Design of an optical element forming an axial line segment for efficient LED lighting systems Emil R. Aslanov,1 Leonid L. Doskolovich,1,2,* Mikhail A. Moiseev,1,2 Evgeni A. Bezus,1,2 and Nikolay L. Kazanskiy1,2 1
Technical Cybernetics Department, Samara State Aerospace University, 151 Molodogvardeyskaya Street, Samara 443001, Russia 2 Image Processing Systems Institute of the Russian Academy of Sciences, 151 Molodogvardeyskaya Street, Samara 443001, Russia * [email protected]
Abstract: An LED optical element is proposed as an alternative to coldcathode fluorescent lamps. The optical element generates two symmetric uniformly illuminated line segments on the diffuse reflector. The illuminated segments then act as secondary linear light sources. The calculation of the optical element is reduced to the integration of the system of two explicit ordinary differential equations. The results of the simulation of an illumination system module consisting of a set of optical elements generating a set of line segments on the surface of the diffuse reflector are presented. The elements are located directly on the surface of the reflector. The simulation results demonstrate the uniform illumination of a rectangular area at a distance of 30–40 mm from the light source plane. The lighting efficiency of the designed system exceeds 83%. ©2013 Optical Society of America OCIS codes: (220.0220) Optical design and fabrication; (220.2945) Illumination design; (220.4298) Nonimaging optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
C.-F. Lin, Y.-B. Fang, and P.-H. Yang, “Optimized micro-prism diffusion film for slim-type bottom-lit backlight units,” J. Disp. Technol. 7(1), 3–9 (2011). M. Doshi, R. Zane, and F. J. Azcondo, “Low frequency architecture for multi-lamp CCFL systems with capacitive ignition,” J. Disp. Technol. 5(5), 152–161 (2009). R.-S. Chang, J.-Z. Tsai, T.-Y. Li, and H.-L. Liao, “LED backlight module by lightguide-diffusive component,” J. Disp. Technol. 8(2), 79–86 (2012). Labsphere, “Labsphere: Spectralon Targets,” http://www.labsphere.com/products/reflectance-standards-andtargets/spectralon-reflectance-standards/default.aspx. A. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. 50(16), 2431–2442 (2011). V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997). L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik (Stuttg.) 101, 37–41 (1995). M. A. Moiseev and L. L. Doskolovich, “Design of TIR optics generating the prescribed irradiance distribution in the circle region,” J. Opt. Soc. Am. A 29(9), 1758–1763 (2012). Future Lighting Solutions, “Future Lighting Solutions: Remote Phosphor,” http://www.futurelightingsolutions.com/en/technologies/Pages/remote_phosphor.aspx. H.-T. Huang, Y.-P. Huang, and C.-C. Tsai, “Planar Lighting System Using Array of Blue Leds to Excite Yellow Remote Phosphor Film,” J. Disp. Technol. 7(1), 44–51 (2011). A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illumination systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009). L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005). L. L. Doskolovich, A. Y. Dmitriev, E. A. Bezus, and M. A. Moiseev, “Analytical design of freeform optical elements generating an arbitrary-shape curve,” Appl. Opt. 52(12), 2521–2526 (2013).
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28651
1. Introduction Currently, liquid-crystal displays (LCD) are commonly used in mobile devices, computer monitors and TVs. Thus, the development of compact and energy-efficient LCD backlight systems is a relevant task. Cold-cathode fluorescent lamps are widely used in the backlight systems of LCDs [1, 2]. In Fig. 1(a), typical configuration of a CCFL-based backlight system including the light sources, a diffuse reflector and a diffuser is shown.
Fig. 1. Configuration of a backlight system with fluorescent lamps (a); configuration of an LED-based backlight system module (b).
