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Abstract
When a beam of light excites a single spot on a thin luminescent layer embedded in a planar waveguide, a concentric re-emission pattern is observed. An analytical expression is formulated by following the series of events in the waveguide: generation of angle-dependent photoluminescence spectra, reflection at the waveguide–air boundary, absorption by the luminescent layer, and generation of next-generation photoluminescence. The formula reproduces the peak radii observed in the experiments with some organic dyes. It provides insights for the re-emission events in a luminescent solar concentrator and the cross talk in an energy-harvesting display based on photoluminescence.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
A luminescent solar concentrator (LSC) consists of a waveguide containing luminescent materials and the solar cells attached to at least one of its edges [1]. The luminescent materials convert sunlight to photoluminescence (PL) photons. Most of them are trapped in the waveguide and are transported to the solar cells via total internal reflection (TIR) at the air–waveguide interface. During this process, some of them are absorbed by the luminescent materials. This self-absorption phenomenon has been recognized as a serious problem since the early days of research [2,3]. Despite the various efforts to develop an ideal material without the overlap between the absorption and emission spectra [4–6], it still hinders large-scale deployment of an LSC. According to a recent review article on LSCs [7], the highest power conversion efficiency under the standard condition of AM1.5 remains to be 7.1% recorded by a ${5} \times {5}\;{{\rm cm}^2}$ device [8]. It is noteworthy that the absorbed energy can be partially harvested via re-emission events. Theoretical works on the optical efficiency of an LSC consider these higher-generation PL photons [2,3,9]. However, few experiments for observing re-emission events have been reported. The difficulty arises from the fact that the excitation light is incident on an LSC uniformly and that the re-emitted PL photons are spectrally indistinguishable from the original ones. The assumption of uniform incidence might not be applicable for a building-integrated photovoltaic system [10], where surrounding structures could partially block the sunlight in an urban environment. It is certainly invalid for an energy-harvesting display where green images are displayed on an LSC by projecting intensity-modulated blue light on it [11]. Re-emission events pose an inherent problem in this application because they can degrade the contrast ratio of a displayed image. In fact, a disk-shaped cross-talk pattern is generated when a single spot on the screen containing ceramic phosphors is excited [12]. This is in part caused by the granular morphology of these materials, which scatters the excitation light. In addition, the PL photons propagating inside the waveguide can also excite the luminescent material if they are energetic enough. The latter mechanism might limit the image quality of such a display.
Whether it is beneficial or detrimental, quantitative understanding of the re-emission phenomenon is essential for designing energy-harvesting devices based on PL. In this paper, we derive an analytical expression for the re-emission event in a planar waveguide with a thin, nonscattering luminescent layer, observe its emission pattern in experiment, and discuss how the contrast ratio of an energy-harvesting display can be improved.
2. THEORY
A. Model
As illustrated in Fig. 1, a homogeneous luminescent layer of thickness $ d $ is placed in the middle of a waveguide of thickness $ \ell $. This is the configuration of the light-conversion structure for a thin-film LSC [7] as well as the screen for an energy-harvesting projector [11]. We assume that the luminescent layer and the waveguide have the same index of refraction and that the waveguide is nonabsorbing. Although the process of self-absorption and re-emission continues multiple times, the first round of this process is of significance [9]. We consider only the first generation of re-emission events and neglect higher generations in the following analysis.
Suppose that an excitation light is incident at the origin and that the PL emission is isotropic and unpolarized. The PL photons can reach the luminescent layer at the distance $ r $ via multiple paths. For example, the photons with the emission angle $ {\theta _1} $ as defined in the figure can reach there after one reflection. Those with $ {\theta _2} $ reach there after two reflections and one transmission. We account for each of these paths to estimate the intensity of the flux incident at $ r $. The flux absorbed there is calculated from this incident flux. The re-emission spectrum is given by multiplying it by the quantum efficiency and the emission spectrum of the luminescent material.
