Design and Analysis of Free-Space Optical Interconnects

Transcription

1 Design and Analysis of Free-Space Optical Interconnects By Eng-Swee Goh School of Information Technology and Electrical Engineering The University of Queensland Brisbane, Australia Submitted for the Degree of Bachelor of Engineering (Honours) In the Division of Electrical Engineering May 2003

2 i 23 rd of May 2003 Eng Swee Goh 3/63 Warren St St Lucia, Qld 4067 Australia The Dean School of Information Technology and Electrical Engineering The University of Queensland St Lucia, QLD 4072 Dear Sir, In accordance with the requirements of the degree of Bachelor of Engineering (Honours) in the division of Electrical Engineering, I hereby present the following thesis entitled Design and Analysis of Free-Space Optical Interconnects. This work was performed under the supervision of Dr. Aleksandar Rakic. I declare that the work submitted in this thesis is my own, except as acknowledged in the text, and has not been previously submitted for a degree at the University of Queensland or any other institutions. Yours sincerely, Eng-Swee Goh

3 ii Acknowledgements I would like to express my gratitude to my supervisor, Dr. Aleksandar Rakic, for his dedication, guidance, advice and encouragement that have contributed to the completion of this thesis. Special thanks to Feng-Chuan Frank Tsai and James Lindley as well for their previous work in this area and providing a starting point for my research, especially Frank for his endless support and guidance to providing valuable information during my course of work. Finally, I would like to acknowledge my family for their full financial and moral support throughout my study at The University of Queensland.

4 iii Abstract The continual development of more complex and efficient communication and computing systems has increased the demand of faster routing of information between elements such as gates, chips, and boards. With electronic interconnects fast approaching their limits, optical techniques seemed to be the best viable alternative. Free-Space Optical Interconnects (FSOIs) seemed to be able to solve the present data transfer bottleneck in high-speed computers and provide a key to the creation of a new architecture for the next generation of information processing systems. The aim of this thesis is to design and analyse different FSOI architectures using commercial optical design software Code V. Through the software, the analysis such as the effects of aberration on different lenses and optical signal-to-noise ratio (OSNR) at various interconnection distances are performed. One of the proposed interconnect solutions incorporate the use of macroscopic optics which provides a large field of view to image the entire emitter array, and the use of micro optics provides a small f-number to reduce the source divergence. This combination of both macroscopic and micro optics is known as Hybrid system. Aberration could be reduced or eliminated by improving the system geometry and steering the beams across the optical axis at the centre of the optical system. The introduction of 45 prisms can be used to fold the beams for board to board interconnections within a bay in hybrid designs. Two different macroscopic stock lenses were compared as relay macrolens in FSOI design. It was found that both designs had the highest encircled energy at a distance of 56 mm using a receiver size of 40 µm semi-aperture Finally, optical crosstalk, a factor affecting channel OSNR and channel density has been found to be higher in the centre channel than the edge channels in a three by three VCSEL array through diffraction analysis in Code V. Results have also revealed that by designing a symmetric model with the location of the effective system aperture at the centre of the optical system, distortion could be eliminated.

5 iv Acronyms Each acronyms is defined the 1 st time it appears in a chapter, so the reader does not have to search the entire text to find its meaning. As a further help, the author lists all the acronyms here in alphabetical order. BPR BFL CMOS EFL FFL FFT FOV FSOI LSA MSM OSNR PSF TSA VCSEL Diffraction-Based Beam Propagation Back Focal Length Complementary Metal-Oxide Semiconductors Effective Focal Length Front Focal Length Fast Fourier Transform Field of View Free Space Optical Interconnect Longitudinal Spherical Aberration Metal-Semiconductor-Metal Optical Signal-to-Noise Ratio Point Spread Function Transverse Spherical Aberration Vertical Cavity Surface Emitting Laser

6 v Contents Letter to the Dean i Acknowledgements...ii Abstract...iii Acronyms...iv List of Figures...viii List of Tables...xi Introduction Comparison between Optical and Electrical Interconnects Main Features of FSOI Current Developments in FSOI Technology Objective of Thesis Thesis Organization... 5 Optical theory Cardinal (Gauss) Points of an Optical System Paraxial Lens Formulas Geometrical Optics Ray Tracing Geometrical Third-order Aberrations Spherical Aberration Coma Astigmatism Field Curvature Distortion Diffraction Theory Characteristic of Gaussian Beams Code V Simulation Tool Beam Propagation Method System performance through Gaussian Beam Analysis... 20

7 vi Discussion and Analysis of Macroscopic lens FSOI Design using Bi-convex Macroscopic Lens Aberration Analysis Using Diffraction-Based Gaussian Beam Analysis FSOI Design using Plano-convex Macroscopic Lens Aberration Analysis Using Diffraction-Based Gaussian Beam Analysis Analysis of Off-Axis Performance of Macroscopic Lens General Design of Hybrid Systems Off-axis Channel Geometry Optical Crosstalk Analysis Design Optimisation Adjustment of Design Space for Bi-convex Lens Adjustment of Design Space for Plano-convex Lens Analysis of Hybrid Systems using Bi-convex Lens Aberration Analysis Using Diffraction-based propagation of Gaussian Beam Analysis of Hybrid Systems using Plano-convex Lens Aberration Analysis Using Diffraction-Based Propagation of Gaussian Beam Board to Board Level Free-Space Optical Interconnects Hybrid Systems with Folded Geometry Adjustment of Design Space for Minilens Aberration Analysis Using diffraction-based propagation of Gaussian Beam Gaussian Beam profile pattern Design Comparisons...56 Conclusions and Recommendations...58

8 vii 7.1 Conclusions Recommendation for future work Appendix A Code V simulated results...62 A.1 Biconvex and Plano-convex Macroscopic Lens A.2 Hybrid Systems using Bi-convex Macroscopic Lens A.3 Hybrid Systems using Plano-convex Macroscopic Lens A.4 Hybrid Systems with Folded Geometry Appendix B Code V Lens Data Manager...66 B1 Hybrid Systems using Bi-convex Macroscopic Lens B2 Hybrid Systems using Plano-convex Macroscopic Lens B3 Hybrid Systems with Folded Geometry Appendix C...68 C.1 BK 7 Glass Specification References...69

9 viii List of Figures Figure 1-1: Aspect ratio limit of electrical interconnects... 1 Figure 1-2: Basic block diagram of FSOI... 3 Figure 1-3: Illustration of free-space optical link in a shelf to shelf and board to board configuration... 4 Figure 1-4: Three by three array optical system... 4 Figure 2-1: Illustration of cardinal points of a generalized optical system... 6 Figure 2-2: Nodal points... 7 Figure 2-3: Surface sagittal and radius of curvature... 8 Figure 2-4: Ray trace determining the front and back focus distance... 8 Figure 2-5: Diagram of spherical aberration Figure 2-7: Focal properties of an astigmatic lens Figure 2-8: Field curvature by an optical system Figure 2-9: Different distortion effects of a symmetric object Figure 2-10: Gaussian intensity distribution Figure 2-11: Variation of Gaussian beam diameter Figure 2-12: Structure of Code V Figure 2-13: Graph of calculated encircled energy versus simulated encircled energy. 20 Figure 3-1: General layout for macroscopic lens analysis Figure 3-2: Bi-convex macrolens ray tracing plot Figure 3-3: Field curvature and distortion plot for bi-convex macrolens Figure 3-4: Distortion grid for bi-convex macrolens Figure 3-5: OSNR versus interconnection distance for (KBX013) Figure 3-6: Plano-convex macrolens ray tracing plot Figure 3-7: Field curvature and distortion plot for plano-convex macrolens Figure 3-8: Distortion grid plot for plano-convex macrolens Figure 3-9: OSNR versus interconnection distance for (KPX013) Figure 3-10: Off-axis performance of bi-convex and plano-convex lens Figure 4-1: Illustration of basic hybrid system 30 Figure 4-2: Off-axis beam propagation geometry Figure 4-3: Off-axis beam ray tracing.31

10 ix Figure 4-4: Graphical structure of three by three VCSEL lens array Figure 4-5: Diffraction noise signal affecting channel Figure 4-6: Illustration of hybrid relay macrolens spacing optimisation Figure 4-7: Performance of bi-convex (KBX013) lens at different focal length Figure 4-8: Performance of plano-convex (KPX013) lens at different focal length Figure 4-9: Layout of optical system for hybrid system using bi-convex lens Figure 4-10: Field curvature for bi-convex lens hybrid system Figure 4-11: Distortion grid for bi-convex lens hybrid system Figure 4-12: Graph of encircled energy versus interconnection distance of hybrid System using bi-convex lens Figure 4-13: OSNR versus interconnection distance for different channels of hybrid system using bi-convex macrolens Figure 4-14: Layout of optical system for hybrid system using plano-convex lens Figure 4-15: Field curvature for plano-convex lens hybrid system Figure 4-16: Distortion grid for plano-convex lens hybrid system Figure 4-17: Graph of encircled energy versus interconnection distance for hybrid system using plano-convex macrolens Figure 4-18: OSNR versus interconnection distance for different channels in hybrid system using plano-convex macrolens Figure 5-1: Diagram of board to board Level FSOI Figure 5-2: Layout of optical system for adjustment of minilens focal length Figure 5-3: Performance of minilens at different focal length Figure 5-5: Distortion grid for board to board design Figure 5-6: Graph of encircled energy versus interconnection distance for board to board design Figure 5-7: OSNR versus interconnection distance for different channels Figure 5-8: Beam intensity stretches Figure 5-9: Beam intensity leaving prism Figure 5-10: Diagram of Hybrid Systems with Folded Geometry Figure 5-11: Beam intensity at microlens surface Figure 5-12: Beam intnesity at minilens..52 Figure 5-13: Diagram of beam falling on the 45 prism surface Figure 5-14: Beam intensity diagram at the receiver for 3 µm beam half-waist... 54

