DESIGN OF RING-BASED OPTICAL FILTERS WITH A GAP IN THE LOOP

Transcription

1 Faculty of Engineering Laurea Magistrale in Telecommunications DESIGN OF RING-BASED OPTICAL FILTERS WITH A GAP IN THE LOOP Student Teacher Josep Viñals Barturò Gabriella Cincotti Roma, July, 2011 Academic year 2010/2011

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3 INDEX 1. INTRODUCTION FILTERS Introduction to filters Digital filters Optical filters RSOFT RSOFT CAD Environment FullWAVE THE GAP Theoretical study of the gap RSOFT simulations of the gap SINGLE-STAGE RING FILTER Ring filter without gap Ring filter with gap Theoretical analysis Losses of the filter Relation between bandwidths and the transmittance / reflectance RSOFT simulations Comparison Matlab RSOFT MULTI-STAGE GAP FILTER Multi-stage filter with 2 stages Theoretical equations RSOFT simulations Multi-stage filter with 2 stages with a distance with L between stages Theoretical analysis RSOFT simulations Multi-stage filter with 3 stages Theoretical analysis RSOFT simulations... 51

4 6.5 Comparison between filters Comparison using Matlab Comparison using RSOFT CONCLUSIONS REFERENCES ANNEXES Single stage ring Single ring without gap equations Single stage ring without gap MATLAB code Single stage ring with gap equations Single stage filter with gap Matlab code file Multi-stage ring filter Multi-stage ring filter with 2 stages equations Multi-stage Matab code Multi-stage filter with a difference between stages different to 2L equations Multi-stage filter with 3 stages equations Matlab comparison code... 81

5 1. INTRODUCTION In the world of communications, optical networks are playing an important role and the bandwidth growth is many related to the Wavelength Division Multiplexing (WDM) technology, where channels at different frequencies are transmitted in the same optical fiber, allowing an efficient use of the optical spectrum. Optical filters are key devices for WDM systems as they implement some key roles like: - Multiplexing, demultiplexing and add/drop sub-channels. - Gain equalization - Dispersion compensation In general, ring-filter architectures have only one degree of freedom that is the coupling ratio between the waveguide and the ring. To add flexibility to the design of the filter we consider adding a gap in the loop, in this way have two degrees of freedom. The main goal of this thesis is to analyze the performance of a ring-based filter with a gap in the loop and then to design a multi-stage architecture, composed of ring-based filters with gap. 1

6 The thesis is organized as follows: - First Chapter: This chapter describes the filters; it starts with a basic introduction on filters and an explanation about digital filter. Finally there is a description of optical filters, defining their applications and properties. - Second chapter: In this chapter the software RSOFT used for the simulations is described. Among different programs of this software suite, we used: RSOFT CAD Environment and FullWave. - Third chapter: This chapter presents an investigation of the gap. First with a theoretical study of the equations of the gap and a numerical simulation with MATLAB and RSOFT to calculate the transmittance and reflectance parameters. - Fourth chapter: In this chapter there is the study of single-stage ring filters, where the analytical formulation is followed by RSOFT simulations. The first filter studied is a standard ring filter, and then we analyze a ring-filter with a gap in the loop. For the filter with a gap there is also a study of the behaviour of the filter bandwidth as a function of the transmittance of the gap. - Fifth chapter: The final chapter reports the studies of multi-stage filters, that are investigated theoretically and with RSOFT simulations. We consider multi-stage filters of 2 and 3 stages, where each stage ring has a gap in a loop. Finally there is a comparison between all filters studied in the thesis. 2

7 2. FILTERS 2.1 Introduction to filters A filter is a device that is processes an input signal x(t) to obtain an output signal y(t) with a determined and desired frequency characteristic. The filter changes the amplitude and the phase of the input signal depending on the frequency. The behaviour of the filter can be defined by its impulse response h(t), that is the response of the filter when the input is a delta function. The generic block diagram of a filter is the following: Figure 2.1: Filter s generic block diagram The filter considered in this thesis are assumed to be linear and time-invariant, therefore the impulse response satisfies these two properties: - Linearity: A linear system has to satisfy two conditions. If in the input there are two signals, the output is the superposition of the corresponding outputs. Moreover, if the input signal is multiplied by a 3

8 constant, the corresponding output is also multiplied by the same constant. Both properties can be expressed as: - Time Invariance: In a time invariant system the output does not change, although the input is applied at different times. The condition can be defined as: So, an input shifted, will give an output also shifted the same time interval. There are two more conditions that a filter has to fulfill: - Causality: A causal system doesn t give any response (output) before the arrival of the input. So, if an input is applied at, the system will not give any response before. The impulse response of a causal system is zero for. - Stability: A stable system has a bounded output if the input is also bounded. 4

9 This condition is fulfilled if the impulse response function is absolutely summable: 2.2 Digital filters A digital filter is an algorithm which implements over an input discrete-time signal some mathematical operations, depending on the impulse response. The study of digitals filters is useful to analyze and design optical filters; this is because lot of algorisms developed for digital filters are useful to design optical filters. The impulse response of a generic digital filter can be defined using the Z- transform: Where, and w is the angular frequency.this expression can also be defined by the roots of the two polynomials: Where, are the zeroes and are the poles. The gain is, and in a passive filters its modulus cannot be larger than 1. These poles and zeros can be represented in a graph called pole-zero diagram: 5