For the moment, we consider PL photon fluxes emitted at a specific wavelength $ \lambda $. The emission angle $ \theta $ is one of $ {\theta _1},\,{\theta _2},\cdots$, as defined in the figure. Let us denote the strength of the monochromatic excitation light at this wavelength $ {\lambda _{\rm ex}} $ as $ {S_0} $, its transmittance through the luminescence layer as $ {T_{\rm ex}} $, and the quantum efficiency of the material as $ {\eta _{\rm QE}} $. Then, the emission spectrum at the site of generation is expressed as $ {\eta _{\rm QE}}{S_{\rm em}}( \lambda )( {1 - {T_{\rm ex}}} ){S_0} $, where $ {S_{\rm em}}( \lambda ) $ is the true emission spectrum. We assume that the emission is isotropic and name the PL photon flux exiting the layer upward as forward flux $ {F_f} $, as shown in Fig. 1. The flux propagating downward is named as backward flux $ {F_b} $. Due to the self-absorption inside the luminescent layer, the spectrum of the PL photons exiting depends on the direction of the emission. The simple analysis based on the Lambert–Beer law gives the following expressions for the fluxes exiting the layer forward and backward, respectively [13]:
The parameter $ \alpha $ in these equations is defined as follows:
The transmittance $ {T_{\rm ex}} $ is related to the absorption coefficient of the luminescent layer $\mu( \lambda ) $ by
Referring to Fig. 1, let us follow the PL photon flux $ {F_f} $, which is emitted forward at $ {\theta _1} $ as an example. It is reflected at the air–waveguide interface, where the reflectance $ {R_F} $ is given by the Fresnel equations. The flux reaches the luminescent layer at $ r $ after propagating the distance of $ \sqrt {{r^2} + {\ell ^2}} $ inside the waveguide. Taking into account the isotropic spreading over this distance and the spreading due to the oblique incidence, the intensity of the PL photons incident on the luminescent layer at $ r $ is given by $ {R_F} \times \frac{{{F_f}}}{{4\pi ( {{r^2} + {\ell ^2}} )}} \times \cos {\theta _1} $. Because the factor $ \cos {\theta _1} $ is equal to $ \ell /\sqrt {{r^2} + {\ell ^2}} $, the above expression can be simplified. The distance $ r $ is related to the emission angle by $ r = \ell \tan {\theta _1} $.
Next, let us follow the flux emitted forward at $ {\theta _2} $. It goes through two reflections and one transmission before reaching the position $ r = 2\ell \tan {\theta _2} $. The total path length is equal to $ \sqrt {{r^2} + {{( {2\ell } )}^2}} $. In addition to the geometrical spreading effect, the intensity is reduced by the factor $ {P_t}{R_F}^2 $, where $ {P_t} $ is the transmittance through the luminescent layer. Invoking the Lambert–Beer law, the transmittance $ {P_t} $ and the absorptance $ {P_a} $ are given by the following equations:
Hence, the intensity of the flux incident on the luminescent layer at $ r $ is expressed as $ {P_t}R_F^2 \times \frac{{{F_f}}}{{4\pi }} \times \frac{{2\ell }}{{{{\{ {{r^2} + {{( {2\ell } )}^2}} \}}^{3/2}}}} $.
Let $ i $ be the number of reflections that the flux needs to go through before reaching the luminescent layer at $ r = i\ell \tan {\theta _i} $. Then, the flux is decreased by the factor $ {P_t}^{i - 1}{R_F}^i $ in addition to the geometrical effect. The backward flux $ {F_b} $ is treated in the same manner. At the end, these contributions are added together to give the following expression for the intensity of the flux incident at $ r $:
Note that all the variables in the right-hand side of Eq. (7) depend on the emission angle $ {\theta _i} $. For a small value of $ {P_t} $, the contributions of the fluxes emitted at smaller $ {\theta _i} $ diminish quickly as the number $ i $ increases.