11 x Figure 5-15: Point spread function. A two-dimensional plot of energy with approximately 64 µm spot on the detector Figure B-1: Code V lens data manager for hybrid systems using bi-convex macroscopic lens Figure B-2: Code V lens data manager for hybrid systems using plano-convex macroscopic lens Figure B-3: Code V lens data manager for hybrid systems with folded geometry... 67

12 xi List of Tables Table 3-1: Bi-convex lens (KBX013) parameters Table 3-2: Plano-convex lens (KPX013) Parameters Table 4-1: Parameters of hybrid design using bi-convex macroscopic lens Table 4-2: Parameters of hybrid design using plano-convex macroscopic lens Table 5-1: Lens array specifications Table 6-1: Comparison of bi-convex versus plano-convex lens performance Table 6-2: Comparison of hybrid Systems using different macroscopic lens Table 6-3: Summary of board to board interconnection design performance Table A-1: Numerical results for OSNR versus Interconnection distance Table A-2: Numerical results for OSNR versus Interconnection distance Table A-4: Numerical results for OSNR versus Interconnection distance Table A-5: Numerical results for diffraction noise analysis Table A-6: Numerical results for OSNR versus Interconnection distance Table A-7: Numerical results for diffraction noise analysis of folded geometry Table C-1: Specifications of BK 7 Glass lens... 68

13 Chapter 1 Introduction 1 Chapter 1 Introduction With the growing complexity of current communication and computing systems, the need for more efficient and faster routing of information between elements such as gates, chips, boards, and networks becomes more crucial than ever. In relation, electrical interconnects are rapidly advancing to their limits, thus, it is suggested that optical techniques would be a viable alternative [1-6]. Free-space optical interconnects (FSOI) have been the subject of much research due to their great potential for use as high bandwidth interconnects over short distances such as chip to chip or shelf to shelf levels. However, a key issue that needs to be addressed is the integration of these devices with free-space optics into a system environment. 1.1 Comparison between Optical and Electrical Interconnects The rapid growth of interconnection problems is mainly due to the fast amassing of data channels and bandwidth in modern digital machines such as telecommunication switching systems and computers. Traditionally, most interconnects are made from metallic wires and tracks which are referred as the electrical interconnects [6-8]. These conventional electrical lines possess capacitance that results in very substantial problems of signal distortion at high modulation frequencies. As a result, high frequency carriers are never used. Using a relatively general formula for characterizing the numbers of bits per second, B, that can be sent down an electrical interconnect given the inter-symbol interference from the frequency- dependent loss and distortion. l A Figure 1-1: Aspect ratio limit of electrical interconnects [7]

14 Chapter 1 Introduction 2 This limit is set by the ratio of the length l to the cross-sectional area A. This ratio is called the aspect ratio. The limit relates to the above ratio can be approximated by B ~ B 0.A / l 2 bits per second, where B 0 is a constant (with units bits/s) characterizing the electrical line. At high frequency this will certainly become a problem as machines approach Tb/s information bandwidth. Though the usage of different techniques might enable the exceeding of the limit, its mere existence pose a big disadvantage of electrical interconnects. Optical interconnects on the other hand simply do not have this aspect ratio problem when it comes to implementing high-speed communication links. The second difference between optics and electronics is the wavelength corresponding to the frequency of the signal. Electronics generally have a large wavelength compared with the cross-sectional size of the wiring that route the signals within the system. This effect causes multiple interconnecting beams to not be able to focus on different points on the chip. By contrast, optics could simultaneously image multiple sources on a plane to many receivers onto another plane. The fundamental reason that makes this possible is due to the short wavelength of light even with relatively simple optics [5]. Another advantage of optical interconnects is that they provide a possibility of getting rid of the guiding medium all together, thus producing the so-called free-space optical interconnects (FSOI), hence, making this technology worth looking into, as an alternative way of implementing short-distance digital communication links. 1.2 Main Features of FSOI FSOI use the imaging properties of lenses rather than the guided-wave technique used by fibres and waveguides to control the propagation of signals. The basic FSOI design consists of two main components - the transmitter plane on one side and the receiver plane on the opposite side. The middle channel consists of the optical system that directs the light beam from the source to the designated detector as illustrated in Figure 1-2.

15 Chapter 1 Introduction 3 Optical system Source Detector Figure 1-2: Basic block diagram of FSOI FSOI modules perform one-to-one imaging between the VCSEL at the transmitter and the detector arrays at the receiver. Different types of optical systems can be used to achieve this, including diffractive or refractive microlens arrays or hybrid systems that will be introduce in later sections. 1.3 Current Developments in FSOI Technology Recent developments in the integration of large Vertical Cavity Surface Emitting Laser (VCSEL) arrays on Complementary Metal-Oxide Semiconductors (CMOS) have made it the preferred choice for the transmitter of optical interconnects [9,11,12]. In addition, VCSEL can produce a single longitudinal mode due to their short cavity length, low light beam divergence (around 15-deg) and high efficiency low-threshold lasers leading to low power consumption with an operating voltage of less than 2 volts [6]. As for potential receivers, different detectors such as Metal-Semiconductor-Metal (MSM) and Back-Side-Illuminated Double-Pass Schottky Photodiodes are being used in the design of FSOI [8-10]. At present, microlens, minilens and macroscopic lens are being tested for use in wouldbe optical systems such as micro-channel and hybrid designs. Prisms have also been utilised to bend the laser beams for the design for shelf to shelf or board to board levels as illustrated in Figure 1-3 [13-15].

16 Chapter 1 Introduction 4 Figure 1-3: Illustration of free-space optical link in a shelf to shelf and board to board configuration [15] In order to increase the capacity of a system, more channels have to be clustered within an optical system. However, since a single lens images each channel, having more channels clustered together effectuates problems of aberration, diffraction and misalignment. Figure 1-4 shows a typical three by three array optical system. Transmitter plane Receiver Plane Microlens Figure 1-4: Three by three array optical system [16] 1.4 Objective of Thesis The primary objective of the thesis is to look into hybrid optical models and analyse their system performance and effect of diffraction and aberration issues on free-space optical interconnect designs using the commercial simulation software Code V. The secondary objective is to determine the performance of plano-convex lens and biconvex lens. Through the use of the simulation program, a more realistic understanding of the optical design can be known before manufacturing the design.

17 Chapter 1 Introduction Thesis Organization Chapter 1 presents a brief introduction of the current free-space optical interconnects technology and a quick overview of the thesis. Chapter 2 introduces the important theoretical aspects of optical system design, background theory of Geometrical optics and also the characteristic of Gaussian beams that emits from the source are explained. Chapter 3 analyses bi-convex and plano-convex macroscopic lenses. These two lenses are investigated for their performance in aberrations and beam propagation analysis for use promising FSOI system. Chapter 4 discussed the various issues that are taken into account for the general designs of hybrid systems. Methods used to examine the system performance are presented. One of the design optimisation methods to improve the overall design is look into. Finally bi-convex and plano-convex macroscopic relay lenses are used in different hybrid systems and their performance in the designs and investigated based on aberration and beam propagation analysis. Chapter 5 examines a symmetric hybrid system with the insertion of prisms to reorient the beam from one plane to another separate plane for board to board level FSOI designs. Beam intensity pattern of different surface is also investigated in this design. Chapter 6 shows the outcomes of different FSOI designs and the analysis are discussed through a comparison table to evaluate the results. Chapter 7 concludes the thesis and highlights recommendations for future work.