CCFL-based backlight systems can provide high uniformity of illumination, but have several disadvantages. Nowadays, LED-based lighting systems are considered to be more promising [3]. Compared with CCFLs, LEDs consume less energy, are more durable, resistant to vibration and shockproof. In the simplest case, the CCFLs are replaced by linear arrays of LEDs. In the present work, a system consisting of an LED with a special refractive optical element is proposed as an alternative to CCFL (Fig. 1(b)). The optical element mimics a CCFL and generates two symmetric uniformly illuminated line segments on the diffuse reflector. The illuminated segments then act as secondary light sources. It is worth mentioning that the diffuse reflector can be made from materials with high reflectivity (95–99%) and the scattering law close to Lambert's cosine law. One of the available materials of this type is spectralon manufactured by Labsphere Corporation [4, 5]. To achieve high lighting efficiency, it is proposed to use an optical element utilizing the total internal reflection effect. The designed optical element is axially symmetric (the axis of symmetry coincides with the illuminated segment). This ensures the ease of manufacture of the element. Thus, the set of CCFLs in Fig. 1(a) can be replaced by a set of LEDs with optical elements focusing to the line segments on the diffuse reflector. The proposed approach allows to create compact backlight and illumination systems of a new type. According to the simulation results presented below, the proposed approach allows to generate uniform irradiance distributions with large angular size (more than 150° in the direction along the segments) with high lighting efficiency of 83.5%. Let us note that the design of a refractive optical element for focusing into a line segment in the geometry depicted in Fig. 1(b) is for the first time presented in the present work. Previously, the elements of this type were calculated only using the approximation of a thin optical element [6, 7]. 2. Design of the optical element Let us consider the design of a refractive optical element focusing the radiation emitted by a point light source into two symmetrically located line segments of the Ox axis (Fig. 1(b)). Due to symmetry, it is sufficient to design only the right half of the element located at x > 0 (Fig. 2). The designed element has rotational symmetry, the rotation axis coinciding with the Ox axis. The surface of the element consists of the two parts: part a utilizing the total internal
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28652
reflection (TIR) effect, and part b refracting the light rays reflected from the a part of the surface. We assume that the angular size of the part b is relatively small and therefore neglect the rays that impinge on it directly.
Fig. 2. Optical element for focusing into a line segment on the Ox axis.
Let us denote by ρ (α ) the distance from the origin to a point on the a surface, where
α ∈ [0, α max ] is the angular coordinate of the incident ray (Fig. 2). In [8], the following
differential equation was obtained for ρ (α ) :
d ρ (α ) π 2 − α − β (α ) = ρ (α ) ⋅ cot , dα 2
(1)
where β (α ) describes the ray direction after the total internal reflection. From the geometric considerations (Fig. 2) one can obtain the coordinate of the intersection point of the outgoing ray and the Ox axis in the following form: x (α ) = x0 + ρ (α ) cos (α ) − ( x0 − ρ (α ) sin (α ) ) tg ( β (α ) ) γ (α ) n ⋅ sin ( β (α ) ) , (2)
where γ (α ) = 1 − n ⋅ sin ( β (α ) ) , n is the refractive index of the element material. To find 2
the function ρ (α ) describing the element surface, let us obtain another expression for x (α )
using the light flux conservation law. Let us introduce a spherical coordinate system (α , ϕ ) , where ϕ is the polar angle in the Oyz plane perpendicular to the plane of Fig. 2. Since the optical element is located at z > 0 , the polar angle takes values in the range [0, π ] . In these
coordinates, the solid angle element corresponding to the ϕ ∈ [ 0, π ] range can be expressed as
d Ω (α ) = π sin (π / 2 − α ) d α . To write the light flux conservation law, it is necessary to
equate the light flux incident on the element dx of the illuminated line segment to the light flux emitted by the light source into the solid angle element d Ω (α ) . Thus, the following equation can be written: π sin (π / 2 − α ) I (α , ϕ ) dϕ dα = E ( x ) dx, 0
(3)
where E ( x ) , x ∈ [ x1 , x2 ] is the desired irradiance distribution along the segment, and I (α , ϕ ) is the light source intensity. Let us note that for the irradiance E ( x ) to be correctly
defined, the following normalization condition should hold: α max
0
x2 π sin (π / 2 − α ) I (α , ϕ ) dϕ d α = E ( x ) dx. x1 0
(4)
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28653
Assuming a constant irradiance E ( x ) = E0 , x ∈ [ x1 , x2 ] and integrating Eq. (4), we obtain: x (α ) = x1 +
α π 1 cos α I (α , ϕ )dϕ dα . E0 0 0
(5)
In the case of a Lambertian light source, the source intensity in the spherical coordinates has the form I (α , ϕ ) = I 0 cos α sin ϕ .
