Optical propagation and refraction in silicon complementary metal–oxide–semiconductor structures at 750 nm: toward on-chip optical links and microphotonic systems

Currently, optical communications and opto-electronic integrated circuits are realized by utilizing “III-V” group semiconductors, such as gallium arsenide and associated compounds. They form the basis of a mature and quite complex technology. A deficiency is that these technologies cannot be integrated into silicon (Si) and mainstream complementary metal–oxide–semiconductor (CMOS) integrated circuit technology with ease.

A series of viable Si light-emitting technologies have recently become available that enable the integration of light sources directly into Si CMOS technology.^1,2 The creation of large-scale opto-electronic integrated circuits and optical data “highways” in CMOS integrated circuitry utilizing Si CMOS compounds have been envisioned and hold much promise.^3–8 The latest attempts for realizing optoelectronic systems in CMOS technology have until now mainly been focused utilizing wavelengths at 1550 nm,^9–11 mainly because of the ease of design and fabrication of waveguides in this regime. However, no effective optical sources and Si detectors are available at this wavelength.

If optical source, detector, waveguides, and sensors could be realized on the same CMOS chip at, say, 750 nm wavelength, various on-chip-based microphotonic systems can be realized. Achieving these goals can lead to diverse low-cost “all-silicon” opto-electronic systems, which will be the “smarter” and more “intelligent” CMOS chips of the future. These systems could lead to new products for especially the medical and bio world. Such a new field could be appropriately named “Si CMOS photonic microsystems.” These systems also do not require ultrahigh frequency bandwidths and the emission powers of these Si light-emitting diodes (LEDs) may be sufficient to sustain the operation of such systems. This can lead to many new products and open up new markets.

The main objective of our research was, therefore, to directly integrate the sources and detectors for optical sensors on chip. This would have benefits in terms of cost reduction, miniaturization, the simplification of packaging requirements and perhaps also in total power requirements.

Substantial progress has been made in the field of Si $p−n$ junctions emitting visible light when operating in reverse breakdown avalanche mode.^12–18 Recently, Si CMOS light-emitting sources appeared with much higher efficiencies by implementing two and three junction Si CMOS injection-avalanche LEDs that emit at 450 to 750 nm.^19,20 They can be integrated with ease in standard CMOS circuits. Various prototypes have been realized. Their operating voltage (8 V), and current (80 μA to 1 mA) are low. They emit from an area of $1×1μm$ between 10 and 100 nW. The internal light emission levels are much higher. A particularly promising technology that uses defect and surface-assisted transitions in these types of LEDs has recently been identified and developed by our specific group.^20–22

Our Si avalanche-based Si LEDs operate optimally at 650 to 850 nm. Although the optical emission power of Si avalanche LEDs is low, they can reach modulation speeds into the GHz range. Since Si LEDs and Si detectors are both available and can be integrated with relative ease into CMOS technology, the most challenging part of following this route is, hence, to develop appropriate CMOS waveguide technology in this wavelength regime in CMOS integrated circuitry.

The purpose of this paper is, then, (1), to present some detailed simulation results for realization of microphotonic structures in 1.2 and 0.35 μm CMOS integrated circuitry at mainly 750 nm operating wavelengths; (2) to demonstrate the possible performance of CMOS waveguides at 750 nm; (3) to demonstrate a realized first iteration optical link in Si-CMOS integrated circuitry at 650 nm; and (4) to propose some first iteration CMOS-based microphotonic systems and micro-opto-electro-mechanical systems (MOEMS) applications.

Figure 1 shows a typical CMOS integrated circuit structure using 1.2-µm field oxide-based technology. Important features are the transparency of some of the over-layers, such as the field oxide layer, the inter-metallic oxide layers, and the Si nitride passivation layer between 450 and 750 nm wavelengths.^23,24 Other transparent layers such as Si oxi-nitride and Si dioxide can be added as further over-layers.^25–27 The optical transparency of the different layers varies slightly. The dimensions and optical variables as associated with above 0.35 μm CMOS integrated circuit technology are presented in Table 1. The advantage of using this type of technology for microphotonic systems is that the field oxide introduces curvatures into the over-layers that are beneficial for generating microphotonic structures.