18 Chapter 2 Optical Theory 6 Chapter 2 Optical theory The process of designing a lens system could be broken into three parts. The first being the requirements on construction parameters that ensure the image fall in the proper location and contain aberrations within an acceptable level. The second part deals with the need to use the lens in a defined environment. The last part specifies the acceptable irregularity and randomness that can be allowed on the surface of the lens to control both aberration and scattered light. This chapter will briefly explore the Cardinal points of an optical system, Geometrical optics and characteristics of Gaussian beam that needs to be considered in the design and analysis process of lens system. 2.1 Cardinal (Gauss) Points of an Optical System A well-corrected optical system can be likened to a black box with characteristics defined by its cardinal points consisting of the first and second focal points, its first and second principal points and its first and second nodal points. Figure 2-1: Illustration of cardinal points of a generalized optical system [22]

19 Chapter 2 Optical Theory 7 From Figure 2-1 the focal points are points on the axis where light rays parallel to optical axis are brought to a common focus. In the paraxial region, when the incident and emerged rays are extended till they intersect, the plane defined by the points of intersection would be called the principal plane. Points where the principal plane intersects the optical axis are known as principal points. The distance from the principal point to the focal point of a system is the effective focal length (EFL) and the back focal length (BFL) is the distance from the vertex of the last surface of the system to the second focal point. Front focal length (FFL) is the distance from the front surface to the first focal point. These are clearly shown in Figure 2-1[21]. Figure 2-2 shows the first nodal point H (also known as the primary principal point) and the secondary nodal point H" (also known as the secondary principal point) defined as the intersection of the optical axis with the primary and secondary principal surfaces respectively [21] Paraxial Lens Formulas Figure 2-2: Nodal points [21] This section presents the fundamental equations for lenses in air. The following formulas are valid for both thick and thin lens [24]. The focal length, f of any lens is ( n 1) tc = ( n 1) + (1) f r1 r2 n r1 r2 where n is the refractive index, t c is the centre thickness and r 1 and r 2 are the radius of curvature of each side of the lens. For thin lens, t c =0 and for plano-convex lens, either r 1 or r 2 is infinite.

20 Chapter 2 Optical Theory 8 Figure 2-3: Surface sagittal and radius of curvature [21] From Figure 2-3 the surface sagittal and radius of curvature of any lens is given by s = r 2 d 2 r ( ) > 0 2 (2) 2 s d r = + 2 8s (3) where s is the sag height (lens thickness), r is the radius of curvature and d is the diameter of the lens. The radius of curvature could also be found for plano-convex lens using the focal length, f and the refractive index, n as given [21]. r = f ( n 1) (4) Figure 2-4: Ray trace determining the front and back focus distance [22]

21 Chapter 2 Optical Theory 9 Many optical systems consist of two separate components as in Figure 2-4. The following expressions could be used to handle both FFL and BFL denoted by B of all two-component systems. FFL BFL f ( f d) ab b = (5) f f a b ( f d) ab a = (6) f where f a and f b are the reciprocal focal length of the components. f ab is the effective focal length and d is the separation distance; if the lens is thick, d is the separation of their principal points [22]. 2.2 Geometrical Optics Geometrical optical models define the properties of image formation by the lens and relate them to the construction parameters of the lens. Geometrical optics is based upon the fundamental assumption that light propagates along rays. Each ray passing through an optical system follows the shortest path between the object space and image space. A basic set of propositions of which Geometrical optics is based on, are listed below [19]. Rays are normal to the wavefronts and vice versa. Rays satisfy the laws of reflection and refraction. The optical path lengths along any rays between two wavefronts are equal. The optical path is stationary with respect to the variables that specify it. The irradiance at any point is proportional to the ray intensity at that point. Fermat s principal also govern that Light travels along a straight line in a uniform medium. Optical path from point A to A 1 can also travel reversible from A 1 to A. The optical path difference between two neighbouring rays is zero.

22 Chapter 2 Optical Theory Ray Tracing Ray tracing is a technique used to solve problems in optical applications. Data from ray tracing could be used to compute the geometrical path of light through lens and define the aberration performance of the image in any optical system. Through this method, information such as clear apertures, radius of curvature of lens, tolerances and image quality could also be examined. Local coordinates associated with each surface in the lens are frequently used in ray tracing. The rays are traced between the coordinate systems with the positive direction for light propagation being from left to right [20]. By describing the lens of an optical system in Code V, it computes the tracing of rays from the object surface to the image surface in order. The only drawback of ray tracing is its inability to model diffraction. 2.3 Geometrical Third-order Aberrations Aberration occurs in real systems where the angles of incidence are larger than the paraxial approximation. This causes the rays not to converge to a single point image [20]. Aberration could be summarized into five different types, they are Spherical aberration, Coma, Astigmatism, Field curvature and Distortion. With a well-chosen combination of optical parameters such as lens, shapes and different optical materials, these aberrations can be minimized to the level of diffraction limit. In the following sections, these aberrations will be further elaborated.

23 Chapter 2 Optical Theory Spherical Aberration Figure 2-5: Diagram of spherical aberration [21] For a spherical lens, the further the entering rays are from the optical axis, the nearer the focus would be to the lens (crosses the optical axis), as shown in the Figure 2-5 above. The distance along the optical axis between the intercept of the rays that are nearly on the optical axis and the rays that go through the edge of the lens is known as Longitudinal Spherical Aberration (LSA). The height at which these rays intercept the focal plane is called Transverse Spherical Aberration (TSA). Spherical aberration is dependent on the lens shape, orientation, and the index of refraction of the material. Generally, positive lenses are likely to have under-corrected spherical aberration while negative lenses usually have over-corrected spherical aberration. A combination of positive lens made from low-index glass and negative lens made from high-index glass, makes it possible for the spherical aberrations to cancel out [21].

24 Chapter 2 Optical Theory Coma Different degrees of magnification are exhibited at different parts of the spherical lens surface, giving rise to an aberration known as coma. Figure 2-6 below, each concentric zone of a lens forms a ring-shaped image called a comatic circle. This results in blurring in the image plane of off-axis object points. Figure 2-6: Coma pattern having different magnification zones [21] Astigmatism Astigmatism occurs when a spherical lens focuses an off-axis object. Illustrated in Figure 2-7 the plane containing both the optical axis and the object point is called the tangential plane and rays that lie in this plane are called tangential rays. The chief rays passes from the object point through the centre aperture of the lens system. Sagittal plane is also perpendicular to the tangential plane that contains the chief ray. An astigmatic image is formed when tangential rays comes to a focus at the tangential plane from the object than do rays in the sagittal plane. The amount of astigmatism in a lens depends on lens shape.

25 Chapter 2 Optical Theory 13 Figure 2-7: Focal properties of an astigmatic lens [21] Field Curvature There is a tendency of optical systems to image better on curved surfaces than on flat planes even in the absence of astigmatism. This phenomenon is known as field curvature as illustrated in Figure 2-8. In the presence of astigmatism, this problem is compounded, as there are two separate astigmatic focal surfaces. Figure 2-8: Field curvature by an optical system [21] Positive lens elements will have inward curving fields, and negative lenses have outward curving fields. Thus, field curvature can be corrected to some extent through a combination of positive and negative lens elements. Flat-field lenses are lenses with virtually no field curvature [21].

26 Chapter 2 Optical Theory Distortion In any lens system, an image field may not only have curvature but also be distorted. The image of an off-axis point may be formed at a location on the surface other than that predicted by the simple paraxial equations. This distortion is different from coma where rays from an off-axis point fail to meet perfectly in the image plane. Distortion means that the location on the image plane is not correct even if a perfect off-axis point image is formed. The effect of this can be seen in Figure 2-9 below. Figure 2-9: Different distortion effects of a symmetric object [21] Comparison of the shape of image and object plane can be done using the distortion-free object grid with the positive or pincushion distortion and the negative or barrel distortion grid. It should be noted that distortion does not lower system resolution. It simply means that the image shape does not match exactly to the shape of the object.

27 Chapter 2 Optical Theory Diffraction Theory Diffraction is a phenomena or effect resulting from the interaction of light with the sharp limiting edge or aperture of an optical system. The key process that must be considered here is derived from the diffraction integral obtained by using the Huygens- Fresnel approach, which leads to a diffraction integral [20]: i Ae U ( P) = λ R ikr e ik ( Φ+ s) s dx p dy p (7) In this integral, the amplitude, of the wave disturbance U at the point P is given by integration over a reference sphere that fills the exit pupil with coordinates (x p, y p ). Phase error in the pupil given by Φ, which is a function of x p and y p, the coordinates of the pupil. The distance s is the distance from the pupil coordinates to the location on the image surface where the amplitude is computed. The distance R is the radius of the reference sphere from the pupil to the image reference location. This integral can be divided into two regions, the Fraunhofer region and the Fresnel region. The distinction of these two is determined by the rate of change of phase over the aperture with distance of the computation point from the reference point. Fraunhofer diffraction is considered far field diffraction in that a linear approximation to the relative phase change with the propagation distance R separation between points on the image surface is adequate. 2 r R >> (8) 4πλ Fresnel diffraction or near field is used when R does not meet the above requirement. The geometry of the diffracting aperture relative to the point of observation is such that a higher-order change at least quadratic must be considered [20].