(6)
Substituting Eq. (6) into Eq. (5), we obtain a closed-form expression for x (α ) : x (α ) = x1 + ( x2 − x1 )
2α + sin ( 2α ) . 2α max + sin ( 2α max )
(7)
Substituting Eq. (7) into Eq. (2) and differentiating the obtained equation by α , we get: d β (α ) =2 dα
n sin ( β (α ) ) x ′ (α ) γ (α ) + sec ( β (α ) ) γ (α ) sin (α − β (α ) ) ρ (α ) − cos (α − β (α ) ) ρ ′ (α ) , × n 2 sec ( β (α ) ) sin ( 2 β (α ) ) x0 sin ( β (α ) ) − cos (α − β (α ) ) ρ (α ) − 2γ (α ) g (α )
(8)
where g (α ) = x0 tan 2 ( β (α ) ) + cos (α ) cot ( β (α ) ) − sin (α ) tan 2 ( β (α ) ) ρ (α ) .
The derivative ρ ′ (α ) in the right-hand side of Eq. (8) is defined in Eq. (1). The derivative
x ′ (α ) can be obtained from Eq. (5) or Eq. (7). After the substitution of the expressions for
these derivatives into Eq. (8) the right-hand side depends only on the variable α and the functions β (α ) and ρ (α ) . Thus, the design of the optical element for generating a line segment with the desired irradiance is reduced to the solution of a system of two explicit ordinary differential Eqs. (1), (8). This system can be integrated using standard numerical methods (e.g. the Runge-Kutta method). Note that despite the large number of articles dedicated to design of axially-symmetric LED optics, the analytical design of optical element generating an axial line segment is for the first time presented in this work (Eqs. (1), (5), (7) and (8)). Within the considered approach, it is assumed that the designed optical element generates the illuminated segments on a diffuse reflector, which acts as a secondary light source. 3. Numerical simulation results
The developed method for the design of the optical element was implemented in MATLAB numerical computing environment. On the basis of Eqs. (1), (7), (8), an optical element was designed for the uniform illumination of two symmetrical line segments with a Lambertian light source. For the numerical integration of the system of differential Eqs. (1), (8), a standard MATLAB routine ode45 implementing variable-step fourth-order Runge-Kutta method was used. The calculation was performed using the following parameters: distance from the source to the element surface along the Oz axis ρ ( 0 ) = 6 mm , angular size of the a part of the profile (Fig. 2) α max = 50° , coordinates of the endpoints of the segment x1 = 50 mm , x2 = 150 mm , refractive index of the element material n = 1.493 . The designed optical element is shown in Fig. 3. The dimensions of the right half of the element along the Ox, Oy, and Oz coordinate axes are 17.6 mm, 29.4 mm and 14.7 mm, respectively. To check the correctness of the calculations, the designed element was simulated using a commercial optical engineering software TracePro. A LED chip EZ290 with Lambertian
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28654
radiation pattern and 290 × 290 μ m size (Cree Inc.) was used as the light source in the simulations. We assume that the designed optical element is intended for lighting systems employing the remote phosphor technology [9,10] and therefore simulate the blue LED chip instead of a packaged white LED. This technology supposes that the conversion into white light is performed by a remote phosphor layer (RPL) located at the output of a backlight system (Fig. 1(b), dotted line under the diffuser).
Fig. 3. Cross section (a) and wireframe model (b) of the optical element for the uniform illumination of two symmetrical line segments on the Ox axis.
Figure 4(a) shows the grayscale irradiance distribution generated by the optical element in the plane z = 0 containing the segments. The irradiance distribution was obtained by tracing 200 000 rays using the TracePro software and fully meets the design requirements (uniform illumination of two line segments with 100 mm lengths). The lighting efficiency of the optical element (the fraction of the emitted light flux coming to the plane z = 0 ) exceeds 83.5%. In Fig. 4(b), the irradiance distribution generated by a matrix of 10 optical elements (two linear arrays consisting of 5 elements each) is presented. The distance between elements in a linear array is 40 mm. The adjacent linear arrays are shifted by 20 mm along the Ox axis and by 100 mm along the Oy axis relatively to each other.
