Function (PSF) for describing the performance of the GRIN planar structure and the ... Optical interconnects provide a solution for high speed interconnects ...
Design of reconfigurable GRIN planar optical interconnects C.Gomez-Reino, M.T.Flores-Arias, M.V.Perez, C.Bao, A.Castelo and D.Nieto GRIN Optics Group, Department of Applied Physics, Faculty of Physics and Optics and Optometry School, University of Santiago de Compostela, E15782 Santiago de Compostela, Spain.
ABSTRACT Design of all-optics reconfigurable GRIN (Gradient-Index) planar structure for crossover and parallel interconnects will be presented. Design represents a unique combination of GRIN materials, simple geometry optics and waveguide technology for both parallel and distributed processing and communication networks. The optical analysis is based onaxis and off-axis multiple imaging property of GRIN components. The analysis includes the study of the Point Spread Function (PSF) for describing the performance of the GRIN planar structure and the evaluation of the Space Bandwidth Product (SBP) for estimating the number of channels which can be handled. The dependence of the number of channels on the wavelength of the light and the aperture of the planar interconnect is shown. The results are given for five working wavelengths of Laser Diode (LD) and for four transverse aperture of reconfigurable optical interconnect. Keywords: Gradient-index components, optical interconnect, multiple image formation, channels, optical switching, prototypes, selfoc micro-lenses, optical waveguides.
1. INTRODUCTION. Planar optics is a concept for the microintegration of free-space components upon the surfaces of a single glass substrate in a compact and robust manner1. It can be implemented with diffractive and refractive microelements. The light signal travels within the homogeneous substrate from one element to the next one, along a folded path reflected at the surfaces of the substrate. Optical interconnects provide a solution for high speed interconnects between chips or fiber systems to solve the inherent limits of the existing electrical interconnects in operating speed, dense packaging and power dissipation2-4. Optical interconnects are also an important application of planar integrated systems and by using on-axis and off-axis imaging properties of planar integrated microoptics, parallel and crossover interconnects can be implemented5-14. For optical interconnects, the planar system requires a large SBP (space-bandwidth product) for space invariant operations employing the same connection15-17. The SBP, as a figure of merit, provides an efficient way to estimate the number of spatial channels that can be supported by substrate with interconnection purposes. Likewise, PSF is commonly used to describe the performance of the space-invariant system18-19 and it permits one to analyze the imaging properties of the planar integrated system to be applied in parallel and crossover optical interconnect. Optical operations in homogeneous planar systems are carried out by microcomponents integrated upon glass substrates as above mentioned. The light transmission fidelity through these systems is influenced by the noise coming mainly from diffractive optical elements (DOEs) that can be fabricated with either analogical or digital technique in order to get greyscale, multilevel and binary structures19. The most DOEs are currently produced with a binary etching process, resulting in less than ideal shapes and the mismatch of the fabrication and the inherent discretisation gives rise to most of the noise diffracted optical power. However, inhomogeneous planar systems can be used for implementing optical operation, without micro-components integrated upon substrates, taking advantage of inherent imaging and transforming properties from their GRIN nature but to reduce also the background noise. Hemispherical-rod microlenses were used for optical interconnections20 and as fractional Fourier transformers21. Recently, PSF and SBP in GRIN planar optics have been studied for designing crossover interconnects by selfoc semi-rod lenses 22. This work is
Micro-Optics 2008, edited by Hugo Thienpont, Peter Van Daele, Jürgen Mohr, Mohammad R. Taghizadeh, Proc. of SPIE Vol. 6992, 699211, (2008) · 0277-786X/08/$18 · doi: 10.1117/12.779505
Proc. of SPIE Vol. 6992 699211-1 2008 SPIE Digital Library -- Subscriber Archive Copy
organized as follows: the starting point is a short study on light propagation through GRIN media in which we deal with multiple image transmission. After that, a study on a GRIN planar structure and its application in parallel and crossover interconnections for designing all-optic reconfigurable interconnects will be presented.
