algebraic equations yields E1 Ei
E2 Ei
1 12 βF ; 1 βF 14 β2F − βC βB
− 12 βC ; 1 βF 14 β2F − βC βB
(11)
(12)
Z βF η βB η βC η
∞
0
Z
∞
0
Z
∞
0
daρaS0; daρaS j;a π − 2θi exp2ikiz a; daρaS j;a π − 2θi exp−2ikiz a:
(13)
As already said, the subscript j takes the value of 1 for an incident TE-polarized wave and the value of 2 for a TMpolarized incident wave. The integrals in Eq. (13) can be efficiently p evaluated numerically, with integration limits given by a σ 18 a∘ . Where a∘ is the radius of the particle most probable (provided by the manufacturer). Using the amplitudes of the effective plane waves exciting the particles in Eqs. (9) and (10) and dot multiplying both sides of the first equation with eˆ i and the second one with eˆ r yield, E t Ei − E 1 βF − E 2 βB and Er −E1 βC − E 2 βF . The next step is to use the expressions in Eqs. (11) and (12) in the latter two equations and solve for the amplitude of the reflected and transmitted fields, i.e., r coh Er ∕Ei and tcoh Et ∕Ei . We get −βC ; 1 βF β2F − βC βB
(14)
1 − 14 β2F − βC βB : 1 βF 14 β2F − βC βB
(15)
r coh
tcoh
1 4
In the integrals in Eq. (13) the number-density distribution can be written as ρa ρST na where ρST is the number surface density of particles regardless of their radius and na is the so-called particle size distribution function. In many cases the particle size distributions can be assumed to be a lognormal distribution. Such is the case in our experiments we present below. The corresponding expressions are 1 ln2 a∕a∘ na p exp − ; 2 ln2 σ 2π a ln σ
ρST
Θ exp−2 ln2 σ: πa2∘
(16)
(17)
Here, Θ is the surface fraction covered by the particles (the surface coverage), a∘ is the radius of the particle most probable, and σ is the width parameter of the particle size distribution function. Equations (14) and (15) together with the log-normal size distribution given in Eqs. (16) and (17) reduce to the case for a monodisperse monolayer of particles with radius a∘ simply by taking the width parameter σ very small so that the result does not depend on it. B. Supported Monolayer To introduce in the model an interface at the plane were the particles are sitting z 0, let us suppose the refractive index for z > Δ, being Δ a small distance, is n2 and is different from that of incidence medium z < Δ given by n1 . Then we can
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add up all the multiple reflections between the monolayer and the interface at z Δ. Next we let z → 0 and obtain the compound reflection coefficient of a monolayer sitting on top of a substrate of refractive index n2 , see Fig. 3. In this case we also take into account an additional reflection of the coherent beam transmitted into the slab supporting the monolayer at the interface between the medium 2 and medium 3 (green line in Fig. 3), but we ignore multiple reflections with this interface. We get
r coh θi r 12 θi t2coh θi expβ1
r coh θi t2coh θi r 212 θi exp2β1
t2coh θi r 21 θi t12 θi t21 θi expβ2 ;
(18)
where β1 2ik∘ Δn1 cos θi and β2 2ik∘ dn2 cos θ2 , d being the width of the slab (see Fig. 3) and θ2 the angle of refraction of the coherent beam in the slab, r 12 and t12 are the Fresnel transmission and reflection coefficients, respectively, of the interface between refractive indices, n1 and n2 evaluated at an angle of incidence θi . Similarly, t21 and r 21 are the Fresnel reflection and transmission coefficients of the interface between refractive indices, n2 and n1 . r coh and tcoh are the coherent reflection and transmission coefficients of a freestanding monolayer given by Eqs. (14) and (15). Note that since d ≫ λ we can drop the a phase factor expβ2 on the last term on the right-hand side of Eq. (18) since it oscillates very rapidly with the angle of incidence and such oscillations will be averaged out by the detector. Then the sum is simplified when Δ → 0, to
supporting the monolayer. Subsequent multiple reflection with this interface are neglected. Similarly, according to Fig. 3, the transmission coefficient for a monolayer supported by a flat interface is given by tθi t01 t02 tcoh θi t12 θi t21 θi expβ∘ r coh θi tcoh θi r 12 θi t12 θi t21 θi expβ∘ expβ1 (20)
rθi r 01 r 02 r 03 r 001
rθi r coh θi
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Here, β∘ 0.5β1 . Note that we obviated the phase factor exp12 β2 common to all terms in Eq. (20) since it is cancelled out when computing the transmittance. If we now take the limit Δ → 0, the previous sum simplifies to tθi
tcoh θi t12 θi t θ : 1 − r 12 θi r coh θi 21 i
(21)
The expression within the square brackets takes into account all multiple reflections of the coherent beam between the monolayer and the interface supporting it. The transmission coefficient t21 outside the square brackets is due to the interface of the glass slide behind the one supporting the monolayer. Again, subsequent multiple reflections with this interface are neglected. The coherent reflectance and transmittance of a glass slide supporting a monolayer of particles is given by the magnitude square of the coherent reflection and transmission coefficients given in Eqs. (19) and (21), respectively. That is, Rcoh θi jrθi j2
and T coh θi jtθi j2 :
(22)
r 12 θi t2coh θi
1 − r 12 θi r coh θi
t2coh θi r 21 θi t12 θi t21 θi :
(19)
The second term on the right-hand side of this last equation includes all the multiple reflections between the monolayer and the interface supporting it, whereas the last term on the right-hand side is due one reflection of the coherent beam at the bottom surface (exposed to air) of the glass slide
Fig. 3. Schematic illustration of multiple reflections of a monochromatic beam incident on a monolayer of particles immersed in air and sitting on a flat substrate.
3. EXPERIMENTAL METHODOLOGY Previous to testing our model against experimental data, we prepared monolayers of alumina and titanium dioxide particles (rutile) on a flat glass surface. The nominal values of the refractive indices of the particles are np 1.76 for alumina particles and np 2.73 for titanium dioxide particles. The most probable radius a∘ of the alumina particles as indicated by the manufacturer is about 1.5 μm and for the titanium dioxide particles is about 220 nm. As substrate we used standard 1 mm thick microscope glass slides 27 mm × 75 mm of refractive index n2 1.5. The procedure to deposit random monolayers of the particles was as follows. First, the slides were extensively washed and cleaned with detergent, acetone, isopropyl alcohol, and tridistilled water to thereby remove dust and grease from the surface. The next step was to prepare a dispersion of the particles in a solvent that evaporates rapidly after deposition. The alumina particles were obtained in powder from the manufacturer (Alumina Buehler No. 40-6603-030-080) and we simply mixed 1.74 g of the alumina powder with 39 mL of isopropyl alcohol. The titanium dioxide particles were provided to us in a slurry consisting of a mixture of deionized water (999.0 g), with dispersant (0.99 g), and titanium dioxide particles (0.99 g). We mixed 9.0 mL of the titanium dioxide slurry with 36 mL of isopropyl alcohol, thus obtaining a dispersion of TiO2 particles. The two particle dispersions were immersed in an ultrasonic bath for 5 min to reduce particle clusters during the deposition
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of the monolayers on the slide’s surface. Sonication helped depositing isolated particles at random on the slide surface. The particles are, of course, not spherical, but we will consider them as spherical as an approximation. To deposit a monolayer, the clean slide was immersed in the particle dispersion in isopropyl alcohol at constant speed and at a fixed angle of inclination of about 85° (with respect to the horizontal), until 90% of the glass slide was covered. Subsequently, we withdrew at the same speed and angle of inclination the glass slide from within the particle suspension. This technique is commonly known as dip coating. Actually, a monolayer of particles was deposited on both sides of the cover slide, but after the deposition, we cleaned the side of the slide that was facing down during the immersionwithdrawal process. The surface coverage by the particles achieved by the described dip-coating process depended on the time it took us to immerse and remove the slide from the particle dispersion as well as the angle at which the slide was immersed. With this in mind, we prepared three monolayers, with different cover surface fractions, all deposited on a glass slide with alumina particles and three more with TiO2 particles. We adjusted the immersion and withdrawal time to deposit the particles to about of 5, 3, and 1 s, and the angle in immersion to about 85°, for each type of monolayer. These parameters of the deposition process produce sparse monolayers that reflect specularly a measurable amount of light at near normal incidence. Then we proceeded to measure carefully the reflectivity and transmissivity of TEpolarized laser light as a function of the angle of incidence, θi . In Figs. 4(a) and 4(b), we show our experimental setup for coherent reflectance and transmittance measurements. We used a linearly polarized laser diode λ 670 nm and adjusted its polarization (by simply rotating the laser) to be TE polarization in the experiment. The sample (with the deposited monolayer) was fixed on top of a goniometer with its surface perpendicular to the plane of rotation of the goniometer, as depicted in Fig. 4. Special care was taken in the alignment of the experimental setup to ensure that the plane of the monolayer/glass-slide was perpendicular to the surface of the goniometer and the point of reflection of the laser beam on the monolayer coincided with the rotational axis of the goniometer. The reflectivity and transmissivity of the laser beam was measured at different angles of incidence, in a range about 0 to about 86 deg. It is important to remark that the region of the sample used to measure the coherent reflectance was the same used to measure the coherent transmittance. The incident and reflected optical power was collected by a silicon photodetector connected to a transimpedance
Fig. 4.