Fig. 4. (a) Irradiance distribution generated by the designed optical element for the uniform illumination of two line segments in the plane z = 0; (b) Irradiance distribution generated by two linear arrays consisting of 5 optical elements each.
For utilizing the considered optical elements in a backlight or illumination system, a diffuse reflector should be placed in the plane z = 0 . Let us note that direct backlight systems usually contain an optical diffuser which additionally improves spatial and angular uniformity of output light (Fig. 1). In particular, the RPL itself acts as a diffuser with a high haze value [9,10]. Nonetheless, the degree of uniformity of irradiance distribution generated by the system without diffuser is of great importance. It defines both the complexity of the required diffuser and the total thickness of the backlight system. Thus, it is important to study the uniformity of irradiance distribution generated by the matrix of optical elements (Fig. 4(b)) in a certain exit plane z = d over the optical elements (Fig. 2). We assume that the diffuse reflector scatters incident light according to the Lambert’s cosine law [4, 5]. In this case, the illuminated line segments generated in the plane z = 0 act as secondary Lambertian sources. The distributions shown in Fig. 5 were calculated for the considered set of optical elements (two linear arrays of 5 elements each) using TracePro software. 200 000 rays were traced for each light source (Cree EZ290 chip). Figure 5 #198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28655
demonstrates the possibility to obtain highly uniform irradiance distribution over a large area. The irradiance distribution in the plane z = 30 mm in Fig. 5(a) is sufficiently uniform in a rectangular area with the size 400 × 200 mm . At the same time, the secondary light sources (the segments generated on the diffuse reflector) are still resolved in Fig. 5(a). The distribution in the plane z = 40 mm shown in Fig. 5(b) becomes more uniform and the line segments are not distinguishable anymore. Let us note that the distance between the segments along the Ox axis in Fig. 4(b) is also 40 mm. This fact is consistent with the results of [11], where it is shown that the greatest uniformity of illumination is achieved when the distance between the Lambertian sources is equal to the distance to the illuminated plane. Thus, varying the distance d between the base plane z = 0 and the exit plane one can obtain the irradiance distribution with required uniformity degree.
Fig. 5. Irradiance distributions in the planes z = 30 mm (a) and z = 40 mm (b) generated by a set of 10 optical elements (two linear arrays of 5 elements each) with a Lambertian reflecting surface (dashed curves show irradiance plots along the lines passing through the center of the image).
The considered configuration of the optical elements on a diffuse reflector can be considered as a fragment of a backlight or illumination system. Increasing (or decreasing) the number of elements, it is possible to illuminate a rectangular area of a given size with required degree of uniformity. 4. Conclusion
In the present work, a refractive optical element for uniform LED illumination of two line segments on the optical axis is proposed. The designed element is axially symmetric and easy to manufacture. The calculation of the element is reduced to the solution of a system of two explicit ordinary differential equations. The proposed optical element is promising for the applications in LED-based backlight and illumination systems. In this case the illuminated segments are generated on the surface of a diffuse reflector and act as secondary sources. The presented simulation results demonstrate the possibility of achieving highly uniform irradiance distribution in a prescribed rectangular area with lighting efficiency exceeding 83%. The proposed approach allows for several generalizations. First, a reflective optical element can be used in the system. Second, the utilized optical element can generate a line segment perpendicular to the optical axis or a curve of a given shape [12,13] instead of the line segment of the optical axis. This will provide the possibility to use different layouts of the optical elements and to control the shape of the illuminated area. Acknowledgments
This work was supported by the Ministry of Education and Science of Russia, Russian Foundation for Basic Research grants 13-07-97001, 12-07-13113 (joint grant with JSC Russian Railways), 12-07-31055, and 12-07-31193, and a Russian Federation State contract (agreement 8027).
