Abstract

We propose a reverse functional design of modified Fresnel lens (MFL) with discontinuous refractive surfaces to achieve cost-effective high optical performance with thin lenses. The reverse-geometry design process was optimized to control the spatial illuminance distribution (SID) of light-emitting diodes (LEDs). Analysis results based on non-sequential ray-tracing simulations for flat SIDs indicated that the illuminance uniformity of LEDs with optimum MFL with different groove angles increased about 22 times, from 0.348 to 0.016, compared with the normalized standard deviation (NSD) of the general Fresnel lenses (GFL) with groove angles of 0°. In addition, the proposed method enhanced the color uniformity by reducing the circular yellow pattern. Tolerance analysis was carried out to determine tolerance limits for applying the optimum MFL in the assembly process. Finally, the feasibility of the reverse design process was verified by optical measurements of the optimum MFL.

© 2011 OSA

1. Introduction

Light-emitting diodes (LEDs) have been increasingly used in optical applications such as electric devices, traffic and automotive lighting, and general lighting because of their environmental benefits, low power consumption, high brightness, reliability, and long life compared with traditional light sources [13]. The commercially available form has a characteristic disc-like emission shape. In addition, there is a demand for circular, uniform illuminance in general lighting. Nevertheless, there are still some limitations to their direct use in various optical applications: the output light of most LEDs still has a non-uniform Lambertian distribution with circular symmetry, whereas the desired spatial illuminance distribution (SID) may have a different shape in specific applications, such as a projector illumination system [4,5]. Although secondary optical components are usually incorporated to improve the desired SID, they cannot provide an ideal solution if the optical design of those components is based on the traditional forward process in which the SID is determined by the secondary components. Therefore, more advanced design methods have been applied to achieve ideal optical performance [59]. However, these advanced design methods can result in bulky optical components that are difficult to fabricate due to complicated solutions. The LED illuminance system with the optics also has the disadvantage of large size. This results in a large, heavy heat sink that is inconvenient to assemble. Moreover, because the trend of LED illuminance systems has been continuously expanding toward more compact, material-saving, and low-cost designs, it is important and essential for most LED luminaries to design compact secondary optics that can simultaneously be fabricated easily and satisfy the target optical performance. In this study, we present an improved lens design method that applies a free-form surface [4,9,10]. With this method, Fresnel-like circular segmented lenses for a compact system are designed to obtain target performance that involves a uniform, circular light pattern, which is common in commercial LED illuminance systems. We previously studied the optimization of compact Fresnel lenses to concentrate light with uniform SIDs in a specified area with a high degree of freedom for structural constraints [11]. The design of general Fresnel lens (GFL) with groove angles of 0° was carried out according to the traditional forward process using a continuous refractive surface with constant optical power from ray matrices based on edge rays of the point source. Then, the continuous refractive surface was divided into grooves of equal height, N, and each separate groove surface was connected to groove facets that were parallel to the optical axis (groove angle of 0°). In this case, the change in the optical path of rays by multirefraction at the grooves caused a great loss of the total flux in the target area, as shown in Fig. 1(a) . Normalized standard deviation (NSD) was used to evaluate the uniformity of SIDs by optical lenses [11,12].

 

Fig. 1 Forward-geometry design based on ray matrices and special illuminance distributions (SIDs) by ray-tracing simulations of (a) a general Fresnel lens (GFL) with a normalized standard deviation (NSD) of 0.348 by groove angles of 0° and (b) a modified Fresnel lens (MFL) with a NSD of 0.164 by equal optimum groove angles.

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For the GFL, the NSD was calculated as 0.348, indicating very low uniformity. Thus, to improve the uniformity of SID by minimizing flux loss, we proposed modified Fresnel lens (MFL) with equal optimum groove angles, as shown in Fig. 1(b). NSD decreased to 0.164, indicating that the uniformity of SIDs increased more than two times compared with that of the GFL. However, these results confirmed that the forward design process has fundamental limits and cannot achieve a perfectly uniform SID. Therefore, we applied a reverse functional design process in which the shape of an optical component is determined by the desired SID on the target area (Fig. 2 ). In the initial design phase, we defined a target area and a desired SID. The total energy of the point source was divided into energy segments with functional relationships to achieve the desired SID. After input and output vectors of on-axis rays were determined by energy-mapping relationships, surface-normal vectors and coordinates of each point were calculated by the vector formulation of refraction [9,10]. Then, we obtained a closed-form solution to numerically draw an initial seed profile of MFL with different groove angles for the point source. In the reverse design phase, we analyzed ray-tracing simulations of the initial MFL design for a real LED module and an extended source, and we revaluated the uniformity based on the NSD. Finally, to minimize the NSD, input-energy segments were compensated cumulatively by reverse functions of the previous output energy. Therefore, the optimum shape of the MFL was finalized effectively by the reverse functional design process applied in this study.