The below 0.35 μm CMOS technology is in principle similar, except that the Si field oxide layer is thinner (250 nm) and a planarization process is used after transistor fabrication in order to accommodate a higher number of metal over-layers. The transparency of Si is poor, while the transparency of the field oxide layer is excellent. Si nitride and Si oxi-nitride are transparent for wavelengths longer than 600 nm.^25–27 This is substantially lower than the absorption edge of Si, which lies at 1100 nm. If Si LEDs and Si $p−n$ detectors are used, Si CMOS technology does not have to rely on III-V or Si–Ge technology. In particular this technology utilizes trench isolation technology that can be used for generating in-line opto-couplers and waveguides to even below the Si substrate-overlayer interface.

Si avalanche light-emitting devices in the 450 to 650 nm regimes have been in use for a long time.^12–18 The fabrication of these devices is high-temperature compatible and can be used in standard Si designs. CMOS-compatible avalanche Si LEDs (Si CMOS Av LEDs) have emerged since the early 1990s. Kramer and Zeits²⁸ were the first to propose the utilization of Si Av LEDs inside CMOS technology. They first illustrated the potential of this technology. Snyman et al., have realized a series of light-emitting devices in standard CMOS technology, such as micro displays and electro-optical interfaces with higher emission efficiencies and higher emission radiances (intensities).^19–21 Particularly promising results have been obtained regarding efficiency and intensity, when a combination of current density confinement, surface layer engineering, and injection of additional carriers of opposite-charge density into the avalanching junction, were implemented.¹⁹ These devices showed three orders of increase in optical output as compared with previous similar work. However, increases in efficiency seemed to be compromised by higher total device currents because of loss of injected carriers, which do not interact with avalanching carriers.

Latest analyses of the work of Kramer et al.²⁸ and Snyman et al.,²⁰ shows that longer wavelength emissions up to 750 nm can be achieved by focusing on the electron relaxation techniques in the purer n-side of the Si $p−n$ avalanching junctions. This development has a very important implication. The spectral radiance of this device compares extremely well with the spectral detectivity of the Si reach through avalanche photo detector (RAPD) technology. A particularly good match is obtained between the emission spectrum of this device and the absorption spectrum, which can be detected with an RAPD. More information about the design and performance aspects of our latest $E$-field and momentum shift assisted 750 nm Si LEDs can be found from Snyman and Bellotti.²⁰ By controlling the defect density in this device, one can favor either the 650 or 750 nm emissions. Total emission intensities of up to 1 μW per $5μm2$ area at the $Si-SiO2$ interface have recently been observed.

Figure 2 shows a design of a top-down as well as cross-section layout of a Si CMOS and avalanche operation-based LED. The LED consists of an $n+$ bar placed opposite to two $p+$ bars and diagonally away from the $n+$ bar. All were embedded in a p-well containing $2×1016$ ions. The $n+$ and $p+$ regions define the peripheries of the $SiO2$ field oxide layer and define the center $SiO2$ field oxide layer. Upon biasing the structure with positive bias voltage $+V$ on the $n+$ and negative bias voltage $−V$ on the $p+$ region, the depletion region extends from the $n+$ edge facing the $p+$ region in the $SiO2$ field oxide layer.

The intensity of the emitted light was approximately 1 nW per micron length at an operating voltage of 15 V. A current of 80 μA to 1 mA was emitted between 450 and 750 nm with a tail reaching deep into the near infrared region (Fig. 3).

Si avalanche-based LEDs can be integrated into CMOS circuits with ease, and offer the following advantages:

Figure 4 shows the layout of a simple $n+p$ well Si avalanche LED and interface that can feed light into the adjacently lying optically transparent field oxide layer. The light is emitted slightly below the apex of the $SiO2$ “field oxide” region. The positioning of the light-emitting region ensures optimum coupling of light into the layer.

Special detectors have been designed to ensure optimum detection efficiency for laterally incident radiation.¹⁵

It follows, consequently, that much more advanced and more favorable electro-optical structures can be designed and fabricated utilizing CMOS technology.