2. PLANAR MULTIPLE IMAGING IN GRIN OPTICS: PSF As a prototype for designing optical interconnects in planar optics we consider a GRIN substrate, limited by perfect flat top, that can be regarded as a hemicylinder of radius a that was made by cutting a selfoc rod lens along the longitudinal axis (fig.1a). The refractive index of the selfoc semi-rod lens is given by 23-24
n 2 (r ) = n 02 [1 − g 2 r 2 ] ; r = (x 2 + y 2 )
0 ≤ x ≤ a ; − a ≤ y ≤ +a
where n0 is the index at the z axis for which the transverse parabolic refractive index profile is a maximum, g is a constant gradient parameter and a the radius of the semi-rod lens. For the GRIN structure, a light source located at any point (0,y0) on the surface is imaged on points of this surface if the imaging condition 24,
H a (z m ) = 0
is fulfilled at lengths
(m = 1,2,3,...)
where Ha is the axial ray defined by Luneburg to facilitate the analysis of paraxial light propagation in GRIN materials25. At zm, the PSF is represented mathematically as the 2D Dirac delta function24
y0 ⎞ ⎛ ⎟ δ⎜⎜ x , y − H f (z m ) ⎟⎠ ⎝
H f (z m ) = (− 1)
where the transverse magnification is given by m
for selfoc materials, being Hf the field ray25. From eq.(5) it follows that the GRIN planar substrate makes, on the flat top, images of the input signal with unit magnification represented by δ(x,y±y0) at lengths given by eq.(4). Spatial inversion between successive images is achieved. The figure 1b depicts the ray trajectory for imaging in GRIN planar optics. The GRIN substrate replaces the lenses for imaging that are used in homogeneous planar optics. Every part of the GRIN substrate of length π/g acts a refractive lens. Therefore, an input signal placed upon a surface of the GRIN planar structure is imaged through it onto the surface without use of any other imaging lens, e.g., DOEs integrated on the surface of a slab homogeneous substrate are not required, providing a performance advantage at inhomogeneous case.
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tIg 0) Fig. 1. a) GRIN planar substrate and b) ray trajectory in the GRIN planar substrate
Fig. 2. Photographs of planar optical imaging in selfoc semi-rod lenses: a) on-axis and ) and c) off-axis multiple imaging.
Figure 2 shows three photographs of planar optical imaging in selfoc semi-rod lenses. The GRIN planar configuration was made by using four commercially available GRIN rod lenses of 0.25 pitch. The diameter and the length of each selfoc lens were 2.0mm and 5.17mm, respectively. The gradient parameter of lenses was 0.304 mm-1 and the on-axis refractive index was 1.608 at 632.8nm. The four lenses were embedded in a glass block and the successive lenses are
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glued with an optical epoxy. The selfoc rod lenses are polished until a hemicylindrical GRIN configuration is obtained. A He-Ne laser beam was coupled to the GRIN substrate by a right angle prism of 1.785 refractive index. Figure 2a shows planar on-axis multiple imaging and figures 2b and 2c depict off-axis multiple imaging. In figures 2b and 2c, the images of the input spot are shifted up or down from the axis with unit magnification due to the spatial inversion between the successive images. The multiple imaging in GRIN planar optics offers the potential for a single dual implementation of crossover and parallel optical interconnects. The sequence crossover/parallel interconnect is repeated along the planar substrate, and, for instance, the first stage of the network configuration serves as crossover interconnect and the second stage provides the parallel interconnect and it could be implemented optically by placing a movable transverse mirror on the output ports of the first stage in order to reflect the images to the output ports of the second stage providing reconfigurability to the photonic planar component (fig.1b). Thus, if the idea of GRIN planar optical imaging is extended to optical interconnections, the GRIN structure of figure 1 can be applied for crossover as well as parallel interconnects of unit magnification.
3. SPATIAL BANDWIDTH PRODUCT We now estimate the maximum number of data channels that can be implemented in parallel and crossover interconnects with a GRIN planar structure. The number of spatial channels in optical interconnects is limited by diffraction determining the minimum size of the resolvable spot or pixels of the signal. In order to estimate the number of channels we use as a figure of merit the SBP of the diffraction-limited planar imaging system. The SBP is the number of pixels that are transformed by the planar system and it is given by the product of the number of resolvable spots Ny and Nz in the two dimensions of the planar system15-17, 19,22, that is
SBP = N y N z =
Wy Wz w ywz