 

Fig. 2 Reverse functional design process of an optimum MFL with different groove angles for a real LED module by compensation of the input energy.

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2. Initial Design of MFLs for the point source

2.1 Design solution for the point source

In the first design phase of a MFL with different groove angles, the functional relationship between the intensity distribution of a point source of light and the SID in the target area should be formulated based on the law of energy conservation, as expressed in Eq. (1), with the assumption that the optical system has no losses [13]. The basic concept used to achieve the desired SID in a target area is that the total flux emitted from the source is divided into m equal fluxes, and each separate flux enters each target concentric circle area between ti and ti + 1, as shown in Fig. 3 . The circle areas have functional relationships for i, and vice versa. In this paper, the discrete flux, Φi(i), for a flat SID is expressed by Eq. (2), in which fluxes enter the concentric circles with equal areas.

 

Fig. 3 Schematic of functional relationship for MFL with different groove angles based on a discrete energy distribution for the reverse-geometric design.

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sourceI(θ)dΩ=targetE(R)dA,
φinput(i)=θi1θi2πcosθsinθdθ=πR2m,(i=1,2,,m,θ0=0),

In Eq. (1), I(θ) and E(R) are the intensity distribution of a point source and the SID in the target area, respectively. Here, dΩ and dA are defined as 2πsinθdθ and 2πRdR, respectively. Therefore, if R, m, and E(R) are determined, the initial incident angles, θi, of ray vectors that are emitted from a point source can be calculated easily by Eq. (2). In the next phase, the directions of incident rays with separate fluxes were changed to propagate toward calculated target points for illuminance redistribution by controlling ray vectors at each point of the groove surface based on the vector formulation of refraction [1416]. This can be expressed as Eq. (3) by multiplying through the incident ray vector, Ii, output ray vector, Oi, and the normal vector, Ni, of a point, Pi, on the groove surface, Zj(x), by n and n’ (Fig. 3). In the case of rotational symmetry, the slopes of each point on the groove surface of the MFL with flat first surfaces can be calculated by Eq. (4).

n'Oi=nIi+(n'OiNinIiNi)Ni,where  Oi=(tixPix,0,tizPiz),         Ii=(Pixstanθihtan(sin1(1nsinθi)),0,Pizsh),       Ni=(dPiz,0,dPix),
dPizdPix=((Pixstanθihtan(sin1(1nsinθi)))(Pixstanθihtan(sin1(1nsinθi)))2+(Pizsh)2)n((tixPix)(tixPix)2+(tizPiz)2)n'((Pizsh)(Pixstanθihtan(sin1(1nsinθi)))2+(Pizsh)2)n((tizPiz)(tixPix)2+(tizPiz)2)n',

In Eq. (3) and (4), n and n’ are the refractive index of each medium, (tix, tiz) and (Pix, Piz) are the coordinates of ti on the target area and Pi on the groove surface, respectively, s is the distance between the source and the first surface of the MFL, and tix is the x-coordinate of the radius of a concentric circle divided into m equal areas and can be defined as R(i/m)1/2, where R is the radius of the entire target area. Initial coordinates (Pix, 0) of point Pi were calculated by the law of refraction using incident angles based on separate energies, s and h. The starting points of each groove surface among the Pi were determined by Pi = (m/N)xj, where j is from 1 to the groove number, N. Therefore, we can numerically draw the discontinuous initial groove profile of the MFL using slopes of each Pi with Eq. (2)(4) from the coordinates of the starting point, Pi = (m/N)xj. In the final design phase, these groove surfaces were connected to groove facets with groove angles to prevent optical loss, as mentioned for Fig. 1. However, the design process for the groove angles in this paper was fundamentally different from applying equally optimum groove angles to all grooves, as described in Ref [11]. As shown in Fig. 3 and 4 , the final initial seed profile of the MFL, with its own groove angle, θgj, was completed by connecting Pi = (m/N)xj to each intersection point between the output rays on Pi = (m/N)xj and the groove surfaces, Zj(x), because the direction vectors of the output ray and the surface normal vectors were calculated exactly. Therefore, the void effect in the SID found with the previous MFL with equal optimum groove angles in Ref [11]. could decrease under a point light source.

 

Fig. 4 Construction concept of groove facets (a) with equal groove angle in general optimized MFL of Ref [11]. and (b) with different groove angles in functional designed MFL based on a discrete energy distribution with the reverse-geometric relationship.