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28656
Technical Cybernetics Department, Samara State Aerospace University, 151 Molodogvardeyskaya Street, Samara 443001, Russia 2 Image Processing Systems Institute of the Russian Academy of Sciences, 151 Molodogvardeyskaya Street, Samara 443001, Russia * [email protected]
Abstract: An LED optical element is proposed as an alternative to coldcathode fluorescent lamps. The optical element generates two symmetric uniformly illuminated line segments on the diffuse reflector. The illuminated segments then act as secondary linear light sources. The calculation of the optical element is reduced to the integration of the system of two explicit ordinary differential equations. The results of the simulation of an illumination system module consisting of a set of optical elements generating a set of line segments on the surface of the diffuse reflector are presented. The elements are located directly on the surface of the reflector. The simulation results demonstrate the uniform illumination of a rectangular area at a distance of 30–40 mm from the light source plane. The lighting efficiency of the designed system exceeds 83%. ©2013 Optical Society of America OCIS codes: (220.0220) Optical design and fabrication; (220.2945) Illumination design; (220.4298) Nonimaging optics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
C.-F. Lin, Y.-B. Fang, and P.-H. Yang, “Optimized micro-prism diffusion film for slim-type bottom-lit backlight units,” J. Disp. Technol. 7(1), 3–9 (2011). M. Doshi, R. Zane, and F. J. Azcondo, “Low frequency architecture for multi-lamp CCFL systems with capacitive ignition,” J. Disp. Technol. 5(5), 152–161 (2009). R.-S. Chang, J.-Z. Tsai, T.-Y. Li, and H.-L. Liao, “LED backlight module by lightguide-diffusive component,” J. Disp. Technol. 8(2), 79–86 (2012). Labsphere, “Labsphere: Spectralon Targets,” http://www.labsphere.com/products/reflectance-standards-andtargets/spectralon-reflectance-standards/default.aspx. A. Bhandari, B. Hamre, Ø. Frette, L. Zhao, J. J. Stamnes, and M. Kildemo, “Bidirectional reflectance distribution function of Spectralon white reflectance standard illuminated by incoherent unpolarized and plane-polarized light,” Appl. Opt. 50(16), 2431–2442 (2011). V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997). L. L. Doskolovich, N. L. Kazanskiy, V. A. Soifer, and A. Y. Tzaregorodtzev, “Analysis of quasiperiodic and geometric optical solutions of the problem of focusing into an axial segment,” Optik (Stuttg.) 101, 37–41 (1995). M. A. Moiseev and L. L. Doskolovich, “Design of TIR optics generating the prescribed irradiance distribution in the circle region,” J. Opt. Soc. Am. A 29(9), 1758–1763 (2012). Future Lighting Solutions, “Future Lighting Solutions: Remote Phosphor,” http://www.futurelightingsolutions.com/en/technologies/Pages/remote_phosphor.aspx. H.-T. Huang, Y.-P. Huang, and C.-C. Tsai, “Planar Lighting System Using Array of Blue Leds to Excite Yellow Remote Phosphor Film,” J. Disp. Technol. 7(1), 44–51 (2011). A. J.-W. Whang, Y.-Y. Chen, and Y.-T. Teng, “Designing uniform illumination systems by surface-tailored lens and configurations of LED arrays,” J. Disp. Technol. 5(3), 94–103 (2009). L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, P. Perlo, and S. Bernard, “Designing reflectors to generate a line-shaped directivity diagram,” J. Mod. Opt. 52, 1529–1536 (2005). L. L. Doskolovich, A. Y. Dmitriev, E. A. Bezus, and M. A. Moiseev, “Analytical design of freeform optical elements generating an arbitrary-shape curve,” Appl. Opt. 52(12), 2521–2526 (2013).
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28651
1. Introduction Currently, liquid-crystal displays (LCD) are commonly used in mobile devices, computer monitors and TVs. Thus, the development of compact and energy-efficient LCD backlight systems is a relevant task. Cold-cathode fluorescent lamps are widely used in the backlight systems of LCDs [1, 2]. In Fig. 1(a), typical configuration of a CCFL-based backlight system including the light sources, a diffuse reflector and a diffuser is shown.
Fig. 1. Configuration of a backlight system with fluorescent lamps (a); configuration of an LED-based backlight system module (b).
