## Abstract

A simple and efficient solution for coupling a collimated light beam into a thin light guide is presented. The approach is based on two gratings, with their grating lines perpendicular to each other, fabricated into the opposite surfaces of the light guide. The presented numerical simulation shows that an optimized double-sided solution for unpolarized light enables around 2–7 times higher incoupling efficiencies than what is possible with conventional solution based on only one grating. Experimental verification is made by using UV-replicated binary gratings on both sides of a PMMA foil.

©2007 Optical Society of America

## 1. Introduction

Light coupling into light guides is one of the central issues in several mobile optical applications such as backlights and virtual displays [1, 2, 3, 4]. The light is typically fed through the edge of the light guide which gives a high incoupling efficiency especially when the dimensions of the light guide are larger than the dimensions of the light source [5, 6, 7].

However, in many applications it is more desirable to couple light into the light guide through the facet rather than through the edge. The simplest solution for such a purpose is to employ a single diffraction grating with its period suitable for incoupling [8, 9, 10]. This method is applicable as far as the lateral dimension of the grating is in optimal case less than 2*L* tan *θ*, where *L* is the thickness of light guide and *θ* is the angle of propagating beam. As a consequence, dimension can not be more than twice the light guide thickness when the propagation angle is less than 45 degree. When the diameter of the incoming light beam, which is directly related to the size of the grating coupler, is large compared to the thickness of the light guide, the incoupled light is reflected back from the backside of the light guide onto the grating area and hence it will be efficiently outcoupled. Thus, especially in the case of thin light guides, a simple grating coupler is not feasible. In some special cases it is possible to use demagnifying optics in order to reduce the size of the incoming light beam, but typically these kind of solutions add the number of components in applications and therefore diminish cost efficiency. Previously Siitonen *et al*.[11] used a radial grating geometry for non-collimated LED light incoupling and they were able to reduce the light guide thickness to 0.6 mm. This solution was recently extended to white light incoupling [12].

In this article we concentrate on a case in which the diameter of the light beam is 8–16 times larger than the thickness of the light guide.We shall show that in such a situation the incoupling efficiency can be greatly improved by using double-sided diffraction gratings. In the proposed solution the grating lines are perpendicularly oriented at the opposite sides of the light guide. The main idea of this incoupling method is to deflect the ray propagation angle inside the light guide, such that the outcoupling efficiency is reduced when the incoupled light ray is reflected back to the incoupling grating. In this way it is possible to trap more light inside thin light guides. A minor drawback of the proposed method is the change of the propagation angle of light inside of the light guide but, nevertheless, it can be accepted in numerous applications, especially in the enlightenment of backlights and keypads.

The paper is organized as follows: In section 2, we introduce the incoupling geometry and give the numerical simulation results for the grating lateral dimension/light guide thickness ratios of 8 and 16. In section 3 we give the experimental results with UV replicated gratings on both sides of a PMMA foil and show that the system provides also experimentally improved incoupling efficiencies.

## 2. Numerical simulations

The double-sided incoupler geometry with perpendicularly oriented binary gratings on both of the surfaces is shown in Fig. 1. In the figure, *L* is the thickness of the light guide, *d* is the grating period, *h* is the height of the grating and the filling factor of the grating *f* = *c*/*d*. The subscripts t and b denote the gratings at top and bottom surfaces, respectively.

The origin of the coordinate system is fixed in the middle of the lower grating. As an example,part of the propagating rays, generated by the incident ray hitting the element at position (*x*, *y*, *z*) = (0,0,*L*), are also drawn in the Fig. 1. The incoupled ray “1” hits the lower grating in a conical mounting and therefore it is totally reflected and simultaneously splitted into two rays by the grating. When the ray “2” hits the upper grating, a part of it is coupled out and the remaining part is splitted between the reflected diffraction orders. We point out that since the ray “2” reaches the upper grating in a direction other than the diffraction orders of the grating, the grating does not couple it efficiently out. The most of the further reflected rays, like rays “3”–“6”, are also coupled out inefficiently because they do not reach the top grating in the directions of diffraction orders. Finally the rays leave the grating region and will propagate inside the light guide purely by total internal reflections.