10 Figure 2.2: Pole-zero diagram Using this diagram it s easy to analyze certain properties of the transfer function, for example: stability, causality... Depending on the type of transfer function, digital filters can be classified as: - Finite Impulse Response (FIR): If the input is finite the output is also finite, namely the output go to zero in a finite time. Another feature is that the output only depends on the input. The most common type of FIR filters is the Moving Average filters. o Moving Average (MA): the impulse response has only zeroes. The transfer function equation is: In the following plot there is the magnitude response behaviour of a MA filter with different root values: 6

11 Figure 2.3: Magnitude response of a MA filter - Infinite Impulse Response (IIR): These filters are recursive; the output depends on the current and previous input, and also on the previous output. There are two types of filters IIR: o AutoRegressive (AR): the impulse response of AR filters has only poles. The transfer function can be expressed: The behaviour of the magnitude response for an AR filter in function of the root magnitude is shown in the following plot: 7

12 Figure 2.4: magnitude response of an AR filter o AutoRegressive Moving Average (ARMA): The transfer function of this type of filters has both poles and zeroes. Digital filters have many advantages respect to analog filters, as an example: - Larger noise immunity - High accuracy - Easy parameters modification - Very low cost 8

13 2.3 Optical filters An optical filter is a device that let pass some frequency bands (wavelengths) and attenuates or erases the others. Optical filters play an important role in WDM systems. The main characteristics for an optical filter are: - Flexibility to implement different pole/zero diagrams - Low pass-band losses - Low sensitivity to fabrication variations - Low dependency to environmental conditions - Long-term stability However, depending of the application, the priority of these features can vary. For example, a filter for multiplexing will requires low loss, while a filter for demultiplexing requires a large stop band rejection to minimize cross-talk interferences. In general, the operation of filtering is obtained by the interference of two or more waves with different relative delays between them. The incoming optical field is divided in different waves, each wave travels a path with a different length, and finally they recombine causing the interference. The different phase of each wave determines if the interference is constructive or destructive. The phase introduced by each path is: where: 9

14 So, the equation of the output field is the sum of the fields for each path: The frequency behaviour of the filter has a periodicity of 1/T, and is known as the FSR (Free Spectral Range): In this thesis, the basic optical filter analyzed is constituted by a bus waveguide with a coupled ring. The input field is divided by a directional coupler of the rectilinear waveguide and the ring, so that interfering field will go into the ring and returns to the waveguide thanks to the coupler. The layout of this filter is shown in the following figure: 10

15 Figure 2.5: layout of a standard ring-based filter To modify the filter performance a gap has been added in the ring. This gap gives to the filter a behaviour more selective in frequency. This is the single stage filter that will be analyzed in the next chapters. The layout of the filter is shown in the following figure: Figure 2.6: layout of a ring-based filter with a gap in the loop To improve the performance of a single-stage filter we have considered a multistage architecture composed of some stages of the ring with gap filter. In this 11

16 thesis, we have investigated an architecture with a series of ring filters. With a multi-stage filter it is possible to better approximate the behaviour of the transfer function to the desired one. Of course, in the multi-stage optical filters a desired characteristic will be minimize the number of stages used to implement the filter, this is to reduce the complexity, the device size and sensibility to fabrication variations. 12

17 3. RSOFT RSOFT product family is a software suite for optical communications design and simulation. These products have applications for: - Component design: analyze and design optical components, in this group RSOFT has products for passive and active components. The products of RSOFT used in this thesis are from this group. - System simulation: simulate optical telecom systems at a signal propagation level. - Network modelling: RSOFT has a software suite of optical networks design with the capability of simulate discrete events. In this thesis it s used mainly two programs of the RSOFT suite: RSOFT CAD Environment and FullWAVE. 3.1 RSOFT CAD Environment RSOFT CAD environment is used to design the layout of optics devices and circuits, and it s also used to control the simulation software. We can define the material and geometric characteristics of the filter circuit. 13

18 The main screen of the program: Figure 3.1: Main screen of RSOFT CAD Environment To design correctly the circuit, we have first to define the global settings: the free-space wavelength, background index, index difference (the index difference between the background and the waveguides), waveguide width and select the simulation tool. After defining these parameters, we can draw the circuit waveguides, and finally define the simulation options in the FullWAVE simulation screen. 3.2 FullWAVE FullWAVE is an integrated CAD tool and simulation engine that calculates the electromagnetic field as function of time and space. The simulation engine that implements FullWAVE is based on FDTD (Finite Difference Time Domain). 14