28 Chapter 2 Optical Theory Characteristic of Gaussian Beams Some basic understanding of the characteristic of laser beams is required in order to understand the effects of diffraction in Gaussian beams. Propagation of Gaussian beams through an optical system can be treated almost as simply as geometric optics. Due to the unique self-fourier transform characteristic of the Gaussian we do not need an integral to describe the evolution of the intensity profile with distance. Figure 2-10: Gaussian intensity distribution [18] The beam emitted from a laser in TEM 00 shown in Figure 2-10, has a perfect plane wavefront at its beam waist position and the Gaussian irradiance profile that varies radially from the axis and along a line perpendicular to the direction of propagation and through the centre of the beam can be described by 2 2d 2 / d1 I ( d ) = I e o (9) where I 0 is the axial irradiance of the beam, d 1 is the diameter of the beam where 2 irradiance is 1/ e I 0, or 13.5% of its maximum intensity value on the beam axis [18].

29 Chapter 2 Optical Theory 17 Z Figure 2-11: Variation of Gaussian beam diameter [26] The transverse distribution intensity remains Gaussian at every point in the system with only the radius of the Gaussian and the radius of curvature of the wavefront change. Looking at Figure 2-11, as the distance along axis of propagation from beam waist (located at z=0) increases, the beam radius w 0 grows according to equation (11). The beam size and wavefront curvature will then vary with z. The beam size will increase, slowly at first, then faster, eventually increasing proportionally to z. The beam radius increases from its smallest value, known as the beam waist, w 0. The equation to work out beam radius is given in equation (10) also known as Rayleigh range below [24]. ( λ0z = ) (10) π 2 0 ω0 As the beam travels along the z-axis, it is possible to express the wavefront radius of curvature R(z) given in equation 11 and Gaussian beam radius w(z), in equation (12) as functions of z [24]. 2 z 0 R ( z) = z 1 + (11) z 1/ z w ( z) = w (12) 0 z

30 Chapter 2 Optical Theory 18 The amount of encircled energy that falls on the detector from the beam emitted from a laser with the lowest order mode TEM 00 could also be calculated using the equation below [11]: S 2 b = 1 exp 2w 00 2 where b denotes the diameter of the detector size, w is the beam radius along z plane. (13) In free-space between lenses or other optical elements, the position of the beam waist and the waist diameter (or Rayleigh range) completely describes the beam. When a beam passes through a lens, the diameter is unchanged but the wavefront curvature changes, resulting in a new value of waist position and waist diameter on the output side of interface [24]. In this thesis, only Gaussian beam is used to model diffraction in an optical system analysis. 2.6 Code V Simulation Tool Figure 2-12: Structure of Code V [25] Figure 2-12 shows the structure of Code V. This software is structured around a lens database that includes all the data defining your lens, including radii, thicknesses, glass data, apertures and much more. This program originated as a ray-tracing program, most of the analysis is based in geometrical optics such as field curvature and distortion grid.

31 Chapter 2 Optical Theory 19 However there is other analysis available such as Diffraction-Based Beam Propagation (BPR) that is mainly going to be used in this thesis for analysis of FSOI designs. Gaussian beam propagation comprise of calculating the beam size at each surface in the optical system. From these calculations, it is possible to determine the amount of optical energy that will be coupled on each surface Beam Propagation Method In code V three beam propagators are used namely, Fraunhofer, Fresnel and Sphere to Sphere propagators [25]. The selection of the propagator used in the analysis could be done manually or automatically. The computational tool used for all the diffraction propagators is implemented using a standard Fast Fourier Transform (FFT) algorithm. A propagator operates on the complex amplitude of the field distribution and produces the complex amplitude of the field distribution at a different position. The intensity is the square of the complex amplitude. The FFT operates on the square grid, producing another square grid with same number of points but different grid spacing. In BPR, it is the Fresnel number that determines the type of propagator used at each surface. The Fraunhofer, or far field, propagator is the Fourier transform of the complex amplitude of the field distribution with the possible inclusion of a quadratic phase term for focusing. It is used in regions where the Fresnel number is very small such as when a collimated beam is propagated a large distance or a converging beam is propagated to a focus. The Fresnel, or near field, propagator has a three-step process. In the first step, the field distribution is Fourier transformed. This decomposes the field distribution into an angular spectrum of plane waves, which is equivalent to the far field pattern. The second step is to propagate each plane wave to the plane of interest. Since a plane wave does not change its form as it propagates, a phase proportional to the propagation distance is added. There is a different phase value for each plane wave since each propagates at different angles. The third step is to combine the plane waves by inverse Fourier transforming the angular spectrum.

32 Chapter 2 Optical Theory 20 Lastly, sphere to sphere propagator only differs from the near field propagator as the distance is modified to take into account of spherical reference frame. The field distribution is propagated between two concentric spheres. The range of surfaces for which sphere to sphere propagation has been automatically selected by CODE V or manually specified will defines the location of the spheres [25]. 2.7 System Performance through Gaussian Beam Analysis Channel density proves to be a crucial factor for the performance of FSOI system. Hence, system optical signal-to-noise ratio (OSNR) is largely dependent on channel capacity. In this thesis, only optical properties of interconnection distance are looked into. A Gaussian beam is used to propagate through free-space from the source to analyse the amount of encircled energy received at the detector. In all the encircled energy analysis in this thesis, the temperature was set at 20º C and the wavelength is 850 nm with a beam waist of 3 µm. Using equation (13) in section 2.5, the encircled energy of the Gaussian beam could be calculated as interconnection distance increases. Figure 2-13 shows that the simulated encircled energy in Code V is similar to the calculated results. ` Encircled energy (%) Encircled energy vs interconnection distance Calculated Simulated Interconnection distance (mm) Figure 2-13: Graph of calculated encircled energy versus simulated encircled energy

33 Chapter 3 Discussion and Analysis of Macroscopic Lens 21 Chapter 3 Discussion and Analysis of Macroscopic lens In this section, off-the-shelf optics, or so call stock lens was analysed and the performance was evaluated. In considering stock lens for FSOI, the advantages are the ease of getting the required lens in a shorter time and lower cost issues comparing to custom-built optics. In this following section, two different spherical lenses planoconvex and biconvex macrolens from Newport Resource catalogue 2003 will be analysed and the results presented. Macrolens VCSEL Receiver Figure 3-1: General layout for macroscopic lens analysis Figure 3-1 shows the general layout used in the analysis of different macroscopic lens. Through aberrations analysis and BPR the performance of these two lenses are examined. This analysis incorporates a 1 x 3 array of small aperture VCSEL s with a 250 µm pitch operating at a wavelength of 850 nm with a beam waist of 3 µm. Three optical channels at 0 mm, 0.5 mm and 1 mm spacing along the y-axis are analysed to see the variation in performance of different channels.

34 Chapter 3 Discussion and Analysis of Macroscopic Lens FSOI Design using Bi-convex Macroscopic Lens Ray tracing is initially used to determine the behaviour of the lens as illustrated in Figure 3-2. Data obtained from ray tracing provides the geometrical path of light through the lens and defines the aberration content of the image. Figure 3-2: Bi-convex macrolens ray tracing plot The starting point of a design is to specify the parameters in the optics used. Table 3-1 below, shows the specification of this selected lens. A numerical aperture of is used in this analysis. Rays are traced for three object field points with the furthest offaxis location at 1 mm and the performance of this macrolens is analysed based on the effects of spherical aberration, astigmatism, distortion and diffraction analysis. BK7: Bi-convex lens (KBX013) Specification Aperture mm Lens thickness, t c mm Y-radius mm Focal length mm Table 3-1: Bi-convex lens (KBX013) parameters

35 Chapter 3 Discussion and Analysis of Macroscopic Lens Aberration Analysis Figure 3-3: Field curvature and distortion plot for bi-convex macrolens From the plot in Figure 3-3, the LSA curve shows that the focal point shifts with increasing field height. The presence of a slight focal point shift of 0.8 mm at the edge of the field is detected. On the middle plot, the astigmatic field curves shows the edge of the bi-convex macrolens for sagittal surface (solid line) will be defocused by 0.2 mm while for the tangential surface (dotted line) will be defocused by more than 0.5 mm. This results show that astigmatism affects the quality of image at the receiver end. The right plot shows an ideal distortion plot with no distortion in this bi-convex lens even with increasing field height in this analysis.

36 Chapter 3 Discussion and Analysis of Macroscopic Lens 24 Figure 3-4: Distortion grid for bi-convex macrolens Figure 3-4 shows the image quality at the receiver after the beam has pass through the pair of macrolens. The actual field of view (FOV) shows that it matches with the paraxial FOV. This distortion grid verifies with the previous distortion plot that there is no distortion in this lens design. The overall result of this system shows a slight spherical aberration and astigmatism and no distortion. It can be concluded that there is insignificant aberration in the bi-convex lens Using Diffraction-Based Gaussian Beam Analysis Diffraction analysis was performed on this bi-convex macroscopic lens by propagating a Gaussian beam with a wavelength of 850 nm and a beam waist of 3 µm from the source in the design. A BFL distance of mm separates the pair of lenses. By developing a symmetrical system in an attempt to reduce spherical aberrations, the spacing between the transmitter and the first macrolens was varied equally with the distance between the second relay macrolens and the detector at the receiver. Using a 40 µm semi-aperture detector size the performance of encircled energy falling on the detector was analysed. By increasing the interconnection distance from 32 mm to 80 mm, the collected data are plotted in Figure 3-5 below, results shows that generally, the OSNR falls around the 18 db regions throughout the analysed distance.

