Precise optical modeling of quantum dots for white light-emitting diodes

Quantum dots (QDs)-based white light-emitting diodes (QDs-WLEDs) have been attracting numerous attentions in lighting and flat panel display applications, by virtue of their high luminous efficacy and excellent color rendering ability. However, QDs’ key optical parameters including scattering, absorption and anisotropy coefficients for optical modeling are still unclear, which are severely against the design and optimization of QDs-WLEDs. In this work, we proposed a new precise optical modeling approach towards QDs. Optical properties of QDs-polymer film were obtained for the first time, by combining double integrating sphere (DIS) system measurement with inverse adding doubling (IAD) algorithm calculation. The measured results show that the typical scattering, absorption and anisotropy coefficients of red emissive QDs are 2.9382 mm−1, 3.7000 mm−1 and 0.4918 for blue light, respectively, and 1.2490 mm−1, 0.6062 mm−1 and 0.5038 for red light, respectively. A Monte-Carlo ray-tracing model was set-up for validation. With a maximum deviation of 1.16%, the simulated values quantitatively agree with the experimental results. Therefore, our approach provides an effective way for optical properties measurement and precise optical modeling of QDs for QDs-WLEDs.

QDs layer. Previous studies have presented some optical models of QDs to simulate their optical performances in various optoelectronic devices [14][15][16][17] . In an approach, QDs are treated as dipoles and transfer matrix formalism 11 is utilized to calculate the light out coupling and angular performance of QDs-based device 14,15 . While there are difficulties in directly measuring the QDs' microcosmic parameters such as power dissipation density, wave vector and intrinsic quantum yield et al., making this method impractical to simulate real QDs-polymer film. In another approach, Monte-Carlo ray-trace method is performed to theoretically evaluate the performance of QDs-based materials 16,17 . Unlike the first method, this approach is conducted by inputting the absorption, PL spectra of QDs, the attenuation coefficient and QY of QDs, which are easier to be measured and are practical to QDs devices. In previous literatures, before processing the Monte-Carlo ray-trace simulation, some assumptions have to be made to conduct the calculation, including assuming that light scattering won't occur in the QDs-polymer film due to the small size of QDs 16,18,19 . However, as far as we are concerned, this assumption is not always reasonable because the scattering coefficient of QDs-polymer film is non-negligible, especially when scattering particles are doped in [20][21][22][23] . Therefore, a more precise optical model of QDs, including coefficients of absorption, scattering and anisotropy, need to be developed for QDs-WLED design and optimization.
Here, we reported on a precise optical model of QDs by combining double integrating spheres (DIS) system with inverse adding-doubling (IAD) algorithm. The proposed model takes into account all the fate of incident photons, including transmittance, absorption and scattering, which is compatible with the real situation. A series of QDs-polymer films samples with varying QDs concentrations were fabricated, and their transmittance, reflectance and collimating transmittance were measured by the DIS system. Then their optical properties of scattering coefficient, absorption coefficient and anisotropy coefficient were calculated by the IAD algorithm. Finally, these parameters were adopted in Monte-Carlo ray-tracing model to conduct the simulation, and the ray-tracing results were compared with the experimental results to validate the feasibility of our model.

Experimental setup
Definitions. As depicted in Fig. 1, the optical properties of QDs layer can be characterized by quantities as follows: where the scattering coefficient µ s and the absorption coefficient µ a are defined as the reciprocal of the average free path between two scattering events (I s ) or two absorption events (I a ), respectively. If the QDs layer is highly absorbing, the number of absorption events will be big, leading to a small average free path I a , which yields a high absorption coefficient µ a . Therefore, the higher the scattering or the absorption of a QDs layer is, the higher the respective coefficient would become. In addition to the above two optical quantities, there is another parameter, single-scattering phase function p(θ), which characterizes the amount of light scattered at an angle θ from the incoming direction. It is often expressed in terms of the cosine of the scattering angle θ: where dω is a differential solid angle, and v is the cosine of the scattering angle. The functional form of p(v) is usually unknown. While Jacques et al. 24 and Yoon et al. 25 have demonstrated that Henyey-Greenstein function approximates single-particle light scattering in human dermis and aorta at 633 nm. Therefore, this phase function is adopted in our calculation: The Henyey-Greenstein phase function depends on the anisotropy coefficient g, which is defined as: The anisotropy coefficient g ranges from −1 to 1. When g = −1, scattering is all directed into reverse direction, when g = 1, scattering is all directed into forwarding direction, and when g = 0, scattering is equally probable in all directions. Using the physical sample thickness d, the coefficients µ s and µa can be expressed by two dimensionless quantities, the albedo a and the optical thickness τ 26 : The albedo varies between 0 and 1, a = 0 indicates that no scattering occurs in the sample, while a = 1 indicates the absence of absorption. The optical thickness is defined as the product of the physical sample thickness and the sum of the scattering and the absorption coefficient. For a sample with the optical thickness τ = 1, there is a probability of e −1 = 37% that light will travel through it without being scattered or absorbed. In the following contents, the optical properties will be calculated in terms of three dimensionless quantities a, τ, and g. Measurement setup. The measurement of the optical properties of QDs samples can be achieved by the use of a double integrating sphere set-up, which is schematically depicted in Fig. 2. The QDs sample is placed between two integrating spheres so that it is situated at the exit port of the reflectance sphere (used to measure the reflectance P r ), and on the entrance port of the transmittance sphere (used to measure the transmittance P t ). These two ports have identical sizes. Besides, the third integrating sphere (collimating transmittance sphere) is situated directly behind the transmittance sphere and is used to measure the light power P ct that is specularly transmitted through the sample. The entering light is incident perpendicularly upon the sample.