19 FDTD algorithm is a very common tool because it s robust and easy to implement. All these characteristics make this algorithm one of most used to solve problems of interaction between electromagnetic wave and material structures. The FDTD algorithm solves the Maxwell s equations discretizing the equations on a time and space grid and then solving these equations with numerical methods. To perform a FullWAVE simulations we have to set some parameters, which can be classified between physical and numerical. The physical parameters are: - The refractive index distribution of the circuit as function of the space coordinates. - The electromagnetic input field as function of time and space. The numerical parameters: - A finite computational domain (x, y, z): physical region where the simulation will be performed. - Boundary conditions. - Spatial grid sizes (Δx, Δy and Δz). - Temporal grid (Δt). The physical parameters are defined on the design of the circuits; on the other hand, numerical parameters are related to on the simulation parameters screen: 15

20 Figure 3.2: FullWAVE simulation parameters screen It s important to define also some FullWAVE parameters: first set the default launch excitation to pulsed, and then change the parameters TimeStep and StopTime. The parameters Timestep and StopTime define the desired resolution in the simulation; if the StopTime is increased the resolution will increase too, but the estimated time of the simulation also increases. Therefore a trade off between the resolution and simulation time has to be accomplish. 16

21 4. THE GAP 4.1 Theoretical study of the gap To study the ring filter with a gap and simulate it, it s useful to study the gap, analyzing the field transmitted and reflected. We start defining the model and the gap equations of the transmitted and the reflected field. We can model the gap as: Figure 4.1: Gap model Where: A1+ = Incident field A1- = Reflected field A2+ = Output field 17

22 Figure 4.2: Symbolism of the fields in a gap Where: In the case there are no losses in the gap this condition is satisfied: 18

23 4.2 RSOFT simulations of the gap To obtain the correct values of t 1 and r 1, some simulations of a gap have been done with RSOFT, changing the distance of the gap and for different refractive index of the waveguide. Figure 4.3: Layout of the gap Three different situations were taken into account. Shown in the table I have specified the different parameters of each situation. Situation 1 Situation 2 Situation 3 n g n ext d (µm) n g = waveguide refractive index 19

24 n ext = refractive index of the exterior of the substrate. 2d = waveguide width. The waveguide width (2d) has been selected as the minimum value to have a single mode propagation in the waveguide. The single mode condition is: Where, V is the normalized frequency and is defined as: The wavelength ( ) used in all simulations is 1.55um. Situation 1 gap graph: Figure 4.4: Graph of the gap in the situation 1 20

25 Situation 2 gap graph: Figure 4.5: Graph of the gap in the situation 2 Situation 3 gap graph: Figure 4.7: Graph of the gap in the situation 1 21

26 5. SINGLE-STAGE RING FILTER 5.1 Ring filter without gap The first filter we have considered is the ring filter without gap. The layout of this filter is shown in the following figure: Figure 5.1: Layout of the ring filter without gap The coupler between the ring and the bus waveguide divides the input field in two parts. Part of the field is retarded by the ring and then it couples again into the bus. The equations of the directional coupler can be defined: 22

27 Figure 5.2: Layout of the ring filter without gap with fields symbolism With these equations we can calculate the output of the filter as: The development used to obtain these equations is described in Annex The behaviour of this filter has been simulated with the software Matlab (the code used for the simulation is Annex 9.1.2): 23

28 Figure 5.3: Amplitude response of a ring filter without gap This filter has an all-pass behaviour, so the amplitude of the transfer function is equal to 1 at every frequency. 24

29 5.2 Ring filter with gap The single stage filter has the same layout of the filter studied before but with a gap in the ring: Figure 5.4: Layout of a ring filter with a gap in the loop Theoretical analysis In this case, the theoretical investigation of the filter requires the equations of the gap: 25

30 Using these symbols described in the figure 5.5: Figure 5.5: Layout of the ring filter with gap with fields symbolism These equations can be theoretically solved (In Annex 9.1.3), the transmitted field and reflected field of the filter are: These equations have been numerically simulated with MATLAB by setting the FSR equal to 1, the transmittance equal to 0.6 and a reflectance defined to fulfill the lossless condition; the code used to simulate the single stage filter is in Annex The graphs of the reflected and transmitted fields are plotted in the next figure: 26

31 Figure 5.6: Plot of the transmitted and reflected field of a ring filter with gap The transfer function of the filter (on db): 27 Figure 5.7: Plot of the transfer function of a ring filter with gap

32 5.2.2 Losses of the filter The total power in the filter is the sum of the power transmitted and reflectd: For a lossless filter the total power must be equal to 1. The transmittance and reflectance for a lossless gap device, has to satisfy the condition: Other losses could be introduced by an imperfect coupling between the waveguides, but in this thesis we have assumed that the couplings ratios k 1 and k 2 satisfy the condition: Therefore, the graph of the losses is: 28 Figure 5.8: Plot of the losses of a ring filter with gap in the loop

33 5.2.3 Relation between bandwidths and the transmittance / reflectance The bandwidth is a very important characteristic of the filters and it is defined as the frequency range where the signal passes without attenuation. If the transfer function is expressed in a logarithmic scale (db), the bandwidth is the frequency range where the attenuation is less than 3 db. It s interesting to study the behaviour of the filter bandwidth with the changes of the gaps transmittance and reflectance. The simulations have been carried out using MATLAB and changing the transmittance from 0.5 to 0.95 but always satisfying the lossless condition. The results of the simulations are expressed in the table: t R BW [1/um]

