## A double-sided grating coupler for thin light guides

Optics Express, Vol. 15, Issue 5, pp. 2008-2018 (2007)

http://dx.doi.org/10.1364/OE.15.002008

Acrobat PDF (340 KB)

### 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

1. I. Fujieda, “Theoretical considerations for arrayed waveguide display,” Appl. Opt. **41**, 1391–1399 (2002). [CrossRef] [PubMed]

2. M. Parikka, T. Kaikuranta, P. Laakkonen, J. Lautanen, J. Tervo, M. Honkanen, M. Kuittinen, and J. Turunen, “Deterministic diffractive diffusers for displays,” Appl. Opt. **40**, 2239–2246 (2001). [CrossRef]

3. T. Levola, “Diffractive optics for virtual reality displays,” J. Soc. Inf. Disp. **15**, 467–475 (2006). [CrossRef]

4. K. Chien and H. D. Shieh, “Time-multiplexed three-dimensional displays based on directional backlights with fast-switching liquid-crystal displays,” Appl. Opt. **45**, 3106–3110 (2006). [CrossRef] [PubMed]

5. K. Chien, H. D. Shieh, and H. Cornelissen, “Polarized backlight based on selective total internal reflection at microgrooves,” Appl. Opt. **43**, 4672–4676 (2004). [CrossRef] [PubMed]

6. K. Chien and H. D. Shieh, “Design and fabrication of an integrated polarized light guide for liquid-crystal-display illumination,” Appl. Opt. **43**, 1803–1834 (2004). [CrossRef]

7. S. Aoyama, A. Funamoto, and K. Imanaka, “Hybrid normal-reverse prism coupler for light-emitting diode backlight systems,” Appl. Opt. **45**, 7273–7278 (2006). [CrossRef] [PubMed]

## 2. Numerical simulations

*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.

*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.

*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).

13. E. Noponen and J. Turunen, “Eigen method for electromagnetic synthesis of diffractive elements with three-dimensional profile,” J. Opt. Soc. Am. A **11**, 2494–2502 (1994). [CrossRef]

*z*-axis and is defined as

**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

14. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. **40**, 553–573 (1993). [CrossRef]

15. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structure,” J. Opt. Soc. Am. A **13**, 1870–1876 (1996). [CrossRef]

*S*-matrix-based propagation algorithm [16

16. L. Li,“Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A **20**, 655–660 (2003). [CrossRef]

**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.

*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}*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}.

*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:

*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*.

*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*= 0.95λ

_{t}_{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.

*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*, where

_{i}*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 %.

*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.

*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.

*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.

*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.

## 3. Experimental verification

_{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

12. S. Siitonen, P. Laakkonen, P. Vahimaa, M. Kuittinen, and N. Tossavainen, “White LED light coupling into light guides with diffraction gratings,” Appl. Opt. **45**, 2623–2630 (2006). [CrossRef] [PubMed]

_{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.

_{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.

*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

8. M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, and E. Cambril, “Design and fabrication of slanted binary surface relief gratings for a planar optical interconnection,” Appl. Opt. **36**, 5717–5727 (1997). [CrossRef] [PubMed]

17. B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating couplers,” Opt. Express **12**, 3313–3326 (2004). [CrossRef] [PubMed]

10. P. Laakkonen, S. Siitonen, and M. Kuittinen, “Double-groove two-depth grating coupler for light guides,” J. Opt. Soc. Am. A **23**, 3156–3161 (2006). [CrossRef]

## Acknowledgments

## References and links

1. | I. Fujieda, “Theoretical considerations for arrayed waveguide display,” Appl. Opt. |

2. | M. Parikka, T. Kaikuranta, P. Laakkonen, J. Lautanen, J. Tervo, M. Honkanen, M. Kuittinen, and J. Turunen, “Deterministic diffractive diffusers for displays,” Appl. Opt. |

3. | T. Levola, “Diffractive optics for virtual reality displays,” J. Soc. Inf. Disp. |

4. | K. Chien and H. D. Shieh, “Time-multiplexed three-dimensional displays based on directional backlights with fast-switching liquid-crystal displays,” Appl. Opt. |

5. | K. Chien, H. D. Shieh, and H. Cornelissen, “Polarized backlight based on selective total internal reflection at microgrooves,” Appl. Opt. |

6. | K. Chien and H. D. Shieh, “Design and fabrication of an integrated polarized light guide for liquid-crystal-display illumination,” Appl. Opt. |

7. | S. Aoyama, A. Funamoto, and K. Imanaka, “Hybrid normal-reverse prism coupler for light-emitting diode backlight systems,” Appl. Opt. |

8. | M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, and E. Cambril, “Design and fabrication of slanted binary surface relief gratings for a planar optical interconnection,” Appl. Opt. |

9. | S. Wu, T. K. Gaylord, J. S. Maikisch, and E. N. Glytsis, “Optimization of anisotropically etched silicon surfacerelief gratings for substrate-mode optical interconnects,” Appl. Opt. |