O. Vázquez-Estrada and A. García-Valenzuela
preamplifier converting the photocurrent signal to a voltage signal, which in turn, was registered by a digital multimeter (Agilent 34401A), as depicted in Fig. 4. The position of the detector was adjusted manually, ensuring the coherently reflected or transmitted beam was captured completely by the detector. Also, the amount of diffuse light entering the detector during the measurements was estimated for several angles of incidence by measuring the diffuse power captured by the detector at viewing angles around the coherent beam. In general, the contribution of diffuse light to the specular reflectance and transmittance was negligible, but when noticeable it was subtracted from the detector’s readings. The coherent reflectance was obtained simply dividing the coherently reflected optical power by the incident optical power. Similarly, the coherent transmittance was obtained dividing the coherently transmitted power by the incident optical power.
4. RESULTS In Figs. 5 and 6, we show the graphs corresponding to the measurement of the coherent transmittance and reflectance versus the angle of incidence for the alumina and TiO2 monolayers (red points). In insets of Figs. 5 and 6, we show images taken by an optical microscope of the alumina and TiO2 monolayers used to obtain the reflectivity and transmissivity data shown in the figures. The incidence medium was air with a refractive index of n1 1.00. In these figures we also include theoretical curves calculated with the formulas given above assuming a monodisperse monolayer (dashed blue lines) and a polydisperse (full black lines). To calculate the amplitude scattering coefficients S0, S 1 π − 2θi and S 2 π − 2θi , we used the so-called Mie theory [21,22]. To obtain the curves corresponding to monodisperse monolayers we simply considered a very narrow size distribution and used a width parameter of σ 1.01. The theoretical curves were calculated assuming a lognormal size distribution and supposing that the refractive index of the particles and their most probable radius are given by their nominal values, that is, 1.76 and 1.5 μm for the alumina particles, and 2.73 and 220 nm for the TiO2 (rutile) particles, respectively. For the polydisperse curves we adjusted the width parameter to σ 1.3 for the alumina particles and σ 1.4 for the TiO2 particles for best fitting the experimental data. We used these values of σ for all reflectance and transmittance curves. Then we fitted the theoretical curves to the experimental data by adjusting the surface coverage fraction Θ so as to reproduce best the experimental data. We obtained Θ 3.8%, Θ 5.7%, and Θ 10.6% for the
(a) Schematics of the experimental setup for measuring the coherent reflectance and (b) the coherent transmittance.
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Fig. 5. Coherent reflectance (left) and coherent transmittance (right) for TE-polarized light of three monolayers of alumina with different surface cover fractions. The red dots are experimental measurements, the full (black) curves are theoretical curves assuming a polydisperse monolayer, and the dashed (blue) curves are the corresponding theoretical curves assuming a monodisperse monolayer. The values of the surface coverage fraction that fitted best the polydisperse curves were: (a) Θ 3.8%; (b) Θ 5.7%; and (c) Θ 10.6%.
Fig. 6. Coherent reflectance (left) and coherent transmittance (right) for TE-polarized light of three monolayers of TiO2 with different surface cover fractions. The red dots are experimental measurements, the full (black) curves are theoretical curves assuming a polydisperse monolayer, and the dashed (blue) curves are the corresponding theoretical curves assuming a monodisperse monolayer. The values of the surface coverage fraction that fitted best the polydisperse curves were: (a) Θ 1.3%; (b) Θ 4.9%; and (c) Θ 10.7%.
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alumina particles and Θ 1.3%, Θ 4.9%, and Θ 10.7% for the titanium dioxide particles, respectively. The uncertainty of the values of Θ obtained upon fitting the model to the experimental data while keeping all other parameters in the model fixed was less than 0.02%. We can appreciate in these figures that the theoretical curves reproduce quite well the experimental data. It is interesting to note the strong qualitative difference between the reflectivity curves for the alumina and that for the titanium dioxide particles. In the former case the reflectivity has a dip at large angles of incidence that is not present in the latter case. The multiple-scattering model reproduces very well this phenomenon. Also, note that the monodisperse multiplescattering model actually reproduces quite well the experimental data. In the case of the alumina particles the curve for the monodisperse multiple-scattering model shows some oscillations (due to Mie resonances) which are not present in the experimental data, but the polydisperse model averages out these oscillations and reproduces better the experimental curve. Some discrepancies between theory and experiment can be appreciated, which may be due to the fact that the particles are not spherical and may be also to particle clusters, which is not taken into consideration for the theoretical model. In addition, to the geometric shape of projection of the laser beam on the sample is affected when the angle of incidence increases. At oblique angles of incidence, the laser projects an elliptic footprint that increases along the plane of incidence and covers a larger section of the sample, especially for grazing incidence angles. Illuminating larger areas of the sample may result in seeing different surface coverage fractions at different angles of incidence, since the deposited monolayers do not have a perfectly uniform value of the surface coverage fraction across the whole sample. Nevertheless, we find rather surprising how well the multiple-scattering model fits the experimental data. As an additional test to the multiple-scattering model we obtained transmittance spectra at normal incidence for two specific monolayers using a white light source (LED XR-C) and a minispectrophotometer (Oceanoptics Model USB4000). We chose the alumina monolayer with surface coverage fraction of Θ ≈ 5.7% and the TiO2 particles monolayer with a surface coverage of Θ ≈ 1.3%. The reflectance spectra could not be measured because the reflected power was too little to be measured by the spectrophotometer. Both transmittance spectra, between 450 and 700 nm, are shown in Fig. 7. We also include in the graphs of Fig. 7 theoretical spectra calculated with the polydisperse and monodisperse multiple-scattering models assuming the same values of the parameters used to fit the experimental data in Figs. 5(b) and 6(a). Note that the spectra predicted by the polydisperse model are flat and reproduce very well with the experimental spectra. The corresponding theoretical spectra predicted for a monodisperse monolayer also coincide well on the average with the experimental spectra, but they show a fine structure consisting of irregular oscillations about an horizontal line. This graph illustrates well that the transmittance spectra do not give us more information of these monolayers than the angular reflectivity and transmissivity curves. Nevertheless, it is worth to emphasize that the parameters used to obtain the theoretical curves of reflectance and transmittance of these monolayers in Figs. 5(b) and 6(a) (coverage
O. Vázquez-Estrada and A. García-Valenzuela
Fig. 7. (a) Transmittance spectrum at normal incidence for a monolayer of particles of alumina with surface coverage factor of Θ ≈ 5.7% and (b) transmittance spectrum at normal incidence for a monolayer of particles of TiO2 with surface coverage factor of Θ ≈ 1.3%.
fraction, particles’ size distribution, and refractive index) were the same as those used to obtain the theoretical spectra in Fig. 7.
