Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 666 2009-10-2 666 Model-Based Design for Embedded Systems Fourier transform can be implemented by one of the numerous fast Fourier transform (FFT) techniques. The computational order of the FFT for a 2D input is O(N 2 log 2 N), obviously more efficient when compared to the direct integration method. We show this speed increase later through an example. In continuous theory, the angular spectrum method is an exact solution of the Rayleigh–Sommerfeld formulation. However, when solving the algo- rithm on a digital computer, a discrete Fourier transform (DFT)mustbe used, resulting in the accuracy of the angular spectrum method being dependent on the resolution, or spacing, of the aperture and observation plane meshing. We call the physical size of the aperture and observation planes the “bound- ing box,” defining the size of the optical wave front being propagated. Since the complex wave function is only nonzero for a finite space in the bounding box, the signal is not always bandwidth limited, and the Nyquist sampling theory does not always apply. It can be shown, however, that the resolu- tion of the aperture and observation meshing must be λ/2 or smaller [39]. For many simulation systems without large degrees of tilt and hard diffrac- tive apertures, the resolution can be coarser. In systems with high tilts, the resolution is most sensitive. With a mesh spacing of λ/2, the angular spec- trum decomposition will model plane waves propagating from the aperture to the observation plane in a complete half circle, that is, between –90 and +90 degrees. Other inaccuracies that can occur when using a DFT are aliasing and win- dow truncation. Aliasing occurs when frequencies exist greater than the criti- cal sampling frequency. In this case, these high frequencies are “folded over” into the sampled frequency range [40]. The effect of this is seen in our simu- lations as optical power “reflecting” off of the walls of the bounding box. If significant optical power reflects off the wall, interference between the prop- agating beam and these reflections can occur, resulting in inaccurate optical waveforms. The same effect can be seen when the bounding box truncates the signal. Truncation occurs when the waveform propagates into the edges of the bounding box. The simplest solution to ensure accurate results is having sufficient zero padding around the optical waveform, reducing the chance the waveform is aliased or truncated by walls of the bounding box. In Chatoyant, the user can choose between using the Gaussian or scalar diffractive (angular spectrum) methods during simulation. The components in the optical library support both representations in the optical signal message class. Using these models we can simulate and analyze a variety of heterogeneous systems as presented in the next section. 20.2.7 Simulations and Analysis of Optical MEM Systems In this section, we show how Chatoyant can model and simulate complete mixed-signal systems. The first system uses both electrical and optical sig- nals to simulate a complete “4f” optoelectronic link which uses a four focal length image relaying optical system. The second example, building from the Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 667 2009-10-2 CAD Tools for Multi-Domain Systems on Chips 667 two signal 4f link, adds mechanical signals for simulation and analysis of an optical MEM system. This set of example systems is centered on an optical MEM scanning mirror. With this device we are able to simulate an optical scanning system and a self-aligning optical detection system. These systems show the ability to model a mixed system of mechanical MEMs, optics, and electronic feedback. The last example shows the power of the angular spec- trum technique to model diffractive optical systems with the speed and accu- racy required to perform system-level design. 20.2.7.1 Full Link Example A complete optoelectronic simulation of a 4f optical communication link in Chatoyant is presented in Figure 20.12. The distance between the vertical cavity surface emitting laser (VCSEL) array and the first lens and the dis- tance between the second lens and the detector array are both 1 mm. The distance between the lenses is 2 mm, with both lenses having a focal length of 1 mm, giving a 4f system. The top third of the figure shows the system as represented in Chatoyant. Each icon represents a component model, and each line represents a signal path (either optical or electrical) connecting the outputs of one component to the inputs of the next. Several of the icons, such as the VCSELs and receivers, model the optoelectronic components them- selves, while others, such as the output graph, are used to monitor and dis- play the behavior of the system. The input to the system is an electrical signal with speed varying from 300 MHz to 1.5 GHz. A Gaussian noise with vari- ance of 0.5 V has been added to the multistage driver system to show the ability of our models to respond to arbitrary waveforms. In the center of the figure, three snapshots (before the VCSEL, after the VCSEL, and after the detector) show the behavior of the CMOS drivers under Digital Driver Gaussian waist analysis Power analysis VCSEL 4f optical system PGM + Receiver FIGURE 20.12 Chatoyant analysis of optoelectronic 4f communications link. Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 668 2009-10-2 668 Model-Based Design for Embedded Systems a 300 MHz noisy signal. In these snapshots, one can see the amplification of the system noise through the CMOS drivers, the clipping of subthreshold noise in the VCSEL, and the frequency response on the quality of the received signal. This last observation is better seen in the three eye diagrams, shown at the bottom of Figure 20.12, analyzed at 300 MHz, 900 MHz, and 1.5 GHz. For the component values chosen, the system operates with reasonable BER up to about 1 GHz. For this 4f system, the VCSEL and driver circuits explicitly model the effects of bias current and temperature on the optoelectric conversion, L-I efficiency, of the lasers. Figure 20.13 shows the effects of temperature, T,and current bias, I b , on the bit error rate (BER) of the link. Generally, the fre- quency response of the link is dominated by the design of the receiver circuit; however it is interesting to note that both the VCSEL temperature and bias have a significant effect on system performance, because of their impact in the power through the link. Perhaps most interesting is the fact that increas- ing bias current does not always correspond to better performance over the whole range of frequencies examined. Note that the curve for 1 mA bias offers the best performance below 600 MHz; however, the 0.5 mA bias (the nominal threshold of the VCSEL) crosses the curve for 1 mA and achieves the best performance at higher frequencies. As an example of mechanical tolerancing, we analyze the system with varying-sized photodetectors (50, 30, and 20 μm). The detectors are displaced from + 10 μmto+100 μm in detector position along the axis of optical prop- agation. This results in defocusing of the beam relative to the detector array. We calculate both the insertion loss and the worst case optical crosstalk as the detectors are displaced. The results are shown in Figure 20.14. Systems can be further analyzed for their sensitivity to mechanical tolerances using a Monte Carlo tolerancing method described in [8,9]. Two additional analyses are also shown in the Chatoyant representation in Figure 20.12. The first is the beam profile analysis, which graphically dis- plays one beam’s waist as it propagates between components, showing the possibility of clipping at the lenses. The second analysis shows the optical signals as they strike the detector array. This analysis also gives the user the amount of optical power captured on each of the detectors. From this analy- sis, optical crosstalk and system insertion loss can be calculated. 20.2.7.2 Optical Beam Steering/Alignment System A torsion-scanning mirror is a micromachined 2D mirror built upon a micro-elevator by self assembly (MESA) structure [41,42]. The mirror and MESA structures are shown in Figure 20.15a and b, respectively. The scan- ning mirror can tilt along the torsion bars in both the x and y directions and is controlled electrostatically through four electrodes beneath the mirror, outlined in Figure 20.15a by the dashed boxes. For example, the mirror tilts in the positive x direction when voltage is applied to electrodes 1 and 2, and the Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 669 2009-10-2 CAD Tools for Multi-Domain Systems on Chips 669 BER vs. frequency at VCSEL temperatures 1.E–20 1.E–16 1.E–12 1.E–08 BER 1.E–04 1.E+00 100 300 500 700 900 1100 1300 1500 BER (T =40 C) BER (T =70 C) BER (T =100 C) 1.E –20 1.E –16 1.E –12 1.E –08 1.E –04 1.E +00 BER vs. frequency at various current bias BER 100 300 500 700 900 1100 1300 1500 BER (lb =0.1 mA) BER (lb =0.25 mA) BER (lb =1.0 mA) BER (lb =1.5 mA) BER (lb =0.5 mA) Frequency (MHz) Frequency (MHz) FIGURE 20.13 BER versus frequency at different VCEL temperatures and current biases. 