Photonic Characterization and Modeling of Highly Efficient Color Conversion Layers With External Reflectors

We propose a new method for fabricating arrays of colloidal quantum dots for color conversion layers (CCLs) with atomic layer deposited (ALD) encapsulation. Semiconductor processes were used to transfer the CCL onto a transparent glass substrate. This method enables pixels as small as 5 μm, and the pixel's size and pitch can be controlled through the semiconductor process and mask layout. A dual-color array with a red-and-green CCL was also demonstrated to be suitable for microdisplay applications in experiments. The best color conversion efficiency of a single color CCL reached as high as 20.7%. A set of distributed Bragg reflector mirrors was used to increase photon recycling, and these mirrors yielded an increase in peak intensity of 35.7%. We formulated a model of incoherent reflection and transmission to calculate the changes in CCL properties when mirrors are added; this model had close fit with results from experiments in which mirrors of different reflection coefficients were used. Through the use of ALD, the CCL can be stored at a normal room-temperature environment for more than 9000 hours, and the projected lifetime is linearly extrapolated to be 44,041 hours.

should have a high resolution and be highly color accurate. Thus, micron-scale pixels that display narrow-linewidth RGB colors are required. At present, this is achieved using 1) individual red-green-blue (RGB) microchips or 2) a monochromatic micro-LED array with a color conversion layer (CCL). When RGB microchips are used, miniaturized LED chips with emission peaks at red, green, and blue must be placed on a backplane to form a full-color display [1], [2], [3], [5], [6]. When a CCL is used, multiple CCLs are placed on top a monochromatic micro-LED array, whose high-energy photons (in the blue and ultraviolet region) can excite the photoluminescent materials, such as colloidal quantum dots (CQDs, sometimes simply QDs), in the CCL [7], [8], [9].
Although individual chips can deliver excellent results in large monitors [1], which tend to have a relatively low pixel per inch (PPI) count, these chips may struggle to perform at >6000 PPI because they must be smaller than 5 μm. These difficulties stem from the poor external quantum efficiency of small chips and the difficulty of transferring small chips from the native substrate to the final backplane By contrast, CCL-based devices do not require chips to be transferred in its manufacture and have a highly efficient GaN monochromatic LED array as the pumping source. With low sidewall recombination, GaN-based LEDs that are smaller than 5 μm have been fabricated in several studies [10], [11], [12], [13]. These LEDs enable increases in overall efficiency as long as the conversion efficiency of the layer is sufficiently high. CQDs, whose quantum yield can approach 100% in many cases, are a promising material for CCLs [14], [15]. However, the pixel size of the CCL is limited by the dispense method and the characteristics of the CCL material. There are many methods to pattern CQD into micron-scale pellets, such as photolithography type [7], [8], direct dispense [16], [17], or nano-imprint type [18], [19]. In the photolithography method, the CQDs are mixed with photo-sensitive resin to form a quantum dot photoresist (QDPR). The patterning can be achieved via regular exposure and the development of the QDPR. Because of its compatibility with current semiconductor processes, high throughput and good yield can be expected from the QDPR method. However, achieving the finest pixel and best color conversion efficiency is complex, and it requires tremendous effort in chemistry to correctly produce the QDPR. In the nano-imprint method, a nano/micro scale stamp must be fabricated first, and then the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ CQD layer can be transferred via stamping procedure onto the proper location [18], [19]. In the direct dispense type, the CQD is directly placed into the pixel location by a controlled flow of color conversion materials. Different methods can achieve different pixel sizes and thicknesses of deposited materials. To the best of our knowledge, only a few studies [20], [21] have achieved a pixel size in the range of 5-10 μm. The extraction of converted photons is key to CCL performance after its installation. However, most designs only have a single pass for pumping photons, which is far from ideal. As a solution, an optical layer, usually a highly reflective mirror, can be placed on top of the conversion layer [8], [22], [23] to reflect unused high-energy photons back to the CCL to give these photons multiple chances, or at least a second chance, to be absorbed by the color-converting material. The feasibility of this design has been demonstrated in several studies [23], [24], but models that comprehensively describe this effect have not been formulated.
In this study, we 1) developed an incoherent reflectiontransmission model to calculate how much CCL emission is enhanced under an external mirror and 2) fabricated a novel and highly reliable CCL for use in micro displays.