5. DISCUSSION The above results show that the multiple-scattering model for polydisperse monolayers reproduces very well the experimental data for monolayers of highly scattering particles. The fact that a single value of the particle size distribution can be used to fit all curves, for both reflectance and transmittance versus the angle of incidence, gives us some assurance that the assumption of a log-normal size distribution is valid. On the other hand, the fact that after fixing the width of the size distribution for each type of particle and then only adjusting the surface coverage factor the theoretical model fits rather well and simultaneously both the reflectance and transmittance curves for each sample, clearly supports the validity of the multiple-scattering model for these types of samples. Further confidence in the model would be attained if we could corroborate by other means the surface coverage fraction values obtained from fitting the theoretical curves to the experimental data. To this end we took microphotographs of the samples using an optical microscope with 400× amplification. Then we processed the digital images with a simple program that counted the number of pixels that registered an intensity below certain threshold and divided it by the total number of pixels in the image. The result is a rough estimation of the surface coverage fraction in each sample. An example of an actual micrograph image and its processed image with the software is shown in the Fig. 8 (alumina monolayer, Θ 3.8%). The value of the threshold was adjusted and kept within a range in which the result was less dependent on the exact value of the threshold. In Table 1, we show the values of the surface coverage fraction for the three monolayers of alumina derived from fitting the theoretical model to the experimental coherent reflectance and transmittance curves,
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Table 2. TiO2 Particles TiO2 Particles ΘFit ΘEst
Fig. 8. In the upper figure is shown a micrograph of a monolayer of alumina particles for the sample with a surface coverage fraction of Θ 3.8%. In the lower figure is shown the reconstructed image with the software, for this same sample, of the monolayer of particles without the substrate.
ΘFit , and the one estimated from processing the corresponding microphotographs, ΘEst . Similarly, in Table 2 we show the values of the surface coverage fraction for the three monolayers of TiO2 derived from fitting the theoretical model to the experimental coherent reflectance and transmittance curves, ΘFit , and the one estimated from processing the corresponding microphotograph for these monolayers, ΘEst . The values of ΘFit and ΘEst in Tables 1 and 2 are close to each except for the case of sample 2 of TiO2 particles. Taking into consideration that the microphotographs only capture a very small area of the sample whereas the laser illuminates a much larger area when the coherent reflectance is measured, we believe the agreement between ΘFit and ΘEst is actually quite good, and thus, does support the validity of the theoretical model at least in a qualitative manner.
ΘFit ΘEst
Sample 2 (%)
Sample 3 (%)
1.30 1.44
4.90 1.66
10.70 10.32
As already noted, the curves of coherent reflectance versus the angle of incidence for the alumina particles present an interesting dip at rather large angle of incidence, whereas the curves for the TiO2 monolayers do not. Actually these dips can be readily explained in simple terms referring to Eq. (19). Plotting independently the magnitudes of r coh θi ; t2coh θi , and r 12 θi reveals the reason of the presence of a dip in coherent reflection curve for alumina monolayers. When the particle size is very large as is the case of the alumina particles, these scatter mostly in the forward direction. Thus, the coherent reflection coefficient of the free-standing monolayer r coh is basically zero except for large angles of incidence where it increases rapidly to unity at grazing incidence. On the other hand, the coherent transmission coefficient tcoh , is not much less than unity; it remains nearly constant up to some value of the angle of incidence and then it decreases monotonously to zero for grazing incidence. Now, let us take a look to Eq. (19). For simplicity we can ignore the term corresponding to the first reflection from the back of the glass slide (the third term on the right-hand side). Then, for small and moderate angles of incidence, the first term on the right-hand side of Eq. (19), r coh θi , is very small and can be ignored, whereas t2coh θi is nearly constant. Thus, the coherent reflection coefficient is basically the reflection coefficient of the substrate’s interface. It is nearly constant for small angles of incidence and as the angle of incidence increases it starts to increase at moderate angles of incidence. At some point t2coh θi starts to decrease rapidly and brings the net reflection coefficient to zero. Just after this has happened we are left with rθi r coh θi , which starts to increase as the angle of incidence increases further and the net coherent reflection coefficient, rθi , goes to unity, as it should always do at grazing incidence, drawing a curve with a dip where both t2coh θi and rθi are simultaneously negligible. The dip is not present in the curves for TiO2 monolayers because the particles are smaller and scatter more efficiently in all directions, not only near the forward direction. Then, r coh θi starts to increase well before t2coh θi drops to negligible values. Note that in our experimental examples the absorption by the particles is very small. Thus, at the dip in the reflectance curves for the alumina particles, we have that the coherent transmission and reflection are both very small. Thus, this means that at the angle of incidence where the dip is located the supported monolayer scatters all incident radiation. A possible application of this phenomenon could be in the manufacture of surfaces with reduced gloss at high angles of incidence.
6. SUMMARY AND CONCLUSION
Table 1. Alumina Particles Alumina Particles
Sample 1 (%)
Sample 1 (%)
Sample 2 (%)
Sample 3 (%)
3.80 4.31
5.70 5.20
10.60 13.83
We provided an approximate multiple-scattering model for the coherent reflection and transmission from a monolayer of disordered polydisperse spherical particles. We performed experimental measurements of the coherent reflectance and transmittance from disordered polydisperse monolayers
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of alumina and titanium dioxide particles with small values of the surface coverage. It was found that the model fits very well the experimental data, the surface covered being the only parameter adjusted independently for each sample. We compared the values of the surface coverage fraction obtained from fitting the theoretical model to the experimental data with rough estimation derived from optical microphotographs of the monolayers and found a reasonable agreement. Since we do not know with certainty all the parameters involved in the sample monolayers, the fact that the theoretical model reproduces the coherent reflectance and transmittance versus the angle of incidence shows that the theoretical model is valid at least qualitatively, and thus takes into account the main physical phenomena determining the coherent reflection and transmission of light by highly scattering particles. A direct application of the theoretical model developed in this paper is to aid in the monitoring of the deposition of highly scattering particles on a substrate. Other applications could be in the biological realm. For instance, one could try to characterize living cells on a substrate and monitor their interaction with their surroundings through a change in the refractive index contrast between the cells and the liquid around them. Pursuing such possible applications would require contemplating in the model nonspherical particles with size parameters very large compared to one and a low refractive index contrast. Finally, the reflectivity versus angle of incidence curve shows a clear dip around an angle of incidence of 85° that was not expected. However, the theoretical model reproduces very well this dip in reflectivity. We provided a simple physical explanation of this effect. A possible application of this phenomenon could be in the manufacture of surfaces with reduced gloss at high angles of incidence.
ACKNOWLEDGMENTS We acknowledge financial support from Dirección General de Asuntos del Personal Académico from Universidad Nacional Autónoma de México through grant PAPIIT IN-106712.
REFERENCES 1.
2. 3. 4.
A. García-Valenzuela, E. Gutiérrez-Reyes, and R. G. Barrera, “Multiple-scattering model for the coherent reflection and transmission of light from a disordered monolayer of particles,” J. Opt. Soc. Am. A 29, 1161–1179 (2012). T. Yamaguchi, H. Takahashi, and A. Sudoh, “Optical behavior of a metal island film,” J. Opt. Soc. Am. 68, 1039–1044 (1978). T. Okamoto, I. Yamaguchi, and T. Kobayashi, “Local plasmon sensor gold colloid monolayers deposited upon glass substrates,” Opt. Lett. 25, 372–374 (2000). D.-S. Wang and C.-W. Lin, “Density-dependent optical response of gold nanoparticle monolayers on silicon substrates,” Opt. Lett. 32, 2128–2130 (2007).