To measure the reflectance and the transmittance of a sample, the photomultiplier is usually used as detectors 27 . Theoretically, the voltage V δ recorded by a detector is proportional to the light power P δ incident upon it: where K is the constant proportionality, which depends on the detector characteristics. Due to the background signal V 0 , which occurs in all measurements and originates from noise in the system and the presence of stray light, expression (9) has to be expanded as follows: Therefore, the detected power can be expressed as: The constant K can be eliminated by using reference measurements which conducted with a single integrating sphere set-up. For a reference plate in the sample position, P ref can be determined by: Therefore, by conducting the reference measurements, the relative values V r %, V t % and V ct % can be calculated as: where the values V r,0 , V t,0 and V ct,0 represent the background signals corresponding to the respective measurements V r , V t and V ct .
Calculation theory. As mentioned above, the final aim is to obtain the optical properties given by a, τ, and g, respectively. While the DIS measurements do not yield a, τ, and g directly, several measurement values that depend on the reflectance and the transmittance of the sample. Therefore, we have to find a connection between the optical properties and the measurement results. This can be done by the use of an appropriate model of radiative transfer. One of these models is used in the Adding Doubling (AD) Method 27 , which was introduced by van de Hulst as one-dimensional numerical solution of the radiative transport equation. For given optical properties (albedo a, optical thickness τ, and anisotropy coefficient g), the reflection factors (R d , R cd , R c ) and the transmission factors (T d , T cd , T c ) can be calculated by using this algorithm: where the reflection factor R and the transmission factor T are defined relative to the irradiance on the sample surface. They vary between 0 and 1 and denote the fraction of the total incident light that is reflected or transmitted by the sample, respectively. For diffuse light incident upon the sample, the diffuse reflection (transmission) factor R d (T d ) denotes the fraction of light that is reflected (transmitted) diffusely by the sample. For collimated light incident upon the sample, the diffuse reflection (transmission) factor R cd (T cd ) denotes the fraction of light that is reflected (transmitted) diffusely by the sample. The collimated reflection (transmission) factor R c (T c ) denotes the fraction of light that is reflected (transmitted) specularly.
The AD method assumes that the reflection function R(ν, ν') and transmission function T(ν, ν') for light incident at an angle ν and exiting at an angle ν' are known for one layer. For two similar and adjacent slabs, the light traveling from left to right across the boundaries 0, 1 and 2. The reflection and transmission of the two slabs can be derived from the results of the thin slab by using the AD equations 28,29 : 02 20 1 where R 01 and T 01 are defined as the reflection and transmission operators for light incident upon boundary 0 and moving towards boundary 1, respectively, and vice-versa. E is the unity matrix. Beginning with the starting thickness τ start , R start and T start , this doubling process repeatedly doubles the slab thickness by using the expressions (14) and (15), until the desired slab thickness τ has been reached. τ start has to be far smaller than the thickness of QDs sample d (τ start = 2 −n d, n = 100~1000). After decided the τ start , the R start and T start can be determined by the initialization routine 29 . The resulting R and T are taken as the reflection matrix and the transmission matrix of the sample. The correlation between the measurement data and the reflection/transmission factor can be calculated by the 'double integrating sphere function': With the expressions (13) and (16), we can deduce that, the measurement results V r %, V t % and V ct % can be predicted, if the optical properties a, τ and g of the sample are given, by a combination of the functions f AD and f DIS :

DIS AD r t ct
In our case, we want to perform the calculation in the other direction. An iterative solution of the problem is given by the Inverse Adding Doubling (IAD) method, the function can be expressed as: IAD D IS AD r t ct 1 Figure 3 shows the correlations between the three levels of quantities. Consequently, the procedure to obtain the optical properties of a sample by the use of DIS set-up and IAD algorithm is as follows: (1) Conduct the reference measurements to calibrate the measurement set-up.