CCFL-based backlight systems can provide high uniformity of illumination, but have several disadvantages. Nowadays, LED-based lighting systems are considered to be more promising [3]. Compared with CCFLs, LEDs consume less energy, are more durable, resistant to vibration and shockproof. In the simplest case, the CCFLs are replaced by linear arrays of LEDs. In the present work, a system consisting of an LED with a special refractive optical element is proposed as an alternative to CCFL (Fig. 1(b)). The optical element mimics a CCFL and generates two symmetric uniformly illuminated line segments on the diffuse reflector. The illuminated segments then act as secondary light sources. It is worth mentioning that the diffuse reflector can be made from materials with high reflectivity (95–99%) and the scattering law close to Lambert's cosine law. One of the available materials of this type is spectralon manufactured by Labsphere Corporation [4, 5]. To achieve high lighting efficiency, it is proposed to use an optical element utilizing the total internal reflection effect. The designed optical element is axially symmetric (the axis of symmetry coincides with the illuminated segment). This ensures the ease of manufacture of the element. Thus, the set of CCFLs in Fig. 1(a) can be replaced by a set of LEDs with optical elements focusing to the line segments on the diffuse reflector. The proposed approach allows to create compact backlight and illumination systems of a new type. According to the simulation results presented below, the proposed approach allows to generate uniform irradiance distributions with large angular size (more than 150° in the direction along the segments) with high lighting efficiency of 83.5%. Let us note that the design of a refractive optical element for focusing into a line segment in the geometry depicted in Fig. 1(b) is for the first time presented in the present work. Previously, the elements of this type were calculated only using the approximation of a thin optical element [6, 7]. 2. Design of the optical element Let us consider the design of a refractive optical element focusing the radiation emitted by a point light source into two symmetrically located line segments of the Ox axis (Fig. 1(b)). Due to symmetry, it is sufficient to design only the right half of the element located at x > 0 (Fig. 2). The designed element has rotational symmetry, the rotation axis coinciding with the Ox axis. The surface of the element consists of the two parts: part a utilizing the total internal
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28652
reflection (TIR) effect, and part b refracting the light rays reflected from the a part of the surface. We assume that the angular size of the part b is relatively small and therefore neglect the rays that impinge on it directly.
Fig. 2. Optical element for focusing into a line segment on the Ox axis.
Let us denote by ρ (α ) the distance from the origin to a point on the a surface, where
α ∈ [0, α max ] is the angular coordinate of the incident ray (Fig. 2). In [8], the following
differential equation was obtained for ρ (α ) :
d ρ (α ) π 2 − α − β (α ) = ρ (α ) ⋅ cot , dα 2
(1)
where β (α ) describes the ray direction after the total internal reflection. From the geometric considerations (Fig. 2) one can obtain the coordinate of the intersection point of the outgoing ray and the Ox axis in the following form: x (α ) = x0 + ρ (α ) cos (α ) − ( x0 − ρ (α ) sin (α ) ) tg ( β (α ) ) γ (α ) n ⋅ sin ( β (α ) ) , (2)
where γ (α ) = 1 − n ⋅ sin ( β (α ) ) , n is the refractive index of the element material. To find 2
the function ρ (α ) describing the element surface, let us obtain another expression for x (α )
using the light flux conservation law. Let us introduce a spherical coordinate system (α , ϕ ) , where ϕ is the polar angle in the Oyz plane perpendicular to the plane of Fig. 2. Since the optical element is located at z > 0 , the polar angle takes values in the range [0, π ] . In these
coordinates, the solid angle element corresponding to the ϕ ∈ [ 0, π ] range can be expressed as
d Ω (α ) = π sin (π / 2 − α ) d α . To write the light flux conservation law, it is necessary to
equate the light flux incident on the element dx of the illuminated line segment to the light flux emitted by the light source into the solid angle element d Ω (α ) . Thus, the following equation can be written: π sin (π / 2 − α ) I (α , ϕ ) dϕ dα = E ( x ) dx, 0
(3)
where E ( x ) , x ∈ [ x1 , x2 ] is the desired irradiance distribution along the segment, and I (α , ϕ ) is the light source intensity. Let us note that for the irradiance E ( x ) to be correctly
defined, the following normalization condition should hold: α max
0
x2 π sin (π / 2 − α ) I (α , ϕ ) dϕ d α = E ( x ) dx. x1 0
(4)
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28653
Assuming a constant irradiance E ( x ) = E0 , x ∈ [ x1 , x2 ] and integrating Eq. (4), we obtain: x (α ) = x1 +
α π 1 cos α I (α , ϕ )dϕ dα . E0 0 0
(5)
In the case of a Lambertian light source, the source intensity in the spherical coordinates has the form I (α , ϕ ) = I 0 cos α sin ϕ .