37 Chapter 3 Discussion and Analysis of Macroscopic Lens 25 OSNR (db) OSNR vs Interconnection distance (KBX013) 3um beam half-w idth Interconnection distance (mm) Figure 3-5: OSNR versus interconnection distance for (KBX013) 3.2 FSOI Design using Plano-convex Macroscopic Lens Figure 3-6: Plano-convex macrolens ray tracing plot Same ray tracing process was also performed for the Plano-convex lens and it can be seen in Figure 3-6. Similarly a numerical aperture of was also used. Rays are again traced for three object field points with the furthest off-axis location 1 mm away. Results from the analysis of spherical aberration, astigmatism, distortions and Gaussian beam analysis are presented below. Table 3-2 shows the parameters used in this planoconvex lens.

38 Chapter 3 Discussion and Analysis of Macroscopic Lens 26 BK7: Plano-convex lens (KPX013) Specification Aperture Lens thickness, t c Y-radius Front Focal length (FFL) Back Focal length (BFL) mm mm mm 12.7 mm mm Table 3-2: Plano-convex lens (KPX013) Parameters Aberration Analysis Figure 3-7: Field curvature and distortion plot for plano-convex macrolens The LSA plot on the left in Figure 3-7 clearly shows the presence of a significant amount of focal shift at the edge of the field. About 2.5 mm was indicated on the chart at the edge of the field. Looking at the middle plot, the edge of the bi-convex macrolens for sagittal surface (solid line) is defocused by 0.4 mm while for the tangential surface (dotted line) there is a defocused by 1 mm. This result shows that astigmatism and the field curvature will influence the image quality at the receiver. The distortion plot on the right shows no considerable distortion in this plano-convex lens.

39 Chapter 3 Discussion and Analysis of Macroscopic Lens 27 Figure 3-8: Distortion grid plot for plano-convex macrolens It is also agreed in Figure 3-8, the actual FOV (red lines) matches with the paraxial FOV. This verifies there is no distortion in this design Using Diffraction-Based Gaussian Beam Analysis 9 OSNR vs Interconnection distance (KPX013) 8 OSNR (db) um beam w aist Interconnection distance (mm) Figure 3-9: OSNR versus interconnection distance for (KPX013)

40 Chapter 3 Discussion and Analysis of Macroscopic Lens 28 Similarly, diffraction analysis was performed on this plano-convex macroscopic lens as well by propagating a Gaussian beam with a wavelength of 850 nm and a beam waist of 3 µm. Separation distance between the transmitter and the first macrolens was also varied equally with the distance between the second relay macrolens and the detector at the receiver to create a symmetric system. The BFL used in this lens was mm and also a 40 µm semi-aperture detector size was employed to analyse the performance of this lens. The data between 32 mm to 80 mm interconnection distances were collected as well. Figure 3-9 above, shows that generally, the OSNR falls around 7 db to 8 db regions with some fluctuations which may be due to inconsistent aberrations on the macrolens surface.

41 Chapter 3 Discussion and Analysis of Macroscopic Lens Analysis of Off-Axis Performance of Macroscopic Lens Encircled energy vs Interconnect distance 0 Encircled energy Interconnection distance (mm) Figure 3-10: Off-axis performance of bi-convex and plano-convex lens Diffraction analysis was performed using a VCSEL source with a wavelength of 850 nm and a beam waist of 3 µm. Interconnection distances from 35 mm to 74 mm were simulated. An off-axis performance of both bi-convex and plano-convex lens were examined with off-axis channel spacing along y-axis starting from central channel at 0 mm and increasing to 0.5 mm, 1 mm and the furthest channel at 2.5 mm distance from the centre optical axis. The (solid lines) shows the performance of plano-convex lens at different off-axis channel while the (dotted lines) shows the performance of bi-convex lens based on the encircled energy received on the 40 µm semi-aperture detector. Results show that this bi-convex lens (KBX013) has a slightly higher encircled energy falling on the detector with the same interconnection distance between 0 mm to 1.5 mm channel spacing. While at 2 mm to 2.5 mm channel spacing the plano-convex lens seems to perform better than bi-convex macroscopic lens in BPR.

42 Chapter 4 General Design of Hybrid Systems 30 Chapter 4 General Design of Hybrid Systems FSOI involves the relay of a set of data channels from the transmitter plane to the receiver plane by employing focusing lens. One of the proposed interconnect solutions incorporate the use of macroscopic optics which provides a large field of view to image the entire emitter array, and the use of micro optics provides a small f-number to reduce the beam divergence. This combination of both macroscopic and micro optics is known as Hybrid system as illustrated in Figure 4-1. Throughout the analysis, a three by three VCSEL array are analysed and its performance in terms of aberration analysis, encircled energy and OSNR of individual channels are examined. Microlens Macrolens VCSEL Receiver Figure 4-1: Illustration of basic hybrid system

43 Chapter 4 General Design of Hybrid Systems Off-axis Channel Geometry Transmitter plane Off-axis Geometry Receiver plane Figure 4-2: Off-axis beam propagation geometry Figure 4-2, shows the transmitter and receiver plane geometries. As Gaussian beam propagates through free-space, the beam refract at an angle at the macrolens surface due to the radius of curvature of the lens. Notice from the figure the channel numbering is different between the transmitter and the receiver plane. Figure 4-3: Off-axis beam ray tracing Figure 4-3 further illustrates the off-axis ray trace of a set of relay macrolens, which also demonstrates the way the beam, will travel as it reaches the lens surface and gets refracted from the original parallel path. Figure 4-4 shows the dimensions used in a three by three VCSEL array. Each microlens has a 250 µm aperture size and the spacing between each lens is 250 µm apart, which is also known as the pitch.

44 Chapter 4 General Design of Hybrid Systems 32 The use of microlens is to limit the beam radius. This is important as clipping reduces the efficiency of the system and can lead to increased levels of optical crosstalk Pitch 250 µm µm Aperture Figure 4-4: Graphical structure of three by three VCSEL lens array In all the following analysis, the encircled energy at channel one, three, seven and nine are assumed symmetry (green circles), where their performances of encircled energy are identical. Similar assumptions are also assumed for the energy level at channel two, four, six and eight (orange circles). In the rest of the analysis in this thesis, only channel one, two and channel five, the centre channel of the three by three VCSEL array will be look into to analyse the performance of three by three VCSEL array system. 4.2 Optical Crosstalk Analysis 5 Figure 4-5: Diffraction noise signal affecting channel 5

45 Chapter 4 General Design of Hybrid Systems 33 Generally an optical system contains many channels. This may creates diffraction spreading of the signals propagating to other detectors, which is known as crosstalk noise or diffraction noise. This phenomenon is one of the main problems impairing the performance of the whole optical design. In the investigation of OSNR of a particular channel, the effects of surrounding diffraction noise (orange circles) that may overlap on neighbouring channels is illustrated in Figure 4-5. In analysing any individual channel, the amount of diffraction spreading to neighbouring detectors should be considered when studying the performance of individual channels in the FSOI design. Using BPR in Code V the amount of encircled energy falling on the detector could be analysed, thus a general formula is used in the calculation of signal-to-noise ratio: S SNR = 10 log (14) 1 S where S is the received encircled energy. Another more accurate formula that could be used to estimate the SNR by including the optical noise signal is given in equation below: S SNR = 10log (15) N where S is the received encircled energy and N is the total diffraction noise energy from neighbouring detectors.