Fig. 3. Forward flux (left half) and backward flux (right half) for three settings of the transmittance of the excitation light ($ {T_{\rm ex}} $). In each color-coded image, the color bar is at the far right, and the range of values is as indicated. (a) and (b) $ {T_{\rm ex}} = 0.01 $; (c) and (d) $ {T_{\rm ex}} = 0.1 $; (e) and (f) $ {T_{\rm ex}} = 0.5 $.
Fig. 4. Re-emission intensity for the two settings of the waveguide thickness ($ \ell $). The transmittance of the excitation light at 450 nm for normal incidence ($ {T_{\rm ex}} $) is varied.
The analysis so far is limited to the specific wavelength $ \lambda $. Integrating the product $ {P_a}{I_{\rm in}} $ with respect to the wavelength gives the absorbed optical power at the distance $ r $. The spectrum of the re-emitted PL photons is formulated as follows:
B. Numerical Example
As a luminescent material, we take Lumogen F Red 305, a standard organic dye applied for an LSC as an example. The absorption and emission spectra in Ref. [9] are normalized by each peak value and are shown in Fig. 2. Once the transmittance $ {T_{\rm ex}} $ is set, the product $\mu ( \lambda )d $ in Eq. (3) is known for every wavelength. Because the quantum efficiency of this material is reported to be close to unity [14], we set $ {\eta _{\rm QE}} $ to 1. We set other input parameters as follows: the index of refraction $ n = 1.5 $, $ {\lambda _{\rm ex}} = 450\;{\rm nm} $, and $ \ell = 4\;{\rm mm} $ or 10 mm. These numbers are close to those in the experiment to be described in the next section.
First, the two fluxes are calculated as a function of the distance $ \ell \tan \theta $ by Eqs. (1) and (2). Each color-coded image in Fig. 3 represents the spectrum emerging from the luminescent layer at $ r = \ell \tan \theta $ for the case of $ \ell = 10\;{\rm mm} $. The emission spectrum suffers from self-absorption more at a larger emission angle $ \theta $. The transmittance $ {T_{\rm ex}} $ is varied, as indicated. In Fig. 3(a), for example, the intensity at a shorter wavelength decreases as $ r $ increases. This spectral change is known as redshift. It is more severe at a smaller $ {T_{\rm ex}} $. The redshift in the forward flux $ {F_f} $ is more pronounced than the backward flux $ {F_b} $ because the PL photons traverse longer distances in the luminescence layer on average [13].
Second, the intensity of the flux incident on the luminescent layer at $ r = \ell \tan \theta $ is calculated by Eq. (7) for $ i $ up to 10. Setting this upper limit ($ {i_{\max }} $) to 10 might seem arbitrary, and one might wonder if this is justified. We will address this convergence issue after we present the re-emission intensity distributions in Fig. 4. The result is shown in the left half of Fig. 5. The calculation is repeated for the case of $ \ell = 4\;{\rm mm} $, and the result is shown in the right half of Fig. 5.
Fig. 5. Incident flux for the case of $ \ell = 10\;{\rm mm} $ (left half) and for the case of $ \ell = 4\;{\rm mm} $ (right half). In each color-coded image, the color bar is at the far right, and the range of values is as indicated. (a) and (b) $ {T_{\rm ex}} = 0.01 $; (c) and (d) $ {T_{\rm ex}} = 0.1 $; (e) and (f) $ {T_{\rm ex}} = 0.5 $.
The spectra in Fig. 5 have peaks at the distances around the integer multiples of $ \ell \tan {\theta _c} $, where $ {\theta _c} $ is the critical angle for TIR. This feature is caused by the PL photons emitted around $ {\theta _c} $. Namely, the factor $ {R_F} $ rapidly approaches unity as the emission angle becomes closer to $ {\theta _c} $ [15]. The concentric radiation pattern observed by the experiment in the next section is caused by this dependency of $ {R_F} $ on the distance via Eq. (8). Furthermore, some of these PL photons are transmitted at $ r = \ell \tan {\theta _c} $, and they trigger re-emission events at $ r = 2\ell \tan {\theta _c} $. This scenario results in the smaller second peaks in Fig. 5.