O. Vázquez-Estrada and A. García-Valenzuela 5. A. Bagchi, R. G. Barrera, and A. K. Rajagopal, “Perturbative approach to the calculation of the electric field near a metal surface,” Phys. Rev. B 20, 4824–4838 (1979). 6. V. V. Truong, G. Bosi, and T. Yamaguchi, “Optical behavior of granular metal films: single-image versus multiple-image approaches in the treatment of substrate effects,” J. Opt. Soc. Am. A 5, 1379–1381 (1988). 7. J. Toudert, D. Babonneau, L. Simonot, S. Camelio, and T. Girardeau, “Quantitative modelling of the surface plasmon resonances of metal nanoclusters sandwiched between dielectric layers, the influence of nanoclusters size, shape and organization,” Nanotechnology 19, 125709 (2008). 8. M. L. Protopapa, A. Rizzo, M. Re, and L. Pilloni, “Layered silver nanoparticles embedded in a BaF2 matrix: optical characterization,” Appl. Opt. 48, 6662–6669 (2009). 9. M. L. Protopapa, “Surface plasmon resonance of metal nanoparticles sandwiched between dielectric layers: theoretical modeling,” Appl. Opt. 48, 778–785 (2009). 10. T. Menegotto, M. B. Pereira, R. R. B. Correia, and F. Horowitz, “Simple modeling of plasmon resonances in Ag/ (SiO2) nanocomposite monolayers,” Appl. Opt. 50, C27–C30 (2011). 11. S. Yoshida, T. Yamaguchi, and A. Kinbara, “Changes of the optical properties of aggregated silver films after deposition,” J. Opt. Soc. Am. 61, 463–469 (1971). 12. R. G. Barrera and A. García-Valenzuela, “Coherent reflectance in a system of random Mie scatterers and its relation to the effective medium approach,” J. Opt. Soc. Am. A 20, 296–311 (2003). 13. R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007). 14. E. A. van der Zeeuw, L. M. Sagis, G. J. M. Koper, E. K. Mann, M. T. Haarmans, and D. Bedeaux, “The suitability of angle scanning reflectometry for colloidal particle sizing,” J. Chem. Phys. 105, 1646–1653 (1996). 15. M. C. Peña-Gomar, M. L. González-González, A. GarcíaValenzuela, J. Anto-Roca, and E. Perez, “Monitoring particle adsorption by use of laser reflectometry near the critical angle,” Appl. Opt. 43, 5963–5970 (2004). 16. M. C. Peña-Gomar, J. J. F. Castillo, A. García-Valenzuela, R. G. Barrera, and E. Pérez, “Coherent optical reflectance from a monolayer of large particles adsorbed on a glass surface,” Appl. Opt. 45, 626–632 (2006). 17. V. P. Dick, V. A. Loiko, and A. P. Ivanov, “Angular structure of radiation scattered by monolayers of particles: experimental study,” Appl. Opt. 36, 4235–4240 (1997). 18. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-ofscattering expansion, and coherent field,” Opt. Express 16, 2288–2301 (2008). 19. L. Tsang and J. A. Kong, Scattering of Electromagnetic Waves; Advanced Topics (Wiley, 2001). 20. J. J. H. Wang, “A unified and consistent view on the singularities of the electric dyadic Green’s function in the source region,” IEEE Trans. Antennas Propag. 30, 463–468 (1982). 21. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, 1983). 22. H. C. van de Hulst, Light Scattering by Small Particles, 1st ed. (Dover, 1981).
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Optical reflectivity of a disordered monolayer of highly scattering particles: coherent scattering model versus experiment Omar Vázquez-Estrada1,2 and Augusto García-Valenzuela1,* 1
Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Apartado Postal 70-186, Distrito Federal 04510, Mexico 2 e-mail: [email protected] *Corresponding author: [email protected] Received December 19, 2013; accepted January 23, 2014; posted February 7, 2014 (Doc. ID 203481); published March 17, 2014
Recently a multiple-scattering model for the reflectivity of a disordered monolayer of scattering particles on a flat surface was put forth [J. Opt. Soc. Am. 29, 1161 (2012)]. The approximate theoretical model provides relatively simple formulas for the reflectivity, but it was developed for a monodisperse monolayer. Here we extend the model to the case of a polydisperse monolayer and derive the appropriate formulas to calculate the optical transmissivity of the monolayer supported by a flat interface. A second objective of this paper is to test the approximate theoretical model against experimental data with highly scattering particles. We prepared monolayers of three different surface coverage fractions of polydisperse alumina and titanium dioxide particles deposited randomly on a glass slide. We measured their optical reflectivity and transmissivity versus the angle of incidence using a laser with a wavelength of 670 nm. Using the nominal values for the particles’ most probable radius and refractive index, we fitted the theoretical model to the experimental curves and found that it reproduces very well the experimental curves. Interestingly, a dip in the reflectivity curves at large angles of incidence is present for the alumina monolayers but not in the titanium dioxide monolayers. The dip corresponds to a maximum in the scattering efficiency by the alumina monolayers. The theoretical model reproduces very well this behavior. © 2014 Optical Society of America OCIS codes: (290.5825) Scattering theory; (290.4210) Multiple scattering; (290.5850) Scattering, particles; (240.0240) Optics at surfaces; (120.5700) Reflection; (120.7000) Transmission. http://dx.doi.org/10.1364/JOSAA.31.000745
1. INTRODUCTION The optical reflectivity of disordered monolayers of particles has been of interest for many years now [1–17]. Particular attention was paid in the past to the case of monolayers of small metallic particles whose size is much smaller than the incident wavelength adsorbed on a flat surface (see, for instance, [2–11]). The problem of light reflection from monolayers of particles with size comparable to the wavelength has been addressed in more recent studies (see, for instance, [12–17]). However, comparatively fewer works have been devoted to this problem. In this case the particles may scatter the incident light efficiently and thus, in addition to a coherently reflected and/or transmitted optical beam, diffuse light propagating in all directions is also present. Nevertheless, the coherent reflectance, also referred to as the reflectivity, can be easy to measure when the particles are supported by a flat interface and the surface coverage fraction is small or moderate. The coherent transmittance or transmissivity of this type of monolayers can also be easily measured if the particles are supported by a transparent substrate. Although this problem has been studied in relatively few occasions it is not difficult to foresee some interesting applications. In particular, detecting and characterizing the optical properties of air or waterborne particles could be done by forcing their adsorption on a substrate (maybe selectively, according to their size 1084-7529/14/040745-10$15.00/0
or composition) and then analyzing the optical reflectivity and transmissivity of the partially covered surface. In order to eventually develop such type of applications it is necessary to devise theoretical models and verify they can actually describe real measurements. Such a model or models would be essential aids in pursuing applications of optical reflectivity measurements. Recently a multiple-scattering model for coherent reflectance of a sparse, random, and monodisperse monolayer of spherical particles was developed in [1]. Formulas were derived for the coherent reflectance of disordered monolayers only and no comparison with experimental measurements was attempted at the time. Thus, the objectives of this work are: (i) to extend the model to the case of polydisperse monolayers and derive also the formulas for the coherent tansmittance; (ii) to perform measurements of the coherent reflectance and transmittance of random monolayers of highly scattering particles supported on a transparent substrate; and (iii) to compare the predictions of the theoretical model with experimental data. We make no attempt to extend the model to nonspherical particles. Although in many experiments and potential applications particles are not spherical, when the particles’ shape is on the average spheroidal, we may expect they behave as equivalent spherical particles to some degree of approximation. © 2014 Optical Society of America
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When particles in a flat monolayer are very small compared to the incident wavelength, the beam is reflected, or transmitted, only in the specular direction. However, as already said, when the particles are not small, the beam will not only be reflected and transmitted partially in the specular direction, but also, it will be scattered in multiple directions. The light scattering efficiency of the particles is determined not only by their size, but also by their refractive index contrast. When particles are not small compared to the wavelength of light, the field reflected and transmitted by a monolayer of these particles, can be divided into two components, a coherent component and a diffuse component. When the incident light comes in a well-defined optical beam and impinges on a disordered monolayer on a flat substrate, the coherently reflected beam travels in the specular direction whereas the coherently transmitted beam refracts into the substrate following Snell’s law. The coherent fields correspond to the average of the reflected or transmitted fields over all the permitted configurations of the particles in the monolayer, that is to the so-called configurational average of the electromagnetic fields. Fluctuations of the electromagnetic fields about their average give rise to what we call diffuse light. The diffuse reflected and transmitted electromagnetic fields have a null average but they do transport energy, since the average of the intensity associated with these fields is not zero. If the incident beam is well collimated, the energy carried by the coherent reflected and transmitted beams is also well collimated, and thus their energy fluxes are concentrated within small solid angles. On the other hand, diffuse light is scattered in all directions, so the energy flux is spread over large solid angles and in many cases contributes very little within the solid angles subtended by the coherently reflected and/or transmitted beams. We illustrate in Fig. 1 the coherent and diffuse light components when measuring the reflectivity of a disordered monolayer of particles. In practice, the reflectivity or specular reflectance of a monolayer will directly correspond to the coherent reflectance of the system, if the incident light beam illuminates a large area of the surface with a very large number of particles and if the diffuse light contribution within the solid angle
O. Vázquez-Estrada and A. García-Valenzuela
subtended by the coherently reflected beam is negligible. When the latter condition is not entirely satisfied we may still measure the coherent reflectance by measuring the power captured by the detector at angles near the direction of the coherent reflected beam just outside the solid angle subtended by the coherent beam and subtract this value from the specular measurement done with the same detector. In doing this we would assume the diffuse light is distributed over solid angles much broader than the coherent beam, and thus it could be considered constant within and around the solid angle of the coherent beam. In general, the relative amount of average power carried by the coherent and the diffuse components depends on the angle of incidence as well as on the size and refractive index of the particles. This work is organized as follows. First, in Section 2 we present an outline of the derivation of the coherent reflectance of a free-standing, polydisperse monolayer of spherical particles referring to the mathematical steps presented in detail in [1] for a monodisperse monolayer. Then we derive the formulas for the coherent reflection and transmission coefficients of an interface partially covered by a random monolayer. We include additional terms to take into account the contribution of a second interface behind the one supporting the monolayer, to model the case when the monolayer is deposited on the surface of a (optically) thick slab. In Section 3 we briefly discuss our experimental methodology to deposit monolayers of alumina and titanium dioxide particles on the surface of a glass slide and describe our measurements of the coherent reflectance and transmittance of the samples with a laser beam. In Section 4 we present a selection of experimental curves of the coherent reflectance and transmittance versus the angle of incidence along with theoretical curves fitted to the experimental curves by adjusting the surface coverage fraction. In Section 5 we discuss the results and compare the surface coverage fraction obtained from adjusting the model to the experimental curves with rough estimates obtained from a relatively simple image analysis of optical micrographs of the monolayers. Finally, in Section 6 we give our conclusions.