(2) Determine the three measurement values V r %, V t % and V ct % for the sample.
(3) Calculate the optical properties a, τ and g with the IAD algorithm, by using the measurement data as the input of the program. (4) Calculate µ a and µ s from the returned values a and τ.

Results and Discussion
To verify the accuracy and effectiveness of our IAD algorithm, we performed an AD calculation for an 'imaginary' sample with the optical properties a = 0.8, τ = 1, g = 0.8 and a refractive index n s = 1.5. The sample was situated between two 'imaginary' glasses with reflection factor R g1 = R g2 = 0.04 and transmission factor T g1 = T g2 = 0.96 (n g = n s = 1.5). By using the AD function described in expression (13), we can get the 'measurement values' = .
In another direction, if we use above three values in equation (19) as the inputs for the IAD algorithm, we should obtain the optical properties a = 0.8, τ = 1, g = 0.8 as a result. As all three measurement values are known, all optical properties can be determined. Figure 4 shows the development of a and g during the iteration. In our IAD program, the routine terminates after 145 iteration steps, returning the values of a = 0.7999 and g = 0.8001. Therefore, our IAD algorithm has high accuracy and fast convergence speed.  Without loss of generality, CdSe/ZnS core/shell QDs were used as the QDs-polymer film samples to validate our model. High-quality red-emissive QDs with an optimized peak wavelength of 626 nm were prepared by the Tri-n-Octylphosphine (TOP)-assisted successive ionic layer adsorption and reaction (SILAR) method 11,30 . According to our previous work 31 , adding QDs with a peak wavelength of 626 nm into phosphor-converted LEDs can greatly improve the color rendering index. Polymethyl methacrylate (PMMA) was chosen as the polymer matrix of QDs film samples. The QDs-PMMA film was prepared with the in-situ polymerization method 32 . Firstly 25 mL of MMA monomer and 0.2% wt/wt of azodiisobutyronitrile (AIBN) with respect to MMA were mixed in a three-neck flask and kept stirring until the AIBN was dissolved completely. Then 3 mL of MMA-AIBN solution was transfer to a cleaned centrifuge tube and 10 µL of QDs solution was added into the tube, and the mixture was homogeneously dispersed by the ultrasound treatment. After that, the tube was placed into the thermostatic water bath at 70 °C for 15 min until the mixture reached certain viscosity, and then cooled to temperature. Then, the viscous liquid was introduced into the tailored mould. The mould was placed in the vacuum oven at 45 °C and kept for 24 h. Finally, the resulting film was cut into circle and labeled as sample No. 1. Similarly, Nos 2-6 samples were prepared by adding 20, 30, 40, 50 and 60 µL of QDs solution into 3 mL of MMA-AIBN solution. Figure 5(a) shows the high-resolution transmission electron microscope (HRTEM) images of the as-prepared QDs, from which the average diameter of QDs is measured as 6.8 nm with uniform size distribution. Figure 5(b) depicts the absorption and PL spectra of the CdSe core and CdSe/ZnS core-shell QDs. The narrow width at half maximum (FWHM) of near 31 nm from PL spectra has confirmed their uniform size distribution. Due to the TOP-assisted SILAR method, which removes the surface lattice imperfections by the surface ions re-dissolution and lattice re-arrangement during the whole ZnS shell formation process, the absolute PL quantum yields (PLQY) was enhanced from 45% to 69%. Figure 5(c) and (d) show the photographs of the QDs-PMMA film under daylight and UV light, respectively. The as-prepared film samples maintain high transparency and uniformity, and the thickness of these films are measured as 0.41 to 0.44 mm.