Third, the intensity of the re-emitted PL photons $ {I_{\rm re}} $ is calculated by Eq. (9). As shown in Fig. 4, $ {I_{\rm re}} $ peaks at the distance given by Eq. (8). When the parameter $ {T_{\rm ex}} $ becomes larger, $ {I_{\rm re}} $ becomes smaller and extends to a longer distance.
Going back to the convergence issue, we increased the upper limit for the number of reflections ($ {i_{\max }} $) from 1 to 10 for the case of 4-mm-thick luminescent waveguide (LWG), with $ {T_{\rm ex}} = 0.1 $. Every time $ {i_{\max }}$ increases, a small peak appears at $ r = i\ell \tan {\theta _c} $ and the intensity for $ r \gt i\ell \tan {\theta _c} $ increases slightly. But there is almost no change in the distribution for $ r $ slightly smaller than $ i\ell \tan {\theta _c} $. For example, the intensity at $ r = 3.6\;{\rm mm} $ ($ \approx \ell \tan {\theta _c} $) changes from 0.047698 for $ {i_{\max }} = 1 $ to 0.047718 for $ {i_{\max }} = 2 $. It remains 0.047718 for $ {i_{\max }} = 10 $. Therefore, setting $ {i_{\max }} $ to 10 is justified.
3. EXPERIMENT
We call the configuration illustrated in Fig. 1 an LWG in this section. Fabrication of LWGs, measurements of radiation patterns and emission spectra are described in this order.
A. Fabrication of LWGs
The LWGs shown in Fig. 6 were used in this experiment. Fabrication procedure is as follows. An organic dye was mixed with ultraviolet curable resin (NOA81, Norland Products). Lumogen F Red 305 (BASF) and Coumarin 6 (Sigma Aldrich) were used as purchased. The solution was dispersed on an acrylic plate. After another plate was placed on it, the resin was cured with an ultraviolet lamp. The acrylic plate was either 2-mm- or 5-mm-thick, and its area was $ 50\;{\rm mm} \times 50\;{\rm mm}$. The index of refraction is 1.49 for the acrylic plates and 1.56 for the ultraviolet curable resin.
Fig. 6. Photographs of the LWGs. The luminescent layers in (a) and (c) contain Lumogen F Red 305 and (b) and (d) contain Coumarin 6. Nominal thickness of each LWG ($ \ell $) is either 4 mm or 10 mm.
Table 1 lists the properties of these LWGs. The thickness of the cured LWG ($ \ell $) was measured by a caliper. The transmittance of the LWG for normal incidence at 450 nm ($ {T_{\rm ex}} $) was measured with a laser (Z-Laser GmbH, Z30M18H-F-450-pe) and a power meter (Ophir Optronics Solutions Ltd., PD300-SH).
Table 1. Properties of the LWGs
B. Measurement of Radiation Patterns
The beam from the same laser was focused on a single spot on each LWG. The incident angle was 20° from the plane normal. The incident optical power was fixed at 105 µW for the two Lumogen LWGs and reduced to 36.2 µW for the two Coumarin 6 LWGs. A complementary metal-oxide-semiconductor (CMOS) camera (The Imaging Source Asia Co., Ltd., DMK23UP1300) was placed 20 cm away from the LWG. It faced the LWG squarely and captured its radiation pattern. Because its dynamic range exceeded that of the camera (12 bit), multiple images were acquired with exposure times ($ {T_E} $), varying over 4 orders of magnitude. This allowed us to record the intensity distribution around the excited spot as well as the surrounding region.
Alternatively, one might try to place the camera on the other side of the light source. In principle, a high-pass filter would block the excitation light passing through the LWG. We tried this measurement setup but ended up taking the picture of the laser module, which was illuminated by the PL photons emitted by the LWG. A caution in this regard is not to place anything around the LWG that could reflect the PL photons emitted from the LWG surface. In our experiment, our darkroom wall was about 50 cm away from the LWG. At this long exposure time, a worn-down black curtain would transmit the ambient light and disturb the measurement.