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2.2 Illuminance simulation of the initial MFL

The analysis of SIDs by LightTools® software [17] using non-sequential Monte Carlo ray tracing was necessary to validate the initial design based on a point source for the MFL with different groove angles. The MFL applied for the simulation was designed under input conditions, including a non-absorbing lens material with a wavelength-independent refractive index (n) of 1.5, the refractive index of air (n’) of 1, a distance (s) of 1 mm, an initial lens thickness (h) of 1 mm, and a normalized target area of 0.6. Here, normalized target area means that target area was dependent on the location of the target plane. That is, the value 0.6 represents target areas on every location within a viewing angle of 30°, meaning that the optical condensing power from secondary optical components increased about two times compared with the point source of the Lambertian distribution with a viewing angle of 60°. As a result, the final initial seed profile of the MFL with a total thickness of 1.5 mm, lens diameter of 10 mm, and 50 grooves was applied to a 3D model for ray-tracing simulations, as shown in Fig. 5(a) . Figure 5(b) shows the vertical image of the SID in the target area, which is the rotation symmetry result that reduced the noise level of the ray-tracing simulation of 1,000,000 rays. The SID profile for the x-axis shows a nearly flat result, even when there was a slight difference between the desired SID (dotted red line) around the outer region (x/z > 0.4), as shown in Fig. 5(c). The NSD value of 0.011 also indicates an excellent, flat SID compared to the GFL with groove angles of 0° (0.348) and to the MFL with equal optimum groove angles (0.164) in Section 1. The initial design of MFL with different groove angles based on a point source satisfied the desired flat SID without additional compensation processes

 

Fig. 5 Results of (a) initial MFL with different groove angles modeling for the ray-tracing simulation, (b) spatial illuminance distribution image for the point source, and (c) relative illuminance distribution of the x-axis with a NSD of 0.011.

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3. Reverse Design for MFLs with an extended source

3.1 Illuminance simulation of an optimum MFL

The initial seed profile of the MFL with different groove angles designed using a distance of 1 mm between the first surface of the MFL and an ideal point source could not form a flat SID in the target area for real LED modules because their final emission areas were an extended surface rather than a point [18]. Therefore, considering the geometric data for a real LED module and the material and optical properties of each surface, the flux ratio of each position over the entire target area changed considerably compared with that of the initial design due to the influence of rays emitted from the off-axis position, deviation of the refractive index, chromatic aberration, and Abbe number, although the ray-tracing simulation was performed under the same conditions. The vertical image of the SID in the target area and the SID profile for the x-axis of the initial MFL for a real LED module show that the illuminance value decreased in the direction opposite the optical axis (Fig. 6(b) and 6(d)). The NSD was 0.073, indicating a more uniform SID compared with that of the GFL with groove angles of 0° (0.348) and the MFL with equal optimum groove angles (0.164), but a less uniform SID compared with that of the point source (0.011). Therefore, the initial MFL with different groove angles designed from the point source should be optimized by adding an energy-compensation process, as shown in Fig. 2, to satisfy the desired SID for the extended source, such as LED modules. In this paper, we defined the NSD of E(Zi(x)|j) deviation [11,12] in Eq. (5) as a merit function based on various methods for design optimization [1921].

E(Zj(x)|n1)deviation=1Eavgi=1m(E(tix;Zj(x)|n1)Eavg)2m,             (i=1,2,,m),(j=0,1,,N1),(n=1,2,)
where Zj(x)|n-1 is the n-1th-compensated geometric configuration of each groove surface of the MFL according to the n-1th-compensated separate input flux, m is the total number of measured points, E(tix ; Zj(x)|n-1) is the illuminance value on the x-coordinate of point ti in the target area, and Eavg is the average illuminance of the entire target area. In the general case, it is necessary to modify the geometric variables of each groove surface directly to calculate the merit function to evaluate uniformity. Therefore, to find variable conditions with optimum values, the modification process should be performed repetitively, and a numerical approach is necessary to calculate these variables automatically. Therefore, in this study, we proposed an efficient iterative optimum design method in which the partial derivative or the approximate gradient of the merit function is not used, but the only illuminance values based on LightTools® software are applied for simple and efficient optimization.

 

Fig. 6 Results of (a) the final MFL with different groove angles modeling for a ray-tracing simulation, (b) SID image for the initial MFL, (c) SID image for the final compensated MFL, and (d) relative illuminance distributions of the x-axis with a NSD of 0.073 before optimization and with a NSD of 0.016 after optimization for a real LED module.