2. MULTIPLE-SCATTERING MODEL
Fig. 1. Coherent and diffuse components for a light beam reflected and transmitted by a polydisperse monolayer of particles.
Here we derive a multiple-scattering model to compute the coherent reflection and transmission coefficients of an electromagnetic plane wave incident on a monolayer of polydisperse particles sitting on a flat interface between media of refractive indices n1 and n2 . We use vector electrodynamics theory and take into account in an approximate way multiplescattering among the particles and the substrate. The model also takes into account approximately the exclusion volume of the particles, and the reflection and transmission coefficients for either TE (s) or TM (p) polarized light are given in terms of the elements of the scattering amplitude matrix of an isolated particle. The approximations incurred in deriving the model limit its validity to low surface coverage fractions of the monolayer but it can be used for particles of sizes either small or comparable to the wavelength of radiation. In [1], a detailed derivation of the model for the reflection of light from a monodisperse monolayer of spherical particles sitting on the interface is given. The corresponding details are omitted in this paper.
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A. Free-Standing Monolayer First we consider a free-standing disordered monolayer of polydisperse particles in which an incident electric field, given by Ei r; t E ∘ expiki · r − iωtˆei , excites the particles. By a free-standing monolayer of particles we mean an ensemble of particles embedded in a homogeneous medium of refractive index n1 , randomly placed on an imaginary plane. Our analysis is restricted to spherical particles and we assume that all particles have the same refractive index, np . Additionally, we assume that all the particles touch a specific plane surface, but since the particles have different radii, their centers are no longer aligned on a plane as it was the case for a monodisperse monolayer. We will put origin of the coordinate system to the tangent plane that supports all the particles, see Fig. 2. The field scattered by any of the particles, say the nth particle, can be written as [18,19] Z E r s
3 0 3 00
↔
00
0
00
d r d r Gr; r · Ta r − rn ; r − rn ·
00 Eexc n r ;
(1)
where rn is the position vector of the nth particle with respect to its center, Eexc is the field that excites the nth particle, n
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where ρa is the density of probability of a particle to have a radius of a, pn rn is the density of probability of finding the nth particle of radius an at rn , and hEexc n in is the configurational average of the field exciting the nth particle while keeping its position fixed at rn . Remember, we will assume all particles sit on the plane z 0. Then the radius of a particle of radius a is located at z −a, see Fig. 2. Thus, the probability density pn rn depends on the particle radius. We assume it is uniform over the area of xy-plane covered by the monolayer and depends on zn as δzn − an . As in the case for monodisperse monolayers we can start with the so-called quasi-crystalline approximation (QCA) to set up an integral equation for the field exciting any of the particles. The difference here is that we must also average over the particle size distribution. We can write this equation as Z Z∞ Z ↔ i r 3r Eexc d3 r 0 Gr; r0 r; r E daρa d j n j 0 Z ↔ 00 · d3 r 00 prn jrj Ta r0 − rn ; r00 − rn · Eexc n r ; rn ; (4)
↔
Gr; r00 is the dyadic Green’s function of the vector wave ↔
equation in the background medium [20], Ta r0 − rn ; r00 − rn is the transition operator (T matrix) used in scattering theory for a particle of radius a [18,19]. For a collection of N particles we simply add up the field scattered by all the particles: Er Ei r
N Z X
↔
d3 r 0 d3 r 00 Gr; r0
n1 ↔
00 · Ta r0 − rn ; r00 − rn · Eexc n r :
(2)
The coherent reflected and transmitted fields correspond to the configurational average of the field scattered by the particles on either side of the monolayer. Since we are assuming the position of the particles is random and the particles have a size distribution, we must take the average over the position of the particle and over their size. We can write Z∞ Z Z ↔ hEri Ei r daρa d3 r n pn rn d3 r 0 Gr; r0 0 Z ↔ 3 00 00 (3) · d r Ta r0 − rn ; r00 − rn · hEexc n in r ; rn ;
where daρa is the number density of particles with radius between a and a da, prn jrj is the conditional density of probability of finding the nth particle around rn given that the jth particle is around rj . prn jrj depends on the radii of the jth particle and the nth particles. Clearly, it must be zero when jrn − rj j < l, where l is the sum of radii of the nth particle and jth particles and should approach a constant when the particles are far away from each other. To obtain an approximate solution to the integral Eq. (4), we replace prn jrj by Ujrn − rj j − lprn where U is the step function and we assume prn is given by A−1 δzn − an , where A is the area extended by the monolayer on the xy plane and an is the radius of the nth particle. Moreover, we assume the exciting field is in the form of two effective plane waves, one traveling in the incidence direction and the other in the specular direction [1]. We assume that all particles, regardless of their size, see the same effective field. This approximation may be regarded as equivalent to the effective field approximation used in the coherent reflection of light from a half space of a random distribution of particles [18,19]. It should be a good approximation for low surface coverage fractions. Then we write i Eexc ei E 2 expikr · rˆer ; p r; rp E 1 expik · rˆ
Fig. 2. Illustration of the monolayer supported by a flat interface. The particles are sitting on the interface and embedded in medium 1, with refractive index n1 .