Since most optical simulation tools, such as those available on the free market today, are designed for application in the macro environment, we opted to design our own initial software tool for specific application in CMOS structures and dimensions. The commercial tools are also mostly designed for application in the macro-optical environments while a custom-designed tool would offer better customization and adaptation to targeted needs. This also offered additional design flexibility and better understanding of our analysis, as compared to merely utilizing available “ray tracing” software.

Our group subsequently used MATLAB²⁹ to geometrically develop specific CMOS structures of specific optical properties and to evaluate the optical refraction and optical propagation phenomena in such structures.

A special ray tracing program was subsequently developed, which could specifically analyze the refraction and optical propagation phenomena in CMOS integrated circuit structures and identify the geometrical positioning of the optical sources within these structures. Furthermore, we could include a series of iterative propagation and scattering paths using the Monte Carlo randomization techniques. The refractive indices that were used in the optical simulation at 750 nm were: 1.00 for air, 2.00 for Si nitride, 1.48 for Si plasma, 1.45 for Si oxide and 3.76 for the Si substrate.^10,30

The domain of the modeled Si CMOS structure was defined by geometrical algebraic statements generated with MATLAB in the code called “K-Dedicated Code A.” Matrix geometrical algebraic statements were used to generate the graphical preliminary matrix elements look-up table for a typical CMOS structure (see Appendix ¹).

Subsequently, specific substructures were developed with mathematical formulas and boundaries. This approach subsequently led to:

The geometric blocks used to build up the structure were limited to basic triangular and rectangular shapes, because CMOS structures can be generated and modeled by these shapes effectively. As for the triangles, two sides (TSIDE_1 and TSIDE_2), one angle (TANGLE12) in between the sides, and one vertex (TVERTEX1) were the parameters used to construct the various triangles. Likewise for the rectangles one, vertex (RVERTEX1) and two sides (RSIDE1 and RSIDE2) were used to construct the various rectangular shapes involved in the generation of the CMOS structure.

Figure 5(b) subsequently illustrates how a complex new CMOS structure was eventually generated using a number of the pre-defined sub-block structures, each defined with particular dimensions, refractive index, and color information.

A stepping algorithm was developed as a subroutine within the “K-dedicated code B” and applied to multiple rays, each with a different initial launch angle. Snell’s and Fresnel laws were applied as basic physical laws governing refraction, reflection, and even for determining polarization characteristics.³¹

For the total internal reflection of the rays within the structure, the critical angle for internal reflection was used as Display Formula

For detailed reflectance calculations, the following Fresnel equations were used: Display Formula

The concepts of Monte Carlo-based randomization³² were subsequently also incorporated in the stepping algorithm. This algorithm progressively performs a stepping function in which a new propagation position and new launch angle is defined based on the local matrix information values. If a change of refractive index did occur, the incident angle to the interface was determined, Snell’s law for refraction and Fresnel laws for reflection were applied, and new rays with new propagation positions and launch angles were defined within the larger matrix. An example of the program code for determining refraction when encountering an interface is shown in Program Snippert 2 (See appendix ¹). Display Formula

Figure 7 shows the propagation of the optical rays when the optical source is shifted slightly to the left along the Si-plasma deposited oxide interface. Improved coupling was observed into the Si nitride layer. Total internal reflection is observed for some of the waves in the Si nitride layer.

Figure 8 shows the ray propagation when the thickness of some of the field oxide and plasma deposited oxide layers was increased. The optical propagation in a variety of CMOS structures was subsequently investigated. The vertical thickness of the layers was varied. Figure 8 also demonstrates that the optical power could be separated into two different paths. A certain percentage of the radiation escaped out of the design, while the rest was emitted laterally into the Si nitride layer. Optical splitters of this kind have diverse applications in MOEMS and can be used for interference and phase contrast.

Figure 9 shows a structure in which the curvature caused by the field oxide layer was utilized. The optical source is positioned in the middle of the hemispherical structure. The launch angles for the rays range from $30degt$ to 150 deg. The radiation is emitted under a solid angle of 120 deg.

The estimated optical emission factor of the $Si-SiO2$ interface source is approximately 0.40. Minor scattering is observed due to surface bending irregularities at the Si nitride-air interface and absorption losses occur in the Si nitride layer at shorter wavelengths.