34 Figure 5.9: Bandwidth vs gap transmittance In this graph, we see that the bandwidth of the filter increases when the transmittance increases too. The shape of the transfers function also changes, for example if we compare the transfer function with a transmittance of 0.55 and 0.90, we see that the side lobe of the transfer function reduces when the transmittance of the gap increases. 30

35 Graph with a transmittance of 0.55: Figure 5.10: Plot of the transmitted and reflected fields with a transmittance of 0.55 Plot with a transmittance of 0.90: Figure 5.11: Plot of the transmitted and reflected fields with a transmittance of

36 5.2.4 RSOFT simulations We have also carried out some numerical simulations with RSOFT. The parameters used are the same that in the gap simulations, and are summarized in the following table: Situation 1 Situation 2 Situation 3 n g n ext d (µm) Situation 1: Figure 5.12: Situation 1 RSOFT graph of a ring filter with a gap in the loop 32

37 Situation 2: Figure 5.13: Situation 2 RSOFT graph of a ring filter with a gap in the loop Situation 3: 33 Figure 5.14: Situation 3 RSOFT graph of a ring filter with a gap in the loop

38 These simulations, confirm the behaviour obtained with MATLAB. In the case of different situations, the gap presents different transmittance and reflectance, therefore the losses and behaviour of the filter changes. The FSR of the filter is: In the simulations the changes of n g and n ext cause a variation in the effective refractive index (n eff ), so that the FSR in each situation also changes Comparison Matlab RSOFT To compare the simulations, it has been used the RSOFT simulation of the situation 1, and used a Matlab code to adapt the FSR of the filter to the one simulated. The main problem to adapt MATLAB simulations to RSOFT is to calculate a suitable n eff, to do this it has been used the Compute fundamental mode engine of RSOFT software, the result is: 34 Figure 5.15: RSOFT compute fundamental mode

39 Therefore, the n eff of the filter is equal to With this value, the dimensions of the ring and the transmittance and reflectance values it has been done a MATLAB simulation to compare with RSOFT. The Matlab result is shown in the Figure 5.16: Figure 5.16: MATLAB plot to compare with RSOFT 35

40 If we compare with the RSOFT graph: Figure 5.17: RSOFT graph to compare with MATLAB The shape of the graphs is the same; the only difference is that the RSOFT results have a lower amplitude value. A reason to this behaviour could be that the RSOFT engine uses a model that introduces more losses, in the Matlab simulation the only losses that have been taken into account are the losses introduced by the gap. Another reason for the different shapes of the graphs is that the resolution of the RSOFT is reduced with respect to MATLAB. 36

41 6. MULTI-STAGE GAP FILTER 6.1 Multi-stage filter with 2 stages The multi-stage filter studied here uses 2 stages in series with the same characteristics. Each stage has a ring filter with a gap with the same dimensions. The distance between the stages is equal to 2L, where L is half of the ring perimeter Theoretical equations We define each stage as a ring filter with a gap, similar to the one studied in the previous Chapters, and we model each stage as: Figure 6.1: Model of each stage of the multi-stage ring filter The equations of each stage can be written as: 37

42 The block diagram of the filter is: Figure 6.2: Block diagram of a multi-stage filter with 2 stages Using the equations for each stage, the output equations of the filter are: The mathematical framework where we calculated these equations is described in Annex These equations have been simulated with the Matlab considering the FSR equal to 1, a transmittance of the gap of 0.6 and a reflectance selected to satisfy the lossless condition. 38

43 The resulting graphs (Matlab code in Annex 9.2.2) are shown in the following figures: Figure 6.3: Plot of the transmitted and reflected field of a Multi-stage filter with 2 stages 39 Figure 6.4: Plot of the transfer function of a Multi-stage filter with 2 stages

44 The loss graph: Figure 6.5: Plot of the losses of a Multi-stage filter with 2 stages RSOFT simulations Using RSOFT it has been simulated this multi-stage filter for the 3 situations that have been used previously to simulate the gap and the single stage filter. The parameters of each stage are the same as used in the single stage ring filter. 40

45 The layout used for the simulations is shown in the following figure: Figure 6.6: Layout of a Multi-stage filter with 2 stages The distance between each stage is equal to 2L, where L is half of the ring perimeter. 41

46 The results of these simulations are reported in the following figures: Situation 1: Situation 2: Figure 6.7: Situation 1 RSOFT graph of a Multi-stage filter with 2 stages 42 Figure 6.8: Situation 2 RSOFT graph of a Multi-stage filter with 2 stages

47 Situation 3: Figure 6.9: Situation 3 RSOFT graph of a Multi-stage filter with 2 stages 43

48 6.3 Multi-stage filter with 2 stages with a distance with L between stages In the previous simulations, the distance between stages was equal to 2L where L is the half of the ring perimeter. This condition assures that there are no phase changes between the stages. In this case: On the other hand, if the distance between the two stages is L we have: Where β is the propagation constant in the waveguide. In this case there is a phase change between the output field of the first stage and the input of the second one Theoretical analysis If the distance between stages is equal to L: We can solve the equations of the multi-stage filter, and the transmitted and reflected fields are: 44