A light beam is coupled into the light guide by a grating if the propagation angle of the *m*th diffraction order (usually *m* = 1) is *θ _{m}* ≤

*θ*

_{tot}= arcsin(1/

*n*), where

*θ*is the angle between the light ray and the surface normal of the grating,

_{m}*θ*

_{tot}is the critical angle of total internal reflection, and

*n*is the refractive index of the light guide (in this article we will use

*n*= 1.49 corresponding to PMMA material, as well as the UV-curable resin used in the experimental part of this article, for the visible light).

In the tracing of the propagation of various rays inside the light guide we used the geometrical notation introduced in Ref. [13]: The propagation angle, i.e. the angle between the light beam and the *z*-axis and is defined as

where **k** = *k _{x}*

**x**+

*k*

_{y}**y**+

*k*

_{z}**z**is the wave vector and (

**x**,

**y**,

**z**) are the Cartesian unit vectors. The conical angle, i.e. the angle between the

*x*axis and the projection of the wave vector onto the (

*x*,

*y*) plane, is defined as

In order to evaluate the diffraction efficiencies of all of the rays, a rigorous diffraction
simulation based on Li’s article [14] was constructed. Naturally, correct Fourier-factorization rules [15] as well as stable *S*-matrix-based propagation algorithm [16] were employed in the implementation of the theory. Instead of typical input parameters (wavelength, angle of incidence, conical angle, and polarization angle) we used the wave vectors **k** and the corresponding vectorial complex amplitudes **E**= *E _{x}*

**x**+

*E*

_{y}**y**+

*E*

_{z}**z**of rays. This selection was done because we need to exactly know the polarization state of the diffracted rays with respect to the gratings. After the first reflection the rays are propagating in an oblique angle with respect the

*x*axis and their polarization states are elliptical, even though the input polarization is linear. The knowledge of polarization and propagation direction, given by the rigorous computations, was then used to solve the behavior of the rays at subsequent hits in the gratings.

Furthermore, for tracing purposes, a simple ray-tracing code was build in order to gather total input and output efficiencies and for storing the ray propagation information. The lateral shift of the ray in (*x*, *y*) plane during propagation can be easily solved by using the wave vector components as follows Δ*x* = *Lk _{x}*/

*k*and Δ

_{z}*y*=

*Lk*/

_{y}*k*.

_{z}When examining the energy distribution between the rays, two different efficiencies can be distinguished for each ray: the first one, denoted by *η*
_{g}, is the standard diffraction efficiency giving the portion of energy diffracted into the direction of the examined ray in one hit. On the other hand, the second one, which we shall call the ray efficiency and denote by *η*
_{r}, is the total portion of the energy carried by the ray, i.e. the energy is compared to the total input energy. Thus, for example, if we denote a ray efficiency of the ray “1” by *η*
_{r,1} and the diffraction
efficiency of the ray “2” by *η*
_{g,2}, the ray efficiency of the ray “2” is simply *η*
_{r,2} =*η*
_{r,1}
*η*
_{g,2}.

The incoupling efficiency *η*
_{ic} (i.e. the energy of light that propagates outside of the grating aperture compared to the input energy) can be calculated as follows:

where *R* is the sum of the efficiencies of outcoupled rays at *z* < −*L*, *T* is the sum of outcoupled efficiencies at *z*>*h*
_{b}, and *J* is a tracing loss factor. Owing to the nature of the problem, the number of rays is increasing exponentially as they are bouncing between the gratings. Therefore, we have to limit the amount of computation steps. We performed this by simply neglecting the rays whose efficiencies fall below certain threshold value. When the ray is neglected, its efficiency is summed to *J* and hence, in the end of computations, we know total computational loss. In reality, the energy in *J* is distributed between *η*
_{ic}, *R*, and *T*.