The images acquired with $ {T_E} = 1000\;{\rm ms} $ are shown in Fig. 7. For this inspection, the gamma value is set to 0.5 to reveal the features with low pixel values. For the 10-mm-thick LWGs, a clear inner ring and a faint outer ring are visible. There are more rings for the 4-mm-thick LWGs. The pixel values in the central region are saturated. This saturated image region is elongated along the vertical line to which the incident plane is perpendicular. It is likely that this deformation is caused by the obliquely incident excitation beam: it passes through the dye layer, is reflected by the air–plate interface, and excites the nearby region. In the analysis below, we avoid this effect of oblique incidence by extracting an intensity profile along the horizontal line.
Fig. 7. Images of the LWGs acquired with the exposure time of 1000 ms. (a) Lumogen F Red 305, $ \ell = 10\;{\rm mm} $; (b) Coumarin 6, $ \ell = 10\;{\rm mm} $; (c) Lumogen F Red 305, $ \ell = 4\;{\rm mm} $; (d) Coumarin 6, $ \ell = 4\;{\rm mm} $.
The intensity profile along the horizontal line, including the excited spot, is extracted from the original image without the gamma correction. The profiles for the two Lumogen LWGs are shown in Fig. 8. The inset shows the magnified distribution around the excited spot for the case of $ {T_E} = 0.025\;{\rm ms} $. The full width at half-maximum of this primary peak is about 0.15 mm for each LWG. The secondary peaks are clearly visible, although the outer peaks are not in this presentation.
Fig. 8. Intensity profiles of the radiation patterns for the two Lumogen LWGs. The exposure time ($ {T_E} $) is as indicated. The inset is a magnified view of the profile around the excited spot for $ {T_E} = 0.025\;{\rm ms}$. The thickness of the waveguide is (a) 10 mm and (b) 4 mm.
The right-hand side of the intensity distributions in Fig. 8 are reproduced in Fig. 9. Also plotted in each graph is the model prediction obtained by inputting the measured thickness ($ \ell $) and the transmittance at 450 nm ($ {T_{\rm ex}} $) and by scaling the distribution such that it matches the measured pixel values in the outer region ($ r \gt 12\;{\rm mm} $).
Fig. 9. Measured intensity distributions are compared to the model predictions. The measured values for $ \ell $ and $ {T_{\rm ex}} $ are used for the calculation. The thickness of the waveguide is (a) 10 mm and (b) 4 mm.
For the thicker LWG, the model reproduces the experiment in the range $ r \gt 10\;{\rm mm} $: the position of the innermost side peak is well matched at $ r = \ell \tan {\theta _c} $, as shown in Fig. 9(a). The second side peak is only identified at $ r = 2\ell \tan {\theta _c} $ in the experiment. The positions of the two side peaks are well reproduced by the model for the thinner LWG, as shown in Fig. 9(b). However, the model deviates from the experiment significantly at $ r $ smaller than about 8 mm for both LWGs. We suspect that some of the PL photons scattered by the material around the excited spot are captured by the camera. Another possibility is that a long tail distribution of the laser Gaussian beam would excite this region and that the resultant PL was captured by the camera with 1000 ms exposure time. But in this case, the intensity level for the two LWGs had to be similar. Instead, they are about 3 times different for the two LWGs, as shown in Fig. 9. Therefore, this scenario is unlikely.
Table 2 lists the contrast ratio, defined as the intensity ratio of the primary peak to the innermost secondary peak. Note that the intensity is proportional to the pixel value divided by the exposure time. The contrast ratio of the 10-mm-thick LWGs is approximately 6 times larger than that of the 4-mm-thick LWGs. The use of Coumarin 6 leads to slightly smaller contrast ratios than does Lumogen.
Table 2. Contrast Ratio of the LWGs
C. Measurement of Emission Spectra
A spectral study on the PL photons exiting the edge surface of an LSC provides information on how much they suffer from self-absorption on their way to the edge [16]. A similar study on the PL photons exiting the top and bottom surfaces of an LSC indicate the degree of scattering [17]. These studies help us understand how the optical efficiency of an LSC is degraded by self-absorption and scattering events. In both cases, an optical fiber collects the PL photons and guides them to a spectrometer.