In QDs-LEDs packaging, the main concerns are the absorption and scattering properties of QDs film for blue light emitted from the LED chip and converted red light emitted from itself. Therefore, in the DIS measurement, two diode lasers with peak wavelength of 450 nm and 650 nm were used as the light sources. The optical power of these lasers is 20 mW, and their beam diameter is expanded to 8 mm to illuminate a sufficiently large area of QDs-PMMA film. Diameters of the reflectance, transmittance and collimating transmittance spheres are 300 mm, 300 mm and 150 mm, respectively. Diameters of the apertures are 18 mm. For blue light (450 nm) excitation, the spectra of transmitted and reflected light are divided into two parts at 500 nm. Light with spectra from 380 nm to 500 nm is defined as unconverted blue light. For red light (650 nm) excitation, since no converted light is emitted from the QDs film, both the transmitted light and reflected light are sorted as the red light. Figure 6 shows the measured reflectance and transmittance of QDs-PMMA samples based on blue and red light incidence. For simplicity, the reflectance, transmittance and collimating transmittance of the blue light and the red light are denoted as V RB , V TB , V CTB , V RR , V TR and V CTR . In the case of blue light incidence, V RB only takes a very small proportion (1.04%~1.45%), and it varies little against the QDs concentration. This is mainly because, during the light propagation, the backward scattered blue light is further absorbed by the QDs particles. It can be found that the QDs present remarkable high absorption and scattering for the blue light, and cause rapid decrease of V CTB and small reduction of V TB when the QDs concentration increased.
In the case of red light incidence, the results are quite different from those of blue light. The V RR and V TR increased with the increasing QDs concentration, which depicts that the increase of QDs particles leads to the enhancement of multiple scattering effects, and the backward scattered red light can emit out due to the weak absorption of the QDs. Consequently, the V CTR decreased with the increasing QDs concentration, because more and more light changed propagation direction.
According to the measured reflectance and transmittance values, the optical properties (µ s , µ a and g) of QDs-PMMA film samples are calculated by IAD algorithm, and the results are listed in Table 1. Measured results showed that the QDs present strong absorption for blue light, high reflection for converted red light, and an anisotropic emission pattern of converted red light. It is seen that the scattering and absorption coefficients increase with the increasing QDs concentration, while the anisotropy coefficient changes slightly with the QDs concentration. Under the same QDs concentration, the scattering and absorption coefficients toward blue light incidence are large than that of red light incidence, which agrees with the measured results.
To validate our calculated optical properties of QDs film, an optical model of DIS system, which imitate the real DIS measurement system, was built up. The blue light (450 nm) is emitted out perpendicular to the laser aperture, with a beam divergence angle of 1.2 mrad. The inner surfaces of the three integrating spheres are coated with a diffuse white material having optical properties of 11.1% absorption and 89.9% scattering, and are used as the receivers to collect the transmitted and reflected light.
The Monte Carlo method is applied for ray-tracing of the optical model. Total number of the traced rays is 2 million, of which 1 million rays are the blue light and the other 1 million rays are the converted-red light. The threshold of light rays is 10-4. During the ray-tracing, the energy of light rays is calculated by optical power in watts. Light rays with single wavelength are used to represent the blue light and the converted-red light, which are set to be 450 nm and 626 nm, respectively. This simplification has been proven to be effective for the ray-tracing of light-emitting process [33][34][35][36][37] . Figure 7(a) shows the comparisons between the ray-tracing and experimental results of reflectance, transmittance and collimating transmittance for the incident blue light. It can be seen that the simulated V TB and V CTB values agree well with the experimental results, with a maximum deviation of 0.26%. The V RB values in the ray-tracing are slightly larger than those in the experiments, with a maximum deviation of 0.32%, respectively. This larger difference is mainly due to the relatively low value of V RB . Figure 7(b)   are not displayed here due to their weak signals that can hardly be collected by the photodetector. With a maximum deviation of 1.16%, the simulated values quantitatively agree with the experimental results. Therefore, the QDs optical properties can be measured and calculated precisely by our approach, which provides wide guidance for QDs-WLEDs design and fabrication.

Conclusion
In summary, this work provides a precise optical modeling approach towards QDs-polymer materials. The optical properties of QDs-PMMA film were measured by a double integrating sphere system and calculated by the inverse adding doubling algorithm. Measured results showed that the QDs present strong absorption for blue light, high reflection for converted red light, and an anisotropic emission pattern for both blue light and converted red light. Based on the calculated results, the absorption coefficient, scattering coefficient, and anisotropy coefficient were imported into the Monte-Carlo ray-tracing model for validation. With a maximum deviation of 1.16%, the simulated values quantitatively agree with the experimental results. Therefore, our optical modeling approach can obtain the necessary optical properties of QDs-polymer film, and the parameters can be utilized to conduct the ray-tracing simulation of QDs-based devices.
Data availability. All data generated or analyzed during this study are included in this published article.