II. SAMPLE FABRICATION
The CCL of this study was designed to be prepared as follows. In general, CQDs are to be used as the main color-converting material. Although native CQD solutions are not photolithographically compatible and thus cannot be patterned easily, our method of fabrication can yield micron-scale resolutions and accuracy through semiconductor-grade processes.
The fabrication involves the following steps. First, an InP wafer is patterned in arrays of square-shaped pockets. Subsequently, an inductively coupled plasma (ICP) system is used to dry etch the InP material in the open area. The reaction gases to be used in the process are Cl 2 , CH 4 , and Ar. The substrate temperature is 20°C, and the chamber pressure is maintained at approximately 6 mTorr. The approximate etch rate is 0.4 μm/min. After ICP etching, an atomic layer deposition (ALD) system is used to deposit a thin layer of Al 2 O 3 (approximately 5 nm) for encapsulation. A plasma-enhanced chemical vapor deposition (PECVD) system is also used to deposit 1 μm of SiO 2 ; this layer reinforces the structure after patterning fabrication. CQD ink is then directly dispensed to fill the pockets created by the fabrication process. In this study, the dispensing system was an Aerosol Jet 300 (Optomec), which we used in a previous study for direct patterning at 35-μm pixels [23]. After the ink is dispensed, the extra quantum dots can be wiped or lifted off by an extra photoresist layer. Then, this InP wafer with CQD arrays is bonded to a glass substrate by using a polymer layer (such as that made from PDMS or SU8 photoresist). The whole structure is then submerged in a HCl-H 2 O solution for the InP wafer to be etched away selectively. After the InP is removed, QD patterns form on the top surface. These steps are illustrated in Fig. 1(a). Because the glass is transparent, we can excite the CQD layer from the bottom of the structure and place the mirror on the top to encourage photon recycling.
The CQDs used in this study were purchased from Unique Materials Co. Ltd. The green and red emission peaks were at 525 and 620 nm, respectively, and the linewidth of the emission spectrum varied between 24 and 28 nm [25], [26]. The finished CCL can be placed on top of the blue or ultraviolet LED lead frame and measured in an integrating sphere. The LED chip was purchased from Genesis Photonics, and its emission peak was 397 or 454 nm, depending on the application. Fig. 1(b) illustrates the CCL with the pumping LED and mirror. The photonic characteristics of the CCL can be measured using an integrating sphere (Isuzu Optical SLM-12), and the fine pixel arrays can be viewed under a fluorescence optical microscope (FLOM, Olympus BX15C). Fig. 2(a)-(d) presents FLOM images of arrays of 5-and 20-μm green pixels, 10-μm red pixels, and 25-μm red and green pixels. The finished structure was also examined in detail under a scanning-electron microscope (Figs. 2(e), (f)). It can be noticed that there are deposits on the structure surface in Fig. 2(e), while the surface is clean in Fig. 2(f). The deposits were left-over from the incomplete etch mask removal and transferred to the final CCL. In the second version of the process, we were able to eliminate the deposits via adjustment of process parameters, and thus a much cleaner surface can be obtained in Fig. 2(f).

A. Color Conversion Efficiency Without a Mirror
Color conversion efficiency (CCE) is a key photonic property of a CCL. The pumping LEDs output high-energy photons (in the blue or ultraviolet region), and the active material inside the CCL can then absorb these photons and re-emit them as lower-energy photons. Our proposed CCL is made of CQDs (or sometimes called QDs) whose emitted photons have a wavelength that is determined by the size of the individual dot. This process can be quantitatively described in terms of the emission spectra of a CQD-coated and non-CQD sample, and the difference in these spectra can be used to calculate the CCE [27], [28] as follows.

CCE = # of QD emitted photons # of absorbed blue or U V photons
where I ex QD and I ex ref are the integrated intensities of the pumping LED (or the excitation source) with and without the CCL, respectively, and I em QD and I em ref are the intensities of the emission bands of the QD layer with and without the CCL, respectively. This formula serves as a sufficient description of cases where no other optical reflecting layer is present on the CCL; this is because only a single pass of photons is considered in (1).