In our measurement described so far, we used a CMOS camera to record the radiation pattern induced by single-spot excitation. This does not provide any spectral information of the PL photons. By placing a fiber-optic probe with small numerical aperture in the vicinity of the top surface of an LWG, we can measure the spectrum of the PL photons emitted from a small circular region on the dye layer.
The spectra in Fig. 10 are obtained with the 4-mm-thick Lumogen LWG by this technique. The position of the optical fiber is varied from 5 to 20 mm with a 5 mm step. The inset shows each spectrum normalized by its peak intensity. They coincide with each other almost completely. This fact confirms that the PL photons exiting the top surface are indeed those re-emitted by the small area near the optical fiber rather than those generated by the excitation beam at the origin. Integration of each spectrum by the wavelength gives the PL intensity averaged over each small circular area determined by the numerical aperture of the optical fiber. Its dependency on the distance from the origin roughly agrees with the intensity profile in Fig. 8(b). However, the small peaks become less recognizable because the probe tip can only be placed on the LWG surface.
Fig. 10. Emission spectra measured at various distances from the excited spot. The inset shows that each spectrum coincides after normalization by its peak intensity.
4. DISCUSSION
Re-emission of PL photons away from the excited spot sets an upper limit on the contrast ratio in display applications. In this section, we first derive an expression for the contrast ratio by extending the model described in Section 2 and show that it roughly reproduces the measured contrast ratio and its dependency on the thickness of the waveguide. Note that this is an order-of-magnitude discussion, as will become clear below. Second, we discuss possible solutions for alleviating the contrast degradation.
A. Contrast Ratio
Contrast ratio of a display (CR) is defined as ${L_{{\rm white}}}/{L_{{\rm black}}}$, where ${L_{{\rm white}}}$ and ${L_{{\rm black}}}$ are the luminance of a white and a black region, respectively. Luminance (measured in $ {{\rm cd/m}^{2}} $ or nit) is a photometric quantity and is calculated by integrating the product of radiance and the standard relative luminous efficiency function $ V( \lambda ) $ with respect to the wavelength $ \lambda $ [18].
The excited spot on an LWG represents white pixels. Denoting ${r_0}$ as the radius of this spot and $\Omega $ as the solid angle for observation, the luminance of this region is given by
The region around the excited spot represents black pixels. Its luminance is expressed as follows:
Hence, calculating CR requires integration by the wavelength over the emission range of the dye. Note that the re-emission distribution peaks at $ r = \ell \tan {\theta _c} $ and $ {R_F} $ in Eq. (7) is equal to 1. Assuming that the summation in Eq. (7) is approximated by the first component only, CR is given by the following equation for the case of normal observation ($ \theta = 0$):
The function $ A( {{T_{\rm ex}}} ) $ is defined below:
As a numerical example, we plot CR as a function of $ {T_{\rm ex}} $ in Fig. 11. The material parameters in Fig. 2 are used for this calculation, and other parameters are assumed as follows: $ {\eta _{\rm QE}} = 1 $, ${r_0} = 0.03\;{\rm mm}$, $\ell = 4\;{\rm mm}$, and 10 mm, $ V( \lambda ) = 1 $.
Fig. 11. Contrast ratio CR calculated for the two Lumogen LWGs with different thicknesses. See text for the assumed parameters.
As shown in Fig. 11, the model reproduces the orders of the experimental values in Table 2. According to Eq. (12), the ratio of CR for the two LWGs with $\ell = 10\;{\rm mm}$ and 4 mm should be equal to $ {( {10/4} )^2} = 6.25 $.