0 10 +μm displaced in optical axis Crosstalk (dB) Crosstalk vs. detector displacement 30 50 70 –25 –50 –75 –100 50 um Det 20 um Det 30 um Det Insertion loss vs. detector displacement 10 30 50 70 0 –3 –6 –9 –12 –15 –18 Insertion loss (db) ±μm displaced in optical axis 50 um Det 30 um Det 20 um Det FIGURE 20.14 Insertion and crosstalk versus mechanical tolerancing. (From Kurzweg, T.P. et al., J. Model. Simul. Micro-Syst., 2, 21, 2001. With permission.) mirror tilts in the negative y direction when voltage is applied to electrodes 1and4. The MESA structure is shown in Figure 20.15b. The mirror is elevated by four scratch drive actuator (SDA) sets pushing the support plates together, allowing for the scanning mirror to buckle and rise up off the substrate [43]. The MESA structure’s height is required to be large enough such that the tilt of the mirror will not cause the mirror to hit the substrate. Post fabrication system alignment can also be performed by the MESA structure. Figure 20.16 shows a drawing of the torsion-scanning mirror system. On the left one can see one VCSEL emitting light vertically through a lenslet, and a prism that reflects off a plane mirror. The light is then reflected off of the optical MEM scanning mirror, back to the plane mirror, and captured through a lenslet and prism onto detectors on the right. With the flexibility of the scanning mirror, this system could act as a switch, an optical scanner, Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 670 2009-10-2 670 Model-Based Design for Embedded Systems (a) 23 14 (b) x y FIGURE 20.15 (a) Scanning torsion mirror, (b) MESA structure. (From Kurzweg, T.P. et al., CAD for optical MEMS, Proceedings of the 36th IEEE/ACM Design Automation Conference (DAC’99), New Orleans, LA, June 20–25, 1999. With permission.) FIGURE 20.16 Scanning mirror system. (From Kurzweg, T.P. et al., CAD for optical MEMS, Proceedings of the 36th IEEE/ACM Design Automation Conference (DAC’99), New Orleans, LA, June 20–25, 1999. With permission.) or a reconfigurable optical interconnect. We have simulated systems using this scanning mirror configuration for switching and self-alignment through optical feedback. We first demonstrate an optical scanning system. In this scanning system, we simulate a single source beam propagating through the 3 × 3 subsystem seen in Figure 20.16. With the appropriate volt- age levels applied to the four electrodes, the scanning mirror tilts and directs the source to any of the nine detectors. This system, as represented in Chatoy- ant, is shown in Figure 20.17. The SDA arrays move the mirror to the correct height for alignment. We control the electrodes with a waveform generator, which applies the appropriate voltages on the four electrodes for the beam to scan or switch in a desired pattern. As an example, we are able to scan a diamond pattern with the wave- forms shown in Figure 20.18. The desired pattern is shown by the white arrow trace on the first output image. The other nine images show snapshots of the detector plane as the diamond pattern is scanned. Dashed lettered lines correspond to time intervals in the waveforms and in the snapshots. Mechanical alignment is critical in this system. For example, the lenslets in this simulation are only 100 μm in diameter. Therefore, when steering the Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 671 2009-10-2 CAD Tools for Multi-Domain Systems on Chips 671 UCSEL Prism Mirror Mirror Prism Powergrid Const SDA FIGURE 20.17 Scanning system as represented in Chatoyant. (From Kurzweg, T.P. et al., CAD for optical MEMS, Proceedings of the 36th IEEE/ACM Design Automation Conference (DAC’99), New Orleans, LA, June 20–25, 1999. With permission.) Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 672 2009-10-2 672 Model-Based Design for Embedded Systems A xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a:xv 3.10a:xv 3.10a:xv 3.10a:xv 3.10a: BCDE Electrode 4 Electrode 3 Electrode 1 Electrode 2 ABCDE FIGURE 20.18 Scanning waveforms and scanned diamond pattern. (From Kurzweg, T.P. et al., CAD for optical MEMS, Proceedings of the 36th IEEE/ACM Design Automation Conference (DAC’99), New Orleans, LA, June 20–25, 1999. With permission.) beam, precision in the voltage waveforms is needed so that the light, bend- ing through the prism, hits the desired detector’s lenslet. We next simulate a self-aligning system using optical feedback, using the same system setup as seen in Figure 20.16. Such a system could be used as a noise suppression system. The scanning mirror is used to actively align the system, with the electrodes now being controlled by a waveform generator with a programmed control algorithm. The waveform generator receives the power values detected on each of the detectors, determines where the beam is, and which electrodes to apply voltage to in order to steer the beam onto the center detector. The system is considered aligned when the power detected on the center detector matches a threshold value set by the user. The user also specifies, in the control algorithm, the size of the voltage step that will be placed on the corresponding electrodes. With active feedback, the system will keep step- ping enough voltage to the electrodes until the beam is steered onto the cen- ter detector and the system is aligned. The system, as displayed in Chatoyant, is shown in Figure 20.19. Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 673 2009-10-2 CAD Tools for Multi-Domain Systems on Chips 673 Const FIGURE 20.19 Self-aligning system using optical feedback. (From Kurzweg, T.P. et al., J. Model. Simul. Micro-Syst., 2, 21, 2001. With permis- sion.) Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 674 2009-10-2 674 Model-Based Design for Embedded Systems (c) Time (b) Time (a) Time xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a:xv 3.10a:xv 3.10a:xv 3.10a:xv 3.10a:xv 3.10a:xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: xv 3.10a: FIGURE 20.20 Self-alignment results. (From Kurzweg, T.P. et al., J. Model. Simul. Micro-Syst., 2, 21, 2001. With permission.) To simulate this self-aligning system, we introduced random offsets in the lenses and in the VCSEL position and observe as the beam moves toward focus on the center detector. Snapshots of the image at the detectors are given in Figure 20.20 for three cases. The first results, shown in Figure 20.20a, are when the second lens is offset 35 μminthex-direction. Figure 20.20b shows the results of the second lenslet offset in both the −x-andy-direction by 35 μm. The final case has both lenses offset. The first is offset by 5 μminthe x-direction, and the second lens is offset by 35 μminthe−x-direction and 5 μminthey-direction. The results are seen in Figure 20.20c. Notice that the beam on the final images is not exactly in the center of the middle detector. This is because of the power being detected at this point exceeding the power threshold (98.6%) we set for alignment. 20.2.7.3 Angular Spectrum Optical Simulation of the Grating Light Valve In this section, we simulate and analyze a grating light valve (GLV) sys- tem in Chatoyant. This device has many display applications, including digital projection, HDTV, and vehicle displays. The GLV is simply a MEM Nicolescu/Model-Based Design for Embedded Systems 67842_C020 Finals Page 675 2009-10-2 CAD Tools for Multi-Domain Systems on Chips 675 (micro-electrical-mechanical) phase grating made from parallel rows of reflective ribbons. When all the ribbons are in the same plane, incident light that strikes normal to the surface reflects 180 degrees off the GLV. However, if alternating ribbons are moved down a quarter of a wavelength, a “square- well” diffraction pattern is created, and the light is reflected at an angle from that of the incident light. The angle of reflection depends on the width of the ribbons and the wavelength of the incident light. Figure 20.21 shows the rib- bons, from both a top and side view, and also the reflection patterns for both positions of the ribbons. The GLV component is fabricated using standard silicon VLSI technol- ogy, with ribbon dimensions approximately 3–5 μm wide and 20–100 μm long [44]. Each ribbon moves through electrostatic attraction between the ribbon and an electrode fabricated underneath the ribbon. This electrostatic attraction moves the ribbons only a few hundred nanometers, resulting in an approximate switching time of 20 ns. Since the GLV depends on a diffrac- tive phenomenon to direct the light beam, a rigorous modeling technique is required for modeling the GLV system. For the simulation of the GLV, we examine one optical pixel. A projected pixel is diffracted from a GLV composed of four ribbons, two stationary and two that are movable [44]. Each ribbon has a length of 20 μmandawidth of 5 μm. Ideally, there is no gap between the ribbons, however, in reality, a gap is present and is a function of the feature size of the fabrication. Although this gap can be modeled in our tool, in these simulations, we provide an ideal GLV simulation with no gap. The GLV is modeled as a phase grating, where the light that strikes the down ribbons propagates a half of a wavelength more than the light that strikes the up ribbons. In our model, light reflecting from the down ribbons is multiplied by a phase term. The phase term is similar to a propagation term through a medium: U down_ribbon = U exp(j2kd), where d is the distance that the ribbon is moved toward the substrate, typically λ/4 for the GLV. Far-field diffraction theory states that the diffracted angle reflected from the square-well grating is [36]: θ = qλ/a, where q is the diffraction mode Down ribbons (a) Up ribbons (b) Incident Reflected (c) Incident Reflected Reflected 1/4 λ Ribbons FIGURE 20.21 GLV device (a) top view and side view operation for, (b) up ribbons and, (c) down ribbons. . Nicolescu /Model-Based Design for Embedded Systems 67842_C020 Finals Page 666 2009-10-2 666 Model-Based Design for Embedded Systems Fourier transform can be implemented by one. optoelectronic 4f communications link. Nicolescu /Model-Based Design for Embedded Systems 67842_C020 Finals Page 668 2009-10-2 668 Model-Based Design for Embedded Systems a 300 MHz noisy signal. In. as a switch, an optical scanner, Nicolescu /Model-Based Design for Embedded Systems 67842_C020 Finals Page 670 2009-10-2 670 Model-Based Design for Embedded Systems (a) 23 14 (b) x y FIGURE 20.15 (a)