(5)
where ki kix aˆ x kiy aˆ y kiz aˆ z and kr kix aˆ x kiy aˆ y − kiz aˆ z , eˆ i and eˆ r are unitary polarization vectors, and E 1 and E 2 are the amplitudes of the effective planes waves, which we must solve from a consistency requirement. Introducing Eq. (5) in the right-hand side of the QCA integral equation, and using the momentum representation of the transition operator, performing the integral over d3 r n on different portions of the monolayer with appropriate plane wave expansions for the dyadic Green’s function as it was done for the monodisperse case [1], and ignoring all integrals of only evanescent waves, yields
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η Eind j r; rj ≈ − E 1 2
Z 0 r
∞
O. Vázquez-Estrada and A. García-Valenzuela
where
daρaSa ki ; ki expki · r
Sa k ; ki exp−2ikiz a expikr · r Z∞ η − E2 daρaSa ki ; kr exp2ikiz a expki · r 2 0 Sa kr ; kr expikr · r;
(6)
where the phase factors exp−2ikiz a and exp2ikiz a take into account that the center of particles of radius a are at z −a (and not at the origin), and η
Sa q; p
2π ; k2m cos θi
(7)
↔ km ↔ ˆ · Ta q; p · eˆ p : I − qˆ q 4πi
(8)
We added the subscript a to the transition operator and to the vector S to indicate they are functions of the particle radius. If we demand the left-hand side of Eq. (6) to be equal to Eq. (5) we obtain two consistency equations from which we can solve for the amplitude of the effective exciting planes waves. We get Z∞ η E 1 eˆ i E i eˆ i − E1 daρaSa ki ; ki 2 0 Z∞ η daρaSa ki ; kr exp2ikiz a; − E2 2 0 Z∞ η daρaSa kr ; ki exp−2ikiz a E 2 eˆ r − E 1 2 0 Z∞ η daρaSa kr ; kr ; − E2 2 0
(9)
(10)
where Sa kr ; ki and Sa kr ; kr are given by Eq. (8) above. As mentioned in [1], these latter vectors are related to the elements of the amplitude scattering matrix of an isolated particle. In fact one can show that, Sa ki ; ki S a 0ˆei , Sa kr ; kr S a 0ˆer , Sa kr ;ki S j;a π −2θi ˆer , and Sa ki ; kr S j;a π − 2θi ˆei , where in the last two identities the subscript j takes the value 1 or 2 when the incident wave to the monolayer is TE (s) or TM (p) polarized, respectively. S 1;a θ and S 2;a θ are the elements of the amplitude scattering matrix of an isolated spherical particle of radius a as defined in the book by Bohren and Huffman [21]. S0 S 1;a 0 S 2;a 0 is the so-called forward scattering amplitude of the particle. Taking the scalar product on both sides of the expressions (9) and (10) with eˆ i and eˆ r , respectively, and solving the obtained algebraic equations yields E1 Ei
E2 Ei
1 12 βF ; 1 βF 14 β2F − βC βB
− 12 βC ; 1 βF 14 β2F − βC βB
(11)
(12)
Z βF η βB η βC η
∞
0
Z
∞
0
Z
∞
0
daρaS0; daρaS j;a π − 2θi exp2ikiz a; daρaS j;a π − 2θi exp−2ikiz a:
(13)
As already said, the subscript j takes the value of 1 for an incident TE-polarized wave and the value of 2 for a TMpolarized incident wave. The integrals in Eq. (13) can be efficiently p evaluated numerically, with integration limits given by a σ 18 a∘ . Where a∘ is the radius of the particle most probable (provided by the manufacturer). Using the amplitudes of the effective plane waves exciting the particles in Eqs. (9) and (10) and dot multiplying both sides of the first equation with eˆ i and the second one with eˆ r yield, E t Ei − E 1 βF − E 2 βB and Er −E1 βC − E 2 βF . The next step is to use the expressions in Eqs. (11) and (12) in the latter two equations and solve for the amplitude of the reflected and transmitted fields, i.e., r coh Er ∕Ei and tcoh Et ∕Ei . We get −βC ; 1 βF β2F − βC βB
(14)
1 − 14 β2F − βC βB : 1 βF 14 β2F − βC βB
(15)
r coh
tcoh
1 4
In the integrals in Eq. (13) the number-density distribution can be written as ρa ρST na where ρST is the number surface density of particles regardless of their radius and na is the so-called particle size distribution function. In many cases the particle size distributions can be assumed to be a lognormal distribution. Such is the case in our experiments we present below. The corresponding expressions are 1 ln2 a∕a∘ na p exp − ; 2 ln2 σ 2π a ln σ
ρST
Θ exp−2 ln2 σ: πa2∘
(16)
(17)
Here, Θ is the surface fraction covered by the particles (the surface coverage), a∘ is the radius of the particle most probable, and σ is the width parameter of the particle size distribution function. Equations (14) and (15) together with the log-normal size distribution given in Eqs. (16) and (17) reduce to the case for a monodisperse monolayer of particles with radius a∘ simply by taking the width parameter σ very small so that the result does not depend on it. B. Supported Monolayer To introduce in the model an interface at the plane were the particles are sitting z 0, let us suppose the refractive index for z > Δ, being Δ a small distance, is n2 and is different from that of incidence medium z < Δ given by n1 . Then we can
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add up all the multiple reflections between the monolayer and the interface at z Δ. Next we let z → 0 and obtain the compound reflection coefficient of a monolayer sitting on top of a substrate of refractive index n2 , see Fig. 3. In this case we also take into account an additional reflection of the coherent beam transmitted into the slab supporting the monolayer at the interface between the medium 2 and medium 3 (green line in Fig. 3), but we ignore multiple reflections with this interface. We get
r coh θi r 12 θi t2coh θi expβ1
r coh θi t2coh θi r 212 θi exp2β1
t2coh θi r 21 θi t12 θi t21 θi expβ2 ;
(18)
where β1 2ik∘ Δn1 cos θi and β2 2ik∘ dn2 cos θ2 , d being the width of the slab (see Fig. 3) and θ2 the angle of refraction of the coherent beam in the slab, r 12 and t12 are the Fresnel transmission and reflection coefficients, respectively, of the interface between refractive indices, n1 and n2 evaluated at an angle of incidence θi . Similarly, t21 and r 21 are the Fresnel reflection and transmission coefficients of the interface between refractive indices, n2 and n1 . r coh and tcoh are the coherent reflection and transmission coefficients of a freestanding monolayer given by Eqs. (14) and (15). Note that since d ≫ λ we can drop the a phase factor expβ2 on the last term on the right-hand side of Eq. (18) since it oscillates very rapidly with the angle of incidence and such oscillations will be averaged out by the detector. Then the sum is simplified when Δ → 0, to
supporting the monolayer. Subsequent multiple reflection with this interface are neglected. Similarly, according to Fig. 3, the transmission coefficient for a monolayer supported by a flat interface is given by tθi t01 t02 tcoh θi t12 θi t21 θi expβ∘ r coh θi tcoh θi r 12 θi t12 θi t21 θi expβ∘ expβ1 (20)
rθi r 01 r 02 r 03 r 001
rθi r coh θi
749
Here, β∘ 0.5β1 . Note that we obviated the phase factor exp12 β2 common to all terms in Eq. (20) since it is cancelled out when computing the transmittance. If we now take the limit Δ → 0, the previous sum simplifies to tθi
tcoh θi t12 θi t θ : 1 − r 12 θi r coh θi 21 i
(21)
The expression within the square brackets takes into account all multiple reflections of the coherent beam between the monolayer and the interface supporting it. The transmission coefficient t21 outside the square brackets is due to the interface of the glass slide behind the one supporting the monolayer. Again, subsequent multiple reflections with this interface are neglected. The coherent reflectance and transmittance of a glass slide supporting a monolayer of particles is given by the magnitude square of the coherent reflection and transmission coefficients given in Eqs. (19) and (21), respectively. That is, Rcoh θi jrθi j2
and T coh θi jtθi j2 :
(22)
r 12 θi t2coh θi
1 − r 12 θi r coh θi
t2coh θi r 21 θi t12 θi t21 θi :
(19)
The second term on the right-hand side of this last equation includes all the multiple reflections between the monolayer and the interface supporting it, whereas the last term on the right-hand side is due one reflection of the coherent beam at the bottom surface (exposed to air) of the glass slide
Fig. 3. Schematic illustration of multiple reflections of a monochromatic beam incident on a monolayer of particles immersed in air and sitting on a flat substrate.