B. Possible Solutions for Contrast Degradation
Because CR in Eq. (12) is proportional to $ {\ell ^2} $, increasing $ \ell $ is effective for enhancing the contrast ratio. Incidentally, this parameter has a direct impact on the optical efficiency of energy-harvesting devices based on wave-guiding PL photons: a thicker LWG has a larger optical efficiency. Qualitatively, this is understood because the PL photons traverse a shorter distance in the self-absorbing luminescent layer in a thicker LWG. Quantitatively, the effective absorption coefficient is inversely proportional to $ \ell $ in the analytical expression for this optical efficiency [19]. On the other hand, a thicker LWG requires a larger solar cell to be attached to its edge surface. For portable display applications, a thinner device is preferred [20]. When there are conflicting requirements, a figure of merit is often defined to optimize a design parameter. However, its definition requires careful considerations for a specific application. Here, for the moment, let us assume that $ \ell $ is constant. We still have the option of where to place the luminescent layer within the waveguide. The middle of the waveguide is not the right place because the forward and backward fluxes meet at the same position on the luminescent layer after reflection on each interface with air and increase the peak re-emission intensity. In order to move the position of the peak re-emission position as far as possible from the incident spot, the best position should be either the very top or bottom of the waveguide.
In addition to $ \ell $, the transmittance $ {T_{\rm ex}} $ is another important design parameter that determines the characteristics of an energy-harvesting device. A smaller value of $ {T_{\rm ex}} $ means that more PL photons are generated by the luminescent layer, as shown in Fig. 4. This is beneficial for energy-harvesting, but the re-emission events also increase. It turns out that $ CR $ is a slowly varying function of $ {T_{\rm ex}} $, as shown in Fig. 11. Hence, the choice of $ {T_{\rm ex}} $ does not seriously affect the contrast ratio.
Note that the analysis so far has assumed single-spot excitation. In practice, an image contains a number of excited spots, and the radiation pattern associated with each spot is superimposed. For example, an image of the stars in the night sky will be reproduced on an LWG with a faint concentric pattern around each star. If the density of the stars is high, the surrounding region will have a more or less uniform background luminance. One might be tempted to call this the “Milky Way” effect. Incidentally, single-spot excitation is equivalent to inputting a delta function to a system. Its response is a point spread function and is unique to a specific system. Therefore, if the effect is not eye-pleasing, one might wish to eliminate it by some image-processing techniques in advance [21].
A display is almost always viewed in a lit environment, and ambient light reflected at its surface degrades CR. For a quantitative discussion, an ambient contrast ratio (ACR) is used to compare different display technologies. For example, assuming a typical office lighting level (320–500 lux), peak luminance of 1200 nits for a liquid crystal display (LCD) and 600 nits for an organic light-emitting diode display (OLED), ACR is calculated to be below 1000 for both displays [22]. Therefore, characterizing and enhancing ACR is of interest for display applications of LWGs.
5. CONCLUSION
When a single spot on a thin, nonscattering luminescent layer embedded in a planar waveguide is excited by a narrow beam of light, a concentric radiation pattern is observed. An analytical expression for this intensity distribution is derived by considering a series of events in the waveguide: generation of PL photon fluxes, reflections at the air–waveguide interfaces, absorption by the luminescent layer, and re-emission of higher-generation PL photons.
In experiment, we fabricated some LWGs with organic dyes and recorded their radiation patterns over a wide dynamic range. The analytical expression reproduced the measured intensity distributions. Spectral studies on the PL photons emerging from the top surface of the LWGs with a fiber-optic probe placed at different radii confirmed that these were indeed the photons re-emitted at these points.
The re-emission phenomenon is beneficial for an LSC because a part of the optical power, otherwise lost by self-absorption, is recovered. The model for this mechanism might be applied for evaluating luminescent materials in a form close to a final implementation of an LSC. On the other hand, the re-emission event is detrimental for an energy-harvesting display application because it raises the black level of a displayed image. Possible solutions for alleviating this contrast degradation include increasing the thickness of the waveguide, moving the luminescent layer to the top of a waveguide, and processing the image data beforehand.
Funding
Japan Society for the Promotion of Science (17K07032).
Disclosures
The authors declare no conflicts of interest.
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