Since the optimization of the whole system would be a cumbersome and time consuming problem, we optimized the top and bottom surface gratings separately. The optimization of the top surface incoupling grating is a straightforward task—the maximization of the first diffraction order efficiency for the selected state of polarization of incoming ray is all what is needed. By assuming unpolarized illumination which can be evaluated as an average of the TE- and TM-contributions *η*
_{unpo}. = (*η*
_{TE} +*η*
_{TE})/2 and optimizing the grating for *n* = 1.49 and *θ* ±1 = 45° we obtained the following parameters: period *d _{t}* = 0.95λ

_{0}, where λ

_{0}is the assumed wavelength in vacuum, fill-factor

*f*=

_{t}*c*

_{t}/

*d*

_{t}= 0.3787, and depth

*h*

_{t}= 0.7673λ

_{0}. Note that when the conical angle φ is changed, also the angle

*θ*is altered. We decided to restrict to

*θ*< 55° which, in first reflection, fixes the grating period to

*d*

_{b}= 1.8λ

_{0}. The other grating parameters were then optimized to maximize the diffraction efficiency of the first reflection orders. The following parameters were obtained:

*f*

_{b}= 0.3435 and

*h*

_{b}= 0.2997λ

_{0}. It is noticeable that, according to our numerical simulations, the bottom surface grating is almost independent of polarization for all the propagating rays.

After the optimization of the gratings we calculated the incoupling efficiencies by using the above-introduced ray-tracing method. Owing to the symmetry of the problem, it is enough to consider just one quarter of the grating area. If we assume a PMMA light guide of *L* = 0.75 mm, vacuum wavelength λ_{0} = 632.8 nm, and that incident ray enters the element at (*x*, *y*, *z*) = (0,0,−0.75) mm, we need to consider the area 0 < *x* < 3 mm and 0 < *y* < 3 mm. Note that the thicknesses of the gratings are negligible compared to *L*. The rays propagating in that area are exactly the rays “1”–“6” illustrated in Fig. 1. The components of wave vectors of the rays, *k _{i}*, where

*i*= (

*x*,

*y*,

*z*), the electric-field (vectorial) complex amplitudes

*E*, and the diffraction efficiencies and ray efficiencies are given in Table 1 for TE polarized incoming light. Note that as the input wave vector is parallel to the

_{i}*z*direction, the terms TE and TM polarization must be understood as in grating theory, i.e. TE polarization means a field with the electric field parallel to the (top) grating lines. In addition, the angles

*θ*and φ are also given in Table 1. Naturally, the values of

*k*,

_{i}*θ*, and φ are quantized. By examining Table 1 and recalling that only one quarter of the element is considered, we notice that in this case the total incoupling efficiency

*η*

_{ic}is four times the sum of efficiencies carried by rays 4–6, i.e.

*η*

_{ic}= 56 %.

In the ray-tracing algorithm we decided to neglect rays whose efficiencies fall below 0.2 %. Although this threshold limit is rather low, these rays can still carry a remarkable amount of energy. In Fig. 2(a) the rays whose efficiencies are higher than 1%are drawn within the 6mm × 6mm grating region and, in Fig. 2(b) the rays whose efficiencies fall below 0.2%are illustrated. For clarity, the rays with efficiencies from 0.2 % to 1 % are not presented. In this case, the number of neglected rays is 45 and the total energy carried by them is *J* = 7.5 %. Based on the Fig. 2(b), most of the energy in *J* is coupled in *R* and *T* and therefore the incoupling efficiency *η*
_{ic} does not depend strongly on the threshold limit.