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φinput(i)|n=w(i)|n×φinput(i)|n11mi=1mw(i)|n×φinput(i)|n1,(i=1,2,,m),(n=1,2,),where  w(i)|n=(E(tix;Zj(x)|n1)11mi=1m(E(tix;Zj(x)|n1)1,(j=0,1,,N1),
Pix|n=stan(θi|n)+htan(sin1(1nsin(θi|n))),(i=1,2,,m),(n=1,2,)where  θi|n=12cos1(cos(2θi1|n2φinput(i)|nπ)),θ0|n=θ0=0,

To minimize the illuminance deviation, geometric configurations of the initial MFL were iteratively changed according to the nth-compensated input flux, Φinput(i)|n. That is, Φinput(i)|n can be calculated by multiplying the n-1th-compensated input flux, Φinput(i)|n-1, by the nth-compensation coefficient, w(i)|n, as expressed in Eq. (6), including reversed values of illuminance results on points equivalent to each number i for the n-1th-compensated MFL. The initial points, Pix, on the groove surface of the MFL are redefined simultaneously by Eq. (7), considering new incident angles of ray vectors for the nth-compensated input flux. Therefore, we solved the feedback loop to determine the cumulative distribution function of the input energy for minimizing the NSD by iterative ray-tracing simulations. The vertical image of the SID in the target area and the SID profile of the x-axis for the 5th-compensated MFL with different groove angles by Eq. (5)(7) show that the uniformity of the SID increased about five times from the initial NSD of 0.073 to the final NSD of 0.016 (Fig. 6(c) and 6(d)). This value indicates that the optimum MFL for an extended source can form a nearly flat, uniform SID with a level equal to that of the initial MFL for the point source. Note that the reverse compensation process of the input energy is a more efficient approach for achieving the optimally shaped MFL with different groove angles compared with the general parameterization process of geometric objects, which statistically samples response results with a direct change in geometric variables within the coordinate intervals [11,12], because the final calculated NSD increased about 10 times, from MFL = 0.164 with an equal optimum groove angle [11] to 0.016.

To further analyze the contribution of energy compensation for an extended source, we carried out separate ray-tracing simulations of 1,000,000 rays for two groove profiles of the initial and final MFL with different groove angles (Fig. 7 ) based on the assumption that the final emission area of the LED module is an extended source area. This area was divided into m areas from the center point source of S0 to the final concentric circle source of Sr (Fig. 8(a) ). As a result of each simulation, we verified that the initial MFL, with a nearly flat SID for the point source, could be designed by considering only S0 from among the divided sources (Fig. 8(b)). We also confirmed that the total SID of the initial MFL for an extended source (Fig. 6(d)) became non-uniform because as the concentric circle source extended to Sr, the uniformity of the simulated SID for those sources decreased (Fig. 8(b)). However, each SID of the final MFL for the divided sources from S0 to Sr/2 had reversed distribution tendencies against the total SID of Fig. 8(b) under the same simulation conditions, even if the final MFL was redesigned only using the S0 source. The SID results based on point source play an important role in the reverse compensation of each SID of the final MFL for the divided sources from Sr/2 to Sr to form the total flat SID (Fig. 8(c)) because the source area from S0 to Sr/2 can be regarded as the major emission area for the real LED module.

 

Fig. 7 Comparison of geometry for the initial and final compensated MFL with different groove angles.

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Fig. 8 (a) Schematic for analysis of an extended source, (b) analysis of a SID by the initial MFL with different groove angles for discrete sources, and (c) analysis of a SID by the final compensated MFL for discrete sources.

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3.2 Color simulation of an optimum MFL

The optimum MFL with different groove angles by reverse functional design showed superior color uniformity. By applying spherical lenses with continuous refractive surfaces on a general white LED based on a yellow phosphor, a strong yellow ring pattern was generated around the target area (Fig. 9(a) ). However, it had diminished greatly at the final MFL (Fig. 9(b)). The pattern can be evaluated quantitatively using the flux ratio between yellow rays (wavelength range of 490–780 nm) and blue rays (wavelength range of 380–490 nm), the yellow–blue ratio (YBR), in the target area [22,23]. The distribution of the YBR for the spherical lens increased farther from the optical axis, whereas that of the final MFL was uniform (Fig. 9(c)). This can be explained using the distribution of rays emitted from sources between S0 and Sr/2 of Fig. 8(a). Those rays can be considered bluish top-emitting rays due to passage through a thin-layer yellow phosphor, as reported for a white LED module in Ref [11]. Therefore, variation in the YBR around the target area was reduced because the SID of the bluish rays increased farther from the optical axis using the optimum design method to achieve uniform SIDs for an extended source. This reverse functional design can be applied as a more advanced solution to improve the color uniformity compared with the moderate mixing method applied to yellow and blue rays by optimized groove facets of GFLs with groove angles of 0° and MFLs with equal optimum groove angles in Ref [11]. The optimum MFL with different groove angles by reverse functional design showed superior color uniformity.

 

Fig. 9 Comparison of (a) the color distribution by a spherical lens, (b) the color distribution by an optimal MFL with a real extended white LED source module, and (c) the yellow–blue ratio (YBR) of each lens.