Figure 10 shows a similar structure, but here the light source emitter position is shifted. When the source is placed at position A, near to the “bird’s beak” field oxide layer, the rays emitted vertically from the structure are slightly focused. When the source is placed at position B, the radiation is emitted toward a 45-deg slanted angle.

Figure 11 demonstrates optimized wave-guiding of the total emitted Si LED radiation inside the Si nitride layer. The plasma deposited oxide layer thickness has been reduced and the optical emission point was positioned near the “bird’s beak” point in the Si oxide layer. Ray launch angles ranging from 34 deg to 76 deg couple effectively the optical radiation into the Si nitride layer. Because of the lower refractive indices of both the plasma-deposited and field-oxide layers below and above, the optical rays can effectively propagate inside the Si nitride layer. Si nitride is transparent for wavelengths above 600 nm. Si oxi-nitride ($SiOxNx$ is also transparent at lower wavelengths.^25,30 In both cases, the wavelength of the wave-guided radiation is much lower than the absorption edge wavelength for Si detectors (1100 nm). The combination of an Si LED light source, an optical waveguide, and an Si detector can hence be incorporated with ease into the same structure. A particular CMOS microphotonic structure can hence be realized in the CMOS integrated circuitry.

The current total optical emission levels from our Si CMOS LEDs are in the order of 1 to 100 nW. These power emission levels are much higher than the low power detection level of Si $p−n$ detectors.

CMOS technology below 0.35 μm can also be utilized to design lateral waveguides. CMOS designs below 350 nm are planarized after the MOSFET transistor fabrication. Up to six metal layers are deposited on top of these layers, where sloping of the outer layers is caused by the thicker outer metal layers.²³ This technology uses trench isolation for electrically isolating $n$- and $p$ MOSFETS laterally in the CMOS structure. The trench-isolation technology opens up interesting optical properties. Trenches are spatially defined by RF etching followed by oxide lining and subsequent plasma deposition of Si oxides. If these trenches could be filled with an optical material of higher refractive index, optical radiation emitted from the Si-overlayer interface, could then be coupled with high efficiency directly into the adjacent optical channels. The current CMOS technology can create a thin oxidation layer that is used as an isolation layer in the trench technology. If this layer is enhanced and followed by a layer of high refractive index material, such as Si nitride, interesting lateral optical conductors and waveguides can be constructed at the Si–overlayer interface. Optical sources positioned at the edges of the trenches enable optimum coupling of optical radiation into the trench waveguide. In a similar way waveguides can be fabricated in the outer CMOS layers. Figure 12 schematically illustrates these concepts.

We used advanced optical simulation software (RSOFT Beam PROP)³⁴ to design and simulate specific CMOS-based waveguides operating at 750 nm based on CMOS materials and parameters. This software uses an advanced finite analysis method where the volume is divided up into a large number of matrix elements. Helmholtz’s equation predicts the optical parameters along the matrix based on the initially defined optical fields.

We performed a number of optical simulations in CMOS structures with 0.2 to 0.6 μm-wide channels, embedded in 1.5-µm diameter Si oxide. The operating wavelength was 750 nm. A flat emission surface with wide-angle emission were used at the edge structure of the Si nitride, in order to simulate the emission characteristics of our Si avalanche LED source best.

Our preliminary results show that multimode, as well as single mode wave-guiding can be achieved. Figure 13 shows a three dimensional view of the electrical field along the waveguide of a 0.6 μm wide Si nitride waveguide. Multimode propagation is demonstrated with almost zero loss up to a distance of 20 μm. Multimode propagation in CMOS micro systems has the advantage of having a large acceptance angle for coupling optical radiation from a Si LED into and out of a waveguide. Our calculations show that a coupling efficiency of 0.38 can be achieved for a flat Si-emitting surface positioned at the edge of the Si nitride core of the waveguide. This is mainly due to the better refractive index matching between Si (3.76 at 750 nm wavelength) to Si nitride (2.00 at 750 nm).

Figure 14 shows the predicted optical simulation for a 0.3-µm diameter Si oxi-nitride waveguide embedded in Si oxide. The two-dimensional plot of the electrical field propagation along the waveguide shows clearly single-mode propagation. The calculated loss curve in the adjacent figure shows almost zero loss over a distance of up to 20 μm. The coupling efficiency into such a waveguide is, however, much reduced to values of about 0.05. Multimode to single-mode converter structures can, however, be used to increase overall coupling efficiency into the waveguide.