B. Incoherent Reflection and Transmission Model
In many applications, the LED source does not pump sufficient photons to the CCL. An extra optical layer (usually a mirror) is required to boost photon recycling at the excitation wavelength while promoting CQD emission from the CCL. Optical models are useful for the design of such systems.
This section details our optical model based on incoherent reflection and transmission. In our model, the amplitude of the traveling wave is considered in the absence of information regarding its phase [29]. We established a basic three-layer structure ( Fig. 3(a)), where each layer (denoted A, B, and b) is made of an absorbing material (such as the CCL) or occupied by air. In this article, A and b denote layers occupied by air, and B denotes a layer made of the CCL. The interfacial layer between A and B is the glass substrate of the CCL, and the interfacial layer between B and b is the mirror layer placed on the top of the CCL. The mirror in our design is a modified version of a distributed Bragg reflector (DBR) mirror [30]. Adopting the method proposed by Harbecke [29], we modeled a single-pass reflection and transmission under the incoherent condition by using the following equations: where R Ab, incoh and T Ab, incoh are the overall incoherent reflection and transmission coefficients of the material stack in Fig. 3(a). R AB , T AB , and T BA are the levels of interfacial reflection and transmission between the media A and B. If we assume no energy loss in this interface, then R AB + T AB = 1. The coefficient A B describes the remaining intensity of the passing wave, and R Bb and T Bb are the levels of reflection and transmission between layer B and layer b. In our case, R Bb is the level of reflection of the mirror, and T Bb = 1 − R Bb . A more complex model was established ( Fig. 3(b)) for realworld cases when the pumping LED and the reflecting bottom part of the lead frame substrate are present. To simplify the model, we lumped two components (LED and the bottom substrate) into a single surface that can reflect both pumping photons and CCL photons.
On the same theoretical basis ( Fig. 3(c), (d)), the incoherent reflection and transmission for the UV photons and QD photons can be described using the following equations. Specifically, the UV transmission, reflection, and absorption coefficients in the presence of a mirror (R DBR, UV ) on the CCL are as follows.
where A UV is the number of remaining UV photons after a single pass through the QD layer (analogous to the A B term in (2)), R glass is the level of reflection of the glass substrate, R DBR, UV is the reflectivity of the DBR mirror at the UV photon band, and r UV is the lumped reflectivity at UV band of the bottom surface of the lead frame combined with the LED chip. The term Abs UV, incoh is the absorbed UV intensity in the presence of the mirror. A UV values of 1 and 0 indicate that UV photons in the QD layer undergo no absorption and complete absorption, respectively. The ratio of the peak UV intensity between the sample with a CCL and a mirror and the sample without any mirror and a CCL (only an LED) is: After the UV light from the pumping source is absorbed, the QD CCL can emit photons with longer wavelengths. For the incoherent model, reflecting planes at the top and bottom can be established ( Fig. 3(d)) with the light source between reflecting planes that include the mirror and glass substrate. We set the surviving ratio of the QD photons, Y QD , for the layer sandwiched between the glass and bottom surface. Y QD can be defined as the percentage of the QD photons that are not absorbed by the medium between the glass and bottom surface of the lead frame ( Fig. 3(d)). The corresponding incoherent reflection in the QD transmission is described as follows: where r QD is similar to r UV but evaluated at the CQD emission band. We found that Y QD can be set as 1 in our cases. The peak intensity ratio at the QD emission wavelength of the CCL with and without the (DBR) mirror is as follows: , where UV abs,DBR and UV abs, no DBR are the levels of UV absorption of the QD layer with and without the DBR mirror, respectively. The parameter X QD describes the surviving ratio of the QD photon intensities after passing through the CCL. The enhanced absorption at the UV band due to the presence of the DBR is as follows: UV abs,DBR UV abs, no DBR = Abs mirror enhanced We list all the detailed mathematical procedures in the appendix section of this article.
These equations can be used to plot the QD intensity ratio under different absorption conditions (A UV ). The calculation of this intensity ratio based on (9) is illustrated in Fig. 4(a)-(c), and the ideal condition is one where the UV reflection of the mirror is close to 1 and its QD reflection at the CQD emission wavelength approaches 0; this condition is marked by a red circle in Fig. 4. This intensity ratio can be termed the enhancement factor, which varies with UV absorption level in Fig. 4. In cases of weak UV absorption (i.e., a high A UV value), the influence of the DBR mirrors (or the cavity effect) is stronger and can be approximately two times higher than that when a mirror is absent. However, if all UV photons are absorbed in a single pass (i.e., A UV is low), then the mirror does not increase the level of photon recycling in the device because most of the UV photons are absorbed before they can come into contact with the mirror. These principles can be generalized to the CCL and the optics layer on top of it.