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3.3 Tolerance analysis of uniformity

Deviations in the horizontal (decenter) and vertical directions (defocus), as well as rotational deviation (tilt), of the lens occur in most assembly processes. Therefore, tolerance analysis should be carried out to determine the tolerance limits for applying the optimized MFL with different groove angles because assembly errors have negative effect on the optical performance. In this study, the decenter was analyzed based on the movement relative to the x-axis, and the tilt was considered for only the rotation around the y-axis because the optimized MFL was a rotationally symmetric lens. The defocus moved in both directions based on the distance of 1 mm between the LED and the lens. Figure 10 shows that the NSD increased proportionally with increasing decenter relative to the x-axis. If the lens was attached with a decenter of 200 µm relative to the x-axis, the NSD could be calculated as ~0.1, indicating that the illuminance uniformity decreased to 90%. Therefore, we determined that the tolerance range was ± 50 µm for the decenter based on the NSD of 0.04 with a uniformity of 96%, which is about twice as likely to have uniform illuminance compared with the initial MFL for an extended source with a NSD of 0.073 (Fig. 6(d)).

 

Fig. 10 Analysis of illuminance uniformity variation for the decenter tolerance between the LED module and the designed MFL (e.g., tolerance range for the decenter was ± 50 µm for an illuminance uniformity of 96%).

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For the defocus relative to the z-axis, the NSD increased with an increasing absolute value of deviation (Fig. 11 ). The total tolerance range based on a uniformity of 96% was ~450 µm, indicating that the variation of the NSD for the defocus was insensitive compared with the decenter. However, the illuminance area increased or decreased with increasing or decreasing defocus at an illuminance uniformity of 96%, as shown in the illuminance-distribution image in Fig. 11. Therefore, the tolerance range for the defocus was analyzed based on the viewing angle of the LED module, which can determine the illuminance area. As a result, the tolerance range for the defocus was ± 100 µm for a viewing angle deviation of ± 2° and an illuminance uniformity of 96%. The NSD variation for the tilt was similar to that of the decenter, as shown in Fig. 12 . That is, the NSD was calculated as ~0.1 for the tilt of 1° on the y-axis. For a uniformity of 96%, the optimum MFL should be attached within a tilt tolerance of about ± 0.35°. Therefore, for a uniformity of 96%, the assembly ranges of an optimally designed MFL for the decenter, defocus, and tilt were allowed to be ± 50 µm, ± 100 µm, and ± 0.35°, respectively.

 

Fig. 11 Analysis of illuminance uniformity variation for the defocus tolerance between the LED module and the designed MFL (e.g., tolerance range for the defocus was ± 100 µm for a viewing angle deviation of ± 2° and an illuminance uniformity of 96%).

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Fig. 12 Analysis of illuminance uniformity variation for the tilt tolerance between the LED module and the designed MFL (e.g., tolerance range for tilt was ± 0.35° for an illuminance uniformity of 96%).

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4. Results and discussion

To verify the feasibility of the reverse design process of MFL with different groove angles, the optimum MFL was fabricated by an ultraviolet (UV)-imprinting process [11] (Fig. 13(a) ). In this process, a relatively low pressure was applied to ensure the replication quality of each groove during UV exposure. The original mold, with the inverted shape of the optimal MFL, was fabricated by diamond turning (Fig. 13(b)). Figure 13(c) shows the MFL fabricated by this process. To evaluate the quality of illuminance and color distribution, a diffuser plate (130 × 130 mm) was placed 100 mm above the LED module, including the lens. Within the target region for a viewing angle of 30° (Fig. 14(a) and (b) ), the values of NSD for the x-axis illuminance distribution were 0.218 for spherical lenses and 0.039 for the fabricated MFL, respectively. Definitely, it is very difficult to fabricate Fresnel lens facets perfectly on the same level of a design profile. In addition, the sharp tips of the Fresnel grooves have rounded edges after the mold fabrication and replication process. The larger the radii of the rounded edges are, the more the optical efficiency and illuminance uniformity deteriorate. Nevertheless, the fabricated MFL was attached on the LED within the tolerance range, although the NSD was calculated to be higher than that of the final simulation. Therefore, we verified that the reverse functional design of MFL achieved the desired flat illuminance distribution of the lighting system. The MFL fabricated by UV imprinting also showed superior uniformity in the color distribution compared with the simulation result. For the spherical lens, the yellow ring pattern was strong around the target area (Fig. 14(a)), as expected. However, it had diminished greatly at the final MFL, as shown in Fig. 14(b). The distribution of the YBR for the spherical lens increased rapidly around the target area, whereas that for the final MFL was uniform (Fig. 14(c)). Thus, we verified that the more specific analysis for an extended source was correct, and the reverse functional design of MFL can be applied as an advanced solution to improve color uniformity.

 

Fig. 13 The main UV-imprinting process and images of the fabricated original mold and the fabricated MFL.

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Fig. 14 Comparison of (a) the color distribution image with the NSD of 0.218 by a spherical lens, (b) the color distribution image with the NSD of 0.039 using the fabricated MFL with a real, extended white LED source module, and (c) the YBR of each lens.