where Wy and Wz are the finite extent of the input or output signals along y and z and wy and wz are the image spot size in x and y, respectively. The simple imaging for one-to-one connection, in a space-invariant optical system such as the GRIN substrate, is described mathematically as the convolution of light complex amplitude at the input with the PSF. The same operation is performed on each pixel in the input, so the number of types of operations, or, equivalently, the number of different PSFs on each pixel is one16,26. The SBP is then the same for any stage of multiple image formation in sequence. This explains why the successive stages, corresponding to crossover and parallel interconnects, preserve identical values for the SBP of both interconnects. The planar optics integration yield geometrical constraints since the light originating from the input signal must not overlap with the light of the output signal along longitudinal axis after one sinusoidal propagation of period π/g and the input and output signals must be limited by the transverse size of the GRIN substrate. Both constraints can be written as
Wy ≤ 2a
Wz ≤ π g
Likewise, the size wx and wy of the diffraction-limited image spot can be evaluated as the separation between +1st and 1st minima of the diffraction pattern produced by a rectangular pupil of Wy and Wz dimensions taking into account the inherent imaging capability due to the GRIN nature. The number of resolvable spots along y and z axes can be represented, in compact form, as
n 0 gW⎛2y ⎞ N⎛ y⎞ = ⎜⎜ z ⎟⎟ ⎝ ⎠
⎜⎜ ⎟⎟ ⎝z⎠
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With this in mind, the SBP is expressed as22
⎡ n 0 gWy2 Wz2 cos θ ⎤ SBP = ⎢ ⎥ 2πλ ⎣ ⎦
The optimum value of SBP of the GRIN structure for optical interconnections is achieved as the number of resolvable spots in the y and z directions Ny and Nz are maximized and they are determined from the geometrical constraints to give
⎛ πn sin θ max max SBP opt = N max = ⎜⎜ 0 y Nz 2gλ ⎝
N max = y
⎞ ⎟⎟ ⎠
πn 0 tan θ max sin θ max 2gλ
πn 0 cos θ max 2gλ
N max = z
θmax being the maximal angle for interconnection (fig.3)
π/g C Fig.3 Maximal angle θmax for crossover (C) and parallel (P) interconnects
Eqs.(11-13) can be written in terms of the transverse aperture 2a of GRIN substrate as follows
⎛ πn 0 a =⎜ ⎜ λ π 2 + ( 2ag) 2 ⎝
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⎞ ⎟ ⎟ ⎠
N max = y
N max = z
2gn 0 a 2 λ π 2 + (2ag) 2 π2n 0
2gλ π 2 + (2ag) 2
In eqs.(9) and (12) there is an explicit dependence of the optimum value of SBP on the wavelength as well as an implicit dependence by means of n0 and g. Of particular interest, is the wavelength dependence of the GRIN planar interconnects. This is important, since, owing to telecommunications window, the wavelength of the sources operating in optical interconnection vary. Two effects are of important: first, a change of the wavelength means a different diffraction scale and then different SBP. The second effect is the variation of the length of the planar interconnects, since the period π/g of the sinusoidal propagation of light inside GRIN structure has also wavelength dependence.
4. RESULTS Figures 4 and 5 depict optimum SBP and maximum number of resolvable spots in the transverse direction y versus transverse aperture and wavelength. Results are given for five working wavelengths 850, 1310, 1490, 1550 and 1625nm of LD (Laser Diode) and for four transverse apertures from 1 to 4 mm of the optical interconnector. For a given λ, the optimum SBP (fig 4a) and the number of resolvable spots along the y direction (fig. 5a) increase with 2a. On the contrary, for a given 2a, SBPopt (fig 4b) and Nymax(fig 5b) decrease with λ. Sufficiently large SBP is achieved for interconnection purposes and for small transverse apertures. In this context, the GRIN planar optics for crossover and parallel interconnects can be designed with a selfoc semi-rod lens. For that, the value of on-axis refractive index n0 and gradient parameter g are taken from dispersion equations for a typical SLW selfoc microlens fabricated by Nippon Sheet Glass Co.27. In particular, for 850nm and for a selfoc semirod-lens of 4mm diameter, a crossover and parallel network configuration of 719×719 channels can be designed for 20.9 and 41.7 mm lengths of semi-rod lens, respectively. However, for the same wavelength and for a semi-rod of 1 mm diameter, the number of crossover and parallel channels reduces to 177×177 for 5.2 and 10.4 mm lengths of semi-rod lens. In the same way, for the highest wavelength of 1625nm and for 4mm diameter of semi-rod, a planar crossover and parallel interconnection configuration between 367 inputs to 367 outputs is obtained with semi-rod lengths of 21.3 and 42.6 mm, respectively. For 1625nm and for a semirod of 1 mm diameter, the number of crossover and parallel channels becomes 91×91 for 5.3 and 10.6 mm lengths of semi-rod, respectively. Finally, for specific applications in which it may be necessary to perform crossover and parallel interconnects of variable magnification a hemycilinder GRIN substrate with axial variation of the gradient parameter can be used. Depending on the functional form of g(z), different magnifications can be obtained. The scaling factor can be evaluated by the field ray. Likewise, for higher maximal tilt angles and thick GRIN substrates, the chromatic and offaxis aberrations are competing with diffraction effects and both subjects must be considered for the estimation of SBP.
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SBPOP 108 15 —*— A=BSOnm