Figure 15 displays the mode field pattern in the waveguide perpendicular to the axis of propagation. With the values derived from the simulations for the real part of the propagation constant, an accurate energy loss could be calculated with conventional optical propagation physics. Low loss of $0.65dB·cm−1$ is predicted at 650 nm. At shorter wavelengths from 450 to 650 nm much higher losses of more than $1dB·cm−1$ are predicted because of the higher absorption coefficient in Si nitride at these wavelengths. The bandwidth-length product could be derived from the calculated effective refractive index and predicted modal dispersion values calculated by RSoft BeamProp (Fig. 14). The calculations predict a bandwidth-length product of approximately 200 GHz-cm.

With a short wavelength and high refractive index difference between core and cladding (1 μm in diameter), a high level of micro bending can be achieved. Since with CMOS technology many different over-layers can be deposited on top of each other, optical energy can easily be coupled between waveguides and optical sources near the Si over-layer interface to layers higher up in the structure.

Another feature that was investigated was the positioning of a two-dimensional Si nitride lens at the end of the waveguides. Such a design can be used to obtain two- and three-dimensional control of the optical radiation at distances of up to 10 μm away from the end of the waveguide. Figure 16 demonstrates “focusing” of the optical radiation from a single-mode Si nitride waveguide.

These initial simulations of Si nitride waveguides in CMOS technology are extremely promising. All the latter designs can be applied to CMOS-based microphotonic systems.

Figure 17(a) and 17(b) display an optical emission micrograph as observed vertically upward into the microscope objective and as emitted from a 0.35 μm CMOS structure as shown in Fig. 17(b). This CMOS structure closely resembled the cross-section over-layer structure presented in Fig. 18. The emission occurred at a $n+p$ junction interface adjacently to the field oxide layers. Interesting features as observed are the following:

A very bright vertical emission was observed. This emission is a result of the vertical emission characteristics as simulated in Fig. 7. The total optical emission power of 100 nW was observed for these structures using a calibrated Si $p−i−n$ detector. Red light was observed at one side next to the vertical emission. This indicates that more red light were refracted into the acceptance angle of the microscope, while the blue light was refracted toward the surface of the structure. This correlates with the simulation results as obtained in Fig. 6, which demonstrated less dispersion of emitted light vertically outward from CMOS structures with a silcon nitride top layer and high dispersion along the surface of the chip for shorter wavelengths. This observation, therefore, indicates good correlations between the optical simulation results as predicted by our optical simulation techniques.

Figure 18(a) and 18(b), demonstrates a first iteration optical link as was realized in Si 1.2-µm CMOS integrated technology over a distance of 150 μm. The optical source was of the type as demonstrated in Figs. 2 and 4. The CMOS waveguide design was similar to the one presented in Figs. 11 and 12(a). Optical coupling was realized into the top Si nitride layer similar to what was demonstrated in Fig. 8. The detector was a custom-designed Si detector, of $10×10μm$ dimension, and optimized for lateral incidence. Some first iteration results with regard to this realization have been published elsewhere already.²¹

This particular realization was achieved in the Carl and Emily Funch Institute of Microelectronic Laboratories University of Pretoria in South Africa. We have recently achieved similar results in ESIEE Labs in France (Europe). This is currently the topic of further research.

When the Si LEDs were pulsed at low frequencies (below 1 kHz), signal levels were ranging from 6 nA to 100 nA. When phototransistor detectors were used, signal levels were detected of about 1 μA. No signals were observed at the detector when the voltage pulses were repeated at these voltage levels, but without optical emission. It should be appreciated that these results are first iterations. The photomicrograph shows high losses at the optical source. Assuming a detector efficiency of $0.3W/A$, and coupling efficiencies of 0.1 from the source to the waveguide, as well as a loss of $0.75dB·cm−1$ at 750 nm for wave guiding in the Si nitride layer itself, indicate source optical power in the 100 nW range. This correlates with our earlier optical measurements for vertical emission from our Si Av LEDs as in Fig. 17. The loss characteristics for the waveguide correlate with loss characteristics for Si nitride at 750 nm as determined by Gorin et al.³⁰ These analyses provide an important basis for further developments.