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5. Conclusions

We proposed a reverse functional design of MFL with different groove angles for an extended source as an effective way to achieve cost-effective high optical performance with thin lenses. The reverse-geometry design process was used for the point source and optimized using a feedback loop to compensate for the SID of a real LED module. To further analyze the contribution of energy compensation for an extended source, we carried out non-sequential ray-tracing simulations for separate concentric-circle sources. Analysis results for the flat SID indicated that the illuminance uniformity of LEDs with the optimum MFL with different groove angles increased about 22 times, from 0.348 to 0.016, compared with the NSD of the GFL with groove angles of 0°. In addition, it enhanced the color uniformity due to the reduced circular yellow pattern of the uniform distribution, which resulted from the YBR of the design process. This process is an advanced solution compared with the passive method in which yellow and blue rays are mixed by optimized groove facets of the GFL. Tolerance analysis was also carried out to determine the tolerance limits for applying the optimized MFL in an assembly process. For a uniformity of 96%, the ranges for the decenter, defocus, and tilt were ± 50 µm, ± 100 µm, and ± 0.35°, respectively. Finally, the feasibility of the proposed design process was verified by optical measurements using the fabricated optimum MFL. Further studies are currently underway to analyze the variation in the optical performance with a change in the radius of rounded edges based on a ray-tracing simulation and to optimize the fabrication process for accurate replication of a sharp groove tip.

Acknowledgments

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (MEST) (NRF-2009-R1A5A001-2009-0080932).

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19. J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998). [CrossRef]  

20. R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004). [CrossRef]  

21. S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

22. H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005). [CrossRef]  

23. Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008). [CrossRef]  

References

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  1. E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
    [Crossref] [PubMed]
  2. M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
    [Crossref]
  3. J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
    [Crossref] [PubMed]
  4. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
    [Crossref] [PubMed]
  5. Y. Honguh, Y. Takashima, H. Mihara, and M. Kitamura, “Illumination apparatus and a projection type image display apparatus using it,” US Patent 6264332.
  6. I. Moreno, “Spatial distribution of LED radiation,” in The International Optical Design Conference, G.Gregory, J. Howard, J. Koshel, eds., Proc. SPIE 6342, 634216:1–7 (2006).
  7. H. Yang, J. W. Bergmans, T. C. W. Schenk, J. P. Linnartz, and R. Rietman, “An analytical model for the illuminance distribution of a power LED,” Opt. Express 16(26), 21641–21646 (2008).
    [Crossref] [PubMed]
  8. W. Tai and R. Schwarte, “Design of an aspherical lens to generate a homogenous irradiance for three-dimensional sensors with a light-emitting-diode source,” Appl. Opt. 39(31), 5801–5805 (2000).
    [Crossref]
  9. B. Parkyna, and D. Pelkaa, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” in Nonimaging Optics and Efficient Illumination Systems III, R. Winston, P. Benítez, eds., Proc. SPIE 6338, 633808:1–7 (2006).
  10. F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
    [Crossref]
  11. B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
    [Crossref] [PubMed]
  12. B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
    [Crossref]
  13. W. R. Boyd, Radiometry and Dectection of Optical Radiation (Wiley, New York, 1983).
  14. W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
    [Crossref]
  15. R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, Nonimaging Optics (Elsevier, 2005).
  16. J. Charves, Introduction to nonimaging optics, (CRC Press, New York, 2008).
  17. Optical Research Associates, http://www.opticalres.com
  18. C.-C. Sun, W.-T. Chien, I. Moreno, C.-C. Hsieh, and Y.-C. Lo, “Analysis of the far-field region of LEDs,” Opt. Express 17(16), 13918–13927 (2009).
    [Crossref] [PubMed]
  19. J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
    [Crossref]
  20. R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
    [Crossref]
  21. S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).
  22. H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
    [Crossref]
  23. Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
    [Crossref]

2010 (1)

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[Crossref]

2009 (3)

2008 (4)

2007 (1)

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

2005 (3)

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[Crossref] [PubMed]

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

2004 (1)

R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
[Crossref]

2000 (1)

1998 (2)

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[Crossref]

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
[Crossref]

Beck, J. V.

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[Crossref]

Bergmans, J. W.

Chen, F.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[Crossref]

Chien, W.-T.

Cho, J.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

Choi, M.

Craford, M. G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Ding, Y.

Gu, P. F.

Harbers, G.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Hsieh, C.-C.

Kang, S.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[Crossref]

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[Crossref] [PubMed]

Kim, B.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[Crossref]

B. Kim, M. Choi, H. Kim, J. Lim, and S. Kang, “Elimination of flux loss by optimizing the groove angle in modified Fresnel lens to increase illuminance uniformity, color uniformity and flux efficiency in LED illumination,” Opt. Express 17(20), 17916–17927 (2009).
[Crossref] [PubMed]

Kim, H.