3. EXPERIMENTAL METHODOLOGY Previous to testing our model against experimental data, we prepared monolayers of alumina and titanium dioxide particles (rutile) on a flat glass surface. The nominal values of the refractive indices of the particles are np 1.76 for alumina particles and np 2.73 for titanium dioxide particles. The most probable radius a∘ of the alumina particles as indicated by the manufacturer is about 1.5 μm and for the titanium dioxide particles is about 220 nm. As substrate we used standard 1 mm thick microscope glass slides 27 mm × 75 mm of refractive index n2 1.5. The procedure to deposit random monolayers of the particles was as follows. First, the slides were extensively washed and cleaned with detergent, acetone, isopropyl alcohol, and tridistilled water to thereby remove dust and grease from the surface. The next step was to prepare a dispersion of the particles in a solvent that evaporates rapidly after deposition. The alumina particles were obtained in powder from the manufacturer (Alumina Buehler No. 40-6603-030-080) and we simply mixed 1.74 g of the alumina powder with 39 mL of isopropyl alcohol. The titanium dioxide particles were provided to us in a slurry consisting of a mixture of deionized water (999.0 g), with dispersant (0.99 g), and titanium dioxide particles (0.99 g). We mixed 9.0 mL of the titanium dioxide slurry with 36 mL of isopropyl alcohol, thus obtaining a dispersion of TiO2 particles. The two particle dispersions were immersed in an ultrasonic bath for 5 min to reduce particle clusters during the deposition
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of the monolayers on the slide’s surface. Sonication helped depositing isolated particles at random on the slide surface. The particles are, of course, not spherical, but we will consider them as spherical as an approximation. To deposit a monolayer, the clean slide was immersed in the particle dispersion in isopropyl alcohol at constant speed and at a fixed angle of inclination of about 85° (with respect to the horizontal), until 90% of the glass slide was covered. Subsequently, we withdrew at the same speed and angle of inclination the glass slide from within the particle suspension. This technique is commonly known as dip coating. Actually, a monolayer of particles was deposited on both sides of the cover slide, but after the deposition, we cleaned the side of the slide that was facing down during the immersionwithdrawal process. The surface coverage by the particles achieved by the described dip-coating process depended on the time it took us to immerse and remove the slide from the particle dispersion as well as the angle at which the slide was immersed. With this in mind, we prepared three monolayers, with different cover surface fractions, all deposited on a glass slide with alumina particles and three more with TiO2 particles. We adjusted the immersion and withdrawal time to deposit the particles to about of 5, 3, and 1 s, and the angle in immersion to about 85°, for each type of monolayer. These parameters of the deposition process produce sparse monolayers that reflect specularly a measurable amount of light at near normal incidence. Then we proceeded to measure carefully the reflectivity and transmissivity of TEpolarized laser light as a function of the angle of incidence, θi . In Figs. 4(a) and 4(b), we show our experimental setup for coherent reflectance and transmittance measurements. We used a linearly polarized laser diode λ 670 nm and adjusted its polarization (by simply rotating the laser) to be TE polarization in the experiment. The sample (with the deposited monolayer) was fixed on top of a goniometer with its surface perpendicular to the plane of rotation of the goniometer, as depicted in Fig. 4. Special care was taken in the alignment of the experimental setup to ensure that the plane of the monolayer/glass-slide was perpendicular to the surface of the goniometer and the point of reflection of the laser beam on the monolayer coincided with the rotational axis of the goniometer. The reflectivity and transmissivity of the laser beam was measured at different angles of incidence, in a range about 0 to about 86 deg. It is important to remark that the region of the sample used to measure the coherent reflectance was the same used to measure the coherent transmittance. The incident and reflected optical power was collected by a silicon photodetector connected to a transimpedance
Fig. 4.
O. Vázquez-Estrada and A. García-Valenzuela
preamplifier converting the photocurrent signal to a voltage signal, which in turn, was registered by a digital multimeter (Agilent 34401A), as depicted in Fig. 4. The position of the detector was adjusted manually, ensuring the coherently reflected or transmitted beam was captured completely by the detector. Also, the amount of diffuse light entering the detector during the measurements was estimated for several angles of incidence by measuring the diffuse power captured by the detector at viewing angles around the coherent beam. In general, the contribution of diffuse light to the specular reflectance and transmittance was negligible, but when noticeable it was subtracted from the detector’s readings. The coherent reflectance was obtained simply dividing the coherently reflected optical power by the incident optical power. Similarly, the coherent transmittance was obtained dividing the coherently transmitted power by the incident optical power.
4. RESULTS In Figs. 5 and 6, we show the graphs corresponding to the measurement of the coherent transmittance and reflectance versus the angle of incidence for the alumina and TiO2 monolayers (red points). In insets of Figs. 5 and 6, we show images taken by an optical microscope of the alumina and TiO2 monolayers used to obtain the reflectivity and transmissivity data shown in the figures. The incidence medium was air with a refractive index of n1 1.00. In these figures we also include theoretical curves calculated with the formulas given above assuming a monodisperse monolayer (dashed blue lines) and a polydisperse (full black lines). To calculate the amplitude scattering coefficients S0, S 1 π − 2θi and S 2 π − 2θi , we used the so-called Mie theory [21,22]. To obtain the curves corresponding to monodisperse monolayers we simply considered a very narrow size distribution and used a width parameter of σ 1.01. The theoretical curves were calculated assuming a lognormal size distribution and supposing that the refractive index of the particles and their most probable radius are given by their nominal values, that is, 1.76 and 1.5 μm for the alumina particles, and 2.73 and 220 nm for the TiO2 (rutile) particles, respectively. For the polydisperse curves we adjusted the width parameter to σ 1.3 for the alumina particles and σ 1.4 for the TiO2 particles for best fitting the experimental data. We used these values of σ for all reflectance and transmittance curves. Then we fitted the theoretical curves to the experimental data by adjusting the surface coverage fraction Θ so as to reproduce best the experimental data. We obtained Θ 3.8%, Θ 5.7%, and Θ 10.6% for the
(a) Schematics of the experimental setup for measuring the coherent reflectance and (b) the coherent transmittance.
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Fig. 5. Coherent reflectance (left) and coherent transmittance (right) for TE-polarized light of three monolayers of alumina with different surface cover fractions. The red dots are experimental measurements, the full (black) curves are theoretical curves assuming a polydisperse monolayer, and the dashed (blue) curves are the corresponding theoretical curves assuming a monodisperse monolayer. The values of the surface coverage fraction that fitted best the polydisperse curves were: (a) Θ 3.8%; (b) Θ 5.7%; and (c) Θ 10.6%.
Fig. 6. Coherent reflectance (left) and coherent transmittance (right) for TE-polarized light of three monolayers of TiO2 with different surface cover fractions. The red dots are experimental measurements, the full (black) curves are theoretical curves assuming a polydisperse monolayer, and the dashed (blue) curves are the corresponding theoretical curves assuming a monodisperse monolayer. The values of the surface coverage fraction that fitted best the polydisperse curves were: (a) Θ 1.3%; (b) Θ 4.9%; and (c) Θ 10.7%.
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alumina particles and Θ 1.3%, Θ 4.9%, and Θ 10.7% for the titanium dioxide particles, respectively. The uncertainty of the values of Θ obtained upon fitting the model to the experimental data while keeping all other parameters in the model fixed was less than 0.02%. We can appreciate in these figures that the theoretical curves reproduce quite well the experimental data. It is interesting to note the strong qualitative difference between the reflectivity curves for the alumina and that for the titanium dioxide particles. In the former case the reflectivity has a dip at large angles of incidence that is not present in the latter case. The multiple-scattering model reproduces very well this phenomenon. Also, note that the monodisperse multiplescattering model actually reproduces quite well the experimental data. In the case of the alumina particles the curve for the monodisperse multiple-scattering model shows some oscillations (due to Mie resonances) which are not present in the experimental data, but the polydisperse model averages out these oscillations and reproduces better the experimental curve. Some discrepancies between theory and experiment can be appreciated, which may be due to the fact that the particles are not spherical and may be also to particle clusters, which is not taken into consideration for the theoretical model. In addition, to the geometric shape of projection of the laser beam on the sample is affected when the angle of incidence increases. At oblique angles of incidence, the laser projects an elliptic footprint that increases along the plane of incidence and covers a larger section of the sample, especially for grazing incidence angles. Illuminating larger areas of the sample may result in seeing different surface coverage fractions at different angles of incidence, since the deposited monolayers do not have a perfectly uniform value of the surface coverage fraction across the whole sample. Nevertheless, we find rather surprising how well the multiple-scattering model fits the experimental data. As an additional test to the multiple-scattering model we obtained transmittance spectra at normal incidence for two specific monolayers using a white light source (LED XR-C) and a minispectrophotometer (Oceanoptics Model USB4000). We chose the alumina monolayer with surface coverage fraction of Θ ≈ 5.7% and the TiO2 particles monolayer with a surface coverage of Θ ≈ 1.3%. The reflectance spectra could not be measured because the reflected power was too little to be measured by the spectrophotometer. Both transmittance spectra, between 450 and 700 nm, are shown in Fig. 7. We also include in the graphs of Fig. 7 theoretical spectra calculated with the polydisperse and monodisperse multiple-scattering models assuming the same values of the parameters used to fit the experimental data in Figs. 5(b) and 6(a). Note that the spectra predicted by the polydisperse model are flat and reproduce very well with the experimental spectra. The corresponding theoretical spectra predicted for a monodisperse monolayer also coincide well on the average with the experimental spectra, but they show a fine structure consisting of irregular oscillations about an horizontal line. This graph illustrates well that the transmittance spectra do not give us more information of these monolayers than the angular reflectivity and transmissivity curves. Nevertheless, it is worth to emphasize that the parameters used to obtain the theoretical curves of reflectance and transmittance of these monolayers in Figs. 5(b) and 6(a) (coverage
O. Vázquez-Estrada and A. García-Valenzuela
Fig. 7. (a) Transmittance spectrum at normal incidence for a monolayer of particles of alumina with surface coverage factor of Θ ≈ 5.7% and (b) transmittance spectrum at normal incidence for a monolayer of particles of TiO2 with surface coverage factor of Θ ≈ 1.3%.