It follows that the described technologies can be integrated either in hybrid or monolithic form in standard CMOS designs, realizing various microphotonic systems and/or MOEMS. Si LEDs and associated electronic processing circuitry can be integrated into standard CMOS technology allowing mass production at low cost. Optical modules and CMOS technology can be integrated in hybrid form using RF recess technology. With waveguide technology, whole MOEMS structures can be grown monolithically on CMOS circuits. MOEMS ranging from vibration sensors, motion and accelerometer sensors to more advanced chemical, physical, and biosensor as well as nanotechnology devices can be constructed. These applications are illustrative at this stage, but nevertheless demonstrate some interesting possibilities and applications.

In Fig. 19 a hypothetical microphotonic system is demonstrated that consists of an Si LED, a Si detector together with waveguides integrated monolithically in a CMOS structure.³⁵ Wide area Si LEDs are used in order to increase the total optical emission power into the waveguide systems. Appropriate filtering by means of ring resonators, and enhanced phase contrast detection can be obtained by utilizing unbalanced Mach-Zehnder interferometers. Near the end of the layout, an opening is integrated in the CMOS over-layers by post processed RF etching, that enables gas or liquids to interact with the evanescent field of a waveguide section and introduces intensity and/or phase contrast changes. Adjacent digital and analog circuitry processes the observed information. Hence, a complete microphotonic sensor system can be integrated into standard CMOS circuitry. The added intelligence component, the lowering in cost and the increase in reliability of such systems can be significant.

Figure 20 demonstrates a more explicit and hypothetical MOEMS system. The system consists of a Si CMOS LED, an array of Si detectors and a series of CMOS-based waveguides. A mechanical arm is RF etched into the structure by postprocessing. Radiation is directed at an angle onto the mechanical arm, and the reflected light is sensed by an array of CMOS detectors. Such a system can detect vibration, acceleration, and axial rotation and relate these to electronic output signals.

Similarly, in the cavity optical absorption measurements, fluid flow and gas flow as well as fluorescence experiments of samples can be performed by arranging source and detector and waveguide arrays on the side surfaces of the cavity. The potential applications are diverse and can be linked to emerging fields, such as bio- and nanotechnology.

PROGRAM SNIPPERT 1 //Pseudo-code/block of flow diagram for domain definition//

$maxX=160;$

$sizeX=maxX*10;$

$maxY=116.4;%Fixed$

$sizeY=maxY*10;$

$x=0∶sizeX;$

$y=171−50+(x−50)*(x>50)−(x−100)*(x>100);$

$z=134−50+(x−50)*(x>50)−(x−100)*(x>100);$

$q=98−50+(x−50)*(x>50)−(x−100)*(x>100);$

$s=53−50+(x−50)*(x>50)−(x−100)*(x>100);$

$yn=−y;$

$zn=−z;$

$qn=−q;$

$sn=−s;$

$refractive_indices=[12.41.51.43.53.53.5];$

$forj=1:sizeX%Column$

 $m=1;$

  $fork=(−201+0.345:0.345:201−0.345)$

   $ifk>y(j)$

    $Body1(j,m)=refractive_indices(1);$

   $else if(k>z(j))&&(k<y(j))$

    $Body1(j,m)=refractive_indices(2);$

   $else if(k<q(j))&&(k<z(j))$

    $Body1(j,m)=refractive_indices(3);$

   $else if(k<s(j))&&(k<q(j))$

    $Body1(j,m)=refractive_indices(4);$

   $else if(k<sn(j))&&(k<qn(j))$

    $Body1(j,m)=refractive_indices(6);$

   $else if(k<qn(j))&&(k<zn(j))$

PROGRAM SNIPPERT 2 //Pseudo code or flow diagram for structure shape generation//