Kim, J.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[Crossref]

Kim, J. K.

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
[Crossref] [PubMed]

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[Crossref] [PubMed]

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

Koshel, R. J.

R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
[Crossref]

Krames, M. R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Kudaev, S.

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

Lim, J.

Linnartz, J. P.

Liu, S.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[Crossref]

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[Crossref]

Liu, X.

Liu, Z.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[Crossref]

Liu, Z. Y.

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[Crossref]

Lo, Y.-C.

Luo, H.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

Luo, X.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[Crossref]

Luo, X. B.

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[Crossref]

Moreno, I.

Mueller, G. O.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Mueller-Mach, R.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Ohm, W.-S.

B. Kim, J. Kim, W.-S. Ohm, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8596–8604 (2010).
[Crossref]

Park, Y.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

Parkyn, W. A.

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
[Crossref]

Rietman, R.

Schenk, T. C. W.

Schreiber, P.

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

Schubert, E. F.

J. K. Kim and E. F. Schubert, “Transcending the replacement paradigm of solid-state lighting,” Opt. Express 16(26), 21835–21842 (2008).
[Crossref] [PubMed]

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[Crossref] [PubMed]

Schwarte, R.

Shchekin, O. B.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Sone, C.

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

Sun, C.-C.

Tai, W.

Wang, K.

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[Crossref]

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[Crossref]

Woodbury, K. A.

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[Crossref]

Yang, H.

Zheng, Z. R.

Zhou, L.

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Luo, J. K. Kim, E. F. Schubert, J. Cho, C. Sone, and Y. Park, “Analysis of high-power packages for phosphor-based white-light-emitting diodes,” Appl. Phys. Lett. 86(24), 234505 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Z. Y. Liu, S. Liu, K. Wang, and X. B. Luo, “Optical analysis of color distribution in white LEDs with various packaging methods,” IEEE Photon. Technol. Lett. 20(24), 2027–2029 (2008).
[Crossref]

J. Disp. Technol. (1)

M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Disp. Technol. 3(2), 160–175 (2007).
[Crossref]

Meas. Sci. Technol. (1)

J. V. Beck and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis,” Meas. Sci. Technol. 9(6), 839–847 (1998).
[Crossref]

Opt. Eng. (1)

F. Chen, S. Liu, K. Wang, Z. Liu, and X. Luo, “Free-form lenses for high illumination quality light-emitting diode MR16 lamps,” Opt. Eng. 48(12), 123002 (2009).
[Crossref]

Opt. Express (6)

Proc. SPIE (3)

R. J. Koshel, “Aspects of illumination system optimization,” Proc. SPIE 5529, 206–217 (2004).
[Crossref]

S. Kudaev and P. Schreiber, “Optimization of symmetrical free-shape non-imaging concentrators for LED light source applications,” Proc. SPIE 5942(594209), 1–10 (2005).

W. A. Parkyn, “Illumination lenses designed by extrinsic differential geometry,” Proc. SPIE 3428, 389–396 (1998).
[Crossref]

Science (1)

E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005).
[Crossref] [PubMed]

Other (7)

R. Winston, J. C. Miñano, and P. Benítez, with contributions by N. Shatz and J. C. Bortz, Nonimaging Optics (Elsevier, 2005).

J. Charves, Introduction to nonimaging optics, (CRC Press, New York, 2008).

Optical Research Associates, http://www.opticalres.com

Y. Honguh, Y. Takashima, H. Mihara, and M. Kitamura, “Illumination apparatus and a projection type image display apparatus using it,” US Patent 6264332.

I. Moreno, “Spatial distribution of LED radiation,” in The International Optical Design Conference, G.Gregory, J. Howard, J. Koshel, eds., Proc. SPIE 6342, 634216:1–7 (2006).

B. Parkyna, and D. Pelkaa, “Free-form illumination lenses designed by a pseudo-rectangular lawnmower algorithm,” in Nonimaging Optics and Efficient Illumination Systems III, R. Winston, P. Benítez, eds., Proc. SPIE 6338, 633808:1–7 (2006).

W. R. Boyd, Radiometry and Dectection of Optical Radiation (Wiley, New York, 1983).

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Figures (14)

Fig. 1
Fig. 1

Forward-geometry design based on ray matrices and special illuminance distributions (SIDs) by ray-tracing simulations of (a) a general Fresnel lens (GFL) with a normalized standard deviation (NSD) of 0.348 by groove angles of 0° and (b) a modified Fresnel lens (MFL) with a NSD of 0.164 by equal optimum groove angles.

Fig. 2
Fig. 2

Reverse functional design process of an optimum MFL with different groove angles for a real LED module by compensation of the input energy.