C. Reflecting Mirrors
The reflecting mirror is a key component in the CCL structure. Regular and modified DBR mirrors were used in the experiments of this study [30]. The regular DBR mirrors had a quarterwavelength stack of dielectric layers that were used to produce the Bragg resonance, and the TiO 2 -SiO 2 quarter-wavelength stacks were deposited by an e-beam evaporator system to form uniform reflecting mirrors. However, in addition to a major reflection band, the regular DBR mirrors have relatively strong reflection sidelobes, which are undesirable in many applications.
In the modified DBR mirrors, the thicknesses of some highindex-low-index pairs in the DBR mirror were adjusted to deviate from the quarter wavelengths [30] for reflection sidelobes to be suppressed; this was done to achieve flat high transmission/low reflection in the QD emission wavelength region.
Usually, the normal incidence of reflectivity (0°) is used to model a structure's optical characteristics. However, in the case of LED pumps, the pumping photons can come from various directions. Thus, in addition to normal reflectivity, the angular variation of reflectivity must be considered. For this reason, we adopted the effective reflectivity (R eff ) as a measure: where θ I is the different angles at which the mirror reflectivity is measured. This calculation is applied to all mirrors once their angular reflectivity value is obtained, and all quantities, such as R DBR, UV and R DBR, QD , will take on the values from these effective results instead of the normal incident reflectivity. We aimed to make our model more realistic by implementing this adjustment.

A. Color-Converted Spectrum
Fig. 5 shows our measured spectra of several samples, including single-colored and dual-colored CCLs. For the green CQD (at 525 nm), a 397 nm UV LED source was used, as indicated in Fig. 5(a). A blue LED will be used if the CQD emits red photons (as shown in Fig. 5(b)). If we subtract the spectral intensity of a pure UV LED from the one with CCL of the same pumping LED at the same current level, we can extract the effective QD emission and UV absorption and calculate the CCE by using (1). This method can be also applied to our red-color CCL when a blue LED served as a light source, as indicated in Fig. 5(b). In this figure, the difference in spectral magnitudes of the LED-only case and the LED+CCL case was obtained by subtracting the two spectral intensity profiles illustrated in the inset of Fig. 5(b). The resultant areas 1 and 2 can be used to calculate the CCE on the basis of (1). Fig. 5(c) illustrates these results for the green QD sample with different dosages of CQD ink applied using the Aerosol Jet system. Fig. 5(d) depicts the results on the LED-current-dependent QD emission for the red and green arrays on the same substrate, whose images are presented in Fig. 2(d). One will notice that the green QD peak in Fig. 5(a) is weaker than in Fig. 5(b). The reason is that the number of CQD pixels is small, and the pixel density needs to be higher. Most of the incident blue/UV photons passed through the CCL without being absorbed. Consequently, the excited green QD peak is weak. When we fabricated the red CCL the second time, the number of QD pixels increased significantly, leading to a much stronger QD peak.
The efficiency can then be calculated using (1). Initially, most of the samples were made using the Aerosol Jet apparatus, which was used to deposit the QD layers. The resultant efficiency was within 4%-5%. This low CCE was due to the low coverage of the QD pattern and high percentages of scattered and unabsorbed UV photons in the package, which led to non-radiative loss. The CCE can be increased by increasing the concentration of QD (Fig. 5(b)), where the CCE could reach as high as 20.7% under a low current (20 mA) condition. The conversion efficiency of the blue or UV photons to green and red photons may also differ because of the different absorption coefficients of the quantum dots [31], [32].

B. Mirror Reflection Measurements
The spectral response of reflectivity of the mirrors were also measured in this experiment. The reflection spectra of the traditional and modified DBR mirrors are presented in Fig. 6(a) and (b), respectively. The emission wavelengths of pumping LED and QD emissions are marked in purple and green in the spectra, respectively. Prominent reflection sidelobes were observed in the reflection spectrum of the traditional DBR mirror; by contrast, the reflection spectrum of the modified DBR mirror ( Fig. 6(b)) was flat and indicated low reflection near QD emission wavelengths. As presented in Fig. 6(b), the modified DBR mirror had reflection coefficients of 4.61% and 97.37% at 525 and 397 nm, respectively, whereas the traditional DBR in Fig. 6(a) had reflection coefficients of 16.77% and 93.38% at the same wavelengths. The low flat reflection (and thus high flat transmission) near the QD emission wavelength facilitates the efficient transmission of QD photons from the complex optical structure of micro displays in practical applications.