49 Mathematical description of the calculations is in Annex If the distance between the stages is different from 2L there is a phase change; but it does not affect the simulations where the absolute value of the field is shown. However, if the distance between the two stages is not a multiple of L, the transmitted and reflected fields will have a different frequency behaviour RSOFT simulations To analyze the differences between the two filter architectures, where the distance are L and 2L, we have used the RSOFT simulator considering a filter of 2 stages and 3 stages. The first filter simulated is the multi-stage filter with 2 stages, which has been analyzed previously. For a distance between stages equal to L, the results of the output field (blue) and the reflected field (green) are shown in the following figures: 45

50 Figure 6.10: Situation 2 RSOFT graph of a filter with 2 stages and distance equal to L For a distance between stages equal to 2L: Figure 6.11: Situation 2 RSOFT graph of a filter with 2 stages and distance equal to 2L 46

51 The next filter analyzed is a multi-stage filter with 3 stages, for a distance between stages equal to L: Figure 6.12: Situation 2 RSOFT graph of a filter with 3 stages and distance equal to L For a distance between stages equal to 2L: Figure 6.13: Situation 2 RSOFT graph of a filter with 3 stages and distance equal to 2L 47

52 With this comparison we see that the behaviour of the multi-stage filter in amplitude is the same with a distance between stages of L or 2L. For a 3 stages filter, these is a little difference in the band pass, this could be related to the fact that RSOFT needs more memory for the simulations, so the resolution of the simulation is worse than the case when the distance is L. To get better results with the simulations of the 3 stage filter, the distance between stages will be set equal to L. 6.4 Multi-stage filter with 3 stages Now we pass to consider a multi-stage filter with 3 stages. Similar to the filter with 2 stages, each stage has the same parameters Theoretical analysis For the theoretical study of this filter the distance between each stage is equal to 2L. Referring to the following definitions: Figure 6.14: Block diagram of a multi-stage filter with 3 stages And using the equations for each stage of section 6.2.1, we can obtain the equations of the filter: 48

53 The mathematical calculations are reported in Annex To analyze the behaviour of the filter, these equations have been simulated with MATLAB, considering the FSR of the filter equal to 1 and a transmittance of the gap to 0.8, the reflectance has been defined to accomplish the lossless condition. The code used for the simulations is reported in Annex The results of the simulations are shown in the following figures: Figure 6.15: Plot of the transmitted and reflected field of a Multi-stage filter with 3 stages 49

54 Figure 6.16: Plot of the transfer function of a Multi-stage filter with 2 stages The total losses are shown in Figure 6.17: Figure 6.17: Plot of the losses of a Multi-stage filter with 3 stages 50

55 6.4.2 RSOFT simulations In these simulations, we have considered the distance between stages equal to L to obtain better resolution and decrease the time required for each simulation. The others parameters are the same as those used for the single stage filter simulations. Also in this case we have considered three situations with the same parameters as described in the Section 4.2. The layout used for the simulations is shown in Figure 6.18: Figure 6.18: Layout of a Multi-stage filter with 3 stages 51

56 The results of the simulations are reported in the following figures: Situation 1: Situation 2: Figure 6.19: Situation 1 RSOFT graph of a Multi-stage filter with 3 stages 52 Figure 6.20: Situation 2 RSOFT graph of a Multi-stage filter with 3 stages

57 Situation 3: Figure 6.21: Situation 2 RSOFT graph of a Multi-stage filter with 3 stages To investigate the effect of increasing the number of stages we have considered of a multi-stage filter with 4 stages, using the same parameters as before. We have analyzed only situation 2, because the time needed to perform each simulation was too much. 53

58 The layout used for the simulations is reported in Figure 6.22: Figure 6.22: Layout of a Multi-stage filter with 4 stages The result of the simulation is shown in Figure 6.23: 54 Figure 6.23: Situation 2 RSOFT graph of a Multi-stage filter with 4 stages

59 6.5 Comparison between filters To analyze the differences between the different filter architectures considered in this thesis we have compared the single stage filter and the multi-stage filter with 2, 3 and 4 stages. This comparison has been performed using either MATLAB and RSOFT Comparison using Matlab Defining the FSR equal to 1, we have compared the behaviour of the output field, the reflected field and the transfer function of each filter. We have considered filters with a single stage, 2 and 3 stages. The equations used for the comparisons are the ones that have been described in the previous sections, and the MATLAB code is a combination of the codes used in the previous simulations. The MATLAB code used for the comparisons is reported in Annex

60 The output fields of each filter are shown in the Figure 6.24: Figure 6.24: Comparison of output fields 56

61 The transfer function of each filter is reported in Figure 6.25: Figure 6.25: Comparison of transfer functions 57

62 The reflected fields of each filter are shown in Figure 6.26: Figure 6.26: Comparison of reflected fields 58

63 The reflected fields in db of each filter are shown in Figure 6.27: Figure 6.27: Comparison of reflected fields in db 59