In the case of the real application the optimal size of the incoupling grating is equal to the incoming light beam size. Due to limiting factors, e.g., accuracy of alignment, thickness of light guide, size of light source etc., the the incoupling grating is reasonable to be larger than the size of the beam. In the ideal case the size of the bottom grating could be twice the thickness of the light guide smaller than the top side grating. In practice this does not give any benefit. We next turn to consider the incoupling efficiency as a function of the input ray position. Such a consideration is one of the central issues if the dimensions of the light source are several times the thickness of the element. It is clear that, in this kind of situation, the size of the gratings must be as large as the input beam. We first examine the case corresponding to the situation in which the ratio between the grating extension and the thickness of the light guide is 8:1. For this purpose we solved *η*
_{ic} as explained above for input ray positions (*x*, *y*) = (0…3mm,0…3mm) for *L* = 0.75 mm. The calculated incoupling efficiencies for TE and TM polarized light are shown in Fig. 3. It is remarkable that the incoupling efficiency for TE polarized light in Fig. 3(a) is 47–62 %, i.e. quite independent from the incoming ray position. The incoupling efficiencies for TM polarized light are much lower, only from 24 % to 29 % but, again, the efficiency does not depend much on the incoming ray position.We note that the given values are solved for the gratings which were optimized for unpolarized light. If the incoming beam is fully polarized, the coupling efficiencies could be improved by optimizing the gratings for the actual input polarization state.

To compare the results given above to the common solution of one-sided grating coupler, we optimized the local incoupling efficiencies also for such a type of an element. In this case we obtained the smallest incoupling efficiency, 17.7 %, for TE polarization when the ray is hitting in the middle of the grating, whereas the maximum value, 37.7 %, is reached at (*x*, *y*) = (3.0,0) mm. For TM polarized light we obtained 16.2 % and 19.8 %, respectively. We may conclude that, for TE polarized light, the new two-sided solution at least doubles the incoupling efficiency compared to common one-sided grating. For TM polarized light the incoupling efficiency is improved by a factor ∼ 1.5. All in all, we can almost double the incoupling efficiency for unpolarized light by using perpendicularly oriented, two sided gratings.

We also solved the incoupling efficiencies for *L* = 0.375 mm, i.e. for the ratio 16:1. In the computation we again assumed unpolarized incoming light and used the same, optimized, gratings as in previous example. The results are shown in Fig. 4. The achieved incoupling efficiencies are from 20 % to 50 % for TE-polarized light and from 13 % to 25 % for TM-polarized light. The corresponding incoupling efficiencies for one sided grating coupler are 2.0 % at (*x*, *y*) = (0,0) mm and 36.7 % at (*x*, *y*) = (3.0,0) mm for TE polarized light and, for TM polarized light, 8.5 % and 16.8 % at (*x*,*y*) = (0,0) mm and (*x*,*y*) = (3.0,0) mm, respectively. Thus, the incoupling efficiency is increased by factor 10 for TE polarized light and by factor more than two for TM-polarized light, and thus approximately by factor ∼ 7 for unpolarized light.

Our simulations thus show that the benefits of the two-sided solution become more and more obvious when the ratio between the grating extension and the thickness of the light guide is increases. This is particularly important in the cases often encountered in practice, as the dimensions of the light source are usually fixed and, at the same time, the thickness of the light guide is desired to be reduced.

## 3. Experimental verification

In order to verify the function of the double-sided incoupler, we fabricated two master gratings
of 6 mm × 6 mm with electron beam lithography into a SiO_{2} substrate. These master
gratings were fabricated with filling factor of 1 − *f* because the element will be inverted in the copying process. The fabrication process is described in Ref. [12]. With the SiO_{2} master gratings we made UV-copies in SK9 UV-curable resin on both side thin poly(methyl methacrylate) (PMMA) polymer foil. After hardening, we cut the copies and measured the grating parameters using a scanning electron microscope. We obtained the following values *d*
_{t,m} = 0.95λ_{0}, *f*
_{t,m} = 0.43, *h*
_{t,m} = 0.72λ_{0}, *d*
_{b,m} = 1.8λ_{0}, *f*
_{b,m} = 0.37, and *h*
_{b,m} = 0.35λ_{0} for λ_{0} = 632.8 nm. The cross section of the replicas are shown in Fig. 5. As a core of the light guide we used a 0.25 mm and 0.5 mm thick PMMA foils. Since we were not able to adjust the thickness of the SK9 layer very accurately, we measured the total thicknesses of the foils after copying. We obtained 0.38 mm and 0.63 mm for thinner and thicker foil, respectively. Thus, each of the SK9-layers were ∼ 65 *μ*m thick after curing.