46 Chapter 4 General Design of Hybrid Systems Design Optimisation Before any system can be analysed and compared, some optimisation must be made to the initial design in order to improve the image quality. Optimisation is the process of modifying the optical systems to reduce aberrations and improve the overall performance. In hybrid design, two different lenses are used - microlens and macroscopic lens. A crucial factor of any optical design was setting the optimum focus of the relay macrolens that will determine the performance of the optical design. Illustrated in Figure 4-6, this process involves fixing the distance spacing between the transmitter microlens and the macrolens at 20 mm. Moreover, the distance between the macrolens and the receiver microlens will be set to 20 mm as well. The reason for this is for the design to obtain close to a symmetrical system. By using BPR and propagating a collimated beam of 3 µm beam half-width, and a nominal far field half-angle divergence of ~0.6º as a result of the microlens aperture size. Each VCSEL beam can be modelled as a patch of point sources, emitting with a 0.6º half angle. Based on the encircled energy from the analysis of beam propagation, the results are presented in graphs in the next two sections. Microlens Macrolens VCSEL Increase the lens spacing 20 mm spacing Receiver Figure 4-6: Illustration of hybrid relay macrolens spacing optimisation

47 Chapter 4 General Design of Hybrid Systems Adjustment of Design Space for Bi-convex Lens In Figure 4-7, from results of focal length optimisation, all the channels seemed to intersect at a point where the distance is deemed to be at 24 mm or two focal length distances. This point also shows all the three channels have near matching encircled energy levels. Thus the best focal length of the bi-convex is assumed to be 12 mm that is near to the theoretical calculated value of mm. Encircled energy Encircled energy vs bi-convex lens focal length channel 5 channel 2 channel bi-convex lens focal length (mm) Figure 4-7: Performance of bi-convex (KBX013) lens at different focal length

48 Chapter 4 General Design of Hybrid Systems Adjustment of Design Space for Plano-convex Lens Same analysis was performed on the plano-convex lens. The results of focal length optimisation is shown in Figure 4-8, the optimum distance is deemed to be at 22 mm or two focal length distances, where all the three channels seemed to have the identical encircled energy level. Thus the BFL of the plano-convex lens is found to be 11 mm that is close to the theoretical calculated value of mm. Encircled energy Encircled energy vs plano-convex lens focal length channel 5 channel 2 channel Plano-convex lens Focal length (mm) Figure 4-8: Performance of plano-convex (KPX013) lens at different focal length

49 Chapter 4 General Design of Hybrid Systems Analysis of Hybrid Systems using Bi-convex Lens Microlens Macrolens VCSEL Increasing the Interconnection distance Receiver Figure 4-9: Layout of optical system for hybrid system using bi-convex lens The basic hybrid FSOI design consists of a pair of small field microlens and another pair of large field macrolens as presented in Figure 4-9 above. The system is designed to incorporate a three by three square array of small-aperture VCSEL s with a 250 µm pitch operating at a wavelength of 850 nm. The VCSEL s are collimated by microlens that has a diameter of 250 µm. During propagation, these collimated beams have a halfangle divergence, ~0.6 as a result of the microlens aperture size. Thus, each VCSEL beam can be modelled accurately as a small patch of sources each emitting with a 0.6 half-angle. The performance of three different channels is investigated and the results are then analysed and plotted. Table 4-1 below, shows the parameters used in this analysis. Microlens Macrolens (KBX013) Aperture 250 µm Aperture mm Lens thickness 0.03 mm Lens thickness 3.82 mm Y radius mm Y radius mm Focal length 0.8 mm Front focal length mm Back focal length mm Table 4-1: Parameters of hybrid design using bi-convex macroscopic lens

50 Chapter 4 General Design of Hybrid Systems Aberration Analysis Figure 4-10: Field curvature for bi-convex lens hybrid system In this section, rays are traced for three object field points with the farthest off-axis location at 2 mm, which corresponds to the corner element of a 17 x 17 VCSEL array. From Figure 4-10, the LSA curve shows the presence of a slight focal point shift at the edge of the field with increasing field height. The centre plot of field curvature, illustrates that on the sagittal surface (solid line) the image will be defocused by about 0.8 mm. While on tangential surface (dotted line) the image will be defocused by 2 mm. As the beams travel off-axis once it passes the first relay macrolens, these values of defocus shows that astigmatism and field curve will affect the image quality at the receiver plane. The distortion plot on the right shows about 0.5 % of distortion at the edge of the lens.

51 Chapter 4 General Design of Hybrid Systems 39 Figure 4-11: Distortion grid for bi-convex lens hybrid system Figure 4-11 shows the actual FOV is slightly smaller and its corners are curved inwards comparing to the distortion-free paraxial FOV (black lines). The paraxial is in place to show the extent of distortion. This result shows negative or barrel distortion at the image on the receiver plane. The amount of distortion is relatively insignificant Using Diffraction-based propagation of Gaussian Beam Encircled energy Encircled energy vs Interconnection distance channel 5 channel 2 channel Interconnection distance (mm) Figure 4-12: Graph of encircled energy versus interconnection distance of hybrid system using bi-convex lens

52 Chapter 4 General Design of Hybrid Systems 40 From Figure 4-12, a detector size of 40 µm semi-aperture was used to analyse the FSOI design. The interconnection distance of this design was increased from 34 mm to 76 mm to analyse the amount of encircled energy at the detector. Results shows that all the channels perform equally well over an interconnect distance between 38 mm to 66 mm. The amount of encircled energy that emits from the source plane and received at the detector plane was generally over 97 % of the total signal transmitted. At 56 mm interconnection distance, this FSOI design shows the encircled energy of over 99% falling on the receiver plane at all channel detectors. OSNR (db) OSNR vs Interconnection distance Channel Channel 2 Channel Interconnection distance (mm) Figure 4-13: OSNR versus interconnection distance for different channels of hybrid system using bi-convex macrolens Figure 4-13, shows the OSNR versus interconnection distance of this hybrid system with two bi-convex macroscopic lenses. The plotted results shows that the channel situated at the corner of the array, for example channel 1, has a higher OSNR than other channels, which is due to the fact that less neighbouring channels are surrounding channel 1. In this analysis the surrounding diffraction noise that was caused by other channels were analysed and take into account the performance of the entire system. Overall, the centre channel (channel 5) that is surrounded by eight other neighbouring channels suffers more crosstalk. The plot shows a peak point at 56 mm interconnection distance with OSNR of over 40 db, which indicates that the system will perform better as compared to other interconnection distances.

53 Chapter 4 General Design of Hybrid Systems Analysis of Hybrid Systems using Plano-convex Lens Microlens Macrolens VCSEL Increasing the Interconnection distance Receiver Figure 4-14: Layout of optical system for hybrid system using plano-convex lens In this section, another hybrid system of FSOI design illustrated in Figure 4-14 is analysed using a pair of large field plano-convex macroscopic lens. The system is also designed to incorporate a three by three square array of small-aperture VCSEL s with a 250 µm pitch operating at a wavelength of 850 nm. The VCSEL s are collimated again by microlens that has a diameter of 250 µm. During propagation, these collimated beams have a half-angle divergence, ~0.6. Similarly three same optical channels are analysed to see the variation in performance of different channels. In Table 4-2, the parameters are specified in this analysis. Microlens Macrolens (KPX013) Aperture 250 µm Aperture mm Lens thickness 0.03 mm Lens thickness 3.82 mm Y radius mm Y radius mm Focal length 0.8 mm Front focal length 12.7 mm Back focal length mm Table 4-2: Parameters of hybrid design using plano-convex macroscopic lens

54 Chapter 4 General Design of Hybrid Systems Aberration Analysis Figure 4-15: Field curvature for plano-convex lens hybrid system Rays are traced similarly for three object field points with the farthest off-axis location at 2 mm, which corresponds to the corner element of a 17 x 17 VCSEL array. From the LSA curve, it shows the presence of mm focal point shift at the edge of the field. A plot of field curvature on the centre of Figure 4-15, illustrates that on the sagittal surface (solid line) the image will be defocused by about 1.25 mm. While on tangential surface (dotted line) the image will be defocused by 5 mm. Once the off-axis beam passes the first relay macrolens, these values of defocus shows that astigmatism and field curve will have an effect on the image quality at the receiver plane. The plot on the right shows approximately 0.6 % of distortion occurs at the edge of the lens. Therefore, distortion is another factor that may influence the performance of hybrid FSOI design.

55 Chapter 4 General Design of Hybrid Systems 43 Figure 4-16: Distortion grid for plano-convex lens hybrid system Figure 4-16 shows the actual FOV is smaller and its edges are also curved inwards comparing to the distortion-free paraxial FOV (black lines). This shows negative distortion occurs at the image on the receiver plane Using Diffraction-Based Propagation of Gaussian Beam Encircled energy Encircled energy vs Interconnection distance channel 5 channel 2 channel Interconnection distance (mm) Figure 4-17: Graph of encircled energy versus interconnection distance for hybrid system using plano-convex macrolens

56 Chapter 4 General Design of Hybrid Systems 44 Similar to the design in the previous section, diffraction analysis was again performed in this hybrid system with plano-convex lens. In the analysis a detector of 40 µm semiaperture size was used to analyse the FSOI design. From Figure 4-17, the interconnection distance was increased from 32 mm to 80 mm and the results showed all the channels perform equally well over an interconnect distance between 40 mm to 66 mm. In addition, within this 26 mm range, the amount of encircled energy that is received at the detector plane maintained above 98% of total optical power emitted from the VCSEL source. At 56 mm interconnection distance, this FSOI has an overall encircled energy of over 99.6 % falling on the designated detector OSNR vs interconnection distance OSNR (db) Channel 1 Channel 2 Channel Interconnection distance (mm) Figure 4-18: OSNR versus interconnection distance for different channels in hybrid system using plano-convex macrolens Looking at Figure 4-18, the graph shows the OSNR versus interconnect distance for the hybrid system using a pair of plano-convex macroscopic lenses. The plot shows once more the farthest off-axis location, for instance channel 1, has a higher OSNR than other channels, since fewer neighbouring channels are adjacent to this channel. In this analysis the surrounding diffraction noise created by other channels are also analyse and

57 Chapter 4 General Design of Hybrid Systems 45 taken into account of the performance of the entire system. In general, the centre channel (channel 5) suffers from more crosstalk due to more neighbouring channels. The plot shows a highest point at 56 mm interconnection distance where all channels exhibit an OSNR of around 40 db, which reflects that at this interconnection distance, the design performs better than any other interconnection distance.