64 When the number of stages of the filter increases, there are some changes in the shape of the output field. We observe that, the side lobes of the filter decrease and the attenuation in the rejection band increases. On the other hand, the gain on the pass band of the filter diminishes; this is mostly due to the fact that there are more gaps in the whole architecture. In the case of the reflected field, when the number of stages increases the shape of the field gets flatter, so the pass band of the reflection field gets flat, while the attenuation decreases Comparison using RSOFT Using the results of the previous simulations and enlarging some graphs, we can compare the behaviour, from the simulations of RSOFT of these filters. The results of the simulations in the situation 1 are shown in the following figures. Single-stage: 60 Figure 6.28: Situation 1 RSOFT Comparison single-stage filter

65 Multi-stage 2 stages: Figure 6.29: Situation 1 RSOFT Comparison multi-stage filter with 2 stages Multistatge 3 stages: 61 Figure 6.30: Situation 1 RSOFT Comparison multi-stage filter with 3 stages

66 The results of the simulations in the situation 2 are reported in the following figures. Single-stage: Multi-stage 2 stages: Figure 6.31: Situation 2 RSOFT Comparison single-stage filter 62 Figure 6.32: Situation 2 RSOFT Comparison multi-stage filter with 2 stages

67 Multi-stage 3 stages: Figure 6.33: Situation 2 RSOFT Comparison multi-stage filter with 3 stages Multi-stage 4 stages: Figure 6.34: Situation 2 RSOFT Comparison multi-stage filter with 4 stages 63

68 The results of the simulations in the situation 3 are shown in the following figures. Single-stage: Multi-stage with 2 stages: Figure 6.35: Situation 3 RSOFT Comparison single-stage filter 64 Figure 6.37: Situation 3 RSOFT Comparison multi-stage filter with 2 stages

69 Multi-stage with 3 stages: Figure 6.38: Situation 3 RSOFT Comparison multi-stage filter with 3 stages The behaviour of the transmitted and reflected fields with the increase of the number of the stages is the same seen as that one from in the MATLAB simulations. In the simulations of the transmitted field, we observe that the side lobes decrease when the number of the stages increases. The behaviour of the reflected field it is not very clear in these simulations mainly because the amplitude of the fields are lower and the RSOFT resolution don t allow us to observe it.. 65

70 7. CONCLUSIONS The goal of this thesis was to analyze the performance of ring-based filters with a gap in the loop considering also a multi-stage architecture composed of a set of ring-filters with a gap. The single-stage ring filter with a gap shows a selective frequency behaviour, with a flat pass band and a larger side lobe compared to standard ring resonators. Using a set of single-stage filters, we have designed a multi-stage filter, that presents similar performance in the frequency behaviour, but the side lobes decrease and the attenuation of the rejection band increases. This behaviour is amplified by incrementing the number of stages. On the other hand, the losses introduced by the filter increases as well because each gap introduces some losses. Possible future work is related to the adaption of ring-based filters with a gap to some practical applications. Among the possible applications, it is very interesting the use of ring-filter for an optical modulator. Nowadays, these modulators use Mach-Zehnder Interferometer; these devices have a large footprint, a high power consumption and require a high driving voltage. Therefore, the utilization of ring-based filters will be a good option with a small footprint and low power consumption like it was demonstrated in [3]. 66

71 8. REFERENCES [1] Maria Liberata Cozzolino. Sintesi di Filtri Ottici ad Anello con Gap. Università degli Studi di Roma Tre, Facoltà di Ingegneria, Roma, [2] Christi K. Madsen and Jian H. Zhao. Optical Filter Design and Analysis: A Signal Processing Approach. John Wiley & Sons, Inc. New York, [3] Po Dong, Roshanak Shafiiha, Shirong Liao, Hong Liang, Ning-Ning Feng, Dazeng Feng, Guoliang Li, Xuezhe Zheng, Ashok V. Krishnamoorthy, and Mehdi Asghari. Wavelength-tunable silicon microring modulator. Optics Express, Vol.18, No 11, 24 May

72 9. ANNEXES 9.1 Single stage ring Single ring without gap equations According the symbolism used in the next figure, it has been calculated the equations of the filter: Figure 9.1: Layout of the ring filter without gap with fields symbolism The equations for a directional coupler: If we develop the equations to obtain the output field: 68

73 9.1.2 Single stage ring without gap MATLAB code The code used in the MATLAB simulations of the single stage filter without gap: clear close all k1=0.5; %Coupling v=linspace(0,10,1000); ex=exp(i*pi*v); %Output field A2plus=(k1-ex)./(1-ex*k1); %Output field absa2=abs(a2plus); plot(v,absa2); title('plot of A2 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') axis([ ]) 69

74 9.1.3 Single stage ring with gap equations The symbolism used to denominate the fields: Figure 9.2: Layout of the ring filter with a gap in the loop with fields symbolism Gap equations: Lossless condition: Coupler equations: Calculations to get the output and reflected fields: 70