fraction, particles’ size distribution, and refractive index) were the same as those used to obtain the theoretical spectra in Fig. 7.
5. DISCUSSION The above results show that the multiple-scattering model for polydisperse monolayers reproduces very well the experimental data for monolayers of highly scattering particles. The fact that a single value of the particle size distribution can be used to fit all curves, for both reflectance and transmittance versus the angle of incidence, gives us some assurance that the assumption of a log-normal size distribution is valid. On the other hand, the fact that after fixing the width of the size distribution for each type of particle and then only adjusting the surface coverage factor the theoretical model fits rather well and simultaneously both the reflectance and transmittance curves for each sample, clearly supports the validity of the multiple-scattering model for these types of samples. Further confidence in the model would be attained if we could corroborate by other means the surface coverage fraction values obtained from fitting the theoretical curves to the experimental data. To this end we took microphotographs of the samples using an optical microscope with 400× amplification. Then we processed the digital images with a simple program that counted the number of pixels that registered an intensity below certain threshold and divided it by the total number of pixels in the image. The result is a rough estimation of the surface coverage fraction in each sample. An example of an actual micrograph image and its processed image with the software is shown in the Fig. 8 (alumina monolayer, Θ 3.8%). The value of the threshold was adjusted and kept within a range in which the result was less dependent on the exact value of the threshold. In Table 1, we show the values of the surface coverage fraction for the three monolayers of alumina derived from fitting the theoretical model to the experimental coherent reflectance and transmittance curves,
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Table 2. TiO2 Particles TiO2 Particles ΘFit ΘEst
Fig. 8. In the upper figure is shown a micrograph of a monolayer of alumina particles for the sample with a surface coverage fraction of Θ 3.8%. In the lower figure is shown the reconstructed image with the software, for this same sample, of the monolayer of particles without the substrate.
ΘFit , and the one estimated from processing the corresponding microphotographs, ΘEst . Similarly, in Table 2 we show the values of the surface coverage fraction for the three monolayers of TiO2 derived from fitting the theoretical model to the experimental coherent reflectance and transmittance curves, ΘFit , and the one estimated from processing the corresponding microphotograph for these monolayers, ΘEst . The values of ΘFit and ΘEst in Tables 1 and 2 are close to each except for the case of sample 2 of TiO2 particles. Taking into consideration that the microphotographs only capture a very small area of the sample whereas the laser illuminates a much larger area when the coherent reflectance is measured, we believe the agreement between ΘFit and ΘEst is actually quite good, and thus, does support the validity of the theoretical model at least in a qualitative manner.
ΘFit ΘEst
Sample 2 (%)
Sample 3 (%)
1.30 1.44
4.90 1.66
10.70 10.32
As already noted, the curves of coherent reflectance versus the angle of incidence for the alumina particles present an interesting dip at rather large angle of incidence, whereas the curves for the TiO2 monolayers do not. Actually these dips can be readily explained in simple terms referring to Eq. (19). Plotting independently the magnitudes of r coh θi ; t2coh θi , and r 12 θi reveals the reason of the presence of a dip in coherent reflection curve for alumina monolayers. When the particle size is very large as is the case of the alumina particles, these scatter mostly in the forward direction. Thus, the coherent reflection coefficient of the free-standing monolayer r coh is basically zero except for large angles of incidence where it increases rapidly to unity at grazing incidence. On the other hand, the coherent transmission coefficient tcoh , is not much less than unity; it remains nearly constant up to some value of the angle of incidence and then it decreases monotonously to zero for grazing incidence. Now, let us take a look to Eq. (19). For simplicity we can ignore the term corresponding to the first reflection from the back of the glass slide (the third term on the right-hand side). Then, for small and moderate angles of incidence, the first term on the right-hand side of Eq. (19), r coh θi , is very small and can be ignored, whereas t2coh θi is nearly constant. Thus, the coherent reflection coefficient is basically the reflection coefficient of the substrate’s interface. It is nearly constant for small angles of incidence and as the angle of incidence increases it starts to increase at moderate angles of incidence. At some point t2coh θi starts to decrease rapidly and brings the net reflection coefficient to zero. Just after this has happened we are left with rθi r coh θi , which starts to increase as the angle of incidence increases further and the net coherent reflection coefficient, rθi , goes to unity, as it should always do at grazing incidence, drawing a curve with a dip where both t2coh θi and rθi are simultaneously negligible. The dip is not present in the curves for TiO2 monolayers because the particles are smaller and scatter more efficiently in all directions, not only near the forward direction. Then, r coh θi starts to increase well before t2coh θi drops to negligible values. Note that in our experimental examples the absorption by the particles is very small. Thus, at the dip in the reflectance curves for the alumina particles, we have that the coherent transmission and reflection are both very small. Thus, this means that at the angle of incidence where the dip is located the supported monolayer scatters all incident radiation. A possible application of this phenomenon could be in the manufacture of surfaces with reduced gloss at high angles of incidence.
6. SUMMARY AND CONCLUSION
Table 1. Alumina Particles Alumina Particles
Sample 1 (%)
Sample 1 (%)
Sample 2 (%)
Sample 3 (%)
3.80 4.31
5.70 5.20
10.60 13.83
We provided an approximate multiple-scattering model for the coherent reflection and transmission from a monolayer of disordered polydisperse spherical particles. We performed experimental measurements of the coherent reflectance and transmittance from disordered polydisperse monolayers
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of alumina and titanium dioxide particles with small values of the surface coverage. It was found that the model fits very well the experimental data, the surface covered being the only parameter adjusted independently for each sample. We compared the values of the surface coverage fraction obtained from fitting the theoretical model to the experimental data with rough estimation derived from optical microphotographs of the monolayers and found a reasonable agreement. Since we do not know with certainty all the parameters involved in the sample monolayers, the fact that the theoretical model reproduces the coherent reflectance and transmittance versus the angle of incidence shows that the theoretical model is valid at least qualitatively, and thus takes into account the main physical phenomena determining the coherent reflection and transmission of light by highly scattering particles. A direct application of the theoretical model developed in this paper is to aid in the monitoring of the deposition of highly scattering particles on a substrate. Other applications could be in the biological realm. For instance, one could try to characterize living cells on a substrate and monitor their interaction with their surroundings through a change in the refractive index contrast between the cells and the liquid around them. Pursuing such possible applications would require contemplating in the model nonspherical particles with size parameters very large compared to one and a low refractive index contrast. Finally, the reflectivity versus angle of incidence curve shows a clear dip around an angle of incidence of 85° that was not expected. However, the theoretical model reproduces very well this dip in reflectivity. We provided a simple physical explanation of this effect. A possible application of this phenomenon could be in the manufacture of surfaces with reduced gloss at high angles of incidence.
ACKNOWLEDGMENTS We acknowledge financial support from Dirección General de Asuntos del Personal Académico from Universidad Nacional Autónoma de México through grant PAPIIT IN-106712.
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