58 Chapter 5 Board to Board Level Free-Space Optical Interconnects 46 Chapter 5 Board to Board Level Free-Space Optical Interconnects 5.1 Hybrid Systems with Folded Geometry This design combines two commonly used relay configurations in FSOI. It consists of microlens and minilens array. The purpose of microlens was to collimate the VCSEL emitting from the transmitter and the beam at the receiver plane of the two separate modules. The relay module uses a pair of minilens for the interconnection. The two right-angle BK7 prisms were fixed at each end of the relay blocks to bend the beam by 90. In this application, the prisms are used to reorient a beam or an image. These prisms have a face length of 5 mm, which were 2.75 mm larger to enable for alignment of the link. Table 5-1 shows the rest of the geometry of this design. Minilens Prism Microlens VCSEL Receiver Figure 5-1: Diagram of board to board Level FSOI

59 Chapter 5 Board to Board Level Free-Space Optical Interconnects 47 Specifications lens type (BK7) Microlens (BK7) Plano-convex minilens Focal length 800 µm 15.7 mm Array size 3 x 3 1 x 1 Aperture 250 µm 1.5 mm Y-Radius mm 8 mm Table 5-1: Lens array specifications 5.2 Adjustment of Design Space for Minilens The focal length of the minilens is first optimised using BPR. This process involves adjusting the lens spacing of the system. An illustration of focal length optimisation for the relay plano-convex minilens is shown in Figure 5-2. Microlens Minilens Increase the lens spacing Transmitter 20 mm spacing Receiver Figure 5-2: Layout of optical system for adjustment of minilens focal length In this process, other module parameters have to be fixed while altering the required focal length of the lens. To reduce distortion in a FSOI design, all the beams have to cross the optical axis at the centre of the optical system. Thus, the relay minilens spacing has to be optimised in an attempt to focus the beams to create a symmetric

60 Chapter 5 Board to Board Level Free-Space Optical Interconnects 48 system. This process involves fixing the distance spacing between the transmitter microlens and the minilens at 20 mm. Moreover, the distance between the minilens and the receiver microlens will be set to 20 mm as well. The reason for this is for the design to achieve close to a symmetrical system. After that, the distance between the pair of relay minilenses is varied from a distance of 0 to 40 mm. Based on the encircled energy from the analysis of BPR, the results are plotted in Figure 5-3. Encircled energy vs minilens focal length Encircled energy Channel 5 channel 1 Channel minilens focal length (mm) Figure 5-3: Performance of minilens at different focal length From the results of the plot, the best focal length is deemed to be at 32 mm or two focal distances where all the channels seemed to have the highest encircled energy. Thus the focal length of the minilens is found to be at 16 mm that is close to the theoretical calculated value of 15.7 mm. 5.3 Aberration Analysis In this analysis, rays are traced for three object field points with the furthest off-axis location at 0.5 mm. The left plot of Figure 5-4 shows there is a maximum focal point shift of 0.02 mm at the edge of the field height. The field curvature plot illustrates that on the sagittal surface (solid line) the image will be defocused by about 0.1 mm. While

61 Chapter 5 Board to Board Level Free-Space Optical Interconnects 49 on tangential surface (dotted line) the image will be defocused by 0.23 mm. Once the off-axis beam passes the first relay minilens, these values of defocus shows that astigmatism and field curve affects the image quality at the receiver plane. The distortion plot on the right shows about 0.01 % of distortion at the edge of the lens. This result is relatively small, as it does not cause any significant impact on the image. Figure 5-4: Field curvature for board to board design Figure 5-5: Distortion grid for board to board design

62 Chapter 5 Board to Board Level Free-Space Optical Interconnects 50 The distortion grid in Figure 5-5 shows the actual FOV (red lines) is slightly larger than the Paraxial FOV. This result is relatively satisfactory as it match with the 0.01 % distortion curve in Figure 5-4. Therefore there is only slight distortion at the edge of the distortion grid. 5.4 Using diffraction-based propagation of Gaussian Beam Diffraction analysis was again performed in this hybrid system with folded geometry. In the analysis a detector size of 40 µm semi-aperture was used again to analyse the performance of this FSOI design. From Figure 5-6, the interconnection distance of this design was varied from 34 mm to 76 mm, the results showed that all the channels perform just as well over an interconnect distance between 38 mm to 63 mm. In addition, within this 25 mm range, the amount of encircled energy that is received at the detector plane maintained above 97% of total optical power emitted from the VCSEL source. The distance at 54 mm of this FSOI obtain an overall encircled energy of over 99.6 % falling on the receiver plane Encircled energy vs Interconnection distance Encircled energy channel 5 channel 2 channel Interconnection distance (mm) Figure 5-6: Graph of encircled energy versus interconnection distance for board to board design

63 Chapter 5 Board to Board Level Free-Space Optical Interconnects 51 OSNR (db) OSNR vs Interconnection distance 50.0 Channel Channel Channel Interconnection distance (mm) Figure 5-7: OSNR versus interconnection distance for different channels Looking at Figure 5-7, the graph shows the OSNR versus interconnect distance of this board to board level FSOI. Similarly, channel 1, has a higher OSNR than other channels, which is due to the fact that fewer neighbouring channels are surrounding channel 1. In this similar analysis, the surrounding diffraction noise were analysed and take into consideration of the performance of the entire system. On the whole, the centre channel (channel 5) suffers from more crosstalk. The plot also shows a peak at 54 mm interconnection distance, which demonstrates that the system will perform better at this distance as compared to other interconnection distance.

64 Chapter 5 Board to Board Level Free-Space Optical Interconnects Gaussian Beam profile pattern Figure 5-8: Beam intensity stretches Figure 5-9: Beam intensity leaving prism Minilens Prism Microlens Array VCSEL Figure 5-10: Diagram of Hybrid Systems with Folded Geometry Figure 5-11: Beam intensity at microlens surface Figure 5-12: Beam intensity at minilens surface

65 Chapter 5 Board to Board Level Free-Space Optical Interconnects 53 Refer to previous page in Figure 5-10, shows the Gaussian beam profile pattern at different surface. In this analysis the VCSEL transmits a beam with 3 µm half-width. Figure 5-11, shows a perfect airy disk where the central maximum of diffraction forms a circular aperture at the microlens surface. The plot also shows majority of the beam intensity is contained in the centre spot. In Figure 5-8, the Gaussian beam contour shows an elliptical shape. This result was due to the difference in travel distance where the beams reaches the 45 surface of the prism. Further illustration is shown in Figure 5-13 below. As the VCSEL propagates towards the 45 inclination surface of the prism, one edge of the beam arrives at this surface (point A) earlier while the other edge of the beam will travel a further distance before reaching (point B) on the same surface. Due to this reason, the beam appears to be stretched on that surface. Figure 5-9 illustrates the beam intensity at the surface where the beam exits the prism surface. It shows there is a larger beam radius due to beam divergence although majority of the beam intensity is contained in the central spot. This also highlights that Gaussian beam diverge and increases in beam radius as it travels through free-space. Figure 5-12 shows the beam pattern on the relay minilens surface. The beam intensity performs normally as majority of the beam intensity falls inside the central spot. B A 45 VCSEL Figure 5-13: Diagram of beam falling on the 45 prism surface

66 Chapter 5 Board to Board Level Free-Space Optical Interconnects 54 Figure 5-14: Beam intensity diagram at the receiver for 3 µm beam half-waist In Figure 5-14, the beam pattern seems to be normal. Most of the energy falls in the central spot of the beam intensity and the contour of the beam is circular in shape. There are some very fade rings surrounding the beam pattern, which may suggest some minor aberration or diffraction exists on the image surface.

67 Chapter 5 Board to Board Level Free-Space Optical Interconnects 55 Figure 5-15: Point spread function. A two-dimensional plot of energy with approximately 64 µm spot on the detector Figure 5-15 shows a two-dimensional surface plot of point-spread distribution (PSF) where the relative intensity of this point of encircled energy were plotted as a function of distance. This plot is created by fraunhofer diffraction at a circular aperture at the detector. Simulations from code V showed that for VCSEL s with a 15-deg divergence angle, and no misalignments in the system, about 99 % of the total energy is encircled in a spot no bigger than 64 µm at the receiver plane. The remaining 1 % is spread out over on the image surface. As there are no visible other peaks outside the central maximum spot size, diffraction is not evident.