75 71

76 9.1.4 Single stage filter with gap Matlab code file The Matlab code used to simulate the single stage filter is: clear close all v=linspace(0,10,10000); t=0.8;%transmitance of the gap r=sqrt(1-t^2) %r for a lossless device %r=0.1;%r for comparision with RSOFT k1=sqrt(0.5);%coupling factor 1 k2=sqrt(1-k1^2); %coupling factor n= ;%efective index field ra=1.7; %radius in um L=2*pi*ra; %perimeter of the ring Beta=2*pi*n*v; t1=t*exp(i*2*pi*v);%fsr=1 r1=r*exp(i*2*pi*v);%fsr=1 %t1=t*exp(i*beta*l); %Comparision with RSOFT %r1=r*exp(i*beta*l); %Comparision with RSOFT A1=1; %A1=gaussmf(v,[ ]); %incident field for comaprision with RSOFT A1min=(j*((k2^2)*(r1))./(1-2*t1*k1+(k1^2)*(t1.^2+r1.^2))).*A1;%A1- A2piu=((k1*(k1^2+k2^2)*(t1.^2+r1.^2)+k1-2*t1*k1^2-(k2^2)*t1)./(1-2*t1*k1+(k1^2)*(t1.^2+r1.^2))).*A1;%A2+ A1min_dB=20*log10(abs(A1min)); A2piu_dB=20*log10(abs(A2piu)); 72

77 N_loss=abs(A1min).^2+abs(A2piu).^2;%filter loses figure(1) subplot(2,1,1), plot(v,abs(a1min),'-r') %plot of A1- title('plot of A1 minus') axis([ ]) %axis([ ])%Axis for comparision with RSOFT ylabel('amplitude') subplot (2,1,2), plot(v,abs(a2piu),'-b') %plot of A2+ title('plot of A2 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') axis([ ]) %axis([ ])%Axis for comparision with RSOFT figure(2) plot(v,n_loss)%plot of A2+^2+A1-^2 title('device loss') xlabel('freqüency (f/c, 1/um)') %axis([ ]) axis([ ])%Axis for comparision with RSOFT figure(3) subplot(2,1,1), plot(v,a1min_db,'-r') %plot of A1- in db title('plot of A1 minus in db') ylabel('amplitude in db') axis([ ]) %axis([ ])%Axis for comparision with RSOFT subplot (2,1,2), plot(v,a2piu_db,'-b') %plot of A2+ in db title('plot of A2 plus in db') ylabel('amplitude in db') xlabel('freqüency (f/c, 1/um)') axis([ ]) %axis([ ])%Axis for comparision with RSOFT figure(4) plot(v,a1,'-g')%plot of the input field hold on plot(v,abs(a1min),'-r') %plot of A1- hold on plot(v,abs(a2piu),'-b') %plot of A2+ title('plot of A1 minus and A2 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') axis([ ]) %axis([ ])%Axis for comparision with RSOFT 73

78 9.2 Multi-stage ring filter Multi-stage ring filter with 2 stages equations To analyze the multi-stage filter, first we remember the equations of a single stage that have been seen in Section Therefore, we define each stage like: Figure 9.3: Definition of a stage If the input is : Because the device is passive and reversible, if the input is : If we have both inputs: 74

79 In the lossless case we consider: Now we consider a multi-stage filter with 2 stages, the block diagram is: Figure 9.4: Block diagram of a multi-stage filter with 2 stages If the separation between stages is equal to 2L or a multiple of 2L, where L is the half of the ring perimeter, we can consider: The equations for the each stage will be: 75

80 9.2.2 Multi-stage Matab code The matlab code used to simulate the filter with 2 and 3 stages: clear close all v=linspace(-2,2,10000); t=0.5; r=sqrt(1-t^2); %tau for a lossless device k1=sqrt(0.5); k2=sqrt(1-k1^2); t1=t*exp(i*2*pi*v); r1=r*exp(i*2*pi*v); f2=r1*k2^2./(1-2*t1*k1+k1^2*(t1.^2+r1.^2)); f1=(k1*(k1^2+k2^2)*(t1.^2+r1.^2)+k1-2*t1*k1^2-k2^2*t1)./(1-2*t1*k1+k1^2*(t1.^2+r1.^2)); A22piu=f1.^2./(1+abs(f2).^2); A11min=i*f2.*(1+abs(f1).^2+abs(f2).^2)./(1+abs(f2).^2); A23piu=(f1.^3)./(1+2*abs(f2).^2+abs(f1).^2.*abs(f2).^2+abs(f2).^4); A11min3=i*f2.*(1+2*abs(f2).^2+abs(f1).^2+2*abs(f1).^2.*abs(f2).^2+abs( f2).^4+abs(f1).^4)./(1+2*abs(f2).^2+abs(f1).^2.*abs(f2).^2+abs(f2).^4) ; figure(1) subplot(2,1,1),plot(v,abs(a22piu)) axis([ ]) title('m2 Plot of A22 plus') ylabel('amplitude') 76