10. | P. Laakkonen, S. Siitonen, and M. Kuittinen, “Double-groove two-depth grating coupler for light guides,” J. Opt. Soc. Am. A |

11. | S. Siitonen, P. Laakkonen, P. Vahimaa, K. Jefimovs, M. Kuittinen, M. Parikka, K. Mönkkönen, and A. Orpana,“Coupling of light from an LED into a thin light guide by diffractive gratings,” Appl. Opt. |

12. | S. Siitonen, P. Laakkonen, P. Vahimaa, M. Kuittinen, and N. Tossavainen, “White LED light coupling into light guides with diffraction gratings,” Appl. Opt. |

13. | E. Noponen and J. Turunen, “Eigen method for electromagnetic synthesis of diffractive elements with three-dimensional profile,” J. Opt. Soc. Am. A |

14. | L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. |

15. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structure,” J. Opt. Soc. Am. A |

16. | L. Li,“Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A |

17. | B. Wang, J. Jiang, and G. P. Nordin, “Compact slanted grating couplers,” Opt. Express |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(230.7400) Optical devices : Waveguides, slab

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 12, 2006

Revised Manuscript: January 24, 2007

Manuscript Accepted: January 29, 2007

Published: March 5, 2007

**Citation**

Samuli Siitonen, Pasi Laakkonen, Jani Tervo, and Markku Kuittinen, "A double-sided grating coupler for thin light guides," Opt. Express **15**, 2008-2018 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-5-2008

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### References

- I. Fujieda, "Theoretical considerations for arrayed waveguide display," Appl. Opt. 41, 1391-1399 (2002). [CrossRef] [PubMed]
- M. Parikka, T. Kaikuranta, P. Laakkonen, J. Lautanen, J. Tervo, M. Honkanen, M. Kuittinen, and J. Turunen, "Deterministic diffractive diffusers for displays," Appl. Opt. 40, 2239-2246 (2001). [CrossRef]
- T. Levola, "Diffractive optics for virtual reality displays," J. Soc. Inf. Disp. 15, 467-475 (2006). [CrossRef]
- K. Chien and H. D. Shieh, "Time-multiplexed three-dimensional displays based on directional backlights with fast-switching liquid-crystal displays," Appl. Opt. 45, 3106-3110 (2006). [CrossRef] [PubMed]
- K. Chien, H. D. Shieh, and H. Cornelissen, "Polarized backlight based on selective total internal reflection at microgrooves," Appl. Opt. 43, 4672-4676 (2004). [CrossRef] [PubMed]
- K. Chien and H. D. Shieh, "Design and fabrication of an integrated polarized light guide for liquid-crystal-display illumination," Appl. Opt. 43, 1803-1834 (2004). [CrossRef]
- S. Aoyama, A. Funamoto, and K. Imanaka, "Hybrid normal-reverse prism coupler for light-emitting diode backlight systems," Appl. Opt. 45, 7273-7278 (2006). [CrossRef] [PubMed]
- M. Miller, N. de Beaucoudrey, P. Chavel, J. Turunen, and E. Cambril, "Design and fabrication of slanted binary surface relief gratings for a planar optical interconnection," Appl. Opt. 36, 5717-5727 (1997). [CrossRef] [PubMed]
- S. Wu, T. K. Gaylord, J. S. Maikisch, and E. N. Glytsis, "Optimization of anisotropically etched silicon surface relief gratings for substrate-mode optical interconnects," Appl. Opt. 45, 15-21 (2006). [CrossRef] [PubMed]
- P. Laakkonen, S. Siitonen, and M. Kuittinen, "Double-groove two-depth grating coupler for light guides," J. Opt. Soc. Am. A 23, 3156-3161 (2006). [CrossRef]
- S. Siitonen, P. Laakkonen, P. Vahimaa, K. Jefimovs, M. Kuittinen, M. Parikka, K. M¨onkk¨onen, and A. Orpana, "Coupling of light from an LED into a thin light guide by diffractive gratings," Appl. Opt. 43, 5631-5636 (2004). [CrossRef] [PubMed]
- S. Siitonen, P. Laakkonen, P. Vahimaa, M. Kuittinen, and N. Tossavainen, "White LED light coupling into light guides with diffraction gratings," Appl. Opt. 45, 2623-2630 (2006). [CrossRef] [PubMed]
- E. Noponen and J. Turunen, "Eigen method for electromagnetic synthesis of diffractive elements with three dimensional profile," J. Opt. Soc. Am. A 11, 2494-2502 (1994). [CrossRef]
- L. Li, "A modal analysis of lamellar diffraction gratings in conical mountings," J. Mod. Opt. 40, 553-573 (1993). [CrossRef]
- L. Li, "Use of Fourier series in the analysis of discontinuous periodic structure," J. Opt. Soc. Am. A 13, 1870-1876 (1996). [CrossRef]
- L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003). [CrossRef]
- B. Wang, J. Jiang, and G. P. Nordin, "Compact slanted grating couplers," Opt. Express 12, 3313-3326 (2004). [CrossRef] [PubMed]

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