68 Chapter 6 Design Comparisons 56 Chapter 6 Design Comparisons Summaries of the different analysed models are compared from Table 6-1 to Table 6-3. In the first table, the results show that plano-convex lens performs slightly poorer in terms of aberration analysis. There is also one major factor that bi-convex lens out performs the plano-convex lens: the optical signal-to-noise ratio. This could be due to the difference of radius of curvature of the two macroscopic lenses. In the hybrid systems, both designs using different macroscopic lenses perform equally well in terms of interconnection distance and also in aberration analysis, although both designs suffer from some aberration effects. Both plano-convex and biconvex macrolens used in the hybrid designs could achieve an average OSNR of 40 db and 44 db respectively with below 1 % of distortion levels. These results are quite satisfactory. Finally, a folded geometry systems is analysed for its performance for board to board interconnection designs. An interconnection distance of 54 mm could be achieved with a relatively OSNR of 42 db and a distortion level of 0.01 % could be attained in this design. Throughout the comparisons, all FSOI designs would suffer from some degrees of aberrations that include spherical aberration, astigmatism field curve and distortion. This means that the image quality at the receiver would be affected. The focal length of the lenses used also plays an important role in reducing the aberration effects.

69 Chapter 6 Design Comparisons 57 Relay Macroscopic Lens Bi-convex lens Plano-convex lens (KBX 013) (KPX 013) Focal Length mm mm Aberration Analysis Longitudinal Spherical Aberration 0.8 mm 2.5 mm Astigmatic Sagittal Surface 0.2 mm 0.4 mm Field Curves Tangential Surface 0.5 mm 1 mm Distortion Not visible Not visible Diffraction-Based Gaussian Beam propagation using 3µm Beam Half-Width Average OSNR over Interconnection Distance between 32 mm to 80 mm 18 db 7.5 db Table 6-1: Comparison of bi-convex versus plano-convex lens performance Hybrid Systems Bi-convex Plano-convex macroscopic lens macroscopic lens Focal Length 12 mm 11 mm Aberration Analysis Longitudinal Spherical Aberration mm mm Astigmatic Sagittal Surface 0.8 mm 1.25 mm Field Curves Tangential Surface 2 mm 5 mm Distortion 0.5 % 0.6 % Diffraction-Based Gaussian Beam propagation using 3µm Beam Half-Width Optimum Interconnection Distance 56 mm 56 mm Average OSNR at Optimum Interconnection Distance 44 db 40 db Table 6-2: Comparison of hybrid Systems using different macroscopic lens Hybrid Systems with Folded Geometry Minilens Focal Length 16 mm Aberration Analysis Longitudinal Spherical aberration 0.02 mm Astigmatic Sagittal Surface 0.1 mm Field Curves Tangential Surface 0.23 mm Distortion 0.01 % Diffraction-Based Gaussian Beam propagation using 3µm Beam Half-Width Optimum Interconnection Distance 54 mm Average OSNR at Optimum Interconnection Distance 42 db Table 6-3: Summary of board to board interconnection design performance

70 Chapter 7 Conclusion and Recommendations 58 Chapter 7 Conclusions and Recommendations 7.1 Conclusions This thesis presented the process for designing free-space optical interconnects. The design process was broken down into various steps. Starting from type of laser beam followed by the interaction of beams with different optical lenses was look into. Consequently the calculation of signal-to-noise ratio, design capacity and interconnection length are analysed. In each step of the design process, the effect of design parameters was examined. Diffraction effects, that limit the performance of FSOI in the most significant way, were also investigated and accounted for in the analysis. During this thesis, it has been realised that there are actually many methods and approaches that can be explored to improve the performance of a FSOI design. Nonetheless, the aim of this thesis concerning the analysis and design of free-space optical link has been fulfilled. Results from two different macroscopic lenses were compared and found that having a smaller radius of curvature for bi-convex lens; the performance could be improved and resulting in similar or better performance than plano-convex lens that has a larger radius of curvature. In both hybrid designs, the use of bi-convex and plano-convex relay macroscopic lens could be used to achieve a longer interconnection distance in a FSOI system. Another advantage of having macrolens as it provides a large field of view to image the entire emitter array. Both designs have an interconnection distance of 56 mm with a maximum of 99.6 % of the encircled energy detected on the 40 µm detector semi-aperture size.

71 Chapter 7 Conclusion and Recommendations 59 The final design introduces prism optical elements to redirect or fold the laser beams for board to board interconnections. This design was analysed with a three by three VCSEL array and the performance of this system was found to be a visible FSOI design with over 99.6% of encircled energy could be achieved on the 40 µm semi-aperture detector over an interconnection distance of 54 mm. This thesis has shown that by designing a VCSEL-based optical link using lasers and detectors with commercially available optical components such as refractive lenses, could be used to achieve a high-density interconnects by forming closely spaced parallel beams between the transmitter and receiver. Finally the Gaussian beam intensity pattern is looked into to further determine the spot size at different surfaces and also the diffraction pattern of the image on the detector. Some of the approaches to improve FSOI systems are discussed below: - By designing a symmetric FSOI system with the location of the effective system aperture at the centre of the optical system, distortion could be eliminated. - Effects of aberration could be reduced or eliminated by improving the system geometry and steering the beams across the optical axis at the centre of the optical system.

72 Chapter 7 Conclusion and Recommendations Recommendation for future work FSOI proves to be a promising technology that could emerge as a new architecture for the next generation of information processing systems. Much work still has to be done to make this a suitable technology. Although in this thesis, only a small 3 by 3 VCSEL array was analysed, the optical design discussed here could be extended to 17 by 17 interconnects. This array of FSOI could be further expanded to examine the maximum number of channels that could be achieved in a specified macrolens aperture size in a hybrid system. Besides that other optimisation improvements such as the curvature of lens surfaces could be varied to further improve aberrations in FSOI design. The general requirements for FSOI are small spot size, low crosstalk, large field of view, low cost, large misalignment tolerance and capable to be cascaded. Since this thesis is based on design and analysis of different FSOI systems, another possible future work can be the study of other FSOI designs and possible fabrication of potential FSOI designs to further evaluate the FSOI system through practical experimental testing process.

73 Appendices 61 Appendices

74 Appendices 62 Appendix A Code V simulated results A.1 Biconvex and Plano-convex Macroscopic Lens bi-convex lens plano-convex lens Interconnection distance (mm) channel 5 OSNR (db) channel 5 OSNR (db) Table A-1: Numerical results for OSNR versus Interconnection distance

75 Appendices 63 A.2 Hybrid Systems using Bi-convex Macroscopic Lens Encircled Energy Interconnection distance (mm) Channel 5 Channel 2 Channel 1 Channel 1 OSNR (db) Channel 2 OSNR (db) Channel 5 OSNR (db) Table A-2: Numerical results for OSNR versus Interconnection distance Dist (mm) Diffraction noise of channel 5 falling on channel 1 Diffraction noise of channel 2 falling on channel 1 Diffraction noise of channel 1 falling on channel 5 Diffraction noise of channel 4 falling at channel 2 Diffraction noise of channel 5 falling on channel 2 Diffraction noise of channel 1 falling on channel E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 Table A-3: Numerical results for diffraction noise analysis

76 Appendices 64 A.3 Hybrid Systems using Plano-convex Macroscopic Lens Encircled Energy Interconnection distance (mm) channel 5 channel 2 channel 1 Channel 1 OSNR (db) Channel 2 OSNR (db) Channel 5 OSNR (db) Table A-4: Numerical results for OSNR versus Interconnection distance Dist (mm) Diffraction noise of channel 5 falling on channel 1 Diffraction noise of channel 2 falling on channel 1 Diffraction noise of channel 1 falling on channel 5 Diffraction noise of channel 4 falling on channel 2 Diffraction noise of channel 5 falling on channel 2 Diffraction noise of channel 1 falling on channel E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 Table A-5: Numerical results for diffraction noise analysis

77 Appendices 65 A.4 Hybrid Systems with Folded Geometry Interconnection Encircled Energy Channel 1 Channel 2 Channel 5 distance (mm) channel 5 channel 2 channel 1 OSNR (db) OSNR (db) OSNR (db) Table A-6: Numerical results for OSNR versus Interconnection distance Diffraction noise of channel 5 falling on channel 1 Diffraction noise of channel 2 falling on channel 1 Diffraction noise of channel 1 falling on channel 5 Diffraction noise of channel 4 falling on channel 2 Diffraction noise of channel 5 falling on channel 2 Diffraction noise of channel 1 falling on channel 2 Dist (mm) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-02 Table A-7: Numerical results for diffraction noise analysis of folded geometry

78 Appendices 66 Appendix B Code V Lens Data Manager B1 Hybrid Systems using Bi-convex Macroscopic Lens Figure B-1: Code V lens data manager for hybrid systems using bi-convex macroscopic lens B2 Hybrid Systems using Plano-convex Macroscopic Lens Figure B-2: Code V lens data manager for hybrid systems using plano-convex macroscopic lens

79 Appendices 67 B3 Hybrid Systems with Folded Geometry Figure B-3: Code V lens data manager for hybrid systems with folded geometry