TRENDS IN OPTICS AND PHOTONICS SERIES Vol.90
Investigation of the gradient index conical lens for increased angular tolerance in free space optical interconnections D. Song, M. Sanchez, M. Gross, and S. Esener UC San Diego, Dept. of ECE, 9500 Gilman Dr., La Jolla, CA 92093-0407 Tel.: (858) 822 1295, FAX: (858) 534 1225, email:
[email protected]
Abstract: A method of using gradient index conical lenses (GRIN-CL) to increase the angular tolerance of free space optical interconnects is presented. Simulation results show that GRIN-CL has 3 times higher angular tolerances than simple microlenses.
©2003 Optical Society of America
OCIS codes: (080.3620) Lens design; (220.1770) Concentrators
1. Introduction In a typical free space optical interconnect (FSOI) system [1][2] the light beams at the detector surface have both lateral and angular misalignments because of misalignments at different stages, including light source, microlens array, relay lens, etc. These misalignments will decrease the detection efficiency and increase crosstalk between adjacent channels, degrading the signal to noise ratio (SNR) and increasing the bit error rate (BER). Generally, a microlens is used in front of the detector to increase the lateral tolerance of the light beams. However, the use of a microlens also decreases the angular tolerance from 180 degree to a finite number [3]. This angular tolerance is directly proportional to the diameter of the detector and inversely proportional to the focal length of the microlens. For a focal length of 1000 µm (which is the lowest practical limit of a singlet lens, assuming the microlens a diameter of 500 µm) and detector diameter of 100 µm (which is the upper limit for high speed detector), the acceptance angle is about 3 deg. For some applications such as board to board FSOI, this tolerance might be too small. In this paper, a method to improve the angular tolerance by using a gradient index conical lens (GRIN-CL) is presented, as well as its simulation method in commercial optical design software. Also we will discuss the simulation results of GRIN-CL with different shapes and index profiles. 2. Method of using GRIN-CL to increase tolerance 2.1 Introduction of GRIN-CL A GRIN-CL is a lens with a tapered shape and a gradient refractive index. Several studies have addressed ray propagation in normal conical GRIN lenses, especially with Gaussian, or parabolic index profiles [4][5]. For normal conical GRIN lenses with a parabolic index profile, the ray trajectories can be obtained analytically by solving the eikonal equation [4]. The solution shows that the behavior of a ray in a normal conical lens is a sinusoidal in radius, decreasing as the ray moves toward the conical tip [4]. Conical lenses can focus light while maintaining a larger acceptance angle than a simple lens, because of the gradient index and total inner reflection at the side walls. Thus if we employ a conical lens array before detectors instead of a microlens array, the large acceptance angle may help to increase the angular misalignment tolerance of the optical system. 2.2 Simulation models for GRIN-CL Based on current commercial optical simulation software, GRIN lenses with a conical shape cannot be simulated directly, but several models can provide an approximate simulation. The three models are: sequential surfaces model (SS model, as in Fig. 1a), nested cones model (NC model, as in Fig. 1b), and segmented nested cones model (SNC model, as in Fig. 1c).