(6)
Substituting Eq. (6) into Eq. (5), we obtain a closed-form expression for x (α ) : x (α ) = x1 + ( x2 − x1 )
2α + sin ( 2α ) . 2α max + sin ( 2α max )
(7)
Substituting Eq. (7) into Eq. (2) and differentiating the obtained equation by α , we get: d β (α ) =2 dα
n sin ( β (α ) ) x ′ (α ) γ (α ) + sec ( β (α ) ) γ (α ) sin (α − β (α ) ) ρ (α ) − cos (α − β (α ) ) ρ ′ (α ) , × n 2 sec ( β (α ) ) sin ( 2 β (α ) ) x0 sin ( β (α ) ) − cos (α − β (α ) ) ρ (α ) − 2γ (α ) g (α )
(8)
where g (α ) = x0 tan 2 ( β (α ) ) + cos (α ) cot ( β (α ) ) − sin (α ) tan 2 ( β (α ) ) ρ (α ) .
The derivative ρ ′ (α ) in the right-hand side of Eq. (8) is defined in Eq. (1). The derivative
x ′ (α ) can be obtained from Eq. (5) or Eq. (7). After the substitution of the expressions for
these derivatives into Eq. (8) the right-hand side depends only on the variable α and the functions β (α ) and ρ (α ) . Thus, the design of the optical element for generating a line segment with the desired irradiance is reduced to the solution of a system of two explicit ordinary differential Eqs. (1), (8). This system can be integrated using standard numerical methods (e.g. the Runge-Kutta method). Note that despite the large number of articles dedicated to design of axially-symmetric LED optics, the analytical design of optical element generating an axial line segment is for the first time presented in this work (Eqs. (1), (5), (7) and (8)). Within the considered approach, it is assumed that the designed optical element generates the illuminated segments on a diffuse reflector, which acts as a secondary light source. 3. Numerical simulation results
The developed method for the design of the optical element was implemented in MATLAB numerical computing environment. On the basis of Eqs. (1), (7), (8), an optical element was designed for the uniform illumination of two symmetrical line segments with a Lambertian light source. For the numerical integration of the system of differential Eqs. (1), (8), a standard MATLAB routine ode45 implementing variable-step fourth-order Runge-Kutta method was used. The calculation was performed using the following parameters: distance from the source to the element surface along the Oz axis ρ ( 0 ) = 6 mm , angular size of the a part of the profile (Fig. 2) α max = 50° , coordinates of the endpoints of the segment x1 = 50 mm , x2 = 150 mm , refractive index of the element material n = 1.493 . The designed optical element is shown in Fig. 3. The dimensions of the right half of the element along the Ox, Oy, and Oz coordinate axes are 17.6 mm, 29.4 mm and 14.7 mm, respectively. To check the correctness of the calculations, the designed element was simulated using a commercial optical engineering software TracePro. A LED chip EZ290 with Lambertian
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28654
radiation pattern and 290 × 290 μ m size (Cree Inc.) was used as the light source in the simulations. We assume that the designed optical element is intended for lighting systems employing the remote phosphor technology [9,10] and therefore simulate the blue LED chip instead of a packaged white LED. This technology supposes that the conversion into white light is performed by a remote phosphor layer (RPL) located at the output of a backlight system (Fig. 1(b), dotted line under the diffuser).
Fig. 3. Cross section (a) and wireframe model (b) of the optical element for the uniform illumination of two symmetrical line segments on the Ox axis.
Figure 4(a) shows the grayscale irradiance distribution generated by the optical element in the plane z = 0 containing the segments. The irradiance distribution was obtained by tracing 200 000 rays using the TracePro software and fully meets the design requirements (uniform illumination of two line segments with 100 mm lengths). The lighting efficiency of the optical element (the fraction of the emitted light flux coming to the plane z = 0 ) exceeds 83.5%. In Fig. 4(b), the irradiance distribution generated by a matrix of 10 optical elements (two linear arrays consisting of 5 elements each) is presented. The distance between elements in a linear array is 40 mm. The adjacent linear arrays are shifted by 20 mm along the Ox axis and by 100 mm along the Oy axis relatively to each other.