C. Verification of Incoherent Reflection Model by Using Different Mirrors
The CQD peak became higher when a proper mirror was placed on top of the CCL. Intuitively, a mirror with high reflectivity in the pumping photon wavelength region should be used, and the reflectance at the CQD emission wavelength should be lowered; however, we did not systematically explore other reflectivity combinations. This would require several mirrors with different reflection coefficients at the UV (397.5 nm) and QD (525 nm) emission bands. Three groups were formed that were each characterized by 1) high reflectivity at the UV band and low reflectivity at the QD band, 2) low reflectivity at the UV band and high reflectivity at the QD band, or 3) similar levels of reflectivity at the UV and QD bands. The spectra of the samples with and without mirrors can be measured to calculate the ratios of the QD peaks and UV peaks; these ratios can be compared with the theoretical results calculated using (7) and (9), displayed in Fig. 7(a) and (b). If the parameters in (7) and (9) varies, the scattering of the data points in Fig. 7(a) and (b) can be observed. The theoretical calculations should ideally be close to the empirical results.
The difference between the theoretical and empirical results may have stemmed from the random scattering of the photons inside the package and from the possible loss of photons through the glass substrate due to the light guide effect.   Table I presents the measured reflectance values of various mirrors in this study and the ratio of the measured CQD emission peak with versus without a mirror. The reflectivity values in Table I are effective ones that were weighted by the different angles of incidence, as described in (11). The fitted parameters of R glass , A UV , r UV , r QD , and X QD used for the results in Fig. 7(a) and (b) are listed in Table II. The highest increase of 35.7% was observed when mirror #8, whose normal reflectivity is presented in Fig. 6(b), was placed on the CCL.

D. Test of Long-Term Storage Life
Our CCL design also offers longer lifetimes relative to existing designs because of the encapsulated CQD layer. The surrounding ALD and PECVD dielectric layers shield the CCL from ambient oxygen and moisture, which can erode the QD material and degrade its illuminative efficiency (indicated by the CCE). To verify its durability, we subjected the samples to a long-term storage-life test. In this test, the samples were exposed to air without any protection (such as that from darkness, temperature controls, or a vacuum chamber), and their emission spectra were measured in the integrating sphere using the same LED under the same injection current applied regularly over a long period. The results are illustrated in Fig. 8(a) and (b). The overall test lasted for over 9000 hours (for single color samples), and a higher than usual oscillation was observed at the first 500 hours (inset of Fig. 8(a)). Both single-color and dual-color CCLs exhibited an initial increase in CCE, consistent with the results of other studies [33], [34]. This may be because of the passivation of the surface traps of the quantum dots by ambient moisture and gases [35], [36], and the CCL exhibited a clear decrease in efficiency after a peak in efficiency at several hundred hours of exposure to the environment. The lifetime of such degradation was calculated on the basis of steady decreases in efficiency after the CCL was exposed to the environment. For both single-color and dual-color samples, efficiency decreased in a slow and steady manner. Extrapolating the data, we determined that the single-color and dual-color samples have expected lifetimes of 44041 and 29059 hours, respectively; this expected lifetime is the time taken for efficiency to degrade to 50% of its initial value (termed LT50). These lifetimes are much longer than one year (i.e., 8750 hours). As indicated by the inset of Fig. 8(b), the peak emission wavelength remained unchanged during the tests. The shifts of the red and green peaks were at 0.646 and 0.611 nm, respectively. The full width at half-maximum values were also stable; the difference between the 0th and 8064th peaks was 1.65 nm for the green peak and 1.03 nm for the red peak. Therefore, our CCL is not only efficient but also has a stable spectral response over time when exposed to the environment.

V. CONCLUSION
Our novel method is based on semiconductor processes that are used to pattern the CQDs into large-scale arrays whose pixel size can be as small as 5 μm. Our method can be used to fabricate CCLs that can reach a CCE of 20% if sufficient CQDs are provided. The peak of QD is 35.7% higher value when a mirror is used. We also developed a theoretical model of incoherent reflection and transmission to explain our results, and our model agreed well with the experimental results. CCLs with well-designed optical mirrors are a key part of small displays, which are expected to become increasingly prevalent with the development of IoT.

APPENDIX
This appendix is used for our detailed derivation of essential equations in the article. In the first section, the color conversion efficiency of the CCL without a mirror is explained. The case with a DBR mirror becomes more complicated and we need to apply incoherent model to find out the proper expressions, which will be illustrated in the second section.