$RT=1;$

figure(1)

axis equal

$INCLINATION_CHECK=45;%67.5;$

$ANGLE_CHECK=90;%45;$

$ANGLE_CHECK_HALF=45;%22.5;$

$VERTEX_MATRIX_X=□eros(4,1);$

$VERTEX_MATRIX_Y=□eros(4,1);$

$BLOCKSHAPE=[];$

$BLOCKNUMBER=1;$

$BLOCKSHAPE(BLOCKNUMBER)=0;%0for triangle$

$TVERTEX1=[0;0];$

$TSIDE1=RT;$

$TSIDE2=RT;$

$TANGLE12=ANGLE_CHECK;$

$TROTATION=INCLINATION_CHECK;$

$TCOLOR=‘y’;$

$REFRACTIVE_INDEX(BLOCKNUMBER)=1.4;$

$[ab]=trblock(TVERTEX1,TSIDE1,TSIDE2,TANGLE12,TROTATION,TCOLOR,REFRACTIVE_INDEX);$

$VERTEX_MATRIX_X(:,BLOCKNUMBER)=[a;0];$

$VERTEX_MATRIX_Y(:,BLOCKNUMBER)=[b;0];$

$fill(a,b,TCOLOR)$

hold on

$BLOCKNUMBER=BLOCKNUMBER+1;$

$BLOCKSHAPE(BLOCKNUMBER)=1;%1for rectangle$

$[xs,ys]=getstart(a,b,2);$

$RVERTEX1=[xs;ys];$

$RSIDE1=RT*3;$

$RSIDE2=RT*cosd(INCLINATION_CHECK)*2;$

$RROTATION=90;$

$RCOLOR=‘y’;$

PROGRAM SNIPPERT 3 //Fresnel formula for refraction and Snell’s law of reflection//

$bb=angle_next_point(f);$

$ifabs(BODY2(yy(step,f),xx(step,f))/BODY2(yy(step+1,f),xx(step+1,f))*sind(90+inc(f)−angle_next_point(f)))<=1$

  $angle_next_point(f)=90+inc(f)−$$asind(BODY2(yy(step,f),xx(step,f))/BODY2(yy(step+1,f),xx(step+1,f))*sind(90+inc(f)−angle_next_point(f)));$

  $signR=1;$

  $FLAG=0;$

else

$disp(‘REFLECTION’)$

$ifangle_next_point(f)−inc(f)>90&angle_next_point(f)>90$

  $angle_next_point(f)=−angle_next_point(f)+2*inc(f);$

elseif

$angle_next_point(f)−inc(f)<90&angle_next_point(f)>90$

  $angle_next_point(f)=angle_next_point(f)−2*inc(f);$

elseif

$angle_next_point(f)−inc(f)>90&angle_next_point(f)<90$

  $angle_next_point(f)=angle_next_point(f)−2*inc(f);$

elseif

$angle_next_point(f)−inc(f)<90&angle_next_point(f)<90$

  $angle_next_point(f)=−angle_next_point(f)+2*inc(f);$

   $endangle_next_point(f)$

$xx(step+1,f)=xx(step,f);$$yy(step+1,f)=yy(step,f);$

 $a(f)=BODY2(yy(step,f),xx(step,f));$

  $FLAG=1;$

En

This work has been made possible with the financial support of Tshwane University of Technology (TUT), the Laboratory for Innovative Systems in Electronics of the Faculty of Engineering and the Built Environment Electrical Department at TUT and the National Research Foundation of South Africa. All the simulation work and concepts as presented in this article were generated by means of South African National Research Foundation grants FA200604110043 (2007-2009), NRF KISC grant 69798 (2009-2011) and SANRF travel block grants (2007, 2008, 2009). The utilization of device design and characterization facilities at the Carl and Emily Fuchs Institute for Microelectronics at the University of Pretoria, South Africa, in realizing some of the experimental devices and results is gratefully acknowledged. Mr. Gustave Udahemukha is thanked for his technical contribution to the MATLAB code used in the first iteration simulations. The technical content as contained in this article currently forms the subject of the current PCT patent applications PCT/IB2010/050356; PCT/ZA2010/00032, PCT/ZA2010/00031, PCT/ZA2010/00033, and PCT/ZA2011/000090 of 2010 and 2011. It is now also the subject of the following USA Patent Applications: 20120153864, 20120154812, 20120170942, 20120175642. Several provisional applications on Si LEDs, MOEMS, waveguide design, matrix and array interfaces, modulators and electro-optical couplers serve as background priority patents to these patents. The authors wish to thank the anonymous reviewers for their constructive contributions towards finalising the article.