Fig. 4. (a) Irradiance distribution generated by the designed optical element for the uniform illumination of two line segments in the plane z = 0; (b) Irradiance distribution generated by two linear arrays consisting of 5 optical elements each.
For utilizing the considered optical elements in a backlight or illumination system, a diffuse reflector should be placed in the plane z = 0 . Let us note that direct backlight systems usually contain an optical diffuser which additionally improves spatial and angular uniformity of output light (Fig. 1). In particular, the RPL itself acts as a diffuser with a high haze value [9,10]. Nonetheless, the degree of uniformity of irradiance distribution generated by the system without diffuser is of great importance. It defines both the complexity of the required diffuser and the total thickness of the backlight system. Thus, it is important to study the uniformity of irradiance distribution generated by the matrix of optical elements (Fig. 4(b)) in a certain exit plane z = d over the optical elements (Fig. 2). We assume that the diffuse reflector scatters incident light according to the Lambert’s cosine law [4, 5]. In this case, the illuminated line segments generated in the plane z = 0 act as secondary Lambertian sources. The distributions shown in Fig. 5 were calculated for the considered set of optical elements (two linear arrays of 5 elements each) using TracePro software. 200 000 rays were traced for each light source (Cree EZ290 chip). Figure 5 #198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28655
demonstrates the possibility to obtain highly uniform irradiance distribution over a large area. The irradiance distribution in the plane z = 30 mm in Fig. 5(a) is sufficiently uniform in a rectangular area with the size 400 × 200 mm . At the same time, the secondary light sources (the segments generated on the diffuse reflector) are still resolved in Fig. 5(a). The distribution in the plane z = 40 mm shown in Fig. 5(b) becomes more uniform and the line segments are not distinguishable anymore. Let us note that the distance between the segments along the Ox axis in Fig. 4(b) is also 40 mm. This fact is consistent with the results of [11], where it is shown that the greatest uniformity of illumination is achieved when the distance between the Lambertian sources is equal to the distance to the illuminated plane. Thus, varying the distance d between the base plane z = 0 and the exit plane one can obtain the irradiance distribution with required uniformity degree.
Fig. 5. Irradiance distributions in the planes z = 30 mm (a) and z = 40 mm (b) generated by a set of 10 optical elements (two linear arrays of 5 elements each) with a Lambertian reflecting surface (dashed curves show irradiance plots along the lines passing through the center of the image).
The considered configuration of the optical elements on a diffuse reflector can be considered as a fragment of a backlight or illumination system. Increasing (or decreasing) the number of elements, it is possible to illuminate a rectangular area of a given size with required degree of uniformity. 4. Conclusion
In the present work, a refractive optical element for uniform LED illumination of two line segments on the optical axis is proposed. The designed element is axially symmetric and easy to manufacture. The calculation of the element is reduced to the solution of a system of two explicit ordinary differential equations. The proposed optical element is promising for the applications in LED-based backlight and illumination systems. In this case the illuminated segments are generated on the surface of a diffuse reflector and act as secondary sources. The presented simulation results demonstrate the possibility of achieving highly uniform irradiance distribution in a prescribed rectangular area with lighting efficiency exceeding 83%. The proposed approach allows for several generalizations. First, a reflective optical element can be used in the system. Second, the utilized optical element can generate a line segment perpendicular to the optical axis or a curve of a given shape [12,13] instead of the line segment of the optical axis. This will provide the possibility to use different layouts of the optical elements and to control the shape of the illuminated area. Acknowledgments
This work was supported by the Ministry of Education and Science of Russia, Russian Foundation for Basic Research grants 13-07-97001, 12-07-13113 (joint grant with JSC Russian Railways), 12-07-31055, and 12-07-31193, and a Russian Federation State contract (agreement 8027).
#198210 - $15.00 USD Received 24 Sep 2013; revised 25 Oct 2013; accepted 11 Nov 2013; published 14 Nov 2013 (C) 2013 OSA 18 November 2013 | Vol. 21, No. 23 | DOI:10.1364/OE.21.028651 | OPTICS EXPRESS 28656