In the measurements of the incoupling efficiency, we used a HeNe laser (λ_{0} = 632.8 nm) as a light source. Both the reflection *R*
_{m} and the transmission *T*
_{m} values were measured by using an integrating sphere, as illustrated in Fig. 6. After the measurement of *R*
_{m} and *T*
_{m} the incoupled efficiency *η*
_{ic,m} can be solved simply from *η*
_{ic,m} = 1−*R*
_{m} − *T*
_{m}. The thicker plastic foil illuminated by the laser beam is shown in Fig. 7. The incoupled beam propagation inside the light guide is clearly visible and four main propagation directions can be easily distinguished.

The measured incoupling efficiencies *η*
_{ic,m} are presented in Table 2, along with the corresponding computed values *η*
_{ic,c}. By examining the table, we notice that the measured values are higher than their theoretical counterparts. This can be explained by the factor *J*, i.e. the total efficiency of neglected rays in the simulation. Taking the effect of neglected rays into account, we can conclude that the theoretical values correspond fairly well to the experimental ones. We also measured efficiencies of one-sided grating coupler and these results are given in Table 3. Also in these results the essential correspondence between computed and measured results is noticed.

## 4. Conclusion

In conclusion, we have investigated the concept of two perpendicularly oriented gratings on the opposite sides of a thin light guide for incoupling of a collimated light beam. The purpose of the first (top surface) grating is to couple the incoming collimated beam into the thin light guide. Parametric optimization was used to design this grating with high incoupling efficiency for unpolarized light. The purpose of the second (bottom surface) grating is to deflect propagating rays such that they are coupled out from the light guide inefficiently by the first grating. The second grating was designed for this purpose by selecting suitable grating parameters based on numerical simulations.

The numerical analysis of the system was based on ray tracing in which the ray behavior on grating surfaces was solved rigorously. In numerical simulations we noticed that the incoupling efficiency mostly depends on the ratio between the grating extension and the thickness of the light guide. As an example we analyzed two different cases, namely, ratios 8:1 and 16:1, for unpolarized light. In the first case we obtained two times higher incoupling efficiency than what was possible with a similar, but one-sided, grating coupler. In the second case the incoupling efficiency was 6–7 times higher. Thus, our numerical simulations showed that by adding the second grating with perpendicular orientation, we can remarkably improve the coupling efficiency.

We also fabricated the designed gratings and copied them on both sides of thin plastic foils. We then measured the experimental coupling efficiencies and found them to be close to the numerical predictions.

The proposed system can be easily mass manufactured with UV replication and hot embossing methods. Therefore the two sided grating coupler is very attractive in mass products where thin light guides are needed.

We expect that the light incoupling efficiency may be improved by using slanted [8, 17] or double groove binary [10] gratings, especially with light guide materials with high refractive index. Furthermore, in addition to higher coupling efficiency, by using asymmetric gratings, the incoupled beam can be directed in just one direction along the light guide. The proposed idea can also be expanded for the white light incoupling and partly diverging light sources (LEDs) by careful design of the incoupling grating.

## Acknowledgments

The financial support of Finnish Funding Agency for Technology and Innovation (TEKES), Employment and Economic Development Centre for North Karelia, and the Academy of Finland (project 111701) is gratefully acknowledged. The authors acknowledge the Network of Excellence on Micro-Optics (NEMO), http://www.micro-optics.org.

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