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TRENDS IN OPTICS AND PHOTONICS SERIES Vol.90
a)
b)
c)
Fig. 1 Three available models for GRIN-CL a) Sequential surface model (SS model) b) nested cones model (NC model) c) segmented nested cones model (SNC model)
The SS model works very well for paraxial beam propagations, but has difficulty in dealing with reflections on the side boundary. The NC model is suitable to side boundary reflection, but only for a normal cone shape. The SNC model is an improved NC model. Since each segment has the property of NC model, it has no difficulty in dealing with reflections at the boundary. Now that the whole conical lens consists of several segments, the shape profile can be implemented by designating different cone radii for different sections. 2.3 Simulation procedure In order to build a GRIN-CL in CODE V as accurately as possible, a large number of surfaces are used. Generally the surface number is several hundred. Due to the fact that these surfaces follow some common rules, we used a helper program (written in Visual C++) to generate these surfaces in CODE V. After this model is built in CODE V, tolerance tests can be implemented. 3. Simulation Results The purpose of our simulations is to investigate the acceptance angles of GRIN-CL with different refractive index profiles and shape profiles for various beam diameters. The parameters for the GRIN-CL are as following. The basic refractive index is n0 = 1.50, which is a practical value for commercial plastic. For Gaussian index, the refractive index change from the center to the boundary is ∆n = 0.05 , which is possible for most GRIN lens fabrication. The cone length is 5mm, and the diameter of the entry and exit faces are 0.5mm and 0.1mm respectively. 3.1 Comparison of solid index profile and Gaussian index profile The transmission efficiency, defined as the percentage of incident intensity which is directed to the detector surface, is used to compare the different structures. The efficiency with different normalized diameters and at different incident angles are plotted in Fig.2, where the normalized beam diameter is defined as the ratio of the diameter of the incident beam to that of the lens entrance. 1.0
Efficiency
0.8 0.6 Normalized beam diameters , and : 0.1 , and : 0.3 , and : 0.5 , and : 0.7 , and : 0.9
0.4 0.2 0.0 0
5
10
15
20
Incident angle (Deg.) Fig. 2 Efficiency of exit beams with various normalized beam diameters for solid and Gaussian indexes Solid lines: solid refractive index, dashed lines: Gaussian refractive index
It is seen that for both uniform and Gaussian refractive index profiles and a given beam diameter, the efficiency drops with increasing incident angles. From the results we can see that the efficiency of Gaussian index profile lens drops more slowly than uniform index profile lens. Specifically, when the beam diameter is 0.5 (250µm), for a threshold of 80%, the acceptance angle for the Gaussian index lens is more than 15° and more than 10° for the solid index lens. In either case the acceptance angle is much larger than that with a simple lens, which is about 3° as mentioned earlier. So we can conclude here that the Gaussian index profile has better performance than solid index profile in tolerance increasing. 3.2 Comparison of different shape profiles 138
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Since the Gaussian index profile has a better performance than uniform index profile, we further investigated the shape effects for Gaussian index profile. Three typical shapes are explored: normal, concave, and convex cones. The normal cone has straight side walls, while the convex and concave cones are given an outward or inward curvature at the side walls. As shown in Fig. 3, the concave shape has the worst performance, at both 10 and 15 degree incident angles, while the convex and normal conical lenses have similar performance. Although the convex cone shape performance marginally better than the normal cone shape, the normal cone shape may be more practical because it is much easier to fabricate while has almost the same performance. 1.0
Efficiency
0.9 0.8 0.7 0.6
Normalized beam diameters , , and : 0.3 , , and : 0.5 , , and : 0.7 , , and : 0.9
0.5 0
5
10
15
Incident angle (Deg.) Fig. 3 Efficiency of exit beams with various normalized beam diameters for three cone shapes Solid lines: normal cone shape, dotted lines: convex cone shape, dashed lines: concave cone shape
4. Conclusion By using practical models to simulate GRIN-CLs, we verified that the GRIN-CL has a much larger angular misalignment tolerance than a simple lens in front of detectors in FSOI. The shapes and refractive index profiles both have effects on the acceptance angles of a GRIN-CL. A Gaussian index profile has a better transmission efficiency than a solid index profile, and a convex shape provides the best performance. Both convex and normal cones are better than a concave cone shape. For a FSOI system, a Gaussian index profile GRIN-CL can provide a 15 degree angular tolerance for a threshold of 50%, and a 10 degree tolerance for a threshold of 80%; In the same system, a simple lens can only provide about 3 degrees of angular tolerance. Thus the GRIN-CL is an attractive alternative to a simple microlens. 5. Reference: 1. 2. 3. 4. 5.
M. Ayliffe, and D. Plant, “On the design of misalignment-tolerant free-space optical interconnects”, SPIE, Vol. 4089, pp905-916, 2000 F. Lacroix, et al, “Experimental and numerical analyses of misalignment tolerances in free-space optical interconnects”, Applied optics, Vol.39, No. 5, pp704-713, 2000 D. Tsang, “Alignment and performance tradeoffs for free-space optical interconnections”, Optical computing 1989 Technical digest series, Vol.9, Postconference Edition, p146-149, 1989 A. Safaai-Jazi, and V. Suppanitchakij, “A tapered graded-index lens: analysis of transmission properties and applications in fiber-optic communication systems”, IEEE journal of quantum electronics, Vol.33, No.12, p2159-2166, 1997 O. Latry, et al, “Optimization of the coupling between a tapered fibre and a p-i-n photodiode”, J. Phys. D: Appl. Phys. Vol.28, p15621572, 1995
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