A. The Color Conversion Efficiency for CCL Without a Mirror
Equation (1) originates from calculating the ratio of the excited QD photons over the absorbed blue/UV photons, which can be found in various pieces of literatures [27], [37], [38], [39]. The number of photons can be calculated via the detected power reading from the spectrometer by the formula: The difference between the spectrum with CCL and the one without CCL at the blue/UV band is caused by the CQD absorption of the CCL. As shown in Fig. 9, the difference between "No CCL" and "with CCL" in the blue/UV band is the portion of blue/UV photons absorbed by CCL. The CQD emission peak is at the longer wavelength and no CQD emission for the "No CCL" spectrum. So the color conversion efficiency for a single layer of CCL becomes:

B. Incoherent Reflection and Transmission Model
In the following, we will focus on the derivation of (9) in the article, which describes the QD peak emission ratio between a CCL sample with and without a mirror (or a DBR). The derivation of (2)-(8) will also be revealed in this process.
First, we will derive the incoherent reflection and transmission of a stack of the medium as shown in Fig. 10(a). To evaluate the incoherent condition, one can turn off the phase change during the wave propagation [29]. By adding the field amplitudes of all waves reflected back and forth between the interfaces in Fig. 10(b) (a similar method has been applied in [29], [40], [41]), we can write down these reflected components as: The rest of R i can be derived in the same procedure. The term A B is related to the one-way absorption of layer B, which can be regarded as the color conversion layer in our study. The definition of A B can be seen in Fig. 10(c). The total summation of these reflected terms can become the incoherent reflection of the structure: The summation of the geometric series eventually converges because the common ratio R Bb R BA A 2 B is smaller than 1. Similarly, the first few transmission terms are: These results are the same as 16 and 17 in ref. [29]. Moreover, they are also our (2) and (3). In that reference, the E-field reflection was used, and we have to know that the intensity of the light field inside the structure is proportional to |E| 2 , and the reflectance will need to take the squared value from the E-field value [42]. To obtain (4) to (6), we can use the same formula in (2) and (3) by replacing T BA and T AB with (1-R glass ), A B with A UV (which stands for the surviving ratio of the UV photons when passing through the CCL), and R Bb with R DBR,UV . For absorption, we can invoke the rule R+T+A = 1, and (6) can be obtained. Now, we can deal with the CQD emission and its incoherent model. A detailed diagram is provided in Fig. 11, where two components for CQD emissions exist: upward and downward. We can assume that the total amount is Q and half of the QD photons choose to go up (Q/2), and the other half goes down initially. The QD CCL is sandwiched by two mirrors: DBR and the combination of the glass substrate of CCL and the bottom part of the LED package. The reflectance of the latter mirror is named R QD, incoh as it should be treated as an incoherent component. We apply the same technique of multiple reflections (as in Fig. 10) and calculate the upward and downward QD components that eventually go out of the top DBR. In this calculation, we introduce a new parameter of X QD , similar to A UV in the previous equations. X QD describes the surviving ratio of the QD photon intensities after passing through the CCL.
For the first few upward QD components: For the first few downward QD components: The total transmitted QD photons will equal to upward + downward components: The term R QD, incoh is another incoherent reflection considering the glass substrate and the bottom package surface, and its complete expression is: where r QD is the reflectance of the bottom package surface at QD emission band (similar to r UV ). Meanwhile, the without DBR case will yield QD emission as: One thing one should note here is that the amount of Q when DBR is present will differ from that without a mirror. The reason is the extra mirror reflection; thus, extra absorption of blue/UV photons happens when there is a DBR mirror on the top of the CCL. The actual value of Q is determined by η × (absorbed UV photons), where η is the photon conversion efficiency of the CCL material.
Finally, we have to consider the combination of UV excitation and QD emission. Fig. 12 shows the schematic diagram when all components, like the excitation source, CCL, and DBR, are in place. (4) to (6) in the article can be regarded as the onepass expression for the CCL+DBR structure. Once we have the bottom surface of the package, we increase the complexity of the circulating EM field by adding another effective bottom reflection, r UV . Using the same model shown in Fig. 10(b), we can conclude that the enhanced transmission and absorption can be expressed as: The enhanced absorption of UV photons can eventually increase the CQD CCL output photons proportionally. One has to remember that both Abs UV, incoh and R UV, incoh are functions of R DBR,UV . Previously, we have the QD emission when there are UV photons of quantity Q as: Meanwhile, if there is no DBR at the top of the structure, meaning R DBR,UV = R DBR,QD = 0, we can also get a "no-DBR" QD emission : where the R UV,incoh = R glass according to (5). Thus the final ratio of the QD emission with and without DBR can be seen as: Hence, we obtain the fully-expanded (9), and this is the core equation when we compare different mirror designs.