81 xlabel('freqüency (f/c, 1/um)') subplot(2,1,2),plot(v,abs(a11min)) axis([ ]) title('m2 Plot of A11 min') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') figure(2) subplot(2,1,1),plot(v,abs(a23piu)) axis([ ]) title('m3 Plot of A22 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') subplot(2,1,2),plot(v,abs(a11min3)) axis([ ]) title('m3 Plot of A11 min') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') figure(3) subplot(2,1,1),plot(v,20*log10(abs(a22piu))) axis([ ]) title('m2 Plot of A22 plus in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') subplot(2,1,2),plot(v,20*log10(abs(a11min))) axis([ ]) title('m2 Plot of A11 min in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') figure(4) subplot(2,1,1),plot(v,20*log10(abs(a23piu))) axis([ ]) title('m3 Plot of A22 plus in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') subplot(2,1,2),plot(v,20*log10(abs(a11min3))) axis([ ]) title('m3 Plot of A11 min in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') figure(5) plot(v,abs(a22piu).^2+abs(a11min).^2) title('losses of a Multi-stage filter with 2 stages') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') axis([ ]) figure(6) plot(v,abs(a23piu).^2+abs(a11min3).^2) title('losses of a multi-stage filter with 3 stages') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') axis([ ]) 77

82 9.2.3 Multi-stage filter with a difference between stages different to 2L equations This case will be studied in a multi-stage filter with 2 stages, so the symbolism is the same as it s used in the previous calculations. If the distance between 2 stages is L, then: To simplify the calculation, it s been defined: If we develop the equations of 2 stages: 78

83 9.2.4 Multi-stage filter with 3 stages equations This is the calculations of the equations of a filter with 3 stages, where each stage is a ring resonator with a gap. The block diagram of a multi-stage filter with 3 stages is: Figure 9.5: Block diagram of a multi-stage filter with 3 stages If the separation between stages is equal to 2L or a multiple of 2L, we can consider: The equations of each stage: 79

84 Thanks to the previous results, we can define: 80

85 9.2.5 Matlab comparison code The Matlab code used to compare the different filters (single stage and multistage with 2 and 3 stages) analyzed in this thesis is: clear close all v=linspace(-2,2,10000); t=0.6; r=sqrt(1-t^2); %tau for a lossless device k1=sqrt(0.5); k2=sqrt(1-k1^2); t1=t*exp(i*2*pi*v); r1=r*exp(i*2*pi*v); f2=r1*k2^2./(1-2*t1*k1+k1^2*(t1.^2+r1.^2)); f1=(k1*(k1^2+k2^2)*(t1.^2+r1.^2)+k1-2*t1*k1^2-k2^2*t1)./(1-2*t1*k1+k1^2*(t1.^2+r1.^2)); A2plus=f1; A1min=j*f2; A22piu=f1.^2./(1+abs(f2).^2); A11min=i*f2.*(1+abs(f1).^2+abs(f2).^2)./(1+abs(f2).^2); A23piu=(f1.^3)./(1+2*abs(f2).^2+abs(f1).^2.*abs(f2).^2+abs(f2).^4); 81

86 A11min3=i*f2.*(1+2*abs(f2).^2+abs(f1).^2+2*abs(f1).^2.*abs(f2).^2+abs( f2).^4+abs(f1).^4)./(1+2*abs(f2).^2+abs(f1).^2.*abs(f2).^2+abs(f2).^4) ; figure(1) subplot(3,1,1),plot(v,abs(a2plus)) axis([ ]) title('sg Plot of A2 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') subplot(3,1,2),plot(v,abs(a22piu)) axis([ ]) title('m2 Plot of A22 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') subplot(3,1,3),plot(v,abs(a23piu)) axis([ ]) title('m3 Plot of A23 plus') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') figure(2) subplot(3,1,1),plot(v,abs(a1min)) axis([ ]) title('sg Plot of A1 min') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') subplot(3,1,2),plot(v,abs(a11min)) axis([ ]) title('m2 Plot of A11 min') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') subplot(3,1,3),plot(v,abs(a11min3)) axis([ ]) title('m3 Plot of A11 min') ylabel('amplitude') xlabel('freqüency (f/c, 1/um)') figure(3) subplot(3,1,1),plot(v,20*log10(abs(a2plus))) axis([ ]) title('sg Plot of A2 plus in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') subplot(3,1,2),plot(v,20*log10(abs(a22piu))) axis([ ]) title('m2 Plot of A22 plus in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') subplot(3,1,3),plot(v,20*log10(abs(a23piu))) axis([ ]) title('m3 Plot of A23 plus in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') figure(4) subplot(3,1,1),plot(v,20*log10(abs(a1min))) axis([ ]) title('sg Plot of A1 min in db') ylabel('amplitude (db)') 82

87 xlabel('freqüency (f/c, 1/um)') subplot(3,1,2),plot(v,20*log10(abs(a11min))) axis([ ]) title('m2 Plot of A11 min in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)') subplot(3,1,3),plot(v,20*log10(abs(a11min3))) axis([ ]) title('m3 Plot of A11 min in db') ylabel('amplitude (db)') xlabel('freqüency (f/c, 1/um)' 83

88