Fig. 3
Fig. 3

Schematic of functional relationship for MFL with different groove angles based on a discrete energy distribution for the reverse-geometric design.

Fig. 4
Fig. 4

Construction concept of groove facets (a) with equal groove angle in general optimized MFL of Ref [11]. and (b) with different groove angles in functional designed MFL based on a discrete energy distribution with the reverse-geometric relationship.

Fig. 5
Fig. 5

Results of (a) initial MFL with different groove angles modeling for the ray-tracing simulation, (b) spatial illuminance distribution image for the point source, and (c) relative illuminance distribution of the x-axis with a NSD of 0.011.

Fig. 6
Fig. 6

Results of (a) the final MFL with different groove angles modeling for a ray-tracing simulation, (b) SID image for the initial MFL, (c) SID image for the final compensated MFL, and (d) relative illuminance distributions of the x-axis with a NSD of 0.073 before optimization and with a NSD of 0.016 after optimization for a real LED module.

Fig. 7
Fig. 7

Comparison of geometry for the initial and final compensated MFL with different groove angles.

Fig. 8
Fig. 8

(a) Schematic for analysis of an extended source, (b) analysis of a SID by the initial MFL with different groove angles for discrete sources, and (c) analysis of a SID by the final compensated MFL for discrete sources.

Fig. 9
Fig. 9

Comparison of (a) the color distribution by a spherical lens, (b) the color distribution by an optimal MFL with a real extended white LED source module, and (c) the yellow–blue ratio (YBR) of each lens.

Fig. 10
Fig. 10

Analysis of illuminance uniformity variation for the decenter tolerance between the LED module and the designed MFL (e.g., tolerance range for the decenter was ± 50 µm for an illuminance uniformity of 96%).

Fig. 11
Fig. 11

Analysis of illuminance uniformity variation for the defocus tolerance between the LED module and the designed MFL (e.g., tolerance range for the defocus was ± 100 µm for a viewing angle deviation of ± 2° and an illuminance uniformity of 96%).

Fig. 12
Fig. 12

Analysis of illuminance uniformity variation for the tilt tolerance between the LED module and the designed MFL (e.g., tolerance range for tilt was ± 0.35° for an illuminance uniformity of 96%).

Fig. 13
Fig. 13

The main UV-imprinting process and images of the fabricated original mold and the fabricated MFL.

Fig. 14
Fig. 14

Comparison of (a) the color distribution image with the NSD of 0.218 by a spherical lens, (b) the color distribution image with the NSD of 0.039 using the fabricated MFL with a real, extended white LED source module, and (c) the YBR of each lens.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

s o u r c e I ( θ ) d Ω = t arg e t E ( R ) d A ,
φ i n p u t ( i ) = θ i 1 θ i 2 π cos θ sin θ d θ = π R 2 m , ( i = 1 , 2 , , m , θ 0 = 0 ) ,
n ' O i = n I i + ( n ' O i N i n I i N i ) N i , where   O i = ( t i x P i x , 0 , t i z P i z ) ,           I i = ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) , 0 , P i z s h ) ,         N i = ( d P i z , 0 , d P i x ) ,
d P i z d P i x = ( ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) ) ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) ) 2 + ( P i z s h ) 2 ) n ( ( t i x P i x ) ( t i x P i x ) 2 + ( t i z P i z ) 2 ) n ' ( ( P i z s h ) ( P i x s tan θ i h tan ( sin 1 ( 1 n sin θ i ) ) ) 2 + ( P i z s h ) 2 ) n ( ( t i z P i z ) ( t i x P i x ) 2 + ( t i z P i z ) 2 ) n ' ,
E ( Z j ( x ) | n 1 ) d e v i a t i o n = 1 E a v g i = 1 m ( E ( t i x ; Z j ( x ) | n 1 ) E a v g ) 2 m ,               ( i = 1 , 2 , , m ) , ( j = 0 , 1 , , N 1 ) , ( n = 1 , 2 , )
φ i n p u t ( i ) | n = w ( i ) | n × φ i n p u t ( i ) | n 1 1 m i = 1 m w ( i ) | n × φ i n p u t ( i ) | n 1 , ( i = 1 , 2 , , m ) , ( n = 1 , 2 , ) , where   w ( i ) | n = ( E ( t i x ; Z j ( x ) | n 1 ) 1 1 m i = 1 m ( E ( t i x ; Z j ( x ) | n 1 ) 1 , ( j = 0 , 1 , , N 1 ) ,
P i x | n = s tan ( θ i | n ) + h tan ( sin 1 ( 1 n sin ( θ i | n ) ) ) , ( i = 1 , 2 , , m ) , ( n = 1 , 2 , ) where   θ i | n = 1 2 cos 1 ( cos ( 2 θ i 1 | n 2 φ i n p u t ( i ) | n π ) ) , θ 0 | n = θ 0 = 0 ,

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